Formulation and Applications of Consistent Shell and Beam Elements

Abstract

<p>A new shell element for the analysis of thin and thick plate and shell structures has been formulated along with the compatible beam element. The new elements are referred to as the consistent shell element and the consistent beam element. By combining the consistent shell element and the consistent beam element a finite element model for the analysis of reinforced concrete structures has been formulated. The scope of the study is broadened further through the modification of the new elements to allow for the analysis of laminated fiber-reinforced composite beams and shells.</p> <p>Many of the currently available beam and shell elements exhibit spurious variations of the transverse shear stress(es). To obtain improved responses when thin beams or shells are analysed the reduced integration technique has typically been employed. This approach is not acceptable since the reduced integration technique cannot be applied with complete confidence. In the present study it is found that the unsatisfactory behaviour of these elements is due to an inconsistency in their formulation. In the formulation of the new elements a consistent formulation has been ensured. The new elements behave very well in the analysis of both thin and thick beams and shells and contain no spurious zero energy modes. In addition, they provide a quadratic variation of the transverse shear stress(es), and a cubic variation of the displacement(s) through their thicknesses. Therefore, the shear correction factor k, which is usually required to correct for the assumption of constant transverse shear strain through the thickness is not required. Both elements include material non-linearity. Special attention has been given to the efficient implementation of the consistent shell element by employing a sub-matrix formulation in conjunction with a modified frontal solution algorithm. The numerical results show the new elements to be highly accurate and computationally efficient.</p> <p>The reinforced concrete finite element model employs a rational elasto-plastic constitutive relationship for concrete, discrete bar elements for modelling of reinforcement, joint elements for bond slip between concrete and reinforcement, beam elements for supporting girders and shear connector elements along the concrete/girder interface. The numerical results show that the model accurately predicts the behaviour of reinforced concrete slabs, including punching shear failure under point loads. The constitutive model employed has been found to be reasonably objective with respect to refinement of the finite element mesh.</p> <p>The consistent laminated beam element and the consistent laminated shell element, have been formulated for the analysis of laminated fiber-reinforced composites. Special attention has been given to the approximation of stresses through the thickness of the laminate because of their importance in predicting delamination failures. This has been achieved by allowing the transverse shear strain(s) to be discontinuous at the interface of two layers while still maintaining continuity of the global displacements across the interface. The numerical results show that the elements provide very accurate predictions of stresses through the thickness of both thin and thick laminates.</p>