Mechanics Of Materials - Stress Risers In A Cantilever Flexure Test

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............................................................4 1.1. General Background..............................................................4 1.2. Mathematical Derivation for Equations Used.........................4 2. Equipment and Procedure............................................................7 2.1. Equipment..............................................................................7 2.2. Experiment Setup...................................................................7 2.3. Initial Calibration.....................................................................8 2.4. Procedure...............................................................................8 3. Data, Analysis & Calculations.......................................................8 3.1. Known information..................................................................8 3.2. Location for Reference Gage.................................................8 4. Results..........................................................................................9 5. Verification and Recalculation by ANSYS® ...............................13 6. Conclusions.................................................................................16 7. References..................................................................................17 8. Raw Notes...................................................................................17

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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. Kt =

σ max imum σ no min al

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.

σx =

6 ⋅ P ⋅ ( L − x) b ⋅t 2

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

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•

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:

σ no min al =

6 ⋅ P ⋅ ( L − x) (b − d ) ⋅ t 2

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: L − xhole b−d = L − xreference b Equation 4 To find the maximum stress at the hole, Equation 1 and 3 can be combined to form Kt =

σ max 6 ⋅ P ⋅ ( L − x) (b − d ) ⋅ t 2

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

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Kt =

ε max imum εA

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 2

R R ε = A + B  + C   Z  Z 

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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, 2

4

2

4

R R ε1 = A + B   + C    Z1   Z1  2 4 R R ε 2 = A + B  + C    Z2   Z2  R R ε 3 = A + B  + C    Z3   Z3  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

ε max imum = A + B[1] 2 + C [1] 4 = A + B + C 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 Beam Dimensions (inches) b= 1.000 t= 0.250 d= 0.250 x= 3.282 L= 10.000

Gage Factors ( (ΔR/R) / Strain ) Sg1 = 2.08 Sg2 = 2.08 Sg3 = 2.08 Sg4 = 2.095

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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L − xhole b−d 10 − 3.282 1.000 − 0.250 = ⇒ = L − xreference b 10 − xreference 1.000 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. Gage 1 2 3 4

Initial Reading (με) 0 740 -196 394

Final Reading (με) 961 1551 564 1194

Table 2

4. Results To find the strain induced by the deflection, the net strain was found by,

ε n = ε n , final − ε n ,initial 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. Gage 1 2 3 4

Net Strain (με) 961 811 760 800

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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ε 4,corrected = ε measured ⋅

S g ,1 S g ,4

= 800

2.080 = 794.2 µε 2.095

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. 2

4

2

4

2

4

2

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R R  0.125   0.125  ε 1 = A + B   + C   ⇒ 961 = A + B ⋅  +C⋅   0.145   0.145   Z1   Z1  2 4 2 4 R R  0.125   0.125  ε 2 = A + B   + C   ⇒ 811 = A + B ⋅  +C ⋅  0.185   0.185   Z2   Z2  R R  0.125   0.125  ε 3 = A + B   + C   ⇒ 760 = A + B ⋅  +C ⋅   0.325   0.325   Z3   Z3  Equation 13 Solving for A, B, and C simultaneously, A = 601.56 B = −198.37 C = 776.33 Equation 14 Therefore, using Equation 9, the value for εmaxiumum laterally adjacent to the discontinuity is equal to,

ε max imum = A + B + C = (601.56) + (−198.37) + (776.33) = 1179.52 µε Equation 15 The stress concentration factor, as expressed in Equation 1, can now be found. Kt =

ε max imum 1179.52 = = 1.485 ε 4,corrected 794.20 Equation 16

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

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Estimated Strain vs Location 1400

1200

Strain (με)

1000

800

Hole

600

400

200

75 0 0. 3

5 31 2 0.

0 50 0. 2

87 5 0. 1

0 12 5 0.

62 5 0. 0

0 00 0 0.

.0 62 5 -0

0 25 -0 .1

.1 87 5 -0

00 -0 .2 5

5 12 .3 -0

-0

.3 75 0

0

Lateral Distance from the Center of the Hole (inches)

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 K t ,theory − K t ,exp eriment K t ,theory

× 100 = 17.5% error

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 14

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 16

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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