SHEAR STRENGTH OF STRUCTURAL CONCRETE MEMBERS USING A UNIFORM SHEAR ELEMENT APPROACH

Transcription

1 SHEAR STRENGTH OF STRUCTURAL CONCRETE MEMBERS USING A UNIFORM SHEAR ELEMENT APPROACH by Ashin Esandiari B.S., BIHE, 1997 M.A.S., Carleton University, 2001 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in The Faulty o Graduate Studies (Civil Engineering) THE UNIVERSITY OF BRITISH COLUMBIA (Vanouver) April 2009 Ashin Esandiari, 2009

2 Abstrat The simplest shear problem involves a two-dimensional retangular element with uniormly distributed reinorement parallel to the element sides, and subjeted to uniorm normal stresses and shear stress. Suh a uniorm shear element will have uniorm average stresses in reinorement and onrete. The simplest model or elements subjeted to shear ore and bending moment that leads to ode provisions uses one uniorm shear element. Shear ore is assumed to be resisted by a entral portion o the ross-setion ating as a uniorm shear element, while bending moment is assumed to be resisted by the lexural tension reinorement and onrete ompression zone at the rosssetion ends. In this thesis, the shear strength o bridge girders and squat shear walls are evaluated using a uniorm shear element approah. Current ode shear design provisions or beams are neessarily simpliied proedures that are generally onservative. While the extra osts are small or new design, it may lead to unneessary load restritions on bridges or unneessary retroitting when used or shear strength evaluation. A new shear strength evaluation proedure or strutural onrete girders is proposed. The proedure aounts or the inluene o more parameters and provides more insight into the ailure mode than ode design methods. To veriy the proedure, predited trends are ompared with Modiied Compression Field theory (MCFT) or uniorm shear elements, and Response-2000 or beam elements subjeted to ombined shear and bending moment. Shear strength preditions are also ompared with results rom strength tests on reinored and prestressed onrete beams, together with preditions rom urrent ode shear design provisions. The urrent Canadian building ode CSA A ontains new provisions or the ii

3 seismi design o squat walls that were developed using a uniorm shear element approah. These new ode provisions are rigorously evaluated or the irst time in this study. A new method to aount or the lexure-shear interation at the base o squat shear walls is proposed as well as reinements to the 2004 CSA A23.3 shear strength provisions or squat shear walls. These are veriied by omparing the predited trends with the preditions o MCFT-based nonlinear inite element program VeTor 2. iii

4 Table o Contents Abstrat... ii Table o Contents... iv List o Tables... vii List o Figures...viii List o Symbols... xvi Aknowledgements... xxi Chapter 1. Introdution Shear in Conrete Strutures Modiied Compression Field Theory Simpliied Shear Analysis Shear Strength Evaluation o Bridge Girders Design o Squat Shear Walls Researh Objetives Thesis Organization Chapter 2. Literature Review: Beam Shear Strength Review o Previous Studies Reent Code Approahes Reent Studies Conluding Remarks Chapter 3. Uniorm Shear Elements General Uniorm Shear Element Modiied Compression Field Theory (MCFT) AASHTO LRFD Method CHBDC Method Proposed Evaluation Method or Members With at Least Minimum Transverse Reinorement Proposed Evaluation Method or Members Without Transverse Reinorement Chapter 4. Beam Elements General Exat Solution Simpliied Proedures or Design Proposed Evaluation Method or Members With At Least Minimum Transverse Reinorement iv

5 4.5. Proposed Evaluation Method or Members Without Transverse Reinorement Members With Less than Minimum Transverse Reinorement Example Evaluations o Bridge Girder With at Least Minimum Transverse Reinorement Example Evaluation or Bridge Girder With Less than Minimum Transverse Reinorement Chapter 5. Comparison with Beam Test Results General Members With at Least Minimum Transverse Reinorement Members Without Transverse Reinorement Eet o Important Parameters Minimum Transverse Reinorement and Transition Between Members With and Without Minimum Transverse Reinorement Chapter 6. Reined 2006 CHBDC Method or Shear Design General Reined CHBDC Approah or Members With at Least Minimum Transverse Reinorement Bridge Examples Comparison with Experimental Results Chapter 7. Literature Review: Squat Shear Walls Shear Strength o Squat Shear Walls Summary o Observed Behaviour Reent Code Approahes Chapter 8. Comparison o NLFE Preditions with Experimental Results o Squat Shear Walls General Finite Element Program Comparison with Wall Test Results Chapter 9. Analytial Study o Flexural and Shear Resistane o Squat Shear Walls General Traditional Approah or Flexural Resistane o Deep Beams CSA A23.3 Approah or Flexural and Shear Resistane o Squat Shear Walls Finite Element Analysis o Squat Shear Walls Failing in Flexure Proposed Setional Model or Flexural Capaity Comparison o Finite Element Results with the Preditions o Proposed Method or Flexural Capaity o Squat Shear Walls Finite Element Analyses o Squat Shear Walls Failing in Shear v

6 Chapter 10. Summary and Conlusions General Shear Strength Evaluation o Bridge Girders Strength o Squat Shear Walls Reommendations or Future Work Reerenes Appendies Appendix A: Exel Spreadsheets or the Proposed Evaluation Methods and the Reined CHBDC Method..276 Appendix B: Detailed Steps in Proposed Evaluation Proedures Appendix C: Detailed Examples o Proposed Evaluation Proedure or Members With Stirrups Appendix D: Rating Truks Used or Example Evaluations o Bridge Girder. 294 Appendix E: Tested Beams Used or Comparison with Experimental Results Appendix F: Comparison o Preditions with Beam Test Results vi

7 List o Tables Table 4-1 Summary o preditions or example bridge girders with at least minimum transverse reinorement Table 4-2 Comparison o nominal shear strength preditions (kn) or bridge girder example ignoring shear resisted by inlined lexural ompression Table 9-1 Summary o walls analyzed to investigate lexural apaity o squat shear walls Table 9-2 Summary o walls analyzed to investigate shear strength o squat shear walls vii

8 List o Figures Fig. 1-1 Reinored onrete simply supported beam subjeted to shear and moment... 1 Fig. 1-2 Uniorm shear elements Fig. 1-3 Prediting beam shear behaviour with one uniorm shear element Fig. 1-4 Prediting squat shear wall behaviour with one uniorm shear element... 5 Fig. 1-5 An example o an existing onrete bridge (BC Ministry o Transportation) Fig. 3-1 Element o reinored onrete subjeted to uniorm shear and normal stresses Fig. 3-2 Conrete average stress-strain relationship in ompression (Vehio and Collins, 1986) Fig. 3-3 Conrete average stress-strain relationship in tension (Vehio and Collins, 1986) Fig. 3-4 Mohr irle o stress or raked reinored onrete (adopted rom Vehio and Collins, 1986) Fig. 3-5 Free body diagram o a uniorm shear element in the rak diretion or average stresses and loal stresses at the raks (Collins et. al 1996) Fig. 3-6 Mohr irle o strains or reinored onrete (adopted rom Vehio and Collins, 1986) Fig. 3-7 Comparison o predited and observed shear stress shear strain response o six uniorm shear elements (Bentz et al. 2006) Fig. 3-8 MCFT preditions o shear response o uniorm shear elements or: (a) members with transverse reinorement, (b) members without transverse reinorement Fig. 3-9 Developing proedure o CHBDC 2006/CSA A equation or angle o inlination o prinipal ompression (Bentz et al., 2006) Fig Shear strength relation with transverse reinorement ratio (Bentz et al., 2006) Fig Comparison o θ and β values given by CHBDC 2006/ CSA A with values determined rom MCFT or elements without transverse reinorement (Bentz and Collins, 2006) viii

9 Fig Comparison o θ and β values given by CHBDC 2006/ CSA A with values determined rom MCFT or elements with at least minimum transverse reinorement (Bentz and Collins, 2006) Fig Inluene o longitudinal strainε x and transverse strainε z on: (a) shear stress, (b) angle o inlination o diagonal ompression, () longitudinal ompression stress in onrete or an element with ρ z = 0.005, ' = 40 MPa, y = 400 MPa Fig Comparison o predited angleθ with MCFT result at: (a) yielding o transverse reinorement, (b) rushing o onrete or ' = 40 MPa, y = 400 MPa Fig Comparison o predited angleθ with MCFT result at: (a) yielding o transverse reinorement, (b) rushing o onrete or ' = 30 MPa, y = 400 MPa Fig Comparison o predited angleθ with MCFT result at: (a) yielding o transverse reinorement, (b) rushing o onrete or ' = 60 MPa, y = 400 MPa Fig Comparison o predited angleθ with MCFT result at: (a) yielding o transverse reinorement, (b) rushing o onrete or ' = 40 MPa, y = 250 MPa Fig Comparison o predited angleθ with MCFT result at: (a) yielding o transverse reinorement, (b) rushing o onrete or ' = 40 MPa, y = 600 MPa Fig Comparison o predited β with MCFT result at: (a) yielding o transverse reinorement, (b) rushing o onrete or ' = 40 MPa, y = 400 MPa Fig Comparison o predited β with MCFT result at: (a) yielding o transverse reinorement, (b) rushing o onrete or ' = 30 MPa, y = 400 MPa Fig Comparison o predited β with MCFT result at: (a) yielding o transverse reinorement, (b) rushing o onrete or ' = 60 MPa, y = 400 MPa Fig Comparison o predited β with MCFT result at: (a) yielding o transverse reinorement, (b) rushing o onrete or ' = 40 MPa, y = 250 MPa Fig Comparison o predited β with MCFT result at: (a) yielding o transverse reinorement, (b) rushing o onrete or ' = 40 MPa, y = 600 MPa Fig Comparison o predited shear stress with MCFT result at: (a) yielding o transverse reinorement, (b) rushing o onrete or ' = 40 MPa, y = 400 MPa ix

10 Fig Comparison o predited shear stress with MCFT result at: (a) yielding o transverse reinorement, (b) rushing o onrete or ' = 30 MPa, y = 400 MPa Fig Comparison o predited shear stress with MCFT result at: (a) yielding o transverse reinorement, (b) rushing o onrete or ' = 60 MPa, y = 400 MPa Fig Comparison o predited shear stress with MCFT result at: (a) yielding o transverse reinorement, (b) rushing o onrete or ' = 40 MPa, y = 250 MPa Fig Comparison o predited shear stress with MCFT result at: (a) yielding o transverse reinorement, (b) rushing o onrete or ' = 40 MPa, y = 600 MPa Fig Bilinear approximation o: (a) ot2θ, (b) ot 2 θ used to approximate Eq. [3-39] Fig Comparison o predited longitudinal onrete ompression stress with MCFT result at: (a) yielding o transverse reinorement, (b) rushing o onrete or ' = 40 MPa, y = 400 MPa Fig Variation o: (a) angle, (b) shear stress with onrete ontribution rom variable angle truss model Fig Comparison o predited axial ompression stress ratio n ' (in MPa v / units) with MCFT or members without transverse reinorement Fig Comparison o predited onrete ontribution β (in MPa units) with MCFT or members without transverse reinorement Fig Comparison o predited angle θ with MCFT or members without transverse reinorement Fig. 4-1 Applition o MCFT to beam elements using multi-layer analysis (Vehio and Collins, 1986) Fig. 4-2 Equilibrium o the dual setions in MCFT exat solution or beam elements (Vehio and Collins, 1986) Fig. 4-3 Algorithm o MCFT exat solution proedure or beam elements (Vehio and Collins, 1986) Fig. 4-4 Some output plots rom omputer program Response Fig. 4-5 Some output plots rom omputer program Response x

11 Fig AASHTO LRFD and 2006 CHBDC approximate setional model or beams subjeted to shear and moment Fig. 4-7 Variation o shear response over depth o prestressed I-girder with omposite dek slab: (a) ross-setion, (b) longitudinal strain, () angleθ, (d) shear low, (e) normal stress multiplied by width Fig. 4-8 Approximate beam strain proile and ores in the proposed method Fig. 4-9 Typial shear stress-strain relationships or beams Fig Response 2000 preditions or variation o shear response over the depth o a beam, M/V = 2.0 m, y = 550 MPa, ' = 20 MPa Fig Cross-setions o girders in example bridges Fig Moment and shear envelopes o evaluated bridges with minimum stirrups Fig Comparison o predited shear strengths along span o I-girder bridge: (a) Response 2000 and proposed method, (b) Response 2000 and ode design methods Fig Comparison o Response 2000 predited mid-depth strain along span o I-girder bridge with proposed and ode design methods Fig Comparison o predited shear strengths along span o box-girder bridge: (a) Response 2000 and proposed method, (b) Response 2000 and ode design methods Fig Comparison o Response 2000 predited (a) mid-depth strain, (b) lexural tension reinorement strain along span o box-girder bridge with proposed and ode design methods Fig Comparison o predited shear strengths along span o hannel-girder bridge: (a) Response2000 and proposed method, (b) Response 2000 and ode design methods Fig Comparison o Response 2000 predited mid-depth strain o I-girder bridge with proposed and ode design methods Fig Cross-setion o the evaluated bridge girder example at 4.67 m rom midsupport (M u = knm, V u =662.4 kn) Fig. 5-1 Cumulative requeny o test-to-predited ratios o proposed and ode methods: (a) RC beams, (b) PC beams Fig. 5-2 Cumulative requeny o test-to-predited ratios o proposed and ode methods: (a) 132 reinored onrete (RC) beams, (b) 131 prestressed onrete (PC) beams xi

12 Fig. 5-3 Test-to-predited ratios o proposed and ode methods versus shear stress ratio v / ' or 88 prestressed onrete beams with transverse reinorement Fig. 5-4 Test-to-predited ratios o proposed and ode methods versus eetive depth or 132 reinored onrete beams without transverse reinorement Fig. 5-5 Test-to-predited shear strength ratios or 76 lightly reinored tested beams: (a) assuming no stirrups, (b) using linear interpolation approah, or members with less than minimum stirrups Fig. 6-1 Comparison o ode predited axial ompression stresses n v with MCFT or dierent onrete ontributions to shear stress Fig. 6-2 Linear approximation o otθ Fig. 6-3 Comparison o I-girder predited mid-depth strain along the bridge span Fig. 6-4 Comparison o box-girder predited (a) mid-depth strain (b) lexural tension reinorement strain along the bridge span Fig. 6-5 Comparison o hannel-girder predited mid-depth strain along the bridge span Fig. 6-6 Cumulative requeny o test-to-predited ratios o reined CHBDC 2006 and CHBDC 2006 methods: (a) 80 RC beams, (b) 88 PC beams, with stirrups Fig. 6-7 Cumulative requeny o test-to-predited ratios o reined CHBDC 2006 and CHBDC 2006 methods or 132 RC beams without stirrups Fig. 7-1 Squat walls shear ailure modes Fig. 8-1 Load-deormation responses o wall DP1: (a) observed, (b) alulated (Palermo and Vehio, 2004) Fig. 8-2 Details o the three previously tested walls in the literature examined to ompare experimental results with inite element preditions Fig. 8-3 Comparison o experimental load-top displaement urve with inite element predition or wall tested by Wiradinata and Saatioglu (1986) Fig. 8-4 Comparison o inite element preditions o shear stress proiles along (a) top setion, (b) mid-height setion, () base setion o the wall tested by Wiradinata and Saatioglu (1986) and the same wall without the top beam and distributed ore applied at top xii

13 Fig. 8-5 Comparison o inite element preditions o vertial reinorement stresses at the base o the wall tested by Wiradinata and Saatioglu (1986) and the same wall without the top beam and distributed ore applied at top Fig. 8-6 Comparison o experimental load-top displaement urve with inite element predition or wall tested by Kuang and Ho (2008) Fig. 8-7 Comparison o inite element preditions o shear stress proiles along (a) top setion, (b) mid-height setion, () base setion o the wall tested by Kuang and Ho (2008) and the same wall without the top beam and distributed ore applied at top Fig. 8-8 Comparison o inite element preditions o vertial reinorement stress at the base o the wall tested by Kuang and Ho (2008) and the same wall without the top beam and distributed ore applied at top Fig. 8-9 Comparison o experimental load- top displaement urve with inite element predition or wall tested by Leas et al. (1990) Fig Comparison o inite element preditions o shear stress proiles along (a) top setion, (b) mid-height setion, () base setion o the wall tested by Leas et al. (1990) and the same wall without the top beam and distributed ore applied at top Fig. 9-1 Comparison o a deep beam and a squat shear wall Fig. 9-2 Uniormly distributed ore low in squat shear walls Fig. 9-3 Horizontal reinorement yielding shear ailure o squat shear walls as in 2004 CSA A Fig. 9-4 Typial detail o walls analyzed to investigate lexural apaity o squat shear walls Fig. 9-5 Conrete stress ontour diagrams based on inite element analysis o a squat shear wall with 0.5% distributed reinorement in both diretions and height-to-length ratio o 0.5 immediately prior to lexural ailure Fig. 9-6 Steel stress ontour diagrams based on inite element analysis o a squat shear wall with 0.5% distributed reinorement in both diretions and height-to-length ratio o 0.5 immediately prior to lexural ailure Fig. 9-7 Finite element preditions or shear stress distributions at base o our squad shear walls immediately prior to lexural ailure xiii

14 Fig. 9-8 Finite element preditions or total normal stress distributions at base o our squad shear walls immediately prior to lexural ailure Fig. 9-9 Predited moment apaities o 16 squat shear walls Fig Ratios o inite element predited moment apaity to the plane setion analysis predited moment apaity or the 42 squat walls ailing in lexure Fig Truss model or a squat wall with height-to-length ratio o Fig Truss model or a squat wall with height-to-length ratio o Fig Proposed setional model or lexural apaity o squat shear walls Fig Comparison o inite element preditions or shear ore at lexural apaity o squat shear walls with h w /l w =2.0, with 2004 CSA A23.3 and proposed method preditions Fig Comparison o inite element preditions o shear ore at lexural apaity o squat shear walls with h w /l w =1.0, with 2004 CSA A23.3 and proposed method preditions Fig Comparison o inite element preditions or shear ore at lexural apaity o squat shear walls with h w /l w =0.5, with 2004 CSA A23.3 and proposed method preditions Fig Comparison o inite element preditions or shear ore at lexural apaity o squat shear walls with h w /l w =0.3, with 2004 CSA A23.3 and proposed method preditions Fig Proposed simple model or lexural apaity o squat shear walls Fig Variation o the portion o distributed vertial reinorement α that ontributes to the lexural apaity o walls Fig Typial details o walls analyzed to investigate shear strength o squat shear walls Fig Comparison o inite element preditions or shear strength with ode preditions or squat walls with: (a) ρ v =ρ h, (b) ρ v =3ρ h Fig Finite element preditions or shear stress distribution in squat walls with ρ v =ρ h =0.005 immediately prior to shear ailure: (a) at base o wall, (b) at mid-height. 235 xiv

15 Fig Conrete shear stress ontour diagrams based on inite element analysis o a squat shear wall with 0.5% distributed reinorement in both diretions and height-tolength ratio o 0.5 prior to diagonal tension shear ailure Fig Diagonal tension ailure o low rise shear walls aounting or ompression zone ontribution Fig Loalized sliding eet on load-displaement urve o squat shear walls with ρ z = ρ v = and height-to-length ratio o: (a) 0.3, (b) 0.5, () Fig Loalized sliding eet on base shear stress distribution o squat shear walls with ρ z = ρ v = and height-to-length ratio o: (a) 0.3, (b) 0.5, () Fig Loalized sliding eet on mid-height shear stress distribution o squat shear walls with ρ z = ρ v = and height-to-length ratio o: (a) 0.3, (b) 0.5, () Fig Ratios o inite element analysis-to-2004 CSA A23.3 reined method predited shear strength or the 44 walls ailing in shear xv

16 List o Symbols A g, A v = Gross ross-setion area o squat shear wall, A s = Total reinorement area o ompression olumn in squat shear wall, A p, A s = Area o prestressed, nonprestressed lexural tension reinorement; A sw, A pw = Area o prestressed, nonprestressed longitudinal reinorement entered in web; A pwi = Area o i th layer o prestressed longitudinal reinorement in the web, A s = Area o nonprestressed longitudinal reinorement in the ompression hord, A swj = Area o j th layer o nonprestressed longitudinal reinorement in the web, A t = Area o onrete surrounding lexural tension reinorement, A v = Area o transverse reinorement (stirrups) spaed at s, a a g b b w C C = Shear span, = Maximum onrete grain size, = Width o beam ompression ae width, = Width o shear area (web), =Compression olumn strength in squat shear wall, = Fore in the lexural ompression hord, d = Depth rom ompression ae to entroid o lexural tension reinorement, d nv = Depth o uniorm ompression stress n v over diagonally raked web, d pw = Depth rom ompression ae to entroid o prestressing tendons in web, d w d v E = Depth rom ompression ae to entroid o longitudinal reinorement in web, = Depth o uniorm shear stress, = Modulus o Elastiity o onrete, E, E = Modulus o Elastiity o prestressed, nonprestressed reinorement; p s F l = Required tension ore in longitudinal reinorement on lexural or ompression side o member, xvi

17 F p = Prestressing tendons ore, F px = Prestressing tendons ore omponent in the x-diretion, F t = Fore taken by onrete tension stiening in the raked tension hord. ' = Speiied ompressive strength o onrete, 1, 2 = Conrete tensile, ompressive strain in the prinipal diretion; x, z = Conrete normal stress in x-dir., z-dir.; p = Eetive stress in prestressing tendons, pr = Stress in prestressing tendons at maximum resistane, sx, sz = Stress in x-dir., z-dir. reinorement; sxr, szr = Stress in x-dir., z-dir. reinorement at the loation o diagonal raks; x, z = Applied normal stress in x-diretion, z-diretion; y = Yield strength o reinorement, h = Thikness o squat shear wall, height o beam, H, h w = Height o squat shear wall, jd = Internal lexural lever-arm, l = Span o beam, L, l w = Length o squat shear wall, M = Bending moment at the setion o interest, N * =Axial ore in squat shear wall, n v = Axial ompression stress in onrete required to resist applied shear, n = Predited value o n vo v at ε x = 0, n v * = Axial ompression stress at loation o diagonal raks required to resist applied shear, n v = Maximum n v as limited by reserve apaity o longitudinal reinorement, N, N = Resultant o n v, n v stress assumed uniorm over eetive depth d nv ; P P s v v = Wall panel strength in squat shear wall, =Axial load is squat shear wall, xvii

18 s = Spaing o transverse reinorement, s max = Maximum spaing o transverse reinorement, s x, s z = Crak spaing parameter in x-diretion, z-diretion; S xe = Crak spaing parameter dependent on rak ontrol harateristis o longitudinal reinorement and aggregate size, s θ = Crak spaing, T T d t v = Fore in the lexural tension hord, = Resultant ore in the distributed vertial reinorement in web o squat shear wall, = Thikness o squat shear wall, = Total shear stress, v, v s = Shear stress attributed to onrete, stirrups; v i = Conrete stress transerred along the raks by aggregate interlok, v i2 = Shear stress on raks required to ahieve biaxial yielding o reinorement, v i max = Maximum onrete stress that an be transerred by aggregate interlok, V = Total shear resistane, V biaxial = Shear resistane when both longitudinal and transverse reinorement yield, V, V = Shear resistane attributed to onrete, stirrups; s V rush = Shear resistane at onrete rushing, V yield = Shear resistane at yielding o transverse reinorement, V p V u w z α α = Shear ore resisted by inlined prestressing tendons, = Total shear resistane at ultimate limit state, = Crak width, = Bending moment lever-arm in deep beam, = Conrete tension strength ator, portion o the total amount o distributed vertial reinorement that ontributes to the lexural apaity in squat shear wall, = Coeiient deining the relative ontribution o onrete to shear resistane o xviii

19 β onrete shear wall, = Conrete shear ontribution ator, n v = Predited rate o hange o n v per unit ε x, θ = Predited rate o hange o θ per unit ε x, δ s = Shear stain aused by rak slip, ε p ε x ε y ε z = Strain o prestressing tendons, = Average longitudinal strain over depth o member, = Yield strain o transverse reinorement, = Strain o transverse (z-diretion) reinorement, ε 1, ε 2 = Conrete tensile, ompressive strain in the prinipal diretion; ε ' = Conrete strain orresponding to onrete peak ompressive stress, φ φ p φ s = Conrete resistane ator, = Prestressed reinoring steel resistane ator, = Nonprestressed reinoring steel resistane ator, φ 0w =Overstrength ator in New Zealand Standards, γ, γ xz = Shear strain in x-z plane, λ, λ = Ratio d pw / d, p λ s = Ratio d w / d, λ pi = Ratio d pwi / d, λ j = Ratio d wj / d, θ µ = Dutility Fator in New Zealand Standards, = Angle o inlination o priniple ompression stress (diagonal raks), θ o = Predited value o θ at ε x = 0, θ p ρ h = Angle o prestressing tendons to horizontal axis (x-diretion), = Ratio o distributed horizontal reinorement to onrete area in squat shear wall, xix

20 ρ v ρ x ρ z = Ratio o distributed vertial reinorement to onrete area in squat shear wall, = Ratio o distributed longitudinal (x-dir.) reinorement area to onrete area, = Ratio o transverse (z-dir.) reinorement area to onrete area, ρ z min = Minimum transverse reinorement ratio, σ y = Yield stress o squat shear wall reinoring steel, xx

21 Aknowledgements I would like to express my sinere appreiation to my supervisor, Dr. Perry Adebar, or his valuable advie, onstant guidane, and enouragement during every stage o this researh. First part o this researh on shear strength evaluation o existing bridge girders was unded by the British Columbia Ministry o Transportation. Their support is grateully aknowledged. Speial thanks to the aulty, sta, and my ellow students at the University o British Columbia who provided me with a pleasurable and peaeul environment during this work. I am deeply grateul to my parents, Nasrin and Iraj, who have ontinuously supported me throughout my years o eduation. At last but not the least, I would like to express my deepest appreiation to my wie, Noushin, or her love, understanding, patiene, and never altering support during this study. I should also appreiate the new addition to the amily, my son Parsa, who motivated me in the last six months to work harder. xxi

22 Chapter 1. Introdution 1.1. Shear in Conrete Strutures The shear behaviour o strutural onrete is a omplex phenomenon. One approah to developing a theory or shear behaviour is to use the results o beam tests suh as the one shown in Figure 1-1. While suh tests are appropriate or pure lexure behaviour, as the region between the point loads is subjeted to uniorm bending moment, they annot be easily used to develop a general shear theory. The reason is that the setions in the shear spans between the point loads and supports are subjeted to varying bending moment even though they are subjeted to onstant shear as shown by the ree body diagram in Figure 1-1(b). As the applied ore P is inreased, shear ore together with bending moment and bending moment gradient along the shear span all inrease, whih makes it hard to extrat shear deormations rom total deormations. The at that transverse reinorement strain is not uniorm over the beam depth ompliates the matter urther. Fig. 1-1 Reinored onrete simply supported beam subjeted to shear and moment. 1

23 An alternate approah or developing a general shear theory is to use idealized elements with uniorm distributed reinorement in two diretions (e.g., vertial and horizontal) subjeted to uniorm shear and normal stresses (uniorm strains) and no bending moments (Fig. 1-2). Suh elements are simpler than beams and their omplete behaviour an be more easily investigated rom experimental results. Fig. 1-2 Uniorm shear elements Modiied Compression Field Theory The Modiied Compression Field Theory (MCFT, Vehio and Collins 1983) is a smeared rak rotating angle model that was developed rom tests perormed on uniorm shear elements. It predits the behaviour o uniorm shear elements throughout the whole range o loading rom irst raking until ailure. MCFT equations inlude equilibrium equations, ompatibility equations, and material onstitutive relationships. MCFT has also been used or the setional analysis o beams under ombined axial load, bending moment and shear, as well as the nonlinear inite element analysis o onrete strutures. For setional analysis, the beam setion is divided into layers, and 2

24 eah o these layers is assumed to at as a uniorm shear element. The layers are then linked by satisying global equilibrium o the setion in addition to ompatibility requirements suh as the well known assumption o plane setions remain plane. The proedure was implemented in omputer programs suh as Response 2000 (Bentz 2000) and veriied against signiiant number o experimental results o beams. Response 2000 and equivalent programs are sophistiated researh tools that provide onsiderable inormation suh as stress proiles, strain proiles, and ailure mehanisms. VeTor 2 (Wong and Vehio, 2002) is a nonlinear inite element program that employs the MCFT as onstitutive relationships or uniorm stress uniorm strain elements. VeTor2 an be used to analyze a variety o dierent onrete strutures inluding beams and squat shear walls. MCFT has also been used or simpliied shear design proedures that utilize a single uniorm shear element to approximately desribe the omplete setional shear behaviour. The Amerian Assoiation o State Highway and Transportation Oiials (AASHTO) Load and Resistane Fator Design (LRFD) bridge odes and the Canadian Highway Bridge Design Code (CHBDC) CSA S6 and the Canadian ode or onrete building strutures CSA A23.3 use MCFT-based methods in their shear design provisions Simpliied Shear Analysis Simpliied shear analysis an be done by using a single uniorm shear element to represent the behaviour o the shear resisting portion o a onrete struture. This approah an be applied to a beam as shown Fig The web o the beam is assumed to resist uniorm shear stress over the shear depth o the beam (the uniorm shear element), while ompression and tension hords are assumed to resist the applied bending 3

25 moment. The idealized ompression hord is the lexural ompression zone o the beam, while the tension hord is the zone ontaining onentrated lexural reinorement. The single element shear analysis an also be applied to onrete shear walls suh as the squat shear wall shown in Fig A ertain length o the wall is assumed to resist uniorm shear stress (the uniorm shear element), while the ends o the wall are assumed to resist the overturning. The appliation o single uniorm shear analysis to these two types o onrete strutures are investigated in the urrent thesis. Fig. 1-3 Prediting beam shear behaviour with one uniorm shear element. 4

26 Fig. 1-4 Prediting squat shear wall behaviour with one uniorm shear element Shear Strength Evaluation o Bridge Girders The strength o existing onrete bridge girders (see Fig. 1-5) need to be evaluated in order to determine the load apaity rating o bridges beause o inreased trai loads, or deterioration o bridges. The shear strength o onrete bridge girders oten limits the load apaity ratings o bridges. Current bridge design odes suh as 2007 AASHTO LRFD and the 2006 CHBDC inlude simpliiations that generally result in sae designs. The additional onstrution osts are justiied by the redued hane o a design error. On the other hand, the onsequene o these same simpliiations may be greater when a simpliied shear design method is used to evaluate existing girders that annot be made a little stronger. The simpliiations may result in unneessary load restritions on bridges or unneessary 5

27 repairs o bridge girders. Thus more omplex proedures are justiied or shear strength evaluation. Aording to shear design proedures based on the MCFT, suh as AASHTO LRFD and 2006 CHBDC, the shear strength o a girder is a untion o axial strain, whih depends on a number o ators inluding the applied shear ore. Using suh a design proedure to evaluate strength requires trial-and-error as the applied shear ore at ailure is needed to alulate shear strength. An alternative approah or shear strength evaluation is a omputer program suh as Response 2000, whih applies the MCFT. This would provide onsiderable insight suh as whether the ailure mode will be dutile (due to reinorement yielding), or brittle (due to diagonal rushing o onrete). While suh omputer methods are very useul or speial investigations, the omplexity o data input and output makes it diiult to use suh programs to hek numerous setions along a bridge, the program options may result in dierent users reahing dierent onlusions or the same girder, and it is not possible to onirm results o suh programs using hand alulations. There is a need or a shear strength evaluation proedure that aounts or more o the omplexities than a design method does and gives some insight into the shear ailure mode; but yet is simple enough that a user an implement the proedure into a small omputer program or heking numerous setions along a bridge, and an onirm the results o the omputer program by hand alulations. Developing suh a method is the objetive o the urrent thesis. 6

28 Fig. 1-5 An example o an existing onrete bridge (BC Ministry o Transportation) Design o Squat Shear Walls Clause 21 o the 2004 edition o the Canadian onrete ode CSA A23.3 ontains new provisions or the seismi shear design o squat walls that were developed using the MCFT to desribe the behaviour o a single uniorm shear element. These new design provisions have not been rigorously evaluated by omparing designs resulting rom the 7

29 proedure with the results o tests on squat walls or the results o nonlinear inite element analysis. The Amerian Conrete Institute model building ode ACI 318 and New Zealand Conrete ode NZS 3101 use empirial equations or the shear strength o squat shear walls that were developed rom test data. Test results may not represent the true lowerbound strength that is desired in odes. For example, in almost all squat wall tests, the shear ore was applied at the top o the wall to a load transer beam that an signiiantly enhane the shear apaity o the wall. The diaphragm that transers the ore in a real squat wall may provide muh less horizontal restraint at the top o the wall. Squat walls are signiiantly restrained at the base due to the large oundation that typially supports the wall. Treating the wall as a uniorm shear element may result in overly onservative designs. For example, Clause o the 2004 CSA A23.3 states that the vertial tension ore required to resist overturning at the base o squat walls shall be provided by onentrated reinorement and vertial distributed reinorement in addition to the amount required to resist shear. This requirement has greatly inreased the required amount o vertial reinorement in squat walls ompared to traditional designs. I an ation an develop in squat walls, this requirement is too onservative. The shear behaviour o squat walls needs to be investigated using a state-o-the-art nonlinear inite element analysis program suh as VeTor 2. O partiular interest is the horizontal restraint provided at the top o the wall by loading beams used during testing, and the horizontal restraint provided at the base o the wall by real oundations. Suh an analysis an be used to identiy any reinements that should be made to the seismi shear design provisions in the 2004 CSA A23.3 developed rom a single shear element 8

30 analysis Researh Objetives This thesis onsists o two parts. The objetive o the irst, and larger o two parts, is to develop a shear strength evaluation proedure or onrete bridge girders. The evaluation proedure should give insight into the shear ailure mode o onrete bridge girders and yet be simple enough that a user an implement it into a small omputer program or heking numerous setions along a bridge, and an onirm the results o suh a omputer program by hand alulations. The proedure must be validated by omparing preditions with results rom strength tests on a signiiant number o reinored and prestressed onrete beams. Preditions rom the proedure or atual existing bridge girders should also be ompared with preditions rom the more omplex evaluation proedure that results rom using Response In addition to developing a standalone shear strength evaluation proedure, any reinements that an be made to the shear design proedure in the 2006 CHBDC so that the proedure is more appropriate or shear strength evaluation should also be made. An obvious possibility is to develop a reined proedure to alulate the axial strain used in the uniorm shear element o bridge girders. Any suh proposals must also be validated by omparing with test results and Response 2000 preditions. The objetive o the seond part o the thesis is to investigate the shear behaviour o squat shear walls using nonlinear inite element analysis. The investigation should inlude the inluene o the horizontal restraint provided at the top o the wall by loading beams used during testing, and the horizontal restraint provided at the base o the wall by real oundations. The analysis an be used to identiy reinements that an be made to the 9

31 seismi shear design provisions or squat walls in the 2004 CSA A23.3 suh as the requirement or vertial reinorement or shear all along the base o the wall. Typial designs rom the 2004 CSA A23.3 method need to be ompared with designs using ACI 318 and NZS Thesis Organization This thesis ontains 10 hapters and six appendies. Part 1 o the thesis, whih involves the shear strength evaluation o onrete bridge girders, is presented in Chapters 2 to 6. Part 2, whih involves the shear design o squat shear walls, is presented in Chapters 7 to 9. The detailed organization o the thesis is given below. Chapter 2 presents a brie review o general literature on the shear strength o beams and bridge girders and a summary o the relevant researh that has reently been done. Chapter 3 disusses uniorm shear elements and the MCFT, as well as MCFT-based design methods in the 2006 CHBDC and 2007 AASHTO LRFD shear design provisions. It presents new MCFT-based methods or shear evaluation o members with and without transverse reinorement and ompares preditions rom these methods with preditions o the 2006 CHBDC and the 2007 AASHTO LRFD shear design methods. Chapter 4 presents the appliation o the proposed uniorm shear element method presented in Chapter 3 to beam elements. To validate the method, results rom the proedure are ompared with the results rom Response 2000, the 2006 CHBDC and 2007 AASHTO LRFD design methods or three existing onrete bridges. Chapter 5 ompares results rom large-sale tests with the proposed method preditions or both beams with and without transverse reinorement. Chapter 6 presents the proposed reined CHBDC shear design method and its validation against test data. 10

32 Chapter 7 begins Part 2 o the thesis by presenting a summary o the previous work done on shear strength o squat shear walls. Chapter 8 briely explains the inite element program used to predit nonlinear behaviour o squat shear walls and ompares its preditions with three tests rom the literature. The eet o a rigid loading beam as typially used in previous experimental work is also investigated. Chapter 9 presents a new method to alulate lexural strength o squat walls aounting or shear lexure interation at base o the walls as well as the 2004 CSA A23.3 reined method or shear strength o those walls, while Chapter 10 presents summary and onlusions o the thesis. Two Exel Spreadsheets are inluded with this report (Appendix A). The irst is or shear strength evaluation aording to the method desribed in Chapter 3 and 4. The seond is or the reined shear design proedure desribed in Chapter 6. Appendix B ontains detailed instrutions on how to apply the shear strength evaluation proedures desribed in Chapter 3. Suiient detail is provided so that anyone an write their own spreadsheet to apply the method, or hek any step in the spreadsheet provided in Appendix A. Appendix C presents three worked examples o shear strength evaluation o atual onrete bridge girders. Appendix D provides details o rating truks used in the bridge examples presented in Chapter 4. Appendix E presents inormation on the tests used to veriy the proposed methods in Chapter 5 and 6. In addition, proposed method preditions as well as other ode preditions are given. Appendix F provides additional plots or omparison o test data with the proposed method and ode design methods preditions. The plots illustrate the 11

33 trend o test-to-predited shear strength ratios with important parameters in shear suh as shear stress ratio, eetive depth, onrete ompressive strength, and longitudinal reinorement ratio. 12

34 Chapter 2. Literature Review: Beam Shear Strength 2.1. Review o Previous Studies Ritter (1899) and later Mörsh (1920, 1922) introdued the 45 degree truss angle to predit shear behaviour o onrete beams. The model assumes that ore low an be idealized by a truss in whih onrete lexural ompression hord is the horizontal ompression member, longitudinal tensile reinorement is the horizontal tensile element, ompression elements ormed between the raks are the inlined elements, and stirrups are vertial elements and the angle o inlined raks is assumed to be 45 degrees. In this method, vertial omponents o inlined ompression ores in the diagonal struts are assumed to be balaned by stirrup ores only and no post raking resistane is onsidered or onrete at the raks. Withey (1906, 1907) and Talbot (1909) perormed experimental investigation and showed that 45 degree truss model is too onservative. This led to evaluation o 45 degree truss model by a number o researhers and the idea o ontribution o onrete to shear resistane o beams and the traditional ormulation o: [2-1] V = V + Vs [2-2] V s = A v y d v s otθ in whih V is onrete ontribution to shear resistane, V s is stirrup ontribution to shear resistane, A v is the area o transverse shear reinorement within a distane s along the member, y is the yield strength o the transverse reinorement, d v is the eetive shear depth o the member traditionally assumed equal to depth rom ompression ae to entroid o lexural tension reinorement d, and θ is the angle o inlination (measured 13

35 rom the longitudinal axis) o the onrete prinipal diagonal ompressive stress, i.e., angle o ritial diagonal rak traditionally assumed equal to 45 degrees. 45 degree truss model plus onrete ontribution to shear V is also reerred to as modiied truss analogy in the literature. Consequently, many researhers suh as Morrow and Viest (1957), Bresler and Pister (1958), Hanson (1958), Guralnik (1959), Viest (1959) investigated raked onrete ontribution to shear strength o the beams and proposed dierent equations to quantiy V. ACI-ASCE shear ommittee 326 (1962), based on available experimental investigations, suggested the ollowing ormula or shear strength o reinored onrete beam elements. Vd [2-3] V = ( 1.9 ' ρ w ) bwd 3.5 ' bwd in psi units M where ' is ompressive strength o onrete, ρ w is ratio o lexural tension reinorement area to onrete area, b w is web width, d is depth rom ompression ae to entroid o lexural tension reinorement, and M and V are atored bending moment and shear ore ating at the setion o interest. This ormulation is still in AASHTO standards as well as ACI 318. For ordinary RC beams, Bresler and Sordelis (1963) proposed the ollowing simple equation or V. [2-4] V = 2 ' b d in psi units ( V = 0.17 ' b d in MPa unis) w In 1962, Leohardt and Walters explained the shear ailure modes o beam ation and arh ation through a series o tests. They showed that deep beams an transer shear to the supports by ompressive stresses in onrete whose low is like an arh. Thus deep beams have a higher shear resistane ompared to shallow beams. Later, Kani (1964) w 14

36 introdued his omb theory and tooth ailure mehanism to predit the shear strength o beams. In his model, inlined shear raks along a beam would orm a omb whose base is the beam top ompression hord and teeth are reinored onrete between the raks. Shear ailure then happens due to raking o the root o one tooth as a onsequene o the ore aused by longitudinal reinorement due to shear. For prestressed onrete beams, MGregor (1960) introdued the idea o using the lesser o web shear raking load V w and lexural shear raking load V i or onrete ontribution to shear strength by testing prestressed beams subjeted to moving loads. The idea was to identiy the probable raking mode and determine onrete shear ore aordingly. Two types o raking were notied in experimental results. Some raks alled web-shear raking initiated in an angle at almost mid-depth o web and extended toward both ompression and tension hords. Some others alled lexural shear raking initiated rom lexural raks and inlined toward the web. This was urther investigated and reined by other researhers suh as Hernandez et al. (1960) and Mattok and Kaar (1961), MaGregor et al. (1965 and 1966), Oleson and Sozen (1967), and MaGregor and Hanson (1969). These studies led to the ollowing V i, V w ormulas, whih are semi theoretial/experimental and are still in the urrent AASHTO Standard Speiiations and ACI 318. ViM r [2-5] Vi = 0.6 ' bwd + Vd ' bwd in psi units M max [2-6] V w = ( 3.5 ' pe ) bwd + V p in psi units where V d is dead load shear ore at the setion o interest, M max is maximum atored moment at the setion due to externally applied load, V i is atored shear ore at the 15

37 setion due to externally applied load orresponding to M max, M r is the external moment whih auses initial lexural raking o the setion, pe is the eetive stress in the prestressing steel ater losses, and V p is the vertial omponent o inlined prestressing ore. European researhers introdued the variable angle truss model. The model is similar to the traditional 45 degree truss model but assumes angle o inlination o diagonal ompression is variable. Angle o inlination o raked onrete ould not be determined by equilibrium equations beause there were three equilibrium equations versus our unknowns. Thereore, Kuper (1964) and some others solved the problem using the priniple o minimum energy while other researhers suh as Leonhardt and Walter (1964), Kuyt (1972), Nielsen and Braestrup (1975), Thürlimann (1979), Thürlimann et al. (1983), and Nielsen (1984) took the plastiity approah or this purpose. It is worth mentioning that solving truss model using equilibrium approah is a lower-bound approah based on the theory o plastiity; thus, a range o possible angle o inlination may be used to determine a onservative predition or shear strength o a onrete member. Mithell and Collins (1974) and Collins (1978) developed the Compression Field Theory (CFT). The theory solves equilibrium equations making use o additional equations o material onstitutive relationships and ompatibility equations and is apable o prediting the shear behaviour o onrete beam elements in the entire range o loading up to ailure. Collins (1978) veriied CFT by an experimental investigation. It should be mentioned that CFT assumes that onrete is inapable o arrying tension stresses ater raking. 16

38 Vehio and Collins (1983) developed the Modiied Compression Filed Theory (MCFT) or uniorm shear elements (retangular shear elements with uniorm stresses and strains in every diretion). Similar to CFT, MCFT aounts or ompatibility o strains, material onstitutive relationships and equilibrium equations, but it also aounts or onrete ontribution in tension ater raking (tension stiening eet) as well as the eet o biaxial strains on the onrete stress-strain relationship when onrete is in ompression. The theory was developed based on experimental results o the tests perormed on uniorm shear elements. MCFT inludes two sets o equilibrium equations namely average stress equations and equations o stresses at raks to ensure that average stresses an be transerred at raks by aggregate interlok. It is worth mentioning that both CFT and MCFT assume that the diretions o prinipal average stresses and prinipal average strains in onrete oinide; thus, rak angle hanges throughout the loading ater raking. Later MCFT was used to develop more pratial and simpler equations or shear design. Amongst those are the AASHTO LRFD and the 2006 CHBD (Canadian Highway Bridge Design) ode approahes whih were developed by Collins et al. (1996) and Bentz et al. (2006), respetively. The 2006 CHBD and 2007 AASHTO LRFD shear design methods are explained later in this hapter and Chapter 3. Vehio and Collins (1986) explained how uniorm shear elements an be used to predit the behaviour o beam elements under shear. They used multi-layer analysis and linked the layers by the assumptions o plane setions remain plane and satisying equilibrium at dual setions o the beam whih are losely spaed. They also explained that by assuming a ertain shape or shear low proile (retangular, paraboli, et) the 17

39 proedure is simpliied to one setion rather than dual setions; however the results are approximate. Vehio and Collins (1988) method was later employed in omputer programs Response (Felber, 1990) and Small (Ho, 1994) to predit the behaviour o onrete setions under shear ore, bending moment and axial ore. Bentz (2000) developed Response 2000 that also predits the response o beam setions. Response 2000, whih is the most omplete program o its kind, is signiiantly improved in numerial tehniques and is a sophistiated researh tool. It provides detail inormation about the behavior o onrete beam setions subjeted to bending moment, shear and axial ore. The inormation, whih is provided throughout the entire range o loading up to ailure, inlude load-deormation urves, steel and onrete stress and strain proiles, shear stress on raks, rak diretions and widths, and et. Bentz (2000) veriied Response 2000 against the 534 tests reported in the literature. More details about Response 2000 are provided in Chapter 4. Between 1987 and 1997, Hsu together with other researhers developed two methods alled the Rotating Angle Sotened-Truss Model (RA-STM) and the Fixed Angle Sotened-Truss Model (FA-STM) to predit shear behaviour o uniorm shear elements. Both methods predit the behaviour o uniorm shear elements by solving equilibrium equations making use o ompatibility and onstitutive relations. Like MCFT, RA-STM assumes that the diretions o prinipal stresses and strains oinide while FA-STM assumes that ater raking rak angle does not hange; thus, prinipal ompression stress does not oinide with rak diretion. Both methods use dierent material 18

40 onstitutive laws rom the ones used in MCFT and aount or loal stress eet at raks by reduing the average strength o reinorement. Other researhers suh as Loov (1978, 1998), Gambarova (1987), Reinek and Hardjasaputra (1990), Reinek (1991), and Loov and Patnaik (1994) took the truss model with rak rition approah to solve shear problems o beams. In this model, two ores parallel (rition ore) and perpendiular to rak surae in addition to stirrup ores are assumed to at on the shear rak plane o the beam. The summation o the vertial omponents o these two ores is onrete ontribution to shear V. The equilibrium equations are solved using the onstitutive laws or the transer o ores aross the raks by rition whih depend on shear load level, strain level and rak spaing. The simpliied rak rition models use a onstant V whih is mostly a untion o longitudinal reinorement ratio, beam size and material properties Reent Code Approahes In this setion some o the reent ode provisions or shear design o onrete beams are reviewed ACI and 2002 AASHTO Standards ACI and 2002 AASHTO Standards assume that shear ailure happens at rak angle o 45 deg and onrete ontribution to shear at ailure is equal to the load at whih diagonal raking is expeted to our. The method assumes shear strength is proportional to the member depth thus does not inlude size eet. Both odes use Eqs.[2-1] and [2-2] or shear strength o beams where θ =45 deg, d v =d, V is alulated rom Eq. [2-4] or reinored onrete members and the lesser o Eqs.[2-5] and [2-6] or 19

41 prestressed onrete members. Both odes limit the shear strength o a beam to V + 8 ' b d (in psi units) to prevent onrete rushing. w AASHTO LRFD AASHTO LRFD shear design method, whih is still present in the 2008 AASHTO LRFD as an alternative method, was derived rom MCFT assuming uniorm shear low along the beam eetive shear depth d v assumed to be 0.9 d. While the longitudinal strain varies over the beam depth, the method uses one longitudinal strain ε x at a ertain depth o the beam to determine the shear strength using a uniorm shear element approah. The shear strength is given by: Av ydv [2-7] V = V + Vs + Vp = β ' bwd v + otθ + Vp s where onrete ontribution ator β and angle o prinipal ompression θ are untions o longitudinal strain ε x and shear stress ratio v ' or members with transverse reinorement, while they are untions o ε x and rak spaing parameter S xe (a untion o beam size and maximum onrete aggregate size) or members without transverse reinorement. The values o β and θ are given in tables. The longitudinal strain ε x is determined rom: M / d v + 0.5V otθ p Ap [2-8] ε x = : or members with transverse reinorement 2( E A + E A ) s s p p M / d v + 0.5V otθ p Ap [2-9] ε x = : or members without transverse reinorement ( E A + E A ) s s p p 20

42 where: M = bending moment at setion o interest; V= shear strength at the setion o interest; p = eetive prestressing ore; A s, A p = area o nonprestressed and prestressed lexural tension reinorement; and E s, E p = Modulus o Elastiity o nonprestressed and prestressed reinorement, respetively. The ε x equations are dierent or members with and without transverse reinorement by a ator o 2 as the longitudinal strain is taken at mid-depth or members with transverse reinorement, and at the entroid o lexural tension reinorement or members without transverse reinorement. I the longitudinal strain is negative in the 2007 AASHTO LRFD method, the onrete ompression stiness must be added to the denominator o Eqs. [2-8] and [2-9] AASHTO LRFD method requires trial-and-error or design sine ε x, whih is not known initially, is needed to determine β and θ values rom the tables. In this method, it is also neessary to hek i there is suiient strength in the longitudinal reinorement to resist the extra demand aused by shear. Also shear stress is limited to 0.25 ' to avoid onrete rushing in the web prior to stirrup yielding. More details about this method is given in Chapter CSA A23.3 and CHBDC CSA A23.3 method or shear design o beams, whih is the same as 2000 Canadian Highway Bridge Design Code (CHBDC) method, is essentially the same as 2007 AASHTO LRFD method exept it uses the longitudinal strain at the entroid o lexural tension reinorement Eq. [2-9] or both members with and without transverse 21

43 reinorement. In addition, the method does not allow negative longitudinal strains unlike 2007 AASHTO LRFD that allows negative ε x values CSA A23.3, CHBDC 2006, and AASHTO LRFD 2008 Similar to 2007 AASHTO LRFD method, this method is based on MCFT and uses Eq. [2-7] to determine shear strength o beams, but it provides equations or β and θ rather than tabulated values. Further simpliiations in this method resulted in equations or β and θ that are not untions o shear stress ratio v ' or members with transverse reinorement. β and θ are untions o longitudinal strain ε x only. In addition, the equations or β and θ are the same or both members with and without transverse reinorement. The equations are: [2-10] β = in psi units ε x s xe sxe [2-11] θ = ( ε x )( ) 75deg 2500 where s xe is the size eet parameter taken as 300 mm or members with transverse reinorement and varies or members without transverse reinorement depending on depth o elements, vertial spaing o longitudinal reinorement layers, and onrete maximum aggregate size. In 2008 AASHTO LRFD and 2004 CSA 23.3, Eq. [2-11] is onservatively simpliied to θ = ε or members without transverse x reinorement. The method assumes that mid-depth strain is the reerene strain or both members with and without transverse reinorement. In addition, it simpliies Eq. [2-8] by 22

44 substituting V osθ with 2V in the numerator to avoid trial-and-error proedure or design purposes. ε x is given by: [2-12] ε x = M / d s v 2( E A + V s + E p p A A p ) p The strain ε x used in 2008 AASHTO LRFD equations or β and θ is exatly twie the value used in 2004 CSA A23.3 and 2006 CHBDC (Eq. 2-12), and thus1500 in Eq. [2-10] is replaed by 750 εx, and 7000ε x in Eq. [2-11] is replaed by 3500 ε x. As a result, the methods inal preditions or shear strength o beams remain the same. This method uses the same equations and limits as 2007 AASHTO LRFD method to hek the suiieny o longitudinal reinorement or shear and avoid onrete rushing. More details are provided in Chapter 3. The 2008 AASHTO LRFD also has a simple design proedure that was proposed by Hawkins et al. (2005). This method is disussed in Setion 2-3. ε x 2.3. Reent Studies Oh and Kim (2004) They tested two ull-sale post-tensioned prestressed onrete girders that were mm long and 1200 mm deep, and had web transverse reinorement and web width o 180 mm. The geometry o the beams were the same but one o them was high strength onrete ( = 60 MPa) while the other one was normal strength onrete ( ' = 40 ' MPa). 12 seven strand prestressing tendons with 12.7 mm diameter and the prestressing ore o MPa were used in the girders. Girders were loaded by an asymmetrial point load up to ailure. They alulated prinipal strains, shear strain and angle o 23

45 inlination based on the measured strains in the horizontal, vertial and 45 degree diretions. Based on experimental results, they onluded that the onept o ompatibility o strains and rotation o rak angle throughout loading was well suited or prediting prestressed onrete beam behaviour in shear or both normal strength and high strength onrete. Cladera and Marı (2004 and 2005) They proposed simpliied shear design proedures or onrete beams with and without transverse reinorement. For beams with transverse reinorement, they used available shear test results o 123 reinored onrete beams in the literature and employed Artiiial Neural Networks (ANNs) to develop their proposed shear design proedure. The seleted database had shear span-to-depth ratio a/d equal to or greater than 2.49, ' rom 21 to 125 MPa (less then 80 MPa in 80% o the tests), eetive depth rom 198 to 925 mm (depth o less than 600 in. in 90% o the tests), transverse reinorement ρ z y rom 0.33 to 3.57 MPa, and longitudinal reinorement rom 0.5% to 5.8%. It was observed that AASHTO LRFD preditions were the losest to ANNs results ompared to EC2 and ACI preditions. As a result, they used AASHTO LRFD tabulated values or θ to develop the ollowing equation or members with web reinorement, whih was believed to be onservative ompared to the AASHTO LRFD tabulated values: [2-13] θ = ε + 45 x τ ' In the above equation, ε x is mid-depth strain, and τ is shear stress equal to V/b w d v. They used the same ormula as in the 2004 CSA A23.3 or ε x. 24

46 For onrete ontribution V in members with web reinorement, they developed a ormula with the same ormat as in EC2 but the inluene o some parameters were hanged based on ANNs results. Their proposed ormula or V was: [2-14] V = [0.17ξ (1000ρl ) 1/ ' τ 1/ σ p ] bwd v where ρ l is ratio o lexural tension reinorement area to onrete area, σ p is normal stress due to prestressing or axial load, and ξ is the size eet parameter equal to: 200 [2-15] ξ = s x in whih s x is the lesser o d v or spaing o longitudinal vertial reinorement in the web. Finally, they veriied their proposed method or members with stirrups against 162 reinored onrete beams and 40 prestressed onrete beams reported in the literature. Kuhma et al. (2005) (NCHRP Projet 12-56) As part o a National Cooperative Highway Researh Program (NCHRP) Projet 12-56, Kuhma et al. tested 6 high-strength prestresssed onrete girders to investigate the appliability o available shear design proedures espeially the AASHTO LRFD method or high strength prestressed onrete. They tested speimens that had 50 t lear span, ' rom 10 to18 ksi, depth o 73 in., and 26 to 42 prestressing strands o 0.6 inh diameter. All girders had transverse reinorement and were subjeted to uniorm load (see Kim 2004 or more details). Some o the important onlusions rom the study are: Draped strands (6 draped strands were used) improved the raking shear strength 16%-23%, as well as the ultimate shear strength 15%-16%. Draped strands and horizontal web strands provided good rak ontrol. 25

47 Welded wire reinorement (WWR) improved the ability o tested beams in redistribution o shear ores between stirrups. This is beause WWR ould sustain large strains. A sudden inrease in transverse strain was notied immediately ater raking. A number o the speimens ailed due to web rushing at the base o the web over the support. The ailure indiated that the shear stresses were not uniorm over the depth; but were onentrated in a ompression strut. The AASHTO LRFD approah o limiting the shear stress assumed to be uniorm over the shear depth to 0.25 results in a sae predition o these tests as well. AASHTO LRFD method provided aurate preditions or the shear strength o the speimens. The ratios o measured to predited strengths or the 20 tests ranged rom 0.97 to 1.29, had an average value o 1.12 and a COV o Hawkins et al. (NCHRP Report 549, 2005) This projet involved evaluating the most prevalent shear design proedures inluding the shear provisions in AASHTO LRFD, 2004 CSA A23.3, ACI 318 as well as JSCE (Re. 44), EC2 (Re. 28) and DIN (Re. 27) against 1359 test results available in the literature. They ound that among all the shear design proedures in building odes, AASHO LRFD and the 2004 CSA A23.3 provide the best preditions with only 10% probability o being unonservative. Based on their evaluation results, they reommended that one use either the 2004 CSA A23.3 provisions or the ollowing modiied ACI proedure: For reinored onrete beams, θ = 45 deg. and [2-16] V = 1.9 ' bwd v (in psi units) For prestressed onrete beams,θ = 45 deg. i Vi Vw or M u M r otherwise: ' 26

48 [2-17] pe otθ = 1+ 3 (in psi units) ' and V is the lesser o: Vi M r [2-18] V i = ' bwd v + Vd ' bwd v (in psi units) M [2-19] V w = ( 1.9 ' pe) bwdv + Vp (in psi units) max where: d v is the eetive shear depth o the member whih may be taken as 0.9d. V w in Eq. [2-25] is signiiantly redued rom the traditional ACI 318 web shear raking expression. Hawkins et al. ompared preditions rom the proposed simpliied method above with the results o 147 tests they seleted rom a database o 1359 shear tests. They seleted test data o members that ontained minimum transverse reinorement ( ρ z y >50 psi [0.35 MPa]), had an overall depth o at least 20 inhes (500 mm), and were ast rom onrete having a ompressive strength o at least 4 ksi (28 MPa). They exluded tests in whih anhorage or lexural ailure ourred. The results showed that the 2006 CHBD and AASHTO LRFD shear design provisions gave the best preditions ompared to the test results; however, their proposed method had onsiderably better preditions than those rom ACI 318. Hawkins et al. also ompared the preditions o Response 2000 with test results and reported that Response 2000 preditions were better than the preditions o evaluated odes Conluding Remarks Reent studies (Hawkins et al. 2005, Kuhma et al. 2005, Cladera and Marı 2004 and 2005) have shown that the 2007 AASHTO LRFD and 2006 CHBDC shear design 27

49 methods whih were developed based on MCFT are two o best methods ompared to other evaluated ode methods when their preditions are ompared with experimental results. Response 2000 whih perorms sophistiated MCFT-based setional analysis or beams and expliitly aounts or moment shear interation has also been evaluated and ound to be better than MCFT-based simpliied methods in 2007 AASHTO LRFD and the 2006 CHBDC (Hawkins et al. 2005). On the other hand, modiied truss analogy whih uses the 45 angle truss model in addition to onrete ontribution to shear suh as shear design methods in ACI 318 and EC2 was ound not to be as onsistent with experimental results (Hawkins et al. 2005, Cladera and Marı 2004 and 2005). As a result, MCFT as well as MCFT-based setional analysis or beams (Response 2000) are used in this study to develop a new proedure or shear evaluation o onrete bridges. In addition, the preditions o the proposed proedure are ompared to 2007 AASHTO LRFD, 2006 CHBDC, and ACI 318, whih is one o the modiied truss analogy methods in the literature that has been used or many years. Finally, the proposed equations are veriied against numerous test results reported in the literature. 28

50 Chapter 3. Uniorm Shear Elements 3.1. General A uniorm shear element has uniormly distributed reinorement in two diretions parallel to the element sides and is subjeted to uniorm shear and normal stresses. Suh elements are simpler than beams and their omplete behaviour an be more easily investigated rom experimental results. This hapter presents methods that an be used to predit the behaviour o uniorm shear elements. How the shear behaviour o beams an be approximated using a single uniorm shear element is disussed in Chapter 4. The Modiied Compression Field Theory (MCFT, Vehio and Collins 1983) is one o the theories used that an predit the behaviour o uniorm shear elements. It is a smeared rotating rak angle model that was developed rom tests. MCFT is disussed in this hapter. In addition, simpliied MCFT-based methods available in odes and the literature are reviewed and disussed. A new shear evaluation proedure or uniorm shear elements is proposed. The proposed evaluation proedure is ompared with MCFT and the preditions rom the ode simpliied proedures Uniorm Shear Element Uniorm shear element is a retangular membrane element (subjeted to in-plane shear and axial stresses) with uniormly spaed longitudinal (x-diretion) and transverse (zdiretion) reinorement, no inlined prestressing tendons, and subjeted to uniorm applied normal stresses x and z and shear stress v. Suh an element will have uniorm reinorement stresses sx and sz, and uniorm onrete stresses x, z and v =v as shown in Fig Figure 3-1 also shows that this element may be used to model a portion o an I-girder subjeted to in-plane shear and moment. 29

51 Fig. 3-1 Element o reinored onrete subjeted to uniorm shear and normal stresses Modiied Compression Field Theory (MCFT) Modiied Compression Field Theory (MCFT) was developed by Vehio and Collins (1986) rom testing reinored onrete elements subjeted to uniorm shear stress. It is a smeared, rotating rak model where the inlination o diagonal raks is determined by ombining equilibrium requirements, strain ompatibility assumptions and empirial average stress average strain relationships or raked onrete and reinorement. The MCFT an be used to predit the shear stress shear strain relationships o strutural onrete membrane elements with dierent amounts o transverse (z-diretion) and longitudinal (x-diretion) reinorement. Any struture in whih ores are primarily transerred through in-plane ation an be idealized as a ombination o uniorm shear elements. In the ollowing setions, MCFT equations or material onstitutive 30

52 relationships, equilibrium equations, and ompatibility are explained Material Constitutive Relationships Paraboli stress-strain relationship as shown in Fig. 3-2 is onsidered or onrete in ompression in the prinipal diretion. Craked onrete sotens when subjeted to biaxial strains ompared to onrete uniaxial stress-strain relationship. As a result, the prinipal ompressive strength (peak stress) may be signiiantly lower than the uniaxial strength when onrete is subjeted to signiiant tension strain transverse to the prinipal ompression. Vehio and Collins (1986) showed that the redution in onrete strength (peak stress) in suh ases an be predited by the ollowing equation: ' [ 3-1] 2 max = ' ε / ε ' where 1 2 max is onrete peak stress under biaxial strains, ε 1 is onrete prinipal tensile strain, is onrete peak stress under uniaxial ompression and ε ' is onrete strain ' orresponding to onrete peak ompressive stress. Thereore, a paraboli stress-strain relationship o onrete in the prinipal ompressive diretion an be expressed as: 2 [ 3-2] 2 ε 2 ε 2 2 = 2 max ' ' ε ε in whih 2 is onrete ompressive stress in the prinipal diretion and ε 2 is onrete ompressive strain in the prinipal diretion. 31

53 Fig. 3-2 Conrete average stress-strain relationship in ompression (Vehio and Collins, 1986). One o the important aspets o MCFT is the model or prinipal onrete tension stresses. It aounts or onrete ontribution to reinoring bar stiness ater raking, whih is alled tension stiening and has signiiant inluene on onrete ontribution to shear strength o a reinored onrete element. In MCFT onrete model, onrete tension stress inreases linearly until raking. Ater raking, onrete ontinues to resist an average tension stress but it redues as the prinipal tensile prinipal strain inreases. The original onrete tensile stress-strain relationship in MCFT is shown in Fig. 3-3 and is given by: [ 3-3] 1 = E.ε 1 ε1 ε r [ 3-4] r 1 = 1 r ε 1 ε > ε 32

54 Fig. 3-3 Conrete average stress-strain relationship in tension (Vehio and Collins, 1986). Eq. [3-4] was later hanged to a more onservative equation (Collins et. al 1996, Rahal and Collins 1999, Bentz et. al 2006, and Bentz and Collins 2006) as: [ 3-5] 1 r = ε 1 The stress-strain relationship expressed by Eqs. [3-3] to [3-5] aounts or average tensile stress in onrete in the prinipal diretion and is valid i aggregate interlok in addition to stress inrease in the reinoring steel at the raks are apable o equilibrating average stresses. Otherwise, tensile stress must be redued aordingly. The maximum shear stress that an be resisted by aggregate interlok along a rak is given by: [ 3-6] 0.18 ' v i max = (in MPa units) w( a + 16) g where ag is onrete maximum aggregate size, and w is rak width determined rom: [ 3-7] w = s θ ε1 33

55 in whih s θ is rak spaing and is assumed equal to: [ 3-8] s θ 1 = sinθ osθ + s x s z where sx is rak ontrol parameter o x-diretion reinorement, and sz is rak ontrol parameter o z-diretion reinorement. For members with at least minimum amount o reinorement, rak spaing may be onservatively assumed as s θ =300 mm (Collins et. al 1996, Rahal and Collins 1999, Bentz et. al 2006, and Bentz and Collins 2006). A bilinear stress-strain relationship is used or reinoring steel. Prior to yielding o reinorement, the steel stress is assumed to be E ε where E s is the Modulus o s s Elastiity o steel ( MPa) and ε s is the average strain o reinorement. Ater yielding, the steel stress remains onstant and equal to y (steel yield stress) no strain hardening is assumed Equilibrium Equations Equilibrium equations o MCFT an be expressed by Mohr irles o stresses as shown in Figure 3-4. Mohr irle o raked reinored onrete stresses is a summation o Mohr irles o reinorement stresses and onrete stresses. Reinorement is onsidered to take axial load in the diretion o the reinorement, and thus does not resist shear stress. As a result, the shear stress in Mohr irle o onrete stresses is equal to the one in Mohr irle o reinored onrete stresses. Mohr irles o stresses shown in Fig. 3-4 an be ormulated as: [ 3-9] = x ρ x + sx 1 v otθ 34

56 [ 3-10] = z ρ z + sz 1 v tanθ [ 3-11] v = ( ) /(tan θ + otθ ) where x and z are normal stresses in x and z-diretions, ρ x and ρ z are reinorement ratios in x and z-diretions, sx onrete angle o prinipal diretion o stresses. and sz are reinorement stresses in x and z-diretions, and θ is The equilibrium equations above are in terms o average stresses. It is also neessary to hek stresses at the raks. Fig. 3-5 ompares the ree body diagram o a uniorm shear element on average and at the raks. At the raks, onrete tension stress in the prinipal diretion 1 beomes zero and aggregate interlok stress v i ontributes to equilibrium instead. In addition, reinoring steel stresses may be higher at the raks ompared to the average stresses. Equilibrium equations at the raks are: [ 3-12] x = ρ x sxr vi otθ v otθ [ 3-13] z = ρ z szr + vi tanθ v tanθ Reinorement Conrete Reinored Conrete Fig. 3-4 Mohr irle o stress or raked reinored onrete (adopted rom Vehio and Collins, 1986). 35

57 Fig. 3-5 Free body diagram o a uniorm shear element in the rak diretion or average stresses and loal stresses at the raks (Collins et. al 1996). where v i is stress along the raks due to aggregate interlok, and sxr and szr are reinorement stresses at the raks in x and z-diretions, respetively. The ollowing onditions need to be heked in order to determine whether the average tensile onrete stress an be equilibrated at the raks or the average tensile stress in onrete needs to be redued (Bentz 2000). [ 3-14] vi2 = ρ x ( sxr sx ) ρ z ( szr sz ) sinθ osθ 2 2 [ 3-15] 1 ρ x ( sxr sx )os θ + ρ z ( szr sz ) sin θ [ 3-16] 1 ρ x ( sxr sx ) + min( vi max, vi2 ) otθ [ 3-17] 1 ρz ( szr sz ) + min( vi max, vi2) tanθ 36

58 In the equations above, vi2 yielding o reinorement and Equation [3-6]. is the shear stress on raks required to ahieve biaxial vi max is the aggregate interlok apaity determined rom Compatibility Equations Although raks in onrete represent disontinuities, the average strains over a length ontaining a number o raks are onsidered to satisy requirements o ontinuous materials in MCFT. Consequently, like all other ontinuous materials, ompatibility in reinored onrete is expressed by Mohr irle o strains as shown in Figure 3-6. Some important ompatibility equations are: Fig. 3-6 Mohr irle o strains or reinored onrete (adopted rom Vehio and Collins, 1986). [ 3-18] 2 tan θ ε x = ε z + ε + ε 2 2 [ 3-19] ε 1 ε + ε + ε 2 = x z 37

59 [ 3-20] γ ( ε + ε ) otθ xz = 2 x [ 3-21] ε1 ε ( 1+ ot θ ) + ε 2 ot θ = x where ε 1 and ε 2 are strains in prinipal diretions, ε x andε z are strains in x and z- diretions, γ xz is shear strain, and θ is prinipal ompression strain diretion to x axis. It should be mentioned that MCFT assumes the diretion o prinipal strain oinides with the diretion o prinipal average stress. In other words, MCFT assumes θ = θ Solution o MCFT Equations Determining stresses given strains using MCFT equations is an easy task; however, alulating strains rom given ores is tedious and requires trial-and-error. For the latter ase, two unknowns are estimated in the beginning and solving equations veriies whether the estimated values are orret or need to be hanged. For example, one solution strategy is to estimate onrete stresses in the x and z-diretions. Then, alulate steel stresses in the x and z-diretions in addition to onrete angle o inlination θ and prinipal stresses rom equilibrium equations. Subsequently, determine strains in the prinipal, x and z-diretions using material onstitutive laws or onrete and steel. At this stage, rak equilibrium onditions should be heked and prinipal tension strain should be adjusted aordingly. The alulated strains should then satisy ompatibility equations otherwise the estimated onrete stresses should be revised and the proedure should be repeated until ompatibility equations are satisied Experimental Veriiation MCFT was developed (Vehio and Collins, 1986) based on experimental tests perormed on 30 uniorm shear elements under the variety o uniorm biaxial stresses. 38

60 Reently, Bentz et al.(2006) have ompared the experimental results o 100 uniorm shear element tests available in the literature with MCFT preditions and onluded that MCFT predits the behaviour o suh elements with an average test-to-predited shear strength ratio o 1.01 and oeiient o variation o 12.2%. As shown in Fig. 3-7, Bentz et al. (2006) also ompared experimental results o shear stress-shear strain relationship rom six tested elements with the preditions o MCFT. The six elements onsisted o two elements tested at the University o Toronto (Kirshner and Collins 1986, and Khalia 1986) and our elements tested at the University o Houston (Pang and Hsu, 1995). University o Toronto tests are labeled by SE and University o Houston tests are labeled by A and B in Figure 3-7. The reasonable agreement o MCFT preditions with the experimental results is evident. Fig. 3-7 Comparison o predited and observed shear stress shear strain response o six uniorm shear elements (Bentz et al. 2006). 39

61 MCFT and Traditional Shear Design Formulation In traditional shear design approah, the shear resistane o a strutural onrete member is expressed as the sum a onrete ontribution V, a stirrup ontribution V s and the vertial omponent o inlined prestressing ore V p in the general orm: Av yd v otθ [ 3-22] V = V + Vs + V p = β ' bwd v + + V p s where β is the onrete ontribution ator aounting or the shear resistane o raked onrete, ' is the ompressive strength o onrete, b w is the web width, d v is the eetive shear depth o the member, A v is the area o transverse shear reinorement within a distane s along the member, y is the yield strength o the transverse reinorement and θ is the angle o inlination (measured rom the longitudinal axis) o the onrete prinipal diagonal ompressive stress, i.e., angle o ritial diagonal rak. The onrete ontribution is traditionally taken equal to the shear ore at irst diagonal raking. For prestressed onrete members, β depends on the moment-to-shear ratio and level o prestress in the traditional equations. For a small member (no sizeeet redution), a onservative lower-bound value or V results rom assuming β = 2 psi (0.17 MPa) and d v = d. The transverse reinorement ontribution V s has traditionally been alulated assuming θ = 45 deg (ot θ = 1.0), and d v = d. For a uniorm shear element, Equation [3-22] an be modiied by eliminating V p and dividing the remaining shear ore omponents by the shear area b w d v : [ 3-23] v = v + v = β ' + ρ otθ s z y where ρ z is the transverse reinorement ratio A v /b w s. Assuming lamping stress z is equal to zero and transverse reinorement has yielded, MCFT equilibrium equation 40

62 Eq.[3-10] an also be rearranged in the same ormat as Eq. [3-23] where: [ 3-24] β = otθ 1 MCFT assumes =.33 '; thus, rom average stresses (Eq. 3-5), β an be r 0 alulated rom: [ 3-25] 0.33otθ β = ε 1 But β is also limited depending on the maximum stresses that an be transerred at the raks. The onrete ontribution ator β and angle o inlination θ hange rom element to element as well as throughout the loading stages o one element ater raking up to ailure. Figure 3-8(a) depits how MCFT predits the shear stress shear strain relationships o strutural onrete elements with dierent amounts o transverse and longitudinal reinorement in addition to the orresponding onrete ontribution ators βs and rak angles θs ( rom horizontal axis) o onrete or dierent loading stages. Notie that until onrete raks, the inlination o diagonal ompression θ in elements subjeted to pure shear is 45 degrees. Ater raking, θ redues depending on the relative amount o reinorement ρ z /ρ x, and the shear stress ratio v '. Ater transverse (z-dir.) / reinorement yields, shear strain γ inreases, longitudinal strain ε x inreases, inlination o diagonal ompression θ redues, and onrete ontribution ator β redues. As v redues and v s inreases, the total shear may redue, stay onstant or inrease, depending on the amount o transverse reinorement as shown in Fig. 3-8(a). The point at whih diagonal ompression stresses in onrete reahes the rushing strength o onrete is also shown on eah urve. Ater this point, the shear strength o the element redues. 41

63 For the element (with ρ z = 0.005, ρ x = 0.02) where all the reinorement is already yielding, the strength redution is sudden. For members without transverse reinorement, ρ z beomes zero thus MCFT Eq.[3-23] an be presented by onrete ontribution only as v = β ' similar to traditional method. The MCFT an predit the omplete shear stress shear strain relationships o suh elements as well. It was used to predit the response o elements with no transverse (z-diretion) reinorement and 3% distributed longitudinal (x-diretion) reinorement subjeted to dierent levels o longitudinal axial stress x, and the results are shown in Fig. 3-8(b). One element was subjeted to onstant axial ompression stress x o 3 MPa, the seond element was subjeted to pure shear ( x =0), and the third element was subjeted to a onstant axial tension stress o 1.5 MPa. The element subjeted to shear and axial tension an be onsidered to represent the lower portion o a web subjeted to shear and bending moment the lexural stresses ause axial tension in the lower part o the web. The raking point (indiated by a round dot in Fig. 3-8b) is very strongly inluened by the magnitude o axial stress. Axial tension redues the shear ore at raking, while axial ompression inreases the shear ore at raking. Aording to the MCFT, the onrete ontribution V is not the shear ore at diagonal raking; but is the additional shear ore beyond that resisted by yielding stirrups that an be transerred aross diagonal raks by interlok o rough rak suraes. For a member without transverse reinorement subjeted to ombined shear and axial tension (representing the inluene o bending moment), Adebar and Collins (1996) showed that the ritial point is when transer o shear aross diagonal raks initially limits the applied stresses. In other words, the ritial point is when θ and β 42

64 orrespond to the solution in whih β alulated rom average stresses Eq. [3-25] is equal to the upper limit due to aggregate interlok apaity alulated rom Equation [3-6]. This point is indiated by a square dot in Figure 3-8(b). Shear stress ratio v/ ' ε x = θ=37.0 β=0.183 ε x = θ =38.7 β=0.173 Conrete raking θ=45.0 ε x = θ=34.9 β=0.193 ε x = θ=30.9 β=0.208 ρ z =0.01, ρ x =0.03 ρ z =0.005, ρ x =0.03 ρ z =0.005, ρ x =0.02 ρ z =0.002, ρ x =0.03 ε x = θ=32.2 β=0.069 z = x = 0 ' = 40 MPa) y = 400 MPa ε x = θ=25.9 β=0.049 ε x =0.002 ε z = θ=26.8, β=0.007 ε x = θ=26.7 β=0 Transverse Transverse reinorement yielding Longitudinal Longitudinal reinorement yielding Conrete rushing θ in deg. β in MPa units ε x = θ=19.6 β= (a) Shear strain γ 0.08 θ = 33.6 x = -3 MPa ε x = , ε y = θ = 29.7 o, β = Conrete ρz=0.005 raking Crak ρz slip =0.005, Shear stress ratio v/ ' x = 0 x = 1.5 MPa θ = 59.0 ' = 40MPa, y = 400 MPa ρ x = 0.03, s x e = 300 mm z = 0 β in MPa units (b) θ = 45.0 ε x = , ε y = θ = 34.7 o, β = ε x = , ε y = θ = 36.8 o, β = Shear strain γ Fig. 3-8 MCFT preditions o shear response o uniorm shear elements or: (a) members with transverse reinorement, (b) members without transverse reinorement. 43

65 It is worth mentioning that at the ritial point, β and angle θ are not untions o ' beause ε 2 is relatively small and negligible ompared to ε 1 in Eq. [3-21] and thereore both βs assoiated with average stress ondition (Eq. 3-25) and aggregate interlok apaity (Eq. 3-6) are only untions o θ at a given longitudinal strain ε x AASHTO LRFD Method The shear design method given in the 2007 and older versions o AASHTO LRFD was developed based on MCFT analysis o uniorm shear elements. The derivation o this method was presented by Collins et al. (1996). The method uses the traditional shear design ormula (Eq. 3-22) and inludes tables with values o β and θ. MCFT analysis assumptions used in the derivation o the method are: lamping stress z =0, the rak spaing s θ or members with transverse reinorement is 300 mm, onrete maximum aggregate size is 19 mm, and the raking stress o onrete is r = 0.33 ' in MPa units. Analyses were perormed at a given longitudinal strains ε x assuming that longitudinal reinorement had suiient apaity to provide the required axial ompression ore to transer shear at the raks. Another hek as part o the method is perormed to onirm this assumption. For members with at least minimum transverse reinorement, it was assumed that transverse reinorement had yielded. Thus, the rak hek equation or both members with at least minimum and without transverse reinorement is Eq. [3-17] whih an be rewritten as: 0.18 [ 3-26] β [24ε /( 1 s θ a g + 16)] 44

66 β is obtained rom Eq.[ 3-25] but redued i Eq.[3-26] is not satisied. For members with at least minimum transverse reinorement, the tabulated values o β and θ depend on longitudinal strain ε x and shear stress ratio v ' aording to MCFT. As or design o members with transverse reinorement, the desired shear strength o an element an be ahieved with dierent relative amount o transverse and longitudinal reinorement, the values o θ and β provided in the tables or suh members orrespond to one partiular solution. This solution uses a ost untion to determine θ and orresponding β that are assoiated with minimum ost or the amount o longitudinal and transverse reinorement needed or shear. Aording to the AASHTO LRFD tables, or ε x rom to , and / v / rom to 0.250, θ varies rom 22 to 37 deg., and β varies rom 0.13 to 0.53 in MPa units or members with transverse reinorement. To ensure that transverse reinorement v yields prior to onrete rushing, the shear strength is limited to 25 ' = 0.. For members without transverse reinorement, as disussed earlier, ritial shear strength is reahed when θ and β orrespond to the solution in whih β alulated rom average stresses Eq. [3-25] is initially limited by aggregate interlok apaity expressed by Eq [3-26]. The tabulated values o θ and β orrespond to the ritial point mentioned above. They are untions o longitudinal strain ε x and eetive rak spaing parameter S xe that is given by: ' [ 3-27] S xe = a 35s g x

67 It should be noted that rak spaing o members without transverse reinorement is s determined by s θ = x sine rak spaing parameter in z-diretion is assumed to be sinθ ininity in Eq. [3-8] or suh members. Aording to AASHTO LRFD tables or members without transverse reinorement, ε x varies rom to , and or small members (S xe = 300 mm), θ varies rom 25 to 37 deg and β varies rom 0.53 to 0.17 (in MPa units) over that range. For very large members (S xe = 2000 mm), θ is about double, and β is about hal. Speiially, θ varies rom 44 to 72 deg and β varies rom 0.26 to 0.05 (in MPa units) over the same range. As disussed earlier, MCFT analyses were done at a given longitudinal strain assuming that longitudinal reinorement had suiient strength to transer shear at raks. To hek the validity o this assumption, 2007 AASHTO LRFD method uses Eq. [3-28], whih is derived rom MCFT Eqs. [3-12] & [3-13] when = 0. [ 3-28] n = (2v + v )otθ = 2( v 0.5v ) otθ v* s s In the equation above, n v * is the required axial strength (ore per unit area) provided by longitudinal reinorement to transer shear at the rak and reinoring steel ore per unit area in the z-diretion v = ρ otθ. n * should be less than available longitudinal s z y reinorement strength per unit area or shear to ensure that the assumption explained above is valid. v x = y 46

68 CHBDC Method The 2006 CHBDC shear design proedure, whih is the same as the 2004 CSA A23.3 shear design proedure, was developed by Bentz et al. (2006) and Bentz and Collins (2006) and is again based on MCFT with the same assumptions as the ones used or 2007 AASHTO LRFD method. The method does not inlude tables but provides equations or θ and β instead. These equations are: [ 3-29] β = ε x S xe S xe [ 3-30] θ = ( ε x )( ) 75deg 2500 Similar to 2007 AASHTO LRFD method, θ and β equations are untions o rak spaing parameter and longitudinal strain or members without transverse reinorement. For members with at least minimum transverse reinorement, θ and β equations are untions o ε x only and are independent o shear stress ratio v / ' unlike the 2007 AASHTO LRFD method. This is beause the equation or β was developed or no transverse reinorement (low shear stress ratio), and the solution or θ was developed or high shear stress ratio v / ' = Combining these θ and β equations results in a simpler design proedure than using the tables in 2007 AASHTO LRFD. 0.4 First part o Eq. [3-29] ( ) was derived rom Eq. [3-6] assuming ε x S xe = 300 mm and w = ε x mm, whih is in good agreement with the MCFT preditions o rak width w when no transverse reinorement is present. The seond part o Eq. [3-29] is a orretion ator when S xe is dierent than 300 mm. 47

69 As mentioned earlier, Eq. [3-30] was developed or heavily reinored setions where v / ' = Figure 3-9 presents the variation o angle θ with longitudinal strain ε x or suh elements at yielding o transverse reinorement and rushing o onrete based on MCFT. As shown, the 2006 CHBDC equation or angle θ alls in the region where v transverse reinorement has yielded but onrete has not rushed when 25 ' = 0.. Bentz et al. (2006) explained that Eqs. [3-29] and [3-30] is a linear relationship between shear strength and transverse reinorement ratio assuming onstant longitudinal strain while in reality this relationship is onave downwards based on plasti analyses as shown in Figure As a result, they onluded that using Eqs. [3-29] and [3-30] or members with traditional amount o transverse reinorement is onservative and appropriate or design. Fig. 3-9 Developing proedure o CHBDC 2006/CSA A equation or angle o inlination o prinipal ompression (Bentz et al., 2006). 48

70 Fig Shear strength relation with transverse reinorement ratio (Bentz et al., 2006). Bentz et al. (2006) ompared the 2006 CHBDC equations or θ and β with MCFT or members without transverse reinorement and with dierent rak spaing parameters or longitudinal strains varying rom to as shown in Figure The MCFT results orrespond to the point when v is maximum. The 2006 CHBDC equation or β gives a good estimate o β at large longitudinal strains but does not give a good estimate o β at low longitudinal strains. The 2006 CHBDC equation or θ is not in good agreement with MCFT preditions over a wide range o longitudinal strains. However, θ is less important than β or members without transverse reinorement beause shear apaity o these elements is a untion o β not θ. The angle o prinipal ompression θ is only used to determine the demand on longitudinal reinorement imposed by shear. Thereore, the lower the angle, the more onservative the results. 49

71 Fig Comparison o θ and β values given by CHBDC 2006/ CSA A with values determined rom MCFT or elements without transverse reinorement (Bentz and Collins, 2006). 50

72 Shown in Fig. 3-12, Bentz et al. (2006) also ompared the 2006 CHBDC equation or θ and β with MCFT or members with dierent amount o transverse reinorement or longitudinal strains ranged rom to One again, the MCFT preditions orrespond to maximum v ontribution to shear strength. Figure 3-12 shows that the 2006 CHBDC equation or θ is onservative ompared to MCFT preditions exept or a ew ases. The 2006 CHBDC equation or β is also onservative exept or the ases where ε x is low. For suh ases, however, Bentz et al. (2006) explained that onservative estimate o θ ould ompensate or the unonservative estimate o β. v In the 2006 CHBDC method, shear strength is also limited to 25 ' = 0. to avoid onrete rushing prior to transverse reinorement yielding. Moreover, the same equation as in 2007 AASHTO LRFD method (Eq. 3-28) is used to hek the suiieny o longitudinal reinorement strength to transer shear at raks Proposed Evaluation Method or Members With at Least Minimum Transverse Reinorement 2007 AASHTO LRFD and the 2006 CHBDC methods are intended or the design o new strutures. The methods assume that the required shear apaity o a setion is given and the amount o transverse reinorement needs to be determined. As a result, the 2007 AASHTO LRFD is based on a partiular MCFT solution or a given required shear apaity aimed at the ost eetive ombination o longitudinal and transverse reinorement. The 2006 CHBDC uses simpliied equations or angle o inlination θ and the axial stress required to transer shear that are independent o the amount o transverse reinorement to avoid iteration. 51

73 Fig Comparison o θ and β values given by CHBDC 2006/ CSA A with values determined rom MCFT or elements with at least minimum transverse reinorement (Bentz and Collins, 2006). 52

74 In ontrast to design, shear evaluation deals with ases where amount o reinorement is known and shear apaity o the setion needs to be determined. As a result, applying the shear design provisions o 2007 AASHTO LRFD or 2006 CHBDC to evaluation problems require trial-and-error. Moreover, the relative amount o longitudinal and transverse reinorement annot be hanged; thereore, the θ and β that result in a ertain relative amount o reinorement may not be the best solution. Ignoring the eet o some o parameters suh as transverse reinorement ratio on angle o prinipal ompression and axial ompression required or shear ould result in onservative estimate o shear apaity whih is aeptable or design but ould lead to unneessary posting o bridge load limit, retroit or replaement o existing bridges. In this setion, a new proedure that is speiially intended or shear strength evaluation problems is presented. The method presented here is or members with at least minimum transverse reinorement only. Setion 3-7 presents a similar method or members without transverse reinorement. As was done to develop the shear design methods in AASHTO LRFD and 2006 CHBDC, it is assumed that in members with at least minimum transverse reinorement, the diagonal rak spaing is 300 mm, onrete maximum aggregate size is 19 mm, and onrete raking stress is 0.33 ' in MPa units. To develop equations that an be applied to beams, it is assumed that the longitudinal strain ε x o the uniorm shear element is onstant, and the required longitudinal reinorement is available in the beam to equilibrate the longitudinal onrete ompression. A separate hek is done in the evaluation proedure to ensure that this is the ase. 53

75 Figure 3-13 presents a variety o results rom MCFT. Speiially, it presents the inluene o axial strain ε x and transverse strain ε z on (a) shear stress v, (b) angle o inlination o prinipal average ompressive stress θ and () axial ompression stress in onrete x all or an element with ρ z = Figure 3-8(a) demonstrates how the axial strain ε x inreases as the shear stress applied to an element with a given amount o longitudinal reinorement ρ x. The urves shown in Fig are or dierent onstant ε x values, and are plotted over the range o ε z rom transverse reinorement yielding to onrete rushing. Figure 3-13(a) illustrates how a larger ε x results in a lower shear stress at irst yielding o transverse reinorement (ε z =0.002) and lower shear stress at onrete rushing (maximum ε z whih orresponds to ε 2 = ). A larger ε x also results in a higher shear stress inrease ater yielding o transverse reinorement. For example, when ε x = 0, the shear stress at ε z =0.002 is the maximum shear stress, while when ε x = 0.001, the shear stress at ε z =0.002 is about 80 % o the maximum shear stress. Figure 3-13(b) suggests the reason or this is that the higher ε x results in a larger ompression angle θ at irst yielding o the transverse reinorement and hene less stirrup ontribution. While θ varies rom 24.4 to 36.7 deg at irst yielding o transverse reinorement, it varies rom only 21.3 to 24.3 deg at onrete rushing. Figure 3-13() gives the onrete longitudinal ompression stress x required to maintain the speiied axial strain ε x. For an element with large ε x (e.g., 0.001), there is lower x at transverse reinorement yielding (ε z =0.002) and a larger inrease in x as ε z inreases. For an element with small ε x (e.g., ε x = 0), there is higher x at ε z =0.002 and a 54

76 6.0 ε x = 0 Shear stress v (MPa) ε x = ε x = ε x = ε x = ε x = (a) Transverse reinorement strain ε z Angle θ (degree) 30 ε x = ε x = ε x = ε x = ε x = 0 ε x = (b) Transverse reinorement strain ε z Compression stress x (MPa) ε x = 0 ε x = ε x = ε x = ε x = ε x = () Transverse reinorement strain ε z Fig Inluene o longitudinal strainε x and transverse strainε z on: (a) shear stress, (b) angle o inlination o diagonal ompression, () longitudinal ompression stress in onrete or an element with ρ = 0.005, = 40 MPa, y = 400 MPa. z ' 55

77 small inrease in x as ε z inreases. As shown in Figure 3-13 (a), maximum shear strength o a member is generally lose to the shear strength that orresponds to either yielding o transverse reinorement or onrete rushing. However, the MCFT analyses are based on the assumption that longitudinal reinorement does not yield as explained beore. In some ases the element does not reah its maximum apaity at yielding o transverse reinorement or rushing o onrete due to yielding o longitudinal reinorements at raks. For instane, the element with ρ z = and ρ x = 0.02 shown in Fig. 3-8(a) reahes the maximum shear stress when the longitudinal reinorement yields at ε x = As a result, in the proposed evaluation method, the strength is evaluated at three possible ailure points. Yielding o transverse reinorement and onrete rushing are two o the ailure modes and the third ailure mode involves yielding o both the transverse and longitudinal reinorement Proposed Equations or Angle o Inlination o Prinipal Compression Figure 3-14 shows the relationship between θ and the axial strain ε x or dierent quantities o transverse reinorement ρ z, ' = 40 MPa, and y = 400 MPa at yielding o transverse reinorement and onrete rushing stage. The solid lines show the relationships given by MCFT, whih is approximately linear or a onstant ρ z. The relationships are very dierent at yielding o transverse reinorement (Fig. 3-14a), and onrete rushing (Fig. 3-14b). 56

78 40 ρ z = Angle θ (degree) CHBDC 2006 ρ z = ρ z = Proposed MCFT 20 ρ z = (a) Longitudinal strain ε x 35 Angle θ (degree) CHBDC 2006 Proposed MCFT ρ z = ρ z = ρ z = ρ z = (b) Longitudinal strain ε x Fig Comparison o predited angleθ with MCFT result at: (a) yielding o transverse reinorement, (b) rushing o onrete or = 40 MPa, y = 400 MPa. ' 57

79 The 2006 CHBDC method has one equation or θ as a untion ε x, and this is shown in Fig as a dotted line. As the 2006 CHBDC approximate value or θ is generally larger than the atual value, it results in a smaller V s than atual and is generally onservative. The 2006 CHBDC approximate value or θ is smaller than the atual value when ε x is large at the point o transverse yielding (right-hand side o Fig. 3-14a); but as disussed with reerene to Fig. 3-13(a), there will be a signiiant shear strength inrease ater yielding or large ε x, and the 2006 CHBDC approximate angle is very onservative or all onrete rushing ases (Fig. 3-14b). The 2006 CHBDC approximate angle is also very onservative at transverse reinorement yielding or members with low amounts o transverse reinorement. More aurate equations or θ were developed or the proposed method. This was done by looking at a signiiant number o MCFT preditions o shear strength or uniorm shear elements with dierent amount o transverse reinorement and material properties at traverse reinorement yielding and onrete rushing. As indiated by the AASHTO LRFD method, the angle θ depends on the shear stress ratio v / '. In design, the shear stress v is known; but not during strength evaluation. Thus the parameter ρ ' was used in plae o v '. It was also ound that angle θ at yielding o z y / / transverse reinorement is a untion o transverse steel strain at yielding ε y. The proposed equation or θ is: [ 3-31] θ θ + θ ε 45 = o x where at yielding o transverse reinorement: ρ z y [ 3-32] θ o = ( )( 50ε y + 1.1) ; ε y ' 58

80 [ 3-33] θ = 1000 [37.5( 200ε y + 1.4) θo] in whih ε y is the reinoring steel yield strain and shall not be taken greater than and at onrete rushing: ρ z y [ 3-34] θ o = ' ρz y [ 3-35] θ = ' The angles predited by these equations are also shown in Figs. 3-14(a) and (b) as dashed lines, and they are learly in good agreement with MCFT. When the longitudinal strain equals the transverse strain, the MCFT angle is equal to 45 deg. For 400 MPa grade reinorement, the transverse reinorement yields at a strain o 0.002, thus the angles onverge to 45 deg. in Fig. 3-14(a) at a longitudinal strain o The largest longitudinal strain used in the shear analysis o beams is 0.001, and at this strain, the MCFT angles have almost onverged. For simpliity, Eqs. [3-32] and [3-33] predited angles onverge at a longitudinal strain o 0.001, and or 400 MPa grade reinorement, that angle is 37.5 deg. Figures 3-15 to 3-18 are similar to Fig exept they involve steel grades 250 MPa and 600 MPa, or onrete ompression strengths 30 MPa and 60 MPa. Figures 3-14 to 3-18 illustrate that the proposed equations adequately apture the eet o onrete strength ' and steel grade on the angle o inlination θ at both yielding o transverse reinorement and rushing o onrete. This validates the approah o using the parameters ρ ' and ε y in the proposed equations or angle θ. z y / 59

81 40 ρ z = MCFT 35 CHBDC 2006 Angle θ (degree) ρ z = Proposed ρ z = ρ z = (a) Longitudinal strain ε x ρ z = Proposed CHBDC 2006 Angle θ (degree) ρ z = ρ z = MCFT 15 ρ z = (b) Longitudinal strain ε x Fig Comparison o predited angleθ with MCFT result at: (a) yielding o transverse reinorement, (b) rushing o onrete or = 30 MPa, y = 400 MPa. ' 60

82 40 35 Angle θ (degree) MCFT CHBDC 2006 ρ z = ρ z = ρ z = ρ z = Proposed (a) Longitudinal strain ε x Angle θ (degree) CHBDC 2006 ρ z = ρ z = ρ z = Proposed ρ z = MCFT (b) Longitudinal strain ε x Fig Comparison o predited angleθ with MCFT result at: (a) yielding o transverse reinorement, (b) rushing o onrete or = 60 MPa, y = 400 MPa. ' 61

83 45 40 Angle θ (degree) ρ z = ρ z = ρ z = MCFT Proposed CHBDC ρ z = (a) Longitudinal strain ε x Angle θ (degree) CHBDC 2006 ρ z = ρ z = ρ z = Proposed ρ z = MCFT (b) Longitudinal strain ε x Fig Comparison o predited angleθ with MCFT result at: (a) yielding o transverse reinorement, (b) rushing o onrete or = 40 MPa, y = 250 MPa. ' 62

84 45 40 ρ z = Angle θ (degree) CHBDC 2006 ρ z = ρ z = MCFT 20 ρ z = Proposed (a) Longitudinal strain ε x Proposed Angle θ (degree) CHBDC 2006 MCFT ρ z = ρ z = ρ z = ρ z = py=0.01 Yielding and rushing oinidene (b) Longitudinal strain ε x Fig Comparison o predited angleθ with MCFT result at: (a) yielding o transverse reinorement, (b) rushing o onrete or = 40 MPa, y = 600 MPa. ' 63

85 Figure 3-17(a) indiates that the predited angles at ε x =0 rom the proposed equations are slightly smaller than the MCFT results or lower grade steel 250 MPa ( y = 250 MPa ) and thus are slightly unonservative; this ompensates or the onservative estimate o proposed β equation or suh steel as will be disussed later in this hapter. MCFT results showed that angle θ is not signiiantly sensitive to steel grade at yielding o transverse reinorement one steel grade is higher than 400 MPa; thus ε y shall not be taken greater than in the proposed method equations. This is, however, ignored in the proposed method when applied to beam elements due to the at that onrete rushing is normally the governing ailure mode at larger strains. Figure 3-18 shows that the proposed method predited angles at yielding o transverse reinorement are slightly onservative or steel grade 600 MPa. It will be shown later in this hapter that the unonservative predition o onrete ontribution ator β ompensates or it. Notie in Fig that MCFT preditions had to stop or lower strains and higher transverse reinorement ratios beause onrete rushing happened prior to yielding o transverse reinorement. Shown in Figs. 3-17(b) and 3-18(b), the proposed method equation o angle at onrete rushing is in good agreement with MCFT preditions or dierent steel grades. Figure 3-15 to 3-18 show that the 2006 CHBDC preditions o angle are always smaller and thus onservative ompared to the MCFT preditions at rushing o onrete while they are mostly onservative but sometimes unonservative ompared to MCFT preditions at yielding o transverse reinorement. 64

86 Proposed Equations or Conrete Contribution Fator The solid lines in Figs show the onrete ontribution ator β aording to MCFT at yielding o transverse reinorement (Fig. 3-19a) and onrete rushing (Fig. 3-19b) or 40 MPa onrete and 400 MPa reinoring steel. The 2006 CHBDC approximate β value is shown as a dotted line. As this equation was developed or members without transverse reinorement, it gives unonservative values o β or low ε x values in members with at least minimum transverse reinorement. Bentz et al. (2006) explained that the unonservative estimate o V in these members, ompensates or the onservative estimate o V s due to the larger than atual value o θ. Figure 3-19(a) indiates that the onrete ontribution ator β does not vary signiiantly at transverse reinorement yielding. It will be shown later that the onrete ontribution ator β does vary with steel grade or steel grades lower than 400 MPa. As was done to develop the equations or θ, the proposed equations or β were developed by looking at a signiiant number o MCFT results or uniorm shear elements with varying amount o transverse reinorement and dierent material properties. The proposed equation or onrete ontribution ator at yielding o transverse reinorement is: [ 3-36] β 0.18( 300ε + 1.6) in MPa units = y Figure 3-19(b) indiates that β does vary somewhat more at onrete rushing. The 2007 AASHTO LRFD method assumes β is a untion o both ε x and v '. As / desribed above in reerene to the proposed expression or θ, using v / ' would require iteration in evaluation, thus the parameter ρ ' was substituted or v '. For z y / / simpliity, β was not made a untion o ε x. 65

87 Conrete ontribution β (MPa) CHBDC 2006 MCFT Proposed ρ z = ρ z = ρ z = ρ z = (a) Longitudinal strain ε x Conrete ontribution β (ksi) CHBDC 2006 MCFT ρ z = ρ z = Proposed ρ z = ρ z = (b) Longitudinal strain ε x Fig Comparison o predited β with MCFT result at: (a) yielding o transverse reinorement, (b) rushing o onrete or ' = 40 MPa, y = 400 MPa. 66

88 The proposed expression or onrete ontribution ator at onrete rushing is: ρ z y [ 3-37] β = in MPa units ' Figure 3-19 shows the proposed expressions are onservative and agree well with MCFT. Figures 3-20 to 3-23 examine the proposed equations or onrete ontribution ator β and ompare them with MCFT and the 2006 CHBDC method when onrete ompressive strength is 30 and 60 MPa, or steel grade is 250 and 600 MPa. The same trend as in Fig is notied in Figs.3-20 and 3-21 in whih the onrete ompressive strength is 30 and 60 MPa, respetively. One again, the urrent CHBDC preditions are unonservative or low longitudinal strains and the proposed method is onservative and onsistent with MCFT results. For steel grade 250 MPa shown in Fig. 3-22(a), the trend is somewhat dierent at yielding o transverse reinorement as the variation o onrete ontribution ator with longitudinal strain is more signiiant. As a result, the proposed method equation is onservative or low longitudinal strains. However, this is ompensated or by the slightly unonservative estimate o angle as explained in the previous setion. The 2006 CHBDC preditions are more onsistent with MCFT at yielding o transverse reinorement or 250 MPa reinoring steel ompared to those or 400 and 600 MPa steel grades. As shown in Figs. 3-23, the onrete ontribution ators given by the proposed equation or steel grade 600 MPa are unonservative but this is ompensated or by the onservative estimate o orresponding angles as mentioned beore. Figs. 3-22(b) and 3-23(b) illustrate that the proposed equation or onrete ontribution ator at onrete rushing is onsistent with the MCFT. 67

89 Conrete ontriburion β (MPa) CHBDC 2006 MCF ρ z = ρ z = ρ z = Proposed ρ z = (a) Longitudinal strain ε x Conretee ontribution β (MPa) CHBDC MCFT ρ z = ρ z = Proposed ρ z = ρ z = (b) Longitudinal strain ε x Fig Comparison o predited β with MCFT result at: (a) yielding o transverse reinorement, (b) rushing o onrete or ' = 30 MPa, y = 400 MPa. 68

90 Conrete ontriburion β (MPa) CHBDC 2006 ρ z = ρ z = ρ z = Proposed ρ z = MCFT (a) Longitudinal strain ε x Conretee ontribution β (MPa) CHBDC 2006 Proposed MCFT ρ z = ρ z = ρ z = ρ z = (b) Longitudinal strain ε x Fig Comparison o predited β with MCFT result at: (a) yielding o transverse reinorement, (b) rushing o onrete or ' = 60 MPa, y = 400 MPa. 69

91 0.4 CHBDC 2006 Conrete ontriburion β (MPa) ρ z = Proposed ρ z = ρ z = ρ z = MCFT (a) Longitudinal strain ε x Conretee ontribution β (MPa) CHBDC 2006 Proposed MCFT ρ z = ρ z = ρ z = ρ z = (b) Longitudinal strain ε x Fig Comparison o predited β with MCFT result at: (a) yielding o transverse reinorement, (b) rushing o onrete or ' = 40 MPa, y = 250 MPa. 70

92 Conrete ontriburion β (MPa) CHBDC 2006 ρ z = ρ z = Proposed ρ z = ρ z = MCFT (a) Longitudinal strain ε x Conretee ontribution β (MPa) 0.4 Pv=0.006 Yielding and rushing oinidene 0.3 CHBDC MCFT ρ z = ρ z = ρ z = Proposed ρ z = (b) Longitudinal strain ε x Fig Comparison o predited β with MCFT result at: (a) yielding o transverse reinorement, (b) rushing o onrete or ' = 40 MPa, y = 600 MPa. 71

93 Predited Total Shear Stress Total shear stress v is the sum o the onrete ontribution v, whih depends on β and the stirrup ontribution v s, whih depends on the angle θ. As dierent ombinations o θ and β an result in the same total shear stress, dierent methods an be really ompared by omparing the preditions or total shear stress. Figures 3-24 to 3-28 ompare MCFT preditions o total shear stress with the results rom the proposed method at transverse reinorement yielding and onrete rushing or onrete ompressive strength o 30, 40, and 60 MPa and steel grade o 250, 400, and 600 MPa. In all ases, the proposed method agrees better with MCFT results than the 2006 CHBDC. Generally, the 2006 CHBDC preditions are unonservative or low longitudinal strains but it beomes less unonservative or onrete strength o 60 MPa. The 2006 CHBDC preditions are mostly onservative ompared to MCFT results at onrete rushing exept or low longitudinal strains. As explained beore, the 2006 CHBDC limits the maximum shear strength to 0.25 ' to ensure that onrete rushing ours ater yielding o transverse reinorement. Figures 3-28 proves that this is onsistent with MCFT results, and thus this is adopted in the proposed method. Notie that or 40 MPa onrete and 600 MPa reinoring steel, onrete rushing and steel yielding oinide at shear stress o 9.3 MPa and 10.3 MPa or transverse reinoring steel ratios o ρ z = and 0.010, respetively. This is lose to.25 ' = 10 MPa predited by the 2006 CHBDC. 0 72

94 12 Shear stress v (MPa) MCFT Proposed CHBDC 2006 ρ z = ρ z = ρ z = ρ z = (a) Longitudinal strain ε x 12 Shear stress v (MPa) CHBDC 2006 Proposed ρ z = ρ z = ρ z = ρ z = MCFT (b) Longitudinal strain ε x Fig Comparison o predited shear stress with MCFT result at: (a) yielding o transverse reinorement, (b) rushing o onrete or = 40 MPa, y = 400 MPa. ' 73

95 10 Shear stress v (MPa) MCFT CHBDC 2006 Proposed ρ z = ρ z = ρ z = ρ z = Maximum shear stress allowed by CHBDC (a) Longitudinal strain ε x Shear stress v (Mpa) MCFT CHBDC 2006 Proposed ρ z = ρ z = ρ z = Maximum shear stress allowed by CHBDC 2006 ρ z = (b) Longitudinal strai ε x Fig Comparison o predited shear stress with MCFT result at: (a) yielding o transverse reinorement, (b) rushing o onrete or = 30 MPa, y = 400 MPa. ' 74

96 10 Shear stress v (MPa) CHBDC 2006 Proposed MCFT ρ z = ρ z = ρ z = ρ z = (a) Longitudinal strain ε x 10 ρ z = Shear stress v (MPa) ρ z = CHBDC 2006 ρ z = MCFT Proposed ρ z = (b) Longitudinal strai ε x Fig Comparison o predited shear stress with MCFT result at: (a) yielding o transverse reinorement, (b) rushing o onrete or = 60 MPa, y = 400 MPa. ' 75

97 8 CHBDC 2006 MCFT Shear stress v (MPa) Proposed ρ z = ρ z = ρ z = ρ z = (a) Longitudinal strain ε x 8 Shear stress v (MPa) MCFT Proposed ρ z = ρ z = ρ z = ρ z = CHBDC (b) Longitudinal strai ε x Fig Comparison o predited shear stress with MCFT result at: (a) yielding o transverse reinorement, (b) rushing o onrete or = 40 MPa, y = 250 MPa. ' 76

98 14 12 CHBDC 2006 Maximum shear stress allowed by CHBDC 2006 Shear stress v (MPa) Proposed MCFT ρ z = ρ z = ρ z = ρ z = (a) Longitudinal strain ε x Shear stress v (MPa) Maximum shear stress allowed by CHBDC 2006 MCFT Proposed Crushing Yielding and rushing oinidene ρ z = ρ z = ρ z = ρ z = CHBDC (b) Longitudinal strai ε x Fig Comparison o predited shear stress with MCFT result at: (a) yielding o transverse reinorement, (b) rushing o onrete or = 40 MPa, y = 600 MPa. ' 77

99 Proposed Equations or Longitudinal Conrete Compression Stress The resultant o the longitudinal onrete ompression stress x is the axial ompression ore N v required in a beam web to resist shear, thus this stress is also reerred to as n v. In order to estimate ε x, whih inluenes β and θ, an estimate o n v is needed. As the longitudinal ompression n v in onrete must be balaned by tension in steel, n v is equal to ρ x s where ρ x is the reinorement ratio in the x-diretion and s is the stress in the horizontal reinorement. As = ε, ε x or a uniorm shear element is given by: s E s x [ 3-38] ε = x n x v ρ E s In the 2007 AASHTO LRFD shear design method, it is assumed or simpliity that n v = v ot θ, while in the 2006 CHBDC shear design method, this has been urther simpliied to n v = 2v. Aording to MCFT, substituting or 1 in Eq. [3-10] rom Eq. [3-9] and assuming x = z =0, the atual relationship is: 2 [ 3-39] n = v ot 2θ + ρ ot θ v 2 z y To simpliy this equation in the proposed method, ot 2θ and ot 2 θ have been approximated as linear untions or θ >25 deg and θ 25 deg as illustrated in in Fig Substituting these linear untions, as well as v = β ' untion o ε x gives: [ 3-40] nv = nvo + nv ε x, and Eq. [3-31] orθ as a 78

100 1.6 ot 2θ θ θ (a) Angle θ (degrees) ot 2 θ θ θ (b) Angle θ (degrees) Fig Bilinear approximation o: (a) ot2θ, (b) ot 2 θ used to approximate Eq. [3-39]. 79

101 where or θ > 25 deg: [ 3-41] nv = ( 0. 09β ' 0. 20ρ z y ) θ nv [ 3-42] nvo = θ o β ' ρ z y θ and or θ 25 deg: [ 3-43] nv = ( 0. 15β ' 0. 77ρ z y ) θ nv [ 3-44] nvo = θ o β ' ρ z y θ A simpliiation that avoids trial-and-error is to only use Eqs. [3-41] and [3-42] or transverse reinorement yielding, use Eqs. [3-43] and [3-44] i θ 23deg and use Eqs. [3-41] and [3-42] i θ > 23 deg or onrete rushing. Although this simpliiation o results in slightly unonservative estimate o axial ompression stress at reinorement yielding or members with low amount o transverse reinorement and low longitudinal strain (members with angle o prinipal ompression o less than 25 deg), this unonservatism is ompensated or by onservative estimate o β or suh elements as shown in Figures 3-19 to At onrete rushing, angle rate o hange with longitudinal strain is not high (see Figs to 3-18); thus θ o is a good representative o θ. For average strain o ε x = 0.005, θ o is almost 2 deg smaller than θ (see Figs to 3-18). In addition, approximated linear untions o ot 2θ and ot 2 θ or θ >25 deg shown in Fig are still onsistent with atual values at θ = 23 degrees. The n v alulated rom MCFT as well as estimated by Eqs. [3-40] to [3-44] together with the simpliiation explained above are ompared at yielding o transverse reinorement in Fig. 3-30(a) and onrete rushing in Fig. 3-30(b). o 80

102 25 Compression stress x (MPa) ρ CHBDC 2006 z = Proposed MCFT ρ z = ρ z = ρ z = (a) Longitudinal strain ε x 25 Compression stress x (MPa) ρ z = ρ z = MCFT CHBDC 2006 Proposed ρ z = ρ z = (b) Longitudinal strain ε x Fig Comparison o predited longitudinal onrete ompression stress with MCFT result at: (a) yielding o transverse reinorement, (b) rushing o onrete or ' = 40 MPa, y = 400 MPa. 81

103 The 2006 CHBDC approximation o n v = 2v is generally onservative (larger estimate o n v than atual) at yielding o transverse reinorement. Thus the estimated axial tension ore in the longitudinal reinorement that balanes the axial ompression ore N v will be larger than atual, and the estimated axial strain ε x will be larger than atual. When this trend is ombined with the trend shown in Fig. 3-24, the 2006 CHBDC method generally gives onservative preditions o shear strength or a member with a given amount o longitudinal reinorement. The preditions o the proposed equations or n v are not ompared with the MCFT preditions or other material properties sine the proposed equation has been diretly derived rom the MCFT atual equation Longitudinal Reinorement Yielding There are some ases where an element annot reah the apaity at yielding o transverse reinorement or onrete rushing beause the longitudinal reinorement does not have the apaity beyond that needed to resist the applied bending moment to equilibrate the longitudinal ompression stress in onrete due to shear. The average longitudinal ompression stress n v whih inluenes the strain o the member was disussed above. This average stress is redued by the ability o raked onrete to resist some average tension stresses. The axial ompression stress required loally at diagonal raks to resist shear is onsiderably larger and is given by Eq. [3-28] as disussed beore. The onrete ontribution ator β and angle o prinipal ompression θ determined at transverse reinorement yielding and onrete rushing an be used in Eq. [3-28] to determine the longitudinal ompression at diagonal raks that must be balaned by tension in longitudinal reinorement; but as shown in Fig. 3-8(a), yielding o longitudinal reinorement may our somewhere between the point o transverse 82

104 reinorement yielding and onrete rushing. Rather than determine a third pair o β and θ values assoiated with the point o longitudinal reinorement yielding, a simpler approah is to ignore the onrete ontribution when determining the shear to ause yielding o both the transverse and longitudinal reinorement. Setting v =0 in Eqs. [3-23] and [3-28], and solving these two equations or v (eliminating θ ) results in the ollowing expression or shear strength given the maximum n v * ontrolled by the apaity o distributed longitudinal reinorement denoted n v : [ 3-45] v = ρ z ynv Atual onrete ontribution annot be determined by equilibrium equations only and require more ompliated analysis using material onstitutive relations and ompatibility equations. However, onrete ontribution is small in ommon ases at biaxial yielding o reinorement as both transverse and longitudinal strains are large at this stage and thus the stress that an be resisted by aggregate interlok is very small. Figure 3-31 illustrates the variation o angle θ and shear strength per unit area with shear stress resisted by onrete v ranging rom 0% to 40% o the shear strength when the longitudinal ompression stress n v is 3 MPa and 6 MPa. θs and βs shown in Fig were determined rom equilibrium equations. From equilibrium, v is given by Eq. [3-23] and n v is given by Eq. [3-28]. Substituting or v and n v, the equations an be solved or angle θ and shear stress v. 83

105 60 Angle (degree) n v * = 3 MPa n v * = 6 MPa (a) v / v 4 n v * = 6 MPa v (MPa) 3 2 n v * = 3 MPa (b) v / v Fig Variation o: (a) angle, (b) shear stress with onrete ontribution rom variable angle truss model. 84

106 Figure 3-31 shows that angle is steeper when onrete ontribution is higher; thereore shear does not hange signiiantly with onrete ontribution. Notie that there is only an inrease o less than 10% in shear strength one onrete ontribution hanges rom 0 to 40% Proposed Evaluation Method or Members Without Transverse Reinorement Members without transverse reinorement behave dierently than members with at least minimum transverse reinorement. As a result, the proposed evaluation proedure or members without transverse reinorement is dierent rom the proedure or members with at least minimum transverse reinorement. Figures 3-8(a) and 3-8(b) show the predited shear behaviour o both members with at least minimum and without transverse reinorement, and these are learly dierent. Figure 3-8(b) illustrates that ater raking o onrete, there is a sudden drop in shear stress; however, the shear stress normally inreases as the stress resisted by aggregate interlok inreases. Adebar and Collins (1996) showed that the ritial point is when transer o shear aross diagonal raks initially limits the applied stresses. The orresponding shear stress is taken as the shear strength per unit area o a uniorm shear element without transverse reinorement. The proposed evaluation proedure to determine the shear strength at this ritial point is presented here. At the ritial ailure points (shown as square dots in Fig. 3-8b), a diagonally raked element with only longitudinal reinorement no stirrups resists shear by a ombination o diagonal ompression and tension stresses that must satisy two requirements. The irst is that the resultant normal stress on any horizontal plane must be 85

107 zero as there is no transverse reinorement to balane onrete normal stress. The seond requirement is that the resultant stresses ating on a plane parallel to the assumed rak diretion θ must be the required ombination o shear and ompression stress needed to transmit the shear aross the rak. As a result o these two requirements, axial ompression stress will exist on vertial planes in the element, and the resultant o the axial ompression n v must be equilibrated by tension in the longitudinal reinorement. Aording to the MCFT n v is given by Eq.[3-39]. As disussed earlier, this has been simpliied to n v = v ot θ in the AASHTO LRFD shear design method, whih is a reasonable simpliiation or members with signiiant transverse reinorement, and in the 2006 CHBDC shear design method, this has been urther simpliied to n v = 2v. For members without transverse reinorement, v s = 0 and v = v. As the axial strain o a member depends on the magnitude o n v, whih is a untion o the applied shear stress, shear strength evaluation requires trial-and-error or the 2007AASHTO LRFD and 2006 CHBDC methods. Fig ompares the axial ompression stress n v rom the two ode approximate expressions, and the MCFT or two dierent sized members (S xe = 300 and 2000 mm) without transverse reinorement. As v is proportional to, the ratio n ' plotted ' v / in Fig is independent o '. The two ode approximate expressions or n v, whih are used or both members with and without transverse reinorement, give muh higher ompression stress than MCFT preditions or members without transverse reinorement. In the proposed evaluation method or members without transverse reinorement, it is assumed that n v = 0. This is a better approximation and greatly simpliies the shear strength evaluation o suh members and also removes the need or 86

108 trail-and-error proedure. 0.8 Axial ompression ratio ((ksi) S xe = 2000 Proposed S xe = 300 mm CHBC AASHTO MCFT Longitudinal strain ε x Fig Comparison o predited axial ompression stress ratio units) with MCFT or members without transverse reinorement. n ' (in MPa v / The expression or onrete ontribution ator β was seleted to ompensate or the assumption o zero longitudinal ompression stress n v, and was adjusted to give a good it with test results on girders. The proposed expression is: [ 3-46] β = (in MPa units) 1+ ( S ) ε ( S ) xe x xe The limit imposed on Eq. [3-46] is a ut-o or large members with very low longitudinal strain as shown or S xe = 2000 in Figure This prevents Eq. [3-46] preditions rom being unonservative ompared to MCFT or suh members. Fig ompares β rom Eq. [3-46] with that rom the MCFT and the 2007 AASHTO LRFD table and 2006 CHBDC Eq. [3-29]. As the 2006 CHBDC method was primarily developed or S xe = 300 mm, its preditions are in good agreement with MCFT 87

109 or that size member. The 2007 AASHTO LRFD values or β are almost exatly the same as the MCFT as the tabulated values were derived rom the MCFT AASHTO LRFD urves shown in Fig are based on linear interpolation between tabulated values. 0.4 Conrete ontribution β (MPa) S xe = 300 mm 2007 AASHTO CHBC 2006 MCFT S xe = 2000 mm Proposed Longitudinal strain ε x Fig Comparison o predited onrete ontribution β (in MPa units) with MCFT or members without transverse reinorement. For members without transverse reinorement, the angle θ is only used to determine the demand imposed on longitudinal reinorement by shear. As shown in Fig. 3-34, the 2007 AASHTO LRFD tabulated values are essentially idential to the MCFT. As the 2006 CHBDC expression or θ is a simple linear untion o axial strain, it does not it nearly as well to the MCFT values. The proposed expression or inlination o average prinipal ompression stress in onrete, whih is assumed to be parallel to the ritial diagonal raks is: 88

110 [ 3-47] θ = ε )( S ) (in MPa units) ( x xe The preditions o Eq. [3-47] are also shown in Figure The preditions are loser to MCFT results ompared to the 2006 CHBDC preditions or longitudinal strains o greater than strain is lower. ε x = Longitudinal reinorement unlikely yield at raks one the As in members with transverse reinorement, in some ases, yielding o longitudinal reinorement will limit the shear apaity o an element beause the longitudinal reinorement does not have the apaity to equilibrate the longitudinal ompression stress in onrete due to shear. The average longitudinal ompression stress n v that inluenes the strain o the member was disussed above. This average stress is redued by average tension stresses in raked onrete. The axial ompression stress required loally at diagonal raks to resist shear is onsiderably larger. For members without transverse reinorement it is given by: [ 3-48] n = 2v otθ v* The v and θ determined or maximum post-raking apaity o an element an be used in Eq. [3-48] to determine the longitudinal ompression at diagonal raks that must be balaned by tension in longitudinal reinorement. How the demand on longitudinal reinorement rom shear is ombined with the demand rom bending moment is explained in Chapter 4. 89

111 AASHTO S xe = 2000 mm Angle θ (degree) CHBC 2006 Proposed MCFT S xe = 1000 mm S xe = 300 mm Longitudinal strain ε x Fig Comparison o predited angle θ with MCFT or members without transverse reinorement. 90

112 Chapter 4. Beam Elements 4.1. General A rigorous analysis o a beam subjeted to axial load, bending moment and shear ore an be done using a multi-layer analysis where the response o eah layer is modeled as a uniorm shear element. Response 2000 is a omputer program that does suh analyses. A simpler shear analysis an be done by using a single uniorm shear element to approximate the omplete shear behavior o a beam and this is onsistent with shear design provisions in 2007 AASHTO LRFD and 2006 CHBDC. In this hapter, the multi-layer analysis proedure or beams subjeted to shear is briely reviewed, and the simpliied proedures used in 2007 AASHTO LRFD and 2006 CHBDC are summarized. The proposed proedure to apply the uniorm shear element model rom Chapter 3 is introdued and the preditions rom the proposed proedure are ompared with preditions using the shear design provisions in 2007 AASHTO LRFD and 2006 CHBDC, and the preditions rom Response Exat Solution The proedure explained here was developed by Vehio and Collins (1986). To aount or moment-shear interation at a beam setion, a beam ross-setion is divided to multiple layers as shown in Figure 4-1. Eah layer is assumed to behave like a uniorm shear element i.e. undergoes uniorm stresses and strains in every diretion. As a result, the MCFT ormulation explained in Chapter 3 an be applied to eah individual layer. Eah layer may have dierent stresses, strains and angle o prinipal ompression. To link the layers in this proedure, the well known assumption o plane setions remain 91

113 plane is used. In addition, global equilibrium should be satisied meaning that stresses ating on the layers should balane bending moment, shear, and axial ore ating at the ross-setion. It is worth mentioning that any shear low distribution at the setion an result in a set o stresses in beam layers whih satisy both setion equilibrium and ompatibility assumption o plane setions remain plane. Consequently, one other hek is needed to ensure that the assumed shear low distribution would not violate equilibrium between the beam setions. This hek is done by looking at the seond setion lose to the primary one assuming that the seond setion has to have the same shear low distribution as in the primary setion. As shown in Figure 4-2, the normal stresses alulated at individual layers should then be onsistent with the assumed shear low distribution determined rom the equilibrium o eah layer between the dual setions. Shown in Fig. 4-2, Vehio and Collins (1986) suggested that the seond setion should be spaed about d/6 rom the primary one. Fig. 4-1 Applition o MCFT to beam elements using multi-layer analysis (Vehio and Collins, 1986). 92

114 Fig. 4-2 Equilibrium o the dual setions in MCFT exat solution or beam elements (Vehio and Collins, 1986). 93

115 Figure 4-3 presents the algorithm o the proedure as presented by Vehio and Collins (1986). In summary, a shear low distribution is assumed in the beginning. Then, longitudinal strain proiles (whih are linear) are adjusted at the dual setions in a way that MCFT equations or individual layers, as well as global setional equilibrium at both setions, are satisied. Note that dual setions might have dierent longitudinal strain proiles but they have the same shear low proile. In addition, shear ore remains the same at both setions but moment varies to omply with equilibrium. One longitudinal strain proiles and normal stresses are determined at all layers or both setions, shear low distribution may then be alulated by looking at equilibrium o eah layer between the setions as illustrated in Figure 4-2. The proedure should be repeated until the alulated shear low proile is the same as the assumed one. The proedure is ompliated sine it involves multi-layer MCFT analysis o two setions and trial-and-error. Vehio and Collins (1986) explained that the proedure might be simpliied by assuming an approximate shear low proile or approximate shear strain proile. Common assumptions are onstant shear low (uniorm shear low) or paraboli shear strain proiles. This redues the analysis proedure to one setion only but the results are approximate. Vehio and Collins (1986) ompared the shear-moment interation preditions rom the exat method with preditions assuming onstant shear low and paraboli shear strain proile or three dierent onrete setions. In all ases, the approximate preditions using the assumption o onstant shear low gave similar results to the exat proedure. As it will be shown later in this thesis, the onstant shear low is the basi assumption o the 2007 AASHTO LRFD and the 2006 CHBDC shear provisions or beam elements. 94

116 Fig. 4-3 Algorithm o MCFT exat solution proedure or beam elements (Vehio and Collins, 1986). 95

117 Response 2000 Response 2000 (Bentz, 2000) is a omputer program that applies the MCFT to beams using a multi-layer analysis. The unique aspet o Response 2000 is that instead o using dual setions to predit the shear low distribution over the depth o a beam setion, it uses tangent stiness o layers meaning that it looks at two setions with ininitesimal distane apart instead. Response begins the analysis with an assumed shear low distribution to determine the strains and stresses in the layers. Knowing strain ondition o layers, stiness o eah layer an be determined using material onstitutive laws o steel and onrete explained in Chapter 3. The layers tangent stiness matrix will then be integrated over the depth to determine global stiness matrix o the setion whih is then used to alulate shear strain and shear low proiles rom external ores ating at the setion. The alulated shear strain proile is the assumed shear strain proile or the next iteration and the proedure is repeated until onvergene is reahed. Response 2000 is a sophistiated researh tool that provides detailed output o results inluding onrete and steel stresses and strains, shear on raks, onrete angle o prinipal ompression, rak spaing at all layers, and ore deormation plots as well as shear-moment interation diagrams. Figures 4-4 and 4-5 present some example output plots rom Response 2000 or a typial prestressed I-girder ross-setion used in bridges. The depth with darker olor in the ross-setion is an unraked depth and the portion o transverse reinorement that is yielding is also shown in darker olor. Among other plots, Figure 4-4 shows the setional proiles or prinipal ompression stress and shear stress on raks. 96

118 Fig. 4-4 Some output plots rom omputer program Response

119 Fig. 4-5 Some output plots rom omputer program Response

120 The allowable stresses are also shown in a dierent olor on the same plots. Two shear stress proiles in dierent olors are also provided. One is determined rom MCFT equations or individual layers and the other one is determined rom tangent stiness method explained previously. As shown, the two proiles should be lose i the analysis proedure has reahed onvergene. Bentz (2000) veriied Response 2000 against 534 tests reported in the literature on a variety o member types, and ound a mean ratio o measured to predited shear strength o 1.05 and a COV o 12%. In the report on the shear strength o bridge girders, Hawkins et al. (2005) ompared Response 2000 preditions with the results o 149 tests they seleted rom a database o 1359 shear tests. They seleted members that ontained minimum transverse reinorement ( ρ z y >50 psi [0.35 MPa]), had an overall depth o at least 500 mm, and were ast rom onrete having a ompressive strength o at least 28 MPa. Also, tests in whih anhorage or lexural ailure ourred were not inluded. The 149 tests they seleted inluded 85 prestressed onrete girders. They ound a mean ratio o measured to predited shear strength o 1.02 and 1.11, and a COV o 11% and 17% or reinored and prestressed girders, respetively. Response 2000 is not as appropriate or design and evaluation sine dierent users may get dierent results by adjusting parameters suh as material onstitutive models and rak spaing options available in the program. Also, as the program was developed or investigating a number o setions in detail, it is not ideal or evaluating many dierent setions in a normal design oie environment. 99

121 4.3. Simpliied Proedures or Design The shear design proedures in both the 2007 AASHTO LRFD and 2006 CHBDC are based on a single-layer shear analysis. The shear stress is assumed to be uniorm over the shear depth d v, whih is estimated as 0.9d. Similarly, the prinipal average ompression stress (diagonal rak) angle θ and the longitudinal ompression stress in onrete due to shear stress n v are also both assumed to be onstant over the shear depth d v. For members with transverse reinorement, the longitudinal strain ε x used in the shear analysis is taken as the average value over the setion, and or simpliity, this is estimated as hal the strain o the lexural tension reinorement. The longitudinal strain ε x used in the shear analysis or members without transverse reinorement is dierent in the two odes. It is equal to the maximum longitudinal strain in the 2007 AASHTO LRFD, while it is taken as the average longitudinal strain over the setion in the 2006 CHBDC. Both odes assume that bending moment is arried by the onrete ompression hord and the tension hord reinorement and the lexural internal lever-arm is equal to eetive shear depth ( jd = d v ). As a result o the assumption o uniorm longitudinal ompression stress over d v, hal o longitudinal ompression ore required or shear N v is arried by the tension hord reinorement and the other hal is arried by the ompression hord. These assumptions are shown in Figure 4-6. As mentioned previously, both 2007 AASHTO LRFD and 2006 CHBDC methods assume that the eetive shear depth an be idealized as one uniorm shear element thus one longitudinal strain is needed to determine the maximum shear ore taken by the eetive shear depth. In 1994 AASHTO LRFD and the 2000 CHBDC, this strain was the maximum longitudinal strain over the shear depth, whih is equal to the longitudinal 100

122 strain at the enter o the lexural reinorement, but this was later ound to be too onservative or members with at least minimum transverse reinorement. Consequently, the mid-depth strain was hosen as the longitudinal strain in the 2007 AASHTO LRFD and the 2006 CHBDC or members with at least minimum transverse reinorement. In both odes, the mid-depth strain is approximated by hal the maximum strain at the enter o lexural tension reinorement. Cross-setion Strain Fores Fig AASHTO LRFD and 2006 CHBDC approximate setional model or beams subjeted to shear and moment. The maximum longitudinal strain is given by: [ 4-1] M + 0.5N v jd ε x = ( E A + E A s s p p p A ) p 101

123 in whih M/jd+0.5N v is the ore in the lexural tension reinorement, p A p is prestressing ore at zero longitudinal strain, and (E s A s +E p A p ) is the stiness o lexural tension reinorement. The mid-depth strain, whih is taken as hal the maximum longitudinal strain, is alulated rom: [ 4-2] ε x M + 0.5N v p A jd = 2( E A + E A ) s s p p p where: M = bending moment at setion o interest; jd = internal lexural lever-arm (M/jd = lexural tension ore); N v = axial ompression ore needed to resist shear in web; A s, A p = area o nonprestressed and prestressed lexural tension reinorement; and E s, E p = Modulus o Elastiity o nonprestressed and prestressed reinorement, respetively. As explained in Chapter 3, 2007 AASHTO LRFD and the 2006 CHBDC use Eqs. [4-3] and [4-4] or N v, respetively. These equations are presented in the orm o stresses in Chapter 3. [ 4-3] N v = V otθ [ 4-4] N v = 2V For members without transverse reinorement, the 2006 CHBDC still uses the middepth strain thus uses Eq. [4-2] to alulate the longitudinal strain. In ontrast, 2007 AASHTO LRFD uses the maximum longitudinal strain over the setion depth given by Eq. [4-1]. To design a setion by 2007 AASHTO LRFD, mid-depth strain is estimated to determine θ and β values rom tables. The longitudinal strain ε x an then be alulated rom Eqs. [4-1] or [4-2]. This is repeated until the alulated ε x is equal or less than the 102

124 estimated ε x. Knowing θ and β, the required amount o transverse reinorement an be alulated using Eq.[3-22]. In ontrast, the 2006 CHBDC shear provisions do no require trial-and-error or design sine ε x is not a untion o θ. ε x is alulated rom Eq. [4-2] where N v is obtained rom Eq. [4-4], θ and β are determined rom Eqs. [3-30] and [3-29], and the required amount o transverse reinorement is then given by Eq.[3-22]. Both methods require trial-and-error or shear strength evaluation as the longitudinal strain ε x is a untion o shear resistane (shear ore at ailure level) and shear resistane is not known until the evaluation is omplete. As stated earlier, θ s and β s in the 2007 AASHTO LRFD and the 2006 CHBDC are based on the assumption that there is enough apaity provided in the axial diretion by longitudinal reinorement at the raks. To hek the validity o this assumption, the odes require that: M [ 4-5] Fl = + ( V 0.5V s ) otθ jd be equal to or greater than the top and the bottom hord longitudinal reinorement apaity. The bending moment is negative i the equation is used or ompression hord. In Eq. [4-5], M/jd is the demand on the lexural reinorement due to the bending moment and V 0.5V )otθ is the demand on the lexural reinorement due to the ( s shear ore. The demand V 0.5V )otθ due to the shear ore is hal the demand in ( s uniorm shear elements given by Eq. [3-28] as hal N v is assumed to be resisted by the lexural reinorement in beam elements (see Fig. 4-6). 103

125 4.4. Proposed Evaluation Method or Members With At Least Minimum Transverse Reinorement Figure 4-7 shows the Response 2000 preditions or a typial prestressed I-girder at the point o transverse reinorement yielding (dotted line) and onrete rushing (solid line). Inormation about the I-girder is provided in Setion 4-7. The longitudinal strains vary linearly over the depth as shown in Fig. 4-7(b). The transverse shear low (Fig. 4-7d) varies in a omplex nonlinear way with the maximum value being in the dek slab, whih is modeled omposite with the girder. The inlination o the prinipal ompression stress (Fig. 4-7) varies rom 0 on the top o the dek slab to 90 deg at the bottom ae o the girder. Over the height o the web, the angle generally varies between 27 and 39 deg at transverse reinorement yielding and between 24 and 26 deg at onrete rushing. The longitudinal onrete normal stress is multiplied by width o member (analogous to shear low) in Fig. 4-7(e) to ailitate omparison with the proposed method predition, whih assumes a onstant web width. The proposed method uses a single shear analysis and thus assumes uniorm shear stress over the shear depth d v. The mid-depth strain is used as the longitudinal strain in the shear analysis. The atual shear stress is not uniorm as shown in Fig. 4-7(d); however, the shear stress at mid-height is a reasonable estimate o the average shear stress. In the 2007 AASHTO LRFD and 2006 CHBDC simpliied design proedures, the longitudinal onrete ompression stress n v required to resist shear is also assumed to be uniorm over the shear depth d v. Figure 4-7(e) indiates that n v at mid-height is a reasonable estimate o n v over the web region o the member; but is not a good estimate o the average n v over the omplete shear depth. The shear depth extends well into the dek slab, the maximum shear low ours in this region; however this portion o the 104

126 Fig. 4-7 Variation o shear response over depth o prestressed I-girder with omposite dek slab: (a) ross-setion, (b) longitudinal strain, () angleθ, (d) shear low, (e) normal stress multiplied by width. 105

127 member does not experiene any diagonal raking and thus does not develop the additional longitudinal onrete ompressive stresses n v due to shear. There is lexural ompression in the dek slab that is balaned by lexural tension and thereore should not be part o N v. The longitudinal onrete ompressive stresses due to shear n v also do not extend down into the lexural tension lange. The onrete in this region is in tension, whih helps to stien the reinorement. Thus the n v estimated at setion mid-height is assumed to be uniorm over a redued depth d nv rom top o bottom lange to the bottom o top lange (see Fig. 4-7e). As a result, N v is determined by longitudinal ompression stress alulated rom Eq. [3-40] multiplied by the eetive area o b w d nv. The proposed equation or N v is: [ 4-6] N v = ( nv0 + nvε x ) bwd nv The tension stiening eet o onrete in the tension lange o the girder, whih has a ross-setional area o A t, is also aounted or in the proposed method. This is done by assuming an average onrete tension ore o F t = α ' A over this area; where α = in MPa units ( in psi units). α is alulated rom Eq. [3-5] assuming longitudinal strain o at the enter o lexural tension reinorement. Another reinement in the proposed method is to aurately aount or reinorement in the web, whih may be in any number o layers and be loated at any elevations. Figure 4-8 shows the beam approximate setional model that is used in the proposed t method. One layer o web nonprestressed reinorement distaned d w rom the ompression ae and one layer o web prestressed reinorement distaned d pw rom the ompression ae are shown. The longitudinal strain at the enter o web nonprestressed 106

128 Cross-setion Strain Fores Fig. 4-8 Approximate beam strain proile and ores in the proposed method. reinorement is ε sw and at the enter o web prestressed reinorement is ε pw. Thus the ores in the web nonprestressed and prestressed reinorement are: [ 4-7] Fsw = Es Aswε sw [ 4-8] F pw = E p Apwε pw + Apw p where A sw is web nonprestressed reinorement area, A pw is web prestressed reinorement area, and p is prestressing ore at zero strain. From strain ompatibility and assuming extreme ompression iber strain is equal to zero: d [ 4-9] ε w sw = 2 ε x d d pw [ 4-10] ε pw = 2 ε x d A ator o 2.0 in Eqs. [4-9] and [4-10] omes rom the assumption o mid-depth longitudinal strain ε x is hal the longitudinal strain at the enter o the lexural tension 107

129 reinorement. Substituting or ε sw and ε pw rom Eqs. [4-9] and [4-10] in Eqs. [4-7] and [4-8]: [ 4-11] Fsw = 2λs Es Aswε x [ 4-12] F pw = 2 λ p E p Apwε x + Apw p where d w d pw λ = s d and λ = p d. Fore T in the lexural tension reinorement shown in Fig. 4-8 is determined rom taking moments about the point o appliation o ompression ore C : M [ 4-13] T = N v λs Fsw λ p Fpw jd In deriving Eq. [4-13], it is assumed that the distane rom the point o appliation o C to the ompression ae is very small and almost equal to zero. Substituting or F sw and F pw rom Eqs. [4-11] and [4-12] and or N v rom Equation [4-6]: M 2 2 [ 4-14] T = + 0.5( nv0 + nvε x ) bwd nv 2 λs Es Aswε x 2λ p E p Apwε x λ p Apw p jd The mid-depth longitudinal strain, whih is assumed to be equal to hal the longitudinal strain at the enter o lexural tension reinorement, is determined rom: [ 4-15] ε x T α ' At = 2( E A + E A s s p p p ) A p where, as explained previously, α ' is the ore in the tension hord resisted by A t onrete tension stiening and p A p is the ore in the prestressed reinorement at zero longitudinal strain. Substituting or T rom Eq. [4-14] and solving or ε x, the proposed equation or mid-depth strain is: 108

130 [ 4-16] ε x = 2 M / jd n [ E (A s s 2 s vo + λ A b sw w d nv ) + E α p (A p ' A + λ t 2 p A pw p ( A + λ A ) p p )] 0. 5 n b v pw w d nv The longitudinal strain at setion mid-height is assumed to be hal the strain o the lexural tension reinorement. This is generally a onservative assumption as the strain on the opposite ae is usually ompressive (see Fig. 4-7b). I the bending moment is small and the shear ore is large, the setion may be subjeted to tension strains over the ull depth. This would be the ase i total ore in ompression hord C shown in Fig. 4-8 is greater than zero. In that ase, the setion mid-height strain alulated by Eq. [4-16] should be multiplied by 2. From Equilibrium and strain ompatibility using the same proedure used to determine T, C is given by: [4-17] C = M jd + 0.5( n 0 + n ε ) b d 2(1 λ ) λ E A ε 2(1 λ ) λ E A ε (1 λ ) A v v x w nv s For typial problems, nonprestressed reinorement may be assumed to be uniormly distributed over the ull web entered at mid-depth. For suh ases, Eqs. [4-16] and [4-17] an be simpliied to: [ 4-18] [ 4-19] ε x = 2 M / jd n s s vo b [ E (A A w sw d nv α ) + E p (A p ' A + λ t 2 p A s pw p s sw x ( A + λ A ) p p v pw )] 0. 5 n b w d nv p p p pw x p pw p C = M jd + 0.5( n 0 + n ε ) b d 0.5E A ε 2(1 λ ) λ E A ε (1 λ ) A v v x w nv s sw x p p p pw x p pw p Note area o lexural tension reinorement A s in denominator o Eq. [4-18] is multiplied by a ator o 4 ompared to area o longitudinal reinorement A sw entered in the web. A ator o 2 omes rom the assumption that mid-height strain is hal the strain o the lexural tension reinorement and a seond ator o 2 omes rom the need to provide 109

131 twie as muh reinorement at setion mid-depth to resist bending ompared reinorement on the lexural tension ae. Unlike the other simpliied proedures, ε x equation in the proposed method is not a untion o shear resistane but is a untion o setion geometry, amount o reinorement and material properties. These variables are known in evaluation problems thus ε x an be determined without a need or trial-and-error. One ε x is known β and θ are known rom Eq. [3-31] to [3-37]; thereore setion shear strength an be alulated rom Eq. [3-22] without trial-and-error. Note that ε x and the orresponding shear strength should be evaluated at yielding o transverse reinorement and onrete rushing ailure modes using the same equations or ε x and θ ( Eqs and 3-31), but the parameters β, n v0, n v, θ 0, θ have dierent values at eah ailure mode (see Eqs to 3-37 and 3-41 to 3-44). For yielding o longitudinal reinorement, another approah explained in the next setion is used. In the ase o multiple layers o nonprestressed and prestressed reinorement in the web, Eqs. [4-16] and [4-17] an be expressed in a more general orm as: [ 4-20] ε x = 2 M / jd + 0.5n [ E ( A + s s m j= 1 v0 2 j b w λ A d swj nv α ) + E p ( A ' A p + t n i= 1 p λ A 2 pi ( A p pwi + n i= 1 λ A pi pwi )] 0.5 n b v w ) d nv C = M / jd + 0.5n n i= 1 (1 λ ) pi p A pwi v0 b w d nv + ε [0.5 n b d x v w nv 2 m j= 1 λ (1 λ ) E A j j s swj 2 n i= 1 λ (1 λ ) E A pi pi p pwi ] where m and n are number o layers o nonprestressed and prestressed reinorement in the web, A pwi and A swj are total area o the i th and j th layer o prestressed and 110

132 nonprestressed reinorement. The parameters λ = d and λ = d are used to pi d pwi / j d wj / aount or the loation o prestressed and nonprestressed reinorement in the web where d pwi is the distane rom the lexural ompression ae to entroid o i th layer o web prestressed reinorement and d wj is the distane rom the lexural ompression ae to entroid o j th layer o web nonprestressed reinorement Evaluation at Yielding o Longitudinal Reinorement In the proposed method, shear strength o a setion is limited to the shear ore ausing biaxial yielding o reinorement at the rak. The stress orresponding to this shear ore is determined rom Eq. [3-45]. Expressing Eq. [3-45] in terms o ores, the shear ore whih auses biaxial yielding o reinorement is given by: [ 4-21] V ρ z y ( bwd v ) N v where N v is the longitudinal ompression ore reserved or shear. Assuming all longitudinal nonprestressed reinorement has yielded and all longitudinal prestressed reinorement has reahed their apaity pr in Figure 4-8: [ 4-22] T = As y + Ap pr [ 4-23] F sw = Asw y [ 4-24] F pw = Apw pr Substituting or T, F sw, and F pw in Eq. [4-13] and solving or N v, the axial ompression ore reserved or shear in the lexural tension reinorement is given by: [ 4-25] N = 2 [ ( A + λ A ) + ( A + λ A ) M / jd] v y s s sw pr p p pw 111

133 Similarly, the same proedure or the ompression side results in Eq. [4-26] or the axial ompression ore reserved or shear in the lexural reinorement in the ompression hord. [ 4-26] N = 2 [ ( A + (1 λ ) A ) + (1 λ ) A M / jd] v y s s sw pr p pw + Equation [4-21] must be evaluated separately or the lexural tension and lexural ompression sides o the member and the smaller value ontrols the shear strength. In Eqs. [4-25] and [4-26], the internal lexural lever-arm jd has a diret inluene on longitudinal reinorement ore and thereore the reserved apaity in longitudinal reinorement to resist shear. In the 2006 CHBDC, jd is estimated as 0.9d. Response 2000 preditions or numerous setions that ailed due to biaxial yielding o reinorement showed that a more aurate estimate is given by: [ 4-27] jd = d pr A p + y 1.2 ' b A s where b is the ompression ae width o the setion. In Eq. [4-27], the average lexural ompression stress over the onrete lexural ompression zone is assumed to be.6 ' Governing Failure Mode Figure 4-9 depits typial shear stress-shear strain relationships or reinored onrete beams. Solid lines show the typial urves i yielding o longitudinal reinorement does not limit the shear strength. Suh ases involve yielding o transverse reinorement and onrete rushing ailure modes and the higher o these two is the governing ailure mode that determines the shear strength. Dashed lines illustrate possible hanges in the urves i longitudinal reinorement yields prior to yielding o transverse reinorement or ater yielding o transverse reinorement. In the upper urve shown in Fig. 4-9 the shear stress 112

134 inreases ater yielding o transverse reinorement until onrete rushes. In the lower urve the shear stress redues ater yielding o transverse reinorement. When longitudinal reinorement yields beore yielding o transverse reinorement in both urves, the shear stress inreases up until yielding o transverse reinorement and then remains onstant. In suh ases, biaxial yielding o reinorement ailure mode is the highest shear stress throughout the loading and thus is the governing ailure mode. Fig. 4-9 Typial shear stress-strain relationships or beams. Another possibility is that longitudinal reinorement yields ater transverse reinorement yielding. I this happens, shear stress remains almost unhanged ater longitudinal reinorement yielding until onrete rushes (see Fig. 4-9). However, this shear stress is only the highest and governing shear stress throughout the load deormation urve i the onrete rushing shear stress is higher than the shear stress 113

135 orresponding to yielding o transverse reinorement (urve type 1). For the type 2 urve, on the other hand, yielding o transverse reinorement is the governing ailure mode sine it orresponds to the highest shear stress in the load deormation urve. I the shear strength orresponding to the biaxial yielding o reinorement is lower than yielding o transverse reinorement shear stress or urve type2, the problem would be how to determine whether longitudinal reinorement yielding has happened prior to or ater yielding o transverse reinorement. This an be examined by heking the ondition given by Eq. [4-5] at transverse reinorement yielding stage; longitudinal reinorement will yield ater transverse reinorement yielding i the ondition is satisied. Sine urve type 2 orresponds to setions with low amount o transverse reinorement lose to minimum in whih onrete rushing shear stress is not signiiantly lower than transverse reinorement yielding shear stress, the proposed method uses the biaxial reinorement yielding shear strength as the governing shear strength i it is lower than any o the other two ailure modes. In summary, the governing ailure mode in the proposed method is: - V yield i greater than V rush and less than V biaxial - V rush i greater than V yield and less than V biaxial - V biaxial i less than Max (V yield, V rush ) where V yield, V rush, and V biaxial are the beam shear strength orresponding to yielding o transverse reinorement, rushing o onrete, and biaxial yielding o reinorement, respetively. As explained in Chapter 3, the governing ailure mode should not be greater than 0.25 'b d to avoid onrete rushing prior to transverse reinorement yielding. w v 114

136 4.5. Proposed Evaluation Method or Members Without Transverse Reinorement Fig shows the Response 2000 preditions or a reinored onrete beam (Fig. 4-10a) without transverse reinorement at the maximum applied shear ore. The longitudinal strains are assumed to vary linearly over the depth (Fig. 4-10b), while the transverse strain (Fig. 4-10) is highly nonlinear and the maximum value is mid-way between the lexural tension reinorement and the setion mid-depth. The shear stress distribution (Fig. 4-10d) is not uniorm; but an be reasonably approximated as uniorm over the shear depth d v. Fig. 4-10(e) examines the distribution o longitudinal normal stress over the beam depth. The lexural ompression over the top third o the beam is very prominent. The tension ore needed to equilibrate this ompression is alulated as part o the lexural analysis. Immediately below the lexural ompression zone, there is a peak tension stress, and below that, the beam is diagonally raked. Over the diagonally raked portion o the beam, there are small longitudinal ompression stresses. In at, the tension stresses above the diagonal raks and the tension stiening (average tension stresses in raked onrete) around the longitudinal reinorement are larger than the longitudinal ompression. In Chapter 3, it is shown that the longitudinal ompression stress n v is small and an be assumed equal to zero or a uniorm shear element without transverse reinorement. The assumption o no longitudinal ompression ore due to shear in a diagonally raked member without transverse reinorement is even more valid or a beam than a uniorm shear element as shown in Figure 4-10(e). Numerous analyses o members without stirrups (similar to that shown in Fig. 4-10) indiated that the maximum transverse strain typially ours mid-way between the level 115

137 Fig Response 2000 preditions or variation o shear response over the depth o a beam, M/V = 2.0 m, y = 550 MPa, = 20 MPa. ' 116

138 o the lexural tension reinorement and the setion mid-depth as shown in Fig. 4-10(). Thus in the proposed proedure, the longitudinal strain ε x used in the shear analysis or members without stirrups is the maximum longitudinal strain divided by 1.5. As in the 2006 CHBDC and 2007 AASHTO LRFD methods, the shear low is uniorm over the eetive shear depth d v. Assuming the axial ompression ore due to shear and tension stiening ore in the tension hord are zero, aounting or the tension ore resisted by distributed nonpressed reinorement, and aounting or the prestressing tendons in the web, the same proedure used to derive Eq. [4-18] results in the ollowing proposed longitudinal strain: [ 4-28] ε x = 1.5 [ E s M / jd ( A s p A ( A + λ A ) sw p ) + E p p ( A pw p + λ A Sine ε x is not a untion o shear demand or angle o prinipal ompression, the proposed method does not require trial-and-error or evaluation. While the inluene o longitudinal ompression due to shear is negligible in the proposed method when alulating the axial strain o the member, it annot be ignored when heking whether the longitudinal reinorement yields at a diagonal rak. The longitudinal ompression stress required loally at diagonal raks to resist shear is muh larger than the average longitudinal ompression stress, and is given by MCFT as: [ 4-29] N = 2V otθ v* Thus the longitudinal reinorement yields i this axial ompression ore ( N v *) is greater than reserved axial ompression ore in the tension or ompression hords given by Equations [4-25] and [4-26]. For ases in whih longitudinal reinorement yields, the shear strength is taken as: 2 p pw )] 117

139 [ 4-30] V = N v 2 whih is determined by solving Eq. [4-29] or V with θ equal to 45 degrees. MCFT results shown in Fig indiate that angle o prinipal ompression is mostly greater than 45 degrees thus 45 deg is generally a onservative assumption. Note that the higher the angle, the more onservative the preditions or members with no stirrups. This is beause the angle or suh members is only used to determine shear demand on longitudinal reinorement Members With Less than Minimum Transverse Reinorement In the 2006 CHBDC and 2007 AASHTO LRFD shear design provisions, minimum amounts o transverse reinorement are dierent. Thus one o these minimum amounts was seleted to be used in the proposed method. The 2006 CHBDC requirements or minimum amount o transverse reinorement ρ zmin and maximum transverse reinorement spaing s max are: [ 4-31] ρ = min z ' y [ 4-32] s 0.75d 600 mm or v <.10 ' max = v u 0 [ 4-33] s 0.33d 300 mm or v.10 ' max = v u 0 The minimum perentage o transverse reinorement is 38% larger in 2007 AASHTO LRFD but the maximum permitted spaing o this reinorement is slightly larger. Comparison o shear strength preditions rom the proposed method with beam test results in Chapter 5 shows that the proposed evaluation proedure an reasonably predit the shear strength o beams with transverse reinorement amount as low as the

140 CHBDC minimum amount. Consequently, the 2006 CHBDC minimum amount o transverse reinorement was adopted in the proposed method. In the 2006 CHBDC and 2007 AASHTO LRFD shear design provisions, i a setion has less than the minimum amount o transverse reinorement, the shear strength is alulated assuming no transverse reinorement. Angelakos et al. (2001) ound that this approah results in a onservative estimate o shear strength or members with slightly less than the minimum transverse reinorement. Based on the results o 21 large reinored onrete beam tests, they proposed that the shear strength inrease linearly rom the shear strength o a member with no stirrups to the shear strength o the member with minimum stirrups. In the evaluation setion, the 2006 CHBDC reommends the same approah; but the linear inrease in shear strength starts when the member has more than one-third o the minimum transverse reinorement. Members with less than one third o minimum transverse reinorement are assumed to have the same shear strength as members with no transverse reinorement. In the proposed method, the 2006 CHBDC linear approah was adopted or members with less than minimum transverse reinorement based on omparison o the shear strength preditions rom the proposed method with the experimental results that is presented in Chapter Example Evaluations o Bridge Girder With at Least Minimum Transverse Reinorement The trends predited by the proposed expressions or inlination o diagonal raks θ, onrete ontribution ator β, total shear stress v, and longitudinal ompression stress n v or an element subjeted to uniorm shear were ompared with MCFT in Chapter 3. In order to veriy the omplete shear strength evaluation proedure inluding the 119

141 assumptions used to apply the uniorm shear approah to bridge girders, the results obtained or three example bridge girders are ompared with the results obtained using omputer program Response The predited shear strengths determined rom the 2007 AASHTO LRFD, 2006 CHBDC, and ACI 318 shear design methods are also presented. Note that the 2008 AASHTO LRFD shear design provisions are similar to the 2006 CHBDC shear design provisions. Figure 4-11 shows the ross-setions o the three dierent girders in the three bridges. The ollowing material properties were assumed in all ases: ' = 40 MPa, y = 400 MPa, pu = 1860 MPa, and E p = E s = MPa. In order to ompare resistanes rom dierent odes, the nominal resistanes (without resistane ators) were alulated. The I-girder bridge is a 21 m single span bridge with six prestressed onrete I-girders spaed at 2 m. The ross-setion o the girders near the support is shown in Fig. 4-11(a). The our prestressing tendons near the top o the web are draped toward the bottom lange at 7.75 m rom the support. The bridge transverse reinorement ratios hange rom ρ z = 0.834% to 0.437% and rom ρ z = 0.437% to 0.327% at loations o 7.32 m and 8.69 m rom the supports. The box-girder bridge has a single span o m and onsists o nine prestressed onrete box girders. The ross-setion o the girders near the support is shown in Fig. 4-11(b). The 16 prestressing tendons distributed over the web are draped toward the bottom hal o the web at m rom the support. The hannelgirder (Fig. 4-11) bridge onsists o 14 preast nonprestressed hannel girders interonneted by reinoring bars grouted in plae. Eah o the simple spans o this multi-span bridge is 8.40 m. 120

142 (a) I-girder (b) Box-girder () Channel-girder Fig Cross-setions o girders in example bridges. 121

143 The bridge girder amounts o transverse reinorement hange rom ρ z = 0.588% to 0.294% and rom ρ z = 0.294% to 0.098% at loations o 1.39 m and 1.75 m rom the supports. All bridges had two trai lanes. Bridge live load (truk load) or I-girder Bridge onsisted o CHBDC standard truk with ive axels and total weight o 625 kn in addition to one speial permit truk with eight or 11 axles and total weight o 839 kn (85.5 tonnes) or 1699 kn (173.2 tonnes). For the other two bridges, however, the CHBDC standard truk or 839 kn (85.5 tonnes) permit truk were only used. Lane load and dynami allowane ator, as well as a multilane redution ator were inluded as per the 2006 CHBDC. The load ators were based on the 2006 CHBDC or Level 2 Inspetion. Live load was transversely distributed aording to the 2006 CHBDC. Fatored moment and shear envelope diagrams or all three bridges are shown in Figure Appendix D inludes inormation o the truks used or evaluation in this thesis. The bridge girders were evaluated at a number o setions along the span. At loations where the spaing o transverse reinorement hanges, the shear strength was assumed to linearly vary rom shear strength o the setion with lower amount o transverse reinorement to shear strength o the setion with higher amount o transverse reinorement over the length d v entered at the loation o hange in the amount o transverse reinorement. This is permitted in the 2006 CHBDC evaluation setion. At eah setion, evaluation was done at a onstant moment equal to the atored moment demand orresponding to maximum shear demand ating at that setion. To evaluate the example bridge girders by the 2006 CHBDC method, shear apaity was estimated and ε x was alulated aordingly. 122

144 Shear envelope (kn) Moment Shear Moment envelope (knm) Distane rom let support (m) (a) I-girder shear and moment envelopes Shear envelope (kn) Moment Shear Moment envelope (knm) Distane rom let support (m) (b) Box-girder shear and moment envelopes Shear envelope (kn) Moment Shear Moment envelope (knm) Distane rom let support (m) () Channel-girder shear and moment envelopes Fig Moment and shear envelopes o evaluated bridges with minimum stirrups. 123

145 Using ε x, θ and β were determined, and shear apaity was alulated. The same proedure was repeated until the estimated shear apaity was equal to the alulated shear apaity. To evaluate example bridge girders by the 2007 AASHTO method, θ and β were irst estimated. Shear apaity was then alulated using the estimated θ, β. Subsequently, ε x was determined substituting the alulated shear apaity and estimated angle in the equation or longitudinal strain ε x. Entering the tables with ε x and shear apaity ratio ( v / ' ), θ and β were determined; linear interpolation was used or intermediate values. The same proedure was repeated until the estimated θ and β were equal to the θ and β extrated rom the tables. To evaluate prestressed bridge girders by the ACI 318 shear design method, the well known V i, V w approah was used to determine the onrete ontribution. For V i alulations, it was assumed that top dek onrete weight as well as girder weight is supported by girders only while other added load (wearing and live loads) are supported by omposite ation o dek and girders. For the hannel-girder bridge, whih was nonprestressed, V was assumed to be 2 b d in psi units ( 0.17 b d in MPa units) ' w ' w as speiied by ACI Comparison o Results or I-girder Bridge The variation o predited strengths over about hal the span o the I-girder is shown in Fig Both the proposed method and Response 2000 an predit the shear strength at yielding o transverse reinorement and rushing o onrete. 124

146 Shear strength (kn) Response (Crushing) Response (Yielding) Proposed (Crushing) 200 Proposed (Yielding) (a) Distane rom let support (mm) Shear strength (kn) Response (Crushing) Response (Yielding) CHBDC 2006 AASHTO 2007 ACI (b) Distane rom let support (mm) Fig Comparison o predited shear strengths along span o I-girder bridge: (a) Response 2000 and proposed method, (b) Response 2000 and ode design methods. 125

147 Figure 4-13(a) ompares the shear strength determined rom Response 2000 and the proposed method or the onrete rushing mode (solid lines) and transverse reinorement yielding mode (dashed lines). Generally, there is very good agreement along the span. Note that the inormation presented in Fig. 4-7 is or the I-girder explained here at 7.92 m rom the support. Fig. 4-13(b) ompares the shear strength rom Response 2000 with shear strengths aording to the 2007 AASHTO LRFD, 2006 CHBDC and ACI 318 shear design methods AASHTO LRFD gives a sae predition almost all along the span; the preditions beome slightly unonservative near the support. The 2006 CHBDC shear design method gives an unsae predition near the support where the axial strains are very low. On the other hand, the method gives a very low predition at 4.21 m rom the support, where the predited strength is 778 kn while the shear strength aording to Response 2000 is 938 kn at transverse reinorement yielding and 1032 kn at onrete rushing. Near mid-span, the 2006 CHBDC predition orresponds well with transverse reinorement yielding. The ACI 318 shear design method gives very unsae preditions where V w ontrols the shear strength rom the support up to 6.10 m rom the support. This equation was previously reognized as being unsae (Hawkins et al. 2005). Figure 4-14 ompares the Response 2000 preditions or mid-depth strain over about hal the span with the preditions rom 2007 AASHTO LRFD, 2006 CHBDC, and the proposed method. Similar to Response 2000, the proposed method an predit the middepth strains at yielding o transverse reinorement and rushing o onrete. The 2006 CHBDC and 2007 AASHTO LRFD an only predit one mid-depth strain at a setion. The proposed method preditions agree well with the preditions rom Response

148 both at yielding o transverse reinorement and rushing o onrete. The 2006 CHBDC and 2007 AASHTO LRFD preditions are signiiantly larger than Response Strain (mm/m) Distane rom let support (mm) Response (Crushing) Response (Yielding) Proposed (Crushing) Proposed (Yielding) CHBDC 2006 AASHTO 2007 Fig Comparison o Response 2000 predited mid-depth strain along span o I- girder bridge with proposed and ode design methods. preditions at yielding o transverse reinorement; however, the predited shear ores at ailure (shear strength) rom the 2006 CHBDC and 2007 AASHTO LRFD are slightly lower than the Response 2000 preditions at yielding o transverse reinorement. This suggests that both 2007 AASHTO LRFD and 2006 CHBDC methods onservatively estimate the mid-depth longitudinal strain. The preditions rom the 2007 AASHTO LRFD method are less onservative ompared to the preditions rom the 2006 CHBDC method. Notie that shear strength at yielding o transverse reinorement rom the proposed method (Fig. 4-13a) is greater than the shear strength at onrete rushing near 127

149 the support where the predited longitudinal strain (Fig. 4-14) is small and almost zero. This was also notied in MCFT preditions or uniorm shear elements presented in Chapter 3 (see Fig. 3-13a) Comparison o Results or Box-girder Bridge In Fig. 4-15(a), the proposed method preditions o shear strength or the box-girder along almost hal the span are ompared with the Response 2000 preditions at yielding o transverse reinorement and onrete rushing. The proposed method preditions ompare well with the preditions rom Response In the proposed method preditions, shear strength at transverse reinorement yielding governs at setions that are near the support as it is greater than the shear strength orresponding to the rushing o onrete. This is onsistent with Response 2000 preditions. Figure 4-15(b) ompares the shear strength preditions rom Response 2000 with the preditions rom 2007 AASHTO LRFD, 2006 CHBDC, and ACI 318. As also notied in the I-girder, ACI 318 method results in unonservative shear strength preditions or setions that are lose to the support (up to 3.7 m rom support). The ACI 318 method preditions beome onservative or setions lose to mid-span ompared to the preditions rom Response The 2006 CHBDC method preditions are unonservative lose to the support; however, beome onservative near mid-span ompared to the Response 2000 preditions at rushing o onrete. Figures 4-16(a) and 4-16(b) present Response 2000 predited mid-depth strains and lexural reinorement strains (strain at the enter o lexural tension reinorement) or the box-girder over about hal the span with the preditions rom the proposed method. 128

150 2000 Shear strength (kn) Response (Crushing) Response (Yielding) Proposed (Crushing) Proposed (Yielding) (a) Distane rom let support (m) Shear strength (kn) Response (Crushing) Response (Yielding) CHBDC 2006 AASHTO 2007 ACI (b) Distane rom let support (m) Fig Comparison o predited shear strengths along span o box-girder bridge: (a) Response 2000 and proposed method, (b) Response 2000 and ode design methods. 129

151 Strain (mm/m) Response (Crushing) Response (Yielding) Proposed (Crushing) Proposed (Yielding) CHBDC 2006 AASHTO (a) Distane rom let support (m) Strain (mm/m) Response (Crushing) Response (Yielding) Proposed (Crushing) Proposed (Yielding) CHBDC 2006 AASHTO (b) Distane rom let support (m) Fig Comparison o Response 2000 predited (a) mid-depth strain, (b) lexural tension reinorement strain along span o box-girder bridge with proposed and ode design methods. 130

152 The preditions rom 2007 AASHTO LRFD, and the 2006 CHBDC methods are also presented. The proposed method preditions o mid-depth strain (Fig. 4-16a) are the losest to those predited by Response 2000 but the dierene is still signiiant. The dierene, however, is due to the assumption o mid-depth strain is hal the strain at the enter o lexural tension reinorement. As shown in Fig. 4-16(b), the preditions o lexural tension reinorement strain rom the proposed method agree well with Response 2000 preditions. The 2007 AASHTO LRFD preditions or lexural tension reinorement stain ompare well with the Response 2000 preditions at yielding o transverse reinorement but are lower than Response 2000 preditions at onrete rushing. The 2006 CHBDC preditions o lexural tension reinorement strain are onservative and even larger than Response 2000 preditions at onrete rushing along a signiiant portion o the span Comparison o Results or Channel-girder Bridge Shear strength preditions o Response 2000 or almost hal the span o the hannelgirder bridge are shown with the preditions rom the proposed method in Fig. 4-17(a), and with the preditions rom 2007 AASHTO LRFD, 2006 CHBDC, and ACI 318 in Figure 4-17(b). Figure 4-17(a) illustrates that Response 2000 preditions o shear strength are ompatible with those predited by the proposed method both at yielding o transverse reinorement and rushing o onrete. Near mid-span, the proposed method shear strength preditions at yielding o transverse reinorement is higher than those at onrete rushing sine the amount o transverse reinorement is low (0.098%) and slightly higher than the 2006 CHBDC minimum amount (0.095%). Similar trend is also notied but is less signiiant in the Response 2000 preditions. 131

153 700 Shear strength (kn) Response (Crushing) Response (Yielding) Proposed (Crushing) Proposed (Yielding) (a) Distane rom let support (m) 700 Shear strength (kn) Response (Crushing) Response (Yielding) CHBDC 2006 AASHTO LRFD ACI (b) Distane rom let support (m) Fig Comparison o predited shear strengths along span o hannel-girder bridge: (a) Response2000 and proposed method, (b) Response 2000 and ode design methods. 132

154 The 2006 CHBDC preditions are slightly lower than the 2007AASHTO LRFD preditions and ACI 318 preditions are too onservative exept or the portion o the span that is near mid-span AASHTO LRFD preditions as well as preditions rom the proposed method are slightly unsae (with Response 2000-to-predited shear strength ratio o about 0.95) near mid-span. Response 2000 results showed that yielding o transverse reinorement does not extend over the ull eetive shear depth as the amount o transverse reinorement is low. This does not ause signiiant redution in shear strength or members ontaining minimum transverse reinorement as shown in Fig. 4-17(b). Notie that the preditions rom the proposed method are not signiiantly higher than Response 2000 preditions near mid-span. In Fig. 4-18, Response 2000 preditions o hannel girder mid-depth strain over about hal the span with the preditions rom the proposed method, and rom the 2007 AASHTO LRFD and the 2006 CHBDC methods are presented. The proposed method preditions are the losest to Response 2000 preditions. The 2007 AASHTO LRFD preditions are better than the 2006 CHBDC preditions ompared to Response 2000 preditions. It should be mentioned that the large mid-depth strains predited by the odes as well as the proposed method near the support is beause the axial ore N v due to shear has aused tension in the ompression hord. The same trend is also notied in Response 2000 preditions. 133

155 Strain (mm/m) Response (Crushing) CHBDC 2006 Proposed (Yielding) Proposed (Crushing) Response (Yielding) AASHTO LRFD Distane rom let support (m) Fig Comparison o Response 2000 predited mid-depth strain o I-girder bridge with proposed and ode design methods Summary o Results or Example Bridge Girders The predited shear strengths at three setions along eah o the three bridge girders are summarized in Table 4-1. For eah bridge, one setion is loated in the low-moment region lose to the support, another setion is loated in the high-moment region near mid-span, and the third setion was loated between the other two. The shear strength at transverse reinorement yielding and onrete rushing aording to Response 2000 are both shown, and the lower (ritial) one is identiied (*). Unlike the shear design methods, the proposed method also gives two shear strengths, and these generally agree well with Response 2000 results. 134

156 Table 4-1 Summary o preditions or example bridge girders with at least minimum transverse reinorement. Bridge I-girder Boxgirder Channelgirder Setion inormation Dist. rom supp. (m) ρ z (%) V u (kn) M u (knm) Predited shear strength (kn) (ratio Response 2000 shear to predited shear) AASHTO (1.01) (1.22) (1.40) (1.04) (1.12) (1.17) (1.22) (1.14) (0.95) CHBDC (0.92) (1.30) (1.49) (0.98) (1.16) (1.25) (1.30) (1.22) (1.03) ACI (0.95) (0.91) (1.05) (0.97) (1.19) (1.25) (1.29) (1.40) (0.99) Proposed Response 2000 shear strength (kn) Yielding Crushing Yielding Crushing * (1.01) * (1.05) * (0.96) * 959.4* (1.05) 587.4* (1.12) * * * * (1.05) * (1.03) 603.6* (1.00) 405.4* (1.06) * * * * * Mean COV (%) The ratios o Response 2000 shear strengths to predited shear strengths are shown within brakets. For the proposed method, the ratios vary rom 0.96 to 1.12, and have a mean o 1.04 and a COV o 4.3%. For the 2007 AASHTO LRFD shear design method the ratios vary rom 0.92 to 1.40, with a mean o 1.14 and a COV o 11.8%. The ratios rom the 2006 CHBDC shear design method vary rom 0.92 to 1.49, with a mean o 1.18 and COV o 15.4%. Finally, or the ACI 318 method, the ratios vary rom 0.91 to 1.40; have a mean o 1.11, and a COV o 15.9%. Note that 2007 AASHTO LRFD and the proposed method preditions are slightly unsae only or the third setion o the hannel girder bridge that ontains very low amount o transverse reinorement lose to the

157 CHBDC minimum amount. The CHBDC and ACI 318 methods, however, resulted in unsae estimate o shear strength at the irst two setions o the I-girder and hannel girder bridges where the moment shear ratios were low Example Evaluation or Bridge Girder With Less than Minimum Transverse Reinorement To demonstrate the proposed evaluation method, it was also applied to T-girders with less than minimum transverse reinorement in a two-span ontinuous bridge with a span o 24.7 m. The 8.8 m wide bridge has two lanes o trai, and our girders. The girders are 1067 mm deep at the ends o the bridge, and are haunhed to 2286 mm deep over a m length near the middle support. The girders were strengthened by post-tensioned 32 mm Dywidag bars loated near the top lange and attahed to the girders by steel diaphragms. The results rom the shear strength evaluation at the setion loated 4.67 m rom the middle support are disussed below. The details o the setion at that loation are shown in Fig At this setion, the amount o transverse reinorement in the girders was ρ z = 0.042%, whih is 62% o the 2006 CHBDC minimum transverse reinorement ρ zmin = 0.068%. The 610 mm spaing o the stirrups is exatly the maximum allowed spaing. The onrete ylinder ompression strength ' = 21 MPa and reinorement yield strength y = 400 MPa. The longitudinal rak spaing parameter at the setion o interest is S xe = 1469 mm. Bridge live load (truk load) onsisted o CHBDC standard truk with total weight o 625 kn as well as 3 speial permit truks with 7, 8 and 9 axles and total weights o 750 kn (76.5 ton), 839 kn (85.5 ton) and 819 kn (83.5 ton). In addition, a 6-axle mobile 136

158 rane with 118 kn (12 ton) axle load was onsidered. Lane load and dynami allowane ator, as well as a multi-lane redution ator were inluded as per the 2006 CHBDC. The load ators were based on the 2006 CHBDC or Level 2 Inspetion. Live load was transversely distributed aording to the 2006 CHBDC. Fatored moment and shear envelope values at the setion o interest were M u = knm, V u =662.4 kn. Truk details are presented in Appendix D. Fig Cross-setion o the evaluated bridge girder example at 4.67 m rom midsupport (M u = knm, V u =662.4 kn). Table 4-2 ompares the shear strength preditions or the ritial setion o the bridge girders. The irst row o preditions are assuming the amount o transverse reinorement is less than minimum, and thereore V is alulated assuming no transverse reinorement and V s = 0. The seond row o preditions was made assuming the setion had minimum stirrups ρ zmin = 0.068% even though the atual stirrups were only 62% o this amount. The third row gives the predited strength using the

159 CHBDC linear interpolation method. As the atual amount o transverse reinorement (ρ z = 0.042%) is a 43% inrease rom one-third minimum to minimum (ρ z = 0.068%), the shear strength in row three is equal to the value in row one plus 43% o the dierene between rows one and two. The predition given in the third row o the olumn or Response 2000 is the predition using the atual amount o reinorement. For the ACI 318 preditions, the external prestressing was treated as an axial ompression. Table 4-2 Comparison o nominal shear strength preditions (kn) or bridge girder example ignoring shear resisted by inlined lexural ompression. AASHTO CHBDC ACI 318 Proposed Response 2000 Assuming no stirrups Assuming min. stirrups Using 2006 CHBDC Interpolation (1) (1) Based on atual amount o stirrups not CHBDC interpolation method. The 2006 CHBDC predition or the setion without transverse reinorement is 787 kn, whih is signiiantly lower than all other preditions, and the 2006 CHBDC predition or the setion with minimum transverse reinorement is 1729 kn, whih is signiiantly higher than all other preditions. There are two reasons or this. First, the predited longitudinal strain ε x that inluenes the shear response is very small or whih the 2006 CHBDC values o β are onservative or large members without stirrups (see Fig. 3-33). Seondly, the 2006 CHBDC uses the same β equation or members with transverse reinorement (Eq. 3-29) exept S xe beomes 300 mm, whih results in a 138

160 signiiantly higher β value or a large member with transverse reinorement ompared to the same member without transverse reinorement. The preditions rom the proposed method are similar to the Response 2000 results in all ases. It is interesting to note that the evaluated setion was within the hunhed portion o the girder. As a result, the inlined lexural ompression ore ontributes to the shear strength o the bridge. Aounting or this eet results in 222 kn additional shear strength using simple hand alulations assuming jd=0.9d, and 547 kn additional shear strength using Program Response The Response 2000 predition is muh higher beause a onsiderable portion o the setion is in ompression and thereore jd is smaller and hene the lexural ompression is larger. 139

161 Chapter 5. Comparison with Beam Test Results 5.1. General In this hapter, the proposed shear evaluation method is veriied by omparing preditions o shear strength with the measured shear strengths o beams that ailed in shear. The shear strengths o the beams were also predited using the shear design provisions o 2007 AASHTO LRFD, 2006 CHBDC, and ACI 318. The minimum amount o transverse reinorement required in the shear design provisions o 2006 CHBDC and 2007 AASHTO LRFD are dierent. Results rom tests on beams with a low perentage o transverse reinorement are used in this hapter to show the 2006 CHBDC limit is appropriate or use with the proposed method. The 2006 CHBDC has an interpolation proedure that an be used to evaluate the shear strength o a member with less than the minimum perentage o transverse reinorement. The results rom tests on beams with a low perentage o transverse reinorement are also used to show this proedure is appropriate or use with the proposed method Members With at Least Minimum Transverse Reinorement To veriy the proposed method or members with at least minimum transverse reinorement, preditions or 80 reinored onrete beams and 88 prestressed onrete beams with at least minimum transverse reinorement as speiied by 2006 CHBDC ' ( ρ z 0.06 in MPa units ) were ompared with the measured test results. The y seleted tests were mostly extrated rom the shear database olleted or National Cooperative Highway Researh Program (NCHRP) Projet (Hawkins et al. 2005) 140

162 and presented by Kim (2004). The shear database inludes 160 reinored onrete beams with transverse reinorement, and 164 prestressed onrete beams with transverse reinorement. A total o 156 o these beams that had a depth o at least 300 mm and transverse reinorement ratio not greater than were seleted rom the database. Among the tests seleted or omparison are ten prestressed I-girders that were 73 in. (1854 mm) deep tested or NCHRP projet at the University o Illinois (Kim 2004, Kuhma et al. 2005) as well as seven prestressed I-girders that were 44 in. (1118 mm) tested by the Strutural Researh Center in Florida (Shahawy and Bathelor, 1996). In addition to the results rom the database, seven reinored onrete beams tested by Mphonde and Frantz (1985) and ive reinored onrete beams tested by Rahal and Al- Shaleh (2004) were inluded. To predit the shear strength o the tested beams using the proposed method, 2007 AASHTO LRFD, and 2006 CHBDC, trial-and-error is needed as the applied moment-toapplied shear ore ratio is known at the ritial setion. Evaluation o a bridge girder using the proposed method does not need trial-and-error as the aompanying bending moment is known in this ase. Trail-and-error is not needed to predit the shear strength o tested beams using ACI 318 as the shear strength is a untion o applied moment-toapplied shear ore ratio in the method. Linear interpolation was used to determine β and θ values rom the tables provided in the 2007 AASHTO LRFD shear design provisions. The loading or most o the tests onsisted o one or two onentrated loads. The ritial shear setion or these was assumed to be at a distane d (eetive depth) rom the onentrated load toward the support. For tests with uniormly distributed loading, the ritial setion was assumed to be at distane d rom the support or rom the loation o 141

163 hange in the amount o transverse reinorement toward mid-span. For beams with inlined tendons, the vertial omponent o the eetive prestressing ore V p was added to the shear strength. Figures 5-1(a) and (b) present the umulative requeny diagrams o test-topredited shear strength ratios o 80 reinored onrete (RC) beams and 88 prestressed onrete (PC) beams, respetively. These diagrams show the number o ratios (as a portion o the total number) that are equal to or less than the orresponding test-topredited ratio on the horizontal axis. A peret model would have a umulative requeny o zero or a ratio o test-to-predited less than 1.0, and a umulative requeny o 1.0 or a ratio o test-to-predited greater than 1.0. The loser the atual result is to this, the better is the predition. Figure 5-1(a) shows that the proposed method has the largest (saest) minimum testto-predited shear strength ratio o 0.88 ompared to 0.84, 0.72, and 0.80 whih are the minimum test-to-predited shear strength ratios assoiated with ACI 318, 2007 AASHTO LRFD, and 2006 CHBDC preditions, respetively. Figure 5-1(a) also shows that about 20% o preditions o the proposed method, as well as the 2006 CHBDC and AASHTO LRFD methods have test-to-predited ratios below 1.0. However, only about 10% o proposed method and 2006 CHBDC test-to-predited ratios are below 0.95 while about 15% o test-to-predited ratios rom 2007 AASHTO LRFD are less than 95% o the atual test results. Beyond test-to-predited shear strength ratio o 1.0, the proposed method and the 2007 AASHTO LRFD ratios have the losest umulative requeny diagram to the peret predition. ACI 318 preditions are the saest as only 7% o the predited 142

164 1.1 1 RC beams Cumulative requeny Proposed 80 tests CHBDC 2006 & AASHTO AASHTO 2007 ACI tests (ρ z >2ρ zmin ) (a) Test-to-predited shear strength ratio PC beams 22 tests(ε x >0.0001) Cumulative requeny tests (b) Test-to-predited shear strength ratio Proposed CHBDC 2006 & AASHTO 2008 AASHTO 2007 ACI318 Fig. 5-1 Cumulative requeny o test-to-predited ratios o proposed and ode methods: (a) RC beams, (b) PC beams. 143

165 values have a test-to-predit shear strength ratio less than 1.0, but the method is generally overly onservative ompared to the other methods. For example, ACI 318 has a maximum test-to-predited ratio o 2.1, while all other methods have a maximum ratio less than about 1.8. The proposed method has a mean value o test-to-predited shear strength ratios o 1.15 and COV o 16.5%. The mean values o test-to-predited shear strength ratios are 1.15, 1.19, and 1.32 or 2007 AASHTO LRFD, 2006 CHBDC, and ACI 318, and the orresponding COV o ratios are 18.0%, 17.6%, and 18.7%, respetively. Figure 5-1(b) ompares the umulative requeny distribution o test-to-predited shear strength or 88 PC beams, and one again, the proposed method gives the losest to the peret predition. For the ratios o less than 1.0, the 2006 CHBDC and the proposed method test-to-predited ratios show almost the same umulative requeny distribution while 2007 AASHTO LRFD preditions are saer. However, only 10% o the 2006 CHBDC and the proposed method preditions are less than atual test results, whih is reasonably aeptable given the at that their minimum atual test-to-predited shear strength ratios are about About 50% o the ACI 318 preditions are unsae (have test-to-predited ratios less than 1.0). The mean values o test-to-predited shear strength ratios are 1.07, 1.27, 1.32, 1.31 or ACI 318, the proposed method, 2007 AASHTO LRFD, and CHBDC 2006 preditions. Also, the orresponding COV o test-to-predited shear strength ratios are 24.8%, 16.7%, 16.0%, 15.8%. The dierene o the proposed method preditions with the 2006 CHBDC and 2007 AASHTO LRFD preditions ould be more signiiant in real bridge girders than what is shown in Figure 5-1 as was shown in the bridge examples in Chapter 4. This is due to the 144

166 at that a signiiant number o the tested reinored onrete beams had transverse reinorement amount o less than two times the minimum amount speiied by 2006 CHBDC whereas in real pratie, reinored onrete bridge girders oten ontain signiiantly more transverse reinorement. There were only 26 RC members among the examined tests that had at least two times the minimum amount o transverse reinorement or whih the umulative requeny distribution o test-to-predited shear strength ratios rom the proposed method as well as rom the ode methods are shown in Figure 5-1(a). The proposed method preditions are the losest to a peret model preditions. The mean o ratios are 1.14, 1.18, 1.23 while COV o ratios are 9.1%, 7.8%, 8.4% or the proposed method, the 2007 AASHTO LRFD and the 2006 CHBDC, respetively. Most tested prestressed onrete beams had a very low predited mid-depth strain (almost zero) but many real prestressed onrete bridge girders are expeted to experiene larger strains at mid-depth espeially or setions that are lose to mid-span. The reason is that in the tested beams, ailure usually ours at setion lose to the support whereas in bridge girders ailure an also happen lose to mid-span at loations where amount o transverse reinorement hanges. There were only 22 tests whih had the predited mid-depth strain higher than rom the proposed method. The umulative requeny distribution o test-topredited ratios o those tests is also shown in Figure 5-1(b). Notie that the median o the proposed method ratios is 1.24 (umulative requeny value orresponding to test-topredited ratio o 0.5) while it is 1.34 and 1.36 rom the 2007 AASHTO LRFD and the 2006 CHBDC methods. The mean o test-to-predited ratios are 1.20, 1.30, and 1.33 or 145

167 the proposed method, 2007 AASHTO LRFD, and 2006 CHBDC preditions. The orresponding COV o ratios are 11.0%, 10.5%, and 9.8%, respetively. In summary, the proposed method preditions or the tested beams with at least minimum amount o transverse reinorement are in better agreement with the test results ompared to the preditions rom the 2006 CHBDC shear design proedure, whih is the same as 2008 AASHTO LRFD shear design proedure. The test-to-predited shear strength ratios rom the proposed method are on average about 4% lower than the preditions rom the 2006 CHBDC method or both RC and PC beams and they are still reasonably onservative. As the amount o transverse reinorement inreases to two times the minimum amount and more, the proposed method preditions are even better and the dierene with the preditions rom the 2006 CHBDC method beomes more signiiant. In ase o PC beams, as the mid-depth predited strain gets larger, the proposed method preditions ompare better with the test results. In real bridge girders, shear ailure an happen lose to mid-span at loations where the amount o transverse reinorement hanges. At these loations, the mid-depth longitudinal strain is large Members Without Transverse Reinorement To veriy the proposed method or members without transverse reinorement, preditions or 132 reinored onrete beams and 131 prestressed onrete beams without transverse reinorement were ompared with the measured test results. The seleted tests were extrated rom the same database used or members with transverse reinorement (Kim 2004). The reinored onrete members that were seleted had a minimum depth o 380 mm, while the prestressed members had a minimum depth o 300 mm and a minimum eetive prestressing stress o 550 MPa. The reinored onrete 146

168 members inluded a number o tests with a depth o about 1000 mm and up to a depth o 2000 mm. In ontrast, the depth o the prestressed members ranged rom 300 mm to about 460 mm. The shear span-to-depth ratios o all the members ranged rom 2.5 to 8.0. As in members with minimum transverse reinorement, the loading or most o the tests onsisted o one or two onentrated loads, and again, the ritial shear setion or these was assumed to be at a distane d (eetive depth) rom the onentrated load toward the support. For test with uniormly distributed loading, the ritial setion was assumed to be at distane d rom the support or rom the loation o hange in transverse reinorement amount toward mid-span. For beams with inlined tendons, the vertial omponent o the eetive prestressing ore V p was inluded in the alulated shear strength. Fig. 5-2 presents the umulative requeny distribution o the test-to-predited shear strength ratios. As explained previously, a peret predition would have a umulative requeny o zero or test-to-predited shear strength ratios less than 1.0, and a umulative requeny o 1.0 or test-to-predited shear strength ratios greater than 1.0. Fig. 5-2(a) shows that ACI 318 preditions are unsae (test-to-predited shear strength ratios less than 1.0) or about 45% o the tests (umulative requeny o 0.45) on reinored onrete beams without stirrups. This well known issue is beause ACI 318 ignores size eet in members without stirrups. The proposed method and 2006 CHBDC preditions are similar the predited shear strengths or about 15% o the tests are somewhat unsae. The 2007 AASHTO LRFD shear strength preditions are a little lower. The proposed method has a mean value o test-to-predited strength o 1.17 and COV o 17.3%. The mean values o test-to-predited shear strength are 1.05, 1.26, and 1.16 or 147

169 RC beams Cumulative requeny Proposed CHBDC 2006 & AASHTO 2008 AASHTO 2007 ACI (a) Test-to-predited shear strength ratio PC beams Cumulative requeny Proposed CHBDC 2006 & AASHTO 2008 AASHTO 2007 ACI (b) Test-to-predited shear strength ratio Fig. 5-2 Cumulative requeny o test-to-predited ratios o proposed and ode methods: (a) 132 reinored onrete (RC) beams, (b) 131 prestressed onrete (PC) beams. 148

170 ACI 318, 2007 AASHTO LRFD, and 2006 CHBDC, and the COV are 30.3%, 18.5%, and 17.8%, respetively. Fig. 5-2(b) indiates that the ACI 318 method gives the best preditions or the 131 prestressed onrete beams that are not very large. The proposed method does better than 2006 CHBDC and 2007 AASHTO LRFD. Mean o test-to-predited shear strength ratios is 1.17 rom ACI 318, 1.57 rom 2007 AASHTO LRFD, 1.37 rom the proposed method, 1.55 rom the 2006 CHBDC. The COV o these ratios is 16% rom ACI 318, 17% rom the proposed method, and about 23.1% and 23.2% rom 2007 AASHTO LRFD and the 2006 CHBDC Eet o Important Parameters To ensure that the proposed method aurately aptures the eet o important parameters in shear, the test-to-predited shear strength ratios presented in Setions 5-2 and 5-3 were plotted versus a number o parameters and also ompared with ratios rom the odes. 80 plots are presented in Appendix F and only 6 o these are presented here in Figures 5-3 and 5-4. Appendix F inludes plots or RC and PC beams both with and without transverse reinorement. Five parameters inluding beam depth, shear stress ratio v, shear span-to-depth ratio, longitudinal reinorement ratio, and eetive ' prestressing ore were looked at and the results are presented in Appendix F. Figure 5-3 shows test-to-predited shear strength ratios rom ACI 318, 2006 CHBDC and the proposed method versus shear stress ratios ( onrete beams with minimum amount o transverse reinorement. v ) or 88 prestressed ' 149

171 Test-to-predited shear strength ratio 2.0 ACI (a) Shear stress ratio V test /( 'b w d v ) Test-to-predited shear strength ratio 2.0 CHBDC (b) Shear stress ratio V test /( 'b w d v ) Test-to-predited shear strength ratio 2.0 Proposed () Shear stress ratio V test /( 'b w d v ) Fig. 5-3 Test-to-predited ratios o proposed and ode methods versus shear stress ratio v / ' or 88 prestressed onrete beams with transverse reinorement. 150

172 To alulate atual shear stress ratios, shear strength rom the atual test results was divided by the beam eetive shear area ( b ) and onrete ompressive strength '. d w v As shown in Fig. 5-3(a), ACI 318 is unonservative over the whole range o pratial v shear stress ratio ( 25 ' 0. ), but onservative or higher shear ratios. The reason is that ACI 318 assumes that onrete ontribution to the shear strength is equal to raking strength o onrete whereas onrete ontribution dereases ater raking due to inrease in strains and thereore redution in aggregate interlok apaity. Figures 5-3(b) and () illustrate that both 2006 CHBDC and the proposed method reasonably predit shear strength o the tested beams over the entire range o shear stress ratios. Notie that as shear stress ratio inreases (transverse reinorement ratio inreases), the proposed method preditions are loser to atual test results ompared to the preditions rom the 2006 CHBDC. Figure 5-4 presents the variation o test-to-predited shear strength ratios o the 131 reinored onrete beams with beam depth or ACI 318, 2006 CHBDC, and the proposed method. As shown in Fig. 5-4(a) and explained beore, ACI 318 preditions beomes highly unonservative as beam depth inreases sine it ails to aount or beam size eet in shear. Both 2006 CHBDC and the proposed method reasonably apture the eet o beam size on shear strength o the tested beams as shown in Figures 5-4(b) and (). 151

173 Test-to- predited shear strength ratio ACI (a) Eetive depth d (mm) Test-to- predited shear strength ratio 2.5 CHBDC (b) Eetive depth d (mm) Test-to- predited shear strength ratio 2.5 Proposed () Eetive depth d (mm) Fig. 5-4 Test-to-predited ratios o proposed and ode methods versus eetive depth or 132 reinored onrete beams without transverse reinorement. 152

174 5.5. Minimum Transverse Reinorement and Transition Between Members With and Without Minimum Transverse Reinorement As explained previously in Chapter 4, the 2006 CHBDC minimum transverse reinorement ratio is ρ = 0.06 min z ' y. This minimum amount is 30% higher than the minimum amount speiied by the 2007 AASHTO LRFD method. Codes normally treat members with less than minimum transverse reinorement as members with none. Based on the experimental study by Angelakos et al. (2001), the 2006 CHBDC uses a dierent approah in the evaluation setion. Aording to the 2006 CHBDC evaluation setion, shear strength o lightly reinored members with less than one third o minimum amount o transverse reinorement is the same as members with no transverse reinorement. The shear strength inreases linearly rom the shear strength o the member with no stirrups to the shear strength o the member with minimum stirrups as the transverse reinorement amount inrease rom one third o minimum to minimum. To investigate i the 2006 CHBDC minimum transverse reinorement ratio (ρ zmin ) may be adopted in the proposed method and i the 2006 CHBDC linear approah or members with less than minimum transverse reinorement is appropriate or use with the proposed method, preditions o proposed method were ompared with reinored onrete beam test results ontaining transverse reinorement ratio ranging rom 0.49ρ zmin to 1.8ρ zmin. 76 beams were seleted rom the shear database (Kim 2004) and 9 reinored onrete beams tested by Rahal and Al-Shaleh (2004) were inluded. 50 o the seleted beams ontained more than the 2006 CHBDC minimum transverse 153

175 reinorement and the remaining 26 tests ontained less than minimum transverse reinorement. Figure 5-5 examines the variation o test-to-predited shear strength ratios rom the proposed method with transverse reinorement amount as a ratio o minimum transverse reinorement speiied by the 2006 CHBDC. In Fig. 5-5(a) members with less than minimum transverse reinorement is treated as members without transverse reinorement, while in Fig. 5-5(b) the linear interpolation approah permitted by 2006 CHBDC is used or members with less than minimum transverse reinorement. For members with more than minimum transverse reinorement, the preditions in Figs. 5-5(a) and 5-5(b) are the same. For those members, 12 tests out o 50 tests have ratios o less than 1.0 or whih the preditions are unsae; however, only 5 o them (10%) have ratios o less than This is also onsistent with preditions rom the proposed method or the 80 reinored onrete beams in Fig. 5-1(a) meaning that the saety level o the preditions would not hange signiiantly as the transverse reinorement amount dereases to the 2006 CHBDC minimum transverse reinorement ratio. As a result, the 2006 CHBD minimum transverse reinorement is adopted in the proposed method. In Figs. 5-5(a) and 5-5(b) members with less than minimum amount o transverse reinorement have ρ z /ρ min <1.0. Figure 5-5(a) shows treating these members as members without stirrups leads to onservative estimates o shear strength or members with transverse reinorement amount o less but lose to minimum amount. In Fig. 5-5(b), in whih the linear interpolation approah is used, a more uniorm trend is observed. Notie that the overall onsisteny o the preditions with the atual test results has been improved onsiderably. Seven preditions out o 26 preditions (28%) are unsae but 154

176 2.5 Test-to-predited shear strength ratio (a) ρ z / ρ min Test-to-predited shear strength ratio (b) ρ z / ρ min Fig. 5-5 Test-to-predited shear strength ratios or 76 lightly reinored tested beams: (a) assuming no stirrups, (b) using linear interpolation approah, or members with less than minimum stirrups. 155

177 only two preditions (7.7%) are less than 95% o the atual test results. This is again onsistent with the saety level o the proposed method preditions or the 80 reinored onrete tested beams with more than minimum transverse reinorement. 156

178 Chapter 6. Reined 2006 CHBDC Method or Shear Design 6.1. General The 2006 CHBDC shear design provisions inlude an equation or mid-depth longitudinal strain ε x that is simpliied and onservative; however, the ode permits the use o more aurate proedures to determine mid-depth longitudinal strain. The proposed method or shear strength evaluation presented in Chapters 3 and 4 inludes a more aurate equation or mid-depth longitudinal strain that is not a untion o applied shear ore. In this hapter a similar equation or mid-depth longitudinal strain or shear design, whih is a untion o applied shear ore, is presented. It is investigated whether the more aurate equation is appropriate or use in shear design using the 2006 CHBDC shear design provisions. The predited longitudinal strains in bridge girders are ompared with strains determined using Response In addition, preditions o shear strength rom the 2006 CHBDC with the proposed equation or mid-depth longitudinal strain ε x are ompared with available test results Reined CHBDC Approah or Members With at Least Minimum Transverse Reinorement Some o the reinements introdued in Chapter 3 or the proposed method namely the assumption o n v ats over a depth o d nv that an be dierent rom eetive shear depth d v, aounting or tension stiening eet in the tension hord, and how to inlude the eet o web nonprestressed and prestressed reinorement an well be implemented in ε x equation o the 2006 CHBDC without any hanges. However, the proposed N v equation used to derive ε x equation in Chapter 4 is or evaluation problems and thus is a untion 157

179 o transverse reinorement amount. For design, ε x equation and thus N v equation should be a untion o applied shear ore to avoid iteration. As explained in Chapter 3, the 2006 CHBDC uses the approximation o 2v or axial ompression stress needed to transer shear along the raked depth n v, whih is too onservative. In ontrast, 2007 AASHTO LRFD uses a more aurate approximation votθ. Figure 6-1 ompares the 2006 CHBDC and 2007 AASHTO LRFD preditions o n v with MCFT or varying onrete ontribution and angle o inlination o 29 and 36 deg, whih are the orresponding angles to ε x = 0 and ε x = in the 2006 CHBDC method, respetively. As illustrated, 2007 AASHTO LRFD approximate equation or n v is more aurate; thus is used to develop a reined equation or ε x. Using votθ to approximate n v in the ε x equation requires trial-and-error in design sine θ is not initially known. To overome this problem, as shown in Figure 6-2, otθ may be replaed with a linear untion o θ given below or the range o θs between 28 and 40 deg, whih is the suggested range o θ by the 2006 CHBDC equation (Eq. 3-30). [6-1] otθ = θ Substituting or otθ rom Eq. [6-1] and θ rom Eq. [3-30], votθ an be approximated as: [6-2] nv = v otθ = 1.8v 392vε x Using the same proedure as used to derive Eq. [4-18] but using the n v equation above results in the ollowing ε x equation or design. [6-3] ε = x 2 M / jd + 0.9Vd [ E ( A s s nv A / d sw v α ) + E p ( A ' A p t + λ A 2 p pw p ( A p + λ A p )] + 196Vd nv pw ) / d v 158

180 2.5 2 CHBDC 2006 (or all θ ) 1.5 θ = 29 AASHTO 2007 n v /v 1 θ = 36 MCFT v /v Fig. 6-1 Comparison o ode predited axial ompression stresses n v with MCFT or dierent onrete ontributions to shear stress ot θ θ Angle θ (degrees) Fig. 6-2 Linear approximation o otθ. 159

181 where V is the applied shear ore. The more general orm o Eq. [6-2] or m layers o web nonprestressed reinorement and n layers o web prestressed reinorement is: [6-4] ε x = 2 M / jd + 0.9Vd [ E ( A + s s m j= 1 2 j nv λ A / d swj v α ) + E p ( A ' A p + t n i= 1 p λ A 2 pi ( A p pwi + n i= 1 λ A pi )] + 196Vd pwi nv ) / d v in whih A swj is total area o the j th layer o web nonprestressed reinorement, A pwi is total area o the i th layer o web prestressed reinorement. λ j and λ pi are the ratios o d pwi / d and d wj / d in whih d pwi and d wj are the distanes rom the entroids o the i th layer o web prestressed reinorement and the j th layer o web nonprestressed reinorement to the lexural ompression ae, respetively Bridge Examples Figures 6-3 to 6-5 ompare Eq. [6-2] and the 2006 CHBDC equation or ε x with Response 2000 preditions or the same three bridge girders that were evaluated with the proposed evaluation method in Chapter 4. Comparisons are made at yielding o transverse reinorement stage sine, as shown in Chapter 4, the 2006 CHBDC preditions o shear strength is lose to that stage. In other words, shear strength at yielding o transverse reinorement given by Response 2000 is used in Eq. [6-2] and the ε x equation o 2006 CHBDC and the results are ompared with Response 2000 preditions o mid-depth strain at yielding o transverse reinorement. 160

182 Strain (mm/m) Response CHBDC 2006 Reined CHBDC Distane orm let support (mm) Fig. 6-3 Comparison o I-girder predited mid-depth strain along the bridge span. As shown in Figures 6-3 to 6-5, the reined ε x equation is loser to the Response preditions in all ases. The reined equation improves the preditions o strains up to 40%, 35%, and 15% or I-girder, box-girder, and hannel-girder bridges ompared to the 2006 CHBDC ε x equation, respetively. Figure 6-4 also inludes the omparison o Response 2000 preditions o lexural tension reinorement strain with those rom the 2006 CHBDC and the reined ε x equation or the box-girder. Note that the lexural tension reinorement strain is two times the mid-depth strain in the 2006 CHBDC method. The reined ε x equation or mid-depth strain is also hal the reined ε x equation or lexural tension reinorement strain. 161

183 Strain (mm/m) Response CHBDC 2006 Reined CHBDC (a) Distane orm let support (m) Response CHBDC 2006 Reined CHBDC 2006 Strain (mm/m) (b) Distane orm let support (m) Fig. 6-4 Comparison o box-girder predited (a) mid-depth strain (b) lexural tension reinorement strain along the bridge span. 162

184 Strain (mm/m) Response CHBDC 2006 Reined CHBDC Distane orm let support (m) Fig. 6-5 Comparison o hannel-girder predited mid-depth strain along the bridge span. Notie that the reined equation adequately estimates the lexural tension reinorement strain ompared to the Response 2000 results (Fig. 6-4b). The reined equation predition o mid-depth strains shown in Fig. 6-4(a) is not as good due to the approximation o middepth strain is hal the lexural tension reinorement strain Comparison with Experimental Results To veriy i the reined ε x equation an be used in the 2006 CHBDC shear design proedure, the shear strength preditions o the reined 2006 CHBDC method, in whih the original ε x equation is replaed with Eq. [6-2], are ompared with experimental results rom the 80 reinored onrete beams and 88 prestressed onrete beams. The results are also ompared with the shear strength preditions rom the 2006 CHBDC method. 163

185 RC beams 0.9 Cumulative requeny Reind CHBDC 2006 CHBDC (a) Test-to-predited shear strength ratio PC beams Cumulative requeny Reined CHBDC 2006 CHBDC (b) Test-to-predited shear strength ratio Fig. 6-6 Cumulative requeny o test-to-predited ratios o reined CHBDC 2006 and CHBDC 2006 methods: (a) 80 RC beams, (b) 88 PC beams, with stirrups. 164

186 Note that the same experimental results were also used to veriy the proposed evaluation method in Chapter 5. Figure 6-6 ompares the umulative requeny diagram o the test-to-predited shear strength ratios rom the reined 2006 CHBDC method with the one rom the 2006 CHBDC method. Fig. 6-6(a) shows that the reined ε x equation does not have a signiiant eet on the preditions or reinored onrete beams. As also explained in Chapter 5, the reason is there were not many tested beams with transverse reinorement amount more than twie the minim amount speiied by the 2006 CHBDC while bridge girders generally ontain higher amount o transverse reinorement. Note that more amount o reinorement results in higher shear strength thus higher longitudinal strain due to shear. As a result, the higher the transverse reinorement, the more signiiant the inluene o the proposed reined ε x equation on the predited shear strength. As shown in Fig. 6-6(b), the reined ε x equation does improve the 2006 CHBDC preditions or prestressed members. Using the reined ε x equation in the 2006 CHBDC shear design proedure improves the average test-to-ode predited shear strength ratio rom 1.31 to 1.26 while the COV o ratios remains about the same. Although Figure 6-6(b) shows that reined ε x equation inreases the umulative requeny values o the testto-predited ratios o less than 1.00, 14% o the reined method preditions are unsae and only less than 5% o the ratios are below This is still reasonable and even saer than the 2006 CHBDC predition or the 80 RC members shown in Figure 6-6(a). To more aurately predit the longitudinal strain o members with no stirrups in the 2006 CHBDC method, one an replae the simpliied equation o N v = 2V with the one rom MCFT N v = 2V ot 2θ in the 2006 CHBDC equation or ε x. This was done to 165

187 predit shear strength o the 132 reinored onrete tested beams with no stirrups, whih were also used to veriy the proposed evaluation method in Chapter 5. The umulative requeny diagram o the test-to-predited shear strength ratios rom the reined 2006 CHBDC method is shown together with the one rom the original 2006 CHBDC in Figure RC beams 0.9 Cumulative requeny Reined CHBDC 2006 CHBDC Test-to-predited shear strength ratio Fig. 6-7 Cumulative requeny o test-to-predited ratios o reined CHBDC 2006 and CHBDC 2006 methods or 132 RC beams without stirrups. As shown, the reined ε x equation inreases the number o unsae preditions rom 18% to 32% o the tested beams. It is due to the reason that the 2006 CHBDC uses the middepth longitudinal strain to predit the shear strength o a beam while Response 2000 shows that ailure usually ours at loations loser to lexural tension hord (see Fig. 4-10). The 2006 CHBDC uses a simpliied equation or axial ompression whih is overly 166

188 onservative or members without stirrups and this ompensates or the unonservative approah o using the mid-depth longitudinal strain in the shear analysis. Consequently, it is not reommended to use a more aurate ε x in the 2006 CHBDC shear design provisions or members with no stirrups. 167

189 Chapter 7. Literature Review: Squat Shear Walls 7.1. Shear Strength o Squat Shear Walls Benjamin and Williams ( ) studied the behaviour o reinored onrete squat walls at the Stanord University. In their experimental program, they tested walls surrounded by onrete rames and applied a monotoni onentrated lateral load at the tension side o the wall (top o tensile olumn). Some o their indings were size eet was not signiiant, vertial reinorement had more inluene on shear strength than horizontal reinorement, and shear strength was onsiderably higher or the walls with smaller height-to-length ratios. Their last version o proposed empirial equation to estimate shear strength o squat shear walls is: 0.1 [ 7-1] V u = C P P C where: [ 7-2] C = As ' L H 2 [ 7-3] P = σ tl y ρ in whih L is wall length, H is wall height, ρ is distributed reinorement ratio whih is the same or both vertial and horizontal diretions, t is wall thikness, A s is ompression olumn total reinorement area, σ y is wall reinorement yield stress, P is wall panel strength, and C is ompression olumn strength. The limitations they identiied or their equations are as ollows: ρ 1.5%, 42 σ 52 ksi, 0.9 L / H 3, P / C y Later an extension o Benjamin and William s researh was onduted at the Massahusetts Institute o Tehnology (MIT) whose results were reported by Antebi et 168

190 al. (1960). The primary aim o the program was to predit the shear strength o low rise walls under blast loading. One again, the walls were surrounded by onrete rames. The majority o walls were tested under blast loading while a ew o them were tested under monotoni stati loading. Using results o the tests done at Stanord University and MIT, Antebi et al. (1960) proposed two sets o empirial equations to predit the shear strength o low rise walls under stati and dynami (blast) loading. Based on the proposed equations, shear strength o low rise walls was a untion o height-to-length ratio, rame reinorement area and yield stress, and amount o equal vertial and horizontal distributed reinorement in the wall. They also identiied limitations in their method; however, these were less restriting than those identiied by Benjamin and Williams. depaiva and Seiss (1965) onduted a series o tests o simply supported deep beams with applied onentrated load at the top. They ound that in deep beams, stirrups would not engage appreiably in shear ore transer sine load was diretly transerred to the supports by onrete ompression struts (arh ation). These results on deep beams led to a number o reommendations in the 1960s or shear strength o shear walls. For example, Uniorm Building Code (UBC 1967) shear wall provisions onsidered no steel ontribution in walls with height-to-length ratio equal to or less than 1.0 and limited total shear stress o walls with height-to-length ratio equal to or greater than 2.0 to vu 10 ' in psi units. On the other hand, it onsidered onrete ontribution o v =.4 ' in psi units or walls with height-to-length ratio equal to or less than and v = ' in psi units (as in shallow beams) or walls with height-to-length ratio o 2 equal to or greater than 2.7. Linear interpolation was used or intermediate values. 169

191 Additional tests on deep beams by Crist (1966) and Leonhardt and Walther (1966) showed that arh ation (a diret strut rom the load to the support) depends upon how the load is applied. In deep beams, the load is usually applied on the ompression ae whih results in signiiant arh ation. When a deep beam is loaded indiretly using a transverse beam, the arh ation is greatly redued and the quantity o stirrups beomes very important. In 1971, shear wall provisions irst appeared in ACI 318. The bakground to these provisions was presented by Cardenas et al. (1973). Steel ontribution in shear was developed based on truss analogy whih resulted in the same ormula as or steel ontribution in shallow beams. Conrete ontribution was developed to be the lesser o shear ores that resulted in web raking and lexural shear raking. Below is ACI shear provisions or shear walls in psi units. [ 7-4] v N u 3.3 ' + 4l h w [ 7-5] v 0.6 l ' + w (1.25 M u ' + 0.2N / V u l w u / 2 / l w h) [ 7-6] v ( N / A ) ' 2 u g [ 7-7] vs = ρ h y [ 7-8] v + vs 10 ' [ 7-9] V = ( v + v hd u s ) in whih V u is ultimate shear strength, h is wall thikness, l w is wall length, ρ h is horizontal reinorement ratio, d is distane rom extreme ompression iber to entroid 170

192 o onentrated lexural reinorement, and A g is wall ross-setion gross area. The above equations are still in the urrent ACI non-seismi shear design provisions or walls. Equation [7-4] is the shear stress ausing web raking, whih almost always governed or low rise walls, while Eq. [7-5] is shear stress ausing lexural shear raking. In ACI , the onrete ontribution to shear strength in walls is not taken less than the onrete ontribution in shallow beams as given by Eq. [7-6]. Total shear stress was limited to avoid diagonal rushing o onrete (Eq. 7-8). Cardenas et al. (1973) ompared the preditions rom ACI with previous test data and ound that ACI preditions were sae and satisatory. Barda (1972) onduted tests on eight squat shear walls whih had langes and top onrete beam to transer the load to the top edged o the wall. Six walls were tested under yli loding while the remaining two were subjeted to monotoni loading. Barda investigated the eet o wall aspet ratio, lexural reinorement, horizontal and vertial distributed reinorement ratios on shear strength o squat walls and proposed the ollowing equation: [ 7-10] V = ( 8.4 ' ρ hd (in psi units) u n y ) where h is wall thikness, and ρ n is vertial reinorement ratio. Later Barda et al. (1977) proposed another equation that aounted or the inluene o wall aspet ratio and axial ore. hw N u [ 7-11] Vu = ( 8 ' 2.5 ' + + ρ n y ) hd (in psi units) l 4l h w w where h w is wall height, l w is wall length and N u is wall axial ompression ore. 171

193 Park and Paulay (1975) and Paulay et al. (1982), based on walls tested at the University o Canterbury, reported that dutile behaviour o squat shear walls were ahievable i horizontal shear reinorement was suiient to avoid shear ailure at lexural apaity level. They also proposed no reliane on onrete ontribution to shear strength o squat shear walls should be made i dutile behaviour was desired. From their experimental program, Paulay et al. (1982) onluded that diagonal bars at the base o squat shear walls avoided sliding shear ailure under load reversals thus improved the dutile behaviour o squat shear walls. Hernandez (1980) tested 23 walls all with top slab and some with intermediate slabs under yli loading. He investigated the eet o wall aspet ratio, horizontal and vertial reinorement ratios, onrete ompressive strength, axial load, and boundary onditions. Based on the results, Hernandez proposed the ollowing equations to determine shear strength o walls. hw 2 [ 7-12] V0 = ( ( ) ) ' 0.5 ' l [ 7-13] σ σ V = A V 0 1+ ; 5 V 0 V0 [ 7-14] V = A ρ ) s ( y w where σ is wall ompressive stress (axial ore/ gross onrete area), A is area o wall ross-setion, ρ is vertial reinorement ratio i h / 1 or horizontal reinorement ratio otherwise. In these equations, all ores are in kg, stresses are in kg/m 2, and lengths are in m. w l w 172

194 Seven retangular squat walls with aspet ratio o 1.0 were tested by Cardenas et al. (1980). The load was transerred to the top edge o the wall by means o a sti beam at the top. All walls were subjeted to monotoni load exept one, whih was subjeted to yli loading. They onluded that ACI design provisions or squat shear walls were reasonably sae or monotoni loading as well as yli loading. Furthermore, both horizontal and vertial reinorement were ound to ontribute to shear strength o the tested walls. Maier and Thürlimann (1985) tested ten langed squat shear walls with height-tolength ratio o 1.0 and onluded horizontal reinorement inluene on shear resistane o squat walls was negligible, and yli loading did not redue shear apaity o tested walls onsiderably. The dominant mode o ailure in their test was rushing o onrete at the base. Between 1985 and 1994, Saatiuglu together with other researhers tested 8 squat shear walls under yli loading to investigate the eet o vertial and horizontal reinorement, aspet ratio, and reinorement detailing at the base. All walls had top rigid beams and did not have langes. The results were presented in a number o publiations inluding Wirandianta (1985), Wirandianta and Saatiuglu (1986), Pielette (1987), Wasiewiz (1988), Mohammadi-Doostdar (1994). They onluded that shear strength o squat shear walls were mostly inluened by the wall aspet ratio and base sliding due to shear was more ritial or squat walls with height-to-length ratio o They also onluded that speial detailing at the base ould improve shear sliding behaviour that would result in a more dutile behaviour o walls. 173

195 Wirandianta (1985) proposed a modiiation to ACI speial provisions or earthquake (whih is still the same in ACI ) to expliitly aount or aspet ratio when determining onrete ontribution to shear strength o walls. His proposed equation was: [ 7-15] v = α ' (in psi units) where hw [ 7-16] α = 6 2 l w Wood (1990) ompared ACI 318 shear strength preditions or 143 previously tested squat shear walls with test results. She onluded that ACI 318 preditions were sae or walls with horizontal reinorement ratio o ρ 2. 1 MPa, whih ould be as little as 1.5 times the minimum reinorement, while they beame unsae or other walls. She also onluded that modiied truss analogy whih assumes a onstant onrete ontribution in addition to steel ontribution orresponding to a onstant angle i.e. 45 degrees was not onsistent with test results. She showed that shear strength o tested walls was onsiderably less sensitive to the amount o horizontal reinorement ompared to the preditions rom the modiied truss analogy. Based on experimental data and shear rition model, Wood proposed a lower-bound semi-empirial equation or shear strength predition o squat walls: h y [ 7-17] V n Av y = ; ' vn 10 ' (in psi units) 4A v where A v is shear reinorement area rossing the shear plane at the base, and A v is wall ross-setion gross area. 174

196 Leas et al. (1990) and Leas and Kotsovos (1990) onduted a series o testes on seventeen reinored onrete squat walls with height-to-length ratio o 1.0 and 2.0 under yli and monotoni loading. Walls inluded dierent amount o onentrated lexural reinorement as well as vertial and horizontal distributed reinorement and load was transerred to the walls by means o a rigid beam built on eah wall top edge. From their experimental program, they onluded that dominant ailure mode was vertial tension splitting o the onrete ompression zone near the wall ompression ae at the base. Vertial reinorement did not have a signiiant eet. 26 squat shear walls tested under yli loading were tested by Hidalgo et al. (1998, 2002). They evaluated the method proposed by Wood (1990) as well as ACI provisions by omparing the preditions with the results they obtained. Their study showed that horizontal reinorement had insigniiant inluene on the wall shear resistane. ACI preditions were onservative and satisatory, but Wood s method preditions were slightly better. Gule et al. (2008) ompared test results o 120 retangular squat shear walls reported in the literature with preditions rom Barda (1977) method, Wood (1990) method, and ACI shear design provisions. The walls had minimum thikness o 51 mm, no diagonal reinorement or additional reinorement to ontrol sliding shear, and heightto-length ratio o less than 2.0. It was onluded that sliding o the wall base was the dominant ailure mode in the tested walls. The satter in shear strength test-to-predited ratios was substantial or all methods and Wood (1990) preditions were the best ompared to experimental results. While ACI 318 preditions were onservative, Barda (1977) method preditions onsistently overestimated the shear apaity o walls. It was 175

197 also seen that most o the onservative preditions o shear strength were assoiated with lightly reinored walls Summary o Observed Behaviour In all available test results reviewed in the literature, load was introdued at the top edge o the wall by a rigid beam. As a result, it is not surprising that many researhes suh as Benjamin and Williams ( ), Barda (1972), Leas et al. (1990), Wood (1990), and Hidalgo et al. (1998, 2002) have ound the eet o horizontal reinorement to be insigniiant in shear and others suh as Gule et al. (2008) identiied that base sliding was the dominant ailure mode. I the top beam had been removed in the tests and load was distributed at top o the wall, it ould have resulted in a larger ontribution o horizontal reinorement, as well as ailure at lower load levels due to diagonal tension rather than at the base at higher load levels. Although suh a rigid beam inreases the wall shear strength mostly against diagonal tension ailure, suh rigidity is not always available in real strutures espeially in the ase o lexible diaphragms. The eet o suh beams is investigated in the next hapter. In ontrast to diagonal tension ailure mode, base sliding ailure mode is not sensitive to how the load is applied at top o the wall. Based on omparisons made with experimental results o tested walls whih generally had a top loading beam, urrent ACI ode provisions have been ound to onservatively apture this ailure mode by many researhers suh as Cardenas et al. (1980), Hidalgo et al. (1998, 2002), and (Gule et al. 2008); thus should also be appropriate or walls without a top loading beam and with distributed load at top. 176

198 The most ommon shear ailure modes o squat shear walls reported in the literature are diagonal tension ailure mode, diagonal onrete rushing ailure mode, and base sliding shear ailure mode. These ailure modes, whih are explained in Paulay et al. (1982), are shown in Figure 7-1. Diagonal tension ailure mode (Fig. 7-1a) ours when there is an insuiient amount o horizontal reinorement to balane the diagonal ompression ore that does not go diretly to the base. The diagonal tension ailure plane that will develop is shown in Figure 7-1(a). When there is suiient distributed reinorement to transer shear, squat shear walls may ail in diagonal rushing o onrete whih mostly happens near the ompression ae o wall base as shown in Figure 7-1(b). This usually happens in walls with large horizontal reinorement ratio and large lexural apaity. Diagonal onrete rushing ailure is undesirable sine it is a sudden and brittle ailure mode. Third ailure mode whih is alled base sliding ailure mode is aused by yli loading. As a onsequene o yli loading, ompression ae o the wall whih had been in signiiant tension when the load was reversed, is raked and thus the wall slides along the base (Fig. 7-1) when it experienes high shear ore and a signiiant number o load yles. The last two ailure modes are ontrolled in the odes by limiting the total shear stress. Some odes suggest using diagonal reinorement at the base- oundation interae to avoid base sliding ailure. 177

199 (a) Diagonal tension ailure (b) Conrete rushing ailure () Base sliding shear ailure Fig. 7-1 Squat walls shear ailure modes. 178

200 7.3. Reent Code Approahes ACI The non-seismi provisions or the shear strength o squat walls in ACI 318 are the same approah as in ACI given by Eqs. [7-4] to [7-9]. It also requires the same minimum amount o distributed vertial reinorement as in the 1971 edition: hw [ 7-18] ρ n = (2.5 )( ρ h ) l w where ρ n is distributed vertial reinorement ratio and ρ h is distributed horizontal reinorement ratio. ρ n need not to be taken greater than ρ h. In the additional requirements or seismi design, ACI 318 provides the ollowing equation or shear strength o squat shear walls. [ 7-19] V = A α ' + ρ ) (in psi units) n v ( h y where A v is the total area o wall ross-setion, α =3.0 or h / 1. 5, and α =2.0 or h / 2.0. For intermediate values, linear interpolation is used. Based on ACI 318, w l w w l w vertial reinorement ratio may not be less than horizontal reinorement ratio. To avoid onrete rushing, the total shear ore is limited to A ' or all wall 8 v piers sharing a ommon lateral ore and to 10Av ' or individual wall piers. A wall pier reers to a vertial wall segment between two openings New Zealand Standards (NZS ) Aording to NZS 3101, the shear strength o squat shear walls is: [ 7-20] V = V + V = vb d = ( v + v b d s w s ) w 179

201 N * [ 7-21] v = ( ' + ) (in MPa units) A g [ 7-22] vs = ρ h y where N* is axial ore, A g is gross area o the wall ross-setion, b w is wall thikness, and eetive depth d is equal to 0.8 l w in whih l w is wall length. Moreover, the total shear stress v is limited to the lesser o.2 ',.1 ', and 9 MPa. 0 1 In the seismi provisions, shear stress provided by onrete is given by 5 µ N * [ 7-23] v = ( )(0.27 ' + ) 0 4 4A g (in MPa units) [ 7-24] v N * 0.6 (in MPa units) A g where µ is dutility ator and limited to 3.0 or squat shear walls. Conrete rushing is avoided by: φ [ 7-25] ( 0 w v ) ' µ (in MPa units) where φ0w is overstrength ator CSA A23.3 In the 2004 CSA A23.3, the same equation as the one given or beams is used to determine shear strength o squat shear walls: [ 7-26] V = β ' b d + ρ b d otθ w v h y w v where shear length d v = 0.8 l w in whih l w is wall length, onrete ontribution ator β = 0, b w is wall thikness, ρ h is horizontal reinorement ratio, and angle o prinipal ompression θ an be reely hosen between 30 and 45 deg. The hosen angle is then 180

202 used to determine the amount o distributed vertial reinorement rom the ollowing equation. [ 7-27] ρ v 2 = ρ ot θ h P y s A g where ρ h is the horizontal reinorement ratio, P s is axial load, and A g is wall gross rosssetion area. To avoid onrete rushing in the 2004 CSA A23.3, shear strength is limited to: [ 7-28] V 0.15 ' bwd v Aording to the 2004 CSA A23.3, distributed vertial reinorement needed or shear given by Eq. [7-27] does not ontribute to the wall lexural apaity. As a result, additional vertial reinorement or lexure must be provided in addition to the distributed vertial reinorement needed or shear. 181

203 Chapter 8. Comparison o NLFE Preditions with Experimental Results o Squat Shear Walls 8.1. General In this hapter, experimental results o three squat shear wall tests reported in the literature are ompared with nonlinear inite element preditions. The walls had dierent aspet ratios and dierent reported ailure modes. A brie introdution to the nonlinear inite element program VeTor 2 used in this study is given. The inite element program predited behaviour or the squat shear walls is then presented. The implementation o the program is veriied by omparing the inite element preditions with the wall test results. In addition, the inluene o the top loading beam on the wall strength is studied. This is done by presenting the inite element results or the same three walls in whih the top loading beam was removed and the load was distributed along the wall top edge Finite Element Program Program VeTor2 developed by Wong and Vehio (2002) at the University o Toronto was used or nonlinear inite element analysis. The program perorms nonlinear inite element analysis o onrete strutures and expliitly aounts or interation o moment and shear. Although VeTor 2 has simple elements and employs simple numerial tehniques, it uses the state o the art material models to relate biaxial element strains to biaxial element stresses. In-plane elements o uniorm stress and strain ield are used in the program. 182

204 Two models or raked reinored onrete subjeted to biaxial strains have been implemented in VeTor 2. One o the models is MCFT, whih was explained in Chapter 3, and the other one is Disturbed Stress Field Model DSFM (Vehio 2000). DSFM is oneptually similar to MCFT but it allows dierent orientations or the prinipal stress and strain diretions. DSFM determines the dierene between prinipal stress and prinipal strain orientations by alulating additional strains aused by rak slip. MCFT and DSFM preditions o shear strength are basially the same or ordinary strutures. DSFM beomes more aurate or strutural onrete that is heavily or very lightly reinored in the orthogonal diretions (Vehio 2000). Palermo and Vehio (2004) veriied VeTor 2 or shear walls. They ompared the program preditions or our slender walls and two squat walls with the experimental results. All walls were subjeted to yli loadings. The walls inluded two slender barbell shaped walls with height-to-length ratio o 2.4 (height = 4570 mm, width = 1910 mm) tested by Portland Cement Assoiation (Oesterle et al. 1976), two retangular slender walls with onealed end olumns and height-to-length ratio o 2.0 (height = 1200 mm, length= 600 mm) tested by Pilakoutas and Elnashai (1995), and two langed squat walls with height-to-length ratios o 0.70 and 0.67 (height = 2020 mm, width = 2885 mm and 3045 mm) tested at the University o Toronto (Palermo and Vehio, 2002). The inite element preditions were in good agreement with the test results, As an example, omparison o the predited load displaement urve with the test results or one o the squat walls (height = 2020 mm, width = 2885 mm) is shown in Figure

205 Fig. 8-1 Load-deormation responses o wall DP1: (a) observed, (b) alulated (Palermo and Vehio, 2004). 184

206 8.3. Comparison with Wall Test Results The squat shear walls that were previously tested generally had a rigid loading beam at the top. While suh a rigid beam allows redistribution o shear ore at top and thereore an signiiantly inrease the shear resistane o a wall, it is not guaranteed that suh a top rigidity or strength is available in real squat walls. Codes, however, should provide lower-bound preditions, whih are not unonservative in any ases, thus or shear design purposes it is reasonable to ignore any top rigidity in squat walls. The main purpose o this setion is to show how the top loading beam in the tested walls may have inluened the shear apaity o walls and what type o ailure is mostly aeted by suh a beam. Experimental results o three previously tested walls are ompared with the inite element preditions or the walls with and without the top loading beam. In the walls without the top beam, the load is uniormly applied along the top o the wall. Wall details as well as material properties are presented in Figure 8-2. The walls were tested by Wiradinata and Saatioglu (1986), Kuang and Ho (2008), and Leas et al. (1990) and had height-to-length ratios o 0.5, 1.0, and 2.0, respetively. The irst two walls were tested under yli loading while the third one was tested under monotoni loading. They were seleted sine they had dierent reported ailure modes i.e. diagonal tension ailure (yielding o horizontal reinorement), lexural ailure (yielding o vertial reinorement), and diagonal rushing o onrete. In the inite element analysis, all walls were monotonially loaded and shear ore was applied in the let-to-right diretion. The results presented hereater are separated based on the wall reported ailure modes. 185

207 - Geometry and Material Properties Researher Name Wiradinata, Saatioglu (1986) Kuang, Ho (2008) Leas et al. (1990) Speimen h w (mm) l w (mm) l (mm) t (mm) h b (mm) w b (mm) ρ v (%) ρ h (%) ρ l (%) vy (MPa) hy (MPa) ' (MPa) Wall U SW Fig. 8-2 Details o the three previously tested walls in the literature examined to ompare experimental results with inite element preditions. 186

208 Diagonal Tension Failure This wall tested by Wiradinata and Saatioglu (1986) had a height-to-length ratio o 0.5. Figure 8-3 presents the load-displaement urves or the wall rom the experimental results (dotted line) as well as inite element preditions (solid line). Finite element preditions are in reasonably good agreement with the experimental results. Setional analysis or pure lexure assuming plane setions remain plane (plane setion analysis) predits the lexural apaity o the wall is at V= 544 kn. Atual wall maximum strength was reahed at V=575 kn and the reported ailure mode was diagonal tension. In omparison, inite element predition o wall maximum strength is V= 527 kn Experimental results Shear ore (kn) FE (Without top beam) FE (With top beam) Top wall displaement (mm) Fig. 8-3 Comparison o experimental load-top displaement urve with inite element predition or wall tested by Wiradinata and Saatioglu (1986). As was mentioned above, wall strength rom test results is higher than those predited by the plane setion analysis or pure lexure and inite element analysis. The reason 187

209 might be the strain hardening in the vertial reinorement. As the steel strain hardening inormation was not reported, inite element analysis as well as plane setion analysis were done assuming no strain hardening or steel. Finite element predition o load-displaement urve or the same wall in whih the top beam was removed and shear ore was uniormly distributed over the wall top edge is also presented (dashed line). Displaements orrespond to the wall top edge mid-point as the top wall displaement varies over the wall top edge beause there is not a rigid beam at top to make the top displaement uniorm over the wall top edge. Notie that the wall strength is redued to 408 kn due to diagonal tension shear ailure, whih is 23% redution in strength ompared to inite element preditions or the wall with the top beam. The reason is examined in Figure 8-4, whih shows setional shear stress distributions o both walls (with and without a top beam) at top, mid-height, and base o the walls prior to ailure. Shear is redistributed by the top beam toward the let side o the wall (Fig. 8-4a) in order to be transerred through diagonal struts that go diretly to the base without a need or horizontal reinorement. This is not the ase in the wall without the top beam and thereore it reahes its strength earlier due to yielding o horizontal reinorement (diagonal tension shear ailure). Figures 8-4(b) and 8-4() show that shear stress distributions at the mid-height and the base are more uniorm or the wall without the top beam. 188

210 Shear stress (MPa) stress With top beam Without top beam (a) Distane rom let edge (mm) 6 5 With top beam Shear stress (MPa) Without top beam (b) Distane rom let edge (mm) Shear stress (MPa) Without top beam With top beam () Distane rom let edge (mm) Fig. 8-4 Comparison o inite element preditions o shear stress proiles along (a) top setion, (b) mid-height setion, () base setion o the wall tested by Wiradinata and Saatioglu (1986) and the same wall without the top beam and distributed ore applied at top. 189

211 In Figure 8-5 the vertial steel stress proiles at the base immediately prior to ailure are ompared. The wall with the top beam reahes the lexural apaity while the other wall does not. Notie that all vertial reinorement has yielded in the lexural tension side o the wall with the top beam. 1.5 Steel stress normalized to yield stress With top beam Without top beam Distane rom let edge (mm) Fig. 8-5 Comparison o inite element preditions o vertial reinorement stresses at the base o the wall tested by Wiradinata and Saatioglu (1986) and the same wall without the top beam and distributed ore applied at top Yielding o Vertial Reinorement Failure The wall tested by Kuang and Ho (2008) had a height-to-length ratio o 1.0. It was expeted to ail in lexure sine a suiient amount o horizontal reinorement was provided or shear. As opposed to the other two walls disussed in this hapter whih did not have axial ompression, this wall was subjeted to an axial ompression o 300 kn. Figure 8-6 ompares the atual load-deormation urve with the inite element preditions. Setional analysis or pure lexure assuming plane setions remain plane predits a shear ore at lexural apaity equal to 321 kn. The atual wall strength and 190

212 the inite element predited strength are also about 330 and 325 kn. Finite element preditions or the same wall with no top beam and distributed ore at top are also shown in Figure 8-6 (dashed line). It is shown that the top beam does not signiiantly inluene the lexural apaity and load deormation urve o the wall i shear reinorement is suiient. 350 Shear ore (kn) FE (Without top beam) FE (With top beam) Experimental results Top wall displaement (mm) Fig. 8-6 Comparison o experimental load-top displaement urve with inite element predition or wall tested by Kuang and Ho (2008). Figure 8-7 shows inite element preditions o setional shear stress distributions or walls with and without top beam at top, mid-height, and base o the walls prior to their ailure. One again, shear is redistributed toward let side o the wall at top o the wall 191

213 4 3.5 Without top beam Shear stress (MPa) With top beam (a) Distane rom let edge (mm) 6 Without top beam 5 With top beam Shear stress (MPa) (b) Distane rom let edge (mm) Shear stress (MPa) With top beam 2 Without top beam () Distane rom let edge (mm) Fig. 8-7 Comparison o inite element preditions o shear stress proiles along (a) top setion, (b) mid-height setion, () base setion o the wall tested by Kuang and Ho (2008) and the same wall without the top beam and distributed ore applied at top. 192

214 with the top beam (Fig. 8-7a). This does not inluene the shear distribution at mid-height and the base as shown in Figures 8-7(b) and 8-7() thus does not aet the lexural strength. Figure 8-8 illustrates the vertial steel stress proiles at base o the walls prior to ailure. As shown, both walls are predited to ail in lexure sine a signiiant portion o vertial reinorement at the tension side o the wall is yielding. Steel stress normalized to yield stress Without top beam With top beam Distane rom let edge (mm) Fig. 8-8 Comparison o inite element preditions o vertial reinorement stress at the base o the wall tested by Kuang and Ho (2008) and the same wall without the top beam and distributed ore applied at top Conrete Crushing Failure Letas et al. (1990) reported that their wall speimen SW26 ailed due to rushing o the onrete at ompression side o the wall base. Finite element analysis also predits the same ailure mode or the wall. The experimental load-deormation urve and the inite element predition are presented in Fig.8-9. Finite element preditions and experimental 193

215 results are in good agreement. Finite element predition o shear strength is V=110 kn. This is less than the shear ore at lexural ailure predited by the plane setion analysis or pure lexure, whih is equal to 124 kn. The reason is that onrete rushes in the prinipal diretion at the ompression side o the base. Finite element preditions or the same wall without the top beam and distributed ore at top are also shown in Figure 8-9. Finite element predited ailure mode or the latter wall is also rushing o onrete and as shown shear strength is not highly aeted by the top beam. 120 FE (With top beam) 100 Shear ore (kn) Experimental results FE (Without top beam) Top wall displaement (mm) Fig. 8-9 Comparison o experimental load- top displaement urve with inite element predition or wall tested by Leas et al. (1990). Figure 8-10 ompares the inite element preditions o setional shear proiles at top, mid-height, and base or the walls with and without the top beam. Notie again that the wall with the top beam arries signiiant portion o shear by redistributing the shear toward the let side at top and thereore making use o diagonal onrete struts that go 194

216 5 Shear stress (MPa) With top beam Without top beam (a) Distane rom let edge (mm) 5 4 Without top beam Shear stress (MPa) With top beam (b) Distane rom let edge (mm) With top beam Shear stress (MPa) Without top beam () Distane rom let edge (mm) Fig Comparison o inite element preditions o shear stress proiles along (a) top setion, (b) mid-height setion, () base setion o the wall tested by Leas et al. (1990) and the same wall without the top beam and distributed ore applied at top. 195

217 diretly to the support. On the other hand, the shear distribution at top is more uniorm or the wall without the top beam (see Fig. 8-10a). Top beam, however, does not signiiantly inluene the shear stress distribution at mid-height or the base o the wall (see Fig. 8-10b and 8-10) thus the wall shear strength is not signiiantly inluened by the top beam. Setional models or shear predit insigniiant ontribution o lexural ompression hord in shear. This is not true at the base o a squat wall as shown in Fig. 8-10(). Compression stress in the prinipal diretion is maximum in the ompression zone at the base o the wall due to the ombination o stresses aused by lexure and shear. This results in onrete rushing in the wall ompression zone at the base or highly reinored walls. In onlusion, omparison o inite element program (VeTor 2) preditions with the atual behaviour o three previously tested walls presented in this hapter show that the shear strength at diagonal tension shear ailure (yielding o horizontal reinorement) may be greatly inreased by the top rigid loading beam ommonly used in the squat shear wall tests to transer shear to the wall top edge. In ontrast, lexural ailure as well as onrete rushing shear ailure is not signiiantly inluened by the top beam. It an also be onluded that base sliding shear ailure is not inluened by the top beam. In highly reinored walls where horizontal reinorement does not yield, shear distribution at the base is not signiiantly inluened by the top loading beam. 196

218 Chapter 9. Analytial Study o Flexural and Shear Resistane o Squat Shear Walls 9.1. General This hapter involves lexural and shear resistane o squat shear walls. The brie bakground o some available methods to predit the shear strength o squat shear walls is presented. In addition, nonlinear inite element analysis (VeTor 2) is used to determine shear and lexural apaity o suh walls and the results are ompared with the ode preditions. A new method to determine the lexural resistane o squat shear walls whih aounts or lexure-shear interation at the wall base is presented and the preditions are ompared with the inite element results. Proposed reinements to the 2004 CSA A23.3 provisions or squat shear walls are also introdued. As explained earlier in Chapters 7 and 8, tested squat shear walls generally had a top loading beam whih inreases the shear resistane o suh walls i they ail due to horizontal reinorement yielding. In the walls analyzed here, shear ore is uniormly distributed over the wall top edge and no top loading beam is inluded sine the purpose is to determine the lower-bound resistane o the walls Traditional Approah or Flexural Resistane o Deep Beams Leonhardt and Walther (1966) perormed linear inite element analysis o unraked onrete to determine lexural apaity o single span deep beams. They onluded that internal lexural lever-arm or a deep beam is less than what is predited by the plane setion analysis. This phenomenon resulted in lower lexural apaity o suh beams ompared to the plane setion analysis preditions. They proposed equations or internal lexural lever-arm o a deep beam. In the proposed equations, internal lexural lever-arm 197

219 is equal to 0.6 times beam height or a deep beam with span-to-height ratio less than or equal to 1.0 and is equal to 0.8 times the beam height or a beam with span-to-height ratio o 2.0 and linearly varies rom a beam with span-to-height ratio o 1.0 to a beam with span-to-height ratio o 2.0. Hal o a single span deep beam has a similar geometry and similar boundary onditions as a squat shear wall. A deep beam and a squat shear wall are shown in Figs. 9-1(a) and (b). While parameters shown in Fig. 9-1(a) are the ones used by Leonhardt and Walther (1966) equations, parameters shown in Fig. 9-1(b) are the parameters ommonly used or squat shear walls. To apply the Leonhardt and Walther (1966) equations to a squat shear wall, internal lexural lever-arm z in Fig. 9-1(a) beomes jd shown in Fig. 9-1(b) (z = jd), beam span l in Fig. 9-1(a) beomes two times wall height h w shown in Fig. 9-1(b) ( l = 2h w ), and beam height h in Fig. 9-1(a) beomes wall length l w shown in Fig. 9-1(b) ( h = l w ). Leonhardt and Walther (1966) equations or internal lexural lever-arm applied to a squat shear wall are: hw [ 9-1] jd = 0.4( h w + lw ) where l hw [ 9-2] jd = 1. 2hw where < 0. 5 l w w in whih l w is wall length, h w is wall height, and jd is internal lexural lever-arm at the base o the wall. Leonhardt and Walther (1966) also proposed that the lexural tension zone o a deep beam is small; thereore, lexural tension reinorement should be onentrated at the lexural tension ae. 198

220 (a) Deep beam (b) Squat shear wall Fig. 9-1 Comparison o a deep beam and a squat shear wall. 199

221 CSA A23.3 Approah or Flexural and Shear Resistane o Squat Shear Walls The 2004 CSA A23.3 requires that a squat shear wall behaves as a single uniorm shear element. Figure 9-2 shows a truss model or uniorm shear stress distribution in a squat shear wall when the axial ore is small and thereore negleted. Fig. 9-2 Uniormly distributed ore low in squat shear walls. The web is assumed to resist shear along the length d v taken as 0.8 l w in the 2004 CSA A23.3. Thus vertial reinorement distributed over the shear length is required to balane the vertial omponent o diagonal ompression ore needed or shear as shown 200

222 in Figure 9-2. In the model shown in Fig. 9-2, the overturning moment is resisted by the onentrated vertial reinorement ore T and the ompression ore C. The 2004 CSA A23.3 requires that the distributed vertial reinorement needed or shear not ontribute to the lexural apaity o squat shear walls. The requirement or additional vertial reinorement or shear an also be expressed as a redued internal lexural lever-arm as was done or deep beams by Leonhardt and Walther (1966). When shear stress is uniormly distributed, there is an ininite number o possible ailure planes with horizontal reinorement yielding; however, all planes orrespond to the same total ore applied to the wall. One suh ailure plane is shown in Figure 9-3(a). Fig.9-3(b) illustrates the ree body diagram o the element bounded by the ailure plane. The horizontal ore ρ b h and vertial ore ρ b h tanθ are the resultant ores h y w w h y w w resisted by distributed horizontal and vertial reinorement, respetively. ρ h and ρ v are distributed horizontal and vertial reinorement ratios, and b w is wall thikness. In the ree body diagram shown in Fig. 9-3(b), the ore due to aggregate interlok at the rak is not inluded as it is ignored by the 2004 CSA A23.3. As shear stress is uniormly distributed (Fig. 9-3a), the wall shear strength V is determined rom: [ 9-3] V = vb d w v where v is shear stress. From equilibrium in the horizontal diretion in Fig.9-3(b): [ 9-4] v = ρ otθ h y 201

223 (a) Failure plane (b) Free body diagram o wall element bounded by ailure plane Fig. 9-3 Horizontal reinorement yielding shear ailure o squat shear walls as in 2004 CSA A

224 Substituting v rom Eq. [9-4] in Eq. [9-3] results in the ollowing shear strength equation or squat shear walls given in the 2004 CSA A23.3. [ 9-5] V = ρ b d otθ h y w v The relationship between the amount o distributed horizontal and vertial reinorement an be obtained by taking moments about the point o appliation o onrete ompression ore C in Figure 9-3(b). 2 [ 9-6] ρ ρ ot θ v = h 2004 CSA A23.3 allows any angle θ between 30 and 45 deg in Eq. [9-5] as long as distributed vertial reinorement needed or shear, given by Eq. [9-6], is alulated using the same angle. When θ = 45 deg, distributed vertial reinorement ratio is equal to the distributed horizontal reinorement ratio while when θ = 30 deg, distributed vertial reinorement ratio is 3 times the distributed horizontal reinorement ratio. In the above derivations wall is not subjeted to axial ore or simpliity. For walls with gross ross-setion area A g and subjeted to axial ore P s, the amount o distributed vertial reinorement given by Eq. [9-6] is redued by P y s A g. The reason is part o axial ore needed or shear is provided by the applied axial ore and the remaining part is resisted by the distributed vertial reinorement. As explained in Chapter 7, the 2004 CSA A23.3 limits the shear strength to 0.15 'b d to avoid onrete rushing. w v 203

225 9.4. Finite Element Analysis o Squat Shear Walls Failing in Flexure A total o 42 walls in our groups with aspet ratios o h w /l w =0.3, 0.5, 1.0, and 2.0 were analyzed. Conrete ompressive strength o ' = 40 MPa and steel yield stress o y = 400 MPa without strain hardening were assumed. The walls were designed to ail due to yielding o vertial reinorement at the base o the wall aording to the 2004 CSA A23.3 preditions. Typial details o the walls that were analyzed in this study are shown in Figure 9-4. Wall ross-setions were uniorm along the wall height and no top loading beam was assumed. All walls were monotonially loaded and the load was applied rom let-to-right uniormly distributed over the wall top edge. To ahieve a lower-bound solution, the ontribution o the ompression zone was minimized by plaing the minimum amount o onentrated reinorement permitted by the 2004 CSA A23.3 in 10% o the wall length on the ompression side. In order to inrease the lexural apaity o the walls, a large amount o onentrated vertial reinorement was plaed on the tension side o the wall. Table 9-1 presents the onentrated and distributed vertial reinorement ratios as well as the distributed horizontal reinorement ratios o the 42 walls that were analyzed. The onentrated vertial reinorement ratios ρ in the tension zone, whih is the ratio o the total amount o onentrated reinorement over the tension zone, ranged rom 0.5% to the maximum o 3.0% depending on the height-to-length ratios. The ratio o the total amount o onentrated reinorement over the ompression zone ρ was kept onstant and equal to the minimum amount aording to the 2004 CSA A23.3 ( ρ = 0.5%). l ' l ' l 204

226 Fig. 9-4 Typial detail o walls analyzed to investigate lexural apaity o squat shear walls. Horizontal reinorement ratios were varied rom 0.25% to 1.0% or every ombination o aspet ratio and onentrated vertial reinorement exept or a ew in whih distributed horizontal reinorement ratios o 0.25% and slightly higher aused shear ailure. For walls with height-to-length ratio o 2.0, distributed vertial 205

227 Table 9-1 Summary o walls analyzed to investigate lexural apaity o squat shear walls. h w /l w ρ l (%) ρ h (%) ρ v (%) ρ ' l (%) ρ l = onentrated lexural reinorement ratio, ρ' l =onentrated reinorement ratio in onrete ompression zone, ρ v = distributed vertial reinorement ratio, ρ h = distributed horizontal reinorement ratio 206

228 reinorement ratios were assumed to be either equal to the distributed horizontal reinorement ratio (ρ v /ρ h =1.0) or 3 times the horizontal reinorement ratio (ρ v /ρ h =3.0). For all other walls, the amount o distributed vertial reinorement was equal to the amount o distributed horizontal reinorement as distributed reinorement amounts o ρ v /ρ h =3.0 aused shear ailure Finite Element Model Uniorm shear elements disussed in Chapter 8 were used. All walls were modeled with 30 square elements along the length while number o elements along the height was varied depending on the wall aspet ratio. 60 elements were used over the height o walls with height-to-length ratio o 2.0, while nine elements were used over the height o walls with height-to-length ratio o 0.3. All nodes along the wall base were onstrained in both vertial and horizontal diretions and no other onstraint was provided. As was explained previously, the applied shear ore at the top was uniormly distributed over the wall length and a top loading beam was not provided Finite Element Results Figure 9-5 presents the shear stress and prinipal ompression stress distributions immediately prior to lexural ailure in the wall with 0.5% distributed horizontal and vertial reinorement ratio and height-to-length ratio o 0.5. At that load level, the wall had experiened lots o raking and reinorement was yielded over a signiiant portion o the wall in both horizontal and vertial diretions. Stress low is shown by stress ontours whih onnet the equal stress points in the wall. 207

229 (a) Shear stress (MPa) (b) Prinipal ompression stress (MPa) Fig. 9-5 Conrete stress ontour diagrams based on inite element analysis o a squat shear wall with 0.5% distributed reinorement in both diretions and height-to-length ratio o 0.5 immediately prior to lexural ailure. 208

230 As shown, shear stress distribution is not uniorm at the base even though shear is applied uniormly at the top. Shear stress at the base is almost zero along the tension side to midlength beyond where it inreases as it approahes the lexural ompression zone. The same trend is also notied in the distribution o onrete prinipal stress. Conrete prinipal stress is almost zero in the triangular area in the tension side o the wall separated by a line rom let top orner to the wall base mid-length. The reason an be examined in Fig. 9-6, whih illustrates the stress in the vertial and horizontal reinorement at raks or the same wall. The 2004 CSA A23.3 assumes shear ore arried by horizontal reinorement is transerred to the onentrated lexural reinorement and then direted to the base by onrete diagonal struts (see Fig. 9-1). In ontrast, Fig. 9-6(a) presents a dierent distribution predited by the inite element analysis. Horizontal reinorement transers the shear to onrete diagonal struts beore it reahes the onentrated lexural reinorement. Notie that horizontal reinorement stress is less than 100 MPa on let side o the wall over the onsiderable portion o the length exept lose to the wall base where ores are redistributed to provide more lexural apaity. Figure 9-6(b) examines stress distribution in the vertial reinorement o the same wall. Vertial distributed reinorement has mostly yielded. Flexural onentrated reinorement has also yielded at the base and yielding has extended well over almost hal o wall height. This is again due to the at that horizontal reinorement is anhored in the onrete diagonal struts and does not transer the load to the onentrated vertial reinorement. 209

231 (a) Horizontal steel stress at raks (MPa) (b) Vertial steel stress at raks (MPa) Fig. 9-6 Steel stress ontour diagrams based on inite element analysis o a squat shear wall with 0.5% distributed reinorement in both diretions and height-to-length ratio o 0.5 immediately prior to lexural ailure. 210

232 The shear stress distributions at the base o our walls are examined in Figure 9-7. The walls all had a same ross-setion with horizontal and vertial reinorement ratio o 0.5% and had dierent aspet ratios o h w /l w = 0.3, 0.5, 1.0, and 2.0. The results shown in Fig. 9-7 orrespond to the load level immediately prior to lexural ailure. As the heightto-length ratio dereases, shear is arried by a larger portion o wall length at the base. For example, or the wall with h w/ l w =0.3, about 60% o the wall length is subjeted to signiiant shear. For the wall with h w/ l w =0.5, shear is resisted over about 40% o the wall length. For walls with height-to-length ratios o 1.0 and 2.0, almost all shear ore is resisted by the ompression zone. This means that the demand on distributed vertial reinorement due to shear is not signiiant, thus redution o lexural apaity due to shear is also not signiiant ρ v =ρ h = 0.005, ρ l = Shear stress (MPa) h w /l w =0.3 h w /l w =0.5 h w /l w =2.0 h w /l w = Relative distane rom let edge Fig. 9-7 Finite element preditions or shear stress distributions at base o our squad shear walls immediately prior to lexural ailure. 211

233 Figure 9-8 shows total normal stress (vertial ore per unit length divided by thikness o wall) distributions o the same walls at the base. The total normal stress is equal to the vertial ompression stress in onrete n v plus the stress ρ v s that is resulted rom steel stress s. When vertial reinorement is yielding and there is no vertial ompression stress due to shear, the total normal stress is equal to ρ v y. Vertial reinorement ratio is 0.5%, thus total normal stress o ρ v y = 2.0 MPa is the maximum tensile stress whih orresponds to yielding o vertial reinorement. For walls with height-to-length ratios o 1.0 and 2.0, normal stress in a signiiant portion o the wall rom the tension ae to the lexural ompression zone reahes the maximum value o 2.0 MPa whih again suggests an insigniiant redution in lexural apaity due to shear. 5 0 Total normal stress (MPa) Yield stress o vertial rein. h w /l w =0.3 h w /l w =1.0 h w /l w =0.5 h w /l w = ρ v =ρ h = 0.005, ρ l = Relative distane rom let edge Fig. 9-8 Finite element preditions or total normal stress distributions at base o our squad shear walls immediately prior to lexural ailure. 212

234 In the wall with height-to-length ratio o 0.3, the total shear stress equals ρ (2.0 v y MPa) at about 0.4l w rom the tension ae. For the wall with an aspet ratio o 0.5, the total normal stress equals ρ (2.0 MPa) at about 0.6l v y w rom the tension ae. This is the onsequene o shear being present in the rest o wall portion at the base that results in normal ompression stresses that balanes diagonal ompression stresses. This results in a redution o the moment apaity o the walls with low height-to-length ratios ompared to the walls with height-to-length ratios o 1.0 or higher. Finite element predited moment apaities o 16 walls with ρ = 0.5% are presented in Figure 9-9 (solid lines with markers). The preditions o plane setion analysis (Response 2000) are also shown as a dotted line. The amount o distributed vertial reinorement in these walls was equal to the amount o distributed horizontal reinorement and ranged rom 0.25% to 1.0%. The predited lexural apaity rom plane setion analysis depend only on the amount o distributed vertial reinorement, whih is equal to the amount o distributed horizontal reinorement. Walls with heightto-length ratios o 1.0 and 2.0 almost reah the apaity predited by plane setion analysis. In ontrast, walls with height-to-length ratios o 0.3 and 0.5 have relatively less lexural apaity due to the inluene o shear. The lexural strength redution is more signiiant as the wall height-to-length ratio dereases. l 213

235 Pure lexure (Response 2000) h w /l w = 1.0 h w /l w = 2.0 Moment (knm) Finite element h w /l w = 0.5 h w /l w = ρ v =ρ h, ρ l = Horizontal reinorement ratio ρ h Fig. 9-9 Predited moment apaities o 16 squat shear walls. The ratios o inite element preditions o moment apaity to the plane setion analysis preditions o moment apaity or the 42 walls are illustrated in Figure One again the redution o moment due to shear is insigniiant or the walls with height-to-length ratios o 1.0 and 2.0 while it beomes more signiiant or the walls with height-to-length ratio o 0.3. For example, the redution in moment apaity with respet to the plane setion analysis preditions is about 9% or the walls with height-to-length ratio o 0.5 and about 27% or the walls with height-to-length ratio o

236 Moment ratio points 7 points 11 points 20 points Finite element Leonhardt & Walter Height-to-length ratio Fig Ratios o inite element predited moment apaity to the plane setion analysis predited moment apaity or the 42 squat walls ailing in lexure. Leonhardt and Walther (1966) presented equations or the internal lexural lever-arm in deep beams that an be applied to squat shear walls. As Leonhardt and Walther solution does not inlude the ontribution o distributed vertial reinorement, the preditions o lexural apaity rom their equations ould be very dierent ompared to the nonlinear inite element preditions. Finite element analysis, as explained previously, shows that distributed vertial reinorement ontributes to the wall lexural strength. However, i the preditions o both methods are onverted to the ratio o lexural apaity with respet to the apaity predited by the plane setion analysis, there are similarities between nonlinear inite element preditions or squat walls and Leonhardt and Walther linear solution or deep beams. Assuming that internal lever-arm is about 80% o the wall 215

237 length rom the plane setion analysis, the redution in moment apaity with respet to the plane setion analysis an be assumed to be proportional to the redution in the lexural lever-arm and this is presented in Figure Both Leonhardt and Walther equations and inite element predit no redution in lexural apaity or walls with h w /l w greater than or equal to Presentation o NLFE with Simple Truss Model This setion explains how the ore low in squat shear walls predited by the inite element analysis an be presented using simple truss models in whih reinoring steel is assumed to resist all tension and onrete resists diagonal ompression. The ores in a wall with height-to-length ratio o 0.5 are presented in Figure Fig. 9-11(a) presents the ores in the horizontal reinorement in addition to the horizontal omponents o the diagonal member ores, while the ores in the vertial reinorement and vertial omponents o the diagonal member ores are shown in Figure 9-11(b). Horizontal elements are the distributed horizontal reinorement, tensile vertial elements are the onentrated vertial reinorement and distributed vertial reinorement, and elements in ompression represent onrete. Uniormly distributed shear ore is applied over the eetive shear length d v = 0.8l w along the top. Unit horizontal ore is applied on eah node o the truss along the top. All ores in truss members relate to this unit ore, whih represents shear resisted by eah member i the shear is uniormly distributed. 216

238 (a) Horizontal omponents o diagonal ompression ores and tension ores in horizontal reinorement (b) Vertial omponents o diagonal ompression ores and tension ores in vertial reinorement Fig Truss model or a squat wall with height-to-length ratio o

239 As was notied in the inite element results, the truss model shows that ores arried by horizontal reinorement ould be anhored in the diagonal struts and do not need to be transerred to the tensile onentrated vertial reinorement. This is possible beause the diretion o diagonal struts hange to balane the horizontal ore that is arried by the horizontal reinorement. Notie in Fig that the irst two diagonal struts at the top o the wall on the let side as well as the one below them hange diretion to balane the horizontal ores as they ross the vertial elements. This results in an inrease in the diagonal strut ore due to the inrease o its horizontal omponent only. Figure 9-11 also shows that rom 6 vertial elements in the wall web representing distributed vertial reinorement, 4.8 o them (80%) ontribute to the lexural apaity o the wall. Furthermore, as was also seen in inite element results, shear at the base is resisted by only a portion o the wall length on the ompression side. In Fig. 9-12, another truss example or a wall with height-to-length ratio o 0.3 is provided. The same trend is notied; however, shear is resisted by a larger portion o the wall length at the base and less ontribution o the distributed vertial reinorement to the lexural apaity is notied ompared to the wall with height-to-length ratio o 0.5 shown in Figure About 50% o the distributed vertial reinorement ontributes to the lexural apaity o the wall based on the truss model shown in Figure It is worth mentioning that the magnitude o vertial omponent o diagonal struts remains onstant throughout the height o the wall even though the diagonal struts, whih go diretly to the base, hange diretion as they ross the horizontal reinorement. 218

240 (a) Horizontal omponents o diagonal ompression ores and tension ores in horizontal reinorement (b) Vertial omponents o diagonal ompression ores and tension ores in vertial reinorement Fig Truss model or a squat wall with height-to-length ratio o

241 9.5. Proposed Setional Model or Flexural Capaity This setion presents a proposed setional model that inludes the inluene o shear on the lexural apaity o squat shear walls. As was notied in the inite element results and was later presented by simple truss models, distribution o normal ompression stress at the base o a squat shear wall is over a longer length than predited by plane setion analysis. This phenomenon, whih results in a redution o wall moment apaity ompared to what is predited by the plane setion analysis, is more signiiant when the wall height-to-length ratio is small. It beomes insigniiant or walls with height-tolength ratios lose to 1.0 or greater. The proposed model an apture this behaviour by inluding an axial ore N v in addition to the other ores that at at the wall base. N v, whih is the ore needed or shear, is zero or walls with greater height-to-length ratios and beomes signiiant as the wall height-to-length ratio beomes smaller. For N v equal to zero, the model beomes the same as setional analysis under pure lexure and thus no redution due to shear is alulated. Figure 9-13 shows the proposed setional model at the base to alulate lexural apaity o squat shear walls. T in Figure 9-13 is the ore in the onentrated reinorement and T d is the resultant ore in the distributed vertial reinorement. At lexural apaity, T is equal to the area o onentrated reinorement times steel yield stress. Assuming distributed reinorement yields over the wall entire length, T d an also be reasonably approximated by total area o distributed vertial reinorement times steel yield stress. Axial ore N v is the resultant ore o the normal stress ating over a portion o wall length d nv at the base. As was presented by the truss model in Fig. 9-12, rotation o 220

242 diagonal struts that go diretly to the wall base does not aet the magnitude o N v per unit length. Notie in Fig. 9-12(b) that the magnitude o the vertial omponents o all diagonal struts that go diretly to the base is equal to 1.0 and it does not inrease as they rotate. This suggests that normal stress at the base is equal to the normal stress at the top o the wall. One good approximation, as explained in previous hapters, or the normal stress needed or shear is [ 9-7] N = v otθ ( b d ) v w nv v otθ and thus: Fig Proposed setional model or lexural apaity o squat shear walls. In whih angle θ is the angle o diagonal ompression and V v = b d w v. In the proposed model, angle o inlination is determined rom the 2004 CSA A23.3 equilibrium-based equation Eq. [7-27] given the amount o distributed vertial and horizontal reinorement and wall axial ore. The model beomes similar to the 2004 CSA A23.3 when d nv =d v 221

243 and it beomes a typial setional analysis under pure lexure when d nv =0. Based on the inite element results, d nv is a untion o wall aspet ratio. The proposed d nv shown in Fig is given by: [ 9-8] d d h 0 nv = v w In the proposed model shown in Fig. 9-13, the magnitude and loation o C is determined rom equilibrium in the vertial diretion using the equivalent stress blok or onrete ompression stress in the lexural ompression zone. Moment apaity is then determined rom moment equilibrium at the base. This is an iterative proedure or a given wall with a given amount o distributed reinorement beause the wall lexural apaity as well as shear ore orresponding to the wall lexural apaity is unknown. Notie that lexural apaity is a untion o shear ore at lexural apaity in the proposed model. For design, however, the proedure is not iterative as the applied bending moment and shear ore are known Comparison o Finite Element Results with the Preditions o Proposed Method or Flexural Capaity o Squat Shear Walls Figures 9-14 to 9-17 ompare the inite element preditions or the lexural apaity o the 42 squat shear walls ailing in lexure with the preditions o the 2004 CSA A23.3 as well as the proposed method preditions. In these igures, the relationship between the average shear stress over the wall length V b l w w immediately prior to lexural ailure with the horizontal reinorement ratio is presented. 222

244 2.4 h w / l w = 2.0 ρ v =3ρ h, ρ l = ρ v Finite element Shear stress V /b w l w (MPa) CSA 2004 ρ v =ρ h, ρ l = 0.02 ρ v =ρ h, ρ l = 0.01 Proposed ρ v =ρ h, ρ l = ρ v =ρ h, ρ l = Horizontal reinorement ratio ρ h Fig Comparison o inite element preditions or shear ore at lexural apaity o squat shear walls with h w /l w =2.0, with 2004 CSA A23.3 and proposed method preditions. 2.8 h w / l w = 1.0 Finite element Shear stress V /b w l w (MPa) CSA 2004 ρ v =ρ h, ρ l = 0.01 Proposed ρ v =ρ h, ρ l = ρ v =ρ h, ρ l = Horizontal reinorement ratio ρ h Fig Comparison o inite element preditions o shear ore at lexural apaity o squat shear walls with h w /l w =1.0, with 2004 CSA A23.3 and proposed method preditions. 223

245 4.0 h w / l w = 0.5 ρ v =ρ h, ρ l = 0.01 Shear stress V /b w l w (MPa) Finite element ρ v =ρ h, ρ l = CSA 2004 Proposed Horizontal reinorement ratio ρ h Fig Comparison o inite element preditions or shear ore at lexural apaity o squat shear walls with h w /l w =0.5, with 2004 CSA A23.3 and proposed method preditions h w / l w = 0.3 Shear stress V /b w l w (MPa) Finite element ρ v =ρ h, ρ l = Pure lexure (Response 2000) CSA 2004 Proposed Horizontal reinorement ratio ρ h Fig Comparison o inite element preditions or shear ore at lexural apaity o squat shear walls with h w /l w =0.3, with 2004 CSA A23.3 and proposed method preditions. 224

246 Finite element preditions are shown with thik solid lines with markers, while the proposed method preditions are presented by the thinner solid lines. The preditions o the 2004 CSA A23.3 method are shown by dashed lines. To determine the lexural apaity o the walls aording to the 2004 CSA A23.3 method, only the portion o distributed vertial reinorement that is not needed or shear was inluded in the setional analysis. This was an iterative proedure beause the wall lexural apaity was not known and thereore shear at lexural apaity was also unknown. Figures 9-14 and 9-15 ompare the inite element preditions with those rom the 2004 CSA A23.3 and the proposed method or the walls with height-to-length ratios o 2.0 and 1.0. The proposed method preditions ompare well with the inite element results. Note that the proposed method preditions are so lose to the inite element preditions in some ases that the lines annot be distinguished. The 2004 CSA A23.3 preditions are onservative as it totally exludes the ontribution o distributed vertial reinorement needed or shear. Figures 9-16 and 9-17 present the inite element preditions with the 2004 CSA A23.3 and the proposed method preditions or the walls with height-to-length ratios o 0.5 and 0.3. The 2004 CSA A23.3 preditions are onservative. The proposed method preditions are in reasonably good agreement with the inite element preditions Simpliied Proposed Method A simpler model than the proposed model presented in Setion 9.5 is to use a redued amount o distributed vertial reinorement when alulating the lexural apaity at the base o a squat shear wall. For example, the urrent CSA A23.3 does not use any o the distributed reinorement needed or shear. The model is shown in Fig where T d is 225

247 again the resultant tension ore in the distributed vertial reinorement assumed to be yielding over the wall length and α is the portion o the total amount o distributed vertial reinorement that ontributes to the lexural apaity o the wall. While α =1.0 means that all o the distributed vertial reinorement ontributes to the lexural apaity, α = 0 means none o the distributed vertial reinorement ontributes to the lexural apaity. In order to get the same results rom the model presented here in Fig and the proposed model presented beore in Fig. 9-13, taking moments about the point o appliation o ompression ore C in both models should yield equal moments. Thus: [ 9-9] α T 0.5d ) = T (0.5d ) N (0.5d ) d ( v d v v nv Substituting or T d = ρ b d and solving or α : v y w v N v d nv [ 9-10] α = 1 ( ) ρ b d d v y w v v Fig Proposed simple model or lexural apaity o squat shear walls. 226

248 Substituting or N v rom Eq. [9-7], Eq. [9-10] an be written as: ot [ 9-11] 1 v θ α = ρ v y d d nv v 2 Assuming lexural and shear ailure our at the same load level and v = 0 as assumed by the urrent CSA A23.3, rom equilibrium: [ 9-12] v = ρ tanθ v y Substituting v rom Eq. [9-12] in Eq. [9-11] results in the ollowing simpliied equation: [ 9-13] d α = 1 d nv v 2 in whih d nv is determined rom Eq. [9-8] and α is the portion o distributed vertial reinorement needed or shear that ontributes to the lexural apaity. d nv an be expressed as a untion o wall aspet ratio h l w w assuming d v = 0.8l w as in the 2004 CSAA23.3. Substituting d v = 0.8l w in Equation [9-8]: d nv hw [ 9-14] = ( 0.8 ) 0 l l w w As h l w w inreases d d nv v dereases thus the redution o lexural apaity due to shear also hw dereases. For walls with 0. 8, no redution in lexural apaity due to shear is l w predited by the proposed simpliied method sine d nv beomes zero. Substituting d nv h w rom Eq. [9-14] in Eq. [9-13], α is given as a untion o wall aspet ratio : l w 227

249 hw hw [ 9-15] α = lw lw This untion is plotted in Figure 9-19 (solid line). As shown, about 80% and 40% o distributed vertial reinorement needed or shear ontribute to the lexural apaity or walls with h l w w =0.5 and 0.2, respetively. No redution in lexural apaity due to shear hw or walls with 0. 8 l w is predited. Equation [9-15] an be onservatively approximated by Eq. [9-16] whih is also plotted in Figure hw [ 9-16] α = l w 1.2 Portion o distributed vertial rein. α h w /l w h w /l w Fig Variation o the portion o distributed vertial reinorement α that ontributes to the lexural apaity o walls. 228

250 A simpler approah or ode provisions is to use all o the distributed vertial hw reinorement in walls with 0. 8 l w and use none o vertial distributed reinorement hw needed or shear in walls with < 0. 8 l w when alulating lexural apaity o a squat shear wall Finite Element Analyses o Squat Shear Walls Failing in Shear 44 walls were designed to ail in shear and analyzed by VeTor 2. The walls, whih were monotonially loaded at the top rom the let-to-right diretion, had aspet ratios o h w /l w =0.3, 0.5, 1.0, and 2.0, with onrete ompressive strength o ' = 40 MPa and steel yield stress o y = 400 MPa. Typial wall details are shown in Figure Two types o ross-setions were used. Type 1 ross-setion, whih was retangular, was used or height-to-length ratios o 0.3 and 0.5 while Type 2 ross-setion, whih inluded a lange on the lexural tension side, was used or the remaining taller walls to avoid lexural ailure. A wall with a lange on one side only is perhaps unrealisti but gives a lowerbound shear strength. Previous studies have shown that i a squat shear wall has a lange on the ompression side, the ompression zone ontribution to the wall shear resistane signiiantly inreases and thus the shear resistane o the wall inreases. All walls had uniorm ross-setions over the height, and no top beam was provided. Table 9-2 presents the wall ross-setion details inluding onentrated and distributed vertial reinorement ratios and distributed horizontal reinorement ratios. One again, to minimize the ontribution o the lexural ompression zone to the 229

251 Fig Typial details o walls analyzed to investigate shear strength o squat shear walls. 230

252 Table 9-2 Summary o walls analyzed to investigate shear strength o squat shear walls. h w /l w Wall type t / l w b / b w ρ l (%) ρ h (%) ρ v (%) ρ ' l (%) ρ l = onentrated lexural reinorement ratio, ρ' l =onentrated reinorement ratio in onrete ompression zone, ρ v = distributed vertial reinorement ratio, ρ h = distributed horizontal reinorement ratio. 231

253 shear resistane, onentrated vertial reinorement ratio in the ompression zone was taken equal to the 2004 CSA A23.3 minimum ratio o 0.5%. In addition, onentrated vertial reinorement was plaed over only 10% o the wall length on the wall ompression side. Distributed horizontal reinorement ratios were varied rom 0.25% to 1.5%. Aording to the 2004 CSA A23.3 provisions, diagonal ompression angle θ is reely hosen rom 30 to 45 degrees. I θ =45 deg is hosen and there is no axial ompression, the amount o distributed vertial reinorement needed or shear is equal to the amount o distributed horizontal reinorement (see Eq. [9-6]). I θ =30 deg is hosen, the amount o distributed vertial reinorement needed or shear is 3.0 times the amount o distributed horizontal reinorement (see Eq. [9-6]). Thus distributed vertial reinorement ratios were either equal to the distributed horizontal reinorement ratios (ρ v /ρ h =1.0) or 3.0 times the horizontal reinorement ratios (ρ v /ρ h =3.0). The same inite element mesh disussed or walls ailing in lexure (Setion 9.4.1) was used Comparison o Finite Element Results with Code Preditions Figure 9-21 ompares the inite element preditions (solid lines with markers) or shear strength o the walls ailing in shear with the preditions o ACI 318 (dashed line), NZS 3101 (dashed-dotted line), and the 2004 CSA A23.3 (dotted line). While Fig. 9-21(a) presents the results or walls with equal amount o distributed reinorement in both diretions, Fig. 9-21(b) shows the results or the walls with distributed vertial reinorement ratios o 3 times horizontal distributed reinorement ratios. NZS 3101 preditions were determined assuming dutility ator o 2.0 and overstrength ator o 232

254 1.0. NZS 3101 method is explained in Setion Note that the 2005 National Building Code o Canada speiies a dutility ator o 2.0 or squat shear walls. 8 7 ρ v =ρ h h w / l w = 0.5 Shear stress V /b w l w (MPa) h w /l w =2.0 h w /l w <=1.0 h w / l w = 0.3 ACI 318 h w /l w =2.0 NZS 1301 Finite element h w /l w =1.0 CSA (a) Horizontal reinorement ratio ρ h 8 7 ρ v =3ρ h Finite element h w /l w =1.0 Shear stress V /b w l w (MPa) h w /l w = 0.3 h w /l w = 0.5 CSA 2004 NZS 1301 h w /l w =2.0 ACI (b) Horizontal reinorement ratio ρ h Fig Comparison o inite element preditions or shear strength with ode preditions or squat walls with: (a) ρ v =ρ h, (b) ρ v =3ρ h. 233

255 The end horizontal lines in the ode predition urves in Figs. 9-21(a) and (b) orrespond to the ode total shear ore limits to avoid onrete rushing and base sliding due to shear. These limits are explained in Chapter 7. The hange in the slopes o inite element urves beyond horizontal reinorement ratio o about in Fig. 9-21(b) is also due to the hange o ailure mode rom diagonal tension shear ailure to onrete rushing ailure. ACI 318 and NZS 3101 preditions are the same in Figs. 9-21(a) and 9-21(b) as they are not inluened by ρ v /ρ h ratios. In ontrast, the 2004 CSA A23.3 preditions are higher in Fig. 9-21(b) ompared to Fig. 9-21(a) that is onsistent with the inite element results. ACI 318 preditions are onsistently unonservative or ρ v /ρ h =1.0 until they reah onrete rushing limit. NZS 3101 preditions are less unonservative or ρ v /ρ h =1.0; they are slightly unonservative or the taller walls with horizontal reinorement ratios o less than For walls with ρ v /ρ h =3.0 (Fig. 9-21b), ACI 318 preditions are in better agreement with the inite element results ompared to those or the walls with ρ v /ρ h =1.0 (Fig. 9-21a), but they are still unonservative in some ases. NZS 3101 preditions are highly onservative or walls with ρ v /ρ h =3.0. The 2004 CSA A23.3 trend o preditions is similar to that rom the inite element results, but they are onservative in all ases. While the 2004 CSA A23.3 preditions are loser to the inite element results or walls with height-to-length ratios o 1.0 and 2.0, they beome more onservative or walls with height-to-length ratio o 0.3. Shear stress distributions at the wall base and mid-height are shown in Figs (a) and (b) or the our walls with ρ v =ρ z = Shear stress at base (Fig. 9-22a) is not 234

256 ρ h =ρ v =0.005 h w /l w = 0.3 Shear stress (MPa) h w /l w = 0.5 h w /l w =1.0 2 h w /l w = (a) Relative distane rom let edge 4 ρ h =ρ v =0.005 h w /l w = 0.3 h w /l w =1.0 Shear stress (MPa) h w /l w =2.0 h w /l w = (b) Relative distane rom let edge Fig Finite element preditions or shear stress distribution in squat walls with ρ v =ρ h =0.005 immediately prior to shear ailure: (a) at base o wall, (b) at mid-height. 235

257 uniorm; shear is mostly arried by ompression side o the wall espeially in the taller walls. Maximum shear stress in the ompression side o the walls is as high as 17 MPa in the wall with height-to-length ratio o 0.3. Mid-height shear stress distributions (Fig. 9-22b) are loser to uniorm distribution. As the wall height-to-length ratios derease, ontribution o the ompression side o the walls to shear resistane is more signiiant. The 2004 CSA A23.3 assumes that shear is resisted by 80% o the wall length (d v =0.8l w ) whih is onservative sine the ompression zone signiiantly ontributes to the wall shear strength espeially or walls with a heightto-length ratio o 0.3. Aggregate interlok ontribution to shear resistane was ound to be negligible in the inite element results. This is shown in Fig. 9-23(a), whih illustrates the stress ontour diagram o shear stress on raks together with approximate ailure plane based on the inite element results. Notie that shear stress on raks is less than 0.1 MPa in the region that ailed in shear. This onirms that the assumption o V =0 in the 2004 CSA A23.3 is reasonable. Figure 9-23(b), whih illustrates the stress ontour diagram o total shear stress, shows that the shear low pattern is the same as the one or the wall ailing in lexure shown in Figure 9-5(a). Fig. 9-23(b) also illustrates that shear stress in the ompression zone is signiiant, thus it signiiantly ontributes to the shear strength o squat walls. The 2004 CSA A23.3 ignores the ompression zone ontribution to shear strength as it assumes that shear ore is resisted by 80% o the wall length. It was ound that the signiiant inrease in the inite element predited shear strength o squat 236

258 (a) Shear stress on raks v (MPa) (b) Shear stress (MPa) Fig Conrete shear stress ontour diagrams based on inite element analysis o a squat shear wall with 0.5% distributed reinorement in both diretions and height-tolength ratio o 0.5 prior to diagonal tension shear ailure. 237

259 walls with height-to-length ratio o 0.3 ompared the shear strength o the taller walls was due to the ompression zone ontribution to shear resistane. To inlude the eet o ompression zone in shear resistane, the ree body diagram shown in Fig. 9-3(b) an be revised as shown in Fig. 9-24(b). (a) Failure plane (b) Free body diagram o wall element bounded by ailure plane Fig Diagonal tension ailure o low rise shear walls aounting or ompression zone ontribution. 238

260 Aording to equilibrium equation in the horizontal diretion: [ 9-17] vb whw tan θ = Vsh + Fh = ρ h ybwhw + Fh F h is a untion o ompression zone length whih in turn is a untion o wall length and onrete raking stress. Conrete raking stress is a untion o ', thus F h an be estimated as: [ 9-18] F h = κ ' bwlw where κ is a oeiient. Substituting F h rom Eq. [9-18] in Eq. [9-17] and solving or v: lw [ 9-19] V vbwd [ v h y ot ' ] bwd v h = = ρ θ + α w in whih α is another oeiient equal to κ ot θ. Eq. [9-17] is similar to the ACI 318 shear strength equation or squat shear walls and other proposed methods in the literature suh as the method proposed by Wirandianta (1985). It inludes a steel ontribution V s that is equal to ρ otθb d and is independent o wall aspet ratio, and onrete h y w v ontribution l = α b d whih is a untion o wall aspet ratio. α = 0.1 in w V '( ) hw w v Eq. [9-19] results in preditions that are in good agreement with the inite element results or the 44 walls ailing in shear when θ is determined rom the 2004 CSA A23.3 equation Eq. [7-27]. As it will be shown later, ompression zone ontribution to shear is signiiantly redued i loalized sliding along previously existing raks at the base is 239

261 modeled in the inite element analysis, and thus the model explained above is not advoated in this thesis Sliding along Previously Existing Craks As desribed earlier, it was assumed in the inite element analysis that the wall base was ully onstrained in the horizontal diretion. As a result o this assumption, inite element preditions o shear stress arried at the base in the ompression zone region was as high as 17 MPa in some ases. Suh a high shear stress is appropriate or reinored onrete that is monotonially loaded; however, under load reversals, the ompression zone will have horizontal raks as it was the tension zone in the reverse diretion o loading thus may not be able to resist suh a high shear stress. Equation [3-6], whih was developed rom experimental results, gives the maximum shear stress that an be transerred aross raks. This equation limits the shear stress o raked onrete to 0.58 ' (3.7 MPa or 40 MPa onrete) when the rak width is zero. NZS 3101, on the other hand, limits the shear stress on walls to 0.65 ' (4.1 MPa or 40 MPa onrete) or a wall with overstrength ator o 1.0 and dutility ator o 2.0 (see Eq. 7-25), while the 2004 CSA A23.3 limits the shear stress to 0.15 ' (6 MPa or 40 MPa onrete). A similar limit in ACI 318 is 0.83 ' or individual wall piers, whih is 5.25 MPa or 40 MPa onrete. Based on these numbers, it was deided to limit the shear stress o walls to the onservative value o 4 MPa or 40 MPa onrete to aount or sliding along previously existing raks. Only three walls with horizontal and vertial distributed reinorement ratios o were analyzed in this way. 240

262 VeTor 2 is not apable o limiting the horizontal shear stress in an element to a speiied value. Thus the analysis was done using VeTor 2 together with an event-toevent proedure that was manually arried out. The analysis started or the ase where all nodes along the wall base were onstrained both in vertial and horizontal diretions. The analysis was then stopped when reation at any node at the base reahed the magnitude that orresponds to 4 MPa shear stress in the horizontal diretion. The node onstraint in the horizontal diretion was then removed and the ore orresponding to 4MPa shear stress was applied at the node. The analysis ontinued until the reation at the next node reahed the magnitude orresponding to 4MPa shear stress. The proedure was repeated until all reations at the nodes had the magnitude o less than or equal to the magnitude orresponding to 4 MPa shear stress. Figure 9-25 shows the inite element preditions o load-displaement relationships or the walls with height-to-length ratios o 0.3 (Fig 9-25a), 0.5 (Fig. 9-25b), and 1.0 (Fig. 9-25) when horizontal shear stress is limited to 4MPa. The inite element preditions when shear stress is not limited are also presented. The results show that the shear strength o the wall with height-to-length ratio o 0.3 is signiiantly aeted by limiting the shear stress in the ompression zone, while the shear strength o the wall with heightto-length ratio o 1.0 is not aeted signiiantly. When the shear stress in the ompression zone is limited, the inite element predited shear strength or the wall with height-to-length ratio o 0.3, shown in Fig. 9-25(a), is only 10% greater than the inite element predited shear strength o the wall with height-to-length ratio o 1.0 shown in Figure 9-25(). This indiates that wall aspet ratio does not signiiantly inluene the 241

263 4 ACI 318 shear strength Shear stress V /b w l w (MPa) Limit shear in omp. zone Without limit Reined 2004 CSA A23.3 (d v = 0.9 l w ) 2004 CSA A23.3 shear strength h w /l w = (a) Right edge top displaement (mm) 4 ACI 318 shear strength Shear stress V /b w l w (MPa) Without limit Reined 2004 CSA A23.3 (d v = 0.9 l w ) 2004 CSA A23.3 shear strength Limit shear in omp. zone h w /l w = (b) Right edge top displaement (mm) 4 ACI 318 shear strength Shear stress V /b w l w (MPa) Reined 2004 CSA A23.3 (d v = 0.9l w ) Limit shear in omp. zone Without limit 2004 CSA A23.3 shear strength h w /l w = () Right edge top displaement (mm) Fig Loalized sliding eet on load-displaement urve o squat shear walls with ρ z = ρ v = and height-to-length ratio o: (a) 0.3, (b) 0.5, ()

264 shear strength o squat shear walls when the shear stress in the ompression zone is limited. This is onsistent with the 2004 CSA A23.3 provisions. Figure 9-26 illustrates base shear stress distributions o the walls when shear stress is limited together with the distributions or the walls when shear stress is not limited. As expeted, shear stress is more uniorm or walls in whih shear stress is limited. The midheight shear stress distributions o the walls are examined in Figure It is evident that the ompression zone ontribution to shear resistane is highly redued due to limiting the shear stress in the ompression zone or the wall with height-to-length ratio o 0.3 while it does not have an inluene or the wall with height-to-length ratio o 1.0. As shown in Fig. 9-27, shear stress is still onsiderable in the ompression zone or all walls, but this shear stress is not aounted or in the 2004 CSA A23.3; it assumes only 80% o the wall lengths ontribute to shear resistane. To aount or the inluene o ompression zone on the shear resistane, it is proposed to take d v =0.9l w rather than d v =0.8l w as in the 2004 CSA A23.3. The resulted preditions o shear strength are ompared with inite element preditions in Figure The good agreement between the inite element preditions o shear strength when shear stress in the ompression zone is limited and the preditions rom the 2004 CSA A23.3 method with the proposed reinement is evident Comparison o Finite Element Results with the 2004 CSA A23.3 Reined Method As explained in the previous setion, the proposed reinement to the CSA A23.3 provisions or shear strength o squat shear walls is to use an eetive shear length d v =0.9l w. In the urrent provisions, d v =0.8l w. In this setion, the inite element preditions 243

265 18 16 h w /l w = 0.3 Shear stress (MPa) Limit shear in omp. zone Without limit (a) Relative distane rom let edge 12 h w /l w = Shear stress (MPa) Limit shear in omp. zone Without limit (b) Relative distane rom let edge 12 h w /l w = Shear stress (MPa) Limit shear in omp. zone Without limit () Relative distane rom let edge Fig Loalized sliding eet on base shear stress distribution o squat shear walls with ρ z = ρ v = and height-to-length ratio o: (a) 0.3, (b) 0.5, ()

266 4 h w /l w = 0.3 Without limit Shear stress (MPa) Limit shear in omp. zone (a) Relative distane rom let edge 4 h w /l w = 0.5 Without limit Shear stress (MPa) Limit shear in omp. zone (b) Relative distane rom let edge 4 h w /l w = 1.0 Shear stress (MPa) Without limit Limit shear in omp. zone () Relative distane rom let edge Fig Loalized sliding eet on mid-height shear stress distribution o squat shear walls with ρ z = ρ v = and height-to-length ratio o: (a) 0.3, (b) 0.5, ()

267 o shear strength or the 44 walls ailing in shear are ompared with the preditions rom the 2004 CSA A23.3 with the proposed reinement. Figure 9-28 ompares the inite element preditions o shear strength with the 2004 CSA A23.3 preditions using d v =0.9l w or the 44 walls ailing in shear in whih shear stress is not limited. The horizontal axis is the wall height-to-length ratio while the vertial axis is the ratio o inite element predited shear strength to the 2004 CSA A23.3 predited shear strength using d v =0.9l w. Clearly using d v =0.9l w in the 2004 CSA A23.3 is onservative as the ratios vary rom about 1.0 to about The inite element preditions ompare well with the revised 2004 CSA A23.3 method or walls with height-to-length ratios o 1.0 and 2.0. The shear strength ratios are below 1.25 in most ases. Note that in walls with height-to-length ratios o 1.0 and 2.0 limiting the shear stress in the ompression zone has insigniiant inluene on shear strength as shown in the previous setion. Only one wall with a height-to-length ratio o 1.0 and one wall with a height-to-length ratio o 2.0 has a high shear strength ratio lose to Those two walls had a low amount o distributed reinorement 0.25% in both horizontal and vertial diretions. The strength o the two walls are signiiantly higher than the 2004 CSA A23.3 reined method preditions beause their shear strength at onrete diagonal raking was higher than shear strength at yielding o horizontal reinorement. The 2004 CSA A23.3 revised method beomes more onservative or walls with smaller height-to-length ratios espeially or those with height-to-length ratio o 0.3. For those walls, limiting the shear stress in the ompression zone dereases the shear strength about 15% as shown in the previous setion. Thus, it ould be onluded that the

268 CSA A23.3 method with the proposed reinement an reasonably predit the shear strength o squat shear walls. Limiting the shear stress in the ompression zone is very time onsuming and thereore it was not possible to repeat the preditions or all walls aounting or this Shear strength ratio Wall height-to-length ratio Fig Ratios o inite element analysis-to-2004 CSA A23.3 reined method predited shear strength or the 44 walls ailing in shear. 247

269 Chapter 10. Summary and Conlusions General This thesis involves two related topis whih are shear strength o onrete bridge girders and strength o squat shear walls. The summary and onlusions or the two topis are reported separately below Shear Strength Evaluation o Bridge Girders Beam shear design provisions o AASHTO LRFD and the 2006 CHBDC are based on simpliied versions o Modiied Compression Field Theory (MCFT, Vehio and Collins 1983) whih is one o the theories developed to predit the behaviour o uniorm shear elements. As a result o simpliiations or design, these methods are onservative whih is preerred or design; but may ause unneessary bridge load restritions or retroit when these methods are used or evaluation. Response 2000 (Bentz 2000), whih perorms beam setional analysis employing a smeared layered approah, provides a more aurate predition or beam shear strength and is able to predit the behaviour o beams throughout the entire range o loading. While suh a omputer program is a powerul researh tool, it is not onvenient or engineering pratie sine it requires advaned knowledge o shear and high level o judgment. It requires a signiiant number o input parameters and inludes dierent material models and the results annot be easily heked by hand alulations. Use o Response 2000 or evaluation o numerous setions o a bridge is also time onsuming sine it requires detailed inormation o eah ross-setion and provides signiiant amount o inormation rom whih user should extrat the needed inormation. 248

270 Proposed Evaluation Method A new method or shear evaluation o beams was developed and is presented in this thesis. The new method aounts or the eet o more parameters in shear, provides more insight than other simpliied approahes, and still is simple enough to be easily implemented in an Exel spreadsheet. Also, the results an be easily heked by hand alulations. The proposed evaluation proedure, whih is dierent or members with and without stirrups, was developed so that trial-and-error is not required; but also inludes a number o reinements suh as aounting or: (i) inluene o V (onrete tension stresses) on average longitudinal ompression ore N v required to resist shear in a diagonally raked web (V redues average tension strain o member), (ii) dierene between total shear depth d v and depth o diagonally raked web d nv, (iii) tension ore resisted by distributed longitudinal reinorement in web, (iv) loation o prestressed tendons in web, and (v) tension-stiening provided by raked onrete in tension hord. The proedure was veriied by omparing preditions o shear strength or members with and without stirrups with the MCFT preditions or a single uniorm shear element. The proposed method preditions were in better agreement with MCFT preditions ompared to the 2006 CHBDC method or both members with and without transverse reinorement. To urther validate the method, shear strength preditions or our existing bridge girders were ompared with the results rom Response The girders in three o these bridges had more than minimum transverse reinorement while the girders in the ourth had less than minimum transverse reinorement. Three o the bridges had prestressed onrete girders. Preditions o the 2007 AASHTO LRFD, 2006 CHBDC, 249

271 and ACI were also ompared with the results rom the proposed method. The proposed method preditions were ound to be loser to Response 2000 preditions ompared to the ode preditions. For example, Response 2000 preditions or the nine evaluated setions o the three girders with more than minimum transverse reinorement were on average only 4% higher than the results rom proposed method. The COV o the ratios o Response 2000 predited shear strength to the predited shear strength rom the proposed method was only 4%. Among the evaluated ode methods, the 2007 AASHTO LRFD proedure was the most onsistent with Response The preditions were on average 13% higher than the Response 2000 preditions and COV o predited shear strength ratios was 11%. ACI had the largest deviation rom Response 2000 with COV o predited shear strength ratios equal to 16%. Further veriiation o the method by omparing with experimental results is summarized in Setion Proposed Reinement or the 2006 CHBDC Shear Design Method In the 2006 CHBDC and 2004 CSA A23.3 odes, the shear resistane o a beam with or without transverse reinorement is a untion o mid-depth longitudinal strain ε x. These odes provide a simple equation to estimate ε x, and allow the use o a more sophistiated proedure to determine ε x ; but do not desribe how this proedure is to be done. One approah that ould be used to estimate the mid-depth longitudinal strain is a setional analysis using Response A more omplex version o the ode equation or ε x was developed as part o this thesis. This equation inludes a better estimate o the axial ompression ore needed in the web o a girder to resist shear. It also rigorously aounts or the inluene o 250

272 distributed longitudinal reinorement in the web and tension stiening o the lexural tension reinorement. A omparison with the mid-depth longitudinal strain determined with Response 2000 indiates the proposed equation or ε x is more aurate than the ode simpliied equation. The mid-depth longitudinal strains predited by the simple equation in the odes were up to 40% larger than predited by the proposed equation and Response It is reommended that when the 2006 CHBDC method is used to evaluate the shear strength o an existing bridge girder with transverse reinorement, the proposed equation or ε x be used to obtain a higher shear strength estimate. While 2006 CHBDC and 2004 CSA A23.3 odes permit the use o a more sophistiated proedure to determine ε x in members without transverse shear reinorement, a omparison with test results has shown that this may be unsae. This is explained in Setion Comparison o Preditions with Experimental Results Members with at least minimum transverse reinorement The proposed shear evaluation method or members with transverse reinorement was veriied against atual shear strengths determined rom tests o 80 reinored onrete beams and 88 prestressed onrete beams reported in the literature. Comparisons with 80 reinored onrete beam tests showed that the preditions rom the proposed evaluation method are the losest to the test results ompared to the 2006 CHBDC, 2007 AASHTO LRFD, and ACI 318 design methods. Only 26 o the 80 reinored onrete beams that have been previously tested had transverse reinorement ratios more than twie the minimum, while most existing bridge girders have more than twie the minimum transverse reinorement. For the 26 beams 251

273 with more than twie the minimum transverse reinorement, preditions rom the proposed method ompare even better with the test results than the ode methods. The proposed evaluation method preditions were also in better agreement with experimental results than the ode design methods or the 88 prestressed onrete beam tests with stirrups. As was also determined in previous studies (Hawkins et. al, 2005), ACI 318 method is unsae with about 50% o the shear strength preditions being greater than the atual test results. O the 88 prestressed onrete beams, only 22 beams had a predited mid-depth longitudinal strain higher than While the proposed method shear strength preditions or those members were in good agreement with atual test results, the 2006 CHBDC gave on average 13% more onservative results or those beams ompared to the proposed method preditions. The 2006 CHBDC shear strength preditions with the proposed reined ε x equation was validated against the same 80 reinored onrete and 88 prestressed onrete tested beams. Using the proposed ε x equation in the 2006 CHBDC shear design provision improved the average test-to-ode predited shear strength ratio rom 1.31 to 1.26 in prestressed onrete members while the COV o shear strength ratios remained about the same. Members without transverse reinorement The proposed proedure or members without transverse reinorement was validated by omparing shear strength preditions with test results or 132 reinored onrete and

274 prestressed onrete beams. The proposed evaluation method gave the best overall agreement with test results. For the reinored onrete beams, the proposed method had an average value o test-to-predited shear strength ratio o 1.17 and COV o 17.3%. The average values o test-to-predited shear strength ratios were 1.26 and 1.16 or the 2007 AASHTO LRFD and 2006 CHBDC methods, respetively. The COV o these ratios were 18.5% and 17.8%, respetively. The ACI 318 preditions or the shear strength o beams without transverse reinorement were very unsae due to the well known size eet phenomenon. For the prestressed onrete beams, the average test-to-predited shear strength ratios were 1.37 or the proposed method, 1.55 or 2006 CHBDC, 1.57 or 2007 AAHTO LRFD, and 1.17 or ACI 318. The orresponding COV o these ratios were 17%, 23.2%, 23.1%, and 16%, respetively. Using a more reined equation or ε x to predit the shear strength o 132 reinored onrete beams without stirrups inreased the number o unsae preditions rom the 2006 CHBDC rom 18% to 32% o the tests. The reason is that the 2006 CHBDC method uses the mid-depth longitudinal strain to predit shear strength, but Response 2000 results show that shear ailure initiates loser to lexural tension reinorement where the longitudinal strain is larger. The onservative simple equation given or the mid-depth longitudinal strain in the 2006 CHBDC ompensates or the unonservative assumption o using mid-depth strain to determine shear strength. Thereore, the use o more aurate ε x in the 2006 CHBDC shear design provision may result in unonservative shear strength preditions or members without transverse reinorement. It is reommended 253

275 that the 2006 CHBDC and the 2004 CSA A23.3 ode shear design provisions be modiied so as not to permit the use o a reined proedure to alulate ε x or members without transverse reinorement. Members with less than minimum transverse reinorement Although the 2006 CHBDC and 2008 AASHTO LRFD shear design provisions are similar proedures that are both based on MCFT, the 2006 CHBDC minimum transverse reinorement is 30% lower than the 2008 AASHTO LRFD minimum transverse reinorement. An investigation was made to determine whih minimum transverse reinorement should be used in the proposed evaluation method. This was done by omparing preditions rom the proposed method with the test results or 76 tested beams whih were lightly reinored or shear. The preditions o the propose method were still onsistent with test results or beams with transverse reinorement as low as the 2006 CHBDC minimum. Thus, the 2006 CHBDC lower minimum amount o transverse reinorement was adopted in the proposed evaluation method. In the ode shear design proedures, beams with less than minimum stirrups are oten assumed to have the same shear strength as members with no stirrups. In the setion on evaluation, the 2006 CHDBC assumes that members with less than one third o minimum transverse reinorement shall have the same shear strength as members with no stirrups. For higher amount o transverse reinorement, it assumes that shear strength o a setion inreases linearly rom the strength with no transverse reinorement to the strength with minimum transverse reinorement as transverse reinorement amount varies rom one third o minimum reinorement to the minimum amount. This proedure was examined 254

276 and ound to be onsistent with experimental results rom the 76 beams with light amount o stirrups Strength o Squat Shear Walls ACI 318 building ode and New Zealand onrete ode (NZS 3101) shear design provisions or squat shear walls are empirial proedures determined rom squat shear wall tests. Suh tests might not represent the lower-bound shear strength o atual squat shear walls in buildings beause the test speimens typially had very large load transer beams at the top o the walls. In this study, it was examined how the top load transer beams may have inluened the shear strength o suh walls. This was done by omparing the behaviour o three previously tested walls with the nonlinear inite element preditions. In one ase, the walls were analyzed with the top load transer beam as in the test, while in the other ase, the top beam was removed and the shear ore was uniormly distributed over the wall length at the top o the wall. Finite element results showed that the top load transer beams ould onsiderably enhane the shear apaity o suh walls where diagonal tension ailure mode is the governing ailure mode. In ontrast, it does not have a signiiant eet on the wall lexural apaity, and the wall shear apaity when diagonal onrete rushing is the governing shear ailure mode. The 2004 CSA A23.3 uses a single uniorm shear element to predit the shear behaviour o squat shear walls. As a result, the vertial distributed reinorement needed or shear should be provided in addition to the distributed vertial reinorement onsidered to resist lexure in the 2004 CSA A23.3 provisions. In other words, the 2004 CSA A23.3 provisions do not allow using vertial distributed reinorement needed or 255

277 shear to resist lexure at the wall base. This was investigated using nonlinear inite element method and was ound to be onservative espeially or walls with height-tolength ratios equal to and greater than 1.0. Finite element analyses were perormed on 42 dierent walls that were shear dominated but the apaity o the wall was limited by yielding o the vertial reinorement at the base. These walls did not have a top loading beam. The results showed all or part o vertial distributed reinorement is available to resist lexure depending on the wall aspet ratio. A truss model to explain why all or part o distributed vertial reinorement is not needed or shear was presented. A method to determine lexural strength o squat shear walls aounting or lexure shear interation at the wall base was proposed. The method aounts or the eet o wall height-to-length ratio and allows ull ontribution o vertial distributed shear reinorement in lexure or walls with height-to-length ratios o equal to or greater than 0.8. The proposed method was veriied against inite element preditions or the 42 shear dominated walls where the apaity was limited by yielding o vertial reinorement. The walls had height-to-length ratios o 2.0, 1.0, 0.5 and 0.3 and had varying amounts o distributed horizontal reinorement, distributed vertial reinorement, and onentrated vertial reinorement. The shear design provisions or squat shear walls o ACI 318, NZS 3101, 2004 CSA A23.3 were evaluated by omparing preditions with the inite element preditions o 44 walls ailing in shear, whih were monotonially loaded and subjeted to uniormly distributed horizontal load at top. The walls had height-to-length ratios o 2.0, 1.0, 0.5 and 0.3, did not ontain a top load transer beam, and had varying amounts o distributed 256

278 horizontal reinorement, distributed vertial reinorement, and onentrated vertial reinorement. The results showed that ACI shear strength preditions were unonservative espeially or walls with the same perentage o distributed horizontal and vertial reinorement. NZS 3101 preditions were also unonservative but loser to the inite element results ompared to ACI The 2004 CSA A23.3 preditions were always onservative and were inreasingly onservative or walls with lower height-to-length ratios. The 2004 CSA A23.3 preditions were in reasonably good agreement or walls with height-to-length ratios equal to and greater than 1.0. Finite element results indiated that the inrease in the shear strength o walls ompared to the single uniorm shear predition, whih is the basis or the 2004 CSA A23.3 equations or shear strength o squat shear walls, is due to the ontribution o the lexural ompression zone in shear. The lexural ompression zone in one diretion o loading will be the lexural tension zone in the reverse diretion o loading. Thus the lexural ompression zone will likely have previously existing horizontal raks that are losed by the vertial ompression. Under high shear stress, these raks may slip loally and thus the shear resisted by the ompression zone will be redued. This is a omplex phenomenon that is not modeled by the nonlinear inite element program that was used. In order to investigate the inluene o loal slip along previously existing raks in the ompression zone a simple model was used. The shear stress at any point was limited to 10% o the onrete ompressive strength. As CSA A23.3 limits the average shear stress aross the shear length o a squat wall to 15% o the onrete ompressive strength, the lower limit on loal shear stress is learly onservative. Three walls were analyzed and the results were ompared with the 257

279 inite element results or the same walls in whih shear stress at base was not limited as well as the preditions rom the 2004 CSA A23.3. Loalized sliding resulted in a signiiant redution in shear strength o the walls with low height-to-length ratios, while it did not inluene the shear strength o the wall with height-to-length ratio o 1.0. When loalized sliding was aounted or, the 2004 CSA A23.3 preditions were in better agreement with the inite element preditions but they were still onservative as the ontribution rom the ompression zone was still signiiant. A reinement to the 2004 CSA A23.3 shear strength method or squat shear walls was proposed. The proposed reinement aounts or the ontribution o lexural ompression zone in shear by assuming shear is resisted by 90% o the wall length while the urrent CSA A23.3 assumes shear is resisted by only 80% o the wall length Reommendations or Future Work Bridge Girders As part o this study, many test results were reviewed and it was observed that the available results are rom beams that are generally very similar. Thus, there is a need or additional tests to veriy new shear design methods and shear strength evaluation proedures or bridge girders. Many tested reinored onrete beams with stirrups had an amount o transverse reinorement lose to the minimum. Available test results or shear strength o reinored onrete beams that ontain more than two times the minimum transverse reinorement are very limited. In real bridges, on the other hand, transverse reinorement is oten more than twie the minimum amount. O 720 tests available or 258

280 reinored onrete beams in the literature only 26 had a depth equal to or greater than 300mm and had more than twie the minimum amount o stirrups. Available test results or prestressed onrete beams are rom beams in whih the predited mid-depth longitudinal strain is very small. O 88 prestressed onrete beams only 23 beams had a predited mid-depth longitudinal strain greater than The reason or this was that the ritial setion or shear was generally lose to the support where the moment-to-shear ratio was small and thus the longitudinal strain was small. In real bridge girders, ailure ould happen lose to mid-span at loations where the amount o transverse reinorement hanges. At these loations, the moment-to-shear ratio may be muh larger and thus the mid-depth longitudinal strain will be muh larger Squat Shear Walls Squat shear walls that were tested generally had a large loading beam that introdued the load on the wall top. As shown in this study, suh a load transer beam will strengthen the wall top and thus signiiantly inrease the shear strength o the walls. In real buildings, load is transerred to the shear walls by means o diaphragms. Diaphragms may not have the same strength and stiness as the loading beams used in the tests. The urrent study looked at the lower-bound strength when the load is assumed to be uniormly distributed over the wall top edge. A study is needed to investigate the inluene o dierent types o diaphragms on the load distribution over the wall top edge as well as the wall shear strength. This ould be done making use o nonlinear inite element analysis and squat shear wall tests. Conrete slabs, whih are assumed to be rigid diaphragms, are very sti and strong when they remain unraked. However, they might be raked and thus their stiness 259

281 ould be greatly redued beause they are lightly reinored. Craking o slabs need not always be due to the externally applied loads and ould also be due to other eets suh as shrinkage. As a result o raking, onrete slabs might not have the same stiness and strength as the large top loading beams used in the tests. Flexible diaphragms suh as steel dek diaphragms do not have suiient strength to transer the loads to the top o shear wall. The load is normally transerred to the wall by a steel angle that is onneted to the wall. Depending on the stiness o this angle and the spaing o the onnetions, the load distribution ould be very dierent rom the distribution in the top loading beam ommonly used in the wall tests. 260

282 Reerenes 1- AASHTO LRFD Bridge Design Speiiations (2007) inluding interim revision or 2008, Amerian Assoiation o State and Highway Transportation Oiials, Washington D.C. 2- ACI 318 (2005), Building Code Requirements or Strutural Conrete, Amerian Conrete Institute, Farmington Hills, MI. 3- ACI-ASCE Committee 326 (1962) Shear and Diagonal Tension, Proeeding o ACI Journal, Vol. 59, January, February and Marh, pp. 1-30, , Angelakos, D., Bentz, E. C., and Collins, M. P. (2001). Eet o Conrete Strength and Minimum stirrups on Shear Strength o Large Members, ACI Strutural Journal, Vol. 98, No.3, May, pp Antebi, J., Utku, S., and Hansen, R. J. (1960). The Response o Shear Walls to Dynami Loads, Department o Civil Engineering, Massahusetts Institute o Tehnology, Cambridge, MA. 6- Barda, F. (1972). Shear Strength o Low-Rise Walls with Boundary Elements, PhD thesis, Lehigh University, Bethlehem, 278pp. 7- Barda, F., Hanson, J. M., and Corley, W. G. (1977). Shear Strength o Low-Rise Walls with Boundary Elements, Reinored Conrete Strutures in Seismi Zones, SP-53, Amerian Conrete Institute, Farmington Hills, MI, pp Benjamin, J. R., and Williams, H. (1954). Investigation o Shear Walls, Part 6 Continued Experimental and Mathematial Studies o Reinored Conrete Walled Bents Under Stati Shear Loading, Department o Civil Engineering, Stanord University, 59pp. 261

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297 Appendix A: Exel Spreadsheets or the Proposed Evaluation Methods and the Reined CHBDC Method 276

298 Required Inormation 1- Loading inormation: Appendix B: Detailed Steps in Proposed Evaluation Proedures M = Fatored bending moment at the setion o interest, N 2- Material properties: = Fatored axial ore at the setion o interest. ' = Speiied ompressive strength o onrete, = Yield stress o reinoring steel, y pu = Ultimate strength o prestressing tendons, pr = Stress in prestressing tendons at maximum lexural resistane. φ φ φ s p = Conrete resistane ator = Nonprestressed reinoring steel resistane ator = Prestressed reinoring steel resistane ator 3- Conrete setion geometry: H d v = Height o beam = Depth o uniorm shear stress = jd (may be taken as 0.9d), d nv = Depth o uniorm ompression stress n v over diagonally raked web = (setion height) (tension lange depth) (unraked ompression hord depth). It is reommended to take d nv as distane rom top o tension lange to bottom o top dek or I-girders having omposite ation with top dek and distane rom top o tension lange to bottom o ompression lange or boxgirders. For retangular setions d nv should be taken equal to d v. = Width o web b w A t = Area o onrete surrounding lexural tension reinorement (tension lange area), equals to zero or retangular setions. d = Depth rom ompression ae to entroid o web prestressing tendon. pw 4- Reinorement: A s A A A s s sw v = Area o longitudinal reinorement in lexural tension hord, = Area o longitudinal reinorement in lexural ompression hord, = Total area o distributed longitudinal reinorement entered in web, = Area o transverse reinorement (spaed at s), = Spaing o transverse reinorement, 277

299 Appendix B: Detailed Steps in Proposed Evaluation Proedures (ont.) S xe = Eetive rak spaing parameter as in 2006 CHBDC and AASHTO LRFD. 5- Prestressing tendons: A p = Area o prestressed reinorement in lexural tension hord, A pw = Area o prestressed reinorement in web, θ p p = Angle o inlination o draped prestressed reinorement, = Eetive stress o prestressing. Evaluation Proedure or Members With at Least Minimum Stirrups 1- Calulate additional parameters. Calulate jd = d v orm jd = d v = min( 0.9d = 0.72H ) Calulate ρ z rom ρ z = A v b w s y Calulate V p rom V p = φ p p A pw sinθ p Calulate λ rom λ = d pw d Set α = in MPa units (2.0 in psi units) 2- Calulate shear strength at yielding o transverse reinorement V yield. Calulate β rom β 0.18( 300ε + 1.6) 0.18 in MPa units = y 278

300 Appendix B: Detailed Steps in Proposed Evaluation Proedures (ont.) Calulate θ 0 and θ ρ z (85 ' θ rom y 0 = )( 50ε y + 1.1) θ = 1000[37.5( 200ε y + 1.4) θo] Calulate nv and n v0 rom n = 0.09φ β ' 0.20φ ρ ) θ n v vo Calulate ε x rom ( s z y nv = θ o + 4.0φ sβ ' + 9.4φ s ρ z θ M / jd + 0.5nvobwd nv φα ' At p ε x = 2 2 [ E ( A A ) + E ( A + λ A s s Calulate ore in the ompression hord C rom C = M / jd + 0.5n (1 λ ) p A pw vo + 0.5N sw b w d nv p x p v y ( A + λa ) pw + ε [0.5 n b w p pw )] 0.5 n b d nv 0.5E + 0.5N v s A w d sw nv 2λ(1 λ) E p A pw ] Multiply ε x by 2 i there is tension in omp. hord (i C is positive). Calulate angle θ rom θ = θ + θ o ε x Calulate transverse reinorement yielding shear strength rom Av yd v otθ V yield = V + Vs + V p = φβ ' bwd v + φs + V s 3- Calulate shear strength at rushing o onrete V rush. Calulate β rom ρ z y β = in MPa units ' p Appendix B: Detailed Steps in Proposed Evaluation Proedures (ont.) 279

301 Calulate θ 0 and θ rom ρ z y θ o = ' ρz y θ = ' Calulate and n 0 rom nv v For θ > 0 23 : n = 0.09φ β ' 0.20φ ρ ) θ n vo v ( s z y nv = θ o + 4.0φ sβ ' + 9.4φ s ρ z θ For θ 0 23 : n = 0.15φ β ' 0.77φ ρ ) θ n vo Calulate ε x rom v ( s z y nv = θ o + 5.5φ β ' φ s ρ z θ M / jd + 0.5nvobwd nv αφ ' At p ε x = 2 2 [ E ( A A ) + E ( A + λ A s Calulate ore in the ompression hord C rom C = M / jd + 0.5n (1 λ ) p A pw s vo b + 0.5N w d sw nv x p p + ε [0.5 n b v y y w d nv ( A + λa ) pw p pw )] 0.5 n b 0.5E s A sw + 0.5N v w d nv 2λ(1 λ) E p A pw ] Multiply ε x by 2 i there is tension in omp. hord (i C is positive). Calulate angle θ rom θ = θ + θ o ε x Calulate shear strength rom Av yd v otθ V = V + Vs + V p = φβ ' bwd v + φs + V s p 280

302 Appendix B: Detailed Steps in Proposed Evaluation Proedure (ont.) 4- Calulate shear strength at biaxial yielding o reinorement V biaxial. Calulate jd orm jd φ p = d pr A p + φ A N s y 1.2φ ' b s Calulate longitudinal reinorement apaity reserved in the omp. hord N v rom N v = 2 [ φ s y( As + 0.5Asw) + pr (1 λ ) Apw) + M / jd 0.5N ] Calulate longitudinal reinorement apaity reserved in the tension hord N vt rom N vt = 2 [ φ s y ( As + 0.5Asw) + pr ( Ap + λ Apw) M / jd 0.5N ] Determine axial ore reserved apaity N v rom N = min( N v v, N vt Calulate shear strength rom ) V = φ ρ ( b d ) N + V s z y w v v p 5- Determine governing shear strength. V yield is governing ailure mode i greater than V rush, and less than V biaxial. V rush is governing ailure mode i greater than V yield and less than V biaxial. V biaxial is governing ailure mode i less than Max (V yield, V rush ). V max = 0.25φ ' bwd v + V p 281

303 Appendix B: Detailed Steps in Proposed Evaluation Proedures (ont.) Evaluation Proedure or Members Without Stirrups 1- Calulate additional parameters. Calulate jd = d v orm jd = d v = min( 0.9d = 0.72H ) Calulate V p rom V p = φ p p A pw sinθ p Calulate λ rom λ = d pw d 2- Calulate shear strength at maximum onrete ontribution V. Calulate ε x rom ε x = 1.5 Calulate β rom [ E s M / jd ( A s p A ( A + λ A ) sw p ) + E p ( A pw p 2 + λ A β = in MPa units 1+ ( S ) ε ( S ) Calulate shear strength rom xe x xe pw )] V = V + V = φ β ' b d + V p w v p 3- Chek or yielding o long. reinorement and determine shear strength. Calulate θ rom θ = ε )( S ) in MPa units ( x xe Calulate demand on the longitudinal reinorement due to shear 282

304 Appendix B: Detailed Steps in Proposed Evaluation Proedures (ont.) N = 2V otθ v* Calulate longitudinal reinorement apaity reserved in the omp. hord N v rom N v = 2 [ φ s y( As + 0.5Asw) + pr (1 λ ) Apw) + M / jd 0.5N ] Calulate longitudinal reinorement apaity reserved in the tension hord N vt rom N vt = 2 [ φ s y ( As + 0.5Asw) + pr ( Ap + λ Apw) M / jd 0.5N ] Determine axial ore reserved apaity N v rom N = min( N v v, N vt ) I N v N v *, shear strength is equal to V, otherwise determine shear strength rom V = N v 2 283

305 Appendix B: Detailed Steps in Proposed Evaluation Proedures (Summary Sheet or Proposed Method or Members With at Least Min Stirrups) YIELDING OF TRANSVERSE REINFORCEMENT AND CONCRETE CRUSHING Yielding o stirrups: Conrete rushing ater stirrup yielding: ρ z y β = 0.18( 300ε y + 1.6) 0.18 (MPa units) β = (MPa units) ' ρ z y θ o = ( )(1.1 50ε y ) ' θ = 1000[37.5( ε y ) θ o ] ρ z y θ o = ' ρz y θ = ' Yielding o stirrups, and onrete rushing i Conrete rushing i θ 23 deg: θ > 23 deg: 0 0 n = 0.09β ' 0.20ρ ) θ n vo v ( z y nv = θ o + 4.0β ' + 9.4ρ z θ ε x = 2 M / d s v [ E ( A s y + 0.5n vo A b sw w d nv ) + E 2 p ( A p n = 0.15β ' 0.77ρ ) θ n vo v ' A ( z y nv = θ o + 5.5β ' ρ z θ t 2 + λ A pw p ( A + λa ) p v pw )] 0.5 n b w d nv y C v vo w nv Multiply ε x by 2.0 i C > 0 where = M / d + 0.5n b d + ε [0.5 n b d 0.5E A 2λ(1 λ) E A ] (1 λ ) x v w nv s sw p pw p A pw n s p θ = θ + θ o ε x V = V + V + V = β ' b d + ( A d / s) otθ + V w Governing V n is the greater o stirrups yielding and onrete rushing shear strength v v y v p CHECKS Longitudinal reinorement yielding PARAMETERS (TOP CHORD IN COMP.) N V ρ ( b d ) N + V n z y where N = min( N, N ) v v vt v = [ y ( As + 0.5Asw) + pr (1 λ ) Apw) + w v 2 M / jd] v p N vt = 2[ ( A + 0.5A ) + ( A + λ A ) M / jd] y s sw pr p pw where jd = d pr A p + y 1.2 ' b Conrete rushing beore stirrups yield V n 0.25 ' b d w v A s d v = max( 0.9d, 0.72H ) λ =d pw /d ρ z = A v/( b w s) 284

306 Appendix C: Detailed Examples o Proposed Evaluation Proedure or Members With Stirrups Example 1: Prestressed Conrete I-girder Bridge at 7.91 m rom the support. M V = 3720 knm = 415 kn N = 0 y = 400 MPa ' = 40 MPa pu = 1860 MPa θ = 0 p 285

307 Appendix C: Detailed Examples o Proposed Evaluation Proedure (ont.) Solution o Example 1: 286

308 Appendix C: Detailed Examples o Proposed Evaluation Proedure (ont.) Solution o Example 1 (ont.): 287

309 Appendix C: Detailed Examples o Proposed Evaluation Proedure (ont.) Example 2: Prestressed Conrete Box-girder Bridge at 8.94 m rom the support. M V = knm = kn N = 0 y = 400 MPa ' = 40 MPa pu = 1860 MPa θ = 1.3 deg. (20 web tendons are draped) p 288

310 Appendix C: Detailed Examples o Proposed Evaluation Proedure (ont.) Solution o Example 2: 289

311 Appendix C: Detailed Examples o Proposed Evaluation Proedure (ont.) Solution o Example 2 (ont.): 290

312 Appendix C: Detailed Examples o Proposed Evaluation Proedure (ont.) Example 3: Reinored Conrete hannel-girder Bridge at 1.39 m rom the support. M V = knm = kn N = 0 y = 400 MPa ' = 40 MPa Average assumed b w = 381 mm 291

313 Appendix C: Detailed Examples o Proposed Evaluation Proedure (ont.) Solution o Example 3: 292

314 Appendix C: Detailed Examples o Proposed Evaluation Proedure (ont.) Solution o Example 3 (ont.): 293