A New Rectangular Finite Element Formulation Based on Higher Order Displacement Theory for Thick and Thin Composite and Sandwich Plates

Affiliation(s)

Shaktibad Apartment, Phase-2, Ground Floor, Garia Station Road, Calcutta, India.

Department of Mechanical Engineering, Institute of Structural Mechanics, Technical University of Darmstadt, Darmstadt, Germany.

Shaktibad Apartment, Phase-2, Ground Floor, Garia Station Road, Calcutta, India.

Department of Mechanical Engineering, Institute of Structural Mechanics, Technical University of Darmstadt, Darmstadt, Germany.

ABSTRACT

A new displacement based higher order element has been formulated
that is ideally suitable for shear deformable composite and
sandwich plates. Suitable functions for displacements and rotations for each
node have been selected so that the element shows rapid convergence, an
excellent response against transverse shear loading and requires no shear correction
factors. It is completely lock-free and behaves extremely well for thin to
thick plates. To make the element rapidly convergent and to capture warping effects
for composites, higher order displacement terms in the displacement kinematics
have been considered for each node. The element has eleven degrees of freedom
per node. Shear deformation has also been considered in the
formulation by taking into account shear strains ( *r*_{xz} and *r*_{yz}) as nodal unknowns. The element is very simple to
formulate and could be coded up in research software. A small Fortran code has
been developed to implement the element and various examples of isotropic and
composite plates have been analyzed to show the effectiveness of the element.

Cite this paper

S. Goswami and W. Becker, "A New Rectangular Finite Element Formulation Based on Higher Order Displacement Theory for Thick and Thin Composite and Sandwich Plates,"*World Journal of Mechanics*, Vol. 3 No. 3, 2013, pp. 194-201. doi: 10.4236/wjm.2013.33019.

S. Goswami and W. Becker, "A New Rectangular Finite Element Formulation Based on Higher Order Displacement Theory for Thick and Thin Composite and Sandwich Plates,"

References

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[2] R. D. Mindlin, “Influence of Rotary Inertia and Shear on Flexural Motions of Isotropic Elastic Plates,” Journal of Applied Mechanics, Transaction of American Society of Mechanical Engineers, Vol. 18, No. 1, 1951, pp. 31-38.

[3] O. C. Zienkiewicz and R. L. Taylor, “The Finite Element Method,” McGraw-Hill, New York, 1989.

[4] S. P. Timoshenko and S. Winowsky-Krieger, “Theory of Plates and Shells,” 2nd Edition, McGraw-Hill, New York, 1959.

[5] J. M. Whitney, “The Effect of Transverse Shear Deformation on the Bending of Laminated Plates,” Journal of Composite Materials, Vol. 3, No. 3, 1969, pp. 534-547. doi:10.1177/002199836900300316

[6] N. J. Pagano, “Exact Solutions for Composite Laminates in Cylindrical Bending,” Journal of Composite Materials, Vol. 3, No. 3, 1969, pp. 398-411. doi:10.1177/002199836900300304

[7] N. J. Pagano, “Exact Solutions for Rectangular Bidirectional Composites and Sandwich Plates,” Journal of Composite Materials, Vol. 4, No. 1, 1970, pp. 20-34.

[8] N. J. Pagano and S. J. Hatfield, “Elastic Behavior of Multilayered Bidirectional Composites,” AIAA Journal, Vol. 10, No. 7, 1972, pp. 931-933. doi:10.2514/3.50249

[9] A. L. Dobyns, “Analysis of Simply Supported Orthotropic Plates Subjected to Static and Dynamic Loads,” AIAA Journal, Vol. 19, No. 5, 1981, pp. 642-650. doi:10.2514/3.50984

[10] D. G. Ashwell, A. B. Sabir and T. M. Roberts, “Further Studies in the Application of Curved Finite Elements to Circular Arches,” International Journal of Mechanical Sciences, Vol. 13, No. 6, 1971, pp. 507-517. doi:10.1016/0020-7403(71)90038-5

[11] A. H. Sheikh and P. Dey, “A New Triangular Element for the Analysis of Thick and Thin Plates,” Communications in Numerical Methods in Engineering, Vol. 17, No. 9, 2001, pp. 667-673. doi:10.1002/cnm.440

[12] J. J. Engblom and O. O. Ochoa, “Finite Element Formulation Including Interlaminar Stress Calculations,” Computers & Structures, Vol. 23, No. 2, 1986, pp. 241-249. doi:10.1016/0045-7949(86)90216-6

[13] M. Guenfoud, “Presentation de l’Element DSTM pour le Calcul Lineaire des Coques d’Epaisseur Quelconque,” Ann l’ITBTP, Vol. 515, 1993, pp. 25-52.

[14] L. Belounar and M. Guenfoud, “A New Rectangular Finite Element Based on the Strain Approach for Plate Bending,” Thin-Walled Structures, Vol. 43, No. 1, 2005, pp. 47-63. doi:10.1016/j.tws.2004.08.003

[15] B. N. Pandya and T. Kant, “A Consistent Refined Theory for Flexure of a Symmetric Laminate,” Mechanics Research Communications, Vol. 14, No. 2, 1987, pp. 107-113. doi:10.1016/0093-6413(87)90026-7

[16] S. Goswami, “A C0 Plate Bending Element with Refined Shear Deformation Theory for Composite Structures,” Composite Structures, Vol. 72, No. 3, 2006, pp. 375-382. doi:10.1016/j.compstruct.2005.01.007

[17] G. R. Bhashyam and R. H. Gallagher, “An Approach to the Inclusion of the Transverse Shear Deformation in the Finite Element Plate Bending analysis,” Computers and Structures, Vol. 19, No. 1-2, 1984, pp. 35-40. doi:10.1016/0045-7949(84)90200-1

[18] S. Goswami, “A Finite Element Investigation on the Effects of Cross-Sectional Warping on Flexural Response of Laminated Composites and Sandwiches using Higher Order Shear Deformation Theory,” Journal of Reinforced Plastics and Composites, Vol. 24, No. 15, 2005, pp. 1587-1604. doi:10.1177/0731684405050398

[19] C. W. Pryor Jr. and R. M. Barker, “A Finite Element Analysis Including Transverse Shear Effects for Applications to Laminated Plates,” AIAA Journal, Vol. 9, No. 5, 1971, pp. 912-917. doi:10.2514/3.6295

[20] J. N. Reddy, “A Penalty Plate Bending Element for the Analysis of Laminated Anisotropic Composite Plates,” International Journal for Numerical Methods in Engineering, Vol. 15, No. 8, 1980, pp. 1187-1206. doi:10.1002/nme.1620150807

[21] N. D. Phan and J. N. Reddy, “Analysis of Laminated Composite Plates using a Higher Order Shear Deformation Theory,” International Journal for Numerical Methods in Engineering, Vol. 21, No. 12, 1985, pp. 2201-2219. doi:10.1002/nme.1620211207

[22] R. K. Kapania and S. Raciti, “Recent Advances in Analysis of Laminated Beams and Plates,” AIAA Journal, Vol. 27, No. 7, 1989, pp. 923-946. doi:10.2514/3.10202

[23] J. N. Reddy and D. H. Robbins, “Theories and Computational Models for Composite Laminates,” Applied Mechanics Review, Vol. 47, No. 6, 1994, pp. 147-165. doi:10.1115/1.3111076

[24] A. K. Noor, S. Burton and C. W. Bert, “Computational Models for Sandwich Panels and Shells,” Applied Mechanics Review, Vol. 49, No. 3, 1996, pp. 155-199. doi:10.1115/1.3101923

[25] X. Y. Zhang and C. H. Yang, “Recent Developments in Finite Element Analysis for Laminated Composite Plates,” Composite Structures, Vol. 88, No. 1, 2009, pp. 147-157. doi:10.1016/j.compstruct.2008.02.014

[1] E. Reissner, “The Effect of Transverse Shear Deformation on the Bending of Elastic Plates,” Journal of Applied Mechanics, Transaction of American Society of Mechanical Engineers, Vol. 12, No. 2, 1945, pp. A66-A77.

[2] R. D. Mindlin, “Influence of Rotary Inertia and Shear on Flexural Motions of Isotropic Elastic Plates,” Journal of Applied Mechanics, Transaction of American Society of Mechanical Engineers, Vol. 18, No. 1, 1951, pp. 31-38.

[3] O. C. Zienkiewicz and R. L. Taylor, “The Finite Element Method,” McGraw-Hill, New York, 1989.

[4] S. P. Timoshenko and S. Winowsky-Krieger, “Theory of Plates and Shells,” 2nd Edition, McGraw-Hill, New York, 1959.

[5] J. M. Whitney, “The Effect of Transverse Shear Deformation on the Bending of Laminated Plates,” Journal of Composite Materials, Vol. 3, No. 3, 1969, pp. 534-547. doi:10.1177/002199836900300316

[6] N. J. Pagano, “Exact Solutions for Composite Laminates in Cylindrical Bending,” Journal of Composite Materials, Vol. 3, No. 3, 1969, pp. 398-411. doi:10.1177/002199836900300304

[7] N. J. Pagano, “Exact Solutions for Rectangular Bidirectional Composites and Sandwich Plates,” Journal of Composite Materials, Vol. 4, No. 1, 1970, pp. 20-34.

[8] N. J. Pagano and S. J. Hatfield, “Elastic Behavior of Multilayered Bidirectional Composites,” AIAA Journal, Vol. 10, No. 7, 1972, pp. 931-933. doi:10.2514/3.50249

[9] A. L. Dobyns, “Analysis of Simply Supported Orthotropic Plates Subjected to Static and Dynamic Loads,” AIAA Journal, Vol. 19, No. 5, 1981, pp. 642-650. doi:10.2514/3.50984

[10] D. G. Ashwell, A. B. Sabir and T. M. Roberts, “Further Studies in the Application of Curved Finite Elements to Circular Arches,” International Journal of Mechanical Sciences, Vol. 13, No. 6, 1971, pp. 507-517. doi:10.1016/0020-7403(71)90038-5

[11] A. H. Sheikh and P. Dey, “A New Triangular Element for the Analysis of Thick and Thin Plates,” Communications in Numerical Methods in Engineering, Vol. 17, No. 9, 2001, pp. 667-673. doi:10.1002/cnm.440

[12] J. J. Engblom and O. O. Ochoa, “Finite Element Formulation Including Interlaminar Stress Calculations,” Computers & Structures, Vol. 23, No. 2, 1986, pp. 241-249. doi:10.1016/0045-7949(86)90216-6

[13] M. Guenfoud, “Presentation de l’Element DSTM pour le Calcul Lineaire des Coques d’Epaisseur Quelconque,” Ann l’ITBTP, Vol. 515, 1993, pp. 25-52.

[14] L. Belounar and M. Guenfoud, “A New Rectangular Finite Element Based on the Strain Approach for Plate Bending,” Thin-Walled Structures, Vol. 43, No. 1, 2005, pp. 47-63. doi:10.1016/j.tws.2004.08.003

[15] B. N. Pandya and T. Kant, “A Consistent Refined Theory for Flexure of a Symmetric Laminate,” Mechanics Research Communications, Vol. 14, No. 2, 1987, pp. 107-113. doi:10.1016/0093-6413(87)90026-7

[16] S. Goswami, “A C0 Plate Bending Element with Refined Shear Deformation Theory for Composite Structures,” Composite Structures, Vol. 72, No. 3, 2006, pp. 375-382. doi:10.1016/j.compstruct.2005.01.007

[17] G. R. Bhashyam and R. H. Gallagher, “An Approach to the Inclusion of the Transverse Shear Deformation in the Finite Element Plate Bending analysis,” Computers and Structures, Vol. 19, No. 1-2, 1984, pp. 35-40. doi:10.1016/0045-7949(84)90200-1

[18] S. Goswami, “A Finite Element Investigation on the Effects of Cross-Sectional Warping on Flexural Response of Laminated Composites and Sandwiches using Higher Order Shear Deformation Theory,” Journal of Reinforced Plastics and Composites, Vol. 24, No. 15, 2005, pp. 1587-1604. doi:10.1177/0731684405050398

[19] C. W. Pryor Jr. and R. M. Barker, “A Finite Element Analysis Including Transverse Shear Effects for Applications to Laminated Plates,” AIAA Journal, Vol. 9, No. 5, 1971, pp. 912-917. doi:10.2514/3.6295

[20] J. N. Reddy, “A Penalty Plate Bending Element for the Analysis of Laminated Anisotropic Composite Plates,” International Journal for Numerical Methods in Engineering, Vol. 15, No. 8, 1980, pp. 1187-1206. doi:10.1002/nme.1620150807

[21] N. D. Phan and J. N. Reddy, “Analysis of Laminated Composite Plates using a Higher Order Shear Deformation Theory,” International Journal for Numerical Methods in Engineering, Vol. 21, No. 12, 1985, pp. 2201-2219. doi:10.1002/nme.1620211207

[22] R. K. Kapania and S. Raciti, “Recent Advances in Analysis of Laminated Beams and Plates,” AIAA Journal, Vol. 27, No. 7, 1989, pp. 923-946. doi:10.2514/3.10202

[23] J. N. Reddy and D. H. Robbins, “Theories and Computational Models for Composite Laminates,” Applied Mechanics Review, Vol. 47, No. 6, 1994, pp. 147-165. doi:10.1115/1.3111076

[24] A. K. Noor, S. Burton and C. W. Bert, “Computational Models for Sandwich Panels and Shells,” Applied Mechanics Review, Vol. 49, No. 3, 1996, pp. 155-199. doi:10.1115/1.3101923

[25] X. Y. Zhang and C. H. Yang, “Recent Developments in Finite Element Analysis for Laminated Composite Plates,” Composite Structures, Vol. 88, No. 1, 2009, pp. 147-157. doi:10.1016/j.compstruct.2008.02.014