An improved estimate for the error in the classical, linear theory of plate bending
Author:
James G. Simmonds
Journal:
Quart. Appl. Math. 29 (1971), 439-447
DOI:
https://doi.org/10.1090/qam/99753
MathSciNet review:
QAM99753
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Abstract: The relative mean square error in the three-dimensional stress field predicted by classical plate theory is shown to be $O{\left ( {h/{L_ * }} \right )^2}$, where $h$ is the plate thickness and ${L_ * }$ is a mean square measure of the wavelength of the midplane deformation pattern. This improves a recent result of Nordgren who obtained a relative error estimate of $O\left ( {h/{L_ * }} \right )$. The improved error estimate, which, like Nordgren’s, is based on the Prager—Synge hypercircle theorem in elasticity, is obtained by constructing a kinematically admissible three-dimensional displacement field that depends on the solution of the classical plate equations but which yields an accurate, nonzero distribution of the transverse shearing strain through the thickness.
- R. P. Nordgren, A bound on the error in plate theory, Quart. Appl. Math. 28 (1971), 587–595. MR 280051, DOI https://doi.org/10.1090/S0033-569X-1971-0280051-4
- W. Prager and J. L. Synge, Approximations in elasticity based on the concept of function space, Quart. Appl. Math. 5 (1947), 241–269. MR 25902, DOI https://doi.org/10.1090/S0033-569X-1947-25902-8
- J. L. Synge, The method of the hypercircle in elasticity when body forces are present, Quart. Appl. Math. 6 (1948), 15–19. MR 25904, DOI https://doi.org/10.1090/S0033-569X-1948-25904-0
W. T. Koiter, On the foundations of the linear theory of thin elastic shells, Proc. Nederl. Akad. Wetensch. B73, 169–195 (1970)
- D. A. Danielson, Improved error estimates in the linear theory of thin elastic shells, Nederl. Akad. Wetensch. Proc. Ser. B 74 (1971), 294–300. MR 0302014
R. P. Nordgren, A bound on the error in plate theory, Quart. Appl. Math. 28, 587–595 (1971)
W. Prager and J. L. Synge, Approximations in elasticity based on the concept of function space, Quart. Appl. Math. 5, 241–269 (1947)
J. L. Synge, The method of the hypercircle in elasticity when body forces are present, Quart. Appl. Math. 6, 15–19 (1948)
W. T. Koiter, On the foundations of the linear theory of thin elastic shells, Proc. Nederl. Akad. Wetensch. B73, 169–195 (1970)
D. A. Danielson, Improved error estimates in the linear theory of thin elastic shells, Proc. Nederl. Akad. Wetensch. 874, 294–300 (1971)
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Article copyright:
© Copyright 1971
American Mathematical Society