Topic 8: Torsion
Torque - is a moment that tends to twist a member about its longitudinal axis.
τ = Shearing stress
T = Torque
ρ = Distance
J = Polar moment of Inertia
G = Modulus of rigidity/Shear modulus
L = Length
Ans:
AB = 6.88 ksi
BC = 3.82 ksi
Ѳ = 2.63 deg
The pipe shown has an inner diameter of 80 mm and an outer diameter of 100 mm. If its end is
tightened against the support at A using a torque wrench at B, determine the shear stress
developed in the material at the inner and outer walls along the central portion of the pipe when the
80-N forces are applied to the wrench.
Ans:
Outer walls = 0.345 MPa
Inner walls = 0.276 MPa
Ans:
Ѳ = 1.62 deg
A solid constant-diameter shaft is subjected to the torques shown. The bearings shown allow the
shaft to turn freely. (a) Plot a torque diagram showing the internal torque in segments (1), (2), and
(3) of the shaft. (b) If the allowable shear stress in the shaft is 80 MPa, determine the minimum
acceptable diameter for the shaft.
Ans:
Minimum diameter = 24.1 mm
A 2-in-diameter steel shaft rotates at 240 rpm. If the shearing stress is limited to
12 ksi, determine the maximum horsepower that can be transmitted.
1.0 hp = 550 lb-ft/s
1.0 hp = 746 watts = 1 N-m/s
Ans:
71.78 hp
A solid 3-in.-diameter bronze [G = 6,000 ksi] shaft is 7-ft long. The allowable shear stress in the
shaft is 8 ksi and the angle of twist must not exceed 0.03 rad. Determine the maximum
horsepower that this shaft can deliver.
(a) when rotating at 150 rpm. (b) when rotating at 540 rpm.
1.0 hp = 550 lb-ft/s
1.0 hp = 746 watts = 1 N-m/s
Ans:
a.) 40.6 hp
b.) 146.0 hp
The solid rod AB has a diameter dAB = 60 mm. The pipe CD has an outer diameter of 90 mm and a
wall thickness of 6 mm. Knowing that both the rod and the pipe are made of steel for which the
allowable shearing stress is 75 MPa, determine the largest torque T that can be applied at A.
Ans:
3.18 kN-m
Ans:
Aluminum = 8.63 ksi
Steel = 3.45 ksi
A compound shaft consists of an aluminum alloy [G = 26 GPa] tube (1) and a solid bronze [G = 45 GPa] shaft
(2). Tube (1) has a length of L1 = 900 mm, an outside diameter of D1 = 35 mm, and a wall thickness of t1 = 4
mm. Shaft (2) has a length of L2 = 1,300 mm and a diameter of D2 = 25 mm. If an external torque of TB = 420
N-m acts at pulley B in the direction shown, calculate the torque TC required at pulley C so that the rotation
angle of pulley C relative to A is zero.
Ans:
136.8 N-m
SHEAR AND MOMENT
DIAGRAM
Additional/Supplementary
Exercises
(a) Determine the torque T that causes a maximum shearing stress of 45 MPa in the hollow
cylindrical steel shaft shown. (b) Determine the maximum shearing stress caused by the
same torque T in a solid cylindrical shaft of the same cross-sectional area.
Ans:
The torques shown are exerted on pulleys A, B, and C. Knowing that both shafts are solid,
determine the maximum shearing stress in (a) shaft AB, (b) shaft BC.
Ans:
Ans:
a.) 3080 psi
b.) 0.0169 rad
c.) 0.00749 rad
Ans:
d = 51.9 mm
The two solid shafts are connected by gears as shown and are made of a steel for which the
allowable shearing stress is 7000 psi. Knowing the diameters of the two shafts are, respectively,
dBC = 1.6 in. and dEF = 1.25 in. determine the largest torque TC that can be applied at C.
Hint: Utilize the formula for Torsion, which is F x distance
Ans:
4.30 kip-in
Knowing that for each shaft the allowable shearing stress is 60 MPa, determine the largest
torque T that can be applied. (Medyo difficult boss, pag nasolve mo torsion gods ka na)
Ans:
73.6 kN-mm
Ans:
dbc = 67.4 mm
dbolt = 22 mm
A 5-m steel shaft rotating at 2 Hz has 70 kW applied at a gear that is 2 m from the left end where 20 kW
are removed. At the right end, 30 kW are removed and another 20 kW leaves the shaft at 1.5 m from the
right end. (a) Find the uniform shaft diameter so that the shearing stress will not exceed 60 MPa. (b) If a
uniform shaft diameter of 100 mm is specified, determine the angle by which one end of the shaft lags
behind the other end. Use G = 83 GPa.
1.0 hp = 550 lb-ft/s
1.0 hp = 746 watts = 1 N-m/s
Ans:
a.) 69.66 mm
b.) 0.448 degrees
For the simply supported beam subjected to the loading shown.
a.) Plot the shear-force and bending-moment diagrams.
b.) Find the maximum positive bending moment, the maximum negative
bending moment, and their respective locations.
For the simply supported beam subjected to the loading shown.
a.) Plot the shear-force and bending-moment diagrams.
b.) Find the maximum positive bending moment, the maximum negative
bending moment, and their respective locations.
Draw the shear and bending-moment diagrams for the beam and loading shown, and
determine the maximum absolute value (a) of the shear, (b) of the bending moment.
(a) 3.00 kN. (b) 0.800 kN-m.
(a) 150.0 lb. (b) 1500 lb-in.
Determine the maximum absolute value (a) of the shear, (b) of the bending moment.
(a) 3.45 kN. (b) 1125 N-m.
(a) 2000 lb. (b) 19200 lb-in.