COMPOUND STRAINS AND STRESSES 155 
- bending action tending to increase the curvature of the ring at A. There 
must Pscefore be some intermediate point F in AFB at which there is 
no tendency to alter the curvature of the ring, and therefore at F there 
_ is no bending action ; consequently at F the ring may be jointed without 
altering the stresses in the other parts. There are obviously four points 
at which joints may be introduced, these being symmetrically situated with 
reference to the vertical and horizontal diameters AC and BE, as shown 
in Fig. 217. The ring now consists of four links connected by pin joints. 
_ The lower link FK, shown detached in Fig. 218, is in the condition 
_ of a beam supported at the ends and loaded in the middle. The maxi- 
- mum tensile stress f due to the bending moment will be at L, and will 
16Wa 
| be equal to =H 
diameter d. 
The right-hand link FH, shown detached in Fig. 219, is subjected to 
a& maximum bending moment at the horizontal section at B amounting 
, assuming the cross section of the ring to be a circle of 
to Wer-2), causing a maximum tensile stress f, at M equal to . 
' ew ol . In addition there is a uniform tensile stress f, on the hori- 
- zontal section at B equal to chalk due to the load ua The total stress 
ae, _16W(r-2) , 2W 
at M is therefore /, +/,= 7s — ra?" 
__ Now it seems reasonable to suppose that the points F and H will be 
80 situated that the total stress at M will equal the stress at L, because 
when the first permanent set takes place, say at M, if there is not a 
simultaneous permanent set at L, the line FH would shift towards M, 
causing the bending moment, and therefore the stress, at L to increase. 
_ The line FH will therefore adjust itself so that permanent set takes place 
simultaneously at M and L, and therefore the stress at M must be the 
- same as that at L. Making use of this, 
-_= 16Wa 16W(r-2) 2W 
PIs 90 Ti 
16Wz 16W/r d W/sr 1 
ati fae sat tie) - sata): 
—_—__—_— 
If r=nd, thend= /|"* 
| v/zfint) 
The foregoing results are only roughly approximate, because of the 
_ assumption that the moment of resistance to bending of the curved piece 
4 FBH is the same as for a straight piece. In the case of a curved bar 
subjected to bending the neutral axis of a cross section does not pass 
1 through its centre of gravity, and the stress does not vary uniformly from 
_ the neutral axis, as in the case of a straight bar. The errors in the 
_ formule deduced above are on the wrong side for safety. 
For a full discussion of the theory of bending of curved bars the 
_ student is referred to Morley’s Strength of Materials. 
; 151. Poisson’s Ratio.—When a bar is subjected to direct stress, either . 
_ tensile or compressive, there is not only a longitudinal strain, but also 
ol & 
from which, x =; + 
