The Finite Element Method - Mechanics of Structures: Variational and Computational Methods - page 336

— Find the element displacements in the local coordinate system:

v i T i v i

= (6.4)

— Calculate the nodal element forces in the local coordinate system. Use either

− p i 0 = k i p i v i

(6.5)

v i

with from Eq. (6.4) or

p i k i v i p i 0 p i T i p i

= − and = (6.6)

— Compute the variables, such as cross-sectional forces and corresponding stresses,
along the element.
— Display graphically the responses.
7. Controls for the calculations
— Scrutinize the results for physical plausibility.
— Check the overall system equilibrium.
— Check the equilibrium at particular nodes.
— Check the equilibrium and compatibility of particular elements.

6.3 A Simple Finite Element Calculation

We will consider an analysis of a beam lying on an elastic foundation as an initial illustration
of the finite element method. We choose a simple problem that follows directly from the
displacement method for frameworks of Chapter 5, yet contains the essential ingredients of
the finite element method. For the simple Euler-Bernoulli beam of Chapter 4, use of a poly-
nomial as the (assumed) trial function can lead to “exact” element matrices and solution.
However, the same polynomial leads to approximate element matrices for a beam on an
elastic foundation. The approximate element matrices for the beam on elastic foundation
are developed in Example 4.4.
Summation of the virtual work for all of the elements leads to, upon assembly of the
element matrices, the system equations for the displacement method. These equations
are the nodal equilibrium conditions for the system. Let

δ

W k represent the work of the
concentrated forces and moments applied to a system node. Then, introducing Eq. (13) of
Example 4.4,

W i − δ W = − δ − δ W k

Elements Nodes

v iT k i B +

v i

p i 0 − δ V T P ∗ k i

w

= δ −

Elements
= δ V T KV P ∗ = P 0 − − 0 (6.7)

or

P ∗ P 0

= +

KV (6.8)