{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,2]],"date-time":"2025-11-02T10:40:59Z","timestamp":1762080059813,"version":"build-2065373602"},"reference-count":49,"publisher":"MDPI AG","issue":"24","license":[{"start":{"date-parts":[[2022,12,7]],"date-time":"2022-12-07T00:00:00Z","timestamp":1670371200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Applied Sciences"],"abstract":"<jats:p>This study aimed to provide a static solution to the boundary value problem presented by symmetric (0\u00b0\/90\u00b0\/0\u00b0) and antisymmetric (0\u00b0\/90\u00b0) cross-ply composite, moderately thick shallow shells and plates (a special case of the shells) subjected to mixed-type unsolved boundary conditions. The boundary-discontinuous double Fourier series (BDM) method, in which displacements are expressed in trigonometric functions, is employed in a well-established framework. The analytical solution obtained using the BDM is compared with the successful integration of the generalized differential quadrature (GDQ) method for the static analysis of composite shells with a roller skate-type boundary condition prescribed on two opposite edges, while the remaining two edges are subjected to simply supported constraints. Comprehensive results are presented in order to show the effects of curvature on the deflections and stresses of moderately thick shallow shells made up of symmetric and antisymmetric cross-ply laminated composite materials. The validity of the proposed model is authenticated through the available HSDT-based literature review, and the convergence characteristics are demonstrated. The changing trends of displacements and stresses are explained in detail by investigating the effect of various parameters such as lamination, material properties, the effect of curvature, etc. Based on the results obtained using the proposed static solution, analytical BDM results were found to be in very close agreement with the numerical GDQ method, especially for symmetric lamination. However, the results obtained using the BDM and GDQ methods for antisymmetric lamination show differences, possibly due to the presence of a discontinuity in the derivatives originating from the bending\u2013stretching matrix in antisymmetric lamination. Important numerical results presented include the sensitivity of the predicted response quantities of interest to material properties, lamination, and thickness effects, as well as their interactions. The results presented here may also serve as benchmark comparison points with numerical solutions such as finite elements, boundary elements, etc.<\/jats:p>","DOI":"10.3390\/app122412547","type":"journal-article","created":{"date-parts":[[2022,12,8]],"date-time":"2022-12-08T01:44:31Z","timestamp":1670463871000},"page":"12547","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Analytical and Numerical Solutions to Static Analysis of Moderately Thick Cross-Ply Plates and Shells"],"prefix":"10.3390","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-9974-2538","authenticated-orcid":false,"given":"\u0130lke","family":"Alg\u00fcl","sequence":"first","affiliation":[{"name":"Department of Mechanical Engineering, Gebze Technical University, Gebze 41400, Turkey"}]},{"given":"Ahmet Sinan","family":"Oktem","sequence":"additional","affiliation":[{"name":"Department of Mechanical Engineering, Gebze Technical University, Gebze 41400, Turkey"}]}],"member":"1968","published-online":{"date-parts":[[2022,12,7]]},"reference":[{"key":"ref_1","unstructured":"Love, A.E.H. (2013). A Treatise on the Mathematical Theory of Elasticity, Cambridge University Press."},{"key":"ref_2","unstructured":"Donnell, L.H. (1938, January 12\u201316). A Discussion of Thin Shell Theory. Proceedings of the 5th International Congress of Applied Mechanics, New York, NY, USA."},{"key":"ref_3","unstructured":"Sanders, J.L. (1960). An Improved First-Approximation Theory for Thin Shells."},{"key":"ref_4","unstructured":"Timoshenko, S., and Woinowsky-Krieger, S. (1959). Theory of Plates and Shells, McGraw-Hill."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"177","DOI":"10.2307\/2371288","article-title":"A New Derivation of the Equations for the Deformation of Elastic Shells","volume":"63","author":"Reissner","year":"1941","journal-title":"Am. J. Math."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"61","DOI":"10.4236\/ojcm.2012.23009","article-title":"Review of Recent Literature on Static Analyses of Composite Shells: 2000\u20132010","volume":"2","author":"Qatu","year":"2012","journal-title":"Open J. Compos. Mater."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"2031","DOI":"10.1016\/j.compstruct.2011.02.018","article-title":"A Review of Meshless Methods for Laminated and Functionally Graded Plates and Shells","volume":"93","author":"Liew","year":"2011","journal-title":"Compos. Struct."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.paerosci.2015.07.001","article-title":"Recent Development in Modeling and Analysis of Functionally Graded Materials and Structures","volume":"79","author":"Gupta","year":"2015","journal-title":"Prog. Aerosp. Sci."},{"key":"ref_9","first-page":"1586388","article-title":"On the Finite Element Model of Rotating Functionally Graded Graphene Beams Resting on Elastic Foundation","volume":"2021","author":"Tho","year":"2021","journal-title":"Math. Probl. Eng."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Nam, V.H., van Vinh, P., van Chinh, N., van Thom, D., and Hong, T.T. (2019). A New Beam Model for Simulation of the Mechanical Behaviour of Variable Thickness Functionally Graded Material Beams Based on Modified First Order Shear Deformation Theory. Materials, 12.","DOI":"10.3390\/ma12030404"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"400","DOI":"10.1016\/j.jsamd.2016.08.005","article-title":"High Frequency Modes Meshfree Analysis of Reissner\u2013Mindlin Plates","volume":"1","author":"Bui","year":"2016","journal-title":"J. Sci. Adv. Mater. Devices"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"20","DOI":"10.1177\/002199837000400102","article-title":"Exact Solutions for Rectangular Bidirectional Composites and Sandwich Plates","volume":"4","author":"Pagano","year":"1970","journal-title":"J. Compos. Mater."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1338","DOI":"10.1061\/(ASCE)0733-9399(1992)118:7(1338)","article-title":"Analytical Solutions for Thick, Doubly Curved, Laminated Shells","volume":"118","author":"Fan","year":"1992","journal-title":"J. Eng. Mech."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"949","DOI":"10.1016\/j.compstruct.2018.09.035","article-title":"Static Behavior of Arbitrarily Supported Composite Laminated Cylindrical Shell Panels: An Analytical 3D Elasticity Approach","volume":"207","author":"Kumari","year":"2019","journal-title":"Compos. Struct."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"129","DOI":"10.1016\/j.compstruct.2016.12.026","article-title":"Exact L\u00e9vy-Type Solutions for Bending of Thick Laminated Orthotropic Plates Based on 3-D Elasticity and Shear Deformation Theories","volume":"163","author":"Atashipour","year":"2017","journal-title":"Compos. Struct."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"2253","DOI":"10.1007\/s12206-009-0511-4","article-title":"Analytical Solutions for Bending Analysis of Rectangular Laminated Plates with Arbitrary Lamination and Boundary Conditions","volume":"23","author":"Nik","year":"2009","journal-title":"J. Mech. Sci. Technol."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"28","DOI":"10.2514\/3.5601","article-title":"Analysis of a Simply Supported Laminated Anisotropic Rectangular Plate","volume":"8","author":"Whitney","year":"1970","journal-title":"AIAA J."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"319","DOI":"10.1016\/0020-7225(85)90051-5","article-title":"A Higher-Order Shear Deformation Theory of Laminated Elastic Shells","volume":"23","author":"Reddy","year":"1985","journal-title":"Int. J. Eng. Sci."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"667","DOI":"10.2514\/1.15286","article-title":"Static Analysis of Symmetric Angle-Ply Laminated Plates by Analytical Method","volume":"44","author":"Huang","year":"2006","journal-title":"AIAA J."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"4535871","DOI":"10.1155\/2018\/4535871","article-title":"An Accurate Solution Method for the Static and Vibration Analysis of Functionally Graded Reissner-Mindlin Rectangular Plate with General Boundary Conditions","volume":"2018","author":"Li","year":"2018","journal-title":"Shock Vib."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"261","DOI":"10.1006\/jsvi.2001.3913","article-title":"On the Roles of Complementary and Admissible Boundary Constraints in Fourier Solutions to the Boundary Value Problems of Completely Coupled r Th Order PDEs","volume":"251","author":"Chaudhuri","year":"2002","journal-title":"J. Sound Vib."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"1005","DOI":"10.1016\/0020-7225(89)90080-3","article-title":"On Boundary-Discontinuous Double Fourier Series Solution to a System of Completely Coupled PDE\u2019s","volume":"27","author":"Chaudhuri","year":"1989","journal-title":"Int. J. Eng. Sci."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"1325","DOI":"10.1016\/0020-7225(89)90057-8","article-title":"On Analytical Solutions to Boundary-Value Problems of Doubly-Curved Moderately-Thick Orthotropic Shells","volume":"27","author":"Chaudhuri","year":"1989","journal-title":"Int. J. Eng. Sci."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"263","DOI":"10.1016\/0020-7683(93)90065-F","article-title":"Sensitivity of the Response of Moderately Thick Cross-Ply Doubly-Curved Panels to Lamination and Boundary Constraint-I. Theory","volume":"30","author":"Chaudhuri","year":"1993","journal-title":"Int. J. Solids Struct."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"1551","DOI":"10.1016\/0020-7225(93)90031-O","article-title":"Boundary-Discontinuous Fourier Analysis of Doubly-Curved Panels Using Classical Shallow Shell Theories","volume":"31","author":"Chaudhuri","year":"1993","journal-title":"Int. J. Eng. Sci."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"587","DOI":"10.1016\/0020-7225(88)90056-0","article-title":"Exact Solution of Shear-Flexible Doubly Curved Anti-Symmetric Angle-Ply Shells","volume":"26","author":"Chaudhuri","year":"1988","journal-title":"Int. J. Eng. Sci."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/0020-7683(91)90044-G","article-title":"Static Analysis of Moderately-Thick Finite Antisymmetric Angle-Ply Cylindrical Panels and Shells","volume":"28","author":"Chaudhuri","year":"1991","journal-title":"Int. J. Solids Struct."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"243","DOI":"10.1016\/j.compstruct.2005.09.012","article-title":"Levy Type Analysis of Cross-Ply Plates Based on Higher-Order Theory","volume":"78","author":"Oktem","year":"2007","journal-title":"Compos. Struct."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"489","DOI":"10.1016\/j.compstruct.2006.05.028","article-title":"Fourier Analysis of Thick Cross-Ply Levy Type Clamped Doubly-Curved Panels","volume":"80","author":"Oktem","year":"2007","journal-title":"Compos. Struct."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"475","DOI":"10.1016\/j.compstruct.2006.05.020","article-title":"Levy Type Fourier Analysis of Thick Cross-Ply Doubly Curved Panels","volume":"80","author":"Oktem","year":"2007","journal-title":"Compos. Struct."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"539","DOI":"10.1016\/j.compstruct.2007.02.001","article-title":"Boundary Discontinuous Fourier Analysis of Thick Cross-Ply Clamped Plates","volume":"82","author":"Oktem","year":"2008","journal-title":"Compos. Struct."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"37","DOI":"10.1016\/j.compstruct.2011.07.020","article-title":"Static and Dynamic Analysis of Laminated Composite and Sandwich Plates and Shells by Using a New Higher-Order Shear Deformation Theory","volume":"94","author":"Mantari","year":"2011","journal-title":"Compos. Struct."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"2675","DOI":"10.1177\/0021998311431641","article-title":"Higher Order Theory Based Fourier Analysis of Cross-Ply Plates and Doubly Curved Panels","volume":"46","author":"Oktem","year":"2012","journal-title":"J. Compos. Mater."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"85","DOI":"10.1016\/j.compstruct.2004.01.001","article-title":"A Combined Theoretical and Experimental Investigation on Free Vibration of Thin Symmetrically Laminated Anisotropic Plates","volume":"67","author":"Chaudhuri","year":"2005","journal-title":"Compos. Struct."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"59","DOI":"10.1016\/j.compstruct.2013.01.002","article-title":"Static Analysis of Completely Doubly-Curved Laminated Shells and Panels Using General Higher-Order Shear Deformation Theories","volume":"101","author":"Viola","year":"2013","journal-title":"Compos. Struct."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"76","DOI":"10.1016\/j.tws.2011.11.004","article-title":"Static Analysis of Thick Laminated Shells with Different Boundary Conditions Using GDQ","volume":"51","author":"Asadi","year":"2012","journal-title":"Thin-Walled Struct."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"605","DOI":"10.1016\/j.compstruct.2015.02.045","article-title":"Geometrically Nonlinear Transient Analysis of Moderately Thick Laminated Composite Shallow Shells with Generalized Differential Quadrature Method","volume":"125","author":"Kurtaran","year":"2015","journal-title":"Compos. Struct."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"241","DOI":"10.1016\/j.compstruct.2015.03.060","article-title":"Geometrically Nonlinear Transient Analysis of Thick Deep Composite Curved Beams with Generalized Differential Quadrature Method","volume":"128","author":"Kurtaran","year":"2015","journal-title":"Compos. Struct."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"265","DOI":"10.1016\/j.compositesb.2016.02.010","article-title":"The Local GDQ Method for the Natural Frequencies of Doubly-Curved Shells with Variable Thickness: A General Formulation","volume":"92","author":"Tornabene","year":"2016","journal-title":"Compos. B Eng."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"115740","DOI":"10.1016\/j.compstruct.2022.115740","article-title":"Higher Order Theories for the Free Vibration Analysis of Laminated Anisotropic Doubly-Curved Shells of Arbitrary Geometry with General Boundary Conditions","volume":"297","author":"Tornabene","year":"2022","journal-title":"Compos. Struct."},{"key":"ref_41","doi-asserted-by":"crossref","unstructured":"Allahkarami, F., Tohidi, H., Dimitri, R., and Tornabene, F. (2020). Dynamic Stability of Bi-Directional Functionally Graded Porous Cylindrical Shells Embedded in an Elastic Foundation. Appl. Sci., 10.","DOI":"10.3390\/app10041345"},{"key":"ref_42","doi-asserted-by":"crossref","unstructured":"Zare Jouneghani, F., Dimitri, R., Bacciocchi, M., and Tornabene, F. (2017). Free Vibration Analysis of Functionally Graded Porous Doubly-Curved Shells Based on the First-Order Shear Deformation Theory. Appl. Sci., 7.","DOI":"10.3390\/app7121252"},{"key":"ref_43","doi-asserted-by":"crossref","unstructured":"Kiarasi, F., Babaei, M., Asemi, K., Dimitri, R., and Tornabene, F. (2021). Three-Dimensional Buckling Analysis of Functionally Graded Saturated Porous Rectangular Plates under Combined Loading Conditions. Appl. Sci., 11.","DOI":"10.3390\/app112110434"},{"key":"ref_44","doi-asserted-by":"crossref","unstructured":"Tornabene, F., Fantuzzi, N., Bacciocchi, M., Viola, E., and Reddy, J.N. (2017). A Numerical Investigation on the Natural Frequencies of FGM Sandwich Shells with Variable Thickness by the Local Generalized Differential Quadrature Method. Appl. Sci., 7.","DOI":"10.3390\/app7020131"},{"key":"ref_45","doi-asserted-by":"crossref","unstructured":"Reddy, J.N. (2003). Mechanics of Laminated Composite Plates and Shells: Theory and Analysis, CRC Press.","DOI":"10.1201\/b12409"},{"key":"ref_46","unstructured":"Hobson, E.W. (1921). The Theory of Functions of a Real Variable and the Theory of Fourier\u2019s Series, The University Press."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"222","DOI":"10.1017\/S0305004100018375","article-title":"Double Fourier Series and Boundary Value Problems","volume":"40","author":"Green","year":"1944","journal-title":"Math. Proc. Camb. Philos. Soc."},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"837","DOI":"10.1016\/S0020-7683(96)00056-X","article-title":"A Generalized Approach for Implementing General Boundary Conditions in the GDQ Free Vibration Analysis of Plates","volume":"34","author":"Shu","year":"1997","journal-title":"Int. J. Solids Struct."},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.compstruct.2007.03.002","article-title":"Effect of Inplane Boundary Constraints on the Response of Thick General (Unsymmetric) Cross-Ply Plates","volume":"83","author":"Oktem","year":"2008","journal-title":"Compos. Struct."}],"container-title":["Applied Sciences"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2076-3417\/12\/24\/12547\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:36:00Z","timestamp":1760146560000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2076-3417\/12\/24\/12547"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,12,7]]},"references-count":49,"journal-issue":{"issue":"24","published-online":{"date-parts":[[2022,12]]}},"alternative-id":["app122412547"],"URL":"https:\/\/doi.org\/10.3390\/app122412547","relation":{},"ISSN":["2076-3417"],"issn-type":[{"type":"electronic","value":"2076-3417"}],"subject":[],"published":{"date-parts":[[2022,12,7]]}}}