Laboratory Experiment #2 Column Buckling · 3 This resulted in the critical load equations for the...
Transcript of 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.
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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
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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
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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
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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.
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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
Figure 5. Load vs. Displacement of Simply Supported – 21 inch
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Figure 6. Load vs. Displacement of Simply Supported – 24 inch
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.
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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
Figure 8. Load vs. Displacement for Clamped – 21 inch
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Figure 9. Load vs. Displacement for Clamped – 24 inch
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
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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
Figure 11. Displacement vs. Displacement/Force – 21 inch
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Figure 12. Displacement vs. Displacement/Force – 24 inch
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
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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.
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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.
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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
Force (lbs) Displacement (in) w/P (in/lb)
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
Table 4. Simply Supported – 21 inch
16
Table 5. Simply Supported – 24 inch
Force (lbs) Displacement (in) w/P (in/lb)
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
Force (lbs) Displacement (in) w/P (in/lb)
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
Table 7. Clamped – 21 inch
Force (lbs) Displacement (in) w/P (in/lb)
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
Table 8. Clamped – 24 inch
Force (lbs) Displacement (in) w/P (in/lb)
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