into its two components, Tcp, TCE exactly as in the previous case.

By considering the joint A we can determine the thrust along AB, the axis of the screw and the upward pull on its bearings being respectively the horizontal and vertical components of TAC

K

Tension of Lifting Chain. We have hitherto considered the load W as simply hanging to C, but this is not the case in practice, W being always supported by a tackle, the rope or chain of which passes over a pulley at C and hence to a chain barrel.

TAC

TBC

Now C is loaded with two loads, W and W/n, and we must first find the resultant load at C.

these being parallel to W and CD respectively. Then joining EG, EG is the resultant load at the joint; and we now draw EK, GK parallel to AC, BC respectively, these giving TAC, TBC, and proceed with the diagram of forces as before.

In many cases the chain runs on pulleys on the stay, so that its direction is that of the stay. In this case the ordinary diagram for W alone is drawn (Fig. 335), and

then TAC is composed of the real tension in AC and the tension of the chain which assists it. Hence

W

Tension in AC=TAC (from diagram)· 22

If the chain were led down the jib, its tension would increase the thrust on the jib, and we should have

Thrust in BCT BC (from diagram) +

W
12

Bending of Crane Posts.—In small cranes backstays are often omitted, the crane post being made stiff and supported in bearings. Fig. 343 shows the post prolonged, B and D being the bearings.

The preceding considers the crane as a whole; but now consider the post alone. Then the overturning forces are the horizontal components of TAC, TBC, these being equal (Fig. 335), and each equal to Q say.

Then (Fig. 344) we have two couples, Q. AB, P. BD, keeping the post in equilibrium. So that

Q.AB=P.BD=Wa.

The post is now subjected to bending and shearing, due to a load P+Q at B, supported by P at D and Q

at A.

Whence

and the figures (a) and (b) show the curves of B. M. and S. F.

Fig. 345 shows another method of supporting the post, there being a bearing at each end. In this case there is practically no bending or shearing.

There is a very large variety of types of crane, but space will not permit of our examining any more

examples; in all cases the methods used in the present chapter will be found to apply.

EXAMPLES.

1. A simple triangular truss, 24 feet span, 3 feet deep, is supported at the ends, and carries a load of 3 tons concentrated in the middle. Find the stress on each member.

Ans. 6.2; 6 tons.

2. The span of a roof is 15 ft., length of rafters 9 ft. and 12 ft. respectively. The rafters are 2 ft. apart along the roof, and the roofing material weighs 15 lbs. per sq. ft. Find by con

struction the thrust of each rafter and the stress in the tie bar. Ans. 189; 252; 151 lbs.

3. A pole 40 ft. long is used as a derrick. One end rests on the ground, the other is supported at a height of 30 ft. from the ground by a chain at right angles to the pole. If the chain will safely bear a pull of 3 tons, what weight can be lifted by the Ans. 4 tons.

derrick?

4. The jib of a derrick is inclined at 45° to the vertical, and

the topping lift is attached to a point vertically over the foot of the jib at a height equal to its length. Find by construction the pull on the topping lift and thrust of the jib when lifting 4 tons. Ans. 3.1; 4 tons.

5. The horizontal member of a simple triangular truss 10 ft. span, 3 feet deep, is loaded with 2 tons, uniformly distributed over half the span, and is supported at the ends. Find the stresses in the bars. Ans. 1; .83; .97 tons.

6. A crane has a vertical crane post AB 8 ft. long, and a horizontal tie BC 6 ft. long, AC being the jib; it turns in bearings at A and B, and the chain supporting the load passes over pulleys at C and A, passing from A to the chain barrel at an angle of 30° to AB. Find the stresses in the bars and thrusts on the bearings when raising I ton at a uniform rate.

Ans. Thrust on jib=1 ton.

7. The crane post AC of a crane is 8 ft. high, AB the jib is 21 feet long, and there are two tie bars CB each 16 ft. long. The load of 5 tons is supported by a pair of two-sheaved blocks, the chain passing to the chain barrel bisecting AC. The backstay is inclined at 45° to AC. Find the stresses in all the bars, and the magnitudes of the balance weight at the foot of the backstay.

Ans. Jib, 13; stay, 93; post, 4 tons; backstay, 11.8; counter-balance, 83.

8. In a crane, the jib is twice as long, and the stay one and a half times as long, as the crane post. The backstay is parallel to the jib. The crane post turns in bearings at its foot, and is produced below the ground to a distance equal to its height above where it turns in the foot step bearing. The end of the platform, where the backstay meets it, is stayed to the end of the crane post at the footstep, there being no balance weight. Find all the stresses and thrusts on the bearings, when loaded with 10 tons.

9. A fixed crane post is 20 feet high, the stay is 20 ft. long, and the jib 32 feet. There are two backstays each 24 feet long, their plan being a right-angled triangle. Find the stresses when lifting 10 tons, the plane of jib being midway between the stays, and show how the stress on the backstays changes when the jib has swung through 45° and 90° from its first position.

Ans. Jib, 16; stay, 10 tons.

10. A pair of sheer-legs are 60 ft. high when upright, each leg being 52 ft. long. The back leg is 90 ft. long. The front legs swing round pivots 3 ft. from the dock side. Find the greatest stress in each leg when placing a boiler weighing 20 tons on board a vessel of 40 ft. beam, the hatch being in the middle line of the ship. Ans. Back leg, 16; front leg, 163 tons.

PART III

HYDRAULICS