Improved shear correction factors for deflection of simply supported very thick rectangular auxetic plates
- T. C. Lim^{1}Email author
https://doi.org/10.1186/s40712-016-0065-z
© The Author(s). 2016
Received: 22 June 2016
Accepted: 3 November 2016
Published: 25 November 2016
Abstract
Background
The first-order shear deformation theory (FSDT) for plates requires a shear correction factor due to the assumption of constant shear strain and shear stress across the thickness; hence, the shear correction factor strongly influences the accuracy of the deflection solution; the third-order shear deformation theory (TSDT) does not require a correction factor because it facilitates the change in shear strain across the plate thickness.
Methods
This paper obtains an improved shear correction factor for simply supported very thick rectangular plates by matching the deflection of the Mindlin plate (FSDT) with that of the Reddy plate (TSDT).
Results
As a consequence, the use of the exact shear correction factor for the Mindlin plate gives solutions that are exactly the same as for the Reddy plate.
Conclusions
The customary adoption of 5/6 shear correction factor is a lower bound, and the exact shear correction factor is higher for the following: (a) very thick plates, (b) narrow or long plates, (c) high Poisson’s ratio plate material, and (d) highly patterned loads, while the commonly used shear correction factor of 5/6 is still valid for the following: (i) marginally thick plates, (ii) square plates, (iii) negative Poisson’s ratio materials, and (d) uniformly distributed loadings.
Keywords
Aspect ratio Auxetic materials Incompressible materials Shear deformation Thick platesBackground
It is well known that the shear correction factors of plates are simpler than those for beams (Dong et al., 2010; Puchegger et al., 2003; Hlavacek and Chleboun, 2000; Pai and Schultz, 1999; Popescu and Hodges, 2000; Yu and Hodges, 2004; Chan et al., 2011; Pai et al., 2000; Hutchinson, 1980; Hutchinson, 2001; Han et al., 1999); this is due to the cross-sectional geometry in beams being more varied than for plates. For plates, the commonly adopted shear correction factor is typically 5/6; in some instances, Poisson’s ratio is taken into account (e.g., Rössle, 1999; Lee et al., 2002). Exact shear correction factors for vibrating Mindlin plates have been proposed by Stephen (1997) and Hull (2005, 2006). In this paper, exact shear correction factors for simply supported very thick rectangular Mindlin plates are derived by comparing its deflection against that of Reddy plates. The Mindlin plate, which adopts the first-order shear deformation theory (FSDT), requires a correction factor due to its assumption of uniform shear across the plate thickness while the Reddy plate, which adopts the third-order shear deformation theory (TSDT), does not require any correction as it caters for the varying shear strain across the plate thickness. The rigor of the Reddy plate, therefore, forms the justification for its use as a benchmark for evaluating the accuracy of Mindlin plate deflection—this has been done for triangular plates (Lim, 2016a, b). Following a recent preliminary analysis (Lim, 2016c) to evaluate the ratio of maximum deflection of Reddy plate to that of Kirchhoff plate or the classical plate theory (CPT), the TSDT is now being employed for extracting the exact shear correction factor of rectangular plates in the FSDT.
Methods
General consideration
with G being the shear modulus of the plate.
Perusal to Eq. (5a) suggests that a meaningful exact shear correction factor can be obtained if both the Reddy plate and Kirchhoff plate deflections are known. Neglecting the higher order term in Eq. (5a) gives a shear correction factor of κ = 14/17. The two constant shear correction factors of 5/6 and 14/17 have been discussed by Wang et al. (2000).
Uniform load
Computed results of Eq. (15)
Plate aspect ratio \( \frac{a}{b} \) | Numerator \( {\displaystyle \sum_{m=1}^{\infty }{\displaystyle \sum_{n=1}^{\infty}\frac{{\left(-1\right)}^{\frac{m+n}{2}\kern0.5em -1}}{mn\left(\frac{b}{a}{m}^2+\frac{a}{b}{n}^2\right)}}} \) | Denominator \( {\displaystyle \sum_{m=1}^{\infty }{\displaystyle \sum_{n=1}^{\infty}\frac{{\left(-1\right)}^{\frac{m+n}{2}\kern0.5em -1}}{mn{\left(\frac{b}{a}{m}^2+\frac{a}{b}{n}^2\right)}^2}}} \) | Ratio \( f\left(a,b,\frac{a}{2},\frac{b}{2}\right) \) |
---|---|---|---|
1.0 | 0.44895 | 0.24409 | 1.839281 |
1.2 | 0.44059 | 0.23578 | 1.868649 |
1.5 | 0.40575 | 0.20627 | 1.967082 |
2.0 | 0.34749 | 0.15215 | 2.283865 |
3.0 | 0.24622 | 0.08167 | 3.014816 |
5.0 | 0.15405 | 0.031175 | 4.941460 |
7.5 | 0.10389 | 0.013907 | 7.470339 |
10.0 | 0.07859 | 0.007824 | 10.04473 |
with a statistical accuracy of R ^{2} = 0.9998.
Sinusoidal load
the former is instructive for showing the effect of relative plate thickness \( h/\sqrt{ab} \) and aspect ratio a/b, in addition to Poisson’s ratio. Unlike the previous section on uniform load, this section on sinusoidal load allows one to observe the interlacing effect of load waviness pattern and plate aspect ratio on the shear correction factor.
Shear correction factor expressions for special cases of rectangular plates under sinusoidal loads
Simple sinusoidal load distribution (m = n = 1) | General sinusoidal load distributions (m, n ≥ 1) | |
---|---|---|
Square plates a = b | \( \kappa =\frac{5}{6}\left[1+\frac{\pi^2}{210\left(1-v\right)}\frac{h^2}{a^2}\right] \) | \( \kappa =\frac{5}{6}\left[1+\frac{\pi^2}{420\left(1-v\right)}\frac{h^2}{a^2}\left({m}^2+{n}^2\right)\right] \) |
Rectangular plates a ≠ b | \( \kappa =\frac{5}{6}\left[1+\frac{\pi^2}{420\left(1-v\right)}\frac{h^2}{ab}\left(\frac{b}{a}+\frac{a}{b}\right)\right] \) | \( \kappa =\frac{5}{6}\left[1+\frac{\pi^2}{420\left(1-v\right)}\frac{h^2}{ab}\left(\frac{b}{a}{m}^2+\frac{a}{b}{n}^2\right)\right] \) |
and consequently, the relative thickness is governed by the ratio of the plate thickness to its shorter side.
Results and discussion
In determining the range of relative thickness that is applicable for the shear deformation theories, one may refer to Steele and Balch (2009) who classified the plate thickness into four categories: (i) a/h > 100, (ii) 20 < a/h < 100, (iii) 3 < a/h < 20, and (iv) a/h < 3. This implies that one may then adopt the membrane theory for h/a < 0.01, CPT for h/a < 0.05, shear deformation theories for h/a < 0.3333, and elasticity theory for h/a > 0.3333. It therefore follows that the TSDT-based shear correction factor for FSDT problems are therefore applicable for relative thickness range of h/a < 0.3333. As such, the following results were computed for relative thickness up to 0.2 since shear deformation theories are not applicable for relative thickness of 1/3 and above. As with the CPT and FSDT, the TSDT is applicable for auxetic materials since the development of these theories are not confined to cases where Poisson’s ratio is positive.
Uniform load
Sinusoidal load
Comparison with other cases
Summary of shear correction factors of simply supported and uniformly loaded square and triangular plates with h/a = 0.2
Schematics of plates | Extremely auxetic material (v = − 1) | Typical conventional material (v = 0.3) | Incompressible material (v = 1/2) | Remarks |
---|---|---|---|---|
κ = 0.8351 | κ = 0.8384 | κ = 0.8405 | Isosceles right triangular plate (Lim 2016a) | |
κ = 0.8344 | κ = 0.8364 | κ = 0.8376 | Equilateral triangular plate (Lim 2016b) | |
κ = 0.8341 | κ = 0.8354 | κ = 0.8362 | Square plate (this paper) | |
κ = 0.8349 | κ = 0.8377 | κ = 0.8394 | Rectangular plate with aspect ratio of 4 (this paper) |
The descriptions of shear correction factors developed herein apply only for rectangular plates of simply supported boundary condition, and are therefore not applicable for thick rectangular plates of clamped and/or free edges. Nevertheless, it is worthy to note that the FSDT for Levy plates, as reviewed by Wang et al. (2000), adopts a shear correction factor of 5/6. Therefore, on this basis and on the basis of the results obtained from this paper, it can be said that the shear correction factor of 5/6 could well continue to be a tight lower bound and that Poisson’s ratio, alongside the plate’s relative thickness, exerts influence on the shear correction factor.
Conclusions
- (a)
Very thick plates
- (b)
Very long or narrow plates
- (c)
Plates made from large Poisson’s ratio (especially incompressible materials)
- (d)
Highly patterned loading pattern or sinusoidal load with high load waviness
- (a)
Marginally thick plates
- (b)
Square or almost square plates
- (c)
Plates made from auxetic materials, and
- (d)
Less wavy load pattern, especially uniform loads.
Nomenclature
A = Amplitude of plate deflection
a, b = In-plane dimensions of rectangular plate along x, y axes
D = Plate flexural rigidity
E = Young’s modulus
G = Shear modulus
h = Plate thickness
Μ = Marcus moment
m, n = Load waviness along x, y axes
q = Load intensity
R ^{2} = Coefficient of determination
w = Plate deflection
x, y = In-plane Cartesian coordinates
κ = Shear correction factor
v = Poisson’s ratio
Superscripts
K = Kirchhoff plate
M = Mindlin plate
R = Reddy plate
Subscript
0 = Maximum intensity for load.
Declarations
Competing interests
The author declares that he has no competing interests.
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
References
- Boldrin, L., Hummel, S., Scarpa, F., Di Maio, D., Lira, C., Ruzzene, M., Remillat, C. D. L., Lim, T. C., Rajasekaran, R., & Patsias, S. (2016). Dynamic behaviour of auxetic gradient composite hexagonal honeycombs. Composites Structures, 149, 114–124.View ArticleGoogle Scholar
- Chan, K. T., Lai, K. F., Stephen, N. G., & Young, K. (2011). A new method to determine the shear coefficient of Timoshenko beam theory. Journal of Sound and Vibration, 330(14), 3488–3497.View ArticleGoogle Scholar
- Dong, S. B., Alpdogan, C., & Taciroglu, E. (2010). Much ado about shear correction factors in Timoshenko beam theory. International Journal of Mechanical Sciences, 47(13), 1651–1655.MATHGoogle Scholar
- Han, S. M., Benaroya, H., & Wei, T. (1999). Dynamics of transversely vibrating beams using four engineering theories. Journal of Sound and Vibration, 225(5), 935–988.View ArticleMATHGoogle Scholar
- Hlavacek, I., & Chleboun, J. (2000). Reliable analysis of transverse vibrations of Timoshenko-Mindlin beams with respect to uncertain shear correction factor. Computer Methods in Applied Mechanics and Engineering, 190(8-10), 903–918.MathSciNetView ArticleMATHGoogle Scholar
- Hull, A. J. (2005). An exact analytical expression of the shear coefficient in the Mindlin plate equation. Journal of the Acoustical Society of America, 117(4), 2601–2601.View ArticleGoogle Scholar
- Hull, A. J. (2006). Mindlin shear coefficient determination using model comparison. Journal of Sound and Vibration, 294(1&2), 125–130.View ArticleGoogle Scholar
- Hutchinson, J. R. (1980). Vibrations of solid cylinders. ASME Journal of Applied Mechanics, 47(4), 901–907.View ArticleMATHGoogle Scholar
- Hutchinson, J. R. (2001). Shear coefficients for Timoshenko beam theory. ASME Journal of Applied Mechanics, 68(1), 87–92.View ArticleMATHGoogle Scholar
- Lee, K. H., Lim, G. T., & Wang, C. M. (2002). Thick Levy plates re-visited. International Journal of Solids and Structures, 39(1), 127–144.View ArticleMATHGoogle Scholar
- Lim, T. C. (2010). In-plane stiffness of semiauxetic laminates. ASCE Journal of Engineering Mechanics, 136(9), 1176–1180.View ArticleGoogle Scholar
- Lim, T. C. (2015a). Shear deformation in beams with negative Poisson’s ratio. IMechE Journal of Materials Design and Applications, 229(6), 447–454.Google Scholar
- Lim, T. C. (2015b). Auxetic Materials and Structures. Singapore: Springer.View ArticleGoogle Scholar
- Lim, T. C. (2016a). Refined shear correction factor for very thick simply supported and uniformly loaded isosceles right triangular auxetic plates. Smart Materials and Structures, 25(5), 054001.View ArticleGoogle Scholar
- Lim, T. C. (2016b). Higher order shear deformation of very thick simply-supported equilateral triangular plates under uniform load. Mechanics Based Design of Structures and Machines, 44(4), 514–522.View ArticleGoogle Scholar
- Lim, T. C. (2016c). Combined effect of load waviness and auxeticity on the shear deformation in a class of rectangular plates. IOP Conference Series: Materials Science and Engineering, 157, 012011.Google Scholar
- Lim, T. C. (2016d). Large deflection of circular auxetic membranes under uniform load. ASME Journal of Engineering Materials and Technology, 138(4), 041011.View ArticleGoogle Scholar
- Pai, P. F., Anderson, T. J., & Wheater, E. A. (2000). Large-deformation tests and total-Lagrangian finite-element analyses of flexible beams. International Journal of Solids and Structures, 37(21), 2951–2980.View ArticleMATHGoogle Scholar
- Pai, P. F., & Schultz, M. J. (1999). Shear correction factors and an energy-consistent beam theory. International Journal of Solids and Structures, 36(10), 1523–1540.View ArticleMATHGoogle Scholar
- Popescu, B., & Hodges, D. H. (2000). On asymptotically correct Timoshenko-like anisotropic beam theory. International Journal of Solids and Structures, 37(3), 535–558.MathSciNetView ArticleMATHGoogle Scholar
- Puchegger, S., Bauer, S., Loidl, D., Kromp, K., & Peterlik, H. (2003). Experimental validation of the shear correction factor. Journal of Sound and Vibration, 261(1), 177–184.View ArticleMATHGoogle Scholar
- Rössle, A. (1999). On the derivation of an asymptotically correct shear correction factor for the Reissner-Mindlin plate mode. Comptes Rendus de l'Académie des Sciences - Series I – Mathematics, 328(3), 269–274.View ArticleMATHGoogle Scholar
- C. R. Steele, C. D. Balch (2009) Introduction to the Theory of Plates. Stamford University. http://web.stanford.edu/~chasst/Course%20Notes/Introduction%20to%20the%20Theory%20of%20Plates.pdf. Accessed 24 October 2016.
- Stephen, N. G. (1997). Mindlin plate theory: best shear coefficient and higher spectra validity. Journal of Sound and Vibration, 202(4), 539–553.View ArticleGoogle Scholar
- Ting, T. C. T. (2005). Poisson's ratio for anisotropic elastic materials can have no bounds. Quarterly Journal of Mechanics and Applied Mathematics, 58(1), 73–82.MathSciNetView ArticleMATHGoogle Scholar
- Wang, C. M., Reddy, J. N., & Lee, K. H. (2000). Shear Deformable Beams and Plates. Oxford: Elsevier.MATHGoogle Scholar
- Yu, W., & Hodges, D. H. (2004). Elasticity solutions versus asymptotic sectional analysis of homogeneous, isotropic, prismatic beams. ASME Journal of Applied Mechanics, 71(1), 15–23.View ArticleMATHGoogle Scholar