Aci Code.pdf

Reliability-based Calibration of Design Code for Concrete Structures (ACI 318)

Andrzej S. Nowak and Anna M. Rakoczy

ACI 318 Code • • • • • •

Outline Objectives New material test data Resistance parameters Reliability analysis Resistance factors Further developments

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DESIGN CODES HISTORICAL PERSPECTIVE

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

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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

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Load factors specified by ACI 318 and ASCE 7 The design formula specified by ACI 318-99 Code

The design formula specified by ASCE-7 Standard

1.4 D < f R 1.4 D + 1.7 L < f R 1.2 D + 1.6 L < f R 0.75 (1.4 D + 1.7 L + 1.7 W) < f R 1.2 D + 1.6 L + 0.5 S < f R 0.9 D + 1.3 W < f R 1.2 D + 0.5 L + 1.6 S < f R 0.75 (1.4 D + 1.7 L + 1.87 E) < f R 1.2 D + 1.6 W + 0.5 L + 0.5 S < f R 1.2 D + 1.0 E + 0.5 L + 0.2 S < f R 0.9 D – (1.6 W or 1.0 E) < f R

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

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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

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

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Considered Load Components • • • • • •

D = dead load L = live load S = snow W = wind E = earthquake Load combinations

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Assumed Statistical Data • Dead load l = 1.03-1.05, V = 0.08-0.10

• 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

Load Factor

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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

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Considered Cases • Old – Statistical data for materials from 1970’s – Design according to ACI 318-99 • New – Statistical data for materials from 2001-05 – Design according to proposed ACI 318

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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 318-12

• Provide a basis for selection of resistance factors Department of Civil Engineering

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

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

Parameters of Resistance • The mean value of R is

 R  R n  M  F P • Coefficient of variation

VR 

VM 

2

  VF    VP  2

• Bias factor

l R  l M l Fl P Department of Civil Engineering

2

Resistance Factor

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Material Factor • Available data-base from 1970’s (MacGregor) • Concrete industry provided test results (20002001 and 2003), gathered for this calibration • Code Calibration of ACI 318 (2005) is based on these recent test results

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

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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

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Strength of Ordinary Concrete Ready mix concrete 3,000 psi 3,500 psi 4,000 psi 4,500 psi 5,000 psi 6,000 psi

(21 MPa) (24 MPa) (28 MPa) (31 MPa) (35 MPa) (42 MPa)

Plant-cast concrete 5,000 5,500 6,000 6,500

psi psi psi psi

(35 (38 (42 (45

MPa) MPa) MPa) MPa)

Strength of Concrete Light-weight concrete 3,000 psi 3,500 psi 4,000 psi 5,000 psi

(21 (24 (28 (35

MPa) MPa) MPa) MPa)

High strength concrete 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)

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 (12.5-15 mm) Department of Civil Engineering

Ordinary Concrete – Number of Samples

Lightweight Concrete – Number of Samples

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

Probability Paper Data is plotted on the normal probability paper. A normal distribution function is represented by a straight line.

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Probability Paper

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Ordinary Concrete – CDF of Strength 4

fc’ = 3,000 psi, 21 MPa

2

1

0

-1

-2

-3

Source

10000

9000

8000

7000

6000

5000

4000

3000

2000

1000

-4

0

Inverse Normal Distribution

3

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)

Strength [psi]

Ordinary Concrete – CDF of Strength 4

fc’ = 3,000 psi, 21 MPa

l = 1.33 V = 0.145

2

1

0

-1

 = 4000

-2

s = 580

-3

All sources (samples: 4016)

Approximation

10000

9000

8000

7000

6000

5000

4000

3000

2000

1000

-4

0

Inverse Normal Distribution

3

Strength [psi]

Ordinary Concrete – CDF of Strength 4

fc’ = 3,500 psi, 25 MPa

2

1

0

-1

-2

-3

Source 16 (samples: 339)

Source 19 (samples:

99)

10000

9000

8000

7000

6000

5000

4000

3000

2000

1000

-4

0

Inverse Normal Distribution

3

Strength [psi]

Ordinary Concrete – CDF of Strength 4

fc’ = 3,500 psi, 25 MPa l = 1.24 V = 0.115

2

1

0

 = 4330

-1

s = 500

-2

-3

All sources (samples:

527)

Approximation

10000

9000

8000

7000

6000

5000

4000

3000

2000

1000

-4

0

Inverse Normal Distribution

3

Strength [psi]

Ordinary Concrete – CDF of Strength 4

fc’ = 4,000 psi, 28 MPa

2

1

0

-1

-2

-3

Source

1 (samples: 316)

10000

9000

8000

7000

6000

2 (samples: 156)

Source 13 (samples: 274)

Source 14 (samples: 269)

Source 15 (samples: 220)

Source 16 (samples: 584)

Source 18 (samples:

Source 19 (samples: 533)

99)

Source

5000

4000

3000

2000

1000

-4

0

Inverse Normal Distribution

3

Strength [psi]

Ordinary Concrete – CDF of Strength 4

fc’ = 4,000 psi, 28 MPa

l = 1.21 V = 0.155

2

1

0

 = 4850 s = 750

-1

-2

-3

All sources (samples: 2784)

Approximation

10000

9000

8000

7000

6000

5000

4000

3000

2000

1000

-4

0

Inverse Normal Distribution

3

Strength [psi]

Ordinary Concrete – CDF of Strength 4

fc’ = 4,500 psi

2

1

0

-1

-2

-3

Source 12 (samples: 839)

Source 13 (samples: 298)

Source 15 (samples: 164)

Source 23 (samples: 346)

10000

9000

8000

7000

6000

5000

4000

3000

2000

1000

-4

0

Inverse Normal Distribution

3

Strength [psi]

Ordinary Concrete – CDF of Strength 4

fc’ = 4,500 psi

l = 1.19 V = 0.16

2

1

0

 = 5350

-1

s = 850

-2

-3

All sources (samples: 1919)

Approximation

10000

9000

8000

7000

6000

5000

4000

3000

2000

1000

-4

0

Inverse Normal Distribution

3

Strength [psi]

Ordinary Concrete – CDF of Strength 4

fc’ = 5,000 psi, 35 MPa

2

1

0

-1

-2

-3

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)

10000

9000

8000

7000

6000

5000

4000

3000

2000

1000

-4

0

Inverse Normal Distribution

3

Strength [psi]

Ordinary Concrete – CDF of Strength 4

fc’ = 5,000 psi, 35 MPa

l = 1.22 V = 0.125

2

1

0

-1

 = 6100

-2

s = 760

-3

All sources (samples: 1722)

Approximation

10000

9000

8000

7000

6000

5000

4000

3000

2000

1000

-4

0

Inverse Normal Distribution

3

Strength [psi]

Ordinary Concrete – CDF of Strength 4

fc’ = 6,000 psi, 42 MPa

l = 1.22 V = 0.075

2

1

0

 = 7340

-1

s = 550

-2

-3

All sources (samples: 130)

Approximation

10000

9000

8000

7000

6000

5000

4000

3000

2000

1000

-4

0

Inverse Normal Distribution

3

Strength [psi]

Lightweight Concrete – CDF of Strength l = 1.430 V = 0.155

 = 4290 s = 665 fc’= 3000 psi, 21 MPa

Lightweight Concrete – CDF of Strength l = 1.296 V = 0.122

 = 4535 s = 555 fc’= 3500 psi, 25 MPa

Lightweight Concrete – CDF of Strength l = 1.338 V = 0.123

 = 5350 s = 660 fc’= 4000 psi, 28 MPa

Lightweight Concrete – CDF of Strength l = 1.328 V = 0.117

 = 5975 s = 700

fc’= 4500 psi , 32 MPa

Lightweight Concrete – CDF of Strength l = 1.110 V = 0.076

 = 5550 s = 420

fc’= 5000 psi , 35 MPa

Lightweight Concrete – CDF of Strength l = 1.187 V = 0.121

 = 8070 s = 975

fc’= 6800 psi , 48 MPa

Lightweight Concrete – CDF of Strength l = 1.126 V = 0.100

 = 8000 s =805

fc’= 7100 psi, 49 MPa

Summary of the Statistical Parameters for Concrete

0.16 0.14

0.10

1.0

0.08

0.9

0.06

0.8

0.04

0.7

0.02

fc’ [psi]

fc’ [psi] 10000

9000

7000

0.00

6000

13000

11000

10000

9000

8000

7000

6000

5000

4000

3000

2000

12000

fc’ [psi]

0.6

fc’ [psi]

13000

1.1

12000

0.12

11000

1.2

5000

1.3

Lightweigh Concrete Ordinary, Ready Mix Ordinary, Plant Cast High Strength Recommended V for NWC Recommended V for LWC

V

4000

1.4

0.18

3000

λ

1.5

0.20

Lightweigh Concrete Ordinary, Ready Mix Ordinary, Plant Cast High Strength Recommended λ for NWC Recommended λ for LWC

2000

1.6

V

8000

l

Bias Factor and Coefficient of Variation for Compressive Strength and Shear Strength of Concrete Concrete Grade fc' (psi)

Compressive strength

Shear Strength

l

V

l

V

3000, 21 MPa

1.31

0.17

1.31

0.205

3500

1.27

0.16

1.27

0.19

4000, 28 MPa

1.24

0.15

1.24

0.18

4500

1.21

0.14

1.21

0.17

5000

1.19

0.135

1.19

0.16

5500

1.17

0.13

1.17

0.155

6000, 42 MPa

1.15

0.125

1.15

0.15

6500

1.14

0.12

1.14

0.145

7000

1.13

0.115

1.13

0.14

8000

1.11

0.11

1.11

0.135

9000

1.10

0.11

1.10

0.135

10,000

1.09

0.11

1.09

0.135

12,000, 84 MPa

1.08

0.11

1.08

0.135

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

Reinforcing Steel Bars, Grade 60 – CDF of Yield Stress Bars #3

4

2

1

0

-1

-2

-3

Yield Stress [ksi]

Source 2 (samples: 741)

Source 4 (samples: 123)

110

100

90

80

70

60

50

40

30

20

10

-4

0

Inverse Normal Distribution

3

Reinforcing Steel Bars, Grade 60 – CDF of Yield Stress 4

Bars #3 l = 1.18 V = 0.04

2

1

0

 = 71.0

-1

s = 3.0

-2

-3

All sources (samples: 864)

Approximation

110

100

90

80

70

60

50

40

30

20

10

-4

0

Inverse Normal Distribution

3

Yield Stress [ksi]

Reinforcing Steel Bars, Grade 60 – CDF of Yield Stress 4

Bars #4

2

1

0

-1

-2

-3

Source 1 (samples:

60)

Source 2 (samples: 2369)

Source 3 (samples:

60)

Source 4 (samples:

Source 5 (samples:

90)

106)

110

100

90

80

70

60

50

40

30

20

10

-4

0

Inverse Normal Distribution

3

Yield Stress [ksi]

Reinforcing Steel Bars, Grade 60 – CDF of Yield Stress 4

Bars #4

l = 1.13 V = 0.03

2

1

0

 = 67.5

-1

s = 1.9

-2

-3

All sources (samples: 2685)

Approximation

110

100

90

80

70

60

50

40

30

20

10

-4

0

Inverse Normal Distribution

3

Yield Stress [ksi]

Reinforcing Steel Bars, Grade 60 – CDF of Yield Stress 4

Bars #5

2

1

0

-1

-2

-3

Source 1 (samples:

60)

Source 2 (samples: 3333)

Source 3 (samples:

60)

Source 4 (samples:

Source 5 (samples:

90)

179)

110

100

90

80

70

60

50

40

30

20

10

-4

0

Inverse Normal Distribution

3

Yield Stress [ksi]

Reinforcing Steel Bars, Grade 60 – CDF of Yield Stress 4

Bars #5

l = 1.12 V = 0.02

2

1

0

 = 67.0

-1

s = 1.5

-2

-3

All sources (samples: 3722)

Approximation

110

100

90

80

70

60

50

40

30

20

10

-4

0

Inverse Normal Distribution

3

Yield Stress [ksi]

Reinforcing Steel Bars, Grade 60 – CDF of Yield Stress 4

Bars #6

2

1

0

-1

-2

-3

Source 1 (samples:

60)

Source 2 (samples: 1141)

Source 3 (samples:

60)

Source 4 (samples:

Source 5 (samples:

90)

104)

110

100

90

80

70

60

50

40

30

20

10

-4

0

Inverse Normal Distribution

3

Yield Stress [ksi]

Reinforcing Steel Bars, Grade 60 – CDF of Yield Stress 4

Bars #6 l = 1.12 V = 0.02

2

1

0

 = 67.0

-1

s = 1.5

-2

-3

All sources (samples: 1455)

Approximation

110

100

90

80

70

60

50

40

30

20

10

-4

0

Inverse Normal Distribution

3

Yield Stress [ksi]

Reinforcing Steel Bars, Grade 60 – CDF of Yield Stress 4

Bars #7

2

1

0

-1

-2

-3

Source 1 (samples:

60)

Source 2 (samples: 1318)

Source 3 (samples:

60)

Source 4 (samples:

Source 5 (samples:

90)

79)

110

100

90

80

70

60

50

40

30

20

10

-4

0

Inverse Normal Distribution

3

Yield Stress [ksi]

Reinforcing Steel Bars, Grade 60 – CDF of Yield Stress 4

Bars #7 l = 1.14 V = 0.03

2

1

0

 = 68.5

-1

s = 1.9

-2

-3

All sources (samples: 1607)

Approximation

110

100

90

80

70

60

50

40

30

20

10

-4

0

Inverse Normal Distribution

3

Yield Stress [ksi]

Reinforcing Steel Bars, Grade 60 – CDF of Yield Stress 4

Bars #8

2

1

0

-1

-2

-3

Source 1 (samples:

60)

Source 2 (samples: 1146)

Source 3 (samples:

60)

Source 4 (samples:

Source 5 (samples:

90)

90)

110

100

90

80

70

60

50

40

30

20

10

-4

0

Inverse Normal Distribution

3

Yield Stress [ksi]

Reinforcing Steel Bars, Grade 60 – CDF of Yield Stress 4

Bars #8 l = 1.13 V = 0.025

2

1

0

 = 68.0

-1

s = 1.6

-2

-3

All sources (samples: 1446)

Approximation

110

100

90

80

70

60

50

40

30

20

10

-4

0

Inverse Normal Distribution

3

Yield Stress [ksi]

Reinforcing Steel Bars, Grade 60 – CDF of Yield Stress 4

Bars #9

2

1

0

-1

-2

-3

Source 1 (samples:

60)

Source 2 (samples: 1290)

Source 3 (samples:

60)

Source 4 (samples:

Source 5 (samples:

90)

73)

110

100

90

80

70

60

50

40

30

20

10

-4

0

Inverse Normal Distribution

3

Yield Stress [ksi]

Reinforcing Steel Bars, Grade 60 – CDF of Yield Stress 4

Bars #9 l = 1.14 V = 0.02

2

1

0

 = 68.5

-1

s = 1.5

-2

-3

All sources (samples: 1573)

Approximation

110

100

90

80

70

60

50

40

30

20

10

-4

0

Inverse Normal Distribution

3

Yield Stress [ksi]

Reinforcing Steel Bars, Grade 60 – CDF of Yield Stress 4

Bars #10

2

1

0

-1

-2

-3

Source 1 (samples:

60)

Source 2 (samples:

825)

Source 3 (samples:

60)

Source 4 (samples:

70)

Source 5 (samples:

74)

110

100

90

80

70

60

50

40

30

20

10

-4

0

Inverse Normal Distribution

3

Yield Stress [ksi]

Reinforcing Steel Bars, Grade 60 – CDF of Yield Stress 4

Bars #10 l = 1.13 V = 0.02

2

1

0

 = 68.0

-1

s = 1.4

-2

-3

All sources (samples: 1089)

Approximation

110

100

90

80

70

60

50

40

30

20

10

-4

0

Inverse Normal Distribution

3

Yield Stress [ksi]

Reinforcing Steel Bars, Grade 60 – CDF of Yield Stress 4

Bars #11

2

1

0

-1

-2

-3

Source 1 (samples:

60)

Source 2 (samples: 1019)

Source 3 (samples:

60)

Source 4 (samples:

Source 5 (samples:

90)

87)

110

100

90

80

70

60

50

40

30

20

10

-4

0

Inverse Normal Distribution

3

Yield Stress [ksi]

Reinforcing Steel Bars, Grade 60 – CDF of Yield Stress 4

Bars #11 l = 1.13 V = 0.02

2

1

0

 = 68.0

-1

s = 1.5

-2

-3

All sources (samples: 1316)

Approximation

110

100

90

80

70

60

50

40

30

20

10

-4

0

Inverse Normal Distribution

3

Yield Stress [ksi]

Reinforcing Steel Bars, Grade 60 – CDF of Yield Stress 4

Bars #14 l = 1.14 V = 0.02

2

1

0

 = 68.5

-1

s = 1.5

-2

-3

Yield [ksi]

All sources - Source 3 (samples: 12)

Approximation

110

100

90

80

70

60

50

40

30

20

10

-4

0

Inverse Normal Distribution

3

Reinforcing Steel Bars, Grade 60 – CDF of Yield Stress 4

for simulation: l = 1.13 V = 0.03

3 2 1 0 -1 -2 -3

110

100

90

80

70

60

50

40

30

20

10

0

-4

#3

#4

#5

#6

#7

#8

#9

#10

#11

#14

All Size

Approximation

Yield Stress [ksi]

Reinforcing Steel Bars, Grade 60 (420 MPa) – Statistical Parameters #3

l 1.18

V 0.04

#4 #5 #6 #7 #8 #9 #10 #11 #14

1.13 1.12 1.12 1.14 1.13 1.14 1.13 1.13 1.14

0.03 0.02 0.02 0.03 0.025 0.02 0.02 0.02 0.02

Bar Size

Prestressing Strands Grade 270 (1800 MPa) – Number of Samples

Total Number of Samples 47,421

Prestressing Steel (7-wire strands), Grade 270 CDF of Breaking Stress 4

Strands 0.5 in (12 mm)

2

1

0

-1

-2

-3

Breaking Stress [ksi]

Source 1 (samples:

3908)

Source 2 (samples: 1158)

Source 3 (samples:

268)

Source 4 (samples: 9795)

Source 5 (samples: 18258)

400

380

360

340

320

300

280

260

240

220

200

180

160

140

120

100

80

60

40

20

-4

0

Inverse Normal Distribution

3

Prestressing Steel (7-wire strands), Grade 270 CDF of Breaking Stress 4

Strands 0.5 in (12 mm) l = 1.04 V = 0.015

2

1

 = 280

0

s=4

-1

-2

-3

Breaking Stress [ksi]

All sources (samples: 33387)

Approximation

400

380

360

340

320

300

280

260

240

220

200

180

160

140

120

100

80

60

40

20

-4

0

Inverse Normal Distribution

3

Prestressing Steel (7-wire strands), Grade 270 CDF of Breaking Stress 4

Strands 0.6 in

2

1

0

-1

-2

-3

Source 1 (samples:

700)

Source 2 (samples:

Source 3 (samples:

212)

Source 4 (samples: 3442)

Source 5 (samples: 8889)

785)

400

380

360

340

320

300

280

260

240

220

200

180

160

140

120

100

80

60

40

20

-4 0

Inverse Normal Distribution

3

Breaking Stress [ksi]

Prestressing Steel (7-wire strands), Grade 270 CDF of Breaking Stress 4

Strands 0.6 in (12 mm) l = 1.02 V = 0.015

2

1

0

 = 275

-1

s=4

-2

-3

All sources (samples: 14028)

Approximation

400

380

360

340

320

300

280

260

240

220

200

180

160

140

120

100

80

60

40

20

-4

0

Inverse Normal Distribution

3

Breaking Stress [ksi]

Prestressing Steel – Statistical Parameters 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)

22 83 114 66

1.07 1.11 1.11 1.12

0.01 0.025 0.01 0.02

270 ksi (1900 MPa)

3/8 (9.5 mm) 7/16 (11 mm) 1/2 (12.5 mm) 0.6 (15 mm)

54 16 33570 14028

1.04 1.07 1.04 1.02

0.02 0.02 0.015 0.015

Grade

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

Bending Moment Resistance

a  R  As  f y  d   2  As. f y a ' 0.85 f c .b for beams r = 0.6 and 1.6%, for slabs r = 0.30%. Department of Civil Engineering

Shear Resistance of Flexural Members

R  Vn  Vc  Vs VC  2 f c bw .d '

Vs 

Av . f y .d s

Department of Civil Engineering

Shear Resistance of Slabs in TwoWay Shear  2  R  min  1    c

f

sd    1  2 f c' b0 d , 4   2b0 

   2 f c' b0 d , 

' c simulations

f

 0.95

' c nominal

Department of Civil Engineering

 f b0 d   ' c

Eccentrically Loaded Columns 1. Basic Assumptions

Columns Interaction Diagram

Pn [K] Axial Load 1500

Compression Control 1250

Limit State

1000 750

Balanced Failure

Safe Behavior 500 250

Tension Control Pure Bending 100

200

300

400

500

600

Department of Civil Engineering

M n [K ft]

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.

Simulated Interaction Diagrams • concrete strength of 8 ksi (55 MPa) • tied columns • cast-in-place

red dots = nominal values

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

Interaction Diagrams For Concrete 5 ksi (35 MPa) (tied columns, cast-in-place)

Interaction Diagrams For Concrete 8 ksi (55 MPa) (tied columns, cast-in-place)

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

Bearing Resistance of Concrete R  0.85 f c A1 '

A2 A1 2 1

A1 A2

Department of Civil Engineering

Bias Factor of Resistance for Beams, Flexure Bias factor for resistance - R/C beam, flexure 1.20

1.19

Reinforcement ratio: 1.18

r = 0.006

Bias Factor

1.17

r = 0.016

1.16

1.15

1.14

1.13

1.12

1.11

1.10 0

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

11000

12000

13000

14000

Specified concrete compressive strength [psi]

Coefficient of Variation of Resistance for Beams, Flexure Coefficient of variation for resistance - R/C beam, flexure 0.10

0.09

Coefficient of variation

0.08

0.07

0.06

0.05

0.04

Reinforcement ratio: 0.03

r = 0.006

0.02

r = 0.016

0.01

0.00 0

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

11000

12000

13000

14000

Specified concrete compressive strength [psi]

Bias Factor of Resistance for Beams, Shear Bias factor for resistance - R/C beam shear with and without shear reinforcement 1.34

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

1.32 1.30 1.28

Bias Factor

1.26 1.24 1.22 1.20 1.18 1.16 1.14 1.12 1.10 0

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000 11000 12000 13000 14000

Specified concrete compressive strength [psi]

Coefficient of Variation of Resistance for Beams, Shear Coefficient of variation for resistance - R/C beam shear with and without shear reinforcement

0.32

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

0.30 0.28

Coefficient of variation

0.26 0.24 0.22 0.20 0.18 0.16 0.14 0.12 0.10 0

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000 11000 12000 13000 14000

Specified concrete compressive strength [psi]

Bias Factor of Resistance for One way Slab, Flexure Bias factor for resistance - R/C slab, 1-way flexure 1.10

d = 4 in d = 6 in 1.08

Bias Factor

d = 8 in

1.06

1.04

1.02

1.00 0

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

11000

12000

13000

14000

Specified concrete compressive strength [psi]

Coefficient of Variation of Resistance for One way slab, Flexure Coefficient of varaition for resistance - R/C slab, 1-way flexure 0.20

d = 4 in d = 6 in

Coefficient of variation

0.18

d = 8 in

0.16

0.14

0.12

0.10 0

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

11000

12000

13000

14000

Specified concrete compressive strength [psi]

Bias Factor and Coefficient of Variation of Resistance for Concrete Bearing Strength Bias factor for resistance - concrete bearing

Coefficient of variation for resistance - concrete bearing

1.40

0.20

1.35

0.18

Coefficient of variation

Bias Factor

1.30

1.25

1.20

1.15

1.10

0.16

0.14

0.12 1.05

1.00 0

1000

2000

3000

4000

5000

6000

7000

Specified concrete compressive strength [psi]

0.10 0

1000

2000

3000

4000

5000

6000

7000

Specified concrete compressive strength [psi]

Statistical Parameters of Fabrication Factor (Ellingwood, Galambos, MacGregor, Cornell) l

V

width of beam, b

1.01

0.04

effective depth of beam, d

0.99

0.04

effective depth of one-way slab, d

0.92

0.12

d = 4 in

1.03

0.09

d = 6 in

1.02

0.06

d = 8 in

1.015

0.04

depth and width of column, b1, b2

1.005

0.04

area of reinforcement, As, Av

1.00

0.015

spacing of shear reinforcement, s

1.00

0.04

effective depth of two-way slab, d

Statistical Parameters of Professional Factor (Ellingwood, Galambos, MacGregor, Cornell) l

V

R/C beams - flexure

1.02

0.06

R/C beams - shear without stirrups

1.16

0.11

R/C beams - shear with stirrups

1.075 0.10

Axially loaded columns, tied

1.00

0.08

Axially loaded columns, spiral

1.05

0.06

One way slabs - flexure

1.02

0.06

One way slabs - shear

1.16

0.11

Two way slabs - shear

1.16

0.11

Bearing strength

1.02

0.06

Monte Carlo Simulation Results - Examples Resistance parameters for concrete fc’ = 4000 psi (28 MPa) l

V

R/C beams - flexure

1.14

0.08

R/C beams - shear without stirrups

1.27

0.23

R/C beams - shear with stirrups

1.235 0.15

Axially loaded columns, tied

1.22

0.145

Axially loaded columns, spiral

1.29

0.14

One way slabs - flexure

1.055 0.14

One way slabs - shear

1.165 0.255

Two way slabs - shear

1.305 0.24

Bearing strength

1.275 0.17

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

Reliability Index, 

Department of Civil Engineering

Reliability Index,  For a linear limit state function, g = R – Q = 0, and R and Q both being normal random variables

  

R

 Q 

s s 2 R

2 Q

R = mean resistance Q = mean load sR = standard deviation of resistance sQ = standard deviation of load

Reliability index and probability of failure PF 10-1 10-2 10-3 10-4 10-5 10-6 10-7 10-8 10-9

 1.28 2.33 3.09 3.71 4.26 4.75 5.19 5.62 5.99

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

Reliability Indices for R/C Beams, Flexure, (D+L) Ordinary concrete

Lightweight concrete

R/C Beam in flexure, ρ=0.6%

R/C Beam in flexure, ρ=0.6%

6.0

6.0

β

β

OLD

5.0

5.0

4.0

4.0

3.0

3.0

f=0.85 2.0

ϕ = 0.85 2.0

f=0.90 f=0.95

1.0

D/(D+L)

0.0 0.1

0.2

0.3

ϕ = 0.95

1.0

Old data, f=0.90 0.0

ϕ = 0.90

0.4

0.5

0.6

0.7

0.8

0.9

1.0

NWC, ϕ = 0.95

D/(D+L)

0.0 0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Reliability Indices for R/C Beams, Shear, (D+L) Ordinary concrete

Lightweight concrete

R/C Beam shear, no shear reinforcement, f'c = 4000psi

5.0

R/C Beam shear, no shear reinforcement, f'c = 27.5 MPa (4000 psi)

5.0

β

β

4.0

4.0

3.0

3.0

2.0

2.0

ϕ = 0.70

f=0.80

ϕ = 0.75

f=0.85

1.0

1.0

f=0.90 Old data, f=0.85

ϕ = 0.80 NWC, ϕ = 0.75

D/(D+L)

D/(D+L)

0.0

0.0 0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Reliability Indices for R/C Slab, flexure, (D+L) Ordinary concrete

Lightweight concrete

R/C Slab, one-way flexure

R/C Slab, one-way flexure

5.0

5.0

ϕ = 0.85

β

β

ϕ = 0.90

4.0

4.0

3.0

3.0

2.0

2.0

ϕ = 0.95 NWC, ϕ = 0.90

f=0.85 f=0.90

1.0

1.0

f=0.95 Old data, f=0.90

D/(D+L)

D/(D+L) 0.0

0.0 0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Reliability Indices for R/C Slab, one-way shear, (D+L) Ordinary concrete

Lightweight concrete R/C Slab, one-way shear, f'c = 27.5 MPa (4000psi)

R/C Slab, one-way shear, f'c=4000psi 5.0

5.0

β

f=0.75 f=0.80

4.0

ϕ = 0.70

β

ϕ = 0.75

4.0

ϕ = 0.80

f=0.85 Old data, f=0.85

3.0

NWC, ϕ = 0.75

3.0

2.0

2.0

1.0

1.0

D/(D+L)

D/(D+L) 0.0

0.0

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Reliability Indices for R/C Slab, two-way shear, (D+L) Ordinary concrete

Lightweight concrete R/C Slab, two-way shear, f'c = 27.5 MPa (4000psi)

R/C Slab, two-way shear, f'c=4000psi 5.0

5.0

β

β

f=0.75

f=0.80

4.0

ϕ = 0.70 ϕ = 0.75

4.0

ϕ = 0.80

f=0.85 Old data, f=0.85

3.0

NWC, ϕ = 0.75

3.0

2.0

2.0

1.0

1.0

D/(D+L)

D/(D+L) 0.0

0.0 0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Reliability Indices for Concrete bearing, (D+L) Ordinary concrete

Lightweight concrete Concrete bearing, f'c = 27.5 MPa (4000psi)

Concrete bearing, f'c=4000psi

5.0

5.0

β

β 4.0

4.0

3.0

3.0

2.0

2.0

ϕ = 0.60

f=0.65

ϕ = 0.65

f=0.70

1.0

1.0

ϕ = 0.70

f=0.75 Old data, f=0.65

NWC, ϕ = 0.65

D/(D+L)

D/(D+L)

0.0

0.0 0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

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

Target Reliability • • • • •

Consequences of failure Economic analysis Past practice Human perception Social/political decisions

Department of Civil Engineering

Selected Range of Reliability Indices for Beams, designed according to “old” ACI 318

Reliability Index

Range of Target Reliability Index for Beams 5.0 4.5 4.0 3.5 3.0 2.5 2.0 1.5 1.0 0.5 0.0

Selected Range of Reliability Indices for Slabs, designed according to “old” ACI 318 Range of Target Reliability Index for Slabs 6.0

Reliability Index

5.0 4.0 3.0 2.0 1.0 0.0

Selected Range of Reliability Indices for Columns and Plain Concrete Elements, designed according to “old” ACI 318 Range of Target Reliability Index for Columns and Plain Concrete Elements 7.0

Reliability Index

6.0 5.0 4.0 3.0 2.0 1.0 0.0

Calibration Results ACI 318-99

ACI 318-05

Recommended

f

f

f

T

• R/C beams – flexure

0.90

0.90

0.90

3.5

• R/C beams - shear w/o stirrups

0.85

0.75

0.85

2.5

• R/C beams - shear with stirrups

0.85

0.75

0.85

3.5

• Axially loaded columns, tied

0.70

0.65

0.70

4.0

• Axially loaded columns, spiral

0.75

0.70

0.75

4.0

• One way slabs – flexure

0.90

0.90

0.90

2.5

• One way slabs – shear

0.85

0.75

0.85

2.5

• Two way slabs – shear

0.85

0.75

0.85

2.5

• Bearing strength

0.70

0.65

0.70

3.0

Load factors specified by ACI 318 and ASCE 7 The design formula specified by ACI 318-99 Code

The design formula specified by ASCE-7 Standard

1.4 D + 1.7 L < f R 0.75 (1.4 D + 1.7 L + 1.7 W) < f R 0.9 D + 1.3 W < f R 0.75 (1.4 D + 1.7 L + 1.87 E) < f R

1.4 D < f R 1.2 D + 1.6 L < f R 1.2 D + 1.6 L + 0.5 S < f R 1.2 D + 0.5 L + 1.6 S < f R 1.2 D + 1.6 W + 0.5 L + 0.5 S < f R 1.2 D + 1.0 E + 0.5 L + 0.2 S < f R 0.9 D – (1.6 W or 1.0 E) < f R

American Concrete Institute (ACI)

Department of Civil Engineering

Reliability Indices for Beams, designed according to the “new” ACI 318 Reliability Indices for Beams

Reliability Index

5 4

target value new , ordinary concrete

3 2 1 0

new , high strength concrete new , light w eight concrete

Reliability Indices for Slabs, designed according to the “new” ACI 318 Reliability Indices for Slabs

Reliability Index

5 4 3

target value new , ordinary concrete

2 1 0

new , high strength concrete new , light w eight concrete

Reliability Indices for Columns and Plain Concrete Elements, designed according to the “new” ACI 318 Reliability Indices for Columns and Plain Concrete Elements 8

Reliability Index

7 6 5 4

target value

3

new , ordinary concrete

2 1 0

new , high strength concrete new , light w eight concrete

Load factors specified by ACI 318 and Proposed Design Formula The design formula specified by ACI 318-99 Code 1.4 D + 1.7 L < f R 0.75 (1.4 D + 1.7 L + 1.7 W) < f R 0.9 D + 1.3 W < f R 0.75 (1.4 D + 1.7 L + 1.87 E) < f R

Proposed design formula 1.4 (D + L) < f R 1.2 D + 1.6 L < f R 1.2 D + 1.6 L + 0.5 S < f R 1.2 D + 0.5 L + 1.6 S < f R 1.2 D + 1.6 W + 0.5 L + 0.5 S < f R 1.2 D + 1.0 E + 0.5 L + 0.2 S < f R 0.9 D – (1.6 W or 1.0 E) < f R

Examples of the Reliability Analysis ACI 318-99 Old Statistical Data

ACI 318-05 New Statistical Data

ACI 318-05 with new load factor, 1.4(D+L)

R/C beam, flexure - ACI 318-99 (1.4D+1.7L) Old Statistical Data

R/C beam, flexure - ASCE-7 (1.4D or 1.2D+1.6L) New Statistical Data

R/C beam, flexure - Proposed (1.2D+1.6L or 1.4D+1.4L) New Statistical Data

7

. Reliability Index, 

Reliability Index, 

f0.

6 5 4 3 2

6 5 4 3 2

1

1

0

0 0

0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9

Load Ratio D/(D+L)

1

f0.5 f0. f0.5

7 .

7

8

f0.5 f0. f0.5

Reliability Index, 

8 .

8

6 5 4 3 2 1 0

0

0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9

Load Ratio D/(D+L)

1

0

0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9

Load Ratio D/(D+L)

1

Conclusions for ACI 318 Calibration • Quality of materials (concrete and reinforcing steel) have improved in the last 20-30 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

Department of Civil Engineering

Thank you

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