Multiplied by

MULTIPLIED BY its horizontal distance from d. These conditions give two equa tions which contain two unknown quantities—the thrust and the distance its point of application is above c. After solving these equations, the line of resistance can be drawn by any of the methods already explained.

"If this new line of resistance lies entirely within the prescribed limits, it is plain that it is possible to draw a line of resistance therein; but if the second line does not lie within the prescribed limits, it is not at all probable that a line of resistance can be drawn therein. The possibility of finding, by a third or subsequent trial, a line of resistance within the limits can not, in general, be answered definitely, since such a possibility depends upon the form of the section of the arch ring.

"If the line of resistance drawn through U and L goes outside of [the middle third of] the arch ring beyond the extrados only, as at a, the second line -of resistance should be drawn through c and L; and if, on the other hand, it goes outside below the intrados only, as at b, the second line should be drawn through U and d." BoBErrLER'S TEEO$Y. This theory is the one most fre quently employed. It is based upon the hypothesis of least crown thrust (§ 1215), and assumes that the external forces are vertical.

In Scheffler's theory the joint of rupture is found by trial sub stantially as explained in § 1218-22; and the crown thrust is, for example, the maximum value in the third to the last column of Table 89, page 624. In other words, Scheffler's theory is the same as the Rational Theory (§ 1229-41), except that it neglects the horizontal components of the earth pressure, and therefore uses a smaller crown thrust and usually also a different joint of rupture. For the arch in Fig. 195, page 623, according to Scheffler's theory joint 4 is the joint of rupture, and 8,706 lb. is the crown thrust (see Table89). The line of resistance is determined exactly as for the rational theory except that the load line for Scheffler's theory is straight. The lines of resistances (not the equilibrium polygon) for both the rational and Scheffler's theory for the arch shown in Fig. 195, page 623, are given

in Fig. 202. In this particular case, the difference between the two lines above the joint of rupture is not material; but the dif ference below that joint has an important effect upon the thickness of the arch at the springing, and also upon the thickness of the abutment (§ 1246).

If the maximum ratio of the horizontal to the vertical component of the external forces (see last paragraph on page 616) had been employed in determining the crown thrust and the line of resistance, there would have been a greater difference in the position of both the joint of rupture and the line of resistance. Although the hori zontal components of the external forces can not be accurately deter mined, any theory that disregards them is needlessly inaccurate.

Erroneous Application.

Not infrequently the principle of the joint of rupture is entirely neglected in applying this theory; that is to say, the crown thrust employed in determining the line of resistance is that which would produce equilibrium of rotation about the springing line, instead of that which would produce equilibrium about the joint of rupture. For example, instead of employing the maximum value in the third to the last column of Table 89, page 624, the last quantity in that column is used. The line of resistance ob tained by this method is shown in Fig. 202 by the dotted line; the crown thrust (5,990, as computed in Table 89) being laid off from the lower end of w, to E, to the scale employed in laying off the load line.

The amount of the error is illustrated in Fig. 202. According to this analysis, the line of resistance is tangent to the intrados, which seems to show that the arch can not stand for a moment. However, many such arches do stand, and carry a heavy railroad traffic without any signs of weakness; and further, any reasonable method of analysis shows that the arch is not only safe, but even extravagantly so (4 1234)_ This method of analysis certainly ac counts for some, and perhaps many, of the excessively heavy arches built in the past.