respect to the normal to the point, where each intersects its corresponding surface of contact, are to be considered as important elements of the theory.

d

B

M

a

2

Let then a line ABCDE be taken, which is the locus of the consecutive intersections of the resultants aA, bB, cC, dD, &c. The direction of the resultant pressure upon every section is a tangent to this line; it may therefore with propriety be called the LINE OF PRESSURE. Its geometrical form may be determined under the same circumstances as that of the line of resistance. A straight line cC drawn from the point c, where the LINE OF RESISTANCE abcd intersects any joint 5 6 of the structure, so as to touch the LINE OF PRESSURE ABCD, will determine the direction of the resultant pressure upon that joint: if it lie within the cone spoken of, the structure will not slip upon that joint; if it lie without it, it will.

Lone of Res

Line of Pressure

10

Thus the whole theory of the equilibrium of any structure is involved in the determination with respect to that structure of these two lines-the line of resistance, and the line of pressure one of these lines, the line of resistance, determining the point of application of the resultant of the pressures upon each of the surfaces of contact of the system; and the other, the line of pressure, the direction of that resultant.

The determination of both, under their most general forms, lies within the resources of analysis; and general equations for their determination in that case, in which all the surfaces of contact, or joints, are planes-the only case which offers itself as a practical case-have been given by the author of this work in the sixth volume of the "Cambridge Philosophical Transactions."

THE STABILITY OF A SOLID BODY.

285. The stability of a solid body may be considered to be greater or less, as a greater or less amount of work must be done upon it to overthrow it; or according as the amount

Fig. 1.

Fig. 2.

of work which must be done upon it to bring it into that position in which it will fall over of its own accord is greater or less. Thus the stability of the solid represented in fig. 1. resting on a horizontal plane is greater or less, according as the work which must be done upon it, to bring it into the position represented in fig. 2., where its centre of gravity is in the vertical passing through its point of support, is greater or less. Now this work is equal (Art. 60.) to that

which would be necessary to raise its whole weight, vertically, through that height by which its centre of gravity is raised, in passing from the one position into the other. Whence it follows that the stability of a solid body resting upon a plane is greater or less, as the product of its weight by the vertical height through which its centre of gravity is raised, when the body is brought into a position in which it will fall over of its own accord, is greater or less.

If the base of the body be a plane, and if the vertical height of its centre of gravity when it rests upon a horizontal plane be represented by h, and the distance of the point or the edge, upon which it is to be overthrown, from the point where its base is intersected by the vertical through its centre of gravity, by k; then is the height through which its centre of gravity is raised, when the body is brought into a position in which it will fall over, evidently represented by (+)-h; so that if W represent its weight, and U the work necessary to overthrow it, then

U is a true measure of the stability of the body.

THE STABILITY OF A STRUCTURE.

286. It is evident that the degree of the stability of a structure, composed of any number of separate but contiguous solid bodies, depends upon the less or greater degree of approach which the line of resistance makes to the extrados or external face of the structure; for the structure cannot be thrown over until the line of resistance is. so def ected as tc

intersect the extrados: the more remote is its direction from that surface, when free from any extraordinary pressure, the less is therefore the probability that any such pressure will overthrow it. The nearest distance to which the line of resistance approaches the extrados will, in the following pages, be represented by m, and will be called the MODULUS OF STABILITY of the structure.

This shortest distance presents itself in the wall and buttress commonly at the lowest section of the structure. It is evidently beneath that point where the line of resistance intersects the lowest section of the structure that the greatest resistance of the foundation should be opposed. If that point be firmly supported, no settlement of the structure car take place under the influence of the pressures to which it is ordinarily subjected.*

THE WALL OR PIER.

287. The stability of a wall.

If the pressure upon a wall be uniformly distributed along its length, and if we conceive it to be intersected by vertical planes, equidistant from one another and perpendicular to its face, dividing it into separate portions, then are the conditions of its stability, in respect to the pressures applied to its entire length, manifestly the same with the conditions. the stability of each of the individual portions into which it is thus divided, in respect to the pressures sustained by that portion of the wall; so that if every such columnar portion or pier into which the wall is thus divided be constructed so as to stand under its insistent pressures with any degree of firmness or stability, then will the whole structure stand with the like degree of firmness or stability; and conversely.

In the following discussion these equal divisions of the length of a wall or pier will be conceived to be made one foot apart; so that in every case the question investigated will be that of the stability of a column of uniform or varia

* A practical rule of Vauban, generally adopted in fortifications, brings the point where the line of resistance intersects the base of the wall, to a distance from the vertical to its centre of gravity, of ths the distance from the latter to the external edge of the base. (See Poncelet, Mémoire sur la Stabilité des Revétemens, note, p. 8.)

In the wall of a building the pressure of the rafters of the roof is thus uniformly distributed by the intervention of the wall plates.

ble thickness, whose width measured in the direction of the length of the wall is one foot.

288. When a wall is supported by buttresses placed at equal distances apart, the conditions of the stability will be made to resolve themselves into those of a continuous wall,

if we conceive each buttress to be extended laterally until it meets the adjacent buttress, its material at the same time so diminishing its specific gravity that its weight when thus spread along the face of the wall may remain the same as before. There will thus be obtained a compound wall whose external and internal portions are of different specific gravities; the conditions of whose equilibrium remain manifestly unchanged by the hypothesis which has

been made in respect to it.

THE LINE OF RESISTANCE IN A PIER.

289. Let ABEF be taken to represent a column of uniform dimensions. Let PS be the direction of any pressure P sustained by it, intersecting its axis in O. Draw any horizontal section IK, and take ON to represent the weight of the portion AKIB of the column, and OS on the same scale to represent the pressure P, and complete the parallelogram ONRS; then will OR evidently represent, in magnitude and direction, the resultant of the pressures upon the portion AKIB of the mass (Art. 3.), and its point of intersection Q with IK will represent a point in the line of resistance. Let PS intersect BA (produced if necessary) in G, and let GC=k, AB=a, AK=x, MQ=y, POC=a, weight of each cubic foot of the material of the mass. Draw RI per pendicular to CD; then, by similar triangles,

EX D

QM RL
OMPOL

But QM=y, OM-CM-CO=x-k cot.

a, RL=RN

sin. RNL P sin. a, OL-ON+NL=ON+RN cos. RNL

ax+P cos. a;

which is the general equation of the line of resistance of a pier or wall.

290. The conditions necessary that the stones of the pier may not slip on one another.

Since in the construction of the parallelogram ONRS, whose diagonal OR determines the direction of the resultant pressure upon any section IK, the side OS, representing the pressure P in magnitude and direction, remains always the same, whatever may be the position of IK; whilst the side ON, representing the weight of AKIB, increases as IK descends the angle ROM continually diminishes as IK descends. Now, this angle is evidently equal to that made by OR with the perpendicular to IK at Q; if, therefore, this angle be less than the limiting angle of resistance in the highest position of IK, then will it be less in every subjacent position. But in the highest position of IK, ON=0, so that in this position ROM=a. Now, so long as the inclination of OR to the perpendicular to IK is less than the limiting angle of resistance, the two portions of the pier separated by that section cannot slip upon one another (Art. 141.). It is therefore necessary, and sufficient to the condition that no two parts of the structure should slip upon their common surface of contact, that the inclination a of P to the vertical should be less than the limiting angle of resistance of the common surfaces of the stones. All the resultant pressures passing through the point O, it is evident that the line of pressure (Art. 284.) resolves itself into that point.