bar is known as the modulus of elasticity, and may also be expressed as the tensile force, which, when applied, will double the bar's length, and of course may be different in the same material when subjected to tension, compression or shear,

Permanent Set. If a bar be extended or contracted by the application of a load beyond its elastic limit, it is said to have permanent set. This would take place in mild steel if a load of 17 tons per square inch of section were exceeded.

The Moment of Inertia of a section or body is a mathematical quantity used to calculate the strength of materials, and is taken relatively to the neutral axis or centre of gravity of the section. If the section of a bulb tee beam, as shown in Fig. 56, be centrally loaded on top, the fibres above the line xy (neutral axis) will be compressed, and those below extended, and consequently the arc formed by the table of the beam will be shorter, and that formed by the bulb longer, than the arc through the line NS,

which will be exactly the same length as the original dimension of the beam before the application of the load, the laminæ through this axis being neither in compression nor tension, and are therefore known as the neutral surface of the beam. Hence, if we take very small areas at known distances from the neutral axis to their centres of gravity and multiply these areas by the square of their distances above or below this line, we shall have by adding the products together the moments of inertia (I) of the section; and again by dividing this moment by the distance of the most extreme fibre we shall get the quantity known as the section modulus.

In the example given the result is fairly accurate, although a more absolute result may be obtained by greater subdivision of the areas. This, however, is not necessary for ordinary calculations.

The value of the section modulus depends entirely on the geometrical form of the section. The material of which the beam is made and its ultimate strength known and divided by the factor of safety selected, gives us the safe limiting stress. This stress multiplied by the section modulus produces the moment of resistance of the beam. In the example given let the beam be of steel of 60,000 lbs. ultimate strength and the factor of safety 5, we then have 60000 12,000 lbs. safe limiting stress, and section modulus 16,5 × 12,000 lbs. 198,000 lbs. moment of resistance. Suppose then that this were a 12-foot boat skid beam fixed at both ends and loaded at centre, what weight of steam pinnace would it safely support? The maximum bending moment on a beam so loaded would be WL where W is the weight and L the length between points of support. Equating this bending moment with the moment of resistance, we have

=

Where the figure or section is symmetrical about its centre of gravity the I and other elements may be readily figured from the appended Table of Elements of Usual Sections.

Radius of Gyration. The radius of gyration is that fundamental property of a section used in determining the strength of pillars and struts, and its square or r2 about a given axis is equal to the moment of inertia of the surface about the axis divided by the area, therefore the radius of gyration

inertia

r =

area

13,000 17,000 17,000 21,300

8,500 13,000

4,200 10,700 4,300

TENSION

II

BENDING

SHEARING

II

II 8,500 8,500 11,400 11,400 14,200
III 4,300 4,300 5,700 5,700 7,100
COMPRESSION I 13,000 13,000 17,000 17,000 21,300 13,000 17,000 13,000
8,500
8,500 11,400 11,400 14,200 8,500 13,000
8,500
I 13,000 13,000 17,000 17,000 21,300 10,700 15,000
II 8,500 8,500 11,400 11,400 14,200 7,100 10,000
III 4,300 4,300 5,700 5,700
3,500 5,000
I 10,000
6,800

5,700❘ 8,500 2,800 4,300 1,400

2,800

10,700 4,300

III

3,400

7,100
10,200 13,700 13,700 17,000
6,800 9,100 9,100 11,400
3,400 4,600 4,600 5,700

7,100 2,800
3,600 1,400

Idead load, II live load, III = live load, acting alternately in opposite directions.