Stability of equilibrium



As noted, the buckling propensities of a system are reflected in the shape of the equilibrium path, ie. the load- deflection curve. If a system is in equilibrium then its total potential energy,   U, has a turning value since an infinitessimal disturbance,   dδ, of the system from its equilibrium position does not change the potential energy (PE) -   δ is any convenient characterising displacement. Expressed mathematically, the equilibrium path is thus defined by :

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.



We consider first a typical perfect mechanism which behaves in a stable, neutral or unstable manner depending on the mechanism's geometry.



Effect of imperfections



Perfect structures are figments of the imagination. If the mechanism above were manufactured, it would be impossible to ensure that initially A, B, and C all lay on the load's line of action, or that both springs were completely free before the load was applied.



Small imperfections can have a marked effect on buckling behaviour
as this example demonstrates.


These examples, though dealing with one particular mechanism, confirm the following conclusions which are quite general for imperfect buckle- prone structures :

pressure driven subsea pipe collapse

Submerged pipelines



With the gradual depletion of easily exploitable gas and oil fields, less accessible sub-sea reserves are being increasingly tapped. The distribution pipework of these often lie at depths of around 200 m and is thus subjected to substantial external pressures. It is imperative 

collapse stages of a subsea tube 

that pipe buckling is prevented as it propagates catastrophically, being driven by a longitudinal component of the water pressure at the speed of sound in the metal wall. Cross- sections taken from various positions along a partially collapsed pipe are shown on the left here.

The minimum pressure which maintains buckle travel in a particular pipe is the critical   propagation pressure,   pc 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   Sy. As a result of this non-conservatism, the energy methods of the previous sections cannot be used; however work-energy principles are applicable. model of subsea pipe collapse

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 plastic hinge component of model 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   Sy t/2 ( per unit length). The hinge therefore develops a constant bending moment   My = Sy 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 :
          =   pc ∗ decrease in cross- sectional area   =   pc ∗ square area a-b-c-d   =   pc D2 /2
Strain energy gain of the four plastic hinges, per unit length :
          =   4 My ∗ rotation of each hinge over the process   =   Sy t2 ∗ π/2
Equating work and energy leads to an estimate of the propagation pressure :
          pc   =   π ( t/D )2 Sy
stability characteristic of subsea pipe collapse
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.


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