305W Structure LabReport Template Me1

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AEROSPACE 305W STRUCTURES & DYNAMICS

LABORATORY

Laboratory Experiment #1

Bending of Selectively Reinforced Beams

March 24, 2009

Stephanie Penatzer

Lab Section 14

Lab Partners:

Christopher BerryDennis Larkin

George Wilson

Kevin Kennedy

Nicholas Kesler

Course Instructor: Dr. Stephen Conlon

Lab TA:Patrick Williams


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Abstract

The objective of this experiment is to investigate the flexural behavior of structural beam

elements through experimentation and analysis. Additionally, the goal of the experiment is to

assess the utility of composite materials as reinforcement components. This will be achieved

by testing three beams; an aluminum beam, an aluminum with aluminum caps, and an

aluminum with graphite/epoxy caps. These beams will be loaded, hanging a load hook with

weights at the end of the beam, in an I-beam orientation and in an H-beam orientation. The

beams are loaded; the strain is measured using a strain gage and the displacement is measured

using the LVDT. The theoretical data, found by utilizing the Euler-Bernoulli beam theory,

supports the idea that composite reinforcements resist deflection and strain. When comparing

the theoretical data to the experimental there is a clear similarity between them. However, the

experimental data has more displacement and strain than the theoretical data due to the error in

the real world experimentation. Although there is considerable error, the experimental results

provide support to these claims. The error comes from assumptions made in the theoretical

calculations and human errors during experimentation. Even with error, the experimental data

supports the conclusion that composites make excellent strengthening materials in structural

design.

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I. Introduction

The goal of this experiment is to explore the flexibility of structural beams through

experimentation and analysis. This can be accomplished through the comparison of the Euler-

Bernoulli Beam Theory with the experimental results. Along with assessing flexibility, the

experiment examines the effectiveness of composite materials as reinforcements to the structural

beams. This type of information is useful in aerospace structural design; specifically in regards to

reduction of weight with an increase in strength. The goal is to design an aircraft with minimal

materials that accomplishes such design goals as low weight, high stability, high strength, and low

cost. In order to determine these objectives, three beams were loaded; each with a different

reinforcement, and the strain was measured and recorded. This experiment used three specimens: an

aluminum irregular T-beam, an aluminum irregular T-beam reinforced with aluminum caps, and an

aluminum irregular T-beam reinforced with graphite/epoxy caps. A schematic of the beams

compositions are featured in figure 1.

Figure 1. Test Beam compositions

This experiment utilizes the theory of beam bending to analytically determine the flexibility of the

beams as well as the utility of composites. Figure 2 provides a pictorial representation of the beams

directional axis. This theory makes four assumptions: stress is related to strain, xx, through Hookes

Law, equation 1, deformations due to stresses in the y-direction and z-direction are much smaller

than those in the x-direction, cross-sections remain planar and normal to the centroid, and the load

passes through the neutral axis. Equation 1 utilizes xx, stress in the x-direction, and E, Youngs

modulus of elasticity.

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Aluminum Aluminum with Aluminum Caps Aluminum with Graphite/EpoxyCaps


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Figure 2. Beam Axis

The shear force, Sz(x) produced by a transverse load is given by equation 2, where A is the cross-sectional

area and the bending moment about the y-axis, My, is given by equation 3. These equations are used to

determine the transverse displacement, w(x). Equation 4 provides the transverse displacement of a beam

based on its length (as show in figure 3), properties (such as modulus of elasticity), and the force applied

(P). The limit load for each beam can be determined using the yield stress, found in equation 5, which is

half the load value that will produce a yield stress in the beams. Equations 4 and 5 are dependent upon the

lengths measured in the set-up of the experiment as pictured in figure 3. The derivation of the equation for

strain in the x-direction, xx, as well as the derivation of the transverse displacement, w(x), can be found

in the appendix. Equation 6 describes the cross-sectional moment of inertia for a beam where b is the

base of the beam and t is the thickness. Equation 7 is the adjustment known as the parallel axis theorem,

which gives the moment of inertia of an asymmetrical beam. z_bar is the defined by equation 8 where i is

the specific beam configuration, j is an individual beam section and all z_bar measurements are measured

from the vertical center of the beams. Both z_bar and EI values can be found in table 1.

.

(1)

(2)

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(3)

(4)

(5)

(6)

(7)

=jj

jjj

iAE

zAEz (8)

Figure 3. Important Specimen Dimensions

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Table 1. Measured Dimensions, EI values, and z values

L (in)Lf(in)

Ld(in)

Ls(in) z (in)

EI (lbin^2)

Alum. I-beam

9.812

5 9.375

7.062

5

0.687

5

0.238

92 92347

Alum. OnAlum. I-beam 9.75 9.875

7.812

5 0.875

0.283

94 218448

Graphite/Epoxy onAlum. I-beam

9.437

5

9.187

5 6.125 1

0.277

06 341372

Alum. H-beam

9.687

5 9.375

6.687

5 1

0.238

92 92347

Alum. OnAlum. H-beam 9.875

9.562

5

7.562

5 1

0.283

94 218448

Graphite/Epoxy on

Alum. H-beam 9.375 9.1257.312

5 1

0.277

06 341372

The theoretical data described above can be confirmed through the experimental data. The beam

is clamped at one end and loaded using a load hook at the tip of the beam. The displacement is then

recorded using the LVDT, an apparatus which sits on the top of the beam and measures the drop in

the z-direction. The strain is measured by strain gages placed on the top and side of the beam and is

read with the strain gage box. The beam is loaded from 0 to 5 pounds and then unloaded to record

hysteresis; a process where a lag occurs between the application and removal of a force. In a case

such as this the beam will register different strain and displacement at the beginning and end of

loading.

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II. Experimental Procedure

The basic process encompassed by this experiment is to load three different beams to determine

the flexural behavior and also to determine the value of composites as a strengthening tool in beams.

Before beginning the experiment the strain gage box was powered on and zeroed by depressing the

AMP ZERO button and adjusting the AMP ZERO potentiometer to display plus or minus zero. The

box was then left to sit for five minutes and readjusted in the manor previously described.

Each beam was clamped into the fixture as pictured in figure 4; with the filler blocks flush with

the clamp and the desired length the of the beam outside of the clamp. The filler blocks were used to

clamp the beam without damaging it and were kept flush with the fixture to prevent it from effecting

the beams rigidity and strength.

Figure 4. Beam into fixture configuration

The beams had two strain gages, one on the top and one on the side. When testing the beam as an

I-beam, the top strain gage was used and when testing the beam as an H beam, the side strain gage

was used. These strain gages were wired into the strain gage box with the red wire into the P+, the

black wire into the S-, and the white wire into the D (350) as pictured in figure 5.

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Clamp Configuration


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The load hanger was then suspended from a point near the end of the beam farthest from the

clamp. The LVDT was mounted on the fixture as shown in figure 6 in order to monitor the transverse

displacement of the beam. The distances from the root to all the apparatus were measured as

pictured in figure 3.

Figure 6. LVDT Mount Location and Load Hook Location

The strain gage was zeroed again to account for the load hanger by adjusting the AMP ZERO

potentiometer display to read zero and the recording software (Lab View) was also zeroed by creating a

dummy file and moving the LVDT arm until it recorded the voltage as close to zero as possible. The Load

hanger was then loaded from 0 to 2 pounds at half pound increments and then from 2 to 5 pounds at one

pound increments as well as unloaded from 5 to 0 in the same fashion.

The beam was then turned onto its side and tested as an H beam and the side strain gage was

connected to the strain gage box. After changing the orientation of the beam and zeroing the equipment,

the beam was loaded as before and the strain values were recorded. This procedure was repeated for each

of the three beams in the I and H orientations and all of the strain data was recorded in Lab View.

8

Red Wire-Red Knob (P+)

Black Wire-White Knob (S-)

White Wire-Yellow Knob

(D350)

LVDT

Load Hook

Figure 5. Strain Gage Box Set-Up


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III. Results and Discussion

Analytically speaking, the displacements of the beams decrease with the addition of a stronger

composite. Therefore, the aluminum beam should displace much more than the aluminum cap beam

and the graphite/epoxy cap beam. However, in review of the experimental data, it is important to note

that the displacement of the purely aluminum beam is greater than that of the beam with

graphite/epoxy caps and even greater than the displacement of the beam with aluminum caps. These

results can be seen clearly in figures 7 through 9. The discrepancy between the theoretical data as

compared to the experimental data is great, especially with the graphite/epoxy capped beam. These

differences can be accounted for through various types of error, values presented in table 2, which

occurred during the experiment. The error was calculated by subtracting the experimental data from

the theoretical, dividing that number by the theoretical data and multiplying by one hundred. Sincethe apparatus was being continually changed throughout the experiment the beams were not placed

precisely and the LVDT was not perfectly zeroed because it is so sensitive. This could have thrown

off the reading for the displacements. The addition, the caps also seemed to bring in more error

throughout the experiment. Since the LVDT was placed directly on the caps the bond between the

specimen and the cap could have been loose and the displacement of the cap was measured but not

the total displacement of the beam. Also, the beam could have displaced in the y and z directions as

well which would throw off the displacement measurements in the x-direction.

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Figure 7. Analytical and Experimental Displacement Vs Applied Load for an Aluminum I-

Beam

Figure 8. Analytical and Experimental Displacement Vs Applied Load for an Aluminum I-

Beam with Aluminum Caps

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Figure 9. Analytical and Experimental Displacement Vs Applied Load for an Aluminum I-Beam with Graphite/Epoxy Caps

Table 2. Percent Error of Displacement for I-Beam Specimens

Aluminum I-Beam

Aluminum I-Beam withAluminum Caps

Aluminum I-BeamwithGraphite/EpoxyCaps

0 0 0

45.67025357 48.2650506 359.9499431

57.27324263 48.5603993 371.9198475

59.10159242 48.79011496 372.8536698

59.72569259 48.80652322 379.0508543

54.93506453 48.5603993 378.4566038

52.94849216 49.44644542 373.7025992

51.70380788 50.78535956 382.7182293

56.28435151 51.98152182 392.4851619

60.91236192 52.43274901 409.0180618

68.16423009 55.05807084 408.4662577

70.45845747 53.7125934 417.8469274

70.24749403 59.48830144 442.4658797

70.88038434 65.39527556 486.7799938

0 0 0

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Theoretically, the strain in the beams should have reduced when the caps were applied to the

beams and the reduction of the strain should have coincided with the strength of the caps applied.

Again, the experimental data does not match our theoretical data. However, the experimental strain

more closely adheres to the theoretical strain than the displacements do. This is because the

displacement of the I-beam was more sensitive to the boundary conditions set at the beginning of the

experiment. The graphs in figures 10 through 12 show that there is considerable error, especially inthe graphite/epoxy capped beam. This error can be observed in table 3 and were calculated in the

same manner as the displacement errors. These differences occur from many outside faults. The

strain gages could have been attached with a poor connection to the strain gage box. Also, the beam

could have been clamped incorrectly which produced strain in the y and z directions. The bonding

element between the beam and the caps could have provided more error if the bond was not strong

enough. In addition, more error could have been produced if the strain gage box was not perfectly

zeroed.

Along with errors, the experimental data differs from the actual data due to the assumptions made

when using the beam bending theory as well as the assumptions made to form boundary conditions.

The displacement is based on the assumption that deformations due to stresses in the y-direction and z-

direction are much smaller than those in the x-direction, cross-sections remain planar and normal to the

centroid, and the load passes through the neutral axis. The strain, however, is only bounded by the

assumption that stress is related to strain, xx, through Hookes Law. These assumptions, described

above, can cause inconsistencies in the data attained through experimentation while making the

analytical data easier to work with.

It can be observed that the error in the graphite/epoxy caps is much higher than that of the other

beams. This is due to the fact that the assumed Youngs modulus is incorrect. If the displacement is

graphed linearly and a line is fit to it as displayed in figure 13 the true Youngs modulus (E) can be

determined. In order to solve for E, the equation for slope from the graph is set equal to the equation

of displacement, equation 4 without the applied load, and the equation is rearranged. The resulting E

is equal to 10.2E6 psi.

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Figure 10. Analytical and Experimental Strain Vs Applied Load for an Aluminum I-Beam

Figure 11. Analytical and Experimental Strain Vs Applied Load for an Aluminum I-Beam withAluminum Caps

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Figure 12. Analytical and Experimental Strain Vs Applied Load for an Aluminum I-Beam

with Graphite/Epoxy Caps

Figure 13. Linear fit to a graph of Displacment verses Load for an Aluminum Beam with

Graphite/Epoxy caps

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Table 3. Percent Error of Strain for I-Beam Specimens

Aluminum I-Beam

Aluminum I-Beamwith AluminumCaps

Aluminum I-BeamwithGraphite/EpoxyCaps

0 0 0

5.742963732 11.96059886 66.91121529

6.624154725 1.583927675 81.814012046.917885056 0.67349484 86.78161096

3.539984023 3.495270846 113.1098687

2.481489992 5.752689086 98.70383632

2.848652906 6.034867969 94.48138025

2.716474257 5.52694769 88.96735027

1.306567555 4.341801083 100.4424967

1.012836853 4.059624338 97.71031754

2.218197532 3.495270846 84.79456838

7.181176774 0.455214993 92.74272364

3.068950654 1.583927675 111.6195905

12.79249168 8.356195219 96.7168088

0 0 0

The experiment also included the displacement and strain of each specimen turned on its side and

tested as an H-beam. These tests were not accurate and contained many points of error which can be

viewed in table 4 and 5. Most of the error came from the fact that the H-beam was not symmetrical,

meaning that its left edge was longer than its right edge. When loading such a beam, the specimen

tends to twist which throws off the readings. Also, with the beam twisting, the assumption that there

is only strain in the x-direction is no longer valid and the analytical data is not close to the actual

data. These findings are easily observed in figures 14 through 16. It is clear that the experimental

displacement more closely matches the theoretical displacement than the experimental strain to the

theoretical strain. This is best explained by the afore mentioned error and assumptions. Based on the

assumptions made, the strain is the most sensitive to those conditions. Also, the z_bar is small in the

experiment which will give a small strain that is of the same order as ambient noise and therefore

creates error in the measurements.

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Figure 14. Analytical and Experimental Data for an Aluminum H-beam

Figure 15. Analytical and Experimental Data for an Aluminum H-beam with Aluminum

Caps

Figure 16. Analytical and Experimental Data for an Aluminum H-beam with

Graphite/Epoxy Caps

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Table 4. Percent Error of Displacement for H-Beam specimens

Aluminum H-Beam

Aluminum H-Beam withAluminum Caps

Aluminum H-BeamwithGraphite/EpoxyCaps

0 0 0

19.7838562 19.3340872 144.383741

18.62800398 21.82925447 152.2116577

17.27950973 23.02259535 153.1662817

16.48967738 23.18532365 151.1615713

16.97128247 25.84321923 150.9388257

18.87362258 25.87034061 153.5958625

16.27006546 26.55922375 153.3190215

15.00151765 27.87732298 158.512176

15.93101547 30.29112611 162.7125216

14.23576556 32.18962295 167.1037919

14.69810644 36.33015416 172.0041951

11.2305498 39.62088202 181.232227

11.46172024 50.794892 209.2981722

0 0 0

Table 5. Percent Error of Strain for an H-Beam

Aluminum H-Beam

Aluminum H-Beam withAluminumCaps

Aluminum H-Beamwith Graphite/EpoxyCaps

0 0 0

76.23413815 57.29026107 130.034837

87.20299782 83.98384565 133.0383358

92.07804605 86.94980149 130.0348471

89.94521158 83.98385015 133.0383358

90.249902 85.76342036 132.0371746

93.14446328 94.21638971 133.0383358

90.49365526 93.9494544 131.836939391.31631962 99.11021415 130.0348483

90.85928438 97.62723772 129.0336859

87.20299782 104.4489293 125.529619

86.59361545 113.0501985 108.0092899

87.20299782 121.3548695 127.0313684

79.89042317 156.9463126 112.0139348

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0 0 0

Regardless of the error, it is clear that the composite materials do affect the beams resistance to

strain and displacement. This is because the composites are stronger and more rigid than the beam

itself. When a load is applied the beam distributes it along its upper and lower edges. With the

application of the composites the load is carried by the stronger cap. Even adding a cap of the same

material gives the beam more rigidity to resist the applied load. All of the experimental and

theoretical data can be found in tables 6 through 11 in the appendix.

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IV. Conclusions

Based upon the observed and theoretical data presented in this lab it can be concluded that a

beams flexural characteristic can be changed based on its composition. The beam with the thin

aluminum top and bottom pieces will theoretically deflect more than the beam with the extra

aluminum caps. For example, in the experiment, the aluminum beam deflected approximately 0.0144

inches when fully loaded where as the beam with the aluminum caps only deflected 0.0077 inches

under full load. The beam that has caps made of graphite/epoxy should hypothetically deflect even

less because it adds more rigidity and strength to the aluminum beam. When examining the data

attained by using the Euler-Bernoulli beam theory, the displacement and strain in the aluminum

capped beam should have been 0.0051 inches and 58.49 () respectively while the graphite/epoxy

capped beam should have had a displacement of 0.00196 inches and a strain of 33.23 (). Clearly,this shows that the graphite/epoxy capped beam should have less strain and less displacement.

However, the experimental data does not support the Euler-Bernoulli beam theory entirely. In the

experiment, the aluminum cap beam had a displacement of 0.0077 inches and strain a strain of 55.26

() which is less than the graphite/epoxy capped beam with a displacement of 0.0095 inches and a

strain of 62.78 () when fully loaded. In addition, these numbers do not match with the Euler-

Bernoulli beam theory numbers calculated at the beginning of the lab. The discrepancy can be

attributed to error and assumptions. Such error includes the sensitivity of the LVDT when being

zeroed, the extra rigidity given to the beam by the clamping method, the poor connection between the

strain gage wires and the strain gage box, and the swinging of the load hook after loading. The

assumptions made for the Euler-Bernoulli beam theory, strain and displacement only occur in the x

direction, and that the beam does not deform and stays planar are more sources of error. The

experimental data does however support the second objective of the experiment, which states that

composites are effective as reinforcements for structural beams. As expected, the graphite/epoxy cap

resisted deflection more than the un-capped aluminum beam.

This experiment makes the idea of using composites in everyday structural design a plausible

idea. Using a lighter weight material with lower strength will decrease the weight of the component

and then covering the bulk material in a strong composite skin will to distribute the loads held by the

beam.

In order to minimize the amount of error in the experiment it would be better to do calculations

without assumptions; although this would make the calculations much more difficult. Also, the

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21/25


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5

3

33.07586

7 -0.004527 35.09475939 -0.003047245

4

43.96911

6 -0.006072 46.79301252 -0.004062994

5

55.25848

4 -0.007658 58.49126566 -0.005078742

4

44.76135

3 -0.006175 46.79301252 -0.004062994

3

33.67004

4 -0.004645 35.09475939 -0.003047245

2

22.57873

5 -0.00315 23.39650626 -0.002031497

1.5

17.62725

8 -0.002342 17.5473797 -0.001523623

1

11.88354

5 -0.00162 11.69825313 -0.001015748

0.5 6.337891 -0.00084 5.849126566 -0.000507874

0 0 -7.00E-05 0 0

Table 8. Strain and Displacement Measured and Calculated for an Aluminum I-Beam with

Graphite/Epoxy Caps

Force(lbs)

Strain()

Displacement (in)

TheoreticalStrain ()

TheoreticalDisplacement (in)

0 0 0.000023 0 0

0.5

5.54565

4 -0.000903 3.3225173 -0.0002

1

12.0816

04 -0.001853 6.6450346 -0.00039

1.5

18.6175

54 -0.002785 9.9675519 -0.00059

2

28.3224

49 -0.003762 13.290069 -0.00079

3

39.6118

16 -0.005636 19.935104 -0.00118

4

51.6934

2 -0.00744 26.580138 -0.00157

5

62.7847

29 -0.009477 33.225173 -0.00196

4

53.2778

93 -0.007735 26.580138 -0.00157

339.4137

57 -0.005996 19.935104 -0.00118

2

24.5593

26 -0.003993 13.290069 -0.00079

1.5

19.2117

31 -0.00305 9.9675519 -0.00059

1

14.0621

95 -0.00213 6.6450346 -0.00039

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0.5 6.53595 -0.001152 3.3225173 -0.0002

0

0.59417

7 -0.000203 0 0

Table 9. Strain and Displacement Measured and Calculated for an Aluminum H-Beam

Force(lbs) Strain() Displacement (in) TheoreticalStrain () TheoreticalDisplacement (in)

0

0.19805

9 0.00003 0 0

0.5

2.57476

8 -0.000694 10.833893 -0.00087

1

2.77282

7 -0.001408 21.667786 -0.00173

1.5

2.57476

8 -0.002147 32.501678 -0.0026

2 4.3573 -0.00289 43.335571 -0.00346

3

6.33789

1 -0.00431 65.003357 -0.00519

4

5.94177

2 -0.005615 86.671143 -0.00692

5

10.2990

72 -0.007244 108.33893 -0.00865

4

7.52624

5 -0.005883 86.671143 -0.00692

3

5.94177

2 -0.004364 65.003357 -0.00519

2

5.54565

4 -0.002968 43.335571 -0.00346

1.5 4.3573 -0.002214 32.501678 -0.0026

1

2.77282

7 -0.001536 21.667786 -0.00173

0.5 2.17865 -0.000766 10.833893 -0.00087

0

1.78253

2 -3.50E-05 0 0


with Aluminum Caps

Force(lbs)

Strain()

Displacement (in)



0

-

1.58447

3 0.000008 0 0

0.5

2.37670

9 -0.00055 5.564794 -0.00046

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1

1.78253

2 -0.001123 11.129588 -0.00092

1.5 2.17865 -0.001701 16.694382 -0.00138

2

3.56506

3 -0.002271 22.259176 -0.00184

3

4.75341

8 -0.00348 33.388764 -0.00277

4

2.57476

8 -0.004641 44.518352 -0.00369

5

3.36700

4 -0.005833 55.64794 -0.00461

4

0.39611

8 -0.004715 44.518352 -0.00369

3

0.79223

6 -0.003603 33.388764 -0.00277

2

-

0.99029

5 -0.002437 22.259176 -0.00184

1.5 -2.17865 -0.001885 16.694382 -0.00138

1

-

2.37670

9 -0.001287 11.129588 -0.00092

0.5

-

3.16894

5 -0.000695 5.564794 -0.00046

0

-

4.15924

1 -6.15E-05 0 0


with Graphite/Epoxy CapsForce(lbs)

Strain()

Displacement (in)



0

0.59417

7 -0.000028 0 0

0.5

-

0.99029

5 -0.00064 3.2971546 -0.00026

1 -2.17865 -0.001321 6.5943091 -0.00052

1.5

-

2.97088

6 -0.001989 9.8914637 -0.00079

2 -4.3573 -0.002631 13.188618 -0.00105

3

-

6.33789

1 -0.003943 19.782927 -0.00157

4 -8.7146 -0.005313 26.377237 -0.0021

5 -

10.4971

-0.006634 32.971546 -0.00262

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305W Structure LabReport Template Me1