Buckling Behavior of Aluminum Alloy Thin-Walled Beam with Holes under Compression Loading

Thin-walled members are increasingly used in structural applications, especially in light structures like in constructions and aircraft structures because of their high strength-to-weight ratio. Perforations are often made on these structures for reducing weight and to facilitate the services and maintenance works like in aircraft wing ribs. This type of structures suffers from buckling phenomena due to its dimensions, and this suffering increases with the presence of holes in it. This study investigated experimentally and numerically the buckling behavior of aluminum alloy 6061-O thin-walled lipped channel beam with specific holes subjected to compression load. A nonlinear finite elements analysis was used to obtain the buckling loads of the beams. Experimental tests were done to validate the finite element results. Three factors namely; shape of holes, opening ratio D/DO and the spacing ratio S/DO were chosen to study their effects on the buckling strength of the channel beams. Finite elements results were obtained by using Taguchi method to identify the best combination of the three parameters for optimum critical buckling load, whereas determining the contribution of each parameter on buckling strength was implemented by using the analysis of variance technique (ANOVA) method. Results showed that the combination of parameters that gives the best buckling strength is the hexagonal hole shape, D/DO=1.7 and S/DO = 1.3 and the opening ratio (or size of holes) is the most effective on buckling behavior.


INTRODUCTION
Aluminum alloy members are increasingly used in structural applications, especially in bridges and space structures because of their high strength-to-weight ratio, attractive appearance and the perfect corrosion resistance. Furthermore aluminum alloy members are ease of transportation, extrusion and assembly. Recently, these members are often perforated for reducing weight and to facilitate the building services and inspection. These perforations cause a redistribution of stresses in the member that may change the ultimate strength of the structural member and the elastic stiffness. The buckling behavior of structural members with perforations is significantly influenced by the size, location, shape and number of perforations. There is a lot of investigations have been studied on the design and behavior of aluminum alloy members. (Hopperstad, Langseth and Tryland, 1999) investigated experimentally the overall stability of aluminum alloy members of 6082-T4 and 6082-T6 under compression loading with ten different cross-sections. The results were compared with the design strength. (Mazzolani et al., 2000) studied the behavior of 6060, 6061 and 6082 aluminum alloy members experimentally with square (SHS) and rectangular (RHS) hollow sections under uniform axial compression load. They established a new classification criterion for sections of aluminum alloy. (Zhu and Young, 2006), (Zhu and Ã, 2006a), and(Zhu and Ã, 2006b) investigated buckling behavior of 6063-T5 and 6061-T6 aluminum alloy columns SHS and RHS with five series of specimens in fixed-ended under compression load. Using direct strength method, the design formulae for aluminum alloy columns were proposed by carrying out numerical investigation and parametric study. (Yuan, Yuanqing and Bu, 2015) conducted an experimental tests on 6061-T6 and 6063-T5 aluminum alloy I-section stub columns subjected to compression loading in fixed-ended supports. Local buckling and post buckling behaviors were studied. (Liu et al., 2015) studied experimentally the locally buckling failure of 6063-T5 aluminum alloy members with four stiffened closed sections subjected to compressive force. (Zhao, Zhai and Sun, 2016) investigated experimentally the buckling behavior of 6068-T6 aluminum alloy extruded columns with L-type and box-type sections under eccentric compression load. On the other hand, the structural members with perforations have been widely studied on coldformed steel members. In this work, an experimental and numerical study was presented to investigate the buckling behavior of aluminum alloy 6061-O perforated thin-walled lipped channel beam subjected to compression load. Three factors namely; shape of holes, opening ratio / and spacing ratio / were chosen to study their influence on buckling strength of beam. Taguchi method was applied to identify the optimum set of parameters that afford the best strength of buckling. Furthermore, the analysis of variance technique (ANOVA) was employed to determine the most effective parameter on the ultimate load.

TAGUCHI
Taguchi method is a useful technique and is known as "orthogonal array" for design of experiments. It is the most effective in comparing with the other methods, because it provides an effective and simple way for optimizing and producing high quality products with low cost of manufacturing. This method needs a few numbers of experiments, and so it makes the design of experiments too easy. This method employs a statistical measurement named as the ratio of the mean (called signal) to the standard deviation (called noise) (S/N), which represents a logarithmic function of wanted output. There are three standard types of S/N ratio; higher the better,(HB), lower the better,(LB) and Nominal the best,(NB). In this study, the higher buckling strength is required, therefore, the higher the better formula is used and it can be obtained from Eq.(1) below : Where, is the read data and is the number of observations. This study aim to specify the most effective factors to achieve the higher enhancement of buckling strength of the thin-walled member also to detect the optimum set of factors in a limited number of experiments with using Taguchi method.

ANALYSIS OF VARIANCE (ANOVA)
Analysis of variance (ANOVA) is a powerful statistical technique, which specifies the important parameters and demonstrates the percentage contribution of each parameter. In this research, the ratio (S/N) was used for making the decision. The technique of ANOVA is based on the total sum of squared deviations ( ) which is equal to (Kabe and Gupta, 2010): (2) The percentage contribution P could be calculated as: = . . : Error's variance. . . : A degree of freedom. : -test, which is an indicator of the quality characteristic of the process.

EXPERIMENTAL PROGRAM 4.1 Material properties
Aluminum alloy 6061-O sheets with 1.6 mm thickness were used to manufacture the specimens. Tensile tests were conducted to validate the mechanical properties of sheets. Tests were done on the tensile specimens according to ASTM specifications B557M-02a (International, 2003) and test results were indicated in Table 1. Where is the cross-section width, is the flange width, is the lip width and t is the thickness. Eq. (9) refers to the design limits of width-to-thickness ratios set by the Eurocode (EN-1993-1-3) for the lipped channel cross-sections, whereas Eqs. (10) and (11) provide sufficient stiffness and to prevent the primary buckling of the lip itself according to Clause 5.2.2 of the Eurocode. In this work, three shapes of holes designed to be made in the web as shown in Fig. 2 with dimensions according to the limits of applicability of web holes of cellular beam in Eurocode (BS-5950) by Eqs. (12) and (13). The geometry of web holes must be within the given ranges of Eurocode in order to prevent any unwanted failures like cracks between holes, and to achieve a maximum possible reduction in weight.

Specimens preparing
A total of four specimens were tested in this work to verify the numerical solution. One of them without holes as a reference beam and the other three specimens had five holes with shapes of (hexagonal, circular and square) as shown in Fig. 3. The cross-section of lipped beams was made by flexing the Aluminum alloy 6061-O sheet by using a hydraulic bending press machine. The axes were programed by a control system to achieve accurate positions of the inverter for giving accurate dimensions of cross-section. A water jet process was used to make holes on the web of the beam to get a good surface finishing and to minimize any residual stresses at the area of the holes. Figure 3. Test specimens.

Buckling test
The WDW-200E computer Controlled Electronic Universal Testing Machine was used for testing the specimens for buckling failure under compression loading. Four specimens of thin-walled lipped channels were tested in fixed-fixed end condition by applying an axial compressive force at the upper face of the specimen until failure. Two of identical fixing attachments were designed and manufactured to fix the channel on the machine to prevent any displacement or rotation of cross-section. Two solid cubes of Teflon with dimensions of (96.5*34.5*20) 3 put between the flanges at beam ends in order to prevent any relative motion or distortion of cross-section parts at the beam ends due to loading. Fig.4 shows the ends fixture with the inside and outside attachments. The load is gradually applied at rate of 0.3 mm/min by the controlling computer. The load-displacement curve was obtained and the results were recorded. Fig. 5 shows the buckling test under compression loading.

Modeling
The perforated beams are modeled and analyzed in ANSYS 15 software by finite element analysis using SHELL181. It is defined by four nodes having six degrees of freedom at each node, translations in the x, y, and z directions, and rotations about the x, y, and z-axes. SHELL181 is well-suited for linear, large rotation, and/or large strain nonlinear applications (Madenci and Guven, 2006) as shown in Fig. 6. The mesh convergence was established by increased the mesh density in each part of the model. It was observed that there were no considerable changes in load response between 10 mm and 4 mm element size but the processing time was considerable and any increment in mesh density is unnecessary as shown in Fig. 7. Therefore, Element size of 10 mm was used in subsequent analysis.

Boundary conditions and loads
Boundary conditions are needed to constrain the model to get a unique solution. To achieve this, all translations (UX, UY, and UZ) and rotations (ROTX, ROTY and ROTZ) for all points at the bottom end are restrained in order to obtain a fixed support. The rotations about the three directions and the translation along the axial direction for the points at the top end are also restrained. The force F is applied on the upper face of the model as shown in Fig. 8. The first mode shape from the eigen buckling analysis was used to prepare the initial deformed model which will be used later in the nonlinear buckling analysis as a beam with initial imperfection. The main purpose of using nonlinear analysis is to obtain the ultimate capacity of the beam.

Test Results
The test results showed that the buckling mode of failure of the lipped channels under compression loading is an interaction of local and distortional buckling as shown in Fig. 9. As it is clear from Fig.9, the flanges were distorted in a half-wave for all specimens in the longitudinal axes. It may happen due to the high compression stresses generated on the flanges plane during loading process, Element size making the stiffener has no enough stiffness to prevent the flange from rotating, that lead the flanges to rotate at the flange/web junction, outward or inward depending on the nature of the load, supporting system or imperfections. Many buckles can be seen clearly along the web of the reference channel (i.e. the channel without holes) and near the edges of the holes of the perforated channels. Local buckling often occurs at a lower load than the distortional buckling and both are lower than the ultimate load. During the test, the local buckles began to appear approximately at 40% of the ultimate load and the distortional buckling was occur approximately at 80% of the ultimate load. The load-displacement curves that obtained from the experimental tests were shown in Fig. 10 and the results of ultimate load were indicated in Table 3. It was clear that the presence of perforations causing a decrease in the ultimate loads of the channels with a percentage reduction of 9.14% for the hexagonal hole shape, 13.85% for the circular hole shape and a maximum percentage reduction of 15.24% for the square hole shape. Test results were used to validate the FEM results.

Validation of FEM Results
The ANSYS version (15) analysis gave a good agreement with the experimental results and a comparison of load-displacement curves between the experimental tests and the numerical simulations were done and indicated in Fig. 11. In the nonlinear solution, the load was applied gradually by load steps and sub-steps and the ultimate load was reading at where the nonconverged solution occurs. It was clear from the simulation results that ultimate strength of the thin-walled beam decreases with the presence of holes on the web when subjected to compression loading with a minimum percentage reduction of 9.99% for the hexagonal hole shape and a maximum of 15.86% for the square hole shapes as shown in Table 4 and Fig. 12. The buckling failure mode was in agreement with the actual test failure and the half-waves of the distortional buckling failure for all types of holes were evident in Fig. 13. Figure 11. Comparison of load-displacement curves between experimental and numerical results.

Finite element Results
Nonlinear finite element analyses were performed to calculate the ultimate load. In this study, the strength-to-weight ratio was used for the optimization process due to its importance in the lightweight structures especially with exist of perforators. According to three parameters and three levels, an orthogonal array L27 was established, as presented in Table 5. So, the twenty-seven value of ultimate load was got from different combinations of parameters' levels.

Taguchi and ANOVA results
The orthogonal array and values of S/N was shown in Table 6 and Results of mean based on S/N were presented in Table 7 and Fig. 14:  Table 7 and Fig. 14 it was seen that the optimum set of levels is (A1 B1 C3) which gives the best ultimate buckling strength and best strength to weight ratio. The combination of parameters is the hexagonal hole shape, / = 1.7 and / = 1.3. Delta column refer to the deference between the maximum and minimum level value. It shows that the opening ratio / has more effect on buckling strength than the other parameters. The rank column indicates the order of parameters from the highest to the lowest influential on buckling strength depending on Delta values. Fig. 15 shows the numerical load-displacement curve for the condition (A1 B1 C3).    Table 8 shows the results of ANOVA. These results represent the importance and contribution of parameters as a percentage. The opening ratio has the highest contribution (44.05%) and the lowest effect is the shape of holes (4.56%).

CONCLUSION
This research investigated the buckling behavior of perforated aluminum alloy 6061-O thin-walled lipped channel beam under compression loading and studied the effect of three factors namely shape of holes, opening ratio / and the spacing ratio of / on ultimate strength and the strength-to-weight ratio of the perforated beam. This study also aims to obtain the best set of factors which gives the optimum ultimate buckling strength and the most influential factor on the ultimate load. A finite element method was implemented to investigate the buckling behavior of the beam under a compression load. The finite element models were validated by experimental experiences by using four specimens. Taguchi and ANOVA methods were implemented to confirm the effect of parameters and their combinations. From the results obtained, the following conclusions can be obtained: 1-In the experimental tests, it was found that the hexagonal hole shape causes a less reduction in ultimate strength compared with the other hole shapes (9.14%), while the square hole shape had a larger reduction (15.24%). 2-The combination of parameters that gives the best ultimate strength is the hexagonal hole shape, / = 1.7 and / = 1.3 3-The opening ratio has the highest contribution (44.05%) on the buckling strength and the lowest influential is the shape of holes (4.56%).

NOMENCLATURE
= lip width, mm = width of the flange, mm = width of the web, mm = diameter of the hole, mm E = Young modulus, Gpa = number of levels for each parameter = number of observation (trails) in the orthogonal array. = mean S/N ratio for the ith observation. = mean of all parameters. =contribution's percentage.
= experimental ultimate buckling load, N = numerical ultimate buckling load, N = sum of the squared deviations. = space between two holes centers, mm = thickness of the beam, mm = parameter's variance. = error's variance. = the read data for the higher the better formula of Taguchi ʋ = poisson's ratio σ y = yield stress, Mpa σ u = ultimate stress, Mpa