Mechanics of Materials Laboratory

Stress Concentration

Group C:

David Clark

Jacob Parton

Zachary Tyler

Andrew Smith

10/17/2006

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 calculate the

maximum stress for different 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 of the beam. 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 by Statics and 1.6979 by ANSYS®.

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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 an irregularity is found by the stress concentration factor, K t. 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. The nominal stress refers to the ideal 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. This expression is the same as Equation 2, however

the loss of area due to the hole is accounted for. It should be noted that Equations 2 and 3

are only valid for a rectangular beam with a point load.

A convenient way of determining the nominal stress at the area adjacent to 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 occurrence 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 was 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

were 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 was mounted 1.00 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

1.00 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 εmaximum 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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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 diameter 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 measured 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 general purpose, finite element modeling package for numerically

solving a wide variety of mechanical problems. These problems include: static/dynamic

structural analysis (both linear and non-linear), heat transfer and fluid problems, as well

as acoustic and electro-magnetic problems.

First the geometry of the test specimen must be modeled. The geometry is

recreated in Dassault Systems Catia V5 B14. The point at which the load is applied must

be modeled as a small cylindrical pad. 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 relatively close to real-

world behavior.

The model is then loaded into ANSYS® Workbench 10. There, the model is

divided into different elements to form a "mesh" of the product. This process alone

establishes the name for the software technique. The mesh below models the beam by

creating over 54,000 pieces, or a finite number of elements.

Figure 3

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Figure 4

Running the simulation requires initial conditions to constrain the model and

control the mathematical engine. A fixed constraint on the end face, a gravitational load

across the entire part, and a load on the small pad is added to the model. The gravity was

calculated as a function of altitude and latitude for Huntsville. This is determined by the

formula

Equation 18

where φ is the latitude in degrees and h is the height in meters above sea level. With a

latitude of 38.017º and an altitude of 3,841 meters, the acceleration due to gravity is

approximately 9.77351 meters per second squared or 384.7839 inches per second

squared.

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Figure 5

Next, locations for measuring strain were placed along the beam at the same

locations as the actual beam.

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The force was determined by an iterative process of applying a load and reading

the resulting strain at the reference gage location. According to ANSYS®, the load

applied to create 800 με at the reference gage was 9.915 pounds.

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

adjacent to the hole.

Figure 6

Although the stress is valuable in determining failure modes for a material, the

laboratory focuses on strain. The strain model, as shown below, it the same as the stress

model scaled by the Young's Modulus.

15

Figure 7

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.

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Figure 8

ANSYS® Workbench 10 utilizes strain probes to return normal, shear, equivalent

(von-Mises), maximum principal, middle principal, minimum principal, and strain

intensity. Below is a table with these values for each of the gages, along with the

experimental strain as measured within the experiment.

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The results from ANSYS® provide some insight into possible errors within the

laboratory experiment. The largest cause, the maximum computed strain, has the highest

error. The value for maximum strain as calculated mathematically was 1179.52 με,

whereas the maximum equivalent strain determined by ANSYS® was 1365.3 με. The

error from this discrepancy can be expressed as

Equation 19

With this improved calculation for maximum strain, the stress concentration

factor can be found. To determine Kt, the anticipated inputs and outputs must be clearly

defined. Different stress concentration factors can exist for the same setup. For example,

the stress concentration factor derived from principal strain readings can be different

from the same setup with inputs of equivalent strains. The remainder of the analysis

determines a stress concentration factor for a beam with a rectangular cross section

composed of an isotropic material that is a function of nominal stress at a region,

represented by the maximum principal strain at a reference location, and maximum

equivalent stress due to a hole. This was chosen because the input will closely match the

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value for the strain gage readings. For most cases, the equivalent strain would be the most

critical parameter.

Using previous methods, the stress concentration factor is expressed as

Equation 20

The stress concentration factor found by ANSYS® varies only slightly from the

accepted value.

Equation 21

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.

To better estimate the maximum stress or strain around a hole, a finite element

analysis program such as ANSYS® can be used. The error in the stress concentration

factor from ANSYS® when compared to the accepted value is approximately 5.8%.

ANSYS® proved to be a much more accurate solution than strain gages and can provide

results without physically having to test.

7. References

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

Mechanics of Materials Laboratory Manual. June 2000.

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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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