10-Reliability-based Sensitivity Analysis for Prestressed Concrete Girder Bridges

81PCI Journal | Fal l 2013

The new generation of load and resistance factor design (LRFD) codes is based on probabilistic methods. Structural performance depends on the uncertainty involved in the applied load and in the load-carrying capacity. Therefore, the load and resistance factors represent partial safety factors and their values are a function of bias factors (ratios of mean to nominal) and coefficients of variation.1,2

The primary load combination for highway bridges includes dead load, live load, and dynamic load. Each of the individual components is random in nature. Moreover, the live-load effect also depends on truck position on the bridge and load distribution factor. The statistical param-eters of live load are taken from the recent study presented by Rakoczy3 and Nowak et al.4 based on an extensive weigh-in-motion survey.

The uncertainty in the resistance model is caused by vari-ability in material properties, dimensions, and analytical procedures. Each parameter can be considered as a random variable and can be described by statistical parameters such as the mean value , standard deviation , coefficient of variation V, probability density function, and cumula-tive distribution function (CDF). Assessing the statistical parameters requires extensive material and component testing. However, because of the prohibitive cost, full-scale

Over the past 30 years, both material properties and bridge loads have changed.

The objective of this paper is to present the results of a reli-ability analysis for prestressed concrete girders using the most recent live load and resistance models. Girder spans ranging from 80 to 200ft (24 to 61m) and girder spacings of 8 and 10ft (2.4 and 3m) are included.

The paper also presents a derivation of sensitivity functions for various parameters that affect performance of prestressed concrete girders. The developed procedure is demonstrated on representative girders.

Reliability-based sensitivity analysis for prestressed concrete girder bridges

Anna M. Rakoczy and Andrzej S. Nowak

Fal l 2013 | PCI Journal82

(13mm) and 0.6in. (15mm) diameter. The data were obtained from five U.S. fabricators.

For a more efficient interpretation, the statistical data were plotted on normal probability paper. The properties and use of normal probability paper can be found in textbooks on statistics and probability theory, for example, Nowak and Collins.8 The most valuable characteristic of probability paper is easy evaluation of the statistical parameters as well as type of the distribution function. The vertical axis on normal probability paper is the inverse of the standard cumulative distribution and represents the number of standard deviations from the mean value. If the CDF curve is close to a straight line, then the considered variable has a normal distribution. In this case, the mean value and standard deviation can be read directly from the CDF plot. The mean value is the horizontal coordinate of intersection with the CDF and the slope of the CDF is the inverse of standard deviation. Bias factor is defined as the ratio of the mean to nominal value, and the coefficient of variation is calculated as a standard deviation divided by the mean.1,2

First, the CDF of the ultimate strength of prestressing strands is plotted on normal probability paper. The CDFs of tensile strength of prestressing strands of 0.5 and 0.6in. (13 and 15mm) diameters are close to a straight line (Fig.1 and 2). This is more visible when all of the data points are plotted together.

The statistical parameters such as mean value , bias factor , and coefficient of variation V can be determined directly from the graph representing the CDF. Table1 summarizes the statistical parameters for prestressing strands.

The statistical parameters for ordinary concrete were obtained by Nowak et al.,9 including the test data of 28-day

tests are limited or impossible. Therefore, the behavior of large bridge components must be evaluated using analyti-cal methods and simulations.

The objective of this study is to calculate the reliability in-dices for prestressed concrete girders using new statistical parameters for material and load components. The sensitiv-ity analysis was performed and the results presented for resistance and load parameters that can affect reliability indices of prestressed concrete beams. The analysis is presented for American Association of State Highway and Transportation (AASHTO) type girders5 and Nebraska University (NU) girders.6,7

Research significance

Prestressed, precast concrete beams are widely used in highway multigirder bridges because of lower construction costs and increased span length. Recently, new and updated statistical parameters are available for material properties and live load on highway bridges. Therefore, there is a need to evaluate the performance of prestressed concrete beams based on new load and resistance models. The sensitivity analysis performed establishes a relationship be-tween resistance and load parameters and reliability index and thus helps to identify the most critical parameters for prestressed concrete beams. The results of this study can serve as a basis for verification of the current design code provisions, including review of load and resistance factors.

Statistical parameters of material

The statistical parameters of the material factor for tensile strength of prestressing strands were developed based on new data that include 47,421samples of strands of 0.5in.

Figure 1. Cumulative density functions of tensile strength of 0.5in. diameter prestressing strands. Source: Reproduced from Nowak and Rakoczy (2012). Note: fu = ultimate strength of reinforcing steel. 1in. = 25.4mm; 1ksi = 6.895MPa.

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Source 2 (1158)

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Total (33,387)

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The CDFs for each nominal compressive strength from 3000 to 12,000psi (21 to 84MPa) were plotted on normal probability paper (Fig.3). Table2 lists the recommended bias factor and coefficients of variation V of compressive strength . The statistical parameters are given separately

compressive strength of concrete for standard cylinders, 6in. 12in. (150mm 300mm), ordinary concrete with nominal compressive strength of 3000 to 6500psi (21 to 45MPa), and high-strength concrete with nominal com-pressive strength of 7000 to 12,000psi (49 to 84MPa).

Figure 2. Cumulative density functions of tensile strength of 0.6in. diameter prestressing strands. Source: Reproduced by permission from Nowak and Rakoczy (2012). Note: fu = ultimate strength of reinforcing steel. 1in. = 25.4mm; 1ksi = 6.895MPa.

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Table 1. Summary of the statistical parameters for prestressing

Strand diameter, in. Number of samples Bias factor Coefficient of variation V

0.5 33,387 1.04 0.017

0.6 14,028 1.02 0.015

Source: Reproduced from Nowak and Rakoczy (2012). Note: 1in. = 25.4mm.

Figure 3. Cumulative density functions of compressive strength of concrete. Source: Reproduced by permission from Nowak et al. (2011). Note: = specified compressive strength of concrete at 28days. 1psi = 6.895kPa.

fc = 3000 psi fc = 3500 psi fc = 4000 psi fc = 4500 psi fc = 5000 psi fc = 5500 psi fc = 6000 psi fc = 6500 psi fc = 7000 psi fc = 8000 psi fc = 9000 psi fc = 10,000 psi fc = 12,000 psi

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coefficient of variation V of fy is 0.03.9

Resistance models

The resistance of a structural component R is a random variable related to uncertainties resulting from material properties and dimensions. It is convenient to consider R as a product of nominal value Rn, material factor M, fabrica-tion factor F, and professional factor P as is expressed in Eq.(1).

R = RnMFP (1)

The nominal capacity of prestressed concrete girders was determined according to AASHTO LRFD Bridge Design Specification provisions for flexure. The formula for bend-

for compressive strength and shear strength. It was as-sumed that the bias factor for shear strength is the same as for compressive strength. The coefficient of variation V of shear strength was assumed to be larger than that for the corresponding by 20%.10

Nowak et al9 analyzed the test data for steel reinforcing bars with a yield strength fy of 60ksi (420MPa) and a wide range of bar sizes from no.3 through no.14 (10M through 43M). The CDFs of the yield strength of reinforcing steel fy for various bar sizes were plotted on normal probability paper (Fig.4). The bias factor for reinforcing steel bars is in the range of 1.12 to 1.14 for most diameters, except no.3 reinforcing bars, for which the bias factor is 1.18. The coefficient of variation of fy is small, between 0.02 and 0.04. The recommended bias factor for fy is 1.13, and the

Figure 4. Cumulative density functions of yield strength of reinforcing bars. Source: Reproduced by permission from Nowak et al. (2011). Note: fu = ultimate strength of reinforcing steel. No.3 = 10M; no.4 = 13M; no.5 = 16M; no.6 = 19M; no.7 = 22M; no.8 = 25M; no.9 = 29M; no.10 = 32M; no.11 = 36M; 1ksi = 6.895MPa.

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No. 5 No. 6

No. 7 No. 8

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Table 2. Recommended statistical parameters for compressive strength of concrete

Compressive strength of concrete f 'c , psi

Compressive strength Shear strength

Bias factor Coefficient of variation V Bias factor Coefficient of variation V

4000 1.24 0.150 1.24 0.180

5000 1.19 0.135 1.19 0.160

6000 1.15 0.125 1.15 0.150

7000 1.13 0.115 1.13 0.140

8000 1.11 0.110 1.11 0.135

9000 1.10 0.110 1.10 0.135

10,000 1.09 0.110 1.09 0.135

12,000 1.08 0.110 1.08 0.135

Note: 1psi = 6.895kPa.


ing moment Eq.(2) is based on AASHTO LRFD specifica-tions (Eq.[5.7.3.2.2-1])5 provisions.

(2)

where

Aps = area of prestressing steel

fps = average stress in prestressing steel at nominal bend-ing resistance

dp = distance from extreme compression fiber to centroid of prestressing tendons

a = depth of the equivalent stress block = c1

c = distance from extreme compression fiber to neutral axis

1 = stress block factor

As = area of mild steel tensile reinforcement

fs = stress in mild steel tensile reinforcement at nominal flexural resistance

ds = distance from extreme compression fiber to centroid of mild steel tensile reinforcement

= area of mild steel compressive reinforcement

= stress in mild steel compressive reinforcement at nominal flexural resistance

= distance from extreme compressive fiber to the cen-troid of mild steel compressive reinforcement

b = width of compression face of member; for a flange section in compression, effective width of flange

bw = web width or diameter of a circular section

hf = compression flange depth of an I or T member

The new material test data, described in the previous section, served as a basis for material factor M. It was observed that statistical parameters for material improved over the years.

Variation in dimensions and geometry is represented by fabrication factor F. No new data are available for the fabrication factor; therefore, the recommended statistical

parameters were based on previous studies by Ellingwood et al.11 For the dimensions of cast-in-place concrete com-ponents, the recommended parameters of the F bias factor F and F coefficient of variation VF are 0.92 and 0.12, respectively, for effective depth of slab and F of 1.01 and VF of 0.04 for effective width of beam. For effective depth of prestressed concrete girder, F of 1.0 and VF of 0.025 are recommended.

For steel components, reinforcing bars, and stirrups, the statistical parameters are F of 1.0 and VF of 0.01. The area of reinforcing steel As is treated as a practically determinis-tic value, with F of 1.0 and VF of 0.015; similarly, the area of prestressing strands has a F of 1.0 and VF of 0.01.

The professional (analysis) factor P represents the varia-tion in the ratio between the actual resistance and what can be analytically predicted using accurate material strength and dimension values. The professional factors bias factor P and coefficient of variation VP for a prestressed concrete beam in flexure are 1.01 and 0.06, respectively. Those statistical parameters of P are based on the study by El-lingwood et al.11

In this study, the statistical parameters of resistance were taken from previous research.12 Nominal capacity was calculated according to AASHTO LRFD specifications. Then, each of the design parameters, which were treated as random variables, was simulated 1million times using the Monte Carlo technique, and then values of resistance were calculated accordingly. The resulting CDFs were plot-ted on normal probability paper. The mean resistance R, standard deviation of resistance R, bias factor of resis-tance R, and coefficient of variation of resistance VR were calculated using CDF plots. The analysis was performed for the NU girders and AASHTO girders with different span lengths, different spacing between the beams, and two types of concrete: of 10,000psi (69MPa) for prestressed concrete girders and of 5000psi (34MPa) for the concrete slab. Statistical parameters of resistance were about the same for all types of girders. Statistical parameters of flexural resistance for prestressed concrete beams depended only on the strand diameter. The bias fac-tor for resistance was 1.05 for strands of 0.6in. (13mm) diameter and 1.07 for strands of 0.5in. (15mm) diameter. The coefficient of variation for both cases was low, about 0.07. The bias factor for shear resistance for prestressed concrete beams was 1.20 and coefficient of variation was 0.115 regardless of the size of the strands. More detailed information about the statistical parameters is provided by Nowak and Rakoczy.12

Load and load combination models

The main load components for short- and medium-span highway bridges are dead load, live load, and dynamic


load. Environmental conditions (wind, ice, and tempera-ture) and extreme events (collisions) may have additional load effects. In this study only primary load components were considered.

Dead load is the permanent weight of structural and non-structural members of a bridge. It is convenient to separate precast concrete elements, cast-in-place components (slab), and wearing surface (concrete or asphalt). In this study, the bias factor value of cast-in-place members is 1.05 and coefficient of variation V is 0.10, whereas for precast concrete components, bias factor is 1.03 and coefficient of variation V is 0.08. For wearing surface, is 1.00 and coefficient of variation V is 0.25. The statistical parameters for dead load were taken from the literature.2,11

The most recent live-load model was developed by Ra-koczy3 and Nowak et al.4 based on an extensive weigh-in-motion survey including more than 65million vehicles at 32 different locations. Table3 lists the statistical param-eters. The values for 90, 120, and 200ft (27, 36, and 61m) were taken from research by Rakoczy3 and Nowak et al.4 For other span lengths, the results were linearly interpo-lated or extrapolated.

The parameters were derived for static load only. In the total load model, the dynamic load had to be included, and it was taken as 0.1 of the static load. The coefficient of variation V of total live load is 0.18 for all span length, as recommended in National Cooperative Highway Research Program report368.2

The statistical parameters of the total load also depend on the dead loadtolive load ratio. For longer bridges, dead load can be larger than live load, and for short-span bridges, they can be about the same. Therefore, the reli-ability analysis was performed for span lengths ranging from 80 to 200ft (24 to 61m).

In the AASHTO LRFD specifications, the design load HL-93 is either a three-axle design truck and uniform load equal to 0.64kip/ft (9.3kN/m) or a design tan-dem with two axles of 25kip (110kN) spaced at 4ft (1.2m) and a uniform load of 0.64kip/ft (9.34kN/m). The load case that produces the greater load effect governs.

The design loads are multiplied by load factors to obtain a factored total design load Qd as is presented in Eq.(3).

Qd = 1.75(LL + IM) + 1.25DC + 1.5DW (3)

where

DC = dead load of structural components and nonstruc-tural attachments

DW = dead load of wearing surface and utilities

LL = vehicular live load

IM = vehicular dynamic load allowance

Table 3. Bias factors of maximum moment due to live load in 75 years

Span, ftLL

ADTT250 ADTT1000 ADTT2500 ADTT5000 ADTT10,000

200 1.34 1.36 1.37 1.40 1.40

190 1.35 1.37 1.38 1.41 1.41

180 1.35 1.37 1.39 1.42 1.42

170 1.36 1.38 1.40 1.42 1.43

160 1.37 1.39 1.41 1.43 1.44

150 1.37 1.39 1.41 1.44 1.44

140 1.38 1.40 1.42 1.45 1.45

130 1.38 1.40 1.43 1.45 1.46

120 1.39 1.41 1.44 1.46 1.47

110 1.40 1.41 1.44 1.46 1.47

100 1.40 1.42 1.43 1.45 1.47

90 1.41 1.42 1.43 1.45 1.47

80 1.41 1.42 1.43 1.45 1.47

Source: Data from Rakoczy (2011). Note: ADTT = average daily truck traffic; LL = bias factor of live load. 1ft = 0.305m.


Moreover, the reliability index is related to the probabil-ity of failure (Eq.[6]).

(6)

where

= standard normal distributions function

The reliability index for a limit state function defined in Eq.(4) can be calculated using the general formula Eq.(7).8

(7)

where

Q = mean load effect

Q = standard deviation of load effect

One of the first steps in the reliability procedure is to define the limit state function for the considered structural type. In this study, the flexural limit state of prestressed concrete girders was considered. The reliability index was calculated for various average daily truck traffic values from 250 to 10,000 and for span lengths from 80 to 200ft (24 to 61m). For each considered span length and girder spacing, dead load and live load were estimated and treated as constants. The design of prestressed concrete girders was according to AASHTO LRFD specifications. Then the statistical parameters of load, mean load effect Q, and coefficient of variation for load effect VQ were calculated using the load model parameters.

Reliability analysis

The reliability analysis was used in this study for evaluat-ing the structural performance in terms of a reliability index , defined as a function of the probability of failure.8 The relia bility analysis was performed for the strength limit state functions formulated for the considered struc-tural types and load components. In the reliability analysis, load and resistance were treated as random variables. The statistical parameters for resistance of prestressed concrete beams were developed in this study based on the new data. The statistical parameters of load were taken from the most recent study based on the weigh-in-motion truck data.3,4

Formulation of the limit state function requires a definition of failure representing a boundary between the desired and undesired performance of a structural component. The gen-eral form of the limit state function is expressed by Eq.(4).

g(R,Q) = R Q (4)

where

g = limit state function

Q = load effect (demand)

If function g is greater than or equal to 0, the structure is safe (structural capacity is greater than load effect); if g is less than 0, the structure fails (load effect is greater than structural capacity). Hence, the probability of failure Pf is equal to the probability that the limit state function reaches a negative value (Eq.[5]).

Pf = P(R Q < 0) = P(g < 0) (5)

Figure 5. Reliability index for moment versus span with resistance factor = 1.0. Note: ADTT = average daily truck traffic. 1ft = 0.305m.

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ratio of the reliability index corresponding to either an increased load component or reduced resistance parameter and the reliability index for an intact girder.

The results indicate that reliability depends mostly on the strength fps and cross-sectional area Aps of the prestress-ing steel and effective depth d. The reduction of effective depth by 30% to 35% reduced the reliability index to the negative values. The concrete strength , effective width b, and dynamic load allowance IM are not important.

The reliability of a girder depends mostly on the resis-tance parameters. Reduction of the critical parameters, that is, fps and Aps, reduces the reliability index (Fig.8) for the considered types of girders. The vertical axis repre-sents the ratio of the reliability index corresponding to reduced resistance parameter and the reliability index for an intact girder. The reliability index drops by 100% when prestressing steel fpsAps is reduced about 40%. Similarly, Fig.8 shows the effect of a reduced effective depth d. This parameter is more important because, for example, a de-crease of effective depth d by 30% to 35% causes a 100% decrease in the reliability index.

Conclusion

The statistical parameters of resistance were considered for AASHTO and NU girders designed according to the AASHTO LRFD specifications. Based on a large mate-rial test data set for prestressing strands provided by the industry, new statistical parameters of resistance were derived. The results indicate that concrete and reinforcing steel properties have improved over the past 30years,9 and this can have a positive effect on the load-carrying capacity of structural components.

Figure5 plots the results of the reliability analysis for bending moment with a resistance factor of 1.0 versus span length. The results show that is about 3.6 for lower average daily truck traffic and 3.4 for higher average daily truck traffic. Also, shorter bridges have a slightly higher than longer span bridges.

Sensitivity analysis

The performance of the structure can be underestimated or overestimated because of insufficient data, time-related changes, or human error. Some parameters have a major effect on overall performance, while others do not. The im-portant parameters can be identified by sensitivity analysis. Therefore, sensitivity analysis was conducted for typical prestressed, precast concrete girders. The considered load, resistance, and section geometry parameters were dead load DL (sum of DC and DW), live load LL, dynamic load allowance IM, material strengths and fps, effective depth d, width of the compression face of the member b, and area of prestressing steel Aps.

Load components were treated as separate parameters and were increased one at a time by 10% from the nomi-nal value. Resistance parameters were grouped in three subgroups: fpsAps, b , and d. To check the sensitivity of re-sistance parameters, the reliability indices were calculated for their nominal values and then for resistance parameters reduced by multiples of 10%.

The reliability indices were calculated for the AASHTO and NU concrete girders with spans from 100 to 200ft (30 to 61m). Figures6 and 7 show the drop of the reliability index for reduced/increased load and resistance parameters for all of the considered girders. The vertical axis is the

Figure 6. Sensitivity function for NU2000 girder, 200ft (61m) and NU1600 girder, 160ft (49m). Note: Aps = area of prestressing steel; b = width of the compres-sion face of the member; d = effective depth; DL = total dead load (DC + DW); = specified compressive strength of concrete at 28 days; fps = average stress in prestressing steel at nominal bending resistance; IM = vehicular dynamic load allowance; LL = vehicular live load allowance.

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the reliability is mostly affected by the prestressing steel fpsAps and effective depth d. The reduction of effective depth by 30% to 35% caused a 100% decrease in the reliability index. Reliability indices drop 100% when prestressing steel fpsAps is reduced about 40%. The least im-portant parameters are concrete strength b and dynamic load allowance IM.

References

1. Nowak, A. S., A. S. Yamani, and S. W. Tabsh. 1994. Probabilistic Models for Resistance of Concrete Bridge Girders. ACI Structural Journal 91 (3): 269276.

2. Nowak, A. S. 1999. Calibration of LRFD Bridge De-

On the other hand, the new statistics for live load indicate that traffic volumes and truck weights have increased.3,4 Increased loading has a negative effect on the reliability index.

The reliability analysis was performed for AASHTO and NU prestressed, precast concrete girders with spans from 80 to 200ft (24 to 61m) and average daily truck traffic from 250 to 10,000 vehicles using the new statistics for resistance and load effect.

The sensitivity analysis was performed for typical pre-stressed concrete girders. The considered parameters included load components, material strength, and section geometry. The results of sensitivity analysis indicate that

Figure 7. Sensitivity functions for AASHTO TypeIV girder, 120ft (36m) and AASHTO TypeIII girder, 100ft (30m). Note: Aps = area of prestressing steel; b = width of the compression face of the member; d = effective depth; DL = total dead load (DC + DW); = specified compressive strength of concrete at 28 days; fps = average stress in prestressing steel at nominal bending resistance; IM = vehicular dynamic load allowance; LL = vehicular live load allowance.

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fpsAps fpsAps

AASHTO Type IV girder AASHTO Type III girder

Figure 8. Effect of decrease in prestressing steel fpsAps and effective depth d for prestressed concrete girders. Note: 1ft = 0.305m.

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Notation

a = depth of the equivalent stress block = c1

Aps = area of prestressing steel

As = area of mild steel tensile reinforcement

= area of mild steel compressive reinforcement

b = width of the compression face of the member; for a flange section in compression, the effective width of the flange

bw = web width or diameter of a circular section

c = distance from extreme compression fiber to neutral axis

d = effective depth

dp = distance from extreme compression fiber to the cen-troid of prestressing tendons

ds = distance from extreme compression fiber to the cen-troid of mild steel tensile reinforcement

= distance from extreme compressive fiber to the cen-troid of mild steel compressive reinforcement

DC = dead load of structural components and nonstruc-tural attachments

DL = total dead load (DC + DW)

DW = dead load of wearing surfaces and utilities

= specified compressive strength of concrete at 28 days, unless another age is specified

fps = average stress in prestressing steel at nominal bend-ing resistance

fs = stress in mild steel tensile reinforcement at nominal flexural resistance

= stress in mild steel compressive reinforcement at nominal flexural resistance

fu = ultimate strength of reinforcing steel

fy = yield stress of reinforcing steel

F = fabrication factor

sign Code. National Cooperative Highway Research Project report 368. Washington, DC: TRB (Transporta-tion Research Board), National Academy Press.

3. Rakoczy, P. 2011. WIM Based Load Models for Bridge Serviceability Limit States. PhD diss., Univer-sity of NebraskaLincoln.

4. Nowak, A. S., P. Rakoczy, and J. M. Kulicki. 2012. WIM-Based Live Load for Bridges. In TRB 91st Annual Meeting Compendium of Papers DVD, paper 12-4109. Washington, DC: TRB.

5. AASHTO (American Association of State Highway and Transportation Officials). 2011. AASHTO LRFD Bridge Design Specifications. 6th ed. Washington, DC: AASHTO.

6. NDOR (Nebraska Department of Roads) Bridge Divi-sion. 2009. NDOR Bridge Office Policies and Proce-dures. Lincoln, NE: NDOR.

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R = standard deviation of resistance

= resistance factors

= standard normal distribution function

g = limit state function

hf = compression flange depth of an I or T member

IM = vehicular dynamic load allowance

LL = vehicular live load allowance

M = material factor

P = professional factor

Pf = the probability of failure

R = resistance (capacity)

Rn = nominal value

Q = load effect (demand)

Qd = factored total design load

V = coefficient of variation

VF = coefficient of variation for fabrication factor F

VP = coefficient of variation for professional factor P

VQ = coefficient of variation for load effect Q

VR = coefficient of variation of resistance R

= reliability index, defined as a function of the prob-ability of failure

1 = stress block factor

= bias factor

F = bias factor for fabrication factor F

LL = bias factor of live load

P = bias factor for professional factor P

R = bias factor of resistance R

= mean value

Q = mean load effect

R = mean resistance

= standard deviation

Q = standard deviation of load effect


About the authors

Anna M. Rakoczy is a postdoc-toral research assistant in the Department of Civil Engineering at the University of NebraskaLincoln. She received her PhD from UNL in 2012. Her major research area is structural engi-

neering, and she has studied and worked in reliability and risk analysis, structural analysis, and design code calibration. Her research interests include the develop-ment of reliability-based design criteria for lightweight concrete structures and evaluation criteria for service-ability limit states of railway bridges.

Andrzej S. Nowak has been a professor of civil engineering at the University of NebraskaLin-coln since 2005, after 25 years at the University of Michigan. He is a Fellow of ACI and member of ACI committees 343, Concrete

Bridges and 348, Structural Safety. His major contribu-tion is the development of reliability-based calibration procedure for limit state design codes. The procedure was successfully applied to the development of the AASHTO LRFD code and calibration of resistance factors for ACI 318.

Abstract

Structural performance of bridges depends on the applied loads and the load-carrying capacity, both of which are random in nature. Therefore, probabilistic methods are needed to quantify safety. It has been widely agreed to measure structural reliability in terms of a reliability index.

Material properties have changed over the last 30 years. The new material test data include compres-sive strength of concrete, yield strength of reinforc-

ing steel, and tensile strength of prestressing strands. Average daily truck traffic and gross vehicle weight have also changed. The updated statistical parameters for prestressed concrete girders and live load affect the reliability indices.

The objective of this paper is to present the results of the reliability analysis for prestressed concrete girders using the most recent live load and resistance (flex-ure) models. Several types of American Association of State Highway and Transportation Officials girders and Nebraska University (NU) girders are considered with spans ranging from 80 to 200ft (24 to 61m) and girder spacings from 8 to 10ft (2.4 to 3m).

An important part of the research presented is deriva-tion of sensitivity functions for various parameters that affect performance of prestressed concrete girders. The focus is on strength limit states, in particular bending capacity. The contribution of several resistance param-eters, such as reinforcement area and yield stress, is considered. The procedure is demonstrated on repre-sentative girder bridges.

Keywords

Girder, load and resistance factor design, LRFD, reli-ability analysis, sensitivity analysis, statistical param-eters, strand.

Review policy

This paper was reviewed in accordance with the Precast/Prestressed Concrete Institutes peer-review process.

Reader comments

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10-Reliability-based Sensitivity Analysis for Prestressed Concrete Girder Bridges

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