TORSION
Objectives:
You should be able to…
• Describe the effects of torsional loads
• Compute the deformation and shear stress
of a body subjected to torque
• Compute the power transmitted by a
torque
The figure shows a 2-in.-diameter solid steel cylinder that is
built into the support at C and subjected to the torques TA and
TB.
1. Determine the maximum shear stresses in segments AB
and BC of the cylinder; and
2. Compute the angle of rotation of end A. Use G = 12x106
psi for steel.
Determine the torque in each of the two segments of the cylinder
Polar moment of inertia of the cylinder
Maximum shear in each segment
Rotation of end A of the cylinder
The shaft consists of a 3-in.-diameter aluminum segment that
is rigidly joined to a 2-in.-diameter steel segment. The ends of
the shaft are attached to rigid supports. Calculate the
maximum shear stress developed in each segment when the
torque T = 10 kip in. is applied. Use G = 4 x 106 psi for
aluminum and G = 12 x106 psi for steel.
Equilibrium
Compatibility
Using torsion formula
Solve the two equations
The four rigid gears, loaded as shown in the figure, are
attached to a 2-in.-diameter steel shaft. Compute the angle of
rotation of gear A relative to gear D. Use G = 12 x 106 psi for
the shaft.
Rotation of Gear A relative to Gear D
SEATWORK
Problem 1
(a) For the hollow shaft and loading
shown, determine the maximum
shearing stress. (b) Determine the
diameter of a solid shaft for which
the maximum shearing stress in the
loading shown is the same as in part
(a)
Problem 2
(a) For the solid steel shaft shown
(G=77 GPa), determine the angle of
twist in A. (b) Solve part (a)
assuming that the steel shaft is
hollow with a 30-mm outer diameter
and a 20-mm inner diameter.
The figure shows a steel shaft of length L = 1.5 m and diameter
d = 25 mm that carries a distributed torque of intensity (torque
per unit length) 𝒕 = 𝒕𝑩 𝒙/𝑳 , where t = 200 N m/m. Determine
(1) the maximum shear stress in the shaft; and (2) the angle of
twist of the shaft. Use G = 80 GPa for steel.
B
Equilibrium:
The maximum torque in the shaft is TA which occurs at the fixed support.
Maximum shear stress in the shaft:
Torque acting on a cross section located at the distance x from the fixed end:
Angle of twist:
The tapered, wrought iron shaft carries the
torque T = 2000 lb-in at its free end.
Determine the angle of twist of the shaft. Use
G = 10 x 106 psi for wrought iron.
Each of the two identical shafts is attached to a rigid wall at
one end and supported by a bearing at the other end. The
gears attached to the shafts are in mesh. Determine the
reactive torques at A and C when the torque T is applied to
gear B.
Equilibrium:
෍ 𝑀𝐵 = 0
𝑇 − 𝑇𝐴 − 𝐹𝑅 = 0
𝑒𝑞𝑛. 1
෍ 𝑀𝐷 = 0
2
𝑇𝐶 − 𝐹 𝑅 = 0
3
𝑒𝑞𝑛. 2
Substitute eqn.2 to eqn.1
3
𝑇 = 𝑇𝐴 + 𝑇𝐶
2
𝑒𝑞𝑛. 3
Compatibility:
2
𝑅 ∙ 𝜃𝐴 = 𝑅 ∙ 𝜃𝐶
3
3
𝜃𝐶 = 𝜃𝐴
2
𝑇𝐶 𝐿 3 𝑇𝐴 𝐿
= ∙
𝐽𝐺
2 𝐽𝐺
3
𝑇𝐶 = 𝑇𝐴
𝑒𝑞𝑛. 4
2
Substitute eqn.4 to eqn.3
4
𝑇𝐴 =
𝑇
13
6
𝑇𝐶 =
𝑇
13
POWER TRANSMISSION
A solid steel shaft in a rolling mill
transmits 20 kW of power at 2 Hz.
Determine the smallest safe diameter of the
shaft if the shear stress is not to exceed 40
MPa and the angle of twist is limited to 6o
in a length of 3 meters. Use G = 83 GPa.
Torque :
To satisfy the strength condition:
To satisfy the requirement of rigidity:
To satisfy both requirements, choose 58.7 mm.
The figure shows an inboard engine, 8-ft long steel drive shaft,
and propeller for a motor boat. The shaft is designed to safely
transmit 200 hp at 3500 rev/min. Determine the diameter of
the smallest shaft that can be used and its corresponding angle
of twist. For steel, use a working shear stress of 12,000 psi
and G = 12x106 psi.
1. The solid compound shaft, made of three
di¤erent materials, carries the two torques
shown. (a) Calculate the maximum shear
stress in each material. (b) Find the angle
of rotation of the free end of the shaft. The
shear moduli are 28 GPa for aluminum, 83
GPa for steel, and 35 GPa for bronze.
2. The steel shaft is formed by attaching a hollow shaft to a solid shaft. Determine the
maximum torque T that can be applied to the ends of the shaft without exceeding a
shear stress of 70 MPa or an angle of twist of 2.5 degrees in the 3.5-m length. Use G =
83 GPa for steel.
3.
A tubular shaft is being designed to transmit 225 kW at 1,700 rpm. The maximum
shear stress in the shaft must not exceed 30 MPa. If the outside diameter of the
shaft is D=75mm, determine the minimum wall thickness of the shaft.
1. The solid compound shaft, made of three di¤erent
materials, carries the two torques shown. (a) Calculate the
maximum shear stress in each material. (b) Find the angle
of rotation of the free end of the shaft. The shear moduli are
28 GPa for aluminum, 83 GPa for steel, and 35 GPa for
bronze.
2. The steel shaft is formed by attaching a hollow shaft to a solid shaft. Determine the maximum torque T
that can be applied to the ends of the shaft without exceeding a shear stress of 70 MPa or an angle of
twist of 2.5 degrees in the 3.5-m length. Use G = 83 GPa for steel.
3. Four pulleys are attached to the 50-mm-diameter
aluminum shaft. If torques are applied to the pulleys as
shown in the figure, determine the angle of rotation of
pulley D relative to pulley A. Use G = 28 GPa for
aluminum.
Seatwork:
1.
2.