Poisson's Ratio

Introduction:

The purpose of this experiment is to measure the Poisson's Ratio of an aluminum beam by loading the beam in cantilever bending. Poisson's ratio is one of the two fundamental elastic constants (along with the Modulus of Elasticity) relating to strain in a biaxial stress field.

It is an experimentally observable fact that 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 the absolute value of the ratio of transverse strain to the axial strain in a uniaxially stressed member.

m = e y / e x

where e x is the axial strain and e y is the transverse strain. 

Poisson's Ratio can be measured with a reasonable accuracy on a cantilever 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 the figure below:

Figure 1. Cantilever flexure beam

Equipment and Supplies:

Flexor, cantilever flexure frame

High-strength aluminum alloy beam, 1/4 x 1 x 12-1/2 in

Two strain gages

Model 1011 Portable Strain Indicator

Procedure:

In this experiment, an aluminum beam 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 longitudinal strains at corresponding points on the upper and lower surfaces of the beam are numerically equal, differing only in sign; and the same assumption is made for the lateral strains. Based upon these assumptions, one strain gage will be installed longitudinally on the upper surface of the beam, and a second gage is laterally at the corresponding point on the lower surface.

The beam will be clamped in the Flexor and loaded in bending to an arbitrary strain level. Since the stress state in the beam is uniaxial, the lateral and longitudinal strains will be measured for calculation of Poisson's Ratio.

Acquisition of Data:

The gages will be connected to the strain indicator one at a time. First an initial "reference " reading will be obtained for each gage with the beam undeflected. Then, a final reading will be obtained for each gage with the beam deflected.

 Figure 2

Do not adjust the balance control again during the experiment.

Figure 3. Wiring Diagram

Analysis and Presentation of Data:

Before calculating Poisson's Ratio form the indicated longitudinal and lateral strains, the indicated lateral strain should be corrected for transverse sensitivity. Because the longitudinal strain in the beam is several times as large as the lateral strain, the lateral gage is subjected to a much larger strain in a direction transverse to its primary sensing axis than along that axis. As a result of the finite width of the grid lines in the gages, and the presence of the end loops connecting the grid lines, strain gages are generally sensitive not only to the strain parallel to the grid direction, but also 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 easily from the graph below.

 

Report:

Prepare a brief report, describing in your own words the purpose of the experiment, the equipment and setup used, and the results obtained.

Include all original data and calculations for correcting the lateral strain and computing the Poisson's Ratio.

Discuss the probable sources of error.

Compare your result with the published data and calculate the percent error.

Calculate the Poisson's Ratio for aluminum , stainless steel, gray cast iron, titanium alloy, brass, and copper. Use the Modulus of Rigidity (G) formula to calculate the Poisson's Ratio (m).

WORK SHEET

Strain measurements

 

Longitudinal strain

m e

Lateral strain

m e

Initial (undeflected)

5,000

 

Final (deflected)

 

 

Final reading - Initial reading

 

750

 

Correction of Lateral Strain for Transverse Sensitivity:

Note that the transverse strain sensed by the lateral strain gage is negative. Therefore,

e t / e a = -(-- ) / 750 = ___________

From the package data,

Kt = ____________

From the correction chart,

C = ____________

Corrected lateral strain = 750(C) = ____________

Calculation of Poisson's Ratio:

m = 750( C ) / (...) = _____________

Published Values of Poisson's Ratio:

Using the modulus of elasticity and modulus of rigidity for each material listed below, calculate the Poisson's Ratio.

Material

Modulus of Elasticity

Modulus of Rigidity

Poisson's Ratio

Aluminum alloy (6061-T6)

10 x 10 6

4 x 10 6

 

Stainless steel

(Annealed)

29 x 10 6

11.6 x 10 6

 

Gray cast iron

15 x 10 6

6 x 10 6

 

Titanium alloy

16.5 x 10 6

6.5 x 10 6

 

Brass

14 x 10 6

6 x 10 6

 

Copper

15 x 10 6

6 x 10 6

 

 

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 Last Update: March 8, 2000

Prepared by: Serdar Z. Elgun