Engineering Analysis with Boundary Elements 83 (2017) 1–24 Contents lists available at ScienceDirect Engineering Analysis with Boundary Elements journal homepage: www.elsevier.com/locate/enganabound Efficient analysis of plates on nonlinear foundations Ahmed Fady Farid a , Marina Reda a , Youssef F. Rashed a,b,∗ a Department of Structural Engineering, Cairo University, Giza, Egypt b Supreme Council of Universities in Egypt, Egypt a r t i c l e i n f o Keywords: Boundary element method Finite element method Iterative solution Tensionless Elastic-plastic Plates a b s t r a c t This paper presents efficient analysis of plates on nonlinear foundations. The Reissner plate theory is used to model plates. Foundations are presented as the Winkler springs or the elastic half space. The developed analy- sis is mainly presented for tensionless foundation; however as demonstrated, it is straightforward extended to analysis of elastic-plastic foundations. The plate is analyzed using the boundary element method (BEM). Unlike the traditional BEM which uses equations in form ([H] {u} = [G] {t}), the presented formulation uses finite ele- ment like equations, in the form of ([K] {u} = {P}). An innovative formulation is presented to derive the relevant plate stiffness matrix [K] and load vector {P} from the BEM integral equation. Iterative procedures together with condensation process are used to eliminate degree of freedom at failed zones. Results of the present analysis are more accurate than those obtained from previously published results. The main advantages of the presented technique are its simplicity and accuracy and it gains both advantages of the boundary element and the finite element methods. © 2017 Elsevier Ltd. All rights reserved. 1. Introduction Plates on nonlinear (tensionless or elastic-plastic) foundations are an important problem in mechanics and have several applications in practical structural and geotechnical engineering. Numerical analysis of plates over elastic foundations (Winkler springs or elastic half space) is considered by many authors as follows: (a) Using the finite element method: the work of Cheung and Zienkiewicz [2], Svec and McNeice [3], Svec and Gladwel [4] for thin plates, Rajapaks and Selvadurai [6] for thick plates. (b) Using the boundary element method: the work of Katsikadelis and Armenakas [5], Syngellakis and Bai [7], Paiva and Butter- field [8] for thin plates, Rashed et al. [9–12] for thick plates. Solving plates on tensionless foundation can be categorized into two main categories. The first category involves solution using iterative pro- cedure to consider the miscontact between plate and foundation. The second category, on the other hand, involves solution by transforming the problem into a set of nonlinear equations, which could be solved using optimization algorithms. Among the previous literature which follows the first category are: the work of Cheung and Nag [13] who presented a finite element analy- sis of beams and plates on linear and nonlinear elastic continuum. Weits- man [14] presented an approximate solution for the radius of contact between an elastic plate and a semi-infinite elastic half space subjected to concentrated load. Weitsman [15] presented analysis of tensionless ∗ Corresponding author at: Department of Structural Engineering, Cairo University, Giza, Egypt. E-mail addresses: yrashed@scu.eg, yrashed@hotmail.com (Y.F. Rashed). beams, or plates, and their supporting Winkler or Reissner foundations due to concentrated loads. Svec [16] developed a finite element itera- tive procedure to determine the contact region between the plate and the elastic half space. The continuous contact pressure is approximated in [16] by a set of statically equivalent forces acting at the nodal points of the elements. Hence, the plate could be considered to be resting on a complicated system of springs. Celep [17] presented the behavior of elastic plates of rectangular shape on a tensionless Winkler foundation using auxiliary function. Galerkin’s method is used in [17] to reduce the problem to a system of algebraic equations. Li and Dempsey [18] used an iterative procedure to analyze unbonded contact of a square thin plate under centrally symmetric vertical loading on elastic Winkler or elastic half space foundations. Hu and Hartely [19] solved thin plate on tensionless elastic half space using integral equations. Analysis us- ing the T-element of plates on unilateral elastic Winkler type founda- tion is presented by Jirousek et al. [20]. Nonlinear bending behavior of Reissner–Mindlin plates with free edges resting on tensionless elas- tic foundations of the Pasternak-type using admissible functions is pre- sented by Hui-Shen and Yu [21]. Results of finite element analysis of beam elements on unilateral elastic foundation using special zero thick- ness element designed for foundation modeling is presented by Torbacki [22]. Buczkowski and Torbacki [23] presented finite element analysis of plate on layered tensionless foundation. Kongtng and Sukawat [24] used the method of finite Hankel integral transform to solve the mixed bound- ary value problem of unilaterally supported rectangular plates loaded by uniformly distributed load. The studies based on the second category, on the other hand, could be listed as follows: http://dx.doi.org/10.1016/j.enganabound.2017.07.003 Received 6 December 2016; Received in revised form 23 June 2017; Accepted 6 July 2017 0955-7997/© 2017 Elsevier Ltd. All rights reserved.