many practical engineers of 3-5 tons for compression, and 4.5 tons for tension, the resistance will be 48.5 lbs. per superficial foot, an ample allowance. The last kind of adventitious bracing to which I shall here have to refer is the counterbracing used in tied arches, and although as such it comes properly in this place, the student may miss it until he has mastered the theory of the arch, which is dealt with in a subsequent chapter.

Let Fig. 42 represent a tied arch carried on supports A B, A e B being the arch, Ao B the tie holding the abut

counterbracing is formed by the diagonal bars a k, bj, bm, ck, &c., the object in adding these to the structure being to aid in distributing partial loads so as to avoid distortion of the arch. We find here, as in all other cases of adventitious bracing, that its duty is to resist the forces causing distortion. Suppose a concentrated load to be at m carried up to C, its tendency will be to pull down and flatten that part of the arch about b to e, throwing up the part from f to h; but the elevation of ƒ will immediately bring a tensile force on fn, which, acting upward through n e, will tend to support the point e, and from there it will pass on to take part of the load off m up the bar me; the bars being all ties, e o and cn are not brought into action.

If in an arrangement of this description initial tension is put on the diagonals, there must be initial compression on all the uprights, and the result of this will be that when the weight comes on the structure the first part of it will

be taken up by the diagonals, until they are sufficiently elongated to let out the compression on the uprights, which will then come in to share in carrying the load. The amount of strain that can come upon the counterbracing can be calculated from the maximum load coming on the bottoms of the bars.

In regard to bracing generally, it is very evident that great care should be taken to have the bars of correct length, for if they are not properly proportioned they will be useless, and perhaps worse, as leading to a feeling of security, and thus causing strains to be fearlessly put upon works quite inadequate to sustain them.

It is the more necessary to impress on the mind of the reader the necessity of giving particular attention to the bracing described in this chapter, because there exists an inclination in some to regard it as a matter of guess-work (a matter of practical experience they will perhaps call it), and put in bars without obtaining any data, or making any calculations as to the proper quantities of material in

use.

The arrangement of the bracing in relation to the other elements of the structure must also be considered early in the progress of the general design. We must not go on designing the girders and main load-carrying elements in the faith that the bracing can afterwards be got in somehow, or very likely we may find there is not room enough or convenience of form to make suitable and workmanlike connections between the bracing and the principal girders, and a patchy joint is only less objectionable than one joint that requires another to carry it, or connect it with the main girders; this latter contrivance is indisputable evidence of incompetence or gross carelessness on the part of the designer.

I have thought it necessary here to impress the importance of the proper disposition of these joints, but the

methods of forming these will be fully entered upon in the subsequent chapter devoted to joints of all descriptions.

Where the bracing bars are so long that they will be liable to bend by their own weight, they may be stiffened by riveting or bolting together at the points of intersection; but in short bars this is not necessary, and should be avoided for reasons previously given.

Long bars acting as struts must be regarded as columns, and calculated accordingly, and not taken as exerting the full resistance of their sectional area to the crushing force.

And, finally, all bars used for bracing purposes should be rigorously examined in order to be sure that they are perfectly straight, which is a point of paramount impor

tance.

CHAPTER VI.

DEFLECTION AND DISTORTION.

THE determination of the deflection of a girder is not a matter that is very satisfactorily solved by calculation, by reason of the variations of the modulus of elasticity, for this does not strictly follow the law generally accepted, and is not constant for all intensities of strain.

It was found in a series of experiments on the elasticity of cast-iron bars 1 inch square and 10 feet long, under tensile strains ranging from 0·47 to 6.6 tons, the modulus of elasticity ranged from 14,050,320 lbs. to 9,549,120 lbs. ; there was permanent set from the second load 0·7 ton, so the elasticity appeared to be injured at as light a strain as that, and keeping within the bounds of the extreme load permitted in practice, we find with a load of 2·82 tons per square inch a modulus of 13,166,300 lbs., the modulus decreasing rapidly with an increase of strain. In compression the modulus of elasticity was at 0.92 ton, 13,214,400 lbs., and at 16.56 tons, 10,836,480 lbs.

In twenty experiments on wrought-iron bars the modulus of elasticity varied from 29,119,800 lbs. at 0.56 ton per square inch, to 26,335,080 lbs. at 11.26 tons.

From these figures it is evident that not only does the modulus vary with the load, but also that it is not the same for tension and compression.

In Fig. 43, let A B represent a portion of a bent beam,

the dotted line showing the neutral axis, and O being the centre of curvature from a; draw a ƒ parallel to be, then ef is the extension of the fibre e c. By similar triangles O a Leta = depth e a, E = modulus

is to a b as ae is to eƒ.

of elasticity, 8 = strain

2

per square inch on fibre e c, and R

If the strain be uniform throughout the length of the girder, then the radius of curvature is constant, and the

practice. The proof of this will be found in any elementary work treating of the properties of the circle.

Applying this to girders having uniform strain throughout, where 7 = span of girder, and d = depth, c strain on top flange, and t = do. on bottom flange, eg, g h = D= deflection at centre.

Supposing the girder to be sup