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This report aims to investigate unsymmetrical bending and determine the position of the shear center, along with analyzing the effects of unsymmetrical bending.
The experimental setup for the unsymmetrical bending and shear center experiment is as follows:
It serves as the specimen for the experiment.
The objective of this setup is to investigate unsymmetrical bending behavior by subjecting the U-shaped beam to different loads and measuring its deflections.
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This data will be used to determine the position of the shear center and assess the effects of unsymmetrical bending.
Loads ranged from 120g to 480g.
By employing these methods, the experiment aimed to gain insights into unsymmetrical bending behavior and accurately determine the shear center's position within the U-shaped beam.
During unsymmetrical bending experiments, measurements were recorded as loads were applied to the beam. The initial beam rotation was set to zero, and the experiment continued until the rotation angle reached 180 degrees, with increments of 22.5 degrees. This process was repeated for four different loads: 120g, 240g, 360g, and 480g.
The recorded data from the experiment is presented in Table 1:
| Head Angle (°) | Reading measurements in mm | |||
|---|---|---|---|---|
| 0 | -0.46 | -0.66 | -1.50 | -2.89 |
| 22.5 | -0.42 | -0.78 | -0.81 | -1.97 |
| 45 | -0.75 | -0.19 | -1.56 | -0.66 |
| 67.5 | -1.11 | -0.21 | -2.45 | -0.81 |
| 90 | -1.71 | -0.61 | -2.82 | -0.88 |
| 112.5 | -1.88 | -1.21 | -3.75 | -2.23 |
| 135 | -1.25 | -1.25 | -3.02 | -3.10 |
| 157.5 | -1.10 | -1.28 | -2.83 | -4.02 |
| 180 | -0.60 | -0.77 | -1.61 | -3.68 |
Subsequently, calculations were performed based on the following equations:
U = (left + right) / √2
V = (left - right) / √2
The results of these calculations are presented in Table 2:
| Head Angle (°) | Calculated values of deflections in mm | |||
|---|---|---|---|---|
| 0 | -0.79 | 0.14 | -3.10 | 0.98 |
| 22.5 | -0.85 | 0.25 | -1.97 | 0.82 |
| 45 | -0.66 | -0.40 | -1.57 | -0.64 |
| 67.5 | -0.93 | -0.64 | -2.31 | -1.16 |
| 90 | -1.64 | -0.78 | -2.62 | -1.37 |
| 112.5 | -2.18 | -0.47 | -4.23 | -1.07 |
| 135 | -1.77 | 0.00 | -4.33 | 0.06 |
| 157.5 | -1.68 | 0.13 | -4.84 | 0.84 |
| 180 | -0.97 | 0.12 | -3.74 | 1.46 |
The relationship between deflection and applied load at different angles is presented in Table 3:
| Rotating Angle (°) | dU/dP (mm/g) | dV/dP (mm/g) | dU/dF (mN) | dV/dF (mN) |
|---|---|---|---|---|
| 1 | 0 | -0.017 | 0.0063 | -0.00171 |
| 2 | 22.5 | -0.0102 | 0.0026 | -0.001022 |
| 3 | 45 | -0.0081 | 0.0011 | -0.000814 |
| 4 | 67.5 | -0.0084 | -0.0031 | -0.000841 |
| 5 | 90 | -0.016 | -0.0029 | -0.00160 |
| 6 | 112.5 | -0.0271 | -0.0011 | -0.002712 |
| 7 | 135 | -0.0218 | 0.004 | -0.002182 |
| 8 | 157.5 | -0.0143 | 0.0053 | -0.001433 |
| 9 | 180 | -0.017 | 0.0063 | -0.00171 |
A Mohr’s circle was generated using the data from Table 3.
From the Mohr’s circle, key values were extracted, including OC and r, which were used to calculate Ix and Iy as follows:
Ix = 7.103 × 10-10 m4
Iy = 2.125 × 10-10 m4
To draw another Mohr’s circle, the values of Ix, Iy, and Ixy of the beam were determined.
After calculating the centroid, Ix, Iy, and Ixy were determined as follows:
Ix = 1485.45 mm4
Iy = 965.07 mm4
Ixy = 361.91 mm4
Ix(t) = 1847.36 mm4
Iy(t) = 1123.54 mm4
The shear center of the beam was determined by collecting data from two indicators, as shown in Table 4:
| Eccentricity of Load (mm) | Left-hand Indicator Reading (mm) | Right-hand Indicator Reading (mm) |
|---|---|---|
| -25 | -6.47 | -1.43 |
| -20 | -5.9 | -1.98 |
| -15 | -5.53 | -2.56 |
| -10 | -4.95 | -3.05 |
| -5 | -4.65 | -3.87 |
| 0 | -4.14 | -4.27 |
| 5 | -3.47 | -4.51 |
| 10 | -3.02 | -5.05 |
| 15 | -2.54 | -5.59 |
| 20 | -1.98 | -6.00 |
| 25 | -1.51 | -6.42 |
Due to the accuracy of the experiment and the beam's symmetry, it was determined that the shear center precisely locates at 0mm.
Table 5 summarizes the experimental and theoretical values of the principal second moments of area (Ix and Iy):
| Ix | Iy | |
|---|---|---|
| Experimental | 1485.45 mm4 | 965.07 mm4 |
| Theoretical | 1847.36 mm4 | 1123.54 mm4 |
The percentage error for Ix and Iy is 19.6% and 14.1%, respectively. This discrepancy can be attributed to both systematic and random errors:
To minimize errors in future experiments, it is recommended to conduct more measurements to improve the accuracy of gradient function and utilize indicators with higher precision. Specialized equipment should also be considered to eliminate manual setup inaccuracies.
The percentage error between the experimental and theoretical shear center values is up to 33%. This discrepancy is primarily attributed to:
This report investigated unsymmetrical bending and shear center of a U beam experimentally and theoretically. Shear center was determined both graphically and through calculations, revealing the disparity between the two methods.
Unsymmetrical Bending and Shear Centre Experiment Report. (2024, Jan 05). Retrieved from https://studymoose.com/document/unsymmetrical-bending-and-shear-centre-experiment-report
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