Unsymmetrical Bending and Shear Centre Experiment Report

Introduction

This report aims to investigate unsymmetrical bending and determine the position of the shear center, along with analyzing the effects of unsymmetrical bending.

Apparatus Description

Experimental Setup

The experimental setup for the unsymmetrical bending and shear center experiment is as follows:

1. Components: The setup consists of several key components:
   • One top plate and chuck
   • One bottom plate equipped with two digital indicators used to measure deflection
   • One free end chuck
   • A U-shaped beam
2. Arrangement: The U-shaped beam is positioned vertically between the top and bottom chucks.
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   It serves as the specimen for the experiment.

3. Load Application: A load is applied to the U-shaped beam through a pulley mechanism. This load is systematically varied during the experiment.
4. Data Collection: As the load is applied, the digital indicators measure and display the deflection of the beam. The readings from these indicators are recorded for further analysis.

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.

Methods

1. Specimen Preparation: A U-shaped beam, composed of a material with known properties, was selected as the specimen. It was carefully inspected to ensure its integrity and absence of defects.
2. Experimental Apparatus Setup: The experimental apparatus was configured as follows:
   • Top Plate and Chuck: One end of the U-shaped beam was securely fastened to a top plate and chuck, providing a fixed point of support.
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   • Bottom Plate with Indicators: The opposite end of the beam was connected to a bottom plate equipped with two digital indicators, strategically placed to measure vertical deflection with precision.
   • Free End Chuck: The free end of the U-shaped beam was clamped using a free end chuck, enabling controlled loading.
   • Load Application: Load was incrementally applied to the U-shaped beam through a pulley system. Loads ranged from 120g to 480g.
3. Data Collection: As the load increased incrementally, the digital indicators recorded deflection measurements at various angular positions as the beam rotated due to the applied load.
4. Data Recording: The deflection measurements, both left and right side readings, were meticulously recorded for each load and at each angular position. This data was organized into a tabular format for subsequent analysis.
5. Calculation of Deflections: Utilizing the recorded data, deflections in the vertical direction (U) and perpendicular to the load direction (V) were calculated based on the equations:
   • U = (left + right) / √2
   • V = (left - right) / √2
6. Analysis of Deflection Data: The calculated deflections (U and V) were analyzed to understand the beam's response to applied loads at various angles. This analysis facilitated the determination of the shear center and assessment of unsymmetrical bending effects.
7. Construction of Mohr's Circle: A Mohr's circle was constructed using values of dU/dF and dV/dF derived from the deflection data. This graphical representation aided in visualizing the relationship between deflection and load at different angles.
8. Determination of Shear Center: The shear center was determined both graphically, by inspecting the Mohr's circle, and through calculations based on experimental data. This allowed for precise localization within the beam's cross-section.
9. Comparison with Theoretical Values: Experimental results, including deflections and shear center location, were compared with theoretical values. This comparative analysis provided insights into experimental accuracy and potential sources of error.

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.

Unsymmetrical Bending

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:

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:

The relationship between deflection and applied load at different angles is presented in Table 3:

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

Shear Centre

The shear center of the beam was determined by collecting data from two indicators, as shown in Table 4:

Due to the accuracy of the experiment and the beam's symmetry, it was determined that the shear center precisely locates at 0mm.

Discussion

Table 5 summarizes the experimental and theoretical values of the principal second moments of area (Ix and Iy):

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:

• The gradient graph is based on only four data points of dU/dF and dV/dF, resulting in an imperfect representation of the theoretical function.
• Non-vertical application of force during load addition may have affected indicator readings.
• Imperfect perpendicularity of the indicator to the beam's surface.
• Residual forces from stickiness and friction within the apparatus.
• Simplified calculations and rounding of results introduced significant 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:

• Minor variations in the dimensions of the specimen during apparatus setup.
• Non-vertical force application due to pulley rotation and friction.
• Stickiness and friction within the apparatus.
• Lack of calculations to refine the shear center location, relying solely on visual inspection.

Conclusion

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.