COMPOUND STRAINS AND STRESSES 145 
Consider the indefinitely small square prism HJKL, shown enlarged 
‘in Fig. 196. Let JK =z, and let the depth of the prism at right angles 
_ to the plane of the paper also be z, Cut off from the prism HJKL a 
wedge KNJ, the angle JKN being «. Consider the equilibrium of the 
wedge KNJ. On the face JK there is a normal stress 7,, the resultant 
_ of which is 7,x*. On the face JN there is a normal stress 7,, the re- 
P sultant of which is 7,2? tana. On the face KN there is a stress equiva- 
_ lent to a normal stress r, and a shear stress f,. The resultant of the 
; 2 
normal stress on the face KN is 7, ”—, and the resultant of the shear 
COS a 
stress on this face is /, — Resolving these resultant forces parallel 
to KN, fy = 02 sin a + 7,2 tan a cos a, or 
J,=1, Sin a cos a +7, sin a cos a= }(7, +7.) sin 2a. 
Hence f, is a maximum when a= 45°, then f,=}(7,+7,). If tensile 
‘stress is positive and compressive stress is negative, and if 7, and r, 
earry their proper signs with them, then f,=}(7,-7,). Inserting the 
values of 7, and 7, in terms of p and /, then the maximum value of 
a 32B 16T 16°). as 
2 => => ST 
Js is 4,/p? +477. But p mF and f = therefore f, —2 VB +T? 
and in /B?+T?. But a simple twisting-moment T,= ie? would 
produce the same shear stress f,. Hence a simple twisting moment 
T,= ,/B?+T? will produce the same maximum shear stress as the bend- 
‘ing moment B and twisting moment T acting together. 
A bending moment B,=T,= ,/B?+T? would produce a maximum 
normal stress equal to 2/,, and therefore (Art. 140, p. 138) a maximum 
shear stress 7, at 45° to the direction of the normal stress. 
There is little doubt that in the case of ductile materials, such as mild 
_ Steel, it is the resistance to shear which determines the strength (see Art. 
; 166, p. 175). Hence in designing shafts made of ductile material, and 
_which are subjected to bending and twisting, the formula T,= ,/B* + T? 
! should be~used in preference to the one T,=B+ ,/B?+T?. But, for 
: mild steel shafts, in equating B+ ,/B?+T? to ig? it must be re- 
Guest was the first to demonstrate that mild steel shafts subjected 
_ to bending and twisting gave way by shear,* and his theory and the 
results of his experiments have been confirmed by Hancock, Scoble 
C. A. Smith, and others. 
+ T.= ,/B?+T? is generally called the “Guest” formula, and 
- T.=B+ ./B? + T? is generally called the “Rankine” formula. 
___ Shafts designed by the Rankine formula are weaker than those 
4 
_ designed by the Guest formula. 
is” “Strength of Ductile Materials under Combined Stress,” Phil. Mag., July 1900, 
’ K 
