Mechanics of Materials Laboratory

Cantilever Flexure Test

David Clark

Group C:

David Clark

Jacob Parton

Zachary Tyler

Andrew Smith

9/29/2006

Abstract

Stress risers, geometric irregularities that break the uniformity of a material, cause

a predictable increase in stress. The stress concentration, expressed as the maximum

stress under loading divided by the nominal stress, can mathematically predict the

maximum stress for different nominal loadings. The following exercise explores the

effect of a hole in an otherwise uniform rectangular aluminum cantilever beam. When a

load is applied to the unsupported end of the beam, the stress adjacent to the hole

increases dramatically more than the area closest to the edge. This resulting stress profile

was measured, as well as theoretically determined with Statics. As a second opinion,

ANSYS® was used to verify and visually render the stress concentrations over the surface

of the beam. The stress concentration factor for Aluminum 2024-T6 was ultimately found

to be approximately 1.485.

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Table of Contents

1. Introduction & Background.............................................................................3

1.1. General Background................................................................................3

1.2. Mathematical Derivation for Equations Used.........................................3

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. Location for Reference Gage...................................................................7

4. Results..............................................................................................................8

5. Verification and Recalculation by ANSYS®.................................................12

6. Conclusions....................................................................................................15

7. References......................................................................................................16

8. Raw Notes......................................................................................................16

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1. Introduction & Background

1.1. General Background

Geometric irregularities on loaded members can dramatically change stresses in

the structure. A simple irregularity, a drilled hole, is studied within this experiment such

that the effects of this feature can be analyzed and explored.

Extensive research into the effects of these discontinuities, called "stress-risers"

has been conducted previously. A standard means of computing the maximum theoretical

stress around a irregularity is found in the stress concentration factor, Kt. The stress

concentration factor is a ratio of two stresses, as shown below.

Equation 1

For a hole, the maximum stress is always found at the closest position to the discontinuity

because this is where the material has the least amount of support. The nominal stress

refers to the stress based on the net area of the section.

1.2. Mathematical Derivation for Equations Used

As proven in previous experiments, the value for stress can be calculated with the

following formula.

Equation 2

P is the magnitude of the force applied

L is the longitudinal length from the clamp to the load

x is the longitudinal distance from the clamp to the cross sectional area being

inspected

3

b is the base dimension of the beam

t is the thickness of the beam

The nominal stress for a cross sectional area with a hole can be expressed as:

Equation 3

where d is the diameter of the hole.

It should be noted that Equations 2 and 3 are only valid for a rectangular beam with a

point load.

A convenient way of calcuating the nominal stress at the area with a hole is done by

computing the stress at another location on the beam where there is no discontinuity. To

do this, an initial position for this reference location can be found by setting Equation 2

and 3 equal. After cancelling like terms, the mathematical setup reduces to:

Equation 4

To find the maximum stress at the hole, Equation 1 and 3 can be combined to

form

Equation 5

Using the technique proven with Equation 4, we can substitute the nominal stress

with the stress from a reference gage at point A. This method is convenient since it

utilizes a physical occurance that is easily measured. Therefore, the easiest expression in

this setup to use is

4

Equation 6

The maximum stress at the hole cannot be easily measured due to the physical

limitations of the strain gage. Three gages were placed at known distances away from the

hole such that the maximum stress could be calculated using numerical methods. The

formula used to approximate the distribution adjacent from the hole was expressed by the

form

Equation 7

Where A, B, and C are constant coefficients, Z is the distance from the center of the hole

to the center of the gage, and R is the radius of the hole.

With three gages, the equations can be expressed as,

Equation 8

The coefficients A, B, and C must be found by solving the three equations above

simultaneously.

The maximum stress occurs at the edge of the hole, when R/Z = 1, therefore

Equation 9

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2. Equipment and Procedure

2.1. Equipment

1. Cantilever flexure frame: A simple apparatus to hold a rectangular beam

at 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. Four strain gages:

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. Three strain

gages should be mounted laterally adjacent to the hole such that the long metal traces run

parallel to the length of the beam. The distance from the center of the hole to the gages

was as follows:

Z1 = 0.145 in (3.68 mm)

Z2 = 0.185 in (4.70 mm)

Z3 = 0.325 in (8.26 mm)

A reference strain gage will be mounted 1.043 inches from the clamp. For the

derivation of this value, see Section 3.2.

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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, the gage located nearest to the hole. Since there is no forced deflection,

the indicator should also 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. Do not adjust the gage factor for each gage. The effect of

different gage factors can be calculated later.

After the reference gage is measured, a force, P, was added such that an 800 με

increase was experienced on gage 4. In reverse order, measure and record the increased

strain readings on each of the remaining gages.

3. Data, Analysis & Calculations

3.1. Known information

Table 1

3.2. Location for Reference Gage

The location for the reference gage can be calculated using Equation 4. Inputting

the known dimensions expressed above, the output of the gage returns the nominal stress

along the region with the hole when

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Equation 10

Solving for xreference, the distance from the clamp where the gage should be mounted is

1.043 inches.

The table below displays the initial (undeflected) and final (deflected until gage 4

was increased by 800 με) strain readings.

Table 2

4. Results

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

Equation 11

The table below catalogs the net strains by each gage. Note that gage 4, the

reference gage, experienced an 800 με increase, as per the experiment setup.

Table 3

Strain gage 4 was larger and had a gage factor that was different from the other

gages. To correct for this, the following equation was used:

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Equation 12

The maximum strain is calculated from Equation 8. As expressed in Equation 9,

the maximum strain can be expressed as the sum of the coefficients A, B, and C.

Equation 13

Solving for A, B, and C simultaneously,

Equation 14

Therefore, using Equation 9, the value for εmaxiumum laterally adjacent to the

discontinuity is equal to,

Equation 15

The stress concentration factor, as expressed in Equation 1, can now be found.

Equation 16

The following chart displays the stress versus distance from the hole.

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Estimated Strain vs Location

0

200

400

600

800

1000

1200

1400-0

.375

0

-0.3

125

-0.2

500

-0.1

875

-0.1

250

-0.0

625

0.00

00

0.06

25

0.12

50

0.18

75

0.25

00

0.31

25

0.37

50

Lateral Distance from the Center of the Hole (inches)

Str

ain

(μ

ε)

Hole

Figure 1

A theoretical stress concentration factor can be calculated with data from a chart,

such as Peterson's "Stress concentration factors for bending of a finite-width plate with a

circular hole."

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Figure 2

The chart defines the necessary terms needed to characterize the physical

characteristics of the material being tested. Since the hole dimeter divided by the

thickness is equal to 1, the d/h=1 line was traced until it intersected the d/H value of 0.25.

This corresponds to 1.8 on the y-axis, therefore Kt, theoretical = 1.8.

The comparison of the theoretical and measued values for Kt can be expressed in

percent error as

Equation 17

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5. Verification and Recalculation by ANSYS®

ANSYS® is a common software package that quickly became the standard for

mechanical testing throughout the design, testing, and manufacturing industry. Modeling

and testing utilizing ANSYS® technology is beyond the scope of this exercise, however

the results can be used to graphically represent the rise in stress that was measured.

First the geometry of the test specimen must be modeled. Shown below is the

geometry as modeled in Autodesk® InventorTM 9. For accurate results, the model is

modified to help reenact the actual setup of the flexure frame. The end being clamped

must have small impressions on top and bottom so that ANSYS® can define the areas

being physically constrained.

The point at which the load is applied must be modeled as a small indentation. If

the point were to be applied to a point with an infinitely small area, the stress at that point

would be displayed as infinitely large. With a small pad for an applied area, the stress

along the part remains nominally close to real-world behavior.

Figure 3

Below is the stress gradient over the beam. Note the stress pattern experienced

adjacent to the hole.

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Figure 4

Upon closer inspection of the hole, the neutral axis becomes visible. Also note the

gradient from outside the hole to the edge of the beam.

Figure 5

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Adjusting the color gradient to narrow the range of stress analyzed, a clearer view

of the stress gradient can be seen.

Figure 6

Figure 7

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Figure 8

The nominal stress, represented by the stress at 1.024 inches from the clamp, is

represented in Figure 8. A phenomenon that is not apparent in the original experiment

that acts as a potential cause for error is the fact that the hole disrupts the stress gradient,

causing a non-uniform stress along the 1.024 inch line.

The ultimate stress as reported by ANSYS is 1082 psi. The nominal stress located

1.024 inches from the clamp is reported to be 786.7 psi at the center. Using previous

methods, the stress concentration is calculated to be 1.375.

6. Conclusions

Due to the large margin of error from the measured and calculated results, 17.5%

error, the experimental results are acceptable only as a starting point for design

guidelines. Designs that utilize geometric irregularities and stress risers must be designed

with an acceptable margin of safety, as well as be rigorously tested to ensure sufficient

quality.

Improvements to ensure higher accuracy would be to obtain readings for two bars:

one bar with a drilled hole and a second bar of the same dimensions without a hole. Stress

readings around the hole would be the same for the first bar, however the nominal stress

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read at the reference location would be measured from the bar without the hole. The

equations used in the procedure above assume no effects on the stress distribution along

the bar due to the hole.

7. 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

8. Raw Notes

Figure 9

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Figure 10

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Figure 11

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Figure 12

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