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Optimal design of thin-walled open cross-section column for maximum buckling load
Thin-Walled Structures 141 (2019) 423–434
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Full length article
Optimal design of thin-walled open cross-section column for maximum buckling load
T
Md Intaf Alama, Baburaj Kanagarajana, Prasun Janab,∗ a b
Department of Mechanical Engineering, Indian Institute of Technology (Indian School of Mines), Dhanbad, 826004, India Department of Aerospace Engineering, Indian Institute of Technology, Kharagpur, 721302, India
ARTICLE INFO
ABSTRACT
Keywords: Thin-walled column Open cross-section Torsional and flexural buckling Finite element analysis Optimal design
In this work, a finite element based optimization methodology is developed to obtain the optimal designs of thinwalled open cross-section columns for maximum buckling load. As a constraint for the optimization study, the total material volume of the column is kept constant. At first, an analytical formulation based on Bleich's (1952) approach, which considers the combined effect of both torsional and flexural buckling, is used to validate the finite element buckling load computation in ANSYS. Subsequently, these finite element buckling results are coupled with a Genetic Algorithm (GA) based optimization routine in MATLAB to obtain the optimal design of the cross-section of the columns. Optimal results are compared with a base model of the column having a cruciform cross-section. The optimization of the cross-sections results in remarkable enhancement, up to as high as 236%, in the maximum buckling load capacity compared to the base model.
1. Introduction Thin-walled open cross-section columns are widely used in the design of structural components in various fields of engineering including mechanical, civil, marine, and aerospace structures. These thin-walled sections, compared to their thick-walled counterparts, produce structures that exhibit high strength to weight ratio leading to less material cost, ease of handling and transport, and better fuel effeciency. Due to these advantages, thin-walled sections find their applications in marine, automobile and aerospace industries where weight is a critical factor in the design of structural components. However, one drawback of these columns is that the torsional rigidity of the open cross-sections is significantly less than that of the closed sections. As a consequence, these structural members are prone to buckling failure [1,2]. Therefore, in order to use these thin-walled open cross-section columns effectively and efficiently, one requires to understand the behaviour of these columns against their common mode of failure i.e. the buckling instability. In this context, the objective of the present work is to develop a finite element based optimization scheme for optimizing the crosssection of a thin-walled open cross-section column that produces the maximum strength against the buckling failure. The total material volume of the column is kept constant in this optimization study. The finite element method is used for the computation of the critical buckling loads of the columns. The advantage of the finite element
∗
method is that it considers all possible modes of instability including the torsional and flexural modes, and any local mode of buckling of the column. Furthermore, it can be used for complex geometries which are otherwise intractable. The present work employs a Genetic Algorithm (GA) based optimization routine which iteratively interacts with these finite element buckling results and produces the optimal shape of the cross-section as the final outcome. The outline of the remaining paper is as follows: Section 2 discusses the relevant literature pertaining to this study. In Section 3, the critical buckling load is obtained by performing eigenvalue buckling analysis in finite element analysis software ANSYS. Next, the buckling load is calculated analytically based on Bleich's [24] theory followed by the validation of the critical buckling load already calculated using finite element based computation. In Section 4, the details of the optimization methodology are discussed. The optimization results are presented in Section 5 and compared with that of the base model i.e. a column having cruciform cross-section. Finally, Section 6 concludes this study with recommendations for the design of thin-walled open-section columns against buckling failure. 1.1. Relevant literature The fundamental concept of flexural and torsional buckling of structures can be found in well-known textbooks [3–5]. There are also
Corresponding author. E-mail addresses: [email protected] (M.I. Alam), [email protected] (B. Kanagarajan), [email protected], [email protected] (P. Jana).
https://doi.org/10.1016/j.tws.2019.04.018 Received 8 August 2018; Received in revised form 12 April 2019; Accepted 13 April 2019 Available online 03 May 2019 0263-8231/ © 2019 Elsevier Ltd. All rights reserved.
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Fig. 1. Geometry of a typical open cross-section column: (a) complete column, (b) cross-section of the column (dimensions are in mm).
manufacturing constraints by using annealing algorithm method for the optimization. Madeira et al. [21] adopted the Direct Strength Method (DSM) to determine the optimum cross-section of cold formed steel columns to maximize both distortional and local-global buckling strength. Recently, Ye and co-authors [22,23] used the particle swarm optimization method to get the optimum cross-section of cold formed steel beams with the greatest flexure strength and developed a foldedflange cross section that can be used in practical applications. Above review of the literature reveals that the finite element based optimization for the open cross-section of slender columns for maximizing the buckling strength remains unexplored. This paper uses finite element analysis in ANSYS (v15.0) coupled with a MATLAB (vR2014a) optimization routine to obtain the optimal design of the cross-section. These optimal geometry as shown through the results along with the insights generated from the inferences from this study will be useful in the design phases of thin-walled open-section columns.
many interesting research articles that focus on understanding the buckling behaviour of various open-section columns. For example, Seah and Khong [6] presented a semi-analytical and semi-numerical analysis by adopting the Rayleigh-Ritz method for the prediction of the lateral and torsional buckling of channel beams with unbraced longitudinal edge stiffeners. Schardt [7] used Generalized Beam Theory (GBT) to investigate lateral and torsional buckling for channel and hat-section columns by taking the distortional effect into account. Alwis and Wang [8] focused on the Wagner term in flexural-torsional buckling of thinwalled open-section column and examined its importance in the buckling load computation by using two simple bar models. Li [9] and Cheng et al. [10] presented analytical results by using energy principle for computing the flexural and lateral-torsional buckling of cold-formed beams. Later, Erkmen and Attard [11], and Potier-Ferry and co-authors [12,13] used nonlinear geometrical formulation in order to capture a more realistic estimate of the lateral-torsional buckling load for thinwalled open cross-section columns. Recently, Sahraei and Mohareb [14] used finite element method to develop a family of three finite elements in order to capture warping torsion, load position, and shear deformation effects for the analysis of lateral-torsional buckling of beams with doubly symmetric thin-walled cross-section. There are also research articles which discuss the optimization of open-section columns for maximizing their strength. Some of these relevant work are discussed here. Adeli and Karim [17] adopted a neural network model for optimizing the shape for economical use of coldformed steel taking into account the conventional hat, Ie and Z-shapes. By adopting micro genetic algorithm for the optimization, Lee and coauthors [15,16] optimized the shape of the cold formed beams subjected to uniformly distributed loads. Phan et al. [18] used a real-coded niching genetic algorithm to determine the optimized design of coldformed steel frames to explore the effects of stressed-skin action. Ma et al. [19] used genetic algorithm to determine the optimum crosssection of steel channel for working under compression or bending as stated by Eurocode 3. Leng et al. [20] presented more practical and economical optimization solution to incorporate end-user and
1.2. Buckling analysis of a typical open cross-section column As discussed earlier, the main objective of the paper is to develop an optimization routine for obtaining the optimal buckling strength in open cross-section column. To compute the buckling load, commercially available finite element analysis software ANSYS has been used. It can be noted that a finite element solution for the buckling analysis is essential in this case as obtaining an analytical solution for a parametrically modelled open cross-section column often becomes fairly complex and intractable. To support this statement, an analytical solution based on Bleich's [24] approach is also presented. It serves the following two purposes: (i) it shows the complexities of handling the differential equations for any optimization algorithm; (ii) it validates the finite element computation of critical buckling loads in ANSYS. Therefore, in this section, the critical buckling load computation of a typical thin-walled open cross-section column is demonstrated. For this demonstation, an open cross-section column with the panel widths and the orientation as shown in Fig. 1 is considered. The wall thickness 424
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(t) is taken as uniform throughout and equal to 5 mm. Two different lengths, L = 800 mm and 1000 mm, are selected for calculating the buckling loads. The material is assumed to be isotropic with Young's modulus (E) = 210 GPa and Poisson's ratio ( µ ) = 0.3.
Table 1 First critical buckling loads obtained from ANSYS for clamped-simply supported (CS) and both end simply supported (SS) column. Length, L (mm)
1.3. Finite element computation
Pcr (in KN) using ANSYS
1000 800
Linear (eigenvalue) buckling analysis in ANSYS has been used to compute the buckling load of the column. The geometric model of the column has been meshed with SHELL181 element available in ANSYS. The element has six degrees of freedom at each node (three rotational and three translational) and it is used to model thin to moderately thick plate structures. Mesh convergence study has been carried out and it has been found that converged results are achieved when the element size is taken as A/50, where A is the total panel width (190 mm in this case). Two different end boundary conditions are considered in this analysis: i) one end clamped and the other end simply supported (CS), ii) both ends simply supported (SS). In ANSYS, these boundary conditions have been implemented as follows: i) CS boundary condition: Both rotational and translational degrees of freedoms have been constrained at one end. At the other end where the force is applied, translational degrees of freedoms have been constrained in x and y direction and it remains free to move along the z-direction (see Fig. 2). ii) SS boundary condition: As the model and boundary conditions are symmetric, a half model has been considered and the symmetric boundary condition has been applied at one end. At the end where the force is applied, translational degrees of freedoms have been constrained in x and y direction, keeping the edge free to move along z-direction. In ANSYS, an axial force P (say 1 kN), equally distributed on all edge nodes, is applied at the simply supported end. With this applied axial force, static analysis has been performed initially with pre-stress effect activated and then the eigenvalue buckling analysis is performed. The buckling load factor (BF) obtained from the eigenvalue buckling analysis is multiplied to the applied load P to get first the critical buckling load (Pcr ) which leads to the structural instability of the column. Thus the first critical buckling load is given by Pcr = BF × P . The first critical
Boundary condition: CS
Boundary condition: SS
251.40 327.40
171.11 213.74
buckling loads for these four cases (two boundary conditions and two column lengths) are given in Table 1. The first mode shape for length L = 800 mm and L = 1000 mm with the combined effect of both torsional and flexural buckling modes is shown in Fig. 3. 1.4. Analytical calculation of critical buckling load The analytical solution, presented here, is based on the differential equation of buckling of centrally loaded columns by torsion and flexure given by Friedrich Bleich [24]. Bleich's theory is based on the energy method and the same is discussed briefly for a general column crosssection. Consider a thin-walled open cross-section column made of several flat plates of uniform thickness as shown in Fig. 4. It is subjected to centrally applied axial compressive force i.e. along z-direction, where x and y are the principal axes of the cross-section. For the arbitrary crosssection shown in Fig. 4, in the deformed configuration, the origin O of the cross section is translated and is twisted with angle of twist β. The translation of the cross-section is defined by u and v along ξ and η axes, respectively. Because u, v and β are small, u simply equals the x-direction displacement of O, and v equals the y-direction displacement of O. As shown in Fig. 4, Gk and Gk + 1 are the centroid of the k th and (k + 1)th plates respectively and Pk is the point of intersection of the plate. The total potential energy (U) of the deformed bar can be divided into two parts: the potential energy due to axial compressive stresses and the strain energy V. The strain energy V is further divided into axial strain V1 due to longitudinal stresses and the strain energy V2 produced due to twisting as determined using St. Venant theory. The total strain energy for n number of flat plates for bending and stretching is obtained as:
V1 =
L
1 2
n
(EIk s"2k + EAk k2) dz
(1)
0 k=1
where Ik is the centroidal moment of inertia, k is the longitudinal strain and Ak is the cross sectional area of the k th plate respectively. sk is the displacement of the centroid of the k th plate along the direction vector and given as:
sk = {uiˆ + vjˆ + (kˆ × r k )} tˆk ¯
(2)
where r k is the distance of the centroid from the origin of the cross¯ section. As the successive plates are joined at certain location Pk , the strain in both the plates must be equal. Strain for k th and (k + 1)th plates are given as: k sk
k
where
k
=
k+1
+
k + 1 sk + 1
, k = 1, 2,
., n
1
(3)
is the distance of centroid from Pk . By assumption:
n
Ak k= 1
Fig. 2. Finite element mesh of the clamped-simply supported (CS) column. At the bottom end Ux, Uy, Uz, Rx, Ry, and Rz are constrained and at the top end Ux and Uy are constrained.
k
=0
(4)
Substituting the values obtained from Eqs. (2)–(4) in Eq. (1), one can get the most general form of quadratic expression as: 425
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Fig. 3. First buckling mode (isometric view) for clamped-simply supported (CS) column: (a) for L = 800 mm, (b) for L = 1000 mm.
and B6 do not occur in the conventional theory of bending. The strain energy for the entire length due to twisting is obtained as:
V2 =
L
1 2
2
GK
dz
(6)
0
where G is the shear modulus, is the rate of angle of twist and the value of torsion constant K for a flat plate of width b and thickness t is given by
K=
bt 3 3
(7)
Due to axial compressive stress the rod deforms laterally and the vertical downward motion of the top of the rod can be obtained as:
1 2
=
L
2
u z
0
dz
(8)
By a similar calculation, the squares of the partial derivatives of lateral deflections in two orthogonal directions are added. Given a typical point xiˆ + yjˆ in the cross section, its lateral deflection is
kˆ × (xiˆ + yjˆ ) = (u
uiˆ + vjˆ +
1 (x , y ) = 2
Table 2 Comparison of the analytical buckling results with the corresponding finite element results for clamped-simply supported (CS) and both ends simply supported (SS) column. Length, L (mm)
Pcr (in KN) ANSYS
Analytical
CS
1000 800 1000 800
251.40 327.40 171.11 213.74
254.97 333.03 176.69 220.33
SS
V1 =
1 2
(9)
Hence, the total deflection for entire length is given as:
Fig. 4. Cross-section showing translation and rotation with some key points.
Boundary conditions
y ) iˆ + (v + x ) jˆ.
2
2
L
u
y
+ (v +
x } dz (10)
0
The potential energy due to axial compressive stress the entire length is obtained as:
Ratio of Pcr (ANSYS/ Analytical)
Ua =
at the end for
(x , y ) dA
(11)
Substituting the value of Eq. (10) in Eq. (11),
0.986 0.983 0.968 0.970
1 2
Ua =
L
2
A u +v
2
2
+ Ip
2Ayc u
+ 2Axc v
dz
0
(12)
L
(B1 u
2
+ B2 v
2
+ B3
2
+ B4 u v + B5 u
+ B6 v
where Ip is the polar moment of the cross sectional area about the origin O, and x c and yc are the coordinates of the centroid. The total potential energy due to deformation is given as:
) dz
0
(5) where B1, B2 , B3 , B4 , B5 and B6 depends on the geometric properties of the cross-section. The coefficients B1, B2 and B4 are equal to the moments of inertia of the complete cross section and the coefficients B3 , B5
L
U=
f (u", v", 0
426
"u v,
) dz
(13)
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Fig. 5. Schematic diagram of the cross-sections: (a) Parametric cross-section for the optimization, (b) Base model – a cruciform section. Table 3 Parameters for defining a number of regular cross-sections. The corresponding minimum principal second moment of area (Imin ) is also shown for these cross-sections. Name of section
Parameter values
Imin (mm4 )
C-section H-section I-section T-section Cruciform
a2 = 70 mm; a5 = a7 = 60 mm; a1 = a3 = a4 = a6 = a8 = a9 = 0 mm a2 = 70 mm; a4 = a5 = a6 = a7 = 30 mm; a1 = a3 = a8 = a9 = 0 mm a2 = 110 mm; a4 = a5 = a6 = a7 = 20 mm; a1 = a3 = a8 = a9 = 0 mm a2 = 80 mm; a6 = a7 = 55 mm; a1 = a3 = a4 = a5 = a8 = a9 = 0 mm a1 = a2 = a4 = a5 = 47.5 mm; a3 = a6 = a7 = a8 = a9 = 0 mm
331875 180625 53958 533683 358177
Fig. 6. Flow chart used for obtaining the optimal cross-section.
(13a)
U = Ua + V
d2 dz 2
Substituting the values of Ua and V in Eq. (13a), the following equation is obtained.
f (u”, v”, v,
=
1 2
B1 u
+ B2 v 2
+ GK
2
2
+ B3 2
A u +v
+ B4 u v + B5 u 2
2Ayc u
(14)
+
B4 v 2
+
B5 2
+ A (u
yc
)=0
(15a)
B2 v
+
B4 u 2
+
B6 2
+ A (v + x c
)=0
(15b)
B3
+
B5 u 2
+
B6 v 2
+ B6 v + 2Axc v (13b)
i.e.
=0
B1 u
2
+ Ip
f q
d dz
where q stands for u, v and . Thus the equations of equilibrium are:
”, u 2
f q
Equation (13b) can be solved by the calculus of variation method
GK
+
(Ip
Ayc u + Ax c v ) = 0 (15c)
The differential equilibrium Eqs. (15a) and (15b) and (15c) have been solved in symbolic math package Maple using Galerkin's 427
(x ),
428
44.40 94.40 136.31
34.95 167.41 225.92
373.72 401.07
0.00 657.25 657.25
4.34 830.77 866.82
31.92 833.41 1099.50
147.68 839.35 2078.9
236.16 856.93 2880.65
4000
In this section, the finite element analysis results of thin-walled open cross-section obtained from Section 3.1 have been validated by the analytical results that are obtained in Section 3.2. Table 2 shows this comparison for the two boundary conditions and for two different lengths, L = 800 mm and L = 1000 mm as mentioned in Section 3.1. From Table 2, it is evident that the buckling results are in good agreement. Therefore, it is now established that the finite element analysis in ANSYS can predict the buckling failure of the columns with a very good accuracy. Note that the objective of the present study is to find the optimum cross-section of the open-section column which provides maximum strength against the buckling failure. In order to study this, different cross-section shapes of the column must be obtained by varying the panel widths and the angles of each panel. This results in finding a set of large number of different complex cross-sections and eventually leads to a tedious process of obtaining analytical solutions for all these cross-sections. To overcome this difficulty, the buckling computation has been done using parametric modelling in ANSYS and the optimization has been done by linking these ANSYS computations
CS-8
1.5. Comparison of buckling results
3000
Use of above Galerkin's approach leads to an eigenvalue problem and the associated eigenvalues are solved to compute the critical buckling loads.
CS-7
(16b)
2000
x L
CS-6
(x )= sin
1500
ii) SS boundary condition:
CS-5
(16a)
1250
x2 x sin L2 L
2x L
CS-4
(x )=
1000
i) CS boundary condition:
CS-3
where 1, 2 and 3 are some constants. The shape functions satisfying the two boundary conditions, are taken as follows:
750
(x )
CS-2
3
10.6 mm, a2 = 50.3 mm, a3 = 09.1 mm, a4 = 24.4 mm, a5 = 35.6 mm, a6 = 24.5 mm, 35.7 mm, a8 = 3.6⁰, a9 = 1.5⁰ 2.5 mm, a2 = 43.9 mm, a3 = 2.6 mm, a4 = 31.6 mm, a5 = 37.4 mm, a6 = 41.0 mm, 31.0 mm, a8 = 7.0⁰, a9 = 6.6⁰ 44.9 mm, a2 = 23.4 mm, a3 = 09.4 mm, a4 = 38.7 mm, a5 = 46.7 mm, a6 = 18.9 mm, 07.9 mm, a8 = 2.6⁰, a9 = 15.5⁰ 47.6 mm, a2 = 32.5 mm, a3 = 05.9 mm, a4 = 46.7 mm, a5 = 46.1 mm, a6 = 05.4 mm, 05.9 mm, a8 = 0.4⁰, a9 = 1.1⁰ 47.5 mm, a2 = 47.5 mm, a3 = 0.0 mm, a4 = 47.5 mm, a5 = 47.5 mm, a6 = 0.0 mm, 0.0 mm, a8 = 0.0⁰, a9 = 0.0⁰ 2.5 mm, a2 = 58.8 mm, a3 = 3.0 mm, a4 = 4.1 mm, a5 = 4.9 mm, a6 = 59.4 mm, 57.3 mm, a8 = 17.8⁰, a9 = 0.1⁰ 11.0 mm, a2 = 61.2 mm, a3 = 2.4 mm, a4 = 2.4 mm, a5 = 2.2 mm, a6 = 62.4 mm, 48.4 mm, a8 = 3.4⁰, a9 = 3.4⁰ 7.6 mm, a2 = 66.0 mm, a3 = 2.4 mm, a4 = 2.1 mm, a5 = 1.9 mm, a6 = 59.2 mm, 51.0 mm, a8 = 3.1⁰, a9 = 3.0⁰
=
500
(x )
CS-1
2
Parameter Variables
v=
Length, L (mm)
(x )
Case
1
Table 4 Optimal results and comparison with cruciform section (base model) for clamped-simply supported (CS) column.
u=
= = = = = = = = = = = = = = = =
approach. The displacements trial functions are defined in terms of shape function ‘ϕ( x )’ for both the boundary conditions as
a1 a7 a1 a7 a1 a7 a1 a7 a1 a7 a1 a7 a1 a7 a1 a7
Percentage increase Cruciform results (KN) Optimal results (KN)
Fig. 7. Iteration history of the GA optimization process for the clamped-simply supported (CS) column with L = 750 mm.
7.32
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Table 5 Optimal cross-sections and the mode shapes for CS column with different lengths: L = 500 mm (CS-1), 7500 mm (CS-2) and 1000 mm (CS-3). Case
Optimal cross-section
First buckling mode shape
CS-1
CS-2
CS-3
with a Genetic Algorithm (GA) based optimization scheme in MATLAB. Similar finite element based optimization studies for the optimal design of perforated plate structures, for maximizing the uniaxial in-plane buckling load, can be found in recent works of Jana and co-authors [25,26].
[X, fval] = ga (obj funcn, no. of variables, LB, UB, options). The default options from MATLAB GA simulation tool box are used for this analysis. Some of these options are summarized below:
2. Optimal design of the cross-section This section discusses the details of the optimization procedure adopted in the present study. The input variables, for defining a seven panel open cross-section as shown in Fig. 5 (a), are taken as a = {a1, a2, a3, a4, a5, a6, a7, a8, a9} where, a1-a7 are the panel widths and a8, a9 are the angles. The output variable i.e. the critical buckling load will be the function of these input variables a. As the constraint for this optimization problem, the sum of the panel width variables a1 to a7 are kept as constant and equal to 190 mm. The lower and upper bound for each variable from a1 to a7 are 0 mm and 190 mm, respectively. The angle variables a8 and a9 are varied between 0° and 180°. Wall-thickness (t) of each panel is taken to be 5 mm. The material is isotropic with Young's modulus (E) = 210 GPa and Poisson's ratio ( µ ) = 0.3. Mathematically, the above optimization problem can be written as follows. It can be shown that, with the proper assignments of these nine input variables, the adopted parametric model is able to produce some regular cross-sections such as I-section, H-section, T-section, C-section, Cruciform section etc. See Table 3 for some arbitrarily chosen parameter values for these cross-sections with a total panel width of 190 mm. The objective of this optimization study is to maximize the critical buckling load. In order to use the in-built GA optimization routine (a minimization scheme) in MATLAB, the objective function is defined as, Pcr (a) = Pcr (a) , where Pcr (a) is a non-zero positive number representing the critical buckling load computed using ANSYS. The negative sign is used to convert the maximization problem to a standard minimization problem. The syntax used in MATALB for the GA is given as.
Population type
double vector
Population Size Selection Function Elite Count Crossover fraction Mutation function
50 Stochastic Uniform 2.5 0.8 constraint dependent
The flow chart of the optimization process is shown in Fig. 6. The finite element buckling analysis in ANSYS takes ak as the input parameters and returns the positive buckling load Pcr which is used to calculate the objective function (Pcr ). MATLAB routine links the ANSYS output to the in-built optimization algorithm. The objective function (Pcr ) is calculated and the optimal parameters (a ) are obtained as a final output from these iterations. A sample of ANSYS commands used for the parametric modelling of the open-section column is given in Appendix A. 3. Results and discussions In this section, the results obtained from the above optimization study are presented. The optimal cross-sections for the maximum critical buckling loads are obtained for eight different lengths of the column and two different boundary conditions. Fig. 7 shows the iteration history of the GA optimization process for a typical case of clamped-simply supported (CS) column with L = 750 mm. Note that these eight length parameters are arbitrarily chosen here without any specific prefernces. The optimal buckling results are finally compared with the base model i.e. a cruciform cross-section as schematically shown in Fig. 5 (b). 429
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Table 6 Optimal cross-sections and mode shapes for CS column with different lengths: 1250 mm (CS-4), 1500 mm (CS-5), 2000 mm (CS-6), 3000 mm (CS-7), 4000 mm (CS-8). Case
Optimal cross-section
First buckling mode shape
CS-4
CS-5
CS-6
CS-7
CS-8
Table 7 Buckling loads (Pcr ) for the regular cross-section (shown in Table 3) columns with CS boundary condition and comparison with corresponding optimal buckling loads (Pcr (opt ) ). Length (mm)
500 750 1000 1250 1500 2000 3000 4000
C-section
H-section
I-section
T-section
Cruciform
Pcr (kN)
Pcr Pcr (opt )
Pcr (kN)
Pcr Pcr (opt )
Pcr (kN)
Pcr Pcr (opt )
Pcr (kN)
Pcr Pcr (opt )
Pcr (kN)
Pcr Pcr (opt )
1030.0 815.9 541.4 402.9 324.0 238.9 164.6 99.7
0.36 0.39 0.49 0.46 0.49 0.60 0.73 0.73
2755.0 1304.0 764.7 481.5 335.7 189.3 84.4 47.5
0.95 0.63 0.70 0.55 0.51 0.48 0.37 0.35
842.0 398.3 227.1 146.0 101.6 57.0 25.4 14.3
0.29 0.19 0.21 0.17 0.15 0.14 0.11 0.10
438.8 417.9 405.1 392.0 376.2 332.9 215.2 134.0
0.15 0.20 0.37 0.45 0.57 0.84 0.95 0.98
856.9 839.3 833.4 830.7 657.2 373.7 167.4 94.4
0.30 0.40 0.76 0.96 1.00 0.93 0.74 0.69
3.1. Clamped-simply supported (CS) column
column produce an increase in critical buckling load by 236.16%, 147.68%, 31.92%, 4.33%, 7.32%, 34.95%, and 44.40% for column lengths of 500 mm, 750 mm, 1000 mm, 1250 mm, 2000 mm, 3000 mm, and 4000 mm, respectively, when compared with the base model i.e. the cruciform section. Moreover, for the column length of 1500 mm, the
Table 4 shows the optimal results for the clamped-simply supported (CS) column for eight different length parameters. It is worthy to note that the optimal cross-sections for the clamped-simply supported (CS) 430
Thin-Walled Structures 141 (2019) 423–434
46.13 71.95
55.97
81.90 122.05
49.02
183.42 247.66
35.02
324.68 348.03
7.19
465.01 465.01
0.00
720.00 720.00
0.00
805.05 982.38
22.03
802.49
12.1 mm, a2 = 38.0 mm, a3 = 11.2 mm, a4 = 39.9 mm, a5 = 31.1 mm, a6 = 31.3 mm, 26.6 mm, a8 = 7.5⁰, a9 = 3.3⁰ 48.3 mm, a2 = 23.8 mm, a3 = 08.2 mm, a4 = 44.3 mm, a5 = 45.6 mm, a6 = 11.5 mm, 08.3 mm, a8 = 3.2⁰, a9 = 17.6⁰ 47.5 mm, a2 = 47.5 mm, a3 = 0.0 mm, a4 = 47.5 mm, a5 = 47.5 mm, a6 = 0.0 mm, 0.0 mm, a8 = 0.0⁰, a9 = 0.0⁰ 47.5 mm, a2 = 47.5 mm, a3 = 0.0 mm, a4 = 47.5 mm, a5 = 47.5 mm, a6 = 0.0 mm, 0.0 mm, a8 = 0.0⁰, a9 = 0.0⁰ 04.1 mm, a2 = 49.1 mm, a3 = 09.6 mm, a4 = 53.8 mm, a5 = 53.2 mm, a6 = 11.2 mm, 09.3 mm, a8 = 1.2⁰, a9 = 3.3⁰ 0.0 mm, a2 = 74.71 mm, a3 = 0.0 mm, a4 = 0.0 mm, a5 = 0.0 mm, a6 = 60.80 mm, 54.54 mm, a8 = 0.0⁰, a9 = 0.0⁰ 0.0 mm, a2 = 77.0 mm, a3 = 0.0 mm, a4 = 0.0 mm, a5 = 0.0 mm, a6 = 54.8 mm, 57.4 mm, a8 = 0.0⁰, a9 = 0.0⁰ 0.0 mm, a2 = 77.6 mm, a3 = 0.0 mm, a4 = 0.0 mm, a5 = 0.0 mm, a6 = 55.8 mm, 55.8 mm, a8 = 0.0⁰, a9 = 0.0⁰
1656.90
106.47
cruciform section turns out to be the optimal cross-section. For a better visualization, the schematic diagrams of the optimal cross-sections for the CS column along with their first critical buckling mode shapes are shown in Tables 5 and 6. The mode shape plots show that the first four columns with the optimal cross-section buckle in both flexure and torsional mode. Whereas, the last four columns buckles primarily in flexure mode. Further, the first critical buckling loads with CS boundary condition for five regular cross-sections, as depicted in Table 3, are also computed. The buckling results are shown in Table 7. Table 7 shows that, for a particular length, the buckling loads with the regular cross-sections are always lower than the optimal results reported in Table 4. Hence, this table confirms, at least partially, that the adopted optimization scheme produces optimal cross-sections for which the buckling load is maximum. Table 7 also shows the ratios of the buckling loads of these regular cross-section columns to the corresponding optimal buckling loads. From these load ratios along with the schematic diagrams of the optimal cross-sections shown in Tables 5 and 6, it can be observed that an H-section performs better under compressive loads for relatively shorter columns in which buckling occurs due to both flexure and torsion. However, as the length increses, cruciform sections provides higher buckling loads. For further increase in length, T-section is proved to be a better cross-section. The superior performance of T-section for longer columns can be attributed to the fact that the buckling modes are predominantly due to flexure and T-sections have a higher Imin (see Table 3) to resist the flexural buckling mode. 3.2. Both ends simply supported (SS) column
= = = = = = = = = = = = = = = =
Optimal results for the simply-simply supported (SS) columns are shown in Table 8. The optimal cross-sections for column lengths 500 mm, 750 mm, 1500 mm, 2000 mm, 3000 mm, and 4000 mm produce an increase of 106.47%, 22.03%, 7.19%, 35.02%, 49.02%, and 55.97% in the buckling load, respectively, when compared to the base model. However, for lengths 1000 mm and 1250 mm, the cruciform section turns out to be the optimal cross-section. The schematic diagrams of the optimal cross-sections for these SS columns along with their first buckling mode shapes are shown in Tables 9 and 10. The buckling results for the regular cross-sections with SS boundary conditions are shown in Table 11. From Table 11, we observe that no regular cross-section shows higher buckling resistance compared to the optimal results reported in Table 8 for the SS column. Table 11 also shows the ratios of the buckling loads to the optimal results. From these ratios along with the shapes of the optimal cross-sections shown in Tables 9 and 10, it can be noticed that, similar to clamped-simply supported columns, H-section provides better resistance to buckling for shorter lengths of column. For moderate column lengths, cruciform section turns out to be the optimal cross-section. Furthermore, for even longer columns, T-section provides better buckling resistance as it possesses maximum Imin to prevent the flexural buckling mode. 4. Conclusions
4000
The study began with an objective to obtain the optimal design of thin-walled open cross-section columns for maximum buckling load considering the combined effect of both torsion and flexure buckling modes. The optimal design of the cross-section was obtained by coupling the finite element buckling results in ANSYS with the in-built Genetic Algorithm (GA) optimization algorithm in MATLAB. These optimal designs were compared with the base cruciform model. From this study, the following conclusions are drawn.
• Optimal cross-sections show a significant increase, up to as high as 236%, in maximum buckling load capacity when compared with the base model of the cruciform section.
SS-8
3000 SS-7
2000 SS-6
1500 SS-5
1250 SS-4
1000 SS-3
SS-2
750
a1 a7 a1 a7 a1 a7 a1 a7 a1 a7 a1 a7 a1 a7 a1 a7 500 SS-1
Parameter Variables Length, L (mm) Case
Table 8 Optimal results and comparison with cruciform section (base model) for simply-simply supported (SS) column.
Optimal results (KN)
Cruciform results (KN)
Percentage increase
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Table 9 Optimal cross-section and the mode shapes for SS columns with different lengths: L = 500 mm (SS-1), 7500 mm (SS-2), 1000 mm (SS-3) and 1250 mm (SS-4). Due to symmetry, half of the length is considered. Case
Optimal cross-section
First buckling mode shape
SS-1
SS-2
SS-3
SS-4
• As the total area of the cross-section is kept constant, the amount of •
• • •
compared to these regular sections, have significantly higher resistance against the buckling instability.
material in the base model and in optimal design remains the same. Thus the significant increase in buckling load capacity is achieved without compromising the cost of the material. As the length of the column becomes short the columns buckle predominantly in the torsional mode and the cruciform section is no longer better for resisting the buckling failure. The optimal crosssections obtained in this study, for different boundary conditions, provides substantial increase in the buckling load capacity. It is observed that, among the five regular sections, H-section is closer, in terms of performance, to these optimal cross-sections for shorter columns. For a longer column, buckling mainly takes place in the flexural mode and T-section turns out to be the best design for resisting buckling of the column. For a moderate column length, the cruciform section turns out to be the better performing cross-section as it has substantial resistance to both flexural and torsional buckling modes. Regular sections e.g. I-section, T-section, H-section, etc. have some advantages due to their symmetry, e.g. an ease in manufacturing. However, it has been shown that the optimal cross-sections, as
While designing the thin-walled open cross-section columns the designer may consider the above important points pertaining to their buckling behaviour. Acknowledgements Authors would like to thank Anindya Chatterjee and Jishnu Bhattacharya for discussions and the Science and Engineering Research Board, Government of India, for financial support (Grant No.: ECR/ 2016/000964). Authors also thank anonymous reviewers for their valuable comments and suggestions which helped them in improving the manuscript. Appendix A. ANSYS APDL used for the buckling analysis !This APDL is written in a compact form using ‘$’symbol. !Here, MATLAB routine supplies the a1-a7, theta1 and theta2 values. 432
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Table 10 Optimal cross-section and the mode shapes for SS columns with different lengths: L = 1500 mm (SS-5), 2000 mm (SS-6), 3000 mm (SS-7) and 4000 mm (SS-8). Due to symmetry, half of the length is considered. Case
Optimal cross-section
First buckling mode shape
SS-5
SS-6
SS-7
SS-8
Table 11 Buckling results for the regular cross-section (as shown in Table 3) columns with SS boundary condition. Length (mm)
500 750 1000 1250 1500 2000 3000 4000
C-section
H-section
I-section
T-section
Cruciform
Pcr (kN)
Pcr Pcr (opt )
Pcr (kN)
Pcr Pcr (opt )
Pcr (kN)
Pcr Pcr (opt )
Pcr (kN)
Pcr Pcr (opt )
Pcr (kN)
Pcr Pcr (opt )
709.8 483.9 337.3 264.6 221.6 168.9 86.3 48.5
0.43 0.49 0.47 0.57 0.63 0.68 0.71 0.67
1419.4 648.02 367.71 236.32 154.00 92.6 41.3 23.2
0.86 0.66 0.51 0.50 0.44 0.37 0.34 0.32
429.9 196.4 111.2 71.36 49.62 27.9 12.4 6.7
0.25 0.20 0.15 0.15 0.14 0.11 0.10 0.09
377.7 376.6 370.1 347.7 311.8 229.4 117.9 68.9
0.23 0.38 0.51 0.75 0.89 0.93 0.97 0.96
802.4 805.0 720.0 465.0 324.6 183.4 81.9 46.1
0.48 0.82 1.00 1.00 0.93 0.74 0.67 0.64
FINISH. /CLEAR, START $/FILNAME, buckling_run $/PREP7. $height = 1.25 $thick = 0.005. !This data (parameter values) is fed from MATLAB routine. a1 = 0.0476 $a2 = 0.0325 $a3 = 0.0059 $a4 = 0.0467 $a5 = 0.0461 $a6 = 0.0054. $a7 = 0.0059 $theta1 = 0.0067 $theta2 = 0.0192 $a_sum = 0.19. K,1,0,0,0 $K,2,0,a1,0 $K,3,0,a1+a2, $K,4,0,a1+a2+a3,0. $K,5,-a4*cos(theta1),a1-a4*sin(theta1),0. $K,6,a5*cos(theta1),a1+a5*sin(theta1),0. $K,7,-a6*cos(theta2),a1+a2-a6*sin(theta2),0. $K,8,a7*cos(theta2),a1+a2+a7*sin(theta2),0 $K, 9,0,0, height. $L,1,9 $L,1,2 $L,2,3 $L,3,4 $L,2,5 $L,2,6 $L,3,7 $L,3,8.
$ADRAG,2,1 $ADRAG,3,1 $ADRAG,4,1 $ADRAG,5,1. $ADRAG,6,1 $ADRAG,7,1 $ADRAG,8,1 $nummrg,all, all. MPTEMP, $MPTEMP,1,0 $MPDATA,EX,1, 210e9 $MPDATA, PRXY,1,0.3. $ET,1, SHELL181 $ sect,1,shell, $ secdata, thick,1,0.0,3. $secoffset, MID $seccontrol, $AESIZE,ALL,a_sum/50 $AMESH, ALL. NSEL,S,LOC,Z,0,1E-6 $D,ALL,0,ALL, $ALLS $NSEL,S,LOC,Z,height1E-6, height. $*GET,NODECOUNT, NODE,0,COUNT, $D,ALL, UX $D,ALL, UY $F,ALL,FZ,-1000/NODECOUNT $ALLS. /SOLU $ANTYPE, STATIC $PSTRES, ON $SOLVE $FINISH. /SOLU $ANTYPE, BUCKLE $BUCOPT, LANB,1 $SOLVE $FINISH. /POST1 $RSYS, SOLU $FILE, buckling_run, rst $SET,1,1. 433
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*GET,factor, ACTIVE,0,SET,FREQ,/OUT,buck_fact, dat $*vwrite,1, factor. (F4.1,2X,F10.4) /OUT. SAVE $FINISH.
401–415. [13] F. Mohri, N. Damil, M. Potier-Ferry, Pre-buckling deflection effects on stability of thin-walled beams with open sections, Steel Compos. Struct. 13 (2012) 71–89. [14] A. Sahraei, M. Mohareb, Upper and lower bound solutions for lateral-torsional buckling of doubly symmetric members, Thin-Walled Struct. 102 (2016) 180–196. [15] H. Adeli, A. Karim, Neural network model for optimization, J. Struct. Eng. 123 (1997) 1535–1543. [16] J. Lee, S.-M. Kim, H. Seon Park, B.H. Woo, Optimum design of cold-formed steel channel beams using micro genetic algorithms, Thin-Walled Struct. 27 (2005) 17–24. [17] J. Lee, S.-M. Kim, H. Seon Park, Optimum design of cold-formed steel columns by using micro genetic algorithms, Thin-Walled Struct. 44 (2006) 952–960. [18] D.T. Phan, J.B.P. Lim, T.T. Tanyimboh, A.M. Wrzesien, W. Sha, R.M. Lawson, Optimal design of cold-formed steel portal frames for stressed-skin action using genetic algorithm, Eng. Struct. 93 (2015) 36–49. [19] W. Ma, J. Becque, I. Hajirasouliha, J. Ye, Cross-sectional optimization of coldformed steel channels to Eurocode 3, Eng. Struct. 101 (2015) 641–651. [20] J. Leng, Z. Li, J.K. Guest, B.W. Schafer, Shape optimization of cold-formed steel columns with fabrication and geometric end-use constraints, Thin-Walled Struct. 85 (2014) 271–290. [21] J.F.A. Madeira, J. Dias, N. Silvestre, Multiobjective optimization of cold-formed steel columns, Thin-Walled Struct. 96 (2015) 29–38. [22] J. Ye, I. Hajirasouliha, J. Becque, A. Eslami, Optimum design of cold-formed steel beams using Particle Swarm Optimisation method, J. Constr. Steel Res. 122 (2016) 80–93. [23] J. Ye, I. Hajirasouliha, J. Becque, K. Pilakoutas, Development of more efficient coldformed steel channel sections in bending, Thin-Walled Struct. 101 (2016) 1–13. [24] F. Bleich, Buckling Strength of Metal Structures, McGraw-Hill Book Company, NEW YORK, 1952. [25] P. Jana, Optimal design of uniaxially compressed perforated rectangular plate for maximum buckling load, Thin-Walled Struct. 103 (2016) 225–230. [26] P. Choudhary, P. Jana, Position optimization of circular/elliptical cutout within an orthotropic rectangular plate for maximum buckling load, Steel Compos. Struct. 29 (2018) 39–51.
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Full length article
Optimal design of thin-walled open cross-section column for maximum buckling load
T
Md Intaf Alama, Baburaj Kanagarajana, Prasun Janab,∗ a b
Department of Mechanical Engineering, Indian Institute of Technology (Indian School of Mines), Dhanbad, 826004, India Department of Aerospace Engineering, Indian Institute of Technology, Kharagpur, 721302, India
ARTICLE INFO
ABSTRACT
Keywords: Thin-walled column Open cross-section Torsional and flexural buckling Finite element analysis Optimal design
In this work, a finite element based optimization methodology is developed to obtain the optimal designs of thinwalled open cross-section columns for maximum buckling load. As a constraint for the optimization study, the total material volume of the column is kept constant. At first, an analytical formulation based on Bleich's (1952) approach, which considers the combined effect of both torsional and flexural buckling, is used to validate the finite element buckling load computation in ANSYS. Subsequently, these finite element buckling results are coupled with a Genetic Algorithm (GA) based optimization routine in MATLAB to obtain the optimal design of the cross-section of the columns. Optimal results are compared with a base model of the column having a cruciform cross-section. The optimization of the cross-sections results in remarkable enhancement, up to as high as 236%, in the maximum buckling load capacity compared to the base model.
1. Introduction Thin-walled open cross-section columns are widely used in the design of structural components in various fields of engineering including mechanical, civil, marine, and aerospace structures. These thin-walled sections, compared to their thick-walled counterparts, produce structures that exhibit high strength to weight ratio leading to less material cost, ease of handling and transport, and better fuel effeciency. Due to these advantages, thin-walled sections find their applications in marine, automobile and aerospace industries where weight is a critical factor in the design of structural components. However, one drawback of these columns is that the torsional rigidity of the open cross-sections is significantly less than that of the closed sections. As a consequence, these structural members are prone to buckling failure [1,2]. Therefore, in order to use these thin-walled open cross-section columns effectively and efficiently, one requires to understand the behaviour of these columns against their common mode of failure i.e. the buckling instability. In this context, the objective of the present work is to develop a finite element based optimization scheme for optimizing the crosssection of a thin-walled open cross-section column that produces the maximum strength against the buckling failure. The total material volume of the column is kept constant in this optimization study. The finite element method is used for the computation of the critical buckling loads of the columns. The advantage of the finite element
∗
method is that it considers all possible modes of instability including the torsional and flexural modes, and any local mode of buckling of the column. Furthermore, it can be used for complex geometries which are otherwise intractable. The present work employs a Genetic Algorithm (GA) based optimization routine which iteratively interacts with these finite element buckling results and produces the optimal shape of the cross-section as the final outcome. The outline of the remaining paper is as follows: Section 2 discusses the relevant literature pertaining to this study. In Section 3, the critical buckling load is obtained by performing eigenvalue buckling analysis in finite element analysis software ANSYS. Next, the buckling load is calculated analytically based on Bleich's [24] theory followed by the validation of the critical buckling load already calculated using finite element based computation. In Section 4, the details of the optimization methodology are discussed. The optimization results are presented in Section 5 and compared with that of the base model i.e. a column having cruciform cross-section. Finally, Section 6 concludes this study with recommendations for the design of thin-walled open-section columns against buckling failure. 1.1. Relevant literature The fundamental concept of flexural and torsional buckling of structures can be found in well-known textbooks [3–5]. There are also
Corresponding author. E-mail addresses: [email protected] (M.I. Alam), [email protected] (B. Kanagarajan), [email protected], [email protected] (P. Jana).
https://doi.org/10.1016/j.tws.2019.04.018 Received 8 August 2018; Received in revised form 12 April 2019; Accepted 13 April 2019 Available online 03 May 2019 0263-8231/ © 2019 Elsevier Ltd. All rights reserved.
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Fig. 1. Geometry of a typical open cross-section column: (a) complete column, (b) cross-section of the column (dimensions are in mm).
manufacturing constraints by using annealing algorithm method for the optimization. Madeira et al. [21] adopted the Direct Strength Method (DSM) to determine the optimum cross-section of cold formed steel columns to maximize both distortional and local-global buckling strength. Recently, Ye and co-authors [22,23] used the particle swarm optimization method to get the optimum cross-section of cold formed steel beams with the greatest flexure strength and developed a foldedflange cross section that can be used in practical applications. Above review of the literature reveals that the finite element based optimization for the open cross-section of slender columns for maximizing the buckling strength remains unexplored. This paper uses finite element analysis in ANSYS (v15.0) coupled with a MATLAB (vR2014a) optimization routine to obtain the optimal design of the cross-section. These optimal geometry as shown through the results along with the insights generated from the inferences from this study will be useful in the design phases of thin-walled open-section columns.
many interesting research articles that focus on understanding the buckling behaviour of various open-section columns. For example, Seah and Khong [6] presented a semi-analytical and semi-numerical analysis by adopting the Rayleigh-Ritz method for the prediction of the lateral and torsional buckling of channel beams with unbraced longitudinal edge stiffeners. Schardt [7] used Generalized Beam Theory (GBT) to investigate lateral and torsional buckling for channel and hat-section columns by taking the distortional effect into account. Alwis and Wang [8] focused on the Wagner term in flexural-torsional buckling of thinwalled open-section column and examined its importance in the buckling load computation by using two simple bar models. Li [9] and Cheng et al. [10] presented analytical results by using energy principle for computing the flexural and lateral-torsional buckling of cold-formed beams. Later, Erkmen and Attard [11], and Potier-Ferry and co-authors [12,13] used nonlinear geometrical formulation in order to capture a more realistic estimate of the lateral-torsional buckling load for thinwalled open cross-section columns. Recently, Sahraei and Mohareb [14] used finite element method to develop a family of three finite elements in order to capture warping torsion, load position, and shear deformation effects for the analysis of lateral-torsional buckling of beams with doubly symmetric thin-walled cross-section. There are also research articles which discuss the optimization of open-section columns for maximizing their strength. Some of these relevant work are discussed here. Adeli and Karim [17] adopted a neural network model for optimizing the shape for economical use of coldformed steel taking into account the conventional hat, Ie and Z-shapes. By adopting micro genetic algorithm for the optimization, Lee and coauthors [15,16] optimized the shape of the cold formed beams subjected to uniformly distributed loads. Phan et al. [18] used a real-coded niching genetic algorithm to determine the optimized design of coldformed steel frames to explore the effects of stressed-skin action. Ma et al. [19] used genetic algorithm to determine the optimum crosssection of steel channel for working under compression or bending as stated by Eurocode 3. Leng et al. [20] presented more practical and economical optimization solution to incorporate end-user and
1.2. Buckling analysis of a typical open cross-section column As discussed earlier, the main objective of the paper is to develop an optimization routine for obtaining the optimal buckling strength in open cross-section column. To compute the buckling load, commercially available finite element analysis software ANSYS has been used. It can be noted that a finite element solution for the buckling analysis is essential in this case as obtaining an analytical solution for a parametrically modelled open cross-section column often becomes fairly complex and intractable. To support this statement, an analytical solution based on Bleich's [24] approach is also presented. It serves the following two purposes: (i) it shows the complexities of handling the differential equations for any optimization algorithm; (ii) it validates the finite element computation of critical buckling loads in ANSYS. Therefore, in this section, the critical buckling load computation of a typical thin-walled open cross-section column is demonstrated. For this demonstation, an open cross-section column with the panel widths and the orientation as shown in Fig. 1 is considered. The wall thickness 424
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(t) is taken as uniform throughout and equal to 5 mm. Two different lengths, L = 800 mm and 1000 mm, are selected for calculating the buckling loads. The material is assumed to be isotropic with Young's modulus (E) = 210 GPa and Poisson's ratio ( µ ) = 0.3.
Table 1 First critical buckling loads obtained from ANSYS for clamped-simply supported (CS) and both end simply supported (SS) column. Length, L (mm)
1.3. Finite element computation
Pcr (in KN) using ANSYS
1000 800
Linear (eigenvalue) buckling analysis in ANSYS has been used to compute the buckling load of the column. The geometric model of the column has been meshed with SHELL181 element available in ANSYS. The element has six degrees of freedom at each node (three rotational and three translational) and it is used to model thin to moderately thick plate structures. Mesh convergence study has been carried out and it has been found that converged results are achieved when the element size is taken as A/50, where A is the total panel width (190 mm in this case). Two different end boundary conditions are considered in this analysis: i) one end clamped and the other end simply supported (CS), ii) both ends simply supported (SS). In ANSYS, these boundary conditions have been implemented as follows: i) CS boundary condition: Both rotational and translational degrees of freedoms have been constrained at one end. At the other end where the force is applied, translational degrees of freedoms have been constrained in x and y direction and it remains free to move along the z-direction (see Fig. 2). ii) SS boundary condition: As the model and boundary conditions are symmetric, a half model has been considered and the symmetric boundary condition has been applied at one end. At the end where the force is applied, translational degrees of freedoms have been constrained in x and y direction, keeping the edge free to move along z-direction. In ANSYS, an axial force P (say 1 kN), equally distributed on all edge nodes, is applied at the simply supported end. With this applied axial force, static analysis has been performed initially with pre-stress effect activated and then the eigenvalue buckling analysis is performed. The buckling load factor (BF) obtained from the eigenvalue buckling analysis is multiplied to the applied load P to get first the critical buckling load (Pcr ) which leads to the structural instability of the column. Thus the first critical buckling load is given by Pcr = BF × P . The first critical
Boundary condition: CS
Boundary condition: SS
251.40 327.40
171.11 213.74
buckling loads for these four cases (two boundary conditions and two column lengths) are given in Table 1. The first mode shape for length L = 800 mm and L = 1000 mm with the combined effect of both torsional and flexural buckling modes is shown in Fig. 3. 1.4. Analytical calculation of critical buckling load The analytical solution, presented here, is based on the differential equation of buckling of centrally loaded columns by torsion and flexure given by Friedrich Bleich [24]. Bleich's theory is based on the energy method and the same is discussed briefly for a general column crosssection. Consider a thin-walled open cross-section column made of several flat plates of uniform thickness as shown in Fig. 4. It is subjected to centrally applied axial compressive force i.e. along z-direction, where x and y are the principal axes of the cross-section. For the arbitrary crosssection shown in Fig. 4, in the deformed configuration, the origin O of the cross section is translated and is twisted with angle of twist β. The translation of the cross-section is defined by u and v along ξ and η axes, respectively. Because u, v and β are small, u simply equals the x-direction displacement of O, and v equals the y-direction displacement of O. As shown in Fig. 4, Gk and Gk + 1 are the centroid of the k th and (k + 1)th plates respectively and Pk is the point of intersection of the plate. The total potential energy (U) of the deformed bar can be divided into two parts: the potential energy due to axial compressive stresses and the strain energy V. The strain energy V is further divided into axial strain V1 due to longitudinal stresses and the strain energy V2 produced due to twisting as determined using St. Venant theory. The total strain energy for n number of flat plates for bending and stretching is obtained as:
V1 =
L
1 2
n
(EIk s"2k + EAk k2) dz
(1)
0 k=1
where Ik is the centroidal moment of inertia, k is the longitudinal strain and Ak is the cross sectional area of the k th plate respectively. sk is the displacement of the centroid of the k th plate along the direction vector and given as:
sk = {uiˆ + vjˆ + (kˆ × r k )} tˆk ¯
(2)
where r k is the distance of the centroid from the origin of the cross¯ section. As the successive plates are joined at certain location Pk , the strain in both the plates must be equal. Strain for k th and (k + 1)th plates are given as: k sk
k
where
k
=
k+1
+
k + 1 sk + 1
, k = 1, 2,
., n
1
(3)
is the distance of centroid from Pk . By assumption:
n
Ak k= 1
Fig. 2. Finite element mesh of the clamped-simply supported (CS) column. At the bottom end Ux, Uy, Uz, Rx, Ry, and Rz are constrained and at the top end Ux and Uy are constrained.
k
=0
(4)
Substituting the values obtained from Eqs. (2)–(4) in Eq. (1), one can get the most general form of quadratic expression as: 425
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Fig. 3. First buckling mode (isometric view) for clamped-simply supported (CS) column: (a) for L = 800 mm, (b) for L = 1000 mm.
and B6 do not occur in the conventional theory of bending. The strain energy for the entire length due to twisting is obtained as:
V2 =
L
1 2
2
GK
dz
(6)
0
where G is the shear modulus, is the rate of angle of twist and the value of torsion constant K for a flat plate of width b and thickness t is given by
K=
bt 3 3
(7)
Due to axial compressive stress the rod deforms laterally and the vertical downward motion of the top of the rod can be obtained as:
1 2
=
L
2
u z
0
dz
(8)
By a similar calculation, the squares of the partial derivatives of lateral deflections in two orthogonal directions are added. Given a typical point xiˆ + yjˆ in the cross section, its lateral deflection is
kˆ × (xiˆ + yjˆ ) = (u
uiˆ + vjˆ +
1 (x , y ) = 2
Table 2 Comparison of the analytical buckling results with the corresponding finite element results for clamped-simply supported (CS) and both ends simply supported (SS) column. Length, L (mm)
Pcr (in KN) ANSYS
Analytical
CS
1000 800 1000 800
251.40 327.40 171.11 213.74
254.97 333.03 176.69 220.33
SS
V1 =
1 2
(9)
Hence, the total deflection for entire length is given as:
Fig. 4. Cross-section showing translation and rotation with some key points.
Boundary conditions
y ) iˆ + (v + x ) jˆ.
2
2
L
u
y
+ (v +
x } dz (10)
0
The potential energy due to axial compressive stress the entire length is obtained as:
Ratio of Pcr (ANSYS/ Analytical)
Ua =
at the end for
(x , y ) dA
(11)
Substituting the value of Eq. (10) in Eq. (11),
0.986 0.983 0.968 0.970
1 2
Ua =
L
2
A u +v
2
2
+ Ip
2Ayc u
+ 2Axc v
dz
0
(12)
L
(B1 u
2
+ B2 v
2
+ B3
2
+ B4 u v + B5 u
+ B6 v
where Ip is the polar moment of the cross sectional area about the origin O, and x c and yc are the coordinates of the centroid. The total potential energy due to deformation is given as:
) dz
0
(5) where B1, B2 , B3 , B4 , B5 and B6 depends on the geometric properties of the cross-section. The coefficients B1, B2 and B4 are equal to the moments of inertia of the complete cross section and the coefficients B3 , B5
L
U=
f (u", v", 0
426
"u v,
) dz
(13)
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Fig. 5. Schematic diagram of the cross-sections: (a) Parametric cross-section for the optimization, (b) Base model – a cruciform section. Table 3 Parameters for defining a number of regular cross-sections. The corresponding minimum principal second moment of area (Imin ) is also shown for these cross-sections. Name of section
Parameter values
Imin (mm4 )
C-section H-section I-section T-section Cruciform
a2 = 70 mm; a5 = a7 = 60 mm; a1 = a3 = a4 = a6 = a8 = a9 = 0 mm a2 = 70 mm; a4 = a5 = a6 = a7 = 30 mm; a1 = a3 = a8 = a9 = 0 mm a2 = 110 mm; a4 = a5 = a6 = a7 = 20 mm; a1 = a3 = a8 = a9 = 0 mm a2 = 80 mm; a6 = a7 = 55 mm; a1 = a3 = a4 = a5 = a8 = a9 = 0 mm a1 = a2 = a4 = a5 = 47.5 mm; a3 = a6 = a7 = a8 = a9 = 0 mm
331875 180625 53958 533683 358177
Fig. 6. Flow chart used for obtaining the optimal cross-section.
(13a)
U = Ua + V
d2 dz 2
Substituting the values of Ua and V in Eq. (13a), the following equation is obtained.
f (u”, v”, v,
=
1 2
B1 u
+ B2 v 2
+ GK
2
2
+ B3 2
A u +v
+ B4 u v + B5 u 2
2Ayc u
(14)
+
B4 v 2
+
B5 2
+ A (u
yc
)=0
(15a)
B2 v
+
B4 u 2
+
B6 2
+ A (v + x c
)=0
(15b)
B3
+
B5 u 2
+
B6 v 2
+ B6 v + 2Axc v (13b)
i.e.
=0
B1 u
2
+ Ip
f q
d dz
where q stands for u, v and . Thus the equations of equilibrium are:
”, u 2
f q
Equation (13b) can be solved by the calculus of variation method
GK
+
(Ip
Ayc u + Ax c v ) = 0 (15c)
The differential equilibrium Eqs. (15a) and (15b) and (15c) have been solved in symbolic math package Maple using Galerkin's 427
(x ),
428
44.40 94.40 136.31
34.95 167.41 225.92
373.72 401.07
0.00 657.25 657.25
4.34 830.77 866.82
31.92 833.41 1099.50
147.68 839.35 2078.9
236.16 856.93 2880.65
4000
In this section, the finite element analysis results of thin-walled open cross-section obtained from Section 3.1 have been validated by the analytical results that are obtained in Section 3.2. Table 2 shows this comparison for the two boundary conditions and for two different lengths, L = 800 mm and L = 1000 mm as mentioned in Section 3.1. From Table 2, it is evident that the buckling results are in good agreement. Therefore, it is now established that the finite element analysis in ANSYS can predict the buckling failure of the columns with a very good accuracy. Note that the objective of the present study is to find the optimum cross-section of the open-section column which provides maximum strength against the buckling failure. In order to study this, different cross-section shapes of the column must be obtained by varying the panel widths and the angles of each panel. This results in finding a set of large number of different complex cross-sections and eventually leads to a tedious process of obtaining analytical solutions for all these cross-sections. To overcome this difficulty, the buckling computation has been done using parametric modelling in ANSYS and the optimization has been done by linking these ANSYS computations
CS-8
1.5. Comparison of buckling results
3000
Use of above Galerkin's approach leads to an eigenvalue problem and the associated eigenvalues are solved to compute the critical buckling loads.
CS-7
(16b)
2000
x L
CS-6
(x )= sin
1500
ii) SS boundary condition:
CS-5
(16a)
1250
x2 x sin L2 L
2x L
CS-4
(x )=
1000
i) CS boundary condition:
CS-3
where 1, 2 and 3 are some constants. The shape functions satisfying the two boundary conditions, are taken as follows:
750
(x )
CS-2
3
10.6 mm, a2 = 50.3 mm, a3 = 09.1 mm, a4 = 24.4 mm, a5 = 35.6 mm, a6 = 24.5 mm, 35.7 mm, a8 = 3.6⁰, a9 = 1.5⁰ 2.5 mm, a2 = 43.9 mm, a3 = 2.6 mm, a4 = 31.6 mm, a5 = 37.4 mm, a6 = 41.0 mm, 31.0 mm, a8 = 7.0⁰, a9 = 6.6⁰ 44.9 mm, a2 = 23.4 mm, a3 = 09.4 mm, a4 = 38.7 mm, a5 = 46.7 mm, a6 = 18.9 mm, 07.9 mm, a8 = 2.6⁰, a9 = 15.5⁰ 47.6 mm, a2 = 32.5 mm, a3 = 05.9 mm, a4 = 46.7 mm, a5 = 46.1 mm, a6 = 05.4 mm, 05.9 mm, a8 = 0.4⁰, a9 = 1.1⁰ 47.5 mm, a2 = 47.5 mm, a3 = 0.0 mm, a4 = 47.5 mm, a5 = 47.5 mm, a6 = 0.0 mm, 0.0 mm, a8 = 0.0⁰, a9 = 0.0⁰ 2.5 mm, a2 = 58.8 mm, a3 = 3.0 mm, a4 = 4.1 mm, a5 = 4.9 mm, a6 = 59.4 mm, 57.3 mm, a8 = 17.8⁰, a9 = 0.1⁰ 11.0 mm, a2 = 61.2 mm, a3 = 2.4 mm, a4 = 2.4 mm, a5 = 2.2 mm, a6 = 62.4 mm, 48.4 mm, a8 = 3.4⁰, a9 = 3.4⁰ 7.6 mm, a2 = 66.0 mm, a3 = 2.4 mm, a4 = 2.1 mm, a5 = 1.9 mm, a6 = 59.2 mm, 51.0 mm, a8 = 3.1⁰, a9 = 3.0⁰
=
500
(x )
CS-1
2
Parameter Variables
v=
Length, L (mm)
(x )
Case
1
Table 4 Optimal results and comparison with cruciform section (base model) for clamped-simply supported (CS) column.
u=
= = = = = = = = = = = = = = = =
approach. The displacements trial functions are defined in terms of shape function ‘ϕ( x )’ for both the boundary conditions as
a1 a7 a1 a7 a1 a7 a1 a7 a1 a7 a1 a7 a1 a7 a1 a7
Percentage increase Cruciform results (KN) Optimal results (KN)
Fig. 7. Iteration history of the GA optimization process for the clamped-simply supported (CS) column with L = 750 mm.
7.32
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Table 5 Optimal cross-sections and the mode shapes for CS column with different lengths: L = 500 mm (CS-1), 7500 mm (CS-2) and 1000 mm (CS-3). Case
Optimal cross-section
First buckling mode shape
CS-1
CS-2
CS-3
with a Genetic Algorithm (GA) based optimization scheme in MATLAB. Similar finite element based optimization studies for the optimal design of perforated plate structures, for maximizing the uniaxial in-plane buckling load, can be found in recent works of Jana and co-authors [25,26].
[X, fval] = ga (obj funcn, no. of variables, LB, UB, options). The default options from MATLAB GA simulation tool box are used for this analysis. Some of these options are summarized below:
2. Optimal design of the cross-section This section discusses the details of the optimization procedure adopted in the present study. The input variables, for defining a seven panel open cross-section as shown in Fig. 5 (a), are taken as a = {a1, a2, a3, a4, a5, a6, a7, a8, a9} where, a1-a7 are the panel widths and a8, a9 are the angles. The output variable i.e. the critical buckling load will be the function of these input variables a. As the constraint for this optimization problem, the sum of the panel width variables a1 to a7 are kept as constant and equal to 190 mm. The lower and upper bound for each variable from a1 to a7 are 0 mm and 190 mm, respectively. The angle variables a8 and a9 are varied between 0° and 180°. Wall-thickness (t) of each panel is taken to be 5 mm. The material is isotropic with Young's modulus (E) = 210 GPa and Poisson's ratio ( µ ) = 0.3. Mathematically, the above optimization problem can be written as follows. It can be shown that, with the proper assignments of these nine input variables, the adopted parametric model is able to produce some regular cross-sections such as I-section, H-section, T-section, C-section, Cruciform section etc. See Table 3 for some arbitrarily chosen parameter values for these cross-sections with a total panel width of 190 mm. The objective of this optimization study is to maximize the critical buckling load. In order to use the in-built GA optimization routine (a minimization scheme) in MATLAB, the objective function is defined as, Pcr (a) = Pcr (a) , where Pcr (a) is a non-zero positive number representing the critical buckling load computed using ANSYS. The negative sign is used to convert the maximization problem to a standard minimization problem. The syntax used in MATALB for the GA is given as.
Population type
double vector
Population Size Selection Function Elite Count Crossover fraction Mutation function
50 Stochastic Uniform 2.5 0.8 constraint dependent
The flow chart of the optimization process is shown in Fig. 6. The finite element buckling analysis in ANSYS takes ak as the input parameters and returns the positive buckling load Pcr which is used to calculate the objective function (Pcr ). MATLAB routine links the ANSYS output to the in-built optimization algorithm. The objective function (Pcr ) is calculated and the optimal parameters (a ) are obtained as a final output from these iterations. A sample of ANSYS commands used for the parametric modelling of the open-section column is given in Appendix A. 3. Results and discussions In this section, the results obtained from the above optimization study are presented. The optimal cross-sections for the maximum critical buckling loads are obtained for eight different lengths of the column and two different boundary conditions. Fig. 7 shows the iteration history of the GA optimization process for a typical case of clamped-simply supported (CS) column with L = 750 mm. Note that these eight length parameters are arbitrarily chosen here without any specific prefernces. The optimal buckling results are finally compared with the base model i.e. a cruciform cross-section as schematically shown in Fig. 5 (b). 429
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Table 6 Optimal cross-sections and mode shapes for CS column with different lengths: 1250 mm (CS-4), 1500 mm (CS-5), 2000 mm (CS-6), 3000 mm (CS-7), 4000 mm (CS-8). Case
Optimal cross-section
First buckling mode shape
CS-4
CS-5
CS-6
CS-7
CS-8
Table 7 Buckling loads (Pcr ) for the regular cross-section (shown in Table 3) columns with CS boundary condition and comparison with corresponding optimal buckling loads (Pcr (opt ) ). Length (mm)
500 750 1000 1250 1500 2000 3000 4000
C-section
H-section
I-section
T-section
Cruciform
Pcr (kN)
Pcr Pcr (opt )
Pcr (kN)
Pcr Pcr (opt )
Pcr (kN)
Pcr Pcr (opt )
Pcr (kN)
Pcr Pcr (opt )
Pcr (kN)
Pcr Pcr (opt )
1030.0 815.9 541.4 402.9 324.0 238.9 164.6 99.7
0.36 0.39 0.49 0.46 0.49 0.60 0.73 0.73
2755.0 1304.0 764.7 481.5 335.7 189.3 84.4 47.5
0.95 0.63 0.70 0.55 0.51 0.48 0.37 0.35
842.0 398.3 227.1 146.0 101.6 57.0 25.4 14.3
0.29 0.19 0.21 0.17 0.15 0.14 0.11 0.10
438.8 417.9 405.1 392.0 376.2 332.9 215.2 134.0
0.15 0.20 0.37 0.45 0.57 0.84 0.95 0.98
856.9 839.3 833.4 830.7 657.2 373.7 167.4 94.4
0.30 0.40 0.76 0.96 1.00 0.93 0.74 0.69
3.1. Clamped-simply supported (CS) column
column produce an increase in critical buckling load by 236.16%, 147.68%, 31.92%, 4.33%, 7.32%, 34.95%, and 44.40% for column lengths of 500 mm, 750 mm, 1000 mm, 1250 mm, 2000 mm, 3000 mm, and 4000 mm, respectively, when compared with the base model i.e. the cruciform section. Moreover, for the column length of 1500 mm, the
Table 4 shows the optimal results for the clamped-simply supported (CS) column for eight different length parameters. It is worthy to note that the optimal cross-sections for the clamped-simply supported (CS) 430
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46.13 71.95
55.97
81.90 122.05
49.02
183.42 247.66
35.02
324.68 348.03
7.19
465.01 465.01
0.00
720.00 720.00
0.00
805.05 982.38
22.03
802.49
12.1 mm, a2 = 38.0 mm, a3 = 11.2 mm, a4 = 39.9 mm, a5 = 31.1 mm, a6 = 31.3 mm, 26.6 mm, a8 = 7.5⁰, a9 = 3.3⁰ 48.3 mm, a2 = 23.8 mm, a3 = 08.2 mm, a4 = 44.3 mm, a5 = 45.6 mm, a6 = 11.5 mm, 08.3 mm, a8 = 3.2⁰, a9 = 17.6⁰ 47.5 mm, a2 = 47.5 mm, a3 = 0.0 mm, a4 = 47.5 mm, a5 = 47.5 mm, a6 = 0.0 mm, 0.0 mm, a8 = 0.0⁰, a9 = 0.0⁰ 47.5 mm, a2 = 47.5 mm, a3 = 0.0 mm, a4 = 47.5 mm, a5 = 47.5 mm, a6 = 0.0 mm, 0.0 mm, a8 = 0.0⁰, a9 = 0.0⁰ 04.1 mm, a2 = 49.1 mm, a3 = 09.6 mm, a4 = 53.8 mm, a5 = 53.2 mm, a6 = 11.2 mm, 09.3 mm, a8 = 1.2⁰, a9 = 3.3⁰ 0.0 mm, a2 = 74.71 mm, a3 = 0.0 mm, a4 = 0.0 mm, a5 = 0.0 mm, a6 = 60.80 mm, 54.54 mm, a8 = 0.0⁰, a9 = 0.0⁰ 0.0 mm, a2 = 77.0 mm, a3 = 0.0 mm, a4 = 0.0 mm, a5 = 0.0 mm, a6 = 54.8 mm, 57.4 mm, a8 = 0.0⁰, a9 = 0.0⁰ 0.0 mm, a2 = 77.6 mm, a3 = 0.0 mm, a4 = 0.0 mm, a5 = 0.0 mm, a6 = 55.8 mm, 55.8 mm, a8 = 0.0⁰, a9 = 0.0⁰
1656.90
106.47
cruciform section turns out to be the optimal cross-section. For a better visualization, the schematic diagrams of the optimal cross-sections for the CS column along with their first critical buckling mode shapes are shown in Tables 5 and 6. The mode shape plots show that the first four columns with the optimal cross-section buckle in both flexure and torsional mode. Whereas, the last four columns buckles primarily in flexure mode. Further, the first critical buckling loads with CS boundary condition for five regular cross-sections, as depicted in Table 3, are also computed. The buckling results are shown in Table 7. Table 7 shows that, for a particular length, the buckling loads with the regular cross-sections are always lower than the optimal results reported in Table 4. Hence, this table confirms, at least partially, that the adopted optimization scheme produces optimal cross-sections for which the buckling load is maximum. Table 7 also shows the ratios of the buckling loads of these regular cross-section columns to the corresponding optimal buckling loads. From these load ratios along with the schematic diagrams of the optimal cross-sections shown in Tables 5 and 6, it can be observed that an H-section performs better under compressive loads for relatively shorter columns in which buckling occurs due to both flexure and torsion. However, as the length increses, cruciform sections provides higher buckling loads. For further increase in length, T-section is proved to be a better cross-section. The superior performance of T-section for longer columns can be attributed to the fact that the buckling modes are predominantly due to flexure and T-sections have a higher Imin (see Table 3) to resist the flexural buckling mode. 3.2. Both ends simply supported (SS) column
= = = = = = = = = = = = = = = =
Optimal results for the simply-simply supported (SS) columns are shown in Table 8. The optimal cross-sections for column lengths 500 mm, 750 mm, 1500 mm, 2000 mm, 3000 mm, and 4000 mm produce an increase of 106.47%, 22.03%, 7.19%, 35.02%, 49.02%, and 55.97% in the buckling load, respectively, when compared to the base model. However, for lengths 1000 mm and 1250 mm, the cruciform section turns out to be the optimal cross-section. The schematic diagrams of the optimal cross-sections for these SS columns along with their first buckling mode shapes are shown in Tables 9 and 10. The buckling results for the regular cross-sections with SS boundary conditions are shown in Table 11. From Table 11, we observe that no regular cross-section shows higher buckling resistance compared to the optimal results reported in Table 8 for the SS column. Table 11 also shows the ratios of the buckling loads to the optimal results. From these ratios along with the shapes of the optimal cross-sections shown in Tables 9 and 10, it can be noticed that, similar to clamped-simply supported columns, H-section provides better resistance to buckling for shorter lengths of column. For moderate column lengths, cruciform section turns out to be the optimal cross-section. Furthermore, for even longer columns, T-section provides better buckling resistance as it possesses maximum Imin to prevent the flexural buckling mode. 4. Conclusions
4000
The study began with an objective to obtain the optimal design of thin-walled open cross-section columns for maximum buckling load considering the combined effect of both torsion and flexure buckling modes. The optimal design of the cross-section was obtained by coupling the finite element buckling results in ANSYS with the in-built Genetic Algorithm (GA) optimization algorithm in MATLAB. These optimal designs were compared with the base cruciform model. From this study, the following conclusions are drawn.
• Optimal cross-sections show a significant increase, up to as high as 236%, in maximum buckling load capacity when compared with the base model of the cruciform section.
SS-8
3000 SS-7
2000 SS-6
1500 SS-5
1250 SS-4
1000 SS-3
SS-2
750
a1 a7 a1 a7 a1 a7 a1 a7 a1 a7 a1 a7 a1 a7 a1 a7 500 SS-1
Parameter Variables Length, L (mm) Case
Table 8 Optimal results and comparison with cruciform section (base model) for simply-simply supported (SS) column.
Optimal results (KN)
Cruciform results (KN)
Percentage increase
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Table 9 Optimal cross-section and the mode shapes for SS columns with different lengths: L = 500 mm (SS-1), 7500 mm (SS-2), 1000 mm (SS-3) and 1250 mm (SS-4). Due to symmetry, half of the length is considered. Case
Optimal cross-section
First buckling mode shape
SS-1
SS-2
SS-3
SS-4
• As the total area of the cross-section is kept constant, the amount of •
• • •
compared to these regular sections, have significantly higher resistance against the buckling instability.
material in the base model and in optimal design remains the same. Thus the significant increase in buckling load capacity is achieved without compromising the cost of the material. As the length of the column becomes short the columns buckle predominantly in the torsional mode and the cruciform section is no longer better for resisting the buckling failure. The optimal crosssections obtained in this study, for different boundary conditions, provides substantial increase in the buckling load capacity. It is observed that, among the five regular sections, H-section is closer, in terms of performance, to these optimal cross-sections for shorter columns. For a longer column, buckling mainly takes place in the flexural mode and T-section turns out to be the best design for resisting buckling of the column. For a moderate column length, the cruciform section turns out to be the better performing cross-section as it has substantial resistance to both flexural and torsional buckling modes. Regular sections e.g. I-section, T-section, H-section, etc. have some advantages due to their symmetry, e.g. an ease in manufacturing. However, it has been shown that the optimal cross-sections, as
While designing the thin-walled open cross-section columns the designer may consider the above important points pertaining to their buckling behaviour. Acknowledgements Authors would like to thank Anindya Chatterjee and Jishnu Bhattacharya for discussions and the Science and Engineering Research Board, Government of India, for financial support (Grant No.: ECR/ 2016/000964). Authors also thank anonymous reviewers for their valuable comments and suggestions which helped them in improving the manuscript. Appendix A. ANSYS APDL used for the buckling analysis !This APDL is written in a compact form using ‘$’symbol. !Here, MATLAB routine supplies the a1-a7, theta1 and theta2 values. 432
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Table 10 Optimal cross-section and the mode shapes for SS columns with different lengths: L = 1500 mm (SS-5), 2000 mm (SS-6), 3000 mm (SS-7) and 4000 mm (SS-8). Due to symmetry, half of the length is considered. Case
Optimal cross-section
First buckling mode shape
SS-5
SS-6
SS-7
SS-8
Table 11 Buckling results for the regular cross-section (as shown in Table 3) columns with SS boundary condition. Length (mm)
500 750 1000 1250 1500 2000 3000 4000
C-section
H-section
I-section
T-section
Cruciform
Pcr (kN)
Pcr Pcr (opt )
Pcr (kN)
Pcr Pcr (opt )
Pcr (kN)
Pcr Pcr (opt )
Pcr (kN)
Pcr Pcr (opt )
Pcr (kN)
Pcr Pcr (opt )
709.8 483.9 337.3 264.6 221.6 168.9 86.3 48.5
0.43 0.49 0.47 0.57 0.63 0.68 0.71 0.67
1419.4 648.02 367.71 236.32 154.00 92.6 41.3 23.2
0.86 0.66 0.51 0.50 0.44 0.37 0.34 0.32
429.9 196.4 111.2 71.36 49.62 27.9 12.4 6.7
0.25 0.20 0.15 0.15 0.14 0.11 0.10 0.09
377.7 376.6 370.1 347.7 311.8 229.4 117.9 68.9
0.23 0.38 0.51 0.75 0.89 0.93 0.97 0.96
802.4 805.0 720.0 465.0 324.6 183.4 81.9 46.1
0.48 0.82 1.00 1.00 0.93 0.74 0.67 0.64
FINISH. /CLEAR, START $/FILNAME, buckling_run $/PREP7. $height = 1.25 $thick = 0.005. !This data (parameter values) is fed from MATLAB routine. a1 = 0.0476 $a2 = 0.0325 $a3 = 0.0059 $a4 = 0.0467 $a5 = 0.0461 $a6 = 0.0054. $a7 = 0.0059 $theta1 = 0.0067 $theta2 = 0.0192 $a_sum = 0.19. K,1,0,0,0 $K,2,0,a1,0 $K,3,0,a1+a2, $K,4,0,a1+a2+a3,0. $K,5,-a4*cos(theta1),a1-a4*sin(theta1),0. $K,6,a5*cos(theta1),a1+a5*sin(theta1),0. $K,7,-a6*cos(theta2),a1+a2-a6*sin(theta2),0. $K,8,a7*cos(theta2),a1+a2+a7*sin(theta2),0 $K, 9,0,0, height. $L,1,9 $L,1,2 $L,2,3 $L,3,4 $L,2,5 $L,2,6 $L,3,7 $L,3,8.
$ADRAG,2,1 $ADRAG,3,1 $ADRAG,4,1 $ADRAG,5,1. $ADRAG,6,1 $ADRAG,7,1 $ADRAG,8,1 $nummrg,all, all. MPTEMP, $MPTEMP,1,0 $MPDATA,EX,1, 210e9 $MPDATA, PRXY,1,0.3. $ET,1, SHELL181 $ sect,1,shell, $ secdata, thick,1,0.0,3. $secoffset, MID $seccontrol, $AESIZE,ALL,a_sum/50 $AMESH, ALL. NSEL,S,LOC,Z,0,1E-6 $D,ALL,0,ALL, $ALLS $NSEL,S,LOC,Z,height1E-6, height. $*GET,NODECOUNT, NODE,0,COUNT, $D,ALL, UX $D,ALL, UY $F,ALL,FZ,-1000/NODECOUNT $ALLS. /SOLU $ANTYPE, STATIC $PSTRES, ON $SOLVE $FINISH. /SOLU $ANTYPE, BUCKLE $BUCOPT, LANB,1 $SOLVE $FINISH. /POST1 $RSYS, SOLU $FILE, buckling_run, rst $SET,1,1. 433
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*GET,factor, ACTIVE,0,SET,FREQ,/OUT,buck_fact, dat $*vwrite,1, factor. (F4.1,2X,F10.4) /OUT. SAVE $FINISH.
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