ABSTRACT
Summation formulae are used in the theory of elastic stability so that approximate estimates of the critical load factors of a complex problem are obtained by combining the load factors of subproblems in different ways. If the critical load factors are directly added, then the formula is called a Southwell type formula. If the reciprocals of the critical load factors are added, then the formula is called a Dunkerley type formula. First we consider examples of the Southwell and Dunkerley formulae. Example 2-1: Consider the torsional buckling of a thin-walled bar of height H with the lower end built-in and the upper end free, subjected to a concentrated vertical force P at the free end. The equilibrium of the deformed bar can be described by the differential equation
with the boundary conditions
• • " - ~ f ' ' Here $ is the rotation of the cross section, ip is the radius of gyration, and prime denotes differentiation along the axis of the bar. The torsional stiffness of the cross section is composed from two parts: the warping stiffness EIu, and the Saint-Venant torsional stiffness Git. Let us consider first the case where the bar has warping stiffness only (Git = 0). In this case the critical load is [TlMOSHENKO and GERE, 1961]:
Let us consider now the case where the bar has Saint-Venant torsional stiffness only (EIo) = 0). In this case the critical load is
The Southwell summation approximates the critical load of the original bar by the expression
The exact value of PCr is given by TiMOSHENKO and GERE [1961]:
It can be seen that in this example PS = PCI, that is, the Southwell formula (2-3) provides the exact value of the critical load. Example 2-2: Fig. 2-1 shows a simply supported thin-walled bar of doubly symmetric (Ix > Iy) open cross section, subjected to an axial force N and a couple M whose plane is perpendicular to the x axis. We want to determine the values of the N,M pairs by the Dunkerley formula under which the bar comes into the critical state. Consider flrst the case where the bar is subject to the force N only (M = 0). Suppose that the critical load for torsional buckling is greater than the critical load for flexural buckling about the y axis. In this case the bar buckles in the weak direction, that is, about the y axis, and the Euler critical load is
Let us consider now the case where the bar is subjected to the couple M only (TV = 0). The critical value of the couple can be obtained from TiMOSHENKO and GERE [1961]:
Since N and M are of different dimension, the Dunkerley formula should be written in dimensionless form as follows:
where M and Mcr should be considered with the same sign. To assess the accuracy of the Dunkerley formula (2-4), we have to determine the exact relationship between the critical pah-N, M by solving the set of differential equations
with the boundary conditions u(0) =
M(0 = «"(0) = «"(/) = *(0) = <?(/) = <P"(0) = $"(/) = 0 where u is the displacement of the centroid of the cross section in the x direction. If Mg
denotes the critical load of the bar for pure torsional buckling then the desired expression has the form
[TIMOSHENKO and GERE, 1961] where
Let N < Ny^i < N,pcl < oo, then in the co-ordinate system M/Mcr, N/NyfT, Eq. (2-4) represents a straight line and Eq. (2-5) represents a hyperbola, only one of whose branches should be considered (Fig. 2-lb). In the interval 0 < M/MCT < 1, points of the Dunkerley line (2-4) lie below the curve (2-5) inside the stability domain.
