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; ρ = B/A the ratio of live to dead load; f1 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

f1 (A+B)/(A+2B) = f1(1+ρ)/(1+2ρ).

The following table gives values of f so computed on the assumption that f1 = 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
ρ

1+ρ
——
1+2ρ

Values of f, tons per sq. in.

Iron.

Mild Steel.

All dead load

0

1.00

7.5

9.0

.25

0.83

6.2

7.5

.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

∞

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 Δ = fmax.-fmin., then, by Wöhler's law, the bar will ultimately break, if

fmax. = FΔ, . . . (1)

where F is some unknown function. Launhardt found that, for stresses always of the same kind, F = (t-u)/(t-fmax.) approximately agreed with experiment. For stresses of different kinds Weyrauch found F = (u-s)/(2u-s-fmax.) to be similarly approximate. Now let fmax./fmin. = φ, where φ is + or - according as the stresses are of the same or opposite signs. Putting the values of F in (1) and solving for fmax., we get for the breaking stress of a bar subjected to repetition of varying stress,

fmax. = u(1+(t-u)φ/u) [Stresses of same sign.]