On the mechanics of laminated doubly-curved shells subjected to point and line loads

Abstract It is well-known that the implementation of concentrated forces, such as point and line loads, represents a challenging task, especially from the computational point of view, since a strong discontinuity has to be inserted in the structural model. The present paper aims to solve the static problem of laminated composite doubly-curved shell structures subjected to concentrated loads employing the Generalized Differential Quadrature (GDQ) as numerical tool, according to what has been shown by the authors in their previous works. Its accuracy and reliability features are proven for several grid distributions when the concentrated loads are modeled through the Dirac-delta function. The theoretical framework on which this approach is based is a Unified Formulation, which allows to investigate several Higher-order Shear Deformation Theories (HSDTs). The differential geometry is used to describe accurately the reference surface of various doubly-curved shell structures. The validity of the current approach is shown comparing the GDQ results with the exact and semi-analytical ones available in the literature. A posteriori recovery procedure based on the three-dimensional equilibrium equations for a shell structure is introduced to compute the through-the-thickness variation of strain, stress and displacement components by means of the GDQ method.

[1]  J. N. Reddy,et al.  Active control of laminated cylindrical shells using piezoelectric fiber reinforced composites , 2005 .

[2]  I. A. Jones.  Flügge shell theory and solution for orthotropic cylindrical shells under pinching loads , 1998 .

[3]  E. Viola,et al.  Radial basis functions based on differential quadrature method for the free vibration analysis of laminated composite arbitrarily shaped plates , 2015 .

[4]  Prediction of natural frequencies of laminated curved panels using refined 2-D theories in the spectral collocation method , 2014 .

[5]  Nicholas Fantuzzi,et al.  Effect of agglomeration on the natural frequencies of functionally graded carbon nanotube-reinforced laminated composite doubly-curved shells , 2016 .

[6]  Moshe Eisenberger,et al.  Vibration analysis of variable thickness plates and shells by the Generalized Differential Quadrature method , 2016 .

[7]  Shao Wen Yuan.  Thin cylindrical shells subjected to concentrated loads , 1946 .

[8]  E. Carrera.  Historical review of Zig-Zag theories for multilayered plates and shells , 2003 .

[9]  A.P.S. Selvadurai,et al.  Concentrated loading of a fibre-reinforced composite plate: Experimental and computational modeling of boundary fixity , 2014 .

[10]  Elena Ferretti,et al.  GDQFEM Numerical Simulations of Continuous Mediawith Cracks and Discontinuities , 2013 .

[11]  Erasmo Viola,et al.  Static analysis of functionally graded doubly-curved shells and panels of revolution , 2013 .

[12]  E. Carrera.  Theories and finite elements for multilayered, anisotropic, composite plates and shells , 2002 .

[13]  Sritawat Kitipornchai,et al.  Three-dimensional exact solution for inhomogeneous and laminated piezoelectric plates , 1999 .

[14]  Nicholas Fantuzzi,et al.  Generalized differential quadrature finite element method for cracked composite structures of arbitrary shape , 2013 .

[15]  Nicholas Fantuzzi,et al.  Generalized stress–strain recovery formulation applied to functionally graded spherical shells and panels under static loading , 2016 .

[16]  A. Kalamkarov,et al.  Analysis of Smart Piezo-Magneto-Thermo-Elastic Composite and Reinforced Plates: Part II – Applications , 2014 .

[17]  Nicholas Fantuzzi,et al.  Stress and strain recovery for functionally graded free-form and doubly-curved sandwich shells using higher-order equivalent single layer theory , 2015 .

[18]  S. A. Eftekhari,et al.  A modified differential quadrature procedure for numerical solution of moving load problem , 2016 .

[19]  Erasmo Carrera,et al.  Analysis of thick isotropic and cross-ply laminated plates by generalized differential quadrature method and a Unified Formulation , 2014 .

[20]  N. D. Duc,et al.  Nonlinear dynamic analysis and vibration of shear deformable eccentrically stiffened S-FGM cylindrical panels with metal–ceramic–metal layers resting on elastic foundations , 2015 .

[21]  Nicholas Fantuzzi,et al.  The strong formulation finite element method: stability and accuracy , 2014 .

[22]  J. Romanoff,et al.  Bending deflection of sandwich beams considering local effect of concentrated force , 2015 .

[23]  Khanh Chau Le,et al.  An asymptotically exact theory of smart sandwich shells , 2016, 1606.02172.

[24]  Lorenzo Dozio,et al.  A variable-kinematic model for variable stiffness plates: Vibration and buckling analysis , 2016 .

[25]  Paul M. Weaver,et al.  On displacement-based and mixed-variational equivalent single layer theories for modelling highly heterogeneous laminated beams , 2015 .

[26]  F. Fazzolari.  Natural frequencies and critical temperatures of functionally graded sandwich plates subjected to uniform and non-uniform temperature distributions , 2015 .

[27]  J. N. Reddy,et al.  Exact Solutions of Moderately Thick Laminated Shells , 1984 .

[28]  Ernian Pan,et al.  Size-dependent behavior of functionally graded anisotropic composite plates , 2016 .

[29]  C. Shu,et al.  APPLICATION OF GENERALIZED DIFFERENTIAL QUADRATURE TO SOLVE TWO-DIMENSIONAL INCOMPRESSIBLE NAVIER-STOKES EQUATIONS , 1992 .

[30]  Ted Belytschko,et al.  Implementation and application of a 9-node Lagrange shell element with spurious mode control , 1985 .

[31]  Nicholas Fantuzzi,et al.  A new approach for treating concentrated loads in doubly-curved composite deep shells with variable radii of curvature , 2015 .

[32]  Nicholas Fantuzzi,et al.  Free vibrations of free-form doubly-curved shells made of functionally graded materials using higher-order equivalent single layer theories , 2014 .

[33]  Nicholas Fantuzzi,et al.  Radial basis function method applied to doubly-curved laminated composite shells and panels with a General Higher-order Equivalent Single Layer formulation , 2013 .

[34]  Nicholas Fantuzzi,et al.  The local GDQ method applied to general higher-order theories of doubly-curved laminated composite shells and panels: The free vibration analysis , 2014 .

[35]  M. Dehghan,et al.  Thermo-electro-elastic analysis of functionally graded piezoelectric shells of revolution: Governing equations and solutions for some simple cases , 2016 .

[36]  A. Ceruti,et al.  Free-Form Laminated Doubly-Curved Shells and Panels of Revolution Resting on Winkler-Pasternak Elastic Foundations: A 2-D GDQ Solution for Static and Free Vibration Analysis , 2013 .

[37]  J. N. Reddy,et al.  FGM and Laminated Doubly-Curved and Degenerate Shells Resting on Nonlinear Elastic Foundations: A GDQ Solution for Static Analysis with a Posteriori Stress and Strain Recovery , 2013 .

[38]  F. Tornabene,et al.  Higher-order structural theories for the static analysis of doubly-curved laminated composite panels reinforced by curvilinear fibers , 2016 .

[39]  E. Viola,et al.  General higher-order shear deformation theories for the free vibration analysis of completely doubly-curved laminated shells and panels , 2013 .

[40]  Nicholas Fantuzzi,et al.  Dynamic analysis of thick and thin elliptic shell structures made of laminated composite materials , 2015 .

[41]  Thomas J. R. Hughes,et al.  Nonlinear finite element analysis of shells: Part I. three-dimensional shells , 1981 .

[42]  Yuansheng Cheng,et al.  High-order free vibration analysis of sandwich plates with both functionally graded face sheets and functionally graded flexible core , 2015 .

[43]  Erasmo Carrera,et al.  Static analysis of doubly-curved anisotropic shells and panels using CUF approach, differential geometry and differential quadrature method , 2014 .

[44]  Wenbin Yu,et al.  A variational asymptotic theory of composite laminated plates: Hybrid transformation to Reissner–Mindlin model , 2015 .

[45]  On Radial Deflection of a Cylinder of Finite Length with Various End Conditions , 1958 .

[46]  M. Shariyat,et al.  Three-dimensional magneto-elastic analysis of asymmetric variable thickness porous FGM circular plates with non-uniform tractions and Kerr elastic foundations , 2015 .

[47]  Nicholas Fantuzzi,et al.  Strong Formulation Isogeometric Analysis (SFIGA) for laminated composite arbitrarily shaped plates , 2016 .

[48]  N. Kuruoglu,et al.  On a problem of the vibration of functionally graded conical shells with mixed boundary conditions , 2015 .

[49]  Nicholas Fantuzzi,et al.  A SFEM-based evaluation of mode-I Stress Intensity Factor in composite structures , 2016 .

[50]  A. Ferreira,et al.  MLSDQ based on RBFs for the free vibrations of laminated composite doubly-curved shells , 2016 .

[51]  W. Becker.  Concentrated forces and moments on laminates with bending extension coupling , 1995 .

[52]  G. Kulikov,et al.  Exact geometry solid-shell element based on a sampling surfaces technique for 3D stress analysis of doubly-curved composite shells , 2015 .

[53]  A. Alibeigloo.  Thermo elasticity solution of sandwich circular plate with functionally graded core using generalized differential quadrature method , 2016 .

[54]  A. Kalamkarov,et al.  Analysis of Smart Piezo-Magneto-Thermo-Elastic Composite and Reinforced Plates: Part I – Model Development , 2014 .

[55]  R. Dimitri,et al.  Free vibration analysis of conical shells reinforced with agglomerated Carbon Nanotubes , 2016 .

[56]  Nicholas Fantuzzi,et al.  Higher-order theories for the free vibrations of doubly-curved laminated panels with curvilinear reinforcing fibers by means of a local version of the GDQ method , 2015 .

[57]  Erasmo Viola,et al.  Free vibration analysis of functionally graded panels and shells of revolution , 2009 .

[58]  Salvatore Brischetto,et al.  Numerical and exact models for free vibration analysis of cylindrical and spherical shell panels , 2015 .

[59]  Young-Wann Kim.  Free vibration analysis of FGM cylindrical shell partially resting on Pasternak elastic foundation with an oblique edge , 2015 .

[60]  Salvatore Brischetto,et al.  Exact 3D solutions and finite element 2D models for free vibration analysis of plates and cylinders , 2014 .

[61]  J. N. Reddy,et al.  Winkler–Pasternak foundation effect on the static and dynamic analyses of laminated doubly-curved and degenerate shells and panels , 2014 .

[62]  S. A. Eftekhari.  A Differential Quadrature Procedure with Direct Projection of the Heaviside Function for Numerical Solution of Moving Load Problem , 2016 .

[63]  Francesco Tornabene,et al.  General higher-order layer-wise theory for free vibrations of doubly-curved laminated composite shells and panels , 2016 .

[64]  E. Viola,et al.  General higher-order equivalent single layer theory for free vibrations of doubly-curved laminated composite shells and panels , 2013 .

[65]  E. Carrera.  On the use of the Murakami's zig-zag function in the modeling of layered plates and shells , 2004 .

[66]  Ö. Civalek,et al.  Vibration analysis of FG cylindrical shells with power-law index using discrete singular convolution technique , 2016 .

[67]  Fiorenzo A. Fazzolari,et al.  Stability analysis of FGM sandwich plates by using variable-kinematics Ritz models , 2016 .

[68]  Nicholas Fantuzzi,et al.  A strong formulation finite element method (SFEM) based on RBF and GDQ techniques for the static and dynamic analyses of laminated plates of arbitrary shape , 2014 .

[69]  I. A. Jones.  Approximate solutions to the orthotropic pinched cylinder problem , 1998 .

[70]  C. Hwu,et al.  In-plane/out-of-plane concentrated forces and moments on composite laminates with elliptical elastic inclusions , 2007 .

[71]  S. A. Eftekhari.  A note on mathematical treatment of the Dirac-delta function in the differential quadrature bending and forced vibration analysis of beams and rectangular plates subjected to concentrated loads , 2015 .

[72]  S. A. Eftekhari,et al.  Accurate variational approach for free vibration of simply supported anisotropic rectangular plates , 2014 .

[73]  Liviu Librescu,et al.  A few remarks concerning several refined theories of anisotropic composite laminated plates , 1989 .

[74]  A. Sofiyev.  Large amplitude vibration of FGM orthotropic cylindrical shells interacting with the nonlinear Winkler elastic foundation , 2016 .

[75]  A. Sofiyev.  Influences of shear stresses on the dynamic instability of exponentially graded sandwich cylindrical shells , 2015 .

[76]  E. Viola,et al.  Vibration analysis of multi-stepped and multi-damaged parabolic arches using GDQ , 2014 .

[77]  T. Y. Yang,et al.  High Order Rectangular Shallow Shell Finite Element , 1973 .

[78]  Paul M. Weaver,et al.  Static inconsistencies in certain axiomatic higher-order shear deformation theories for beams, plates and shells , 2015 .

[79]  Salvatore Brischetto,et al.  AN EXACT 3D SOLUTION FOR FREE VIBRATIONS OF MULTILAYERED CROSS-PLY COMPOSITE AND SANDWICH PLATES AND SHELLS , 2014 .

[80]  Laureano José Luis Mantari.  Refined and generalized hybrid type quasi-3D shear deformation theory for the bending analysis of functionally graded shells , 2015 .

[81]  Nicholas Fantuzzi,et al.  New insights into the strong formulation finite element method for solving elastostatic and elastodynamic problems , 2014 .

[82]  Stefano Tornincasa,et al.  Testing and simulation of the three point bending anisotropic behaviour of hazelnut shells , 2015 .

[83]  Erasmo Viola,et al.  Transient dynamic response of generally-shaped arches based on a GDQ-time-stepping method , 2016 .

[84]  Elena Ferretti,et al.  Soft Core Plane State Structures Under Static Loads UsingGDQFEM and Cell Method , 2013 .

[85]  Jongman Kim,et al.  Design of sandwich structures for concentrated loading , 2001 .

[86]  K. M. Liew,et al.  Static Analysis of Reissner-Mindlin Plates by Differential Quadrature Element Method , 1998 .

[87]  N. Kuruoglu,et al.  Domains of dynamic instability of FGM conical shells under time dependent periodic loads , 2016 .

[88]  V. Verijenko,et al.  A higher-order theory for the analysis of laminated plates and shells with shear and normal deformation , 1993 .

[89]  G. Tsamasphyros,et al.  Study and solution of BEM-singular integral equation method in the case of concentrated loads , 2013 .

[90]  Nicholas Fantuzzi,et al.  Accurate inter-laminar recovery for plates and doubly-curved shells with variable radii of curvature using layer-wise theories , 2015 .

[91]  Salvatore Brischetto,et al.  3D exact and 2D generalized differential quadrature models for free vibration analysis of functionally graded plates and cylinders , 2016 .

[92]  Nicholas Fantuzzi,et al.  Four-parameter functionally graded cracked plates of arbitrary shape: A GDQFEM solution for free vibrations , 2016 .

[93]  L. S. D. Morley.  THE THIN-WALLED CIRCULAR CYLINDER SUBJECTED TO CONCENTRATED RADIAL LOADS , 1960 .

[94]  Nicholas Fantuzzi,et al.  Generalized Differential Quadrature Finite Element Method for vibration analysis of arbitrarily shaped membranes , 2014 .

[95]  K. M. Liew,et al.  Differential quadrature element method for static analysis of Reissner-Mindlin polar plates , 1999 .

[96]  K. P. Rao,et al.  A rectangular laminated anisotropic shallow thin shell finite element , 1978 .

[97]  F. Tornabene,et al.  The local GDQ method for the natural frequencies of doubly-curved shells with variable thickness: A general formulation , 2016 .

[98]  E. Carrera,et al.  Static analysis of multilayered smart shells subjected to mechanical, thermal and electrical loads , 2013 .

[99]  W. Müller,et al.  Three-dimensional elastic deformation of functionally graded isotropic plates under point loading , 2014 .

[100]  Francesco Ubertini,et al.  Strong Formulation Finite Element Method Based on Differential Quadrature: A Survey , 2015 .

[101]  Chih‐Ping Wu,et al.  Asymptotic theory of laminated circular conical shells , 1999 .

[102]  L. B. D. Veiga.  Asymptotic study of the solution for pinched cylindrical shells , 2005 .

[103]  Alfredo Liverani,et al.  Static analysis of laminated composite curved shells and panels of revolution with a posteriori shear and normal stress recovery using generalized differential quadrature method , 2012 .

[104]  S. A. Eftekhari.  A DIFFERENTIAL QUADRATURE PROCEDURE WITH REGULARIZATION OF THE DIRAC-DELTA FUNCTION FOR NUMERICAL SOLUTION OF MOVING LOAD PROBLEM , 2015 .

[105]  Alireza Daneshmehr,et al.  Modified couple stress theory applied to dynamic analysis of composite laminated beams by considering different beam theories , 2015 .

[106]  M. Ghadiri,et al.  Nonlinear vibration of axially functionally graded non-uniform nanobeams , 2016 .

[107]  M. Vaghefi,et al.  Thermal effect on free vibration of functionally graded truncated conical shell panels , 2016 .

[108]  Chang Shu,et al.  Parallel simulation of incompressible viscous flows by generalized differential quadrature , 1992 .

[109]  A. Daneshmehr,et al.  Size-dependent thermal buckling analysis of micro composite laminated beams using modified couple stress theory , 2015 .

[110]  E. Viola,et al.  Static analysis of completely doubly-curved laminated shells and panels using general higher-order shear deformation theories , 2013 .

[111]  Fiorenzo A. Fazzolari,et al.  A refined dynamic stiffness element for free vibration analysis of cross-ply laminated composite cylindrical and spherical shallow shells , 2014 .

[112]  F. Fazzolari.  Reissner's Mixed Variational Theorem and variable kinematics in the modelling of laminated composite and FGM doubly-curved shells , 2016 .

[113]  Nicholas Fantuzzi,et al.  Static analysis of functionally graded conical shells and panels using the generalized unconstrained third order theory coupled with the stress recovery , 2014 .

[114]  Erasmo Viola,et al.  Inter-laminar stress recovery procedure for doubly-curved, singly-curved, revolution shells with variable radii of curvature and plates using generalized higher-order theories and the local GDQ method , 2016 .

[115]  R. Dimitri,et al.  Free vibrations of composite oval and elliptic cylinders by the generalized differential quadrature method , 2015 .

[116]  Salvatore Brischetto,et al.  Refined 2D and Exact 3D Shell Models for the Free Vibration Analysis of Single- and Double-Walled Carbon Nanotubes , 2015 .

[117]  Miao Xuhong,et al.  A unified solution for the vibration analysis of FGM doubly-curved shells of revolution with arbitrary boundary conditions , 2016 .