## Introduction

To allow us to meet our emissions reductions targets, concrete structures must be optimised. This means that analysis methods capable of modelling all the complex behaviour of concrete (shear, deflections, cracking) for any geometry must be available. A completely new approach is required to fully realise the potential of reinforced concrete as a sustainable material.

Simple finite element based methods cannot properly predict the behaviour of complex concrete structures at the ultimate limit state, due to the nature of concrete cracking. This is because the underlying mathematics assumes that the body being analysed remains continuous as it deforms (thereby allowing partial differential equations to calculate strains). If a crack is to form the material becomes discontinuous, and special techniques are required to continue the analysis.

As a result, ‘real-world’ concrete design, particularly for shear behaviour, is undertaken using empirical methods developed from tests on prismatic concrete structures. This introduces boundaries to design possibilities, precluding the design and analysis of non-prismatic structures whose forms are outside the limits of the test data. The opportunity that fluid concrete provides to minimise material use is therefore unnecessarily prevented.

Meshfree numerical models for the simulation of continua place nodes more or less randomly within a design space, allowing each node to interact with a certain number of other nodes within a defined horizon. The meshfree method of smoothed particle hydrodynamics (SPH) is powerful in situations where the domain is continually changing, such as pouring fluid into a glass. The term ‘peridynamic’ (from near and force) was coined by Silling for meshfree methods in solids. This approach, in contrast to conventional FE, does not presume a continuous displacement field and instead models solid materials as a collection of particles held together by tiny forces, the values of which are a function of each particle’s relative position. Displacement of a particle follows Newton’s laws of motion. Such models are frequently used where a crack moves through a solid, breaking the arms between particles.

This approach is well suited to reinforced concrete since: 1) concrete really is a random arrangement of cement and aggregate particles; 2) failure is governed by tensile strain criteria, which is ideal as the only real way that concrete fails is in tension (all other failure modes in everyday design situations are a consequence of tensile failure) and the model can therefore accurately predict behaviour, and 3) since the elements fail progressively in tension, the peridynamic approach automatically models cracking behaviour, which is extremely difficult to model conventionally.

Novel methods for casting complex forms capitalise on the fluidity of concrete to allow new architectural expression. Combining these new techniques with design optimisation has the potential to facilitate significant material savings4, but robust analysis techniques that can evaluate all the complex behaviour of concrete structures are not yet available. This fellowship will link the new peridynamic model with optimisation processes that consider constructability to facilitate the design of minimum material use concrete structures without boundaries to their geometry.