Boundary element analysis of multi-thickness shear-deformable slabs without sub-regions Mina Wagdy a , Youssef F. Rashed b,c,n a Dar Al-handasah, Department of Structures, Dokki, Giza, Egypt b Supreme Council of Universities in Egypt, Egypt c Department of Structural Engineering, Cairo University, Giza, Egypt article info Article history: Received 19 January 2014 Accepted 24 March 2014 Available online 25 April 2014 Keywords: Boundary element method Multi-thickness slab Stiffness matrix Stiffness-like domain integrals abstract In this paper, a new boundary element formulation is developed for the analysis of multi-thickness slabs. The shear deformable plate bending theory is employed. The additional thickness is added to the plate using additional stiffness matrix. A new systematic methodology for deriving stiffness matrix of additional thicknesses or drops is presented. The formulation is implemented into a computer code and several examples are considered to demonstrate the validity of the presented formulation. & 2014 Elsevier Ltd. All rights reserved. 1. Introduction Multi-thickness slabs are increasingly used for the design of building slabs due to economic reasons [1]. This including flat slabs with drop panels and thickened rafts. The ability of having more than one thickness in the slab adds flexibility to optimize the suggested statical systems while meeting all architectural requirements. The analysis of building multi-thickness slabs is commonly carried out using traditional finite element (FEM) based computer programs [2]. In such programs the slab has to be discretized into appropriate mesh and representing different thicknesses of the slab using different thicknesses of finite elements. In most cases columns and beams are modeled as skeletal frames. The boundary element method (BEM) [3] is an alternative way to analyze practical slabs. The advantages of using the boundary element method (in comparison to the finite element method) are – No slab internal discretization is needed which saves effort and time. – The real geometry of slab could be modeled. No need to approximate the slab using the surrounding center-lines. This provides a more realistic modeling of the slab and easy link to BIM software such as Autodesk Revit [12]. The application of the BEM to plate bending problems modeled using the thin plate theory was introduced by Bézine [4] and Stern [5]. Vander Weeën [6] derived a BEM for plate bending problem based on the shear deformable plate theory according to Reissner [7]. Rashed [8] extended the formulation of Vander Weeën [6] to treat plates on internal columns. The formulation in [8] considers the real size of internal supports. Also it can be easily extended to model slabs over beams [9] or cores and walls. One of the commonly used methodologies to solve multi- thickness slabs is the sub-region method. In this method a new boundary around the thickened slab is introduced. The slab is solved first with the thickened slab as an opening. Then the thickened slab is solved as an individual slab. Finally both slabs are combined using compatibility and equilibrium equations for the displacements and tractions respectively. In such a multi-region formulation, beams and supporting walls are treated as nodes or lines as those of the FEM [10]. Extending the formulation of Rashed [8] to treat multi-thickness slabs will not advantage from the real geometry modeling pre- sented in [8]. Moreover additional artificial internal boundaries will appear which will cause problems in the nearby points at which stress-resultants are computed (the well-known near- boundary problem [11]). The purpose of this paper is to develop alternative solution to model multi-thickness slabs without sub-region. This will extend the use of already developed boundary element based computer Contents lists available at ScienceDirect journal homepage: www.elsevier.com/locate/enganabound Engineering Analysis with Boundary Elements http://dx.doi.org/10.1016/j.enganabound.2014.03.011 0955-7997/& 2014 Elsevier Ltd. All rights reserved. n Corresponding author at: Department of Structural Engineering, Cairo University, Giza, Egypt. Tel.: þ20 100 5112949. E-mail address: youssef@eng.cu.edu.eg (Y.F. Rashed). Engineering Analysis with Boundary Elements 43 (2014) 86–94