The University of Melbourne 436-291 Engineering Mechanics436-291 Engineering Mechanics Tutorial Five...

The University of Melbourne

436-291 Engineering Mechanics

Tutorial Five Torsion and Bending

Part A (Introductory)

1. (Problem 11-1 from Hibbeler - Statics and Mechanics of Materials)

Figure 1: Shafts

The solid shaft (the left one in Figure 1) made of steel alloy is having an allowable shear stress ofτallow = 120 N/mm2, and diameter of 15 mm. Determine:

(a) the polar moment of inertia J for the shaft; and

(b) the maximum torque T that can be transmitted through the shaft.

(c) Sketch the shear-stress distribution along a radial line.


Figure 2: Steel tube

The steel tube shown in Figure 2 having an outer diameter of 50 mm is used to transmit 25 kWwhen turning at 2700 rev/min. Given that the allowable shear stress is τallow = 70 MPa, determine:

(a) the turning in radian per second;

1

Week 06 Tutorial Six – Torsion and Bending

(b) the torque transmitted; and

(c) the inner diameter d of the tube to the nearest 0.1 mm.


Figure 3: Steel tube

The splined ends and gears attached to the A-36 steel shaft are subjected to the torques shown inFigure 3. Given that the shaft has a diameter of 40 mm and G = 75 GPa, determine:

(a) the polar moment of inertia for the shaft;

(b) the angle of twist of between each gears; and

(c) the angle of twist of end B with respect to end A.


Figure 4: Wooden beam

A wooden beam is shown in Figure 4. Determine:

(a) the second moment of area about the axis which it is bending; and

(b) the moment M that should be applied to the beam in order to create a compressive stress atpoint D of σD = 30 MPa.

(c) Sketch the stress distribution acting over the cross section and compute the maximum stressdeveloped in the beam.

of 6


Figure 5: Aluminium machine part


The aluminium machine part shown in Figure 5 is subjected to a moment of M = 75 Nm. Giventhat the neutral axis is 0.0175 m from the top surface of the top flange, determine:

(a) the second moment of area about the neutral axis; and

(b) the bending stress created at point B.

(c) Sketch the result on a volume element.


Figure 6: Beam

The beam shown in Figure 6 is made up of multiple sections. The beam is subjected to a momentof M = 4 kNm, and its the second moment of area about the neutral axis is INA = 91.73 cm4,determine:

(a) the location of neutral axis (measured from the top surface of the top flange);

(b) the maximum tensile bending stress; and

(c) the maximum compressive bending stress.

of 6


Part B


The solid shaft with 30 mm diameter shown in Figure 3 is used to transmit the torques applied tothe gears. Determine the shear stress developed in the shaft at points C and D. Indicate the shearstress on volume elements located at these points.


Figure 7: Coupling

The coupling shown in Figure 7 is used to connect the two shafts together. Assuming that the shearstress in the bolts is uniform, determine the number of bolts necessary to make the maximum shearstress in the shaft equal to the shear stress in the bolts. Each bolt has a diameter d.


Figure 8: Steel shafts with fillet weld

The steel shafts shown in Figure 8 are connected together using a fillet weld. Determine the averageshear stress in the weld along section a-a if the torque applied to the shafts is T = 60 Nm. Note:The critical section where the weld fails is along section a-a.


The device shown in Figure 9 serves as a compact torsional spring. It is made of A-36 steel andconsists of a solid inner shaft CB which is surrounded by and attached to a tube AB using a rigidring at B. The ring at A can also be assumed rigid and is fixed from rotating. If the allowableshear stress for the material is τallow = 84 MPa and the angle of twist at C is limited to φallow = 3◦,determine the maximum torque T that can be applied at the end C. Given that: G = 70 GPa.

of 6


Figure 9: Torsional spring

Figure 10: Steel shaft


The steel shaft shown in Figure 10 is made from two segments: AC has a diameter of 10 mm, andCB has a diameter of 20 mm. If it is fixed at its ends A and B and subjected to a torque of 500 Nm,determine the maximum shear stress in the shaft.


In Figure 6, given that M = 4 kNm, determine the resultant force the bending stresses produce onthe horizontal top flange plate AB.


The man has a mass of 78 kg and stands motionless at the end of the diving board (see Figure 11).If the board has the cross section shown, determine the maximum normal strain developed in theboard. The modulus of elasticity for the material is E = 125 GPa. Assume A is a pin and B is aroller.


If the internal moment acting on the cross section of the strut has a magnitude of M = 800 Nm andis directed as shown in Figure 12, determine the bending stress at points A and B. The location z̄ ofthe centroid C of the strut’s cross-sectional area must be determined. Also, specify the orientationof the neutral axis.


The 30 mm diameter shaft is subjected to the vertical and horizontal loadings of two pulleys asshown in Figure 13. It is supported on two journal bearings at A and B which offer no resistance to

of 6


Figure 11: Man on diving board

Figure 12: Channel

Figure 13: Shaft with pulleys

axial loading. Furthermore, the coupling to the motor at C can be assumed not to offer any supportto the shaft. Determine the maximum bending stress developed in the shaft.

of 6

The University of Melbourne 436-291 Engineering Mechanics436-291 Engineering Mechanics Tutorial Five...

Documents

Transcript of The University of Melbourne 436-291 Engineering Mechanics436-291 Engineering Mechanics Tutorial Five...