FEM for Trusses (Finite Element Method) Part 3

High Order One-Dimensional Elements

For truss members that are free of body forces, there is no need to use higher order elements, as the linear element can already give the exact solution, as shown in Example 4.1. However, for truss members subjected to body forces arbitrarily distributed in the truss elements along its axial direction, higher order elements can be used for more accurate analysis. The procedure for developing such high order one-dimensional elements is the same as for the linear elements. The only difference is the shape functions.

In deriving high order shape functions, we usually use the natural coordinate ξ, instead of the physical coordinate x. The natural coordinate ξ is defined as

where xc is the physical coordinate of the mid point of the one-dimensional element. In the natural coordinate system, the element is defined in the range ofFigure    4.7 shows a one-dimensional element of «th order with (n + 1) nodes. The shape function of the element can be written in the following form using so-called Lagrange interpolants:

whereis the well-known Lagrange interpolants, defined as

From Eq. (4.82), it is clear that

Therefore, the high order shape functions defined by Eq. (4.81) are of the delta function property.

Figure 4.7. One-dimensional element of nth order with (n + 1) nodes.

Figure 4.8. One-dimensional quadratic and cubic element with evenly distributed nodes. (a) Quadratic element, (b) cubic element.

Using Eq. (4.82), the quadratic one-dimensional element with three nodes shown in Figure 4.8a can be obtained explicitly as

The cubic one-dimensional element with four nodes shown in Figure 4.8b can be obtained as