Indeterminacy

The left hand diagram above shows a tube welded to a flange. We might decide that the attachment is too weak, and so we might weld four brackets to the tube and flange, as in the right hand diagram. What could be wrong with that?

If the system were indeed already over-stressed, we may have made it worse. We might have done the welding badly, so that thermal stresses were not allowed to anneal out. We might have shaped the triangles badly, and forced a distortion when adding them.

These new stresses might nullify completely any possible gain from the new parts. In effect, we may have made the system weaker.

If there is a cyclic stress, from pressure or temperature, the newly added stresses may cause failure by fatigue to happen earlier than it would have without the modifications.

This example is not very realistic, as the tube is thin and the gussets are thick.

When we see an apparently complicated network of struts, can we work out why each one is there, and why no others are there? Two different questions can be asked. Firstly, what is the simplest arrangement that will make the construction rigid? Secondly, which members are in tension and which in compression?

Looking at the second question, we could ask, thirdly, does each member remain always in tension or always in compression as a live load moves across the bridge?

Let's see if we can answer these questions.

If we want to find out which members are in tension and which in compression, we can often get somewhere by imagining them made of something like stiff rubber, that can change visibly when stressed. A member in tension will tend to stretch. A member in compression will tend to shrink.

Or we can imagine a normal rigid bridge, and then imagine removing one member. By working out what would happen we can see whether the member was in tension or in compression.

This has been done in two of the diagrams below. What can you tell about the missing members?

An example of an over-determined structure would be a pair of cantilevers joined together without the usual suspended span. If the ends do not match, one will be pulled up and one will be pulled down. If nobody records the shift and calculates the result, the stresses in each half are unknown, though the average of the two halves would be as per the design.

The next set of diagrams show simple frameworks and the effects of adding extra struts. The diagram at top right is only there to provide a labelling scheme for the joints.

In the top two diagrams there are only enough parts to make the system rigid. This rigidity does not rely on any stiffness at the joints, as long as the bolts fit the holes. But in the third diagram the extra strut BC has been added. What effect does it have?

If all the parts were perfectly made, the whole thing would fit well without problems. But nothing can be perfect. There must be variations in manufacture, however small. There are two simple cases.

Firstly, the slackness of the bolts in the holes exceeds the tolerances in the distances between the holes in each strut. In such a case the framework can be loosely assembled and then the bolts can be tightened.

Secondly, the bolts fit the holes better than the tolerance in length. In this case, adding the sixth strut BC could be impossible without some changes to the lengths. In other words, the struts are strained. Once the framework has been assembled, it will end up in a configuration of minimum energy.

There are two conditions for a cross-braced quadrilateral. Either both diagonals are too short, or both are too long.

The strain energy in each strut is equal to the square of the strain, multiplied by a constant which is particular to each bar. For identical bars, the constant will be the same. So minimum energy is related to the sum of weighted squares of the strains.

If the parts of a structure are so thick that the forces may not be purely longitudinal, it may be impossible to know the internal stresses. This can arise if the treatment has set up residual stresses, as in the case of a block of hot glass that is suddenly cooled.

Why does any of this matter? It matters because the relationship between stress and strain in most structural materials is such that quite small strains can result in stresses which are comparable with the maximum safe stress.

Thermal expansion is often large enough to threaten structures. It is for this reason that railways need gaps between the sections of rail, unless the rails are held firmly in place. Bridges, too, may need provision for expansion. A beam may be fixed at one end, and on rollers at other supports. Curved structures such as suspension cables and arches can deflect, relieving the stress, but the flat deck may still need preventative measures.

Most structural materials do not allow inspection of the interior to measure the stresses and strains, except possibly by methods such as neutron diffraction. Strain of the surfaces can be measured using strain gauges, which are small devices which change their electrical properties such as resistance, when strained.

Using a transparent model, it is sometimes possible to evaluate the average strain along a line by using polarised light. If the molecules are asymmetrical, they may be used to rotate the plane of polarization of light. If the molecules are not asymmetrical, they may become so when the material is strained.

If the specimen is placed between crossed polarizers, virtually no light will get through an unstrained specimen. Strains will show up as transparent regions, which are usually coloured. Because light has wave structure, a graph of brightness against strain is not linear. It oscillates. If enough strain can be applied, coloured bands may be seen, indicating contours of equal strain.

By placing a piece of transparent material between crossed polarizers we can sometimes detect variations in thickness, variations in strain, variations in composition, or variations in orientation of crystal axes. The picture below shows a perfectly good set-square, the function of which is not at all affected by the internal strains revealed by the polarised light.

Strains may be left in components after welding, rolling, injection moulding, extrusion, and other forming processes.

Here is a picture of a part of an adjustable set-square. It does not produce any strong colours, but there are clearly indicated strains around the hole, though not much seems to be happening elsewhere, except at the two rivet holes.

The next picture shows a CD box. Three other boxes produced very similar pictures. The reproduction process does not do justice to the colours, which have been subjected to tri-colour photography, scanning, JPEG, and display on a screen. In any case, the complexity that can be hidden in a simple object is obvious. These strains are quite large. Adding stresses using a vice, a piece of ice, and a soldering iron made very little difference.

The pictures below show a strip of perspex, with and without strain.and without strain. The neutral axis is visible as a darker band in the second picture.

Returning to thermal expansion, it should be made clear that expansion is not in itself a problem. If an isolated piece of metal gets uniformly hotter it expands, but experiences no internal stress. But of course you cannot actually make something uniformly hotter, because you need a thermal gradient in order for the heat to flow into the object.

What you can do is to heat the object so slowly that the differential expansion does not create too much stress.

The cracking of thick glass is not caused by temperature: it is caused by temperature differences in the glass.

These cracks in the glaze of a pot are deliberate. They are caused by differential contraction between pot and glaze. Here are some more.

The behaviour of mechanical parts has a parallel in mathematics. If we imagine a long continuous parallel beam on a number of supports, we can see that if they are not level, there will be extra stresses in the beam. The parallel in maths is that if we make a number of measurements, and we form the mean of the numbers, each individual measurement is seen to be slightly different from the mean.

All these differences correspond loosely to the induced stresses. They arise because we have more measurements than are strictly needed to define the result. Having too many measurements brings two benefits. Firstly, a more accurate result is obtained, because random errors tend to cancel. Secondly, by subtracting the mean from each measurement, we can find out the distribution and range of the errors.

This process can be extended to the case in which there are numerous unknown variables, and numerous equations relating the measurements. If the number of equations is equal to the number of unknowns then we have just enough information. But if there are too many equations, then no set of values will fit them all. We have to find a set that gives the best fit.

For this we can use a simple least-squares fit, minimising the sum of the squares of the differences between the fitted values and the actual values. Better still, we can weight the value according to the expected measurement errors, which may not be the same for all the values. As with the averages, the differences between measured and calculated values represent a "strain".

In the case of a beam, the sum of the squares of the strains is related to the stored energy.

The diagram below shows a table of data that fall roughly on a straight line. If we have reason to believe that that they represent linear behaviour, and that the deviations are purely caused by measuring error, we can calculate a best fit using the least squares method.

The third column shows the differences between the data and the values generated by a linear formula. The fourth column shows the squares of these differences. The least squares method consists in changing the position and the slope of the black line until the sum of the squares is as small as possible. If it is known that the measuring inaccuracy varies along the line, this can be taken into account in making the addition, by weighting the deviations according to the reciprocal of the inaccuracy.

AB missing - A B and C will move down. AB gets shorter, so AB was in compression. The structure might break at C as it falls.

DE missing - supports both hinged. As the supports form a triangle with F, in theory the bridge could survive, if the supports can withstand the outward thrust resulting from the implied flat triangle.

DE missing - one support sliding. D E and F can all fall, and DE will get bigger. So part DE must have been in tension.