Bridges are vulnerable to extreme events such as natural disasters in addition to hazards stemming from negligence and improper maintenance, overloading, collisions, intentional acts of vandalism, and terrorist attacks. These structures must be protected but the current approach to risk is not always rational. Sensitivity analysis will be performed to relate the reliability of bridges and reliability of the transportation network.

1. Risk Analysis and Target Reliability for
Bridges
Andrzej S. Nowak, Ph.D.
University of Nebraska-Lincoln

2. Disclaimer
The 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.

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

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

5. 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

6. Uncertainties
 Loads (natural and created)
 Material properties
 Load carrying capacity (resistance)
 Basic requirement:
Load effect (Q) < Resistance (R)
 Safety margin
R–Q>0

7. Reliability Index, β

8. 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?

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

10. Load Factor

11. Resistance Factor

12. 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

13. Bridge Loads
• Dead load
• Live load (static and dynamic)
• Environmental loads (wind, snow,
earthquake, temperature, ice)
• Special loads (vehicle and vessel
collision, fire, explosion)

16. Two Trucks Side by Side

17. 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

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

24. Reliability Index, β
For a linear limit state function, g = R – Q = 0, and
R and Q both being normal random variables
β=
(µ R − µQ )
σ +σ
2
R
2
Q
µR = mean resistance
µQ = mean load
σR = standard deviation of
resistance
σQ = standard deviation of load

25. Reliability Index and Probability of Failure
PF β
10-1 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

26. 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

27. 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

28. Selection Criteria for the Target Reliability
• Consequences of failure
• Economic analysis
• Past practice
• Human perception
• Social/political decisions

29. 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

30. 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

31. Examples of the Target Reliability
Indices for Bridge Components
Primary component (multiple load path)
βT = 3.5
Primary component (single load path)
βT = 5.0
Secondary component
βT = 2.0

32. β T for Strength vs. Service Limit States
• Consequences of exceeding the limit state are
different
• For decompression, βT = 1
• For deflection, βT = 0
• For fatigue, βT = 1-2

33. 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

34. 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

35. The weakest link
determines the strength

37. 37

38. 38

40. Parallel system

42. Golden Gate Bridge, San Francisco
built in 1933-1935, span of 1280 m

43. Examples of the Target Reliability
Indices for Bridges - Materials
For steel, reinforced concrete, prestressed
concrete girders,
βT = 3.5
For sawn wood bridge components,
βT = 2.0
For girder bridge as a system (all materials),
βT = 5.5-6.5

44. Operational Importance
• Regional and national
economy
• Emergency situations (floods,
earthquakes, fires, hurricanes)

45. Historical Value
• Historical structures can have a
special value for the society
• Preservation of the general features

46. 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

47. 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 βT = 5-6, while
for beams, βT = 3-4

48. 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

49. Recommended β T
TIME PRIMARY COMPONENTS SECONDARY
PERIOD 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

50. 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

51. CREDITS
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