BTABII.ITY or ABu'r i wrs. The stability of the abutments is in a measure indeterminate, since it depends upon the position of the line of resistance of the arch. The stability of the abutment may be determined most easily by treating it as a part of the arch, i.e., by extending the load line so as to include the forces acting upon it and drawing the reactions in the usual way.
In Fig. 202 (page 639) is shown the line of, resistance for the abutment according to the rational theory of the arch (I 1229-41), and also that according to Scheffler's theory (§ 1242-44), the former by the solid line and the latter by the broken one. Since to over estimate the horizontal components of the external forces would be to err on the side of danger, in applying the former theory in Fig. 202 the horizontal component acting against the abutment was disre garded on the assumption that the abutment might be set in a pit without greatly disturbing the surrounding earth and consequently there might not be any appreciable horizontal earth thrust against the abutment. If the horizontal component had been considered, the difference between the lines of resistance for the two theories would have been still greater. Notice that the analysis which recog nizes tic existence of the horizontal forces, i.e., the rational theory, permits a lighter abutment than the theory which assumes the exter nal forces to be entirely vertical.
The omission of the horizontal components assumes that the only object of the abutment is to resist the thrust of the arch; and that consequently the flatter the arch the greater the thrust and the heavier the abutment. Ordinarily the abutment must resist the
thrust of the arch tending to overthrow it and to slide it outward, and must act also as a retaining wall to resist the lateral pressure of the earth tending to overthrow it and to slide it inward. For large arches the former is the more important; but for small arches, particularly under high embankments, the latter is the more impor tant. Hence, for a large arch or for an arch having a light surcharge, the abutment should be proportioned to resist the thrust of the arch; but for a small arch under a heavy surcharge of earth, the abutment should be proportioned as a retaining wall (Chap. XVIII).
Although the horizontal pressure of the earth can not be com puted accurately, there are many conditions under which the hori zontal components should not be omitted. For example, if the abutment is high, or if the earth is deposited artificially behind it, ordinarily it would be safe to count upon the pressure of the earth to assist in preventing the abutment from being overturned out ward. Finally, although it may not always be wise to consider the earth pressure as an active force, there is always a passive resistance which will add greatly to the stability of the abutment, and whose intensity will increase rapidly with any outward movement of the abutment.