COMPOUND STRAINS AND STRESSES 151 
The distribution of a shear stress over a section of a beam may be 
_ found by making use of the section modulus figure which was described 
in Art. 113, p. 105. A little consideration will show that the force R 
(Fig. 201) is equal to a,( f,—/,), where a, is the area of that part of the 
section modulus figure which lies beyond the line at a distance h from 
the neutral axis. Hence it follows that g= bales ; 
Exercises [Xa. 
1. A cube ABCD (Fig. 205) is subjected to compressive stress of 10 tons per 
Square inch on the faces AB and CD, ‘Taking D as the origin, 
draw a ipolat curve showing the intensities of the shear stresses B 
on inclined sections through D. Scale, 1 inch to 1 ton per A 
square inch. 
2. A cube ABCD (Fig. 206) is subjected to a compressive 
stress of 3 tons per square inch on the faces AD and BC, and p' Cc 
_ also to a compressive stress of 5 tons per square inch on the faces 
AB and CD. Determine the shear stresses, in tons per square JF 
inch, on the interfaces BD and BE. oe 
3. On A and B, the opposite faces of a cube, there is no stress, but on 
3 the remaining four 
faces there are nor- 
mal stresses of the B Atiiiiitie & inn 
game kind, and of + a ae ees \7/ me 
intensity p. Show > A) Wr poe j—> 
that there is no ep \ WS oe eh cee 
shear stress on any 77] Nie Tt wie 
interfaces which — EX ee > \ itEe 
are perpendicular sreaag Pea ee pee \\ ex 
to the faces A and CS pre ic” bas 
B, also that the TTTtttt c 
peer of the Fra. 206. Fra, 207. 
these interfaces is equal to p. 
4. ABCD (Fig. 207) is a cube, On the faces AD and BC there is tensile 
stress of intensity p, and on the faces AB and CD there is compressive stress of 
intensity g. Show that there is pure shear stress of intensity f on all interfaces 
inclined at an angle @ to AB, and find fand @in terms of p and g. (Hint.— 
Consider the equilibrium of the element BCE.) 
_ 6. The rhombus ABCD (Fig. 208) is one end of a right prism. There is pure 
shear stress of intensity f on the faces AB, BC, CD, and 
DA, as shown. Prove that the interfaces AC and BD 
are subjected to pure normal stresses of intensities 
cov q respectively, and that interfaces, such as 
, which are perpendicular to BC, are subjected to 
shear stress of intensity f, and a normal stress of 
intensity s. Express p,q, and s in terms of /, and @ 
the angle ABD. 
6. The maximum tensile stress on a shaft due Fra. 208. 
to the bending moment is half the maximum shear ; 
Stress due to the twisting moment. The maximum tensile stress due to 
the above two stresses combined is 12,000 Ibs. per square inch. If the 
eeoae of the shaft is 3 inches, find the twisting and bending moments in 
7. The maximum stress on a shaft 3 inches in diameter is 9000 lbs. per square 
inch, and the shaft is subjected to equal bending and twisting moments. Find 
the twisting moment in inch-Ibs. 
__ 8. A shaft transmits 50 horse-power at 135 revolutions per niinute. There is 
a bending moment on the shaft equal to three-fourths of the twisting moment, 
; 
