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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?**

- 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

- = 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