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.

Materials used in structures have a thermal response character, which is noted but rarely developed further in solving problems. The bimetallic strip is presented as an example of where thermal effects can be useful (in thermostats, shape-changing structures etc) and where the coupling between equilibrium, compatibility and the generalised Hooke's Law is evidently demonstrated analytically.

Practical frames have interconnected beams and columns, where moments and forces are transmitted across a local, indeterminate junction. The prospect of a more complex analysis is, however, reduced in certain cases of layout; moreover, it is shown that the simpler layout of a determinate junction proves to be analytically more challenging than the indeterminate case.

The deflection characteristics of a simply-supported beam are calculated directly and summarised alongside a cantilever, in order to define standard cases of deflection coefficients.

The construction of bending moment (and shear force) diagrams is considered using fundamental equilibrium relationships, rather than resorting to remembering standard loading cases. The importance of sign convention, coordinate direction and behaviour at supports is given explicit attention in several examples.

Several techniques based on exploiting symmetry are introduced for reducing the level of calculation in beams and frames that are symmetrical or anti-symmetrical in their original layout. Attention is paid to the performance of stress resultants and kinematical parameters on the plane of symmetry in order to establish quantities which are zero from the outset. This approach reduces, for example, the number of redundancies in statically indeterminate cases.

Rather than solve the usual governing equation of deformation for a deflected cable, its shape is computed indirectly from the applied loading, noting that the two variations must differ by two orders. Solutions by polynomial subsititution are thus appropriate, where coefficicents depend on the specific boundary conditions of the problem. The non-uniform build-up of tensions in a cable wrapped around a rough drum is determined, which then informs a simple experiment for determining the coefficient of friction.

Designing a structure by the method of Lower Bound rarely considers the supports: that often, sizing the 'main beam' is key. One difficulty is how to interpret the nature of supports themselves in a Lower Bound analysis context. Some clarity is given for a nominal multi-span beam where one of its supports is realised differently in three ways.

Often loads applied to a structure do not depend on the deflections they induce. If there is a dependency, they couple the very forces in equilibrium to the displacements they induce, leading to statical indeterminacy. Solving such problems proceeds by assuming a displacement profile dictated by the characteristic behaviour of the loads they couple to. They may be actually very small, but such purpose allows us to express the loads accurately in terms of them whilst considering equilibrium in the undeformed state - a state of pseudo-equilibrium.

Chapter 1 describes the main objectives of the book. It argues that uncertainties are omnipresent in all aspects of the design, analysis, construction, operation, and maintenance of structures and infrastructure systems. It sets three goals for engineering of constructed facilities under conditions of uncertainty: safety, serviceability, and optimal use of resources. It then argues that probability theory and Bayesian statistics provide the proper mathematical framework for assessing safety and serviceability and for formulating optimal design under uncertainty. The chapter provides a brief review of the history and key developments of the field during the past 100 years. Also described are commercial and free software that can be used to carry out the kind of analyses that are described in the book. The chapter ends with a description of the organization of the book and outlines of the subsequent chapters.

Chapter 4 presents the basic formulation of the structural reliability problem. It starts with the so-called R-S problem with R denoting a capacity (resistance, supply, strength, etc.) value and S denoting a measure of the corresponding demand (load, stress, etc.), both modeled as random variables. Solutions in integral form are presented for the failure probability by conditioning on R or S, or using formulations in terms of the safety margin or safety factor that lead to the introduction of the concept of reliability index. Exact solutions are presented for specific distributions of R or S. This allows examination of the so-called tail-sensitivity problem, i.e., the sensitivity of the failure probability to the selected probability distributions. It is shown that small failure probabilities are sensitive to the shape of the selected distributions in the tail. The formulation of the structural reliability problem is then generalized and presented in terms of a limit-state function of basic random variables. Using this formulation, the probability of failure is expressed as a multifold integral over the outcome space of the basic random variables. Descriptions of several example applications of the generalized structural reliability formulation conclude the chapter.

Chapter 15 describes the use of the Bayesian network (BN) methodology for reliability assessment and updating of structural and infrastructure systems. A brief review of the BN as a graphical representation of random variables and an efficient framework for encoding their joint distribution and its updating upon observations is presented. D-separation rules describing the flow of information within the network upon observation of random variables are described and methods are presented for discretizing continuous random variables, thus allowing the use of efficient algorithms applicable to BNs with discrete nodes. Efficient BN models for components, systems, random fields, and seismic hazard are developed. For time- or space-variant problems, the dynamic Bayesian network is introduced. This model is used in conjunction with structural reliability methods (FORM, SORM, simulation) to develop enhanced BNs to solve reliability problems for structures under time-varying loads. Detailed examples are presented, including post-earthquake risk assessment of a spatially distributed infrastructure system and reliability assessment of a deteriorating structure under stochastic loads. The chapter concludes with a discussion of the potential of the BN as a tool for near-real-time risk assessment and decision support for constructed facilities, and the need for further research and development to realize this potential.