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12.8 Finite element method for continuum structures 521

Hence
- -- +-
6
12x-
L2 L3
4 6x
--+-
= J L L L2 [EI] --+- 12x 4 6x 6
6
--+-
0 6 12x [ L' L3 L L2
L2 L3
2 6x
--+-
- L L2-
6L 1
which gives
12 -6L -12 -6L
EI -6L 4L2 6L 2L2
[R1=z[ 12 (12.77)
-12
6L
-6L 2L2 6L 4L2
Equation (12.77) is identical to the stiffness matrix (see Eq. (12.44)) for the uniform
beam of Fig. 12.6.
Finally, in step seven, we relate the internal 'stresses', {cr}, in the element to the
nodal displacements {@}. This has in fact been achieved to some extent in Eq.
(1 2.69), namely

(0) = [~1[C1[A-'I{Se)
or, from the above
(01 = [Dl[B1{Se) (12.78)
Equation (12.78) is usually written
(4 [II{@l (12.79)
=
in which [HI = [D][B] is the stress-displacement matrix. For this particular beam-
element [D] = EI and [B] is defined in Eq. (12.76). Thus
612
6
4
6
12
'H] = - - + -x - - + --x - - - ] (12.80)
L2 L3x --+--x
[ L2 L3 L L' L L2
12.8.2 Stiffness matrix for a triangular finite element
Triangular finite elements are used in the solution of plane stress and plane strain
problems. Their advantage over other shaped elements lies in their ability to represent
irregular shapes and boundaries with relative simplicity.
In the derivation of the stiffness matrix we shall adopt the step by step procedure of
the previous example. Initially, therefore, we choose a suitable coordinate and node
numbering system for the element and define its nodal displacement and nodal force
vectors. Figure 12.13 shows a triangular element referred to axes Oxy and having