Civil Engineering Reference
In-Depth Information
By assuming displacement fields of stress patterns within an element it is
possible to derive a stiffness matrix relating the nodal forces to the nodal
displacements of an element. The global stiffness matrix of the structure,
which is the assemblage of all the elements, is then obtained by combining
the individual stiffness matrices of all the elements in the proper manner. If
conditions of equilibrium are applied at every node of the idealised
structure, a set of simultaneous equations can be formed, the solution of
which gives all the nodal displacements, which in turn are used to calculate
all the internal stresses (Ghali, Neville and Cheung, 1971).
In applying the finite element method to a problem, it is first necessary to
discretise the continuum, that is to subdivide the continuum into small areas
of triangular or rectangular shapes. Obviously, it is clear and more
straightforward to use triangular elements to model a structure with inclined
or curved edges.
9.3
Triangular plane stress elements
Let us therefore first of all derive the stiffness matrix of a triangular element
which is the simplest element available in two-dimensional stress analysis.
Consider a triangular element ijm with nodal co-ordinates (x
i
, y
i
), (x
j
, y
j
)
and (
x
m
,
y
m
) respectively as shown in Figure 9.2.
Figure 9.2
Triangular element.
