Lab 4: Torsion of Member with Circular Cross-Section: Principal Strains and Stresses

Amer Al-Daraghma

CIVE 302

October 14, 2014

1. OBJECTIVE

The purpose of this experiment is to understand the relationship between torsion and shear. A circular rod under torsion will be analyzed in order to understand the relationship. Having the circular rod under torsion will obtain the principle stresses and strains at given loads. The theoretical values will be calculated using the Plane Stress theory and compared to the experimental values.

2. THEORY

The shear stress along the member is zero and increases linearly when torsion is applied on the member. Shear stress ( is calculated by multiplying Torsion ( to the radius ( of the cross sectional area of the member and dividing that multiplied value by the polar moment of inertia ( of the cross section.

(Eqn. 1)

The polar moment of Inertia ( for a circular cross section is represented by:

(Eqn. 2)

Shear Stress ( and Shear Strain ) rely on a linear relationship involving the Modulus of Rigidity (G) instead of The Modulus of Elasticity (E) when calculating axial stresses and strains.

(Eqn. 3)

The Modulus of Rigidity (G) is calculated using

(Eqn. 4)

Where (E) is the Modulus of Elasticity and (V) is Poisson’s ratio.

Finally, the theoretical values of the principal stresses are calculated using the Tensile and Compressive Strains obtained using the Plane-Stress theory

(Eqn. 5 & 6)

The experimental values of the stresses and strains can be determined by using these equations and by understanding of Mohr’s Circle. Once the experimental values are obtained, they could be compared to the theoretical values.

3. EXPERIMENTAL SET-UP

Shown below is a strain indicator box, which is needed to read the values of the strains when the load is applied on the member.

Figure 4-1: The Strain Indicator Box

In this experiment, a circular rod with a length of 16 in. and a diameter of 0.750 in. is mounted onto a testing machine. Measuring the displacements was determined by two strain gauges with one being in the principle tensile direction while the other in the principle compressive direction. The gauges are perpendicular to each other and placed at 45 degrees from the member’s axis. The torsion applied was 250 Ib-in per reading.

Figure 4-2: Steel bar connected to torsion testing machine

To have an overall perspective of the equipment, look at Figure 4-3 below. The big machine on the left is The Tinius Olsen Machine

Figure 4-3: The equipment in lab during the experiment

4. PROCEDURE

1. Measure the rod’s length and diameter and record them in your notes

2. Connect the strain gauges to the recorder

3. Turn on the Tinius Olsen machine

4. Record the initial strain when no torsion load is applied

5. Measure the strain at increments of 250 Ib-in

6. Once ma the maximum load is reached, begin unloading using increments of 250 Ib-in as well.

5. DATA COLLECTION

Table 1 – Given or measured data (All values below were either given or measured except for the Shear Modulus and Polar Moment of Inertia. The calculation for these values will be on the attached hand-written sheet). (pg. 10)

Diameter (in)

0.75

Radius (in)

0.375

Polar Moment of Inertia, J ()

0.031063111

Poisson's Ratio

0.25

Shear Modulus, G (ksi)

12000

Modulus of Elasticity, E (ksi)

30,000

Length (in)

16

Distributed load (Ib-in)

250

6. CALCULATIONS

Look at the attached hand-written sheet (pg. 10) for all sample calculations and the excel sheet (pg. 9) for a complete summary for all the values.

7. DATA ANALYSIS

Figure 4-4: Torsion along the member’s length (Area under graph)

Figure 4-5: Values of principle tensile stress when a torsional load is applied

Figure 4-6: Principle tensile stress compared to principle compressive stress

Figure 4-7: Principle Tensile strain compared to Principle compressive strain

Figure 4-8: Shear stress when a torsional load is applied

8. DISCUSSION OF RESULTS & CONCLUSION

After calculating all the values and plotting the data, the measured results and theoretical results have minor differences. The theoretical values were greater than the measured results as expected because errors could have still occurred. One error could be from the way the strains were read because the scale would sometimes slide when the strain indicator was being balanced. Another possible error could be that the gages could have not been perfectly positioned on the angles. However, the average percent error was 2.8%, which represents that only a few errors occurred if there were any.

If an axial load were applied to the rod, the orientation of the gages would need to be changed because the 45 degrees orientation would not be valid. One of the reasons is that axial loads change the coordinates of the x and y planes on Mohr’s circle. Furthermore, the gages were applied to read only the strain in the direction they are placed because they cannot read strains in another direction.

9. REFERENCES

Johnson, T. “Tensile Test of Steel Lab 1 images”, 25, February 2014. CIVE 302 spring 2014 www.blackboard.sdsu.edu

Torsion Diagram

11.82 11.82 11.82 11.82 11.82 11.82 11.82 11.82 11.82 11.82 11.82 11.82 11.82 11.82 11.82 11.82 11.82

Length (in)

Stress (ksi)

Torsion vs. Principle Tensile Stress

0.0 2.820000000000031 8.99999999999997 11.81999999999996 8.820000000000035 5.64000000000002 2.939999999999983 0.23999999999999 0.0 250.0 750.0 1000.0 750.0 500.0 250.0 0.0

Principle Tensile Stress (ksi)

Torsional Load (ib-in)

Principle Tensile Stress vs. Principle Compressive Stress

0.0 -2.820000000000031 -8.999999999999966 -11.81999999999996 -8.820000000000035 -5.64000000000002 -2.939999999999984 -0.23999999999999 0.0 2.820000000000031 8.999999999999966 11.81999999999996 8.820000000000035 5.64000000000002 2.939999999999984 0.23999999999999

Principle Comressive Stress (ksi)

Principle Tensile Stress (ksi)

Principle Tensile Strain vs. Principle Compressive Strain

0.0 -0.000117500000000001 -0.000374999999999998 -0.000492499999999998 -0.000367500000000001 -0.000235000000000001 -0.000122499999999999 -9.9999999999996E-6 0.0 0.000117500000000001 0.000374999999999998 0.000492499999999998 0.000367500000000001 0.000235000000000001 0.000122499999999999 9.9999999999996E-6

Principle Compressive Strain (in/in)

Principle Tensile Strain (in/in)

Torsion vs Shear Stress

0.0 3.01804929122409 9.05414787367227 12.07219716489636 9.05414787367227 6.03609858244818 3.01804929122409 0.0 0.0 250.0 750.0 1000.0 750.0 500.0 250.0 0.0

Shear Stress (ksi)

Torsion (ib-in)

4

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