Structural Redundancy of

Prestressed Concrete Box Girder Bridges

Jian Yang1* , O. Murat Hamutcuoglu2, Yingsheng Ni3, Michel Ghosn4

1Bridge Engineer, Ph.D. candidate

*Corresponding author

McLaren Engineering Group 100 Snake Hill Road, West Nyack, NY 10994

Department of Civil Engineering, The City College of New York / CUNY,

E-mail: jyang@mgmclaren.com

Tel. (845)353-6400 Fax. (845)353-6509

2Structural Engineer, Ph.D., P.E. HNTB Corporation

5 Penn Plaza, 6th Floor, New York, New York 10001

E-mail: ohamutcuoglu@hntb.com Tel. (212) 594-9717 Fax (212) 947-4030

3Ph.D. candidate

Department of Bridge Engineering,

Tongji University, Shanghai 200092, P.R. China E-mail: niyingsheng2008@aliyun.com

Tel. +8618818260850

4Professor of Civil Engineering,

The City College of New York / CUNY, 160 Convent Ave. New York, NY, 10031.

E-mail: mghosn@ccny.cuny.edu Tel. (212)650-8002 Fax.(212)650-6965

Word Count Text - 4250

Tables (3) -750 Figures (10) – 2500

Total = 7500 words Final-submission Date: Nov.15, 2015

1

Abstract 1

2

Bridges are usually designed and evaluated based on member strength and serviceability criteria, 3 however, it is expected that they exhibit sufficient levels of reserve strength and multiple paths to 4

resist collapse should any of their members get damaged or exceed its nominal strength capacity. 5 The ability of a structural system to redistribute the load around damaged members is referred to 6

as structural redundancy. To account for bridge redundancy during the design and safety 7 evaluation process, the AASHTO LRFD and LRFR provide a preliminary set of load modifiers 8 or system factors most of which were based on the code writers judgment and experience. 9

This paper evaluates the redundancy of prestressed concrete box girder bridges under vertical 10

loads and lateral loads. Pushover and pushdown analyses are performed using frame and grillage 11 models in SAP2000. The grillage models are validated by more advanced finite element models 12 built in ABAQUS. The effects of different damage scenarios and types of connections between 13

the box girder and columns on the structural redundancy level are investigated. Based on those 14 results, a set of system factors for prestressed concrete box bridges accounting for the system 15

redundancy is proposed. The proposed system factors can be used during the design and safety 16 assessment of prestressed concrete box girder bridges subjected to transient lateral load and 17 vertical vehicle load. 18

Keywords: structural redundancy; system factor; nonlinear behavior; prestressed concrete box 19

girder. 20

2

Introduction 21

22

Traditionally, the design and load capacity evaluation of highway bridges have been executed on 23 a member by member basis. Yet, bridges are expected to have sufficient levels of structural 24

redundancy to sustain the failure of a main member and be still able to carry significant load to 25 allow for the evacuation of the structure and avoid loss of life before the damage is detected and 26 bridge closure or corrective actions are undertaken. However, the quantification of redundancy 27

is not fully formulated in current bridge standards and specifications. A 1985 “State of the Art 28 Report on Redundant Bridge Systems” concluded that although analytical techniques to study the 29

response of damaged and undamaged flexural systems to high loads are available, “little work 30 has been done on quantifying the degree of redundancy that is needed” (ASCE-AASHTO Task 31 Committee; 1985). Since that time, research studies and design guides and specifications have 32

proposed different approaches for evaluating the structural redundancy in b ridges. A 33 comprehensive review and synthesis of issues related to the redundancy ana lysis of bridges was 34

presented by Dexter et al (2005). 35 36 In a first attempt at providing a method to incorporate redundancy criteria in the bridge design 37

specifications, the AASHTO LRFD (2012) proposed the adoption of load modifiers in the design 38 check equations to account for redundancy during the design of new bridges based on the 39

recommendation of Frangopol and Nakib (1991). Specifically, the AASHTO LRFD 40 recommended applying different load modifiers varying between 0.95, 1.00 or 1.05 to reflect the 41 levels of bridge redundancy and ductility. An additional load modifier was related to the 42

importance of the structure for defense/security consideration. However, the specifications d id 43 not explain how to identify which bridges have low and high redundancy or how to define low 44

and high ductility. As explained in the AASHTO LRFD (2012) Commentary, the recommended 45 values had been subjectively assigned pending additional research. An alternate approach 46 adopted by the Canadian code CAN/CSA-S6-06 (2006) directed bridge engineers to use different 47

resistance factors to reflect different target member reliability index values which were selected 48 based on the redundancy of the bridge system expressed in terms of the consequence of a 49

member’s failure and the ductility of the member being evaluated. However, like the AASHTO 50 LRFD, this approach relied on the judgment of the engineer in deciding which bridges are 51 redundant and in judging the consequence of a member’s failure. The LRFR option of the 52

AASHTO Manual for Bridge Evaluation (MBE, 2011) assigned a system factor to be applied on 53 the resistance side of the equation with values ranging between 0.85 and 1.0 for bridge 54

configurations that have been demonstrated to have low levels of redundancy. A table provided 55 some guidelines as to how to assign the appropriate system factor based on bridge geometries 56 and configurations. Following the same line of thinking, some state load rating manuals such as 57

the Florida DOT (2012) had also developed their own sets of system factors. But, as stated by 58 Mertz (2006), these were primarily based on very limited analyses and heavily relied on 59

“engineering judgment”. 60 61 A series of studies to develop approaches and criteria for the quantification of bridge system 62

redundancy were undertaken under the auspices of the National Cooperative Highway Research 63 Program (NCHRP) by Ghosn and his colleagues (Ghosn and Moses, 1998; Liu, Ghosn, et al., 64

2001; Yang and Ghosn, 2015). Many researchers/engineers (Hunley and Harik 2012; Hovell, 65 2007; Mertz, 2006), consulting companies (HNTB) and code writing organizations (AASHTO 66

3

LRFR; Florida DOT; Wisconsin DOT) have adopted the redundancy concepts developed in 67 NCHRP Reports 406 and458 and specified similar system factors as a way to account for 68

redundancy during the design or the safety evaluation of typical short and medium span bridge 69 systems. However, NCHRP 406 and 458 did not provide system factors for some bridge system 70

and subsystem configurations that have become more popular in recent years such as multi-cell 71 box girder bridges. Also, the two reports did not verify the applicability of the system factors 72 that they proposed for analyzing the entire bridge system including the interaction between the 73

superstructure and substructure. 74 75

This paper summarizes and validates the results of prestressed concrete box girder bridges 76 published in the appendices of NCHRP Report 776 (Ghosn & Yang, 2014) and proposes system 77 factors. As prestressed concrete box girder bridges offer a good option when building curved 78

bridges, they are becoming a popular choice for new designs due to their high torsion stiffness. 79 The proposed system factors will be of immediate interest to bridge design engineers. The 80

objective of this paper is to investigate the redundancy level of this type of bridges and direct 81 bridge engineers to use the proposed system factors to account for the structural redundancy 82 under transient lateral and vertical loads. 83

Bridge Redundancy 84

85

Figure 1 gives a conceptual representation of the behavior of a bridge structure and the different 86 levels that should be considered when evaluating member safety, system safety and system 87 redundancy. The load multipliers LF represent the capacity of the system in terms of the number 88

of design trucks that it can carry to reach different limit states. Specifically, LFu represents the 89 ability of the originally intact system to resist system collapse; LFf represents the functionality 90

limit; and LF1 corresponds to first member failure. 91

Figure 1. Representation of typical behavior of bridge systems

First member

failure

LFd

LF1

LFf

LFu

Ultimate

capacity of

damaged system

Loss of

functionality

Ultimate

capacity of

intact system

Load Factor

Bridge Response

Assumed linear

behavior

Intact system

Damaged bridge

4

If redundancy is defined as the capability of a structure to continue to carry loads after the failure 92 of the most critical member, then comparisons between the load multipliers LFu, LFf, LFd and 93

LF1would provide non-subjective and quantifiable measures of system redundancy. Thus, the 94 following three deterministic measures of system redundancy may be defined in terms of the 95

ratio of the system’s capacity as compared to the most critical member’s capacity: 96 97

1

U

uLF

LFR

98

1

f

fLF

LFR

Eq. (1) 99

1

d

dLF

LFR

100 101

where Ru=system reserve ratio for the ultimate limit state, Rf=system reserve ratio for the 102 functionality limit state, Rd= system reserve ratio for the damage condition. Ru, Rf and Rd can 103

thus be used as measures of system redundancy as they represent the ability of a system to carry 104 load beyond the failure of the most critical member. 105

System factors 106

107 NCHRP Reports 406 and 776 proposed that structural system redundancy be considered during 108

the design/safety check process by applying system factors such that: 109

N

s n i iR Q Eq. (2) 110

Where N

nR is the required member capacity accounting for bridge redundancy, s is the system 111

factor,is the member resistance factor as specified in the current AASHTO LRFD Bridge 112

Design Specifications, i is the load factor for load i, Qi is the load effect of load i. 113

The system factor, s of Equation (2) provides a measure of the system reserve strength as it 114

relates to ductility, redundancy and operational importance, and their interaction. The system 115

factor, s is related to two other factors, su and sd which respectively account for system 116

functionality and resistance to collapse conditions after the strength of the most critical member 117 of an originally intact bridge is exceeded, and for the system strength of a bridge in damage state 118

condition. Functionality is defined as the ability of the originally intact system to carry heavy 119 live loads without exceeding vertical deflection limits equal to span length/100 which would 120 render the bridge unfit for use. Damaged conditions include failure of a main girder of multi-121

girder bridges, the failure of an external web of one box of a box girder bridge or the loss of one 122 column of multi-column bents. 123

124 Non-redundant bridges are penalized by requiring their members to provide higher safety levels 125

than those of similar bridges with redundant configurations. The aim of s, is to add reserve 126

capacity for non-redundant systems so that the overall system reliability is increased. If adequate 127

5

redundancy levels are present, a system factor s=1.0 is used. In the instances where the level of 128

redundancy is more than adequate, a value of s greater than 1.0 may be used. An upper limit 129

equal to 1.20 is recommended for s until more experience is gained in the application of these 130

factors in actual design and safety check situations. 131

Bridge Description 132

133

A multi-cell prestressed concrete box girder bridge with two-column bents is shown in Figure 2. 134 The four-cell box girder deck has the top slab width of 58’ 10”. The three-span continuous 135 bridge is 412 ft long and the span lengths are 126 ft, 168 ft and 118 ft. The thicknesses of the top 136

and bottom slabs are 9 1/8” and 8 1/4”, respectively. The depth of the box girder is 6’ 9”. The 137 connection between the superstructures and the substructures are through the integral cap beams 138

with the same depth of the girders and 8 ft width. The original bridge superstructure is connected 139 to two 6 ft diameter round columns through the moment resisting rigid connections. In the 140 further analyses, the effect of the moment-free bearing configuration is also investigated. 141

142 The unconfined concrete strength is assumed to be 4000 psi and the reinforcing steel is taken as 143

Grade 60 with the ultimate stress capacity is 90 ksi. The prestressing tendons have the peak 144 stress of 270 ksi and their yield stress is 230 ksi. The area of tendons in each section is 9 in.2 and 145 the initial prestressing force is set as 1,824 kips without the losses. The total loss in the 146

prestressing force is assumed 20 ksi including the elastic shortening, creep, shrinkage and the 147 steel relaxation stresses. The tendons are considered in parabolic geometry along the girders and 148

all section moment-curvature calculations include the exact location of the tendons in the 149 sections analyses. 150

6

(a) Elevation view

(b) Typical section and connection between girder and columns

Figure 2. Bridge Configurations

Lateral Pushover Analyses 151

Model 1 with Rigid Cap-Column Connections 152

The lateral pushover analysis of the original model is performed using SAP2000. The plastic 153 hinge definitions are applied at the bottom and top of the columns to capture the nonlinear axial-154 flexure interaction, as shown in Figure 3(a). The lateral pushover analysis is initiated with a 155

nonlinear dead load, tendon jacking and live load analysis that provide the initial stresses in the 156 elements. Figure 3(b) shows failure modes and plastic hinge propagation during the pushover 157 analysis. The dead load is calculated through the self-weight feature of SAP2000 and then the 158

prestressing tendon jacking forces are applied to provide the initial condition of the bridge prior 159 to an extreme event case. 160

7

Model 2 with Pinned Connections 161

Model 2 assumes that the cap beam is connected to the columns through bearings applied 162 between the top of the column and the cap beam, as illustrated in Figure 3(c). Elastic links are 163

used to model the pads that serve to release the rotations at the top of the columns with the 164 rotational stiffnesses of these bearings being set at zero. Failure modes and plastic hinge 165

propagation are shown in Figure 3(d). 166 167 The sectional moment curvature response of the column is illustrated in Figure 4 where the 168

extreme concrete fibers fail due to the strain exceeding the limit spalling strains and then the 169 compression fibers in the confined concrete region reach the ultimate compressive strain limits. 170

The bridge is loaded by 20% of the HL93 load along two lanes to represent the regular traffic 171 that may be on the bridge during an earthquake. 172

173

Figure 5 compares the results of the pushover analysis performed for Model 1 and Model 2. 174 According to linear elastic pushover analysis of Model 1, the lateral force that causes the first 175 member to reach its load carrying capacity P1 is 501 kips when the bottom section of the pier 176

reaches its flexure capacity. Following a nonlinear pushover analysis, the ultimate capacity of the 177 system or collapse is reached when the total load Pu=720 kips. According to Equation (1), the 178

redundancy ratio of Model 1, Ru, is equal to 720 kips/ 501 kips=1.44. Similarly, the redundancy 179 ratio of Model 2, Ru, is equal to 345 kips/ 261 kips=1.32, which is 8.3% lower than that of Model 180 1. 181

8

(a) Model 1: Cross Sections with integral connection between cap beam and column

(b) Model 1: column top and bottom sections exceed elastic limits; concrete compressive strains exceed crushing limit.

(c) Model 2: Cross sections with elastic links (pinned bearings) at the pier

(d) Model 2: column bottom sections exceed their elastic limits; concrete compressive strains exceed crushing limit.

Figure 3. Transverse Pushover Analysis

9

Figure 4. Column Section Behavior Showing Compressed Concrete Region in Dark Blue and

Moment Curvature under the DL and 20% LL axial force.

Figure 5. Transverse Pushover Analysis

0

100

200

300

400

500

600

700

800

0 0.2 0.4 0.6 0.8 1

Pie

r To

tal B

ase

Shea

r (k

ips)

Pier Top Displacement (ft)

Model 1 & 2 Pier Transverse Pushover

Model 1: Rigid

Model 2: Bearing

Model 1: Rigid Linear Analysis

Model 2: Bearing- Linear Analysis

10

Vertical Pushdown Analyses 182

Model 3 with SAP Grillage 183

The nonlinear pushdown analysis under the effect of vertical load is not as straightforward as the 184

lateral pushover analyses performed for Models 1 and 2. The nonlinear behavior of the box 185 girder can be simulated by plastic hinge theory if the multi-cell box girder is modeled using a 186 grillage where the transverse and longitudinal stiffnesses of the elements can be approximated. 187

The flexural and torsional stiffnesses of the grillage elements are calculated through existing 188 methodologies proven to capture box girder behavior accurately (Hambly, 1991; Zokaie, et al., 189

1991). Two HS-20 trucks with 4 ft. lateral distance are applied at the mid-span, as shown in 190 Figure 6. 191

(a) 3D grillage model with two side by side HS-20 trucks

(b) Moment-Curvature Curve

Figure 6. Grillage Frame Model 3 with the HS-20 Trucks and Moment-Curvature Curve of

Grillage Segments

11

Comparison of SAP and ABAQUS Results 192

The purpose of this section is to validate the use of grillage models by SAP2000 for the analysis 193 of superstructures under the effect of vertical loads. Bridge superstructure is the same as Model 194

3, but use pinned supports between superstructures and substructures to simplify the modelling in 195 SAP and ABAQUS. Figure 7 shows a 3-D ABAQUS model, meshed section, prestressed 196

tendons layout, reinforcements and load positions of two HS-20 trucks. The vertical force of the 197 nonlinear pushdown analysis is normalized by the total weight of two HS-20 trucks, 144 kips. 198 Figure 8 shows the ultimate collapse load estimated in the grillage analysis is only 6.6% lower 199

than the ABAQUS result. The results confirm that the grillage model does an adequate job of 200 predicting the failure load. 201

12

(a) 3-D model (b) Section mesh

(c) Prestressed tendons (d) Reinforcements layout

(e) Two HS-20 truck loads

Figure 7. ABAQUS Finite Element Model

Figure 8. Comparison of Pushdown Results in SAP2000 and ABAQUS

0

5

10

15

20

25

0 1 2 3 4

Nu

mb

er

of D

esig

n T

ruck

s

Mid-Span Vertical Displacement (ft)

ABAQUS

SAP2000_Plastic hinge length=0.5 girder depth

SAP2000_Plastic hinge length=1.0 girder depth

13

Model 4 with Column loss 202

In Model 4, one column is removed from the system to study the capacity of the system to carry 203 some load following major damage to one of the columns. The damage scenario could represent 204

a situation where one of the column is hit by a truck, ship or debris carried by a flood or when 205 one column foundation is damaged to a major scour or if one of the columns has been exposed to 206

major deterioration or construction errors. 207 208 Two cases are considered. The first case is given in Model 4-a where the columns sections are 209

assumed to behave within linear elastic limits. The second case, Model 4-b, is a more realistic 210 approach that represents the nonlinear material behavior in both superstructures and 211

substructures. The numerical results of the vertical pushdown analyses on the Models 4-a and 4-212 b are compared to the original case (Model 3), as shown in Figure 9. Regarding the results from 213 Model 4-a where the substructure columns is in elastic manner, the prestressed girders are 214

significantly redundant when one of the piers in the structure is subjected to a column loss. 215 Model 4-a experienced the girder failure at the negative moment section of the exterior girder at 216

load factor (aka, number of design trucks) LFd=23.8. The redundancy ratio of Model 4-a, 217 Rd=23.8/18.0=1.32. Following a nonlinear pushdown analysis, the ultimate capacity of the 218 system or collapse for Model 4-b is reached when the total load LFd=19.0. Therefore, the 219

redundancy ratio of Model 4-b, Rd=19.0/18.0 =1.06. 220

Figure 9. Effects of Column Removal on Pushdown Analyses of Prestressed Girders

0

5

10

15

20

25

30

0 1 2 3 4

Nu

mb

er o

f Des

ign

Tru

cks

Mid-Span Vertical Displacement (ft)

Model 4: Vertical Pushdown with Column Losses

Model 3: Original Case Model 4-a: Column Loss

Model 4-b: Column Loss with Column Hinge Model 4 Linear

Model 4: Linear- Column Hinge Model 3 Linear

14

Model 5 with Damaged Web 221

Another damage scenario considered is the failure in the prestressing tendons in one of the webs. 222 The case is represented by setting the flexural stiffnesses theoretically as zero along the length of 223

the bridge. The second girder in the grillage model is selected as the failed girder to simulate a 224 worst case scenario because the maximum demand in the originally intact system is expected on 225

this girder due to the position of the two HS-20 trucks. Due to the interior girder loss, the exterior 226 girder becomes the most critical component when the structure is subjected to the eccentric 227 vehicle load. Figure 10 shows the ultimate capacity of the system or collapse for Model 5 is 228

reached when the total load LFd= 18.2. Thus, the redundancy ratio, Rd, is equal to 18.2/18.0 229 =1.01. 230

Figure 10. Pushdown Analysis of the Bridge with Web Damage

As reported in NCHRP Project 12-86 Interim Report 1 and NCHRP Report 406 Appendix D, 231 grillage modeling methods in this study have been verified by experimental tests of four concrete 232

box-girder bridge models that were tested by Kurian and Menon (2007), a prestressed concrete 233 box, tested by Mirza et al. at McGill University (1990) and a three-column bridge bent tested by 234 McLean et al (1998). The results demonstrate that the grillage model can be used to pred ict the 235

collapse loads of concrete box girders and column bents with a good accuracy. 236

0

5

10

15

20

25

30

0 1 2 3 4

Nu

mb

er o

f Des

ign

Tru

cks

Mid-Span Vertical Displacement (ft)

Model 5: Vertical Pushdown with Girder Losses

Model 3: Original Case Model 5: Int. Girder Loss Model 5: Linear

15

Redundancy Ratios and System Factors 237

238

Table 1 summarizes the redundancy ratios for the different models. According to NCHRP 406, a 239 redundancy ratio for the originally intact bridge subjected to overloading should produce a 240

redudancy ratio LFu/LF1 greater than 1.30 to be considered suffciently redundant. Damaged 241 bridges should give LFd/LF1 ratio of 0.50 or higher. 242

The redundancy ratios compare the maximum capacity of the sys tem to that of the first member 243 to fail. As an example, according to linear elastic pushover analysis of Model 1, the lateral force 244

that causes the first member to reach its load carrying capacity P1 is 501kips when the bottom 245 section of the pier reaches its flexure capacity. Following a nonlinear pushover analysis, the 246 ultimate capacity of the system or collapse is reached when the total load Pu=720 kips. 247

According to linear “pushdown” analysis of Model 3, if two HS-20 trucks are loaded in the 248 middle span, the first member fails in positive bending when the weight of these trucks is 249

incremented by a factor LF1=18.0. When the nonlinear pushdown analysis is executed, collapse 250 takes place when HS-20 load is multiplied by a factor LFu=24.1. The results of Model 1 through 251 3 in Table 1 show that this bridge provides good levels of redundancy for the ultimate limit state 252

due to either vertical or lateral overloading of the originally intact bridge. Although lateral 253 loading of systems with pinned connections between superstructures and substructures shows 8% 254

lower redundancy ratio compared to systems with integral connections, the bridge system with 255 pinned connections still provides adequate level of redundancy. Further studies show that in the 256 longitudinal loading analysis, the integral connection allowed for similar redundancy levels. 257

However, for the cases where the columns are connected to the superstructure through pinned 258 supports, the redundancy level is vastly reduced. The results of Model 4-a, 4-b and Model 5 also 259

show significant higher redundancy ratios than the minimum redundancy value of 0.50 for 260 damaged scenarios under vertical load. This multi-cell bridge satisfies all the redundancy criteria 261 provided in NCHRP 406. However, single cell box girder bridges are non-redundant due to 262

equally loaded girder webs. 263 264

A reliability analysis is used to calibrate the system factors, based on the results of the pushover 265 and pushdown analyses. The reliability calibration details are described in Chapters 2-5 of 266 NCHRP Report 776. Based on the redundancy ratios in Table 1, recommended values for the 267

system factors s for typical single cell and multi-cell prestressed concrete box girder bridges 268

under lateral loads are given in Table 2. In Table 2, the Risk factor, , reflects the uncertainties 269 in estimating the load demand and capacity; 1.10 is a multi-column factor for two-column 270

systems; 0.24 is a curvature factor; u is the ultimate curvature of the weakest column in the 271

bent which can be obtained from a cross section analysis; is the curvature correction factor for 272

cases with weak connecting elements and weak details (see Equation 3); 43.64 10 (1/ in.) is the 273

average curvature for a typical unconfined column and 31.55 10 (1/ in.) is the average 274

curvature for a typical confined column. One column bents and a two-column system with one 275 column loss are considered as a non-redundant system. For non-redundant systems, system 276 factors of 0.75 and 0.85 are proposed for seismic hazards and all other lateral loads, respectively. 277

Details can been found in Chapter 4 of NCHRP Report 776. 278

16

Table 3 lists the recommended system factors su and sd for typical single cell and multi-cell 279

prestressed concrete box girder bridges under vertical loads. In Table 3, a system factor of 0.80 is 280 applied to the resistance side of the AASHTO LRFD Bridge Design checking equation (Equation 281 1) to penalize the single cell box girder bridges by requiring their members to provide higher 282

safety levels than those of similar bridges with redundant configurations. While for the multi-cell 283

box girder bridges, a system factor s=1.0 is used for originally intact bridge due to adequate 284

redundancy levels and a higher system factor s=1.2 is applied for the significantly redundant 285 bridge systems with one column loss or one web damage. 286

Table 1 Summary table for the redundancy ratios of the prestressed box girder bridge

Analysis Case Model Pu/P1 LFu/LF1 LFd/LF1 Loading Initial Damage

Model 1 1.44 --- --- Lateral None

Model 2 1.32 --- --- Lateral None

Model 3 --- 1.34 --- Vertical None

Model 4-a --- --- 1.32 Vertical One column

Model 4-b --- --- 1.06 Vertical One column

Model 5 --- --- 1.01 Vertical One web

Table 2. System factors for single cell and multi-cell box girder under lateral load

Variable Applicability Recommended value

, risk factor Seismic hazards =0.75

All other lateral loads =0.85

s, system factor Non-redundant

systems s

s, system factor Redundant systems

4

3 4

3.64 10 (1/ in.)1.10 0.24

1.55 10 (1/ in.) 3.64 10 (1/ in.)

u

s

17

The correction factor is given as: 287

288

if

if and

1.0

available p column

u column available p column

u column p column

u connection

available u column u connection u

u

M MM M M

M M

M M

if and

system is non-redundant if

available u column u connection u

available p column

M M

M M

Eq. (3) 289

where availableM = moment capacity of the connecting elements such as cap beams and 290

pile caps or the reduced moment that can be supported by the column based on the available 291

shear reinforcement, development length, splice or connection detailing. Details on how to 292 calculate the available moment capacity for a member with weak detailing are available in the 293 FHWA Seismic Retrofitting Manual for Highway Structures, Part 1 Bridges, 294

p columnM = plastic moment capacity of column, 295

u columnM =ultimate overstrength moment capacity of column calculated using 296

nonlinear sectional capacity analysis programs or conservatively estimated to be 1.15 p columnM , 297

u = ultimate curvature of the weakest column in the bent, 298

u connection = minimum ultimate curvature of the connecting elements. 299

Table 3. System factors for single cell and multi-cell box girder under vertical load

Intact system

Bridge cross section type System factor

Single cell box girder bridges 0.80su

Multi-cell box girder bridges 1.00su

System in damaged state condition

Bridge cross section type System factor

Single cell box girder bridges 0.80sd

Multi-cell box girder bridges 1.20sd

18

Conclusions 300

301

This paper investigated the redundancy of single cell and multi-cell prestressed concrete box 302 girder bridges subjected to vertical vehicle loads or lateral loads. For the prestressed concrete 303

box girder bridge systems under vertical loads, the analyses demonstrate that multi-cell box 304 girder bridges can show adequate levels of redundancy in their originally intact conditions and 305

significantly redundant bridge systems with one column loss or one web damage according to the 306 criteria proposed in NCHRP Reports 406 and 776 assuming that the bridge members have been 307 designed to satisfy the applicable specifications. However, single cell box girder bridges are non-308

redundant due to equally loaded girder webs. For the bridge systems under lateral loads, one 309 column bents and a two-column system with one column loss are considered as non-redundant 310

systems. 311 312 This paper proposed a set of system factors for prestressed concrete box bridges accounting for 313

the structural redundancy. The proposed system factors can be used during the design and safety 314 assessment of prestressed concrete box girder bridges subjected to transient lateral loads or 315

vertical vehicle loads. 316

Acknowledgements 317

318

Support for this study by National Cooperative Highway Research Program (NCHRP) through 319 Project NCHRP 12-86 is gratefully acknowledged. The writers are also grateful for the support 320 and guidance provided by Dr. Waseem Dekelbab, senior program officer and the NCHRP 12-86 321

project panel. Special thanks go to Mr. David Beal and Mr. Bala Sivakumar from HNTB for 322 their reviews, advice and valuable comments. 323

19

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325

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