Hypotheses for the Crown Thrust

HYPOTHESES FOR THE CROWN THRUST. From 3f 1195 it is clear that the position of the line of resistance can not be known until the amount, the direction, and the point of application of the crown thrust are known.

Each value for the intensity of the thrust at the crown gives a different line of resistance. For example, in Fig. 186 (page 609), if the thrust T be increased, the point b—where R. intersects the plane of the joint GI—will approach I; and consequently c, d, and e will approach J, K, and A respectively. If T be increased sufficiently, the line of pressure will pass through A or K (usually the former, this depending, however, upon the dimensions of the arch and the values and directions of F„ and and the arch will be on the point of rotating about the outer edge of one of these joints. This value of T is then the maximum thrust at a consistent with stability of rotation about the outer edge of a joint, and the corresponding line of resistance is the line of resistance for maximum thrust at a. Similarly, if the thrust T be gradually decreased, the fine of resistance will approach and finally intersect the intrados, in which case the thrust is the least possible consistent with stability of rotation about some point in the intrados. The lines of resistance for maximum and minimum thrust at a are shown in Fig. 192 (page 618).

If the point of application of the force T be gradually lowered and at the same time its intensity be increased, a line of resistance may be obtained which will have one point in common with the intrados. This is the line of resistance for maximum thrust at the crown joint. Similarly, if the point of application of T be gradually raised, and at the same time its intensity be decreased, a line of resistance may be obtained which will have one point in common with the extrados. This is the line of resistance for minimum thrust at the crown joint. The lines of resistance for maximum and mini mum thrust at the crown are shown in Fig. 193.

Similarly each direction of the thrust T will give a new line of resistance. In short, every different value of each of the several factors, and also every combination of these values, will give a dif ferent position for the line of resistance. Hence, the problem is to determine which of the infinite number of possible lines of resist ance is the actual one. This problem is indeterminate, since there are more unknown quantities than conditions {equations) by which to determine them. To meet these difficulties and make a solution of the problem possible, various hypotheses have been made. Four of these hypotheses will now be considered briefly.

Hypothesis of Least Pressure.

Some writers have assumed the true line of resistance to be that which gives the smallest absolute pressure on any joint. This principle is a metaphysical one, and

leads to results unquestionably incorrect: Of the four hypotheses here discussed this is the least satisfactory, and the least frequently employed. It will not be considered further.

For an explanation of Claye's method of drawing the line of pressure according to this theory, see Van Nostrand's Engineering Magazine, Vol. xv, p. 33-36. For a general discussion of the theory of the arch founded on this hypothesis, see an article by Pro fessor Du Bois in Van Nostrand's Engineering Magazine, Vol. xui, p. 341-46, and also Du Bois's " Graphical Statics," Chapter xv.

Winkler's Hypothesis.

Professor Winkler, of Berlin,—a well known authority—published in 1879 in the Zeitschrift des Architekten and Ingenieur Vereins zu Hannover, page 199, the following theorem concerning the position of the line of resistance: "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 deter mined by the method of least squares."* The only proof of this theorem is that by it certain conclusions can be drawn from the voussoir arch which harmonize with the ac cepted theory of solid elastic arches. The demonstration depends upon certain assumptions and approximations, as follows: 1. It is assumed that the external forces acting on the arch are vertical; whereas in many cases, and perhaps in most, they are inclined. 2. The loads are assumed to be uniform over the entire span; whereas in many cases the arch is subject to moving concentrated loads, and sometimes the permanent load on one side of the arch is heavier than that on the other. 3. The conclusions drawn from the voussoir arch only approximately agree with the theory of elastic arches. 4. A masonry arch does not ordinarily have a constant cross section as required by the above theorem; but it usually, and properly, increases toward the springing. 5. The phrase " as deter mined by the method of least squares" means that the true line of resistance is that for which the sum of the squares of the vertical deviations is a minimum. Since the joints must be nearly perpen dicular to the line of resistance, the deviations should be measured normal to that line. For a uniform load over the entire arch, the lines of resistance are comparatively smooth curves; and hence, if the sum of the squares of the vertical deviations is a minimum, that of the normal also would probably be a minimum. But for eccentric or concentrated loads, it is by no means certain that such a relation would exist. 6. The degree of approximation in this theorem is less the flatter the arch.