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Reliability?
- “BEST bus services are very reliable”
- “BMC water supply is not very reliable”
- “In Mumbai, Western Railway’s service is more reliable than that of the Central Railway”
What is reliability, in technical terms?
- How do we measure it?
- Why is not a system fully reliable?
Civil Engineering Systems
- Structural (Buildings, Bridges, Dams, Fly-overs)
- Transportation (Road systems, Railways, Air traffic)
- Water (Water supply networks, Waste water networks)
Each system is designed differently, but there is a common philosophy
How to Design
|
Requirement |
Provision |
|
Demand |
apacity/Supply |
|
Load |
Resistance |
|
x million liter/day of |
x million liter/day of |
|
water for IITB |
water for IITB |
|
residents |
Residents |
Basic Design Philosophy
Capacity should be more than demand
C ? D
Example: Provide at least x million liter/day of water to a colony residents
How much more than the demand?
- Theoretically, just more
- However, designers provide a lot more
- Why? ->Because of uncertainty
Uncertainty
We are not certain about the values of the parameters that we use in design specifications Sources/reasons of uncertainty:
- Errors/faults/discrepancies in measurement (for demand) or manufacturing (for capacity)
- Approximations/idealizations/assumptions in modeling
- Inherent uncertainty — “Aleatory”
- Lack of knowledge — “Epistemic”
Measurement and Manufacturing Errors
- Strength of concrete is not same at each part of a column or a beam in a building system
- the depth of a steel girder is not exactly same (and not as specified) at each section (Errors in estimating demand/capacity?)
- Weight of concrete is not same at each part of a column or a beam in a building system (Error in estimating demand/capacity?)
- Wheels of an aircraft hit the runway at different speeds for different flights
Moral of the story:
Repeat a measurement/estimate/experiment several times and we do not get exactly the same result each time
IDEALIZATIONS IN MODELING
- Every real system is analyzed through its “model”
- Idealizations/simplifications are used in achieving this model
Example: (modeling live load on a classroom floor)
- Live loads are from non-permanent “occupants”; such as people, movable furnishers, etc.
- We assume live load to be uniform on a classroom (unit?)
- [We also assume the floor concrete to be “homogeneous” (that is, having same properties, such as strength, throughout)]
- Therefore our analysis results are different from the real situation
Example: (modeling friction in water systems)
- Friction between water and inner surface of a pipeline reduces flow
- We assume a constant friction factor for a given pipe material
- In reality, the amount of friction changes if you have joints, bends and valves in a pipe
- If we need to consider these effects, the analysis procedure will be very complicated
- However, we should remember that there is difference between the behaviors of model and the real system
Epistemic and Aleatory Uncertainties
Epistemic
- Due to lack of understanding
- Not knowing how a system really works
- These uncertainties can be reduced over time (enhanced knowledge, more observation) Aleatory
- Due to inherent variability of the parameter
- Unpredictability in estimating a future event
- These uncertainties can be reduced as well, with more observations
The Case of Earthquakes
- Structures have to be designed to withstand earthquake effects
- Earthquakes that a structure is going to face during its life-span are unpredictable
- We do not know when, how big (magnitude), how damaging (intensity)
- This is due to the unpredictability inherent in the physical nature of earthquakes
Aleatory uncertainty
How Earthquakes Occur
Plate Tectonics
Elastic Rebound Theory
AD = Fault line (along which one side of earth slides with respect to the other)
A = Focus of the earthquake (where the slip occurs and energy is released)
C = Epicenter of the earthquake (point on earth surface directly above the focus)
B = Site (location for the structure)
Earthquake waves travel from A to B (body waves) and C to B (surface waves)
Earthquake waves travel from epicenter to the site (site= where the structure is located)
- The shock-wave characteristics are changed by the media it is traveling through
- The earthquake force that is coming to the base of a structure is also determined by the soil underneath
- We need to know accurately these processes by which the ground motion is affected
- Any lack of knowledge in these regards will lead to: Epistemic uncertainty
Effects of Uncertainty
- Analysis results are not exactly accurate (that is, not same as in real life)
- Estimation of demand and capacity parameters is faulty
- We may not really satisfy the C ? D equation
- However, we will not know this
- Solution: apply a factor of safety (F)
C ? FD or C/F ? D
- This factor takes care of the unforeseen errors due to uncertainty
If C ? 2.5D, then even in real situation, it should be C ? D
Deterministic Design: Factor of Safety
- This is the traditional design philosophy
- A deterministic design procedure assumes that all parameters can be accurately measured (determined)
- Thus, there is no uncertainty in estimating either C or D
- So, if we satisfy a design equation, we make the system “ 100% safe” . It cannot fail.
- In addition, we add a factor of safety to account for unforeseen errors
- This factor of safety is specified based on experience and engineering judgement
- The value of the safety factor varies for different cases
Example:
0.447 fcAc + 0.8 fsAs ? P
- This is the design specification for a reinforced concrete column (RC = concrete reinforced with steel bars)
- fc = strength of concrete, fs = strength of steel
- Ac = area of concrete, As = area of steel bars
- 0.447 and 0.8 are for safety factors
- P = Force acting on the column (demand)
Reliability-Based Design
- This is the newly developed design philosophy
- Here, we accept the uncertainties in both demand and capacity parameters
- However, all these uncertainties are properly accounted for
- Uncertainty in estimating each parameter is quantified
- The C ? D equation does not provide a full-proof design
- The design guideline specifies a probability of failure due to those uncertainties
- Load and resistance factors are used in stead of a single factor of safety
- These factors are based on analysis, not on judgement
Old vs. New
|
Deterministic |
Reliability-Based |
|
100% safe |
Less than 100% safe |
|
No uncertainty |
Uncertainties are properly accounted for |
|
Factor of safety is based on judgement |
Factors are calculated from uncertainty |
|
Simple, but claims are not realistic |
More scientific in all aspects, but complex |
Reliability-Based Design
- Reliability-based design equation:
-
= Resistance/Capacity Factor
= Resistance/Capacity Factor- = Load/Demand Factor
- This equation assigns a probability of failure (Pf) for the design
- This Pf is based on the load and resistance factors (also known as “ partial safety factors” )
- Real systems always have some probability of failure (even though deterministic design does not recognize)
Uncertainties are unavoidable; it exists in natural systems and the way we measure and manufacture
- It is not wise to ignore them
- The best way to deal with uncertainties is to quantify them properly (using statistics and probability)
- Reliability-based design accounts for uncertainties scientifically (whereas, deterministic design does not)
- RBD assigns a specific reliability on a design through Pf (probability of failure)
- It is not bad for a system to have probability of failure, but bad not to know how much
- RBD tries to keep Pf within a target level
