Page 545 - Aircraft Stuctures for Engineering Student

526 Matrix methods of structural analysis

leaving
the integration in the expression for [e], only Jd(vo1) which is simply the
area, A, of the triangle times its thickness t. Thus
(12.94)
Finally the element stresses follow from Eq. (12.79), i.e.

where [HI = [D][B] and [D] and [B] have previously been defined. It is usually found
convenient to plot the stresses at the centroid of the element.
Of all the finite elements in use the triangular element is probably the most versatile.
It may be used to solve a variety of problems ranging from two-dimensional flat plate
structures to three-dimensional folded plates and shells. For three-dimensional
applications the element stiffness matrix [Ke] is transformed from an in-plane xy
coordinate system to a three-dimensional system of global coordinates by the use
of a transformation matrix similar to those developed for the matrix analysis of
skeletal structures. In addition to the above, triangular elements may be adapted
for use in plate flexure problems and for the analysis of bodies of revolution.

Example 12.3
A constant strain triangular element has corners 1(0,0), 2(4,0) and 3(2,2) referred to
a Cartesian Oxy axes system and is 1 unit thick. If the elasticity matrix [D] has ele-
ments Dll = DZz = a, D12 = D21 = by Dl3 = 023 = D31 = 032 = 0 and D33 = cy
derive the stiffness matrix for the element.

From Eqs (12.82)

i.e.

i.e.
(ii)

i.e.
u3 = al + 2a2 + 2a3

From Eq. (i)

and from Eqs (ii) and (iv)