laboratory experiment #2 column buckling · 3 this resulted in the critical load equations for the...

AEROSPACE 305W STRUCTURES & DYNAMICS

LABORATORY

Laboratory Experiment #2

Column Buckling

April 1, 2013

Lovedeep Bhela

Lab Partners: James Bement, Sam Dubin, Parth Patel, Karah Oliver

Course Instructor: Dr. Stephen Conlon,

Lab TA: Dwight Brillembourg

Section: 013

Abstract

The objective of this experiment was to understand how changes in length and end fixity

affect the flexural behavior of slender column specimens when subjected to compressive load.

The experiment was conducted on an apparatus that included a load cell to measure force, a

Linear Variable Differential Transformer (LVDT) to measure the buckling displacement, a

load wheel to apply compressive force, and holding blocks to create boundary conditions. The

experimental calculations indicated that as the length of the column increases, the value for

the critical load decreases. Also when compared to the simply supported, the clamped column

validated the theoretical data by showing higher values of critical load for the same length by

a factor of 4. The data collected from this experiment was analyzed using the horizontal load

asymptote method and imperfection accommodation method. It was difficult to determine

which method was more accurate and showed better correlation to the theoretical data. It was

also clear that the simply supported columns produced a lower value of percent error than the

clamped columns when the experimental values were compared against the theoretical values.

This was validated by observing the relationship between critical stress and the effective

slenderness ratio. This experiment confirms that to achieve a high buckling load, a clamped

column of smaller length is more efficient than a column that is simply supported.

2

I. Introduction

The purpose of this laboratory experiment was to evaluate how flexural instability is affected

by column length and end fixity while observing the effects of imperfections on lateral

deflections. Three columns of stainless steel with varying lengths (18 in, 21 in, and 24 in) were

put under compressive load. The behavior of buckling was examined by using different boundary

conditions on the columns such as simply supported and clamped conditions. A Linear Variable

Differential Transformer (LVDT) was used to measure the amount of deflection at the center of

the column under load.

Buckling is a disproportionate increase in displacement from a small increase in load.

However this relationship is not entirely linear. Column buckling occurs once the critical load is

reached. The distributed load in terms of the applied load and column properties can be seen in

Equations 1 and 2 below:

Young’s modulus (E=28x106 psi) and the moment of inertia (I=1.221x10

-4 in

4) are constant and

the fixed value of compressive load is represented by (P). By combining Equations 1 and 2, the

deflection of the column can be found using the following equation with constant coefficients:

The theoretical calculations were completed by using the appropriate boundary conditions for

the three different lengths of the specimens. The applied boundary conditions for each end fixity

that was used in the experiment can be seen below in Figure 1:

Figure 1. Boundary Conditions for end fixity of Simply Supported and Clamped Beam

Eq. 1, 2

Eq. 3

3

This resulted in the critical load equations for the simply supported column and clamped column.

Where (Pcr) is the critical load, (L) is the length of the column, (EI) is the flexural stiffness, and

(c) is a constant that is dependent on the end fixity of the column. The buckling load equations

for the simply supported (c=1) and clamped beam (c=4) can be seen below:

This theoretical data calculated from Equations 4 and 5 was then be used to compare to the

experimental data to see how well the specimens acted under load based on deflection. The

experimental data consisted on two methods. The first was the asymptotic method in which the

buckling load was the asymptote of the load-deflection plot. The second method was the

imperfection accommodation where the buckling load is estimated from the slope of a straight

line fit to the data of the deflection-deflection/load plot.

The experiment was performed to recreate values of the theoretical data calculated so that the

properties of the columns of varying lengths and end fixities could be observed and predicted.

The behavior of the columns under loading can be understood by comparing the experimental

data against the theoretical data using the percent error equation seen below:

The critical stress of each column specimen was also found once the buckling load was

determined for each length and both end fixity configurations. The critical stress was expressed

in terms of the slenderness ratio (L/r), where (r) is the radius of gyration and is defined below:

In Equation 7 and 8 above, the (A=0.0938 in2) is the area of the stainless steel specimens. The

constant is (c=1) for the simply supported and (c=4) for the clamped configuration. The critical

stress with Young’s modulus held constant was then defined as:

Eq. 4, 5

Eq. 6

Eq. 7, 8

Eq. 9

4

II. Experimental Procedure

In this experiment, the apparatus held a column of stainless steel at varying lengths in a

simply supported setup and clamped setup with support blocks. The apparatus and support block

configurations can be seen below in Figure 2:

Figure 2. Experimental Apparatus and Support Block Configuration

Once a certain specimen was chosen for testing, it was placed in the support blocks according to

the setup desired for simply supported or clamped conditions. The blocks could be flipped to

accommodate either condition. After the specimen was installed, the top loading frame was

leveled. The Linear Variable Differential Transformer (LVDT) was place at the center mark of

the column, making it perpendicular to the specimen. An image of the actual lab experiment and

location of instruments can be seen below in Figure 3:

Figure 3: Image of Experiment Setup and Instrument Locations

5

The Lab View program was used to collect raw data from the experiment while the column

specimens were being buckled. A load cell was used in tension which gave values in voltage for

the respective force on the specimen. A load wheel was turned to increase the force applied to

each specimen and the load cell values were displayed using a transducer. The wheel had to be

turned at a different rate for each specimen due to its length and end fixity. This was necessary to

obtain accurate data without achieving immediate failure of the specimen. The Lab View

program was used to monitor the reasonable rate to turn the wheel and gather a distributed range

of data points. The program also made it noticeable to see when the column buckled because the

force values began to decrease after reaching a peak. This could have also been determined by

observing the column while it buckled. It is also important to keep the apparatus leveled at all

times using the adjustments in order to obtain accurate data. After the specimen was tested to

show buckling, another specimen was tested in the same manner. This occurred 6 times in total

with three tests as simply supported and the other three as clamped for the specimens.

6

III. Results and Discussion

After completing the theoretical values for the 6 specimens it was hypothesized that the

column of stainless steel with the shortest length of 18 inches would have the highest Pcr value

and the 24 inch would have the lowest Pcr value. The length of the column is square inversely

related to the critical load, as the length was increased the value of the Pcr decreased while

everything else was held constant. The Pcr is higher for the clamped configuration than the

simply supported one. The experimental data supported the theoretical data, but some

discrepancies were noticeable. The relationship between the applied force and displacement for

the simply supported configuration can be seen below in Figures 4, 5, and 6:

Figure 4. Load vs. Displacement of Simply Supported – 18 inch


7


The three simply supported specimens had clear data curves that showed where the critical

load was reached. The simply supported specimen for each of the lengths either slightly

exceeded or fell short of the load values that were theoretically calculated. For example, the

simply supported 21 inch specimen had an experimental value of 77.2 lbs while the theoretical

value was 76.49 lbs. This was the specimen in the simply supported configuration that held the

closest comparison to the theoretical value because the percent error difference of this specimen

was only 0.92 from the theoretical. The results of the percent difference and both experimental

and theoretical values are shown below in Table 1 using the asymptotic method:

Table 1. Percent Differences using the Horizontal Asymptote Method

Length

(in)

Theoretical

SS Pcr (lbs)

Experimental

SS Pcr (lbs)

SS

% Error

Theoretical

CL Pcr (lbs)

Experimental

CL Pcr (lbs)

CL

% Error

18 104.1173 100.47 3.51 416.4691 261.49 37.21

21 76.4943 77.23 0.92 305.9773 234.48 23.37

24 58.5660 66.07 12.81 234.2639 173.41 25.98

The clamped boundary condition results had higher load values than the simply supported.

However, they were much lower than the theoretical data predicted. The difference gave larger

values for percent error. The values of Pcr were about 4 times larger than the simply supported

configuration, which makes sense because of the theoretical constant seen in Equation 5. The

experimental data however was closer for the simply supported because of lower percent errors.

8

All three specimens for the clamped configuration acted similarly in the beginning of each

loading as well. The overall trend looked close to what was predicted despite the low load

results. The specimens all gradually carried the loads and once reached their critical buckling

load, slowly tapered off. The relationship between the applied force and displacement for the

clamped configuration can be seen below in Figures 7, 8, and 9:

Figure 7. Load vs. Displacement for Clamped – 18 inch


9


The critical load was the highest for the 18 inch specimen and lowest for the 24 inch specimen as

predicted. This was seen in the theoretical data and validated by the experiment, even though the

values were not the same. The 21 inch specimen gave the least amount of percent difference at

23.37% when compared to the theoretical value while the 18 inch specimen gave the greatest

amount at 37.21%. The reason for the experimental results being much lower than the calculated

values could be due to errors in performing the experiment or repeated usage of the specimens.

For both specimens in both configurations, there could have been some sources of error and

could be attributed to several factors. During the experiment, the top arm of the apparatus had to

be leveled to achieve an equal distribution of the load which could have slightly caused the

column to prematurely buckle then suddenly go back to a non-buckled state. The rate at which

the load wheel was turned could have affected the data as well because of the different rate

required for each specimen. Additionally, the specimens could have slight deformations from

being tested before in previous experiments.

Each of the experimental values used to compare the critical load were obtained by looking at

the figures and was chosen at where the curve flat lined. This gave some discrepancies in the

error percentages and was not the only method to obtain the experimental Pcr. An alternative

method used to find the critical load would be through the effects of imperfections. Using the

experimental values obtained and plotting the displacement versus the displacement over the

force, the slope was determined to be equivalent to the Pcr. This way the value obtained is

10

directly from the data and not simply by observing the curve to see where the specimens showed

signs of buckling. The slope for both configurations of all three column specimens was

determined and can be seen below in Figures 10, 11, and 12:

Figure 10. Displacement vs. Displacement/Force – 18 inch


11


The values from the this method were much closer to the theoretical values. The imperfections

method is more accurate since it gives an analytic value where the other method is more of an

observation. This experimental data using the second method was compared to the theoretical

values and percent errors were determined which can be seen below in Table 2:

Table 2. Percent Differences using the Imperfection Accomodation Method

Length

(in)

Theoretical

SS Pcr (lbs)

Experimental

SS Pcr (lbs)

SS

% Error

Theoretical

CL Pcr (lbs)

Experimental

CL Pcr (lbs)

CL

% Error

18 104.1173 109.89 5.54 416.4691 353.11 15.21

21 76.4943 74.85 2.14 305.9773 252.75 17.40

24 58.5660 78.512 34.06 234.2639 205.61 12.23

As seen in Table 2, the percent difference was much greater for the 18 and 21 inch clamped

specimens than the simply supported. However, the percent error for the 24 inch simply-

supported specimen was much greater. The asymptotic method showed better results for the

simply supported configuration but the slope method is noticeably better in comparison to the

theoretical for the clamped configuration. There was a slight increase in percent difference for

the simply supported specimens, which was unusual. The clamped specimens decreased in

percent error from the previous method and closer correlation to the theoretical data. The percent

12

differences vary depending on the methods used to determine the critical load (Pcr). Tables with

each length specimen’s raw data are located in the Appendix.

Both the experimental and theoretical critical stress was plotted against the effective

slenderness ratio for all the specimens to better understand the effects of buckling load. Once the

experimental critical load data was obtained, Equations 8 and 9 were utilized in order to create

the following theoretical and experimental plot seen in Figure 13 below:

Figure 13. Stress vs. Slenderness for All Specimens

As seen above in Figure 13, both methods for the simply supported configuration came close to

the theoretical values. For the clamped configuration, the imperfection accommodation method

showed closer values to the theoretical values than the asymptote method. The imperfection

accommodation method was expected to show better results because it is not just an observation

of the peak critical load. However it is difficult to say which method would be the more accurate

one by observing the data sets acquired from this experiment. This relationship helps understand

the behavior of the column specimens under buckling load with both end fixities.

13

IV. Conclusions

The experiment required a thorough examination of variable that affect the critical load values

so that the buckling behavior could be predicted. These variables include the lengths and the end

fixity. Column specimens of three different lengths were loaded in simply supported and

clamped configurations until they buckled. The critical buckling load was then determined using

the asymptote and imperfection accommodation methods. The effects of imperfections on lateral

deflection due to the perpendicular compressive force were analyzed and the correlation between

the experimental values against the theoretical values was determined.

Throughout the experiment, it became evident that as the length of the column increases the

critical load decreases for both boundary conditions. This relationship can be explained due to

the governing equations in order to determine the critical load in which the length is inversely

proportional to the critical load. The simply supported and clamped configuration differed in

load values for both theoretical and experimental by a factor of 4. The simply supported

configuration, when compared to the clamped, provided less percent error when using the

asymptote method to determine the critical load. However the clamped configuration for the

imperfection accommodation method decreased in percent error when compared to the

asymptote method. Using the imperfection accommodation method provided better

approximation of the critical load for the clamped condition while the asymptote method gave

better results for the simply supported condition. Possible reasons for such errors could be how

the specimens were positioned in the holding blocks of the apparatus and whether or not they

were able to move during the experiment. Another possible source of error could have been from

the pre-bending of the columns due to cyclical loading from previous tests. The critical stress

versus the effective slenderness ratio showed that the simply supported configuration had the

closest correlation to the theoretical data. However, it was not evident enough to say which

method of determining the critical load was best because of variation and discrepancies in data.

It is recommended that for future experiments, the method of applying compressive load to

the columns be changed or improved. For example a computerized force application device

could greatly provide accurate results and less human intervention in the experiment. The LVDT

could also be replaced with a laser displacement sensor as well. Also, it would be better to install

two separate holding blocks so that another column can be experimented on simultaneously to

reduce errors and acquire additional scope on buckling behavior.

14

Appendix

Table 3. Simply Supported – 18 inch

Force (lbs) Displacement (in) w/P (in/lb)

1.420136 0.000413 0.00029082

1.630616 0.000429 0.00026309

1.688679 0.000421 0.00024931

1.683841 0.000438 0.00026012

1.766097 0.000434 0.00024574

2.259637 0.000448 0.00019826

3.887834 0.000458 0.0001178

5.920055 0.000604 0.00010203

6.060375 0.00192 0.00031681

8.525654 0.002704 0.00031716

10.43691 0.003718 0.00035624

12.144944 0.004959 0.00040832

16.906149 0.007517 0.00044463

20.871401 0.009231 0.00044228

25.064067 0.011726 0.00046784

30.846222 0.017617 0.00057112

34.521156 0.020724 0.00060033

39.717837 0.024473 0.00061617

43.346805 0.03337 0.00076984

47.778983 0.038964 0.00081551

51.674075 0.049406 0.00095611

57.13688 0.065584 0.00114784

62.067437 0.077614 0.00125048

65.142382 0.089763 0.00137795

68.282648 0.098661 0.00144489

71.347916 0.105601 0.00148009

75.707516 0.118433 0.00156435

79.517931 0.127871 0.00160808

83.749307 0.141509 0.00168967

87.494401 0.150645 0.00172177

89.50001 0.158181 0.00176739

91.290301 0.166305 0.00182172

93.443488 0.181368 0.00194094

95.79264 0.197241 0.00205904

98.129695 0.211162 0.00215187

99.544992 0.218864 0.00219864

99.994984 0.228796 0.00228807

100.471588 0.233723 0.00232626

100.18369 0.245129 0.0024468

99.980468 0.257603 0.00257653

99.774827 0.275779 0.00276401

99.419188 0.282436 0.00284086

98.957099 0.295239 0.00298351

98.969196 0.311264 0.00314506

98.843392 0.331007 0.0033488

98.630492 0.341167 0.00345904

98.245822 0.351816 0.00358098

98.185339 0.361412 0.00368092

15


0.094353 1.14258E-05 0.0001211

1.040304 1.31836E-05 1.267E-05

1.456426 2.63672E-06 1.81E-06

1.708034 8.78906E-07 5.146E-07

2.158026 1.75781E-06 8.145E-07

2.808821 1.31836E-05 4.694E-06

3.103977 1.75781E-06 5.663E-07

3.241877 3.51563E-06 1.084E-06

3.51284 1.31836E-05 3.753E-06

3.595097 7.03125E-06 1.956E-06

3.747514 6.15234E-06 1.642E-06

3.716063 1.14258E-05 3.075E-06

3.868479 1.23047E-05 3.181E-06

4.04025 8.78906E-07 2.175E-07

4.124926 4.39453E-06 1.065E-06

4.105572 5.27344E-06 1.284E-06

4.260408 2.63672E-06 6.189E-07

4.538628 2.63672E-06 5.81E-07

4.952331 4.39453E-06 8.874E-07

5.150714 8.78906E-07 1.706E-07

5.225713 1.49414E-05 2.859E-06

5.23539 3.07617E-05 5.876E-06

5.274099 2.28516E-05 4.333E-06

5.20152 2.02148E-05 3.886E-06

5.090231 1.66992E-05 3.281E-06

4.966847 2.10938E-05 4.247E-06

4.949911 2.02148E-05 4.084E-06

4.88459 3.07617E-05 6.298E-06

4.686206 2.54883E-05 5.439E-06

4.616046 2.90039E-05 6.283E-06

4.81443 0.000028125 5.842E-06

5.966022 4.39453E-05 7.366E-06

6.670042 4.30664E-05 6.457E-06

7.357126 6.15234E-05 8.362E-06

7.911149 0.001635 0.0002067

8.545008 0.001648 0.0001929

9.130482 0.001651 0.0001808

9.822405 0.002254 0.0002295

10.344976 0.002609 0.0002522

11.194154 0.002881 0.0002574

11.854626 0.003296 0.000278

12.636064 0.003659 0.0002896

13.240892 0.003784 0.0002858

14.036845 0.003938 0.0002805

14.469902 0.004344 0.0003002

14.786832 0.004803 0.0003248

15.389241 0.005623 0.0003654

15.933586 0.006659 0.0004179

17.07792 0.007143 0.0004183

18.560958 0.007906 0.0004259

19.92303 0.008828 0.0004431

21.289941 0.009745 0.0004577

22.373793 0.010687 0.0004777

23.99957 0.0124 0.0005167

25.782603 0.013801 0.0005353

27.33822 0.014205 0.0005196

29.537374 0.015713 0.000532

31.429276 0.016793 0.0005343

33.444562 0.018054 0.0005398

35.544525 0.019262 0.0005419

37.627552 0.020025 0.0005322

39.32349 0.020799 0.0005289

40.992814 0.021609 0.0005271

42.364564 0.022398 0.0005287

43.775023 0.023639 0.00054

44.931454 0.024604 0.0005476

46.854806 0.025744 0.0005494

48.253168 0.026529 0.0005498

49.562016 0.02805 0.000566

50.999087 0.029847 0.0005852

52.228097 0.032019 0.0006131

53.316787 0.03265 0.0006124

55.000628 0.034846 0.0006336

56.40141 0.037392 0.000663

57.857835 0.039321 0.0006796

59.251359 0.041411 0.0006989

60.436821 0.043635 0.000722

61.489222 0.046343 0.0007537

62.771457 0.049891 0.0007948

63.896437 0.0516 0.0008076

64.936741 0.055832 0.0008598

65.916562 0.060451 0.0009171

67.206055 0.063584 0.0009461

68.464097 0.072257 0.0010554

69.526175 0.077062 0.0011084

70.660832 0.082919 0.0011735

71.524526 0.083085 0.0011616

72.390639 0.083785 0.0011574

72.707569 0.089345 0.0012288

72.73902 0.095753 0.0013164

72.833373 0.101717 0.0013966

73.193851 0.104793 0.0014317

73.47691 0.107898 0.0014685

73.943837 0.111496 0.0015078

74.335766 0.118062 0.0015882

75.180106 0.127283 0.001693

76.0438 0.136569 0.0017959

77.004266 0.149524 0.0019418

77.231682 0.160099 0.002073


16



16.122292 0.006691 0.000415

18.294834 0.007648 0.000418

20.261734 0.009264 0.0004572

22.211699 0.010643 0.0004792

24.38666 0.012532 0.0005139

26.414043 0.013954 0.0005283

28.809162 0.015426 0.0005355

31.226054 0.01805 0.000578

33.03328 0.020593 0.0006234

35.03405 0.023723 0.0006771

37.24772 0.028977 0.000778

39.091236 0.034247 0.0008761

40.840398 0.037005 0.0009061

43.087938 0.04114 0.0009548

45.187901 0.045127 0.0009987

47.418506 0.049672 0.0010475

49.866849 0.053299 0.0010688

52.097454 0.056754 0.0010894

53.841778 0.062011 0.0011517

55.83771 0.073312 0.0013129

57.451391 0.074996 0.0013054

58.593306 0.083278 0.0014213

59.858606 0.089346 0.0014926

61.399707 0.092525 0.0015069

62.708555 0.098227 0.0015664

63.84805 0.110504 0.0017307

64.617391 0.12643 0.0019566

65.841563 0.145159 0.0022047

66.071398 0.158074 0.0023925

17

Table 6. Clamped – 18 inch


92.098351 0.026427 0.00028694

100.585296 0.028021 0.00027858

104.29652 0.02829 0.00027125

107.436786 0.028341 0.00026379

108.172257 0.028433 0.00026285

108.26661 0.028591 0.00026408

108.365802 0.028769 0.00026548

109.490782 0.031101 0.00028405

111.029464 0.032254 0.0002905

113.879413 0.0335 0.00029417

120.653485 0.034708 0.00028767

123.256664 0.035622 0.00028901

125.373562 0.037247 0.00029709

128.850112 0.039079 0.00030329

132.602465 0.039457 0.00029756

135.084678 0.040696 0.00030126

137.624955 0.041106 0.00029868

139.536211 0.042512 0.00030467

142.824056 0.044336 0.00031042

145.453847 0.045218 0.00031088

148.129606 0.046854 0.0003163

150.952943 0.048759 0.00032301

153.974663 0.050767 0.00032971

157.712499 0.052603 0.00033354

160.87212 0.054108 0.00033634

164.283349 0.055481 0.00033772

166.80911 0.056835 0.00034072

169.090521 0.057337 0.00033909

170.750169 0.059138 0.00034634

173.111417 0.060813 0.00035129

175.504116 0.062372 0.00035539

177.396017 0.063599 0.00035851

179.783878 0.065008 0.00036159

181.320141 0.066428 0.00036636

183.132205 0.067636 0.00036933

184.932173 0.069014 0.00037319

186.831332 0.070695 0.00037839

188.648235 0.071913 0.0003812

190.290948 0.073595 0.00038675

192.957029 0.075893 0.00039332

195.3086 0.079734 0.00040825

198.405319 0.082518 0.00041591

202.118962 0.085567 0.00042335

205.447935 0.089067 0.00043353

209.343026 0.090445 0.00043204

211.372828 0.094025 0.00044483

213.378438 0.096502 0.00045226

215.550979 0.096974 0.00044989

217.198531 0.101418 0.00046694

217.890454 0.103906 0.00047687

219.037207 0.106649 0.0004869

220.943625 0.11005 0.00049809

223.152456 0.113542 0.00050881

225.900794 0.115909 0.0005131

229.742661 0.117592 0.00051184

231.825688 0.119676 0.00051623

232.648254 0.125057 0.00053754

234.956277 0.127483 0.00054258

237.167528 0.130108 0.00054859

239.018301 0.135106 0.00056525

240.559403 0.138953 0.00057762

243.02952 0.142304 0.00058554

244.844003 0.145913 0.00059594

246.757679 0.149891 0.00060744

248.15846 0.15235 0.00061392

249.346342 0.15633 0.00062696

250.357614 0.15935 0.00063649

251.726945 0.163743 0.00065048

253.222079 0.166961 0.00065935

254.62528 0.171636 0.00067407

256.072028 0.176376 0.00068877

258.38489 0.181188 0.00070123

259.567933 0.183716 0.00070778

260.830814 0.186548 0.00071521

261.428384 0.186337 0.00071276

261.745314 0.186213 0.00071143

261.987245 0.185159 0.00070675

261.481609 0.192637 0.00073671

261.491286 0.20219 0.00077322

18



0.142739 0.000018 0.0001261

1.874966 3.51563E-06 1.875E-06

2.651565 1.49414E-05 5.6349E-06

4.947492 1.49414E-05 3.02E-06

5.806348 1.66992E-05 2.876E-06

6.234566 3.33067E-18 5.3423E-19

6.147471 8.78906E-07 1.4297E-07

6.292629 8.78906E-07 1.3967E-07

6.478916 2.46094E-05 3.7984E-06

6.827297 2.37305E-05 3.4758E-06

7.775667 0.000666 8.5652E-05

9.454669 0.000704 7.4461E-05

10.712711 0.001055 9.8481E-05

12.048171 0.001361 0.00011296

12.989284 0.001513 0.00011648

12.93122 0.00182 0.00014074

12.817512 0.00211 0.00016462

13.112668 0.002414 0.0001841

13.840881 0.002789 0.0002015

14.561836 0.003213 0.00022065

15.88278 0.003237 0.00020381

17.726296 0.003661 0.00020653

18.848856 0.004087 0.00021683

20.583502 0.004592 0.00022309

36.57999 0.009431 0.00025782

47.621728 0.010108 0.00021226

57.41752 0.010586 0.00018437

65.633502 0.011155 0.00016996

68.490709 0.011185 0.00016331

70.462448 0.011714 0.00016624

73.360783 0.012146 0.00016557

76.442986 0.013362 0.0001748

79.334063 0.014024 0.00017677

82.1574 0.014738 0.00017939

85.191217 0.015099 0.00017724

87.88391 0.01692 0.00019253

93.549938 0.018485 0.0001976

103.461857 0.020612 0.00019922

110.013353 0.021679 0.00019706

115.705993 0.02428 0.00020984

123.041345 0.026818 0.00021796

129.038819 0.028244 0.00021888

134.310499 0.029209 0.00021747

136.865292 0.030052 0.00021957

139.22654 0.032105 0.0002306

143.172437 0.033587 0.00023459

146.39496 0.035641 0.00024346

149.692481 0.037912 0.00025327

153.142419 0.039349 0.00025694

156.616551 0.041099 0.00026242

158.958444 0.043306 0.00027244

162.209999 0.044089 0.0002718

164.169641 0.045374 0.00027638

164.493829 0.049279 0.00029958

165.82687 0.053059 0.00031997

168.352631 0.060333 0.00035837

172.059016 0.066197 0.00038473

180.250805 0.073463 0.00040756

190.119177 0.079924 0.00042039

195.76585 0.090323 0.00046138

200.664956 0.10128 0.00050472

205.868895 0.108456 0.00052682

210.424458 0.117333 0.0005576

212.594581 0.121467 0.00057136

215.616301 0.131425 0.00060953

217.225143 0.138292 0.00063663

219.061401 0.148162 0.00067635

221.325876 0.154883 0.0006998

222.862139 0.163894 0.00073541

224.727428 0.170556 0.00075895

225.501608 0.176982 0.00078484

226.353205 0.182968 0.00080833

226.500783 0.190787 0.00084232

226.93384 0.201884 0.00088962

227.504798 0.208542 0.00091665

228.315267 0.222799 0.00097584

230.446681 0.238387 0.00103446

231.368438 0.24719 0.00106838

231.264408 0.251278 0.00108654

231.803914 0.262258 0.00113138

231.999879 0.275034 0.00118549

232.24181 0.284665 0.00122573

232.90954 0.291878 0.00125318

233.826459 0.298084 0.00127481

234.203871 0.302275 0.00129065

234.375642 0.305249 0.00130239

234.477254 0.310684 0.00132501

234.235322 0.322921 0.00137862

234.402255 0.330927 0.00141179

19



52.32487 0.032759 0.00062607

79.389708 0.050432 0.00063525

96.784557 0.064767 0.00066919

107.744039 0.078393 0.00072759

117.626926 0.102704 0.00087313

128.847693 0.113999 0.00088476

135.411285 0.125835 0.00092928

139.105574 0.131738 0.00094704

142.330516 0.143104 0.00100543

144.691764 0.152016 0.00105062

147.41349 0.163393 0.0011084

150.077152 0.170036 0.00113299

152.056149 0.181391 0.00119292

154.192401 0.186669 0.00121062

155.329477 0.186692 0.00120191

155.520603 0.193194 0.00124224

155.789146 0.197841 0.00126993

156.423006 0.203709 0.0013023

157.419762 0.2102 0.00133528

158.873768 0.222989 0.00140356

160.284227 0.234447 0.0014627

162.415641 0.24446 0.00150515

163.978516 0.251267 0.00153232

164.689793 0.25836 0.00156877

164.917209 0.265111 0.00160754

165.15672 0.271691 0.00164505

165.2293 0.274924 0.00166389

165.797838 0.276818 0.00166961

166.947011 0.286247 0.0017146

167.210716 0.292192 0.00174745

167.157491 0.300237 0.00179613

167.626838 0.306541 0.00182871

168.161505 0.311428 0.00185196

168.727624 0.315879 0.00187212

169.627608 0.320545 0.0018897

170.14776 0.329866 0.0019387

170.730814 0.344108 0.0020155

171.524348 0.354444 0.00206644

171.938051 0.363515 0.00211422

172.22111 0.372037 0.00216023

172.625135 0.379812 0.00220021

172.770294 0.386057 0.00223451

173.063031 0.394055 0.00227694

173.408992 0.407364 0.00234915

laboratory experiment #2 column buckling · 3 this resulted in the critical load equations for the...

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