A ‘frame’ resists the action of external loads primarily by bending of its members. Thus the loads V and H applied to the plane rectangular portal frame of fig. 7.1(a) will give rise to certain bending moments. It is the evaluation of this single variable, the bending moment M, that is the objective of the structural analysis of the frame. Further, for the example of fig. 7.1, in which the loads are applied at discrete points and the members are straight between nodes, a knowledge of the values of the bending moment at those nodes (five in number in fig. 7.1) will give a complete description of the state of the frame. In fig. 7.1(b) the general bending-moment diagram has been sketched (with the sign convention that bending moments producing tension on the outside of the frame are positive).

The application of the first of the master equations of the theory of structures, that of statical equilibrium, will give some relations between the values of the bending moments at the nodes, MA to ME in fig. 7.1(b). As usual, a statical analysis may be made straightforwardly by the use of virtual work, and fig. 7.1(c) shows a pattern of deformation involving discontinuities at the five cardinal points. Between these hinges the members of the frame remain straight; the two columns are each inclined at a small angle θ to their original directions, and the two half-beams at an angle φ.

A structure will in practice be acted upon by a number of independent loads (superimposed floor loads, snow, wind, crane loads, and so on). Moreover, each load will vary between limits which will be specified – the wind may blow from east to west, or not at all, or from west to east. The engineer making an elastic check of a given design will arrange for calculations to be made separately for each load; for any particular cross-section, the value of each load is then chosen to give the greatest and least action at that section. The designer is then able to determine the range of stress at the cross-section due to all the loads, and to make an assessment, according to given elastic rules of design, of the safety of the structure.

By contrast, the engineer making an estimate of the static plastic carrying capacity of a framed structure must arrange all the loading in the way expected to be most critical before the single plastic-collapse calculation is made. In practice there is no great difficulty in arriving at the worst combination of loads (although even for the simple portal frame the most critical position of a crane crab is not immediately obvious).

However, if loads on a frame do act randomly and independently within specified limits, then there is the possibility of incremental collapse of the frame.

This chapter first deals with what complex numbers are and how we manipulate them and understand their properties; and then goes on to describe how useful they are in helping to solve a variety of problems. The case of a resonant system will be dealt with in detail. Because the use of complex numbers makes the solution of this type of problem trivial, we will be able to concentrate on the form of the results, rather than on the details of how to get there.

Complex numbers are also helpful in describing waves of all sorts (see Chapter 14 of Volume 2). They are particularly useful in situations involving the diffraction and interference of light waves. In quantum mechanics, it is not so much that they are a calculational aid, but rather that the wave function used to describe objects like electrons involves complex numbers in an essential manner. Neither of these last two applications is described here.

What are complex numbers?

Mathematicians are adept at inventing new types of concepts. Thus while the use of positive numbers to represent the magnitude of actual quantities in the real world gives them an immediate significance, this is not so obvious for negative numbers.

The final draft of a mathematics book contained the sentence: ‘∂f/∂x means the ratio at constant y of δf and δx, where df and δx are vanishingly small’ When the author received the publisher's proofs, this appeared as ‘∂f/∂x means the ratio at constant y of . and ‥’ On closer examination with a powerful magnifying glass, however, it turned out that the first two full stops were in fact the smallest δf and δx that the publisher was able to produce.

Many branches of science involve partial derivatives. The aim of this chapter is to make you understand what they are, and become so fluent at manipulating them that this sort of operation becomes as familiar and as accepted as the arithmetic operations with ordinary numbers. This will then enable you to concentrate on the basic principles of your science problem, rather than battling with the mathematics of the partial derivatives involved.

Introduction

We are often interested in calculating the derivatives df/dx, d2f/dx2, etc for a function f(x) of a single variable x. Similarly, for functions of more than one variable f(x, y, …), we may well also want the derivatives. These are written as, for example, ∂f/∂x, which means ‘the rate of change of the function f with respect to small changes in x, assuming that all the other independent variables are kept constant’.