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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? 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 overstressed, 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 overconstraint. The benefit is the spreading and controlling of loads and stresses. 

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