Introducción

Figure 5.13(a) shows pictorially a simple bar loaded in the middle and Figure 5.13(b) shows the same bar diagramatically.The most obvious point is that the bar is being bent with the top face of the bar in compression and the bottom face in tension. It is less obvious that the bar is also being sheared but consider Fig. 5.13(c) and (d): it is clear that if the loaded bar was suddenly cut in either of the ways shown, there would be a movement

Fig. 5.13 Bending and shear due to bending.

across the cut face similar to the movement shown in Fig. 5.4(b), there-fore the material of the bar must be capable of resisting that movement.

In the case of a plain rectangular bar there is not much doubt that the necessary resistance is there, but in some other beams, particularly those of typical aircraft construction which are built up from sheet metal and have riveted joints, it is only too easy for the designer to lose sight of these shears.

The beam in Fig. 5.14(a) is at first glance a reasonable construction with the load W being resisted by the push and pull at the lugs A and B.

However, because the shear has been ignored, the vertical ‘web’ of the beam will collapse as shown in Fig. 5.14(b). To make the beam completely sound, an attachment at the end of the web is required. This would have been more obvious if the load W had been properly transferred accord-ing to the principle stated in Fig. 4.3, in which case Fig. 5.14(a) would have become like Fig. 5.14(c). Parts (d) and (e) of the diagram illustrate alter-native solutions to the problem of coping with the shear in the web.

Some other points should be noted about this piece of structure, which, as we said before, is typical of aircraft structural components. It is more complex than it appears at first glance.

Firstly, the rivets through the angles attaching the web to the flanges are loaded in shear. We can imagine that if they were made of some very soft plastic instead of metal they would be cut through. Secondly, although in Fig. 5.14(d) we have divided the web into smaller sections, each section can still buckle into waves, as shown in the long unsupported web of (b).

In his battle against weight, the aircraft structures engineer accepts this situation, knowing that even after the thin sheet has fallen into buckles (the post-buckling situation) the web will still carry load. This is a very important concept which was analysed by Professor Wagner in about 1930 and followed by S. Timoshenko in 1936, whose book The Theory of Elastic Stability, has been the inspiration of most of the mathematical work on analysis published since.

Unfortunately, our second point above has an influence on the first point. The load on the flange rivets increases due to the action of the buckled web, and this fact gives a hint of the thoroughness with which structures of this type have to be investigated if they are to be safe and efficient.

Consider now the effect of making the beam deeper. Refer again to Fig.

5.14(c). We will discuss the upper, tension-side flange, although the stages of the argument will apply equally to the compression side, but in the opposite sense.

(a) As the beam is loaded it bends (or deflects) and we can see that the flange length is increased because the length along the curve of the tension side is greater than the length at the centre of the beam (which has not altered).

(b) The flange length is increased because it is loaded, therefore there is a stress (load/area), and a strain (increase in length/original length).

Fig. 5.14 Sheet metal beams.

(c) If the deflection of the beam is kept the same (i.e. the curve of the centre is kept the same) but the depth of the beam is increased, the strain in the flange will increase and, therefore, the stress will increase as we saw in Section 5.5.

(d) Alternatively, if we increase the depth of the beam but keep the same strain in the flange, then the deflection of the beam will have to be less in compensation. In fact, if we double the depth of beam the deflection is halved.

(e) Also, if we increase the depth of the beam, the load in the flange is reduced (see Fig. 4.2). Again, if we double the depth we halve the load.

(f) We know that the flange load is the area of cross-section of flange multiplied by the stress, and we have already said in (d) that we are keeping the stress constant (actually, we said we would keep the same strain but that of course means that the stress will be the same), therefore, with the same stress but a smaller load, the area and hence the weight become less.

(g) Summing up this important argument, we can say that by doubling the beam depth, we halve the deflection and halve the weight (of the flanges). It is usually said that the advantage of adding to the depth of a beam increases as the square of the depth, which is a rule of thumb which should be treated with some caution, but if the argu-ments above are clear and understood the effect can be considered in logical steps.

We will make one more point before leaving the beam in Fig. 5.14. It is not a very important point because, although the type of construction is typical of aircraft structure, there are not many beams of this type which are entirely isolated; they are more usually part of a bigger structure.

However, the principles involved are quite important. If the flange width is large compared with its thickness, the edges of the flange on the tension side will try to take a short cut across the curve of deflection, as shown in Fig. 5.14(f). This means that the strain on the edges of the flange is less than the strain at the point where the flange is attached to the web. As the strain is less we know that the stress is less and therefore the load in the whole flange is not exactly the stress multiplied by the area, or more importantly the maximum stress is rather higher than the average found by dividing the load by the area. The phenomenon is called shear lag, although that term is also used in connection with the transfer of con-centrated loads into aircraft skin. On the compression side of the beam, the situation is worse because the middle of the flange is being forced to reduce its length and the outer edges are unwilling to follow, so they tend to go into waves or buckles, which in turn push their way towards the middle of the flange and reduce its effectiveness. In fact, if we regard the tension and compression flanges as separate members in their own right, we can say that while it is fairly easy to design a tension member, it is more difficult to design a compression member.