Reliability Assessment

CONTENTS

1. INTRODUCTION 2. METHODS OF RELIABILITY ASSESMENT 3. POINT ESTIMATE METHOD-OVERVIEW 4. RELIABILITY ANALYSIS RESULTS AND DISCUSSIONS REFERENCES

INTRODUCTION

The evaluation of the safety of structures is a task of much importance. It has been one of the of interest for engineers. The safety of a Structure depends on the resistance, R of structure and the action, S (load) on the Structure. The performance of a structure is assessed by its safety, serviceability and economy. The information about input variables is never certain, precise and complete. The sources of uncertainties may be: a. Physical Uncertainty: This group consists of uncertainties in material parameters such as modulus or concrete and steel, stability of concrete and steel in different condition such as tension and flexure. B. Statistical uncertainty: This group consists of uncertainties in loads. Under static loading conditions, one is concerned with dead and live load and there are usually some uncertainties in relation to live loads. Structures may also be subjected to dynamic loads from earthquakes wind and waves. Significant uncertainties are associated with such random loads. C. Model uncertainty: This group consists of uncertainties in mathematical modelling and methods of analysis. Each method of analysis or design is based on simplifying assumption and arbitrary factors of safety are often used. d. Gross errors The presence of uncertainties, the absolute safety of the structure is impossible. A. The unpredictability of 1. Loads on structure during its life. 2. In- place material strength. 3.

Human errors.

B. structural idealization in forming the mathematical model of the structure to predict its response or behaviour

C. limitations in numerical methods. Therefore, some risk of unacceptable performance must be tolerated. With respect to risk of life, the structural safety is important. The safety factors provided in the existing codes and standards primarily based on practice, judgement and experience may not be adequate and economical.

Definition:One of the most Common definitions for reliability accepted by all is that “Reliability is the probability of an item performing its intended function over a given period of time under the Operating conditions encountered”. It is important to note that the above definition stresses four significant elements viz., i.

Probability: Due to uncertainties, the reliability is a probability.

ii.

Intended function: Intended function signifies that the reliability is a performance characteristic. For a structure to be reliable, it must perform a certain function satisfactorily for which it has been designed i.e., safety against shear or flexure or torsion etc.

iii.

Time: The reliability is always related to the lifetime of the structure. During the specified life of the structure. it must perform the assigned function satisfactorily.

iv. Operation conditions: The operating conditions indicate the actions Or stresses that will be imposed on the structure. These may be loads, temperature, shock, vibrations.

Methods of assessment of reliability Reliability assessment involves the computation of a numerical reliability measure for a given condition with design quantities described in probability.

First order second moment method (FOSM) In this method, the random variables are characterised by their first and second moments. In evaluating the first and second moments of the failure function, the first order approximation is used. Hence this method is called as FOSM. The FOSM is derived from Taylors formula for the expansion of a function f(x) about a point x=𝑥

F(x)=f(𝑥)+f’(𝑥)(x-𝑥)+𝑓 (x)/2! (𝑥 − 𝑥)2 ……..+Rn

MONTE CARLO SIMULATION TECHNIQUE For problems involving random variables with known probability distributions. Monte carlo simulation is required. This involves repeating a simulation process, using in each simulation a particular set of values of the random variables generated in accordance with the corresponding probability distributions. A sample from Monte Carlo simulation is similar to a sample of experimental observations. Therefore the results may be treated statically and such results may also be presented in the form of histograms, and methods of statistical estimation and inference are also applicable. One of the main tasks of this method is the generation of random numbers, the simulation process is deterministic. But in practise, this method may be limited by constraints of economy and computer capability. Therefore, it may be used only as the last resort, when and if analytical solutions are not possible. To produce a reasonably accurate estimate of the failure probability at least 100/pf trials are required.

Reliability analysis of shear strength of a RCC beam by point estimate method Strength formulation is given by:-

g(x) = 𝝁𝝉𝑺 𝝁𝑻𝑾 𝝁𝑫 − 𝝁𝑺 /𝟐 Given:𝜇𝜏 𝑆 =95 N/mm2 𝜇 𝑇𝑊 =1.25mm 𝜇𝐷 =50 mm 𝜇𝑆 =4000 N

𝜎𝜏 𝑆 =10 N/mm2 𝜎𝑇𝑊 =.0625mm 𝜎𝐷 =2.5mm 𝜎𝑆 =1000 N

Upper and Lower bound values are calculated as:𝜇𝜏 𝑆 95

𝜎𝜏 𝑆 10

𝜇+𝜏𝑆 105

𝜇−𝜏𝑆 85

𝜇 𝑇𝑊 1.25

𝜎𝑇𝑊 .0625

𝜇 + 𝑇𝑊 1.3125

𝜇 − 𝑇𝑊 1.1825

𝜇𝐷 50

𝜎𝐷 2.5

𝜇 +𝐷 52.5

𝜇 −𝐷 47.5

𝜇𝑆 4000

𝜎𝑆 1000

𝜇 +𝑆 5000

𝜇 −𝑆 3000

=8 variables, where n=3 for resistance equation =2 variables, where n=2 for load equation Iterations:=7235.6 =6546 =5922.65 =6546.1

= 1500 AVERAGE=2000(x2)

=5857.1 =5299 =5299 =4794.53 AVERAGE=5937.5(x1) g(x) = 5935.7-2000=3935.7 = 1217.66 = 608 =

14.85

=

60.86

=

80.4

=

638.3

= =

638.3 1143.3

= 3763.4

=500 =500 = 1000

=806.4 , n=8

= 267.26, n=2

=

3935.7 / 539.14 = 7.31

Conclusions  THE RELIABILITY INDEX OBTAINED IS 7.31, WHICH IS SAFE.  POINT ESTIMATE METHOD IS EASIEST METHOD, WHICH INVOLVES SIMPLE MATHE MATICAL CALCULATIONS.  DESIGNS MADE UNDER THE CODE AND OTHER STANDARD PROVISIONS ARE UNECONOMICAL RATHER MORE SAFE.

Reference 1. MILTON E HARR, RELIABILITY – BASED DESIGN IN CIVIL ENGINEERING, MCGRAW-HILL BOOK COMPANY, INC, NEW YORK, 1987. 2. RANGANATHAN R, STRUCTURAL RELIABILITY ANALYSIS AND DESIGN, JAICO PUBLISHING HOUSE, 1999.