Reliability and Redundancy Analysis of Structural Systems with

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

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

Introduction n n Structural systems are designed on a member by member basis. 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 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-35 W 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 Bridges survive initial damage due to system redundancy and reserve safety Collisions Fatigue Fracture Seismic Damage

Definitions n n n 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. 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 n Ultimate Limit State n Functionality Limit State n Damaged Limit State

State of the Art n n n New guidelines to have redundancy in buildings. high levels of 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.

Structural Reliability

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

Reliability Index, b

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

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

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

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

Option 2 to Reduce Risk n Reduce member failure given a hazard: P(L|E) n n n 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 Increase capacity of columns and critical members to improve their ability to resist unusual loads

Option 3 to Reduce Risk n Avoid collapse if one member fails: P(C|LE) n Use structural configurations that have high levels of redundancy. n n n Appropriately spaced large number of columns Trusses that are not statically determinate Ensure that all the members contributing to a mode of failure are conservatively designed n n 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.

Types of Failures

Issues with Reliability Analysis n Realistic structural models involve: Large numbers of random variables n Multiple failure modes n Low probability of failure for members, 10 -4 n Probability of failure for systems, 10 -6 n Computational effort n

Finite Element Analysis

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

Monte Carlo Simulation (MCS) n n n Random sampling to artificially simulate a large number of experiments and observe the results. Can solve problems with complex failure regions. 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 n n n First Order Reliability Method (FORM) approximates limit-state function with a firstorder function. Reliability index is the minimum distance between the mean value to the failure function. 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) n n n RSM approximates the unknown explicit limit state function by a polynomial function. A second order polynomial is most often used for the response surface. The function is obtained by perturbation of variables near design point.

Response Surface Method (RSM)

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

Subset Simulation (SS) n n 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.

Illustration of Subset Simulation Procedure bi 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 n n Analyze a large number of representative bridge configurations. Find the reliability indexes for those that have shown good system performance. Use these reliability index values as criteria for future designs Find the corresponding deterministic criteria

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

Live Load Simulation • Maximum of N • • • Bin II Repeat for N loading events. 75 -yr design life 5 -yr rating cycle ADTT = 5000 = 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 n n Based on bridge member reliability Corresponding system safety, redundancy and robustness criteria:

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

Design Criteria n n n Apply system factor during the design process to reflect level of redundancy fs <1. 0 increases the system reliability of designs with low levels of redundancy. fs > 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

Example Ps/Concrete Bridge ∆βu = βult - βmem = 5. 75 -2. 85 = 2. 90 > 0. 85 ∆βf = βfunct - βmem = 3. 69 -2. 85 = 0. 84 > 0. 25

Steel Truss Bridge

Steel Truss Bridge ∆βu = βult-βmem = 7. 80 -6. 80 = 1. 00 > 0. 85 ∆βf = βfunct-βmem = 7. 60 -6. 80 = 0. 80 > 0. 25

Damaged Bridge Analysis ∆βd = βdamaged – βmem = 1. 90 -2. 85 = -0. 95>-2. 70 for P/C bridge ∆βd = βdamaged – βmem = 2. 42 -6. 80 = -4. 38<-2. 70 for truss bridge • Truss bridge is not robust. • But bdamaged is greater than 0. 80 ; system safety is satisfied • Member reliability index of the truss is βmember=6. 8

Deterministic Analysis of Ps/Concrete Bridge

Twin Steel Box Girder Bridge

Structural Analysis

Reliability Analysis

Redundancy Analysis n n n Dbu = 1. 24 > 0. 85 O. K. Dbf = 0. 14 < 0. 25 N. G. Dbd = -3. 46 < -2. 70 N. G.

System Safety Analysis n n bultimate = 9. 77 > 4. 35 bfunctionality = 8. 67 > 3. 75 bdamaged = 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

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

Probabilistic results Intact structure Damaged structure

Conclusions n n n 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 goal is to ensure that structural systems meets minimum levels of system safety in order to sustain partial failures or structural damage.