To apply Winkler's theorem, it is necessary to (1) construct a line of resistance, (2) measure its deviations from the axis of the arch, and (3) compute the sum of the squares of the deviations; and it is then necessary to do the same for all possible lines of resistances, the one for which the sum of the squares of the deviations is least being the " true " one.
Instead of applying Winlder's theorem as above, many writers employ the following principle, which it is asserted follows directly from that theorem: "If any line of resistance can be con structed within the middle third of the arch ring, the true line of resistance lies within the same limits, and hence the arch is stable." This assertion is disputed by Winkler himself, who says it is not, in general, correct.* It does not necessarily follow that because one line of resistance lies within the middle third of the arch ring, the "true" line of resistance also does; for the "true" line may coin cide very closely with the axis in one part of the arch ring and depart considerably from it in another part, and still the sum of the squares of the deviations be a minimum. This method of applying Winkler's theorem is practically nothing more or less than an appli cation of the conclusions derived from the hypothesis of least pres sure (1 1211), and will not be considered further.
Navier's Principle. Navier's principle is: The tangential stress at any point of a circle pressed by normal forces is equal to the normal pressure per unit of area multiplied by the radius of curvature of the surface. Rankine applied the above principle to voussoir arches as follows: "The condition of an arch of any form at any point where the pressure is normal is similar to that of a cir cular rib of the same curvature under a normal pressure of the same intensity; and hence the following theorem: The thrust at any nor mally pressed point of a linear arch is the product of the radius of curva ture by the intensity of the pressure at that point. Or, denoting the radius of curvature by p, the normal pressure' per unit of length of intrados by p, and the thrust by T, we have At best, the above formula gives only the amount and by impli cation the direction of the crown thrust; but tells nothing about its point of application. Rankine employed the above crown thrust
to find two points of the line of resistance; and assumed that if a line of resistance can be drawn anywhere within the middle third of the arch ring, that the arch was stable (1 1245). The use of this principle determines the line of resistance only within limits; and in general gives no information as to the stability of the arch against sliding or crushing, and gives a result for the stability against over turning at only two joints. Rankine's theory 1245) of the arch is the only one that employs this principle, and hence it will not be considered further here.
Hypothesis of Least Crown Thrust. This is the last of the four hypotheses to be considered, and is the one almost universally employed in theories of the voussoir arch.* According to this hy pothesis the true line of resistance is that for which the thrust at the crown is the least possible consistent with equilibrium. This assumes that the thrust at the crown is a passive force called into action by the external forces; and that, since there is no need for a further increase after it has caused stability, it will be the least possible consistent with equilibrium. This principle alone does not limit the position of the line of resistance; but, if the external forces are known and the direction of the thrust is assumed, thie hypothesis furnishes a condition by which the line of resistance corresponding to a minimum thrust can be found by a tentative process.
To find the crown thrust that will satisfy the above hypothesis proceed as follows: The portion of the arch shown in Fig. 194 is held in equilibrium by (1) the vertical forces, w„ w,, etc., (2) by the horizontal forces h„ h„ etc., (3) by the reaction R at the abutment, and (4) by the thrust T at the crown. The direction of R is imma terial in this discussion. Let a and b represent the points of application of T and R, respec tively, although the location of these points is yet undetermined. Let T = the thrust at the crown; = the horizontal distance from b to the line of action of w,; = the same for w,; etc.; y = the perpendicular distance from b to the line of action of T; k, = the perpendicular distance from b to the line of action of h,; k, = the same for h,; etc.