Statically Determinate Systems

Abstract

In this chapter we shall focus on what may be called simple statically determinate line-like systems. These are structural systems that can be modelled by lines in a Cartesian coordinate system where their three-dimensional cross sectional properties have been concentrated to a line through a spanwise axis at the centroid of pure bending (see Chapter 3). Similar indeterminate systems are handled in Chapter 6. Some special systems (e.g. arches, cables and cylindrical shells) are presented in Chapter 9. Plates are handled in Chapter 11. To enable any safety considerations of structural systems it is necessary first to obtain the overall reaction forces, and then all the inner axial, bending, shear and torsion components. What is available for the determination of reaction forces in two-dimensional systems (e.g. as shown in Figs. 2.1 and 2.2) is the three equilibrium conditions given in Eq. 1.13: \( \sum {F_{x} } = 0 \), \( \sum {F_{z} } = 0 \) and \( \sum {M_{P} } = 0 \). In all the examples shown in Fig. 2.1 there are three reaction forces, and thus, these may uniquely be determined by these three equilibrium conditions. We say that any structural system with three reaction forces is statically determinate. The systems in Fig. 2.2.a and b both have four unknown reaction forces. Thus, none of them can uniquely be determined by the three equilibrium equations that are available. We say that these systems are statically indeterminate of order one. Thus, we will need one more condition to determine the reaction forces of the system. This additional condition is provided by the distribution of stiffness in the system, as stiffer parts carry more load than more flexible parts. E.g., for the gantry in Fig. 2.2.b,\( N_{BD} \) and \( N_{BC} \) depend on the stiffness ratio between cables BD and BC. The condition used to determine \( N_{BD} \) and \( N_{BC} \) is that the stretching of cables BD and BC is coherent with a joint vertical displacement of point B. This is what we call the compatibility condition of the system. The single bay three storey frame in Fig. 2.2.c has six unknown reaction forces. Thus, is it three times statically indeterminate, and hence, to determine all reaction forces we will need three compatibility conditions in addition to the three equilibrium conditions. We shall later show (Chapter 6) that a more general approach is usually adopted for complex systems such as the one shown in Fig. 2.2.c. It is based on the observation that if a system is linear in all its behaviour, then the reaction forces and inner force distribution are uniquely identified if the elastic deformation of all members is known. It should be noted that some systems are statically determinate with respect to reaction forces, but statically indeterminate with respect to their inner force distribution, see for instance the truss in Fig. 2.3, where external reaction forces \( A_{x} \), \( A_{z} \) and \( B_{z} \) may readily be determined, while the axial forces in the individual truss members cannot be determined by simple equilibrium conditions alone (see Chapter 2.3 below). Such systems are usually not recommended for practical design. The reason is that the weakest members may be subject to unwanted deformation induced constraint forces. In this chapter we focus on statically determinate systems.