Stability Analysis and Design of Structures

A structure is meant to withstand or resist loads with a small and definite deformation. In structural analysis problems, the aim is to determine a configuration of loaded system, which satisfies the conditions of equilibrium, compatibility and force-displacement relations of the material. For a structure to be satisfactory, it is necessary to examine whether the equilibrium configuration so determined is stable. In a practical sense, an equilibrium state of a structure or a system is said to be in a stable condition, if a disturbance due to accidental forces, shocks, vibrations, eccentricities, imperfections, inhomogeneities or irregularities do not cause the system to depart excessively from that state. The usual test is to impart a small disturbance to the existing state of the system, if the system returns back to its original undisturbed state when the cause of disturbance is removed, the system is said to be stable.

In this chapter, the basic principles required to analyze the structural stability problems are discussed. Emphasis is laid on energy methods. In the beginning of the chapter, the idealization of the structures, equilibrium equations and rigid body diagrams have been described. The subject matter on energy principles starts with the definition of mechanical work for external and internal forces of an elastic system and establishes relationship between the two.

This section deals with the class of structures consisting of rigid-body-assemblages wherein the elastic deformations are limited entirely to localized spring elements. In these systems, the rigid bodies are constrained by the support hinges so that only one type of displacement is possible. For the systems discussed here the formulation of the stability problem differs from the classical Euler formulation due to its basically discrete nature.

The classical critical load theory of perfect axial members assumes that the member in question is initially straight, slender, of solid cross section with flexural stiffness rigidity EI being constant throughout its length and subjected to an axial compressive force applied along the centroidal axis of the member. Moreover, it is presumed that the material of the member is homogeneous, isotropic and perfectly elastic. The assumption of small deflection theory of bending also holds good for the critical load theory.

In the previous chapters the stability of column, and beam-column was examined by treating them as independent or isolated members with appropriate boundary conditions. The simple frames have been treated as struts or beam-columns with elastically restrained ends wherein the effect of the connecting members has been modelled by end springs. However, in practice the columns, beams, and beam-columns are normally rigidly joined together to make skeletal structure called a frame in which the total structure is called upon to withstand the applied loads. In these rigid jointed frames, the end conditions of a member and hence its effective length depends upon the relative stiffness of the members meeting at the ends and that of member itself. Moreover, in a frame the deflection even in a single member due to buckling causes distortion in all the members. Thus, the response of the frame needs be examined in its totality wherein actual buckling of total frame is considered. In this chapter the stability analysis of the frames using classical differential equation method, semi-geometrical method, matrix method and modified moment distribution method etc. has been described.

Many flexural members are braced by other elements of the structures in such a manner that they are constrained to deflect only in the plane of applied transverse loads, e.g. slab-beam floor systems are extremely rigid in their own plane and the beams can deflect only in a plane perpendicular to the slab. The horizontal and rotational displacements are prevented by the floor system. On the other hand there are numerous instances where the members have no lateral support or bracings over their lengths and members can buckle in lateral direction under transverse loads. Similarly open column sections having only one or no axis of symmetry e. g. a channel section, and T-section or an angle section when subjected to axial compression; simultaneously undergo lateral displacement and rotation. This type of failure occurs because of low torsional rigidity of such sections. Further, in such sections, the critical load lies between the critical load for the torsional mode and that of pure flexural mode. A pure flexural mode exists when the centroidal axis coincides with shear centre axis. Therefore, a member subjected to an axial compressive force can also undergo lateral buckling.

In the preceding chapters, elastic buckling of structures composed of one-dimensional members has been discussed wherein deflections and bending moments are assumed to be the functions of a single independent variable. On the other hand, buckling of plates involves bending in two planes, and thus deflections and bending moments at a point become function of two independent variables. Consequently, the structural behaviour of plates is described by partial differential equations, whereas ordinary differential equations were adequate to describe the behaviour of columns. Further, the number of boundary conditions was four in the columns whereas in plates there are two boundary condition on each of its edges. Another basic difference between a column and a plate lies in their buckling behaviour. Once a column has buckled, it cannot resist any additional axial load i.e. critical load of a column is also its failure load. On the other hand, the plates which are invariably supported at edges or are interconnected to other plate elements continue to resist additional axial loads even after the loads reach their buckling values. This additional load is sometimes as high as 10–15 times the initial elastic buckling load. Thus, for a plate element the post-buckling load is much higher than the initial buckling load. This fact is largely exploited in the minimum weight design of the structures. The components of open section columns with wide flanges behave more like plate elements. The plates making up a column may undergo a form of local failure, thus necessitating the consideration of instability of plate element. In order to enhance buckling load of a plate sometimes longitudinal and transverse stiffeners are provided. The inherent discontinuities in these stiffened structures make their analysis complex.

Arches, rings and shells constitute a very important class of structures in themselves. An arch and a ring are usually considered to be the basic components of a more versatile shell structure. The classical stability analysis of these structures is cumbersome. In general they can be conveniently analysed by finite element method. For the cases where the structure axis follows the pressure curve, shear forces appear only at the stage of collapse and the solution can be obtained in a simple manner by using corresponding differential equations e. g. for a circular curve, this situation is realized for a uniform pressure normal to the axis i.e. radial pressure. The following analysis of a flat arch may serve as simple illustration.

In the elastic stability analysis discussed in the preceding chapters, the material of the structure is presumed to behave according to Hooke’s Law i.e. the stress in the structure does not exceed the initial yield stress in compression and the member undergoes configuration or shape failure. For many real structures the elastic analysis results in flexural buckling load estimation that exceeds the one associated with the yield stress or proportional limit stressof the material. This is especially true for the relatively short or stocky compression members in the framed structures. For this category of members the prorated design stresses based on safety factors, generally, fall in the range of plastic or inelastic behaviour. For steel framed structures many real designs occur in that range and most of the concrete framed structure columns are short. In these shorter columns the elastic limit is exceeded before the inception of buckling, and the modulus of elasticity E, hitherto constant, becomes a function of critical stress σ _cr= P _cr/A.

In a narrow sense, structural design is an art that is concerned with determination of minimum cross-sectional geometries of structural members based upon the results of structural analysis using acceptable performance criteria such as allowable stress, ultimate strength, maximum deformation, stiffness or stability etc. For the stress or strength and deformation the design procedures are straight forward which ensure the realization of a particular desired state for given loading. It is very rare when stability is the controlling condition; therefore objective is to have more than a predetermined reserve in capacity to preclude possibility of instability at the given loading. In the other words a margin of safety has to be provided. When buckling is a controlling factor, the problem can be handled by adding supporting or bracing members or a sufficiently large cross-section can be selected thereby eliminating buckling as a real problem. The choice will be governed by economy and practicality of the solution.