ME2113-2

TORSION OF CIRCULAR SHAFTS

(EA-02-21)

SEMESTER 3

2015/2016

DEPARTMENT OF MECHANICAL ENGINEERING
NATIONAL UNIVERSITY OF SINGAPORE Objectives
The main objective of the experiment is to study how a hollow shaft would behave compared with the solid shaft, in terms of strength and rigidity under torsion.

Sample Calculations
Table 2: Strength and stiffness of hollow and solid shafts having the same outer diameter
Volume of 12mm diameter solid shaft, Vs = Area of cross section x height = = 1.131 x 10-5 m3
For hollow shaft of outer diameter 12mm, inner diameter 6mm,

Experimental % change in torsional stiffness, = (261.09 – 264.63) / 264.63 x 100% = -1.3377 %

Theoretical % change in torsional stiffness, = - (0.006 / 0.012)4 x 100% = -6.25 %

% change in volume,
Vh = = 8.483 x 10-6 m3 = (8.483 x 10-6 - 1.131 x 10-5) / (1.131 x 10-5) x 100% = -25.00 %

Theoretical % change in maximum shear stress, = [12 / (124 – 64) – 1/123] / (1/123) x 100% = 6.67 %

Table 3: Strength and stiffness of hollow and solid shafts having the same volume
For solid shaft of diameter 7.93mm and hollow shaft of outer diameter 12mm, inner diameter 9mm,
Experimental % change in torsional stiffness, = (240.17 – 119.57) / 119.57 x 100% = 100.86 %

Theoretical % change in torsional stiffness, = 2 x [1-(7.93/12)2] / (7.93/12)2 x 100% = 257.98 %

Theoretical % change in maximum shear stress, = [12 / (124 – 94) – 1/7.933] / (1/7.933) x 100% = -57.78 %

Table 1: Experimental data for solid and hollow shafts

Angular Displacements Torque (Nm) for Solid Shafts Torque (Nm) for Hollow Shafts
Degrees Radians 7.93 mm 8.94 mm 9.74 mm 10.39 mm 12.0 mm 12/9 mm 12/8 mm 12/7 mm 12/6 mm
0 0 0 0 0 0 0 0 0 0 0
0.20 0.003491 0.32 0.32 0.46 0.46 0.59 0.49 0.78 0.85 0.49
0.40 0.006981 0.67 0.7 0.93 0.91 1.25 1.05 1.44 1.64 1.06
0.60 0.010472 1.05 1.14 1.46 1.46 1.99 1.66 2.15 2.44 1.74
0.80 0.013963 1.44 1.5 2 2.03 2.82 2.41 2.92 3.2 2.54
1.00 0.017453 1.85 1.93 2.57 2.73 3.7 3.26 3.76 3.96 3.36
1.20 0.020944 2.28 2.38 3.15 3.46 4.7 4.17 4.6 4.77 4.37
1.40 0.024435 2.73 2.82 3.76 4.2 5.74 5.13 5.57 5.65 5.42
1.60 0.027925 3.19 3.3 4.38 5.01 6.83 6.16 6.56 6.62 6.54
1.80 0.031416 3.67 3.82 5.02 5.82 8 7.2 7.57 7.63 7.82
2.00 0.034907 4.15 4.31 5.7 6.62 9.17 8.27 8.59 8.69 9.06
Ip (mm4) 388.23 627.12 883.56 1144.09 2035.75 1391.63 1633.63 1800.03 1908.52
K (Nm/rad) 119.57 123.94 163.66 192.02 264.63 240.17 244.78 243.51 261.09

Table 2: Strength and stiffness of hollow and solid shafts having the same outer diameter

Solid Shaft

Ds = 12 mm

Vs = 1.13×10-5 m3

Ks = 264.63 Nm rad-1

% change in torsional stiffness

% change in volume

Theoretical % change in maximum shear stress

Experimental Theoretical
Hollow Shafts
1. 12/6 mm dh/Dh = 0.5000 -1.3377 -6.25 -25.00 6.67
2. 12/7 mm dh/Dh = 0.5833 -7.98 -11.58 -34.03 13.10
3. 12/8 mm dh/Dh = 0.6667 -7.501 -19.75 -44.44 24.62
4. 12/9 mm dh/Dh = 0.7500 -9.243 -31.64 -56.25 46.29

Table 3: Strength and stiffness of hollow and solid shafts having the same volume

% change in torsional stiffness

Theoretical % change in maximum shear stress

Experimental Theoretical

-57.78

-48.47

-39.52

-30.76

1. solid, 7.93 mm dia. hollow, 12/9 mm

2. solid, 8.94 mm dia. hollow, 12/8 mm

3. solid, 9.74 mm dia. hollow, 12/7 mm

4. solid, 10.39 mm dia. holow, 12/6 mm 100.86

97.499

48.79

35.97
257.98

160.34

103.58

66.78

Graph 1

Graph 2

Graph 3 Graph 4

Discussion
1. Compare the results for shafts having the same outer diameters (Table 2)
As compared to a solid 12mm diameter shaft, the hollow shafts experience a fall in torsional stiffness as their inner diameter increases, as seen from table 2 where the percentage change in torsional stiffness of the solid shaft to the hollow shaft increases with increasing inner diameters. This increasing trend is applicable to the percentage change in volume of the shaft as well. As the percentage change in torsional stiffness increases, the percentage change in volume increases. Also, the theoretical percentage change in maximum shear stress experiences a similar increasing trend, where it increases as percentage change of torsional stiffness increases. This explains that hollow shafts with larger inner diameters will have less volume and will experience a larger maximum shear stress while having less torsional stiffness.

2. Compare the results for shafts having the same volume (Table 3)
When comparing a solid shaft and a hollow shaft with the same volume, it can be inferred from table 3 that hollow shafts with larger inner diameters will have a higher difference in torsional stiffness as compared to their counterparts with the same volume. This is evidenced by the decreasing trend of percentage change in torsional stiffness when comparing a solid shaft with a hollow shaft of the same volume with decreasing inner diameter. Also, the theoretical percentage change in maximum shear stress from hollow shafts to solid shafts with the same volume experiences a decreasing trend as the inner diameter of the hollow shaft decreases. This shows that hollow shafts with larger inner diameters will experience less maximum shear stress as compared to solid shafts with the same volume.

3. Comparing experimental results with theoretical results in table 2 and 3
The experimental and theoretical graphs show a similar trend and the plotting points are close to each other but still differ. This might be because the shafts used in the experiment have been subjected to torsion repeatedly over a long period of time and might have been deformed or damaged. This would result in inaccurate readings received. Also, the machines itself contain certain experimental error which would cause the experimental readings to deviate from the theoretical readings, although maintaining the similar trend. Thirdly, the accuracy limitation (machine only has an accuracy of 0.01) of the machine might result in deviations from the theoretical result.
4. Would a hollow shaft be stronger and more rigid than a solid shaft if Ds = Dh?
No, a hollow shaft will not be stronger and more rigid than a solid shaft. From table 2 and graph 3, we can see that the percentage change in torsional stiffness of a hollow shaft to a solid shaft with the same outer diameter is always negative. The stiffness of a material is indicated by the constant K in the equation: Where J = , ϕ = angle of twist, L = length of shaft, Dh = outer diameter, dh = inner diameter
The polar second moment of area, J, is dependent on the outer and inner diameters. For a solid shaft, there is no inner diameter, hence J of solid shafts will always be larger than a hollow shaft of the same outer diameter and K will be larger correspondingly. Therefore, solid shafts are stronger and more rigid than hollow shafts with the same outer diameter.

5. For a given amount of material, would you fabricate it to a hollow or solid shaft?
As seen from the experiment, hollow shafts can achieve the same torsional stiffness as solid shafts while having a smaller outer diameter, using less material in the process. Therefore, for a given amount of material, fabricating a hollow shaft will lead to a higher torsional stiffness as compared to a solid shaft. However, it must be noted that this experiment only takes into account torsional stiffness. When encountering combined loading situations, one must consider and experiment with uniaxial or bending stresses before making a decision.

Conclusion
This experiment has showcased how a hollow shaft is stronger and more rigid as compared to a solid shaft using the same amount of material, with the difference in torsional stiffness being higher when the inner diameter of the hollow shaft is larger. Also, it proved that a hollow shaft will experience lower torsional stiffness as inner diameter increases.