Structures Lab Report

AEROSPACE 305W STRUCTURES & DYNAMICS LABORATORY

Laboratory Experiment #2

Column Buckling

February 8, 2012

Lab Section 5Room 47

Hammond Building

Eric Kachel

Lab Partners:Marcia CastilloToby FabisiakHsiaoting KoKim Novarina

Course Instructor: Dr. Stephen ConlonLab TA: Kaushik Basu

Abstract

This experiment was carried out in order to observe the effects of geometry and fixture

type on the critical buckling load of a given member. In this experiment three different length

members, with two different types of fixture each, were loaded until buckling occurred. The data

collected from these six tests was then analyzed using two different methods in order to find the

experimental critical buckling load. These results were then compared to calculated theoretical

values. Using this information a final plot was produced which serves as a guideline for structural

designers. This plot shows an inversely proportional relationship between the slenderness of a

beam and its critical buckling stress. This is was an expected outcome of the experiment.

2

I. Introduction

In the aerospace industry more than any other developing the most efficient structures as

possible is very important. One way aerospace structures fail is due to compressive axial loadings

that result in buckling. For this experiment relationships between specimen length and support type

were examined.

In this experiment six specimens were loaded until buckling occurred. They were of lengths 18,

21, and 24 inches and simply supported or clamped on both ends. All of the test specimens had the

same area and were made of stainless steel. This is important because the moments of inertia and

Young’s modulus of the material are important in calculating the theoretical values for buckling.

Before the experiment was carried out theoretical values for the critical buckling load were

calculated. These calculations help to determine were buckling was expected to occur so the

experiment’s operators could more accurately locate the point of buckling. The derivation of the

theoretical equation can be found in the appendix. The theoretical values are represented along with

the experimental data. Below in Figure 1 are the boundary conditions for both the simply supported

and clamped configurations.

3

Figure 1: Boundary Conditions

II. Experimental ProcedureFor this experiment six specimen were tested. They were of lengths 18, 21, and 24 inches, and

either simply supported or mounted in the clamped-clamped configuration. Each specimen was

loaded into the test apparatus. The test apparatus consisted primarily of a load wheel, force sensor,

linear variable differential transformer (LVDT), and support blocks. Other components of the

apparatus such as the balance mass, level adjust, and level were used to ensure that the desired load

was being applied to the specimen. I diagram of the test set-up can be seen in Figure 2.

4

Figure 2: Test Apparatus Diagram

In Figure 2 the support block is also shown in more detail. The notch section was used to load

the simply supported specimens, and the adjustable screw section was used for the clamped

configuration.

For this experiment both 24 inch members were tested, then the apparatus was adjusted to

accommodate the 21 inch members. After they were tested the loading bar was again lowered and

leveled in order to accommodate the 18 inch specimen.

When loading the specimen the load wheel was turned to apply the force. Once the load neared

the theoretical value for buckling the sample rate was increased in order to ensure the point of

buckling was located as accurately as possible.

When setting the LVDT two factors had to be taken into account. The first was to make sure that

the sensor was level and measuring the maximum deflection at the center of the specimen. Secondly,

when the LVDT was set-up it was compressed about half of its effective measuring length against the

specimen. This is important because it is impossible to know which way the column will buckle so

the LVDT must be ready to extend or

contract depending on the deflection direction.

The LVDT coefficient was -0.31 for this

experiment.

Figure 3 below shows an annotated

image of the test apparatus.

5

Leveling Tools

Figure 3: Labeled Test ApparatusIII.

Results and Discussion

The first samples tested for this experiment were the 24 inch samples. The data from both the

simply supported and clamped tests is represented in the Appendix in Figure 6. The data shows how

as the load is applied to the sample it begins to deflect. Eventually the points on the graph reach an

asymptote. This asymptote is located at the critical buckling force. In Figure 6 these asymptotes are

marked by dashed lines in the color corresponding to the series. The grey dashed lines represent

where the theoretical value for the critical lies. It can be seen that in both of these cases the

experimental values for critical buckling load are above the theoretical values, by a similar margin.

For the simply supported column the theoretical value is ~60 lb. Experimentally, it was found using

the asymptotic approach that the critical load was ~80 lb. Similarly the theoretical value for the

clamped configuration was ~230 lb, and the experiment value was measured to be ~250 lb.

In Figure 6 the simply supported data yields negative values for displacement while the clamped

orientation’s values are positive. This is due to the nature of buckling, and is the reason the LVDT

had to be set-up in order to take readings in either direction. For the collected data a positive value

6

Sample

LVDT Sensor

Load Wheel

Force Sensor

Support Block

Support Block

Counter Weight

means that the member buckled away from the LVDT, and a positive value means that it buckled

towards the sensor. Six tests were performed during this experiment. Three buckled toward the

sensor and three buckled away.

Along with the asymptotic approach, another method was used in order to calculate the critical

buckling load. This method, known as the Imperfection Accommodation Technique, works by

plotting the displacement of the column against the displacement divided by the load. A linear best fit

line is then calculated and the slope of this best fit line determines the critical load. Figure 9 and

Figure 10 show this technique for the 24 inch long member. Using this method the critical load for

the simply supported column was ~75 lb, and the clamped column was ~275 lb. Using this method

gave a better approximation for the simply supported value then the asymptotic method. The

imperfection accommodation technique did yield high percent error for the clamped member.

The next columns tested were the 21 inch members. Both types of analysis were also performed

for these members. Figure 7 shows the asymptotic method for finding the critical buckling load. For

the simply supported member the theoretical load was ~75 lb. In the experiment it was found that the

critical load was ~100 lb. For the clamped member the theoretical value was ~305 lb, and the

experimental asymptote was located at ~395 lb. While the error for the simply supported case was

similar to that found in the 24 inch case, the clamped member had much higher error varying by ~90

lb.

The critical loads were found again using the imperfection accommodation technique. Figure 11

and Figure 12 show the data for this method. From this it was found, for the simply supported beam,

that the critical load was ~106 lb. This value is similar to the value found asymptotically (~106 lb).

For the clamped-clamped configuration the critical load was found to be ~369 lb, this value was

closer to the theoretical value, but still varied by ~20%.

7

The last set of columns tested were the 18 inch members. Figure 8 shows the asymptotic analysis.

For the simply supported column the theoretical value was ~104 lb, the experimental value found

using the asymptote was ~330 lb. This is a very large difference. For this test the member did not

gradually start to deflect like in the other 5 tests. For this test the column showed no apparent signs of

buckling and then buckled catastrophically. This is why the displacement instantly jumps and the

force drastically falls. The force drops to ~140 lb which is close to the average percent error in earlier

trials. For the clamped configuration the theoretical value was ~416 lb, and the experimental value

was ~400 lb.

Using the imperfection accommodation technique the critical loads were calculated again. For the

simply supported member the load was calculated to be ~139 lb using Figure 13. This value should

be viewed cautiously due to the strange nature of this columns behavior. For the clamped column the

experimental value was found to be ~422 lb using Figure 14. This value varied from the theoretical

value by ~1.3%. Table 1 in the Appendix displays the theoretical and experimental values, along

with the percent error.

Overall the experimental values mimic what was expected to see. For the simply supported

columns the governing equations stated that the longer the member the lower the critical buckling

load would be. This is exactly what happened, as can be seen in Figure 4.

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Figure 4: Simply Supported Tests Plotted Together

It can be also see how the clamped members behaved as expected in Figure 5 on the next page.

That is, the shorter members had higher critical buckling loads. Also it can be expected that the

clamped members would have higher critical loads than the simply supported members of the same

length. By comparing the values in Figures 16 it can be seen that this is also true.

9

-2 -1.5 -1 -0.5 0 0.5 1 1.5-50

050

100150200250300350

Simply Supported Samples

24SS21SS18SS

Load (lb)

Disp

lace

men

t (in

)

Figure 5: Clamped Tests Plotted Together

Using both the theoretical and experimental data a plot can be generated to show the effects of

slenderness on a columns resistance to buckling. The slenderness ratio takes area, moment of inertia,

length, and effective length into account and assigns one number to describe the geometry of the

column. When slenderness ratio is plotted against critical stress Figure 15 is produced. This plot is

important because it gives a direct relationship between the slenderness of a column (its geometry)

and the amount of stress it can withstand before buckling. This kind of plot can be very useful during

the design process. For instance if the designer knows that he needs to use a column to withstand

~3000 psi he can use Figure 15’s ideal case to see that his column’s slenderness needs to be ~300.

10

-0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5-50

050

100150200250300350400450

Clamped Samples

24CC21CC18CC

Load (lb)

Disp

lace

men

t (in

)

IV. Conclusions

This experiment succeeded in its purpose. It helped to examine the relationships between different

lengths of members, and between different ways of fixing the members. Overall the results were

what were to be expected from the governing equation and theoretical values. It was seen that when

the length of a member decreased there was an increase in critical buckling load. It was also observed

how the different methods of fixture changed the boundary conditions, and therefore the effective

length of the specimen and affected its strength. Although the overall trends were expected there was

some variation from theoretical values. On average the experimental vales were ~30 lb greater than

the theoretical values. This is counter intuitive as the theoretical data represents the ideal case so one

would expect that imperfections in the material would cause the experimental critical loads to be less

than the theoretical values.

The fact that the experimental values were consistently higher than the theoretical values can be

explained by several sources of error. One of the main sources of error came in leveling the

apparatus. Both the loading bar and LVDT had to be perfectly level in order to ensure accurate force

transfer and readings. Even when the loading bar was level to begin the test as the beam began to

deflect the upper support started to tip due to the bar’s hinged design. This condition was not

avoidable with the given test apparatus. This could have caused uneven loading on the specimen an

effected the results. A more sound design would be one that would ensure that the specimen

experienced pure even axial loading by compressing straight down. Another source of error was the

way deflection was measured. In order to measure deflection the LVDT was compressed up against

the member. Although it was not large, this caused a force in the member which may have affected

the force or direction in which the column buckled. Another source of error may be that these

members had been loaded other times in previous laboratory experiments, which may have affected

the results as well.

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In order to improve this experiment one should start with the test apparatus. A redesign that

applies the force purely axially would be the best option. An alteration in the method used to measure

the deflection of the beam would be useful to be sure it wouldn’t affect the results of the experiment.

Also, as with any experiment multiple trials would be beneficial. This is especially true for the 18

inch simply supported specimen which behaved much different than the rest of the tests.

The main result of this experiment is the direct relationship between a columns slenderness ratio

and critical buckling stress. Additional research in this area could include testing different boundary

conditions. An example would be simply supported on one side and clamped on the other.

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Figure 6: Asymptotic Analysis for 24” Members


Appendix:

13

-1.2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2-50

050

100150200250300350400450

Load vs Displacement - 21"

21SS21CC

Displacement (in)

Load

(lb)

-0.5 0 0.5 1 1.5 2 2.5 30

50

100

150

200

250

300


24SS24CC

Displacement (in)

Load

(lb)


Figure 9: Imperfection Accommodation Technique for 24” – Simply Supported Member

14

-0.035 -0.03 -0.025 -0.02 -0.015 -0.01 -0.005 0 0.005-2.5

-2-1.5

-1-0.5

00.5

f(x) = 74.8506836597154 x + 0.00428518479428885

24-SS Imperfection Method

Displacement/Load (in/lb)

Disp

lace

men

t

-0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5-50

050

100150200250300350400450


18SS18CC

Displacement (in)

Load

(lb)

Figure 10: Imperfection Accommodation Technique for 24” – Clamped Member



15

-0.0035 -0.003 -0.0025 -0.002 -0.0015 -0.001 -0.0005 0 0.0005 0.001-1.2

-1-0.8-0.6-0.4-0.2

00.2f(x) = 368.688903135623 x + 0.00458746540172611

21-CC Imperfection Method


Disp

lace

men

t (in

)

-0.002 0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.0160

0.20.40.60.8

11.21.4

f(x) = 106.27661159294 x − 0.104966660271851



Disp

lace

men

t (in

)

0 0.002 0.004 0.006 0.008 0.01 0.0120

0.51

1.52

2.53

f(x) = 273.361519890784 x − 0.316249247607238



Disp

lace

men

t



Table 1: Theoretical and Experimental Data Comparison

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0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01 0.011 0.0120

0.51

1.52

2.53

3.54

4.5

f(x) = 421.79879751828 x − 0.38411091037346



Disp

lace

men

t

-0.012 -0.01 -0.008 -0.006 -0.004 -0.002 0 0.002-1.6-1.4-1.2

-1-0.8-0.6-0.4-0.2

00.2

f(x) = 138.900258975948 x + 0.00277523682672634



Disp

lace

men

t (in

)

Theoretical (lb) Experimental (lb) Percent Error (%)24" Simply Supported 58.57 74.85 27.795799924" Clamped 234.26 273.36 16.6908563121" Simply Supported 76.49 106.26 38.9201202821" Clamped 305.98 368.69 20.4948035818" Simply Supported 104.12 138.9 33.4037648918" Clamped 416.47 421.8 1.279804068

Figure 15: Column Design Plot

Figure 16: All Test Results Plotted Together

Derivation of fundamental buckling equation: Simply supported:

EId4 wd x4 +P

d2 wd x2 =0

K2= PEI

w ' ' ' '+K 2w ' '=0

⋮

17

200 250 300 350 400 450 500 550 600 650 7000

500100015002000250030003500400045005000

Critical Stress vs Slenderness RatioTheoretical

Slenderness Ratio

Criti

cal S

tres

s (ps

i)

-3 -2 -1 0 1 2 3 4 5-50

0

50

100

150

200

250

300

350

400

450

All Samples

24CC21CC18CC24SS21SS18SS

Load (lb)

Disp

lace

men

t (in

)

w (x )=C1 sin ( K x )+C2cos (K x )+C3 x+C4

Boundry Conditions :o w (0 )=0o M (0 )=EI w' '=0o w (L )=0o M (L )=EI w ' '=0

w (0 )=00+C2+0+C4=0

C2=−C4

w ' ' (0 )=0w '=C1 Kcos ( Kx )−C2 Ksin ( Kx )+C3

w ' '=−C1 K2sin ( Kx )−C2 K2 cos ( Kx )

0−C2 K2=0

C2=0

C4=0

w (L )=0C1sin ( KL )+C2 cos ( KL )+C3 L+C4=0

C1sin ( KL )+C3 L=0

C3=−C1 sin ( KL)

L w ' ' (L )=0

−C1 K2 sin ( KL)−C2 K2cos ( KL)=0

−C1 K2 sin ( KL)=0

sin ( KL )=0

KL=n π

↓

K=n πL

n=1(smallest node)

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K= πL

K2= π2

L2

PEI

=π 2

L2

Pcr=π 2 EI

L2

Solvethe same way using clamped−clamped B .Cs:

Boundry Conditions :o w (0 )=0o S (0 )=EI w'=0o w (L )=0o S ( L )=EI w'=0

Pcr=4 π2 EI

L2

Where :o Pcr=Critical Buckling Loado E=Youn g' s Moduluso I=Smallest Moment of Inertiao L=Length of Member

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