Flexural-torsional buckling behavior of aluminum alloy beams

Abstract

This paper presents an investigation on the flexural-torsional buckling behavior of aluminum alloy beams (AAB). First, based on the tests of 14 aluminum alloy beams under concentrated loads, the failure pattern, load-deformation curves, bearing capacity and flexural-torsional buckling factor are studied. It is found that all the beam specimens collapsed in the flexural-torsional buckling with excessive deformation pattern. Moreover, the span, loading location and slenderness ratio influence the flexural-torsional buckling capacity of beams significantly. Secondly, besides the experiments, a finite element method (FEM) analysis on the flexural-torsional buckling behavior of AAB is also conducted. The main parameters in the FEM analysis are initial imperfection, material property, cross-section and loading scheme. According to the analytical results, it is indicated that the FEM is reasonable to capture mechanical behavior of AAB. Finally, on the basis of the experimental and analytical results, theoretical formulae to estimate the flexural-torsional buckling capacity of AAB are proposed, which could improve the application of present codes for AAB.

Keywords

flexural-torsional buckling / aluminum alloy beams (AAB) / finite element method (FEM) / theoretical formula

Introduction

The flexural-torsional buckling of beams subjected to transverse force always occurs before the material reaches its yield strength, especially in the case of that the moment of inertia about beam major axis is larger than the one about its minor axis. Recently, the flexural-torsional buckling of beams has been extensively investigated. Lots of investigations have focused on the flexural-torsional buckling of steel beams by experimental and numerical approaches [ 1– 8]. Compared with the researches on the steel beam, the studies on the flexural-torsional buckling behavior of aluminum, alloy beams (AAB) are still limited. The reasons for this state of research are as follows: on the one hand, the number of aluminum alloy structures is not sufficient to meet the demand of a generalized understanding of some particular behavioral aspects [ 9]; on the other hand, the material behavior of aluminum alloy is significantly different from that of steel, because the inelastic behavior of aluminum alloy is much more complicated [ 10, 11]. For all these reasons, the flexural-torsional buckling of AAB has become one of the major research topics.

To investigate the flexural-torsional buckling behavior of AAB, extensive experiments and finite element method (FEM) analyses have been performed. Faella et al. [ 12] set up a new method for predicting the stable part of the rotation capacity of AAB in earlier experiments. Wu [ 13] carried out extensive experimental and FE studied on the flexural-torsional buckling capacity of I-shaped and C-shaped AAB subjected to a concentrated load placed at the midspan. Cheng et al. [ 14] performed a numerical analysis program on the flexural-torsional buckling response of I-shaped AAB subjected to a concentrated, uniformly loading and pure bending action. Subsequently, to calibrate these numerical models, they conducted tests on 40 aluminum alloy beam specimens under pure bending moment [ 15]. On the whole, most of the achievements mentioned above are related to the flexural-torsional buckling behavior of I-shaped and C-shaped AAB under concentrated load placed at the midspan or under pure bending action. However, Eurocode 9 (EC9) [ 16] indicates that both the loading scheme and the section type have a significant influence on the flexural-torsional buckling behavior of AAB. Therefore, in order to develop the study, more experiments and finite element method (FEM) analyses on the flexural-torsional buckling behavior of AAB with different loading schemes and different section types are required.

This paper is mainly aimed at achieving tools and methods to characterize and analyze the flexural-torsional buckling behavior of AAB. In the first section, 8 I-shaped beam specimens and 6 T-shaped beam specimens are tested to investigate the flexural-torsional buckling behavior of AAB. In the second section, 480 numerical models are established, which take initial imperfection, material hardening, cross-section and loading scheme into account. In the final section, theoretical formulae are developed to estimate the flexural-torsional buckling factor of beams on the basis of numerical results.

Experimental program

Materials

The experiments were mainly focused on the flexural-torsional buckling behavior of I-shaped and T-shaped AAB. The tensile tests were carried out to investigate the actual mechanical properties of beam specimens. Six tensile coupons were cut directly from the flange of I-shaped beams, as shown in Fig. 1. Other six tensile coupons were cut directly from the web of T-shaped beams, as shown in Fig. 2 as well. In addition, Table 1 presents the elastic modulus E, nominal yield strength f_0.2 and ultimate tensile strength f_u of the tensile coupons.

Specimens and testing arrangements

Specimens

The 6061-T6 aluminum alloy was selected in the experimental study. Fourteen beam specimens, including eight I-shaped beam specimens and six T-shaped beam specimens, were tested under different loading schemes. The configurations of beam specimens are presented in Fig. 3. Detailed information of beam specimens is listed in Table 2. It can be observed that the principal parameters in this study are cross-section type, span and loading position. For the T-shaped beam specimens, two vertical concentrated loads were applied at the one-fourth span of the specimen respectively, as shown in Fig. 4. The span of the T-shaped beam specimens includes 1700 mm, 2300 mm and 2900 mm. For the beam specimens with an I-shaped section, a vertical concentrated force was placed at the middle of the specimen, as shown in Fig. 5. The span of the I-shaped beam specimens includes 1500 mm, 2800 mm, 2100 mm and 2400 mm. It should be mentioned that all the vertical concentrated loads were placed at the centroid of cross-section or the top flange.

Testing arrangements

All the beam specimens in these experiments were regarded as simple beams. Both ends of the specimens were hinged, as shown in Fig. 6. Hinged supports were placed at the bottom of beam specimens due to the restriction of experimental conditions. As a result, the ends of specimens would rotate around the edge of the bottom instead of the centroid of cross-section. This phenomenon leads to the inaccurate flexural-torsional buckling resistance of specimens. It is noteworthy that the influence of the support on the buckling of specimens would be considered in the numerical analysis.

Figure 7 presents the experimental setup of the T-shaped beam specimens. The hinged support was set at the bottom of the beam specimens. The distance from the end of the beam specimens to the hinged support is 50mm. Owing to the low flexural-torsional buckling capacity of the T-shaped beams, pensile loading scheme was employed. Two loading baskets were placed at the one-fourth span and the geometrical center of cross-section. The mass blocks were placed in the loading baskets.

The loading apparatus of the I-shaped beam specimens which are subjected to a concentrated force applied at the top flange are shown in Fig. 8. The concentrated force acting at the middle of the I-shaped beam specimens was produced by a hydraulic jack, due to the high flexural-torsional buckling capacity of the I-shaped beams. To eliminate the enhanced influence of the loading apparatus on the flexural-torsional buckling capacity of the specimens, the dished plate and the convex plate placed between the top flange and the loading plate were designed.

To investigate the flexural-torsional buckling capacity of the I-shaped beam specimens which are subjected to a concentrated force acting at the geometrical center of cross-section, the experimental equipments were designed, as shown in Fig. 9. To eliminate the enhanced influence of the experimental equipments on the flexural-torsional buckling capacity of the specimens, the swiveling mechanism was designed as well.

Arrangements of measuring point

The flexural-torsional buckling loads could be measured by mass blocks or hydraulic jack. To monitor the displacements and the strains during the test process, both linear variable differential transducers (LVDTs) and strain gauges were used. The arrangements of LVDTs and strain gauges are illustrated in Fig. 10 and Fig. 11. Eight strain gauges were set at the middle of each beam specimen. For the T-shaped specimens, the LVDTs were divided into 5 groups, which were placed at the midspan, the one-fourth span and the end of the beam specimen respectively. For the I-shaped specimens, the LVDTs were divided into 3 groups, which were placed at the midspan and the end of the beam specimen respectively.

Test results and discussion

Experimental phenomena

For those T-shaped beam specimens, two concentrated loads increased simultaneously. During the test, each load increment lasted for at least 5 min to reveal the complete response of the beam specimens. All those beam specimens collapsed in flexural-torsional buckling with excessive deformation pattern. In this kind of failure, the deformation of the beam specimens continued to increase without further loading. For that reason, the specimens were not suitable for further loading. The failure pattern of the specimens is shown in Fig. 12.

Load-strain curves and the load-displacement curves

The load-strain curves and the load-displacement curves of the specimens are plotted in Figs. 13 and 14 respectively. For the T-shaped beam specimens, both the strain and the displacement are proportional to the load at the beginning of the loading process. It is implied that the specimens are in the elastic stage. The lateral displacement was so small that it could be neglected. As the load increasing gradually, the increase of the vertical and lateral displacements at the middle of the beam specimens was accelerated. Meanwhile, the strain on the one side of the top flange increased, but on the other side the strain decreased. It is signified that the lateral torsion was produced before the peak load. After the peak load, the displacement kept increasing without further loading until the specimens buckled suddenly. After unloading, only part of the beam specimens’ deformation recovered. Compared with the T-shaped beam specimens, the performance of the I-shaped beam specimens was different distinctly because of the different loading condition. The lateral deformation was unobvious when the failure of the beam specimens occurred due to the small slenderness ratio. The sloping portion in the load-displacement curve was visible.

Buckling capacity

Table 2 lists the measured ultimate load P_u and the measured ultimate moment M_u. From the Table 2, it could be observed that the span of the beam specimens plays an important role in their flexural-torsional buckling capacity. With respect to the same cross-section specimens, the longer the span is, the lower the buckling capacity is. The experimental results of I-shaped beam specimens indicate that the load which is placed at the top flange reduces the buckling capacity. The main reason is that the load placed at the top flange raises the lateral torsion so that the buckling failure occurs earlier.

Summary of test data

The flexural-torsional buckling characteristics of those simple beam specimens could be described by the {\phi }_{\mathrm{b}}-\overline{\lambda } relationship. According to the reference [ 17], the flexural-torsional buckling factor φ_b and the relative slenderness ratio \overline{\lambda } can be determined as follows:

(1) {\varphi }_{\mathrm{b}}=M_{\mathrm{u}}/M_{\mathrm{p}},

(2) \overline{\lambda }=\sqrt{M_{\mathrm{p}}/M_{\mathrm{c}\mathrm{r}}},

(3) M_{\mathrm{p}}=Z{f}_{\mathrm{0.2}},

(4) M_{\mathrm{cr}}={\beta }_{\mathrm{1}}{\displaystyle \frac{{\mathrm{\pi }}^{\mathrm{2}}E{I}_{\mathit{y}}}{L^{\mathrm{2}}}}[{\beta }_{\mathrm{2}}a+{\beta }_{\mathrm{3}}{\beta }_{\mathit{y}}{\displaystyle \frac{I_{\mathrm{\omega }}}{I_{\mathit{y}}}}+\sqrt{{\left({\beta }_{\mathrm{2}}a+{\beta }_{\mathrm{3}}{\beta }_{\mathit{y}}\right)}^{\mathrm{2}}+{\displaystyle \frac{I_{\mathrm{\omega }}}{I_{\mathit{y}}}}\left(\mathrm{1}+{\displaystyle \frac{G{I}_{\mathrm{t}}{L}^{\mathrm{2}}}{{\mathrm{\pi }}^{\mathrm{2}}E{I}_{\mathrm{\omega }}}}\right)}],

where M_p is the plastic hinged moment for flexural-torsional buckling, Z is the plastic modulus of gross section, M_cr is the elastic critical moment for flexural-torsional buckling, β₁, β₂ and β₃ are the modified constants depending on the loading schemes, a is the distance between the loading position and the shear center at the cross-section, β_y is the constant to reflect the unsymmetrical level of the cross-section, I_y is the moment of inertia about y-axis, I_ω is the warping constant, I_t is the torsion constant, G is the shear modulus, L is the length of span. The values of β₁, β₂ and β₃ could be obtained in reference [ 17]. These features about the flexural-torsional buckling of the specimens are listed in Table 4. It could be concluded that the flexural-torsional buckling factor decreases with an increase of the relative slenderness ratio. If the flexural-torsional buckling collapse of AAB occurs in the elastic range, then M_cr is equal to M_u. Therefore, combining the Eq.(1) with the Eq. (2), Euler formula is obtained, as shown in Eq. (5). The Euler curve and the test data are plotted in Fig. 15. The test results are close to the Euler curve.

(5) {\varphi }_{\mathrm{b}}={\displaystyle \frac{\mathrm{1}}{{\left(\overline{\lambda }\right)}^{\mathrm{2}}}}.

Numerical analysis

Calibration of FE models

With the aim of further study, the numerical analysis program that was devoted to the evaluation of the flexural-torsional buckling capacity of AAB was performed by means of the nonlinear code ANSYS. Fourteen finite element (FE) models with the same characteristics as the specimens have been carried out. It should be noted that the shell 181 element was selected to model the flange and the web of the beam due to the unsymmetrical characteristics of cross-section.

To simulate the material properties of aluminum alloy, the Ramberg-Osgood model [ 18] and the Steinhardt suggestion [ 19] were employed, as shown in Eq. (6) and Eq. (7).

(6) ϵ={\displaystyle \frac{\sigma }{E}}+\mathrm{0.002}{\left({\displaystyle \frac{\sigma }{f_{\mathrm{0.2}}}}\right)}^n,

(7) n={\displaystyle \frac{f_{\mathrm{0.2}}}{\mathrm{10}}},

where σ is stress, ϵ is strain and n is constant. The parameters of Eq. (6) and Eq. (7) are obtained from the coupon text, as listed in Table 1.

The boundary and loading conditions of the FE models were similar to those of the specimens (see Fig. 16).

The first linear buckling mode of the FE models is taken as the initial imperfection mode. The constant D is defined as the maximum value of initial imperfection, as shown in Fig. 17. The value of D in each FE model could be calculated as follows: 1) Assuming a value of D, and then analyzing the FE model; 2) making a comparison between the numerical results and the experimental results. According to the comparison, setting a new value of D; 3) repeating step 1 and step 2 until the numerical P_u and the experimental P_u are close. The final value of D is the initial imperfection of the specimen. The initial imperfection of each specimen is listed in Table 5.

The comparison between the ultimate load of FE models and that of specimens is presented in Table 5. On average, the FE predictions are 2.5% lower than the experimental results, with a low standard deviation of 0.05. Although the numerical P_u and the experimental P_u are close due to the determination of D, the model for the correspondent specimen captures the load-rotation and load-displacement relationships with quite accuracy, as illustrated in Fig. 18. Therefore, it is concluded that the aforementioned FE method is available to predict the flexural-torsional buckling behavior of AAB.

Parameters of FE models

After validating the FE models by the experimental results, a systemic FE study on the flexural-torsional buckling capacity of aluminum alloy beams was developed. Extensive FE models were established, taking the influence of initial imperfection, material property, cross-section and loading scheme into account.

Initial imperfection

In Table 5, all the initial imperfections are no more than L/1000. The numerical results indicate that the larger the initial imperfection is, the lower the buckling resistance of the beam is. So in the FE models, the initial imperfection is assumed to be L/1000 conservatively.

Materials

Four aluminum alloys which were widely used in building structures were selected in these FE models. There were 6061-T6, 6063-T6, 6061-T4, and 6063-T5. The properties of these materials could be obtained from EC9 [ 16]. However, considering the results of the tensile tests, the elastic modulus E is equal to 68000.0 MPa. The constitutive equation for aluminum alloy is expressed as Eq. (6) and Eq. (7).

Cross-sections

The cross-section could strongly affect the flexural-torsional buckling behavior of AAB. Therefore, 12 cross-sections frequently used in building structures were contained in these FE models. The configurations of 12 cross-sections are illustrated in Fig. 19.

Loading schemes

Ten loading schemes are applied to these FE models. These loading schemes could be classified into three categories. For the first category, one concentrated load is placed at the midspan of the beam. There are three loading positions in the first category: at the top flange, at the shear center and at the bottom flange. For the second category, two concentrated loads are placed at the one-fourth span of the beam respectively. The loading positions in the second category are the same with these in the first category. For the third category, two bending moments are applied at two ends of the beam respectively. In the third category, four ratios of the two bending moments are 1:1, 1:0.5, 1:0 and 1:-0.5.

Numerical results

Based on the aforementioned parameters, 480 FE models were established. Before the numerical analysis, it could be assumed that the residual stress in aluminum alloy extrusions is so small that it could be neglected [ 20]. The numerical results are plotted in Fig. 20. It is highlighted that: 1) The larger the relative slenderness ratio is, the lower the flexural-torsional buckling factor is. 2) When the relative slenderness ratio is larger than 1.5, the flexural-torsional buckling factor is close to the Euler curve. It is implied that the flexural-torsional buckling collapse of AAB occurs in the elastic range. 3) When the relative slenderness ratio is less than 1.5 and larger than 1.0, the flexural-torsional buckling factor is below the Euler curve. The main reason is the influence of the plastic performance of aluminum alloy on the flexural-torsional buckling behavior of AAB. 4) When the relative slenderness ratio is less than 1.0, some flexural-torsional buckling factors are larger than 1.0. The main reason is that the material property of aluminum alloy has no yield platform. Therefore, the stress may exceed the nominal yield strength f_0.2. As a consequence, the ultimate moment M_u of the FE model may be larger than the plastic hinged moment M_p.

Theoretical formulae

Recently, Perry formula adopted in EC9 [ 16] and British code BS8118 [ 21] has been widely used in the calculation of the flexural-torsional buckling factor. To better model the flexural-torsional buckling behavior of AAB, theoretical formulae developed from the Perry formula are proposed. Therefore, the theoretical formulae are expressed as follows:

(8{{\rm\tf="fp1"\char"0061}}) {\phi }_{\mathrm{b}}={\displaystyle \frac{\mathrm{1}}{\mathrm{2}{\overline{\lambda }}^{\mathrm{2}}}}\left[\left({\overline{\lambda }}^{\mathrm{2}}+\mathrm{1}+D_{\mathrm{0}}\right)-\sqrt{{\left({\overline{\lambda }}^{\mathrm{2}}+\mathrm{1}+D_{\mathrm{0}}\right)}^{\mathrm{2}}-\mathrm{4}{\overline{\lambda }}^{\mathrm{2}}}\right],

(8{{\rm\tf="fp1"\char"0062}}) D_{\mathrm{0}}=-\mathrm{0.16}{\overline{\lambda }}^{\mathrm{2}}+\mathrm{0.47}\overline{\lambda }-\mathrm{0.14},

where D₀ is the equivalent imperfection. To calculate the equivalent imperfection, the polynomial model is employed. First, the flexural-torsional buckling factors and the correspondent relative slenderness ratios are obtained based on the FE results. Secondly, the equivalent imperfections could be calculated though Eq. (8a). Finally, according to the equivalent imperfections and the correspondent relative slenderness ratios, Eq. (8a) is completed by means of statistical regression method. Figure 20 plots the fitting curve and numerical data. It is found that the theoretical curve agrees well with the numerical data.

The text data and the fitting curve are plotted in Fig. 21. The comparisons of experimental data with the theoretical data are listed in Table 6. It can be obviously pointed out that: 1) All the test data are above the fitting curve, which signifies that the Eq. (8) is a reliable method to estimate the flexural-torsional buckling behavior of AAB. 2) On average, the theoretical predictions are 21.3% higher than the experimental results, with a standard deviation of 0.139. The main reason is that the initial imperfection of the FE model is much larger than that of the experimental specimen. 3) The flexural-torsional buckling factor calculated by the Eq. (8) could be the low limitation of AAB.

Conclusions

In this paper the flexural-torsional buckling behavior of AAB has been investigated, involving experimental program, numerical simulation and theoretical analysis. On the whole, the main conclusions are summarized as follows:

1) In the experimental program, all the specimens were failed with flexural-torsional buckling. The results indicate that both the slenderness ratio and the loading location play an important role in their flexural-torsional buckling capacity. The smaller the slenderness ratio is, the higher the buckling capacity is. Meanwhile, the load which is applied on the top flange could accelerate the buckling failure by raising the lateral torsion. The test data are close to the Euler curve.

2) In the numerical simulation, 14 finite element (FE) models with the same characteristics as the specimens have been carried out. The model for the corresponding specimen captures the load-rotation and load-displacement relationships with quite accuracy. Taking into account the influence of initial imperfection, material property, cross-section and loading scheme on the buckling capacity of AAB, 480 FE models were established. When the relative slenderness ratio is larger than 1.5, the flexural-torsional buckling factor is close to the Euler curve, which implies that the flexural-torsional buckling collapse of AAB occurs in the elastic range. When the relative slenderness ratio is less than 1.5 and larger than 1.0, the flexural-torsional buckling factor is below the Euler curve, due to the influence of the plastic performance of aluminum alloy on the flexural-torsional buckling behavior of AAB. When the relative slenderness ratio is less than 1.0, some flexural-torsional buckling factors are larger than 1.0. The main reason is that the stress may exceed the nominal yield strength f_0.2. As a result, the ultimate moment M_u of the FE model may be larger than the plastic hinged moment M_p.

3) In the theoretical analysis, the theoretical formulae are developed to predict the flexural-torsional buckling factor on the basis of the Perry formula. All the test data are above the fitting curve, which signifies the reliability of the theoretical formulae. The flexural-torsional buckling factor calculated by the theoretical formulae could be the low limitation of AAB.

References

[1] Larue B, Khelil A, Gueury M. Elastic flexural–torsional buckling of steel beams with rigid and continuous lateral restraints. Journal of Constructional Steel Research, 2007, 63(5): 692–708

[2] Andrade A, Providencia P, Camotim D. Elastic lateral-torsional buckling of restrained web-tapered I-beams. Computers & Structures, 2010, 88(21–22): 1179–1196

[3] Larue B, Khelil A, Gueury M. Evaluation of the lateral-torsional buckling of an I beam section continuously restrained along a flange by studying the buckling of an isolated equivalent profile. Thin-walled Structures, 2007, 45(1): 77–95

[4] Challamel N, Wang C M. Exact lateral–torsional buckling solutions for cantilevered beams subjected to intermediate and end transverse point loads. Thin-walled Structures, 2010, 48(1): 71–76

[5] Zhang L, Tong G S. Lateral buckling of web-tapered I-beams: A new theory. Journal of Constructional Steel Research, 2008, 64(12): 1379–1393

[6] Rebelo C, Lopes N, Simões da Silva L, Nethercot D, Vila Real P M M. Statistical evaluation of the lateral–torsional buckling resistance of steel I-beams, Part 1: Variability of the Eurocode 3 resistance model. Journal of Constructional Steel Research, 2009, 65(4): 818–831

[7] Simões da Silva L, Rebelo C, Nethercot D, Marques L, Simões R, Vila Real P M M. Statistical evaluation of the lateral–torsional buckling resistance of steel I-beams, Part 2:Variability of steel properties. Journal of Constructional Steel Research, 2009, 65(4): 832–849

[8] Taras A, Greiner R. New design curves for lateral-torsional buckling-Proposal based on a consistent derivation. Journal of Constructional Steel Research, 2010, 66(5): 648–663

[9] De Matteis G, Mandara A, Mazzolani F M. T-stub aluminum joints: Influence of behavioral parameters. Computers & Structures, 2000, 78(1–3): 311–327

[10] De Matteis G, Landolfo R, Manganiello M, Mazzolani F M. Inelastic behavior of I-shaped aluminum beams numerical analysis and cross-sectional classification. Computers & Structures, 2004, 82(23–26): 2157–2171

[11] Manganiello M, De Matteis G, Landolfo R. Inelastic flexural strength of aluminum alloys structures. Engineering Structures, 2006, 28(4): 593–608

[12] Faella C, Mazzolani F M, Piluso V, Rizzano G. Local Buckling of Aluminum Members: Testing and Classification. Journal of Structural Engineering, 2000, 126(3): 353–360

[13] Wu Y G. Studies on local buckling of alunimiun alloy plate and flexural-torsional buckling resistance of bending members. Dissertation for the Doctoral Degree. Shanghai: Tongji University, October, 2005 (in Chinese)

[14] Cheng M, Shi Y J, Wang Y Q. Analysis of lateral stability of I-section aluminum beams. Science in China Series E: Technological Sciences, 2006, 49(6): 742–751

[15] Wang Y Q, Yuan H X, Shi Y J, Cheng M. Lateral-torsional buckling resistance of aluminum I-beams. Thin-walled Structures, 2012, 50(1): 24–36

[16] EN1999–1-1. Eurocode 9: Design of aluminum structures. Part 1–1: General structural rules. 2007

[17] Chen J. Stability of Steel Structures: Theory and Design. Beijing: Science Press, 2011 (in Chinese)

[18] Ramberg W, Osgood W R. Description of stress–strain curves by three parameters. Technical Note No. 902, National Advisory Committee for Aeronautics, 1943

[19] SteinHardt O. Aluminum constructions in civil engineering, Aluminum, 47, 1971, 31–9; 254–61

[20] Federico M. Mazzolani, Tan Zhumei translated. Aluminum alloy structure. Metallurgical Industry Press, 1992

[21] BS8118–1:1991, Structural use of aluminum. Code of practice for design

RIGHTS & PERMISSIONS

Higher Education Press and Springer-Verlag Berlin Heidelberg