Reference no: EM131008863

Test object

You will be measuring strains in a thick cylinder with closed ends made from aluminium alloy (E = 73GPa and ν = 0.33) of outer diameter 150mm and inner diameter 37mm. The cylinder has been specially adapted to obtain the radial distributions of the circumferential and radial stresses. This has been achieved by cutting it midway along the axis and milling a shallow, eccentric rectangular slot into one of the walls as shown below, before reassembling the two halves. Strain gauges have been mounted at various radial and circumferential positions in the groove oriented so as to measure the circumferential strain, εθθ, and the radial strain, εrr. In addition, a gauge is mounted on the outer surface to measure the axial strain, εzz, although this will not be used. The gauge positions are labelled approximately in the diagram below (solid, hatched and shaded fill for circumferential, radial and axial, respectively). The exact radial positions of the gauges are shown in the table, along with the associated channel numbers.

Cylinder experimental arrangement

Channel number	Strain direction	Radial position (mm)
1	Hoop	28
2	Radial	28
3	Hoop	36
4	Radial	36
5	Hoop	45
6	Radial	45
7	Hoop	56
8	Radial	56
9	Hoop	63
10	Radial	63
11	Hoop	18.5
12	Longitudinal	75
13	Hoop	75

Strain gauges

A strain gauge is essentially a long piece of rectangular section metal foil folded back on itself and bonded to a plastic (usually polyimide) backing:

To install a gauge, the polyimide is glued to the surface of the object under test in its unstressed state and wires are soldered onto the pads. The other ends of the wires are attached to a Wheatstone bridge, which is an arrangement for measuring resistance. When the test object experiences a strain in the direction to which it is sensitive (see above), the total foil length in the sensitive direction, L, increases and its cross- sectional area, A, reduces, both of which increases its resistance, R, because R = ρLA, where ρ is the material resistivity. Note that a strain gauge cannot directly measure a shear strain.

If the directions of the principal strains are known on a given surface, strain gauges can be aligned along these directions, such as is the case for uniaxial tension below:

Experimental procedure

Before going into the lab, equip yourself to take readings, as time will be very limited. Use the table on the previous page, and add columns for the relevant strains at your chosen pressure increments.

After obtaining a briefing from the demonstrator, ensure that the internal pressure in the cylinder is 0 barg, and take zero readings on all 10 channels. Decide on some pressure increments (up to a maximum of 70 barg) and take readings on all 10 channels at each increment. Do not exceed the pressure of 70 barg, as this will cause the pressure relief valve to lift with consequent loss of hydraulic fluid, for which the technicians will not thank you.

You are advised to repeat the pressure increments at least once (usually once on the way up and once on the way down) and take zero readings again at the end of your test. The accuracy of strain gauges is limited and you will want to assess this as part of your analysis. You may make a reading of 639µε, which has an apparent precision of 1µε but this does not mean that your strain gauge is accurate to 1µε (it isn't).

Data treatment and analysis

The software will provide you directly with values of strain, so you do not need to calibrate the gauges (normally done by applying a known stress to a well characterised material).

Strain gauges are normally used to measure stresses (because there is no direct way to measure a stress), so the strains need to be converted to stresses using a constitutive relationship. In the case of a linear elastic material where the gauges are oriented along the principal directions, this is particularly easy to do using:

σ₁ = (vE/((1+v)(1-2v))).(ε₁ + ε₂ + ε₃) + E/(1+v) ε₁

σ₂ = (vE/((1+v)(1-2v))).(ε₁ + ε₂ + ε₃) + E/(1+v) ε₂

σ₃ = (vE/((1+v)(1-2v))).(ε₁ + ε₂ + ε₃) + E/(1+v) ε₃

It should be noted that the groove introduces a stress concentration which affects only the radial stress. This means that the measured radial stress needs to be corrected by a factor using: σ_rr,true = 0.67 x σ_rr,measured. You will need to calculate the axial strain from:

ε₃ = σ_zz/E - v/E (σ_θθ + σ_rr)

using the equations in your notes for the three principal stresses.

Report

Submit your report on Vision, and prepare it with your lab group (the same as your project group). A formal laboratory report is not required, so do not repeat material from this laboratory sheet, although you may, of course, refer to it. Your report should contain at least the following:

A brief summary of what you measured, tables of your raw data and an account of the data handling and analysis

Plots of your measured stresses as a function of pressure and radial position Comparison between the measured stresses and the calculated stresses using a thick cylinder analysis and a thin cylinder approximation

An assessment of the accuracy of your strain gauge measurement An estimate of the burst pressure of the cylinder