MAAE2202_Lab B

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Carleton University *

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Jan 9, 2024

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Carleton University Laboratory Report Course #: MAAE 2202 Lab #: B Lab Section #: Stresses in a Thin-Walled Pressure Vessel

2022/10/25 Summary The main objectives of this experiment are to obtain the internal pressure of a thin-walled pressure vessel by measuring the strains on the surface, and to gain practical experience in strain gauge application techniques. In this experiment, the pressure vessel used was a pop can made of aluminum 3004. The experiment was performed twice, once with an undisturbed can, and once with a disturbed, shaken can. The carbon dioxide bubbles formed in the disturbed can varied the results of the experiment. The strain in the longitudinal and circumferential directions was measured using strain gauges. From these values, the stress and pressure in both directions was calculated. For the unshaken can, the pressure in the hoop and longitudinal direction was 227kPa and 105kPa, respectively. For the shaken can, the pressures were 244kPa in the hoop direction and 98kPa in the longitudinal. Nomenclature Table 1: Nomenclature Used in the Report Symbol Parameter Unit F Internal Force kN σ Stress GPa ε Strain Unitless E Elastic/Young’s Modulus GPa P Internal Pressure GPa t wall Vessel Wall Thickness MPa d Vessel Nominal Diameter GPa R Hoop to Longitudinal Ratio Unitless SF Safety Factor Unitless %error Error Percentage % Theory and Analysis Thin-walled cylindrical pressure vessels such as the pop can used in this experiment develop stresses in both circumferential and longitudinal directions when subjected to internal pressure. These stresses are called the hoop and longitudinal stress and can be seen in Figure 1 . Figure 1: Diagram showing stresses in the circumferential and longitudinal directions of a cylinder P a g e 2 | 12

Like other stresses, the stresses in the longitudinal and circumferential direction of the vessel adhere to the Young’s Modulus relation: E = σ long ε long →σ long = E×ε long →ε long = σ long E E = σ hoop ε hoop →σ hoop = E ×ε hoop →ε hoop = σ hoop E To find the hoop stress, the forces in the circumferential direction across the can must be analyzed. This will be done using force equilibrium of Figure 2 . Figure 2: Diagram to show force equilibrium in the circumferential direction Assuming the body is in equilibrium, the force F shown in the diagram can be written in 2 ways: 1. F = PA = P hoop × L×d 2. F = 2 ×σ hoop ×t wall ×L If these equations are set equal, solving for the hoop stress we obtain: σ hoop = P hoop d 2 t wall Equation 1 By substituting hoop stress from the Young’s modulus relation into Equation 1 and solving for pressure: E×ε hoop = P hoop d 2 t wall → P hoop = 2 Eε hoop t wall d Equation 2 P a g e 3 | 12

To find the longitudinal stress, the same method is used. This time we will use force equilibrium for Figure 3 . Figure 3: Diagram to show force equilibrium in the longitudinal direction Assuming the body is in equilibrium, similar to the hoop force, the force F can be expressed in 2 ways: 1. F = P long ×π× d 2 4 2. F = σ long ×π ×d×t wall Setting the equations equal and solving for longitudinal stress, we obtain: σ long = P long d 4 t wall Equation 3 In the same way as the hoop pressure was found, by substituting in stress from Young’s modulus relation to Equation 3, we receive: E×ε long = P long d 4 t wall →P long = 4 E ε long t wall d Equation 4 The safety factor of the can uses the ultimate stress of the can and the experimental stress in the form: SF = σ ult σ exp Equation 5 P a g e 4 | 12

The experimental ratio is the ratio of experimental hoop stress to experimental longitudinal stress: R exp = σ hoop σ long Equation 6 The experimental ratio will be used to compare to the theoretical ratio to gain an error percentage: %error = | R exp − R theo R theo | × 100% Equation 7 Experimental Setup and Procedure P a g e 5 | 12

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