Please copy and paste this embed script to where you want to embed

AEROSPACE 305W STRUCTURES & DYNAMICS LABORATORY

Laboratory Experiment #2

Column Buckling

April 6, 2010

Chris Cameron Section 18

Lab Partners: Joseph O’Leary Zeljko Raic Michael Young Jonathan Hudak Gregory Palencar Course Instructor: Dr. Stephen Conlon Lab 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 (1.1)

EId4wdx4+Pd2wdx2=0

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 (1.2)

k2=PEI

Applying (1.2), (1.1) becomes (1.3)

w''''+k2w''=0

The general solution to the differential equation (1.3) becomes wx=c1sinkx+c2coskx+c3x+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=cn2π2EIL2

(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=cπ2EIL2

(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 clamped at both ends. The eigenvalue problems yielded c values of c=1 for simply supported and c=4 for clamped.

4

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=bh312

(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. Another value of interest is the critical stress, σcr, of the column. This is defined in the equation σcr=PcrA

5

(1.8)

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

(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 σcr=π2Es2

II.

(1.11)

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

6

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.

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 8

Column Length (in) 18 21 24

Clamped Pcr (lb) 445.96 327.64 250.85

Simply Supported Pcr (lb) 111.5 81.9 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) 18 21 24

Clamped Pcr (lb) 401 293 247

Simply Supported Pcr (lb) 101 87.5 54.5

Table 3.3- Percent error, asymptotic vs. theoretical Length (in) 18 21 24

% Error Clamped 10.082 10.573 1.535

% Error Simply Supported 9.417 6.838 13.078

Figure 3.1- Clamped-Clamped asymptotic graph

Figure 3.2- Simply Supported asymptotic graph 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 9

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) 18 21 24

Clamped Pcr (lb) 452.2 381.4 284

Simply supported Pcr (lb) 96.8 90.2 56.5

Table 3.5- Percent error imperfection accommodation vs. theoretical Length (in) 18 21 24

Percent error Clamped 1.399 16.408 13.215

Percent error Simply Supported 13.184 10.134 9.889

Table 3.6- Percent error imperfection accommodation vs. asymptotic Length (in) 18 21 24

Percent error Clamped 12.768 30.171 14.98

Percent error Simply Supported 4.158 3.086 3.67

Figure 3.3- Clamped imperfection accommodation graph 10

Figure 3.4- Simply supported imperfection accommodation graph 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.) 18 21 24

Slenderness ratio 498.974 582.137 665.3

Theoretical stress (psi) 1189.333 873.6 668.8

Asymptotic stress (psi) 1077.333 933.333 581.333

Imperfection stress (psi) 1032.533 962.133 602.667

Table 3.8- Clamped critical stress Length (in.) 18 21 24

Slenderness ratio 498.974 582.137 665.3

Theoretical stress (psi) 4756.907 3494.827 2675.733

Asymptotic stress (psi) 4277.333 3125.333 2634.667

Imperfection stress (psi) 4823.467 4068.267 3029.333

Table 3.9- Simply supported critical stress percent error Length (in.) 18 21 24

Percent error asymptotic 0.094170404 0.068376068 0.130781499

Percent error Imperfection 0.131838565 0.101343101 0.098883573

Table 3.10- Clamped critical stress percent error

11

Length (in.) 18 21 24

Percent error asymptotic 0.100816217 0.105725797 0.015347817

Percent error Imperfection 0.013992286 0.16408253 0.132150688

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

Figure 5.6- Clamped critical stress vs. slenderness ratio graph

12

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.

13

View more...
Laboratory Experiment #2

Column Buckling

April 6, 2010

Chris Cameron Section 18

Lab Partners: Joseph O’Leary Zeljko Raic Michael Young Jonathan Hudak Gregory Palencar Course Instructor: Dr. Stephen Conlon Lab 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 (1.1)

EId4wdx4+Pd2wdx2=0

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 (1.2)

k2=PEI

Applying (1.2), (1.1) becomes (1.3)

w''''+k2w''=0

The general solution to the differential equation (1.3) becomes wx=c1sinkx+c2coskx+c3x+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=cn2π2EIL2

(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=cπ2EIL2

(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 clamped at both ends. The eigenvalue problems yielded c values of c=1 for simply supported and c=4 for clamped.

4

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=bh312

(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. Another value of interest is the critical stress, σcr, of the column. This is defined in the equation σcr=PcrA

5

(1.8)

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

(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 σcr=π2Es2

II.

(1.11)

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

6

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.

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 8

Column Length (in) 18 21 24

Clamped Pcr (lb) 445.96 327.64 250.85

Simply Supported Pcr (lb) 111.5 81.9 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) 18 21 24

Clamped Pcr (lb) 401 293 247

Simply Supported Pcr (lb) 101 87.5 54.5

Table 3.3- Percent error, asymptotic vs. theoretical Length (in) 18 21 24

% Error Clamped 10.082 10.573 1.535

% Error Simply Supported 9.417 6.838 13.078

Figure 3.1- Clamped-Clamped asymptotic graph

Figure 3.2- Simply Supported asymptotic graph 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 9

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) 18 21 24

Clamped Pcr (lb) 452.2 381.4 284

Simply supported Pcr (lb) 96.8 90.2 56.5

Table 3.5- Percent error imperfection accommodation vs. theoretical Length (in) 18 21 24

Percent error Clamped 1.399 16.408 13.215

Percent error Simply Supported 13.184 10.134 9.889

Table 3.6- Percent error imperfection accommodation vs. asymptotic Length (in) 18 21 24

Percent error Clamped 12.768 30.171 14.98

Percent error Simply Supported 4.158 3.086 3.67

Figure 3.3- Clamped imperfection accommodation graph 10

Figure 3.4- Simply supported imperfection accommodation graph 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.) 18 21 24

Slenderness ratio 498.974 582.137 665.3

Theoretical stress (psi) 1189.333 873.6 668.8

Asymptotic stress (psi) 1077.333 933.333 581.333

Imperfection stress (psi) 1032.533 962.133 602.667

Table 3.8- Clamped critical stress Length (in.) 18 21 24

Slenderness ratio 498.974 582.137 665.3

Theoretical stress (psi) 4756.907 3494.827 2675.733

Asymptotic stress (psi) 4277.333 3125.333 2634.667

Imperfection stress (psi) 4823.467 4068.267 3029.333

Table 3.9- Simply supported critical stress percent error Length (in.) 18 21 24

Percent error asymptotic 0.094170404 0.068376068 0.130781499

Percent error Imperfection 0.131838565 0.101343101 0.098883573

Table 3.10- Clamped critical stress percent error

11

Length (in.) 18 21 24

Percent error asymptotic 0.100816217 0.105725797 0.015347817

Percent error Imperfection 0.013992286 0.16408253 0.132150688

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

Figure 5.6- Clamped critical stress vs. slenderness ratio graph

12

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.

13

Thank you for interesting in our services. We are a non-profit group that run this website to share documents. We need your help to maintenance this website.