A numerical approach to the yield strength of shell structures

Abstract

International audienceThis work investigates the formulation of lower and upper bound finite elements for the yield design (or limit analysis) of shell structures. The shell geometry is first discretized into triangular planar facets so that previously developed lower bound equilibrium and upper bound kinematic plate finite elements can be coupled to membrane elements. The other main novelty of this paper relies on the formulation of generalized strength criteria for shells in membrane-bending interaction via an implicit upscaling procedure. This formulation provides a natural strategy for constructing lower and upper bound approximations of the exact shell strength criterion and are particularly well suited for a numerical implementation using second-order cone programming tools. By combining these approximate strength criteria to the previously mentioned finite elements, rigorous lower and upper bound ultimate load estimates for shell structures can be computed very efficiently. Different numerical examples illustrate the accuracy as well as the generality and versatility of the proposed approach

Numerical Yield Design Analysis of High-Rise Reinforced Concrete Walls in Fire Conditions

“…The generalized strength criterion is obtained as the solution of a yield design problem with membrane forces and bending moments acting as macroscopic loading parameters, the resolution of which can be realized resorting to a 3D discretization [18]. Since we aim at modelling the deformed panel as a curved shell, we choose to take advantage of the construction of generalized strength criteria for shells developed in [19]. Indeed, if steel rebars were absent, we would have been in presence of a shell with strength properties homogeneous in its own local plane but heterogeneous across its thickness.…”

“…Indeed, if steel rebars were absent, we would have been in presence of a shell with strength properties homogeneous in its own local plane but heterogeneous across its thickness. Knowing the concrete plane stress strength criterion at all point across the thickness, it is possible to obtain a semi-analytical expression of the shell strength criterion, which is particularly suited for a numerical implementation [19].…”

“…As mentioned earlier, the panel in its deformed configuration will be viewed as a shell modelled by an assembly of planar facets in membrane-bending interaction as described in [19]. In particular, numerical strategies for approximating the generalized strength criterion G either from the inside or from the outside are employed (respectively for the lower bound static approach and the upper bound kinematic approach) to ensure the strict bounding status of the computed critical load factor [11,19]. Such strategies are, moreover, particularly suited for formulating the corresponding discrete optimization problems as second-order cone programs.…”

“…The SOCP formulation of the global shell yield design problem, both for the upper and lower bound approaches, follows the procedure described in [19,11].…”

“…n = 11) has been used for the approximation of the generalized strength criterion for the static (resp. kinematic) approach (see [19]). Table 1 Bracketing of the stability factor after 120 min (H = 12 m) Table 1 collects the numerical estimates of the stability factor in terms of lower and upper bounds for the different geometries after 120 min of fire exposure.…”