I have chosen these values of N and e wholly at random; and it will be perceived that the ratio of L to D corresponds very nearly with the results for the panel system for the same value of N.

Approximation.

When N is large we may drop all the terms below N2, and we shall have

(7+6c) (tan20-1) + 8 N (1 + c) tan2 0 0. (24.) In this, when N = 9 and c = 1, the formula differs from the exact one, less than a minute in the value of 9. For larger values of N the difference will be inappreciable, as shown from the following table:

This approximation may be made for the panel system to get the relation of L to D, since the relation is nearly the same in both systems. The peculiar value of the factor n, in the panel system forbade such an approximation in the general equation.

In like manner, we may determine the relation between the length and depth, and the proper inclination of the braces, in any truss in which the strains may be expressed by a continuous function.

The relation of p to w1 in any practical case cannot be accurately known beforehand, but as all possible inclinations are found from the equation between that resulting when p = 0, and that when w1 = 0, and as these values do not differ more than two degrees, a very close approximation may be very readily arrived at. The function varies slowly about its minimum so that a degree even will make little differWe should diminish the number of bays as much as possible consistently with the liability to rupture by transverse strains and flexure by too great length, and having fixed upon N, determine the relation of

L

p to w1 as nearly as possible, and then make correspond to the equa

tion for minimum material.

D

It will be seen that the results which I have obtained for the minimum material, all approximate the value of a required for the minimum depression of a single triangle composed of bars of uniform section, and sustaining a weight at its apex; that is, they are nearer it, (30°,) than they would be if inclined so that the material in the ties and braces above should be a minimum. Let me suggest to the patient reader that a triangle sustaining compression in both its braces, is not in the same condition as one sustaining compression in one and extension in the other, as in the case of all bridges. Let me also inquire what effect the compression in the upper chord has upon the rigidity of the whole? Is 30° then an approximation to the inclination for the greatest rigidity in a trussed girder?

Multiple Systems or Lattice Trusses.

In adhering to the simple systems I have shown that the minimum material in the whole is obtained by increasing that in the braces above the minimum amount, and so increasing the depth of the truss. Cannot both advantages be combined by retaining the depth of the truss while increasing the inclination of the braces by passing them across two or more panels? This course, while it increases the length of the braces, so much reduces their section as to require less material if we neglect flexure. But both these modifications leave them much more liable to flexure; moreover, the strains cannot be rigidly analyzed, so that much more will have to be added for safety, as well as to prevent the increased liability to flexure, so that we should have little gain. This criticism does not apply to Whipple's truss, or to any other in which the inclined bars are ties. In them I think the multiple system must be advantageous, although it cannot be rigidly calculated.

On Uniform Stress in Girder Work; illustrated by reference to two bridges recently built. By Mr. CALLCOTT REILLY, Assoc. Inst. Civ. Eng.

From Newton's London Journal of Arts, June, 1865.

This communication was suggested by a previous discussion at the Institution, when Mr. Phipps (M. Inst., C.E.,) condemned the troughshaped section commonly adopted for the top and bottom members of truss girders, because the intensity per square unit of the stress upon any vertical cross section was necessarily variable when the connexion of the vertical web with the trough was made in the usual manner. In the construction of the iron work of the two bridges under consideration, attention was invited chiefly to those details which were designed with the object of carrying out as nearly as possible in every part of the girders the condition of uniform stress.

After alluding to the distinction drawn by Prof. Rankine between the words "strain" and "stress," and to his definition of "uniform stress," in which the "centre of stress or "centre of pressure" must be coincident with the centre of gravity of the surface of action, and of "uniformly varying stress" when the centre of gravity deviated from the "centre of stress" in a certain known direction, it was remarked that the failure of any member of a girder would begin where the resistance to strain was really the least, that was, where the intensity of the stress was greatest; from which it followed that the opinion which upheld as right in principle the trough-shaped section, as applied in the usual manner, must be a mistake. And, moreover, every form of section of any member of a girder, or other framework, which did not admit of the approximate coincidence of the centre of stress with the centre of gravity, was liable in degree to the same objection.

The two bridges illustrated different conditions of loading; one carrying the platform on the top, the other having the platform between the main girders near the bottom. Both were of wrought iron, and both exhibited an economy of material in the main girders that, so far as the author was aware, was not common at least in this country. In order to determine the causes of this economy, a comparison was made with two other forms of truss more generally adopted. In one bridge over the river Desmochado, on the line of the Central Argentine Railway, the pair of trusses, 93 feet 4 inches span between the centres of bearings, was designed to carry, in addition to the fixed load, a moving railway load of 1 ton per foot of span for a single line of way, with a maximum intensity of stress of 5 tons per square inch of tension, and of 3 tons per square inch of compression; and the total weight of wrought iron in the framework of the pair of trusses was 18 tons. The cast iron saddles rivetted on at the ends weighed 9 cwt.; if these were included the weight of iron, both wrought and cast, in the pair of trusses was under 4 cwt. per lineal foot of span.

The other bridge, over the Wey and Arun Canal, on the Horsham and Guilford Railway, was 80 feet span between the centres of bearings; it was designed to carry, in addition to the fixed load, a moving load of 1.875 ton per foot of single line of way, at the same maximum intensity of stress as in the other case; and the total weight of wrought iron in the pair of trusses was 20 tons 18 cwt. The cast iron saddles weighed 5 cwt. each; bringing up the weight of both wrought and cast iron in the pair of trusses to 5 cwt. per lineal foot of span. This weight was greater than in the first bridge, although the span was less; but the intensity of the moving load was 87 per cent. greater, and the roadway lying between the trusses instead of on the top, its weight was necessarily much greater. The cross girders were also heavier, each being adapted to support, separately, the heaviest load that could be brought on by a driving axle weighted with 16 tons; the moving load thus brought upon each cross girder, and to which its

strength was proportioned, was 18 tons, equal to 21 tons per foot of span of bridge.

The particular form of truss chosen for these two bridges was that extensively known in the United States as the Murphy-Whipple truss. Each of these trusses was minutely compared, according to the plan adopted on a previous occasion by Mr. Bramwell (M. Inst. C.E.), with two equivalent trusses of the types generally used in this country, viz : the Warren truss, with bars making an angle of 63° 26′ with the horizon, and the simple diagonal truss with two sets of triangles, the bars crossing each other at the angle of 45°, the various circumstances of ratio of depth to span, which was as 1 to 10, and of application and distribution of load, and consequently the number and position of the loaded joints, being common to the three trusses.

The details of the comparison were fully given in the paper, and the proportionate results arrived at in the two cases were exhibited in the following tables, relating to the trusses of the two bridges contrasted respectively with the other equivalent trusses:

From this it appeared that the least practicable weight of No. 1 truss was less than that of No. 1 A by only 1.7 per cent. It might, therefore, be said that practically the two trusses were equal in point of economy; and that there could be no motive for preferring one to the other, except such as might arise from considerations of workshop convenience and facility of construction. The advantage in point of economy of weight of No. 1 over No. 1 B was more decided, being 10 per cent.,-sufficient, it was admitted, speaking generally and without denying that special circumstances might in particular cases justify a choice of the heavier truss, to entitle No. 1. to a preference over No. 1 в.

VOL. L.-THIRD SERIES. - No. 2.- AUGUST, 1865.

8

It thus appeared that No. 2 truss was lighter than either of the others by 4-15 and 11.87 per cent. respectively.

With regard to the peculiarities of detail of the two bridges, it was remarked, that in order that the stress might be uniformly distributed over the surface of any cross section of either "boom," it was necessary that the two halves of the double web of each truss should each support exactly one-half the load upon that truss. This, it was urged, could not be realized by the ordinary modes of fixing the cross girders; but, in the cases under consideration, it was arrived at by supporting the cross girders in the middle of the width of the truss. Thus, in bridge No. I, each cross girder rested upon a light cast iron saddle or bridge, which spanned the width of the top boom, and had its bearing partly upon the top edges of the vertical struts, and partly upon rivets passing through it, the struts, and the vertical side plates of the top boom, in such a way that the line of action of the vertical force transmitted from the cross girder to the truss, coincided exactly with the vertical centre line of its width. In bridge No. II a different arrangement was necessary. In that case each vertical strut consisted of two pairs of angle irons separated in the plane of the truss by a space just wide enough to permit the end of the cross girder to pass in between the pairs. At the same level as the cross girder a plate was riveted to each pair of angle irons; and to the centres of these plates the cross girder was also riveted, so that the weight was equally divided between the four vertical angle irons, and the resulting stress was equally distributed between the two halves of each boom. In both bridges the centrelines of the vertical struts, the diagonal ties, and the top and bottom booms, intersected each other at the centre of gravity of the group of rivets which attached each strut and tie to the boom, and, in order to satisfy the condition of uniform stress, all the centre lines were axes of symmetry. In the top booms of both bridges a section had been adopted which was believed to be new. It was somewhat like an elongated capital letter H, or like a common plate