Page 244 - Finite Element Modeling and Simulations with ANSYS Workbench

Three-Dimensional Elasticity 229

where d is the nodal displacement vector, that is:

ε = B d (7.16)

Strain energy is evaluated as

1 1
U = ∫ σε dV = ∫ (E εε dV) T
T
2 2
V V
1
= ∫ ε E ε dV
T
2
V
 
1
∫
= d T  B EB dV  d (7.17)
T
2  
V 
That is, the element stiffness matrix is
T
k= ∫ B EB dV (7.18)
V
In ξηξ coordinates:

ξη
J
dV = (det ) dd dζ (7.19)

Therefore,
1 1 1
ξη
T
k= ∫ ∫ ∫ B EB(det J dd dζ (7.20)
)
−1 −1 −1
It is easy to verify that the dimensions of this stiffness matrix is 24 × 24.
Stresses:
To compute the stresses within an element, one uses the following relation once the
nodal displacement vector is known for that element:

σ = E ε = EBd

Stresses are evaluated at selected points (Gaussian points or nodes) on each element.
Stress values at the nodes are often discontinuous and less accurate. Averaging of the
stresses from surrounding elements around a node is often employed to smooth the stress
field results.
The von Mises stress for 3-D problems is given by

1
σ e = σ VM = ( σ − σ 2 ) 2 + ( σ − σ 3 ) 2 + ( σ − σ 1 ) 2 (7.21)
3
1
2
2
where σ , σ , and σ are the three principal stresses.
1
2
3