Two-Way Concrete Floor Slab with Beams Design and Detailing

Transcription

1 Two-Way Conrete Floor Slab with Beams Design and Detailing Version: Apr

2 Two-Way Conrete Floor Slab with Beams Design and Detailing Design the slab system shown in Figure 1 for an intermediate floor where the story height = 1 ft, olumn rosssetional dimensions = 18 in. x 18 in., edge beam dimensions = 14 in. x 7 in., interior beam dimensions = 14 in. x 0 in., and unfatored live load = 100 psf. The lateral loads are resisted by shear walls. Normal weight onrete with ultimate strength (f = 4000 psi) is used for all members, respetively. And reinforement with Fy = 60,000 psi is used. Use the Equivalent Frame Method (EFM) and ompare the results with spslab model results. Figure 1 Two-Way Slab with Beams Spanning between all Supports Version: Apr

3 Contents 1. Preliminary Slab Thikness Sizing Two-Way Slab Analysis and Design Using Equivalent Frame Method (EFM) Equivalent frame method limitations Frame members of equivalent frame Equivalent frame analysis Design moments Distribution of design moments Flexural reinforement requirements Column design moments Design of Interior, Edge, and Corner Columns Two-Way Slab Shear Strength One-Way (Beam ation) Shear Strength Two-Way (Punhing) Shear Strength Two-Way Slab Defletion Control (Servieability Requirements) Immediate (Instantaneous) Defletions Time-Dependent (Long-Term) Defletions (Δ lt) spslab Software Program Model Solution Summary and Comparison of Design Results Conlusions & Observations Version: Apr

4 Code Building Code Requirements for Strutural Conrete (ACI ) and Commentary (ACI 318R-14) Minimum Design Loads for Buildings and Other Strutures (ASCE/SEI 7-10) International Code Counil, 01 International Building Code, Washington, D.C., 01 Referenes Notes on ACI Building Code Requirements for Strutural Conrete, Twelfth Edition, 013 Portland Cement Assoiation. Conrete Floor Systems (Guide to Estimating and Eonomizing), Seond Edition, 00 David A. Fanella Simplified Design of Reinfored Conrete Buildings, Fourth Edition, 011 Mahmoud E. Kamara and Lawrene C. Novak Design Data Floor-to-Floor Height = 1 ft (provided by arhitetural drawings) Columns = 18 x 18 in. Interior beams = 14 x 0 in. Edge beams = 14 x 7 in. w = 150 pf f = 4,000 psi f y = 60,000 psi Live load, L o = 100 psf (Offie building) ASCE/SEI 7-10 (Table 4-1) Solution 1. Preliminary Slab Thikness Sizing Control of defletions. ACI (8.3.1.) In lieu of detailed alulation for defletions, ACI 318 Code gives minimum thikness for two-way slab with beams spanning between supports on all sides in Table Beam-to-slab flexural stiffness (relative stiffness) ratio (α f) is omputed as follows: E I I b b b f Es I s I ACI ( b) s The moment of inertia for the effetive beam and slab setions an be alulated as follows: lh ba Is and Ib f Then, 1

5 f b a l h 3 f For Edge Beams: The effetive beam and slab setions for the omputation of stiffness ratio for edge beam is shown in Figure. For North-South Edge Beam: l in. a h 6 b h 6 f 1.47 using Figure 3. f Figure Effetive Beam and Slab Setions (Edge Beam) For East-West Edge Beam: l in. a h 6 b h 6 f 1.47 using Figure 3. f Figure 3 Beam Stiffness (Edge Beam)

6 For interior Beams: The effetive beam and slab setions for the omputation of stiffness ratio for interior beam is shown in Figure 4. For North-South Interior Beam: l 1 64 in. a h 6 b h 6 f 1.61 using Figure 5. f For East-West Interior Beam: l in. Figure 4 Effetive Beam and Slab Setions (Interior Beam) a h 6 b h 6 f 1.61 using Figure 5. f Figure 5 Beam Stiffness (Interior Beam) Sine α f >.0 for all beams, the minimum slab thikness is given by: h min f y ln ,000 greater of ACI (8.3.1.) Where: l lear span in the long diretion measured fae to fae of olumns 0.5 ft 46 in. n lear span in the long diretion 18 /1 1.8 lear span in the short diretion / 1 3

7 h min 60, ,000 greater of in Use 6 in. slab thikness.. Two-Way Slab Analysis and Design Using Equivalent Frame Method (EFM) ACI 318 states that a slab system shall be designed by any proedure satisfying equilibrium and geometri ompatibility, provided that strength and servieability riteria are satisfied. Distintion of two-systems from oneway systems is given by ACI (R & R8.3.1.). ACI 318 permits the use of Diret Design Method (DDM) and Equivalent Frame Method (EFM) for the gravity load analysis of orthogonal frames and is appliable to flat plates, flat slabs, and slabs with beams. The following setions outline the solution per EFM and spslab software. The solution per DDM an be found in the Two-Way Plate Conrete Floor System Design example. EFM is the most omprehensive and detailed proedure provided by the ACI 318 for the analysis and design of two-way slab systems where the struture is modeled by a series of equivalent frames (interior and exterior) on olumn lines taken longitudinally and transversely through the building. The equivalent frame onsists of three parts: 1) Horizontal slab-beam strip, inluding any beams spanning in the diretion of the frame. Different values of moment of inertia along the axis of slab-beams should be taken into aount where the gross moment of inertia at any ross setion outside of joints or olumn apitals shall be taken, and the moment of inertia of the slab-beam at the fae of the olumn, braket or apital divide by the quantity (1- /l ) shall be assumed for the alulation of the moment of inertia of slab-beams from the enter of the olumn to the fae of the olumn, braket or apital. ACI (8.11.3) ) Columns or other vertial supporting members, extending above and below the slab. Different values of moment of inertia along the axis of olumns should be taken into aount where the moment of inertia of olumns from top and bottom of the slab-beam at a joint shall be assumed to be infinite, and the gross ross setion of the onrete is permitted to be used to determine the moment of inertia of olumns at any ross setion outside of joints or olumn apitals. ACI (8.11.4) 3) Elements of the struture (Torsional members) that provide moment transfer between the horizontal and vertial members. These elements shall be assumed to have a onstant ross setion throughout their length onsisting of the greatest of the following: (1) portion of slab having a width equal to that of the olumn, braket, or apital in the diretion of the span for whih moments are being determined, () portion of slab speified in (1) plus that part of the transverse beam above and below the slab for monolithi or fully omposite onstrution, (3) the transverse beam inludes that portion of slab on eah side of the beam extending a distane equal to the projetion of the beam above or below the slab, whihever is greater, but not greater than four times the slab thikness. ACI (8.11.5) 4

8 .1. Equivalent frame method limitations In EFM, live load shall be arranged in aordane with whih requires slab systems to be analyzed and designed for the most demanding set of fores established by investigating the effets of live load plaed in various ritial patterns. ACI ( & 6.4.3) Complete analysis must inlude representative interior and exterior equivalent frames in both the longitudinal and transverse diretions of the floor. ACI ( ) Panels shall be retangular, with a ratio of longer to shorter panel dimensions, measured enter-to-enter of supports, not to exeed. ACI ( ).. Frame members of equivalent frame Determine moment distribution fators and fixed-end moments for the equivalent frame members. The moment distribution proedure will be used to analyze the equivalent frame. Stiffness fators k, arry over fators COF, and fixed-end moment fators FEM for the slab-beams and olumn members are determined using the design aids tables at Appendix 0A of PCA Notes on ACI These alulations are shown below. a. Flexural stiffness of slab-beams at both ends, K sb. N1 18 N , (17.5 1) (1) 1 For stiffness fators, k k 4.11 PCA Notes on ACI (Table A1) F1 F NF FN E Isb E Isb Thus, K k 4.11 PCA Notes on ACI (Table A1) sb NF 1 1 Where I sb is the moment of inertia of slab-beam setion shown in Figure 6 and an be omputed with the aid of Figure 7 as follows: 3 3 bh w 14 0 Isb C t in Figure 6 Cross-Setion of Slab-Beam K sb E 5, E Carry-over fator COF = PCA Notes on ACI (Table A1) Fixed-end moment FEM 0.084w u 1 PCA Notes on ACI (Table A1) 5

9 Figure 7 Coeffiient C t for Gross Moment of Inertia of Flanged Setions b. Flexural stiffness of olumn members at both ends, K. Referring to Table A7, Appendix 0A: For Interior Columns: t a 0 6 / 17 in., t 3 in. b ta H H 1 ft 144 in., H in., 5.67, 1.16 t H Thus, k 6.8 and k 4.99 by interpolation., top, bottom 4 4 (18) I 8,748 in ft 144 in. k E I 4 b K PCA Notes on ACI (Table A7) K K, top, bottom E 414E E 303E 144 For Exterior Columns: t a 7 6 / 4 in., t 3 in. b ta H H 1 ft 144 in., H in., 8.0, 1.3 t H b 6

10 Thus, k 8.57 and k 5.31 by interpolation., top, bottom 4 4 (18) I 8,748 in ft 144 in. k E I K PCA Notes on ACI (Table A7) K K, top, bottom E 51E E 33E

11 . Torsional stiffness of torsional members, K t. K t 9EsC 3 [ (1 ) ] For Interior Columns: 9E 11, 698 Kt 493E 3 64(0.93) ACI (R ) Where: x x y C ( )( ) ACI (Eq b) y 3 x 1 = 14 in x = 6 in x 1 = 14 in x = 6 in y 1 = 14 in y = 4 in y 1 = 0 in y = 14 in C 1 = 4738 C =,75 C 1 = 10,6 C = 736 C = ,75 = 7,490 in 4 C = 10, x = 11,698 in 4 Figure 8 Attahed Torsional Member at Interior Column 8

12 For Exterior Columns: K t 9E 17,868 75E 3 64(0.93) Where: x x y C ( )( ) ACI (Eq b) y 3 x 1 = 14 in x = 6 in x 1 = 14 in x = 6 in y 1 = 1 in y = 35 in y 1 = 7 in y = 1 in C 1 = 11,141 C =,48 C 1 = 16,68 C = 1,40 C = 11,141 +,48 = 13,389 in 4 C = 16,68 + 1,40 = 17,868 in 4 Figure 9 Attahed Torsional Member at Exterior Column 9

13 d. Inreased torsional stiffness due to parallel beams, K ta. For Interior Columns: K ta Where: I sb Kt Isb 493E 5, E I 475 s 3 3 l h in. 1 1 For Exterior Columns: 4 K ta Kt Isb 75E 5, E I 475 s Figure 10 Slab-Beam in the Diretion of Analysis e. Equivalent olumn stiffness K e. K e K K K ta K ta Where K ta is for two torsional members one on eah side of the olumn, and K is for the upper and lower olumns at the slabbeam joint of an intermediate floor. For Interior Columns: K e (303E 414 E )(634 E ) 631E (303E 414 E ) ( 634 E ) For Exterior Columns: Figure 11 Equivalent Column Stiffness K e (33E 51 E )(4017 E ) 764E (33E 51 E ) ( 4017 E ) f. Slab-beam joint distribution fators, DF. At exterior joint, 497E DF (497E 764 E ) 10

14 At interior joint, 497E DF (497E 497E 631 E ) COF for slab-beam =0.507 Figure 1 Slab and Column Stiffness.3. Equivalent frame analysis Determine negative and positive moments for the slab-beams using the moment distribution method. With an unfatored live-to-dead load ratio: L D (150 6 /1) 4 The frame will be analyzed for five loading onditions with pattern loading and partial live load as allowed by ACI ( ). a. Fatored load and Fixed-End Moments (FEM s). Fatored dead load qdu 1.(75 9.3) 101 psf Where (9.3 psf = (14 x 14) / 144 x 150 / is the weight of beam stem per foot divided by l ) Fatored live load qlu 1.6(100) 160 psf Fatored load q q q 61 psf u Du Lu FEM's for slab-beam mnf qu 1 PCA Notes on ACI (Table A1) FEM due to qdu qlu (0.61 ) ft-kip 3 4 FEM due to qdu qlu (0.1 ) ft-kip FEM due to qdu (0.101 ) ft-kip b. Moment distribution. Moment distribution for the five loading onditions is shown in Table 1. Counter-lokwise rotational moments ating on member ends are taken as positive. Positive span moments are determined from the following equation: M u( midspan) ( MuL MuR ) Mo Where M o is the moment at the midspan for a simple beam. 11

15 When the end moments are not equal, the maximum moment in the span does not our at the midspan, but its value is lose to that midspan for this example. Positive moment in span 1- for loading (1): M u 17.5 ( ) (0.61 ) 89.4 ft-kip 8 Positive moment span -3 for loading (1): M u 17.5 ( ) (0.61 ) 66. ft-kip 8 Table 1 Moment Distribution for Partial Frame (Transverse Diretion) Joint Member DF COF Loading (1) All spans loaded with full fatored live load FEM Dist CO Dist CO Dist CO Dist CO Dist M Midspan M

16 Loading () First and third spans loaded with 3/4 fatored live load FEM Dist CO Dist CO Dist CO Dist CO Dist M Midspan M Loading (3) Center span loaded with 3/4 fatored live load FEM Dist CO Dist CO Dist CO Dist CO Dist M Midspan M Loading (4) First span loaded with 3/4 fatored live load and beam-slab assumed fixed at support two spans away FEM Dist CO Dist CO Dist CO Dist CO Dist M

17 Midspan M Loading (5) First and seond spans loaded with 3/4 fatored live load FEM Dist CO Dist CO Dist CO Dist CO Dist M Midspan M Max M Max M Design moments Positive and negative fatored moments for the slab system in the diretion of analysis are plotted in Figure 13. The negative design moments are taken at the faes of retilinear supports but not at distanes greater than from the enters of supports. ACI ( ) 18in ft < =3.1 ft (use fae of support loation) 14

18 Figure 13 Positive and Negative Design Moments for Slab-Beam (All Spans Loaded with Full Fatored Live Load.5. Distribution of design moments Exept as Noted) a. Chek whether the moments alulated above an take advantage of the redution permitted by ACI ( ): Slab systems within the limitations of ACI (8.10.) may have the resulting redued in suh proportion that the numerial sum of the positive and average negative moments not be greater than the total stati moment M o given by Equation in the ACI : ACI ( ) Chek Appliability of Diret Design Method: 1. There is a minimum of three ontinuous spans in eah diretion ACI ( ). Suessive span lengths are equal ACI (8.10..) 3. Long-to-Short ratio is /17.5 = 1.6 <.0 ACI ( ) 4. Column are not offset ACI ( ) 5. Loads are gravity and uniformly distributed with servie live-to-dead ratio of 1.33 <.0 ACI ( and 6) 15

19 6. Chek relative stiffness for slab panel: ACI ( ) Interior Panel: 3.16, l 1 64 in. f , l in. f 1 l f l f 1 Interior Panel: 3.16, l 1 64 in. f , l in. f 1 l f l f 1 O.K. O.K. ACI (Eq a) ACI (Eq a) All limitation of ACI (8.10.) are satisfied and the provisions of ACI ( ) may be applied: qu n (16) M o ft-kip ACI (Eq ) 8 8 ( ) End spans: ft-kip ( ) Interior span: ft-kip To illustrate proper proedure, the interior span fatored moments may be redued as follows: Permissible redution = 183.7/188.8 = Adjusted negative design moment = = ft-kip Adjusted positive design moment = = 69.3 ft-kip M o = ft-kip b. Distribute fatored moments to olumn and middle strips: The negative and positive fatored moments at ritial setions may be distributed to the olumn strip and the two half-middle strips of the slab-beam aording to the Diret Design Method (DDM) in 8.10, provided that Eq (a) is satisfied. ACI ( ) Sine the relative stiffness of beams are between 0. and 5.0 (see step.4.1.6), the moments an be distributed aross slab-beams as speified in ACI ( and 6) where:

20 f C 17,868 t 1.88 I 4, 75 s Where I s 4, 75 in. 1 4 C 17,868 in. (see Figure 9) Fatored moments at ritial setions are summarized in Table. End Span Interior Span Exterior Negative Table - Lateral distribution of fatored moments Fatored Moments (ft-kips) Perent* Column Strip Beam Moment Strip (ft-kips) Moment (ft-kips) Column Strip Moment (ft-kips) Moments in Two Half-Middle Strips** (ft-kips) Positive Interior Negative Negative Positive *Sine α 1l /l 1 > 1.0 beams must be proportioned to resist 85 perent of olumn strip per ACI ( ) **That portion of the fatored moment not resisted by the olumn strip is assigned to the two half-middle strips.6. Flexural reinforement requirements a. Determine flexural reinforement required for strip moments The flexural reinforement alulation for the olumn strip of end span interior negative loation is provided below: M 1.9 ft-kip u Assume tension-ontrolled setion (φ = 0.9) Column strip width, b = (17.5 x 1) / = 91 in. Use average d = / = 5 in f ' b M u As d d f 0.85 ' y f b A s , in. 60,

21 bh (14)(19) A s,min max max max in. < in bh (14)(19) in A s in. Maximum spaing s h 6 1 in. < 18 in. ACI (8.7..) max Provide 8 - #4 bars with A s = 1.60 in. and s = 91/8 = in. s max The flexural reinforement alulation for the beam strip of end span interior negative loation is provided below: M 73.1 ft-kip u Assume tension-ontrolled setion (φ = 0.9) Beam strip width, b = 14 in. Use average d = / = 19 in f ' b M u As d d f 0.85 ' y f b A s , in. 60, ' f bd (14)(19) f 60, A y s,min max max max in. < in bd (14)(19) f 60, 000 y in. A s Provide 5 - #4 bars with A s = 1.00 in. All the values on Table 3 are alulated based on the proedure outlined above. 18

22 Table 3 - Required Slab Reinforement for Flexure [Equivalent Frame Method (EFM)] Span Loation Mu (ft-kip) b * (in.) d ** (in.) As Req d for flexure (in. ) Min As (in. ) Reinforement Provided As Prov. for flexure (in. ) Beam Strip Column Strip Middle Strip End Span Exterior Negative #4 0.8 Positive #4 1.0 Interior Negative Exterior Negative # #4 1.6 Positive #4 1.6 Interior Negative #4 1.6 Exterior Negative #4.8 Positive #4.8 Interior Negative #4.8 Interior Span Beam Strip Positive #4 0.8 Column Strip Positive #4 1.6 Middle Strip Positive #4.8 * Column strip width, b = (17.5 1)/ - 14 = 91 in. * Middle strip width, b = *1-(17.5*1)/ = 159 in. * Beam strip width, b = 14 in. ** Use average d = / = 5.00 in. for Column and Middle strips ** Use average d = / = 18.5 in. for Beam strip Positive moment regions ** Use average d = / = 19 in. for Beam strip Negative moment regions Min. As = b h = b for Column and Middle strips ACI ( ) Min. As = min (3(f')^0.5/fy*b*d, 00/fy*b*d) for Beam strip ACI (9.6.1.) Min. As = As Req'd if As provided >= As Req'd for Beam strip ACI ( ) smax = h = 1 in. < 18 in. ACI (8.7..) b. Calulate additional slab reinforement at olumns for moment transfer between slab and olumn by flexure Portion of the unbalaned moment transferred by flexure is γ f x M u Where: 1 f ACI ( ) 1 ( / 3) b / b 1 b 1 = Dimension of the ritial setion b o measured in the diretion of the span for whih moments are determined in ACI 318, Chapter 8. 19

23 b = Dimension of the ritial setion b o measured in the diretion perpendiular to b 1 in ACI 318, Chapter 8. b o = Perimeter of ritial setion for two-way shear in slabs and footings. b Effetive slab width 3 h ACI ( ) b For Exterior Column: d 5 b in., b d in., bb 3 h 18 3 (6) 36 in. 1 f ( / 3) 0.5 / 3 f M, ft-kip u net Figure 14 Critial Shear Perimeters for Columns 0.85 f ' b b f M u, net As, req ' d f d d 0.85 ' y f b b A s, req ' d , in. 60, bh (14)(19) As,min max max max in. >.973 in bh (14)(19) A s, req ' d.973 in. A ( A ) ( A ) A s, provided s, provided ( beam) s, provided ( b b b beam ) in. < A.973 in. 91 s, provided s, req ' d Additional slab reinforement at the exterior olumn is required. Areq ' d, add in. 0

24 Use 10 - #4 A in. < A in. provided, add req ' d, add Table 4 - Additional Slab Reinforement at olumns for moment transfer between slab and olumn [Equivalent Frame Method (EFM)] Span Loation Effetive slab width, b b (in.) d (in.) γ f M u * (ft-kip) γ f M u (ft-kip) A s req d within b b (in. ) A s prov. for flexure within b b (in. ) Add l Reinf. End Span Column Strip Exterior Negative #4 Interior Negative *M u is taken at the enterline of the support in Equivalent Frame Method solution. b. Determine transverse reinforement required for beam strip shear The transverse reinforement alulation for the beam strip of end span exterior loation is provided below. Figure 15 Shear at ritial setions for the end span (at distane d from the fae of the olumn) dstirrup 0.5 d h lear in. (using #4 stirrups) The required shear at a distane d from the fae of the supporting olumn V u_d= kips (Figure 15). V f ' b d ACI (.5.5.1) v v V kips V kip Stirrups are required. v u _ d Distane from the olumn fae beyond whih minimum reinforement is required: Vu _ d vv Vs ACI ( ) v Vs 9.85 kip Vs, max kip O.K

25 V 8 f ' bd kip ACI ( ) s, max Av, req ' d Vs in. /in. s f d 60, yt ACI ( ) A v,min s 0.75 ' f b max f yt 50 b f yt ACI ( ) (14) max max in. /in (14) 60, 000 A v,min 60, 000 s Av, req ' d Av,min As, req ' d A use s s s s s req ' d n Astirrup in. Av, req ' d s v,min V 9.85 kips 4 f ' bd kips s d smax Lesser of Lesser of Lesser of 9.13 in Sine s s use s req ' d max max ACI ( ) Selet s provided = 8 in. #4 stirrups with first stirrup loated at distane 3 in. from the olumn fae. The distane where the shear is zero is alulated as follows: l 17.5 x VuL, ft 93.4 in. V V u, L u, R The distane from support beyond whih minimum reinforement is required is alulated as follows: x 7.78 x 1 x vv ft 43 in. V u The distane at whih no shear reinforement is required is alulated as follows: x vv x x ft 68 in. V u s 1 provided 18 8 x # of stirrups use 8 stirrups s 8 provided All the values on Table 5 are alulated based on the proedure outlined above.

26 Span Loation Table 5 - Required Beam Reinforement for Shear Av,min/s in /in Av,req'd/s in /in End Span sreq'd in smax in Reinforement Provided Exterior in * Interior in Interior Span Interior in * Minimum transverse reinforement governs 3

27 .7. Column design moments The unbalaned moment from the slab-beams at the supports of the equivalent frame are distributed to the atual olumns above and below the slab-beam in proportion to the relative stiffness of the atual olumns. Referring to Fig. 9, the unbalaned moment at joints 1 and are: Joint 1 = ft-kip Joint = = ft-kip The stiffness and arry-over fators of the atual olumns and the distribution of the unbalaned moments to the exterior and interior olumns are shown in Fig 9. Figure 16 - Column Moments (Unbalaned Moments from Slab-Beam) 4

28 In summary: Design moment in exterior olumn = ft-kip Design moment in interior olumn = 4.91 ft-kip The moments determined above are ombined with the fatored axial loads (for eah story) and fatored moments in the transverse diretion for design of olumn setions. A detailed analysis to obtain the moment values at the fae of interior, exterior, and orner olumns from the unbalaned moment values an be found in the Two-Way Flat Plate Conrete Floor Slab Design example. 3. Design of Interior, Edge, and Corner Columns The design of interior, edge, and orner olumns is explained in the Two-Way Flat Plate Conrete Floor Slab Design example. 4. Two-Way Slab Shear Strength Shear strength of the slab in the viinity of olumns/supports inludes an evaluation of one-way shear (beam ation) and two-way shear (punhing) in aordane with ACI 318 Chapter One-Way (Beam ation) Shear Strength One-way shear is ritial at a distane d from the fae of the olumn. Figure 17 shows the V u at the ritial setions around eah olumn. Sine there is no shear reinforement, the design shear apaity of the setion equals to the design shear apaity of the onrete: V V V V ACI (Eq ) n s Where: V f ' b d ACI (Eq ) w λ = 1 for normal weight onrete 5 V (1 14) kips 1000 Beause φv > V u at all the ritial setions, the slab is o.k. in one-way shear. 5

29 Figure 17 One-way shear at ritial setions (at distane d from the fae of the supporting olumn) 4.. Two-Way (Punhing) Shear Strength Two-way shear is ritial on a retangular setion loated at d slab/ away from the fae of the olumn. The fatored shear fore V u in the ritial setion is alulated as the reation at the entroid of the ritial setion minus the self-weight and any superimposed surfae dead and live load ating within the ritial setion. The fatored unbalaned moment used for shear transfer, M unb, is alulated as the sum of the joint moments to the left and right. Moment of the vertial reation with respet to the entroid of the ritial setion is also taken into aount. For the exterior olumn: Vu kips / Munb ft-kip 1 For the exterior olumn in Figure 18, the loation of the entroidal axis z-z is: AB AB moment of area of the sides about AB area of the sides (14 6 ( / ) (6.5 / )) 9.09 in. ( ) A ( ) in. The polar moment J of the shear perimeter is: Figure 18 Critial setion of exterior support of interior frame 6

30 J 3 3 bbeam, Ext dbeam, Ext dbeam, Extbbeam, Ext bbeam, Ext bbeam, Ext dbeam, Ext b1 bbeam, Ext AB b b d d b b b b beam, Ext slab, Ext slab 1 beam, Ext 1 beam, Ext b1 bbeam, Ext dslab AB b d b b d beam, Int beam. Int beam, Int slab AB J J v ,338 in. f 4 γ 1 γ ACI (Eq ) The length of the ritial perimeter for the exterior olumn: bo (18 5/ ) (18 5) 64 in. v v V γm u v unb AB u A J ACI (R ) u psi , 338 ' 4 v ' ' min 4 f, f, s d f b o ACI (Table.6.5.) v min , , v min 53, 379.5, 74.7 psi = 53 psi v psi v 76.8 psi O.K. u 7

31 For the interior olumn: 3 3 Vu kips 144 Munb (0) ft-kip For the interior olumn in Figure 19, the loation of the entroidal axis z-z is: b 3 1, Int AB 11.5 in. A 4 ( ) 144 in. The polar moment J of the shear perimeter is: Figure 19 Critial setion of interior support of interior frame J bbeam, Int dbeam, Int dbeam, Intbbeam, Int bbeam, Int b1 bbeam, Int bbeam, Int dbeam, Int AB 3 b1 bbeam, Int 3 b1 bbeam, Int d d slab, Int slab b1 bbeam, Int b1 bbeam, Int 4 dslab AB 1 1 b d b b d beam, Int beam. Int beam, Int slab AB J

32 J v 114,993 in. f 4 γ 1 γ ACI (Eq ) The length of the ritial perimeter for the exterior olumn: bo 4 (18 5) 9 in. v v u u V γm A J u v unb AB psi , 993 ' 4 v ' sd ' min 4 f, f, f bo ACI (Table.6.5.) v min , , v min 53, 379.5, 74.7 psi = 53 psi v psi > v 91.0 psi O.K. 5. Two-Way Slab Defletion Control (Servieability Requirements) u Sine the slab thikness was seleted based on the minimum slab thikness tables in ACI , the defletion alulations are not required. However, the alulations of immediate and time-dependent defletions are overed in this setion for illustration and omparison with spslab model results Immediate (Instantaneous) Defletions The alulation of defletions for two-way slabs is hallenging even if linear elasti behavior an be assumed. Elasti analysis for three servie load levels (D, D + L sustained, D+L Full) is used to obtain immediate defletions of the two-way slab in this example. However, other proedures may be used if they result in preditions of defletion in reasonable agreement with the results of omprehensive tests. ACI (4..3) The effetive moment of inertia (I e) is used to aount for the raking effet on the flexural stiffness of the slab. I e for unraked setion (M r > M a) is equal to I g. When the setion is raked (M r < M a), then the following equation should be used: 3 3 Mr Mr I I 1 I I Ma M a e g r g ACI (Eq a) Where: M a = Maximum moment in member due to servie loads at stage defletion is alulated. The values of the maximum moments for the three servie load levels are alulated from strutural analysis as shown previously in this doument. These moments are shown in Figure 0. 9

33 Figure 0 Maximum Moments for the Three Servie Load Levels For positive moment (midspan) setion of the exterior span: M Craking moment. r M f r r fi r g ft-kips y t = Modulus of rapture of onrete. ACI (Eq b) f r I g 7.5 f ' psi ACI (Eq ) = Moment of inertia of the gross unraked onrete setion. I 5395 in. for T-setion (see Figure 1) g 30

34 y t = Distane from entroidal axis of gross setion, negleting reinforement, to tension fae, in. y 15.9 in. (see Figure 1) t Figure 1 I g alulations for slab setion near support I = Moment of inertia of the raked setion transformed to onrete. PCA Notes on ACI (9.5..) r As alulated previously, the positive reinforement for the end span frame strip is #4 bars loated at 1.0 in. along the slab setion from the bottom of the slab and 4 #4 bars loated at 1.75 in. along the beam setion from the bottom of the beam. Five of the slab setion bars are not ontinuous and will be exluded from the alulation of I r. Figure shows all the parameters needed to alulate the moment of inertia of the raked setion transformed to onrete at midspan. Figure Craked Transformed Setion (positive moment setion) E s = Modulus of elastiity of slab onrete s E w 33 f psi ACI ( a) E s n 7.56 PCA Notes on ACI (Table 10-) E s b 1 a 13 in. b n A, n A, in. s beam s slab 3 s, beam s, beam s, slab s, slab 1 n A d n A d in. 31

35 b b a kd in. a 13 b( kd) I na d kd na d kd 3 3 r s, slab ( slab ) s, beam ( beam ) I r 3 1 (3.999) in. 3 4 For negative moment setion (near the interior support of the end span): The negative reinforement for the end span frame strip near the interior support is 7 #4 bars loated at 1.0 in. along the setion from the top of the slab. M f r r fi r g ft-kips y t ACI (Eq b) 7.5 f ' psi ACI (Eq ) I 9333 in. g y 10.0 in. t Figure 3 I g alulations for slab setion near support s E w 33 f psi ACI ( a) E s n 7.56 PCA Notes on ACI (Table 10-) E s b beam 14 B 0.34 in. na s, total 1 PCA Notes on ACI (Table 10-) db kd 8.03 in. PCA Notes on ACI (Table 10-) B

36 3 b ( kd) beam I na ( d kd) PCA Notes on ACI (Table 10-) r 3 s, total I r 3 14 (8.03) in. 3 4 Figure 4 Craked Transformed Setion (interior negative moment setion for end span) The effetive moment of inertia proedure desribed in the Code is onsidered suffiiently aurate to estimate defletions. The effetive moment of inertia, I e, was developed to provide a transition between the upper and lower bounds of I g and I r as a funtion of the ratio M r/m a. For onventionally reinfored (nonprestressed) members, the effetive moment of inertia, I e, shall be alulated by Eq. (4..3.5a) unless obtained by a more omprehensive analysis. I e shall be permitted to be taken as the value obtained from Eq. (4..3.5a) at midspan for simple and ontinuous spans, and at the support for antilevers. ACI (4..3.7) For ontinuous one-way slabs and beams. I e shall be permitted to be taken as the average of values obtained from Eq. (4..3.5a) for the ritial positive and negative moment setions. ACI (4..3.6) For the exterior span (span with one end ontinuous) with servie load level (D+LL full): 3 3 M M r r I I 1 I, sine M r ft-kips < M a = ft-kips e M g M r a a ACI (4..3.5a) - Where I e is the effetive moment of inertia for the ritial negative moment setion (near the support) I e in I I M M e g in., sine r ft-kips > a =59.43 ft-kips Where I e + is the effetive moment of inertia for the ritial positive moment setion (midspan). 33

37 Sine midspan stiffness (inluding the effet of raking) has a dominant effet on defletions, midspan setion is heavily represented in alulation of I e and this is onsidered satisfatory in approximate defletion alulations. The averaged effetive moment of inertia (I e,avg) is given by: I e, avg 0.85 I 0.15 I for end span PCA Notes on ACI (9.5..4(1)) e e I, in. e avg Where: I e I e 4 = The effetive moment of inertia for the ritial negative moment setion near the support. = The effetive moment of inertia for the ritial positive moment setion (midspan). For the interior span (span with both ends ontinuous) with servie load level (D+LL full): 3 3 M M r r I I 1 I, sine M r ft-kips < M a = ft-kips e M g M r a a ACI (4..3.5a) I e in I I M M e g in., sine r ft-kips > a = ft-kips The averaged effetive moment of inertia (I e,avg) is given by: e, avg e e, l e, r I 0.70 I 0.15 I I for interior span PCA Notes on ACI (9.5..4()) I, in. e avg Where: 4 I el, = The effetive moment of inertia for the ritial negative moment setion near the left support. I er, = The effetive moment of inertia for the ritial negative moment setion near the right support. Table 6 provides a summary of the required parameters and alulated values needed for defletions for exterior and interior equivalent frame. It also provides a summary of the same values for olumn strip and middle strip to failitate alulation of panel defletion. 34

38 Span Ext Int zone I g, in. 4 I r, in. 4 Table 6 Averaged Effetive Moment of Inertia Calulations M a, ft-kip For Frame Strip M r, k-ft D + D + D + D + D D LL Sus L full LL Sus L full Left Midspan Right Left Mid Right I e, in. 4 I e,avg, in. 4 D D + LL Sus D + L full Defletions in two-way slab systems shall be alulated taking into aount size and shape of the panel, onditions of support, and nature of restraints at the panel edges. For immediate defletions two-way slab systems the midpanel defletion is omputed as the sum of defletion at midspan of the olumn strip or olumn line in one diretion (Δ x or Δ y) and defletion at midspan of the middle strip in the orthogonal diretion (Δ mx or Δ my). Figure 5 shows the defletion omputation for a retangular panel. The average Δ for panels that have different properties in the two diretion is alulated as follows: ( ) ( ) x my y mx PCA Notes on ACI ( Eq. 8) Figure 5 Defletion Computation for a retangular Panel To alulate eah term of the previous equation, the following proedure should be used. Figure 6 shows the proedure of alulating the term Δ x. same proedure an be used to find the other terms. 35

39 For exterior span - servie dead load ase: Figure 6 Δ x alulation proedure 4 wl frame, fixed PCA Notes on ACI ( Eq. 10) 384EI Where: frame, averaged frame, fixed = Defletion of olumn strip assuing fixed end ondition (0 6) 14 w slab weight + beam weight = () 1854 lb/ft E w 33 f psi ACI ( a) I frame,averaged = The averaged effetive moment of inertia (I e,avg) for the frame strip for servie dead load ase from Table 6 = 761 in (1854)( / 1) (1) frame, fixed in ( )(761) I frame, fixed LDF frame, fixed I g PCA Notes on ACI ( Eq. 11) Where LDF is the load distribution fator for the olumn strip. The load distribution fator for the olumn strip an be found from the following equation: 36

40 LDF LDF LDF l LDF R And the load distribution fator for the middle strip an be found from the following equation: LDF m 1 LDF For the end span, LDF for exterior negative region (LDF L ), interior negative region (LDF R ), and positive region (LDF L + ) are 0.75, 0.67, and 0.67, respetively (From Table of this doument). Thus, the load distribution fator for the olumn strip for the end span is given by: LDF I,g = The gross moment of inertia (I g) for the olumn strip (for T setion) = 0040 in. 4 I frame,g = The gross moment of inertia (I g) for the frame strip (for T setion) = 5395 in , fixed in ( M ) net, L frame L, PCA Notes on ACI ( Eq. 1) K Where: e L, = Rotation of the span left support. ( M ) ft-kips = Net frame strip negative moment of the left support. net, L frame K e = effetive olumn stiffness for exterior olumn. = 764 x E = 99 x 10 6 in.-lb (alulated previously) rad L, 6 I l PCA Notes on ACI ( Eq. 14) g, L, L 8 I e frame Where: L, = Midspan defletion due to rotation of left support. 37

41 I I g e frame = Gross-to-effetive moment of inertia ratio for frame strip. ( / 1) L, in ( M ) ( ) rad net, R frame R, 6 Ke Where R, = Rotation of the end span right support. ( M ) Net frame strip negative moment of the right support. net, R frame K e = effetive olumn stiffness for interior olumn. = 631 x E = 419 x 10 6 in.-lb (alulated previously). I l g ( / 1) , R, R in. 8 I e frame Where: R, = Midspan delfetion due to rotation of right support. x fixed PCA Notes on ACI ( Eq. 9) x, x, R x, L x in. Following the same proedure, Δ mx an be alulated for the middle strip. This proedure is repeated for the equivalent frame in the orthogonal diretion to obtain Δ y, and Δ my for the end and middle spans for the other load levels (D+LL sus and D+LL full). Assuming square panel, Δ x = Δ y= in. and Δ mx = Δ my= 0.01 in. The average Δ for the orner panel is alulated as follows: ( ) ( ) x my y mx x my y mx in. 38

42 Table 7 - Instantaneous Defletions Column Strip Middle Strip Span LDF Δ frame-fixed, in Δ -fixed, in θ 1, rad D θ, rad Δθ 1, in Δθ, in Δ x, in LDF Δ frame-fixed, in Ext Int Δ m-fixed, in θ m1, rad D θ m, rad Δθ m1, in Δθ m, in Δ mx, in Span LDF Δ frame-fixed, in Δ -fixed, in θ 1, rad D+LL sus θ, rad Δθ 1, in Δθ, in Δ x, in LDF Δ frame-fixed, in Ext Int Δ m-fixed, in θ m1, rad D+LL sus θ m, rad Δθ m1, in Δθ m, in Δ mx, in Span LDF Δ frame-fixed, in Δ -fixed, in θ 1, rad D+LL full θ, rad Δθ 1, in Δθ, in Δ x, in LDF Δ frame-fixed, in Ext Int Δ m-fixed, in θ m1, rad D+LL full θ m, rad Δθ m1, in Δθ m, in Δ mx, in Span LDF LL Δ x, in Ext Int LDF LL Δ mx, in 39

43 5.. Time-Dependent (Long-Term) Defletions (Δlt) The additional time-dependent (long-term) defletion resulting from reep and shrinkage (Δ s) may be estimated as follows: ( ) PCA Notes on ACI ( Eq. 4) s sust Inst The total time-dependent (long-term) defletion is alulated as: ( ) ( ) (1 ) [( ) ( ) ] CSA A (N9.8..5) total lt sust Inst total Inst sust Inst Where: ( ) Immediate (instantaneous) defletion due to sustained load, in. sust Inst 1 50 ' ACI ( ) ( ) Time-dependent (long-term) total delfetion, in. total lt ( ) Total immediate (instantaneous) defletion, in. total Inst For the exterior span =, onsider the sustained load duration to be 60 months or more. ACI (Table ) ' = 0, onservatively in. s in. total lt Table 8 shows long-term defletions for the exterior and interior spans for the analysis in the x-diretion, for olumn and middle strips. 40

44 Table 8 - Long-Term Defletions Column Strip Span (Δsust)Inst, in λδ Δs, in (Δtotal)Inst, in (Δtotal)lt, in Exterior Interior Middle Strip Exterior Interior spslab Software Program Model Solution spslab program utilizes the Equivalent Frame Method desribed and illustrated in details here for modeling, analysis and design of two-way onrete floor slab systems. spslab uses the exat geometry and boundary onditions provided as input to perform an elasti stiffness (matrix) analysis of the equivalent frame taking into aount the torsional stiffness of the slabs framing into the olumn. It also takes into aount the ompliations introdued by a large number of parameters suh as vertial and torsional stiffness of transverse beams, the stiffening effet of drop panels, olumn apitals, and effetive ontribution of olumns above and below the floor slab using the of equivalent olumn onept (ACI (R8.11.4)). spslab Program models the equivalent frame as a design strip. The design strip is, then, separated by spslab into olumn and middle strips. The program alulates the internal fores (Shear Fore & Bending Moment), moment and shear apaity vs. demand diagrams for olumn and middle strips, instantaneous and long-term defletion results, and required flexural reinforement for olumn and middle strips. The graphial and text results will be provided from the spslab model in a future revision to this doument. For a sample output refer to Two-Way Flat Plate Conrete Floor Slab Design example. 41

45 X Z Y spslab v5.00. Liensed to: StruturePoint. Liense ID: C6B6-C6B6 File: C:\...\TSDA-spSlab-Two-Way Slab with Beams Spanning Between Supports.slb Projet: Two-Way Slab With Beams Spanning Between Supports Frame: Interior Frame Engineer: SP Code: ACI Date: 03/31/17 Time: 14:47:17

46 100 lb/ft 100 lb/ft 100 lb/ft 100 lb/ft 100 lb/ft CASE/PATTERN: Live/All 84.3 lb/ft 84.3 lb/ft 84.3 lb/ft 84.3 lb/ft 84.3 lb/ft spslab v5.00. Liensed to: StruturePoint. Liense ID: C6B6-C6B6 File: C:\...\TSDA-spSlab-Two-Way Slab with Beams Spanning Between Supports.slb Projet: Two-Way Slab With Beams Spanning Between Supports Frame: Interior Frame Engineer: SP Code: ACI Date: 03/31/17 Time: 14:48:17

47 Shear Diagram - kip Moment Diagram - k-ft spslab v5.00. Liensed to: StruturePoint. Liense ID: C6B6-C6B6 File: C:\...\TSDA-spSlab-Two-Way Slab with Beams Spanning Between Supports.slb Projet: Two-Way Slab With Beams Spanning Between Supports Frame: Interior Frame Engineer: SP Code: ACI Date: 03/31/17 Time: 14:49:5

48 Beam Strip Moment Capaity - k-ft Middle Strip Moment Capaity - k-ft Column Strip Moment Capaity - k-ft spslab v5.00. Liensed to: StruturePoint. Liense ID: C6B6-C6B6 File: C:\...\TSDA-spSlab-Two-Way Slab with Beams Spanning Between Supports.slb Projet: Two-Way Slab With Beams Spanning Between Supports Frame: Interior Frame Engineer: SP Code: ACI Date: 03/31/17 Time: 14:51:8

49 Beam Shear Capaity - kip spslab v5.00. Liensed to: StruturePoint. Liense ID: C6B6-C6B6 File: C:\...\TSDA-spSlab-Two-Way Slab with Beams Spanning Between Supports.slb Projet: Two-Way Slab With Beams Spanning Between Supports Frame: Interior Frame Engineer: SP Code: ACI Date: 03/31/17 Time: 14:54:08

50 -0.04 Instantaneous Defletion - in 0.04 LEGEND: Dead Load Sustained Load Live Load Total Defletion spslab v5.00. Liensed to: StruturePoint. Liense ID: C6B6-C6B6 File: C:\...\TSDA-spSlab-Two-Way Slab with Beams Spanning Between Supports.slb Projet: Two-Way Slab With Beams Spanning Between Supports Frame: Interior Frame Engineer: SP Code: ACI Date: 03/31/17 Time: 14:54:54

51 1-#4(9.0) -#4(9.0) 1-#4(9.0) 4-#4(5.0) 5-#4(88.0) -#4(48.1) -#4(48.1) -#4(10.0) 1-#4(31.0) 1-#4(31.0) 5-#4(88.0) 4-#4(5.0) -#4(9.0) 1-#4(9.0) 1-#4(9.0) 4-#4(10.0) 4-#4(10.0) 4-#4(10.0) Beam Strip Flexural and Transverse Reinforement 14-#4(9.0) 14-#4(45.0) 14-#4(81.0) 14-#4(10.0) 14-#4(81.0) 14-#4(45.0) 14-#4(9.0) 9-#4(10.0) 5-#4(178.5) 9-#4(10.0) 5-#4(147.0) 9-#4(10.0) 5-#4(178.5) Middle Strip Flexural Reinforement 3-#4(1.0) 8-#4(9.0) 5-#4(45.0) 5-#4(81.0) 3-#4(1.0) 8-#4(10.0) 3-#4(1.0) 5-#4(81.0) 3-#4(1.0) 8-#4(9.0) 5-#4(45.0) 8-#4(10.0) 8-#4(10.0) 8-#4(10.0) Column Strip Flexural Reinforement spslab v5.00. Liensed to: StruturePoint. Liense ID: C6B6-C6B6 File: C:\TSDA\TSDA-spSlab-Two-Way Slab with Beams Spanning Between Supports.slb Projet: Two-Way Slab With Beams Spanning Between Supports Frame: Interior Frame Engineer: SP Code: ACI Date: 03/31/17 Time: 14:56:

52 spslab v5.00 StruturePoint , 0:57:01 PM Liensed to: StruturePoint, Liense ID: C6B6-C6B6 C:\TSDA\TSDA-spSlab-Two-Way Slab with Beams Spanning Between Supports.slb Page 1 oooooo o o oo oo oo oo ooooo oooooo oo oo ooooo oo oo o oo oo oo oo o oo oo oo oo oo ooo oo oooooo oooooo ooooo oo oo ooo oo oo oo oo oo oo oooooo oo oo oo oo oo oo o oo oo oo oo oo o oo oo oo oo ooooo oo oooooo ooo ooooo o ooooo (TM) ================================================================================================= spslab v5.00 (TM) A Computer Program for Analysis, Design, and Investigation of Reinfored Conrete Beams, One-way and Two-way Slab Systems Copyright , STRUCTUREPOINT, LLC All rights reserved ================================================================================================= Liensee stated above aknowledges that STRUCTUREPOINT (SP) is not and annot be responsible for either the auray or adequay of the material supplied as input for proessing by the spslab omputer program. Furthermore, STRUCTUREPOINT neither makes any warranty expressed nor implied with respet to the orretness of the output prepared by the spslab program. Although STRUCTUREPOINT has endeavored to produe spslab error free the program is not and annot be ertified infallible. The final and only responsibility for analysis, design and engineering douments is the liensee's. Aordingly, STRUCTUREPOINT dislaims all responsibility in ontrat, negligene or other tort for any analysis, design or engineering douments prepared in onnetion with the use of the spslab program. ================================================================================================================== [1] INPUT ECHO ================================================================================================================== General Information =================== File name: C:\TSDA\TSDA-spSlab-Two-Way Slab with Beams Spanning Between Supports.slb Projet: Two-Way Slab With Beams Spanning Between Supports Frame: Interior Frame Engineer: SP Code: ACI Reinforement Database: ASTM A615 Mode: Design Number of supports = 4 + Left antilever + Right antilever Floor System: Two-Way Live load pattern ratio = 75% Minimum free edge distane for punhing shear = 4 times slab thikness. Cirular ritial setion around irular supports used (if possible). Defletions are based on raked setion properties. In negative moment regions, Ig and Mr DO NOT inlude flange/slab ontribution (if available) Long-term defletions are alulated for load duration of 60 months. 0% of live load is sustained. Compression reinforement alulations NOT seleted. Default inremental rebar design seleted. User-defined slab strip widths NOT seleted. User-defined distribution fators NOT seleted. One-way shear in drop panel NOT seleted. Distribution of shear to strips NOT seleted. Beam T-setion design NOT seleted. Longitudinal beam ontribution in negative reinforement design over support NOT seleted. Transverse beam ontribution in negative reinforement design over support NOT seleted. Material Properties =================== Slabs Beams Columns w = lb/ft3 f' = 4 4 ksi E = ksi fr = ksi fy = 60 ksi, Bars are not epoxy-oated fyt = 60 ksi Es = 9000 ksi Reinforement Database ====================== Units: Db (in), Ab (in^), Wb (lb/ft) Size Db Ab Wb Size Db Ab Wb # # # # # # # #