Sheet1 Data sheet of Lab#3 Pressure Gauge 1 90 degree Gauge 2 0 degree Gauge 3 90 degree Gauge 4 0 degree Gauge 5 90 degree Gauge 6 0 degree Gauge 7 90 degree Gauge 8 0 degree Gauge 9 90 degree Gauge 10 0 degree 10 Psi Δp 0 Δεy 1 Δεx 300 Δεy -1530 Δεx 716 Δεy 183 Δεx 166 Δεy 530 Δεx -301 Δεy 88 Δεx 20 Psi 10 Psi Psi 10 Psi Psi 10 Psi Psi 10 Psi average Δεy 25.75 average Δεx 5.25 average Δεy 26.25 average Δεx 6.25 average Δεy 27.5 average Δεx 4.75 average Δεy 7.5 average Δεx 6.5 average Δεy 24.25 average Δεx 5.0

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The mechanics of materials laboratory experiment aims to analyze the stresses experienced by a thin-walled pressure vessel under internal pressure, and to compare the experimental stress measurements with theoretical predictions. Understanding stress distribution in pressure vessels is crucial for ensuring safety and integrity in engineering applications such as pipelines, tanks, and pressure containers. The experiment leverages strain gauges to measure strain at various points on the vessel's surface, and subsequently computes the corresponding stresses, validating the theoretical models based on the classical thin-walled pressure vessel theory.

The theoretical foundation of the experiment is grounded in the stress analysis of thin-walled cylinders and spherical pressure vessels. For cylindrical sections, the hoop (circumferential) stress \(\sigma_\theta\) and longitudinal stress \(\sigma_z\) are given by the formulas:

• \(\sigma_\theta = \frac{p r}{t}\)
• \(\sigma_z = \frac{p r}{2 t}\)

where \(p\) is the internal pressure, \(r\) is the radius, and \(t\) is the wall thickness. For spherical sections, the hoop and meridional stresses are equal and are derived from:

• \(\sigma = \frac{p r}{2 t}\)

The experimental approach employs strain gauges placed at specified locations on the pressure vessel to measure surface strains in two principal directions: hoop and longitudinal. Using the strain gauge outputs, the experimental stresses are calculated via Hooke’s law for plane stress conditions:

\( \sigma = \frac{E}{1 - \nu^2} (\varepsilon + \nu \varepsilon_{other}) \)

assuming in-plane strains and known material properties (modulus of elasticity \(E\) and Poisson’s ratio \(\nu\)). The strain readings are converted into stress values for each measurement point, and these are compared to the theoretical values to evaluate model accuracy and identify any discrepancies caused by manufacturing tolerances, material inhomogeneity, or experimental inaccuracies.

The experiment involves gradually increasing the internal pressure of the vessel from 10 psi up to a predetermined maximum (not exceeding 60 psi), while recording strain gauge outputs at each pressure increment. System calibration steps, such as zero adjustment and gauge factor correction, ensure accurate strain measurements. The collected data include strain values in microstrain (μɛ), which are then translated into stresses using the known elastic constants:

\( \sigma_{exp} = \frac{E}{1 - \nu^2} (\varepsilon + \nu \varepsilon_{other}) \)

where \(\varepsilon\) and \(\varepsilon_{other}\) denote principal strains measured by the gauges.

The comparison between theoretical and experimental stresses reveals the fidelity of the thin-walled pressure vessel theory for the specific material and geometric conditions. Deviations can be attributed to factors such as non-uniform wall thickness, localized imperfections, or measurement errors. Understanding these differences enhances the confidence in applying theoretical models to real-world scenarios and informs design improvements for safety margins.

In analyzing the experimental data, emphasis is placed on consistency across multiple gauges and pressure levels. Graphical representations, such as stress versus pressure plots, facilitate visual comparison and trend analysis. Additionally, the average stresses obtained from multiple gauges help identify systematic errors or biases.

The discussion also covers auxiliary considerations, such as the optimal design of pressure vessel ends—hemispherical versus flat—in relation to stress distribution and failure modes. Furthermore, the calculated maximum permissible pressure based on the material’s ultimate stresses and a safety factor provides insight into safe operating limits. Anticipated failure points are examined, emphasizing the importance of stress concentration areas and material fatigue considerations.

This comprehensive analysis enhances understanding of pressure vessel behavior under internal pressure, blending experimental measurements with theoretical modeling to support safer and more reliable engineering designs.

References

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• Shigley, J. E., & Mischke, C. R. (2004). Mechanical Engineering Design. McGraw-Hill.
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• Meirovitch, L. (2010). Analytical Mechanics. Cambridge University Press.
• American Society of Mechanical Engineers (ASME). (2019). Boiler and Pressure Vessel Code (BPVC), Section VIII.
• Roark, R. J., & Young, W. C. (2011). Formulas for Stress and Strain. McGraw-Hill.
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