1. Find the diameter of a lifting pump, 7 feet long, capable of raising a weight W, including the weight of the pump.

If r be the unknown radius of the pump in inches, and w its weight per cubic foot, the whole weight to be raised will be

Hence if S be the coefficient of resistance of the iron of the pump, we have by (3), Art. 6,

Solving this equation with respect to r, we get

In this formula I must be taken in inches.

S =

Let W 326,600 lbs., w
=

60,000 lbs.; then

= 480 lbs., l = 8 feet, and

2 r = 2.6 inches, the required diameter.

2. Determine the diameter of the rings of an iron chain capable of raising a weight of 90 cwt., the coefficient of absolute strength of the iron being taken at 23 tons.

Ans. Diameter of each ring 35 of an inch.

3. Find the diameter of an iron bar 33 feet long which is stretched of an inch in the direction of its length, by a weight

of 98 tons, 12 lbs., the weight of the bar per cubic foot being 480 lbs. and the modulus of elasticity 29,000,000 lbs.

4. Determine the dimensions of a square column of cast iron in order that it may support a load of 40,568 lbs. in the direction of its axis, the length being inconsiderable, and the coefficient of resistance 16,500 lbs.

Ans. Side of square section = 1.56 inch.

5. One side of a rectangular column of cast iron is 11⁄2 inch find its breadth in order that it may sustain a weight of 18 tons, in the direction of its length, the coefficient of resistance being the same as in the last example. Ans. 1.63 inch nearly.

6. An iron rod, 2 feet long, and 5 inches in circumference, is suspended from one extremity; find what force applied vertically at the other extremity would produce fracture, the coefficient of ultimate strength being taken at 27 tons, and the weight per cubic foot of the rod at 400 lbs.

Ans. 120,309 lbs.

7. A g-inch iron wire is just capable of sustaining a pressure of 7,037 lbs. in the direction of its length; find the coefficient of ultimate strength. Ans. 25 tons.

8. If the modulus of elasticity of wrought iron be 24,900,000 lbs., find what force will stretch a bar of that material, which is 10 feet long and of a square inch in section, half an inch, the weight of the bar per cubic foot being 480 lbs. Ans. 17,286 lbs.

9. Find what extension will be produced in a bar of iron 3 inches in diameter and 400 inches long, by a weight of 84,000 lbs. suspended from one end; the modulus of elasticity being taken at 28,000,000 lbs., and the weight of a unit of the iron the same as in the last example.

Ans. 171 of an inch.

10. A pressure of 1 ton, 14 lbs. breaks a 4-inch iron rod

;

when applied in the direction of its length; find what pressure will break a inch rod of the same material.

Ans. 1.788 tons.

11. A wire rope consists of 15 wires; find the thickness of each wire in order that the rope may safely sustain a load of 6 tons, the coefficient of safety of the wire being taken at 16,000. Ans. 266 of an inch.

12. A wrought iron bar is to sustain safely a load of 6,000 lbs. in the direction of its axis. The dimensions of its rectangular section are to be in the ratio of 1 to 4; find these dimensions, the coefficient of safety to be taken one-sixth of that of the resistance to crushing, which is 112,000.

CHAPTER III.

STEAM BOILERS.

Section 1.

CYLINDERS AND SPHERES.

12. PROP. To find the resistance of a cylinder subjected to an interior pressure.

A cylinder is filled with steam at pressure p (p being the pressure in pounds avoirdupois per square inch), in order to find the total pressure P, estimated perpendicularly to a section of the cylinder passing through its axis, we proceed as follows.

The normal pressure on an element a of the interior surface of the cylinder will be pa; and if x be the distance of this element from the plane of section, and r the radius of the cylinder, the component of pa, estimated perpendicularly to the cutting plane, will be

The sum P of all the pressures corresponding to (1) for all the points of the half surface of the cylinder, will be

the symbol Σ denoting the sum of all the expressions similar to that before which it is prefixed, for the half surface of the cylinder.

Let I be the height of the cylinder, and x, the distance of the cutting plane from the centre of gravity of the half surface of the cylinder; then, by the principle of the centre of gravity,

Wherefore (2) becomes, by means of the last expression,

Lett be the thickness of sheet-iron of the cylinder; then the section of the cylinder which resists the force P being the double

* This value of 1 is readily obtained from the theory of moments. Let A B (Fig. 1) be the intersection of the plane passing through the axis of the cylinder with a cross circular section of the cylinder ACB. Then it is clear that the centre of gravity of the arc of the semicircle ACB I will be at the same distance from AB as the centre of gravity of the half surface of the cylinder. Refer the elements of the semi-circumference ACB to rectangular coordinates which have their origin in O, the centre of the semicircle, the axis of y coinciding with AB.

Now the moment of an element ds of the semicircle about A B as an axis of moments is

x ds,

B

Fig. 1.

"

and the sum of the moments of all the elements of the semicircle about the same axis is

hence, differentiating this equation with respect to a, and keeping in mind that