Hypotheses for the Crown Thrust

The Vertical Forces. The weight of the several sections of the arch and of the earth above them can readily be computed. The sums of these weights, w„ w„ w„ etc., are given in Table 89, page 624.

The center of gravity of each voussoir may be determined as explained in $ 935; but the center of gravity of the prism of earth resting upon each arch stone may, without material error, be taken as acting through its medial vertical line. The center of gravity of the combined weights can be determined by moments. The position of each of the resultants of the several combined weights is shown in Fig. 195.

The moment arm of each of the combined weights with reference to the several origins of moments is measured in Fig. 195, and then entered in Table 89. Far example, the 1.00 in the column headed x,, is the arm of w, about the lower end of the middle third of joint 1; similarly 3.07 is the arm of w, about the lower end of the middle third of joint 2; and 0.70 in the column headed x, is the arm of w, about the origin of moments in joint 2, etc.

The Horizontal Forces. The horizontal thrust of the earth can be computed as stated in * 1209. The values of h,, h,, etc., the horizontal components of the earth thrust, are given in Table 89.

The forces h„ h,, ptc., are applied at the middle of the vertical projections of the upper ends of the respective voussoirs. The mo ment arms of h„ h,, etc., are measured in Fig. 195, and then entered in Table 89. For example, 1.08 in the column headed k, is the arm of h, about the origin of moments in joint 1; 1.76 is the arm of about the origin of moments in joint 2; etc.

The Value of y. The crown thrust T is assumed to be applied at the upper end of the middle third of the crown joint. The arm y is the distance from T to the origin of moments for the several joints. For example, 0.76 in the column headed y in Table 89 is the vertical distance from the upper end of the middle third of the crown joint to the lower end of the middle third of joint 1; and sim ilarly for the other values.

The Joint of Rupture. The crown thrust according to equation 6, page 621, for the several joints is given in the last column of Table 89. An inspection of the results shows that the crown thrust for joint 5 is greater than that for any other joint; and there fore joint 5 is the joint of rupture, and all the arch masonry below joint 5 is virtually only part of the abutment.

The angular distance of the joint of rupture from the crown is called the angle of rupture.* In Fig. 195, page 623, the angle of rum ture is 46° 30'. The angle of rupture is usually between 45° and 60°. Any increase in the assumed intensity of the horizontal components increases the computed value of the angle of rupture. For example, if the quantities in the next to the last column of Table 89 be doubled, the thrust for joint 7 will be the maximum.

Probably this condition could be realized by tightly tamping the earth spandrel-filling.

Notice that the preceding discussion of the position of the joint of rupture is for a uniform stationary load. The angle of rupture for a concentrated moving load will differ from the results found above; but the mathematical investigation of the latter case is too complicated and too uncertain to justify attempting it.

Incorrect Method of Finding Joint of Rupture.

There is a method of determining the joint of rupture in somewhat common use which is inaccurate because of the omission of the horizontal components of the earth pressure and also because of two errors in the mathematical work. This method virtually assumes that the crown thrust is correctly given by the first term of the right-hand side of equation 6, page 621; and proceeds to find the joint of rup ture as follows: Let W = the total weight resting on any. joint; x = the hori zontal distance of the center of gravity of this weight from the origin of moments; and y = the arm of the crown thrust. Then equation 6 becomes To determine the condition for a maximum, it is assumed that W, and y are independent variables. Differentiating equation 7, regarding T, y, and W i as independent variables, ['herefore the condition for a maximum crown thrust is A common method of differentiating equation 7 is more simple but less accurate, and arrives at the same conclusion as above.* The usual interpretation of equation 9 is: "The joint of rupture is that joint at which the tangent to the intrados passes through the intersection of T and the resultant of all the vertical forces above the joint in question." The position of the joint of rupture can be found by the above principle only by trial; and this method possesses no advantage over the one explained in 4 1218-23, and is less con venient to apply.