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6.2 Finite Element Method 101

The six strain components are written in terms of the
three displacement components based on the small deformation
theory as,
   u ;   u   v 
x
x  xy y  x 
  v  y  ;   xz u  z    w
y
x 
   w ;   v    w
z
z  yz z  y 
The six strain-displacement relations are substituted

into the six stress-strain relations, so that the stress components can
be written in terms of the displacement components. These stress
components are then further substituted into the three governing
differential equations. The final three governing differential
equations are now in forms of the three displacement components.
The three displacement components thus can be solved from the
three differential equations.

6.2 Finite Element Method

6.2.1 Finite Element Equations
Finite element equations can be derived by applying the
method of weighted residuals to the three partial differential
equations. Detailed derivation can be found in many finite element
textbooks including the one written by the same author. It is noted
that the finite element equations can also be derived by using the
variational method. The method is based on the minimum total
potential energy principle. This later method was often used to
derive the finite element equations for solid problems in the past.
The derived finite element equations are written in
matrix form as,
K 
     
F

K
where   is the element stiffness matrix;   is the element
vector containing the nodal displacements ,u v and w in the x, y