Kaveh - Computational Structural Analysis and Finite Element Methods

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A. Kaveh Computational Structural Analysis and Finite Element Methods 2014 10.1007/978-3-319-02964-1_1

Springer International Publishing Switzerland 2014

1. Basic Definitions and Concepts of Structural Mechanics and Theory of Graphs

A. Kaveh 1

(1)

Centre of Excellence for Fundamental Studies in Structural Engineering School of Civil Engineering, Iran University of Science and Technology, Tehran, Iran

Abstract

This chapter consists of two parts. In the first part, basic definitions, concepts and theorems of structural mechanics are presented. These theorems are employed in the following chapters and are very important for their understanding. For determination of the distribution of internal forces and displacements, under prescribed external loading, a solution to the basic equations of the theory of structures should be obtained, satisfying the boundary conditions. In the matrix methods of structural analysis, one must also use these basic equations. In order to provide a ready reference for the development of the general theory of matrix structural analysis, the most important basic theorems are introduced in this chapter, and illustrated through simple examples.

In the second part, basic concepts and definitions of graph theory are presented. Since some of the readers may be unfamiliar with the theory of graphs, simple examples are included to make it easier to understand the presented concepts.

1.1 Introduction

This chapter consists of two parts. In the first part, basic definitions, concepts and theorems of structural mechanics are presented. These theorems are employed in the following chapters and are very important for their understanding. For determination of the distribution of internal forces and displacements, under prescribed external loading, a solution to the basic equations of the theory of structures should be obtained, satisfying the boundary conditions. In the matrix methods of structural analysis, one must also use these basic equations. In order to provide a ready reference for the development of the general theory of matrix structural analysis, the most important basic theorems are introduced in this chapter, and illustrated through simple examples.

In the second part, basic concepts and definitions of graph theory are presented. Since some of the readers may be unfamiliar with the theory of graphs, simple examples are included to make it easier to understand the presented concepts.

1.1.1 Definitions

A structure can be defined as a body that resists external effects such as loads, temperature changes, and support settlements, without undue deformation. Building frames, industrial building, bridges, halls, towers, dams, reservoirs, tanks, retaining walls, channels, pavements are typical structures of interest to civil engineers.

A structure can be considered as an assemblage of members and nodes. Structures with clearly defined members are known as skeletal structures . Planar and space trusses, planar and space frames, single and double-layer grids are examples of skeletal structures, Fig..

Fig. 1.1

Examples of skeletal structures. ( a ) A foot bridge truss ( b ) A planar frame. ( c ) A space frame. ( d ) A space truss. ( e ) A single-layer grid. ( f ) A double-layer grid. ( g ) A single-layer dome. ( h ) A double-layer barrel vault

Structures which must artificially be divided into members (elements) are called continua . Concrete dams, plates, and pavements are examples of continua, Fig..

Fig. 1.2

Examples of continua. ( a ) A plate. ( b ) A dam

The underlying principles for the analysis of other structures are more or less the same. Airplane, missile and satellite structures are of interest to the aviation engineer. The analysis and design of a ship is interesting for a naval architect. A machine engineer should be able to design machine parts. However, in this book only structures of interest to structural engineers are studied.

1.1.2 Structural Analysis and Design

Structural analysis is the determination of the response of a structure to external effects such as loading, temperature changes and support settlements. Structural design is the selection of a suitable arrangement of members, and a selection of materials and member sections, to withstand the stress resultants (internal forces) by a specified set of loads, and satisfy the stress and displacement constraints, and other requirements specified by the utilized code of practice. The diagram shown in Fig. is a simple illustration for the cycle of structural analysis and design.

Fig. 1.3

The cycle of analysis and design of a structure

In optimal design of structures this cycle should be repeated hundred and sometime thousands of times to reduce the weight or cost of the structure.

Structural theories may be classified from different points of view as follows:

• Static versus dynamic;
• Planar versus space;
• Linear versus non-linear;
• Skeletal versus continua;
• Statically determinate versus statically indeterminate.

In this book, static analyses of linear structures are mainly discussed for the statically determinate and indeterminate cases. Here, both planar and space skeletal structures and continua models are of interest.

1.2 General Concepts of Structural Analysis

1.2.1 Main Steps of Structural Analysis

A correct solution of a structure should satisfy the following requirements:

Equilibrium : The external forces applied to a structure and the internal forces induced in its members should be in equilibrium at each node.

Compatibility : The members should deform so that they all fit together.

Force-displacement relationship : The internal forces and deformations satisfy the stressstrain relationships of the members.

For structural analysis two basic methods are in use:

• Force method : In this method, some of the internal forces and/or reactions are taken as primary unknowns, called redundants. Then the stressstrain relationship is used to express the deformations of the members in terms of external and redundant forces. Finally, by applying the compatibility conditions that the deformed members must fit together, a set of linear equations yield the values of the redundant forces. The stress resultants in the members are then calculated and the displacements at the nodes in the direction of external forces are found. This method is also known as the flexibility method and compatibility approach .
• Displacement method : In this method, the displacements of the nodes necessary to describe the deformed state of the structure are taken as unknowns. The deformations of the members are then calculated in terms of these displacements, and by use of the stressstrain relationship, the internal forces are related to them. Finally, by applying the equilibrium equations at each node, a set of linear equations is obtained, the solution of which results in the unknown nodal displacements. This method is also known as the stiffness method and equilibrium approach .

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