AEROSPACE 305W STRUCTURES & DYNAMICS LABORATORY

Laboratory Experiment #2

Column Buckling

April 6, 2010

Chris CameronSection 18

Lab Partners:Joseph O’Leary

Zeljko RaicMichael YoungJonathan Hudak

Gregory Palencar

Course Instructor: Dr. Stephen ConlonLab TA: Mike Thiel

Abstract

Columns are commonly used in engineering and specifically in aerospace in situations

such as the ribbing in wings. They support a load but most often their critical load is determined

by when buckling occurs. This buckling is caused either by imperfections in the column or the

loading. This experiment was designed with the objectives of confirming the theoretical

predictions for when columns buckle and how to increase their critical load. It was assumed that

the longer columns would buckle sooner and also the simply supported vs. clamped end columns

would also buckle sooner by a factor of 4. Another assumption was made that the increasing

slenderness ratio of the columns would decrease their critical stress. The experiment was set up

by loading varying lengths of beams with both simply supported and clamped fixities. A load

was applied until the central displacement, measured using a linear variable differential

transformer, began to increase without increase in load. The data supported the assumptions

with an error being at all points below 20%.

2

. I. Introduction

Columns are commonly used engineering structures that are used to carry compressive loads.

A common aerospace column is the ribbing found within the airfoils on a plane. However

instabilities cause columns to not only compress, but to buckle under loading. Buckling is a

disproportionate increase in displacement with an additional applied load. This buckling reduces

the columns ability to carry loads and must be understood in order to determine the maximum

load of a column.

The objective of this lab will be to determine if shorter or longer columns buckle under

different loads and if the method if fixing the ends also affects the buckling load. Also the

slenderness ratio effect on the critical stress will be examined. The experimental data will be

compared to theoretical data to find if the theory behind column buckling predicts the data

collected. Error between the theoretical and experimental data will give insight to improper

assumptions about boundary conditions, as well as other sources of error within the experiment.

Columns instabilities are due to both imperfections in the column as well as imperfections in

the loading. Columns imperfections can be due to imperfections in the material, as well as the

shape of the column being imperfect. The loading imperfections occur when loads are applied

that are not along the centerline of the beam, creating a moment on the end of the column.

Columns can buckle in different ways and this is mainly dependant on the method of fixing

the ends of the column. There are three common types being clamped, simply supported and

free. Each of these types of fixities corresponds to a set of boundary conditions at the end of the

beam. Free fixing allows for both displacement and rotation, simply supported will not allow

displacement, and clamped will not allow displacement or rotation. These boundary conditions

will be used to derive the governing equations for columns.

3

Uniform columns are governed by the differential equation

EId4wd x4 +P d

2wd x2 =0 (1.1)

where E is the modulus of elasticity of the column, I is the cross-sectional area, and P is the load

applied to the column. EI together is the columns stiffness. Assuming that the stiffness and load

are constant, k can be defined as

k2= PEI (1.2)

Applying (1.2), (1.1) becomes

w ' ' ' '+k2w' '=0 (1.3)

The general solution to the differential equation (1.3) becomes

w (x )=c1sin (kx )+c2 cos (kx )+c3 x+c4 (1.4)

where c1, c2, c3, and c4 are constants of integration. The four boundary conditions corresponding

to a given columns displacement and rotation at the each end create four equations. The four

equations and five unknowns create an eigenvalue problem. Solving for the eigenvalues yields a

family of loads causing the column to buckle

P=c n2π2 EIL2 (1.5)

where c is a constant relating to how the ends are fixed, n is the mode shape, and L is the length

of the column. The first mode shape (n=1) gives the critical load of the column

Pcr=cπ2 EIL2 (1.6)

In this experiment columns were loaded until the critical load, corresponding to the lowest load

where buckling occurs, was reached for both columns simply supported at both ends and

4

clamped at both ends. The eigenvalue problems yielded c values of c=1 for simply supported

and c=4 for clamped.

Six different columns were used in this experiment, each having the same cross section. This

allowed for one calculation to be made for I for all the beams. For a rectangular cross section the

moment of inertia is found using the equation

I=bh3

12(1.7)

Figure 1.1 shows the cross section of the columns with measurements for both base and height.

Figure 1.1- Cross section of columns

This orientation of the beam yields a smaller moment of inertia, which will cause the column to

buckle in the direction parallel to the short edge. The moment of inertia corresponding to this

orientation is 0.000122 in4.

By use of equation (1.6) and assuming that EI will be constant for all columns predictions

about the critical load can be made. For columns with the same end fixity the only variable

remaining is L2 in the denominator. This will cause the shortest beams to have the highest

critical loads. Also by comparing the values of c it is found that the clamped beams will have a

higher critical load.

5

Another value of interest is the critical stress, σcr, of the column. This is defined in the

equation

σ cr=PcrA

(1.8)

The slenderness ratio, s, is the value of interest for determining the critical stress and is defined

as

s=L

r √c (1.9)

The radius of gyration, r, is defined as

r=√ IA (1.10)

Applying equations (1.6), (1.9), and (1.10) to equation (1.8) yields

σ cr=π2Es2

(1.11)

6

II. Experimental Procedure

Data was acquired for this lab use of two different instruments. The first was a linear variable

differential transformer (LVDT), which was used to measure the deflection at the midpoint of the

column. The second was a load cell force gauge which was attached to a load wheel and the

horizontal beam. A converter box was used to convert the load from the load wheel to the load

being applied to the column specimen. All data was collected using Labview software, which

for averaged each acquired data point from a hundred instantaneous values. This average

corrected for vibrations. The set up of the experiment is shown in figure 2.1.

Figure 2.1- Experimental set up

Two different fixing methods were used for this experiment. The first, clamped, was created

by inserting the column into a bracket which was then tightened using a screw and hex key. The

simply supported case was created by flipping the clamping brackets so that the ends of the

column were placed into a notch in the bracket. The ends of the simply supported column were

angled into a point in order to have only one contact point in the bracket. These fixing methods

are shown in figure 2.2.

7

Figure 2.2- End fixing methods

Once each column was fixed into the apparatus the horizontal beam was lowered into place.

Balance mass was used to counteract the moment of the beam about the fixed spring end. The

beam was leveled using the level adjust spring. The load cell was then attached to the load

wheel and the horizontal beam. The LVDT was then mounted onto the side of the specimen at

the center to measure the maximum displacement. Ensuring that the load was at zero the LVDT

was also zeroed.

Once the set up was complete data was then gathered. The load wheel was turned to increase

the load on the column. Data points were collected using the Labview software and were largely

spaced for the beginning loads. Once the load began to near the expected critical load, the data

points were measured at smaller load increments. Data was collected until the column began to

deflect greatly without a change in the load. The column was then unloaded and the test was

repeated for each of the six samples.

8

III. Results and Discussion

The collected data was expected to show an asymptotic behavior as the applied load neared

the critical load. The critical load of the shorter specimens was expected to be much higher than

that of the longer specimens. Also due to boundary conditions the clamped specimens were

expected to have critical loads four times greater than their simply supported counterparts. The

theoretical values for the critical loads of the six experiments are shown in table 3.1.

Table 3.1- Theoretical critical buckling loads

Column Length (in) Clamped Pcr (lb) Simply Supported Pcr (lb)18 445.96 111.521 327.64 81.924 250.85 62.7

The horizontal asymptote of the load vs. displacement was found for each specimen and these

load values were used as the experimental critical buckling loads. The error using the asymptotic

method was low for this experiment. All values were under 15% error but also all values with

one exception were lower than the expected theoretical values, most likely caused by fatigue on

the specimens. These values are shown in table 3.2 while their error is shown in table 3.3. The

data is shown in figures 3.1 and 3.2.

Table 3.2- Experimental critical buckling loads

Column Length (in) Clamped Pcr (lb) Simply Supported Pcr (lb)18 401 10121 293 87.524 247 54.5

Table 3.3- Percent error, asymptotic vs. theoreticalLength (in) % Error Clamped % Error Simply Supported

18 10.082 9.41721 10.573 6.83824 1.535 13.078

9

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.180

50

100

150

200

250

300

350

400

450

Force vs. DisplacementClamped - Clamped

18"

21"

24"

18" Asymptote

21" Asymptote

24"AsymptoteDisplacement (in)

Forc

e (lb

)

293

401

247

Figure 3.1- Clamped-Clamped asymptotic graph

0 0.01 0.02 0.03 0.04 0.05 0.06 0.070

20

40

60

80

100

120

Force vs. DisplacementSimply Supported-Simply Supported

18"21"24"18" Asymptote21" Asymptote24" AsymptoteDisplacement (in)

Forc

e (lb

s)

54.5

78.5

101

Figure 3.2- Simply Supported asymptotic graph

10

A second method of experimentally determining the critical buckling load was used. This

method of finding the slope between the displacement and the displacement-load ratio is referred

to as the imperfection accommodation method. Values for the critical load found using this

method were compared to both the theoretical values and the values found using the asymptotic

method. The percent error between the imperfection accommodation method and the theoretical

data was very similar to the percent error between the asymptotic method and the theoretical

data. However there was a large error between the two methods for the Clamped-Clamped

column but not for the Simply Supported column. The values for the critical load and percent

error are shown in tables 3.4 – 3.6 and the data for the critical load is found in figures 3.3 and

3.4.

Table 3.4- Imperfection accommodation critical loads

Column Length (in) Clamped Pcr (lb) Simply supported Pcr (lb)18 452.2 96.821 381.4 90.224 284 56.5

Table 3.5- Percent error imperfection accommodation vs. theoretical

Length (in) Percent error Clamped Percent error Simply Supported18 1.399 13.18421 16.408 10.13424 13.215 9.889

Table 3.6- Percent error imperfection accommodation vs. asymptotic

Length (in) Percent error Clamped Percent error Simply Supported18 12.768 4.15821 30.171 3.08624 14.98 3.67

11

0 0.00005 0.0001 0.00015 0.0002 0.000250

0.01

0.02

0.03

0.04

0.05

0.06

f(x) = 283.969594782373 x − 0.0090920604652454

f(x) = NaN x + NaNf(x) = NaN x + NaN

Displacement vs. Displacement/LoadClamped-Clamped

18"Linear (18")21"Linear (21")24"Linear (24")

Displacement/Load (in/lb)

Disp

lace

men

t (in

)

Figure 3.3- Clamped imperfection accommodation graph

0 0.0002 0.0004 0.0006 0.0008 0.001 0.00120

0.01

0.02

0.03

0.04

0.05

0.06

0.07

f(x) = 56.4977135349544 x − 0.00485000237508319

f(x) = 90.1531676418197 x − 0.0163812197798266

f(x) = NaN x + NaN

Displacement vs. Displacement/LoadSimply Supported-Simply Supported

18"Linear (18")21"Linear (21")24"Linear (24")

Displacement/Load (in/lb)

Disp

lace

men

t (in

)

Figure 3.4- Simply supported imperfection accommodation graph

12

The slenderness of the beams was also measured to find its effect on critical buckling stress.

The experimental values of the buckling stress were determined from both the asymptotic and

the imperfection accommodation methods. There was a reasonably small amount of error found

when comparing the theoretical data with the experimental data and all data supported the

assumption that the higher slenderness ratios would support a lower critical stress. Values for

the data and error can be found in tables 3.7 – 3.10 and graphs of the critical stress vs. the

slenderness ratio can be found in figures 3.5 and 3.6.

Table 3.7- Simply supported critical stress

Length (in.) Slenderness ratio Theoretical stress (psi) Asymptotic stress (psi) Imperfection stress (psi)18 498.974 1189.333 1077.333 1032.53321 582.137 873.6 933.333 962.13324 665.3 668.8 581.333 602.667

Table 3.8- Clamped critical stress

Length (in.) Slenderness ratio Theoretical stress (psi) Asymptotic stress (psi) Imperfection stress (psi)18 498.974 4756.907 4277.333 4823.46721 582.137 3494.827 3125.333 4068.26724 665.3 2675.733 2634.667 3029.333

Table 3.9- Simply supported critical stress percent error

Length (in.) Percent error asymptotic Percent error Imperfection18 0.094170404 0.13183856521 0.068376068 0.10134310124 0.130781499 0.098883573

Table 3.10- Clamped critical stress percent error

Length (in.) Percent error asymptotic Percent error Imperfection18 0.100816217 0.01399228621 0.105725797 0.1640825324 0.015347817 0.132150688

13

450 500 550 600 650 7000

200

400

600

800

1000

1200

1400

Critical Stress vs. Slenderness RatioSimply Supported-Simply Supported

Theoretical AsymptoticImperfection Tech.

Slenderness Ratio

Criti

cal S

tres

s (ps

i)

Figure 3.5- Simply supported critical stress vs. slenderness ratio graph

450 500 550 600 650 7000

1000

2000

3000

4000

5000

6000

Critical Stress vs. Slenderness RatioClamped-Clamped

TheoreticalAsymptoticImperfection Tech.

Slenderness Ratio

Criti

cal S

tres

s (ps

i)

Figure 5.6- Clamped critical stress vs. slenderness ratio graph

14

IV. Conclusions

The objectives of this lab were to determine how varying lengths of beam and end fixity

affected the critical loads. The experimental data supported the theoretical calculations that

the inverse square of the length of the column directly relates to the critical load. Also the

clamped method of fixing the ends was found to produce a critical load roughly four times

greater than the simply supported method. As for the slenderness ratio’s effect on the critical

stress, the theoretical data was again supported although with slightly more error than the

critical loads for the clamped condition. Each experiment did follow the trend that increased

slenderness ratio decreased the critical stress. This shows that when designed columns to

resist buckling they should be kept as short as possible, and also they should be clamped at

the ends. One other method not tested in this experiment would be to increase the stiffness of

the column either using geometry or material properties.

While the overall amount of error was small for each trial there was a common trend that

most buckled at a lower than expected applied load. There are two sources of error that most

likely are responsible for this. One is that the columns have been used repeatedly to repeat

this experiment and can be suffering from fatigue. Newly manufactured columns being used

for each experiment could reduce this error. Another source could be the sideways force

being applied by the LVDT. The spring that holds the device against the column applied a

force that could cause buckling to occur earlier than predicted. By mounting the LVDT with

a glue or tape instead of a spring the error here could be reduced.

15