Linear buckling analysis of perforated steel storage rack columns with the Finite Strip Method

Thin-Walled Structures 61 (2012) 71–85

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Linear buckling analysis of perforated steel storage rack columns with the Finite Strip Method ¨ b Miquel Casafont a,n, Magdalena Pastor a, Jordi Bonada a, Francesc Roure a, Teoman Pekoz a b

E.T.S. d’Enginyeria Industrial de Barcelona, Universitat Polite cnica de Catalunya, Av. Diagonal, 647, 08028 Barcelona, Spain Cornell University, School of Civil and Environmental Engineering, Ithaca, NY, USA

a r t i c l e i n f o

abstract

Available online 19 August 2012

An investigation on the use of the Finite Strip Method (FSM) to calculate elastic buckling loads of perforated cold formed storage rack columns is presented. Nowadays, this calculation can be accurately performed by means of the Finite Element Method (FEM), because the effect of perforations can be explicitly considered in the analysis. However, the FSM is preferred in cold-formed steel design since it is implemented in much convenient and easy to use software. The problem with FSM is that holes cannot be easily modeled. In this paper, the concept of the reduced thickness of the perforated strip is applied to take into account their effect. A formulation is presented for the reduced thickness that has been calibrated with loads obtained in eigen-buckling FEM analyses. Its accuracy has been verified carrying out analyses on real rack columns with different end conditions. & 2012 Elsevier Ltd. All rights reserved.

Keywords: Perforated members Rack structures Linear buckling analysis Finite Strip Method

1. Introduction Buckling analysis plays an important role in the research and design of cold-formed steel members. For example, buckling loads are used as parameters in the determination of ultimate behavior and loads in the Direct Strength Method [1], and other similar procedures, that have been adopted by current cold-formed steel design codes [2,3]. Approaches for buckling analysis include the Finite Element Method (FEM), the Finite Strip Method (FSM), and the Generalized Beam Theory (GBT). The FEM is the most versatile, since it can be easily adapted to complex geometries and different load and member end conditions. However, FSM and GBT analyses can be carried out with more accessible and easy to use programs, such as CUFSM [4] and GBTUL [5]. GBT may be considered the optimum option with applicability to different boundary conditions. FSM as implemented in the program CUFSM is a very complete and robust tool. The Direct Strength Method, which is being used for important design problems, such as design against distortional buckling [6], was developed in combination with CUFSM. Both GBT and FSM deal with unperforated compression members. Special considerations and adjustments have to be developed, as outlined in this paper, to apply them to rack columns with perforations. The investigation presented is aimed at developing approaches to enable the application of FSM to perforated rack columns. In this paper buckling behavior is

n

Corresponding author. Tel.: þ34 934054322; fax: þ 34 934011034. E-mail address: [email protected] (M. Casafont).

0263-8231/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.tws.2012.07.010

considered. The buckling loads obtained are to be used in a Direct Strength Method approach (DSM) as described in Refs. [7,8]. A large number of investigations on the design of perforated coldformed steel members can be found in the scientific literature. Most of these investigations are focused on the holes of the studs and beams that are used in lightweight construction of residential buildings [9–16]. These holes are usually isolated, or far apart from each other, and their size is large. The design procedures produced by the existing research, which are variations of the commonly used procedures for non-perforated members (e.g. Effective Width Method, Direct Strength Method), have been adopted by design codes [2]. Rack columns have smaller perforations that are uniformly distributed all along their length to facilitate the connection with the other members of the structure. There are also several investigations on this type of perforated members: (i) in [17], they are studied by means of the second order GBT analysis; (ii) in [18–21] by means of the isoparametric spline finite strip method; (ii) in [22–24] by means of geometric and material non-linear finite element analysis; and (iv) [25–27] present different calculation procedures that have also been derived from design procedures of non-perforated members. These investigations, however, have not resulted in a generally accepted analytical design method and adopted by the main design specifications of rack structures [28,29]. As a first step in the development of a design method for perforated rack columns, the present paper shows a procedure for the calculation of elastic buckling loads by the Finite Strip Method. It is worth mentioning two recent references that show the two possible ways to tackle the problem: (i) the works by Moen and Schafer [15] that combine the semi-analytical Finite Strip Method and analytical formulas; and (ii) the investigations by Eccher et al. [19]

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using FEM, and the reduced thickness approach using FSM analysis. The cross-sections and perforations analyzed are shown in Fig. 2. The first four cross-sections are similar to those produced by one of the main manufacturers in Europe, while the fifth is chosen to be similar to those produced in North-America. Only rectangular perforations are considered. Their size is similar to the size of rectangular and non-rectangular holes that can be found in real rack columns. Furthermore, calculations are performed on three different sheet thicknesses that are commonly found in the selected type of cross-sections: 1.8, 2 and 2.5 mm. The cross-sections chosen for calibration are very simple. For instance, sharp corners are considered, there are no intermediate web stiffeners, no perforations are included in the flanges, or the same pitch is applied to all perforation patterns. There are many rack columns in the market that can be far complex than the columns shown in Fig. 2. However, since it is not possible to calibrate the model taking into account all existing columns, it was decided to use simple but generic cross-sections and patterns of perforations. After the initial calibration stage, the model is verified with calculations on actual rack columns. Fig. 3 shows the columns used to evaluate the accuracy of the reduced thickness model. They are columns of medium to high load carrying capacity produced by manufacturers in Europe and North-America. The size and actual shape of the perforations are not included, and only the main dimensions of the cross-section are shown to maintain confidentiality.

3. Finite element and finite strip models Fig. 1. CUFSM model of a perforated column: (a) CUFSM model with reduced thickness and (b) stress distribution in a CUFSM with reduced thickness.

with the isoparametric spline Finite Strip Method. The former approach is based on the concept of equivalent properties of the cross-section and equivalent (or reduced) thickness of the strip. Perforations are not directly modeled. On the contrary, in the latter approach, an evolution of the Finite Strip Method is presented that allows to consider the actual perforations in the model of the column. In the present study the concept of reduced thickness (tr) is used to take into account the effect of perforations in the Finite Strip Analysis. Fig. 1 shows an example of CUFSM model with reduced thickness. It can be observed that tr is only applied to the strips where the perforation is located. The goal is to determine the equation that will provide the thickness of the perforated strips to be used in CUFSM. To achieve this objective, finite element linear buckling analyses are carried out on different lengths of the columns presented in Fig. 2 (about 1000 different analyses were performed). Afterwards, for each one of the analyzed columns, CUFSM is run to determine the reduced thickness that results in the same buckling load obtained in the finite element analysis. This is an iterative procedure, since several thicknesses should be tested until the FEM buckling load is obtained (the error allowed is 70.2%). Finally, regression techniques are applied on the resulting values to derive the expression for tr, which will be a function of different geometric parameters of the column.

2. Columns used in this study In the first step of the investigation, 5 different types of rack column sections used in Europe and United States are analyzed

The ANSYS finite element package is used to carry out the linear buckling analyses. The SHELL 63 finite element is chosen to mesh only half of the model, since symmetry boundary conditions are applied at one end (Fig. 4). At the other end, the member is simply supported with respect to local buckling, distortional buckling and torsional–flexural buckling (pinned—warping free condition). The local buckling pinned condition is introduced by forcing the rotations about the z-axis of all nodes at the end section to be equal (this boundary condition is not represented in Fig. 4). The elastic buckling loads are determined for the column buckling in one-half sine wave, as it is usually done for local and distortional buckling modes when DSM is applied. This is the reason why the finite element mesh is forced to deform in a pure sinusoidal shape by means of the following constraint equation, which is imposed to the nodes of the longitudinal dashed lines highlighted in Fig. 4.  pz  pz  s m us ¼ sin =sin um ð1Þ L L where subscript m denotes ‘‘master’’ node. There is one ‘‘master’’ node per constrained line; subscript s denotes ‘‘slave’’ node. A ‘‘slave’’ node is a node that has a subordinate relationship to the ‘‘master’’ node. All nodes of a dashed line in Fig. 4 are ‘‘slave’’ nodes, but the ‘‘master’’ node. Eq. (1) sets the relationship between ‘‘master’’ and ‘‘slave’’ nodes. There are as many constraint equations per line as the number of ‘‘slave’’ nodes; u is a displacement (x or y displacement degree of freedom); and z location with respect to the z-axis. Three remarks should be made concerning the CUFSM models: (i) standard FSM buckling analyses are carried out, without any mode constraint (cFSM option is off); (ii) the analyses are performed with CUFSM version 3.12, which is limited to buckling in one half-sine wave; (iii) no compression stress is applied to the

M. Casafont et al. / Thin-Walled Structures 61 (2012) 71–85

11 mm

14 mm 80 mm

80 mm

80 mm

11 mm

73

80 mm S1

100 mm S2

160 mm S3

14 mm

80 mm

150 mm

15 mm

10

30,5 27,5

27,5

30,5

25 7,5

7,5

22,5

10

7,5

20

10

7,5

25

25

10

7,5

22,5

30,5 27,5

27,5

30,5

12,5

10

20

16

12,5

22,5

16

12,5

20

16

12,5

25

16

80 mm S5 12,5

22,5

16

20

pitch: 50 mm

pitch: 50 mm

pitch: 50 mm

pitch: 50 mm

pitch: 50 mm

80 mm S4

Fig. 2. Cross-sections and perforations of the columns used to calibrate the model of reduced thickness (all dimensions in mm).

perforated strips (Fig. 4b), because this may amplify the effects of localized buckling modes, that excessively increase the effects of cross-section buckling phenomena (local and distortional buckling), and make the model more sensitive to the errors in reduced thickness. In relation to this third point, it is also noted that experimental measurements and non-linear finite element analyses were performed on a column to learn about the crosssection stress distribution. Figs. 5 and 6 show the location of the strain gauges used in the tests and the resulting measurements. It can be observed that the strains at the strip between web perforations are rather smaller than at the other parts of the cross-section. The results of the finite element analyses, that were carried out on columns of different lengths, also confirmed the fact that the stress at the perforated strip is small (see Fig. 7).

The CUFSM buckling load is obtained by multiplying the stress applied in the preprocessor by the resulting buckling factor and by the true net area of the cross-section. The true net area does not include the area of the web perforated strips. For a unit stress used in CUFSM, the buckling load is Pb ¼ Anet Load f actor CUFSM

ð2Þ

It should be pointed out that two small elements are located at each side of the perforated strips to allow for the transition from the value of applied stress to zero. In some cases, when these elements are very small, some incongruent results are obtained. This problem is easily solved by slightly increasing their size. Preliminary analyses on non-perforated columns are carried out with ANSYS and CUFSM, and the results verified to be the

M. Casafont et al. / Thin-Walled Structures 61 (2012) 71–85

74

52

47 54

2

1.50

76 1.8

79 1.8

67

47

61

Cross-section

61

74

1.8

Web perforations

Flange perforations

80

80

1.7

75

2

2

50 Pitch C5 93

75

76

64

Cross-section

50 Pitch C4

50 Pitch C3

93 2

55

50 Pitch C2

69

50 Pitch C1

2

Web perforations

Flange perforations

65 Pitch

C6

C7 100

100

100

120

1.8

120

142

75 Pitch C10

75 Pitch C9

66

Cross-section

50 Pitch C8

67

65 Pitch

1.8

2.5 2.5

Web perforations

Flange perforations

50 Pitch C11

50 Pitch C12

50 Pitch C13

50 Pitch C14

Fig. 3. Cross-sections and perforations of the columns used to verify the model of reduced thickness (all dimensions in mm): (a) geometric model and (b) end boundary conditions and longitudinal constraints.

M. Casafont et al. / Thin-Walled Structures 61 (2012) 71–85

same, to ensure that the models used in both programs are consistent.

4. Considerations regarding the derivation of the reduced thickness formulas Fig. 8 shows the resulting CUFSM reduced thicknesses for different half-sine waves of a column with cross-section S1 and 30.5  16 mm2 perforations. In this figure, various symbols are

75

used to denote different buckling modes: circles are for columns undergoing local buckling (L); triangles for distortional buckling (D); squares for torsional–flexural buckling (TF) and crosses for transition from local to distortional buckling (L–D), and for transition from distortional buckling (symmetric and antisymmetric distortional buckling) to torsional–flexural buckling (D–TF). It can be observed that: (i) Reduced thickness changes depending on the buckling mode. (ii) Within a buckling mode, it shows a dependence on half-sine wave length. (iii) In the transition ranges L–D and D–TF, the reduced thickness significantly changes with the half-sine wave length. (iv) Few results are obtained for local buckling. Columns are generated in ANSYS from the geometric model shown in Fig. 4a. Shorter columns are not modeled because in this case holes may be located at member ends (or very close to them), resulting in very particular buckling modes that are not considered at the current stage of the investigation. As a consequence, few short half-sine waves are analyzed and very poor results for local buckling are obtained. Results similar to those shown in Fig. 8 are observed in the other perforation patterns investigated, see some examples in Fig. 9. In view of these observations, the following considerations are made:

ux=0 uy=0 Symmmetry Constraint equations

y z

x

Fig. 4. FEM model of a perforated column. (a) Geometric model. and (b) End boundary conditions and longitudinal constraints.

(i) One formula of reduced thickness will be set for each buckling mode. (ii) No formula will be defined for the transition ranges. (iii) The formula of reduced thickness for local and distortional modes is calibrated with the thicknesses obtained from the FEM critical elastic buckling loads (minima of the signature curve), which are the loads usually considered in design. The reduced thickness obtained from the other buckling loads are not included in the regression analyses. (iv) The model for global buckling would have to consider the effects of half-wave length. However, it is verified that the sensitivity of the global buckling loads to tr is actually small and, consequently, a constant value, if properly determined, can be used. (v) Since it is not possible to capture the minimum local buckling load from the shortest FEM model shown in Fig. 4a, it is obtained from the analysis of a long column. The column length is chosen

Fig. 5. (a) Location of the strain gauges and (b) specimen ready to be tested.

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Experimental measurements at 150 mm

Experimental measurements at 125 mm 2000

2000 0

1000

-2000 0

Strain

Strain

- 4000 - 1000 - 2000

- 6000 - 8000 -10000

- 3000 -12000 - 4000

-14000 -16000

- 5000

Strain gauge

Strain gauge

Fig. 6. (a) Strains measured at 125 mm (see Fig. 5) for different load levels and (b) strains measured at 150 mm.

so that several local half-waves can develop and the local elastic buckling load becomes similar to the critical local buckling load. Consequently, constraint equations (1) are removed from the FEM model to allow for multiple half-sine waves. On the other hand, the CUFSM model used in the calibration of the reduced thickness for local buckling is the same as that presented in the previous section (buckling in one-half sine wave is considered in the FSM analyses).

5. Regression analyses for the determination of the reduced thickness formulas Before carrying out the correlation and regression analyses, some tentative calculations were performed to set the reduced thickness equations. For instance, in one of these calculations the reduced thickness was defined on the basis of the transverse stiffness of a perforated plate:  1=3 ELnp t 3 Lnp ELt 3r ¼ -t r ¼ t ð3Þ 2 2 L 12ð1n Þ 12ð1n Þ where Lnp is the length of non-perforated sheet between perforations, and L is the pitch length (Fig. 10). The model using this value of tr is designated as the stiffness model. Fig. 11 (stiffness model) shows the ratio of CUFSM buckling loads obtained from Eq. (3) to the ANSYS critical loads for all columns shown in Fig. 1. It can be observed that, although good correlation between tr and t(Lnp/L)1/3 exists (as it is demonstrated below), this simple equation does not lead to accurate values of reduced thickness. Some similar tentative equations were tried, but no definitive result was obtained. This is the reason why, in the end, it was decided to carry out a study applying regression analyses. Correlation and regression analyses are performed in EXCEL. The parameters considered to choose the model of reduced thickness are the coefficient of correlation and the coefficient of determination [30]. First, a correlation analysis was performed between the reduced thickness (tr) derived from the FEM buckling loads and simple mathematical expressions that combine different geometric parameters. It was a systematic study where more than 25 different expressions were investigated. Table 1 shows the seven expressions that resulted in the highest correlation coefficients (R). The geometric parameters of these expressions are described in Fig. 10 (t is the actual thickness of the sheet). Only expressions with physical meaning are considered: (i) X1 and X2 are derived as shown in Eq. (3). (ii) X3 includes the ratio width/height of the cross-section.

(iii) In X4, the non-perforated area of the web is considered. (iv) X5 uses the ratio between width and height of the perforation according to Fig. 10. (v) X6 results from the following product: X6 ¼ t

Lnp Bnp B Lnp Bnp ¼t L B H LH

ð4Þ

(vi) X7 is the ratio of the non-perforated length of the web to the total length. Table 1 shows that the correlation coefficients of the chosen expressions significantly change depending on the mode of buckling, which confirms the fact that three different tr equations should be determined. 5.1. Reduced thickness for local buckling X6 shows the highest correlation coefficient for local buckling, R¼0.884. However, its corresponding coefficient of determination, R2 ¼0.782, is slightly below the commonly considered acceptable limit: 0.8. To improve the model an additional expression is introduced to the equation. X3 is chosen since it shows the second highest correlation coefficient (R¼0.878). The regression analysis that combines X6 and X3 slightly raises the coefficient of determination to 0.793. However, this increase in R2 is too small and it is considered that X3 does not significantly improve the model. This is reasonable because X6 already incorporates X3, as it is demonstrated in Eq. (4). For the same reason, expressions X1, X2 and X4 will not be combined with X6. It only makes sense to combine X5 and X6. The regression analysis performed with them leads to the final tr equation (the values of the geometric parameters should be input in mm): t r ¼ 0:61t

Lnp Bnp Bp þ 0:18t þ0:11, LH Lp

ð5Þ

that results in a coefficient of determination of 0.878. The mean ratio between the reduced thickness of the regression model (5) and the reduced thickness derived from the FEM buckling loads (the correct value) is: m(tr model/tr FEM) ¼1.03; and the standard deviation is: s(tr model/tr FEM)¼0.15, which is considered rather high (see also Fig. 11). The critical buckling loads that result from the reduced thickness of the model show an error with respect the FEM buckling loads in the range from 11% (underestimation) to þ15% (overestimation), which is also considered slightly large; while the error concerning slenderness pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (lL ¼ P y =PcrL ) is in the range from  6.3% to 5.68%. It is interesting to introduce the slenderness in the discussion because it allows evaluating the error that the CUFSM buckling loads will

M. Casafont et al. / Thin-Walled Structures 61 (2012) 71–85

77

50

Longitudinal stresses (MPa)

0 -50 -100 Perforated strip

-150

Web stiffener

-200

Flange

-250 -300 -350 -400 -450 -500 0

20

40

60 Location (mm)

80

100

120

50

Longitudinal stress (MPa)

0 -50 Perforated strip

-100

Web stiffener

-150

Flange

-200 -250 -300 -350 0

20

40

60 Location (mm)

80

100

120

Fig. 7. Stress distribution resulting from a non-linear finite element analysis: (a) paths of displayed stresses, (b) stresses at ultimate load, and (c) stresses at 70% of ultimate load.

cause on member check design procedures (such as the Direct Strength Method). Several additional regression analyses were carried out to improve the model with other expressions not included herein, but the results did not get better (see [31]).

shows rather high coefficients of correlation and determination: R¼0.96 and R2 ¼ 0.92, respectively. The result of the regression analysis with X1 is t r ¼ 0:89t

 1=3 Lnp 0:08, L

ð6Þ

5.2. Reduced thickness for distortional buckling Better results are obtained for the reduced thickness of the distortional buckling mode. Furthermore, in this case the model is also simpler because there is an expression, X1, that already

that is reduced to the following equation:  1=3 Lnp t r ¼ 0:9t L

ð7Þ

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M. Casafont et al. / Thin-Walled Structures 61 (2012) 71–85

B

1.80 L

1.60

L-D

1.40

tr (mm)

H

D

1.20

t

D-TF

1.00

TF

0.80

Cross-section

0.60 0.20

Lp

0.00 0

500

1000 1500 L (mm)

2000

Bp/2

0.40

2500

B

Fig. 8. CUFSM reduced thickness determined from FEM elastic buckling loads of cross-section S5 with 30.5  16 mm2 holes (gross thickness: 1.8 mm).

Lnp=L-Lp 1.40

Bnp=B-Bp L

1.20

Web perforations Bp flange

tr (mm)

1.00 0.80 30.5x16

0.60

Lp flange/2

27.5x12.5 0.40

25x10 22.5x7.5

0.20

L

20x5

Flange perforations

0.00 0

500

1000

1500

Fig. 10. Main geometric parameters of the column.

L (mm) Fig. 9. CUFSM reduced thickness determined from FEM elastic buckling loads of cross-section S5 with different patterns of holes (e.g., 20  5 is a rectangular perforation of 20  5 mm2).

The mean ratio of the reduced thickness of Eq. (7) to the thickness derived from the FEM load is m(tr model/tr FEM)¼1.01, and the standard deviation is s(tr model/tr FEM) ¼0.05. Concerning the elastic buckling loads, this model leads to errors in the range p from  5.3% toþ6.4%, which produces a slenderness error ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (lD ¼ Py =P crD ) that range from  3% to 2.78%. The results obtained with Eq. (7) are considered acceptable and, consequently, there is no need to improve it. However, an additional model that is slightly better (R2 ¼ 0.96) is presented for information purposes only:  1=3  1=3 Lnp Bnp 0:43t ð8Þ t r ¼ 1:34t L B Eq. (7) is preferred to (8) due to its simplicity and because it is in agreement with the mechanics of the cross-section (Eq. (3)). 5.3. Reduced thickness for global buckling First of all, it is pointed out that the model is calibrated to calculate the torsional–flexural elastic buckling load, which is usually the critical global buckling load in rack columns. The global flexural buckling mode is not considered in the calibration stage. According to Fig. 8, the global buckling reduced thickness should be a function of the half-sine wave length. However, in the first steps of the development of the model it was already

observed that when this parameter is not considered, the loss of accuracy is not dramatic from the practical point of view. Actually, when performing the iterative calculations to determine the reduced thicknesses from the FEM buckling loads, it was noticed that the sensitivity of the CUFSM global loads with respect to tr is small. The range of tr values that accomplish the calibration criterion (70.2%) is large. However, it was also verified that the use of simple equations, such as tr ¼t or tr E0, does not lead to accurate results. Consequently, a model should be derived in a similar way as in the previous sections. The first step is to choose the length of calibration. L90G is chosen, which is the first length showing a participation of the global buckling mode above the 90% when performing the CUFSM analysis of the column without holes. If the calibration is performed with longer lengths, significant errors result at the beginning of the global buckling range, where the combination of anti-symmetric distortional buckling and torsional–flexural buckling is still relevant. On the contrary, calibrating with ‘‘short’’ global lengths does not produce loss of accuracy for long halfwave lengths. X7 is the expression with the highest correlation coefficient, R¼0.70. Actually, X7 was included in Table 1 for global buckling instead of X1 because it showed a higher correlation coefficient (the value of R for X1 is 0.61). Although the coefficient of determination corresponding to X7 is too small, R2 ¼0.49, a model is directly tested with no other additional expression. The regression analysis results in the following equation: t r ¼ 0:71t

  Lnp 0:09, L

ð9Þ

M. Casafont et al. / Thin-Walled Structures 61 (2012) 71–85

and maximum errors concerning buckling loads are small,  2.92% andp4.66% respectively. The same occurs to the slenderffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ness (lG ¼ P y =P crG ), with errors in the range from  2.25% to 1.48%. These results do not considerably change if other expressions are added to (10) (see [31]). Therefore, in the end, this equation, which is also very simple, is chosen to be the model for global reduced thickness.

LOCAL BUCKLING 4

tr model/tr FEM

3.5 3 stiffness model

2.5

79

regression model

2 1.5 1

6. Verification analyses

0.5 0 0.2

0.3

0.4 (Lnp·Bnp)/(L·B)

0.5

0.6

DISTORTIONAL BUCKLING 4 stiffness model

3.5 tr model/tr FEM

regression model

3 2.5 2 1.5 1 0.5 0 0.2

0.3

0.4 (Lnp·Bnp)/(L·B)

0.5

0.6

GLOBAL BUCKLING 4 stiffness model

tr model/tr FEM

3.5

regression model

3 2.5 2 1.5 1 0.5 0 0.2

0.3

0.4 (Lnp·Bnp)/(L·B)

0.5

0.6

Fig. 11. Accuracy of the reduced thickness equations.

Table 1 Expressions investigated in the regression analyses. Expression 1/3

X1 ¼t(Lnp/L) X2 ¼t(Bnp/B)1/3 X3 ¼t(B/H) X4 ¼t(LnpBnp) X5 ¼t(Bp/Lp) X6 ¼t(LnpBnp)/(LH) X7 ¼t(Lnp/L)

that is reduced to   Lnp t r ¼ 0:7t L

R local

R distortional

R global

0.484 0.497 0.878 0.833 0.360 0.884 –

0.958 0.780 0.260 0.316 0.656 0.294 –

– 0.394 0.010 0.135 0.153 0.168 0.696

ð10Þ

It leads to rather bad estimates of thickness: m(tr model/tr and s(tr model/tr FEM)¼0.16. However, the minimum

FEM)¼ 0.91,

The development of the reduced thickness equations finishes with a last step of verification where the columns shown in Fig. 3 are analyzed. The aim of this section is to evaluate whether the reduced thickness equations derived above work in real columns. The FEM models used in the verification analyses reproduce precisely the geometry of the columns: rounded corners are modeled, and all stiffeners and perforations are considered (even the small web stiffeners and all flange perforations). The CUFSM models are generated in a similar way, except for the flange perforations. It should be noticed that in real columns flange and web perforations do not usually coincide in the same crosssection. Furthermore, the width of web perforations is usually not constant (see Fig. 3). Consequently, a column contains different cross-sections, while in a FSM analysis only one crosssection can be modeled. The cross section selected for the CUFSM models is the one that contains the widest web perforation, where the smallest net web area is located. Although sometimes the net area that results from removing the width of the flange perforations is smaller than the net area that results from removing the width of the web perforation, flange perforations are usually shorter than web perforations. This is the reason why it is considered that the effect of flange perforations on buckling loads is not so significant. In columns C9 and C10 with inclined slots, the width of the perforated strip introduced in CUFSM is the width of the rectangle that circumscribes the perforation. In this verification stage, it will also be investigated the accuracy of the model recently presented by Moen and Schafer in [15]. This model was calibrated for perforations rather different from those studied in this paper. However, it is an elegant approach, and it is considered worth to investigate its performance when applied to rack columns. 6.1. Verification of the local reduced thickness equation Table 2 (columns 3–5) shows the results of the CUFSM and FEM critical elastic local buckling loads, as well as the ratio of the corresponding CUFSM slenderness to the FEM slenderness. Generally speaking, the results concerning buckling loads are not as good as expected: the error is in the range from  10% to 41%. The worst results are those corresponding to columns C12– C14 which contain large web stiffeners. This is reasonable since no stiffener was considered in the calibration of the tr equation. It should be pointed out that web stiffener buckling, that was critical in most of the short columns, was considered local buckling in these verification analyses. It seems that the current model for local buckling only works for cross-sections with small web stiffeners. A high error ratio can also be observed in column C1 that is attributed to the fact that C1 contains a non-negligible proportion of holes in the flanges: (BpflangeLpflange)/(HL)¼ 0.11. This is verified by repeating the analysis of C1 with smaller flange perforations (see the results of column C10 in Table 2). If the results of the cross-sections mentioned in the above paragraphs are removed from the table, it can be observed that

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M. Casafont et al. / Thin-Walled Structures 61 (2012) 71–85

Table 2 Accuracy of the models for local buckling.

Table 3 Accuracy of the models for distortional buckling (Part 1).

P cr CUFSM lCUFSM Pcr Ref P cr FEM lFEM (N)

Column

Pcr FEM (N)

C1 C10 C2 C3 C4 C5 C6 C7 C8 C9 C10 C11 C12 C13 C14

318,850 434,467 1.36 392,196 434,467 0.90 214,105 208,854 0.97 400,490 428,492 1.07 244,562 272,647 1.11 319,885 305,533 0.96 596,531 598,753 1.00 311,747 318,695 1.02 156,919 156,179 1.00 277,902 288,567 1.04 230,602 250,637 1.09 597,490 537,181 0.90 1,234,395 1,705,932 1.38 1,218,568 1,431,303 1.17 1,113,791 1,572,891 1.41

0.86 1.05 1.01 0.97 0.95 1.02 1.00 1.00 0.99 0.98 0.96 1.05 0.85 0.92 0.84

1.07 0.16

0.97 0.06

Mean Deviation

Pcr CUFSM (N)

[15]

351931 351,931 224,128 427,746 289,506 373,841 639,243 369,896 181,426 346,345 286,479 681,175 1,810,383 1,237,929 1,345,596

Pcr Ref ½15 lRef ½15 P cr FEM lFEM

Column

Pcr FEM (N)

Pcr CUFSM (N)

P cr CUFSM lCUFSM Pcr Ref P cr FEM lFEM (N)

1.10 0.90 1.05 1.07 1.18 1.17 1.07 1.19 1.17 1.25 1.24 1.14 1.47 1.02 1.21

0.95 1.06 0.98 0.97 0.92 0.93 0.97 0.92 0.93 0.90 0.90 0.94 0.83 0.99 0.91

C1 C10 C2 C3 C4 C5 C6 C7 C8 C9 C10 C11 C12 C13 C14

139,111 147,030 48,479 198,888 143,734 143,415 132,695 122,875 122,537 168,190 191,013 151,733 323,191 154,076 167,414

136,672 136,672 44,828 193,212 137,090 134,710 120,717 111,619 121,696 156,875 181,449 148,839 310,328 145,948 157,931

0.98 0.92 0.92 0.97 0.95 0.94 0.91 0.91 0.99 0.93 0.95 0.98 0.96 0.95 0.94

1.01 1.04 1.04 1.01 1.02 1.03 1.05 1.05 0.99 1.04 1.03 1.01 1.02 1.03 1.03

1.15 0.13

0.94 0.05

Mean Deviation

0.94 0.03

1.03 0.01

the performance of the local reduced thickness slightly improves (error in slenderness ratio ranging from  8% to 5%). However, it is considered that the reduced thickness model developed in this paper is limited, and it is not able to cover a wide range of crosssection types. It is also difficult to obtain accurate results with the model developed by Moen and Schafer [15], as can be seen in the last three columns of Table 2. Generally speaking, the problem with this model is that the buckling phenomena considered, namely, local buckling of the gross-cross section and local buckling of the perforated section (or unstiffened strip), do not reproduce the specific buckling features of rack columns. As rack perforations are short, buckling of the perforated section is usually not critical. Consequently, the model is governed by gross-cross section buckling, which leads to overestimations of the local buckling load. 6.2. Verification of the distortional reduced thickness equation The best results are obtained for distortional buckling. The reduced thickness Eq. (7) leads to slightly conservative values, that can be considered acceptable: buckling load error ranging from  9% to  1%, and slenderness error ranging from  1% to 5% (see 4th and 5th columns of Table 3). When the model presented by Moen and Schafer in [15] is used, the results are slightly more conservative and the scatter is higher: buckling load error ranging from 14% to 5%, and slenderness error ranging from  2% to 8% (see three last columns of Table 3). In this model, the reduced thickness approach is applied with Eq. (3) (stiffness model). The main differences with the model presented herein are: (i) the cross-section thickness is modified over the full depth of the web, and (ii) the reduced strips are loaded with non-null stress (uniform stress). The performance of Moen’s model can be improved by reducing the web thickness just at the location of the perforation. See, for instance, the buckling loads of Table 4, where k is a factor modifying the reduced thickness of the stiffness model:  1=3 Lnp t r ¼ kt ð11Þ L Future investigations should properly calibrate the value of k. From the observation of this table, it is concluded that this variation of Moen’s model will probably lead to rather accurate predictions of distortional buckling loads of rack columns. For example, when k equals 0.8, the buckling load error ranges from 7% to 2% (last column of Table 4).

P cr Ref½15 lRef½15 P cr FEM lFEM

[15]

128,103 128,103 47,367 194,202 135,671 131,010 126,910 116,131 128,519 156,289 167,552 137,552 287,346 132,989 143,604

0.92 0.87 0.98 0.98 0.94 0.91 0.96 0.95 1.05 0.93 0.88 0.91 0.89 0.86 0.86

1.04 1.07 1.01 1.01 1.03 1.05 1.02 1.03 0.98 1.04 1.07 1.05 1.06 1.08 1.08

0.93 0.05

1.04 0.03

Table 4 Accuracy of the models for distortional buckling (Part 2). Column

Pcr FEM (N)

Pcr k ¼ 1 (N)

P cr k ¼ 1 Pcr k ¼ 0.9 P cr k ¼ 0:9 Pcr k ¼ 0.8 P cr k ¼ 0:8 P crFEM (N) P crFEM (N) P crFEM

C1 C10 C2 C3 C4 C5 C6 C7 C8 C9 C10 C11 C12 C13 C14

139,111 147,030 48,479 198,888 143,734 143,415 132,695 122,875 122,537 168,190 191,013 151,733 323,191 154,076 167,414

156,332 156,332 52,473 223,317 160,297 160,885 146,920 134,604 147,329 187,528 211,586 167,560 353,894 169,599 182,744

1.12 1.06 1.08 1.12 1.12 1.12 1.11 1.10 1.20 1.11 1.11 1.10 1.09 1.10 1.09

Mean Deviation

146,731 146,731 49,557 209,812 149,808 148,564 138,320 126,135 136,347 178,348 196,615 158,357 332,505 156,693 169,922

1.11 0.03

1.05 1.00 1.02 1.05 1.04 1.04 1.04 1.03 1.11 1.06 1.03 1.04 1.03 1.02 1.01

136,359 136,359 46,393 194,571 138,150 137,101 129,683 117,541 125,047 167,456 180,965 146,495 308,546 142,901 155,924

1.04 0.03

0.98 0.93 0.96 0.98 0.96 0.96 0.98 0.96 1.02 1.00 0.95 0.97 0.95 0.93 0.93 0.96 0.03

Table 5 Accuracy of the global buckling reduced thickness equations. Column GLOBAL BUCKLING L90G

C1 C10 C2 C3 C4 C5 C6 C7 C8 C9 C10 C11 C12 C13 C14 Mean dev.

GLOBAL BUCKLING 2L90G

Pcr FEM Pcr (N) CUFSM (N)

P cr CUFSM lCUFSM Pcr FEM Pcr P cr FEM lFEM (N) CUFSM (N)

P cr CUFSM lCUFSM Pcr FEM lFEM

41,509 49,895 24,366 51,265 38,744 39,880 68,828 56,381 44,915 49,340 66,832 52,744 68,623 54,023 65,325

1.17 0.98 1.02 1.02 1.04 0.93 0.96 0.95 1.05 0.91 0.92 0.99 0.98 0.91 0.91

0.92 1.01 0.99 0.99 0.98 1.04 1.02 1.03 0.98 1.05 1.04 1.01 1.01 1.05 1.05

1.14 0.96 1.02 1.00 1.04 0.94 0.98 0.97 1.07 0.93 0.91 0.99 1.00 1.04 0.96

0.93 1.02 0.99 1.00 0.98 1.03 1.01 1.02 0.97 1.04 1.05 1.01 1.00 0.98 1.02

0.97 0.05

1.02 0.03

0.99 0.05

1.00 0.02

48,750 48,750 24,830 52,356 40,136 37,099 65,820 53,764 47,091 45,061 61,425 52,332 67,064 48,990 59,452

14,979 17,809 8101 16,380 13,547 13,827 22,341 18,404 13,235 18,661 24,758 17,627 21,998 16,741 21,659

17,050 17,050 8247 16,307 14,116 12,984 21,871 17,932 14,124 17,268 22,519 17,520 22,080 17,336 20,854

M. Casafont et al. / Thin-Walled Structures 61 (2012) 71–85

81

geometric properties of the cross-section. The calculation can be easily performed with the CUTWP program [32], which reads CUFSM files. Table 6 shows that the results are similar to those obtained with the Finite Strip Method. The main differences are found in the ‘‘short’’ columns (L90G columns), for which the analytical formulas tend to overestimate the buckling load. They cannot be as accurate as CUFSM because they cannot consider the small participation of the anti-symmetric distortional buckling at L90G. An additional verification of the global buckling model is carried out by comparing it to the ‘‘weighted average’’ crosssection properties approach presented by Moen and Schafer in [15]. In this model, the torsional–flexural elastic buckling load is obtained by means of the classical equations, but with modified cross-sections properties:

6.3. Verification of the global buckling reduced thickness equation The accuracy of the global buckling reduced thickness Eq. (10) is similar to the accuracy of the distortional buckling reduced thickness, except for C1 that contains flange holes rather larger than the other columns. Again, the results of this column improve if the analysis is repeated with smaller flange perforations (see C10 in Table 5). If C1 is not considered, the global buckling load error shown in Table 5 ranges from 9% to 7%, and the slenderness ratio error ranges from  3% to 5%. In this table, the results below ‘‘GLOBAL BUCKLING L90G’’ correspond to the L90G columns length, as defined in Section 5.3; and the results below ‘‘GLOBAL BUCKLING 2  L90G’’ to a column length twice L90G. These two different lengths are analyzed to verify that Eq. (10) works for a wide range of half-wave lengths. The accuracy of the global buckling reduced thickness is confirmed by performing calculations with the classical global buckling equations (torsional–flexural buckling equation). The reduced thickness is considered in the determination of the

(i) ‘‘Weighted average’’ of moment of inertia: Ig Lnp þ Inet Lp L

Iavg ¼

ð12Þ

Table 6 Global buckling loads calculated with analytical formulation. Column

Global buckling L90G Pcr

C1 C10 C2 C3 C4 C5 C6 C7 C8 C9 C10 C11 C12 C13 C14

FEM

(N)

41,509 49,895 24,366 51,265 38,744 39,880 68,828 56,381 44,915 49,340 66,832 52,744 68,623 54,023 65,325

Pcr

Global buckling 2L90G

Formulation

(CUTWP) (N)

50,652 50,652 26,420 54,959 42,302 40,717 71,186 57,884 50,302 48,962 65,950 56,304 72,560 54,553 65,983

Mean Dev.

P cr CUTWP P cr FEM

lCUTWP lFEM

Pcr

1.22 0.97 1.08 1.07 1.09 1.02 1.03 1.03 1.12 0.99 0.99 1.07 1.06 1.01 1.01

0.91 1.02 0.96 0.97 0.96 0.99 0.99 0.99 0.94 1.00 1.00 0.97 0.97 0.99 0.99

14,979 17,809 8101 16,380 13,547 13,827 22,341 18,404 13,235 18,661 24,758 17,627 21,998 16,741 21,659

1.04 0.04

0.98 0.02

FEM

(N)

Pcr

Formulation

(CUTWP) (N)

17,280 17,280 8585 16,741 14,419 13,374 22,468 18,374 14,492 17,795 23,298 17,727 22,312 17,702 21,212

P cr CUTWP P cr FEM

lCUTWP lFEM

1.15 0.97 1.06 1.02 1.06 0.97 1.03 1.00 1.09 0.95 0.94 1.01 1.01 1.06 0.98

0.93 1.02 0.97 0.99 0.97 1.01 0.99 1.00 0.96 1.03 1.03 1.00 0.99 0.97 1.01

1.01 0.05

1.00 0.02

Table 7 Torsional–flexural buckling loads calculated with the ‘‘weighted average’’ cross-section properties approach (Ref. [15]). Column

Global buckling L90G Pcr

C1 C10 C2 C3 C4 C5 C6 C7 C8 C9 C10 C11 C12 C13 C14 Mean Dev.

FEM

41,509 49,895 24,366 51,265 38,744 39,880 68,828 56,381 44,915 49,340 66,832 52,744 68,623 54,023 65,325

(N)

Pcr

Global buckling 2L90G formulation

51,198 51,198 26,618 55,591 42,785 40,758 71,367 58,066 50,637 48,835 65,967 56,241 72,627 54,024 64,826

[15] (N)

P cr Ref½15 P cr FEM

lRef ½15 lFEM

Pcr

1.23 1.03 1.09 1.08 1.10 1.02 1.04 1.03 1.13 0.99 0.99 1.07 1.06 1.00 0.99

0.90 0.99 0.96 0.96 0.95 0.99 0.98 0.99 0.94 1.00 1.00 0.97 0.97 1.00 1.00

14,979 17,809 8101 16,380 13,547 13,827 22,341 18,404 13,235 18,661 24,758 17,627 21,998 16,741 21,659

1.04 0.05

0.98 0.02

FEM

(N)

Pcr

formulation

17,615 17,615 8827 17,124 14,911 13,600 22,920 18,727 14,725 18,118 23,821 17,852 22,506 17,852 21,233

[15] (N)

P cr ref ½15 P crFEM

lRef ½15 lFEM

1.18 0.99 1.09 1.05 1.10 0.98 1.03 1.02 1.11 0.97 0.96 1.01 1.02 1.07 0.98

0.92 1.00 0.96 0.98 0.95 1.01 0.99 0.99 0.95 1.02 1.02 1.00 0.99 0.97 1.01

1.03 0.05

0.99 0.02

82

M. Casafont et al. / Thin-Walled Structures 61 (2012) 71–85

Additional calculations are also carried out with Moen’s approach to determine flexural buckling loads (buckling about the axis perpendicular to the axis of symmetry). The resulting accuracy is similar to that of the torsional–flexural buckling calculations (see Table 8). Buckling loads are slightly more overestimated. Finally, the parametric ranges shown in Table 9 can be adopted as the limits of the reduced thickness model for local, distortional and global buckling, which have been defined from the results of the analysis performed in both the calibration and verification stages. The two subsequent sections are included to verify the performance of the model developed in the present paper when applied to analysis conditions different from those considered in the calibration process: estimation of the buckling load for lengths different from the critical buckling half-wave length, and calculation of the elastic buckling loads of fixed-ended columns.

(ii) ‘‘Weighted average’’ of torsional constant: J g Lnp þ J net Lp J avg ¼ L

ð13Þ

(iii) ‘‘Weighted average’’ shear center x0,avg ¼

x0,g Lnp þ x0,net Lp L

ð14Þ

(iv) Net warping constant: C w,net . where g and net denote gross properties and net properties, respectively. Net section properties can be determined with CUFSM or CUTWP by setting the element thickness to zero at the web holes. The results of the calculation can be observed in Table 7. The resulting accuracy is similar to that obtained with the reduced thickness approach when applied with the classical formulas (Table 6). In Fig. 12, the elastic buckling loads calculated with both approaches are compared for a wide range of lengths. It can be observed that for long column lengths, where torsional–flexural buckling is critical, Moen’s buckling loads are just slightly higher.

6.4. Estimation of the critical elastic buckling load curve by means of the reduced thickness In the previous sections, the local and distortional reduced thickness were calibrated and verified for the critical half-waves, and the global buckling reduced thickness for the L90G and 2L90G length. The aim of the present section is to validate the reduced thickness equations when used to calculate the elastic buckling load of any half-wave, including columns in the transition ranges between buckling modes. For instance, columns in the transition from anti-symmetric distortional buckling mode to torsional– flexural buckling mode are analyzed. Figs. 13 and 14 show the results obtained for column C5. It can be observed that the distortional reduced thickness can be used to estimate the elastic buckling load over a wide range of half-wave lengths: from the transition local–distortional buckling mode to

1.10

Pcr CUTWP/Pcr Ref[15]

1.08 1.06 1.04 1.02 1.00 0.98 0.96

Table 9 Limits of the model of reduced thickness.

0.94 0.92

Parameters

Limits

Parameters

Limits

B/t H/t H/B L Bp/Lp

24–88 26–83 0.48–1.87 50–75 mm r1.6

Lnp/L Bnp/B Bp flange/H Lp flange/L (Bp flangeLp

0.33–0.62 0.51–0.90 o 0.33 o 0.35 o 0.042

0.90 0

5000

10000 15000 Column length (mm)

20000

Fig. 12. Reduced thickness approach (Pcr CUTWP) vs. ‘‘Weighted average’’ crosssection properties approach (Pcr ref[15]).

flange)/(HL)

Table 8 Flexural buckling loads calculated with the ‘‘weighted average’’ cross-section properties approach [15]. Column

Global buckling L90G Pcr

C1 C10 C2 C3 C4 C5 C6 C7 C8 C9 C10 C11 C12 C13 C14 Mean Dev.

FEM

(N)

74,331 77,917 42,141 52,044 78,649 71,518 61,618 74,848 67,932 136,769 68,829 194,177 187,112 57,681 51,658

Pcr

Global buckling 2L90G formulation

87,597 87,597 47,527 55,925 85,652 74,695 66,771 79,422 71,509 137,200 69,680 205,140 195,569 59,890 53,715

[15] (N)

P cr Ref ½15 P cr FEM

l

1.18 1.12 1.13 1.07 1.09 1.04 1.08 1.06 1.05 1.00 1.01 1.06 1.05 1.04 1.04

0.92 0.94 0.94 0.96 0.96 0.98 0.96 0.97 0.97 1.00 0.99 0.97 0.98 0.98 0.98

1.06 0.04

0.97 0.02

Ref ½15

Pcr

FEM

(N)

Pcr

formulation

lFEM 16,704 19,781 10,620 13,136 19,792 17,956 15,560 18,888 17,128 34,498 17,304 48,963 47,141 14,605 13,136

21,899 21,899 11,881 13,981 21,413 18,673 16,692 19,855 17,877 34,300 17,420 51,285 48,892 14,972 13,428

[15] (N)

P cr Ref ½15 P cr FEM

lRef ½15 lFEM

1.31 1.11 1.12 1.06 1.08 1.04 1.07 1.05 1.04 0.99 1.01 1.05 1.04 1.03 1.02

0.87 0.95 0.95 0.97 0.96 0.98 0.97 0.98 0.98 1.00 1.00 0.98 0.98 0.99 0.99

1.05 0.04

0.98 0.02

M. Casafont et al. / Thin-Walled Structures 61 (2012) 71–85

2.00

450000 FEM trL trD trG

350000 300000 250000 200000 150000 100000 50000 0 0

500 1000 1500 Half-wave length (mm)

trD

1.80 CUFSM (or CUTWP) /FEM

400000

Pb (N)

83

2000

trG

1.60

TF equation

1.40 1.20 1.00 0.80 0.60 0.40 0.20 0.00 50

Fig. 13. Result of the linear buckling analysis performed on column C5 with FEM and CUFSM (trL: Local reduced thickness; trD: distortional reduced thickness; trG: global reduced thickness).

60

70 80 % of TF participation

90

100

Fig. 15. Accuracy of DSM when applied to long columns.

1.20 FEM trL

400000

0.80

350000

trL

trD 300000

trD 0.60

trG

Pb (N)

CUFSM (or CUTWP) /FEM

450000 1.00

TF equation 0.40

90% of TF participation

trG

250000 200000 150000

0.20

100000 50000

0.00 0

1000 2000 3000 Half-wave length (mm)

4000

Fig. 14. Accuracy of the analyses performed on column C5.

0 0

500

1000 1500 2000 Length (mm)

2500

3000

Fig. 16. Result of the linear buckling analysis performed on column C5 with fixed ends.

the global buckling mode. Rather good results (just slightly unconservative) are even obtained in the transition range from anti-symmetric distortional buckling to torsional–flexural buckling, and also in this torsional–flexural range. The worst results are concentrated in the local buckling range. On the contrary, the validity of the local and global buckling reduced thickness is limited to their corresponding buckling mode ranges. For example, the global reduced thickness can only be used when the participation of the torsional–flexural buckling mode is high, more than 90%. (It is noted that all percentages of participation shown in the paper correspond to the columns analyzed without perforations.) For lower participations, this thickness leads to very conservative results, that can be considered worse than those obtained with the distortional buckling thickness. Fig. 14 also shows the error ratio resulting from the use of the classical torsion–flexural equations combined with the reduced thickness cross-section properties (circles-TF equation). The resulting loads can be considered even better than those obtained with CUFSM. This may be attributed to the fact that the global buckling reduced thickness introduced in CUFSM is rather small, producing a decrease of the transverse stiffness of the cross-section and, consequently, an amplification of the distortional effects and a reduction of the FSM buckling loads. The performance of the reduced thickness equations when applied to the other columns is similar to that shown for C5. See, for instance, the graph of Fig. 15 that plots the accuracy of the

distortional buckling and global buckling thickness resulting from the analyses performed on the 14 verification columns. Again, it is observed that the global reduced thickness can only be used for columns longer than L90G. 6.5. Analysis of columns with fixed ends A first tentative to extend the reduced thickness approach to columns with general end conditions is presented in this section. The 14 columns of Fig. 3 are analyzed with fixed end supports by means of the new 4.03 version of CUFSM. Figs. 16 and 17 show the results obtained for different lengths of column C5, whose critical buckling mode ranges from local to torsional–flexural buckling. The performance of the reduced thickness equations is similar to that shown in the previous section. The distortional buckling thickness can be used to determine the buckling load in the distortional range of lengths and in the transition ranges between buckling modes (although it can be slightly unconservative when the participation of distortional buckling is small). Local and global reduced thickness should be limited to their corresponding critical ranges. In this case, however, the global reduced thickness only works when applied to very long members with no participation of distortional buckling (the percentage of participation of the

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M. Casafont et al. / Thin-Walled Structures 61 (2012) 71–85

CUFSM (or CUTWP) /FEM

1.2 1

(ii)

0.8 trL trD trG

0.6

(iii)

TF equation

0.4

90% of TF participation

0.2 0 0

1000

2000

3000

4000

Fig. 17. Accuracy of the analyses performed on column C5 with fixed ends.

(iv)

2.00 trD

CUFSM (or CUTWP) /FEM

1.80

trG

1.60

TF equation

1.40 1.20 1.00 0.80 0.60 (v)

0.40 0.20 0.00 50

60

70 80 % of TF participation

90

100

buckling loads. Furthermore, its application should be limited to columns with small web stiffeners and small flange perforations. The distortional tr equation seems to be slightly conservative but, at the same time, is very consistent. It is considered that this model is satisfactory. The results of the tr equation for global buckling are also considered acceptable. This has been verified not only with FEM analyses, but also with complementary calculations applying the ‘‘average weighted’’ cross-section approach and the reduced thickness approach in combination with the classical formulas for torsional–flexural buckling. The model failed just for one of the analyzed columns that showed a high proportion of perforations in the flanges. Nevertheless, the model proposed covers most of the existing types of cross-sections in the market. The use of Moen’s approach [15] does not solve the problem of local buckling loads. On the other hand, when this model is used for distortional buckling and global buckling loads, the results are similar to those of the model developed in the present paper: (iv.1) Moen’s distortional buckling loads are just slightly more conservative and the scatter of the predictions is slightly higher. It is demonstrated that the accuracy of Moen’s approach for distortional buckling can be improved by considering a reduced thickness only at the perforated strips. The reduced thickness equation can be calibrated in a similar way as shown in Section 5.2 of the present paper; (iv.2) when calculated by means of the classical bucking formulas, the reduced thickness approach and the ‘‘weighted average’’ cross-section properties approach lead to similar results. When the reduced thickness model is applied to determine the critical elastic buckling load curve of pinned and fixedended columns, each reduced thickness should be used where its corresponding buckling mode is dominant. The distortional buckling reduced thickness can be used to estimate the buckling load in the transition ranges between dominant buckling modes.

Fig. 18. Accuracy of DSM when applied to long columns with fixed ends.

torsional–flexural mode should be higher than 95%). This has been verified not only for column C5, but also for the other columns (see Fig. 18).

Finally, it is worth pointing out that the reduced thickness approach presented in the paper can be used in practical design, although it does not work for local buckling. This is demonstrated in Refs. [7,8], that are devoted to evaluate different design procedures for rack columns based on the Direct Strength Method.

7. Conclusions

References

An investigation on the calculation of elastic buckling loads of perforated rack columns by means of the Finite Strip Method and the concept of reduced thickness has been presented. After several preliminary tentative models that did not succeed (some of them based on the mechanics of the cross-section), it was decided to carry out a regression analysis to derive the equations for reduced thickness. The outcome of the analysis are three equations, one for each relevant buckling mode: local (critical buckling load), distortional (critical buckling load), and global. Their performance can be summarized with the following comments: (i) The accuracy of the proposed local tr equation is not satisfactory. Local buckling of perforated rack columns is a complex phenomenon that deserves a more sophisticated investigation than what has been presented herein. A simple reduced thickness approach is not able to cover all varieties of perforations and section geometries of rack columns. This is the reason why it is considered that the model presented can only be used to obtain a first approximation to the local

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