Problems Book for Probabilistic Methods for the Theory of Structures with Complete Worked Through Solutions

Isaac Elishakoff

• Publisher: WSPC
• Year: 2017
• ISBN:
  9789813201132
• Topic: Tecnologia e ingegneria
• Subtopic: Ingegneria meccanica

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The first edition of the combined monograph and textbook Probabilistic Methods in the Theory of Structures was published by Wiley-Interscience in 1983. 1n 1999, Dover Publications, Inc. published its second edition under shorter title Probabilistic Theory of Structures. Now, World Scientific has expanded into a 3rd edition to include Problems with Complete Worked-Through Solutions. This compendium of solutions was written in response to requests by numerous university educators around the world, since it has been adopted as a textbook or an additional reading for both undergraduate and graduate courses.

The author hopes that the availability of such solutions manual will further help to establish the courses dealing with probabilistic strength of materials, design, random buckling, and random vibration. The material itself was developed by author for various undergraduate and graduate courses, during years 1972–1989 at the Technion — Israel Institute of Technology, in Haifa, Israel, at the Delft University of Technology in the Netherlands, year 1979/80 at the University of Notre Dame, Indiana, USA and at the Florida Atlantic University, USA since 1994.

Already since mid-eighties, the author was informed that the book was adopted in numerous universities worldwide. Besides complete solutions to more than one hundred problems, additional material and remarks are included as Chapter 12, bringing some ideas down to the "number" level.

It is strongly hoped that this manual will promote much wider dissemination of probabilistic methods' courses at universities, and ultimately, in engineering practice worldwide.

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Contents:

• Preface
• Probability Axioms
• Single Random Variable
• Examples of Probability Distribution and Density Functions. Functions of a Single Random Variable
• Reliability of Structures Described by a Single Random Variable
• Two or More Random Variables
• Reliability of Structures Described by Several Random Variables
• Elements of the Theory of Random Functions
• Random Vibration of Discrete Systems
• Random Vibration of Continuous Structures
• Monte Carlo Method
• Additional Example Problems and Comments

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Readership: Researchers and academics into the theory of structures.
-->Keywords:Mechanical Engineering;Theory of Structures;Strength of Materials;Random Vibrations;Random BucklingReview:0

chapter 5

Reliability of Structures Described by a Single Random Variable

PROBLEM 5.1

The loads acting on different machine components often have a chi-square distribution with m degrees of freedom, as was shown by Serensen and Bugloff,

Find the reliability of the bar under random tensile force. Table 5.1 contains the values of

where

f_N(n)dn = α for α = 0.995, 0.99, 0.975, 0.05, 0.025, 0.01, 0.005, and ν = 1, 2,..., 30 (see figure).

SOLUTION 5.1

Notes to lecturer

Note 1. The chi-square distributed random variable X with

is nondimensional. For it to represent an axial force, we will use the transformation

where n₀ is a constant with the dimension of force. Then (see Equation 4.46)

and

If N is measured in say, kilonewtons, we can choose n₀ = 1 kN, which yields the formula given in the problem.

Note 2. This problem can be designed in three forms; σ ’s the first, the second or the third basic problem of the probabilistic strength of materials.

First problem: Lecturer should specify the cross-sectional area of the bar “a”, yield stress σ_Y and required reliability r, as well as “m” the number of degrees of freedom of x²-distribution. The reliability is:

Table 5.1 lists the values of x²α, ν, so that

Now

Thus if σ_Ya is chosen such that

we have

if 1 − α > r, the strength requirement is met, and violated otherwise.

For the numerical values to be valid, the following set of data should be given to student (as an example):

Then σ_Ya = 6 · 5.695 = 34.17,

which means that strength requ...

Table of contents

• Cover
• Halftitle
• Title
• Copyright
• Contents
• Preface
• Chapter 2. Probability Axioms
• Chapter 3. Single Random Variable
• Chapter 4. Examples of Probability Distribution and Density Functions. Functions of a Single Random Variable
• Chapter 5. Reliability of Structures Described by a Single Random Variable
• Chapter 6. Two or More Random Variables
• Chapter 7. Reliability of Structures Described by Several Random Variables
• Chapter 8. Elements of the Theory of Random Functions
• Chapter 9. Random Vibration of Discrete Systems
• Chapter 10. Random Vibration of Continuous Structures
• Chapter 11. Monte Carlo Method
• Chapter 12. Additional Example Problems and Comments