Line of Pressure.—If an arch ring be divided into a number of voussoirs, and the points of application of the resultant pressures upon the joints between these voussoirs be determined, the broken or curved line joining these points of application is known as the line of pressure for the arch. In Fig. 85 the line abcdef is called the line of pressure for the half arch, when H is the crown thrust and P9, etc., are the external loads coming upon the several divisions. The true line of pressure, or of resistance, is a curve circumscribing the poly gon abcdef. The larger the number of divisions of the arch ring, the more nearly «•i1I the polygon approach this curve.
in determining the line of pressure, the arch ring is divided into a convenient number of parts, usually six to sixteen on each side of the crown, and the external loads
Fig. S
Hypotheses for Line of Fig. 85 represent half of a symmetrically loaded arch, the crown pressure H will be horizontal. Assuming its point of application, a, and that its line of resistance passes through a definite point on one of the other joints as f, the amount of II may be found by taking a center of moments at f and writing the moment equation for all the loads upon the half arch equal to zero. II is then known in amount, direction and point of application and the line of pressure may be drawn, as shown.
Several hypotheses have been proposed for the purpose of fixing the position of the line of thrust. Professor Durand-Clayc assumed that the true line of resistance is that which gives the smallest abso lute pressure upon any joint. This method is outlined in Van Nostrand's Engineering Magazine, Vol. XV, p. 33. Professor Winkler suggested that "for an arch ring of constant cross-section, that line of resistance is approximately the true one which lies nearest to the axis of the arch ring, as determined by the method of least squares." No practicable method of applying this principle to ordi
nary cases of voussoir arches has been devised. _Moseley's hypothesis was that the true line of resistance is that for which the thrust at the crown is the least consistent with stability. This occurs (Fig. 85) when II is at the highest and I? at the lowest point it can occupy on the joint. This hypothesis is the basis of Scheflier's method of drawing the line of resistance.
Scheffler's theory assumes that II is applied at the upper edge of the middle third of the crown joint, and that the value of H is such as to cause the line of pressure to touch the lower edge of the middle third at one of the joints (as d, e, or f) nearer the abutment. The joint at which the line of pressure is tangent to the lower edge of the middle third is known as the joint of rupture. The joint of rupture may be found by taking moments about the lower edge of the middle third of each of several joints and solving for H. All loads acting between the joint considered and the crown should he used in obtain ing the moment, and the one giving the largest value of H is the joint of rupture. The value of 11 so determined is the least consistent with stability, as a less value causes the line of pressure to pass out side the middle third at the joint of rupture.
Should it be found that the line of pressure passes outside the middle third on the upper side of any of the joints between the joint of rupture and the crown, the point of application of H may be lowered without violating the hypothesis. This leads to the usual statement that "if any line of pressure can be drawn within the middle third of the arch ring the arch will be stable." This is justified by common experience.
When the loading upon the arch is not symmetrical, this method of finding the crown thrust cannot be used, and in this case it is usual to select throe points through which to pass the line of pressure, one at the crown and one near each abutment. A line of pressure is then passed through these three points, and if the line so found does not remain within the middle third of the arch ring the positions of the points may be changed and new lines constructed. This may he repeated until it is determined whether any line of pressure can be drawn within the middle third.