"Symmetricability of Pressure Stiffness Matrix for Shells with Loaded Free Edges";
International Journal for Numerical Methods in Engineering, 15 (1980), S. 981 - 990.
For the case of free edges which are loaded, follower forces remaining normal to the middle surface of a shell throughout history do not have a load potential. In finite element analysis, this results in an unsymmetric pressure stiffness matrix. Depending on the structure of the available computer program, implementation of an equation solver permitting solution of unsymmetric simulataneous systems of algebraic equations may be a tedious task. This explains the significance of the topic of symmetracability of pressure stiffness matrices, turning out to be of special importance in the case of static buckling under the assumption of a linear prebuckling path.
At first, incremental equations for tracing the nonlinear load-displacement path are derived. Thereafter, the buckling condition is deduced. Then, it is demonstrated that symmetrization of the pressure stiffness marix is admissible if the so-obtained 'buckliing pressure' differs 'very little' from the 'buckling pressure' resulting from an alternative symmetric 'buckling matrix', as is shown to be the case for a simply supported cylindrical shell with a free upper edge, subjected to hydrstatic external pressure. The alternative symmetric 'buckling matrix' is a consequence of deleting the virtual work term, causing the unsymmetry of the pressure stiffness matrix, in the expression for the external virtual work. A mechanical interpretation of this virtual work term is given. It is shown to be equal to the difference of virtual work of the original pressure load and of a 'substitute pressure-field' of the form of a Fourier series of the former. This explains why, normally, the buckling coefficient resulting from the 'substitute pressure-field' represents a good approximation to the buckling coefficient stemming from the original pressure load.