Closed-form solution for a cantilevered sectorial plate subjected to a tip concentrated force
- Carl W. Christy^{1},
- David C. Weggel^{2}Email author and
- R. E. Smelser^{3}
Received: 18 November 2015
Accepted: 30 May 2016
Published: 21 June 2016
Abstract
A closed-form solution is presented for a cantilevered sectorial plate subjected to a tip concentrated force. Since the particular solution for this problem was not found in the literature, it is derived here. Deflections from the total solution (particular plus homogeneous solutions) are compared to those from a finite element analysis and are found to be in excellent agreement, producing an error within approximately 0.08 %. Normalized closed-form deflections and slopes at the fixed support, resulting from an approximate enforcement of the boundary conditions there, deviate from zero by <0.08 %. Finally, the total closed-form solutions for a cantilevered sectorial plate subjected to independent applications of a tip concentrated force, a tip bending moment, and a tip twisting moment, are compiled.
Keywords
Background
Solutions to sectorial plate problems have been investigated since the early twentieth century. Today, besides being of academic interest, these solutions are useful for verifying finite element analysis results, performing parametric studies, or assisting with preliminary designs. Timoshenko and Woinowsky-Krieger (1959) presented solutions for sectorial plates fixed (clamped) along the circular boundary and simply supported along the (straight) radial edges. However, they acknowledged that sectorial plate solutions containing clamped or free radial edges must be solved using approximate methods. Williams (1952a, b) presented solutions showing the stress singularities that develop due to various boundary conditions for plates in bending and extension; these solutions were developed using eigenfunction expansions.
Barber (1979) investigated the deflections of annular sectorial plates for some problems with concentrated moments and forces on a plate with a straight edge. He also provided a solution for twisting moments applied at the vertex of an infinite sectorial plate. Lim and Wang (2000) developed solutions for annular Mindlin sectorial plates using the Kirchhoff solutions, and Boonchareon et al. (2013) revisited the William’s problem for plate bending, providing solutions for problems that were previously unsolved. Huang et al. (2016) solved the infinite sectorial plate problem subjected to tip loads consisting of two twisting moments and a bending moment in a functionally graded plate, as well as concentrated forces in the plane of the plate; these solutions employed complex variable techniques.
Nadai (1925) provided particular solutions to a sectorial plate subjected to a tip twisting moment and a tip bending moment. These solutions are applicable to a plate of infinite radial extent with free radial edges, implying that any additional boundary conditions are located at a far distance from the plate’s tip. Carrier and Shaw (1950) applied an asymmetric eigenfunction expansion to Nadai’s particular solution for the tip twisting moment problem to account for fixed circumferential boundary conditions for a cantilevered sectorial plate; they enforced the fixed boundary conditions in a relaxed (averaged) manner. Kennedy et al. (2008) corrected mistakes in the presentation of Carrier and Shaw and compared their results to those from a finite element analysis; the agreement was found to be very good. Similarly, Christy et al. (2013) used Nadai’s particular solution for a sectorial plate subjected to a tip bending moment and applied a symmetric eigenfunction expansion for the fixed boundary conditions, similar to the procedure used by Carrier and Shaw (1950), to produce a closed-form deflection solution.
This paper presents the total solution for a cantilevered sectorial plate subjected to a tip concentrated force. However, the particular solution for this problem was not found in the literature; so it is first derived and presented in this paper. Then the particular solution is modified by a symmetric eigenfunction expansion and an “averaged” application of the fixed boundary conditions to produce the closed-form deflection solution. The solution is subsequently compared to the results from a finite element analysis.
Results and discussion
Existing particular solutions for tip twisting moment and tip bending moment
Nadai (1925) presented mixed coordinate solutions (i.e. a function of both Cartesian and polar coordinates) for an infinite sectorial plate subjected to a tip twisting moment and a tip bending moment. Equivalent equations, after being converted to polar coordinates, are presented below.
Derivation of the particular solution for a tip concentrated force
Solution for cantilevered sectorial plates
Example
Example constants
R | 800 mm (31.5 in.) |
α | 0.2 rad (11.44°) |
t | 6.35 mm (0.25 in.) |
E | 70 × 10^{3} MPa (10 × 10^{6} psi) |
ν | 0.33 |
P | 4.45 N (1 lb) |
M _{ b } | 110 N mm (1 lb in.) |
M _{ t } | 110 N mm (1 lb in.) |
Example coefficients
Loading | j | n _{ j } | b _{ j } | a _{ j } |
---|---|---|---|---|
P | 1 | 0.000 | 0.000 | 9.222 × 10^{−2} |
2 | 1.000 | 0.201 | −1.555 × 10^{−1} | |
3 | 7.515 | −0.731 | −2.796 × 10^{−3} | |
M _{ b } | 1 | 0.000 | 0.000 | 5.930 × 10^{−3} |
2 | 1.000 | 0.201 | −4.920 × 10^{−3} | |
3 | 7.515 | −0.731 | −8.140 × 10^{−5} | |
M _{ t } | 1 | 1.000 | −0.201 | 1.826 × 10^{−3} |
2 | 11.171 | −1.753 | −7.722 × 10^{−5} | |
3 | 13.409 | −1.056 | 1.176 × 10^{−5} |
Closed-form and numerical deflections at θ = +α
Normalized position (r/R) | Normalized closed-form deflection | Normalized numerical deflection |
---|---|---|
0.0 | 1.00000 × 10^{0} | 1.00078 × 10^{0} |
0.1 | 8.11252 × 10^{−1} | 8.11479 × 10^{−1} |
0.2 | 6.42049 × 10^{−1} | 6.41955 × 10^{−1} |
0.3 | 4.92393 × 10^{−1} | 4.92091 × 10^{−1} |
0.4 | 3.62288 × 10^{−1} | 3.61865 × 10^{−1} |
0.5 | 2.51757 × 10^{−1} | 2.51288 × 10^{−1} |
0.6 | 1.60864 × 10^{−1} | 1.60426 × 10^{−1} |
0.7 | 8.97613 × 10^{−2} | 8.94631 × 10^{−2} |
0.8 | 3.87709 × 10^{−2} | 3.87370 × 10^{−2} |
0.9 | 8.49806 × 10^{−3} | 8.84308 × 10^{−3} |
1.0 | −1.23480 × 10^{−9} | 0.00000 × 10^{0} |
Closed-form and numerical deflections at r = R/2
Normalized position (θ/α) | Normalized closed-form deflection | Normalized numerical deflection |
---|---|---|
1.00 | 2.51757 × 10^{−1} | 2.51288 × 10^{−1} |
0.92 | 2.50769 × 10^{−1} | 2.50302 × 10^{−1} |
0.84 | 2.49868 × 10^{−1} | 2.49402 × 10^{−1} |
0.76 | 2.49051 × 10^{−1} | 2.48588 × 10^{−1} |
0.68 | 2.48319 × 10^{−1} | 2.47862 × 10^{−1} |
0.60 | 2.47670 × 10^{−1} | 2.47211 × 10^{−1} |
0.52 | 2.47104 × 10^{−1} | 2.46647 × 10^{−1} |
0.44 | 2.46620 × 10^{−1} | 2.46170 × 10^{−1} |
0.36 | 2.46218 × 10^{−1} | 2.45769 × 10^{−1} |
0.28 | 2.45897 × 10^{−1} | 2.45443 × 10^{−1} |
0.20 | 2.45656 × 10^{−1} | 2.45205 × 10^{−1} |
0.12 | 2.45496 × 10^{−1} | 2.45053 × 10^{−1} |
0.04 | 2.45415 × 10^{−1} | 2.44966 × 10^{−1} |
−0.04 | 2.45415 × 10^{−1} | 2.44966 × 10^{−1} |
−0.12 | 2.45496 × 10^{−1} | 2.45053 × 10^{−1} |
−0.20 | 2.45656 × 10^{−1} | 2.45205 × 10^{−1} |
−0.28 | 2.45897 × 10^{−1} | 2.45443 × 10^{−1} |
−0.36 | 2.46218 × 10^{−1} | 2.45769 × 10^{−1} |
−0.44 | 2.46620 × 10^{−1} | 2.46170 × 10^{−1} |
−0.52 | 2.47104 × 10^{−1} | 2.46647 × 10^{−1} |
−0.60 | 2.47670 × 10^{−1} | 2.47211 × 10^{−1} |
−0.68 | 2.48319 × 10^{−1} | 2.47862 × 10^{−1} |
−0.76 | 2.49051 × 10^{−1} | 2.48588 × 10^{−1} |
−0.84 | 2.49868 × 10^{−1} | 2.49402 × 10^{−1} |
−0.92 | 2.50769 × 10^{−1} | 2.50302 × 10^{−1} |
−1.00 | 2.51757 × 10^{−1} | 2.51288 × 10^{−1} |
Conclusions
A closed-form solution for a finite cantilevered sectorial plate subjected to a tip concentrated force is presented. Since the particular solution was not found in the literature, it is derived in this paper. A symmetric eigenfunction expansion is used to augment the particular solution to account for the fixed boundary conditions at the circumferential support. Deflections from the total closed-from solution are found to be in excellent agreement with deflection results from a finite element analysis; the error is always within 0.08 % for the given example. Finally, the total closed-form solutions for a cantilevered sectorial plate subjected to independent applications of a tip concentrated force, a tip bending moment, and a tip twisting moment, are compiled.
Declarations
Authors’ contributions
DW and RS posed the problem and guided its solution. CC performed the detailed closed-form solution, created and ran the finite element model, acquired results to make comparisons, and drafted the manuscript. DW and RS assisted with drafting the manuscript. All authors read and approved the final manuscript.
Acknowledgements
The authors are grateful to Professor J.R. Barber of the University of Michigan, Ann Arbor, for his valuable guidance on deriving the particular solution of a sectorial plate subjected to a tip concentrated force.
Competing interests
The authors declare that they have 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
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