Page:Encyclopædia Britannica, Ninth Edition, v. 4.djvu/352

308 B K I D G E S [ARCHES. same time the rotation of the stones A and C, coupled with the descent of B, tends to push back the abutments N and Q, and therefore to increase their horizontal reaction supposing them to be stable. If the abutments N, Q continue to yield, the stones A and B will continue to turn until the points of contact reach 4 and a 3 , or 5j and b 4 . The horizontal thrust which the abutments require to meet will therefore diminish as the stones turn, and the little structure will only fail to support the weight in case the abutments N and Q are insuificicntly strong or stable to supply the minimum thrust consistent with an equilibrated polygon Fig. 47. cutting the joints inside the ring (or in case the polygon cut the joints at such an angle that the stones slip). If, on the other hand, the abutments were so made as to press in upon A and C with a greater horizontal force than is consistent with two lines of pressure passing through the actual points of contact, then, as in fig. 47, the direction of the couples on the stones A and B would be reversed, and they will roll round so as to bring the points of contact more nearly into the position required to meet an excessive horizontal thrust, and at the same time the changed position of the stones, by allowing N and Q to come forward, will tend to relieve or diminish the original excessive horizontal thrust, where this is due to the elasticity of the stones N and Q, or of the stones supporting these abutments. The structure will not fail unless the points of con tact reach % and 4 , or b. 2 and 6 3 , when the structure would fail by the sides being squeezed in, and the stone B being lifted up out of the arch. This could not happen with stones of the proportions shown in fig. 47, as before the limiting position was reached, the points of contact would lie on a straight line corresponding to an infinite horizontal thrust. In conclusion we see that, whether the hori zontal force supplied by the reaction of the abutting stones be too small or too great, the three voussoirs tend to move so as to adapt the centre of pressure and the actual horizontal force to one another. The equilibrium produced is stable, that is to say, if by some external force the arrangement of the blocks is slightly disturbed, when the force is removed the blocks return to their original position. In the above demonstration it is assumed that the blocks when first put together touch at some point not far from the centre of the bed,- a condition corresponding to reasonably good fitting in the case of the plane joints of a stone arch before the centring is removed. If a model be prepared (fig. 48), having a number of voussoirs of wood with their bed-joints slightly curved and roughened, the result of the above theory will be very clearly and beautifully seen. The action explained in the case of three blocks holds good for any three, and therefore for the whole series. If an additional weight is placed at the crown, as in fig. 48, the crown is a little lowered, but the curve passing through the lines of contact rises at the crown and is lowered at the haunch by the rotation of the blocks, until the lines of contact at the joints arrange themselves, so that the re sultant pressures forming the imaginary polygon pass through these lines of contact. If the extra load be placed at the haunches the ci ovvn rises, but the points or lines of contact between the voussoirs are lowered at the crown and raised at the haunches, as in fig. 4S. If one haunch only is weighted, the curve passing through the lines of contact rises at that haunch and is lowered at the other, as in fig. 485 ; if the model be distorted by the hand it oscillates up and down on each side of the position of equilibrium, as a string similarly loaded would do. Figures 48, 48a, and 486 are taken from photographs of a model. (It should be remarked that the abut- Fig. 48. ments were screwed to the supporting board ; it is obvious that otherwise they would not have been in equilibrium.) The general character of the curve passing through the points of contact may be easily conceived by thinking of a string similarly loaded and inverted. The equilibrated arch will be one of those forms which, a string might take when similarly loaded, but when the load is changed, the length of the curve will not be constant in the arch, whereas it must be constant with any given chain. The curve pass ing through the points of contact corresponds with what Moseley called the line of resistance. The direction of the pressure is not necessarily tangent to this curve, but in the ordinary form of bridge it is nearly so. In the model each voussoir is free to roll, because the bed-joints are curved. In an actual bridge the bed-joints are plane, nevertheless, the stones do turn round to adapt themselves to the pressure, but the result of this rotation is to render the compression along the upper and lower halves of the stone unequal. One edge is more compressed than the other ; the couple tending to turn the voussoir, and actually allowed to do so in the model, is met by an equal and opposite couple, due to the unequal compression of the stone. This couple is the necessary result of a pressure which is not axial, vide 8 ; an equilibrated polygon cutting the joints at various dis tances from the centre is therefore as correct an indication of the actual forces present in a practical arch with flat joints as in the model with curved joints ; but we must remember that where the joints are flat, the pressure will be unequally distributed wherever

the line of the equilibrated polygon does not cut the centre of the