-----+----------+-----------------------------+
| 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 Woehler'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 Woehler'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_. Woehler'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 Woehler's law, the bar will ultimately break, if

f_{max.} = F[Delta], . . . (1)

where F is some unknown function. Launhardt found that