## 80

FIGURE 3.27 p''/p versus L/R. (Reprinted from Ref. [1, p. 229] by courtesy of Marcel Dekker, Inc.)

FIGURE 3.28 Discretized elements: (a) flat plate and (b) framework.

Timoshenko theories with and without elastic media, and coupling torsional and flexural vibrations of gird systems [1, pp. 214-221, 229-258].

3.5 Response Analysis of Finite Element

Systems — Consistent-Mass Formulation

### 3.5.1 Introduction

Mathematical models for structural dynamic analysis may be generally classified into three approaches: lumped mass, dynamic stiffness (frequency-dependent stiffness), and consistent mass (finite element). For computer application, all these are formulated by using the displacement (stiffness) matrix method. Lumped-mass and dynamic-stiffness approaches were presented in previous sections. The characteristics of these two approaches differ in that the motion equation for lumped-mass model consists of independent mass and stiffness matrices while the dynamic stiffness model has mass implicitly combined with stiffness. The lumped-mass and consistent-mass approaches are similar in terms of motion equation: both of them have independent mass and stiffness matrices; their mass matrices, however, are not the same.

Consistent mass may be considered as an alliance of finite elements normally used in continuum mechanics. This method is often used for frameworks as well as plate and shell structures. The fundamental concept of finite elements is able to model a structure or continuum by dividing it into a number of regions. Each region behaves as a structural member with nodes compatible to the nodes of neighboring regions. These regions are called finite elements. A plate shown in Figure 3.28 represents a continuum where two regions are sketched with nodes 1,2, ...,9. the boundaries of neighboring elements at node 5 are compatible at the node but not necessarily compatible along the edges such as 4-5 or 5-8 and so on. Framed structures, however, are automatically discretized by the nature of their members and connections. One may say that finite element analysis dominates structural matrix methods, and frameworks are special cases of finite elements.

### 3.5.2 Formulation of Stiffness and Mass Matrices

Let y(x, t) be the transverse displacement of each point in the direction perpendicular to the axis of a structural element where x denotes the points of the coordinates. If Ni(x) is chosen as coordinate functions of the element and qe(t) represents the element's coordinates, then the dynamic deflection of the element can be expressed as y (x, t ) = J2N> (x)qe(t )

where n is the number of generalized coordinates. For a typical element shown in Figure 3.17, Equation 3.86 may be expressed without time variable as

in which the shape functions are

2x2 x3

3x2 2x3

3x2 2x3

L2 L3

and the stiffness and mass coefficients of the element can be obtained from kij = J EI (x) d^N 2 N dx dx2 dx2

where moment inertia I(x) and mass density r(x), expressed as the function of x, signify that the formulation is applicable to a member with nonuniform cross-section. Equation 3.90 is integration with respect volume dv.

For a prismatic bar, Equations 3.89 and 3.90, respectively, yield Equations 3.91 and 3.92

0 0