Some thoughts about the complexity of design
Hambly’s Paradox is an example which illustrates the difficulty of assessing so called hyperstatic structures i.e. those with redundant members. It was first posited by the eminent structural engineer Edmund Hambly, whose name it bears.
A milkmaid weighing 600 N sits on a three-legged stool. For what basic force should each leg of the stool be designed?
The answer to the question is of course 200 N. We are, of course, assuming the stool is symmetrical and that the milkmaid is sitting in the centre of the stool.
The milkmaid now sits on a square stool with four legs, one at each corner, and again the stool and the loading are symmetrical. For what basic force should each leg of the stool now be designed?
The obvious answer would be 150 N, however as Hambly points out that would not necessarily be the right answer.
If we assume that the milking stool is stiff and so is the floor on which it sits then inevitably the stool will rock and only three of the legs will be in contact with the floor. Even if raised by only a fraction of a millimetre the load in the fourth leg is unquestionably zero.
Since, the stool rocks there must be a point where both the fourth leg and the one diagonally opposite are both free of the ground. In such circumstances both legs carry zero load and the remaining two legs must therefore carry 300 N each.
If the stool is instead placed on rough ground the we have no way of knowing which of the legs will or will not be in contact with the ground and therefore every leg must be designed for 300 N.
This is, of course, the paradox. By increasing the number of legs supporting the stool the design load for each has effectively doubled.
The conclusion that we have reached is based on static analysis i.e. our only concern has been to ensure that the stool remains in equilibrium. Such analysis contains a prior assumption the consequences of which have gone unnoticed until now. That is the assumption that we are dealing with rigid bodies.
This is not how the real world works. All materials deform when subjected to load. The amount of deformation depends on their stiffness. We could consider the stiffness of the milking stool seat and its legs in our assessment, but it is probably sufficient, and somewhat simpler, to consider only the legs.
When the legs are loaded they will shorten a little. The amount they will shorten is proportional to the stiffness of the material they are made of. If the gap between the so called fourth leg and the ground is small it may, as a result of shortening, become in contact with the ground causing the weight of the milkmaid to be redistributed according to the stiffness of each leg. Therefore, depending on the size of the gap the fourth leg may experience a load anywhere between 0 and 300 N.
It follows that thus far geometric imperfections make it almost impossible to predict the actual load in a 4-legged milking stool, even after the elastic properties of each leg has been taken into account.
There is of course one further step we can take, which will restore common sense to milking stool design, but first we must make a further assumption. If we make the milking stool legs relatively stocky then they can reach a state of plasticity [1] without their legs buckling.
In this scenario let us suppose that each leg of the milking stool has been designed to yield at a load P. Let us then suppose that the weight of our milkmaid is 4P. At the outset of our problem the stool is potentially supported on only two legs this means that those two legs are momentarily exposed to a load of 2P each and 4P in total. Each leg shortens elastically until it reaches its yield point whereupon it continues to squash until load is shed into the two adjacent legs. Since the yield point is P load will continue to shed until it is shared equally between all 4 legs. Thus, each leg of the milking stool may be designed for 150 N.
And just like that our paradox vanishes!
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