Pressure Vessel

Objective

This experiment will allow you to investigate hoop and axial stress/strain relations for a pressurized thin-walled cylinder. This is an opportunity to examine a system with a biaxial state of stress, as opposed to the primarily uniaxial stress systems of earlier labs. Furthermore, you will be analyzing transient (as well as steady-state) strain readings from a rosette strain gauge in response to a dynamic event. Finally, you will be able to determine the internal pressure within the vessel from your wall stress measurements and cylindrical pressure vessel relations.

Background

Many components of aerospace vehicles have an internal pressure higher than the surrounding pressure. Perhaps the most obvious examples are the fuselages of manned aircraft and the crew-compartments of spacecraft, which maintain internal pressures close to one atmosphere in low pressure environments. In other applications, the internal pressure can be significantly higher than 1 atm. For example, pressurized structures are required for many gas and liquid storage systems, e.g., propellant tanks in spacecraft, and for the propulsion engines used in aircraft and spacecraft.

We can characterize many of these structures as thin-wall pressure vessels, where the wall is essentially a membrane because its thickness is much less than other characteristic dimensions of the vessel, such as its length or radius.$^1$ In an ideal thin-wall pressure vessel, the stresses are uniform across the thin wall and the bending stresses are negligible over much of the structure. The vast majority of thin-wall pressure vessels are nearly spherical or cylindrical in shape (or composed of combinations of these two geometries). A sphere provides the advantage of minimizing stresses for a given volume and storage pressure compared to other shapes, i.e., it is structurally efficient. While cylinders are not as efficient and have larger stresses at the ends, which require additional structural reinforcement, they often have significant manufacturing and implementation advantages compared to spheres (imagine the drag induced by a spherical aircraft fuselage).

$^1$Structures such as vacuum tanks and submarine hulls may also be characterized as thin-walled pressure vessels, through with internal pressures below the external pressure. For those cases, one must consider wall buckling as a failure mode.

Wall Stress and Pressure Relations

Figure 1. Direction of hoop and longitudinal stresses in a cylindrical thin-wall pressure vessel.

There are two principal stresses for a cylindrical thin-wall pressure vessel (Figure 1): a longitudinal stress in the axial direction$(\sigma_L)$; and a hoop stress$(\sigma_H)$in the circumferential direction. This is a biaxial-stress state since there are two non-zero stress components. For a sufficiently long cylinder, we can relate these stresses to the difference between the internal pressure and the external pressure $(\Delta p)$, sometimes referred to as the gauge pressure.

Figure 2. Section of cylindrical thin-wall pressure vessel of radius R and wall thickness t, subject to a pressure difference Δp.

Ignoring the ends, we can calculate the hoop stress by considering the top half of the cylinder (a section of this is shown in Figure 2). We can use a free body diagram to get the necessary relationships for a cylinder of length L, radius R and wall thickness t. First, from the definition of stress, we have

$$\large F_H=tL\sigma_H$$

(1)

The wall must support the difference in (vertical) forces on the inner and outer sides of the wall due to the pressure difference, so

$$\large 2F_H=\int_{0}^\pi \Delta p \space LR\sin\theta d\theta=LR2\Delta p$$

(2)

Combining (1) and (2), gives us a relation between the hoop stress and the gauge pressure.

$$\large \sigma_H=\frac{R}{t}\Delta p$$

(3)

Figure 3. Apparatus schematic for a four point bending test.

A similar process looking at a cross-sectional cut normal to the longitudinal axis (Figure 3), where the longitudinal force in the wall must balance the pressure difference across the end of the cylinder, gives

$$\large F_L=2\pi Rt\sigma_L=\Delta p \int_{0}^{2\pi}\int_{0}^{R}rdrd\theta=\pi R^2\Delta p$$

(4)

Solving (4) for the longitudinal stress gives

$$\large \sigma_L=\frac{R}{2t}\Delta p$$

(5)

So ideally, the longitudinal stress if one-half the hoop stress for a cylindrical vessel, or $\sigma_H=2\sigma_L$.

Stress-Strain Relations

As you will be measuring strains in our thin-wall vessel, you will need to convert them to stresses. As usual, this relationship is provided by Hooke’s Law for strain, e.g.,

$$\large \varepsilon_x=\frac{\sigma_x}{E}-\nu\frac{\sigma_y}{E}-\nu\frac{\sigma_z}{E} \space\space\space \text{and} \space\space\space \varepsilon_y=-\nu\frac{\sigma_x}{E}+\frac{\sigma_y}{E}-\nu\frac{\sigma_z}{E}$$

(7)

For our geometry, we have $\sigma_x=\sigma_H$, $\sigma_y=\sigma_L$ and $\sigma_z=0$ . Thus we can simplify (7) to

$$\large \sigma_H=\frac{E}{1-\nu^2}(\varepsilon_H+\nu\varepsilon_L) \\ \space \\ \space \\ \sigma_L=\frac{E}{1-\nu^2}(\varepsilon_L+\nu\varepsilon_H)$$

(8)

Including our stress results for the ideal cylindrical vessel ($\sigma_H=2\sigma_L$) gives

$$\large \varepsilon_H+\nu\varepsilon_L=2(\varepsilon_L+\nu \varepsilon_H) \\ \space \\ \space \\ \textrm{or} \\ \space \\ \space \\ \sigma_H=\frac{2E}{2-\nu}\varepsilon_H \\ \space \\ \space \\ \sigma_L=\frac{E}{1-2\nu}\varepsilon_L$$

(9)

Transients

In this lab, you will record the wall strains in your pressure vessel as a function of time while the vessel is depressurized. The strains in the wall will go from one steady-state condition (fully pressurized) to another (depressurized). One way to characterize a transient system like this, moving from an initial state to a final state, is the “rise” or “fall” time, i.e., the time required for a chosen state variable to go from 10% to 90% of the difference from its final value, i.e., $t_{rise}=t_{90}-t_{10}$, where $t_{10}$ is the time required for state variable y to meet the condition $y(t_{10})=y_{initial}+0.1(y_{final}-y_{initial})$. An example of this is shown in Figure 4.

Figure 4. Example of rise time based on a state variable transitioning from an initial zero value to a final value of one.

Experimental Equipment

Figure 5. Equipment used in transient thin-wall strain-stress measurements.

The equipment used in this lab are pictured in Figure 5. The pressure vessel you will investigate is the common soda can. You will mount on the can either a rectangular rosette strain gauge, which measures strain along three directions, or a T-rosette gauge, which measures the strain along two perpendicular directions. You will use the Vishay 7000 acquisition system to record the reading from the strain gauge. Care must be exercised when handling the soda can. Be careful not to indent or drop the can or shake it so as not to increase the internal pressure. Be sure not to excessively strain the lead wires connected to the T-rosette gauges.

At Home Before the Lab

Look up some common aluminum alloys used to manufacture soda cans. Find values for the modulus of elasticity, Poisson's ratio, and 0.2% offset yield stress for these alloys (you might try www.matweb.com).

Procedure

1. Measure the diameter of the can using digital calipers. Zero the calipers with the jaws in the fully closed position. Measure the diameter of the can by laying the calipers over the top of the can with the can standing upright. Carefully close the caliper jaws around the can being sure not to squeeze the can. Iterate between locating the diameter and allowing the jaws to contact the walls of the can. Conduct a total of 3 diameter measurements and record these values.

2. Measure the thickness of the soda can Aluminum material from the provided cutout sheet. Be careful not to close the calipers too tightly and deform the material plastically. Collect a total of 3 thickness measurements and record these values.

3. Note the precise designation of the strain gauges you will use so that you can look up relevant parameters associated with this gauge such as the gauge factor$S_g$and the transverse sensitivity factor$K_t$. Note for T-rosette or rectangular gauges there is a value of$K_t$assigned to each grid.

Transient Can Opening Test

1. Mount the strain gauge on your soda can.

2. If you are using a T-rosette gauge, it should be bonded so that one grid is aligned with the longitudinal axis of the can and the other grid is aligned with the hoop direction. It is best to scribe perpendicular lines on the sanded region of the can to facilitate the alignment of the gauge with the axial and radial directions.

3. If you are using a rectangular rosette gauge, you should bond the gauge in such a way that none of the 3 grids are aligned with the longitudinal and hoop directions.$^1$
4. Properly connect the strain gauge wires to the strain gauge card in the Vishay 7000.

5. Configure the Vishay 7000/Strain smart software interface. Be sure to supply the appropriate values for the strain gauge and transverse sensitivity factors - assign the correct values for the appropriate strain gauge grid. Set the data sampling rate to the maximum possible value (2048 samples/second). The fast sampling rate will allow you to capture some data points associated with the early, sharply rising portion of the strain gauge record when the pressure suddenly changes.

6. The individual strain gauges will already be configured to their respective 1/4-bridge circuits. Zero each 1/4-bridge circuit and perform standard shunt calibrations.

7. Shake the can carefully, being sure not to accidentally damage the strain gauge, or the lead wires.

8. WITHOUT OPENING THE CAN, carefully and gently pry up the soda can tab just a bit so that you will be able to easily get your finger underneath when you are ready to crack open the can.

9. Trigger the Vishay 7000 to start recording data. Grab the top of the can with two fingers while you open the can with your other hand. You should avoid squeezing the can during this step!$^2$Stop recording data and save the strain data to the hard drive.
10. Be sure to note down which strain gauge grid corresponds to each recorded strain gauge signal. This is especially important in the case of the 3-element rectangular rosette gauge if that is the type of gauge you are using in your experiment.

$^1$The whole point is to apply the strain transformation formulas in order to recover the principle strain directions (maximum and minimum) along with the principle angle (qp). Once you calculate these three quantities you should of course recover the hoop and axial directions of the can.

$^2$ You should see the dynamic strain change associated with the hoop and axial strain components on the live computer display. Your curves should feature a sharp change in strain, which then levels off at a steady value.

Data to be Taken

1. Can diameter and can wall material thickness.

2. Gauge factors $S_g$ and transverse sensitivity factors $K_t$ for each of the strain gauges used in your test.
3. Strain gauge outputs for the transient can opening test.

4. Displacement rate and the digital sampling rates for the Vishay system

5. Axial loads and bending strains for loading and unloading runs for short beam-column.

Data Reduction

1. If you used a rectangular rosette for the transient can experiment (and misaligned it as you were supposed to), then you need to transform the measured strains to the principal strains (maximum and minimum) and corresponding principal strain angles using a similar approach to that used in the cantilever beam bending lab. The principal strains should turn out to be the hoop and longitudinal strains.

2. Determine the rise times for the hoop and longitudinal strain (magnitudes) that occurred as the can was first opened.

3. Estimate the dynamic strain rates $d\varepsilon/dt$ by dividing the respective change in strain during the 10% to 90% interval by your rise time estimate. Do this for both the hoop and longitudinal strain signals.
4. Use the value of the Poisson ratio that you found from the material database in the at home exercise and estimate the radial strain in the can wall. Combine with the measured can wall thickness to estimate the change in wall thickness when the can was pressurized.

5. From you hoop and axial strain versus time profiles, identify the steady state portions of each curve (before opening the can and afterwards where the strain finally levels out to a nearly constant value). Use data from each steady state portion to estimate the overall change in hoop and longitudinal strains experienced by the can.

6. Using the material database values of E and $\nu$, use the Hooke’s Law Equations (8) to determine the changes in the can’s $\sigma_H$ and $\sigma_L$ corresponding to the strain changes determined in Step 5.
7. Using the appropriate results for $\sigma_H$ and $\sigma_L$ estimate, from Equations (3) and (5), the gauge pressure within the soda can before it was opened.

Results Needed for Report

1. Table of can dimensions.

2. Table of gauge factors and transverse sensitivity factors for each of the strain gauge.

3. Combined plot of measured strains (in microstrain) as function of time (in seconds) for the can-opening experiment.

4. Plot of maximum principal stress angle as function of time for the can-opening experiment.$^1$

$^1$The plot should exhibit a chaotic and unstable variation during the first several seconds followed by a level, steady state value of $\theta p$. The “average” value of $\theta p$ during the steady region should coincide with the expected direction for a hoop stress.

1. Combined plot of hoop and axial strains (in microstrain) as a function of time (in seconds). You may truncate your plots in time as required – but be sure to clearly display the transient and steady state of your data.

2. Table of estimated rise times (in units of µs) and strain rates (in units of$s^{-1}$) for both hoop and longitudinal strain signals.
3. Table of overall changes in hoop and longitudinal strains and stresses (reported in MPa) from the initial can state to the final state. Provide stress results using both the full (Equations (8)) and simplified (Equations (9)) Hooke’s Law relations.

4. Table of estimated can gauge pressure values (in psi) using each of your results for $\sigma_H$ and $\sigma_L$ (note, you will be reporting at least two pressure estimates).