Automatic yield-line analysis of slabs using discontinuity layout optimization.

Matthew Gilbert,Colin C Smith,Linwei He,Canh V Le

doi:10.1098/rspa.2014.0071

Abstract

The yield-line method of analysis is a long established and extremely effective means of estimating the maximum load sustainable by a slab or plate. However, although numerous attempts to automate the process of directly identifying the critical pattern of yield-lines have been made over the past few decades, to date none has proved capable of reliably analysing slabs of arbitrary geometry. Here, it is demonstrated that the discontinuity layout optimization (DLO) procedure can successfully be applied to such problems. The procedure involves discretization of the problem using nodes inter-connected by potential yield-line discontinuities, with the critical layout of these then identified using linear programming. The procedure is applied to various benchmark problems, demonstrating that highly accurate solutions can be obtained, and showing that DLO provides a truly systematic means of directly and reliably automatically identifying yield-line patterns. Finally, since the critical yield-line patterns for many problems are found to be quite complex in form, a means of automatically simplifying these is presented.

Highlights

The yield-line method is a long established and highly effective means of estimating the ultimate load-carrying capacity of slabs and plates
In the case of slabs, a number of researchers have proposed procedures designed to improve upon the solution obtained using an initial rigid finite-element analysis, by refining this in a subsequent iterative nonlinear optimization phase (e.g. [17,18])
The convex nature of the underlying mathematical optimization problem is preserved, and, even when the adaptive nodal connection procedure is used, the solution obtained will be globally optimal for the prescribed nodal discretization

Summary

Introduction

The yield-line method is a long established and highly effective means of estimating the ultimate load-carrying capacity of slabs and plates. Maintaining precisely the same form of linear optimization problem as given in (2.2), the kinematic yield-line layout optimization formulation for an out-of-plane, quasi-statically loaded, perfectly plastic slab with supported edges and discretized using m nodal connections (yield-line discontinuities), n nodes and a single load case can be defined in equation (2.3) as follows: min E = gTd subject to: Bd = 0. The remaining unshaded area lying ‘above’ potential yield-line CL (i.e. area FGHI) will clearly move in the mechanism postulated, but the work associated with this movement will be accounted for through displacement along edge FG (combined translation and rotation), with the relative displacements at the edge of the slab in effect being absolute displacements It was pointed out earlier in the paper that the layouts of yield-lines in slabs will, like bars in optimal trusses, take the form of Hencky–Prandtl nets, which are orthogonal curvilinear coordinate systems.

Examples

Discussion

Findings

Conclusion