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.

Grillages are out-of-plane structures made from interconnected beams and frames. They thus couple bending and torsional equilibrium, as well as shearing and axial effects. Their displacements and rotations are similarly coupled, and their calculation is amenable to the Force Method when small. This chapter provides suitable demonstration and examines the relative contribution of torsional stiffness towards the overall structural stiffness.

Deflection coefficents are introduced in the context of the Force Method for indeterminate beams and frames. Subtleties of geometrical compatibility are considered in detail, first for simple structures and then for multi-span beams and frames.

Maxwell's Rule conveys a statement of rigidity for a given truss framework. When the number of bars or supports is reduced, the truss will inevitably lose rigidity and collapse as a mechanism. This consequence can also be captured by Maxwell's Rule in a modified statement, which leads to a discussion about its interplay with states of self-stress, illustrated using the simplest pin-jointed arch.

Calculating the fully plastic response of a cross-section is often couched in exact terms, which may prove cumbersome for non-standard cross-sections. This approach is manifestly the method of Lower Bound and may be re-interpreted by searching for simpler but valid equilibrium solutions for the stress-resultants on the critical cross-section; the case of plastic biaxial bending provides an example. Simple joint design according to Lower Bound then follows.