4.2 One-Dimensional Line Elements

A one-dimensional beam element with six displacement degrees of freedom at each node is shown in Fig. 4.1. Structural characteristics of the straight element with uniform cross section A may be obtained from assumed axial, torsional, and flexural deformation components, combining these individual relationships as described next. ^{[1] , [2]} The local coordinate system is assumed such that the x axis corresponds to the axial direction of the element. The six displacement degrees of freedom at any point on the element are shown in Fig. 4.1.

Figure 4.1: One-dimensional line element.

Thus for the axial element the displacement interpolation in the local x direction assumes a linear relationship:

The coefficients c ₁ and c ₂ are evaluated in terms of nodal displacements by setting the boundary conditions u _x = u _x1 at x = 0 and u _x = u _x2 at x = l, so that

where N is the shape function row vector pertaining to the x direction. The strain is given by differentiation of u _x with respect to x,

and the stiffness and inertia matrices are obtained as

using Eqs. (3.59) and (3.61) derived in Chapter 3. In these equations, E is the Young's modulus of elasticity, ? the mass density, A the area of the beam cross section, and l the length of the beam element.

For a...