Reliability and Redundancy Analysis of Structural Systems with Application to Highway Bridges Michel Ghosn The City College of New York / CUNY

Contributors Prof. Joan Ramon Casas Prof. Joan Ramon Casas UPC Construction Engineering Ms. Feng Miao Ms. Feng Miao Mr. Giorgio Anitori Mr. Giorgio Anitori

Introduction Structural systems are designed on a member by member basis. Structural systems are designed on a member by member basis. Little consideration is provided to the effects of a local failure on system safety. Little consideration is provided to the effects of a local failure on system safety. Local failures may be due to overloading or loss of member capacity from fatigue fracture, deterioration, or accidents such as an impact or a blast. Local failures may be due to overloading or loss of member capacity from fatigue fracture, deterioration, or accidents such as an impact or a blast. Local failure of one element may result in the failure of another creating a chain reaction that progresses throughout the system leading to a catastrophic progressive collapse. Local failure of one element may result in the failure of another creating a chain reaction that progresses throughout the system leading to a catastrophic progressive collapse.

I-35W over Mississippi River (2007) Truss bridge Collapse due to initial failure of gusset plate

I-35 Gusset Plate

I-40 Bridge in Oklahoma (2002) Bridge collapse due to barge impact

Route 19 Overpass, Quebec (2006) Box-Girder bridge collapse due to corrosion

Corroded Bridge Deck

Oklahoma City Bombing (1995)

Structural Redundancy Collisions Fatigue Fracture Seismic Damage Bridges survive initial damage due to system redundancy and reserve safety

Definitions Redundancy is the ability of a system to continue to carry loads after the overloading of members. Redundancy is the ability of a system to continue to carry loads after the overloading of members. Robustness is the ability of a structural system to survive the loss of a member and continue to carry some load. Robustness is the ability of a structural system to survive the loss of a member and continue to carry some load. Progressive Collapse is the spread of an initial local failure from element to element resulting, eventually, in the collapse of an entire structure or a disproportionately large part of it. Progressive Collapse is the spread of an initial local failure from element to element resulting, eventually, in the collapse of an entire structure or a disproportionately large part of it.

Structural Performance

Deterministic Criteria Ultimate Limit State Ultimate Limit State Functionality Limit State Functionality Limit State Damaged Limit State Damaged Limit State

State of the Art New guidelines to have high levels of redundancy in buildings. New guidelines to have high levels of redundancy in buildings. Criteria are based on deterministic analyses. Criteria are based on deterministic analyses. Uncertainties in estimating member strengths and system capacity as well as applied load intensity and distribution justify the use of probabilistic methods. Uncertainties in estimating member strengths and system capacity as well as applied load intensity and distribution justify the use of probabilistic methods.

Structural Reliability

Reliability Index,  Reliability index, , is defined in terms of the Gaussian Prob. function: Reliability index, , is defined in terms of the Gaussian Prob. function: If R and S follow Gaussian distributions: If R and S follow Gaussian distributions:  function of means and standard deviations

Reliability Index, 

Lognormal Probability Model If the load and resistance follow Lognormal distributions then the reliability index is approximately If the load and resistance follow Lognormal distributions then the reliability index is approximately  function of coefficients of variation: V =stand. Dev./ mean

System Reliability Probability of structural collapse, P(C), due to different damage scenarios, L, caused by multiple hazards, E: Probability of structural collapse, P(C), due to different damage scenarios, L, caused by multiple hazards, E: P(E) =probability of occurrence of hazard E P(E) =probability of occurrence of hazard E P(L|E) = probability of local failure, L, given E P(L|E) = probability of local failure, L, given E P(C|LE) is probability of collapse given L due to E P(C|LE) is probability of collapse given L due to E

Safety Criteria The probability of bridge collapse must be limited to an acceptable level: The probability of bridge collapse must be limited to an acceptable level: Alternatively, the criteria can be set in terms of the reliability index,, defined as: Alternatively, the criteria can be set in terms of the reliability index, β, defined as:

Option 1 to Reduce Risk Reduce exposure to hazards: lower P(E) Reduce exposure to hazards: lower P(E) Protect columns from collisions through barriers Protect columns from collisions through barriers Set columns at large distances from roadway to avoid crashes Set columns at large distances from roadway to avoid crashes Increase bridge height to avoid collisions with deck Increase bridge height to avoid collisions with deck Build away from earthquake faults Build away from earthquake faults Use steel connection details that are not prone to fatigue and fracture failures Use steel connection details that are not prone to fatigue and fracture failures Increase security surveillance to avoid intentional sabotage Increase security surveillance to avoid intentional sabotage

Option 2 to Reduce Risk Reduce member failure given a hazard: P(L|E) Reduce member failure given a hazard: P(L|E) Increase reliability of connection details by using different connection types, advanced materials, or improved welding, splicing and anchoring techniques Increase reliability of connection details by using different connection types, advanced materials, or improved welding, splicing and anchoring techniques Strengthen columns that may be subject to collisions or sabotage using steel jacketing or FRP wrapping Strengthen columns that may be subject to collisions or sabotage using steel jacketing or FRP wrapping Increase capacity of columns and critical members to improve their ability to resist unusual loads Increase capacity of columns and critical members to improve their ability to resist unusual loads

Option 3 to Reduce Risk Avoid collapse if one member fails: P(C|LE) Avoid collapse if one member fails: P(C|LE) Use structural configurations that have high levels of redundancy. Use structural configurations that have high levels of redundancy. Appropriately spaced large number of columns Appropriately spaced large number of columns Trusses that are not statically determinate Trusses that are not statically determinate Ensure that all the members contributing to a mode of failure are conservatively designed Ensure that all the members contributing to a mode of failure are conservatively designed to pick up the load shed by member that fails in brittle mode to pick up the load shed by member that fails in brittle mode to pick up additional load applied if member that initiates sequence fails in a ductile mode. to pick up additional load applied if member that initiates sequence fails in a ductile mode.

Types of Failures

Issues with Reliability Analysis Realistic structural models involve: Realistic structural models involve: Large numbers of random variables Large numbers of random variables Multiple failure modes Multiple failure modes Low probability of failure for members, Low probability of failure for members, Probability of failure for systems, Probability of failure for systems, Computational effort Computational effort

Finite Element Analysis

Reliability Analysis Methods Monte Carlo Simulation (MCS) Monte Carlo Simulation (MCS) First Order Reliability Method (FORM) First Order Reliability Method (FORM) Response Surface Method (RSM) Response Surface Method (RSM) Latin Hypercube Simulation (LHS) Latin Hypercube Simulation (LHS) Genetic Search Algorithms (GA) Genetic Search Algorithms (GA) Subset Simulation (SS) Subset Simulation (SS)

Monte Carlo Simulation (MCS) Random sampling to artificially simulate a large number of experiments and observe the results. Random sampling to artificially simulate a large number of experiments and observe the results. Can solve problems with complex failure regions. Can solve problems with complex failure regions. Needs large numbers of simulations for accurate results. Needs large numbers of simulations for accurate results.

Monte Carlo Simulation (MCS) Probab. of failure = Number of cases in failure domain/ total number of cases

First Order Reliability Method First Order Reliability Method (FORM) approximates limit-state function with a first- order function. First Order Reliability Method (FORM) approximates limit-state function with a first- order function. Reliability index is the minimum distance between the mean value to the failure function. Reliability index is the minimum distance between the mean value to the failure function. If limit state function is linear If limit state function is linear

First Order Reliability Method Use optimization techniques to find design point = shortest distance between Z=0 to origin of normalized space

Response Surface Method (RSM) Response Surface Method (RSM) RSM approximates the unknown explicit limit state function by a polynomial function. RSM approximates the unknown explicit limit state function by a polynomial function. A second order polynomial is most often used for the response surface. A second order polynomial is most often used for the response surface. The function is obtained by perturbation of variables near design point. The function is obtained by perturbation of variables near design point.

Response Surface Method (RSM)

Subset Simulation (SS) If F denote the failure domain. Subset failure regions F i are arranged to form a decreasing sequence of failure events: If F denote the failure domain. Subset failure regions F i are arranged to form a decreasing sequence of failure events: The probability of failure P f can be represented as the probability of falling in the final subset given that on the previous step, the event belonged to subset F m-1 : The probability of failure P f can be represented as the probability of falling in the final subset given that on the previous step, the event belonged to subset F m-1 :

Subset Simulation (SS) By recursively repeating the process, the following equation is obtained: By recursively repeating the process, the following equation is obtained: During the simulation, conditional samples are generated from specially designed Markov Chains so that they gradually populate each intermediate failure region until they cover the whole failure domain. During the simulation, conditional samples are generated from specially designed Markov Chains so that they gradually populate each intermediate failure region until they cover the whole failure domain..

b i are chosen “adaptively” so that the conditional probabilities are approximately to a pre-set value, p 0. (e.g. p 0 =0.1) Illustration of Subset Simulation Procedure

Development of Reliability Criteria Analyze a large number of representative bridge configurations. Analyze a large number of representative bridge configurations. Find the reliability indexes for those that have shown good system performance. Find the reliability indexes for those that have shown good system performance. Use these reliability index values as criteria for future designs Use these reliability index values as criteria for future designs Find the corresponding deterministic criteria Find the corresponding deterministic criteria

Input Data for Reliability Analysis Dead loadsDead loads Bending moment resistance:Bending moment resistance: Composite steel beams Composite steel beams Prestr. concrete beams Prestr. concrete beams Concrete T-beams Concrete T-beams

Live Load Simulation Bin I Bin II Repeat for N loading events Maximum of N events. Maximum of N events. 75-yr design life 75-yr design life 5-yr rating cycle 5-yr rating cycle ADTT = 5000 ADTT = 5000 = 1000 = 1000 = 100 = 100

Simulated vs. Measured Single event Two-lane 100-ft span

Cumulative Distribution

Maximum Load Effect Max. 5-yr event Two-lane 100-ft span

Reliability-Based Criteria for Bridges Based on bridge member reliability Based on bridge member reliability Corresponding system safety, redundancy and robustness criteria: Corresponding system safety, redundancy and robustness criteria:

Deterministic Criteria Ultimate Limit State Ultimate Limit State Functionality Limit State Functionality Limit State Damaged Limit State Damaged Limit State

Design Criteria Apply system factor during the design process to reflect level of redundancy Apply system factor during the design process to reflect level of redundancy  s <1.0 increases the system reliability of designs with low levels of redundancy.  s <1.0 increases the system reliability of designs with low levels of redundancy.  s > 1.0 allows members of systems with high redundancy to have lower capacities.  s > 1.0 allows members of systems with high redundancy to have lower capacities.

Example Ps/Concrete Bridge 100-ft simple span, 6 beams at 8-ft

Example Ps/Concrete Bridge

∆βu = βult - βmem = = 2.90 > 0.85 ∆βf = βfunct - βmem = = 0.84 > 0.25

Steel Truss Bridge

∆βu = βult-βmem = = 1.00 > 0.85 ∆βf = βfunct-βmem = = 0.80 > 0.25

Damaged Bridge Analysis ∆βd = βdamaged – βmem = = -0.95>-2.70 for P/C bridge ∆βd = βdamaged – βmem = = -4.38<-2.70 for truss bridge Truss bridge is not robust. Truss bridge is not robust. But  damaged is greater than 0.80 ; system safety is satisfied But  damaged is greater than 0.80 ; system safety is satisfied Member reliability index of the truss is β member=6.8 Member reliability index of the truss is β member=6.8

Deterministic Analysis of Ps/Concrete Bridge

Twin Steel Box Girder Bridge

Structural Analysis

Reliability Analysis

Redundancy Analysis  u = 1.24 > 0.85 O.K.  u = 1.24 > 0.85 O.K.  f = 0.14 < 0.25 N.G.  f = 0.14 < 0.25 N.G.  d = < N.G.  d = < N.G.

System Safety Analysis  ultimate = 9.77 > 4.35 O.K.  ultimate = 9.77 > 4.35 O.K.  functionality = 8.67 > 3.75 O.K.  functionality = 8.67 > 3.75 O.K.  damaged = 5.07 > 0.80 O.K.  damaged = 5.07 > 0.80 O.K. Although the system is not sufficiently redundant, the bridge members are so overdesigned by about a factor of 3 that all system safety criteria are satisfied Although the system is not sufficiently redundant, the bridge members are so overdesigned by about a factor of 3 that all system safety criteria are satisfied

Bridge system analysis Multicellular box girder deck Integral design 4 spans (max 48 m)

Probabilistic results Intact structureDamaged structure

Conclusions A method is presented to consider system redundancy and robustness during the structural design and safety evaluation of bridges. A method is presented to consider system redundancy and robustness during the structural design and safety evaluation of bridges. The method is based on structural reliability principles and accounts for the uncertainties in evaluating system strength and applied loads. The method is based on structural reliability principles and accounts for the uncertainties in evaluating system strength and applied loads. The goal is to ensure that structural systems meets minimum levels of system safety in order to sustain partial failures or structural damage. The goal is to ensure that structural systems meets minimum levels of system safety in order to sustain partial failures or structural damage.