DEFLECTIONS.
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:--
S^2
----- × _f_ = deflection in inches.
D × C
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
(3S S^2 )
(-- + -----) × _f_,
(C D × C)
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.
[Illustration: FIGS. 54 to 57.]
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.
_Uniform Stress_: A. B.
Girders of uniform depth, varying flange section 0·27 1·00
Hog-backed girders, ends half of centre depth, varying
flange section 0·24 1·08
_Varying Stress_:
[1]Girders of uniform depth and flange section 0·32 0·87
[1]Hog-backed girders (as above), but uniform flange
section 0·29 0·97
[1]Bow-string girders of uniform flange section 0·16 1·30
[Footnote 1: For uniform loading.]
It is apparent that, if preferred, the scale, instead of being in
inches, divided suitably, may, for each type of girder, be amplified to
the proper degree, so that the amount of the deflection may be read off
at once.
This method of dealing with deflections is quite independent of the
character of the bearings, and is applicable to girders at any height
above ground or over water; but its use would hardly be practicable for
very small beams, or those in an awkward position, or near which it
would be impossible to remain with a running load upon the bridge.
There is a possible source of error in the use of the instrument, most
likely to occur with triangulated girders, with which, if the instrument
is placed at the top of an end post, the reading observed may be the
joint effect of deflection and of local flexure of the members meeting
near the telescope. This may be tested, and, if necessary, allowed for,
by first sighting upon a scale at the next apex, and observing the
effect of the moving load. Again, as girders sometimes cant towards the
running load, if the instrument is placed on one edge of a girder, and
the cantings of the two ends are dissimilar, a false reading will
result, which may be amended by ascertaining the amount of cant at each
end, and correcting for the effect of the difference between the cants
upon the observation. Only in exceptional cases is it likely that either
of these considerations would need attention.
The author has secured with this instrument very promising results,
notwithstanding that under a running load there is a slight haziness of
the scale as seen through the telescope, due to “dither,” largely the
result of imperfections which may be remedied.
Deflections may sometimes be conveniently taken, by a quick-eyed
observer, with a good surveyor’s level and a specially-divided staff
held at the centre of the girder. The divisions preferred by the author
for this purpose are 1/10 inch, plainly marked, which may be seen at 50
feet distance with sufficient clearness to make possible readings by
estimation between the divisions to, say, 1/50 inch. But it is clearly
desirable not to rely upon a single observation only, where all the
evidence is gone so soon as the sight has been taken.
In rail-bearers, or other short girders, it may not be practicable to
adopt such methods, either on account of an inability to find a suitable
place for the instrument, or to read with any telescope with sufficient
promptitude as the load passes rapidly over. The use of rods may also be
out of the question, as the errors attending their manipulation may be
serious where but a small movement has to be noted, this being
complicated in some instances by the bearings being insecure, and
working to an extent which obscures the measurement sought. In such
cases it is preferable to use a stiff slat lying along the girder, which
bears, through short blocks over the girder bearings, upon the flanges;
the deflection is then read by direct measurement of the girder’s
depression at the centre, relative to the slat.
The author is, unfortunately, not able to give any precise information
on the effect of running-load as against a load that is stationary in
connection with girder deflections. It is by no means easy in ordinary
work upon a railway to secure facilities for making such comparative
tests. It may, however, be confidently stated, as a result of such
observations as he has made, that the deflection due to a load coming
rapidly upon a bridge is, as to the main girders of, say, a 50 feet
span, but little greater than that due to the same load stationary; it
may be, perhaps, 5 to 10 per cent. more.
It is evident that to determine the precise difference where the
quantity to be measured is so small needs apparatus of a more delicate
character than that in common use, and the control of an engine, or
engines, for the purpose of making the special tests, conditions which
on a busy line can only be secured by special arrangements previously
made.