Abstract
In this study, a new subparametric strip element is developed to simulate the axially loaded composite cylindrical panel with arbitrary cutout. For this purpose, a code called SSFSM is developed in FORTRAN to analyze the buckling of panels. The first order shear deformation theory is used to form the straindisplacement relations. Spline and Lagrangian functions are used to derive element shape functions in the longitudinal and transverse directions, respectively. The computational cost of the SSFSM is decreased dramatically, as mapping functions of the strip element are very simple. The results obtained from the SSFSM are compared with those of the literature and the results obtained by ABAQUS to show the validity of the proposed approach. A parametric study is performed to show the capability of the SSFSM in calculating the panel buckling load. Results indicate that increasing the panel thickness and panel central angles cause an increase in panel buckling load. The cutout shape is an important factor influencing the panel buckling load. For instance, when the angle between the direction of big chord of the elliptical cutout and compressive load direction are 0 and 90 degrees, the panel buckling load reaches its minimum and maximum magnitude, respectively.
Keywords:
Composite; perforated cylindrical panel; spline finite strip; subparametric
1 INTRODUCTION
In the recent decades, plates and cylindrical composite panels with and without holes are used widely in the aerospace, civil and mechanical industries (Shuohui et al. 2016aShuohui, Y., Tiantang, Y., Tinh, Q.B., Peng, L., Sohichi, H., (2016a). Buckling and vibration extended isogeometric analysis of imperfect graded ReissnerMindlin plates with internal defects using NURBS and level sets. Computers & Structures 177: 2338.bShuohui, Y., Tinh, Q.B., Shifeng, T., Satoyuki, T., Sohichi, H., (2016b). NURBSbased isogeometric analysis of buckling and free vibration problems for laminated composites plates with complicated cutouts using a new simple FSDT theory and level set method. ThinWalled Structures 101: 141 156., Nejati et al. 2017Nejati, M., Dimitri, R., Tornabene, F., Hossein Yas, M. (2017). Thermal buckling of nanocomposite stiffened cylindrical shells reinforced by Functionally Graded Wavy Carbon NanoTubes with temperaturedependent properties. Applied Science 7: 124., Asadi and Qatu 2013Asadi, E., Qatu, M.S., (2013). Free vibration of thick laminated cylindrical shells with different boundary conditions using general differential quadrature. Journal of Vibration and Control 19: 356366., Khalfi et al. 2014Khalfi, Y., Houari, M.S.A., Tounsi, A., (2014). A refined and simple shear deformation theory for thermal buckling of solar functionally graded plates on elastic foundation. International Journal of Computational Methods 11: 1350077., Khetir et al. 2017Khetir, H., Bouiadjra, M.B., Houari, M.S.A., Tounsi, A., Mahmoud, S.R., (2017). A new nonlocal trigonometric shear deformation theory for thermal buckling analysis of embedded nanosize FG plates. Structural Engineering and Mechanics 64: 391402., Asadi et al. 2012). In this matter, buckling analysis of these structures under different load cases has been investigated by several researchers using different methods (Tiantang et al. 2017Tiantang, Y., Shuohui, Y. Tinh, Q.B., Chen, L., Nuttawit W., (2017). Buckling isogeometric analysis of functionally graded plates under combined thermal and mechanical loads. Composite Structures 162: 5469., Tiantang et al. 2016, Patel et al. 2006Patel, B.P., Singh, S., Nath, Y., (2006). Stability and nonlinear dynamic behaviour of crossply laminated heated cylindrical shells. Latin American Journal of Solids and Structures 3: 245 261., Vuong and Dung 2017Vuong, P.M., Dung, D.V., (2017). Nonlinear analysis on buckling and postbuckling FGM imperfect cylindrical shells filled inside by elastic foundations in thermal environment using TSDT. Latin American Journal of Solids and Structures 14: 950 977., Soltani et al. 2019Soltani, K., Bessaim Houari, M.S.A., Kaci, A., Benguediab, M., Tounsi, A., Alhodaly, M., (2019). A novel hyperbolic shear deformation theory for the mechanical buckling analysis of advanced composite plates resting on elastic foundations. Steel and Composite structures 30:1329., Asadi and Qatu 2013). Magnucki and Mackiewicz (2006Magnucki, K., Mackiewicz, M., (2006). Elastic buckling of an axially compressed cylindrical panel with three edges simply supported and one edge free. ThinWalled Structures 44: 387392.) investigated the effects of panel boundary conditions on the elastic buckling of composite cylindrical panel. In their research, an analytical approach was used to calculate the buckling load of a panel without any cutout. Effects of rectangular and circular cutouts on the buckling load of laminated plate were investigated analytically by Masia et al (2005Masia, U., Avalos, D.R., Laura, P.A.A., (2005). Displacement amplitudes and flexural moments for a rectangular plate with a rectangular cutout under a uniformly distributed static load. Journal of Sound and Vibration 280: 433442.) and Ashrafi et al (2013Ashrafi, H., Asemi, K., Shariyat, M., (2013). A three dimensional boundary element stress and bending analysis of transversely/longitudinally graded plates with circular cutouts under biaxial loading. European Journal of Mechanics A Solids 42: 344357.), respectively. Although analytical approaches are simple to calculate the buckling load, but they have some limitations such as cutout shape, panel boundary conditions and the structure shape. To overcome these limitations, the Finite Element Method (FEM) was used extensively to calculate the buckling load of composite cylindrical panels. Anil et al (2007Anil, V., Upadhyay, C.S. and Iyengar, N.G.R., (2007). Stability analysis of composite laminate with and without rectangular cutout under biaxial loading. Composite Structures 80: 92104.) evaluated the effect of rectangular cutout on the critical load of composite plate using FEM. The influence of circular cutout on the buckling load of laminated plate was investigated by Komur and Sonmez (Komur and Sonmez 2008Komur, M.A., Sonmez, M., (2008). Elastic buckling of rectangular plates under linearly varying inplane normal load with a circular cutout. Mechanics Research Communucations 35: 361371.). The effects of cutouts with various shapes (such as circular and elliptical shapes) on the buckling load of composite plates were also evaluated by FEM in the several researches (Arbelo et al. 2015Arbelo, M.A., Herrmann, A., Castro, S.G.P., Khakimova, R., Zimmermann, R., Degenhardt, R., (2015). Investigation of Buckling Behavior of Composite Shell Structures with Cutouts. Applied Composite Materials 22: 623636., Shuohui et al. 2015, Thinh et al. 2016Thinh, T.I., Tu., T.M., Quoc, T.H., Long, N.V., (2016). Vibration and buckling analysis of functionally graded plates using new eightuknown higher order shear deformation theory. Latin American Journal of Solids and Structures 13: 456 477., Rajanna et al. 2016Rajanna, T., Banerhee, S., Desai, M.Y., (2016). Effects of partial loading and fiber configuration on vibration and buckling characteristics of stiffened composite plates. Latin American Journal of Solids and Structures 13: 854 879., Soni et al. 2013Soni, G., Singh, R., Mitra, M., (2013). Buckling behavior of composite laminates (with and without cutouts) subjected to nonuniform inplane loads. International Journal of Structural Stability and Dynamics 13: 120., Pascan et al. 2012Pascan, O.Z., Weihong, Z., Ponthot, J.P., (2012). Mechanical buckling analysis of composite panels with/without cutouts. International Journal of Plant Engineering Management 17: 6576.).
Use of the Finite Strip Method (FSM) is simpler and more economical compared to the FEM, in order to analyze long thinwalled structures such as cylindrical panels (Mirzaei et al. 2015Mirzaei, S., Azhari, M., Bondarabady Rahimi, H.A., (2015). On the use of finite strip method for buckling anaylsis of moderately thick plate by refined plate theory and using new types of functions. Latin American Journal of Solids and Structures 12: 561 582., Fazilati 2017Fazilati, J., (2017). Stability analysis of variable stiffness composite laminated plates with delamination using Spline FSM. Latin American Journal of Solids and Structures 14: 528 543., Balogh and Logo 2015, Rondal 1998Rondal, J., (1998). Coupled Instabilities in Metal Structures: Theoretical and Design Aspects, Springer publisher, New York, USA.). Moreover, while in the FEM the structure is discretized in transverse and longitudinal directions, it is only discretized in transverse direction in FSM (Rondal 1998). FSM can be divided into two categories. In the first one, called semianalytical FSM, the longitudinal shape function of strip element is considered as harmonic function. Also, the transverse shape function of strip element is considered as polynomial function. Buckling load of laminated plates without any cutout subjected to different load cases was evaluated using semianalytical FSM (Yao et al. 2015Yao, L.K., He, B., Zhang, Y., Zhou, W., (2015). Semianalytical finite strip transfer matrix method for buckling analysis of rectangular thin plates. Mathematical Problems in Engineering 15: 111., Ovesy et al. 2015Ovesy, H.R., Totounferoush, A., Ghannadpour, S.A.M., (2015). Dynamic buckling analysis of delaminated composite plates using semianalytical finite strip method. Journal of Sound and Vibration 343: 131143., Dawe et al. 1995Dawe, D.J., Wang, S., Lam, S.S.E., (1995). Finite strip analysis of imperfect laminated plates under end shortening and normal pressure. International Journal for Numerical Methods in Engineering 38: 41934205., Ovesy and Fazilati 2012b). Semi analytical FSM is very useful and efficient for buckling analysis of laminated plate when the boundary conditions of both ends of strip elements are simply supported (Wang et al. 1998Wang, S., Chen, J., Dawe, D.J., (1998). Linear transient analysis of rectangular laminates using spline finite strips. Composites Structures 41: 5766., Dawe and Wang 1994, Cheung and Tham 1997Cheung, Y.K., Tham, L.G. (1997). The Finite Strip Method, CRC Press, New York, USA.). The longitudinal harmonic shape functions of strip element are complicated in case of more complex plate/panel boundary conditions (Wang et al. 1998, Dawe and Wang 1994, Cheung and Tham 1997). Spline FSM (SFSM) is used in the analysis of buckling load of plate/panel with different boundary conditions (Second category of the FSM) (Wang et al. 1998, Dawe and Wang 1994). The spline function is used as the shape function of strip element in the longitudinal direction. Au and Cheung (1996Au, F.T.K., Cheung, Y.K. (1996). Free vibration and stability analysis of shells by the isoparametric spline finite strip method. ThinWalled Structures 24: 5382.) investigated the free vibration and stability of shell without any cutout utilizing isoparametric SFSM. In their work, First Order Shear Deformation plate Theory (FSDT) was used to model the shell. Fazilati and Ovesy (2013) investigated the effects of structure boundary conditions and material on the buckling load of laminated stiffened cylindrical panel with rectangular cutout. They developed a 3D model of cylindrical panel by using SFSM based on Higher order Shear Deformation Theory (HSDT). Finally, Eccher et al. (2008Eccher, G., Rasmussen, K.J.R., Zandonini, R., (2008). Elastic buckling analysis of perforated thinwalled structures by the isoparametric spline finite strip method. ThinWalled Structures 46: 165  191.) evaluated the elastic buckling of perforated thinwalled structures such as cylindrical panels by isoparametric SFSM. In their research, an isoparametric spline finite strip element was developed based on the FSDT. The strip element shape functions of longitudinal and transverse directions have been considered as polynomial and B3 spline respectively (Eccher et al. 2008). Based on the literature, the order of mapping function of subparametric element is lower than that of isoparametric element (Akin 2005Akin, J.E. (2005). Finite Element Analysis with Error Estimators, Elsevier publisher, New York, USA.) and therefore, equilibrium equations obtained in case of subparametric element have lower order than isoparametric ones and could cause lower computational cost (Akin 2005).
A review of the literature reveals that there are two major limitations in buckling analysis of perforated laminated cylindrical panel. While the computational cost of subparametric element is lower than that of the isoparametric element, there is no sign of considering subparametric SFSM in buckling analysis of perforated laminated cylindrical panel. Moreover, although the effects of single cutout on the buckling load of cylindrical panel are investigated in the literature, the effects of cutout with different shapes on the buckling load of composite laminated cylindrical panel are not compared with each other. Therefore, this paper is a response to these needs. In this regard, at first the shape and mapping functions of subparametric spline finite strip element are developed. Next, the displacement functions are derived. Then, the components of Jacobian matrix are calculated and based on the FSDT, the straindisplacement and stressstrain relationships are extended. According to the shape functions, the load vector of strip element is evaluated. Then, components of elastic stiffness and stability stiffness matrices of subparametric element are derived. Finally, according to the developed element, the elastic buckling analysis of laminated perforated cylindrical panel is carried out. To validate the results obtained in this paper, comparison are made between the results obtained in this study and those published in the literature and those obtained from commercial finite element software, namely ABAQUS (ABAQUS 2006).
2 Methodology
The cylindrical panel with arbitrary cutout is shown in Figure 1. As it is shown in Figure 1, R, B,
2.1 Shape functions of strip element
The
The C^{0} continuity is implied in the strip transverse direction (Eccher et al. 2008Eccher, G., Rasmussen, K.J.R., Zandonini, R., (2008). Elastic buckling analysis of perforated thinwalled structures by the isoparametric spline finite strip method. ThinWalled Structures 46: 165  191.). Due to the use of this shape function, it is possible to impose any arbitrary boundary condition on the longitudinal strip edges. Four nodal lines are considered in the longitudinal direction of the strip element and m knots (node) are considered in each nodal line. In other word, each nodal line is divided into m sections. The
where,
The displacement function of the strip element is defined based on the
Spline function in longitudinal direction (a)
2.2 Mapping functions
The subparametric strip element in the local Cartesian coordinate X and Y is shown in Figure 6a. This element is mapped into the parent element as it is shown in Figure 6b. The mapping functions are used to map the element from (X,Y) coordinate to the (
The mapping functions
Same as x, the ycoordinate is estimated by eight nodes laid down on the first and last strip edges. Therefore, y can be calculated by equation 10. In this equation, i is the node number,
Based on the advantages of the strip elements mentioned in this section of the paper, y is simplified and calculated based on the node numbers 1, 2, 3 and 4 (four corner nodes of the element). Therefore, equation 10 can be rewritten as the following equation.
where, ycoordinate of nodes 5, 6, 7 and 8 is calculated based on the ycoordinate of node number 1, 2, 3 and 4. Inserting equations 7 and 8 into equation 11, it can be simplified to get the following relation.
where, ycoordinate of strip element is estimated using four corner nodes of the element. In equation 12,
In equation 13,
Therefore, the computational cost of calculation of Jacobian matrix is decreased dramatically due to the simplicity in evaluation of
2.3 Jacobian matrix
The determinant of the Jacobian matrix (
As it is shown in equation 15, the determinant of Jacobian matrix is related to the
where,
For isoparametric strip element used in the literature, approximation of geometry (x, y) follows the same rule as for displacement shape functions (Eccher et al. 2008Eccher, G., Rasmussen, K.J.R., Zandonini, R., (2008). Elastic buckling analysis of perforated thinwalled structures by the isoparametric spline finite strip method. ThinWalled Structures 46: 165  191.). So, the geometry of isoparametric strip element is approximated by following equations (Eccher et al. 2008).
in which coefficients
2.4 Strain displacement relationships
In this paper, the strain displacement relationship is assumed to be Sanderstype (Moradi et al. 2011Moradi, S., Poorveis, D., Khajehdezfuly, A., (2011). Geometrically nonlinear analysis of anisotropic laminated cylindrical panels with cutout using spline finite strip method. Proceeding of conference on the Advances in Structural Engineering and Mechanics, Seoul, South Korea., Khayat et al. 2016aKhayat, M., Poorveis, D., Moradi, S., Hemmati, M. (2016a). Buckling of thick deep laminated composite shell of revolution under follower forces. Structural Engineering and Mechanics 58: 5991., Khayat et al. 2016b). As it is illustrated in equation 19, the cylindrical panel strain vector
The components of linear part of strain vector (
The relationship between generalized linear stress vector (
where the components of generalized linear stress vector (i.e.
In this equation,
In equation 22,
Note that,
where, c and s stand for the
In equation 27,
2.5 Strip element linear stiffness matrix
The linear stiffness matrix components are calculated based on the internal virtual work (
where
in which,
2.6 Strip element force vector
There are two types of strip element based on the panel loading condition and location of the cutout on the panel. In Figure 7, a circular cutout with free edges is considered in the center of the panel. The figure shows two types of strip elements, extended strip element (i.e. i^{th} element) and truncated strip element (i.e. j^{th} element). The length of the extended strip element is equal to the panel length. The end edges of extended strip element are under compressive distributed load with magnitude of q. The truncated strip element is between the panel transverse edge and cutout edge; therefore, its length is less than the length of the panel. Also as can be seen from Figure 7, for the truncated strip elements only one transverse edge is under compression load of q in u direction. The external virtual work (
In equation 30,
According to Figure 1, ycoordinate is along circumferential direction of the strip and can be related to the natural coordinate
Virtual displacement
It should be noted that, three spline functions (
Finally, equation 30 is simplified as equation 35 to obtain the load vector of cylindrical panel.
where,
2.7 Strip geometrical stiffness matrix
An initial deflection is induced in the cylindrical panel due to the distributed compressive load applied on the edges of the panel. This prebuckling displacement vector is calculated by the following equation.
where,
From the internal virtual work done by the initial linear inplane stresses (
In equation 38,
where,
2.8 Buckling analysis
The linear stiffness matrix (
3 Verification
Based on the formulation presented in the previous sections, a computer program namely SSFSM, was written in FORTRAN to compute the buckling load of a cylindrical panel having cutouts with different shapes. In order to check the validity of the presented formulation, several case studies are carried out and the results obtained from SSFSM are compared with the results available in the literature and the results obtained from the ABAQUS commercial finite element software (ABAQUS 2006). The details are as follows.
3.1 Buckling load of cylindrical panel without cutout
Au and Cheung (1996Au, F.T.K., Cheung, Y.K. (1996). Free vibration and stability analysis of shells by the isoparametric spline finite strip method. ThinWalled Structures 24: 5382.) calculated the buckling load of a square cylindrical panel without any cutout using spline finite strip method. In their research, the length of the panel, panel thickness, modulus of elasticity and poison ratio were considered as 600 mm, 5 mm,
Moreover, in order to verify the convergence ability of the SSFSM in calculating the panel buckling load, the buckling load of the panel for different number of strip elements and knots are calculated and presented in Table 1. As it is indicated in the table, 6 strip elements and 15 knots are sufficient for convergence of the panel buckling load.
In order to evaluate the computational cost of SSFSM, a comparsion is performed between execution time of SSFSM (subparametric element) and the same code with isoparametric element. In this regrad, the panel described in this section (Figure 8) is discretized by six subparametric strip elements and the buckling load of the panel is calculated. Different numbers of konts are considered to simulate the panel. The execution time for both codes is measured for different degrees of freedom. The solution time of code is dependent on the number of degrees of freedom. For instance, when degrees of freedom is 2040, the solution time of SSFSM is about 148 seconds for a 2.8 GHz CPU. Figure 10 shows the computational cost of SSFSM against the isoparametric code. In this figure, the precentage of reduction in execution time for subparametric element is drawn against the number of degrees of freedom. As shown in Figure 10, the execution time of SSFSM is significantly lower than that of the isoparametric one. Two factors are responsible for drastical reduction of SSFSM computational cost. The simplicity of Jacobian determinant in the SSFSM causes decomposition of surface integral encountered in the linear elastic and geometric stiffness matrices. The number of Gauss points requried in the SSFSM to accuratly integrate the stiffness matrices is lower than those of the isoparametric. As the degrees of freedom is increased from 675 to 2040, the precentage of reduction in the execution time is reduced from 62 to 57.5 percent. This reduction is related to the calculation of eigenvalue problem and as the degrees of freedom increases, the time to solve the eigenvalue problem increases.
3.2 Buckling load of cylindrical panel with cutout
A composite cylindrical panel with layup configurations of
The cylindrical panel was modeled in the ABAQUS with 700 of S8R5 elements type. Moreover, the panel was simulated in the SSFSM with 10 strip elements and 20 knots. ABAQUS and SSFSM models are shown in Figure 11. The buckling loads of the panel calculated from the SSFSM and ABAQUS are shown in Figures 12 and 13. As seen from Figures 12 and 13, the maximum and minimum buckling load of the panel is about 490 and 221 N/mm for layup angle 45 and 90 degree, respectively. The maximum difference between results obtained from the SSFSM and those of the ABAQUS is about 4 percent, which shows that there is a good agreement between the result obtained from the SSFSM and the ABAQUS.
4 Evaluation of SSFSM capability in calculation of panel axial buckling load
A parametric study was carried out to clarify the capability of the SSFSM in calculating the buckling load of the composite cylindrical panel with cutout. The effects of panel boundary condition, cutout shape, cutout area, panel thickness, central angle of the panel and layup angle configuration on buckling load of panel were investigated. The details are as follows.
4.1 Effect of cutout shape and panel thickness on panel buckling load
In this section, the effects of cutout shape and panel thickness are investigated on the buckling load of the panel. To this end, the effect of several cutout shapes such as square, diamond, circular and elliptical on the buckling load of the panel are investigated. All cutouts are considered in the center of the panel and their cutout area are considered to be 1296
The parametric study is performed for two crossply layups configuration of
The results presented in Table 3 indicate that the buckling load of the panel increases as its thickness is increased. In case of the panel without any cutout, when its thickness is getting larger by 2 and 4 times, the buckling load of the panel is increased by 3.7 and 26 times, respectively. It means that the buckling load of the panel is changed nonlinearly in terms of its thickness. In all cases, buckling load of the panel with cutout is less than that of the panel without any cutout. Although all cutouts areas are the same, the cutout shape has an important effect on the buckling load.
Cylindrical panel simulated by SSFSM (a) Without cutout (b) Square cutout (c) Diamond cutout (d) Elliptical cutout with ratio of 0.5 (e) Elliptical cutout with ratio of 2/3 (f) Circular cutout (g) Elliptical cutout with ratio of 1.5 (h) Elliptical cutout with ratio of 2.
When the elliptical cutout rotates about the normal axis to the center of the panel, the buckling load varies, dramatically. In all cases, when big chord of the elliptical cutout is normal to the compressive load direction (elliptical cutout with ratio 2), the cutout has negligible effect on the buckling load of the panel compared with other cutout shapes. For these kinds of cutouts, the panel buckling load decreases by 17%, 11% and 3% for thick, medium and thin panels respectively. It indicates that when the panel thickness is decreased, the effect of cutout on changes of the panel buckling load is decreased. In most cases, when the big chord of the elliptical cutout is parallel to compressive load direction (elliptical cutout with ratio 0.5), the maximum reduction is seen in the panel buckling load. The buckling load of the panel with diamond cutout is always larger than that of the panel with square cutout. Moreover, the buckling load of the panel with square cutout is always less than that of the panel with circular cutout. As it is illustrated in Table 3, the layup configuration has an interesting influence on the buckling load of the panel. In the most cases, the buckling load of the panel with layup configuration of
4.2 Effects of cutout size, layup, panel central angle and boundary conditions on the buckling load
In this section, the effects of cutout area, layup configuration and panel central angle are investigated on the buckling load of the panel. A circular cutout is considered at the center of the panel (see Figure 14(f)). The panel length, width and the ratio of panel thickness to its length are considered as 180 mm, 180 mm and 1/50, respectively. The panel is composed of eight layers and its material properties are presented in Table 2. The cutout area is considered as 5%, 10%, 15%, 20% and 25% of panel area. Furthermore, the central angle of the panel is considered as 0.1, 0.3 and 0.5 radian. The layup arrangements of
The results presented in Table 4 indicate that the panel buckling load is decreased as the cutout area is increased or central angle of the panel is decreased. For instance, in case of
Figure 15 shows the buckling modes of laminated composite cylindrical panel having 15% central circular cutout with different layups and central angles. Panel stacking sequence is considered as [90]_{8}, [45]_{8}, [45/45]_{4} and [0/45/90/45]_{s}. The central angle of the panel is 0.1 and 0.5 radians. For unidirectional layups [90]_{8} and [45]_{8} wavelength in the fiber direction is more than that of fiber transverse direction. This is pronounced with increasing in the panel central angle. The local shape of modes in these cases justifies reduction of buckling loads. The buckling mode shapes of multidirectional laminates such as [45/45]_{4} and [0/45/90/45]_{s} behave globally and therefore endure more axial buckling loads.
To show the capability of SSFSM to consider the effects of different cylindrical boundary conditions on the buckling load, the buckling load of the panel was calculated using SSFSM when the loading edges of the panel were fixed. The panel properties and geometry were considered as mentioned at the beginning of this section. All layups presented in Table 4 were considered in the analysis. The central angle of the panel was considered as 0.1 and 0.5 radian. The area ratio of circular cutout was considered as 5%, 10%, 15%, 20% and 25%. The buckling load of the simply supported cylindrical panel (simple panel) is compared with that of fix supported cylindrical panel (fixed panel) in Figures 16 to 23.
The buckling load of the panel with layup
Cylindrical panel mode shape (a) layup [90]_{8} and central angle of 0.1 (b) layup [90]_{8} and central angle of 0.5 (c) layup [45]_{8} and central angle of 0.1 (d) layup [45]_{8} and central angle of 0.5 (e) layup [45/45]_{4} and central angle of 0.1 (f) layup [45/45]_{4} and central angle of 0.5 (g) layup [0/45/90/45]_{s} and central angle of 0.1 (h) layup [0/45/90/45]_{s} and central angle of 0.5.
The effect of boundary condition on the buckling load the panel with layup
5 CONCLUSION
In this paper a subparametric spline finite strip element is developed to study the buckling analysis of perforated composite cylindrical panel. To this end, shape, mapping and displacement functions, Jacobian matrix, straindisplacement relations based on first order shear deformation theory, load vector, and linear and geometrical stiffness matrices of the strip element were derived. The obtained eigenvalue problem was solved by inverse subspace iteration to calculate the buckling load of the cylindrical panel. All of these steps were carried out in a FORTRAN code called SSFSM. The results obtained by the SSFSM were compared with those of the published literature and those obtained from ABAQUS finite element software to check the validity of the results.
A parametric study was performed to show the influences of cutout shape and area, panel thickness, central angle of the panel, boundary conditions of the panel and layup scheme on the axial buckling load of the composite cylindrical panel. The cutouts with the shape of circle, elliptic, square and diamond were considered in the center of the panel. The ratio of big to small chords of the elliptical cutout and the ratio of circular cutout area to panel area were varied. In addition eight layup configurations (angleply and crossply), three panel thicknesses (thick, medium and thin) and three panel central angles were considered in the parametric study. The following conclusions can be extracted based on the current study.

 The simplicity of the Jacobian matrix in the current subparametric study causes the surface integrals encountered in the derivation of stiffness matrices turn into multiplication of two single variable integrals.

 Another advantage of current subparametric formulation over isoparametric one returns to the reduction in the required number of Gauss points to accurately estimate the domain integrals.

 The reduction in execution time of current subparametric code against the standard isoparametric formulation is about 60 percent. With increasing in panel degrees of freedom due to increasing in eigenvalue solvers contribution, this reduction is decreased slightly.

 Increasing in the cutout chord prependicular to the loading direction while unaltering cutout area increases the axial buckling load of cylindrical panel.

 Among the layups considered in the parametric study, quasiisotropic laminate [0/45/90/45]_{s} and unidirectional [0]_{8} laminate carry the maximum and minimum axial buckling loads, respectively.

 Fixed or simply supported curved loaded edges of the panel have not any effect on the axial buckling load of the panel for [90]_{8} layup while they have the most effect in the case of [0]_{8} arrangement.

 Axial buckling mode of unidirectional laminates such as [90]_{8} and [45]_{8} has local expansion in the fibre direction and this is pronounced with increasing in the panel curvature. Unlike the unidirectional, in the multidirectional laminates like [45/45]_{4} and [0/45/90/45]_{s} global expansion is illustrated.
References
 ABAQUS, (2006), User’s manual version 6.5. Hibbit, Karlsson & Sorensen, USA.
 Akin, J.E. (2005). Finite Element Analysis with Error Estimators, Elsevier publisher, New York, USA.
 Anil, V., Upadhyay, C.S. and Iyengar, N.G.R., (2007). Stability analysis of composite laminate with and without rectangular cutout under biaxial loading. Composite Structures 80: 92104.
 Arbelo, M.A., Herrmann, A., Castro, S.G.P., Khakimova, R., Zimmermann, R., Degenhardt, R., (2015). Investigation of Buckling Behavior of Composite Shell Structures with Cutouts. Applied Composite Materials 22: 623636.
 Asadi, E., Qatu, M.S., (2013). Free vibration of thick laminated cylindrical shells with different boundary conditions using general differential quadrature. Journal of Vibration and Control 19: 356366.
 Asadi, E., Wang, W., Qatu, M.S., (2012). Static and vibration analyses of thick deep laminated cylindrical shells using 3D and various shear deformation theories. Composite Structures 94: 494500.
 Ashrafi, H., Asemi, K., Shariyat, M., (2013). A three dimensional boundary element stress and bending analysis of transversely/longitudinally graded plates with circular cutouts under biaxial loading. European Journal of Mechanics A Solids 42: 344357.
 Au, F.T.K., Cheung, Y.K. (1996). Free vibration and stability analysis of shells by the isoparametric spline finite strip method. ThinWalled Structures 24: 5382.
 Balogh, B., Logo, J. (2015). Optimization of curved plated structures with the finite strip and finite element methods. Civil Engeering Series 15: 110.
 Cheung, Y.K., Tham, L.G. (1997). The Finite Strip Method, CRC Press, New York, USA.
 Dawe, D.J., Wang, S., Lam, S.S.E., (1995). Finite strip analysis of imperfect laminated plates under end shortening and normal pressure. International Journal for Numerical Methods in Engineering 38: 41934205.
 Dawe, D.J., Wang, S., (1994). Buckling of composite plates and plate structures using the spline finite strip method. Composites Engineering 4: 10991117.
 Eccher, G., Rasmussen, K.J.R., Zandonini, R., (2008). Elastic buckling analysis of perforated thinwalled structures by the isoparametric spline finite strip method. ThinWalled Structures 46: 165  191.
 Fazilati, J., (2017). Stability analysis of variable stiffness composite laminated plates with delamination using Spline FSM. Latin American Journal of Solids and Structures 14: 528 543.
 Fazilati, J., Ovesy, H.R., (2013). Parametric instability of laminated longitudinally stiffened curved panels with cutout using higher order FSM. Composite Structures 95: 691696.
 Khalfi, Y., Houari, M.S.A., Tounsi, A., (2014). A refined and simple shear deformation theory for thermal buckling of solar functionally graded plates on elastic foundation. International Journal of Computational Methods 11: 1350077.
 Khayat, M., Poorveis, D., Moradi, S., Hemmati, M. (2016a). Buckling of thick deep laminated composite shell of revolution under follower forces. Structural Engineering and Mechanics 58: 5991.
 Khayat, M., Poorveis, D., Moradi, S., (2016b). Buckling analysis of laminated composite cylindrical shell subjected to lateral displacementdependent pressure using semi analytical finite strip method. Steel Composite Structures 22: 4559.
 Khetir, H., Bouiadjra, M.B., Houari, M.S.A., Tounsi, A., Mahmoud, S.R., (2017). A new nonlocal trigonometric shear deformation theory for thermal buckling analysis of embedded nanosize FG plates. Structural Engineering and Mechanics 64: 391402.
 Komur, M.A., Sonmez, M., (2008). Elastic buckling of rectangular plates under linearly varying inplane normal load with a circular cutout. Mechanics Research Communucations 35: 361371.
 Magnucki, K., Mackiewicz, M., (2006). Elastic buckling of an axially compressed cylindrical panel with three edges simply supported and one edge free. ThinWalled Structures 44: 387392.
 Masia, U., Avalos, D.R., Laura, P.A.A., (2005). Displacement amplitudes and flexural moments for a rectangular plate with a rectangular cutout under a uniformly distributed static load. Journal of Sound and Vibration 280: 433442.
 Mirzaei, S., Azhari, M., Bondarabady Rahimi, H.A., (2015). On the use of finite strip method for buckling anaylsis of moderately thick plate by refined plate theory and using new types of functions. Latin American Journal of Solids and Structures 12: 561 582.
 Moradi, S., Poorveis, D., Khajehdezfuly, A., (2011). Geometrically nonlinear analysis of anisotropic laminated cylindrical panels with cutout using spline finite strip method. Proceeding of conference on the Advances in Structural Engineering and Mechanics, Seoul, South Korea.
 Nejati, M., Dimitri, R., Tornabene, F., Hossein Yas, M. (2017). Thermal buckling of nanocomposite stiffened cylindrical shells reinforced by Functionally Graded Wavy Carbon NanoTubes with temperaturedependent properties. Applied Science 7: 124.
 Ovesy, H.R., Totounferoush, A., Ghannadpour, S.A.M., (2015). Dynamic buckling analysis of delaminated composite plates using semianalytical finite strip method. Journal of Sound and Vibration 343: 131143.
 Ovesy, H.R., Fazilati, J., (2012b). Parametric instability analysis of moderately thick FGM cylindrical panels using FSM. Computers and Structures 108: 135143.
 Pascan, O.Z., Weihong, Z., Ponthot, J.P., (2012). Mechanical buckling analysis of composite panels with/without cutouts. International Journal of Plant Engineering Management 17: 6576.
 Patel, B.P., Singh, S., Nath, Y., (2006). Stability and nonlinear dynamic behaviour of crossply laminated heated cylindrical shells. Latin American Journal of Solids and Structures 3: 245 261.
 Rakotonirina, C., (2013). On the Cholesky method. Journal of Interdisciplinary Mathematics 12:875882.
 Rajanna, T., Banerhee, S., Desai, M.Y., (2016). Effects of partial loading and fiber configuration on vibration and buckling characteristics of stiffened composite plates. Latin American Journal of Solids and Structures 13: 854 879.
 Rondal, J., (1998). Coupled Instabilities in Metal Structures: Theoretical and Design Aspects, Springer publisher, New York, USA.
 Shuohui, Y., Tiantang, Y., Tinh, Q.B., Peng, L., Sohichi, H., (2016a). Buckling and vibration extended isogeometric analysis of imperfect graded ReissnerMindlin plates with internal defects using NURBS and level sets. Computers & Structures 177: 2338.
 Shuohui, Y., Tinh, Q.B., Shifeng, T., Satoyuki, T., Sohichi, H., (2016b). NURBSbased isogeometric analysis of buckling and free vibration problems for laminated composites plates with complicated cutouts using a new simple FSDT theory and level set method. ThinWalled Structures 101: 141 156.
 Shuohui, Y., Tiantang, Y., Tinh, Q.B., Shifeng, X., Sohichi, H., (2015). A cutout isogeometric analysis for thin laminated composite plates using level sets. Composite Structures 127: 152 164.
 Soltani, K., Bessaim Houari, M.S.A., Kaci, A., Benguediab, M., Tounsi, A., Alhodaly, M., (2019). A novel hyperbolic shear deformation theory for the mechanical buckling analysis of advanced composite plates resting on elastic foundations. Steel and Composite structures 30:1329.
 Soni, G., Singh, R., Mitra, M., (2013). Buckling behavior of composite laminates (with and without cutouts) subjected to nonuniform inplane loads. International Journal of Structural Stability and Dynamics 13: 120.
 Tiantang, Y., Shuohui, Y. Tinh, Q.B., Chen, L., Nuttawit W., (2017). Buckling isogeometric analysis of functionally graded plates under combined thermal and mechanical loads. Composite Structures 162: 5469.
 Thinh, T.I., Tu., T.M., Quoc, T.H., Long, N.V., (2016). Vibration and buckling analysis of functionally graded plates using new eightuknown higher order shear deformation theory. Latin American Journal of Solids and Structures 13: 456 477.
 Tiantang, Y., Tinh, Q.B., Duc, H.D., Wud, C.T., Thom, V.D., Satoyuki, T., (2016). On the thermal buckling analysis of functionally graded plates with internal defects using extended isogeometric analysis. Composite Structures 136: 684 695.
 Van Ness, J., (1969). Inverse iteration method for finding eigenvectors. IEEE Transactions on Automatic Control 14: 6366.
 Vuong, P.M., Dung, D.V., (2017). Nonlinear analysis on buckling and postbuckling FGM imperfect cylindrical shells filled inside by elastic foundations in thermal environment using TSDT. Latin American Journal of Solids and Structures 14: 950 977.
 Wang, S., Chen, J., Dawe, D.J., (1998). Linear transient analysis of rectangular laminates using spline finite strips. Composites Structures 41: 5766.
 Yao, L.K., He, B., Zhang, Y., Zhou, W., (2015). Semianalytical finite strip transfer matrix method for buckling analysis of rectangular thin plates. Mathematical Problems in Engineering 15: 111.
Publication Dates

Publication in this collection
30 Sept 2019 
Date of issue
2019
History

Received
01 Mar 2019 
Reviewed
19 Aug 2019 
Accepted
02 Sept 2019 
Published
06 Sept 2019