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)
_c_ being the percentage of relief; _t_ the total load per foot run of
the bridge; and P the pitch of cross-girders. The live loads carried by
the cross-girders are for this purpose taken at per foot run, as for the
main girders. With these data it will be easy to construct a diagram of
moments, making it evident whether the relief proposed for the main
girders will give a sufficient percentage of relief to the floor beams.
Granting that this proportion has been decided, and dealing first with
the case in which the centre girder is simply attached to the
cross-girders, and takes no dead load other than its own weight, then
the live load carried by the outside girders, and previously borne
wholly by them, will be reduced by the amount it is intended to transfer
to the centre girder, and will become
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.
[Illustration: FIG. 70.]
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_
----------- = _f_, unit stress on gross section (3)
S^{2}
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_}
---------------- = gross area of flange (4)
8 × D × _f_
_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.
[Illustration: FIG. 71.]
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
_l_ = lift due to unit loads,
_l{t}_ = total lift due to adjustment,
_d_ = drop due to unit loads,
_d{t}_ = total drop due to adjustment,
_s_ = deflection excess = gross adjustment,
there will then be
_d_
--------- × _s_ = _d{t}_,
_l_ + _d_
total drop of centre girder under adjustment,
_l_
--------- × _s_ = _l{t}_,
_l_ + _d_
total lift of centre cross girders under adjustment,
_d{t}_
------ × unit load =
_d_
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.
[Illustration: FIG. 72.]
[Illustration: FIG. 73.]
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.
----------------------------+-----------+------------+------------
| Centre | Centre |
| Girder, | Girder, |Adjustments
---- | Stress | Depth | Unknown.
| Unknown. | Unknown. |
----------------------------+-----------+------------+------------
_Outer Girder._ | | |
| | |
Deflection under modified | | |
live load | ·42 in. | ·42 in. | ·42 in.
Lift of adjustment | _nil_ | _nil_ | ·153 „
| | |
_Cross Girders._ | | |
| | |
Depression under live load | | |
--modified conditions of | | |
support | ·13 in. | ·13 in. | ·13 „
Extreme depression (_m_) | ·55 „ | ·55 „ | ·55 „
Lift of adjustment (cross- | | |
girder only) | _nil_ | _nil_ | ·095 „
Total lift of adjustment | | |
(_l{t}_) | _nil_ | _nil_ | ·248 „
| | |
_Centre Girder._ | | |
| | |
Depth | _3·5 ft._ | 8·2 ft. |_3·5 ft._
Unit stress on gross section| | |
(ex girder’s weight) | 2·14 tons | _5·0 tons_ |_5·0 tons_
Total deflection (ex | | |
girder’s weight) | ·55 in. | ·55 in. |1·28 in.
Deflection excess (_s_) | _nil_ | _nil_ | ·73 „
Depression, or “drop” of | | |
adjustment (_d{t}_) | _nil_ | _nil_ | ·482 „
Gross area of flange |105 sq. in.|19·2 sq. in.|44·5 sq. in.
Weight | 20 tons | 10·4 tons |11·4 tons
Net flange stress (including| | |
girder’s weight) | 3·19 tons | 6·87 tons | 6·94 tons
----------------------------+-----------+------------+------------
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.
[Illustration: FIG. 74.]
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 2-1/2 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.
---------------------------------------+-----------+-----------------
-- |Calculated.| Observed.
---------------------------------------+-----------+-----------------
| in. | in.
| |
No. 1.--56-_Ft. Span._
| |
Depression of centre girder | ·82 | ·84
Lift of cross-girders at centre | ·23 | ·22
Lift of outer girders | ·20 |·10 and ·13
| |
No. 2.--57-_Ft. Span._
| |
Depression of centre girder | ·50 | ·50
Lift of cross-girders at centre | ·18 | ·20
Lift of outer girders | ·11 |·08 and ·10
| |
No. 3.--67-_Ft. Span._
| |
Depression of centre girder | ·70 | ·75
Lift of cross-girders at centre | ·15 | ·17
Lift of outer girders | ·10 | ·09
| |
No. 4.--68-_Ft. Span._
| |
Depression of centre girder | ·70 | ·65
Lift of cross-girders at centre | ·20 | ·18
Lift of outer girders | ·13 | ·14
| |
No. 5.--52-_Ft. and_ 28-_Ft. Spans continuous._
| |
|Long |Short|Long |Short
|Span.|Span.|Span.|Span.
---------------------------------------+-----+-----+-----+-----------
| in. | in. | in. | in.
Depression of centre girder | ·28 | .. | ·29 | ..
Lift of centre girder | .. | ·04 | .. | ·03
Lift of cross-girders (centre of spans)| ·17 | ·09 | ·15 | ·13
Lift of outer girders | ·08 | .. | ·08 | ..
Depression of outer girder | .. | ·01 | .. |negligible.
---------------------------------------+-----+-----+-----+-----------
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.
[Illustration: FIG. 75.]
[Illustration: FIG. 76.]
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 × D
------- = R, radius of curvature in feet,
_f_ × 2
and the radius of curvature for the diagram by
12 × R × F^2 = _r_, radius for plotting, in inches (5)
E being the modulus of elasticity, D the girder’s depth in feet, _f_ the
mean of the extreme flange stresses per square inch of gross area, and F
the fraction indicating scale as 1/48, where 1/4 inch = 1 foot. The
curve, being plotted, shows by direct scaling the movement of any point
relative to its original position. Near the ends of the curve where the
radii may be of considerable length, the arcs may be drawn with the help
of template curves, or even set out as pieces of “straight.”
When the curve is laid down so that its chord equals the span to scale,
the method involves an error of excess in the resulting deflection or
droop which is as much as 7 per cent. when the mean radius for plotting
equals the span as drawn, or when the droop of curve approaches
one-eighth of the span. As the exaggeration of curvature is made less
pronounced, this error rapidly diminishes, till for a droop of about
one-sixteenth the percentage is one-fourth part of that above given.
This excess in the droop of curve may be amended by the following
expression:--
(droop^3 )
droop - (------- × 3·73) = corrected droop, or deflection.
(chord^2 )
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:--
(chord^2 )
_r_ + (------- × ·0625) = corrected plotting radius.
( _r_ )
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.
[Illustration: FIGS. 77 and 78.]
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.