Page 30 - Using ANSYS for Finite Element Analysis A Tutorial for Engineers

IntroductIon to FInIte element AnAlysIs • 17
Where matrix d is the first minor of [a ] and is matrix [a] with row i and
ij
column j deleted.
Finally, determine the inverse

 a  ij   −1 = c [] T
a

1.2.2 elasticity equations

1.2.2.1 stress equilibrium equations

A three-dimensional body occupying a volume V and having a surface S
is shown in Figure 1.8. Points in the body are located by x, y, and z coor-
dinates. The boundary is constrained on some region, where displacement
is specified. On part of the boundary, distributed force per unit area T,
also called traction, is applied. Under the force, the body deforms. The
deformation of a point (x = [x y z] ) is given by the three components of
T
its displacement: u = [u v w] T
The distributed force per unit volume, for example, the weight per
unit volume, is the body force vector f given by:

T
f =   f x f y f 
z 
The body force acting on the elemental volume dV is shown in
Figure 1.8. The surface traction T may be given by its component values
at points on the surface:

y σ
y
x
z
τ yx
τ yz xy
τ zy τ x

τ zx τ xz
Loaded material body
σ z

Figure 1.8. Three-dimensional body.