Laboratory Experiment #2
Column Buckling and Critical Load Analysis
February 9th, 2015
Christian O’Toole
Lab Partners: Steve Himelfarb, David Irizarry, Matt Flaig, Garret Kolmer
Course Instructor: Dr. Stephen Conlon
Lab TA: Ko Basu – Section 2
Location of Experiment: 047 Hammond – Aerospace Engineering Lab
Abstract
Six stainless steel columns were placed under an increasing applied load in clamped-clamped and simply-supported end fixity conditions to test column buckling. The specimens varied in lengths of 18 inches, 21 inches, and 24 inches. The purpose of the experiment was to determine how length and end fixity of a column affects its critical buckling load. The consequence of imperfections on lateral deflections was also studied. Understanding of these concepts is important in the aerospace industry, since columns are present in many applications. Insight on what causes buckling helps determine how to minimize the flexural instability of a structure. To run this experiment a specimen was fitted into the support blocks with the proper end fixity conditions. A compressive load originated from the load cell and was applied at the top of the column. A LVDT measured the displacement at the center of the column which was tabulated in LabVIEW. Additional hardware used was a balance mass, spring loaded controller, bubble level, and load wheel. Once the displacement grew with a decrease in force the critical load had been reached. Trends in the data were analyzed and it was concluded that as the column length increased, the critical load decreased. Additionally, it was observed that ridged end fixity such as the clamped condition withstands higher critical buckling loads compared to a simply-supported case. Finally, column imperfections from lateral deflections were found from analysis of the imperfection accommodation method.
I. Introduction
Six stainless steel specimens with lengths ranging from 18 inches, 21 inches and 24 inches long were placed in two loading arrangements. Half the columns of each length had simply-supported end conditions while the remaining three had clamped end conditions. All the specimens were placed under a compressive load. The purpose of the experiment was to determine how the length and end fixity of a column affects its critical load. The consequence of imperfections on lateral deflections in the column was also studied. It was hypothesized that as the length of a column increased then it would take a lower applied load to buckle. Additionally, it was suspected that the clamped-clamped end fixity would reinforce the column to a greater degree over the simply-supported boundary conditions. These two propositions were based off the analysis of equation 1, the formula for the critical buckling load of a specimen.
For an engineer designing aerospace components, stability of a structure is particularly important. An example of an application related to column buckling, is shell-type structures such as a fuselage on an aircraft. Information from buckling experiments helps determine the amount of material necessary for a design. It also helps evaluate where the material should be placed to get the maximum structural benefit. Ultimately, this will help meet the design criteria of a structure and cut cost while still producing a safe aerospace component.
Column buckling follows a nonlinear relationship between the displacement of the column and the applied load. As the force acting in compression on the column gets closer to the critical buckling load the displacement increases at a greater rate than before. The value of the critical load depends on a variety of a column’s material properties. Flexural stiffness is one property which is a factor of the specimens Young’s Modulus, E, and second area moment of inertia, I. The second area moment of inertia is calculated based off the cross-section of the specimen. Additionally, the length of the column, L, and the end fixity type, c play a role in the determining the critical load. For the experiment in both the simply-supported and clamped end conditions, c is calculated based off the first mode of buckling. Equation 1 displaces the relationship between the critical buckling load of a column and its structural properties which are c, E, I, and L.
Equation 1: Critical Buckling Load
If a column is simply-supported at the ends then c is equal to 1. This value is based off the boundary conditions of the specimen. In a simply-supported cause the displacement at the column ends are equal to zero. Additionally, the moments are equal to zero at the top and bottom of the column as well. With these boundary conditions the displacement, w(x) can be determined. A full derivation of finding the critical buckling load for a simply-supported column is included in Appendix A.
If a column is clamped at the ends, then c is equal to 4. The boundary conditions of a specimen in this arraignment also determine its critical load. The displacement and slope at the ends of the column is equal to zero. With these boundary conditions the displacement, w(x) for a clamped column can be determined. A full derivation of finding the critical buckling load for a clamped column is included in Appendix A.
II. Experimental Procedure
In the experiment six, stainless steel specimens were tested. Each column was the same in cross sectional area of 0.75 inches by 0.125 inches, but varied in lengths ranging from 18 inches, 21 inches and 24 inches as shown in Figure 1. Furthermore, three of the columns had simply-supported end fixity as seen in Figure 2. While the other three had clamped end conditions visible in Figure 3.
Figure 1: Six Stainless Steel Specimens Tested Figure 2: Simply-Supported Figure 3: Clamped
Additionally, an apparatus with various components displayed in Figure 4, was used to perform the test. The apparatus contained a balance mass that operated based of a pulley system seen in Figure 5, which offset the weight of the bar that translated the compressive load onto the specimen. This compressive load originated from the load cell which was increased and decreased in force by the load wheel seen in Figure 6. A digital interpretation of the applied force in the load cell was read by the Omega equipment in Figure 7. One end of the loading beam contained a spring-loaded controller for fine adjustments of the bar as seen in Figure 8. The bubble level shown in Figure 9 determined how much compensation was necessary. A LVDT positioned horizontally with the ground measured displacements at the middle of the specimen as seen in Figure 10. The specimen was held in place using support blocks which contained simply-supported and clamped end fixity conditions on opposite sides as represented in Figure 11. To accurately record data, LabVIEW was utilized as seen in Figure 12.
Figure 4: The Experimental Apparatus
Figure 5: Balance Mass Figure 6: Load Cell and Load Wheel Figure 7: Omega Load Display
Figure 8: Spring-Loaded Controller Figure 9: Bubble Level
Figure 10: LVDT Horizontally Orientated
Figure 11: Support Blocks Figure 12: LabVIEW
To perform the experiment the support blocks were orientated in a clamped end fixity manner and screwed into the top beam and apparatus as seen in Figure 13. Once the top beam was level with the ground, distances from the pin to the counter weight pulley, center of the load wheel and support block were measured using the tape measurer in Figure 14. Based on the moment arm at the pin, the applied compressive load in the column was calculated from the relationship between these distances. This calculation was placed into LabVIEW to accurately measure the applied load of the column. The 18 inch long, clamped end fixity column was secured into the support blocks and the LVDT was aligned perpendicular with its center as seen in Figure 15. Once the top beam was horizontal to the ground according to the bubble level, the LVDT was zeroed in LabVIEW. With the columns theoretical critical load kept in perspective, the load was progressively increased by turning the load wheel. At the same time another lab member recorded data points in short intervals as the compressive force on the column increased. These data points included information about the applied force and the midspan transverse displacement measured by the LVDT. Once the column began to buckle as seen in Figure 16 by showing an increase in displacement for a decrease in applied load the test was complete. The force was unloaded by turning the load wheel in the opposite direction. After unloading, the column was removed and the 21 inch long column was replaced in the same orientation. The experiment was repeated in this same process for the rest of the clamped end fixity specimens and then on the simply-supported specimens. To test the next specimens the support blocks were unbolted, flipped over and reattached to the apparatus as illustrated in Figure 17. The alternative, simply-supported column was placed into the apparatus as seen in Figure 18 and the experiment was repeated in the exact same manner as before.
Figure 13: Clamped End Fixity Support Block Figure 14: Tape Measurer
Figure 15: Clamped End Fixity Test Setup Figure 16: Clamped End Fixity Specimen Buckling
Figure 17: Simply-Supported Support Block Figure 18: Simply-Supported Test Setup
III. Results and Discussion
The measured critical load from each test was compared against the theoretical critical load calculated by equation 1. This data was used to determine the percent error between each specimen in the experiment. The exact critical load values for measured and theoretical data are listed along with their corresponding degree of percent error in Table 1. Since the LVDT’s data acquisition method for determining the column’s displacement is only reliable to a certain degree, the experimental results were truncated to two decimal places. Additionally, the percent error was truncated to two decimal places since it is a function of the measured critical load. The percent error ranged from 4.5% to 22.0%. Although this error was undesirable, it was expected due to the nature of performing the experiment. Many variables went into running the tests and some were difficult to eliminate even in a controlled environment.
Table 1: Percent Error in Measured and Theoretical Critical Load
In order to determine the variation in data between the specimens with clamped end fixity, the load-deflection experimental data was plotted against one another as shown in Figure 19. Similarly, the three simply-supported specimens were plotted by means of displacement vs applied force represented in Figure 20. In each of the six specimens as the applied load got closer to the critical buckling load, the column began to increase in displacement at a greater rate. Once each hit the experimental critical buckling load it represented a logarithmic style in the data and showed increases in displacement for a decrease in applied load. Between both graphs there was a direct relationship between the length of the column and its critical buckling load. As the length increase from 18 inches to 21 inches and eventually 24 inches, it is evident that the 304 stainless steel specimen buckled at a lower critical value. Finally, from the comparison of Figure 19 and Figure 20 the clamped-clamped column was able to resist changes in displacement for a larger applied load. The simply-supported specimens buckled at critical values around a fourth less than the alternative configuration, which was expected.
Figure 19: Clamped-Clamped Displacement vs Force Figure 20: Simply-Supported Displacement vs Force
The asymptotic technique was used to determine the measured critical buckling load. Figure 21 represents the 18 inch specimens, Figure 22 displays all the 21 inch specimens and Figure 23 represents the 24 inch specimens. Each of these plots contains simply-supported and clamped end fixity columns of the same length. The results were plotted as displacement in inches against the applied force in pounds. Dashed lines represent theoretical critical buckling loads are and are included to compare measured to expected results. In five instances the theoretical buckling load wasn’t reached. The simply-supported, 18 inch specimen withstood a value greater than the theoretical buckling load. However, the percent error in this experiment was close to the expected value by a margin of about five percent. As evident in the three figures, the clamped end fixity beam had a critical load, both in theory and as measured in the experiment that was about four times higher than the simply-supported critical load. There were two instances where the theoretical and measured results varied and signified error in the experimental results. This was present in the date plotted by the clamped-clamped test of a 21 inch and 24 inch specimen in Figure 22 and Figure 23, respectively.
Figure 21: C-C and S-S Asymptotic Technique (18 in.) Figure 22: C-C and S-S Asymptotic Technique (21 in.)
The imperfection accommodation technique was plotted for both specimens of the same length. The imperfection accommodation deflection in the column against this deflection over the applied load for each specimen was graphed. The imperfection accommodation deflection in the column is related to the initial displacement, applied force and critical buckling load as seen in equation 2.
Equation 2: Imperfection Accommodation Deflection
As a result, this provided a linear graph where the slope of each simply-supported and clamped end fixity line was the critical buckling load. A straight line of best fit was applied to find the average of the data and the value of the corresponding critical load. The imperfection accommodation of the clamped and simply-supported specimens is represented by Figure 24 for the 18 inch columns, Figure 25 for the 21 inch columns and Figure 27 for the 24 inch columns. Figure 26 includes data on the clamped, 21 inch imperfection accommodation and has been included to clarify Figure 25, since the data is difficult to see. The slope values in all four of the plots below correspond with the measured critical values given by the asymptotic technique above. These two values are similar since they are based off the same calculations with the imperfection accommodation accounting for the initial displacement already in the column to offset the slope.