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A solid steel bar of diameter d 1 = 25.0 mm is enclosed by a steel tube of outer diameter d 3 = 37.5 mm and inner diameter d 2 = 30.0 mm (see figure). Both bar and tube arc held rigidly by a support at end A and joined securely to a rigid plate at end B. The composite bar, which has a length L = 550 mm, is twisted by a torque T = 400 N ·m acting on the end plate. Determine the maximum shear stresses T 1 and T 2 in the bar and tube, respectively. Determine the angle of rotation 0(in degrees) of the end plate, assuming that the shear modulus of the steel is G = 80 GPa. Determine the torsional stiffness k T of the composite bar.

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Mechanics of Materials (MindTap Co...

9th Edition
Barry J. Goodno + 1 other
Publisher: Cengage Learning
ISBN: 9781337093347

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BuyFindarrow_forward

Mechanics of Materials (MindTap Co...

9th Edition
Barry J. Goodno + 1 other
Publisher: Cengage Learning
ISBN: 9781337093347
Chapter 3, Problem 3.8.12P
Textbook Problem
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A solid steel bar of diameter d1= 25.0 mm is enclosed by a steel tube of outer diameter d3= 37.5 mm and inner diameter d2= 30.0 mm (see figure). Both bar and tube arc held rigidly by a support at end A and joined securely to a rigid plate at end B. The composite bar, which has a length L = 550 mm, is twisted by a torque T = 400 N ·m acting on the end plate.

  1. Determine the maximum shear stresses T1and T2in the bar and tube, respectively.

  • Determine the angle of rotation 0(in degrees) of the end plate, assuming that the shear modulus of the steel is G = 80 GPa.
  • Determine the torsional stiffness kTof the composite bar.
  •   Chapter 3, Problem 3.8.12P, A solid steel bar of diameter d1= 25.0 mm is enclosed by a steel tube of outer diameter d3= 37.5 mm , example  1

      Chapter 3, Problem 3.8.12P, A solid steel bar of diameter d1= 25.0 mm is enclosed by a steel tube of outer diameter d3= 37.5 mm , example  2

    (a)

    To determine

    The maximum shear stresses in the bar.

    The maximum shear stresses in the tube.

    Explanation of Solution

    Given information:

    The diameter of steel bar is 25mm , the outside diameter of steel tube is 37.5mm , the inside diameter of the steel tube is 30mm , the length of the composite bar is 550mm , twisted torque is 400Nm and the shear modulus of elasticity is 80×109N/m2 .

    Write the expression for polar moment of inertia of bar.

      Ip1=π32(d14) ...... (I)

    Here, the polar moment of inertia of bar is IP , the diameter of bar is d1 .

    Write the expression for polar moment of inertia of tube.

      Ip2=π32(d34d24) ...... (II)

    Here, the polar moment of inertia of bar is IP2 , the outside diameter of bar is d3 and the inside diameter of the tube is d2 .

    Write the expression for torque in the bar.

      Tb=T(Ip1Ip1+Ip2) ...... (III)

    Here, the torque in the bar is Tb and the total twisted torque is T .

    Write the expression for torque in the tube.

      Tt=T(Ip2Ip1+Ip2) ...... (IV)

    Here, the torque in the tube is Tt and the total twisted torque is T .

    Write the expression for maximum shear stress in bar.

      τ1=Tb(d12)Ip1...... (V)

    Maximum shear stress in the bar is τ1 .

    Write the expression for maximum shear stress in tube.

      τ2=Tt(d32)Ip2...... (VI)

    Maximum shear stress in the bar is τ1 .

    Calculation:

    Substitute 25mm for d1 in Equation (I).

      Ip1=π32(25mm)4=π32(25mm( 1m 1000mm))4=π32(0.025m)4=3.834×108m4

    Substitute 37.5mm for d3 and 30mm for d2in Equation (II).

      Ip2=π32(( 37.5mm)4( 30mm)4)=π32(( 37.5mm( 1m1000mm ))4( 30mm( 1m1000mm ))4)=π32(1.97×106m48.1×107m4)=1.146×107m4

    Substitute 1.146×107m4 for Ip2 , 3.834×108m4 for Ip1 and 400Nm in Equation (III).

      Tb=(400Nm)(( 3.834× 10 8 m 4 )( 3

    (b)

    To determine

    The angle of rotation of the end plate.

    (c)

    To determine

    The torsional stiffness of the bar.

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