shear, and one-fourth the ultimate resistance in compression. It is usual for the iron piers to have such a width of base that, with the above pressures, there will be no tensile stress on the main pillars.

359. Influence of the Form of a Surface as affecting WindPressure. The form of a surface exposed to the wind has a great deal to do in modifying the pressure exerted against it. The pressure, for example, upon convex surfaces is much less than that upon their projected plane surfaces, and that on concave surfaces is much greater.

From theoretical calculations it appears that the pressure upon a sphere is only one-half that on a flat surface, equal in area to a section through its centre, and that upon a solid cylinder is only two-thirds of the pressure on a section through its axis. On the other hand, the pressure on a parachute is nearly double that on its diametral section.

If great nicety of calculation be desired, a coefficient of 0.5 should be taken for all round bars, and a coefficient of 1.5 for channel sections. It is not often, however, that such exactness is necessary.

360. Wind-Bracing.-In order to provide against the windpressure exerted on the superstructure of a bridge, the main girders should be braced together in a horizontal direction, by which means the pressure is transmitted to the abutments. This bracing, except in very deep girders, occurs at the bottom flanges. If the floor of the bridge consists of wrought-iron plates resting on cross girders, or of wrought-iron or steel troughing, such flooring in itself fulfils all the requirements of the wind-bracing. When the floor consists of timber it will be necessary to introduce diagonal bracing of wrought iron. With deep girders which admit of sufficient headroom, the top booms should be connected by arched or diagonal bracing.

Arched bridges do not need so much wind-bracing as those constructed of ordinary girders, as they expose little surface at their centres where the wind-pressure exerts the greatest effect.

361. Stresses on Braced Piers.-Fig. 251 is an example of a braced pier. Under ordinary conditions it will be exposed to two sets of forces, namely:

1. Vertical forces, which consist of the weight of the pier itself together with that of the superstructure and the live load.

2. The horizontal wind-pressure.

In calculating the working stresses produced by these external

forces, it will not be necessary to take into account any live load on the bridge, as with a wind-pressure of from 40 to 50 lbs. per square foot it would not be possible for railway trains or other vehicles to pass over.

R

Fig. 251.

W

The wind-pressure (P) exerted on the superstructure may be represented by a line, Op, which passes through the middle of the depth of the main girders.

The wind-pressure (Q) exerted on the pier may be represented by a line, 0, 9, which passes through the centre of the surface of the pier exposed to the wind.

The resultant (R) of these two pressures is represented by a horizontal line, or, where or Op+0,9. The point of application of this force is at the point o, where 0 0 : 02 01 :: Q: P.

=

The vertical force (W) coming on the pier acts along the central line 0 0 0 Through the point r draw the vertical line rr1, making r1 = W. Join or1; this line will represent both in magnitude and direction the resultant of all the forces acting on the pier. If the line or produced fall between the points A and B the pier will not be overturned, even though it be not anchored down. If or fall outside A B the pier will be liable to be overturned unless anchored down.

In the latter case if the pier be anchored at A, the stress on the anchor-bolts may be found thus

Let R= resultant of the wind-pressures.

0,O=b, the distance of its point of application from the base

of the pier.

W = vertical load on the pier including the weight of the pier itself.

a=AO2=03 B= half the base of the pier.

S= stress on anchor-bolts at A.

Taking moments about B as a fulcrum, we get

In a similar manner may be found the stresses on a A and b B, the main pillars of the pier.

When there is a tensile stress on the anchor-bolts at A, the member A a will be in tension and b B in compression.

When there is no stress on the anchor-bolts at A, there will be compressive stresses both on a A and b B, though not equal in amount.

The stresses on the lattice-bracing of the pier may be found in a similar manner to those on a braced cantilever loaded with a concentrated weight at its extremity, and a practically uniformly distributed load over its entire length.

In order to further illustrate the effects of wind-pressure on braced piers, we cannot do better than give an example of a pier actually in existence-namely, one of the high piers of the Bouble viaduct-an account * of which is given by M. Jules Gaudard of Lausanne.

This viaduct consists of a series of spans of 164 feet each; the main lattice girders are 14 feet 9 inches deep. Each pier consists of four cast-iron columns, which are ballasted with concrete and braced together, as shown in fig. 251; the height of the top of the girders from the base of the pier is 203 feet 5 inches. vertical loads on each pier are as follows:

Dead weight of one span,

Weight of train,

120 tons.
85

The

Weight of pier, .

240 ""

Taking a wind-pressure of 55.3 lbs. per square foot, the total pressure against a girder, allowing for the spaces between the lattices, will be about 40 tons. This pressure acts horizontally midway between the top and bottom of the girder, or at a height of 196-2 feet above the base of the pier.

The pressure on the train is estimated at 32.4 tons, and this acts horizontally at a height of 210-3 feet above the base of the pier. Lastly, the wind-pressure on the pier is estimated at about 40 tons acting horizontally at a height of 92.85 feet above the base.

We have now all the data necessary for determining the stresses on the pier under the above conditions.

As the width of the base of the pier is 67 feet 7 inches, we get— Moment of stability of pier 445 × 33.8 = 15,041 foot-tons.

=

X

Moments of wind-pressure tending to overturn pier

X

=40 × 1962 + 32·4 × 210·3 + 40 × 92·85 = 18,375.72 foot-tons. From which it is seen that the net overturning moment = 18,375-72-15,041 3,334.72 foot-tons.

*Proc. Inst. of C.E., vol. lxix.

It will be necessary, therefore, to anchor the pier down to its foundations. If the anchor-bolts pass through the extremity of the base of the pier, and if S = total tension on the bolts, we get— Sx 67.63,334 72, or S = 49.3 tons.

CHAPTER XXX.

LIFTING TACKLE, ERECTION OF BRIDGES, ETC.

LIFTING TACKLE.

362. Derricks. A derrick is usually a pole or balk of timber placed in an upright position, one end resting on the ground. When the weight to be lifted is great, or when the derrick is long, the latter may be made of wrought iron, either square or circular in cross-section, and formed of continuous plates and angles or open lattice-work. Derricks are kept in a vertical position by means of ropes or chains fastened to their tops, the other ends of the stays being anchored to the ground or made fast to objects in the vicinity. These stays are termed guy ropes or guy chains; and the efficiency of the derrick depends to a large extent upon them. The dimensions of a derrick-pole depend upon its height and the weight to be lifted. For light weights a balk of timber 8 inches to 12 inches square is usual; for heavier weights, 12 to 18 inches square may be required. For anything beyond this it is advisable to use wrought iron. For lifting heavy weights, chains are preferable to ropes for staying, and these should not be less than four for each derrick. The bottom end of the stay may be fastened to stakes driven into the ground. A convenient arrangement for this purpose is a wrought-iron bar with an eye forged on one end and a screw on the other. This can be screwed into the ground by inserting a round bar in the eye and using it as a lever. This is afterwards withdrawn and the guy made fast to the eye. At the top of the derrick a pulley-block is attached, and the chain or rope for hoisting passes round the pulley and is carried down the side of the derrick to a snatch block fastened to its heel, and then passes round the barrel of a crab placed at some distance off and anchored to the ground. When a single derrick

is used for lifting heavy weights, say over 15 tons, it is advisable to employ two sets of pulleys and tackle. For long and heavy girders, two derricks may be used, the girder being slung from two separate points.

363. Sheer Legs, Tripods, and Scotch Cranes.-Other lifting apparatus are sheer legs and tripods. The former consist of two poles and require two or three guy ropes, while the latter consist of three poles and do not require any guys.

These latter are not so convenient for shifting from one place to another as ordinary derricks. What is known as the Scotch

crane is a very useful appliance for lifting purposes. The gib can be raised or lowered in a vertical plane, or swung round in a horizontal plane, so that the work lifted may be placed in its required position. The lifting crabs may be worked by steaminstead of hand-power when heavy weights are being lifted.

The method of determining the stresses on all these lifting appliances is fully explained in Chap. XVI.

364. Ropes and Chains.-Ropes are usually made from " green hemp" or manilla. They are measured by their circumference in inches and are generally sold by weight.

There are several rules for calculating the strength of ropes, but they must be considered only as approximate, there being considerable variation in the strength of pieces even when cut from the same coil. One rule is, that the breaking stress in cuts. is equal to four times the square of the girth in inches; so that if C= circumference in inches,

This is a very approximate rule, and is applicable only to certain qualities of rope.

The following formulæ are more to be relied upon, and can be made applicable to the different kinds of rope by using the proper constants :