 this hypothesis of only downward pressure on the arch-stones is not in accordance with the fundamental principle of stability. In a thorough. investigation this hypothesis must be set aside, and the oblique pressure on the inner ends of the arch-stones must be taken into account. Since the depth of the arch-stones is small in comparison with the whole dimensions of the structure, and since the line of the pressure transmitted from one to another of them must always be within that depth, it is admissible to suppose, for the purpose of analysing the strains, that the arch-stones form an exceedingly thin course, and that their joints are everywhere normal to the curve of the arch. Eventually, however, the depth of the arch-stones must be carefully considered.
We may best obtain a clear view of the whole subject by first assuming that the load presses only downwards on the arch-stones, or that the inner ends of those are cut with horizontal facets.
Let Q'P'APQ (fig. 4) represent a portion of such an arch
placed equally on the two sides of the crown A, then the whole weight of the structure included between the two vertical lines PIT and PH must be supported at P' and P, so that the downward pressure at the point P must be the weight of the building imposed over AP. This pressure downwards is accompanied by a tendency to separate the supporting points P' and P. Now, as this tendency is horizontal its intensity cannot be changed by the load acting only downwards, and must remain the same throughout the structure, wherefore the actual pressure at P must be found by combining this fixed horizontal thrust with the downward pressure equal to the weight of the bridge from A to P. If, then, we draw alt horizontally to represent this constant thrust, and ap upwards to represent the weight of this portion of the arch, the line pis must, according to the law of the composition of pressures, indicate both in direction and in intensity the actual strain at the point P. This pressure must be perpendicular to the joint of the stones, and must therefore be parallel to the straight line drawn to touch the curve at P.
Hence, if the form of the inside of the arch, or the intrados as it is called, be prescribed, we can easily discover the law of the pressures at its various parts ; thus, to find the strain at the point Q, we have only there to apply a tangent to the curve and to draw kg parallel thereto ; kg represents the oblique strain at Q ; eq represents the whole weight from the crown A to Q, and therefore pq is proportional to the weight imposed upon the position PQ of the arc.
Using the language of trigonometry, the horizontal thrust is to the oblique strain at any part of the curve as radius is to the secant of the angle of inclination to the horizon; also the same horizontal thrust is to the weight of the superstructure as radius is to the tangent of the same inclination. And thus, if the intrados be a known curve, such as a circle, an ellipse, or a parabola, we are able without much trouble to compute, on this hypothesis, the load to be placed over each part.
If we use the method of rectangular co-ordinates placing x along OH and z vertically downwards, so that Pr may be the increment of x, irQ that of z, the tangent of the inclination at P is - and therefore if h stand for the horizontal strain, and so for the weight of the arch, we have - , "k to = azs while the oblique strain is h Rx2,• Also the ox?
change of weight from P to a proximate point Q is Szo = h8(Tx) .
Let RST be the outline which the mason-work would have if placed compactly over the arch-stones, in which case RST is called the extrados, then the weight supported at P is proportional to the surface ARSP, and the increment of the weight is proportional to PSTQ, hence if the weights and strains be measured in square units of the vertical section of the structure, and if y be put for PS, the thickness of the mason-work, we have - 8w = y8x, whence y= k - n ‘):2 When the curve APQ is given, the relations of z and of its differentials to x are known, and thus the configurati