==

Strains and Stresses

Back to Bridge Basics  Back to Bridges  Back to Home Page | Find Us on Facebook

Translation  -   Italiano

===

 Stresses in a beam A simple beam, supported at the ends, is a little more complicated than it seems. The top diagram shows lines representing the directions of the compressive and tensile forces. The closeness of the lines represent the magnitude of the stress, in the same way as electric and magnetic lines of force, and their direction shows the directions of the stresses. The stresses are of course not confined to the lines, but are spread through the volume as in the other diagrams, where red represents compression, blue represents tension, and purple and yellow represent shear of opposite polarities. What can you deduce about the relationships between compression, tension and shear? And can you relate the line diagram to the diagrams in the previous panel?           We can see that the stresses are rather localised in a simple beam, though the fading of the colours is a little deceptive. The distribution of energy is even more localised, because the energy is proportional to the product of stress and strain. Where the stress and strain are a half of their maximum values, the energy is a quarter of its maximum value. The simple beam, whether as a solid block or as plate girders is not the most efficient way of using the material. Plate girders almost always have concentrations of material at the top and bottom, as the red and blue picture suggests, and some even have thickening towards the middle of the span, again suggested by the diagram. The next two pictures show I-beams: the second has the thickening towards the middle.    Real engineers would not use coloured diagrams like those shown earlier - they would use something easier to interpret mathematically. Lastly, here are two pictures that show clearly the deflection of a span of the Royal Albert bridge at Saltash when loaded by a train. There is no such thing as a rigid body: every body deflects under the action of external forces until it generates within itself exactly the right forces to balance the external ones. These pictures would have been more convincing had they been taken from the same viewpoint with the same equipment, as the movement of the bridge against the background would have been conclusive. One function of the designer is to achieve an acceptably small deflection with the most economical arrangement of material.

====

 How much stress? How much strain? How can we measure stress and strain? Strain looks easy - because it is simply related to deflection, perhaps you can simply measure an object before and after the application of a force. Yes, but you will only know the overall strains, which is certainly of great importance. The strains that allow the wings of aircraft to bend in flight, and indeed on the ground, would be utterly unacceptable in the fuselage or in a bridge span.   But that is only a part of the story. We also need to be sure that no part of a structure is strained beyond a safe limit. The means of measuring local strain is the subject of another page. On the subject of wings, they may be bent under tests by amounts that look very alarming, but in fact the strain remains quite small. The wing-tips may be a metre or more from their normal positions, but that is irrelevant. Strain is measured by the relative local deflections over very small distances, as a fraction of the distance being measured. For example, suppose a hundred foot wing is bent by a total of six feet, a fairly substantial amount. For simplicity of calculation, let us assume that it is one foot thick throughout, which is of course unrealistic: a real wing tapers. The distance from the neutral axis of the surfaces is half a foot. These surfaces lie on curves which are longer and shorter respectively than the neutral axis.   By how much? The displacement of six feet in a hundred feet means that the angle at the tip is about 12/100 radians, or 0.12 radians. This is a fraction of a complete circle, namely about 0.12/2pi, or 0.06/pi. So the 100 feet is 0.06/pi of the circumference, which is therefore C =100pi/0.06. Now we can calculate the radius, because C = 2piR, or R = C/2pi, which is then 50/0.06, a large value. The radius of the two surfaces differ from this by half a foot, and so the strain is 0.5/(50/0.06), or 0.5 x 0.06/50, which is 0.03/50, or 0.06/100, ie 0.06%, a small value. Thus the appearance of objects does not immediately tell us about the strain, unless we are used to interpreting them. So much for strain. Stress is another matter. Measurement of dimensions tells us nothing. We need to measure forces. Alternatively, we need to know the moduli of elasticity. For example, to bend a strip of steel and a strip of card with identical dimensions, we need vastly different forces, and the stresses differ in proportion.

====

===

 Here are some diagrams which suggest what happens to atomic spacings under various types of strain.       In practice, the strains in any region of an object is likely to be a mixture of two or more types of strain. For example, in the diagram for bending, we see that in any small region, the effect is either compressive or tensile. It is only when we connect the little regions together that we see the overall effect. Einstein faced a similar problem when he was working on the general relativity theory, in which space-time is "distorted" by the presence of objects with mass. The connection of all the small regions and the description of a large region of space time required the use of tensor algebra. The same algebra is used to calculate stresses, by means of the stress tensor. What is a tensor? It is a means of manipulating many equations at the same time, without having to write each one out again and again as it is altered by the steps in the calculation. The nature of the tensor has to be such that the relationships between all the equations is correct. It may be instructive to compare the results of various types of strain and stress. In tension and compression of a bar, the strain will be roughly uniform throughout.  The same is true of shear, and therefore of torsion in a tube. But in bending a beam, the material around the neutral axis is stressed much less than that near the edges, and the same is true for torsion of a bar. Much ingenuity has gone into designing shapes that optimise the use of material, such as I-beams, box-girders and trusses. We must always remember that an object will tend to move to a position of lower energy, which may not be the one we designed for. A strut in compression may buckle rather than shrink, a box girder may buckle rather than bend, and a plate may buckle rather than take shear strain. Much theoretical and practical effort and material goes into combating the tendency to buckle.

===