Stability of equilibrium

( 1a)           U'   ≡   dU/dδ   =   0

The type of equilibrium which exists at any point on the path - stable, neutral or unstable - is also an important consideration; it may be deduced by taking the second derivative :-

( 1b)           U"   ≡   d2U/dδ2	>   0   ;     PE - minimum     stable equilibrium
	=   0   ;     PE - zero slope     neutral equilibrium
	<   0   ;     PE - maximum     unstable equilibrium

It will be appreciated from the sketches above that, compared to the elementary load building blocks, the plate- like components of thin- walled structures are complex in their deflections and in the mathematical descriptions thereof. We shall therefore not attempt to analyse these components - the interested reader is instead referred to the literature on plates and shells.
It is important however to appreciate the general physical behaviour of plate- like components and how this is governed by the energy principles embodied in equations ( 1). We shall therefore apply the principles to various mechanisms to illustrate how the response may be stable or unstable - and significantly how imperfections affect behaviour. The conclusions reached for mechanisms are directly applicable also to plate- like components.

Effect of imperfections

• Imperfections lead to deflections growing from the start of loading, and eliminate any sharp critical bifurcation point. Collapse is therefore not necessarily catastrophic as growing deflections warn of imminent failure.
• As deflections grow, the imperfect structure's behaviour tends to that of the ideal structure.
• Imperfections reduce the load carrying capability of a structure for a given deflection, and also the maximum capacity ( p* above ) when the secondary path is unstable.
• The effects of imperfections are much more pronounced when the post- buckling path is unstable - for example   α = 2 above.
• For structures with neutral or stable post- buckling paths, the main effect of imperfections is usually to introduce high bending stresses at an early stage of loading thus leading to failure through strength limitations rather than geometric collapse. This conclusion influences the treatment of imperfect columns below.
• Different forms of imperfection - for example out-of-straightness compared to load eccentricity - give rise to similar behaviour. Thus if the sole imperfection in the above example were the load eccentricity   e shown here, rather than an initial out- of- straightness, then the resulting performance is very similar to that deduced previously for out- of- straightness. The reader should confirm these results.
• An important corollary of this last conclusion is the impossibility of categorising imperfections solely from the observed behaviour of a structure which is nominally ideal. Indeed such a categorisation is unnecessary in most cases, as, when all said and done, it is the behaviour itself which is important.

Submerged pipelines

The minimum pressure which maintains buckle travel in a particular pipe is the critical   propagation pressure,   p_c for the pipe, and we turn now to the prediction of this   'buckling load.'

The buckled shape indicates gross plastic deformation - indeed elastic effects pale into insignificance and the pipe material assumed to be perfectly plastic, with yield strength   S_y. As a result of this non-conservatism, the energy methods of the previous sections cannot be used; however work-energy principles are applicable.

The adopted model consists of unit length of the thin pipe ( t/D --> 0 ) illustrated here, with four   plastic hinges - a, b, c and d. Deformation occurs only at the hinges, so the four lobes of the cross- section rotate about the hinges without otherwise deforming, thus forming the collapsed shape a', b', c', d'.

The material at a plastic hinge is taken to be perfectly plastic - ie. stresses can only be either zero or yield as shown at ( i) below. If pure bending occurs across the thickness   t then tensile yield extends across half the thickness ( ii) and compressive yield extends across the remaining half thickness. The resultant ( iii) of each stress block is a force   S_y t/2 ( per unit length). The hinge therefore develops a constant bending moment   M_y = S_y t² /4 per unit length as collapse proceeds ( iv).

The work-energy principle may now be applied to the propagation pressure collapsing the plastic hinged model.
Work done by external propagation pressure :
          =   pressure ∗ change in volume   ;     or, per unit length :
          =   p_c ∗ decrease in cross- sectional area   =   p_c ∗ square area a-b-c-d   =   p_c D² /2
Strain energy gain of the four plastic hinges, per unit length :
          =   4 M_y ∗ rotation of each hinge over the process   =   S_y t² ∗ π/2
Equating work and energy leads to an estimate of the propagation pressure :
          p_c   =   π ( t/D )² S_y

The model underestimates the propagation pressure mainly because it neglects strain- hardening and the finite extent of the plastic hinges - this last may be appreciated by comparing the actual buckled shape with the collapsed model. A more realistic empirical expression fitted to experimental results is shown; this tends to the model as   t/D --> 0.

The ability of this simple model (without empiricisms) to provide a ball park estimate of the collapse load should be appreciated.