Deflection, considered only as a fraction of the span, and without regard to other conditions affecting it, is of very little use as an indication of a girder’s fitness for its work; but when taken with reference to the depth of the girder, the nature and amount of the load producing flexure, and, further, with regard to the quality of the workmanship and normal properties of the material of which the beam is constructed, it may be of some little service in helping to form a reliable opinion. This consideration applies with less force, perhaps, to new work than to old, in which there may be unknown influences at work, or unknown defects which by excessive deflection may be betrayed. Though too much importance should not be attached to results of deflection tests in any one instance, yet the practice of observing such movements, and considering them with reference to each case, gives a good general idea of what may be expected in a fresh instance, any material departure from which should be a reason for specific inquiry as to the cause. A further reason with new work is found in the evidence it affords as to whether the loads carried travel to the supports really as intended, or by some route not contemplated; or, in the case of floor beams, in what way the load is distributed amongst them, if, indeed, there be any such distribution.

The author has commonly found that new work gives greater deflections than old—i.e., while calculation gives the same result for each, it does not apply equally well to both. The differences may be accidental, but are probably due to other causes, perhaps to the fact that new work has not by repeated applications of load lost the resilience of parts liable to considerable local stress, such as is very liable to occur at connections, so that the deflection is, whilst new, greater than after many years’ use, by which time such parts may develop a definite “set,” and contribute in a less degree, or not at all, to the total elastic deformation.

It is also possible, as already suggested, that repeated high stress may reduce the ratio of strain to stress, the material gradually becoming more rigid, the modulus of elasticity being, in fact, increased.

In girders of ordinary construction, the major part of the deflection is due to the booms, the remainder to the web; the latter is for plate girders a small amount only, and is commonly neglected, but for open web constructions it may be quite appreciable. For any given type of web arrangement the deflection due to the web will, for all depths, remain a constant quantity for the same span and unit stress; and though a moderate fraction of the whole deflection for a shallow girder, it may be a very considerable part for a girder of great depth, in which that part due to the booms is, of course, smaller, since the deflection due to these varies inversely as the girders’ depths.

Deflection, being dependent upon the elasticity of the material, is of necessity very largely influenced by the value of its modulus E, itself liable to considerable variation, and is increased in a small degree by the yield of joints and rivets, which effect, apart from the initial “set” of the girders, appears to be negligible. The stiffness of members in resisting angular distortion at connections must also, for open-web riveted structures, affect the result, making it somewhat less, and, finally, section excess at joints and gusset attachments has an influence in modifying deflection as compared with that due to the normal gross sections simply.

From these considerations it is apparent that any simple deflection formula must be largely empiric in its nature. For plate girders of uniform depth and flange stress, the writer has found the following to give good results:—

S²D × C × f = deflection in inches.

The span S and depth D are, as a matter of convenience, taken in feet; the constant C is for wrought iron 3500, and for mild steel 4000; f is the mean of the extreme tensile and compressive stresses of the booms, in tons per square inch, estimated upon the gross sections.

This, though satisfactory for plate girders, is not so suited to girders having open webs, in which the deflection will more nearly be

(3SC + S²D × C) × f,

the constant C being 3900 and 4450 for iron and steel respectively. The latter values of C correspond to normal values of the modulus of elasticity of 11,700 and 13,350 tons for iron and for steel, it being assumed that any slight rivet yield is off-set by any small section excess—say, 5 per cent.; it may, however, happen that section excess is greater than assumed, in which case some allowance may properly be made for this by increasing C.

To adapt the formulÆ to girders other than those having parallel booms and uniform stress, the results, as deduced above, may be multiplied by constants given in column B of the Table given on page 93.

The practice of adopting for E in deflection formulÆ a quantity much smaller than its nominal amount, with the object of allowing in riveted girder work for the yield of rivets and of joints, can hardly now be defended, whatever may have been a case at a time when workmanship was much inferior, when there was no machine riveting, and joints were, owing to the small weight of plates and bars, three times as numerous.

The initial “set” of a girder consequent upon first loading is a quantity quite distinct from deflection proper, and may be so small as to be negligible, or read 10 per cent. of the true deflection, varying with design and workmanship.

No estimate of girder deflection can be even approximately true if there is, at the level of the top or bottom flanges, a plated or otherwise rigid floor system which is not taken into account, as this will have the effect of very materially reducing the boom stress. To neglect this influence, where it exists, must necessarily lead to disappointing results, and it is quite practicable in many instances to include it in the calculation.

The influence of angular distortion between the various members has been neglected. It may be pointed out, however, that the resistance accompanying these movements in girders having riveted connections, though unimportant as affecting deflection, is worth some consideration in regard to secondary stress. For girders of similar type and unit stress these angular variations will be the same in amount for any span, but will generally be of less importance in large girders than in small, because in large girders the ratio of the breadth of members to their length is commonly less.

When determining the probable deflection of any girder of exceptional figure, it will be found convenient to make a strain diagram—an old device, in which the actual alterations of length being ascertained for all members, the girder is carefully set out to a suitable scale, with the lengths of members increased or reduced by the actual estimated amounts. The distorted figure resulting will then give the probable deflection. The value of E for this purpose should never be taken at less than the normal amount, and may for a considerable excess of metal in joints and gussets be made as much as 10 per cent. greater, this being a convenient means of making the necessary correction.

The effect of loads quickly applied may here be considered in connection with elastic deformations of girders of the same span, but different depths. If these be designed for similar loads and unit stresses, the deflections due to webs and booms of the girders compared will bear the same relation, each to each, as do the weights, whether in both cases the loads be inert or quickly applied, from which it follows that the mechanical “work” done by the loads in falling through the deflection heights is, neglecting inertia, always in proportion to the girderwork weights, and is a similar amount per ton, which as the total length of members remains substantially unaltered, corresponds to a similar amount of work per unit of section, or similar stress, irrespective of the depth of the girders.

But for a “drop” load, as when there is some obstruction upon a railway bridge, there will be in addition a further amount of work to be absorbed, which is to be considered the same whatever the girder’s depth, and will for deep girders be a larger amount per ton of girderwork than in those that are shallow; this, taking effect on members of the same aggregate length, but lighter, will develop a higher stress than in girders of lesser depth, more particularly in the booms.

The influence of the girder’s inertia in modifying drop-load effects will also be less marked in deep—i.e., light—girders than in girders shallow and heavy.

It is, notwithstanding all this, desirable that the depth of main girders should be liberal for economy’s sake, and also that of floor beams, for reasons already dealt with; the probability of the drop load is somewhat remote, and, though possible, would simply induce, if it occurred, an increment of stress rather more important in deep girders, making it specially desirable in these to give particular attention to the detailing of any connections liable to suffer from impact effects.

It should be remarked that for short and very flexible beams, generally outside the limits of practice, there may also be, under quickly moving loads, a material increase of stress due to the centrifugal effort of the load on running round the deflection curve, and in rising upon the steep part of the curve beyond the girder’s centre. Where advisable, these effects may be modified by cambering the rail.

For pin bridges in which there may be spring in the pins, excess stress in some eye-bars due to inequalities of length, and a want of that rigidity peculiar to riveted structures, the deflection will be greater than above indicated for girders of the ordinary English type.

The method in common use for measuring the deflections of girders but a moderate distance above the ground by means of sliding-rods, though crude, gives, with care, results sufficiently accurate for most practical purposes; but some points necessary to remember may be mentioned with propriety. The lower rod should rest firmly upon something solid, say a stone, well bedded and free from any tendency to rock; the upper end should bear against some part of the girder above, presenting a hard surface, free from dirt or scale, and as the running load approaches the bridge it should be ascertained that there is no slack, that the rods bear hard at the top and bottom. The upper end having been depressed, care is to be exercised to make sure of the reading before the rods alter their relation to each other. These precautions are so self-evident that an apology is almost necessary for mentioning them.

To ascertain deflections with a single pair of rods is only allowable when the girders rest firmly on their bearings; if felt has been placed under the girder ends, or if the bedstones are insecure or rocking, it is necessary to use three pairs of rods, one pair at the middle and a pair at each end, in which case the mean of the two end readings must be deducted from the reading of that at the centre to get the desired result.

In the case of a number of spans in series, each resting upon sill girders common to two sets of bearings, this method also gives results of indifferent reliability, as the depression of each end may be greater as the travelling load comes upon and leaves the span than when it is precisely over the middle, and it is in general out of the question to secure by this mode simultaneous readings for a particular position of the running load, which are what is required.

The author suggests, as a means of ascertaining deflections free from these objections, that it should be done by first measuring the slope at one end, and from this deducing the deflection at the centre.

This is to be accomplished by means of a little instrument, consisting of a telescope with cross-hair sights, and fitted with a reflecting prism at the eye-piece capable of being turned round, so that the observer has a wide choice as to the position he assumes with reference to the instrument, and may look either directly through it, or at right angles to the axis of the telescope. This is clamped at one end of the girder over the bearing, at the other end a scale is secured, to which the telescope is directed, the cross hair being made to sight on the zero of the scale, or the reading noted. For a girder supposed to deflect to uniform curvature (say, with uniform depth and uniform stress, the ordinary case) the reading observed will be four times the deflection; every ¹/₁₀ inch actual reading on the scale will correspond to ¹/₄₀ inch of girder deflection.

Apart from the deflection, this method gives a ready means of observing the end slope, a quantity of equal value for purposes of comparison. As with girders of similar proportions, and similarly stressed, the deflection will at all spans be the same fraction of the span; so should the end slope be a constant quantity under similar conditions, the diagram, Fig. 54, will make the principle quite clear.

Strictly the character of the deflection curve is slightly modified by that part of the deflection due to the web; so that the depression at the centre would, in the case assumed above, be somewhat more than one-fourth part of the end reading, and generally will be a larger fraction of the reading than that deduced from a consideration of flange stress simply. In Figs. 55 to 57, which are intended to explain this, it will be noticed that deflection due to the web is shown straight-lined from the bearings to the centre of the girder; this is strictly true only for a girder correctly designed for an immovable distributed load; but as there should be for girders intended for a travelling load, some excess in web members near the centre under the condition of uniform loading, the point of the figure should be rounded off to be in agreement with this case, though it is left as shown in the diagram for the sake of simplicity.

Suitable constants, including the corrections necessary, are given in column A of the table annexed for a few typical cases, and by these constants the actual readings should be multiplied to find the deflection. The constants have been worked out for depths of one-tenth the span; for greater depths they should be slightly more, and for smaller depths somewhat less, but they may be used between the limits of one-sixth and one-fourteenth, with a maximum error hardly exceeding 5 per cent., and generally much less.

The figures in column B relate to the formulÆ previously stated, and apply equally well to all depths.

Tables of Multipliers for Deflection.

[A] For uniform loading.