Let ri, r, be the radii of its less and greater ends, and x the radius of the fracture plane, at a distance z from the loaded end (radius = r1).

Now (Art. 45) if R be the resistance of the fracture plane,

But if 7 be the length of the frustum, we have by the geometry of the figure,

Leaving out the constant factors, this expression reduces to the following:

Now as the product of the three factors of this expression is constant, their sum will be a minimum when all the factors are equal to each other. Wherefore

Cor. By (2) we can assign limits to r, in terms of r1⁄2. being less than 1, we have

For z

2. A cylindrical axle of fir-wood, 20 feet long, and 12 inches in diameter, is to bear a water wheel 8 feet from one end; find the weight of the wheel, the weight of the axle being 630 lbs., and the coefficient of safety 1,000.

Let y be the unknown weight of the wheel in pounds, R the resultant of y, and the weight of the axle; then if the distance of the point of application of R from one end be denoted by h, we have by taking moments about this end,

Substituting these values and that of h in the equation (1), we get after obvious reductions the equation.

y3 — 1.475y"— 2966827 y 1038736912 = 0.

In order to solve this equation more readily, let us transform it to an equation in z, so that ==

we have

y

100'

or y

=

100 z. Hence

23-14-752-296-6827 z 1038-736912 = 0.

It is easily found by trials, that the value of z in this equation lies between 27 and 28. The solution is therefore readily effected in the following manner by Horner's method:

In the same way the value of 2 might be calculated to any number of decimals.

Wherefore 27 10, and y = 100 z = · required weight of the wheel.

= 2710 lbs., the

3. A solid cylindrical beam of cast-iron 6 inches in diameter, is supported at both ends in a horizontal position, and is loaded uniformly over its length; find what weight it will bear, the supports being 20 feet asunder, and the coefficient of ultimate strength of the iron being taken at 41200.

4. Find the diameter of a cast-iron beam supported as in the last example, capable of sustaining a weight of 10,800 lbs. suspended from its middle, its length and coefficient of resistance to the transverse strain being the same as in Ex. 3.

Ans. 543 inches.

5. An axle of cast-iron which is 18 feet long between its supports in the same horizontal line, is to sustain with safety a wheel weighing 1,200 lbs., 6 feet from one end, together with its own weight; find what must be its diameter, the coefficient of safety of the iron being taken at 5150, and the weight of a cubic foot of the iron 440 lbs.

6. On a hollow cylindrical axle of cast-iron, supported at its extremities in a horizontal position, loads of 500, 300, and 400 lbs. act at distances of 10, 6, and 2 feet from one end. Find the thickness of metal, in order that the loads may work with safety, the coefficient of safety being 5150, the length of the axle 16 feet, and its exterior diameter 6 inches.

Ans. 483 of an inch.

7. Find the thickness of metal in the last example when the weight of the axle is taken into account, its weight per cubic foot being 440 lbs.

Ans. 57 of an inch.

8. Out of a given cylinder a beam with rectangular section is to be cut, whose moment of flexure shall be a maximum.

Ans. The breadth is to the depth as 1 : √2.

9. A beam of length 1, and having a rectangular section throughout its whole length, of constant breadth 6 but variable height, the ends of the variable height forming a straight line, is fixed at one end at which the dimensions of its cross section are 6 and h, in a horizontal position, and is loaded at the other end where the dimensions of its cross section are b and h', with a load of Plbs. Find the section of fracture; and thence sho that h' is necessarily less than 4 h.

Ans.

show

10. In the last example find P in terms of 1, b, h, h, and the coefficient of fracture S.

11. In Ex. 9, find the section of fracture when the height is constant, and the ends of the variable breadth form a straight line; and thence determine the limits of b and b'.

Ans. Distance from one end =

bl
2 (b − b')
- b');

b' L

3

b.

Section 4.

BEAMS OF DIFFERENT KINDS.

In this section we will confine our attention principally to the value of I for beams of different sections.

48. THE SECTION A TRIANGLE.

Denote the base of the triangle ABC (fig. 22) by b, and its perpendicular BD by h. Then if MN be a line parallel to the base AC and at a distance x from it, we have a by similar triangles,

MN: b :: h − x : h, or MN =

b

h

(h — x).

This is the moment of inertia of the triangle about its base. Since a parallel to the base through the centre of gravity of the triangle cuts off from the perpendicular BD a part equal to } h ; hence by Art. 26,

is the moment of inertia of the triangle about an axis through its centre of gravity and parallel to its base.

is the moment of inertia of the triangle about an axis through

its vertex and parallel to its base.

These properties will enable us to find the moment of inertia of a parallelogram and trapezoid.