Lab4

MENG 302L Lab 4: Poisson’s Ratio

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Introduction 1: The purpose of this experiment is to measure Poisson’s Ratio of various materials by

loading beams in cantilever bending. Poisson’s Ratio is one of two fundamental elastic constants (the other is Modulus of Elasticity) relating stress to strain in a biaxial stress field. For example, Hooke’s law for the biaxial stress state can be written as:

(1)

where: σ = stress, psi (N/m2) ϵϵϵϵ = strain, in/in (m/m) = Poisson’s Ratio (dimensionless)

E = Modulus of Elasticity, or Young’s Modulus, psi (N/m2) Both the elastic modulus E and Poisson’s Ratio are required to translate measured strains into stresses. When a test specimen of an isotropically elastic material is subjected to uniaxial stress, the specimen not only deforms in the direction of the stress, but also exhibits deformation of the opposite sign in the perpendicular direction. Poisson’s Ratio is, by definition, the absolute value of the ratio of transverse strain ϵϵϵϵy to the axial strain ϵϵϵϵx in a uniaxially stressed member:

(2)

Thus, Poisson’s Ratio is always a positive number. To obtain the transverse strain in a uniaxial stress field, given the axial strain and ν,

(3)

Poisson’s Ratio can be measured with two strain gages on a uniaxially stressed member. One gage is aligned in the direction of the applied stress, and the second gage perpendicular to the first.2 A tensile test specimen with a uniform stress field is commonly used for this purpose, and the gages are mounted adjacent to each other in the form of a “T”.

Poisson’s Ratio can also be measured with reasonable accuracy on a cantilever beam, even though the strain varies linearly along the beam. In this case, the axial gage can be mounted longitudinally on, say, the upper surface of the beam, and the lateral gage mounted crosswise on the lower surface at the same section. The absolute value of the ratio of the two strains (after the indication of the lateral gage has been corrected for transverse sensitivity) is Poisson’s Ratio. This arrangement is shown in Figure 1 on the following page.

Two different methods of correcting the lateral gage for transverse sensitivity will be compared. The first is a graphical method using a chart. The second is a mathematical formula.

1 This lab is based on E-102 POISSON’S RATIO – FLEXURE, ©Vishay Measurements Group, Inc., 1982, printed March 2002. Portions of the text and Figures 1 & 3 were taken verbatim from that document. 2 Because most strain gages display some degree of sensitivity to strains transverse to the axis of primary sensitivity, it is usually necessary to correct the indication of the perpendicular gage for transverse sensitivity in order to obtain an accurate value of Poisson’s Ratio by this method.


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Figure 1

Pre-lab: Complete the following table:

Material Poisson’s Ratio Source

6061-T6 Aluminum

HR 4140 Steel

AISI 304 Stainless Stl

UNS C38500 Brass

UNS C36000 Brass

Acrylic Plastic

Cork

Balsa Wood

Neoprene Rubber

Equipment and Supplies:

- Vishay Flexor, cantilever flexure frame - (3) ¼ x 1 x 12 ½ in (6.4 x 25.4 x 318 mm) beams of different materials, outfitted with (2)

Vishay Micro-Measurements strain gages each. - Vishay P3 Strain Indicator.


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Procedure: A. General

In this experiment, beams on which two strain gages are mounted will be used to determine Poisson’s Ratio in bending. It is assumed that under flexural loading the strains at corresponding points on the upper and lower surfaces of the beam are numerically equal, differing only in sign. Based upon this assumption, one strain gage is installed axially on the upper surface of the beam, and a second gage laterally (crosswise) at the corresponding point on the lower surface. Since the axial strain varies linearly along the length of the cantilever beam, the average strain (indicated by the axial gage) is equal to the axial strain at the section corresponding to the midpoint of the gage. The lateral strain is virtually uniform over the center half of the beam width.

The beams will be clamped in the Flexor and loaded to an arbitrary strain level. The lateral and axial strains will then be measured for calculation of Poisson’s Ratio.

B. Acquisition of Data 1. Work in teams of three. 2. Select three beams of different materials. 3. Record beam material, gage factor GF, and transverse sensitivity Kt of both gages in the data

table. 4. Back the loading screw out of the way, and insert the beam into the Flexor with the gaged end in

the clamp and the axial gage facing up. Center the free end of the beam between the sides of the Flexor, and firmly clamp the beam in place with the clamping screw.

5. Refer to Figure 2. Connect the strain gage leads to the P3 strain indicator as follows: Common leads from the top (axial) gage to the S- and 120Ω posts of Channel 1. Independent lead from the top gage to the P+ post of Channel 1. Common leads from the bottom (lateral) gage to the S- and 120Ω posts of Channel 2. Independent lead from the bottom gage to the P+ post of Channel 2.

6. Turn on the P3 and set it up as follows: Channels 1 and 2 active, 3 and 4 inactive. Quarter bridge. Set Gage factor and scaling (µϵ) for both gages. Push the balance button to balance both channels. (Don’t save.)

7. Now turn the micrometer until the lateral gage reads 200 µϵ. Record the readings of both gages in the data table.

8. Back the Flexor loading screw away until it clears the beam. The strain indicator readouts should now be close to 0 µϵ. (If the number is more than ±20 µϵ, the source of error should be located, and the experiment performed again.)3 Record both readings in the data table.

9. Partners switch places and repeat steps 3 through 8 for the other two beams.

3If the undeflected readings fail to repeat, the binding post connections may not have been snug enough to avoid small contact resistance changes. Binding post connections should be snug enough to allow a “wiggle test” of the lead wires without a zero balance shift.


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Figure 2: P3 Wiring


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Analysis and Presentation of Data: Before calculating Poisson’s Ratio, the strain reading of the lateral gage should be corrected for

transverse sensitivity. Because the axial strain in the beam is several times as large as the lateral strain, the lateral gage is subjected to a much larger strain perpendicular to its axis than parallel to it. As a result of the finite width of the grid lines in the gages, and the presence of end loops connecting the grid lines, strain gages are generally sensitive not only to the strain parallel to the grid direction, but also (to a much lesser degree) to the strain perpendicular to the grid direction. This property of strain gages is referred to as “transverse sensitivity” and symbolized by K t.

The correction for transverse sensitivity can be made using the attached graph (Figure 3). To use the graph, two quantities are needed: (1) the ratio of the axial strain to the indicated lateral strain, and (2) the transverse sensitivity, K t, of the lateral strain gage (given on the strain gage package data form or on the information panel of pregaged beams).

Enter the graph at the abscissa at the value of K t for the gages used in this experiment. Project a line upward to the sloped line representing the ratio of the strains transverse to and parallel to the axis of the lateral strain gage (note that the ratio is actually negative in this instance). From the intersection of these lines, project horizontally to the correction-factor scale on the ordinate to find “C”. Multiply the indicated lateral strain by “C” to obtain the corrected lateral strain. The indicated axial strain need not be corrected for transverse sensitivity because the transverse strain sensed by the axial gage is small to begin with, and similar in magnitude to the transverse strain in the calibration environment used to measure the gage factor of the strain gage (the gage aligned with the applied stress axis in a uniaxial stress field, with a Poisson’s Ratio of 0.285).

As an alternative procedure, any two strains, measured at right angles to one another, can be corrected for the transverse sensitivity of the strain gages with the following relationship:

(4)

where ϵ1, ϵ2 = two observed (uncorrected) orthogonal strains ϵ1, ϵ2 = corrected strains Kt = transverse sensitivity of strain gages. (In decimal, not percent format!) ν0 = 0.285 (Poisson’s Ratio under which strain gages were calibrated for gage factor) (When using this formula, keep in mind that the sign of the lateral strain is opposite what it

would be if the lateral gage was mounted on the same side as the axial gage.) After correcting the lateral strain indication for transverse sensitivity, divide the result by the

indicated axial strain to obtain Poisson’s Ratio.

Write-up: (Worksheet) - The executive summary should include a brief description of the experiment and your values for

Poisson’s Ratio. - The Results consist of the completed handout. - In the discussion, compare your measured values for Poisson’s Ratio to the published values

found in the pre-lab. Compare the values determined with the chart to those determined with the formula. Explain any differences.

- For the conclusion, recap everything in 50 words or less.


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Worksheet: Data Table

Date: Team Members:

Beam 1 Beam 2 Beam 3 Material

Long. Gage Lat. Gage Long. Gage Lat. Gage Long. Gage Lat. Gage GF

Kt

Strains Axial ϵax Lateral ϵlat Axial ϵax Lateral ϵlat Axial ϵax Lateral ϵlat (Loaded) 200 µϵ 200 µϵ 200 µϵ (Relaxed)

Calculations: 1) Correct lateral strain for transverse sensitivity using Figure 3, pg 7:

a) Determine ϵt/ϵa. Note that the transverse strain sensed by the lateral strain gage (i.e. the axial strain on the bottom of the beam) is negative. Therefore,

ϵϵ

ϵ!"ϵ#!$

Enter ϵt/ϵa in the table. b) Find Correction Factor C from Figure 3 and enter in the table. c) Enter the corrected lateral strain (ϵlat = ϵ lat * C) in the table.

2) Calculate Poisson’s Ratio and enter in the table:

%& 'ϵ()* +!" '

3) Find ϵϵϵϵaxial and ϵϵϵϵlateral using equation 4 (pg 5) and enter in the table. 4) Calculate Poisson’s Ratio using ϵϵϵϵaxial and ϵϵϵϵlateral from step 3 above.

%, '+#!$-.!# +!"/!# '

Calculation Results

Beam 1 Beam 2 Beam 3

1a) ϵt/ϵa

1b) Corr. Factor C

1c) ϵlat

2) νc

3) ϵaxial

3) ϵlat

4) νf


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Figure 3


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Notes:

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Lab4