which is far beyond field 2; for A, we have

3

which is beyond field 3. Hence no apex lies in its own field, and the maximum bending moment is at the weight at which it occurs for the loads W alone, in this example at W.; for, in subtracting W,, the remainder changed sign, that is, beginning at the left hand, the boundary of field 4 is the first which slopes down towards the right.

65. In example No. 61, find when the max. bending moment lies in field 2, when in field 3, and when at W„, for different intensities of the uniform load.

Let u be the intensity of the uniform load, then

Suppose that A, is at the right extremity of field 2, then

2

that is, A, is within, at the right extremity of, or beyond its own field, according as u is greater than, equal to, or less than 25 tons per foot.

Suppose that A, is at the right extremity of field 3, then

again, suppose that A, is at the left extremity of field 3 then

3

3

that is, A, is within its own field if u has any value between and 14, and is nearer the centre the larger u becomes.

3

2

3

Hence the maximum bending moment is at W, if u is less than or equal to 11, between W, and W, if u is between 11 and 11, at W, if u is between 1 and 2ğ, and between W, and the centre of the span if u is greater than 25 tons per

2

foot.

66. The span of a beam is 40 feet, and at a distance of 12 feet to the right of the centre there is a load of 20 tons; there is also an uniform load of half a ton per foot of span. Find the maximum bending moment.

For the concentrated load, the bending moment diagram is a triangle with its apex over the weight; part of the left side of this triangle lies to the right of, and slopes down towards, the centre; so that the apex for field No. 1 may lie in its own field, and the maximum bending moment is either between the centre and the weight or at the weight.

The apex A, is situated to the right of the centre at a distance

For Walone, P=28 x 84 tons; therefore 8,, the slope of the left side of the triangle, is 4 vert. to 1 horiz.;

hence

4

d1 = 8 feet;

and this being within field No. 1, is the point where the

maximum occurs.

For W alone,

M_, P x 28 = 112,

for uniform load, M.

-8

=

-8

67. Find the maximum bending moment in No. 66 if the uniform load is a quarter of a ton per foot of span.

this is beyond field No. 1; hence the maximum is at the weight, that is, at 12 feet to right of centre.

BENDING MOMENTS AND BENDING MOMENT DIAGRAMS FOR MOVING LOADS AND FOR TRAVELLING LOAD SYSTEMS.

In Part First, page 15, the action of a live load when applied to a tie or strut is described; the action is somewhat similar when a live load is applied to a beam. Thus for a beam loaded at the centre, the load W may at one instant be in contact with the central point of the beam, and yet not be resting any of its weight on the beam; the next instant its whole weight may be resting on the beam. It does not follow directly from Hooke's Law but is a matter for demonstration, that for an instant, the strain thus produced is double that which the dead load produces, provided the greatest strain does not exceed the proof strain.

One way of applying the actual weight W to the centre as a dead load is, as in the case of a tie, to put it on bit by bit; another way is to put the whole weight W on the end of the beam, when the strain is zero, and then push it very slowly towards the centre, when the strain gradually increases to the full intensity due to W as a dead load. If W, on the other hand, be pushed from the end to the centre in an indefinitely short time, it will be the same as if it had been applied suddenly at the centre; in this case, then, W is applied as a live load.

DEFINITION. A load which passes along a beam, and which thus occupies at different instants every possible position upon the span, is called a moving or travelling load.

A moving load may be dead or live or of intermediate importance, but not of greater importance than a live load. A travelling crane, which moves very slowly, and so as not to set the suspended weight swinging, is practically a dead moving load. The action of a moving load on a railway bridge is of intermediate importance; when the span of the bridge is short, say less than 20 feet, this importance is about equal to that of a live load; and when the span is long, say more than 40 or 50 feet, it is only a little greater than that of a dead load.

The Commissioners on the Application of Iron to Railway Structures at p. xviii. of their report say, "That as it has appeared that the effect of velocity communicated to a load is to increase the deflection that it would produce if set at rest upon the bridge; also that the dynamical increase in bridges of less than 40 feet in length is of sufficient importance to demand attention, and may even for lengths of 20 feet become more than one-half of the statical deflection at high velocities, but can be diminished by increasing the stiffness of the bridge; it is advisable that for short bridges especially, the increased deflection should be calculated from the greatest load and highest velocity to which the bridge may be liable; and that a weight which would statically produce the same deflection should, in estimating the strength of the structure, be considered as the greatest load to which the bridge is subject.”

CLASSES OF MOVING LOADS. An uniform load coming on at one end of the span, covering an increasing segment till it is all on, then moving to a central position on the span, and passing off at the other end, is called an advancing load. A train of trucks, shorter than the span of a bridge, coming on at one end, travelling across and going off at the other end of the bridge, is an approximate example of, and is generally to be reckoned as, an advancing load. The reason that it is called approximate, is that although the weight of the trucks may be uniform per foot of length, yet they are not continuously in contact with the bridge but transmit the load thereto by means

of wheels at a number of points. An advancing load may be equal in length to the span; in which case, in passing across, it covers the whole span for an instant. If the load be longer than the span, it will continue to cover it for a definite time while passing, but as time does not come into our consideration, it will be included in the advancing load equal in length to the span.

A load concentrated at a point, and which moves backwards and forwards on the span, is called a rolling load; a wheel which rolls along a beam is a practical example of this. In reality the load is distributed over a small area, and if now the load be taken to be uniformly distributed over this small area, it may be considered as an advancing load of small extent; on the diagrams it is represented by a wheel or circle.

A Travelling Load System is a load transmitted to the beam in definite amounts at points fixed relatively to each other, the whole load moving into all possible positions on the span; a locomotive engine is a practical example of such a system, and a rolling load is its simplest form. On the diagrams, the load is represented by a number of circles or wheels with their centres fixed on a frame (see fig. 3), or for ease in drawing by a number of vertical arrows connected by a thick horizontal line (see fig. 43).

It will not be necessary to consider moving loads upon cantilevers as in practice there is seldom such a thing. It is only necessary to suppose the load fixed in the position most remote from the fixed end; this, it is evident, gives the greatest bending moment at each point, the maximum being at the fixed end.

Beam under an advancing load equal in length to the span.

-

Suppose the load to come on from the left end and cover a segment of the span, the bending moment diagram is shown on fig. 33; when the whole span is covered, on fig. 25; and when the load is passing off, by fig. 33 reversed. Since the parabolas in these two figures are the same, it is evident that the apex A is higher on fig. 25 than upon fig. 33, because on the former the base of the parabolic segment is the