Any section not

containing re-en

trant angles (due to St. Venant).

Bb2

=

3 +1.8m

0 =

b

188.5M,(62+B2) b3B3G

B

where m =

4oly

where A area of section;

I, polar moment of inertia of section; y=distance of furthest edge from centre of section.

Twist of Shafts.-In Chapter VIII., we showed that when an element was sheared, the amount of slide x bore the following relation :

x

=

TLD

360

Substituting the value of x in equation (i.), we have

= or 0 =

360f1

#GD

But M1 = fsZ,

for solid circular shafts. Substituting the value of Z, for a hollow shaft in the above, we get

N.B.-The stiffness of a hollow shaft is the difference of the stiffness of two solid shafts whose diameters are respectively the outer and inner diameters of the hollow shaft.

When it is desired to keep the twist or spring of shafts within narrow limits, the stress has to be correspondingly

reduced. Long shafts are frequently made very much stronger than they need be in order to reduce the spring. A common limit to the amount of spring is 1° in 20 diameters; the stress corresponding to this is arrived at thus

In the case of short shafts, in which the spring is of no importance, the following stresses may be allowed:

Steel, f. 10,000 lbs. per sq. inch

Wrought iron, f, = 8000

Cast iron, f. = 3000

Horse-power transmitted by Shafts.-Let a force of

P lbs. act at a distance inches from the centre of a shaft; then

=

Taking the value of Z, o'184D for hollow shafts having the internal diameter equal to half the external, we get

Combined Torsion and Bending.-In Fig. 478 a shaft is shown subjected to torsion only. We have previously seen

Torsion only

FIG. 478.

(Chapter VIII.) that in such a case there is a tension acting

圭筆

Tension only

FIG. 479.

normal to a diagonal drawn at an angle of 45° with the axis of the shaft, as shown by the arrows in the figure. In Fig. 479 a shaft is shown subjected to tension only. In this case the

tension acts normally to a face at 90° with the axis. In Fig. 480 a shaft is shown subjected to both torsion and tension; the face over which there is the greatest tension will therefore lie between the two faces mentioned above, and the tension on this face will be greater than the tension on either of the other faces, when acted upon only by torsion or tension.

We have shown in Chapter VIII. that the stress f normal to the face gh due to combined tension and shear is—

If the tension be produced by bending, we have--