Performance Based Structural Optimisation

Reinforced Concrete (RC) is a cornerstone in current and near-future built environments. Due to the cheapness of concrete and its versatile use, predictions say that an increase in concrete production is foreseeable. However, the environmental impact of concrete proves an increasing challenge, with cement, a component in concrete, contributing to around 8 % of the world’s CO2 emission. Furthermore, concrete is playing a significant role in the depletion of non-renewable resources like sand, gravel, and limestone. Thus, methods that optimise RC structures with regard to material usage can increase sustainability in the built environment.

When designing structures, engineers need to focus on both the structure’s safety and longevity. These two aspects manifest in the design codes as the Ultimate Limit State (ULS) and Serviceability Limit State (SLS) criteria. The ULS considers safety against collapse by ensuring adequate structural capacity. On the contrary, SLS focuses on serviceability and considers, amongst other things, crack-width and deflection requirements.

Material optimisation of RC structures has mainly focused on either linear elastic material response, and in turn, only the serviceability of the structures, or on a rigid plastic model, therefore only being reliable for the safety of the structure. In general, few methods exist that allow for the optimisation of RC structures while considering both the SLS and ULS.

This thesis presents a design-orientated method for material optimisation of RC structures while simultaneously taking several load cases into account, considering both the SLS and ULS criteria. The criteria are implemented as restrictions on the strains of the structure. In the ULS, the strains are limited to the material’s ultimate strains, and the SLS strains are limited as an indirect restraint on the crack width.

The method presented uses sequential convex programming, where convex approximations are solved iteratively until satisfying solutions are identified. The method uses three main analysis steps in solving the problem. The initial layout is determined through the Finite Element Limit Analysis using a rigid plastic material model. The structural response is then calculated based on a non-linear elastoplastic material model. This is done through the use of the principle of minimum potential energy. Lastly, a convex approximation of the governing material optimisation problem is solved.

The method is implemented for truss elements subjected to uniaxial loading and triangular panel elements subjected to in-plane forces. Along with this, different formulations of the convex approximations are investigated. For the truss elements, optimised strut and tie models are found through both a multi-criterion and a tangential stiffness matrix formulation. Both formulations provided similar solutions; however, the stiffness matrix formulation proved to be superior with respect to numerical stability. For the triangular elements, three formulations, denoted interior, penalty and combined formulation, are tested. From these different formulations, structural examples are examined, and the volume s compared to known lower-bound solutions and to more traditional designs. Of the three formulations, the combined resulted in the best behaviour and solutions. Here, optimised designs are determined, which utilise 26-38 % less material than conventional designs while still adhering to strain constraints corresponding to both SLS and ULS criteria.

Along with the focus on optimisation of RC structures, auxiliary work is also presented. This work presents a method for optimising grillages for floor plates subjected to displacement and strain constraints. Here, a convex formulation of a multi-criterion optimisation problem is presented, and the optimal placement of four columns supporting a domain is investigated. Here, a reduction in material usage of 80 % is identified by inserting the supporting columns.