Page:Encyclopædia Britannica, Ninth Edition, v. 4.djvu/351

Rh AKCHES.] BRIDGES 307 forces t and w-^ an: known, those determine the magnitude and direction of the equilibrating force t,, which must act at A to balance them. Let the direction AB and the magnitude of the force t l be Fig. 4 ia. Fie. 44. found by the ordinary parallelogram of forces, as in fig. 44, and let B be the point of intersection of the direction of this force with the line 2 ; then the direction and the magnitude of the equilibrating force t. 2 can be found as for ^ ; similarly the direction of this force gives the point C by its intersection with the line 3, and finally we obtain, by the resolution of forces, the direction, magnitude, and posi tion of the force ( 3, by means of which the reaction of the second abutment will keep the system in equilibrium. When the position and magnitude of t are known, the position and magnitude of all the other forces are determinate ; the conditions of equilibrium are that the lines NA, AB, BC, and CQ, shall not cut the joints above or below the edges a or b, for in that case the blocks would heel over on the edge beyond which the resultant passed ; also, the direction of the lines NA, AB, &c., must be such as not to exceed the angle of repose with the normal to the joints, otherwise one stone will slip on the other. The abutment producing by its reac tion the force t 3 must not yield with a less force than t 3 , and must not be pushed forward so as to produce a greater force than t 3 . The line NABCQ, if inverted, is in form identical with that which a cord would assume, loaded at the points A, B, and 0, with the loads 1, 2, and 3, and having the direction of NA determined. This line will, in the rest of this article, be called an equilibrated polygon, &quot;When the joints are supposed indefinitely near, or the voussoirs thin sheets, the equilibrated polygon becomes a curve called a linear arch. The reasoning applied to three blocks is clearly applicable to any number, and we may therefore say that any series of loaded voussoirs will be in equilibrium when a reaction of known magnitude and direction is applied at one abutment, provided the equilibrated polygon required by this reaction and the given loads can be drawn so that its sides cut all the joints within the ring (or within the Fig. 45d. middle tnird, where strength is an element of the question; at an angle greater than the complement of the angle of repose for the material used. An equilibrated polygon, ABC Q, for a complete arch is shown in fig. 45 Fig. 45a is the diagram giving the slopes K J, JI, &c., as for the loaded chain fig. 30 and 31 39. It will be shown in next paragraph that the arch will be in equilibrium if with any value of the horizontal thrust h an equilibrated polygon can be drawn fulfilling the conditions required. In most arches equilibrated polygons fulfilling these conditions can be drawn with values of h varying between two limits differing by a considerable amount. In that case the smallest value of h will be the true value, and give the true stresses ; for the abutments being inert will not give back a greater thrust than is just required to balance the structure. This would render the thrust h determinate if the equilibrated polygon might actually approach the true edge of the ring, but as this would require infinitely strong materials, we are still left in uncertainty as to the true value of h, but may feel sure that it will be the smallest value consistent with a safe stress on the material. If we provide abutments capable of reacting with a force h sufficient to keep the equilibrated polygon (where it cuts the joints) within the middle third, our abutment will certainly be amply strong enough, and this is the value of h to be adopted in all practical calculations of the stability of an arch. In the example with the three voussoirs it is clear that equilibrium would be obtained with very widely different reactions t at the one abutment. And this fact is also true for a bridge of many voussoirs. The vertical com ponent (which may be called v) of this total thrust t is indeed determinate if we suppose the point where t cuts the joint to be known, being the same as the vertical reac tion from a beam carrying the same weights and supported at the points where t and v cut the abutments ; but the horizontal component or horizontal thrust, which has been called k, cannot be determined by any considerations hitherto mentioned. 40. Experimental Demonstration that the Equilibrium of a series of Voussoirs is stable if any Equilibrated Polygon can be drawn fulfilling the conditions stated above. Let us suppose an arch, fig. 48, to be constructed, the bed-joints of which are not plane but curved, so that each stone touches its neighbour only along a horizontal line, the trace of which in a drawing may be called the point of contact. Such an arch will differ from an ordinary arch in this respect, that the centre of pressure at joints will be shown by the points of contact, while the stones will be able by rolling to alter the points of contact if not in equilibrium. In such an arch the voussoirs in the first place may be put together so as to touch at any desired series of points, but the forces called into play when external support is withdrawn will rearrange the voussoirs so as to bring them into equilibrium, if any equilibrated arch consistent with the loads can be drawn so that the lines forming it cut the joints inside the ring, and a model will show the points of contact, or, in other words, the places where these lines cut the joints. (It is assumed that the obliquity of the sides of the polygon to the joints which they cut is insufficient to produce slipping.) An actual model shows the action very prettily, but the follow ing considerations will easily allow the student to see how it is that the voussoirs always arrange themselves so as to build a time arch. Suppose, first, that the arch consisted merely of three stones, fig. 46, and that the weight on the centre one was so great that the linear arch, or equilibrated polygon, became sensibly two inclined straight lines like rafters. As soon as the voussoirs are left to them selves, the pressure at the surface a t b v and the reaction at the surface, b lt will lie in one straight line, which, meeting a similar straight line from the other abutment, will give one equilibrated polygon, satisfying the required conditions ; but if the horizontal force required for this polygon is not supplied by the abutments, the two forces at joints 1 and 2 will, as shown by the small straight arrows, constitute a couple tending to turn the stone A round, so that the point of contact at joint 1 will be lower, and the point of contact in joint 2 will be higher than before. The same action will occur in stone 0, and the result will be that the weight may be balanced with a smaller horizontal force. At the