Synthesising different collapse mechanisms for indeterminate beams and frames is considered via the prospect of their salient bending moments (without solving for their exact values) as candidate positions for the formation of plastic hinges. The method of Instantaneous Centre is introduced for calculating the relative motion of sub-structures within frames, including those with pitched roofs under distributed loading.

Perfect buckling of an axial rod occurs at the Euler load, which cannot be achieved in practice because of the presence of straightness imperfections. The governing mathematical behaviour is very similar in both cases but with starkly different outcomes. Matters are revised here for the case of an axial load applied eccentrically, which aims to highlight the role of different end loadings, rather than just geometrical imperfections, in the context of buckling.

Analytical shortcuts are introduced for calculating the sectional properties of thin-walled beams for stiffness in bending and in torsion. The performance of a simply-supported beam is then introduced in the context of its optimal stiffness, where deflections are minimised relative to the weight of the beam using a dimensional analysis approach.

Solution of problems with friction are tackled graphically, by assuming slippage is present from the outset. Quasi-statical equilibrium therefore imposes a fixed inclination of frictional force components for reduced working and, thus, a graphical solution.

Loads are often idealised as point forces or distributed intensities for analytical simplicity. More importantly, they are often stipulated for a given problem. This chapter hightlights the need to consider how loads, and their reactions, behave (and form) when geometrical nonlineatrity is evident. Two examples are analysed in detail: the formation of a beam 'ruck' on a rough floor, and the 'snapping-through' of a shallow pin-jointed arch.

Approaches are given for determining the number of analytical redundancies in indeterminate beams and frames, either by removing kinematic freedoms until the structure collapses (and couting backwards) or by subdividing the structure into determinate sub-structures and tallying the number of unknown forces and moments at their original junctions.

Most truss frameworks have almost rigid joints in practice; yet, they are often treated analytically as being pin-jointed. Such a dichotomy is tackled by directly solving for the case of a truss beam with built-in joints and then comparing its response to the pin-jointed case. The two responses are shown to converge when the beam members become very slender, thereby validating the pin-jointed assumption.

The method of Lower Bound analysis is often taught as a curious equilibrium response or with strict adherence to mathematical formalism. The exact behaviour of a truss loaded to failure is presented first so that the character of a Lower Bound approach may be compared later; in particular, why ductility is vital for valid equilibrium solutions and for redistributing bar forces under increasing load.

The method of Virtual Work is applied to three truss examples to highlight directly the various efficacies of the method, in particular, the extra analytical freedom afforded by the nature of indeterminacy.