- 
COMPOUND STRAINS AND STRESSES 141 
The theorem which has just been proved has an important application 
in the case of a shaft subjected to twisting. Let ABCD (Fig. 189) be a 
uare traced on the surface 
as shaft, the sides AD and 5 | 
BC being perpendicular to by 
the axis of the shaft, and 
suppose that this square 
represents an indefinitely 
thin prism of the material. 
The faces AD and BC are 
subjected to pure shear 
ope, and by Art. 141 the ee 
faces AB and CD must be subjected to an equal shear stress. Hence by 
the theorem of the present Article the diagonal face AC is subjected to 
pure tension, and the diagonal face BD is subjected to pure compression, 
also the intensities of the tensile and compressive stresses will be the same 
as that of the shear stress. Now, if the resistance of the material to tension 
be less than its resistance to shearing the shaft will give way along AC, 
which is part of a helix whose inclination to the axis of the shaft is 45°. 
This is what actually occurs when a cast-iron shaft is broken in torsion, 
except that the inclination of the helix is not exactly 45°. Further 
reference to this matter will be found in Art, 167, p. 176. 
As an illustration of the presence of tensile and compressive stresses 
whose directions are inclined at 45° to the directions of the shear stresses 
in a shaft under torsion, it will be found that a spiral spring whose coils 
are close together and inclined at 45° to the axis will be as stiff and 
strong when twisted in one direction as a tube of the same material 
having the same outside and inside diameters, but under torsion in the 
opposite direction the spring will be very weak. When twisted in the 
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a: 
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am ‘ 
ae 
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_ first direction the surfaces of the coils in contact are subjected to pure 
compression, and therefore the fact that the material is divided at the 
surfaces in contact will not affect the power of the coils to resist com- 
pression. When twisted in the opposite direction the coils will separate. 
(r a Ses Stresses—Principal Axes of Stress.—Let ABCD 
ig. 190 an 
badetitel, small ected f. 
cube in a strained 
body, the face 
ABCD and all in- 
terfaces parallel to 
it being free from 
stress. The stresses 
on the faces AB, 
BC, CD, and DA 
ol. ets > 
Ss Tat q 
may be resolved into Fie. 190. 
’ 
] 
= 
‘ 
7 
2 
normal and shear stresses. The normal stress on AD must balance the 
normal stress on BC; let the intensity of these stresses be denoted by p. 
The normal stress on AB must balance the normal stress on CD ; let the 
intensity of these stresses be denoted by 4. 
By Art. 141 the intensities of the shear stresses on AB, BC, CD, and 
DA must be equal; let these be denoted by /. 
