Reliability-based Calibration of Design Code for Concrete Structures (ACI 318) Andrzej S. Nowak and Anna M. Rakoczy

Transcription

1 Reliability-based Calibration of Design Code for Concrete Structures (ACI 318) Andrzej S. Nowak and Anna M. Rakoczy

2 ACI 318 Code Outline Objectives New material test data Resistance parameters Reliability analysis Resistance factors Further developments Department of Civil Engineering

3 DESIGN CODES HISTORICAL PERSPECTIVE

4 ACI 318 Code The basic document for design of concrete (R/C and P/C) buildings in USA ACI 318 specifies resistance factors and design resistance ACI 318 specifies load factors ACI 318 does not specify design load, reference is made to other codes Department of Civil Engineering

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6 Why Calibration of ACI 318? Current load factors were adopted in 1950 s Introduction of the new code with loads and load factors, ASCE 7 (American Society of Civil Engineers) Load factors specified in ASCE 7 are already adopted for steel design (AISC) and wood (NDS) Problems with mixed structures (steel and concrete) Department of Civil Engineering

7 Department of Civil Engineering

8 Load factors specified by ACI 318 and ASCE 7 The design formula specified by ACI Code 1.4 D L < f R 0.75 (1.4 D L W) < f R 0.9 D W < f R 0.75 (1.4 D L E) < f R The design formula specified by ASCE-7 Standard 1.4 D < f R 1.2 D L < f R 1.2 D L S < f R 1.2 D L S < f R 1.2 D W L S < f R 1.2 D E L S < f R 0.9 D (1.6 W or 1.0 E) < f R

9 Objectives of Calibration of ACI 318 Determine resistance factors, f, corresponding to the new load factors (ASCE 7) Reliability of the designed structures cannot be less the predetermined minimum level Maintain a competitive position of concrete structures If needed, identify the need for changes of load factors in the ASCE 7 Department of Civil Engineering

10 Code Calibration Procedure Selection of representative structural types and materials Formulate limit state functions, identify load and resistance parameters Develop statistical models for load and resistance parameters Develop the reliability analysis procedure Select the target reliability level(s) Determine load and resistance factors Department of Civil Engineering

11 Considered Structural Components Beams (reinforced concrete, prestressed concrete) Slabs (reinforced concrete, prestressed concrete) Columns (reinforced concrete, prestressed concrete, tied and spiral, axial and eccentric) Plain concrete Department of Civil Engineering

12 Considered Load Components D = dead load L = live load S = snow W = wind E = earthquake Load combinations Department of Civil Engineering

13 Assumed Statistical Data Dead load l = , V = Live load l = 1.00, V = 0.20 Wind l = 0.80, V = 0.35 Snow l = 0.80, V = 0.25 Earthquake l = 0.65, V = 0.55 Department of Civil Engineering

14 Load Factor Department of Civil Engineering

15 Considered Materials Concrete (cast-in-place and precast) Ordinary concrete Light weight concrete High strength concrete (f c 45 MPa) Reinforcing steel bars Prestressing steel strands Department of Civil Engineering

16 Considered Cases Old Statistical data for materials from 1970 s Design according to ACI New Statistical data for materials from Design according to proposed ACI 318 Department of Civil Engineering

17 Objectives Update the materials strength models using new statistical data Update the resistance models for reliability analysis Calculate reliability indices for components designed using ACI Provide a basis for selection of resistance factors Department of Civil Engineering

18 Parameters of Resistance Material : uncertainty in the strength of material, modulus of elasticity, cracking stresses, and chemical composition. Fabrication : uncertainty in the overall dimensions of the component which can affect the cross-section area, moment of inertia, and section modulus. Analysis : uncertainty resulting from approximate methods of analysis and idealized stress/strain distribution models. Department of Civil Engineering

19 Parameters of Resistance R = R n M F P where : R n = nominal value of resistance M = material factor F = fabrication factor P = professional factor Department of Civil Engineering

20 Parameters of Resistance The mean value of R is Coefficient of variation Bias factor R R n M F P V V V V l R M F P l l l R M F P Department of Civil Engineering

21 Resistance Factor Department of Civil Engineering

22 Material Factor Available data-base from 1970 s (MacGregor) Concrete industry provided test results ( and 2003), gathered for this calibration Code Calibration of ACI 318 (2005) is based on these recent test results Department of Civil Engineering

23 Concrete Strength Compressive strength - cylinders 6 x 12 in (150 x 300 mm) Mostly 28 day strength, for precast concrete also 56 day strength Department of Civil Engineering

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26 Results of Material Tests Cumulative distribution functions (CDF) For an easier interpretation of the results, CDF s are plotted on the normal probability paper CDF of a normal random variable is represented by a straight line Any straight line on the normal probability paper represents a normal CDF Department of Civil Engineering

27 Strength of Ordinary Concrete Ready mix concrete Plant-cast concrete 3,000 psi (21 MPa) 3,500 psi (24 MPa) 4,000 psi (28 MPa) 4,500 psi (31 MPa) 5,000 psi (35 MPa) 6,000 psi (42 MPa) 5,000 psi (35 MPa) 5,500 psi (38 MPa) 6,000 psi (42 MPa) 6,500 psi (45 MPa)

28 Strength of Concrete Light-weight concrete High strength concrete 3,000 psi (21 MPa) 3,500 psi (24 MPa) 4,000 psi (28 MPa) 5,000 psi (35 MPa) 7,000 psi (49 MPa) 8,000 psi (56 MPa) 9,000 psi (62 MPa) 10,000 psi (70 MPa) 12,000 psi (84 MPa)

29 More Materials Data Compressive Strength of Ordinary Concrete, Ready mixed, fc : 3,000 3,500 4,000 4,500 5,000 and 6,000psi (21-42 MPa) Yield Stress of Reinforcing Steel Bars, Grade 60 Bar Sizes: #3, #4, #5, #6, #7, #8, #9, #10, #11 and #14 (9.5mm 44 mm) Breaking Stress of Prestressing Steel (7-wire strands), Grade 270 (1865 MPa), Nominal Diameters: 0.5 in and 0.6 in ( mm) Department of Civil Engineering

30 Ordinary Concrete Number of Samples

31 Lightweight Concrete Number of Samples

32 Presentation of Test Data Cumulative distribution functions (CDF) are plotted on the normal probability paper Vertical axis is the number of standard deviations from the mean value If CDF is close to a straight line, then the distribution is normal The mean and standard deviation can be read directly from the graph Department of Civil Engineering

33 Probability Paper Data is plotted on the normal probability paper. A normal distribution function is represented by a straight line. Department of Civil Engineering

34 Probability Paper Department of Civil Engineering

35 Inverse Normal Distribution Ordinary Concrete CDF of Strength 4 3 f c = 3,000 psi, 21 MPa Strength [psi] Source 1 (samples: 334) Source 12 (samples:1046) Source 13 (samples: 350) Source 14 (samples: 203) Source 15 (samples: 424) Source 16 (samples: 562) Source 17 (samples: 116) Source 18 (samples: 173) Source 19 (samples: 180) Source 23 (samples: 276)

36 Inverse Normal Distribution Ordinary Concrete CDF of Strength f c = 3,000 psi, 21 MPa l = 1.33 V = = 4000 s = Strength [psi] All sources (samples: 4016) Approximation

37 Inverse Normal Distribution Ordinary Concrete CDF of Strength 4 3 f c = 3,500 psi, 25 MPa Strength [psi] Source 16 (samples: 339) Source 19 (samples: 99)

39 Inverse Normal Distribution Ordinary Concrete CDF of Strength 4 3 f c = 4,000 psi, 28 MPa Strength [psi] Source 1 (samples: 316) Source 2 (samples: 156) Source 13 (samples: 274) Source 14 (samples: 269) Source 15 (samples: 220) Source 16 (samples: 584) Source 18 (samples: 99) Source 19 (samples: 533)

41 Inverse Normal Distribution Ordinary Concrete CDF of Strength 4 3 f c = 4,500 psi Strength [psi] Source 12 (samples: 839) Source 13 (samples: 298) Source 15 (samples: 164) Source 23 (samples: 346)

42 Inverse Normal Distribution Ordinary Concrete CDF of Strength f c = 4,500 psi l = 1.19 V = = 5350 s = Strength [psi] All sources (samples: 1919) Approximation

43 Inverse Normal Distribution Ordinary Concrete CDF of Strength 4 3 f c = 5,000 psi, 35 MPa Strength [psi] Source 1 (samples: 138) Source 10 (samples: 206) Source 11 (samples: 294) Source 14 (samples: 263) Source 16 (samples: 100) Source 18 (samples: 133) Source 19 (samples: 422)

46 Lightweight Concrete CDF of Strength l = V = = 4290 s = 665 f c = 3000 psi, 21 MPa

48 Lightweight Concrete CDF of Strength l = V = f c = 4000 psi, 28 MPa = 5350 s = 660

52 Lightweight Concrete CDF of Strength l = V = = 8000 s =805 f c = 7100 psi, 49 MPa

53 Summary of the Statistical Parameters for Concrete λ l Lightweigh Concrete Ordinary, Ready Mix Ordinary, Plant Cast High Strength Recommended λ for NWC Recommended λ for LWC V V Lightweigh Concrete Ordinary, Ready Mix Ordinary, Plant Cast High Strength Recommended V for NWC Recommended V for LWC f c [psi] f c [psi] f c [psi] f c [psi]

54 Bias Factor and Coefficient of Variation for Compressive Strength and Shear Strength of Concrete Concrete Grade f c ' (psi) Compressive strength Shear Strength l V l V 3000, 21 MPa , 28 MPa , 42 MPa , ,000, 84 MPa

55 Reinforcing Steel Bars, Grade 60 (420 MPa) Number of Samples

56 Inverse Normal Distribution Reinforcing Steel Bars, Grade 60 CDF of Yield Stress 4 Bars # Yield Stress [ksi] Source 2 (samples: 741) Source 4 (samples: 123)

57 Inverse Normal Distribution Reinforcing Steel Bars, Grade 60 CDF of Yield Stress Bars #3 l = 1.18 V = = 71.0 s = Yield Stress [ksi] All sources (samples: 864) Approximation

58 Inverse Normal Distribution Reinforcing Steel Bars, Grade 60 CDF of Yield Stress 4 3 Bars # Yield Stress [ksi] Source 1 (samples: 60) Source 2 (samples: 2369) Source 3 (samples: 60) Source 4 (samples: 106) Source 5 (samples: 90)

74 Inverse Normal Distribution Reinforcing Steel Bars, Grade 60 CDF of Yield Stress Bars #14 l = 1.14 V = = 68.5 s = Yield [ksi] All sources - Source 3 (samples: 12) Approximation

75 Reinforcing Steel Bars, Grade 60 CDF of Yield Stress for simulation: l = 1.13 V = Yield Stress [ksi] #3 #4 #5 #6 #7 #8 #9 #10 #11 #14 All Size Approximation

76 Reinforcing Steel Bars, Grade 60 (420 MPa) Statistical Parameters Bar Size l V # # # # # # # # # #

77 Prestressing Strands Grade 270 (1800 MPa) Number of Samples Total Number of Samples 47,421

78 Prestressing Steel (7-wire strands), Grade 270 CDF of Breaking Stress Inverse Normal Distribution Strands 0.5 in (12 mm) Breaking Stress [ksi] Source 1 (samples: 3908) Source 2 (samples: 1158) Source 3 (samples: 268) Source 4 (samples: 9795) Source 5 (samples: 18258)

79 Prestressing Steel (7-wire strands), Grade 270 CDF of Breaking Stress Inverse Normal Distribution Strands 0.5 in (12 mm) l = 1.04 V = = 280 s = Breaking Stress [ksi] All sources (samples: 33387) Approximation

80 Prestressing Steel (7-wire strands), Grade 270 CDF of Breaking Stress Inverse Normal Distribution Strands 0.6 in Breaking Stress [ksi] Source 1 (samples: 700) Source 2 (samples: 785) Source 3 (samples: 212) Source 4 (samples: 3442) Source 5 (samples: 8889)

81 Prestressing Steel (7-wire strands), Grade 270 CDF of Breaking Stress Inverse Normal Distribution Strands 0.6 in (12 mm) l = 1.02 V = = 275 s = Breaking Stress [ksi] All sources (samples: 14028) Approximation

82 Prestressing Steel Statistical Parameters Grade Size Number of samples Bias Factor V 250 ksi (1750 MPa) 1/4 (6.25 mm) 3/8 (9.5 mm) 7/16(11 mm) 1/2 (12.5 mm) ksi (1900 MPa) 3/8 (9.5 mm) 7/16 (11 mm) 1/2 (12.5 mm) 0.6 (15 mm)

83 Structural elements and limit states Reinforced concrete beams - flexure Reinforced concrete beams - shear (w/o stirrups) Reinforced concrete beams - shear (with stirrups) Axially loaded columns, tied Axially loaded columns, spiral One way slabs - flexure One way slabs - shear Two way slabs shear Bearing strength

84 Department of Civil Engineering Bending Moment Resistance 2 a d f A R y s b f f A a c y s '. for beams r = 0.6 and 1.6%, for slabs r = 0.30%.

85 Shear Resistance of Flexural Members R V n V c V s V 2 f c b. d C w ' V s A v. f s y. d Department of Civil Engineering

86 Department of Civil Engineering Shear Resistance of Slabs in Two- Way Shear d b f d b f b d d b f R c c s c c 0 ' 0 ' 0 0 ' 4, 2 1 2, min nominal ' '.95 0 c s simulation c f f

87 Eccentrically Loaded Columns 1. Basic Assumptions

88 P n [K] Columns Axial Load Compression Control Interaction Diagram Limit State Safe Behavior Pure Bending Balanced Failure Tension Control M n [K ft] Department of Civil Engineering

89 Analysis of Possible Cases of Cross Section Behavior (a) (b) Interaction Diagram for Eccentrically Compressed Columns; (a) Cross Sections Type I, (b) Cross Sections Type II.

90 Simulated Interaction Diagrams concrete strength of 8 ksi (55 MPa) tied columns cast-in-place red dots = nominal values

91 Interaction Diagrams For Concrete 3 ksi (21 MPa) (tied columns, cast-in-place)

94 Interaction Diagrams concrete 12 ksi (85 MPa) tied columns, cast-in-place

95 Bearing Resistance of Concrete R 0.85 f ' c A 1 A A A 1 A 2 Department of Civil Engineering

96 Bias Factor Bias Factor of Resistance for Beams, Flexure 1.20 Bias factor for resistance - R/C beam, flexure Reinforcement ratio: r = r = Specified concrete compressive strength [psi]

97 Coefficient of variation Coefficient of Variation of Resistance for Beams, Flexure 0.10 Coefficient of variation for resistance - R/C beam, flexure Reinforcement ratio: r = r = Specified concrete compressive strength [psi]

98 Bias Facto Bias Factor of Resistance for Beams, Shear Bias factor for resistance - R/C beam shear with and without shear reinforcement no shear reinforcement s = 6 in, Av min s = 8 in, Av min s = 12 in, Av min s = 6 in, Av min real (2 #3) s = 8 in, Av min real (2 #3) s = 12 in, Av min real (2 #3) s = 6 in, Av ave s = 8 in, Av ave s = 12 in, Av ave s = 6 in, Av max s = 8 in, Av max s = 12 in, Av max Specified concrete compressive strength [psi]

99 Coefficient of variat Coefficient of Variation of Resistance for Beams, Shear Coefficient of variation for resistance - R/C beam shear with and without shear reinforcement no shear reinforcement s = 6 in, Av min s = 8 in, Av min s = 12 in, Av min s = 6 in, Av min real (2 #3) s = 8 in, Av min real (2 #3) s = 12 in, Av min real (2 #3) s = 6 in, Av ave s = 8 in, Av ave s = 12 in, Av ave s = 6 in, Av max s = 8 in, Av max s = 12 in, Av max Specified concrete compressive strength [psi]

100 Bias Factor Bias Factor of Resistance for One way Slab, Flexure 1.10 Bias factor for resistance - R/C slab, 1-way flexure d = 4 in 1.08 d = 6 in d = 8 in Specified concrete compressive strength [psi]

101 Coefficient of variation Coefficient of Variation of Resistance for One way slab, Flexure 0.20 Coefficient of varaition for resistance - R/C slab, 1-way flexure d = 4 in 0.18 d = 6 in d = 8 in Specified concrete compressive strength [psi]

102 Bias Factor Bias Factor and Coefficient of Variation of Resistance for Concrete Bearing Strength Coefficient of variation 1.40 Bias factor for resistance - concrete bearing 0.20 Coefficient of variation for resistance - concrete bearing Specified concrete compressive strength [psi] Specified concrete compressive strength [psi]

103 Statistical Parameters of Fabrication Factor (Ellingwood, Galambos, MacGregor, Cornell) l V width of beam, b effective depth of beam, d effective depth of one-way slab, d effective depth of two-way slab, d d = 4 in d = 6 in d = 8 in depth and width of column, b 1, b area of reinforcement, A s, A v spacing of shear reinforcement, s

104 Statistical Parameters of Professional Factor (Ellingwood, Galambos, MacGregor, Cornell) R/C beams - flexure R/C beams - shear without stirrups R/C beams - shear with stirrups Axially loaded columns, tied Axially loaded columns, spiral One way slabs - flexure One way slabs - shear Two way slabs - shear Bearing strength l V

105 Monte Carlo Simulation Results - Examples Resistance parameters for concrete f c = 4000 psi (28 MPa) l V R/C beams - flexure R/C beams - shear without stirrups R/C beams - shear with stirrups Axially loaded columns, tied Axially loaded columns, spiral One way slabs - flexure One way slabs - shear Two way slabs - shear Bearing strength

106 Basic questions: How can we measure safety of a structure? How safe is safe enough? What is the target reliability? How to implement the optimum safety level? Department of Civil Engineering

107 Reliability Index, Department of Civil Engineering

108 Reliability Index, For a linear limit state function, g = R Q = 0, and R and Q both being normal random variables R s 2 R Q s 2 Q R = mean resistance Q = mean load s R = standard deviation of resistance s Q = standard deviation of load

109 Reliability index and probability of failure P F

110 Reliability Analysis Procedures Closed-form equations accurate results only for special cases First Order Reliability Methods (FORM), reliability index is calculated by iterations Second Order Reliability Methods (SORM), and other advanced procedures Monte Carlo method - values of random variables are simulated (generated by computer), accuracy depends on the number of computer simulations Department of Civil Engineering

111 Reliability Indices for R/C Beams, Flexure, (D+L) Ordinary concrete Lightweight concrete β R/C Beam in flexure, ρ=0.6% OLD β R/C Beam in flexure, ρ=0.6% f=0.85 f=0.90 f=0.95 Old data, f=0.90 D/(D+L) ϕ = 0.85 ϕ = 0.90 ϕ = 0.95 NWC, ϕ = 0.95 D/(D+L)

112 Reliability Indices for R/C Beams, Shear, (D+L) Ordinary concrete Lightweight concrete 5.0 β R/C Beam shear, no shear reinforcement, f'c = 4000psi 5.0 β R/C Beam shear, no shear reinforcement, f'c = 27.5 MPa (4000 psi) f=0.80 f=0.85 f=0.90 Old data, f=0.85 D/(D+L) ϕ = 0.70 ϕ = 0.75 ϕ = 0.80 NWC, ϕ = 0.75 D/(D+L)

113 Reliability Indices for R/C Slab, flexure, (D+L) Ordinary concrete Lightweight concrete β R/C Slab, one-way flexure β R/C Slab, one-way flexure ϕ = 0.85 ϕ = 0.90 ϕ = NWC, ϕ = f=0.85 f=0.90 f=0.95 Old data, f=0.90 D/(D+L) D/(D+L)

114 Reliability Indices for R/C Slab, one-way shear, (D+L) Ordinary concrete Lightweight concrete 5.0 β R/C Slab, one-way shear, f'c=4000psi f= β R/C Slab, one-way shear, f'c = 27.5 MPa (4000psi) ϕ = f=0.80 f= ϕ = 0.75 ϕ = Old data, f= NWC, ϕ = D/(D+L) D/(D+L)

115 Reliability Indices for R/C Slab, two-way shear, (D+L) Ordinary concrete Lightweight concrete 5.0 β R/C Slab, two-way shear, f'c=4000psi f= β R/C Slab, two-way shear, f'c = 27.5 MPa (4000psi) ϕ = f=0.80 f=0.85 Old data, f= ϕ = 0.75 ϕ = 0.80 NWC, ϕ = D/(D+L) D/(D+L)

116 Reliability Indices for Concrete bearing, (D+L) Ordinary concrete Lightweight concrete 5.0 β Concrete bearing, f'c=4000psi 5.0 β Concrete bearing, f'c = 27.5 MPa (4000psi) f=0.65 f=0.70 f=0.75 Old data, f=0.65 D/(D+L) ϕ = 0.60 ϕ = 0.65 ϕ = 0.70 NWC, ϕ = 0.65 D/(D+L)

117 What is Optimum Reliability? If reliability index is too small there are problems, even structural failures If reliability index is too large the structures are too expensive Department of Civil Engineering

118 Target Reliability Consequences of failure Economic analysis Past practice Human perception Social/political decisions Department of Civil Engineering

119 Reliability Index Selected Range of Reliability Indices for Beams, designed according to old ACI 318 Range of Target Reliability Index for Beams

120 Reliability Index Selected Range of Reliability Indices for Slabs, designed according to old ACI 318 Range of Target Reliability Index for Slabs

121 Selected Range of Reliability Indices for Columns and Plain Concrete Elements, designed according to old ACI 318 Reliability Index Range of Target Reliability Index for Columns and Plain Concrete Elements

122 Calibration Results ACI ACI Recommended f f f T R/C beams flexure R/C beams - shear w/o stirrups R/C beams - shear with stirrups Axially loaded columns, tied Axially loaded columns, spiral One way slabs flexure One way slabs shear Two way slabs shear Bearing strength

123 Load factors specified by ACI 318 and ASCE 7 The design formula specified by ACI Code The design formula specified by ASCE-7 Standard 1.4 D L < f R 0.75 (1.4 D L W) < f R 0.9 D W < f R 0.75 (1.4 D L E) < f R 1.4 D < f R 1.2 D L < f R 1.2 D L S < f R 1.2 D L S < f R 1.2 D W L S < f R 1.2 D E L S < f R 0.9 D (1.6 W or 1.0 E) < f R

124 American Concrete Institute (ACI) Department of Civil Engineering

125 Reliability Index Reliability Indices for Beams, designed according to the new ACI 318 Reliability Indices for Beams target value new, ordinary concrete new, high strength concrete new, light w eight concrete

126 Reliability Index Reliability Indices for Slabs, designed according to the new ACI 318 Reliability Indices for Slabs target value new, ordinary concrete new, high strength concrete new, light w eight concrete 0

127 Reliability Indices for Columns and Plain Concrete Elements, designed according to the new ACI 318 Reliability Index Reliability Indices for Columns and Plain Concrete Elements target value new, ordinary concrete new, high strength concrete new, light w eight concrete

128 Load factors specified by ACI 318 and Proposed Design Formula The design formula specified by ACI Code 1.4 D L < f R 0.75 (1.4 D L W) < f R 0.9 D W < f R 0.75 (1.4 D L E) < f R Proposed design formula 1.4 (D + L) < f R 1.2 D L < f R 1.2 D L S < f R 1.2 D L S < f R 1.2 D W L S < f R 1.2 D E L S < f R 0.9 D (1.6 W or 1.0 E) < f R

129 Reliability Index,. Reliability Index,. Reliability Index,. Examples of the Reliability Analysis ACI Old Statistical Data ACI New Statistical Data ACI with new load factor, 1.4(D+L) R/C beam, flexure - ACI (1.4D+1.7L) Old Statistical Data f R/C beam, flexure - ASCE-7 (1.4D or 1.2D+1.6L) New Statistical Data f 0. 5 f 0. f R/C beam, flexure - Proposed (1.2D+1.6L or 1.4D+1.4L) New Statistical Data f 0. 5 f 0. f ,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 Load Ratio D/(D+L) 0 0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 Load Ratio D/(D+L) 0 0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 Load Ratio D/(D+L)

130 Conclusions for ACI 318 Calibration Quality of materials (concrete and reinforcing steel) have improved in the last years Reliability of structures designed according to old ACI 318 is now higher than the minimum acceptable level Resistance factors can be increased by 10-15%. Therefore, for the new load factors (ASCE 7), old resistance factors are acceptable Department of Civil Engineering

131 Department of Civil Engineering