Stability of Elastic Systems

the property of elastic systems whereby the systems return to a state of equilibrium after small deviations from that state. The concept of the stability of elastic systems is closely connected with the general concept of the stability of motion or equilibrium. Stability is a necessary condition for any engineering structure. Loss of stability may cause failure of either an individual structural element or the structure as a whole. Loss of stability under certain types of loading is characteristic of various flexible elements in a structure, such as bars (subject to bending) and plates and shells (subject to buckling).

Until the second half of the 19th century, the magnitude of the effective stresses was considered the sole criterion for the strength of engineering structures; that is, it was believed that if the stresses did not exceed some limit that was dependent on the mechanical properties of the material, the structure was not in danger. This was valid as long as the building materials used were rock, wood, pig iron, and the like, for which, by virtue of the low stresses allowed, cases of loss of stability were extremely rare. With the appearance of structures in which long bars in compression served as structural members, a number of accidents occurred that prompted a reconsideration of the long-standing consensus. It became evident that the accidents were due to inadequate stability of the bars in compression. For example, as a result of loss of stability caused by wind gusts, the Tacoma Narrows Bridge (USA) collapsed in 1940.

The behavior of a loaded elastic system when the system is deflected from a position of equilibrium by some small amount is a physical indication of the stability or instability of equilibrium. If a system deflected from an equilibrium position returns to the original position after the factor that caused the deflection is removed, the equilibrium is stable. If the deflection does not disappear but continues to increase, the equilibrium is unstable. The load under which stable equilibrium becomes unstable is called the critical load, and the state of the system is the critical state. Establishing critical states is the primary problem of the theory of stability of elastic systems.

For a straight bar compressed along its axis by a force P, the value of the critical force P_cr is determined by Euler’s formula P_cr = π²EI/(μl², where E is the modulus of elasticity of the material, l is the moment of inertia of the cross section, l is the length of the bar, and μ is a factor that depends on the conditions under which the ends are secured. In the case of two hinged supports one of which is stationary and the other is mobile, μ = 1.

For a rectangular plate compressed in one direction, the critical stress is equal to σ_cr = Kπ²D/b²h, where D = Eh³/12 (1 – v)² is known as the flexural rigidity, b and h are the width and thickness, respectively, of the plate, v is Poisson’s ratio for the material, and K is a factor dependent on the conditions under which the edges are secured and on the relation between the dimensions of the plate.

In the case of a circular cylindrical shell compressed along the generatrix, the upper critical stress can be calculated from the equation E(hlR), where h and R are the thickness and radius of curvature, respectively, of the median surface of the shell. The equations for the upper critical stress in the case of transverse pressure or twisting couples have a somewhat different structure. In many cases loss of stability in real shells occurs under a smaller load because of the significant influence of various factors, especially initial irregularities of form.

For complex structures an exact solution is difficult, and it is consequently necessary to resort to various approximate methods. In many cases the energy stability criterion is used. Under this criterion the nature of the variation in the potential energy ∏ of the system is considered when the system is deflected slightly from the equilibrium position (for stable equilibrium ∏ = min). When nonconservative systems are considered, such as a bar compressed by a force whose slope varies during buckling, the dynamic criterion is used, which consists in determining small oscillations of the loaded system. The study of the supercritical behavior of elastic systems is of great importance; it requires the solution of nonlinear boundary value problems. Supercritical deformation of a bar is possible only when the bar has very great flexibility. On the other hand, thin plates may exhibit significant bends in the supercritical stage, provided the edges of the plate are reinforced by rigid bars (stringers). Supercritical deformation of shells is usually followed by a concaving downward loss of the load-bearing capacity of the structure.

The data above apply to cases in which the loss of stability of an elastic system occurs within the elastic limits of the material. Investigation of the stability of elastic systems beyond the elastic limits involves the theory of plasticity. If a load that leads to loss of stability is dynamic, the inertial forces of structural elements that correspond to characteristic displacements must be taken into consideration. The faster the loading, the more pronounced the form of buckling. In the case of loading resulting from impacts, the wave processes of force transfer in a structure are investigated. If a structural material is in a state of creep, the relations of the theory of creep are used to determine the critical parameters.