Table 3 Global formulation

Elastic–plastic analysis [29]

\({\mathbf{d}} = {\mathbf{K}}^{ - 1} {\mathbf{f}}\quad {\text{with}}\quad {\mathbf{K}} = {\mathbf{B}}^{\text{T}} {\mathbf{D}}^{ep} {\mathbf{B}}\)

Limit analysis [27]

\({\text{Min}}\quad \varOmega ({\dot{\varvec{\lambda }}},{\dot{{{\bf d}}}}) = {\mathbf{s}}_{0}^{\text{T}} {\dot{\varvec{\lambda }}}\quad {\text{with}}\quad \left\{ {\begin{array}{*{20}c} {{{{\bf N}\dot{\varvec{\lambda} }}} - {{{\bf B}\dot{\bf {d}}}} = {\mathbf{0}}} \\ {{\bar{\mathbf{f}}}^{\text{T}} {\dot{{{\bf d}}}} > 0} \\ {{\dot{\varvec{\lambda }}} \ge {\mathbf{0}}} \\ \end{array} } \right.\)

Shakedown analysis [27]

\({\text{Min}}\quad \varOmega ({\dot{\varvec{\lambda }}},{\dot{\mathbf{d}}}) = {\mathbf{s}}_{0}^{\text{T}} {\dot{\varvec{\lambda }}}\quad {\text{with}}\quad \left\{ {\begin{array}{*{20}c} {{{{\bf N}\dot{\varvec{\lambda }}}} - {{{\bf B}\dot{{\bf d}}}} = {\mathbf{0}}} \\ {{\mathbf{s}}_{\text{E}}^{\text{T}} {{{\bf N}\dot{\varvec{\lambda} }}} > 0} \\ {{\dot{\varvec{\lambda }}} \ge {\mathbf{0}}} \\ \end{array} } \right.\)

Limit optimization [30]

\({\text{Min}}\;{\text{Z}}({\mathbf{n}}_{\text{p}} ,{\mathbf{s}}) = {\mathbf{n}}_{\text{p}}^{\text{T}} {\mathbf{l}}\quad {\text{with}}\quad \left\{ \begin{aligned} {\mathbf{B}}^{\text{T}} {\mathbf{s}} = {\mathbf{f}} \hfill \\ {\mathbf{N}}^{\text{T}} {\mathbf{s}} \le {\mathbf{s}}_{ 0} \hfill \\ \end{aligned} \right.\)

Shakedown optimization [30]

\({\text{Min}}\;{\text{Z}}({\mathbf{n}}_{\text{p}} ,{\varvec{\uprho}}) = {\mathbf{n}}_{\text{p}}^{\text{T}} {\mathbf{l}}\quad {\text{with}}\quad \left\{ \begin{array}{l} {\mathbf{B}}^{\text{T}} {\varvec{\uprho}} = {\mathbf{0}} \hfill \\ {\mathbf{N}}^{\text{T}} {\mathbf{(s}}_{\text{e}} + {\varvec{\uprho )}} \le {\mathbf{s}}_{ 0} \hfill \\ \end{array} \right.\)