Читать книгу Finite Element Analysis - Barna Szabó - Страница 39

The vector of nodal forces associated with element k, denoted by , is defined as follows:

(1.88)

where is the stiffness matrix, is the solution vector and is the load vector corresponding to traction forces, concentrated forces and thermal loads acting on element k.

The sign convention for nodal forces is different from the sign convention for the bar force: Whereas the bar force is positive when tensile, a nodal force is positive when acting in the direction of the positive coordinate axis.

Exercise 1.18 Assume that hierarchic basis functions based on Legendre polynomials are used. Show that when κ is constant and on Ik then

Figure 1.8 Exercise 1.18. Notation.

independently of the polynomial degree pk. For sign convention refer to Fig. 1.8. Consider both thermal and traction loads. This exercise demonstrates that nodal forces are in equilibrium independently of the finite element solution. Therefore equilibrium of nodal forces is not an indicator of the quality of finite element solutions.