Page 149 - Finite Element Modeling and Simulations with ANSYS Workbench

134 Finite Element Modeling and Simulation with ANSYS Workbench

For quadratic elements (either triangular or quadrilateral), the traction is converted into
forces at three nodes along the edge, instead of two nodes. Traction tangent to the bound-
ary, as well as body forces, are converted into nodal forces in a similar way.

4.4.7 Stress Calculation
The stress in an element is determined by the following relation:

 σ   ε 
x
x
   
 σ  = E  ε  = EBd (4.39)
y
y
   
 τ xy   γ xy 
where B is the strain-nodal displacement matrix and d is the nodal displacement vector,
which is known for each element once the global FE equation has been solved.
Stresses can be evaluated at any point inside the element (such as the center) or at the
nodes. Contour plots are usually used in FEA software packages (during postprocess) for
users to visually inspect the stress results.

4.4.7.1 The von Mises Stress
The von Mises stress is the effective or equivalent stress for 2-D and 3-D stress analysis. For
a ductile material, the stress level is considered to be safe, if

σ≤ σ Y
e
where σ is the von Mises stress and σ the yield stress of the material. This is a generaliza-
Y
e
tion of the 1-D (experimental) result to 2-D and 3-D situations.
The von Mises stress is defined by
1
σ= ( σ− σ 2 ) 2 + σ− σ 3 ) 2 + σ− σ 1 ) 2 (4.40)
(
(
3
2
1
e
2
in which σ , σ and σ are the three principle stresses at the considered point in a structure.
3
1
2
For 2-D problems, the two principle stresses in the plane are determined by
y
x
x
P = σ+ σ y +  σ− σ  2 2
σ 1   +τ xy
2  2 
(4.41)
x
x
y
P = σ+ σ y −  σ− σ  2 2
σ 2 2   2   +τ xy

Thus, we can also express the von Mises stress in terms of the stress components in the
xy coordinate system. For plane stress conditions, we have
2
σ= ( σ+ σ y ) 2 − ( σσ −τ xy ) (4.42)
3
e x x y