COMPOUND STRAINS AND STRESSES 143 
be altered to compressive stresses, this will be equivalent to making p or 
¢q, or both, negative instead of positive. It 
would be well for the student to demon- A B 
strate the proposition, first, taking q as 
ive stress and p as tensile stress, 
and then again, taking both p and q as 
compressive stresses. 
case where p or g is equal to zero L 
is one of great practical importance, and 
will be considered in the next Article. N 
144. Principal Stresses due to Com- 
bined Bending and Twisting —When a 
shaft of diameter d is subjected to a bending M H «K 
moment B, the maximum tensile and com- 
"pressive stresses produced are given by the wae Ps 
Q x 
: 32B 
equation p= = 
of the shaft. If the same shaft is subjected to a twisting moment T, the 
16T 
. Let 
, and these stresses are in direction parallel to the axis. 
maximum shear stress produced is given by the equation f= 
ad 
_ ABCD (Fig. 192) be an indefinitely small square prism of the material of 
the shaft in the neighbour- 
“ais where the tensile or a = —B Rp AE os —B 
pressive stresses are a Ht > t<— 
eal the face ABCD KO 
being on the surface of the p+ +P} fp 
shaft, and AB parallel to its +e 
= Then by the preceding mae: = oe a en 
Article, putting g=0, there — f € 
be pure normal stresses Fia. 192. 
‘on planes at right angles to 
4 one another, the intensities of these normal stresses being given by the 
- equation r=4{p+ ,/p?+4/*}. Inserting the values of p and / given 
above, et ae + a/(Sa) + +4() } -" B+ /B?+T?}. The 
greater of these two values of 7 is JS 4B+ ./B?+T?}, and when p is a 
tensile stress the greater value of 7 is a tensile stress, but when p is a 
_ compressive stress the greater value of 7 is a compressive stress. Now, 
_ Since the resistance of the material of shafts to compression is greater 
than the resistance to tension, the maximum value of 7 should be con- 
_ sidered as a tensile stress. 
The equation r= a {B+ ,/B?+T} may be put in the form 
e* as ee ee 
be. 
Ter=B+ /B?+T?. Now, a simple twisting moment T,= at” will 
3 _ produce : a pure shear stress and also a pure tensile stress (Art. 142) of 
zi _ intensity r, therefore a twisting moment T,= B+ ,/bB* + 'T? will produce 
same maximum normal stress as is produced by the combined action 
: 
‘a 
