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
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
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
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