# Fundamental Concepts of Strength of Materials

DOI: 10.4018/978-1-4666-6379-4.ch002

## Abstract

This chapter presents the concepts of strength of materials that are relevant to the analysis of frames. These are the modified Timoshenko theory of elastic beams (Sections 2.1-2.3) and the Euler-Bernoulli one (Section 2.4). These concepts are not presented as in the conventional textbooks of strength of materials. Instead, the formulations are described using the scheme that is customary in the theory of elasticity and that was described in Chapter 1 (Section 1.1.1) (i.e. in terms of kinematics, statics, and constitutive equations). Kinematics is the branch of mechanics that studies the movement of solids and structures without considering its causes. Statics studies the equilibrium of forces; the basic tool for this analysis is the principle of virtual work. The constitutive model that describes a one-to-one relationship between stresses and deformations completes the formulation of the elastic beam problem. Finally, in Section 2.5, some concepts of the elementary theory of torsion needed for the formulation of tridimensional frames are recalled.
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## 2.1 Kinematics Of Timoshenko Beams

### 2.1.1 Generalized Displacements of Planar Beams

Consider a beam that displaces only in the plane X-Z as shown in Figure 1. The axis X is chosen so that it passes through the centroid of the beam’s cross-section; i.e. X coincides with the centroidal axis of the beam. The fundamental hypothesis of the theory of beams states that a cross-section of the beam moves only as a rigid body.

Figure 1.

Graphical representation of the fundamental hypothesis of the theory of beams

The movement of a cross-section at a distance x of the origin is defined by the matrix of generalized displacements {U(x)}:

(2.1.1) where u(x) is the displacement in the direction of the axis X of the particle in the centroid of the cross-section; w(x) is the displacement of the same point in the direction of the axis Z and β(x) is the rotation of the cross-section.

### 2.1.2 Generalized Deformations of Planar Beams

Consider now a beam differential element as the one shown in Figure 1; the modification of its shape is characterized by the following matrix of generalized deformations:

(2.1.2) where εn is the normal strain of the fiber in the centroidal axis; χ is the curvature and γ the distortion or angular strain (see Figure 2). Notice that the curvature χ measures the amount by which the centroidal axis deviates from being straight; the normal strain εn is related to the elongation of that axis; and the distortion γ measures the modifications in the angle between the centroidal axis and the transversal fibers of the beam differential element.

Figure 2.

Generalized deformations in a beam a) Normal strain b) Curvature c) Angular

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