The stress function basis of the upper bound theorem of plasticity

The discontinuous slip-line form of upper bound plasticity analysis is considered using an equilibrium of forces approach. It is demonstrated that the underlying basis of the approach can be written in terms of stress functions that provide a continuum stress state interpretation of the upper bound solution. An alternative proof of the upper bound theorem, applicable to both associative and non-associative materials, using stress functions is presented. The broader nature of the equilibrium form and the strict conditions under which it is valid are discussed, including examination of the apparent omission of moment equilibrium and associativity in many equilibrium form solutions. Finally, the relationship of the stress function formulation to the output of the computational limit analysis method discontinuity layout optimisation (DLO) and the potential to use the stress function formulation to derive a form of lower bound solution from an upper bound analysis are demonstrated.

Automated summary

This paper discusses the discontinuous slip-line form of upper bound plasticity analysis using an equilibrium of forces approach with stress functions. It presents an alternative proof of the upper bound theorem applicable to both associative and non-associative materials. The broader nature of the equilibrium form and its strict conditions, including the apparent omission of moment equilibrium and associativity in many equilibrium form solutions, are also discussed. Finally, the paper demonstrates the relationship between the stress function formulation and the output of the computational limit analysis method discontinuity layout optimization (DLO), as well as the potential to use the stress function formulation to derive a form of lower bound solution from an upper bound analysis.