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Structural Stability

1.1 Introduction

Structures fail mainly either due to material failure or because of buckling or structural instability. Material failures are governed by the material strength that may be the ultimate strength or the yield point strength of the material. The failure due to structural instability depends on the structural geometry, size, and its stiffness. It does not depend on the strength of the material. It is important to understand the failure due to structural instability, because using a higher strength material will not prevent this type of failure. More and more structures are failing because of stability problems because of the present trend to use high strength materials and large structures. The increase in size increases the slenderness ratio of the members of a structure, and these members reach their stability limit before their material strength. A look at different design codes makes it clear that in many situations the maximum force a system can support is governed by structural instability than by material strength.

An interesting question to ask is, if the material strength is not exceeded, then why does the member fail?. The answer may be that all systems take the path of least resistance when they deform, a basic law of nature. For slender members, it is easier to bend than to shorten under a compressive force resulting in the buckling of the member before it fails by exceeding its material strength. For short members it is easier to shorten than to bend under a compressive force. In practice, there is always a tendency of a slender member to bend sideways even if the intended force is an axial compression. This tendency is due to small accidental eccentricity, unintended lateral disturbing force, imperfections, or other irregularities in the member. For small compressive forces the internal resistance of a member to bending exceeds external action forcing it to bend. As the external forces increase, a limiting load is reached where their overturning effect to bend exceeds the internal resistance to bending of the member. As a result, more and more bending of the system called buckling occurs. The maximum compressive force at which the member can remain in equilibrium in the straight configuration without bending is called the buckling load. A system is called stable if small disturbances cause small deformations of the system configuration. Displaced shape equilibrium and the energy methods are the two most commonly used procedures to solve the buckling loads problem and to study the stability of equilibrium.

Figure 1.1 Types of equilibrium: (a) stable; (b) unstable; (c) neutral.

1.2 General Concepts

Concepts of stability can be explained by considering the equilibrium of a ball resting on three different surfaces [1] shown in Figure 1.1. The ball on the concave surface in Figure 1.1a is in stable equilibrium because any small displacement will increase the potential energy of the ball. The component of the self-weight parallel to the sliding surface will bring the ball back to its original equilibrium position. In Figure 1.1b, the ball rests on a convex surface, a small displacement from its equilibrium position will decrease the potential energy of the ball. The parallel component of the self-weight will slide the ball further from its initial configuration, and the equilibrium is unstable. If the ball is displaced on the flat surface, the potential energy of the ball remains the same, and the ball assumes a new equilibrium position. Thus, potential energy, ?, is a minimum for stable equilibrium, whereas it is a maximum for the unstable equilibrium position, and the potential energy remains the same for the position of neutral equilibrium. Energy methods are based on these concepts for solving the structural stability problems. If ???>?0, the displaced configuration is stable, whereas for ???<?0, the displaced shape is in unstable equilibrium, the transition ?? = 0, which is the position of neutral equilibrium gives critical load at which the system becomes unstable by energy method.

Also, since we are studying the state of equilibrium in the slightly displaced position of the body, the equilibrium equations are written based on the displaced shape of the body in the displaced shape equilibrium method. Both methods can be used to formulate the equilibrium equations and calculate the critical loads. However, the displaced equilibrium approach does not give the nature of equilibrium when the critical load is reached. To answer that question, the second variation of potential energy d2? is to be considered. The potential energy may be expanded into a Taylor series about the equilibrium state and written as

where

d?, d2?, and d3? are called the first, second and third derivatives respectively of the potential energy ?. The critical load Pcr is obtained from the conditions of equilibrium given by d? = 0 for any dqi, or for each i [2]. The equilibrium state is stable if ???>?0. Therefore, the equilibrium state is stable for d2??>?0, and is unstable for d2??<?0.

Because energy is quadratic, it can also be written as

where

For elastic structures, matrix K represents the stiffness matrix of the structure with regard to its generalized displacements, and ? is the potential energy. The stiffness elements are given by

That shows the stiffness matrix is symmetric. The second variation of the potential energy from Eq. (1.1c) is

For d2??>?0, the matrix with elements Kij will be positive definite. A real symmetric matrix is positive definite if and only if all its principal minors are positive, that is,

or

When systems are subjected to compressive forces three types of instabilities can occur: (i) bifurcation of equilibrium; (ii) maximum or limit load instabilities; and (iii) Finite disturbance instability.

1.2.1 Bifurcation of Equilibrium

Equilibrium paths are shown as load displacement plots in Figure 1.2. The equilibrium path starting from the unloaded configuration is called the fundamental or primary path. At a certain load the equilibrium path can continue to be the fundamental path or it could change to an alternate configuration if there is a small lateral perturbation. This alternate path is called the secondary or post-buckling path [3]. The point of intersection between the primary and secondary paths is called the point of bifurcation, and the load corresponding to this point is called the critical load. In Figures 1.2a and 1.2b, the secondary paths are symmetrical. In the symmetric bifurcation the post-buckling load deflection behavior remains the same irrespective of the direction in which the structure bends. It is a stable bifurcation in Figure 1.2a because the load increases with deflection after buckling, axially loaded columns and thin plates subjected to in-plane forces exhibit this behavior. The load decreases below the critical as the deflection increases in the post-buckling stage in Figure 1.2b, and the structure has an unstable bifurcation at the critical load. Guyed towers exhibit this behavior because some of the cables come under compression and are unable to sustain the external forces. If the post-buckling load deflection diagram is affected by the direction of buckling, then the bifurcation is asymmetric as shown in Figure 1.2c. Some framed structures show this kind of behavior.

Figure 1.2 Bifurcation equilibrium paths: (a) Symmetric stable bifurcation; (b) Symmetric unstable bifurcation; (c) Asymmetric bifurcation.

1.2.2 Limit Load Instability

This type of instability is also called snap-through buckling. In this type of buckling, the primary path is nonlinear and once the load reaches a maximum, the point P in Figure 1.3a jumps to Q on another branch of the curve. The load at point P is the critical load in this type of instability. The structure snaps through to a nonadjacent equilibrium position represented by point Q. Spherical caps and shallow arches exhibit this behavior.

1.2.3 Finite Disturbance Instability

This type of instability occurs in cylindrical shells under the action of axial forces shown in Figure 1.3b. The load capacity of the structure drops suddenly at the critical load in Figure 1.3c. The structure takes a non-cylindrical shape after the critical load. The structure continues to take more axial compression in Figure 1.3c after taking another equilibrium configuration. In this type of instability, a finite disturbance of the cylinder or imperfection in the cylinder will lower the critical load considerably and...