The evolution of the contact analysis is strongly related to the analysis of unanchored liquid storage tanks. The successive contact and separation between tank base plate and underlying foundation is a special case of the contact problem between two bodies. However, the developed algorithm handles the contact problem in its general form which make it necessary to present a brief review of contact mechanics history.
Modern contact mechanics begins in 1882 with the publication of the work by Hertz [107]. Analytical solutions to the problem were presented before the appearance of digital computers. These solutions employed the theory of elasticity and were limited to simple linear cases of contact. Moreover, since for most problems the exact contact surfaces are not known a prior, their application is further restricted. Gladwell [68] tackled the problem with the aid of Papkovich-Neuber form of the equilibrium equations. Signorini [257] formulated the general problem of a linear elastic body in frictionless contact with a rigid foundation. Shtaerman [256] showed various solutions using the methods of integral transforms and complex potentials in theory of elasticity. These methods allowed for a more general analytical treatment of certain, still very restricted, classes of contact problems. Fichera [63] presented a formal mathematical treatment of the Signorini problem within the framework of boundary value problems governed by unilateral constraints. In a following work [62], he presented a more complete and general solution of the problem.
The numerical techniques are powerful tools to analyze the contact problem because of their flexibility and ability to model all complications involved with the analysis of such problems. The finite element method was introduced for the problems of structural analysis by Turner [271], Clough, [39], and simultaneously by Argyris [8]. Later the method was extended to structural mechanics, among many other fields, by numerous researchers. Numerical contact algorithms have been proposed in the early 70s to handle the complex nature of the physical and numerical behavior of contact problems. Conry and Seireg [40] treated contact problems as quadratic programming problems. Subsequently, Chan and Tuba [29], Kalker [135] and Panagiotopoulos [214] among others explored the same line of thought. At the same time, the work of Hughes [117], based on a Lagrange Multiplier method, significantly contributed to the development of robust finite element algorithms applicable to large scale computations. Glowinski and coworkers [69] offered a comprehensive exposition of various optimization techniques and their application to the solution of problems characterized by unilateral boundary conditions. Courant [43] used the penalty approximations to solve the contact problem. Luenberger [160] reviewed solutions to the contact problem in his book. Hestenes [108] used augmented Lagrangian methods to solve the problem. Powell [226] improved this method by combining the classical penalty treatment with that of Lagrange multipliers to attaining more satisfactory performance than either one of the above methods when employed separately. A detailed presentation of both theoretical and some numerical aspects of contact problems is contained in [139].
Recently, several contact algorithms have been proposed [215] and incorporated into commercially available Finite Element softwares. Bathe and Chaudhary ([17], [31]) presented a solution methods for the analysis of contact between two or more contact bodies. In there work, they implicitly enforced the contact constraints using an augmented Lagrange multiplier technique which seriously affects the quadratic convergence of the Newton-Rapson method. Eterovic et al [53] proposed a solution technique to enforce these constraints explicitly by means of a gap function. He presented a solution technique that admits the use of a line search procedure to enlarge the region of convergence. He reported that this approach has significantly improved the convergence of the analysis, however, it yields unsymmetrical stiffness matrix. Pian et al [223] derived a modified complementary energy principle by introducing the continuity at the contact surface as the condition of constraint and the reactions at contact surface as additional field variables. He outlined an iterative procedure using a stress hybrid element for incorporation in numerical analyses.