# Non-Linear Structures

## 1st Edition

### Matrix Methods of Analysis and Design by Computers

**Authors:**K. I. Majid

**eBook ISBN:**9781483192512

**Imprint:**Butterworth-Heinemann

**Published Date:**28th June 1972

**Page Count:**360

## Description

Non-Linear Structures: Matrix Methods of Analysis and Design by Computers presents the use of matrix methods of structural analysis suitable for computers.

The book consists of 10 chapters. In the first chapter a brief introduction to the behavior of structures in general is given with reference to the linear elastic and simple plastic methods of structural analysis. Chapter 2 is devoted to linear matrix methods, both force and displacement. Chapter 3 examines the stability of an individual member with various end conditions. It also derives the stability functions used in Matrix force and Matrix displacement methods. Chapter 4 tackles the elastic stability of complete frames. Chapter 5 deals with the elastic instability of frames. The sixth chapter covers the elastic-plastic analysis of frames. This is followed in Chapter 7 by a number of approximate methods for the evaluation of the failure load of frames without following the sequence of hinge formation. The last three chapters are devoted to the design of structures and the non-linear aspects of design problems. A description of non-linear programming by piecewise linearization is included in Chapter 10.

Structural engineers, architects, researchers, and engineering students will find the book useful.

## Table of Contents

Notation

1. Introduction

Behaviour of Idealised Structures

Variation of Generalised Deformation of Structures with the Load Parameter

Load-Displacement Diagrams of a Portal

The Effect of Axial Load on the Behaviour of Idealised Structures

The Real Behaviour of Structures

The Rigid Plastic Theory

The Fully Plastic Hinge Moment

The Shape Factor

The Load Factor

Proportional Loading

The Use of Virtual Work Equation

Collapse of Continuous Beams

Selection of Mechanisms

Design by Rigid-Plastic Theory

Effect of Axial Load on Mp

Exercises

2. Matrix Methods of Structural Analysis

Introduction

Matrix Displacement Method

The Assemblage of Member Stiffness Matrices

Transformation of Displacements

The Overall Stiffness Matrix of a Structure

The Use of Degrees of Freedom in Structural Representation

Worked Examples

Matrix Force Method

The Flexibility Matrix

The Overall Flexibility Matrix of a Structure

Matrix Force Method for the Statically Determinate Structures

Matrix Force Method for Statically Indeterminate Structures

Worked Examples

Automatic Construction of the Load Transformation Matrix—Exercises

3. The Stability Functions

The Effect of Axial Load on the Stiffness of a Member

Rotation of a Member

Sway of a Member

A State of General Sway of a Member with Hinges

The Stability Functions used in the Matrix Displacement Method

The Stability Functions used in Matrix Force Method

Relationships Between the Stability Functions

General Deformation of a Member with Plastic Binges

Effect of Lateral Loading and Gusset Plates

A Computer Routine for the Evaluation of the Stability Functions

Exercises

4. Non-Linear Elastic Frames

Derivation of a Non-Linear Stiffness Matrix

Using s and c Functions

Using φ Functions

Matrix Displacement Method for Non-Linear Analysis of Structures

Worked Examples

Triangulated Frames

Initial Stresses

A Computer Programme for the Non-Linear Analysis of Plane Frames

Exercises

5. Elastic Instability

Elastic Instability of Members

Member with Single Curvature

Member in Double Curvature

Member Pinned at One End and Fixed at the Other

Member in Pure Sway

Elastic Instability of Frames

Determination of the Instability Conditions

Extraction of the Particular Mode from the General

Approximation of the Axial Loads

Elastic Critical Loads of Pitched Roof Portal Frames

Elastic Critical Load of Trusses

A Quick Determination of Instability Condition

Exercises

6. Non-Linearity Due to Plasticity. Elastic-Plasticanalysis of Frames

Introduction

The Displacement Transformation Matrix for a Member with a Hinge

The Overall Stiffness Matrix

Elastic-Plastic Analysis of a Propped Cantilever

Elasticplastic Analysis of Frames

The Nature of the Iteration

Iteration to Constant Mp

Iteration to Reduced Mp

Alterations to the Load Vector

A Computer Programme for the Elastic-Plastic Analysis of General Frames

Worked Examples

Inactive Hinges

Computational Economy

Exercises

7. Failure Load of Frames (Approximate Methods)

Introduction

The Rigid-Plastic Drooping Curve

Calculation of Failure Loads-Approximate Method 1

Assumptions

Analysis of Columns

Analysis of Frames

Rankine's Formula-Approximate Method 2

Experimental Evidence

Exercises

8· Elastic-Plastic Design of Sway Frames

Introduction

Variation of Maximum Moments

Design of Individual Members

Design Criteria

Design Procedure

Economy in Computer Time

Design of Four-Storey Frame

Design of Irregular Frames

Conclusion

9. Optimum Design of Structures using Linear Programming

Introduction

Linear Programming

Equality Constraints

Constraints with Surplus Variables

Minimisation Problems

Minimum Weight Design of Structures

The Rigid Plastic Minimum Weight Design

Design of a Portal by Mechanism Method

Minimum Weight Design using Equilibrium

Exercises

10. Optimum Design of Structures using Non-Linear Programming

Introduction

Some Properties of Convex Function

Piecewise Linearization

Statically Determinate Structures

Worked Example

Statically Indeterminate Structures

Separation of the Variables

Piecewise Linearisation

Design Example

Design of Rigidly-Jointed Frames

Exercises

Appendix I

References

Index

## Details

- No. of pages:
- 360

- Language:
- English

- Copyright:
- © Butterworth-Heinemann 1972

- Published:
- 28th June 1972

- Imprint:
- Butterworth-Heinemann

- eBook ISBN:
- 9781483192512