will be the weight of main girders to carry W_l+W_f and their own weight
(Buck, _Proc. Inst. C.E._ lxvii. p. 331). Hence,
W_g = (W_l+W_f)k/(1-k).
Since in designing a bridge W_l+W_f is known, k(W_l+W_f) can be found from
a provisional design in which the weight W_g is neglected. The actual
bridge must have the section of all members greater than those in the
provisional design in the ratio k/(1-k).
Waddell (_De Pontibus_) gives the following convenient empirical relations.
Let w_1, w_2 be the weights of main girders per ft. run for a live load p
per ft. run and spans l_1, l_2. Then
w_2/w_1 = ½ [l_2/l_1+(l_2/l_1)^2].
Now let w_1', w_2' be the girder weights per ft. run for spans l_1, l_2,
and live loads p' per ft. run. Then
w_2'/w_2 = 1/5(1+4p'/p)
w_2'/w_1 = 1/10[l_2/l_1+(l_2/l_1)^2](1+4p'/p)
A partially rational approximate formula for the weight of main girders is
the following (Unwin, _Wrought Iron Bridges and Roofs_, 1869, p. 40):--
Let w = total live load per ft. run of girder; w_2 the weight of platform
per ft. run; w_3 the weight of main girders per ft. run, all in tons; l =
span in ft.; s = average stress in tons per sq. in. on gross section of
metal; d = depth of girder at centre in ft.; r = ratio of span to depth of
girder so that r = l/d. Then
w_3 = (w_1+w_2)l^2/(Cds-l_2) = (w_1+w_2)lr/(Cs-lr),
where C is a constant for any type of girder. It is not easy to fix the
average stress s per sq. in. of gross section. Hence the formula is more
useful in the form
w = (w_1+w_2)l^2/(Kd-l^2) = (w_1+w_2)lr/(K-lr)
where K = (w_1+w_2+w_3)lr/w_3 is to be deduced from the data of some bridge
previously designed with the same working stresses. From some known
examples, C varies from 1500 to 1800 for iron braced parallel or bowstring
girders, and from 1200 to 1500 for similar girders of steel. K = 6000 to
7200 for iron and = 7200 to 9000 for steel bridges.
iv. _Wind Pressure._--Much attention has been given to wind action since
the disaster to the Tay bridge in 1879. As to the maximum wind pressure on
small plates normal to the wind, there is not much doubt. Anemometer
observations show that pressures of 30 lb per sq. ft. occur in storms
annually in many localities, and that occasionally higher pressures are
recorded in exposed positions. Thus at Bidstone, Liverpool, where the gauge
has an exceptional exposure, a pressure of 80 lb per sq. ft. has been
observed. In tornadoes, such as that at St Louis in 1896, it has been
calculated, from the stability of structures overturned, that pressures of
45 to 90 lb per sq. ft. must have been reached. As to anemometer pressures,
it should be observed that the recorded pressure is made up of a positive
front and negative (vacuum) back pressure, but in structures the latter
must be absent or only partially developed. Great difference of opinion
exists as to whether on large surfaces the average pressure per sq. ft. is
as great as on small surfaces, such as anemometer plates. The experiments
of Sir B. Baker at the Forth bridge showed that on a surface 30 ft. × 15
ft. the intensity of pressure was less than on a similarly exposed
anemometer plate. In the case of bridges there is the further difficulty
that some surfaces partially [v.04 p.0549] shield other surfaces; one
girder, for instance, shields the girder behind it (see _Brit. Assoc.
Report_, 1884). In 1881 a committee of the Board of Trade decided that the
maximum wind pressure on a vertical surface in Great Britain should be
assumed in designing structures to be 56 lb per sq. ft. For a plate girder
bridge of less height than the train, the wind is to be taken to act on a
surface equal to the projected area of one girder and the exposed part of a
train covering the bridge. In the case of braced girder bridges, the wind
pressure is taken as acting on a continuous surface extending from the
rails to the top of the carriages, plus the vertical projected area of so
much of one girder as is exposed above the train or below the rails. In
addition, an allowance is made for pressure on the leeward girder according
to a scale. The committee recommended that a factor of safety of 4 should
be taken for wind stresses. For safety against overturning they considered
a factor of 2 sufficient. In the case of bridges not subject to Board of
Trade inspection, the allowance for wind pressure varies in different
cases. C. Shaler Smith allows 300 lb per ft. run for the pressure on the
side of a train, and in addition 30 lb per sq. ft. on twice the vertical
projected area of one girder, treating the pressure on the train as a
travelling load. In the case of bridges of less than 50 ft. span he also
provides strength to resist a pressure of 50 lb per sq. ft. on twice the
vertical projection of one truss, no train being supposed to be on the
bridge.
19. _Stresses Permitted._--For a long time engineers held the convenient
opinion that, if the total dead and live load stress on any section of a
structure (of iron) did not exceed 5 tons per sq. in., ample safety was
secured. It is no longer possible to design by so simple a rule. In an
interesting address to the British Association in 1885, Sir B. Baker
described the condition of opinion as to the safe limits of stress as
chaotic. "The old foundations," he said, "are shaken, and engineers have
not come to an agreement respecting the rebuilding of the structure. The
variance in the strength of existing bridges is such as to be apparent to
the educated eye without any calculation. In the present day engineers are
in accord as to the principles of estimating the magnitude of the stresses
on the members of a structure, but not so in proportioning the members to
resist those stresses. The practical result is that a bridge which would be
passed by the English Board of Trade would require to be strengthened 5% in
some parts and 60% in others, before it would be accepted by the German
government, or by any of the leading railway companies in America." Sir B.
Baker then described the results of experiments on repetition of stress,
and added that "hundreds of existing bridges which carry twenty trains a
day with perfect safety would break down quickly under twenty trains an
hour. This fact was forced on my attention nearly twenty-five years ago by
the fracture of a number of girders of ordinary strength under a
five-minutes' train service."
Practical experience taught engineers that though 5 tons per sq. in. for
iron, or 6½ tons per sq. in. for steel, was safe or more than safe for long
bridges with large ratio of dead to live load, it was not safe for short
ones in which the stresses are mainly due to live load, the weight of the
bridge being small. The experiments of A. Wöhler, repeated by Johann
Bauschinger, Sir B. Baker and others, show that the breaking stress of a
bar is not a fixed quantity, but depends on the range of variation of
stress to which it is subjected, if that variation is repeated a very large
number of times. Let K be the breaking strength of a bar per unit of
section, when it is loaded once gradually to breaking. This may be termed
the statical breaking strength. Let k_{max.} be the breaking strength of
the same bar when subjected to stresses varying from k_{max.} to k_{min.}
alternately and repeated an indefinitely great number of times; k_{min.} is
to be reckoned + if of the same kind as k_{max.} and - if of the opposite
kind (tension or thrust). The range of stress is therefore
k_{max.}-k_{min.}, if the stresses are both of the same kind, and
k_{max.}+k_{min.}, if they are of opposite kinds. Let [Delta] = k_{max.} ±
k_{min.} = the range of stress, where [Delta] is always positive. Then
Wöhler's results agree closely with the rule,
k_{max.} = ½[Delta]+[root](K²-n[Delta]K),
where n is a constant which varies from 1.3 to 2 in various qualities of
iron and steel. For ductile iron or mild steel it may be taken as 1.5. For
a statical load, range of stress nil, [Delta] = 0, k_{max.} = K, the
statical breaking stress. For a bar so placed that it is alternately loaded
and the load removed, [Delta] = k_{max.} and k_{max.} = 0.6 K. For a bar
subjected to alternate tension and compression of equal amount, [Delta] = 2
f_{max.} and k_{max.} = 0.33 K. The safe working stress in these different
cases is k_{max.} divided by the factor of safety. It is sometimes said
that a bar is "fatigued" by repeated straining. The real nature of the
action is not well understood, but the word fatigue may be used, if it is
not considered to imply more than that the breaking stress under repetition
of loading diminishes as the range of variation increases.
It was pointed out as early as 1869 (Unwin, _Wrought Iron Bridges and
Roofs_) that a rational method of fixing the working stress, so far as
knowledge went at that time, would be to make it depend on the ratio of
live to dead load, and in such a way that the factor of safety for the live
load stresses was double that for the dead load stresses. Let A be the dead
load and B the live load, producing stress in a bar; [rho] = B/A the ratio
of live to dead load; f_1 the safe working limit of stress for a bar
subjected to a dead load only and f the safe working stress in any other
case. Then
f_1 (A+B)/(A+2B) = f_1(1+[rho])/(1+2[rho]).
The following table gives values of f so computed on the assumption that
f_1 = 7½ tons per sq. in. for iron and 9 tons per sq. in. for steel.
_Working Stress for combined Dead and Live Load. Factor of Safety twice as
great for Live Load as for Dead Load._
----------------------+-------+----------+-----------------------------+
| Ratio | 1+[rho] |Values of f, tons per sq. in.|
| [rho] | ------- +-----------------------------+
| | 1+2[rho] | Iron. | Mild Steel.|
----------------------+-------+----------+----------------+------------+
All dead load | 0 | 1.00 | 7.5 | 9.0 |
| .25 | 0.83 | 6.2 | 7.5 |
| .33 | 0.78 | 5.8 | 7.0 |
| .50 | 0.75 | 5.6 | 6.8 |
| .66 | 0.71 | 5.3 | 6.4 |
Live load = Dead load | 1.00 | 0.66 | 4.9 | 5.9 |
| 2.00 | 0.60 | 4.5 | 5.4 |
| 4.00 | 0.56 | 4.2 | 5.0 |
All live load | [inf] | 0.50 | 3.7 | 4.5 |
----------------------+-------+----------+----------------+------------+
Bridge sections designed by this rule differ little from those designed by
formulae based directly on Wöhler's experiments. This rule has been revived
in America, and appears to be increasingly relied on in bridge-designing.
(See _Trans. Am. Soc. C.E._ xli. p. 156.)
The method of J.J. Weyrauch and W. Launhardt, based on an empirical
expression for Wöhler's law, has been much used in bridge designing (see
_Proc. Inst. C.E._ lxiii. p. 275). Let t be the _statical breaking
strength_ of a bar, loaded once gradually up to fracture (t = breaking load
divided by original area of section); u the breaking strength of a bar
loaded and unloaded an indefinitely great number of times, the stress
varying from u to 0 alternately (this is termed the _primitive strength_);
and, lastly, let s be the breaking strength of a bar subjected to an
indefinitely great number of repetitions of stresses equal and opposite in
sign (tension and thrust), so that the stress ranges alternately from s to
-s. This is termed the _vibration strength_. Wöhler's and Bauschinger's
experiments give values of t, u, and s, for some materials. If a bar is
subjected to alternations of stress having the range [Delta] =
f_{max.}-f_{min.}, then, by Wöhler's law, the bar will ultimately break, if
f_{max.} = F[Delta], . . . (1)
where F is some unknown function. Launhardt found that, for stresses always
of the same kind, F = (t-u)/(t-f_{max.}) approximately agreed with
experiment. For stresses of different kinds Weyrauch found F =
(u-s)/(2u-s-f_{max.}) to be similarly approximate. Now let
f_{max.}/f_{min.} = [phi], where [phi] is + or - according as the stresses
are of the same or opposite signs. Putting the values of F in (1) and
solving for f_{max.}, we get for the breaking stress of a bar subjected to
repetition of varying stress,
f_{max.} = u(1+(t-u)[phi]/u) [Stresses of same sign.]
f_{max.} = u(1+(u-s)[phi]/u) [Stresses of opposite sign.]
The working stress in any case is f_{max.} divided by a factor of safety.
Let that factor be 3. Then Wöhler's results for iron and Bauschinger's for
steel give the following equations for tension or thrust:--
Iron, working stress, f = 4.4 (1+½[phi])
Steel, working stress, f = 5.87 (1+½[phi]).
In these equations [phi] is to have its + or - value according to the case
considered. For shearing stresses the working stress may have 0.8 of its
value for tension. The following table gives values of the working stress
calculated by these equations:--
_Working Stress for Tension or Thrust by Launhardt and Weyrauch Formula._
------------------------+-------+-----------+--------------------+
| [phi] | [phi] | Working Stress f, |
| | 1 + ----- | tons per sq. in. |
| | 2 +--------------------+
| | | Iron. | Steel. |
------------------------+-------+-----------+--------------------+
All dead load | 1.0 | 1.5 | 6.60 | 8.80 |
| 0.75 | 1.375 | 6.05 | 8.07 |
| 0.50 | 1.25 | 5.50 | 7.34 |
| 0.25 | 1.125 | 4.95 | 6.60 |
All live load | 0.00 | 1.00 | 4.40 | 5.87 |
| -0.25 | 0.875 | 3.85 | 5.14 |
| -0.50 | 0.75 | 3.30 | 4.40 |
| -0.75 | 0.625 | 2.75 | 3.67 |
Equal stresses + and - | -1.00 | 0.500 | 2.20 | 2.93 |
------------------------+-------+-----------+--------------------+
[v.04 p.0550] To compare this with the previous table, [phi] = (A+B)/A =
1+[rho]. Except when the limiting stresses are of opposite sign, the two
tables agree very well. In bridge work this occurs only in some of the
bracing bars.
It is a matter of discussion whether, if fatigue is allowed for by the
Weyrauch method, an additional allowance should be made for impact. There
was no impact in Wöhler's experiments, and therefore it would seem rational
to add the impact allowance to that for fatigue; but in that case the
bridge sections become larger than experience shows to be necessary. Some
engineers escape this difficulty by asserting that Wöhler's results are not
applicable to bridge work. They reject the allowance for fatigue (that is,
the effect of repetition) and design bridge members for the total dead and
live load, plus a large allowance for impact varied according to some
purely empirical rule. (See Waddell, _De Pontibus_, p.7.) Now in applying
Wöhler's law, f_{max.} for any bridge member is found for the maximum
possible live load, a live load which though it may sometimes come on the
bridge and must therefore be provided for, is not the usual live load to
which the bridge is subjected. Hence the range of stress,
f_{max.}-f_{min.}, from which the working stress is deduced, is not the
ordinary range of stress which is repeated a practically infinite number of
times, but is a range of stress to which the bridge is subjected only at
comparatively long intervals. Hence practically it appears probable that
the allowance for fatigue made in either of the tables above is sufficient
to cover the ordinary effects of impact also.
English bridge-builders are somewhat hampered in adopting rational limits
of working stress by the rules of the Board of Trade. Nor do they all
accept the guidance of Wöhler's law. The following are some examples of
limits adopted. For the Dufferin bridge (steel) the working stress was
taken at 6.5 tons per sq. in. in bottom booms and diagonals, 6.0 tons in
top booms, 5.0 tons in verticals and long compression members. For the
Stanley bridge at Brisbane the limits were 6.5 tons per sq. in. in
compression boom, 7.0 tons in tension boom, 5.0 tons in vertical struts,
6.5 tons in diagonal ties, 8.0 tons in wind bracing, and 6.5 tons in cross
and rail girders. In the new Tay bridge the limit of stress is generally 5
tons per sq. in., but in members in which the stress changes sign 4 tons
per sq. in. In the Forth bridge for members in which the stress varied from
0 to a maximum frequently, the limit was 5.0 tons per sq. in., or if the
stress varied rarely 5.6 tons per sq. in.; for members subjected to
alternations of tension and thrust frequently 3.3 tons per sq. in. or 5
tons per sq. in. if the alternations were infrequent. The shearing area of
rivets in tension members was made 1½ times the useful section of plate in
tension. For compression members the shearing area of rivets in butt-joints
was made half the useful section of plate in compression.
[Illustration: FIG. 37.]
20. _Determination of Stresses in the Members of Bridges._--It is
convenient to consider beam girder or truss bridges, and it is the stresses
in the main girders which primarily require to be determined. A main girder
consists of an upper and lower flange, boom or chord and a vertical web.
The loading forces to be considered are vertical, the horizontal forces due
to wind pressure are treated separately and provided for by a horizontal
system of bracing. For practical purposes it is accurate enough to consider
the booms or chords as carrying exclusively the horizontal tension and
compression and the web as resisting the whole of the vertical and, in a
plate web, the equal horizontal shearing forces. Let fig. 37 represent a
beam with any system of loads W_1, W_2, ... W_n.