Steady Fluid–Structure Coupling Interface of Circular Membrane under Liquid Weight Loading: Closed-Form Solution for Differential-Integral Equations
Abstract
:1. Introduction
2. Membrane Equation and Its Solution
3. Results and Discussions
3.1. Comparison with the Well-Known Hencky Solution
3.2. Verification of Convergence of the Power Series Solution
4. Concluding Remarks
- i.
- The power series method is effective for the analytical solution to differential-integral equations.
- ii.
- When the amount of liquid applied onto the circular membrane is large enough, the difference between the solution presented in this paper and the well-known Hencky solution will become relatively small. If the requirement for calculation accuracy is not too high, the problem of axisymmetric deformation of the circular membrane under liquid self-weight loading may be treated as the well-known Hencky problem; the fluid–structure interaction may be neglected.
- iii.
- When the amount of liquid applied onto the circular membrane is relatively small, the solution presented in this paper will be quite different from the well-known Hencky solution. For a higher calculation accuracy, the fluid–structure interaction should be taken into account.
- iv.
- The numerical example conducted shows that the closed-form solution obtained in this paper has good convergence.
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
Nomenclatures
a | Radius of the circular membrane |
h | Thickness of the circular membrane |
E | Young’s modulus of elasticity |
ν | Poisson’s ratio |
H | Height of the liquid poured into the cup |
ρ | Density of the poured liquid |
g | Acceleration of gravity |
r | Radial coordinate of the cylindrical coordinate system |
φ | Circumferential coordinate of the cylindrical coordinate system |
w | Axial coordinate of the cylindrical coordinate system as well as transverse displacement of a point on the deflected circular membrane |
u | Radial displacement of a point on the deflected circular membrane |
wm | Maximum deflection of the deflected circular membrane |
q(r) | Transverse distributed loads over the circular membrane produced by the gravity of the liquid within radius r |
F(r) | External force produced by q(r) within radius r |
σr | Radial stress |
σt | Circumferential stress |
er | Radial strain |
et | Circumferential strain |
θ | Slope angle of the deflected membrane |
π | Pi (ratio of circumference to diameter) |
W | Dimensionless transverse displacement (w/a) |
Sr | Dimensionless radial stress (σr/E) |
St | Dimensionless circumferential stress (σt/E) |
H0 | Dimensionless height of liquid (H/a) |
G | Dimensionless quantity (ρga2/Eh) |
x | Dimensionless radial coordinate (r/a) |
ci | Coefficients of the power series for Sr |
di | Coefficients of the power series for W |
Appendix A
Appendix B
References
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H | Maximum Deflection [mm] | Maximum Radial Stress [MPa] | ||||
---|---|---|---|---|---|---|
Presented Solution | Hencky Solution | Errors | Presented Solution | Hencky Solution | Errors | |
0.5 | 0.8169 | 0.6089 | 25.5% | 0.0184 | 0.0091 | 50.8% |
50 | 2.8733 | 2.7977 | 2.6% | 0.2005 | 0.1911 | 4.7% |
200 | 4.5161 | 4.5013 | 0.3% | 0.4969 | 0.4947 | 0.4% |
(a) The values of ci at different n | ||||
n | c0 | c2 | c4 | c6 |
4 | 2.5216650 × 10−2 | −4.4711868 × 10−3 | −4.5313948 × 10−4 | - |
6 | 2.5484214 × 10−2 | −4.3775807 × 10−3 | −4.2827528 × 10−4 | −7.5232361 × 10−5 |
8 | 2.5550954 × 10−2 | −4.3544054 × 10−3 | −4.2226218 × 10−4 | −7.3567411 × 10−5 |
10 | 2.5569445 × 10−2 | −4.3479638 × 10−3 | −4.2060234 × 10−4 | −7.3110446 × 10−5 |
12 | 2.5574842 × 10−2 | −4.3460754 × 10−3 | −4.2011702 × 10−4 | −7.2977069 × 10−5 |
14 | 2.5576464 × 10−2 | −4.3455055 × 10−3 | −4.1997075 × 10−4 | −7.2936900 × 10−5 |
16 | 2.5576960 × 10−2 | −4.3453305 × 10−3 | −4.1992585 × 10−4 | −7.2924572 × 10−5 |
18 | 2.5577114 × 10−2 | −4.3452760 × 10−3 | −4.1991189 × 10−4 | −7.2920739 × 10−5 |
20 | 2.5577163 × 10−2 | −4.3452589 × 10−3 | −4.1990751 × 10−4 | −7.2919536 × 10−5 |
22 | 2.5577178 × 10−2 | −4.3452535 × 10−3 | −4.1990612 × 10−4 | −7.2919156 × 10−5 |
24 | 2.5577183 × 10−2 | −4.3452518 × 10−3 | −4.1990568 × 10−4 | −7.2919035 × 10−5 |
(b) The values of ci at different n | ||||
n | c8 | c10 | c12 | c14 |
8 | −1.5488504 × 10−5 | - | - | - |
10 | −1.5357045 × 10−5 | −3.5823498 × 10−6 | - | - |
12 | −1.5318738 × 10−5 | −3.5710160 × 10−6 | −8.9013785 × 10−7 | - |
14 | −1.5307207 × 10−5 | −3.5676061 × 10−6 | −8.8910802 × 10−7 | −2.3216086 × 10−7 |
16 | −1.5303669 × 10−5 | −3.5665600 × 10−6 | −8.8879213 × 10−7 | −2.3206398 × 10−7 |
18 | −1.5302569 × 10−5 | −3.5662349 × 10−6 | −8.8869395 × 10−7 | −2.3203386 × 10−7 |
20 | −1.5302224 × 10−5 | −3.5661328 × 10−6 | −8.8866314 × 10−7 | −2.3202441 × 10−7 |
22 | −1.5302115 × 10−5 | −3.5661005 × 10−6 | −8.8865339 × 10−7 | −2.3202143 × 10−7 |
24 | −1.5302080 × 10−5 | −3.5660903 × 10−6 | −8.8865029 × 10−7 | −2.3202047 × 10−8 |
(c) The values of ci at different n | ||||
n | c16 | c18 | c20 | c22 |
16 | −6.2711514 × 10−8 | - | - | - |
18 | −6.2702167 × 10−8 | −1.7397329 × 10−8 | - | - |
20 | −6.2699234 × 10−8 | −1.7396410 × 10−8 | −4.9288360 × 10−9 | - |
22 | −6.2698306 × 10−8 | −1.7396119 × 10−8 | −4.9287442 × 10−9 | −1.4203606 × 10−9 |
24 | −6.2698011 × 10−8 | −1.7396027 × 10−8 | −4.9287150 × 10−9 | −1.4203513 × 10−9 |
(d) The value of c24 at n = 24 | ||||
n | c24 | - | - | - |
24 | −4.1511468 × 10−10 | - | - | - |
(a) The values of di at different n | ||||
n | d0 | d2 | d4 | d6 |
4 | 1.4389897 × 10−1 | −1.3373387 × 10−1 | −1.0165102 × 10−2 | - |
6 | 1.4380527 × 10−1 | −1.3232658 × 10−1 | −9.7095072 × 10−3 | −1.7991832 × 10−3 |
8 | 1.4373309 × 10−1 | −1.3197584 × 10−1 | −9.5986248 × 10−3 | −1.7643172 × 10−3 |
10 | 1.4368872 × 10−1 | −1.3187818 × 10−1 | −9.5679741 × 10−3 | −1.7547332 × 10−3 |
12 | 1.4367240 × 10−1 | −1.3184954 × 10−1 | −9.5590098 × 10−3 | −1.7519351 × 10−3 |
14 | 1.4366673 × 10−1 | −1.3184090 × 10−1 | −9.5563083 × 10−3 | −1.7510923 × 10−3 |
16 | 1.4366481 × 10−1 | −1.3183824 × 10−1 | −9.5554791 × 10−3 | −1.7508337 × 10−3 |
18 | 1.4366417 × 10−1 | −1.3183741 × 10−1 | −9.5552212 × 10−3 | −1.7507533 × 10−3 |
20 | 1.4366395 × 10−1 | −1.3183716 × 10−1 | −9.5551403 × 10−3 | −1.7507281 × 10−3 |
22 | 1.4366388 × 10−1 | −1.3183707 × 10−1 | −9.5551147 × 10−3 | −1.7507201 × 10−3 |
24 | 1.4366386 × 10−1 | −1.3183705 × 10−1 | −9.5551066 × 10−3 | −1.7507175 × 10−3 |
(b) The values of di at different n | ||||
n | d8 | d10 | d12 | d14 |
8 | −3.9431453 × 10−4 | - | - | - |
10 | −3.9128051 × 10−4 | −9.6550465 × 10−5 | - | - |
12 | −3.9039607 × 10−4 | −9.6267818 × 10−5 | −2.5251202 × 10−5 | - |
14 | −3.9012984 × 10−4 | −9.6182775 × 10−5 | −2.5223804 × 10−5 | −6.8962634 × 10−6 |
16 | −3.9004815 × 10−4 | −9.6156686 × 10−5 | −2.5215401 × 10−5 | −6.8935385 × 10−6 |
18 | −3.9002276 × 10−4 | −9.6148576 × 10−5 | −2.5212789 × 10−5 | −6.8926917 × 10−6 |
20 | −3.9001479 × 10−4 | −9.6146031 × 10−5 | −2.5211969 × 10−5 | −6.8924259 × 10−6 |
22 | −3.9001227 × 10−4 | −9.6145227 × 10−5 | −2.5211710 × 10−5 | −6.8923419 × 10−6 |
24 | −3.9001147 × 10−4 | −9.6144970 × 10−5 | −2.5211627 × 10−5 | −6.8923151 × 10−6 |
(c) The values of di at different n | ||||
n | d16 | d18 | d20 | d22 |
16 | −1.9420161 × 10−6 | - | - | - |
18 | −1.9417401 × 10−6 | −5.5954589 × 10−7 | - | - |
20 | −1.9416535 × 10−6 | −5.5951754 × 10−7 | −1.6411662 × 10−7 | - |
22 | −1.9416261 × 10−6 | −5.5950858 × 10−7 | −1.6411368 × 10−7 | −4.8827855 × 10−8 |
24 | −1.9416174 × 10−6 | −5.5950573 × 10−7 | −1.6411274 × 10−7 | −4.8827547 × 10−8 |
(d) The value of d24 at n = 24 | ||||
n | d24 | - | - | - |
24 | −1.4698279 × 10−8 | - | - | - |
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Li, X.; Sun, J.-Y.; Lu, X.-C.; Yang, Z.-X.; He, X.-T. Steady Fluid–Structure Coupling Interface of Circular Membrane under Liquid Weight Loading: Closed-Form Solution for Differential-Integral Equations. Mathematics 2021, 9, 1105. https://doi.org/10.3390/math9101105
Li X, Sun J-Y, Lu X-C, Yang Z-X, He X-T. Steady Fluid–Structure Coupling Interface of Circular Membrane under Liquid Weight Loading: Closed-Form Solution for Differential-Integral Equations. Mathematics. 2021; 9(10):1105. https://doi.org/10.3390/math9101105
Chicago/Turabian StyleLi, Xue, Jun-Yi Sun, Xiao-Chen Lu, Zhi-Xin Yang, and Xiao-Ting He. 2021. "Steady Fluid–Structure Coupling Interface of Circular Membrane under Liquid Weight Loading: Closed-Form Solution for Differential-Integral Equations" Mathematics 9, no. 10: 1105. https://doi.org/10.3390/math9101105
APA StyleLi, X., Sun, J.-Y., Lu, X.-C., Yang, Z.-X., & He, X.-T. (2021). Steady Fluid–Structure Coupling Interface of Circular Membrane under Liquid Weight Loading: Closed-Form Solution for Differential-Integral Equations. Mathematics, 9(10), 1105. https://doi.org/10.3390/math9101105