Rational

RATIONAL TENCoaY. The following method of determining the line of resistance is based upon the hypothesis of least crown thrust (§ 1215), and recognizes the existence of the horizontal com ponents of the earth pressure. Two forms of loading will be con sidered, viz.: a symmetrical and an unsymmetrical load.

Symmetrical Load. As an example of the application of this theory, let us investigate the stability of the semi-arch shown in Fig. 196 under a symmetrical dead load. Two solutions will be given for a symmetrical load, viz.: I, a general solution; and II, a special solution which is shorter when the location of the joint of rupture can be foretold by inspection.

I.

General Solution. The first step is to determine the line of resistance. The maximum crown thrust was computed in Table 89 (page 624), as already explained (I 1218-22). To construct the force diagram, a line T is drawn to scale to represent the maximum thrust as found in the fifth line of the last column of Table 89. From the right-hand end of T, w, is laid off vertically upwards; and from its extremity, h, is laid off horizontally to the left. Then the line from the left-hand extremity of h, to the lower end of w, (not shown in this particular case) represents the direction and amount of the external force F, acting upon the first division of the arch ring; and the line R, from the upper extremity of F, represents the resultant pressure of the first arch stone upon the one next below it. Similarly, lay off w, vertically upwards from the left hand extremity of h„ and lay off h, horizontally to the left; then a line F, from the lower end of w, to the left-hand end of h, rep resents the resultant of the external forces acting on the second divisions of the arch, and a line R, from the upper extremity of F, represents the resultant pressure of the second arch stone on the third. The force diagram is completed by drawing lines to represent the other values of w, h, F, and the corresponding reactions.

In the diagram of the arch, the points in which the horizontal and vertical forces acting upon the several arch stones intersect, are marked g„ g„ etc., respectively; and the oblique line through each of these points, drawn parallel to F,, F,, etc., of the force diagram, shows the direction of the resultant external force acting on each arch stone.

To construct the line of resistance, draw through the upper limit of the middle third of the crown joint a horizontal line to an intersection with the oblique force through g,; and from this point draw a line parallel to R„ and prolong it to an intersection with the oblique force through g,. In a similar manner continue to the spring

ing line.

Then the intersection of the line parallel to R, with the first joint gives the center of pressure on that joint; and the intersection of R, with the second joint gives the center of pressure for that joint, —and so on for the other joints. A line connecting these centers of pressure would be the line of resistance; but the line is not shown in Fig. 196.

Notice, for example, that to get an intersection of the line parallel to R, with joint 7, the line must be projected backward from its intersection with the oblique line through g,. Care is re quired to prevent mistakes in such cases. Notice also that the line of resistance must pass through the inner end of the middle third of the joint of rupture. This relation affords a valuable check upon the accuracy of the work-of drawing the line of resistance.

Stability against Overturning. Since the line of resistance lies within the middle third of the arch ring, and touches the outer limit of the middle third at the crown and the inner limit at joint 5, the approximate factor of safety against rotation (equation 13, page 468) is 3.

The true factor of safety (equation 12, page 467) is Ag' - rg', Fig. 98, page 466. To apply this formula it is necessary to find g' for a particular joint, i.e., it is necessary to find the center of pressure for the resultant of the normal forces. To find the point g', proceed as follows: From the point of intersection of the F and the R cor responding to the particular joint, draw a line perpendicular to the joint; and then the point in which this normal component pierces the joint is the center of pressure of the normal forces, i.e., the point g'. In this connection, it is important to remember that the point A is the edge of the joint on the opposite side of r (the point in which the resultant of all the forces pierces the joint) from g'. Proceeding as above, the true factor of safety (equation 12, page 467) and also the approximate factor (equation 13, page 468) are computed as below: The above values show that at some points there is a marked difference between the true and the approximate factor of safety.