Occasionally, for long cross-girders, a distributing girder is placed, with the same intent, in the 6 feet way, its function being limited to this use only if the depth and strength are sufficiently small to serve this object alone, as distinct from the case in which it becomes a carrying girder transferring load to the abutments. As a distributor simply, the girder has to equalise the bending moments amongst the cross-girders, to effect which it will be evident that these moments having been ascertained for the several cross-girders previous to alteration, for a position of the wheel loads such that the heaviest comes upon a centre cross-girder, the mean of these moments will, when compared with that for each girder, show the difference to be induced as a result of introducing the distributor. These differences of moment render necessary at the centre of the cross-girders reactions upwards or downwards, as the case may be, of amounts competent to induce moments below the inner rails equal to these differences.

It is these reactions which must be provided by the distributing girder at a moderate stress, and without flexure of such an amount as sensibly to modify the reactions. The greatest section necessary at any one point may then be adopted for the girder throughout. The result will commonly work out to a moderate section, but there will be no harm in a little excess in a case of this kind, the total cost being but little affected by some small addition to the weight, where labour upon the site is so considerable an item as in work of this description. The ends of the distributing girder should be carried on to the abutments or piers to ensure adequate relief of the end cross-girders. It will be found desirable in arranging for distributing girders to ascertain at an early stage, by boning or by levelling, the condition of the cross-girders as to uniformity of heights, as this may affect the length most suitable for separate sections. Between the underside of the distributor and the cross-girder tops there will commonly be spaces of varying amounts, which should be filled by packings to fit, rather than by pulling the work together by force, introducing undesirable stresses of uncertain amount.

In the earlier remarks upon the strengthening of bridgework by the use of new material, it has been assumed that the modulus of elasticity of the new metal is similar to that of the old; it may, however, as in cases where wrought-iron work is reinforced by additions in steel, be necessary to take the difference of elastic properties into account, with which object the new section should be multiplied by a quantity (greater or less than unity) inversely proportional to the higher or lower modulus of the new material, that is to say, by

E of old materialE of new material

CHAPTER XI.
STRENGTHENING OF RIVETED BRIDGES BY CENTRE GIRDERS.

The addition of distributing girders, described in the last chapter, as a means of strengthening a bridge floor, while sufficient in many cases so far as the cross-girders are concerned, does not in any appreciable way assist the main girders. When for a two-line bridge, having outer main girders only, this result also is desired, together with a more complete relief of the floor structure, centre main girders may be used, placed either above or below the cross-girders, on the centre line of the bridge.

There are two principal ways in which such a girder may be brought into use; the easier, but generally less economical, is by making a simple attachment to the cross-girders, the old girder work still taking the whole dead load. By this method the new girder does no work but carry itself till the live load comes upon the bridge, and must be made very stiff to take any sensible portion of the running load; the second method is to make the connection adjustable, so that a part of the floor weights may be imposed upon the new girder as an initial load. In doing this the old outer girders will rise slightly, being relieved of stress, and the cross-girders also lifted at the middle, whilst the new girder is depressed as the load is brought upon it. With some part of the live load a very considerable proportion of the total may in this way be carried by a centre girder of moderate section. The whole question, by either method, turns upon deflections; and it is in determining the relative movements of the girders that the problem chiefly lies.

It is convenient first to determine the percentage of load relief to be effected in the main girders, as to which it is to be observed that as this relief (distributed) is induced by the upward reaction of the new girder acting at the centre of the cross-girders, the stress relief of these will, as a rule, greatly exceed that of the outside girders. For the generality of cases, it may be taken that the relief suitable for the outside girders will be satisfactory in its effects upon the cross-girders, even though it is desired to reduce the stress in these to a greater degree.

If, however, it be thought desirable to check this, it may be done by considering a cross-girder subject to its dead and live loads acting downwards, and to reactions at the centre and ends. At the centre the reaction will be the load of which the two main girders are relieved on a length equal to the pitch of the cross-girders, or as here given:—

c × t × P = reaction at centre (1)

L_l - (c × L_t) = live load on outer girders (2)

L_l being the total live load, and L_t the total dead and live load carried by the bridge. From this the deflection of the outer girders corresponding to this modified live load may be derived.

It is next necessary to ascertain the vertical movement, commonly a depression, of the cross-girders at the centre relative to their ends, when subject to the running load only, and supported at the middle and ends, the centre reaction being obtained as before indicated (1). This movement will be the difference (if any) between the deflection on the whole span of the cross-girder due to the live load, and the upward flexure of the girder due to the centre reaction, considered as separate effects. Stress values having been estimated for the two conditions, these results may readily be deduced by simple flexure formulæ, observing that while the curve of moments due to live load sufficiently approximates to that for a distributed load to justify, for this, the use of a distributed load formula as given in the chapter “Deflections,” the flexure due to the centre reaction will be but 0·80 of that which corresponds to the same stress for distributed loading. Or, the curve assumed by the girder under live load may be plotted by a method to be later explained.

The sum of the movements now determined—that is, the live-load deflection of the outer girders, and depression, as is commonly the case, of the cross-girders—will give the extreme depression (marked m in Fig. 70), from the dead-load condition of the middle cross-girders, when supported to the extent desired by a centre girder whose proportions are not yet known, but which, carrying the required percentage of the total load, must, subject to a reservation presently stated, deflect only this amount. The unit stress in the flanges of the new girder, governed by this flexure, will for a plate girder be

D × C × m S² = f, unit stress on gross section (3)

D and S being, as before (see “Deflections”), the depth and span respectively in feet, C a constant, m the deflection in inches, and f the stress per square inch on the gross section of flange.

The gross area A, of the flange, is given by

S × c × L_t 8 × D × f = gross area of flange(4)

c × L_t, being, as in (2), the load transferred to and carried by the centre girder.

The actual stress in the flanges will, of course, be greater by an amount due to the girder’s own weight; but this does not affect the question of relief. For any ordinary case the stress per square inch will be low; but it will manifestly be useless to assume a greater stress with a view to economy, as the effect of reducing the section will simply be to make the girder too flexible, thus causing it to be less effective than primarily intended. If, as is seldom the case, there is freedom as to the depth of girder permissible, it is evident the unit stress may be made a condition, and the depth deduced by a suitable modification of formula (3); the relief desired being in this way equally well assured. Indeed, in the rare instances in which any depth may be adopted, this method is—contrary to the general rule—distinctly economical, particularly if the girder may be placed below the cross-girders, which simply rest upon it, without elaborate attachments.

Considering now the second method of applying centre girders by which the new girder is made initially to carry part of the dead load, by adjustment, it will at once be recognised as a more complex matter. The measure of relief by which the old girderwork shall benefit need not be affected by the method of applying the centre girder, and may be decided on the principles already considered. The outer girders carrying a reduced load, when the bridge is fully loaded, and the cross-girders being in part supported at their centres in the manner already described, will give a resulting depression m (see Fig. 71) of the centre cross-girders, below the original dead-load position, of a similar amount determined in the same way. This extreme depression determines also the lowest position of the new centre girder, which may be designed to carry the required percentage of the total bridge loads with the maximum stress and depth, as conditions, leaving the initial dead load and necessary adjustments to be ascertained. This is the common case and will be here dealt with, it being assumed to avoid ambiguity in description that the new girder lies above the cross-girders.

The centre girder of fixed depth being then required to carry a definite load at a definite flange stress, will deflect a definite amount at this stress. If this deflection equalled the extreme depression m of the old girder work, no adjustment would be necessary, the centre girder then carrying no initial dead load, as by the first method; but for centre girders designed for economical flange stress the deflection will in ordinary cases greatly exceed this, the depth generally being small, and in order to ensure that the new girder shall do its full work, some dead load must be put upon it. In the act of adjustment the cross-girders must be lifted and the centre girder depressed, till the joint movement equals the excess s of the centre girder deflection over m, when the new girder will carry the proper amount of initial load, and upon further deflection under live load give the full measure of relief. The amount of “lift” or upward flexure of the old girder work, and the depression or “drop” of the new girder, during adjustment, will depend upon relative stiffness, and may be ascertained as follows:—

For unit reactions at the centre of the cross-girders the upward flexure of these may be ascertained, as also the upward flexure of the two outer girders when subject to forces of the same total amount (one-half to each) applied at the cross-girder ends. The sum of these movements will give the total lift of the centre cross-girders, when all are subject to unit lifting forces; similarly, the depression of the centre girder for unit loads applied at the cross-girders may be determined. There will then be known the movements upwards and downwards of the old and new work when being drawn together by unit forces applied as stated.

If

there will then be

d l + d × s = d_t,

total drop of centre girder under adjustment,

l l + d × s = l_t,

total lift of centre cross girders under adjustment,

d_td × unit load =

initial load put upon centre girder at each cross-girder.

The rise of the two outer girders for upward forces together equal to those depressing the centre girder may readily be deduced.

The act of adjustment may conveniently be effected by the arrangement shown in Fig. 72, in which each cross-girder is hung up at its centre by four bolts. At the middle of the centre girder the total amount to be screwed up will be that corresponding to the deflection excess s, but towards the ends this amount decreases, and may advantageously be represented by a diagram as Fig. 73, in which, if s represents to scale the amount to be screwed up at a centre cross-girder, the corresponding amounts for other girders may be read off direct. It will be apparent that it must be necessary to place the centre girder at such a height as to leave a space between the old and the new work greater than the amount to be screwed up, this excess clearance being ultimately filled by a packing.

The precautions to be observed in carrying out this kind of work, and the practical methods of adjustment adopted by the author after some little experience, may here be given.

Great care is necessary at the outset to ascertain the true spacing of the cross-girders, to ensure that the bolt-holes in the bottom flange of the centre girder shall come where desired. The fixing of the cross-girder brackets also needs close attention to avoid after trouble, the bolt-holes in the brackets being preferably drilled on the site after fixing. It will, for masonry abutments, be necessary to fix bedstones to receive the new centre girder, which, being carried out quite possibly under adverse traffic conditions, will perhaps leave the stones liable to settle slightly when the full load is carried. To eliminate the bad effect of this upon the ultimate adjustment, and to take up any initial set of the new girder work, which would be prejudicial in the same way, it is desirable, the centre girder being in place, to screw up the bolts temporarily and leave the work for a week or two. To ensure regularity in the screwing up process, it is convenient to prepare, for use at the bridge, a diagram somewhat similar to Fig. 73, giving the amount by which the new and old work are to be brought together at each cross-girder, with the number of turns for each nut to effect this. With a man at each side of the girder, the whole length is traversed, giving a half-turn to each nut; this is repeated as often as necessary, and so managed as to bring all up proportionately to the final requirement, keeping tally with chalk marks over each cross-girder as a check. The preliminary screwing up should be conducted with little less care than that adopted for the later adjustment, to avoid damage to the old work. This later adjustment having in due course been effected, it is then necessary to measure for packings to fill the spaces remaining between the old cross-girders and the new centre girder. These spaces should be callipered at each of the four corners, care being taken to avoid after-confusion. The measurements ascertained will, however, be too great for the finished packings, as an allowance of not less than ¹⁄₁₀ inch (total), will commonly be wanted to cover irregularities in the surfaces. The packings, having been prepared and checked, may be slipped into place after slacking all the bolts a small amount to permit this to be done, finally screwing up tight and securing the nuts by split-pins, through holes drilled as the last operation.

As a check upon the calculations and adjustment, the “lift” of the outer girders and cross-girders, and the “drop” of the centre girder may be observed by levelling. For this purpose the author has used a staff of inches divided into tenths, with which, and a good level, very accurate readings may be taken for short distances.

No reference has been made to the effect of skew in a bridge on the above methods, the explanation given applying rather to bridges square on plan. The influence of skew on the load distribution will largely be a matter of detailed calculation. The flexure of the girders may also be sensibly affected, but may be arrived at with sufficient accuracy without any great trouble. The chief effect of skew is to modify the amount of screwing up during adjustment, which may be better understood by reference to Fig. 74, and comparing it with Fig. 73, the adjustment diagram for a square bridge.

To illustrate how these methods of strengthening work out, and compare as to weights of centre girders required, the case has been assumed of a wrought iron bridge of 60-feet span, having outer girders 5 feet deep, of 39 square inches gross flange area; and cross-girders, at 8-feet centres, 27-feet span, 1 foot 9 inches deep, with a gross flange area of twenty square inches. The dead load and live load on either road are each 1·75 tons per foot run.

The stress in the outer girders previous to the alteration being 6 tons per square inch gross, it is desired to relieve this to the extent of 33 per cent. by a steel centre girder. In the table here given the quantities given in italics are fixed as primary conditions:—

Centre Strengthening Girders for 60-ft. Span.

Girders subject to distributed load are treated as having uniform stress, but where this is not strictly the case, as in some light girders, it will be necessary to take the fact into account. For centre girders of wrought iron, and a unit stress on the gross section of 4 instead of 5 tons, the girder weights are between 9 and 10 per cent. greater.

In the above treatment of the application of centre strengthening girders there is a source of error which should be touched upon. If, under live load, the centre girder deflects more than the outer girders, as it commonly will, there must be a want of uniformity in the behaviour of the cross-girders, those near the abutments being more relieved than the estimated amount of relief of those at the centre, which will have less than that intended; but the reduction of stress in the cross-girders will generally be so considerable that any such ambiguity of excess or defect is commonly unimportant; the effect of this also upon the main girders is much less than might be supposed, being, for the third of the cases just given, about 2¹⁄₂ per cent. excess for the centre girder, and generally a much smaller error. With this qualification, the method can, however, be regarded as approximate only. It is possible to eliminate some part of the error by lifting the end cross-girders during adjustment, a less amount than that given by the diagrams, Figs. 73 and 74, taking care that the centre girder is depressed its full amount by lifting the centre cross-girders a little more; this refinement is hardly necessary, and unless controlled by calculation cannot be depended upon for precise results.

Particulars are here given of five ordinary cases, comparing the calculated and observed results of adjustment. The operation of levelling was conducted by a quick-eyed and capable assistant, who was not made acquainted with the results expected, in order to avoid any sub-conscious tendency to match the calculated figures:—

Examples of Centre Girder Adjustments.

The method of calculation adopted for these cases was not precisely that given, though depending upon the same broad principles. The first cannot be considered a good example. The last, having continuous girders, of course needed special treatment.

Of about seventeen bridges strengthened in the manner described, the effect generally was satisfactory, in reducing deflection and vibration; but in two cases of small span, owing probably to settlement of bedstones, the results were not so good.

From first to last the work of putting in a centre girder takes some little time, owing to the slow progress generally made in fixing the brackets, preparing packings, etc. The cost of a typical case was about 23 per cent. of the cost of a new superstructure, with a 30 per cent. relief of stress.

A special case of strengthening by a centre girder, having considerable interest, may be here referred to. The primary idea involved was not the author’s. The bridge dealt with has already been noticed under “Bracing” and a section, before alteration, shown in Fig. 26. The span being 85 feet, there was no room for a centre girder of sufficient depth above the cross-girders and between the roads, nor was it considered economical to place the girder wholly below the floor, because of the costly staging this would have necessitated for erection purposes, the height above ground level being very great. A girder was therefore designed, having open latticing at an angle of 60 degrees, with a bottom boom to be below the cross-girders, the top being as high above the rails as could be permitted (see Figs. 75 and 76). A temporary boom was arranged at the intersection of diagonals, the lower boom proper not being fixed till the girder having been lifted into place, with the diagonal members passing between the cross-girders, allowed this to be done. The girder for some time carried itself from bearing to bearing, with the temporary boom in tension, the deflection being then 2 inches. The permanent boom was then put in place, and the girder restored as nearly as was practicable to the camber it was intended to have when complete, but without throwing, during the process, any improper loads upon the old work.

The lower boom being finally riveted up, the cross-girders were made to bear upon it by suitable packings. There were, in addition to the new girder, two stiff frames between the old main girders, to which the new was secured.

The girder was designed with the intention that under dead load only the cross-girders should just rest, but throw no weight, upon the new work, the latter assisting to carry live load only. The floor beams being of small span, and securely riveted to the old girder tops, the centre girder was required to deflect, under its share of live load, the same amount as the old main girders under the remaining portion, the three points of support of the cross-girders thus not altering their relative levels. That this resulted was evident from the fact that, previous to connecting the cross-frames to the centre-girder, the work being otherwise complete, a space between the two of about ¹⁄₂ inch, afterwards filled by a packing, showed no alteration, the closest measurement failing to disclose any relative movement upon the passage of live load. The reduction of vibration was, as might be expected, very marked.

In the conduct of that class of strengthening work which has been dealt with in this chapter, it is essential, in the author’s judgment, that the man responsible for the detailed calculations and design should himself see the operations of adjustment carried out, or delegate it only to one equally familiar with the requirements.

Before dismissing the subject, it will be well to refer to a method of approximately determining flexure curves, of occasional use in dealing with centre girder or similar questions. The figure assumed is plotted to an exaggerated scale, with which object the actual radius of curvature at points along the girder’s length are first ascertained by the formula

E × Df × 2 = R, radius of curvature in feet,

and the radius of curvature for the diagram by

12 × R × F² = r, radius for plotting, in inches (5)

For some purposes it may be preferable to amend the radii for plotting, so that the curve, as laid down, shall be correct, which may be effected by the formula here given, to be applied to each value of r, as first ascertained:—

r + (chord² r × ·0625) = corrected plotting radius.

If, however, the length of curve is made equal to the span (the chord then being less), and the radii for plotting as given by (5) are used, the result will for most purposes be sufficiently precise, though there will now be an error of a contrary kind, which, for a curve having a droop of one-eighth, will be about 2 per cent. too little. A somewhat similar method of setting out deflection curves is described by Professor Fleeming Jenkin in the article “Bridges” of the “Encyclopædia Britannica,” but without corrections.

A careful comparison of results by the above means, with those calculated, shows that with good draughtsmanship they may be relied upon for considerable accuracy. Equally applicable to girders of varying depth and flange stress, they have also a limited use in cases of continuity.

Figs. 77 and 78 illustrate the deflection and stress diagrams for the cross-girders of the bridge supposed to have been strengthened by a centre-girder, when under the influence of live load and a centre reaction of a definite amount. As a matter of convenience, each radius length has been halved, before correction, so that the resulting droop of the curve is twice the true amount.

CHAPTER XII.
CAST-IRON BRIDGES.

Cast Iron as a material for bridges has of late years fallen into disrepute. It is now entirely tabooed by the Board of Trade for railway under-bridges, unless of arched construction. This condemnation of cast iron followed, and was apparently the result of, an accident which occurred to an under-bridge on one of the southern lines, which bridge had already earned for itself an ill repute by breaking down on a previous occasion. The ultimate issue was, however, good, inasmuch as it led to a thorough overhaul of all railway under-bridges in this country, and the renewal of a great number no longer in a condition suited to the carriage of heavy or of passenger traffic; yet there is little doubt that, in the author’s judgment, many excellent cast-iron bridges were then removed at considerable cost, to be replaced by others of wrought iron or steel, which will not last so long as many of those displaced had done, or would still have lasted had they not been dismantled.

The earlier cast-iron bridges were commonly made of cold-blast iron, a material of such strength and toughness as to give an extraordinary amount of trouble in breaking up the heavier parts, when the time arrived to do this, and with which material ordinary hot-blast iron is not to be compared for reliability.