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or closed parts
Thin-Walled Structures 61 (2012) 42–48
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A constrained finite strip method for prismatic members with branches and/or closed parts N. Djafour, M. Djafour n, A. Megnounif, M. Matallah, D. Zendagui RISAM, Department of Civil Engineering, University of Tlemcen, B.P. 119, Tlemcen 13000, Algeria
a r t i c l e i n f o
a b s t r a c t
Available online 19 September 2012
The decomposition of buckling modes of thin-walled members subjected to axial stresses is a topic of great practical interest which can be achieved using the generalised beam theory (GBT) or the constrained finite strip method (cFSM). However, the latter is not general enough to study prismatic members with arbitrary cross-sections and the objective of this paper is to extend the cFSM to allow the buckling modes decomposition for prismatic members with branches and/or closed parts. To define the combined GD buckling mode, two assumptions are used: (i) cylindrical plate bending and (ii) negligible in-plane transverse/shear strains. The corresponding constraint matrix, RGD, is derived in a simple and general way. The methodology used to separate the global and distortional modes is similar to that used in the original cFSM while the derivation of the constraint matrix for local modes remains identical. Some examples are considered and the pure buckling curves are compared to the conventional FSM results. The conclusion is that the new cFSM has successfully computed the GD and the L modes of these sections. & 2012 Published by Elsevier Ltd.
Keywords: Elastic buckling Thin-walled member Modal decomposition Constrained finite strip method
1. Introduction Thin-walled members are very efficient structural elements but their design under compressive loads is rather complex because of the cross-section instability and the great variety of cross-sectional shapes. To calculate the design buckling resistance of thin-walled members, three basic buckling modes are usually distinguished in current design codes: local (plate), distortional, and global (e.g., flexural or lateral-torsional buckling) buckling. This classification results from three different post-critical behaviours. Local buckling can have a significant post-critical reserve, especially for thin elements where the behaviour is mainly elastic. Distortional instability can, also, present a post-critical reserve, though less than local instability. Global buckling does not present such reserve and the strength of the member is less than its elastic critical load. The correct evaluation of the associated elastic critical loads (or moments) is fundamental in calculating the ultimate load carrying capacity of a thin-walled member. Simple analytical models are often available, but are limited in their applicability. An alternative possibility is to perform numerical elastic buckling analyses by using powerful numerical methods such as the finite element method (FEM) or the finite strip method (FSM). However, it is necessary to identify, among a huge amount of results [1], the buckling modes consistent with global, G, distortional, D, and local,
n
DOI of original article: http://dx.doi.org/10.1016/j.tws.2012.04.019 Corresponding author. Tel.: þ213 43 20 56 06; fax: þ213 43 20 41 89. E-mail address: [email protected] (M. Djafour).
0263-8231/$ - see front matter & 2012 Published by Elsevier Ltd. http://dx.doi.org/10.1016/j.tws.2012.04.020
L, modes which are also called the pure buckling modes. Another option is to use numerical methods able to perform modal decomposition, i.e., able to perform eigen-buckling analyses targeting any pure mode or any combination of pure modes. This was first possible by the generalised beam theory (GBT) [2–4] and then by the constrained finite strip method (cFSM) [5–8]. It should be pointed out that the cFSM is based on the fundamentals of GBT but the procedure is completely independent. In addition, GBT and cFSM give very similar results since they use the same mechanical definitions of the different buckling pure modes. Bebiano et al. made available the numerical tool GBTUL [9] to categorise the buckling modes of thin-walled members automatically via GBT. A´da´ny and Schafer [5–7] developed the constrained finite strip method (cFSM) and implemented it in the open source programme CUFSM [10] thus making it possible to separate the deformations of a thin-walled member into those consistent with the required pure modes. It is worth to mention that attempts to perform modal decomposition analyses via the finite element method (FEM) have been recently published. Casfont et al. [11,12] have tried to calculate the pure buckling modes of thin-walled members using the FEM, but GBT analyses, performed at the crosssection level, were used to calculate the necessary constraint matrices. The pure mode shapes from cFSM were also used by A´da´ny et al. [13] to perform modal identification, i.e., to measure the interactions, within a general eigenmode calculated via the FEM. Recently, Djafour et al. [14] proposed simple assumptions to define the combined GD buckling mode class in the context of the cFSM: (i) cylindrical plate bending (if any) and (ii) negligible in-plane transverse strains and shear strains. This new definition
N. Djafour et al. / Thin-Walled Structures 61 (2012) 42–48
Nomenclature a b dG, dD
member length strip length strip width coordinates of Vm in the vector bases hG and hD, respectively hG, hD system of base vectors for G and D spaces m total number of DOFs; m¼4 n n number of nodes nc number of strips connected to a given node ns number of strips r number of half-sine waves ui, vi , wi, yi local node DOFs xyz local coordinate axes E Young’s modulus H, Cek constraint matrices that enforce negligible transverse and shear strains in all the strips IVm, IDk identity matrices having the dimension of Vm and Dk, respectively KE stiffness matrix of the FSM model of the thin-walled member KGD modified stiffness matrix assuming cylindrical plate E bending
was used to formulate the constraint matrix of the combined GD space, RGD, in a simple and general way, preserving the flexibility of the cFSM as a general numerical method. The proposed derivation was successfully used to study ‘‘single branched’’, open or closed, cross-sections (Fig. 1(a)). The objective of this paper is to extend this new approach to allow the buckling modes decomposition for prismatic members with branches and/or closed parts (Fig. 1(b)). Some examples are considered and the computed pure buckling curves are compared to the conventional FSM results. It should be mentioned that the methodology proposed in this paper to separate the global and distortional modes is similar to that used in the original cFSM [5–7] while the derivation of the constraint matrix for local modes remains identical.
43
KG
geometric stiffness matrix of the FSM model of the thin-walled member Kk reduced stiffness matrix resulting from hypotheses 1 and 2 DV KDD k , Kk submatrices of Kk RM constraint matrix for buckling mode class M Ui, V i , W, Yi global node DOFs Vm vector of the warping DOFs in Dk, i.e., the vector of DOFs which defines the combined GD space: Vm DGD XYZ global coordinate axes a angle between local x and global X axes l load parameter n Poisson’s ratio D vector of the DOFs of the FSM mesh De, Dk DOFs to be eliminated and kept, respectively, in order to enforce negligible transverse and shear strains in all the strips DM vector of DOFs in the space of buckling mode class M K diagonal matrix of eigenvalues U matrix of eigenvectors
number of nodes is n which gives a total number of DOFs equal to m¼4 n. For stability analysis, two global matrices are necessary: the stiffness matrix, KE, and the geometric stiffness matrix, KG. To obtain KE and KG, local matrices of individual strips are formed, transformed from local to global coordinates and then assembled. Note, KG is function of the applied reference edge loading and is scaled linearly by a scalar l. The buckling modes of the thinwalled member are calculated by solving the following generalised eigenvalue problem (for more details see [15,16]): KE U ¼ KG UK
ð1Þ
where U ¼ ½ u1 u2 um is the matrix of eigenvectors and K ¼ diag½ l1 l2 lm is the diagonal, m m, matrix of eigenvalues.
2. Brief overview of the FSM and the cFSM
2.2. The constrained finite strip method
2.1. Finite strip method
The main idea of the cFSM [5] is to define constraint matrices for each of the buckling mode classes. These matrices define relationship between original FSM DOFs, D, and those corresponding to a buckling mode class, DM.
In FSM, a typical thin-walled member is divided into ns strips (elements) having all the length, a, of the member. To derive the matrix formulation, two (left-handed) coordinate systems are used: global and local. The global coordinate system is denoted as X–Y–Z, with the Y axis parallel to the longitudinal axis of the member. The local system, which is always associated with a strip, is denoted as x–y–z. Axes y and Y are parallels while axis z is normal to the strip. The intersection line of two connecting strips is called node and has four DOFs: 2 membrane (in-plane) DOFs, ui and vi, and 2 bending (out-of-plane) DOFs, wi and yi (Fig. 2). The
Fig. 1. (a) Unbranched and (b) branched (open and closed) cross-sections.
D ¼ RM DM
ð2Þ
Fig. 2. Strip DOFs, dimensions, and applied edge forces.
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N. Djafour et al. / Thin-Walled Structures 61 (2012) 42–48
The dimension of D is m. It contains the four global DOFs of the n nodes and can be represented as follows: h i DT ¼ UT VT WT HT ð3Þ With UT ¼ U 1 U 2 U n , and, VT, WT and HT are described in the same way. DM has a smaller number of DOFs, mM, and defines a reduced deformation field that satisfies the criterion matching a buckling mode class M. The dimension of the constraint matrix RM is (m mM). The columns of RM may be considered as a set of base vectors, in the space of mode class M. According to general practice, A´da´ny and Schafer [5,7] defined four classes of modes: (i) global, G, (ii) distortional, D, (iii) local, L, and (iv) others, O. Eq. (2) may be employed on all eigenvectors in matrix U.
U ¼ RM UM
ð4Þ
Introducing Eq. (4) into Eq. (1) and pre-multiplying by RTM gives a new generalised eigenvalue problem. RTM KE RM UM ¼ RTM KG RM UM KM
ð5Þ
This equation can be rewritten as KEM UM ¼ KGM UM KM
ð6Þ
All the matrices in this equation have a reduced dimension, (mM mM). This new eigenvalue problem is defined in the constrained DOF spanned by the mode class M. This means that solving Eq. (6) gives solutions in the M space. Derivations of matrices RL and RO are straightforward processes and can be easily completed, even for arbitrary crosssections. They can be derived with exactly the same procedure as that presented in [7]. A much more problematic task is the definition of RG and RD matrices. Until now, the usual definition consists in forming RGD, the constraint matrix of the combined GD space, and, then, separating G and D by using orthogonality conditions. Recently, Djafour et al. [14] have simplified the derivation of RGD. In the context of the FSM, only two assumptions were necessary to define the GD space: (i) cylindrical plate bending (if any) and (ii) negligible in-plane transverse strains and shear strains. The RGD matrix was derived by means of procedures commonly used in numerical methods. This led to a simple and systematic formulation which was successfully used to compute GD and L modes of single branched open or closed cross-sections. The objective of the next section is to modify this formulation to allow the buckling modes decomposition for prismatic members with branches and/or closed parts.
and the warping displacements, Vm, of the main nodes. The idea is to impose the warping DOFs, Vm, as kinematic loading and to calculate the resulting DOFs. In this paper, the second step is revisited in order to allow RGD calculations for prismatic members with branches and/or closed parts. 3.2. Constraint matrix for membrane DOFs In the local coordinate system x–y–z of a strip of length a and width b, the membrane displacements are interpolated from the local membrane DOFs of its two nodes [15,16]. ( ) h i ui u ¼ ð1x=bÞ ðx=bÞ sinðr py=aÞ uj ( ) ð7Þ h i vi v ¼ ð1x=bÞ ðx=bÞ cosðr py=aÞ vj r is the number of half-sine-waves along the y axis. Assumption 2 can be written as
ex ¼ @u=@x ¼ 0 gxy ¼ @u=@yþ @v=@x ¼ 0
ð8Þ
The substitution of Eq. (7) into Eq. (8) leads to the following constraint relations ui ¼ uj ¼ ðvi vj Þ
1 bkr
ð9Þ
With kr ¼rp/a These constraint relations mean that the two local DOFs, ui and uj, are, from now on, defined from the warping (longitudinal) DOFs and are not effective, i.e., instead of the initial eight DOFs of the strip, only six DOFs are independent. It is also important to note that there is one constraint equation per node of the strip. In the finite strip method, it is well-known that the local to global transformation at node i for a strip which is oriented in the global coordinate system X–Y–Z at an angle a is governed by the following transformations: ( ) ) ( ) ) ( ( ui Ui vi Vi 1 0 cos a sin a ¼ and ¼ wi Wi yi Yi 0 1 sin a cos a ð10Þ
3. Derivation of the RGD matrix
Therefore, the local to global transformation of Eq. (9) gives the following two equations which provide a relationship between the transverse DOFs and the warping DOFs of both nodes.
3.1. General
U i cos a þW i sin a ¼ ðV i V j Þ bk1 r
As stated in reference [14], four major steps are necessary to derive RGD. The first one consists in calculating, KGD E , a modified member’s elastic stiffness matrix that takes into account assumption 1, i.e., the assumption of cylindrical plate bending in all strips. This matrix can easily be obtained using an equivalent beam model as done in the original cFSM formulation [5]. In the second step, a constraint matrix, C, for membrane DOFs is obtained from assumption 2. It represents the conditions on the DOFs of the FSM mesh that enforce negligible transverse and shear strains in all the strips. The third step is to apply the constraint matrix C to the modified global stiffness matrix KGD E by using a standard Lagrange multipliers procedure. Finally, RGD is calculated by using its definition: It must give the relationship between the vector D, containing all the DOFs of the FSM mesh,
U j cos a þW j sin a ¼ ðV i V j Þ bk1 r
ð11Þ
To achieve assumption 2 for the whole FSM model, it is necessary to write two constraint equations per strip in the global coordinate system. Consequently the number of DOFs to be eliminated to achieve the constraints brought by all the strips is (2ns). The key point of the new procedure resides in the way of selecting the DOFs to be eliminated. The new analysis is based on a loop on the nodes. For each node i, one needs to write equations and eliminate DOFs as many as strips connected to the node. The decision depends on the orientations of the strips, the number of connections and the currently free/active DOFs. A summary of the different situations is given in Table 1 and the procedure is explained hereafter.
N. Djafour et al. / Thin-Walled Structures 61 (2012) 42–48
Table 1 Summary of the various types of nodes. Number of connected strips
Illustration
DOFs to eliminate
1
Ui or Wi
2 (coplanar)
Ui or Wi and Vi
2 (non coplanar)
Ui and Wi
3
Ui, Wi and Vi
nc 43
Ui, Wi, Vi and (nc 3) Vs of the neighbouring nodes (Vj and Vk in the example)
45
If at least two strips have different directions, Eq. (13) can be solved for the three translational DOFs of node i, which are the three DOFs we have to reduce in this case. The last situation occurs when nc strips, with nc43, are connected to node i which implies that nc DOFs have to be reduced: the three translational DOFs of node i and (nc 3) warping DOFs of the neighbouring nodes. Since the decision goes out of the range of the current node, it is necessary to search and find enough free/active warping DOFs from the nodes connected to node i. It is important to keep the analysis of such ‘‘over constrained’’ node to the end and note that the selection is not unique. To implement the procedure presented above, an idea, to address the problem globally and to deal with the various couplings, consists in rewriting each constraint equation in the following matrix form: 9 8 Ui > > > > > > > > = < Vi > h i> cos a 1=bkr sin a 1=bkr ¼0 ð14Þ W > i > > > > > > > > ; : Vj > Then, Eq. (14) can be expanded to involve all the DOFs of the FSM model. Inside the loop on nodes, the constraint equations are written sequentially and, at the end, a set of (2ns) constraint equations is obtained which can be written in a compact form as follow:
If only one strip is connected to node i, there will be a single equation involving its transverse DOFs. From the constraint equation for node i, i.e., the first line of Eq. (11), it is obvious that if sin a ¼0, there is a unique possibility and the transverse DOF to be reduced is Ui. Similarly, if cos a ¼ 0, the DOF to be eliminated is Wi. Otherwise, it is possible to reduce either Ui or Wi by defining it from the remaining transverse DOF and the warping DOFs. Preferably, the DOF to eliminate is the one whose direction is closest to the local x axis. This preference is rather a precaution to avoid possible singularities that would be caused by the reduction of a DOF nearly perpendicular to the local x axis. If node i is connected to two nodes, j and k, by two strips, s1 and s2, one constraint equation per strip can be written as follows: U i cos as1 þW i sin as1 ¼ ðV i V j Þ bs11kr U i cos as2 þ W i sinas2 ¼ ðV i V k Þ bs21kr
ð12Þ
HD ¼ 0
ð15Þ
The dimension of H is (2ns) m. The other result of this section is that, among the m DOFs that form vector D, a set of (2ns) DOFs, denoted as De, are not effective. They can be deduced from the remaining (m 2ns) DOFs, denoted as Dk. Eq. (15) is partitioned as follows: ( ) De HD ¼ He Hk ¼0 ð16Þ
Dk
which gives
De ¼ He 1 Hk Dk ¼ Cek Dk
ð17Þ
Having matrix Cek, the rest of the procedure remains identical to reference [14] and is briefly presented hereafter. 3.3. Derivation of RGD
where bs1 and bs2 are the respective widths; as1 and as2 are the respective angles. In case of non-coplanar strips, it is obvious that Eq. (12) can be solved for Ui and Wi to define the two transverse DOFs of node i from the warping DOFs of the three connected nodes. Thus, the two DOFs to be eliminated are Ui and Wi. In case of coplanar strips, the left-hand sides of Eq. (12) are identical, at least in absolute value. It becomes clear that the relevant warping DOFs are not independent. One of them must be defined from the others before being eliminated. It is preferable to reduce Vi to limit the elimination process to the DOFs of the current node. The second DOF to be eliminated can be Ui or Wi, depending on the orientation of the strips, as explained above for the case of one connected strip. If node i is connected to three nodes, j, k and l, by three strips, s1 to s3, three constraint equations are involved and can be written as follows: 2
cos as1 6 cos a s2 4 cos as3
1=kr bs1 1=kr bs2 1=kr bs3
9 8 9 38 sin as1 > > V j =bs1 > < Ui > = = 1< 7 sin as2 5 V i V k =bs2 ¼ > > kr > : V =b > ; sin as3 : W i ; s3 l
ð13Þ
asi and bsi are the angle and the width of strip si, respectively.
First, the elastic stiffness matrix in the GD space, KGD E , is partitioned into four matrices by rearranging the DOFs. 2 3( ) Kek Kee De E E GD 4 5 KE D ¼ ð18Þ Dk Kke Kkk E E By using standard Lagrange multipliers procedure [14], Eq. (17) is applied to Eq. (18). The result is Kk, the stiffness matrix of the FSM model of the thin-walled member when supposing cylindrical plate bending and neglecting in-plane transverse and shear strains, i.e., when applying hypotheses 1 and 2. T T ke ek ee Kk ¼ Kkk E þ KE Cek þ Cek KE þ Cek KE Cek
ð19Þ
Kk is a square matrix of dimension equal to the number of the remaining DOFs, (4n 2ns). In the cFSM, the RGD matrix is the constraint matrix that defines the GD space. It defines the vector D, containing all the DOF of the FSM mesh, from the longitudinal – or warping – displacements, Vm, of the main nodes.
D ¼ RGD Vm
ð20Þ
The idea is to impose the warping DOFs, Vm, as kinematic loading and to calculate the resulting DOFs. By rearranging the
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N. Djafour et al. / Thin-Walled Structures 61 (2012) 42–48
DOFs, Matrix Kk and vector Dk can be rewritten as follows: " VV #( ) KVk D Kk Vm Fm ð21Þ ¼ Kk Dk ¼ DV DD Du 0 Kk Kk Note that the forces Fm at warping DOFs are the ‘‘unknown’’ reactions resulting from the kinematic loading; the remaining forces must be zero. The second equation in Eq. (21) gives h i Du ¼ KkDD1 KkDV Vm ð22Þ Eq. (22) can be expanded by an identity matrix IVm having the dimension of Vm to concern all the DOFs in Dk. ( ) " # IVm Vm Dk ¼ ð23Þ ¼ DD1 DV Vm Kk Kk Du Similarly, Eq. (17) can be expanded by an identity matrix IDk having the dimension of Dk to concern all the DOFs in D. ( ) " # Cek De ¼ D¼ Dk ð24Þ IDk Dk Finally, the substitution of Eq. (23) into Eq. (24) gives the required matrix in Eq. (20). " #" # IVm Cek ð25Þ RGD ¼ V KkDD1 KD I Dk k
eight modes in the GD space which can be obtained from the reduced eigenvalue problem defined by Eq. (6). Elastic buckling analyses were conducted by varying the member half-wavelength from 20 to 10,000 mm. The results are shown in Figs. 4 and 5. The curves ‘‘FSM 1’’ to ‘‘FSM 4’’ in Fig. 4 provides bifurcation stresses associated with the first four modes when using conventional FSM. Curves ‘‘Local 1’’ to ‘‘Local 4’’ correspond to plate bending modes and were targeted using a specific RL constraint matrix as defined in [7]. The other four curves in Fig. 4, i.e., ‘‘GD 1’’ to ‘‘GD 4’’, are the first four modes given by the cFSM when targeting the GD buckling mode class. Fig. 5 gives the corresponding buckling mode shapes. It is obvious that the cFSM has successfully decomposed the FSM solution of the E-section into L and GD ‘‘pure’’ modes. Furthermore, the (Local 1) and (GD 1) curves given in Fig. 4 are almost the same as those obtained by Dinis et al. [3] using GBT. 4.2. Buckling mode decomposition of a closed and branched cross-section In this section, an example of closed and branched cross-section is considered. Its shape is an asymmetric 3-cells-section formed by 10 plate elements (drawing 3 connected rectangles) as shown in Fig. 6. The cross-section mesh involves 38 nodes and 40 strips (4 strips per plate). Note that there are 30 nodes which are common to two, and
105 4. Numerical examples In order to validate and illustrate the application and capabilities of the developed method, three examples are considered: (i) an open branched cross-section, (ii) a close branched crosssection and (iii) a cross-section with branches and closed parts. All these examples are simply supported columns (with pinned and free-to-warp end sections) which are loaded by uniform compressive forces and supposed to buckle in one-half sine wave. The modulus of elasticity is E¼210 GPa, and Poisson’s ratio is n ¼0.3. For validation purposes, all the results are compared with values obtained by finite strip analyses.
Ncr (kN)
104 103 102 101
FSM 1 FSM 2 FSM 3 FSM 4
100 101
Local 1 Local 2 Local 3 Local 4
GD 1 GD 2 GD 3 GD 4
102 103 Half-wavelength (mm)
104
4.1. Buckling mode decomposition of open branched cross-section The first illustrative example is a simple column member with an asymmetric E-section which was analysed by Dinis et al. [3] using the GBT. The cross-section is presented in Fig. 3(a) (all dimensions are mid-line for the cross-section.). The FSM mesh (Fig. 3(b)) uses 19 strips and 20 nodes which give a total of 80 DOFs. Eleven nodes are secondary nodes since they are common to two coplanar strips. One node is connected to three strips, which classifies this example as ‘‘branched cross-section’’. The model has eight main nodes and the corresponding RGD matrix has a dimension of (80 8). This means that this E-section has
Fig. 4. FSM Buckling curves of the first four modes of E-section column and their decompositions into GD and L pure modes.
mode 4
mode 3
mode 2 60 8 75
mode 1
t = 1 mm 5
45 20
8 60 (dimensions in mm)
Node
Fig. 3. Example 1: (a) Asymmetric E-section and (b) FSM discretisation.
any a Local
a = 60 mm
a = 600 mm a = 10 000 mm Combined GD mode
Fig. 5. GD and L Buckling mode shapes of example 1.
N. Djafour et al. / Thin-Walled Structures 61 (2012) 42–48
47
105
70
Node Ncr (kN)
t = 2 mm
70
104
103
50 50 (dimensions in mm) Fig. 6. Example 2: (a) Asymmetric 3-cells-section and (b) FSM discretisation.
only two, coplanar strips. In addition, note that, among the remaining eight nodes, two nodes are common to three strips and that one node is connected to four strips. This means that, according to the procedure summarised in Table 1, there are only 4 nodes with effective warping DOFs. Thus, the dimension of RGD matrix is (152 4) and consequently, there are 4 modes in the GD space. It is important to note that the constraint matrix RGD alone is not able to separate the G and D spaces. A further transformation inside the GD space is necessary [5,7]. In cFSM, the idea is to seek a basis which can be used to describe any deformation Vm of this space, and in which the system of base vectors for the G and D spaces are separated; these are denoted hG and hD, respectively. Vm ¼ hG
hD
(
dG dD
)
FSM 1 FSM 2 FSM 3 FSM 4
102 101
Local 1 Local 2 Local 3 Local 4
Dist 1 Global 1 Global 2 Global 3
102 103 Half-wavelength (mm)
104
Fig. 7. FSM Buckling curves of the first four modes of 3-cells-section column and their decompositions into G, D and L pure modes.
mode 4
mode 3
mode 2
ð26Þ
dG and dD are the coordinates of Vm in this new basis. According to cFSM procedure, the system of base vectors hG is ‘‘directly’’ defined and the vectors hD are generated such that they complete the GD space using orthogonality conditions [5]. To target only modes in the D space, the GD space basis is truncated to hD and the constraint matrix is defined by RD ¼RGDhD. Having RD and according to the procedure presented in Section 2, the buckling loads and modes in the D space can be obtained from the reduced eigenvalue problem defined by Eq. (6). For open cross-sections, hG is defined [5] by four special warping functions associated with the longitudinal displacements resulting from special loads, namely: (i) pure axial load, (ii) pure bending moment about the first principal axis of the cross-section, (iii) pure bending moment about the second principal axis, (iv) pure torque. However for closed cross-sections, it is known from literature [14] that the torsional mode cannot occur without membrane shear strains and thus does not belong to the GD space. In this paper, hG for closed cross-sections is obtained from the remaining three special warping functions, i.e., the longitudinal displacements resulting from axial load and bending moments. For the numerical example in progress, which is a closed crosssection with four effective warping DOFs, the dimension of hG is (4 3) and the dimension of hD is (4 1). This 3-cells-section has only one distortional mode according to the definitions used in the present paper. Fig. 7 gives buckling loads versus half-wavelengths for Example 2. Twelve curves are depicted corresponding to: (i) the first four buckling modes resulting from conventional FSM buckling analysis (FSM 1 to FSM 4), (ii) the first four pure local modes (Local 1 to Local 4), (iii) the unique pure distortional mode (Dist. 1) and (iv) the three pure global modes (Global 1 to Global 3). The corresponding buckling mode shapes are drawn in Fig. 8. It is clear that the envelopes of the pure buckling curves, calculated using the developed method, agree with the ‘‘FSM’’ curves for all column lengths and for the four depicted modes, except for curve
mode 1
Local
Dist.
Global
Mode shapes at a = 10 000 mm
"Pure" Modes
FSM
Fig. 8. Buckling mode shapes of example 2.
FSM 3 and for lengths higher than 2 m where the buckling mode is torsional. This validates the proposed method for buckling mode decomposition of closed thin-walled sections. 4.3. Buckling mode decomposition of a cross-section with branches and closed parts The third example in this paper is a 1-cell-section column having branches and closed parts. Its shape is shown in Fig. 9. The FSM model uses 24 nodes and 24 strips (4 strips per plate in the closed part and 2 strips per plate in the open parts). Sixteen nodes are common to two, and only two, coplanar strips and two nodes are common to three strips. This means that the dimension of RGD matrix is (96 6) and consequently, there are six modes in the GD space. To separate global and distorsional modes, the three base vectors defining the G space are obtained from three special warping functions (resulting from axial load and bending moments) and the three base vectors for the D space are generated such that they complete the GD space using orthogonality conditions. Fig. 10 gives buckling loads versus half-wavelengths for Example 3. Fourteen curves are depicted and correspond to: (i) the first four buckling modes resulting from conventional FSM buckling analysis (FSM 1 to FSM 4), (ii) the first four pure local modes (Local 1 to Local 4), (iii) the three pure distortional modes (Dist. 1 to Dist. 3) and (iv) the three pure global modes (Global 1 to Global 3). The corresponding buckling mode shapes are drawn in Fig. 11. The envelopes of the calculated pure buckling curves agree
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N. Djafour et al. / Thin-Walled Structures 61 (2012) 42–48
30
70
30
10
10
80
t = 2 mm
80
Node 70
(dimensions in mm) Fig. 9. Example 3: (a) Monosymetric 1-cell-section and (b) FSM discretisation.
combined GD buckling mode space. Two assumptions are made: (i) cylindrical plate bending (if any) and (ii) negligible in-plane transverse strains and shear strains. The main contribution of this work is a new implementation of the second assumption which allowed RGD calculations for prismatic members with branches and/or closed parts. To validate and illustrate the application and capabilities of the developed method, three examples were considered: (i) an open branched cross-section, (ii) a close branched cross-section and (iii) a cross-section with branches and closed parts. Their buckling curves were compared to the conventional FSM results. The conclusion is that the new cFSM has successfully computed the GD and the L modes of all these sections.
105 Acknowledgments
Ncr (kN)
104
This work has been performed in Tlemcen, at the RISAM research laboratory, Aboubekr Belkaid University of Tlemen. The work was made possible by the financial support of the DGRSDT for which the authors would like to express their thanks.
103 FSM 1 FSM 2 FSM 3 FSM 4
102
Local 1 Local 2 Local 3 Local 4
Dist 1 Dist 2 Dist 3
Global 1 Global 2 Global 3
References
101 101
2
3
4
10 10 Half-wavelength (mm)
10
Fig. 10. FSM Buckling curves of the first four modes of 1-cell-section column and their decompositions into G, D and L pure modes.
mode 4
mode 3
mode 2
mode 1 Local
Dist.
classification at a = 10 000 mm
Global
Mode shapes at a = 10 000 mm
"Pure" Modes
FSM
Fig. 11. Buckling mode shapes of example 3.
with the ‘‘FSM’’ curves for all column lengths and for the four represented modes, except for curve FSM 3 and for long columns where the buckling mode is torsional. These results demonstrate that the proposed method can be used to calculate pure buckling modes of open and/or closed thin-walled sections.
5. Conclusions In this paper, the constrained finite strip method is improved by modifying the derivation of the RGD matrix which defines the
[1] Djafour M, Megnounif A, Kerdal DE, Belarbi AA. Numerical method for buckling analysis of built-up columns with stay plates. Structural Engineering and Mechanics 2007;26(4):441–57. [2] Davies JM, Leach P. First-order generalised beam theory. Journal of Constructional Steel Research 1994;31(2–3):187–220. [3] Dinis PB, Camotim D, Silvestre NGBT. Formulation to analyse the buckling behaviour of thin-walled members with arbitrarily ‘branched’ open crosssections. Thin-Walled Structures 2006;44(1):20–38. ¨ [4] Jonsson J, Andreassen MJ. Distortional eigenmodes and homogeneous solutions for semi-discretized thin-walled beams. Thin-Walled Structures 2011; 49(6):691–707. [5] A´da´ny S, Schafer BW. Buckling mode decomposition of single-branched open cross-section members via finite strip method: derivation. Thin-Walled Structures 2006;44(5):563–84. [6] A´da´ny S, Schafer BW. Buckling mode decomposition of single-branched open cross-section members via finite strip method: application and examples. Thin-Walled Structures 2006;44(5):585–600. [7] A´da´ny S, Schafer BW. A full modal decomposition of thin-walled, singlebranched open cross-section members via the constrained finite strip method. Journal of Constructional Steel Research 2008;64:12–29. [8] Li Z, Hanna MT, A´da´ny S, Schafer BW. Impact of basis, orthogonalization, and normalization on the constrained finite strip method for stability solutions of open thin-walled members. Thin-Walled Structures 2011;49(9):1108–22. [9] Bebiano R, Pina P, Silvestre N, Camotim D. GBTUL1.0b: buckling and vibration analysis of thin-walled members. Technical University of Lisbon; 2008 /http://www.civil.ist.utl.pt/gbtS. [10] Schafer BW CUFSM: elastic buckling analysis of thin-walled members by the finite strip method. Version 4.04. /www.ce.jhu.edu/bschafer/cufsmS (last visited 15 November 2011). [11] Casafont M, Marimon F, Pastor MM. Calculation of pure distortional elastic buckling loads of members subjected to compression via the finite element method. Thin-Walled Structures 2009;47(6–7):701–29. [12] Casafont M, Marimon F, Pastor M, Ferrer M. Linear buckling analysis of thinwalled members combining the generalised beam theory and the finite element method. Computers & Structures 2011;89(21–22):1982–2000. [13] A´da´ny S, Joo´ AL, Schafer BW. Buckling mode identification of thin-walled members by using cFSM base functions. Thin-Walled Structures 2010;48(10–11): 806–17. [14] Djafour M, Djafour N, Megnounif A, Kerdal DE. A constrained finite strip method for open and closed cross-section members. Thin-Walled Structures 2010;48(12):955–65. [15] Schafer BW, A´da´ny S. Buckling analysis of cold-formed steel members using CUFSM: conventional and constrained finite strip methods. In: Proceedings of the Eighteenth International Specialty Conference on Cold-Formed Steel Structures: Recent Research and Developments in Cold-Formed Steel Design and Construction; 2006, pp. 39–54. [16] Cheung YK. Finite Strip Method in Structural Analysis. New York: Pergamon Press; 1997.
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Thin-Walled Structures journal homepage: www.elsevier.com/locate/tws
A constrained finite strip method for prismatic members with branches and/or closed parts N. Djafour, M. Djafour n, A. Megnounif, M. Matallah, D. Zendagui RISAM, Department of Civil Engineering, University of Tlemcen, B.P. 119, Tlemcen 13000, Algeria
a r t i c l e i n f o
a b s t r a c t
Available online 19 September 2012
The decomposition of buckling modes of thin-walled members subjected to axial stresses is a topic of great practical interest which can be achieved using the generalised beam theory (GBT) or the constrained finite strip method (cFSM). However, the latter is not general enough to study prismatic members with arbitrary cross-sections and the objective of this paper is to extend the cFSM to allow the buckling modes decomposition for prismatic members with branches and/or closed parts. To define the combined GD buckling mode, two assumptions are used: (i) cylindrical plate bending and (ii) negligible in-plane transverse/shear strains. The corresponding constraint matrix, RGD, is derived in a simple and general way. The methodology used to separate the global and distortional modes is similar to that used in the original cFSM while the derivation of the constraint matrix for local modes remains identical. Some examples are considered and the pure buckling curves are compared to the conventional FSM results. The conclusion is that the new cFSM has successfully computed the GD and the L modes of these sections. & 2012 Published by Elsevier Ltd.
Keywords: Elastic buckling Thin-walled member Modal decomposition Constrained finite strip method
1. Introduction Thin-walled members are very efficient structural elements but their design under compressive loads is rather complex because of the cross-section instability and the great variety of cross-sectional shapes. To calculate the design buckling resistance of thin-walled members, three basic buckling modes are usually distinguished in current design codes: local (plate), distortional, and global (e.g., flexural or lateral-torsional buckling) buckling. This classification results from three different post-critical behaviours. Local buckling can have a significant post-critical reserve, especially for thin elements where the behaviour is mainly elastic. Distortional instability can, also, present a post-critical reserve, though less than local instability. Global buckling does not present such reserve and the strength of the member is less than its elastic critical load. The correct evaluation of the associated elastic critical loads (or moments) is fundamental in calculating the ultimate load carrying capacity of a thin-walled member. Simple analytical models are often available, but are limited in their applicability. An alternative possibility is to perform numerical elastic buckling analyses by using powerful numerical methods such as the finite element method (FEM) or the finite strip method (FSM). However, it is necessary to identify, among a huge amount of results [1], the buckling modes consistent with global, G, distortional, D, and local,
n
DOI of original article: http://dx.doi.org/10.1016/j.tws.2012.04.019 Corresponding author. Tel.: þ213 43 20 56 06; fax: þ213 43 20 41 89. E-mail address: [email protected] (M. Djafour).
0263-8231/$ - see front matter & 2012 Published by Elsevier Ltd. http://dx.doi.org/10.1016/j.tws.2012.04.020
L, modes which are also called the pure buckling modes. Another option is to use numerical methods able to perform modal decomposition, i.e., able to perform eigen-buckling analyses targeting any pure mode or any combination of pure modes. This was first possible by the generalised beam theory (GBT) [2–4] and then by the constrained finite strip method (cFSM) [5–8]. It should be pointed out that the cFSM is based on the fundamentals of GBT but the procedure is completely independent. In addition, GBT and cFSM give very similar results since they use the same mechanical definitions of the different buckling pure modes. Bebiano et al. made available the numerical tool GBTUL [9] to categorise the buckling modes of thin-walled members automatically via GBT. A´da´ny and Schafer [5–7] developed the constrained finite strip method (cFSM) and implemented it in the open source programme CUFSM [10] thus making it possible to separate the deformations of a thin-walled member into those consistent with the required pure modes. It is worth to mention that attempts to perform modal decomposition analyses via the finite element method (FEM) have been recently published. Casfont et al. [11,12] have tried to calculate the pure buckling modes of thin-walled members using the FEM, but GBT analyses, performed at the crosssection level, were used to calculate the necessary constraint matrices. The pure mode shapes from cFSM were also used by A´da´ny et al. [13] to perform modal identification, i.e., to measure the interactions, within a general eigenmode calculated via the FEM. Recently, Djafour et al. [14] proposed simple assumptions to define the combined GD buckling mode class in the context of the cFSM: (i) cylindrical plate bending (if any) and (ii) negligible in-plane transverse strains and shear strains. This new definition
N. Djafour et al. / Thin-Walled Structures 61 (2012) 42–48
Nomenclature a b dG, dD
member length strip length strip width coordinates of Vm in the vector bases hG and hD, respectively hG, hD system of base vectors for G and D spaces m total number of DOFs; m¼4 n n number of nodes nc number of strips connected to a given node ns number of strips r number of half-sine waves ui, vi , wi, yi local node DOFs xyz local coordinate axes E Young’s modulus H, Cek constraint matrices that enforce negligible transverse and shear strains in all the strips IVm, IDk identity matrices having the dimension of Vm and Dk, respectively KE stiffness matrix of the FSM model of the thin-walled member KGD modified stiffness matrix assuming cylindrical plate E bending
was used to formulate the constraint matrix of the combined GD space, RGD, in a simple and general way, preserving the flexibility of the cFSM as a general numerical method. The proposed derivation was successfully used to study ‘‘single branched’’, open or closed, cross-sections (Fig. 1(a)). The objective of this paper is to extend this new approach to allow the buckling modes decomposition for prismatic members with branches and/or closed parts (Fig. 1(b)). Some examples are considered and the computed pure buckling curves are compared to the conventional FSM results. It should be mentioned that the methodology proposed in this paper to separate the global and distortional modes is similar to that used in the original cFSM [5–7] while the derivation of the constraint matrix for local modes remains identical.
43
KG
geometric stiffness matrix of the FSM model of the thin-walled member Kk reduced stiffness matrix resulting from hypotheses 1 and 2 DV KDD k , Kk submatrices of Kk RM constraint matrix for buckling mode class M Ui, V i , W, Yi global node DOFs Vm vector of the warping DOFs in Dk, i.e., the vector of DOFs which defines the combined GD space: Vm DGD XYZ global coordinate axes a angle between local x and global X axes l load parameter n Poisson’s ratio D vector of the DOFs of the FSM mesh De, Dk DOFs to be eliminated and kept, respectively, in order to enforce negligible transverse and shear strains in all the strips DM vector of DOFs in the space of buckling mode class M K diagonal matrix of eigenvalues U matrix of eigenvectors
number of nodes is n which gives a total number of DOFs equal to m¼4 n. For stability analysis, two global matrices are necessary: the stiffness matrix, KE, and the geometric stiffness matrix, KG. To obtain KE and KG, local matrices of individual strips are formed, transformed from local to global coordinates and then assembled. Note, KG is function of the applied reference edge loading and is scaled linearly by a scalar l. The buckling modes of the thinwalled member are calculated by solving the following generalised eigenvalue problem (for more details see [15,16]): KE U ¼ KG UK
ð1Þ
where U ¼ ½ u1 u2 um is the matrix of eigenvectors and K ¼ diag½ l1 l2 lm is the diagonal, m m, matrix of eigenvalues.
2. Brief overview of the FSM and the cFSM
2.2. The constrained finite strip method
2.1. Finite strip method
The main idea of the cFSM [5] is to define constraint matrices for each of the buckling mode classes. These matrices define relationship between original FSM DOFs, D, and those corresponding to a buckling mode class, DM.
In FSM, a typical thin-walled member is divided into ns strips (elements) having all the length, a, of the member. To derive the matrix formulation, two (left-handed) coordinate systems are used: global and local. The global coordinate system is denoted as X–Y–Z, with the Y axis parallel to the longitudinal axis of the member. The local system, which is always associated with a strip, is denoted as x–y–z. Axes y and Y are parallels while axis z is normal to the strip. The intersection line of two connecting strips is called node and has four DOFs: 2 membrane (in-plane) DOFs, ui and vi, and 2 bending (out-of-plane) DOFs, wi and yi (Fig. 2). The
Fig. 1. (a) Unbranched and (b) branched (open and closed) cross-sections.
D ¼ RM DM
ð2Þ
Fig. 2. Strip DOFs, dimensions, and applied edge forces.
44
N. Djafour et al. / Thin-Walled Structures 61 (2012) 42–48
The dimension of D is m. It contains the four global DOFs of the n nodes and can be represented as follows: h i DT ¼ UT VT WT HT ð3Þ With UT ¼ U 1 U 2 U n , and, VT, WT and HT are described in the same way. DM has a smaller number of DOFs, mM, and defines a reduced deformation field that satisfies the criterion matching a buckling mode class M. The dimension of the constraint matrix RM is (m mM). The columns of RM may be considered as a set of base vectors, in the space of mode class M. According to general practice, A´da´ny and Schafer [5,7] defined four classes of modes: (i) global, G, (ii) distortional, D, (iii) local, L, and (iv) others, O. Eq. (2) may be employed on all eigenvectors in matrix U.
U ¼ RM UM
ð4Þ
Introducing Eq. (4) into Eq. (1) and pre-multiplying by RTM gives a new generalised eigenvalue problem. RTM KE RM UM ¼ RTM KG RM UM KM
ð5Þ
This equation can be rewritten as KEM UM ¼ KGM UM KM
ð6Þ
All the matrices in this equation have a reduced dimension, (mM mM). This new eigenvalue problem is defined in the constrained DOF spanned by the mode class M. This means that solving Eq. (6) gives solutions in the M space. Derivations of matrices RL and RO are straightforward processes and can be easily completed, even for arbitrary crosssections. They can be derived with exactly the same procedure as that presented in [7]. A much more problematic task is the definition of RG and RD matrices. Until now, the usual definition consists in forming RGD, the constraint matrix of the combined GD space, and, then, separating G and D by using orthogonality conditions. Recently, Djafour et al. [14] have simplified the derivation of RGD. In the context of the FSM, only two assumptions were necessary to define the GD space: (i) cylindrical plate bending (if any) and (ii) negligible in-plane transverse strains and shear strains. The RGD matrix was derived by means of procedures commonly used in numerical methods. This led to a simple and systematic formulation which was successfully used to compute GD and L modes of single branched open or closed cross-sections. The objective of the next section is to modify this formulation to allow the buckling modes decomposition for prismatic members with branches and/or closed parts.
and the warping displacements, Vm, of the main nodes. The idea is to impose the warping DOFs, Vm, as kinematic loading and to calculate the resulting DOFs. In this paper, the second step is revisited in order to allow RGD calculations for prismatic members with branches and/or closed parts. 3.2. Constraint matrix for membrane DOFs In the local coordinate system x–y–z of a strip of length a and width b, the membrane displacements are interpolated from the local membrane DOFs of its two nodes [15,16]. ( ) h i ui u ¼ ð1x=bÞ ðx=bÞ sinðr py=aÞ uj ( ) ð7Þ h i vi v ¼ ð1x=bÞ ðx=bÞ cosðr py=aÞ vj r is the number of half-sine-waves along the y axis. Assumption 2 can be written as
ex ¼ @u=@x ¼ 0 gxy ¼ @u=@yþ @v=@x ¼ 0
ð8Þ
The substitution of Eq. (7) into Eq. (8) leads to the following constraint relations ui ¼ uj ¼ ðvi vj Þ
1 bkr
ð9Þ
With kr ¼rp/a These constraint relations mean that the two local DOFs, ui and uj, are, from now on, defined from the warping (longitudinal) DOFs and are not effective, i.e., instead of the initial eight DOFs of the strip, only six DOFs are independent. It is also important to note that there is one constraint equation per node of the strip. In the finite strip method, it is well-known that the local to global transformation at node i for a strip which is oriented in the global coordinate system X–Y–Z at an angle a is governed by the following transformations: ( ) ) ( ) ) ( ( ui Ui vi Vi 1 0 cos a sin a ¼ and ¼ wi Wi yi Yi 0 1 sin a cos a ð10Þ
3. Derivation of the RGD matrix
Therefore, the local to global transformation of Eq. (9) gives the following two equations which provide a relationship between the transverse DOFs and the warping DOFs of both nodes.
3.1. General
U i cos a þW i sin a ¼ ðV i V j Þ bk1 r
As stated in reference [14], four major steps are necessary to derive RGD. The first one consists in calculating, KGD E , a modified member’s elastic stiffness matrix that takes into account assumption 1, i.e., the assumption of cylindrical plate bending in all strips. This matrix can easily be obtained using an equivalent beam model as done in the original cFSM formulation [5]. In the second step, a constraint matrix, C, for membrane DOFs is obtained from assumption 2. It represents the conditions on the DOFs of the FSM mesh that enforce negligible transverse and shear strains in all the strips. The third step is to apply the constraint matrix C to the modified global stiffness matrix KGD E by using a standard Lagrange multipliers procedure. Finally, RGD is calculated by using its definition: It must give the relationship between the vector D, containing all the DOFs of the FSM mesh,
U j cos a þW j sin a ¼ ðV i V j Þ bk1 r
ð11Þ
To achieve assumption 2 for the whole FSM model, it is necessary to write two constraint equations per strip in the global coordinate system. Consequently the number of DOFs to be eliminated to achieve the constraints brought by all the strips is (2ns). The key point of the new procedure resides in the way of selecting the DOFs to be eliminated. The new analysis is based on a loop on the nodes. For each node i, one needs to write equations and eliminate DOFs as many as strips connected to the node. The decision depends on the orientations of the strips, the number of connections and the currently free/active DOFs. A summary of the different situations is given in Table 1 and the procedure is explained hereafter.
N. Djafour et al. / Thin-Walled Structures 61 (2012) 42–48
Table 1 Summary of the various types of nodes. Number of connected strips
Illustration
DOFs to eliminate
1
Ui or Wi
2 (coplanar)
Ui or Wi and Vi
2 (non coplanar)
Ui and Wi
3
Ui, Wi and Vi
nc 43
Ui, Wi, Vi and (nc 3) Vs of the neighbouring nodes (Vj and Vk in the example)
45
If at least two strips have different directions, Eq. (13) can be solved for the three translational DOFs of node i, which are the three DOFs we have to reduce in this case. The last situation occurs when nc strips, with nc43, are connected to node i which implies that nc DOFs have to be reduced: the three translational DOFs of node i and (nc 3) warping DOFs of the neighbouring nodes. Since the decision goes out of the range of the current node, it is necessary to search and find enough free/active warping DOFs from the nodes connected to node i. It is important to keep the analysis of such ‘‘over constrained’’ node to the end and note that the selection is not unique. To implement the procedure presented above, an idea, to address the problem globally and to deal with the various couplings, consists in rewriting each constraint equation in the following matrix form: 9 8 Ui > > > > > > > > = < Vi > h i> cos a 1=bkr sin a 1=bkr ¼0 ð14Þ W > i > > > > > > > > ; : Vj > Then, Eq. (14) can be expanded to involve all the DOFs of the FSM model. Inside the loop on nodes, the constraint equations are written sequentially and, at the end, a set of (2ns) constraint equations is obtained which can be written in a compact form as follow:
If only one strip is connected to node i, there will be a single equation involving its transverse DOFs. From the constraint equation for node i, i.e., the first line of Eq. (11), it is obvious that if sin a ¼0, there is a unique possibility and the transverse DOF to be reduced is Ui. Similarly, if cos a ¼ 0, the DOF to be eliminated is Wi. Otherwise, it is possible to reduce either Ui or Wi by defining it from the remaining transverse DOF and the warping DOFs. Preferably, the DOF to eliminate is the one whose direction is closest to the local x axis. This preference is rather a precaution to avoid possible singularities that would be caused by the reduction of a DOF nearly perpendicular to the local x axis. If node i is connected to two nodes, j and k, by two strips, s1 and s2, one constraint equation per strip can be written as follows: U i cos as1 þW i sin as1 ¼ ðV i V j Þ bs11kr U i cos as2 þ W i sinas2 ¼ ðV i V k Þ bs21kr
ð12Þ
HD ¼ 0
ð15Þ
The dimension of H is (2ns) m. The other result of this section is that, among the m DOFs that form vector D, a set of (2ns) DOFs, denoted as De, are not effective. They can be deduced from the remaining (m 2ns) DOFs, denoted as Dk. Eq. (15) is partitioned as follows: ( ) De HD ¼ He Hk ¼0 ð16Þ
Dk
which gives
De ¼ He 1 Hk Dk ¼ Cek Dk
ð17Þ
Having matrix Cek, the rest of the procedure remains identical to reference [14] and is briefly presented hereafter. 3.3. Derivation of RGD
where bs1 and bs2 are the respective widths; as1 and as2 are the respective angles. In case of non-coplanar strips, it is obvious that Eq. (12) can be solved for Ui and Wi to define the two transverse DOFs of node i from the warping DOFs of the three connected nodes. Thus, the two DOFs to be eliminated are Ui and Wi. In case of coplanar strips, the left-hand sides of Eq. (12) are identical, at least in absolute value. It becomes clear that the relevant warping DOFs are not independent. One of them must be defined from the others before being eliminated. It is preferable to reduce Vi to limit the elimination process to the DOFs of the current node. The second DOF to be eliminated can be Ui or Wi, depending on the orientation of the strips, as explained above for the case of one connected strip. If node i is connected to three nodes, j, k and l, by three strips, s1 to s3, three constraint equations are involved and can be written as follows: 2
cos as1 6 cos a s2 4 cos as3
1=kr bs1 1=kr bs2 1=kr bs3
9 8 9 38 sin as1 > > V j =bs1 > < Ui > = = 1< 7 sin as2 5 V i V k =bs2 ¼ > > kr > : V =b > ; sin as3 : W i ; s3 l
ð13Þ
asi and bsi are the angle and the width of strip si, respectively.
First, the elastic stiffness matrix in the GD space, KGD E , is partitioned into four matrices by rearranging the DOFs. 2 3( ) Kek Kee De E E GD 4 5 KE D ¼ ð18Þ Dk Kke Kkk E E By using standard Lagrange multipliers procedure [14], Eq. (17) is applied to Eq. (18). The result is Kk, the stiffness matrix of the FSM model of the thin-walled member when supposing cylindrical plate bending and neglecting in-plane transverse and shear strains, i.e., when applying hypotheses 1 and 2. T T ke ek ee Kk ¼ Kkk E þ KE Cek þ Cek KE þ Cek KE Cek
ð19Þ
Kk is a square matrix of dimension equal to the number of the remaining DOFs, (4n 2ns). In the cFSM, the RGD matrix is the constraint matrix that defines the GD space. It defines the vector D, containing all the DOF of the FSM mesh, from the longitudinal – or warping – displacements, Vm, of the main nodes.
D ¼ RGD Vm
ð20Þ
The idea is to impose the warping DOFs, Vm, as kinematic loading and to calculate the resulting DOFs. By rearranging the
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N. Djafour et al. / Thin-Walled Structures 61 (2012) 42–48
DOFs, Matrix Kk and vector Dk can be rewritten as follows: " VV #( ) KVk D Kk Vm Fm ð21Þ ¼ Kk Dk ¼ DV DD Du 0 Kk Kk Note that the forces Fm at warping DOFs are the ‘‘unknown’’ reactions resulting from the kinematic loading; the remaining forces must be zero. The second equation in Eq. (21) gives h i Du ¼ KkDD1 KkDV Vm ð22Þ Eq. (22) can be expanded by an identity matrix IVm having the dimension of Vm to concern all the DOFs in Dk. ( ) " # IVm Vm Dk ¼ ð23Þ ¼ DD1 DV Vm Kk Kk Du Similarly, Eq. (17) can be expanded by an identity matrix IDk having the dimension of Dk to concern all the DOFs in D. ( ) " # Cek De ¼ D¼ Dk ð24Þ IDk Dk Finally, the substitution of Eq. (23) into Eq. (24) gives the required matrix in Eq. (20). " #" # IVm Cek ð25Þ RGD ¼ V KkDD1 KD I Dk k
eight modes in the GD space which can be obtained from the reduced eigenvalue problem defined by Eq. (6). Elastic buckling analyses were conducted by varying the member half-wavelength from 20 to 10,000 mm. The results are shown in Figs. 4 and 5. The curves ‘‘FSM 1’’ to ‘‘FSM 4’’ in Fig. 4 provides bifurcation stresses associated with the first four modes when using conventional FSM. Curves ‘‘Local 1’’ to ‘‘Local 4’’ correspond to plate bending modes and were targeted using a specific RL constraint matrix as defined in [7]. The other four curves in Fig. 4, i.e., ‘‘GD 1’’ to ‘‘GD 4’’, are the first four modes given by the cFSM when targeting the GD buckling mode class. Fig. 5 gives the corresponding buckling mode shapes. It is obvious that the cFSM has successfully decomposed the FSM solution of the E-section into L and GD ‘‘pure’’ modes. Furthermore, the (Local 1) and (GD 1) curves given in Fig. 4 are almost the same as those obtained by Dinis et al. [3] using GBT. 4.2. Buckling mode decomposition of a closed and branched cross-section In this section, an example of closed and branched cross-section is considered. Its shape is an asymmetric 3-cells-section formed by 10 plate elements (drawing 3 connected rectangles) as shown in Fig. 6. The cross-section mesh involves 38 nodes and 40 strips (4 strips per plate). Note that there are 30 nodes which are common to two, and
105 4. Numerical examples In order to validate and illustrate the application and capabilities of the developed method, three examples are considered: (i) an open branched cross-section, (ii) a close branched crosssection and (iii) a cross-section with branches and closed parts. All these examples are simply supported columns (with pinned and free-to-warp end sections) which are loaded by uniform compressive forces and supposed to buckle in one-half sine wave. The modulus of elasticity is E¼210 GPa, and Poisson’s ratio is n ¼0.3. For validation purposes, all the results are compared with values obtained by finite strip analyses.
Ncr (kN)
104 103 102 101
FSM 1 FSM 2 FSM 3 FSM 4
100 101
Local 1 Local 2 Local 3 Local 4
GD 1 GD 2 GD 3 GD 4
102 103 Half-wavelength (mm)
104
4.1. Buckling mode decomposition of open branched cross-section The first illustrative example is a simple column member with an asymmetric E-section which was analysed by Dinis et al. [3] using the GBT. The cross-section is presented in Fig. 3(a) (all dimensions are mid-line for the cross-section.). The FSM mesh (Fig. 3(b)) uses 19 strips and 20 nodes which give a total of 80 DOFs. Eleven nodes are secondary nodes since they are common to two coplanar strips. One node is connected to three strips, which classifies this example as ‘‘branched cross-section’’. The model has eight main nodes and the corresponding RGD matrix has a dimension of (80 8). This means that this E-section has
Fig. 4. FSM Buckling curves of the first four modes of E-section column and their decompositions into GD and L pure modes.
mode 4
mode 3
mode 2 60 8 75
mode 1
t = 1 mm 5
45 20
8 60 (dimensions in mm)
Node
Fig. 3. Example 1: (a) Asymmetric E-section and (b) FSM discretisation.
any a Local
a = 60 mm
a = 600 mm a = 10 000 mm Combined GD mode
Fig. 5. GD and L Buckling mode shapes of example 1.
N. Djafour et al. / Thin-Walled Structures 61 (2012) 42–48
47
105
70
Node Ncr (kN)
t = 2 mm
70
104
103
50 50 (dimensions in mm) Fig. 6. Example 2: (a) Asymmetric 3-cells-section and (b) FSM discretisation.
only two, coplanar strips. In addition, note that, among the remaining eight nodes, two nodes are common to three strips and that one node is connected to four strips. This means that, according to the procedure summarised in Table 1, there are only 4 nodes with effective warping DOFs. Thus, the dimension of RGD matrix is (152 4) and consequently, there are 4 modes in the GD space. It is important to note that the constraint matrix RGD alone is not able to separate the G and D spaces. A further transformation inside the GD space is necessary [5,7]. In cFSM, the idea is to seek a basis which can be used to describe any deformation Vm of this space, and in which the system of base vectors for the G and D spaces are separated; these are denoted hG and hD, respectively. Vm ¼ hG
hD
(
dG dD
)
FSM 1 FSM 2 FSM 3 FSM 4
102 101
Local 1 Local 2 Local 3 Local 4
Dist 1 Global 1 Global 2 Global 3
102 103 Half-wavelength (mm)
104
Fig. 7. FSM Buckling curves of the first four modes of 3-cells-section column and their decompositions into G, D and L pure modes.
mode 4
mode 3
mode 2
ð26Þ
dG and dD are the coordinates of Vm in this new basis. According to cFSM procedure, the system of base vectors hG is ‘‘directly’’ defined and the vectors hD are generated such that they complete the GD space using orthogonality conditions [5]. To target only modes in the D space, the GD space basis is truncated to hD and the constraint matrix is defined by RD ¼RGDhD. Having RD and according to the procedure presented in Section 2, the buckling loads and modes in the D space can be obtained from the reduced eigenvalue problem defined by Eq. (6). For open cross-sections, hG is defined [5] by four special warping functions associated with the longitudinal displacements resulting from special loads, namely: (i) pure axial load, (ii) pure bending moment about the first principal axis of the cross-section, (iii) pure bending moment about the second principal axis, (iv) pure torque. However for closed cross-sections, it is known from literature [14] that the torsional mode cannot occur without membrane shear strains and thus does not belong to the GD space. In this paper, hG for closed cross-sections is obtained from the remaining three special warping functions, i.e., the longitudinal displacements resulting from axial load and bending moments. For the numerical example in progress, which is a closed crosssection with four effective warping DOFs, the dimension of hG is (4 3) and the dimension of hD is (4 1). This 3-cells-section has only one distortional mode according to the definitions used in the present paper. Fig. 7 gives buckling loads versus half-wavelengths for Example 2. Twelve curves are depicted corresponding to: (i) the first four buckling modes resulting from conventional FSM buckling analysis (FSM 1 to FSM 4), (ii) the first four pure local modes (Local 1 to Local 4), (iii) the unique pure distortional mode (Dist. 1) and (iv) the three pure global modes (Global 1 to Global 3). The corresponding buckling mode shapes are drawn in Fig. 8. It is clear that the envelopes of the pure buckling curves, calculated using the developed method, agree with the ‘‘FSM’’ curves for all column lengths and for the four depicted modes, except for curve
mode 1
Local
Dist.
Global
Mode shapes at a = 10 000 mm
"Pure" Modes
FSM
Fig. 8. Buckling mode shapes of example 2.
FSM 3 and for lengths higher than 2 m where the buckling mode is torsional. This validates the proposed method for buckling mode decomposition of closed thin-walled sections. 4.3. Buckling mode decomposition of a cross-section with branches and closed parts The third example in this paper is a 1-cell-section column having branches and closed parts. Its shape is shown in Fig. 9. The FSM model uses 24 nodes and 24 strips (4 strips per plate in the closed part and 2 strips per plate in the open parts). Sixteen nodes are common to two, and only two, coplanar strips and two nodes are common to three strips. This means that the dimension of RGD matrix is (96 6) and consequently, there are six modes in the GD space. To separate global and distorsional modes, the three base vectors defining the G space are obtained from three special warping functions (resulting from axial load and bending moments) and the three base vectors for the D space are generated such that they complete the GD space using orthogonality conditions. Fig. 10 gives buckling loads versus half-wavelengths for Example 3. Fourteen curves are depicted and correspond to: (i) the first four buckling modes resulting from conventional FSM buckling analysis (FSM 1 to FSM 4), (ii) the first four pure local modes (Local 1 to Local 4), (iii) the three pure distortional modes (Dist. 1 to Dist. 3) and (iv) the three pure global modes (Global 1 to Global 3). The corresponding buckling mode shapes are drawn in Fig. 11. The envelopes of the calculated pure buckling curves agree
48
N. Djafour et al. / Thin-Walled Structures 61 (2012) 42–48
30
70
30
10
10
80
t = 2 mm
80
Node 70
(dimensions in mm) Fig. 9. Example 3: (a) Monosymetric 1-cell-section and (b) FSM discretisation.
combined GD buckling mode space. Two assumptions are made: (i) cylindrical plate bending (if any) and (ii) negligible in-plane transverse strains and shear strains. The main contribution of this work is a new implementation of the second assumption which allowed RGD calculations for prismatic members with branches and/or closed parts. To validate and illustrate the application and capabilities of the developed method, three examples were considered: (i) an open branched cross-section, (ii) a close branched cross-section and (iii) a cross-section with branches and closed parts. Their buckling curves were compared to the conventional FSM results. The conclusion is that the new cFSM has successfully computed the GD and the L modes of all these sections.
105 Acknowledgments
Ncr (kN)
104
This work has been performed in Tlemcen, at the RISAM research laboratory, Aboubekr Belkaid University of Tlemen. The work was made possible by the financial support of the DGRSDT for which the authors would like to express their thanks.
103 FSM 1 FSM 2 FSM 3 FSM 4
102
Local 1 Local 2 Local 3 Local 4
Dist 1 Dist 2 Dist 3
Global 1 Global 2 Global 3
References
101 101
2
3
4
10 10 Half-wavelength (mm)
10
Fig. 10. FSM Buckling curves of the first four modes of 1-cell-section column and their decompositions into G, D and L pure modes.
mode 4
mode 3
mode 2
mode 1 Local
Dist.
classification at a = 10 000 mm
Global
Mode shapes at a = 10 000 mm
"Pure" Modes
FSM
Fig. 11. Buckling mode shapes of example 3.
with the ‘‘FSM’’ curves for all column lengths and for the four represented modes, except for curve FSM 3 and for long columns where the buckling mode is torsional. These results demonstrate that the proposed method can be used to calculate pure buckling modes of open and/or closed thin-walled sections.
5. Conclusions In this paper, the constrained finite strip method is improved by modifying the derivation of the RGD matrix which defines the
[1] Djafour M, Megnounif A, Kerdal DE, Belarbi AA. Numerical method for buckling analysis of built-up columns with stay plates. Structural Engineering and Mechanics 2007;26(4):441–57. [2] Davies JM, Leach P. First-order generalised beam theory. Journal of Constructional Steel Research 1994;31(2–3):187–220. [3] Dinis PB, Camotim D, Silvestre NGBT. Formulation to analyse the buckling behaviour of thin-walled members with arbitrarily ‘branched’ open crosssections. Thin-Walled Structures 2006;44(1):20–38. ¨ [4] Jonsson J, Andreassen MJ. Distortional eigenmodes and homogeneous solutions for semi-discretized thin-walled beams. Thin-Walled Structures 2011; 49(6):691–707. [5] A´da´ny S, Schafer BW. Buckling mode decomposition of single-branched open cross-section members via finite strip method: derivation. Thin-Walled Structures 2006;44(5):563–84. [6] A´da´ny S, Schafer BW. Buckling mode decomposition of single-branched open cross-section members via finite strip method: application and examples. Thin-Walled Structures 2006;44(5):585–600. [7] A´da´ny S, Schafer BW. A full modal decomposition of thin-walled, singlebranched open cross-section members via the constrained finite strip method. Journal of Constructional Steel Research 2008;64:12–29. [8] Li Z, Hanna MT, A´da´ny S, Schafer BW. Impact of basis, orthogonalization, and normalization on the constrained finite strip method for stability solutions of open thin-walled members. Thin-Walled Structures 2011;49(9):1108–22. [9] Bebiano R, Pina P, Silvestre N, Camotim D. GBTUL1.0b: buckling and vibration analysis of thin-walled members. Technical University of Lisbon; 2008 /http://www.civil.ist.utl.pt/gbtS. [10] Schafer BW CUFSM: elastic buckling analysis of thin-walled members by the finite strip method. Version 4.04. /www.ce.jhu.edu/bschafer/cufsmS (last visited 15 November 2011). [11] Casafont M, Marimon F, Pastor MM. Calculation of pure distortional elastic buckling loads of members subjected to compression via the finite element method. Thin-Walled Structures 2009;47(6–7):701–29. [12] Casafont M, Marimon F, Pastor M, Ferrer M. Linear buckling analysis of thinwalled members combining the generalised beam theory and the finite element method. Computers & Structures 2011;89(21–22):1982–2000. [13] A´da´ny S, Joo´ AL, Schafer BW. Buckling mode identification of thin-walled members by using cFSM base functions. Thin-Walled Structures 2010;48(10–11): 806–17. [14] Djafour M, Djafour N, Megnounif A, Kerdal DE. A constrained finite strip method for open and closed cross-section members. Thin-Walled Structures 2010;48(12):955–65. [15] Schafer BW, A´da´ny S. Buckling analysis of cold-formed steel members using CUFSM: conventional and constrained finite strip methods. In: Proceedings of the Eighteenth International Specialty Conference on Cold-Formed Steel Structures: Recent Research and Developments in Cold-Formed Steel Design and Construction; 2006, pp. 39–54. [16] Cheung YK. Finite Strip Method in Structural Analysis. New York: Pergamon Press; 1997.