Keywords: linear elasticity, 3D finite elements accuracy benchmark, St.Venant, bending with shear, analytical solution, tri-cubic Lagrangian element, rectangular brick element.

The accuracy of finite element solutions have been compared with analytical solutions for plate and shell elements [1]; for 3D elements comparisons with polynomial solutions or thick cylinder solution are known [2]. Here a new accuracy benchmark for 3D FEM-s based on St.Venants analytical solution [3] for bending with shear of prismatic bodies is proposed. Tangential forces are acting on both end bases, their distribution across the base is expressed by nonelementary functions but it is equal for all cross-sections parallel to the end bases. Linearly distributed normal stresses are present only on one end base.

The treatment of distributed tangential forces as loading on the end bases of the prism is here avoided by expressing the corresponding boundary condition in terms of displacements. For the base where also normal distributed forces are applied, the inplane displacements are treated as prescribed. In order to prevent rigid body movements, axial displacements for three noncolinear points of the same base are also restrained. On the other base where the normal distributed forces vanish, no inplane deformation occurs during the deformation under loading, which is utilised as a kinematic constraint. All remaining boundary conditions are expressed in terms of forces; which all vanish with exception of the normal distributed forces on one base.

Numerical experiments are performed with a tri-cubic Lagrangian rectangular brick element with 192 dofs (no previous implementations of this element are known to the author). The element implemented here is capable of exactly reproducing the cubic displacement field in -direction which is characteristic for the St.Venants solution, so for this direction a single element layer will be sufficient. As the bicubic distribution over the base of the prism can consistently approximate the deplanation of all cross sections parallel to the end bases, the simplest reference case is the use of a single element for the whole prism.

Results are presented for the combination of four discretisation cases and three cases of height/width ratios. The relation of the displacement accuracy and the number of kinematic parameters used are similar to those obtained with simpler elements. In the element used here all stress components are represented with at least quadratic functions and the stress differences on element boundaries are practically negligible, good accuracy of the stress field is obtained without any smoothing or postprocessing of the results.

The approximation of the stress field is satisfactory with expected lower values at the corner points that are caused by the characteristic of the stress field and the use of regular meshes. It can be expected that the proposed banchmark could be effective for testing procedures for adaptive mesh generation.

1

O.C. Zienkiewicz, L.T. Taylor, "The Finite Element Method", Butterworth-Heinemann, London, 2000.

2

Florentin, E., Gallimard, L., Ladeveze, P., Pelle, J.P., "Local error estimator for stresses in 3D stractural analysis", Computers and Structures, 81, 1751-1757, 2003. doi:10.1016/S0045-7949(03)00199-8

3

Timoshenko, S.P., Goodier, J.N., "Theory of elasticity", McGraw-Hill, New York, 1951.

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