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If, in addition, the beam itself weighed w' lbs. per foot run, we should have finally,

W= =nb+w' lbs. per foot run,

and we must use this value of w to find the B. M. and S. F.

A

C

E

G

In the general case, where a number of parallel beams support a loaded platform, the load on each beam would be found by supposing it to support the piece of platform which extends on each side of it, half way towards the next beam. Thus in Fig. 207 AB, CD, etc., are the beams shown in plan, and CD would support the piece of platform dotted; and the total load on this piece,

B

D

F

H

Fig. 207.

divided by the length of CD, would give the load per foot run.

Floating Beams-Case VII.-In some cases the distributed loads may act upwards, or, in other words, the supporting forces may be distributed. Such a case is that of a wooden beam floating in water.

Fig. 208.

Taking the beam alone, there is on every foot-length a distributed load w' lbs., where w' is the weight of 1 foot. But the upward pressure of the water will also be uniform, and must therefore be also w' per foot run. At

every point, then, there are equal and opposite forces acting, so there is no tendency to bend or shear.

But now let us place a load W=wl pounds on the centre of the beam, then the beam sinks until the increase in the upward pressure is sufficient to balance W or wl. Since W is put on the middle the beam sinks evenly, and there is therefore a uniform increase of pressure over the whole

length.

[The case in which the loading is not symmetrical is outside our limits.]

This increase of pressure, then, is w lbs. per foot run.

The beam is then in equilibrium under a downward load W lbs., an upward distributed load w lbs. per foot run, and equal and opposite distributed loads w' lbs. per foot run. Since these latter, however, produce no B. M. or S. F., we can omit them, and we are left with the beam loaded, as in Fig. 209.

To make it more easily recognisable we turn it upside down (Fig. 210); this cannot affect the values of M or F, and for their signs

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we can look to the original figure.

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have a case quite different from any of the preceding, but by a little consideration we shall be able to make use of previous results.

A

For consider the part CB, then CB is a beam fixed at C, and loaded uniformly. Now this may at first sight appear a somewhat strange statement, but it is nevertheless perfectly correct. In Fig. 206 AB is fixed to a wall, but then the question as to what sort of wall it was did not enter into our work; what we require in order to be able to fix the end A is something which is capable of standing M and FA without giving way, and so long as it can do that we care nothing else about it. Now CB is fixed at C to the other half of the beam, and the fixing is strong enough to withstand Mc and Fc, because if it were not the beam would break, which we suppose it not to do. Hence then we treat CB as a beam fixed at C, and we can at once draw the curve for CB from Case V. We set up then CD (the bending being plus, see Fig. 191) equal to w.CB2/2, and draw the parabola DB with apex B.

C

Then from symmetry a parabola AD will be the curve for AC, that being a beam also fixed at C.

Hence ADB is the curve of B. M., CD being w/2/8 or W//8.

In BC the shearing force is negative (Fig. 192), therefore from Case V set down

and join BE.

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In AC the shearing is positive (Fig. 192), therefore from Case V set up

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Then AFEB is the full curve.

[For the explanation of the drop at C see page 285.]

The bending and shearing of a ship is a similar case to the preceding; but there is an uneven distribution both of load and of buoyancy or upward pressure,

for the effect of which we must refer to the larger treatise.

Principle of Superposition. When a beam is loaded with more than one load, we can, as in Case IV of the last chapter, obtain the curves of B. M. and S. F. by strict adherence to the definitions; this is also true if one or more concentrated loads be combined with a distributed load. But the process can generally be considerably adopted by the use of what is known as the Principle of Superposition.

This principle may be generally stated thus: The effect due to a combination of causes is the sum of the effects which would be produced by each cause acting separately.

This may, at first sight, appear to be a truism, but it is not so, for it is only true under certain conditions. To show this, and

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length. Similarly in (b) a weight W, bends it, so that

B is distant / from the wall.

2

1

Plainly then if we put on both W1 and W2 the distance in (c) will be less that either 1 or 2.

MA due to W1 and W2=(W1+ W2) 3,

Now

which is not the sum of W11 and W22, the moments at

A produced by W, and W, acting separately, but is

A

A

C

A

Fig. 212.

B

D

D

2

plainly less.

We see, then, that what is necessary for the principle to apply is that 7, 12, lg should be practically identical, and then they will also each be identical with 7.

Generally we may say that all the separate effects should be very small.

Now it is proved both theoretically and practiB cally that in all ordinary cases of bending, the bent lengths 1, 2, and l。 (Fig.

211) are indistinguishable from the original length 1, and so the principle holds. For

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