tremities and loaded at a point, or points, intermediate between them.

A semi-beam or semi-girder is a beam or girder fixed at one extremity only, and free at the other. The term cantilever is also applied to this form of beam.

A continuous beam or girder is one supported at three or more points.

Figs. 22, 23, and 24 represent the three kinds of beams referred to. A B, fig. 22, is the simple beam, which rests on the two supports, A and B, usually termed "abutments," and loaded at an intermediate point with a weight, W.

Fig. 23 represents a semi-beam or cantilever, which is fixed at one end, A, to a wall or other support, termed the abutment, and loaded at the other end, B.

Fig. 24 represents a continuous beam resting on three supports, A, C, and B. As before, the supports, A and B, are called "abutments," while the intermediate support, C, is termed a "pier." The beams are all supposed to rest in a horizontal position, and the horizontal distances between A and B (fig. 22), and between A and C and C and B (fig. 24), are termed the "spans" of the beams.

63. External Forces on Beams.-When a rigid beam, as in fig. 22, is loaded at one or more points, these loads, or weights, act downward in a vertical direction, and they develop forces at A and B, which act upwards in a vertical direction. These upward forces are termed the "reactions" at the points of support, or the supporting forces. All these vertical forces are termed the external forces acting on the beam, in contradistinction to the

forces produced in the fibres of the beam itself, which are termed internal forces, and which will be treated of in a future chapter.

The forces which act downwards may, for the sake of convenience, be considered positive, and those which act upwards negative.

Since all the forces are parallel and act in one plane, there are two conditions which must be fulfilled in order that the beam may be in a condition of equilibrium :

(1) The sum of the upward forces must be equal to that of the downward forces; or, in other words, the algebraic sum of all the forces must be zero.

(2) The sum of the moments of the forces which tend to turn the beam in one direction must be equal to that of those which tend to turn it in the other direction; or, the algebraic sum of the moments of the forces with reference to any point must be

zero.

As regards loads on beams, there are usually recognised three varieties:

(1) Loads concentrated at one or more points, which are known as concentrated loads.

(2) Loads uniformly distributed over the whole or certain parts of the beam. These are known as uniformly-distributed loads and are measured by so many lbs., cwts., or tons, per lineal foot of the beam or span.

(3) Loads made up of a combination of the two former, or those which are partly distributed and partly concentrated.

64. Beam Resting on Two Supports and Loaded with a Single Weight. In fig. 22, let m and n be the segments into which the weight, W, divides the span, and let P and Q represent the reactions at the abutments or the supporting forces; then, by the conditions of equilibrium already given, we have

and taking moments about A; since the force, W, tends to turn the beam about this point in the direction of the hands of a clock, and the force, Q, tends to turn it in the opposite direction, we get

W x m = Q (m + n)

(2).

We have here two equations, from which the values of the two unknown quantities, P and Q, may be determined. Reducing,

we get

m

Q = W.

m + n

(3).

This latter expression, giving the value of P, may also be got directly by taking moments about B.

If W is in the centre of the beam, m = n =

2'

where l

= span.

Substituting these values in the last two equations, we get

which shows that when a weight rests in the centre of the beam, the reactions at the abutments are equal to each other, each being equal to one-half the weight.

It will be noticed that in the above investigation the weight of the beam itself is not taken into consideration, and unless otherwise stated, in all future examples this will also be the

case.

Example 1.-A beam, 20 feet span, supports a load of 30 tons situated at a point 7 feet from the left bearing. Find the supporting forces, or the reactions, at the bearings.

Adopting the same notation which we have just been using, we get

Example 2.-If the supporting forces at the left- and right-hand supports of a beam, 30 feet span, on which is placed a single load, be 27 and 13 tons respectively determine the load and its position on the beam.

The load W - P + Q = 27 + 13

= 40 tons.

The beam is, therefore, loaded with 40 tons, placed at a distance of 9.75 feet from the left support.

65. Beam Resting on Two Supports and Loaded with Two or more Weights. Fig. 25 represents a beam loaded with weights,

W1 W2 W3 W4, whose distances from the left support are respectively x1, X2, X3, X4,

Applying the two conditions of equilibrium, we get

where the symbol W signifies the sum of W,, W2 W3 W4 Taking moments about A, we get

Q × 7 = W11 + W2×2 + W3×3 + W4×4 = Σ W x ;

1

Similarly, by taking moments about B, we get

P × l = W1(l − x1) + W2(l − x2) + W3(l − x3) + W 4(1 − x ̧) = ΣW(1 − x);

P may also be found directly from equation (5) when Q is known, and vice versa.

66. Beam Resting on Two Supports and Loaded with a Distributed Weight.-In the case of a load uniformly distributed over the whole or part of the beam, it is only necessary to find the centro of gravity of the load, and consider it as a concentrated

load acting at this point, and then to find the supporting forces (as explained in the first case).

If a beam of span, l, support a uniformly-distributed load of w tons per foot over its whole length, the total load W = wl, wl 2

and each of the supporting forces ==

If the load only covers part of the beam, as is shown in fig. 26, the supporting forces may be found thus

P

B

Fig. 26.

Let the left-hand portion of the beam for a distance, x, be loaded with w per foot,

Total load W =wx.

Distance of its centre of gravity from the left support

Taking moments about A and B in succession, we get—

Example 3.-The span of a beam is 52 feet, and it supports loads of 10, 15, and 20 tons, resting at points which divide the span into four equal parts. Find the pressures on the two supports.