Rayleigh's Method

As was seen in Sec. 3.6, the stability condition for an equilibrium system is

(3.116)

for all admissible displacement field w( x), in which equality symbol implies the stability limit. The increment in potential energy ? ² V[ w] is the summation of internal strain energy ? ² U[ w] and the external work done ? ² W[ w, P]. Further the latter is usually proportional to the external force P

(3.117)

Thus the condition for the determination of stability limit load is

(3.118)

From which the critical load is obtained

(3.119)

If w _o( x) is the solution which gives the right hand side the minimum value then

(3.120)

and for any other admissible displacement function w ₁( x), which satisfies the kinematic or geometric boundary conditions,

(3.121)

This implies that an arbitrarily assumed displacement function w ₁( x) which satisfies the kinematic or geometric boundary conditions gives a load P ₁ which is always greater than the exact solution P _cr. It is clear from the above discussion that the closer the prescribed function w ₁ is to the exact one w _o, the closer the load P ₁ gets to the exact solution P _cr.

Considering the case of a simple column as shown in Fig. 3.15(a), and using the compressed but undeflected state as reference position...