Risk Analysis and Target Reliability for Bridges

Andrzej S. Nowak, Ph.D.University of Nebraska-Lincoln

DisclaimerThe contents of this report reflect the views of the authors, who are responsible for the facts and the accuracy of the information presented herein. This document is disseminated under the sponsorship of the U.S. Department of Transportation’s University Transportation Centers Program, in the interest of information exchange. The U.S. Government assumes no liability for the contents or use thereof.

Outline Problem Statement Load and Resistance Models Reliability Analysis Procedure Selection of the Target Reliability Load and Resistance Factors

Problem Statement 585,000 highway bridges in USA 30-35% are inadequate 10-15% are structurally deficient How to use the available limited

resources?

Needs New design – how to design with

optimum life cycle costs? Existing structures – how to

assess the actual loads and capacity? How to predict the remaining life?

Select a rational safety margin

Uncertainties Loads (natural and created) Material properties Load carrying capacity (resistance) Basic requirement:

Load effect (Q) < Resistance (R) Safety margin

R – Q > 0

Reliability Index, b

Basic Questions:• How can we measure safety of a

structure?• How safe is safe enough? What is

the target reliability?• How can we implement the

optimum safety level?

Safety Factors• Allowable stress• Load factors• Load combination factors• Resistance factors

Load Factor

Resistance Factor

Code Calibration Procedure• Select representative structures• Develop statistical models for loads• Develop statistical models for resistance• Develop/select reliability analysis

procedure• Determine the target reliability index• Determine load and resistance factors

Bridge Loads• Dead load• Live load (static and dynamic)• Environmental loads (wind, snow,

earthquake, temperature, ice)• Special loads (vehicle and vessel

collision, fire, explosion)

Two Trucks Side by Side

Video Recordings of Traffic Jam Situations FHWA Data• Multiple-presence of trucks occupying three lanes• One lane is almost exclusively occupied by trucks

Video 1, time: 00:18:36

Resistance (Load Carrying Capacity)• Material tests• Component tests• Diagnostic tests• Analytical simulations• Proof load tests

Reliability Index, bFor a linear limit state function, g = R – Q = 0,

and R and Q both being normal random variables

2Q

2R

QR

b

R = mean resistanceQ = mean loadR = standard deviation of resistanceQ = standard deviation of load

Reliability Index and Probability of Failure

PF b10-1 1.28

10-2 2.33

10-3 3.09

10-4 3.71

10-5 4.26

10-6 4.75

10-7 5.19

10-8 5.62

10-9 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

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

Selection Criteria for the Target Reliability

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

Target Reliability Index – Major Considerations

• Primary and secondary components• Multiple and single load paths (redundancy)• Element and system reliability• New design and existing structure• Ductile and brittle materials and components• Important, historical and ordinary structures

Types of Components• Primary component – its failure causes

failure of other components (or total collapse)

• Secondary component – its failure does not affect performance of other components

Examples of the Target Reliability Indices for Bridge Components

Primary component (multiple load path)bT = 3.5

Primary component (single load path)bT = 5.0

Secondary componentbT = 2.0

bT for Strength vs. Service Limit States

• Consequences of exceeding the limit state are different

• For decompression, bT = 1• For deflection, bT = 0• For fatigue, bT = 1-2

System vs. Component• Structures are systems made of

components• Failure of a component may not mean

failure of the system• Ductile and brittle components• Correlation between components

Structural Systems • Series systems – weakest link systems, to be

avoided• Parallel systems – components share the load,

preferred systems• Avoid brittle materials and elements, use

ductile materials and elements

The weakest linkdetermines the strength

37

38

Parallel system

Golden Gate Bridge, San Franciscobuilt in 1933-1935, span of 1280 m

Examples of the Target Reliability Indices for Bridges - Materials

For steel, reinforced concrete, prestressed concrete girders,

bT = 3.5For sawn wood bridge components, bT = 2.0For girder bridge as a system (all materials), bT = 5.5-6.5

Operational Importance• Regional and national

economy• Emergency situations (floods,

earthquakes, fires, hurricanes)

Historical Value• Historical structures can have a

special value for the society• Preservation of the general features

New Design vs. Existing Structure

• For a new design, reliability can be increased with little extra cost

• For an existing structure, any strengthening can be prohibitively expensive

• Current practice accepts lower reliability levels for existing structures

Reliability of Connections• For a bolted connection, the reliability can be

increased with negligible extra cost (extra bolts)

• For a steel component, the increase of reliability is much more costly (heavier section)

• Target reliability index for bolts is bT = 5-6, while for beams, bT = 3-4

How can we implement the target reliability?

• Design and evaluation of existing bridges – by load and resistance factors, safety margins in the design, fool-proof design

• Construction – quality control of materials and work skill, fool-proof construction

• Proper use and operation, maintenance, preventive repairs

Recommended bT

TIME PERIOD

PRIMARY COMPONENTS SECONDARY COMPONENTS

Single Path Multiple Path

5 years 3.50 3.00 2.25

10 years 3.75 3.25 2.50

50 years 4.00 3.50 2.75

Conclusions• Target reliability index varies depending on

consequences of failure, costs, and other considerations

• For new design, bT can be significantly higher than for evaluation of existing structures

• For historical structures, in addition, bT depends on social and political considerations

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