Rational

An equilibrium polygon can be made to pass through any three points; and therefore we may assume three points for a trial equilib rium polygon,—as, for example, (1) the lower limit of the middle third of the joint at the abutment A, (2) the middle, C, of the crown joint, and (3) the upper limit of the middle third of the joint at B.

Construct a force diagram by laying off the external forces suc cessively from 0 in the usual way, selecting a pole, P', at any point, and drawing lines connecting P' with the points of division of the load line. Draw a line from 0 to 8, and then 08 is the resultant of all the loads upon the arch. Then commencing at A, one of the given points, construct a preliminary equilibrium polygon AC'B' by the method explained in § 1231. Draw the closing line AB'; and in the force diagram draw a line HP', through P' parallel to AB'. Then OH and H8 are the reactions at A and B, respectively.

The next step is to find a pole such that the equilibrium polygon will pass through C and B instead of C' and B'. In the force diagram draw a line from 0 to 4, which is the resultant of the four forces to the right of C. Through C draw a line CC' parallel to the line 04. Connect the points A and C, and also A and C'; and then the lines AC and AC' are the closing lines of the equilibrium polygons for the forces to the right of C. Through P' draw a dividing ray parallel to AC' cutting 04 at K; and then OK and K4 represent the reactions of the forces to the right of C, acting at A and C respectively. The point H is common to all force polygons for the given system of forces; and the point K is common to all force polygons for the forces to the right of C. If the polygon is to pass through C, then AC is a closing line; and consequently the pole of the force diagram must lie on a line through K parallel to AC. Similarly, if the polygon

is to pass through B, then AB is the closing line; and consequently the pole must lie on a line through H parallel to AB. Therefore, if from H and K lines be drawn respectively parallel to AB and AC, their intersection, P, is the pole of the polygon which will pass through A, C, and B. This polygon is shown in Fig. 200.

If a line of resistance can not be drawn within the pre scribed limits, then the section of the arch ring must be changed so as to include the line of resistance within the desired limits.

Criterion.

If the line of resistance, when constructed by any of the preceding methods, does not lie within the middle third of the arch ring, the following process may be employed to determine whether or not it is possible to draw a line of resistance in the middle third. This method is strictly applicable only for vertical forces, and hence• is approximate when horizontal forces are considered; but it is sufficiently exact for the purpose.* "Assume, for example, that the line of resistance of Fig. 201 lies outside of the middle third at a and b. Next draw a line of resistance through c and d, the points where normals from a and b intersect the outer and inner boundary of the middle third respectively. To pass a line of resistance through c and d, it is necessary to determine the value and point of application of the corresponding crown thrust. The condition which makes the line of resistance pass through c is: the thrust MULTIPLIED BY the vertical dis tance of its point of application above c Is