Having made these statements, we shall proceed to show how, by the assistance of theoretical results, they may be applied to the wants of the practical engineer.

The absolute strength of ropes or bars, pulled length wise, is in proportion to the squares of their diameters. All cylindrical or prismatic rods are equally strong in every part, if they are equally thick, but if not, they will break where the thickness is least.

The lateral strength of any beam or bar of wood, stone, metal, &c., is in proportion to its breadth × its depth3.— In square beams the lateral strengths are in proportion to the cubes of the sides, and in general of like-sided beams as the cubes of the similar sides of the section.

The lateral strength of any beam or bar, one end being fixed in the wall and the other projecting, is inversely as the distance of the weight from the section acted upon: and the strain upon any section is directly as the distance of the weight from that section.

If a projecting beam be fixed in a wall at one end, and a weight be hung at the other, then the strain at the end in the wall, is the same as the strain upon a beam of twice the length, supported at both ends and with twice the weight acting on its middle. The strength of a projecting beam is only half of what it would be, if supported at both ends.

If a beam be supported at both ends, and a weight act upon it, the strain is greatest when the weight is in the middle; and the strain, when the weight is not in the middle, will be to the strain when it is in the middle, as the product of the weight's distances from both ends, is to the square of half the length of the beam.-Take any two oints in a beam supported at both ends; call one of these ints a and the other b; then a weight hung at a will produce a strain at b, the same as it would do at a if hung at b.

A

P

C

B

In a beam supported at the ends A and B; the strain at C, with the whole weight placed there, is to the strain at C with the whole weight placed equally between C and P, as AC is to AP x PC; and the strain at C by a weight placed equally along AP, is to the strain at C by the same weight placed on C, as AP is to AC.

If beams bear weights in proportion to their lengths, either equally distributed over the beams or placed in similar points, the strains upon the beams will be as their lengths2.

If a beam rest upon two supports, and at the same time be firmly fixed in a wall at each end, it will bear twice as much weight as if it had lain loosely upon the supports; and the strain will be everywhere equal between the supports.

In any beam standing obliquely, or in a sloping direction, its strength or strain will be equal to that of a beam of the same breadth, thickness, and material, but only of the length of the horizontal distance between the points of support.

Similar plates of the same thickness, either supported at

the ends or all round, will carry the same weight either uniformly distributed or laid on similar points, whatever be their extent.

The strength of a hollow cylinder, is to that of a solid cylinder of the same length and the same quantity of matter, as the greater diameter of the hollow cylinder is to the diameter of the solid cylinder; and the strength of hollow cylinders of the same length, weight, and material, are as their greater diameters.

The lateral strength of beams, posts, or pillars, are diminished the more they are compressed lengthwise.

The strength of a column to resist being crushed is directly as the square of the diameter, provided it is not so long as to have a chance of bending. This is true in metals or stone, but in timber the proportion is rather greater than the square.

The strength of homogeneous cylinders to resist being twisted round their axes, is as the cubes of their diameters; and this holds true of hollow cylinders, if their quantities of matter be the same.

PROBLEMS.

To find the strength of direct cohesion :

Area of transverse section in inches x measure of,cohesion strength in lbs. to resist being pulled asunder.

Ex. In a square bar of beech, 3 inches in the side, we have 3 × 3 × 9912 = 89208 lbs.

NOTE. The measure of cohesion for timber is taken from col. C, table A, and for other materials, from tables B or C.

In a beam of English oak, having four equal sides, each side being four inches, we have

4 x 4 x 9836 = 157376 lbs., the strength.

In a rod of cast steel, 2 inches broad and 11⁄2 inch thick, we have 2 × 1 × 134256 =402768 lbs., the strength. What is the greatest weight which an iron wire of an inch thick will bear?

The area of the cross section of such wire will be ⚫007854, hence we have 007854 × 84000 = 659.736 lbs.

To find the ultimate transverse strength of any beam : When the beam is fixed at one end and loaded at the other then the dimensions being in inches,

breadth x depth x transverse strength

length of beam

transverse strength.

the ultimate

NOTE.-In column S, Table A, will be found the transverse strength of timber, and in table E, that of iron, &c., and let it be observed, that when the beam is loaded uniformly, the result of the last rule must be doubled.

What weight will break a beam of Riga fir, fixed at one end and loaded at the other, the breadth being 3, depth 4, and length 60 inches?

What weight uniformly distributed over a beam of English oak would break it, the breadth being 6, depth 9, and its length 12 feet?

If the number be taken from table F, we must use the length in feet.

When the beam is supported at both ends, and loaded in the centre,

tabular value of S, tab. A x depth x breadth × 4

NOTE.-When the beam is fixed at one end and loaded in the middle, the result obtained by the rule must be increased by its half. When the beam is loaded uniformly throughout, the result must be doubled. When the beam is fixed at both ends and loaded uniformly, the result must be multiplied by three.

Ex. What weight will it require to break a beam of English oak, supported at both ends and loaded in the middle, the breadth being 6, and depth 8 inches, and length 12 feet?

1672 × 8 × 6 × 4

By using table E:

144

= 17834.

depth breadth × tabular number
length in feet

Ex.-What weight will a cast iron bar bear, 10 feet ong, 10 inches deep, and 2 inches thick, laid on its edge'

To find the breadth to bear a given weight.

length weight

number in table E x depth

breadth.

What must be the breadth of an oak beam, 20 feet long and 14 inches deep, to sustain a weight of 10000 lbs. ?

In a beam 1 ft. deep and 4 in. broad, the weight being 5000 lbs.; then we have, if the beam be made of Memel fir,

122 x 4 x 130

5000

To find the depth:

14.97 feet, length required.

length x weight

tabular number breadth) = depth.

We wish to support a weight of 2000 lbs. by a beam of American pine; what is its depth, its length being 20 feet and breadth 4 inches?

√(2000 x 20) = √ (145)

2o) = ✓ (145) = 12 inches, nearly.

69 ×

To find the deflection of a beam fixed at one end, and loaded at the other:

length of beam in inches3 × 32 × weight tab. numb. E (in table A) × breadth × depth3 flection in inches.

= de

NOTE.-If the beam be loaded uniformly, use 12 instead of 32 in the rule.

If a weight of 300 be hung at the end of an ash bar fixed