Keywords: software development, geotechnical, plastic limit analysis, finite element, mathematical programming.

The main concern in many geotechnical applications is the determination of a maximum load for a given structure, such as to define a working load that is at the same time safe and economic. Some analytical solutions are available for the lower and upper limits of collapse load for simple problems. However, for more complex problems, there are no analytical solutions or the gap between the lower and upper limits available are too big to be used with confidence (Chen[1]). A full elastoplastic finite element analysis of the structure behaviour provides an effective means to identify yield zones and the propagation of the failure mechanism. However, many finite element codes face numerical instabilities when approaching the collapse load as point out by Farias and Naylor[2].

A combination of finite element discretisation, limit analysis theorems and mathematical programming optimisation algorithms provide and efficient and elegant means to determine the ultimate load of engineering problems. The present study uses the lower limit theorem to maximize the load that satisfies both equilibrium and strength conditions. Mohr-Coulomb strength criterion and an associated flow rule are adopted. Alternatively, one can use the lower limit theorem to find the minimum load that satisfies both compatibility and plastic flow conditions, as described in reference[3].

A program that runs under the Microsoft Windows 9x platform was developed to implement the above techniques. This 32 bits system provides an intuitive graphics interface, which most computer users are well acquainted with. The task of developing applications for Windows environment is facilitate by a great number of software that implement languages with visual resources, such as Delphi, Visual Basic and C++ Builder. These tools are responsible for the growing number of software with graphics interfaces, which allow the development of user-friendly engineering programs. In particular, the Delphi is a RAD (Rapid Application Development) tool for Windows, object oriented and event driven language based on the concept of components. The programming language under Delphi is a version of object oriented Pascal, named Object Pascal, and was used to code the visual part of the program.

The finite element discretisation process used in the present program is based on the approach described by Hinton and Owen[4] for structured meshes. The stress and displacement fields are interpolated using finite element shape functions. In the case of using the Lower Bound Theorem, the program assemblies a system of equations and inequalities, representing the equilibrium conditions and strength restrictions, respectively. The governing system should be optimised, by maximizing a load factor () that multiplies the variable part of the applied external load vector. The strength restrictions for the adopted Mohr-Coulomb failure criterion are non- linear, but may be linearised with the use of a polyhedral representation of the failure surface. In this case, the program calls the commercial code LINDO (Linear Interactive Discrete Optimiser) to perform the optimisation stage. In case of using the original non-linear criterion, the program may call a commercial optimiser (LINGO) or may perform the optimisation stage using its own implementation of a non-linear algorithm proposed by Herkovits[5].

The implementation of the computational routines for the procedures described above, was coded using the Fortan 95 language and later compiled in DLL´s to be linked to the visual interface of the program.

At the end of the solution stage, the collapse load factor is automatically shown in a dialog box. Then the failure mechanism is also automatically displayed by drawing the deformed mesh for the failure velocity field (dual solution for the static problem). An alternative graphic module was also developed using OpenGL routines to display the failure mechanism by means of colour contour zones.

The example of a fill with 8m in height and slope of is presented to illustrate the program use and to validate the implemented procedures. The result is in very good agreement with other available solutions.

1

W. Chen, "Limit Analysis and Soil Plasticity", Elsevier Scientific Publishing Company, Amsterdam, 1975.

2

M.M. Farias, D.J. Naylor, "Safety analysis using finite elements", Computers and Geotechnics, 22, 165-181, 1998. doi:10.1016/S0266-352X(98)00005-6

3

L. Santos da Silva, M.M. Farias, C.L. Sahlit, "Plastic Limit Analysis in Geotechnics using the Finite Element Method plus Linear and Non Linear Mathematical Programming", Proceedings of the Seventh International Symposium on Numerical Models in Geomechanics (VII NUMOG), Graz, 1999.

4

E. Hinton, D.R.J. Owen, "An introduction to finite element computations", Pineridge Press Limited, Swansea, UK, 1979.

5

J.N. Herskovits, "A two-stage feasible directions algorithm for nonlinear constrained optimization". Journal of Mathematical Programming, 36, 19-38, 1986. doi:10.1007/BF02591987

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