Answer:
a) the maximum shear stress τ
the bar is 16T
/πd³
b) the angle of twist between the ends of the bar is 16tL² / πGd⁴
Explanation:
Given the data in the question, as illustrated in the image below;
d is the diameter of the prismatic bar of length AB
t is the intensity of distributed torque
(a) Determine the maximum shear stress tmax in the bar
Maximum Applied torque T_max = tL
we know that;
shear stress τ = 16T/πd³
where d is the diameter
so
τ
= 16T
/πd³
Therefore, the maximum shear stress τ
the bar is 16T
/πd³
(b) Determine the angle of twist between the ends of the bar.
let theta (
) be the angle of twist
polar moment of inertia
= πd⁴/32
now from the second image;
lets length dx which is at distance of "x" from "B"
Torque distance x
T(x) = tx
Elemental angle twist = d
= T(x)dx / G![I_{p}](https://tex.z-dn.net/?f=I_%7Bp%7D)
so
d
= tx.dx / G(πd⁴/32)
d
= 32tx.dx / πGd⁴
so total angle of twist
will be;
=
=
32tx.dx / πGd⁴
= 32t / πGd⁴
= 32t / πGd⁴ [ L²/2]
= 16tL² / πGd⁴
Therefore, the angle of twist between the ends of the bar is 16tL² / πGd⁴