AASHTO LRFD. Reinforced Concrete. Eric Steinberg, Ph.D., P.E. Department of Civil Engineering Ohio University

Transcription

1 AASHTO LRFD Reinforced Concrete Eric Steinberg, Ph.D., P.E. Department of Civil Engineering Ohio University Ohio University (July 2007) 1

2 AASHTO LRFD This material is copyrighted by Ohio University and Dr. Eric Steinberg. It may not be reproduced, distributed, sold or stored by any means, electrical or mechanical, without the expressed written consent of Ohio University. Ohio University (July 2007) 2

3 Topics Day 1 Introduction Flexure Shear Columns Decks Ohio University (July 2007) 3

4 Topics Day 2 Strut and Tie Retaining walls Footings Development (if time permits) Day 3 (1/2 day) Review Quiz Ohio University (July 2007) 4

5 Topics Course covering AASHTO LRFD Bridge Design Specifications, 3 rd Edition, 2004 including 2005 and 2006 interim revisions 4 th edition, 2007 presented where applicable ODOT exemptions also presented Ohio University (July 2007) 5

6 Topics Sections within AASHTO LRFD 5. Concrete (R/C & P/C) 3. Loads and Load Factors 4. Structural Analysis and Evaluation 9. Decks and Deck Systems 11. Abutments, Piers and Walls 13. Railings Ohio University (July 2007) 6

7 Properties - Concrete Compressive Strength ( ) f c > 10 ksi used only when established relationships exist f c < 2.4 ksi not used for structural applications f c < 4 ksi not used for prestressed concrete and decks Ohio University (July 2007) 7

8 Properties - Concrete Modulus of Elasticity ( ) For unit weights, w c to kcf and f c < 15 ksi E c 33,000 K 1 w c 1.5 f c ( ) where K 1 correction factor for source of aggregate, taken as 1.0 unless determined by test. f c compressive strength (ksi) For normal weight concrete (w c kcf) E c 1,820 f c (C ) Ohio University (July 2007) 8

9 Properties - Concrete Modulus of Rupture, f r, ( ) Used in cracking moment Determined by tests or Normal weight concrete (w/ f c < 15 ksi): Crack control by distribution of reinforcement ( ) & deflection / camber ( ) f r 0.24 f c Minimum reinforcement ( ) f r 0.37 f c Shear Capacity, V ci f r 0.20 f c Ohio University (July 2007) 9

10 Properties - Concrete Modulus of Rupture ( ) For lightweight concrete: Sand-lightweight concrete f r 0.20 f c All-lightweight concrete f r 0.17 f c Note: f c is in ksi for all of LRFD including f c Ohio University (July 2007) 10

11 Properties - Reinforcing Steel General ( ) f y 75 ksi for design f y 60 ksi unless lower value material approved by owner Ohio University (July 2007) 11

12 Limit States Service Limit State (5.5.2) Cracking ( ) Deformations ( ) Concrete stresses (P/C) Ohio University (July 2007) 12

13 Limit States Service Limit State (5.5.2) - Cracking Distribution of Reinforcement to Control Cracking ( ) Does not apply to deck slabs designed per Emperical Design (Note: ODOT does not allow Emperical Design) Applies to reinforcement of concrete components in which tension in cross-section > 80 % modulus of rupture (per ) at applicable service limit state load combination per Table f r 0.24 f c Ohio University (July 2007) 13

14 Limit States Service Limit State (5.5.2) - Cracking Distribution of Reinforcement to Control Cracking ( ) Spacing, s, of mild steel reinforcement in layer closest to tension face shall satisfy: 700 γ s e 2d ( ) β f c s ss where: γ e exposure factor (0.75 for Class 2, 1.00 for Class 1) f ss tensile stress in steel reinforcement at service limit state (ksi) d c concrete cover from center of flexural reinforcement located closest to extreme tension fiber (in.) Ohio University (July 2007) 14

15 Limit States Service Limit State (5.5.2) - Cracking Distribution of Reinforcement to Control Cracking ( ) d c in which: where: β s 1 + d c 0.7 h d c h overall thickness / depth of component (in.) Ohio University (July 2007) 15

16 Limit States Service Limit State (5.5.2) - Cracking Class 2 exposure condition (γ e 0.75) - increased concern of appearance and/or corrosion (ODOT - concrete bridge decks. Also 1 monolithic wearing surface not considered in d c and h) Class 1 exposure condition (γ e 1.0) - cracks tolerated due to reduced concerns of appearance and/or corrosion (ODOT all other applications unless noted) For f ss, axial tension considered; axial compression may be considered Effects of bonded prestressing steel may be considered. For the bonded prestressing steel, f s stress beyond decompression calculated on basis of cracked section or strain compatibility Ohio University (July 2007) 16

17 Limit States Service Limit State (5.5.2) - Cracking Min and max reinforcement spacing shall comply w/ & , respectively Minimum Spacing of Reinforcing Bars ( ) Cast-in-Place Concrete ( ) - Clear distance between parallel bars in a layer shall not be less than: 1.5 * nominal bar diameter 1.5 * maximum coarse aggregate size 1.5 in. Multilayers ( ) Bars in upper layers placed directly above those in bottom layer Clear distance between layers 1.0 in. or nominal bar diameter Exception: Decks w/ parallel reinforcing in two or more layers w/ clear distance between layers 6.0 in. Ohio University (July 2007) 17

18 Limit States Service Limit State (5.5.2) - Cracking Maximum Spacing of Reinforcing Bars ( ) Unless otherwise specified, reinforcement spacing in walls and slabs 1.5 * member thickness or 18.0 in. Max spacing of spirals, ties, and temperature shrinkage reinforcement per , , and Ohio University (July 2007) 18

19 Limit States Service Limit State (5.5.2) - Cracking T & S Steel: (5.10.8) A s 1.30 b h 2 (b + h) f y A s where A s area of reinforcement in each direction and each face (in 2 /ft) b least width of component (in) h least thickness of component (in) Ohio University (July 2007) 19

20 Limit States Service Limit State (5.5.2) - Cracking ( ) For flanges of R/C T-girders and box girders in tension at service limit state, flexural reinforcement distributed over lesser of: Effective flange width, per Interior beams ( ) - least of: o ¼ effective span (span for simply supported or distance between permanent load inflection points for continuous spans) o 12* avg. slab depth + greater of (web thickness or ½ of girder top flange width o Avg. spacing of adjacent beams Ohio University (July 2007) 20

21 Limit States Service Limit State (5.5.2) - Cracking ( ) Exterior beams ( ) - ½ of adjacent interior beam effective flange width + least of: o 1/8 effective span o 6* avg. slab depth + greater of (1/2 web thickness or 1/4 of girder top flange width) o Width of overhang width 1/10 of the average of adjacent spans between bearings If effective flange width > 1/10 span, additional longitudinal reinforcement shall be provided in the outer portions of the flange with area 0.4% of excess slab area Ohio University (July 2007) 21

22 Limit States Service Limit State (5.5.2) - Cracking ( ) If d e > 3.0 ft. for nonprestressed or partially P/C members: longitudinal skin reinforcement shall be uniformly distributed along both side faces for distance d e /2 nearest flexural tension reinforcement area of skin reinforcement A sk (in. 2 /ft. of height) on each side face shall satisfy: A sk where: d 30 e A s + A ps 4 ( ) d e effective depth from extreme compression fiber to centroid of tension steel (in.) Ohio University (July 2007) 22

23 Limit States Service Limit State (5.5.2) - Cracking ( ) Max spacing of skin reinforcement d e /6 or 12.0 in. Skin reinforcement may be included in strength computations if strain compatibility analysis used to determine stresses in individual bars / wires Ohio University (July 2007) 23

24 Limit States Example - Skin Reinforcement 48 d e #9 s 36 Ohio University (July 2007) 24

25 Limit States A s 7(# 9 s) 7 in 2 A SK (d e 30) ( ) in 2 /ft A st in Spacing: d e /6 or 12 (44.5)/ or 12 (7.42 controls) Say 6 # 6 (0.22 in 2 /ft) Ohio University (July 2007) 25

26 Limit States 44.5 d e A sk # > d e / Ohio University (July 2007) 26

27 Limit States Service Limit State (5.5.2) Deformations ( ) General ( ) Provisions of shall be considered Deck joints / bearings shall accommodate dimensional changes caused by loads, creep, shrinkage, thermal changes, settlement, and prestressing Deformations ( ) General ( ) Bridges designed to avoid undesirable structural or psychological effects due to deformations While deflection /depth limitations are optional large deviation from past successful practice should be cause for review If dynamic analysis used, it shall comply with Article 4.7 Ohio University (July 2007) 27

28 Limit States Service Limit State (5.5.2) Criteria for Deflection ( ) The criteria in this section shall be considered optional, except for the following: Metal grid decks / other lightweight metal / concrete bridge decks shall be subject to serviceability provisions of Article In applying these criteria, the vehicular load shall include the dynamic load allowance. If an Owner chooses to invoke deflection control, the following principles may be applied: When investigating max. absolute deflection for straight girder system, all design lanes loaded and all supporting components assumed to deflect equally Ohio University (July 2007) 28

29 Limit States Service Limit State (5.5.2) Criteria for Deflection ( ) (cont) For composite design, stiffness of design cross-section used for the determination of deflection should include the entire width of the roadway and the structurally continuous portions of the railings, sidewalks, and median barriers ODOT Do not include stiffness contribution of railings, sidewalks, and median barriers For straight girder systems, the composite bending stiffness of an individual girder may be taken as the stiffness determined as specified above, divided by the number of girders Ohio University (July 2007) 29

30 Limit States Service Limit State (5.5.2) Criteria for Deflection ( ) (cont) When investigating max. relative displacements, number and position of loaded lanes selected to provide worst differential effect Live load portion of Load Combination Service I used, including dynamic load allowance, IM Truck Live load shall be taken from Article For skewed bridges, a right cross-section may be used, and for curved skewed bridges, a radial cross-section may be used Ohio University (July 2007) 30

31 Limit States Service Limit State (5.5.2) Criteria for Deflection ( ) Required by ODOT In the absence of other criteria, the following deflection limits may be considered for concrete construction: Vehicular load, general.span/800 Vehicular and/or pedestrian loads Span/1000 Vehicular loads on cantilever arms.... Span/300 Vehicular and pedestrian loads on cantilever arms...span/375 Ohio University (July 2007) 31

32 Limit States Service Limit State (5.5.2) Optional Criteria for Span-to-Depth Ratios ( ) Required by ODOT Table Traditional Minimum Depths for Constant Depth Superstructures Superstructure Minimum Depth (Including Deck) Material Type Simple Spans Continuous Spans Reinforced Concrete Slabs with main reinforcement parallel to traffic 1.2(S +10) 30 S ft T-Beams L L Box Beams L L Pedestrian Structure Beams L L Ohio University (July 2007) 32

33 Limit States Service Limit State (5.5.2) Optional Criteria for Span-to-Depth Ratios ( ) where S slab span length (ft.) L span length (ft.) limits in Table 1 taken to apply to overall depth unless noted Ohio University (July 2007) 33

34 Limit States Service Limit State (5.5.2) Deflection and Camber ( ) Deflection and camber calculations shall consider dead, live, and erection loads, prestressing, concrete creep and shrinkage, and steel relaxation For determining deflection and camber, (Elastic vs. Inelastic Behavior), (Elastic Behavior), and (nonsegmental P/C) shall apply Ohio University (July 2007) 34

35 Limit States Service Limit State (5.5.2) - Deformations ( ) In absence of comprehensive analysis, instantaneous deflections computed using the modulus of elasticity for concrete as specified in Article and taking moment of inertia as either the gross moment of inertia, I g, or an effective moment of inertia, I e, given by Eq. 1: I e M cr M a 3 I g + 1 M cr M a 3 I cr I g ( ) Ohio University (July 2007) 35

36 Limit States Service Limit State (5.5.2) - Deformations ( ) in which: where: M cr f r cracking moment (kip-in.) ( ) concrete modulus of rupture per f r 0.24 f c (ksi) y t distance from the neutral axis to the extreme tension fiber (in.) M a M cr f r I g y t maximum moment in a component at the stage for which deformation is computed (kip-in.) Ohio University (July 2007) 36

37 Limit States Service Limit State (5.5.2) - Deformations ( ) Effective moment of inertia taken as: For prismatic members, value from Eq. 1 at midspan for simple or continuous spans, and at support for cantilevers For continuous nonprismatic members, average values from Eq. 1 for critical positive and negative moment sections Unless more exact determination made, long-term deflection may be taken as instantaneous deflection multiplied by following factor: If instantaneous deflection based on I g :4.0 If instantaneous deflection based on I e : (A' s / A s ) 1.6 Ohio University (July 2007) 37

38 Limit States Service Limit State (5.5.2) - Deformations ( ) Axial Deformation ( ) Instantaneous shortening / expansion from loads determined using modulus of elasticity at time of loading Instantaneous shortening / expansion from temperature determined per Articles: (Uniform Temp) (ODOT Procedure A for cold climate) (Temp gradient) (μ 6x10-6 / o F) Long-term shortening due to shrinkage and creep determined per Ohio University (July 2007) 38

39 Limit States Fatigue Limit State (5.5.3) General ( ) Not applicable to concrete slabs in multi-girder applications Considered in compressive stress regions due to permanent loads, if compressive stress < 2 * max tensile live load stress from the fatigue load combination (Table and ) Section properties shall be based on cracked sections where tensile stress (due to unfactored permanent loads and prestress and 1.5 * the fatigue load) > f c, Definition of high stress region for application of Eq. 1 for flexural reinforcement, taken as 1/3 of span on each side of section of maximum moment Ohio University (July 2007) 39

40 Limit States Fatigue Limit State (5.5.3) Reinforcing Bars ( ) Stress range in straight reinforcement from fatigue load shall satisfy: f f f min + 8 (r/h) 2006 Interim where f f f min th Edition ( ) f f stress range (ksi) f min min live load stress combined w/ more severe stress from either permanent loads or permanent loads, shrinkage and creep-induced external loads (tension + and compressive -) (ksi) Ohio University (July 2007) 40

41 Limit States Fatigue Limit State (5.5.3) Welded or Mechanical Splices of Reinforcement ( ) Stress range in welded or mechanical splices shall not exceed values below Type of Splice f f for > 1,000,000 cycles Grout-filled sleeve, w/ or w/o epoxy coated bar Cold-swaged coupling sleeves w/o threaded ends and w/ or w/o epoxy coated bar; Integrally forged coupler w/ upset NC threads; Steel sleeve w/ a wedge; One-piece taper-threaded coupler; and Single V-groove direct butt weld All other types of splices 18 ksi 12 ksi 4 ksi Ohio University (July 2007) 41

42 Limit States Fatigue Limit State (5.5.3) Welded or Mechanical Splices of Reinforcement ( ) where N cyc < 1E6, f f may be increased to 24 (6 log N cyc ) ksi but not to exceed the value found in Higher values up to value found in if verified by fatigue test data Ohio University (July 2007) 42

43 Limit States Strength Limit State (5.5.4) Resistance Factors ( ) Ф shall be taken as: Tension-controlled section in RC 0.90 Tension-controlled section in PC 1.00 Shear and Torsion Normal weight concrete 0.90 Lightweight concrete 0.70 Compression-controlled w/ spirals/ties 0.75 Bearing 0.70 Compression in Strut and Tie models 0.70 Ohio University (July 2007) 43

44 Limit States Strength Limit State (5.5.4) Resistance Factors ( ) (cont.) Compression in Anchorage zones Normal weight concrete 0.80 Lightweight concrete 0.65 Tension in steel in anchorage zones 1.00 Resistance during pile driving 1.00 Ohio University (July 2007) 44

45 Limit States Strength Limit State (5.5.4) Resistance Factors ( ) Tension-controlled: extreme tension steel strain w/ extreme compression fiber strain (Φ 0.9 R/C) Compression-controlled extreme tension steel strain its compression controlled strain limit as extreme compression fiber strain For Grade 60 reinforcement and all prestressing steel, compression controlled strain limit can be taken as (Φ 0.75) Transition region ,000 tension strains between the tension and compression controlled limits ( ) ε y Ohio University (July 2007) 45

46 Limit States Strength Limit State (5.5.4) c d t < ε < Compression - controlled Transition Tension -controlled Ohio University (July 2007) 46

47 Limit States Strength Limit State (5.5.4) c d t Tension controlled (Φ 0.9) c d t Compression controlled (Φ 0.75) ε s c < < d t 0.6 Transition Ohio University (July 2007) 47

48 Limit States Strength Limit State (5.5.4) d e d t Ohio University (July 2007) 48

49 Limit States Strength Limit State (5.5.4) Resistance Factors, Φ ( ) R/C sections in transition region (Grade 60 only!): d 0.75 φ t 1 c P/C sections in transition region: d 0.75 φ t 1 c ( ) ( ) Ohio University (July 2007) 49

50 Limit States φ Factor Compression Controlled Grade 60 Transition Extreme Steel Strain Prestressed Reinforced R/C: Strain φ 0.85 Tension Controlled Ohio University (July 2007) 50

51 Limit States Strength Limit State (5.5.4) Stability ( ) Whole structure and its components shall be designed to resist Sliding Overturning Uplift Buckling Ohio University (July 2007) 51

52 Limit States Extreme Event Limit State (5.5.5) Entire structure and components designed to resist collapse due to extreme events (earthquake and vessel/vehicle impact) Ohio University (July 2007) 52

53 Flexure Ohio University (July 2007) 53

54 Flexure Assumptions for Service & Fatigue Limit States (5.7.1) Concrete strains vary linearly, except where conventional strength of materials does not apply Modular ratio, n, is E s /E c for reinforcing bars E p /E c for prestressing tendons Modular ratio rounded to nearest integer Effective modular ratio of 2n is applicable to permanent loads and Prestress Ohio University (July 2007) 54

55 Flexure Assumptions for Strength & Extreme Event Limit States (5.7.2) General ( ) For fully bonded reinforcement or prestressing, strain directly proportional to distance from neutral axis, except in deep members or disturbed regions Maximum usable concrete strain: (unconfined) (as in Std. Spec) (confined, if verified) Factored resistance shall consider concrete cover lost Ohio University (July 2007) 55

56 Flexure Assumptions for Strength and Extreme Event Limit States (5.7.2) General ( ) Stress in reinforcement based on stress-strain curve of steel except in strut-and-tie model Concrete tensile strength neglected Concrete compressive stress-strain assumed to be rectangular, parabolic, or any other shape that results in agreement in the prediction of strength Compression reinforcement permitted in conjunction with additional tension reinforcement to increase the flexural strength Ohio University (July 2007) 56

57 Flexure Assumptions for Strength and Extreme Event Limit States (5.7.2) Rectangular Stress Distribution ( ) f c NA c d e a ε s Ohio University (July 2007) 57 f s

58 Flexure Assumptions for Strength and Extreme Event Limit States (5.7.2) Rectangular Stress Distribution ( ) where (as in Std. Spec) c distance from extreme comp. fiber to neutral axis, NA d e distance from extreme comp. fiber to centroid of tension steel a depth of concrete compression block β 1 c β for f c 4 ksi f c for 4 f c 8 ksi 0.65 for f c 8 ksi Ohio University (July 2007) 58

59 Flexure Flexural Members (5.7.3) Components w/ Bonded Tendons ( ) Depth from compression face to NA, c: For T- sections c A f ps pu + A f s s A ' f ' s s 0.85 β f' b 1 c w 0.85 f' c + k A ps b -b w f pu d p h f ( ) Ohio University (July 2007) 59

60 Flexure Flexural Members (5.7.3) Components w/ Bonded Tendons ( ) For rectangular section behavior: c A f ps pu 0.85 β 1 f' c + A f s s b w + k A ' f s A ps ' s f pu d p ( ) Ohio University (July 2007) 60

61 Flexure Flexural Members (5.7.3) where A ps area of prestressing steel f pu tensile strength of prestressing steel A s area of mild tension reinforcement A s area of compression reinforcement f s stress in mild tension reinforcement at nominal resistance f s stress in mild compression reinforcement at nominal resistance b width of compression flange b w width of web h f height of compression flange d p distance from the extreme compression fiber to the centroid of the prestressing steel Ohio University (July 2007) 61

62 Flexure Flexural Members (5.7.3) Flexural Resistance ( ) Factored resistance, M r, is: M r Ф M n ( ) where M n nominal resistance Ф resistance factor 0.9 Tension Controlled Transition 0.75 Compression controlled Ohio University (July 2007) 62

63 Flexure Flexural Members (5.7.3) Flexural Resistance ( ) Flanged sections where a β 1 c > compression flange depth: M n A ps f ps f' b -b c w where d - p h f a 2 + a 2 A f s s h f 2 d s A ' f s d ' - s ( ) f ps average stress in prestressing steel at nominal bending resistance d s distance from extreme compression fiber to centroid of nonprestressed tensile reinforcement d s distance from extreme compression fiber to the centroid of compression reinforcement Ohio University (July 2007) 63 - a 2 ' s a 2

64 Flexure Flexural Members (5.7.3) General ( ) f s can be replaced with f y in and if c/d s 0.6 f s can be replaced with f y in and if c 3d s Ohio University (July 2007) 64

65 Flexure Flexural Members (5.7.3) Flexural Resistance ( ) Rectangular sections where a β 1 c < compression flange depth, set b w to b Strain Compatibility ( ) Strain compatibility can be used if more precise calculations required Ohio University (July 2007) 65

66 Flexure Flexural Members (5.7.3) Limits for Reinforcement ( ) Max. reinforcement was limited based on: ρ 0.75ρ b c d e 0.45 Std. Spec c d e 0.42 Prior to the 2006 Interim Requirement eliminated because reduced ductility of overreinforced sections accounted for in lower φ factors Ohio University (July 2007) 66

67 Flexure Flexural Members (5.7.3) Limits for Reinforcement ( ) Amount of prestressed / nonprestressed tensile reinforcement sufficient to assure (as in Std. Spec): φm n 1.2 * M cr based on elastic stress distribution and modulus of rupture, f r, determined by: 0.37 f' c If cannot be met, then φm n 1.33 * M u Ohio University (July 2007) 67

68 Flexure Flexural Members (5.7.3) Limits for Reinforcement ( ) where M cr S c f + r f cpe M dnc S c S nc f cpe concrete compressive stress due to effective prestress forces at extreme fiber of section ( ) M dnc total unfactored dead load acting on monolithic or noncomposite section S c composite section modulus 1 S nc monolithic or noncomposite section modulus Note: f cpe, S c, and S nc found where tensile stress caused by externally applied loads Ohio University (July 2007) 68 S c f r

69 Resistance Factor Example Determine the resistance flexure factor, φ, for the beam provided below assuming: f c 4 ksi f y 60 ksi d t 20.5 d e 19.5 A s 8 # 9 Bars Ohio University (July 2007) 69

70 Resistance Factor Example 24 d e d t 18 Ohio University (July 2007) 70

71 Resistance Factor Example A f s y (8 in 2 ) (60 ksi) a 7.84" a 7.84" c 9.23" 0.85 f' b 0.85 (4 ksi) (18 in) β 0.85 c c d t ε s c d t ε s Ohio University (July 2007) 71

72 Resistance Factor Example ε s d t (0.003) c ε s 20.5" (0.003) < " φ 0.9 > Transition or c d > Φ 0.9 < 0.6 Φ 0.75 Therefore, transition Ohio University (July 2007) 72

73 Resistance Factor Example d φ t but c 20.5" " 0.75 Ohio University (July 2007) 73

74 Shear Ohio University (July 2007) 74

75 Shear Design Procedures (5.8.1) Flexural Regions ( ) Shear design for plane sections that remain plane done using either: Sectional model (5.8.3) or Strut-and-tie model (5.6.3) Deep components designed by strut-and-tie model (5.6.3) and detailed per Components considered deep when Distance from zero V to face of support < 2d or Load causing > 1/2 of V at support is < 2d from support face Ohio University (July 2007) 75

76 Shear Design Procedures (5.8.1) Regions Near Discontinuities ( ) Plane sections assumption not valid Members designed for shear using strut-and-tie model (5.6.3) and (Diaphragms, Deep Beams, Brackets, Corbels and Beam Ledges) shall apply Slabs and Footings ( ) Slab-type regions designed for shear per (shear in slabs and footing) or (Strut and Tie) Ohio University (July 2007) 76

77 Shear General Requirements (5.8.2) General ( ) Factored shear resistance, V r, taken as: V r φ V n ( ) where: V n nominal shear resistance per (kip) φ resistance factor (0.9) Ohio University (July 2007) 77

78 Shear Sectional Design Model (5.8.3) Nominal Shear Resistance ( ) The nominal shear resistance, V n, determined as lesser of: V V + V + n c s V p ( ) and V 0.25 f' b d + n c v v V p ( ) Ohio University (July 2007) 78

79 Shear Sectional Design Model (5.8.3) Nominal Shear Resistance ( ) in which: V c β f' c b v d v ( ) if or is used or lesser of V ci and V cw if is used V s A v f y d v (cot θ s + cot α) sin α ( ) Ohio University (July 2007) 79

80 Shear Sectional Design Model (5.8.3) Nominal Shear Resistance ( ) where: b v d v s effective web width within the depth d v per (in.) minimum web width to neutral axis between resultants of tensile and compressive forces due to flexure diameter for circular sections for ducts, 1/2 diameter of ungrouted or 1/4 diameter of grouted ducts subtracted from web width effective shear depth per (in.) distance to neutral axis between resultants of tensile and compressive forces due to flexure (internal moment arm) need not be taken < greater of 0.9 d e or 0.72h spacing of stirrups (in.) Ohio University (July 2007) 80

81 Shear Sectional Design Model (5.8.3) Nominal Shear Resistance ( ) where: A v area of shear reinforcement within s (in. 2 ) V p component in direction of applied shear of effective prestressing force; positive if resisting the applied shear (kip) α angle of transverse reinforcement to longitudinal axis ( ) β factor indicating ability of diagonally cracked concrete to transmit tension as specified in θ inclination angle of diagonal compressive stresses per ( ) Ohio University (July 2007) 81

82 Shear Sectional Design Model (5.8.3) Determination of β and θ ( ) Simplified Procedure for Nonprestressed Sections ( ) For: Concrete footings in which distance from point of zero shear to face of column/pier/wall < 3d v w/ or w/o transverse reinforcement Other non P/C sections not subjected to axial tension and containing minimum transverse reinforcement per , or an overall depth of < 16.0 in. β 2.0 θ 45 o Ohio University (July 2007) 82

83 Shear Sectional Design Model (5.8.3) Nominal Shear Resistance ( ) becomes: V c (2) f' c b v d v ( ) and w/ α 90 o & θ 45 o V s A v f y s d v ( ) Ohio University (July 2007) 83

84 Shear Sectional Design Model (5.8.3) General ( ) In lieu of methods discussed, resistance of members in shear may be determined by satisfying: equilibrium strain compatibility using experimentally verified stress-strain relationships for reinforcement and diagonally cracked concrete where consideration of simultaneous shear in a second direction is warranted, investigation based either on the principles outlined above or on 3-D strut-and-tie model Ohio University (July 2007) 84

85 Shear Sectional Design Model (5.8.3) Sections Near Supports ( ) Where reaction force in the direction of applied shear introduces compression into member end region, location of critical section for shear taken d v from the internal face of the support (Figure 1) Otherwise, design section taken at internal face of support Ohio University (July 2007) 85

86 Shear EFFECTIVE SHEAR DEPTH, dv h 0.72h 0.90de dv x Critical Section for shear* 0.5dv cot(θ) Mn dv (Aps)(fps)+(Asfy) Mn dv (Aps)(fps)+(Asfy) For top bars (Varies) (Varies) Ohio University (July 2007) 86

87 Shear Sectional Design Model (5.8.3) Sections Near Supports ( ) Where beam-type element extends on both sides of reaction area, design section on each side of reaction determined separately based upon the loads on each side of the reaction and whether their respective contribution to total reaction introduces tension or compression into end region For nonprestressed beams supported on bearings that introduce compression into member, only minimal transverse reinforcement needs to be provided between inside edge of the bearing plate / pad and beam end Ohio University (July 2007) 87

88 Shear Sectional Design Model (5.8.3) Sections Near Supports ( ) Minimal Av d v Ohio University (July 2007) 88

89 Shear General Requirements (5.8.2) Regions Requiring Transverse Reinforcement ( ) Except for slabs, footings, and culverts, transverse reinforcement shall be provided where: where: V u V c V p V > 0.5 φ ( V + u c V ) p factored shear force (kip) ( ) nominal concrete shear resistance (kip) prestressing component in direction of shear (kip) φ resistance factor (0.9) Ohio University (July 2007) 89

90 Shear General Requirements (5.8.2) Minimum Transverse Reinforcement ( ) Area of steel shall satisfy: A v f' c b s v f y ( ) where: A v transverse reinforcement area within distance s (in. 2 ) b v width of web (adjusted for ducts per ) (in.) s transverse reinforcement spacing (in.) f y transverse reinforcement yield strength (ksi) Ohio University (July 2007) 90

91 Shear General Requirements (5.8.2) Maximum Spacing of Transverse Reinforcement ( ) Maximum transverse reinforcement spacing, s max, determined as: If v u < f c, then: S max 0.8 d v 24.0 in. ( ) If v u f c, then: S max 0.4 d v 12.0 in. ( ) where v u d v shear stress per (ksi) effective shear depth (in.) Ohio University (July 2007) 91

92 Shear General Requirements (5.8.2) Shear Stress on Concrete ( ) Shear stress on the concrete determined as: where: φ resistance factor (0.9) b v d v v u V φ V u p φ b d v v effective web width (in.) ( ) effective shear depth (in.) Ohio University (July 2007) 92

93 Shear General Requirements (5.8.2) Design and Detailing Requirements ( ) Transverse reinforcement anchored at both ends per Extension of beam shear reinforcement into the deck slab for composite flexural members considered when checking Design yield strength of nonprestressed transverse reinforcement: f y when f y 60.0 ksi strain , but 75.0 ksi when f y > 60.0 ksi Ohio University (July 2007) 93

94 Shear Anchorage of Shear Reinforcement ( ) Single leg, simple or multiple U stirrups ( ) No. 5 or smaller and No. 6-8 w/ f y 40 ksi Standard hook around longitudinal steel No. 6 8 w/ f y > 40 ksi Standard hook around longitudinal bar plus embedment between midheight of member and outside end of hook, l e, satisfying l e 0.44 d b f ' c f y Ohio University (July 2007) 94

95 Shear Anchorage of Shear Reinforcement ( ) Closed stirrups ( ) Pairs of U stirrups placed to form a closed stirrup are properly anchored if the lap lengths > 1.7 l d (l d tension development length) For members > 18 deep, closed stirrup splices w/ tension force from factored loads, A b f y, < 9 k per leg considered adequate if legs extend full available depth of member Ohio University (July 2007) 95

96 Columns Ohio University (July 2007) 96

97 Columns: Compression Members General ( ) Compression members shall consider: Eccentricity Axial loads Variable moments of inertia Degree of end fixity Deflections Duration of loads Prestressing Ohio University (July 2007) 97

98 Columns: Compression Members General ( ) Nonprestressed columns with the slenderness ratio, KL u /r < 100, may be designed by the approximate procedure per where: K effective length factor per L u unbraced length (in.) r radius of gyration (in.) Ohio University (July 2007) 98

99 Columns: Compression Members Limits for Reinforcement ( ) Maximum prestressed and nonprestressed longitudinal reinforcement area for noncomposite compression components shall be such: A s A g + A f ps pu A f g y 0.08 ( ) and A f ps pe A f' g c 0.30 ( ) Ohio University (July 2007) 99

100 Columns: Compression Members Limits for Reinforcement ( ) Minimum prestressed and nonprestressed longitudinal reinforcement area for noncomposite compression components shall be such that: A f s y A f' g c + A f ps pu A f' g c ( ) Ohio University (July 2007) 100

101 Columns: Compression Members Limits for Reinforcement ( ) where: A s nonprestressed tension steel area (in. 2 ) A g section gross area (in. 2 ) A ps prestressing steel area (in. 2 ) f pu tensile strength of prestressing steel (ksi) f y yield strength of reinforcing bars (ksi) f c compressive strength of concrete (ksi) f pe effective prestress (ksi) Ohio University (July 2007) 101

102 Columns: Compression Members Limits for Reinforcement ( ) (as in Std. Spec) Minimum number of longitudinal reinforcing bars in a column: 6 for circular arrangement 4 for rectangular arrangement Minimum bar size: No. 5 Ohio University (July 2007) 102

103 Columns: Compression Members Approximate Evaluation of Slenderness Effects ( ) (as in Std. Spec) Members not braced against sidesway, slenderness effects neglected when KL u /r, < 22 Members braced against sidesway, slenderness effects neglected when: KL u /r < 34 12(M 1 /M 2 ), in which M 1 and M 2 are the smaller and larger end moments, respectively and (M 1 /M 2 ) is positive for single curvature flexure Ohio University (July 2007) 103

104 Columns: Compression Members Approximate Evaluation of Slenderness Effects ( ) Approximate procedure may be used for the design of nonprestressed compression members with KL u /r < 100: Unsupported length, L u, clear distance between components providing lateral support (taken to extremity of any haunches in plane considered) Radius of gyration, r, computed for gross concrete section (0.25*diameter for circular cols) Braced members, effective length factor, K, taken as 1.0, unless a lower value is shown by analysis Ohio University (July 2007) 104

105 Columns: Compression Members Approximate Evaluation of Slenderness Effects ( ) Unbraced members, K determined considering effects of cracking and reinforcement on relative stiffness and taken 1.0 Design based on factored axial load, P u, determined by elastic analysis and magnified factored moment, M c, per b Magnified Moment ( b) ( b-1) M δ M + c b 2b δ M s 2s Ohio University (July 2007) 105

106 Columns: Compression Members Magnified Moment ( b) where δ b C m P 1 u φ P K e 1.0 ( b-3) δ s 1 1 P u φ P K e ( b-4) δ s 1 for members braced against sidesway. Σ is for group of compression members on 1 level of a bent or where compression members are intergrally connected to same superstructure Ohio University (July 2007) 106

107 Columns: Compression Members Magnified Moment ( b) where P u factored axial load (kip) M b C m M ( b-6) 2b for members braced against sidesway and w/o loads between supports C m 1 for all other cases M 1b / M 2b smaller / larger end moment Note: M 1b / M 2b + for single curvature and - for double curvature Ohio University (July 2007) 107

108 Columns: Compression Members Magnified Moment ( b) P e Euler Buckling load (kip) π 2 EI 2 K L u ( b-5) φ K stiffness reduction factor (0.75 for concrete) M 2b moment due to factored gravity loads resulting in negligible sidesway, postitive (kip-ft) M 2s moment due to factored lateral or gravity loads resulting in sidesway > L u /1500, postitive (kip-ft) L u unsupported length (in) Ohio University (July 2007) 108

109 Columns: Compression Members Magnified Moment ( b) K effective length factor ( ) In absence of a more refined analysis, K can be taken as: 0.75 for both ends being bolted or welded for both ends pinned 1.0 for single angles regardless of end conditions Ohio University (July 2007) 109

110 Columns: Compression Members Magnified Moment ( b) The Structural Stability Council provides theoretical and design values for K in Table C1 of the spec. End Conditions Fixed- Fixed Fixed- Pinned Fixed-Lat. Translation Pinned- Pinned Theoretical K Fixed- Free Pinned-Lat. Translation Design K Ohio University (July 2007) 110

111 Columns: Compression Members Magnified Moment ( b) Assuming only elastic action occurs, K can also be found from: π 2 G G 36 a b K 6 G + G a b tan π K π K where a and b represent the ends of the column Ohio University (July 2007) 111

112 Ohio University (July 2007) 112 Columns: Compression Members Columns: Compression Members G can be found by: subscripts c and g represent the column and girders, respectively, in the plane of flexure being considered. Two previous equations result in commonly published nomographs. Magnified Moment ( b) g L g I g E Σ c L c I c E Σ G

113 Columns: Compression Members Magnified Moment ( b) In absence of a refined analysis, ODOT allows following values: Spread footings on rock G 1.5 Spread footings on soil G 5.0 Footings on multiple rows of piles or drilled shafts: End Bearing G 1.0 Friction G 1.5 Footings on a single row of drilled shafts/friction piles Footings on a single row of end bearing piles G 1.0 refined analysis reqd. Ohio University (July 2007) 113

114 Columns: Compression Members Magnified Moment ( b) ODOT - For columns supported on a single row of drilled shafts / friction piles include the depth to point of fixity when calculating effective column length. Refer to Article to determine depth to point of fixity. For drilled shafts socketed into rock, point of fixity should be no deeper than top of rock. List in Table assumes typical spread footings on rock are anchored when footing is keyed 3 in. into rock. Ohio University (July 2007) 114

115 Columns: Compression Members Approximate Evaluation of Slenderness Effects ( ) In lieu of more precise calculation, EI for use in determining P e, as specified in Eq b-5, taken as greater of: EI E c I g E s β d I s ( ) EI E I c g β d ( ) Ohio University (July 2007) 115

116 Columns: Compression Members Approximate Evaluation of Slenderness Effects ( ) where: E c concrete modulus of elasticity (ksi) I g gross moment of inertia of concrete section (in. 4 ) E s steel modulus of elasticity (ksi) I s longitudinal steel moment of inertia about centroidal axis (in. 4 ) β d ratio of maximum factored permanent load moments to maximum factored total load moment; positive (accounts for concrete creep) Ohio University (July 2007) 116

117 Columns: Compression Members Factored Axial Resistance ( ) Factored axial resistance of concrete compressive components, symmetrical about both principal axes, shall be taken as: in which: P r φp n ( ) P n e [ 0.85 f c (A g -A st -A ps ) + f y A st A ps (f pe -E p ε cu ) ] ( &3) with e 0.85 for members w/ spiral reinforcement 0.80 for members w/ tie reinforcement Ohio University (July 2007) 117

118 Columns: Compression Members Factored Axial Resistance ( ) where: P r factored axial resistance, w/ or w/o flexure (kip) P n nominal axial resistance, w/ or w/o flexure (kip) A ps prestressing steel area (in. 2 ) E p prestressing tendons modulus of elasticity (ksi) f pe effective stress in prestressing steel (ksi) ε cu concrete compression failure strain (in./in.) (0.003) Ohio University (July 2007) 118

119 Columns: Compression Members Biaxial Flexure ( ) In lieu of equilibrium and strain compatibility analysis, noncircular members subjected to biaxial flexure and compression may be proportioned using the following approximate expressions: If the factored axial load is 0.10 φ f c A g : 1 P rxy 1 P rx + 1 P ry 1 φp o ( ) in which: ( ) P o 0.85 f' c A g A st A ps + f A y st A ps f pe E ε p cu Ohio University (July 2007) 119

120 Columns: Compression Members Biaxial Flexure ( ) If the factored axial load is < 0.10 φ f c A g : M ux M rx + M uy M ry 1.0 ( ) Ohio University (July 2007) 120

121 Columns: Compression Members Biaxial Flexure ( ) where: P rxy factored axial resistance in biaxial flexure (kip) P rx factored axial resistance based on only e y is present (kip) P ry factored axial resistance based on only e x is present (kip) P u factored applied axial force (kip) M ux factored applied moment about X-axis (kip-in.) M uy factored applied moment about Y-axis (kip-in.) e x eccentricity in X direction, (M uy /P u ) (in.) e y eccentricity in Y direction, (M ux /P u ) (in.) P o nominal axial resistance of section at 0.0 eccentricity Ohio University (July 2007) 121

122 Columns: Compression Members Biaxial Flexure ( ) Factored axial resistance P rx and P ry shall be φ P n where P n given by either Eqs or , as appropriate Spirals and Ties ( ) Area of steel for spirals & ties in bridges in Seismic Zones 2, 3, or 4 shall comply with the requirements of Article Where the area of spiral and tie reinforcement is not controlled by: Seismic requirements Shear or torsion per Article 5.8 Minimum requirements per Article Ohio University (July 2007) 122

123 Columns: Compression Members Spirals and Ties Ratio of spiral reinforcement to total volume of concrete core, measured out-to-out of spirals, shall satisfy: where: ρ s 0.45 A g A c 1 f' c f yh ( ) A g gross area of concrete section (in. 2 ) A c area of core measured to the outside diameter of spiral (in. 2 ) f c 28 day strength, unless another age is specified (ksi) f yh specified yield strength of reinforcement (ksi) Other details of spiral and tie reinforcement shall conform to Articles and Ohio University (July 2007) 123

124 Columns: Compression Members A A r ρ Steel Volume Concrete Volume A 2 π r b π r 2 pitch Pitch Note: The steel volume calc does not account for the pitch but this is minor with smaller pitches Ohio University (July 2007) 124

125 Columns: Compression Members Spirals and Ties ODOT Provision only applies to columns where ratio of axial column capacity to axial column load < 1.5 For all other column designs, spiral reinforcement detailed as specified in BDM Section Ohio University (July 2007) 125

126 Columns: Compression Members Column Example: Take: M Permanent x 113 k-ft M Total x 750 k-ft M Permanent y 83 k-ft M Total y 260 k-ft f c 4ksi 2 36in Diameter of column 36 in (3 ft) A π 1,018in 2 (7.07ft 2 ) g 2 Reinforcement steel 12 # 9 s A s 12 in 2 f y 60 ksi P u 927 kips + self weight of column L ft. Ohio University (July 2007) 126

127 Columns: Compression Members Column Example: Self weight of column 7.07 ft ft 0.15 kcf 1.25 Load Factor 23.4 k ( ) 7.07 ft k P 927 k + u A s A g 12 in 2 1,018 in < 0.08 O.K ( ) A f s y A f' g c 12 in 2 (60 ksi) 1,018 in 2 (4 ksi) O.K.. ( ) Ohio University (July 2007) 127

128 Columns: Compression Members Column Example: Slenderness L Length of column k Effective length factor r Radius of gyration of cross-section of the column L ft 212 in Ohio University (July 2007) 128

129 Columns: Compression Members Column Example: Slenderness Effective length factor (k): Pier Cap Plane of bent ( to bridge) Unbraced w/ k 1.2 (highly rigid pier cap & footing)... (C ) Pier Cap G B G A K Ohio University (July 2007) 129

130 Columns: Compression Members Column Example: Slenderness Plane to bent (// to bridge) Unbraced with k 2.1 free cantilever... (C ) Ohio University (July 2007) 130

131 Columns: Compression Members Column Example: Slenderness r 0.25(d). (C ) where d column diameter r 0.25(36 in) 9 in Plane of Bent kl 1.2 (212in) r 9in 28.3 < 100 O.K, but > 22 Consider slenderness Ohio University (July 2007) 131

132 Columns: Compression Members Column Example: Slenderness Plane to Bent kl r 2.1(212in) 9in 49.5 < 100 O.K but > 22 Consider slenderness Moment Magnification Plane to Bent: M δ M C b 2b + δ M s 2s Ohio University (July 2007) 132

133 Columns: Compression Members Column Example: Moment Magnification where the δ s are moment magnifying factors δ b braced magnifier δ s sway magnifier δ b C m P 1 u φ P K e δ s 1 1 P u φ P K e Ohio University (July 2007) 133

134 Columns: Compression Members Column Example: Moment Magnification where: C m Equivalent moment correction factor C m 1.0 (for all other cases) P u 950 k φ K 0.75 P e π 2 EI (kl ) 2 u Ohio University (July 2007) 134

135 Columns: Compression Members Column Example: Moment Magnification EI max of: E I c g + E I EI 5 s s 1 + β d where: and E c Modulus of elasticity of concrete I g Moment of inertia of the gross section E s Modulus of elasticity of steel E I c g EI β d I s Moment of inertia of the steel section Ohio University (July 2007) 135

136 Columns: Compression Members Column Example: Moment Magnification E c 1,820 f' c 1,820 4 ksi 3,640 ksi I g π r 4 4 π (18 in) ,448 in where r radius of column E s 29,000ksi I s difficult to find and depends on position of bars relative to axis of concern Ohio University (July 2007) 136

137 Columns: Compression Members Column Example: Moment Magnification r D 36 in 3 in 3 in 30 in r 15 in Ohio University (July 2007) 137

138 Columns: Compression Members Column Example: Moment Magnification Assume I Parallel I s(centroid) + Ad 2 30 o r I s 0 since steel bar is small (1.128 in diameter) 1 r 15 d r sin(30) 15"sin(30) 7.5 in I (1in 2 )(7.5 in) in Ohio University (July 2007) 138

139 Columns: Compression Members Column Example: Moment Magnification 60 o r 2 d r sin(60) 15"sin(60) 13 in 2 13 I (1in 2 )(13 in) in 4 2 I (1in 2 )(15 in) in Ohio University (July 2007) 139

140 Columns: Compression Members Column Example: Moment Magnification I 4(56.25in 4 ) + 4(169in 4 ) + 2(225in 4 ) 1,351 in 4 I s A s d 2 1,350 in 4 (from MacGregor s text) M 113 β Permanent 0.15 d M 750 Total EI Maximum of the following values: E I c g + E I EI 5 s s 1 + β d & E I c g EI β d Ohio University (July 2007) 140

141 Columns: Compression Members Column Example: Moment Magnification EI Maximum of the following values: EI 3,640 ksi (82,448 in 4 ) + 29,000 ksi (1,351in 4 ) ,261,864 k - in & EI 3,640 ksi (82,448 in 4 ) ,386,337 k - in (Controls) Ohio University (July 2007) 141

142 Columns: Compression Members Column Example: Moment Magnification π 2 (104,386,337 k - in 2 ) P e (2.1(212in) ) 2 5,198 kips 1 δ 1.32 b 950 k (5,198 k) Due to symmetry of bridge and columns P u P u and P e P e δ δ 1.32 s b M 1.32 (750 k ft) 990 k ft Ohio University (July 2007) 142

143 Columns: Compression Members Column Example: Moment Magnification Plane of Bent: M c// δ b M 2b + δ s M 2s δ b C m P 1 φ u P e where: C m 1.0, P u 950k, φ 0.75 P e π 2 EI (k L ) 2 u Ohio University (July 2007) 143

144 Columns: Compression Members Column Example: Moment Magnification EI 104,386,337 k -in 2 (1+ β ) d (1.15) where 1.15 is from previous (1 + β d ) 83 β d ,386,337 k - in 2 (1.15) EI (1.32) 2 90,942,642 k - in Ohio University (July 2007) 144

145 Columns: Compression Members Column Example: Moment Magnification π 2 (90,942,642 k - in 2 ) P e (1.2 (212 in)) 2 13,869 k 1 δ 1.10 b 950 k (13,869 k) As before, let P u P u and P e P e δ δ s b 1.10 M// 260 k-ft(1.10) 286 k-ft M M 2 + u M 2 // (990 k ft) 2 + (286 k ft) 2 1,030 k ft where M u Factored Moment and P u 950 kips Ohio University (July 2007) 145

146 Columns: Compression Members Table 1: Column Axial & Flexural Capacity Ф φp n (kips)) φm n (k-ft) P u (k) M u (k-ft) , , , , ,617 1, ,142 1, , , Ohio University (July 2007) 146

147 Columns: Compression Members Column Interaction Diagram 3,000 2,500 2,000 φp n (k) 1,500 1, ,000 1, φm n (k-ft) Ohio University (July 2007) 147

148 Columns: Compression Members Column Example: Shear ( ) V c β f' c b d v v where V c nominal concrete shear strength b v web width (the same as bw in the ACI Code) d v effective depth in shear (taken as flexural lever arm) Not subject to axial tension and will include minimum transverse steel β 2.0 By C b v 36 in. Ohio University (July 2007) 148

149 Columns: Compression Members Column Example: Shear (C ) d v A s f y M n + A f ps ps 690 k 2 (6 in ft(12 in/ft) 0.9 ) 60 ksi in Note: A ps f ps 0 where: A ps Area of prestressed reinforcement f ps Stress in prestressed reinforcement Also assuming ½ of the steel is actually in tension (6 in 2 instead of 12 in 2 ) Ohio University (July 2007) 149