The Finite Element Method - Mechanics of Structures: Variational and Computational Methods - page 345

FIGURE 6.11
Plot of u x for u x 1 = 1 with all other nodal displacements equal to zero.

Often, this is expressed as

N 1 v i

N 2 N 3 N 4 0

u = (6.19c)

0 N 1 N 2 N 3 N 4

The displacements in each element of the model are to be represented by these rather
simple polynomials. These assumed displacements contain eight DOF with two DOF per
node.
This trial solution matrix N has “interpolation” polynomials as components. As can be
seen in Eq. (6.19b), the components of N represent the displacement across the element due
to a unit nodal displacement. Thus, a plot of in-plane displacement u , drawn perpendicular
to the plane of the element, would appear for the nodal displacement u x 1 as shown in
Fig. 6.11. The curved lines connect points of the element for which the u x 1 component of
N has the same value. It is important to note that this procedure of establishing N can
be avoided if the displacements due to unit nodal displacements are known beforehand.
Thus, for a variety of elements the inverse required to form N − 1

ux can be avoided if standard
interpolation polynomials such as Hermitian or Lagrangian interpolation polynomials are
introduced.

Principle of Virtual Work
The stiffness matrix for an element will be derived by substituting the assumed displace-
ment field of Eq. (6.19) into the principle of virtual work (form C of Chapter 2, Section 2.3).
In matrix notation, the principle of virtual work can be expressed as

− δ W = δ W i − δ W e = 0

V δ V δ T σ dV

u T p V dV
II

u T p dS
III

− δ W = − −

S p δ

= 0 (6.20)

I

Equation (6.20) is valid for the entire structural system. In order to set up a computational
method in which the system is subdivided into elements, Eq. (6.20) has to be evaluated for
each element. The total virtual work will be obtained as the sum of the internal and external
virtual work, each a scalar quantity, for all the elements.