4.3.1 Constant Strain Triangle

A triangular plane element ^{[3] , [4]} with three nodes is shown in Fig. 4.2. The two displacement functions ( u _x, u _y) at any point within the triangle, expressed in its local coordinate system, have linear variations:

or

Figure 4.2: Triangular plane element.

The unknown coefficients c _i are evaluated in terms of nodal values of the unknown displacements by applying appropriate boundary conditions at the nodes, i.e., at

Further,

or

where the interpolation functions N ₁, N ₂, N ₃ in Eq. (4.39) are obtained from the expressions

and

where A is the area of the triangle. To obtain the strain-displacement matrix, the planar strain vector is expressed as

in which

and the required matrix is obtained from Eqs. (4.39), (4.42), and (4.44) as

For the plane stress problem the stress-strain relationship for a general material can be written as

in which

and where ? _I is the initial strain vector. For homogeneous, isotropic material, ? _x = ? ^y = ?, so that

Then the stiffness and inertia matrices and thermal and initial load vectors are derived from standard relationships such as

and also

For the constant strain triangle (CST), the thermal nodal forces are calculated to be

In the local coordinate system,

in which

and for homogeneous, isotropic material ? _x = ? _y = ?,

For initial strains ? _I