140 lbs. per cubic foot, then the weight of the mass = 140 X 10 × 3 × 3 = 12,600 lbs. If then a force acts at P, say 5 feet from the ground, its value when the mass is on the point of overturning will be 12600 × 1.5

5

= 3,780 lbs. The overturning moment and that of stability may also be compared by the parallelogram of forces. If P represents the horizontal pressure against the mass, produce the horizontal line to intersect gk inj; make jh equal to P, and jl equal to the weight of the mass; complete the parallelogram jhol, then jo will be the resultant thrust on the pillar; if this resultant fall outside the point c, the mass will overturn; but if it fall at its intersection n between c and d, the mass will be stable. For safety in practice this point is made to fall in the middle third of the base, or the least value of cn= economy of material, this value has been made as low as c d

4

cd; but in some works, for the sake of

3

The harder and stronger the material, the nearer c may the resultant be permitted to pass; but if this point should break off, the base is immediately shortened, and the stability correspondingly reduced.

If the mass is on the point of falling, the resultant will pass through c, and the horizontal force and the weight will be in the same ratio as half the thickness to the height jm; if we halve the pressure P, the overturning moment will be halved, and the value cn= ; hence this position

cd

4

of the resultant shows a resistance equal to twice the ex

c d

ternal force. To make cn= P must be reduced to one

" 3

third of its first value, so the resistance will be three times the external force.

In Fig. 76 abji represents a column made of four pieces.

of equal mass; the whole must be stable on ij, and the superincumbent parts must each be stable on their beds

it: the stability will here be greatest on the top joint m n, and gradually decrease to its minimum at st.

Although the stones may not overturn, yet it may happen that one slides upon that beneath it; hence the circum

stances under which this may occur must be investigated. This is the resistance of friction, and a margin of resistance in this, as in other respects, must be provided in any structure, to insure durability.

Let а mass W, Fig. 77, rest upon a bed or surface A B,

and let a force P act upon it in the direction indicated by the arrow. Through the point a, where the force P acts upon the mass W, draw a vertical line, and make a c equal to P, and complete the parallelogram abed; then a d will

represent the pressure acting at right angles to A B, and ab the component acting parallel to it, and tending to slide W upon it. When the inclination of Pa is such that W is on the point of sliding upon A B, the angle o ar is called ba

the limiting angle of friction, and is the coefficient of

a d

friction, being the ratio giving the horizontal force necessary to slide the mass W under any given pressure acting at right angles to A B.

If no extraneous force is in action, then so long as the bed A B remains horizontal, the only force to which it is subject is the pressure or weight W acting vertically, and therefore at right angles to it; but let the bed be inclined, as shown at A' B', to the horizontal A'C, the weight W will now be resolved into two forces, one acting at right angles to A'B', and the other in a direction parallel to A' B'; the latter tending to cause W to slide down the plane. From g, the centre of gravity of the mass, draw the vertical line gh, and from i, its point of intersection with A'B', mark off i equal to W, and complete the parallelogram i klm; then ik is the sliding component of the weight, and im the ik

component pressure on A'B'; will be the coefficient of km

friction as before, and if the mass is on the point of slipping, the angle gin will be the limiting angle of friction, and the angle B' A' C is called the angle of repose, or the natural slope of the material. It will now be shown that the angle of repose is equal to the limiting angle of friction.

A' li being a right angle, as A'C is horizontal and gh vertical, the angle A'il is the complement of the angle i A'l, and nim being at right angles to A'B', A'il is the complement to lim, which is therefore equal to i A'l; but it is also equal to the limiting angle of friction gin; hence the angle of repose is equal to the limiting angle of friction.

It is evident, then, that in any structure, the resultant force upon a joint must not make, with a line at right angles to such joint, an angle greater than the limiting angle of friction, or angle of repose, for the material under treatment, and of course must be sufficiently within it to afford a proper margin of safety.

As we can put the joints at any angle we choose, they can always be made so that the resultant thrusts are at right angles to them, or nearly so.

The following table selected from various experiments shows the angle of repose and coefficient of friction for different materials:

The angles of repose for the earths are those at which

these materials will permanently stand, not the angles at

which they first break away, which, for reasons shortly to be shown, are much steeper.

The structures which depend upon their stability for their safety are arches in masonry, abutments, buttresses, retaining walls of various descriptions, sea-walls and breakwaters, chimneys, towers, lighthouses, and all structures liable from their localities to be washed away or blown over bodily. Large bridges in exposed and stormy places are at times called upon to resist overturning forces of great magnitude. The general force of the wind is given in the following table:

The actual measured force of waves has been found to amount to 4,335 lbs. per square foot at Skerryvore, 3,013 lbs. at Bell Rock, and the highest observed 6,000 lbs.