Torsion Testing Lab

docx

School

Auburn University *

*We aren’t endorsed by this school

Course

3130

Subject

Mechanical Engineering

Date

Feb 20, 2024

Type

docx

Pages

Uploaded by ChancellorKoupreyPerson1035

Enoch Crittenden eic0002@auburn.edu February 5, 2024 Torsion Testing Lab

Objective The objective of the lab is to determine the shear modulus of a cylindrical shaft using two methods: torsion test and torsion theory. Five loads in increments of 5 N (Newtons) are applied to the torque arm of a cylindrical copper shaft, connected to a torsion testing machine. For each load, the strains of the three element (0-45- 90) strain gage rosettes were recorded, as well as the vertical deflection of the torque arm. This data is used to generate a Shear Stress vs. Shear Strain ( Figure 1 ) plot and a Torque vs. Angle of Twist plot and compare the shear modulus to theoretical values. Equations The twisting of an object due to an applied torque is torsion. The shear modulus of a material can be found through two testing methods. The first method consists of determining the applied torque and angle of twist for each load (from 5 – 25 N). The torque can be calculated: T = P∙d ( Eq. 1 ) P is the applied load and d is the length of the torque arm. To determine shear modulus using the applied torque, the angle of twist is also be solved as: tan θ T = h s ( Eq. 2 ) Here, h, is the vertical deflection of the torque arm and s is the distance between the center of the shaft where h is the vertical deflection of the torque arm and s is the distance between the center of the shaft and the dial gage. The vertical deflection can be recorded from the deflection dial gage for each load. The torque and angle of twist calculated for each load were plotted against each other and a slope was found using a linear regression fit for the data using Excel. The shear modulus for this method was then found: G =( T ∗ L θ ∗ J ) ( Eq. 3 ) G is the shear modulus, T θ is the slope of the plot, L is the length of the shaft, and J is the polar moment of inertia. The polar moment of inertia can be found through by the following equation: J = π ∗ D ⁴ 32 ( Eq. 4 )

The diameter of the shaft is denoted as D. This calculated shear modulus for method 1 can then be compared to the published shear modulus for the material to find the percent difference in the following equation: Percent Difference = ¿ G − G th ∨ ¿ G th ∗ 100 ¿ ( Eq. 5 ) Another method to determine shear modulus is to find the shear stress and shear strain for each load. This can be done by a 0-45-90 strain gage rosette. Using the gage normal strains for each of the angles, the shear strain can be calculated: γ xy = 2 ∙Σ ε 45 − Σ ε 0 − Σ ε 90 ( Eq. 6 ) Shear strain ( γ xy ) and ε , the gage normal strain recorded for each of the three given angles. Shear stress can be calculated as: τ xy = T ∗ R J ( Eq. 7 ) Radius, R, is of the shaft. The shear stress and shear strain for each load are plotted against one another and linear regression is used the data to find the slope. The slope of the shear modulus and can be written as the following equation: G = τ xy γ xy ( Eq. 8 ) The slope plotted is τ xy γ xy . This calculated shear modulus for method 2 can then be compared to the published shear modulus for the material to find the difference ( Eq. 9 ). Testing A cylindrical copper shaft was set up to a torsion testing machine. The shaft was clamped on one side and attached to a torque arm on the other. Each load was applied to the torque arm in increments of 5 N (Newtons); from 5 – 25 N. As each load is applied, the vertical deflection is recorded from the gage on the torque arm. During this process, the strain is recorded from the switch and balance unit. Both units can be changed to display the strain for each of the three angles (0-45-90). These recorded values along with the equations above were used to determine the torque, the shear strain, and angle of twist at each load. Once done, plots were then generated, and the slope of the linear regression fit was found; this determines the shear modulus for each method. The experimental values of G are compared to the published value (G = 49000 N mm 2 ) and percent differences for both methods are calculated.

Your preview ends here

Eager to read complete document? Join bartleby learn and gain access to the full version

Access to all documents
Unlimited textbook solutions
24/7 expert homework help

Results Table 1 describes the data given corresponding to the torsion test experiment setup. Table 2 describes the torque, shear strain, shear stress, and angle of twist. Table 1: Measured Data P (N) h ε 0 (x 10 -6 ) ε 45 (x 10 -6 ) ε 90 (x 10 -6 ) 5 0.88 -2 51 0 10 1.96 0 113 0 15 3.01 2 177 0 20 4.14 5 246 -1 25 4.47 5 265 -2 Table 2: Reduced Data P (N) Torque (T) – N*mm Angle of Twist ( θ T ) Shear Stress ( τ xy ) Shear Strain ( γ xy ) 5 500 0.964 79.59 104 10 1000 2.146 159.20 226 15 1500 3.300 238.76 352 20 2000 4.530 318.34 488 25 2500 4.890 397.93 527 The results from Table 1 are reduced to yield the values in Table 2 by using the defined equations above. Each box was calculated and is solved as: T = P ∙d = 5 ∙ 100 mm = 500 N ∙mm – ( Eq. 1 ) Torque θ T = arctan ( h s ) = arctan ( 0.88 52.3 ) = 0.964 ° – ( Eq. 2 ) Angle of Twist J = π D 4 32 = π ¿¿ – ( Eq. 4 ) Polar Moment of Inertia τ xy = TC J = ( 500 )( 4 ) 25.13 = 79.59 – ( Eq. 7 ) Shear Stress γ xy = ( 2 ∙Σ ε 45 ) − Σ ε 0 − Σ ε 90 = ( 51 ) − ( − 2 ) − 0 = 104 – ( Eq. 6 ) Shear Strain Using the formula to determine the percent error between the theoretical and experimental for the Shear Modulus yields: Percent Error = G − G TH G × 100 = 49 − 47.54 49 × 100 = ¿ 2.98% – ( Eq. 9 )

The shear stress values compared to the shear strain values are graphed below in Figures ( 1 ), and Figure ( 2 ) describes the graph between torque and angle of twist. The shear modulus is calculated via linear regression in Excel as shown. Figure 1: Shear Stress vs Shear Strain Figure 2: Torque vs Angle of Twist This lab determines the shear modulus of a material utilizing two different methods: by performing a torsion (shear) test on a shaft with a circular cross- sectional area and application of the torsion theory. This knowledge describes how to use a torsion testing machine, read a strain indicator, use a switch, and balance unit, and properly prepare a cylindrical shaft sample with mounted three element strain rosettes. Based on the obtained percent error, the shear modulus value has

Your preview ends here

Eager to read complete document? Join bartleby learn and gain access to the full version