The preceding investigation is approximate for the following reasons: 1. The effect of the horizontal forces is omitted. 2. TV, x, and y are dependent variables, and not independent as assumed. 3. In the interpretation of equation 9, instead of "the tangent to the intrados," should be employed the tangent to the line of resistance.
In applying this method, a table, computed by M. Petit, which gives the angle of rupture in terms of the ratio of the radii of the intrados and the extrados, is generally employed. The table in volves the assumption that a, Fig. 194 (page 621), is in the extrados and b in the intrados; and also that the intrados and extidos are parallel. According to this table, "a semicircular arch of which the thickness is uniform throughout and equal to the span divided by seventeen and a half is the thinnest or lightest arch that can stand. A thinner arch would be impossible." If the line of resistance is restricted to the middle third, then, according to this theory, the thinnest semicircular arch which can stand is one whose span is five and a half times the uniform thickness. Many arches in which the thickness is much less than one seventeenth of the span stand and carry heavy loads without showing any evidence of weakness. For example, the span of arch No. 23 of Table 90, page 648, is 93 times the thickness of the arch ring, and still it has stood since 1750 without any signs of failure.
Owing to the approximations involved, and also to the limita tions to arches having intrados and extrados parallel, the ordinary tables for the position of the joint of rupture have little, if any, practical value. The only satisfactory way to find the angle of rup ture is by trial by equation 6, page 621, as explained in § 1218-23.
According to M. Petit's table, if the thickness is one fortieth of the span, the angle of rupture is 46° 12'; if the thickness is one twentieth, the angle is 53° 15'; and if one tenth, 59° 41.' In conclusion, notice that the investigations of both this and the preceding section show that an arch of more than about 90° to 120° central angle is impossible.
are not usually stated; and, as a rule, the theory is presented in such a way as to lead the reader to believe that each particular method "is free from any indeterminateness, and gives results easily and accurately." Every theory of the voussoir arch is approximate, ow ing to the uncertainty concerning the amount and distribution of the external forces (§ 1205), to the indeterminateness of the position of the true line of resistance (§ 1210-23), to the neglect of the in fluence of the adhesion of the mortar and of the elasticity of the material, and to the lack of knowledge concerning the strength of masonry; and, further, the stresses in a voussoir arch are indeter minate owing to the effect of variations in the material of which the arch is composed, to the effect of imperfect workmanship in dressing and bedding the stones, to the action of the center—its rigidity, the method and rapidity of striking it,—to the spreading of the abutments, and to the settling of the foundations. These elements are indeterminate, and can never be stated accurately or adequately in a mathematical formula; and hence any theory can be at best only an %pproximation. The influence of a variation in any one of these factors can be approximated only by a clear comprehension of the relation which they severally bear to each other; and hence a thorough knowledge of theoretical methods is necessary for the intelligent design and construction of arches.
Three of the most important theories will now be considered.
To save repetition, it may be mentioned here, once for all, that every theory of the arch is but a method of verification. The first step is to assume the dimensions of the arch outright, or to make them agree with some existing arch or conform to some em pirical formula. The second step is to test the assumed arch by the theory, and then if the line of resistance, as determined by the theory, does not lie within the prescribed limits—usually the middle third, the depths of the voussoirs must be altered, and the design must be tested again.