Structural Analysis 2: Statically Indeterminate Structures

Preface

The main objective of this volume is to provide students of civil, mechanical, aeronautical and marine engineering as well as those interested in structural analysis with the essentials for analyzing statically indeterminate structures. This book allows them to acquire sufficient knowledge to study and analyze statically indeterminate structures. The reader will find a series of exercises at the end of each chapter that can be used to deepen their knowledge and improve their ability to master statically indeterminate structural analysis methods.

This book is the second volume of structural analysis. The first volume is devoted to analyzing statically determinate structures, whereas this volume is exclusively analyzes statically indeterminate structures. Every chapter is designed in a specific way: an illustration of the objectives and the parts covered, a general introduction, a theory of the proposed approach, a numerical study of some examples and a summary at the end. Each chapter concludes with a series of problems. The primary objective is to meet the needs of mechanics students, allowing them to acquire the necessary knowledge and to apply statically indeterminate structural analysis methods. In addition, the numerical examples used are common across different numerical methods.

The book is divided into seven chapters dealing with statically indeterminate structures. It offers a comprehensive presentation of statically indeterminate structural analysis methods.

Chapter 1 is a general introduction to analyzing statically indeterminate structures. Chapter 2 presents the analysis of single span statically indeterminate beams (the method of three moments), the study of continuous beams (the Clapeyron method) and the focus method. In Chapter 3, the method of forces is described in the analysis of statically indeterminate beams, statically indeterminate frames and statically indeterminate trusses. Chapter 4 describes the slope-deflection method for the analysis of flexed elements, such as beams and frames. Chapter 5 illustrates the moment-distribution method. Chapter 6 looks at the influence lines of support reactions and/or internal actions of statically indeterminate structures under moving loads. Chapter 7 is devoted to the analysis of statically indeterminate arches. For this, two categories of arches are used: semicircular arches and curved arches.

Finally, we hope that our approach in this book’s publication will meet the needs of students interested in this scientific and technical subject. Nevertheless, we are very aware that the work presented is not exempt from mistakes. For this reason, we warmly welcome any corrections and comments, which will improve future editions of this book. Comments or suggestions can be sent to the email addresses found on the website www.freewebs.com/khalfallah/index.htm.

Salah KHALFALLAH

July 2018

1

Introduction to Statically Indeterminate Structural Analysis

The teaching objectives of this chapter are as follows:

– the importance and usefulness of statically indeterminate structures;

– calculating the degree of external and internal static indeterminacy of the structures;

– analyzing kinematic static indeterminacy;

– illustrating the strengths and weaknesses of statically indeterminate structures.

In the first part, we give a general introduction to the methods of analyzing statically indeterminate structures. In this context, we describe the external, internal and kinematic static indeterminacies of the structures. In the second part, we illustrate the analysis methods for statically indeterminate structures. Lastly, we list the advantages and disadvantages of statically indeterminate structures.

1.1. Introduction

Structures are grouped into two categories: (1) statically determinate structures and (2) statically indeterminate structures. The static equations are not sufficient for analyzing statically indeterminate structures. In this case, the number of unknowns is strictly greater than the number of independent equilibrium equations.

The primary role of analysis of a statically indeterminate structure is to remove the static indeterminacy of the given structure. This removal means that we can calculate the support reactions and the internal actions when the structure is solicited by mechanical loads, or subjected to deflections, and/or undergoing a support settlement. The analysis methods for statically indeterminate structures are used here to make the number of unknowns equal to the number of equations, which allows the problem to be solved.

This book is particularly devoted to the analysis of statically indeterminate structures. To present the differences between the analysis methods for statically indeterminate structures, the problems we consider generally have a common object across the analysis methods. In this context, each chapter illustrates the theoretical foundation of the analytical method, presented in detail and accompanied by a series of numerical examples.

1.2. External static indeterminacy

The purpose of structural analysis is to determine the support reactions and the variation of internal actions in the elements of a statically indeterminate structure. The static indeterminacy of a structure can be internal, external or internal and external simultaneously. It is called externally statically indeterminate if the number of support reactions exceeds the number of independent equations. The plane structures are externally statically indeterminate if the number of support reactions is greater than 3 (Figure 1.1) and it is greater than 6 if the structure is spatial (Figure 1.2).

From this explanation, we define the degree of static indeterminacy of a system by the difference between the number of support reactions and the number of independent equations that can be constructed. The degree of external static indeterminacy f of a plane structure [1.1] or a space structure [1.2] is deduced by

[1.1] equations

[1.2] equations

We calculate the degree of static indeterminacy of the beam and frame (Figure 1.1).

Beam, f = 5 – 3 =2

Frame, f = 10 – (3 + 1) = 6

Figure 1.1. Statically indeterminate externally of plane structures1

Figure 1.2. Statically indeterminate externally of space structures

For space structures (Figure 1.2), the degree of static indeterminacy is

Frame (1), f = 12 – 6 = 6

Beam (2), f = 10 – 6 = 4

Frame (3), f = 24 – (6 + 12) = 6

The degree of static indeterminacy of trusses is calculated by using relationships [1.1] and [1.2]. Figure 1.3 presents plane and space trusses.

Figure 1.3. Externally statically indeterminate plane structures

The degree of external static indeterminacy of plane structures (Figure 1.3) is

Structure (1): f = 4 – 3 = 1

Structure (2): f = 6 – 3 = 3

In the same way, space truss structures (Figure 1.4) are the most used in the construction of large exhibition halls and sports halls, etc.

Figure 1.4. Statically indeterminate space truss

The degree of static indeterminacy of the structure is

f = 12 – 6 = 6

1.3. Internal static indeterminacy

In this section, we describe how to calculate the degrees of static indeterminacy of trusses, frames, beams and