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5.5: Generalization to Arbitrary Non-linear Problems in Plates and Shells

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  • The previous section felt with the application of the Galerkin method to solve the non-linear ordinary differential equation for the bending/membrane response of beams. Galerkin name is forever attached to the analytical or numerical solution of partial differential equation, such as describing response of plates and shells. In the literature you will often encounter such expression as Galerkin-Bubnov method, Petrov-Galerkin method, the discontinuous Galerkin method or the weighted residual method. The essence of this method is sketched below.

    Denote by \(F(w, \boldsymbol{x})\) the non-linear operator (the left hand side of the partial differential equation) is defined over a certain fixed domain in the 2-D space \(S\). Now, a distinction is made between the exact solution \(w^* (\boldsymbol{x})\) and the approximate solution \(w(\boldsymbol{x})\). The approximate solution is often referred to as a trial function. The exact solution makes the operator \(F\) to vanish

    \[F(w^*, \boldsymbol{x}) = 0\]

    The approximate solution does not satisfy exactly the governing equation, so instead of zero, there is a residue on the right hand of the Equation \ref{6.55}

    \[F(w, \boldsymbol{x}) = R(\boldsymbol{x}) \label{6.55}\]

    The residue can be positive over part of \(S\) and negative elsewhere. If so, we can impose a weaker condition that the residue will become zero “in average” over \(S\), when multiplied by a weighting function \(w(\boldsymbol{x})\)

    \[\int_{S} R(\boldsymbol{x})w(\boldsymbol{x}) dS = 0 \label{6.56}\]

    Mathematically we say that these two functions are orthogonal. In general, there are also boundary terms in the Galerkin formulation. For example, in the theory of moderately large deflection of plates, Equation \ref{6.56} takes the form

    \[\int_{S} (D\nabla^4w − N_{\alpha \beta}w_{,\alpha \beta})w dS = 0 \label{6.57}\]

    The counterpart of Equation \ref{6.57} in the theory of moderately large deflection of beams is Equation (5.4.7) which was solved in the previous section of the notes. The solution of partial differential equations for both linear and non-linear problems is extensively covered in textbooks on the finite element method and therefore will not be covered here.