## U

Evaluation of all the integrals of eqn. (9.84) leads to the beam element flexural stiffness matrix

12/L2 6/L -12/L2 6/L~ £/ 6/L 4 -6/L 2 -12/L2 -6/L 12/L2 -6/L 6/L 2 -6/L 4 J

which is identical to the stiffness matrix of eqn. (9.68) derived using fundamental equations. The same arguments made in Section 9.8.1 apply with regard to including axial terms to give the force/displacement relation, eqn. (9.69), and corresponding element stiffness matrix, eqn. (9.70).

Element stress matrix in local coordinates

Bending and axial stresses are obtained using the same relations as those in §9.8.1.

Transformation of element stiffness and stress matrices to global coordinates

The element stiffness and stress matrices are transformed from local to global coordinates using the procedures of §2.4.8.1 to give the stiffness matrix of eqn. (9.75) and stress matrix of eqn. (9.76).

Formation of structural governing equation and assembled stiffness matrix

The theorem of virtual work used in §9.7.2 to formulate rod element assemblages applies to the present beam elements. It follows, therefore, that the assembled stiffness matrix will be given by eqn. (9.47). The displacement column matrices will, for beams, include rotational dof., not present for rod elements. Further, at the nodes, moment equilibrium, as well as force equilibrium, is now implied by eqn. (9.28).

### 9.9. A simple triangular plane membrane element

The common occurrence of thin-walled structures merits devoting attention here to their analysis. Many applications are designed on the basis of in-plane loads only with resistance arising from membrane action rather than bending. Whilst thin plates can be curved to resist normal loads by membrane action, for simplicity only planar applications will be considered here. Membrane elements can have three or four edges, which can be straight or curvilinear, however, attention will be restricted here to the simplest, triangular, membrane element.

Unlike the previous rod and beam element formulations, with which displacement fields can be represented exactly and derived from fundamental arguments, the displacement fields represented by two-dimensional elements can only be approximate, and need to be derived using an energy principle. Here, the principle of virtual work will be invoked to derive the membrane element equations.

9.9.1. Formulation of a simple triangular plane membrane element using the principle of virtual work equation

With reference to Fig. 9.30, each node of the triangular membrane element has two dof., namely u and v displacements in the global x and y directions, respectively. The total of six dof. for the element limits the u and v displacement to linear interpolation. Hence u(x, y) = a\ + a2x + a^y and v(x, y) = a4 + a5x + ct^y

Fig. 9.30. Triangular plane membrane element.

Fig. 9.30. Triangular plane membrane element.

or, in matrix form

At the nodal points, and

 u(x, y)
0 0