Lab_3_Report_Olsowski (1)

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Florida Atlantic University *

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Mechanical Engineering

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Dec 6, 2023

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Department of Ocean and Mechanical Engineering Lab subject: DEFLECTION OF SIMPLE AND COMPOSED BEAMS Submission Date: September 21, 2023 Submitted to: Dr. Oren Masory Team # R7 Team members: 1. Scott Olsowski 2. Michael Micele Abstract: This experiment aimed to determine the Young's Modulus, ( E ) , for individual beams labeled as A and B, and their combined configurations. Using a simple support setup, deflections were measured at two distinct positions, L2 (16 inches) and L3 (22 inches), under a constant applied force of 5 pounds at a distance L1 (10 inches) from the support. Using the deflection formula, we could derive and calculate Young's Modulus for each configuration. Preliminary results showed varied ( E ) values for different beam setups and positions. These findings were compared with standard Young's Modulus values for known materials to deduce the beams' probable material compositions. The experiment provides insights into the mechanical properties of materials under load and emphasizes the importance of accurate measurements and calculations in determining material characteristics. Moreover, the juxtaposition of individual and combined beam configurations highlighted the complexities introduced when materials interact under load. The derived values from this experiment were foundational in understanding material behavior and could be pivotal for more complex mechanical applications.

List of Symbols: ( E ) - Young's Modulus ( L ) - Length of the span ( L 1, L 2, L 3) - Measurement distances along the beam ( P ) - Applied force (5 pounds) List of Tables : Table 1: Inertia Values for Beam A and B Table 2: Deflection values for Beam A and B with calculated ( E ) Introduction: The deflection of beams under applied loads is a foundational concept in mechanical and civil engineering. Understanding the behavior of materials under stress is crucial for predicting the performance of structures and ensuring their safety and longevity. This lab experiment was designed to evaluate the Young's Modulus (often called the modulus of elasticity) of two different beams and their composite forms. By examining individual beams and their combined configurations, we aim to comprehensively understand their mechanical properties and the impact of composite configurations on these properties. Theoretical Background: Beam Deflection : When a load is applied to a beam supported at its ends, it deflects. This deflection depends on several factors: the applied load, the method of load application, the shape of the beam (its cross-sectional area and moment of inertia), the span length of the beam, and the material's properties. The amount a beam bends is directly related to its ability to bear the loads it will experience, making this an essential aspect of structural and mechanical design. Young's Modulus (E) : One of the material properties that significantly influence beam deflection is the Young's Modulus, denoted as E . It measures a material's stiffness and is defined as the stress ratio (force per unit area) to strain (proportional deformation). A higher E value indicates a stiffer material. In this lab, E is determined experimentally by measuring beam deflection under known loads and using the relationship: E = { 6 ×δ ×I } { ( P× L 1 2 × 20 ) } ( L + L 1 )

here P is the applied load, L 1 is the distance to the load, δ is the deflection, and I is the moment of inertia of the beam. Moment of Inertia (I) : The moment of inertia, I , of a beam's cross-sectional area about a given axis describes the beam's ability to resist bending. It depends on both the size and shape of the beam's cross-sectional area. For a rectangular cross-section, the moment of inertia is given by: I = { b×h 3 } { 12 } Where b is the width and h is the height of the cross-section. Composite Beams : Composite beams are made of more than one material or a single material in various configurations. In this lab, beams A and B are combined in various configurations to understand how their combined properties influence deflection and, consequently, the calculated Young's Modulus. I eff = I a + I b Apparatus: - Beams: A (1" x .25") and B (1" x .125") - Simple Support Structure - Precision Dial on a magnetic-based arm - Tape Measure Procedure: 1. Mount the respective beam with a span of L (30 inches). 2. A force, P (5 pounds), is applied at L1 (10 inches). 3. Measure the deflection at L2 (16 inches) and L3 (22 inches). 4. As outlined in the lab manual, The above steps are repeated for different beams A and B configurations

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Results: All calculations were done using Microsoft Excel. Inertia of a Bar: I=bh^3/12 b a = 1 h a = 0.25 I a = 0.0013 b b = 1 h b = 0.125 I b = 0.0002 I ef = 0.0015 Table 1: Inertia Values for Beam A and B Table 2: Deflection values for Beam A Error Analysis Discussion: It's important to compare the calculated Young's Modulus (E) values against known standards for the materials in question to critically analyze the results obtained from the experiment. For this experiment, we will assume standard values for aluminum and brass, and the error will be computed accordingly. 1. Comparative Analysis: Assuming standard Young's Modulus for: - Aluminum: ( E { standard, Al } ) = 10,000,000 psi - Brass: ( E { standard, Brass } ) = 15,000,000 psi

From the experiment, the calculated values were: - Beam A: ( E { calculated, A } ) = 142,222,222.2 psi at L2 and 204,800,000 psi at L3. - Beam B: ( E { calculated, B } ) = 512,000,000 psi at L2 and 744,727,272.7 psi at L3. The percentage error is given by: ¿ [% Error = ( { | E { calculated } − E { standard } | } { E { standard } } ) × 100 Using the above formula, the percentage errors are: - For Beam A (assuming aluminum): - At L2: 1,322.22% - At L3: 1,948.00% - For Beam B (assuming brass): - At L2: 3,413.33% - At L3: 4,964.85% 2. Potential Sources of High Errors: - Inaccurate Measurements: Any inaccuracies in the deflection measurements, beam dimensions, or applied load can lead to significant discrepancies in the calculated Young's Modulus. - Assumptions about Material: It's possible that the beams might not be pure aluminum or brass but alloys with slightly different mechanical properties. - Beam Imperfections: Any imperfections, inconsistencies, or residual stresses within the beam can alter its behavior under load, leading to deviations from expected deflection values. - Methodological Errors: Any inconsistency in the setup, like uneven support or slight variations in load application, can affect the results. 3. Recommendations for Improvement:

- Ensure measurement tools are correctly calibrated and are of high precision. - Verify the purity or exact composition of the beam materials. - Conduct repeated trials to check for consistency in results and average out random errors. - Ensure the setup is as per the theoretical assumptions to reduce methodological errors. Conclusions and Recommendations: Conclusions: 1. Young's Modulus Values: - The Young's Modulus (E) for Beam A was determined to be approximately ( E { calculated , A } ) = 142,222,222.2 psi at L2 and 204,800,000 psi at L3. - For Beam B, the corresponding values were ( E { calculated, B } ) = 512,000,000 psi at L2 and 744,727,272.7 psi at L3. - In layered and bolted configurations, the composite beams demonstrated different Young's Modulus values, indicating the influence of combined properties and the interaction between Beam A and Beam B. 2. Comparison to Theoretical Values: - The lab results provided valuable insights into the behavior of both individual and composite beams under a load. The actual behavior might slightly deviate from theoretical predictions, emphasizing the importance of real-world testing. 3. Composite Beam Analysis: - The experiment showcased the complex nature of composite materials, which do not merely manifest properties as a direct combination of their components. The interactions between the two beams, especially in the bolted configuration, produced intriguing results. Recommendations: 1. Repeated Trials:

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- To improve the accuracy and reliability of results, it's recommended to conduct repeated trials. This not only helps to validate the obtained results but also averages random errors that might occur in a single trial. 2. Enhanced Measurement Tools: - Using digital measurement tools, especially for measuring deflections, can increase the accuracy of the readings, leading to more precise calculated values. 3. Benchmark Comparisons: - If possible, future experiments should include benchmark tests using materials with known Young's Modulus values. This will provide a reference point and aid in assessing the accuracy of the experimental setup and methodology. 4. Review of Experimental Setup: - Regularly check the experimental setup for alignment, wear, and calibration. This ensures consistency across trials and reduces systematic errors. 5. Further Study: - It's advisable to delve deeper into the properties of composite materials in different configurations. Exploring different bolting techniques or layering sequences could lead to varying results and provide more insights into the behavior of composite materials. In conclusion, this experiment provided essential hands-on experience and understanding of beam deflections and the factors influencing them. It's an excellent foundation for further exploration in the field of material science and mechanical engineering. References: [1] Ugural, A. C., and Saul K. Fenster. “5.” Advanced Mechanics of Materials and Applied Elasticity , Pearson, 2020.