## W

FIGURE 2.89 Degrees of freedom: (a) triangular plane-stress element and (b) triangular bending element.

The various components of strain can be obtained by appropriate differentiation of the displacement function. Thus,

[B] is derived by differentiating appropriately the elements of [L] with respect to x and y. The stresses {s| in a linearly elastic element are given by the product of the strain and a symmetrical elasticity matrix [E]. Thus,

The stiffness and the consistent load matrices of an element can be obtained using the principle of minimum total potential energy. The potential energy of the external load in the deformed configuration of the element is written as

In Equation 2.166 {Q*| represents concentrated loads at nodes and {q| the distributed loads per unit area. Substituting for {f|T from Equation 2.162 one obtains

Note that the integral is taken over the area a of the element. The strain energy of the element integrated over the entire volume v, is given as

Substituting for {e} and {s| from Equations 2.164 and 2.165, respectively,

U = 2 f^d [B]T[E][B] dv) {D*} The total potential energy of the element is

V = 1fD*}^ [B]T[E][B] dv) {D*} - {D*}T{Q*} - {D*}T^ [L]t{q}da Using the principle of minimum total potential energy, we obtain

where and

0 0