ABSTRACT
Consider the unconstrained triangular plate element depicted in Figure 8.1. Suppose that there is no out-of-plane stress (plane stress) or no out-of-plane displacement (plane strain). The displacements u(x,y,t) and v(x,y,t) are to be modeled using the values ue(t), ve(t), ueþ1(t), veþ1(t), ueþ2(t), and veþ2(t). A linear model in x and y suffices for each displacement owing to providing three coefficients to match three nodal values. The interpolation model now is
u x, y, tð Þ v x, y, tð Þ
! ¼ w
" # FTm2 0
" # gu tð Þ gv tð Þ
! (8:1)
in which
gu tð Þ¼ ue tð Þ ueþ1 tð Þ ueþ2 tð Þ
0 B@
1 CA, gv tð Þ¼
ve tð Þ veþ1 tð Þ veþ2 tð Þ
0 B@
1 CA, wm2¼
x
y
0 B@ 1 CA, FTm2¼
1 xe ye 1 xeþ1 yeþ1 1 xeþ2 yeþ2
2 64
3 75 1
In a plate element experiencing bending only, in classical plate theory (e.g., Wang, 1953) the in-plane displacements u and v are expressed by
u x, y, z, tð Þ ¼ z @w @x
, v x, y, z, tð Þ ¼ z @w @y
(8:2)
in which z¼ 0 at the middle (centroidal) plane. The out-of-plane displacement w is assumed to be a function of x and y only. Clearly this model permits no in-plane (membrane) displacements in the middle plane.
