ABSTRACT
Application of the divergence theorem to the first term leads to ∫ψ[xnx+yny]ds, which vanishes since ψ vanishes on S. Finally,
T=2∫ψdxdy. (14.12)
We apply variational methods to the Poisson Equation, considering the stress-potential function ψ to be the unknown. Now,
(14.13)
Integration by parts, use of the divergence theorem, and imposition of the “constraint” ψ=0 on S furnishes
(14.14)
The integrals are evaluated over a set of small elements. In the eth element, approximate ψ as in which vT is a vector with dimension (number of rows) equal to the number of nodal values of ψ. The gradient has a corresponding interpolation model
in which βT is a matrix. The finite-element counterpart of the Poisson Equation at the element level is written as
(14.15)
and the stiffness matrix should be nonsingular, since the constraint ψ=0 on S has already
been used. It follows that, globally, The torque satisfies
(14.16)
In the theory of torsion, it is common to introduce the torsional constant J, for which
T=2µJθ′. It follows that
14.2 BUCKLING OF BEAMS AND PLATES
14.2.1 EULER BUCKLING OF BEAM COLUMNS
14.2.1.1 Static Buckling
Under in-plane compressive loads, the resistance of a thin member (beam or plate) can be reduced progressively, culminating in buckling. There are two equilibrium states that the member potentially can sustain: compression only, or compression with bending. The member will “snap” to the second state if it involves less “potential energy” than the first state. The notions explaining buckling are addressed in detail in subsequent chapters. For now, we will focus on beams and plates, using classical equations in which, by retaining lowest-order corrections for geometric nonlinearity, in-plane compressive forces appear.
For the beam shown in Figure 14.4, the classical Euler buckling equation is
