Elastic Frames


This chapter describes the mathematical models for the analysis of elastic frames. Again, these theories are not presented as in the conventional textbooks of structural analysis but using the scheme of continuum mechanics and the theory of elasticity that was described in Chapter 1 (Section 1.1.1). The reason is that the conventional presentation is not suitable for generalization of the case of inelastic structures, specifically for fracture and damage mechanics of frames. Several classes of elastic frames are described in this chapter: planar (Sections 3.1-3.4), tridimensional (Section 3.5), linear, nonlinear, based on the Euler-Bernoulli beam theory (Section 3.3), based on the formulation of Timoshenko (Section 3.4), and under quasi-static or dynamic forces.
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3.1 Kinematics Of Planar Frames

3.1.1 Tridimensional and Planar Frames

A frame is defined as a set of m beam-column elements connected by n nodes. For instance, a chair like the one shown in Figure 1a can be represented by the tridimensional frame with 26 elements and 20 nodes as indicated in Figure 1b. In the general case, the nodes of the frame can have displacement components in any direction.

Figure 1.

a) A chair as a structure b) Tridimensional frame representation c) Planar frames representation

However, another possibility is the decomposition of structures like this into several two-dimensional frames as shown in Figure 1c; those two-dimensional structures are also called planar frames. It is assumed that this kind of structures can move only in their own plane. Thus, some aspects of the behavior are neglected, for instance torsion effects; but the analysis is considerably simpler. Both kinds of formulations are considered in this book, first the planar frame case, then the general one.

3.1.2 Generalized Displacements of a Planar Frame

Consider a planar frame as the one shown in Figure 2. A set of coordinate axis X Z is introduced. Its initial configuration, i.e. the coordinates of the nodes and the table that indicates how they are connected by beam column elements, is known. Each node and each element of the frame is labeled using natural numbers as shown in Figure 2. Consider now a node i of the frame. The movement of this node is characterized by the displacements ui in the direction X, the displacement wi in the direction Z and the rotation θi as shown in Figure 2. Clockwise rotations of a node are positive, positive displacements follow the conventional rule. The generalized displacement matrix of the frame includes the displacement of all the nodes of the frame:

Figure 2.

Planar frame and generalized displacements of node


The elements of the frame are also labeled using natural numbers as shown in Figure 2. The generalized displacement matrix of a frame element b between nodes i and j is defined as follows:


One of the goals of a structural analysis is the determination of the generalized displacements as a function of time and the external loading; its computation allows the graphical representation of the frame movement.

Some of the components of this matrix are not computed but must be defined by the analyst in order to formulate a problem with a unique solution. The components of the matrix with known values are called restricted displacements; the remaining ones are denoted free displacements. The nodes that have one or more restricted displacements are called supports of the frame. The restricted displacements must be chosen so that arbitrary rigid body movements of the frame are impossible. It is customary to represent graphically the supports and their displacements as shown in the Table 1. Often, the restricted displacements are equal to zero; it is then assumed that the foundations of the structure are rigid. However, this is not always the case; non-nil values are sometimes chosen in order to represent settlement of supports or seismic effects on the structure.

Table 1.
Graphic representation of frame supports

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