Introduction

Last week we talked about bridges and shells. We used a Platonic cube to introduce the concept of internal forces within a structure and principal stresses acting on the face of the cube. And, then we extended the concept to analyse the stress in thin-­walled shells before jumping in a new direction to talk about strain energy. This week we are going to discuss the bending of natural structures, like grasses and trees, as well as man-made structures, such as the Burj Khalifa in Dubai or the Brighton Tower. Once again, we’ll look at the internal forces and stresses in these structures and consider the distribution of shear forces and bending moments inside a beam.

So far, our analysis has been strongly dependent on Newton’s three laws of motion. However, there are cases where these laws do not provide enough information for our structural analyses. And so, this week we will introduce the principle of compatibility of displacements, which allows us to analyse a particular class of structures that have some degree of indeterminacy and redundancy. In this context, redundancy takes its usual meaning a four-legged table has some redundancy because, it has more legs than are strictly necessary, from the perspective of equilibrium. However, it’s indeterminate not because it can’t make decisions or determinations, which is of course true, but because we can’t determine the stresses and strains acting in it from Newton’s three laws alone.