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Static indeterminacy

If it works, don't fix it.

If you provide two load paths, they must share the load, not fight for it.

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Suppose that you think that a structure is too weak. One solution is to add more parts at the presumed weak place. What is the purpose of the gussets in the objects shown below?

lamppostbaseea.jpg (53032 bytes) weldss.jpg (21399 bytes)

The left hand diagram 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.

Example of bridge failure - Hoan bridge

If there is a cyclic variation in stress, caused by pressure or temperature, the newly added stresses may cause failure by fatigue to happen earlier than it would have without the modifications. Worse still is the case where the stresses alternative in polarity, for example from compression to tension, or shears in opposite directions. You can test this by repeatedly bending a paper clip.

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

For another example, imagine that four people have been asked to work together to hold a heavy object. If they all have different ideas about where the object should be, they will waste force in opposing each other, and they may distort the object if it is not very rigid.

Galileo Galilei gives a simple example of someone apparently improving a piece of work. He tells of some workers who support a marble column on two wooden trestles until the building is ready for its installation. See picture below.

Later, another worker sees the column, and he worries that it might crack because of its weight. (This picture differs from the first, but the worker does not know know that. And he doesn't know the saying "If it works, don't fix it.")

So he makes a third support and he places it under the column to take some of the weight.

Some time later, when workers arrive to move the column, they find it broken.

Did you find the difference between the first two pictures? And why did it break?

The worker who added the extra piece should have realised that the column would already have broken if it was too weak, as stone does not creep. On the other hand, an organization in which people never look for and report possible errors or dangers is an organization which could be in some trouble. The electronic analogy is negative feedback. An extreme example of the opposite view is the death sentence for questioning a British naval captain's decision (even with good intentions) in older times.

A Problem

The diagrams above represent a simple beam bridge, which has been affected by subsidence (exaggerated). One response is for the beam  to remain so straight that it is only supported in two places, leading to a bigger effective span. Another is for it to bend. A third would be to break, if either of the first two conditions were unsustainable by the structure.

In practice the designers might include jacks at the base of the piers, to allow for adjustment.

What happens as a result of the movement is that the beam suffers stresses which were not in the design.  In fact the problem exists from the start.

The four support points can never be perfectly aligned, but the alignment is of course made so small that the beam can adjust its shape without absorbing too much energy.

The penalty for a through beam is the over-constraint. The benefit is the spreading and controlling of loads and stresses.

The diagram above represents the response of a simple cantilever bridge to subsidence. In this case the joints allow stress-free movement, so nothing is distorted.

Jacking might still be provided. In a very slender foot-bridge, the slightest error could be noticeable, and so some adjustment may be needed.

After a bridge has been completed, jacks may be concreted over, or they may be left as usable adjusters. The Eiffel tower is a good example of the jacking requirement.  The stresses, and therefore the strains, at the base, changed markedly during construction. Jacking enabled the builders to compensate as the work progressed.

Queensborough Bridge

QueensBoroughB.jpg (75673 bytes)This bridge, one of a vast number in New York City, is very unusual. It has two unequal main spans which comprise cantilevers joined without a suspended span, making the bridge indeterminate. A central anchor span provides anchorage for two two inner cantilevers Do you think that joined cantilevers have any advantages over continuous beams? Over conventional cantilevers with suspended spans? Are there disadvantages? The Middlesbrough transporter bridge is another with no suspended central span.

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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?

In  the third diagram above, is the missing piece needed at all? Think of two cases.  In the first case, the truss is hinged at both supports. In the second case, it is hinged at the left side, but only rests on the right, so that it can slide along when it expands or contracts because of temperature changes.

Now what do you think? In the first case, what does removing the member do to the forces at the supports?  

Answers at bottom of page.

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Sabrina5.jpg (32439 bytes)The deck of the Sabrina bridge at Worcester is built in sections, so that the triangles formed by these and the cables are independent. Therefore any change in the length of one cable has minimal effects on the others.

SSCTiltSmall.jpg (39552 bytes)Most cable-stayed bridges, however, are highly indeterminate because the numerous triangles include common sides - tower and deck. This example, using a compressed picture, shows two cables (the fifth on each side) that appear to be under less tension than the rest. It is essential in most cases to introduce a means of measuring the tension in each cable during construction, and a means of adjusting it. You can see that adding the next section of the deck is likely to alter the tension in at least the previous cable, and so it might be necessary to adjust more than one cable after each placement of a section, unless the tensions are set up to allow for the changes in advance. The procedure of repeating setups is an example of an iterative process. Such a process if likely to be needed whenever there are many elements that interact with one another. A large antenna array in a field is another example where repeated changes might be needed. Early colour TV sets had as many as thirty adjustment potentiometers on the back, which had to be set up in a strict order, to reduce the number of iterations. Accuracy in manufacture has improved to the point where these are no longer needed.

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 shows 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.

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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.

In an over-determined structure, what happens if you remove an old or damaged member? If it was carrying some forces, all the other members will adjust when it is removed, as when one segment of a spider-web is cut. When a new part is put in, will the stresses return to their original state? Not if the new part is made to fit the new situation. Successive replacements might produce a random walk of stress variations in a structure. But will the changes really be random? Might they not systematically move in one direction? Or might they move until certain members consistently had no further effect?

TewkesWeirFB.jpg (57011 bytes)Is this bridge indeterminate if the cross members go right through the little square gusset plates? What if they stop at the corners of the squares? In practice, few bridges consist entirely of short pieces made into triangles. The main top and bottom members usually continue through many panels. Note the transverse beams which extend outside the trusses to provide a means if triangulating in the third dimension.

A question

When you read something that is supposed to be factual, try working out all the consequences of the statements, in other words ask yourself what follows from the given statements. If even one logical consequence of the original statement is wrong, then the original statement must be wrong. We must, however, be careful to note carefully any stated limitations. Many "laws" of physics are subject to limitations. Forgetting them, and using the laws outside their area of competence can lead to problems, or even disasters.

We have seen that having two or more parts sharing a load is not necessarily a good thing. But any part such as a strut or a tie has thickness, and can be considered as two halves that share the load. How do we know that the sharing is perfect?  We don't. If the part is made from a perfectly uniform material, and if the manufacture induces no changes in the material, then all should be well. But manufacturing processes such as bending, forging, rolling, and welding can create residual stresses. These can result in stress distributions under load that are not the ones that were calculated.

A Puzzle

Tessellation.gif (3072 bytes)Here is a puzzle. The picture shows a hexagonal array of equilateral triangles. We see that with all twelve connections present, the construction is over-constrained in the plane. Without one of the connections, the shape is rigid in the plane without redundancy. What if we want to join up a large number of nodes on a triangular grid on a plane surface to make a rigid frame. Which lines need to be removed so that it is only just rigid? What fraction of the total are they if we choose the most efficient selection? From this we see that many space frames are greatly indeterminate.

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.

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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. In interpreting this type of picture we need to consider the possiblity that the colours maybe influenced by the thickness of the material.

Strains may be left in components after welding, rolling, injection moulding, extrusion, and other forming processes. The outwardly simple RSJ (rolled steel joist) may conceal unrelieved stresses.

SetSquare3.jpg (37831 bytes)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.

CDBox.jpg (36092 bytes)This 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.

PolaBox.jpg (23930 bytes)Here is another plastic box. As it shows closer fringes we can conclude that the strains are higher than in the CD box.

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

 

 

Aurora Borealis?

Warners.jpg (27733 bytes)Not an aurora, just the windscreen of a Peugeot 106, viewed between polarizers. The toughened glass has been deliberately made with internal stresses which in fact make it stronger.

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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 from the outside.

What you can do is to heat an object so slowly that the temperature differences, and therefore the differential expansion, are very small, and therefore does not create too much stress.

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

JapanC2.jpg (61603 bytes)   Cracks1.jpg (51959 bytes)

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

FrogCrack2.jpg (47887 bytes) FrogCracks1.jpg (41434 bytes)

JapanC3.jpg (124184 bytes) JapanC4.jpg (102944 bytes)

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FWCrack.jpg (82049 bytes)Many structures are in fact built in ways which include redundancy. Masonry, for example, is often much thicker than it needs to be, as we see from the standing walls after a bombing raid. The truth is that strength isn't everything: structures have to be rigid. A house that was only strong enough to hold you and your belongings would be unacceptably flexible. The penalty for the rigidity is that if the ground should move differentially, cracks may appear in the structure, as in the example at left. One solution is to make the foundations so rigid that the effects of the ground movements are small.  

Cracks are not necessarily disastrous - many a building survives perfectly well with sizable cracks. Indeed the ancient Romans built some superb structures with maximal cracking - they included no mortar. A reinforced concrete bridge designed by Robert Maillart developed cracks. He didn't strengthen his next bridge: he realised that the bridge was working perfectly well with the cracks and he responded by removing material in the area which had cracked in the previous bridge, producing a design of which derivatives are still being built today.

Several possibilities exist for masonry - the stone and mortar can be of equal strength, or one can be stronger than the other. The bond between stone and mortar can also be stronger or weaker than the internal bonds in the stone and mortar. Does any of this matter? If a glue is advertised as being "stronger than the wood", is this always a good thing?

 Click here to skip the next part - a bit of electronics

Suppose that you want to build an electronic circuit, and you find that no device will pass the current that you need. You can use two or more devices to provide parallel paths. There is a problem - the accuracy of manufacture may not be sufficient to allow this practice. Let's take a simple example - putting two batteries or power supplies in parallel. The problems become apparent when we look at the behaviour of an actual system.  

Suppose that because of manufacturing tolerances we have power supplies of 4.9 V and 5.1 V, and we connect them in parallel. Even before we connect an external circuit, we have created a circuit with a driving voltage of 0.2 V. If the internal resistance of each supply is 0.1 ohm, we will have a circulating current of 1 amp. If we now try to draw 2 amps from the system, we shall clearly get more than 2 amps from one supply, and less than two amps from the other.

Or we could have two batteries of exactly the same voltage, but with different resistances, perhaps because of different usage histories. We can illustrate this using a diagram. If you put two cells in series, you get the same current in each, but different voltages. It is even possible in an extreme case for one cell to have its voltage reversed. What is the condition for this? Here is a diagram for two batteries in parallel.

The currents C1 and C2 differ significantly.

Semiconductor components are not easy to match exactly. The next graph shows the result for two zener diodes in parallel.

We can summarize all this with a couple of diagrams. The first one, below, is for a voltage source, which for a limited range of current values, gives a roughly constant voltage. Using mechanical terminology, we could say that it has a high "Ohm's modulus" (cf Young's modulus), that is, it is "stiff" in voltage. Two of these in parallel will probably pass very different currents. We don't actually say Ohm's modulus, we say dynamic resistance, dV/dI. It is common that different fields of activity use different words to express the same type of idea. There is even a case where the same words are used with different pronunciation, as with "sherds".

ZenerShape.gif (2566 bytes)

The second is the opposite: it is stiff in current, ie a current source.  Two of these in series will mean that the devices will probably have very different voltages across them. They have very high dynamic conductances, dI/dV.

ZCurrentSourceShape.gif (2572 bytes)

A good way to make a very constant voltage would be to supply a constant voltage device from a constant current source, and vice versa.

A pretty constant voltage can be provided by a voltaic cell, and a fairly constant current can be provided by a barratter, a specialist filament lamp. Both methods have been superseded by accurate electronic circuits.

A similar result would happen if you wanted to pull or lift something using two ropes. You could tie them together at each end, but you would never get the lengths exactly equal. The longer rope would feel no tension at all until the shorter one had stretched to equal its length. And even after that, the shorter rope would bear most of the force. Most people would, of course, not do it this way: they would  make the two ropes into a loop, and allow the two sides to find their own equality by threading a hook or a karabiner or a pulley through the loop. Other arrangements can be used with rigid ties to allow load sharing.

The provision of constant length is much easier than the provision of constant force. Constant force can be provided by means of weights and pulleys, but this is often inconvenient or impractical.

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Click here to skip the next part - a bit of maths

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.

The method of making too many measurements and then making a least-squares fit or a chi-square fit offers benefits over the minimalist method. Firstly, averaging many results is better than using just one result, provided of course that the many results are consistent. That leads to the second benefit - the extra data enable an estimate of the range of measuring errors to be calculated, and if the expected range of errors can be calculated from a knowledge of the equipment, the consistency of the results can be tested.

Surveyors of land always make more measurements than the bare minimum, enabling the accuracy and consistency of the triangulation to be checked, and more accurate results to be obtained. Anyone who uses a GPS receiver knows that more satellite connections will provide a more accurate position.

We must never forget, when measuring, that no amount of statistical analysis is going to reveal systematic errors, unless we compare our results with those made by other techniques. An examination of published scientific measurements appears to show that many people have underestimated the magnitudes of the uncertainties in their measurements.

Example of bridge failure - Hoan bridge

Static indeterminacy two

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Answers to Questions

Truss with missing parts

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 large outward thrust resulting from the implied flat triangle which has F as its apex.

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.