This line can be completely determined by the methods of analysis, in respect to a structure of any given geometrical form, having its parts in contact by surfaces also of given geometrical forms. And, conversely, the form of this line being assumed, and the direction which it shall have through any proposed structure, the geometrical form of that structure may be determined, subject to these conditions; or lastly, certain conditions being assumed, both as it regards the form of the structure and its line of resistance, all that is necessary to the existence of these assumed conditions may be found. Let the structure ABCD have for its line of resistance the line PQ. Now it is clear that if this line cut the surface MN of any section of the mass in a point n without the surface of the mass, then the resultant of the pressures upon the mass CMN will act through n,

D

Q

B

E

P

and cause this portion of the mass to revolve about the nearest point N of the intersection of the surface of section MN with the surface of the structure.

Thus, then, it is a condition of the equilibrium that the line of resistance shall intersect the common surface of contact of each two contiguous portions of the structure actually within the mass of the structure; or, in other words, that it shall actually go through each joint of the structure, avoiding none: this condition being necessary, that no two portions of the structure may revolve on the edges of their common surface of contact.

THE LINE OF PRESSURE.

284. But besides the condition that no two parts of the structure should turn upon the edges of their common surfaces

of contact, which condition is involved in the determination of the LINE OF RESISTANCE, there is a second condition necessary to the stability of the structure. Its surfaces of contact must no where slip upon one another. That this condition may obtain, the resultant corresponding to each surface of contact must have its direction within certain limits. These limits are defined by the surface of a right cone (Art. 139.), having the normal to the common surface of contact (at the above-mentioned point of intersection of the resultant) for its axis, and having for its vertical angle twice that whose tangent is the coefficient of friction of the surfaces. If the direction of the resultant be within this cone, the surfaces of contact will not slip upon one another; if it be without it, they will.

Thus, then, the directions of the consecutive resultants in 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.

All

5

ь

Q

6

4

N

2

Let then a line ABCDE be taken, which is the locus of the consecutive intersections of the resultants aA, bВ, cС, 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

Line of Best

Resista

a

Line of Pressufe

TO

18

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.

Thus the whole theory of the equilibrium of any structure is involved in the determination with respect to that struc

ture 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 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 Fig. 1. 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

Fig. 2.

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 (h2 + k2)' — 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 deflected as to 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 can 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 of 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 variable 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,

* 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 4ths the distance from the latter to the external edge of the base. (See Poncelet, Mémoire sur la Stabilité des Revêtemens, 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.