When the load on one of the pieces is parallel to the resistances which balance it, the polygon of resistances ceases to be closed, two of the sides becoming parallel to each other and to the load in question, and extending indefinitely. In the polygon of loads the direction of a load sustained by parallel resistances traverses the point O.[2]
§ 8. How the Earth’s Resistance is to be treated.... When the pressure exerted by a structure on the earth (to which the earth’s resistance is equal and opposite) consists either of one pressure, which is necessarily the resultant of the weight of the structure and of all the other forces applied to it, or of two or more parallel vertical forces, whose amount can be determined at the outset of the investigation, the resistance of the earth can be treated as one or more upward loads applied to the structure. But in other cases the earth is to be treated as one of the pieces of the structure, loaded with a force equal and opposite in direction and position to the resultant of the weight of the structure and of the other pressures applied to it.
§ 9. Partial Polygons of Resistance.—In a structure in which there are pieces supported at more than two joints, let a polygon be constructed of lines connecting the centres of load of any continuous series of pieces. This may be called a partial polygon of resistances. In considering its properties, the load at each centre of load is to be held to include the resistances of those joints which are not comprehended in the partial polygon of resistances, to which the theorem of § 7 will then apply in every respect. By constructing several partial polygons, and computing the relations between the loads and resistances which are determined by the application of that theorem to each of them, with the aid, if necessary, of Moseley’s principle of the least resistance, the whole of the relations amongst the loads and resistances may be found.
§ 10. Line of Pressures—Centres and Line of Resistance.—The line of pressures is a line to which the directions of all the resistances in one polygon are tangents. The centre of resistance at any joint is the point where the line representing the total resistance exerted at that joint intersects the joint. The line of resistance is a line traversing all the centres of resistance of a series of joints,—its form, in the positions intermediate between the actual joints of the structure, being determined by supposing the pieces and their loads to be subdivided by the introduction of intermediate joints ad infinitum, and finding the continuous line, curved or straight, in which the intermediate centres of resistance are all situated, however great their number. The difference between the line of resistance and the line of pressures was first pointed out by Moseley.
| Fig. 88. |
§ 11.* The principles of the two preceding sections may be illustrated by the consideration of a particular case of a buttress of blocks forming a continuous series of pieces (fig. 88), where aa, bb, cc, dd represent plane joints. Let the centre of pressure C at the first joint aa be known, and also the pressure P acting at C in direction and magnitude. Find R1 the resultant of this pressure, the weight of the block aabb acting through its centre of gravity, and any other external force which may be acting on the block, and produce its line of action to cut the joint bb in C1. C1 is then the centre of pressure for the joint bb, and R1 is the total force acting there. Repeating this process for each block in succession there will be found the centres of pressure C2, C3, &c., and also the resultant pressures R2, R3, &c., acting at these respective centres. The centres of pressure at the joints are also called centres of resistance, and the curve passing through these points is called a line of resistance. Let all the resultants acting at the several centres of resistance be produced until they cut one another in a series of points so as to form an unclosed polygon. This polygon is the partial polygon of resistance. A curve tangential to all the sides of the polygon is the line of pressures.
§ 12. Stability of Position, and Stability of Friction.—The resistances at the several joints having been determined by the principles set forth in §§ 6, 7, 8, 9 and 10, not only under the ordinary load of the structure, but under all the variations to which the load is subject as to amount and distribution, the joints are now to be placed and shaped so that the pieces shall not suffer relative displacement under any of those loads. The relative displacement of the two pieces which abut against each other at a joint may take place either by turning or by sliding. Safety against displacement by turning is called stability of position; safety against displacement by sliding, stability of friction.
§ 13. Condition of Stability of Position.—If the materials of a structure were infinitely stiff and strong, stability of position at any joint would be insured simply by making the centre of resistance fall within the joint under all possible variations of load. In order to allow for the finite stiffness and strength of materials, the least distance of the centre of resistance inward from the nearest edge of the joint is made to bear a definite proportion to the depth of the joint measured in the same direction, which proportion is fixed, sometimes empirically, sometimes by theoretical deduction from the laws of the strength of materials. That least distance is called by Moseley the modulus of stability. The following are some of the ratios of the modulus of stability to the depth of the joint which occur in practice:—
| Retaining walls, as designed by British engineers | 1 : 8 |
| Retaining walls, as designed by French engineers | 1 : 5 |
| Rectangular piers of bridges and other buildings, and arch-stones | 1 : 3 |
| Rectangular foundations, firm ground | 1 : 3 |
| Rectangular foundations, very soft ground | 1 : 2 |
| Rectangular foundations, intermediate kinds of ground | 1 : 3 to 1 : 2 |
| Thin, hollow towers (such as furnace chimneys exposed to high winds), square | 1 : 6 |
| Thin, hollow towers, circular | 1 : 4 |
| Frames of timber or metal, under their ordinary or average distribution of load | 1 : 3 |
| Frames of timber or metal, under the greatest irregularities of load | 1 : 3 |
In the case of the towers, the depth of the joint is to be understood to mean the diameter of the tower.