Technical Briefs

A Constraint on the Consistence of Transverse Shear Strain Energy in the Higher-Order Shear Deformation Theories of Elastic Plates

[+] Author and Article Information
Guangyu Shi

e-mail: shi_guangyu@163.com

Xiaodan Wang

e-mail: guozhenshile@126.com
Department of Mechanics,
Tianjin University,
Tianjin 300072, China

1 Corresponding author.

Manuscript received Febraury 19, 2010; final manuscript received January 15, 2012; accepted manuscript posted October 8, 2012; published online May 16, 2013. Assoc. Editor: Prof. Krishna Garikipati.

J. Appl. Mech 80(4), 044501 (May 16, 2013) (9 pages) Paper No: JAM-10-1048; doi: 10.1115/1.4007790 History: Received February 19, 2010; Revised January 15, 2012; Accepted October 08, 2012

This paper studies how to improve the third-order shear deformation theories of isotropic plates, which is a question raised by late Reissner in 1985 (ASME Appl. Mech. Rev., 38, pp.1453–1464). It is demonstrated in this paper that a proper displacement field with the higher-order shear deformations given by the method of displacement assumption should satisfy the constraint on the consistence of the transverse shear strain energy a priori in addition to the traction conditions on plate surfaces. This additional constraint on the assumed displacement fields with the higher-order shear deformations is in line with Love's criterion of the consistent first approximation to the strain energy wherein the transverse shear strain energy is included. The constraint on the consistence of the transverse shear strain energy is physically similar to the requirement for the use of the shear coefficients in the first-order shear deformation plate theories proposed by Reissner and Mindlin, respectively. A procedure to improve the assumed displacement field with the third-order shear deformations is presented. The present study shows that the various displacement fields with the simple third-order shear deformations would be identical when the constraint on the consistence of the transverse shear strain energy is enforced.

Copyright © 2012 by ASME
Your Session has timed out. Please sign back in to continue.


Reissner, E., 1985, “Reflection on the Theory of Elastic Plates,” ASME J. Appl. Mech. Rev., 38, pp. 1453–1464. [CrossRef]
Levinson, M., 1980, “An Accurate Simple Theory of Statics and Dynamics of Elastic Plates,” Mech. Res. Commun., 7, pp. 340–350. [CrossRef]
Murthy, V. V., 1981, “An Improved Transverse Shear Deformation Theory for Laminate Anisotropic Plates,” NASA Technical Paper No. 1903.
Reddy, J. N., 1984, “A Simple Higher-Order Theory for Laminated Composite Plates,” ASME J. Appl. Mech., 51, pp. 745–752. [CrossRef]
Shi, G., 2007, “A New Simple Third-Order Shear Deformation Theory of Plates,” Int. J. Solids Struct., 44, pp. 4399–4417. [CrossRef]
Aydogdu, M., 2009, “A New Shear Deformation Theory for Laminated Composite Plates,” Compos. Struct., 89. pp. 94–101. [CrossRef]
Kapania, R. K., and Raciti, S., 1989, “Recent Advances in Analysis of Laminated Beams and Plates, Part I: Shear Effects and Buckling,” AIAA J.27, pp. 923–934. [CrossRef]
Rohwer, K., 1992, “Application of Higher Order Theories to the Bending Analysis of Layered Composite Plates,” Int. J. Solids Struct., 29, pp. 105–119. [CrossRef]
Timoshenko, S. P., and Gere, J., 1972, Mechanics of Materials, Van Nostrand, New York.
Reissner, E., 1945, “The Effect of Transverse Shear Deformation on the Bending of Elastic Plates,” ASME J. Appl. Mech., 12, pp. 66–77.
Mindlin, R. D., 1951, “Influence of Rotatory Inertia and Shear on Flexural Motion of Isotropic, Elastic Plates,” ASME J. Appl. Mech., 18, pp. 31–38.
Pai, P. F., and Schulz, M. J., 1999, “Shear Correction Factors and an Energy Consistent Beam Theory,” Int. J. Solids Struct., 36, pp. 1523–1540. [CrossRef]
Idlbi, A., Karama, M., and Touratie, M., 1997, “Comparison of Various Laminated Plate Theories,” Compos. Struct., 37, pp. 173–184. [CrossRef]
Carrera, E., and Petrolo, M., 2011, “On the Effectiveness of Higher-Order Terms in Refined Beam Theories,” ASME J. Appl. Mech., 78(021013), pp. 1–17. [CrossRef]
Koiter, W. T., 1959, “A Consistent First Approximation in the General Theory of Thin Elastic Shells,” in Proc. IUTAM Symposium on the Theory of Thin Elastic Shells, Delft, Holland, August 24–28, North Holland Publishing, Delft, Holland, pp. 12–33.
Reddy, J. N., Wang, C. M., Lim, G. T., and Ng, K. Y., 2001, “Bending Solutions of Levinson Beam and Plates in Terms of the Classical Theories,” Int. J. Solids Struct., 38, pp. 4701–4720. [CrossRef]
Wang, C. M., Kitipornchai, S., Lim, C. W., and Eisenberger, M., 2008, M., Beam Bending Solutions Based on Nonlocal Timoshenko Beam Theory, ASCE J. Eng. Mech., 134, pp. 475–481. [CrossRef]
Reissner, E., 1975, “On Transverse Bending of Plates, Including the Effect of Transverse Shear Deformation,” Int. J. Solids Struct., 11, pp. 569-573. [CrossRef]
Panc, V., 1975, Theories of Elastic Plates, Noordhoff, Netherlands.
Karama, M., Afaq, K. S., and Mistou, S., 2003, “Mechanical Behavior of Laminated Composite Beam by the New Multi-Layered Laminated Composite Structures Model With Transverse Shear Stress Continuity,” Int. J. Solids Struct., 40, pp. 1525–1546. [CrossRef]
Voyiadjis, G. Z., and Shi, G., 1991, “A Refined Two-Dimensional Theory for Thick Cylindrical Shells,” Int. J. Solids Struct., 27, pp. 261–282. [CrossRef]
Bickford, W. B., 1982, “A Consistent Higher Order Beam Theory,” in Developments in Theoretical and Applied Mechanics Vol. XI, University of Alabama, Huntsville, AL, pp. 137–150.
Levinson, M., 1981, “A New Rectangular Beam Theory,” J. Sound Vib., 74, pp. 81–87. [CrossRef]
Shi, G., and Voyiadjis, G. Z., 2011, “A Sixth-Order Theory of Shear Deformable Beams With Variational Consistent Boundary Conditions,” ASME J. Appl. Mech.78(021019), pp. 1–11. [CrossRef]
Srinivas, C. V., and Rao, A. K., 1970, “Bending, Vibration and Buckling of Simply Supported Thick Orthotropic Rectangular Plates and Laminates,” Int. J. Solids Struct., 6, pp. 1463–1481. [CrossRef]
Reddy, J. N., 1984, “A Refined Nonlinear Theory of Plates With Transverse Shear Deformations,” Int. J. Solids Struct., 20, pp. 881–896. [CrossRef]
Shi, G., and Voyiadjis, G. Z., 2012, “A Sixth-Order Beam Theory for Flexural Vibration Analysis of Beams With the Effects of Shear Flexibility and Rotary Inertia,” J. Sound Vib. (submitted).
Hutchinson, J., 1986, “On the Axisymmetric Vibrations of Thick Clamped Plates,” Proc. Int. Conf. on Vibration Problems in Engineering, Xi'an, China, June 17–20, pp. 75–81.
Hutchinson, J., 1987, “A Comparison of Mindlin and Levinson Plate Theories,” Mech. Res. Commun., 14, pp. 165–170. [CrossRef]
Levinson, M., 1987, “On Higher Order Beam and Plate Theories,” Mech. Res. Commun., 14, pp. 421–424. [CrossRef]


Grahic Jump Location
Fig. 1

The equilibrium of an incremental beam segment in terms of generalized transverse force

Grahic Jump Location
Fig. 2

The shear force distribution along a cantilevered beam given by Bickford's beam theory

Grahic Jump Location
Fig. 3

The shear force distribution along a cantilevered beam given by Shi and Voyiadjis



Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In