Page 31 - Using ANSYS for Finite Element Analysis A Tutorial for Engineers

18 • Using ansys for finite element analysis

T =   T x T y T  T
z 

Examples of traction are distributed contact force and action of
pressure. A concentrated load P acting at a point i is represented by its
three components:

P =   P x P y P  T i
z 
i
The stresses acting on the elemental volume dV are shown in
Figure 1.9. When the volume dV shrinks to a point, the stress tensor is rep-
resented by placing its components in a (3 × 3) symmetric matrix. How-
ever, we represent stress by the six independent components as follows:

s =   x s y s z t xy t yz t  
s
xz
Where s , s , and s are normal stresses and t , t , and t are shear
x
zx
xy
yz
z
y
stresses. Let us consider equilibrium of the elemental volume shown in
Figure 1.9. First, we get forces on faces by multiplying the stresses by
the corresponding areas. Writing ∑ F =0, ∑ F =0, and ∑ F =0 and
y
z
zx
recognizing dV = dx dy dz, we get the equilibrium equations:
∂s x + ∂t xy + ∂t xz + F = 0
∂x ∂y ∂z x
∂t xy + ∂s y + ∂t yz + F = 0
∂x ∂y ∂z y

∂t xz + ∂t yz + s z + F = 0
∂x ∂y ∂z z

y σ yy

σ yx
σ yz
σ xy
σ zy σ xx
σ zx σ xz x
σ zz
z
Figure 1.9. Equilibrium of elemental volume.