Buckling Test of Composite Cylindrical Shells with Large Radius Thickness Ratio

Featured Application: Thin-walled composite cylindrical shells, due to their light weight, high stiffness and strength, can be applied to ultra-light weight rockets, fuselages of airplanes, satel-lites, and spacecraft. Abstract: A buckling test of composite cylindrical shells with a radius–thickness ratio ( r / t ) = 893 under axial compression was conducted to investigate the effects of the radius–thickness ratio ( r / t ). It is known that the buckling load of cylinders shows large differences and scatter between theory and experiment. The ratio of the experimental buckling load and theoretical buckling load is called the knockdown factor (KDF). Many investigations have been conducted to ﬁnd the cause of the degradation and scatter in the KDF, but as yet, no cause has been found. In 1968, NASA’s buckling design criterion, NASA SP-8007, gave an empirical KDF curve that decreased with the increasing r / t (up to 2000) for metal cylinders. The same curve has been applied to composite cylinders. Recently, Takano derived a ﬂat lower-bound KDF in terms of A- and B-basis values (99% and 90% probability with a 95% conﬁdence level) through a statistical analysis of experimental buckling loads. The result, however, based on experimental results up to r / t = 500 and, thus, the dependency on a large range of r / t , is not clear. Thus, the authors focused on a larger range of r / t . Cylindrical shells made from carbon ﬁber-reinforced plastic (CFRP) were tested. The nominal radius, thickness, and length were r = 100.118 mm, t = 0.118 mm, and L = 200 mm and, thus, the r / t = 848 and length-to-radius ratio ( L / r ) = 2.0. Shape imperfections were also measured by using in-house laser displacement equipment. The buckling load was slightly affected by the r / t , but the reduction in the KDF was insigniﬁcant.


Introduction
The scatter and deviation of the buckling load between theory and experiment have been studied since the 1960s, but the cause of the scatter and deviation has yet to be revealed. One potential cause is thought to be shape imperfection. Studies using a nonlinear finite element analysis, including of the measured shape imperfection, have given 5-10% larger results than the experimental buckling loads [1]. The method, however, cannot be used for prediction, because it requires the manufacturing of a cylinder and measurement of the shape imperfection. Regression analyses have further shown that various factors, including layup overlaps or gaps, the radius-to-thickness ratio (r/t), the length-to-radius ratio (L/r) [2], layup stiffness, and oval shape imperfection, are not statistically significant [3]. The authors investigated the effect of axisymmetric shape imperfections on the ratio of the experimental buckling load and theoretical buckling load, which is called the knockdown factor (KDF), and found that it increases with the amplitude of drum-type Figure 1. The knockdown factor (KDF) of the metal cylinders. r/t: radius-to-thickness ratio [5,6].
Composite cylinders, however, show a constant relation between the KDF and r/t in a narrow range of r/t [7] (as shown later). Recently, several studies for the KDF of cylinders have been published; however, there was no study for a composite cylinder with the range of a r/t over 670 [8][9][10][11][12][13][14]. Wagner et al. [8] investigated the KDF of metal cylinders with single boundary perturbation approach. Wagner and Hühne's [9] discussion about that led to new, improved knockdown factors for the design of cylindrical metal shells under axial compression. Tobias and Dieter [10] investigated the relationship between geometric imperfection and the KDF. Wagner et al. [11] presented a numerical design approach based on the reduced stiffness method for the buckling of shell structures. Although a lot of effort has gone into developing novel manufacturing and the production of composites [15][16][17][18], there has been no investigation into the KDF of composite tubes. Thus, the aim of the current study was to investigate the KDF over a large range of a r/t through buckling tests with r = 100 mm, t = 0.112 mm, and r/t = 893. Moreover, the effect of the shape imperfections was measured before and after the buckling tests, and the relationship between the shape imperfections and the KDF was investigated.

Materials and Properties
Two cylindrical specimens (No. 191 and No. 192) were made from carbon fiber-reinforced plastic (CFRP) pre-preg HSX350C075S (Mitsubishi Chemical Co. Ltd.). Table 1 shows the material properties, and Table 2 shows the configurations of the specimens. The knockdown factor (KDF) of the metal cylinders. r/t: radius-to-thickness ratio [5,6].
Composite cylinders, however, show a constant relation between the KDF and r/t in a narrow range of r/t [7] (as shown later). Recently, several studies for the KDF of cylinders have been published; however, there was no study for a composite cylinder with the range of a r/t over 670 [8][9][10][11][12][13][14]. Wagner et al. [8] investigated the KDF of metal cylinders with single boundary perturbation approach. Wagner and Hühne's [9] discussion about that led to new, improved knockdown factors for the design of cylindrical metal shells under axial compression. Tobias and Dieter [10] investigated the relationship between geometric imperfection and the KDF. Wagner et al. [11] presented a numerical design approach based on the reduced stiffness method for the buckling of shell structures. Although a lot of effort has gone into developing novel manufacturing and the production of composites [15][16][17][18], there has been no investigation into the KDF of composite tubes. Thus, the aim of the current study was to investigate the KDF over a large range of a r/t through buckling tests with r = 100 mm, t = 0.112 mm, and r/t = 893. Moreover, the effect of the shape imperfections was measured before and after the buckling tests, and the relationship between the shape imperfections and the KDF was investigated.

Materials and Properties
Two cylindrical specimens (No. 191 and No. 192) were made from carbon fiberreinforced plastic (CFRP) pre-preg HSX350C075S (Mitsubishi Chemical Co. Ltd., Hong Kong, China). Table 1 shows the material properties, and Table 2 shows the configurations of the specimens.

Manufacturing of Cylindrical Specimens
As shown in Figure 2, the cylinders were highly flexible and easily deformed by touch. Thus, to apply a compression load and keep the cylindrical shape, aluminum alloy end rings were bonded to the cylinders by using a bonding fixture, as shown in Figure 3. The bonding fixture aligned the center axis of both end rings without inclination. First, a fixture ring was set on the baseplate of the fixture, and then, the cylindrical specimen was set into the 2-mm-wide slot of the end ring, as shown in Figure 4. The cylindrical specimen was fixed using the alignment ring, as shown in Figure 5, and then, adhesive (AW106/HY953) was applied by syringe. To prevent thermal residual stress, the adhesive was cured at room temperature. After curing, the specimen with the bonded ring was removed, and the other end ring was set on the baseplate of the bonding fixture. Then, the other end of the specimen was inserted into the slot of the ring on the baseplate, and the bonded ring was fixed to the alignment plate, as shown in Figure 6. Adhesive was then applied to the ring on the baseplate by syringe, and it was cured at room temperature. The cylindrical specimen with both end rings is shown in Figure 7.

Manufacturing of Cylindrical Specimens
As shown in Figure 2, the cylinders were highly flexible and easily deformed by touch. Thus, to apply a compression load and keep the cylindrical shape, aluminum alloy end rings were bonded to the cylinders by using a bonding fixture, as shown in Figure 3. The bonding fixture aligned the center axis of both end rings without inclination. First, a fixture ring was set on the baseplate of the fixture, and then, the cylindrical specimen was set into the 2-mm-wide slot of the end ring, as shown in Figure 4. The cylindrical specimen was fixed using the alignment ring, as shown in Figure 5, and then, adhesive (AW106/HY953) was applied by syringe. To prevent thermal residual stress, the adhesive was cured at room temperature. After curing, the specimen with the bonded ring was removed, and the other end ring was set on the baseplate of the bonding fixture. Then, the other end of the specimen was inserted into the slot of the ring on the baseplate, and the bonded ring was fixed to the alignment plate, as shown in Figure 6. Adhesive was then applied to the ring on the baseplate by syringe, and it was cured at room temperature. The cylindrical specimen with both end rings is shown in Figure 7.         Figure 8 shows the shape measurement equipment; the major specifications are listed in Table 3. The equipment consisted of a turntable and a vertical movement head with a laser displacement sensor (LDS). The cylindrical specimen was turned on the turntable and measured by the sensor at different vertical levels. The measurements and movements were controlled by a personal computer. Measurements were taken at 61 points (one point per 6°) on ten vertical levels.  Figure 8 shows the shape measurement equipment; the major specifications are listed in Table 3. The equipment consisted of a turntable and a vertical movement head with a laser displacement sensor (LDS). The cylindrical specimen was turned on the turntable and measured by the sensor at different vertical levels. The measurements and movements were controlled by a personal computer. Measurements were taken at 61 points (one point per 6 • ) on ten vertical levels.  Figure 8 shows the shape measurement equipment; the major specifications are listed in Table 3. The equipment consisted of a turntable and a vertical movement head with a laser displacement sensor (LDS). The cylindrical specimen was turned on the turntable and measured by the sensor at different vertical levels. The measurements and movements were controlled by a personal computer. Measurements were taken at 61 points (one point per 6°) on ten vertical levels.

Buckling Test
The buckling test was conducted on the universal testing instrument (Shimadzu AG-I 100 kN, load accuracy within ±1% for an indicated value). To achieve a uniform load, a steel ball and adjustment fixture were used, and the load was applied via a steel plate, as shown in Figure 9. The alignment between the center of the cylindrical specimen and the crosshead of the universal testing instrument was conducted by the crosshead moving down to alignment fixture 1 and checking the offset between the steel ball and the hole of alignment fixture 1. To easily adjust the cylindrical specimen, fluororesin tape was applied to the baseplate of the universal testing instrument. After adjustment, a 150-N compression load, which is a sufficiently small buckling load, was applied and then removed. The load and displacement curve were then checked, and readjustments were conducted when an anomaly was observed. When an anomaly was not observed, the compression test for buckling was conducted. A crosshead displacement was applied until it reached 1.1 times the buckling displacement; after which, the load was removed. In total, seven buckling load tests were conducted, one without offset (the first test); four with offset (0 • , 90 • , 180 • , and 270 • ); and two without offset (the last test twice for confirming reproducibility). Two offset amplitudes, δ/t = 14.6 and δ/t = 7.3, were tested. The cylindrical specimens were thin enough that they were undamaged after they buckled, which enabled the tests to be repeated on them. For the test without an offset, the alignment fixture in Figure 10 was used, while, for the test with an offset, the alignment fixture in Figure 11 was used. To check that there was no anomaly in the inclination of the top of the cylindrical specimen, the displacements of the four corners of the steel plate were measured by using laser displacement sensors (LDSs), as shown in Figure 12. Figure 13 shows an overview of the buckling test.

Buckling Test
The buckling test was conducted on the universal testing instrument (Shimadzu AG-I 100 kN, load accuracy within ±1% for an indicated value). To achieve a uniform load, a steel ball and adjustment fixture were used, and the load was applied via a steel plate, as shown in Figure 9. The alignment between the center of the cylindrical specimen and the crosshead of the universal testing instrument was conducted by the crosshead moving down to alignment fixture 1 and checking the offset between the steel ball and the hole of alignment fixture 1. To easily adjust the cylindrical specimen, fluororesin tape was applied to the baseplate of the universal testing instrument. After adjustment, a 150-N compression load, which is a sufficiently small buckling load, was applied and then removed. The load and displacement curve were then checked, and readjustments were conducted when an anomaly was observed. When an anomaly was not observed, the compression test for buckling was conducted. A crosshead displacement was applied until it reached 1.1 times the buckling displacement; after which, the load was removed. In total, seven buckling load tests were conducted, one without offset (the first test); four with offset (0°, 90°, 180°, and 270°); and two without offset (the last test twice for confirming reproducibility). Two offset amplitudes, δ/t = 14.6 and δ/t = 7.3, were tested. The cylindrical specimens were thin enough that they were undamaged after they buckled, which enabled the tests to be repeated on them. For the test without an offset, the alignment fixture in Figure  10 was used, while, for the test with an offset, the alignment fixture in Figure 11 was used. To check that there was no anomaly in the inclination of the top of the cylindrical specimen, the displacements of the four corners of the steel plate were measured by using laser displacement sensors (LDSs), as shown in Figure 12. Figure 13 shows an overview of the buckling test.          Figure 18. The origin of the offset angle, 0° was decided arbitrarily. The theoretical buckling load was calculated by the linear bifurcation theory, including layup anisotropy and transverse shear deformation [19]. The theory was for ideal composite cylinders without shape imperfection. A nonlinear finite element analysis, including the measured shape imperfection [1], can evaluate the effect of the shape imperfection; however, it cannot be used for the prediction of the design use, because it requires the manufacture of a cylinder and measurement of the shape imperfection. Thus, the theoretical method for an ideal cylinder [19] was chosen in this study. The calculated theoretical buckling load was 936 N; thus, the KDFs were 0.43-0.52. The offset affected the buckling load; however, it was not the main reason for the reduction in the KDF. The buckling load in the cases without offset showed good reproducibility, except for No. 192 with δ/t = 7.3 ( Figure 17). The first test without offset showed a lower load than in the second and third tests; thus, that was not caused by the degradation of the specimen. Particular angles increased the buckling load: at 180° in the case of No. 191 with δ/t = 14.6 and δ/t = 7.3 (Figures 14 and 15) and at 0° and 270° in the case of No. 192 with δ/t = 14.6 ( Figure 16). Figures 18-21 show the shape imperfections on each vertical level of the cylinders before and after the buckling test. No significant changes can be seen between before and after; this also indicates that no damage occurred during the test. Figures 22-25 show the shape measurements after the offset buckling test together with the corresponding buckling load. Figure 22  3) shows it was obtained at 90°. These results show that the buckling load was affected not only by the offset direction and shape imperfection but, also, by the offset amplitude. In addition, the minimum buckling load was not always obtained in the offset direction in the case of a large-amplitude shape imperfection, as shown in Figure 25 for the 270° direction.   Figure 18. The origin of the offset angle, 0 • was decided arbitrarily. The theoretical buckling load was calculated by the linear bifurcation theory, including layup anisotropy and transverse shear deformation [19]. The theory was for ideal composite cylinders without shape imperfection. A nonlinear finite element analysis, including the measured shape imperfection [1], can evaluate the effect of the shape imperfection; however, it cannot be used for the prediction of the design use, because it requires the manufacture of a cylinder and measurement of the shape imperfection. Thus, the theoretical method for an ideal cylinder [19] was chosen in this study. The calculated theoretical buckling load was 936 N; thus, the KDFs were 0.43-0.52. The offset affected the buckling load; however, it was not the main reason for the reduction in the KDF. The buckling load in the cases without offset showed good reproducibility, except for No. 192 with δ/t = 7.3 ( Figure 17). The first test without offset showed a lower load than in the second and third tests; thus, that was not caused by the degradation of the specimen. Particular angles increased the buckling load: at 180 • in the case of No. 191 with δ/t = 14.6 and δ/t = 7.3 (Figures 14 and 15) and at 0 • and 270 • in the case of No. 192 with δ/t = 14.6 ( Figure 16). Figures 18-21 show the shape imperfections on each vertical level of the cylinders before and after the buckling test. No significant changes can be seen between before and after; this also indicates that no damage occurred during the test. Figures 22-25 show the shape measurements after the offset buckling test together with the corresponding buckling load. Figure 22  These results show that the buckling load was affected not only by the offset direction and shape imperfection but, also, by the offset amplitude. In addition, the minimum buckling load was not always obtained in the offset direction in the case of a large-amplitude shape imperfection, as shown in Figure 25 for the 270 • direction. Figure 26 plots the KDF over a wide range of r/t. The yellow dots at the r/t = 156 and r/t = 670 are the previous results [2,3]. The KDFs at the r/t = 893 are the present results, and they are equal to or lower than the A-basis value obtained by Takano [7], assuming that the KDF is independent from the r/t. The following linear regression was used to evaluate the dependency of the KDF on the r/t by using an analysis of variance (ANOVA):                                        Figure 26 plots the KDF over a wide range of r/t. The yellow dots at the r/t = 156 and r/t = 670 are the previous results [2,3]. The KDFs at the r/t = 893 are the present results, and they are equal to or lower than the A-basis value obtained by Takano [7], assuming that the KDF is independent from the r/t. The following linear regression was used to evaluate the dependency of the KDF on the r/t by using an analysis of variance (ANOVA): Table 4 shows the results of the ANOVA. The coefficient of the r/t is −2.65 × 10 −4 . It seems small, but the p-value is 0.001%, which indicates a statistical significance, because it is much smaller than 1%. Usually, a p-value smaller than 1% indicates a high statistical significance. The linear regression is plotted in Figure 26. The line is higher than the KDFs at the r/t = 893; thus, it cannot be used as the design criterion. A statistically lower-bound curve with linear regression can be easily plotted, but a suitable regression form should reflect the physical reason behind the reduction. A reason could not be obtained in this study, because only two specimens were tested; thus, more tests will be required.   Figure 26 plots the KDF over a wide range of r/t. The yellow dots at the r/t = 156 and r/t = 670 are the previous results [2,3]. The KDFs at the r/t = 893 are the present results, and they are equal to or lower than the A-basis value obtained by Takano [7], assuming that the KDF is independent from the r/t. The following linear regression was used to evaluate the dependency of the KDF on the r/t by using an analysis of variance (ANOVA): Table 4 shows the results of the ANOVA. The coefficient of the r/t is −2.65 × 10 −4 . It seems small, but the p-value is 0.001%, which indicates a statistical significance, because it is much smaller than 1%. Usually, a p-value smaller than 1% indicates a high statistical significance. The linear regression is plotted in Figure 26. The line is higher than the KDFs at the r/t = 893; thus, it cannot be used as the design criterion. A statistically lower-bound curve with linear regression can be easily plotted, but a suitable regression form should reflect the physical reason behind the reduction. A reason could not be obtained in this study, because only two specimens were tested; thus, more tests will be required.   Figure 26. KDF values for a large range of r/t.

Conclusions
Buckling tests were performed on composite cylindrical shells with large radius-tothickness ratios. The shape imperfections before and after the tests were measured, and the buckling loads in the case of an offset load or no offset to the load were measured. The  Table 4 shows the results of the ANOVA. The coefficient of the r/t is −2.65 × 10 −4 . It seems small, but the p-value is 0.001%, which indicates a statistical significance, because it is much smaller than 1%. Usually, a p-value smaller than 1% indicates a high statistical significance. The linear regression is plotted in Figure 26. The line is higher than the KDFs at the r/t = 893; thus, it cannot be used as the design criterion. A statistically lower-bound curve with linear regression can be easily plotted, but a suitable regression form should reflect the physical reason behind the reduction. A reason could not be obtained in this study, because only two specimens were tested; thus, more tests will be required.

Conclusions
Buckling tests were performed on composite cylindrical shells with large radius-tothickness ratios. The shape imperfections before and after the tests were measured, and the buckling loads in the case of an offset load or no offset to the load were measured. The KDFs ranged from 0.43 through 0.52 in the tests; however, the reduction in the KDF by the offset was insignificant. In addition, the relationship between the offset amplitude and offset direction was also insignificant. These results indicate that the major reason for the reduction in the KDF was not load offset or local shape imperfection. The obtained KDFs for the r/t = 893 indicated a dependency between the KDF and r/t. An ANOVA by linear regression indicated that the KDF decreased with the increasing r/t. Thus, to obtain a lower-bound curve of the KDF, the major reason for reduction in the KDF remains to be revealed.

Data Availability Statement:
The data presented in this study are available on request from the corresponding author.

Conflicts of Interest:
The authors declare no conflict of interest.