The Finite Element Method - Mechanics of Structures: Variational and Computational Methods - page 361

Computation of the System Nodal Displacements
The global nodal displacements are obtained from the expression

=

KV P (7)

From a mechanical standpoint, the individual equations represent equilibrium conditions
for a node. The solution of (7) provides for the complete structure an approximation to
the conditions of equilibrium and the static boundary conditions. Since the formulation
utilizes the principle of virtual work, the approximate solution is optimal relative to the
chosen (kinematically admissible) trial displacement functions.
The stiffness matrix K is symmetric and positive definite. These properties permit special
solution algorithms, e.g., Cholesky decomposition, to be employed. Moreover, normally the
system of equations will be banded, a property that depends on the arrangement of the
global node numbers and DOF. For this example, a standard linear equation solver was
employed for numerical results. The nodal displacements were found to be

Node Number u x (m) u y (m)

10 − 4 1 − 0 . 3976 × 0.0

2 − 0 . 2692 × 10 − 4 − 0 . 7152 × 10 − 4

10 − 4

3 0.0 − 0 . 8754 ×

10 − 5 10 − 4 4 0 . 4568 × − 0 . 3287 ×

5 0 . 1913 × 10 − 5 − 0 . 7037 × 10 − 4

(8)

10 − 4 6 0.0 − 0 . 9028 ×

7 0 . 3062 × 10 − 4 − 0 . 4259 × 10 − 4

10 − 4 10 − 4 8 0 . 2309 × − 0 . 7757 ×

10 − 4

9 0.0 − 0 . 9951 ×

The deformed and undeformed systems are sketched in Fig. 6.18.

FIGURE 6.18
Deformation pattern of the model of Fig. 6.14b.