136559 mechanics of materials principal stresses and strains

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Mechanics of Materials Laboratory

Principal Stresses and Strains

David ClarkGroup C:

David Clark

Jacob Parton

Zachary Tyler

Andrew Smith

10/13/2006


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Abstract

Principal stress refers to the magnitudes of stress that occur on certain planes

aligned with a solid body of which they occur. For a rectangular beam, a corner of the

beam could be thought of as the origin of this coordinate system with each of the planes

parallel to the three touching faces of the beam at that point. In the following experiment,

two principal stresses and strains were determined using a rosette, an array of three strain

gauges. When a 3.77 lb load was applied to a beam measuring 1" x 0.125" x 11.250"

demonstrated a principal strain of 1528 and -463 longitudinally and laterally

(respectively) at 1 inch from the clamp. The stress was therefore calculated to be 15.9 and

0.0 ksi longitudinally and laterally (respectively) at the same position.

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Table of Contents

1. Introduction & Background ............................................................4

1.1. General Background ..............................................................4

1.2. Determination of Principal Stresses .......................................4

2. Equipment and Procedure ............................................................6

2.1. Equipment..............................................................................6

2.2. Experiment Setup ...................................................................6

2.3. Initial Calibration .....................................................................7

2.4. Procedure ...............................................................................7

3. Data, Analysis & Calculations .......................................................7

3.1. Known information ..................................................................7

3.2. Gage Readings .......................................................................8

3.3. Further Calculations ...............................................................8

4. Results ........................................................................................10

5. Conclusions .................................................................................10

6. References ..................................................................................10

7. Raw Notes ...................................................................................11

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1. Introduction & Background

1.1.General Background

When a cantilever beam experiences a single load, the magnitude of the stress can

be modeled as a directional result. Since stress is difficult to visualize, an important

relation to help explain this phenomenon is the relation between stress and strain. Since

strain can be seen visually, it is easy to confirm the amount of strain occurring

longitudinally is far greater than the lateral strain. The same is true of stress.

Principal stresses refer to the magnitudes of stress that occur on certain planes

within a solid body. This coordinate system is aligned such that no shear stresses occur

along these principal planes. For a rectangular beam, a corner of the beam could be

thought of as the origin of this coordinate system with each of the planes parallel to the

three touching faces of the beam at that point.

1.2.Determination of Principal Stresses

Principal stresses can be determined simply by aligning strain gages along axis.

The procedure below explains how to use a rosette, an array of strain gages placed in

different orientations around a single point, to determine principal strains.

Figure 1

To perform the required analysis, consider a rectangular rosette with an axis

aligned to one of the gages. The strain at any angle can be found using a set of

expressions in the form of the following.

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)2sin()2cos(22

xyyxyx

+

++

=

Equation 1

Applying Equation 1 to each of the three gages with measured counterclockwise

from the x axis, expressions for 1, 2, 3 can be formed.

For a rectangular rosette, the two principal strains along the two-dimensional

surface is characterized by

( ) ( ) 2322

21

31

,2

1

2

+

+=qp

Equation 2

and

31

3121

,

2tan

2

1

=

qp

Equation 3

Stress can commonly be calculated by using Hooke's Law. Since the strains

measured are not aligned with the load and resulting stress, the generalized form of

Hooke's Law is used.

( )qpp v

v

E +

=2

1

Equation 4

and

( )pqq v

v

E +

=

21

Equation 5

where E is the elastic modulus.

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2. Equipment and Procedure

2.1.Equipment

1. Cantilever flexure frame: A simple apparatus to hold a rectangular beamat one end while allowing flexing of the specimen upon the addition of a

downward force.

2. Metal beam: In this experiment, 2024-T6 aluminum was tested. The beam

should be fairly rectangular, thin, and long. Specific dimensions are

dependant to the size of the cantilever flexure frame and available weights.

3. P-3500 strain indicator: Any equivalent device that accurately translates

to the output of strain gages into units of strain.

4. One rectangular rosette:

5. Micrometers and calipers:

2.2.Experiment Setup

The specimen should be secured in the flexure frame such that an applied force

can be placed perpendicular and opposite of the securing end of the fixture. A rosette

containing three strain gages should be mounted as shown in figure one. In this setup, p

and q are known.

Figure 2

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2.3.Initial Calibration

Record the dimensions of the beam, as well as the gage factor for each strain

gage. Strain gage specifications are usually provided by the manufacturer. Before any

deflection is added on the beam, the strain indicator should be calibrated using the gage

factor the gage 1 (labeled below.) Since there is no forced deflection, the indicator should

be balanced such that a zero readout is achieved.

2.4.Procedure

Utilizing a quarter bridge configuration, measure and record each of the

individual strain gage readings. The gage factor should be readjusted if the gage factor

varies for any of the gages.

After the last gage result has been measured, a known load should be applied to a

point. This location should be at a known length from the clamp of the flexure fixture, as

well as be located in the center of the bar laterally.

The strain for each gage should be measured in reverse order. It is important that

the gage factor be readjusted for each location. Upon completion of recording the

measurements, the load should be removed from the bar and the indicator should return

to the initial zero reading.

3. Data, Analysis & Calculations

3.1.Known information

The applied load was 3.77 pounds.

The following table catalogs other known information within this experiment

setup.

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b = 1.000 Sg1 = 2.060 Tp = 30

t = 0.125 Sg2 = 2.075 Tq = 120

x = 1.000 Sg3 = 2.060

L = 11.250

(degrees)

Orientation Angles

(inches) ((R/R) / Strain)

Beam Dimensions Gage Factors

Table 1

3.2.Gage Readings

Gage Initial Final

1 0 1018

2 -90 1314

3 -401 -343

Table 2

To find the strain induced by the deflection, the net strain was found by,

initialnfinalnn ,, =

Equation 6

The table below catalogs the net strains by each gage.

Gage Net Strain

1 1018

2 1404

3 58

Table 3

3.3.Further Calculations

The principal strains pand q, is found using Equation 2.

( ) ( ) 1528581404140410182

1

2

581018 22=++

+=

p

Equation 7

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( ) ( ) 463581404140410182

1

2

581018 22=+

+=q

Equation 8

Poisson's ratio is defined as the lateral strain divided by the longitudinal strain.

Expressed mathematically,

303.01528

463=

==

p

qv

Equation 9

Equation 3 is used to determine the angle between Gage 1 and the principal axes.

5.30581018

5810181404*2tan21 1 =

=

p

Equation 10

The stress along the principal axes is determined from the generalized Hooke's

Law, as expressed in Equation 4 and 5.

( ) ksip 9.1510463303.0101528303.01

104.10 662

6

=+

=

Equation 11

( ) ksiq 00.0101528303.010463303.01

104.10 662

6

=+

=

Equation 12

As a comparison, theory can be used in calculating the principal stresses.

( ) ( )psi

tb

xLP

allongitudin

3

22

108.14125.0000.1

000.1250.1177.366=

=

=

Equation 13

psilateral 0=

Equation 14

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4. Results

Utilizing Equation 1, p and q were found to be 1528 and -463 respectively.

This corresponds to a Poisson's ratio of 0.303. This corresponds to a 1% error from the

theoretical value 0.3. Possible sources for this error is the uncertainty of the machine, and

imperfections in the adhesive used to secure the strain gage.

The angle at which the gage was placed was calculated to be 30.5, which varies

by 1.67%.

Utilizing Hooke's Law, the longitudinal and lateral strain was calculated to be

15.9 and 0 ksi respectively. Though the lateral stress contains no error, the longitudinal

stress exhibited a 7.43% error from the theoretical 14.8 ksi.

5. Conclusions

The results generated within this experiment demonstrated high integrity when

compared against the theoretical values, and therefore are acceptable for a starting point

in design verification. The rosette is a powerful tool for determining stress along a 2-D

plane.

6. References

Gilbert, J. A and C. L. Carmen. "Chapter 8 Cantilever Flexure Test." MAE/CE 370

Mechanics of Materials Laboratory Manual. June 2000.

Kuphaldt, Tony R. (2003). "Chapter 9 Electrical Instrumentation Signals."

AllAboutCircuits.com. Retrieved September 19, 2006, from Internet:

"http://www.allaboutcircuits.com/vol_1/chpt_9/7.html

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7. Raw Notes

Figure 3

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Figure 4

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Figure 5

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Figure 6

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136559 mechanics of materials principal stresses and strains

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