Problem Statement   

      From the lecture, the principle of virtual work usually forms the foundation for structure mechanics FE analysis. 

     

       For the linear elastic continuum, Hooke’s law relates stress and strain; the convenient matrix form is

 

where x, y, z, xy, yz, xz are the normal and shear stresses and x, y, z, xy, yz, xz are the corresponding strains. 

      Substituting Hooke’s law, the matrix form for virtual work is

      

and the stationary point of P is equivalent to

 

      DP:  ÑT + rg = 0

Augmenting P for initial stress/strain the possibility of point loads yields the final form

     

 The FE approach is to approximate the integral as the sum of integrals over the FE domain We forming the discretization Wh

      

Extremizing the resultant form with respect to all non-constrained FE nodal degrees of freedom produces the algebraic equation 

     

 where:[K] is global “stiffness matrix”and {R} is global “load vector”

 The details of the FE implementation process for plane stress/strain are

 Approximation:

 Kinematics for strain-displacement (linear) is {e}º[D]{u

     

 The FE stiffness matrix contribution to Ph is then

      

 The extremum of Pe for all degree of freedom is not constrained

      

      In the plane stress mode loads are applied in the xy-plane of the plate. The study problem corresponds to a square plate with a hole, which is constrained at one edge and is subjected to a uniformly distributed tensile load on the opposite edge, as shown in Figure 1.

Figure 1. Schematic of the square plate under uniformly distributed tensile load.

 The physical data for the problem are

w

20in.

t

1in.

r

0.35405 lb/in3

r

1 in

n

0.3

E

30E+06 psi

T

100 psi

 where w is width, t is thickness, rho0 is density, r is radius of the hole, n is Poisson ratio, E is Young’s modulus and T is tensile stress

       In structural mechanics von Mises stress is often used as the metric for evaluating design margins. It is defined as the maximum distortion energy, used to predict failure of ductile materials subjected to static loading. von Mises stress is especially useful since it gives a scalar measure of the entire stress tensor field. For deterministic loads both static and dynamic effective von Mises stress is computed as