General Theory of Bridge Construction by Herman Haupt

Author:Herman Haupt
Language: eng
Format: epub, pdf
Publisher: D. Appleton and Co.
Published: 1867-03-25T05:00:00+00:00

Let e d represent the depth of the joint at the crown necessary to resist the horizontal thrust, as determined from assumed dimensions, and let this force be represented by a line o e, equal to c d^ applied at the centre of pressure (o). Let G represent the centre of gravity of the arch A d, and mr ^ length of line that represents the weight. Transfer the force at o to the point m, and make m e' = o e. Construct the parallelogram of forces ms. As wi e' represents the length of joint necessary to resist the horizontal force, m r would be the length sufficient to sustain the weight, and the resultant m s would represent the length of a joint, to resist the combined pressure of the two forces. Draw A p perpendicular to m *, produce and equal in length to ms. Ap will represent both the length of the joint at the point A^ and its proper direction, since it is perpendicular to the line of pressure ms.

By drawing p n parallel, and A n perpendicular to A JBj we find that the triangles Apn and m s r will be equal, hence, An = sr=^cdy and as the same is true at any other point it follows, that the difference of level of the extremities of any Joint of the arch should be equal to the depth at the crown. Also as ^ n = m r = weight of portion of arch A 2),

it follows, {(at the horizontal distance between the extremities of any joint will be proportional to the weight of the portion of the arch between it and the crown, p' being the point of application of the resultant of the pressures upon all parts of the joint A p^ and p' % its line of direction, p' % must be tangent to the curve of equilibrium. By finding the point p' for other joints between A and i>, the curve traced through them will be the line of direction of the pressures.

The manner of finding the point p' for any joint Api& obvious ; it is the intersection of the line A p with the diagonal of the rectangle, one of whose sides e' m is proportional to the horizontal pressure, and i^ constant at every point of the arch; the other, m r, represents the weight of the portion Ad oi the arch, acting through Q- its centre of gravity. The position of Gr. can be readily found for any joint, as {u m') by making a drawing of the arch on pasteboard, cutting it out and balancing the portion, of which the centre of gravity is to be ascertained. The weight can be found either by weighing the pasteboard, or by calculation, and thus we are furnished with an extremely simple and practical method of describing the curve of equilibrium.

The method usually recommended for determining practically the direction of this curve, is to mark off

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