is meant all the external forces acting upon it. These include:

(1) The weight of the structure itself, which is a constant quantity, and acts vertically.

(2) In bridges and similar structures they include the weight of the roadway, which also is constant, and also the live or moving loads coming upon them, such as railway trains, vehicular and pedestrian traffic, &c., which vary in amount. All these forces

act vertically.

(3) Wind pressure, which affects all exposed structures. This is a variable force acting horizontally or nearly so, and in some cases is the most important of all the external forces.

4. Stresses and Strains.-All members of a loaded structure, except those inserted for ornament, are exposed to stresses and strains. These two terms are often used indiscriminately as meaning the same thing; strictly speaking, however, this is not so, and in this work each term will (as far as possible) be used in its proper sense. Generally speaking, they represent cause and effect.

A strain is a change of form. When an external force acts upon a bar of any material, it produces in it a change of form, no matter how minute. It may elongate it, or shorten it, or bend it. This change of form is termed a "strain." Strains may be temporary or permanent. If, after the force be removed, the bar regain its original shape and dimensions, the strain on its fibres will only be temporary, and only last during the application of the force. If, on the other hand, the bar do not regain its original shape and dimensions after the removal of the force, it is said to be permanently strained.

By a stress is meant the internal force or resistance set up in the fibres of the bar in opposing the strain.

The stresses in materials are proportional to the strains, so long as there is no permanent alteration in the form of the body acted upon.

There are three kinds of stresses and strains :

(1) Compressive or positive stresses and strains. (2) Tensile or negative stresses and strains.

(3) Shearing stresses and strains.

If a bar of any material be acted upon by two equal forces applied at its extremities, and acting away from each other in the direction of its length, it becomes extended, and the strains produced in the fibres are said to be tensile or negative strains.

The stresses or resistance to the straining action on the fibres are, in like manner, termed tensile or negative stresses.

If, on the other hand, the bar be acted upon by two equal forces acting towards each other, it becomes shortened, and the strains and stresses generated are termed compressive or positive strains and stresses.

A tensile stress on a bar tends to cause its failure by lengthening and ultimately tearing apart its fibres; a compressive stress induces failure by shortening and ultimately crushing its fibres; and a shearing stress produces failure by causing one part to slide across the other, or by cutting it across.

Besides these three kinds of stresses, there are others which are frequently to be met with, the most common of which are transverse or bending stresses and torsional or twisting stresses, with their corresponding strains; but these and other forms, as will subsequently be shown, may be resolved into one or more of the three elementary forms named.

5. Measurement of Stresses and Strains.-In England and where English standards are adopted, a stress is measured by so many pounds, cwts., or tons. A unit stress is usually measured by so many pounds, cwts., or tons per square inch of sectional area of the body under stress. According to the French standards of measurements, the unit of stress is reckoned as so many kilogrammes per square centimetre. If the stress on a bar of iron 2 inches square be 50 tons, the unit stress, or stress per square inch, will be 12.5 tons. The corresponding unit stress in French measure is 1,968 kilogrammes per square centimetre.

A strain is usually measured in inches or parts of an inch. A unit strain is measured in parts of an inch per lineal foot of the bar under strain, or it may be measured as so much per cent. of the length of the bar. If a bar of steel 2 feet long, under a certain tensile stress, be lengthened by half an inch, the strain produced is equal to inch per foot, or 2.08 per cent. of the length of the bar.

A

6. Tensile Stress.-Fig. 1 is an example of tensile stress. bar, a b, of section A square inches, is suspended at one extremity, a; and a weight of W tons is hung from the other end, b. The bar under these conditions is said to be subjected to a tensile stress of W tons throughout its entire length, or to

7. Compressive Stress.-Fig. 2 is an example of compressive stress. A pillar rests on the ground, and a weight of W tons rests on the top. The pillar is under a compressive stress of W tons, and if A = its sectional area in square inches, the fibres are

W

A

subjected to a compressive stress of tons per square inch

throughout its entire length.

8. Shearing Stress.-Figs. 3 and 4 are examples of shearing stresses. Two links of iron or steel are joined together by a pin of the same material, and are exposed to forces of W tons acting in the directions of the arrows.

In both cases the pin is subjected to a shearing stress. In fig. 3 the pin is exposed to a single shearing stress of W tons at its section, a b, and is said to be in single shear, and if number of square inches in its sectional area, the shearing

=

A

W
A

tons.

In fig. 4 the pin is exposed to a double shearing stress at its sections, a b, and c d, and is said to be in double shear.

W
2

The

shearing stress at each section is equal to tons, and if

sectional area of the pin, the shearing stress per square

2 A,

area of the pin in fig. 4 need only be one half of that in fig. 3, in order to be of equal strength.

9. Transverse Stress.-Fig. 5 is an example of transverse or bending stresses. The beam, A B, rests on two supports at A and B, and is loaded at an intermediate point by a weight, W; the beam in this condition is said to be exposed to a bend

W

Fig. 5.

ing or transverse stress; but, as will be shown in a future chapter, the fibres in the upper portion of the beam are subjected to a compressive stress, and those in the lower portion to a tensile stress, while shearing stresses also

come into operation throughout the beam.

10. Torsional or twisting stresses do not often occur in the members of structural work; but they are to be met with in the shafting of machinery of all descriptions.

It is usually assumed that the stress on a bar is uniformly distributed over the whole cross-section of the bar. This is generally true as regards tensile and compressive stresses so long as the bar is of a compact form. It is only approximately true, however, when applied to shearing stresses, and the form of the pin or bar exposed to this stress has something to do with this want of uniformity. It has been found from experiments that the maximum intensity of shearing stress on a round pin is somewhat greater than the mean intensity, and with a pin of a square or rectangular section, the difference is greater still. For all practical purposes, however, it may be assumed that the stress is uniformly distributed over the section. This being so, if

f= unit stress, or number of lbs. or tons per square

a =

of section of the bar.

inch

number of square inches of sectional area of the bar. F total stress in lbs. or tons on the bar.

Example 1.-If the ultimate tensile strength of mild steel be 32 tons per square inch, what force will be necessary to tear asunder a bar of this material 4 inches in diameter?

Substituting these values of ƒ and a in equation (1), we get

which is the tensile force necessary to rupture the bar.

Example 2.-What force will crush a short column of cast iron, 8 inches in external, and 6 inches internal diameter, the ultimate compressive strength of the metal being 40 tons per square inch?

Example 3.-If the bars shown in fig. 3 be pulled with a

force of 50 tons acting in opposite directions, what must be the diameter of the pin so that the shearing stress on it may be equal to 10 tons per square inch?

From equation (1) we have, by transposing,

d2 = 6.36, or d = 2.5 inches, the required diameter.

Example 4.-In the last example, if the pin be in double shear as shown in fig. 4, what must be its diameter in order to fulfil the same conditions?

Example 5.-If a rectangular tie-beam of oak, 6 inches by 4 inches, be subjected to a tensile stress of 50 tons, what will be the stress per square inch exerted on its fibres ?

Example 6.-A bar of wrought iron of any uniform section is suspended from one end, and hangs vertically, what must be its length so as to break by its own weight, the ultimate strength of the iron being 20 tons per square inch, and the weight of a cubic inch being 0.28 lbs. ?

Let = length of the bar in feet.
its section in square inches.

Breaking weight of)

the bar in pounds

Weight of bar in

pounds