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

The working stress in any case is fmax. 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+½φ)

Steel, working stress, f = 5.87 (1+½φ).

In these equations φ 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.

φ

1+φ/2

Working Stress f, tons per sq. in.

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

To compare this with the previous table, φ = (A+B)/A = 1+ρ. 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, fmax. 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, fmax.-fmin., 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.