In the abutments of arches, in piers and detached buttresses, and in towers and chimneys exposed to the pressure of the wind, it has been found by experience to be advisable so to limit the deviation of the centre of pressure from the centre of figure, that the maximum intensity of the pressure, supposing it to be an uniformly varying pressure (see Article 94), shall not exceed the double of the mean intensity. As in Article 94, let P be the total pressure; S be the mean intensity of the pres

Р

=

S = Po

the area of the joint; let sure, which is also the intensity at the centre of figure of the joint, and at each point in a neutral axis traversing that centre of figure; let x be the perpendicular distance of any point from that axis, and let the pressure at that point be p Poax, so that if x, be the greatest positive distance of a point at the edge of the joint from the neutral axis, the maximum pressure will be

Now, by the condition stated above, p, 2P, and, consequently,

=

Р

..(1.)

If the diameter of the joint is bisected by the centre of figure, and if x (as in Article 94) be the distance of the centre of pressure from the neutral axis, we shall have

and by inserting in this equation the value of x, as given by equation 4 of Article 94, and having regard to the value of a, as given by equation 1 of this Article, we find

an expression whose value depends wholly on the figure of the joint-that is, of the transverse section of the abutment, pier, buttress, tower, or chimney.

Referring to the table at the end of Article 95 for the values of the moment of inertia I, the following results are obtained for joints of different figures. In each case in which there is any difference in the values of q for different directions, the deviation of the centre of pressure is supposed to take place in that direction in which the greatest deviation is admissible-that is to say, at right angles to the neutral axis for which I is a maximum ; so that if h be the diameter in that direction, x,

h

2

When the solid parts of the hollow square and of the circular ring are very thin, the expressions for 9 in Examples VI. and VII. become approximately equal to the following:

which values are sufficiently accurate for practical purposes when applied to square and round factory chimneys.

The conditions of stability of a block supported upon another block at a plane joint may be thus summed up:

Referring to fig. 93, Article 191, let A A represent the upper block, BB part of the lower block, e E the joint, C its centre of pressure, PC the resultant of the whole pressure distributed over the joint, whether arising from the weight of the upper block, or from forces applied to it from without. Then the conditions of stability are the following:

I. The obliquity of the pressure must not exceed the angle of repose, that is to say,

II. The ratio which the deviation of the centre of pressure from the centre of figure of the joint bears to the length of the diameter of the joint traversing those two centres, must not exceed a certain fraction, whose value varies, according to circumstances, from one-eighth to three-eighths, that is to say,

The first of these conditions is called that of stability of friction, the second, that of stability of position.

206. Stability of a Series of Blocks; Line of Resistance; Line of Pressures. In a structure composed of a series of blocks, or of a series of courses so bonded that each may Rbe considered as one block, which blocks P or courses press against each other at plane joints, the two conditions of stability must be fulfilled at each joint.

R

3

4

Fig. 95.

2

Let fig. 95 represent part of such a structure, 1, 1, 2, 2, 3, 3, 4, 4, being some of its plane joints.

Suppose the centre of pressure C, of the joint 1,1, to be known, and also the amount and direction of the pressure, as indicated by the arrow traversing C. With that pressure combine the weight of the block 1, 2, 2, 1, together with any other external force which may act on that block; the resultant will be the total pressure to be resisted at the joint 2, 2, will be given in magnitude, direction, and position, and will intersect that joint in the centre of pressure C. By continuing this process there are found the centres of pressure Č1, C4, &c., of any number of successive joints, and the directions and magnitudes of the resultant pressures acting at those joints.

The magnitude and position of the resultant pressure at any joint whatsoever, and consequently the centre of pressure at that joint, may also be found simply by taking the resultant of all the forces which act on one of the parts into which that joint divides the structure, precisely as in the "method of sections" already described in its application to framework, Article 161.

The centres of pressure at the joints are sometimes called centres of resistance. A line traversing all those centres of resistance, such as the dotted line R, R, in fig. 95, has received from Moseley the name of the "line of resistance;" and that author has also shown

ANALOGY OF BLOCKWORK AND FRAMEWORK.

231

how in many cases the equation which expresses the form of that line may be determined, and applied to the solution of useful problems.

The straight lines representing the resultant pressures may be all parallel, or may all lie in the same straight line, or may all intersect in one point. The more common case, however, is that in which those straight lines intersect each other in a series of points, so as to form a polygon. A curve, such as P, P, in fig 95, touching all the sides of that polygon, is called by Moseley the "line of pressures."

The properties which the line of resistance and line of pressures must have, in order that the conditions of stability may be fulfilled, are the following:

To insure stability of position, the line of resistance must not deviate from the centre of figure of any joint by more than a certain fraction (q) of the diameter of the joint, measured in the direction of deviation.

To insure stability of friction, the normal to each joint must not make an angle greater than the angle of repose with a tangent to the line of pressures drawn through the centre of resistance of that joint.

207. Analogy of Blockwork and Framework.—The point of intersection of the straight lines representing the resultant pressures at any two joints of a structure, whether composed of blocks or of bars, must be situated in the line of action of the resultant of the entire load of the part of the structure which lies between the two joints; and those three resultants must be proportional to the three sides of a triangle parallel to their directions.

Hence the polygon formed by the intersections of the lines representing the pressures at the successive joints in fig. 95, is analogous to a polygonal frame; for the sides of that polygon represent the directions of resistances, which sustain loads acting through its angles, as in the instances of framework described in Articles 150, 151, 153, and 154, and represented in fig. 75. A structure of blocks is especially analogous to an open polygonal frame, like those in Articles 151 and 154, represented by fig. 75, with the piece E omitted because of the absence of ties.

The question of the stability of a structure composed of blocks with plane joints may therefore be solved in the following manner :(1.) Determine and lay down on a drawing of the structure the line of action and the magnitude of the resultant of the external forces applied to each block, including its own weight. Either one or two of those resultants, as the case may be, will be the supporting force or forces.

(2.) Draw a polygon of external forces, like that in fig. 75* or 75**. Two contiguous sides of that polygon will represent the external for

acting on the two extreme blocks of the series, of which one may be a supporting pressure and the other a load, or both may be supporting pressures. In either case their intersection gives the point O, from which radiating lines are to be drawn to the angles of the polygon of external forces, to represent the directions and magnitudes of the resistances of the several joints.

(3.) Draw a polygon having its angles on the lines of action of the external forces, as laid down in step (1.) of the process, and its sides parallel to the radiating lines of step (2). This polygon will represent the equivalent polygonal frame of the given structure, and will have a side corresponding to each joint; and each side of the polygon (produced if necessary) will cut the corresponding plane joint in its centre of pressure, and will show the direction of the resultant pressure at the joint.

Then if each centre of pressure falls within the proper limits of position, and the direction of each resultant pressure within the proper limits of obliquity, as prescribed in Article 205, the structure will be balanced; and the conditions of stability will be fulfilled under variations of the distribution of the load, which will be the greater, the greater is the diameter of each joint; for every increase in the diameters of the joints increases the limits within which the figure of the equivalent polygonal frame may vary, and every variation of that figure corresponds to a variation in the distribution of the load.

208. Transformation of Blockwork Structures.-THEOREM. If a structure composed of blocks have stability of position when acted on by forces represented by a given system of lines, then will a structure whose figure is a parallel projection of the original structure have stability of position when acted on by forces represented by the corresponding parallel projection of the original system of lines; also, the centres of pressure and the lines representing the resultant pressures at the joints of the new structure will be the corresponding projections of the centres of pressure and the lines representing the resultant pressures at the joints of the original structure.

For the relative volumes, and consequently the relative weights, of the several blocks of which the structure is composed, are not altered by the transformation; and if those weights in the new structure be represented by lines, parallel projections of the lines representing the original lines, and if the other forces applied externally to the pieces of the new structure be represented by the corresponding parallel projections of the lines representing the corresponding forces applied to the pieces of the original structure, then will each external force acting on the new structure be the parallel projection of a force acting on the corresponding point of the original structure; therefore the resultant pressures at the