# Large Deflection Analysis of Axially Symmetric Deformation of Prestressed Circular Membranes under Uniform Lateral Loads

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## Abstract

**:**

## 1. Introduction

## 2. Membrane Equations and Closed-Form Solution

#### 2.1. Reformulation of the Generalized Föppl–Hencky Membrane Problem

#### 2.2. Power Series Solution

## 3. Results and Discussions

#### 3.1. Regression of the Solution Presented in Second Section

#### 3.2. Comparison with Existing Solutions

#### 3.3. Convergence of the Power Series Solution Obtained in Second Section

## 4. Concluding Remarks

## Author Contributions

## Funding

## Conflicts of Interest

## Nomenclature

$a$ | Radius of the circular membrane |

$h$ | Thickness of the circular membrane |

$E$ | Young’s modulus of elasticity |

$\nu $ | Poisson’s ratio |

$q$ | Uniformly distributed transverse loads |

$r$ | Radial coordinate |

$\phi $ | Circumferential coordinate |

$w$ | Transverse coordinate and transverse displacement of the deflected membrane |

$u$ | Radial displacement of the deflected membrane |

${u}_{0}$ | Radial plane displacement |

${\sigma}_{r}$ | Radial stress |

${\sigma}_{t}$ | Circumferential stress |

${e}_{r}$ | Radial strain |

${e}_{t}$ | Circumferential strain |

${\sigma}_{0}$ | Initial stress |

${e}_{0}$ | Initial strain |

$\theta $ | Slope angle of the deflected membrane |

$\pi $ | Pi (ratio of circumference to diameter) |

$Q$ | Dimensionless loads $(aq/hE)$ |

$W$ | Dimensionless transverse displacement $(w/a)$ |

${S}_{r}$ | Dimensionless radial stress $({\sigma}_{r}/E)$ |

${S}_{t}$ | Dimensionless circumferential stress $({\sigma}_{t}/E)$ |

${S}_{0}$ | Dimensionless initial stress $({\sigma}_{0}/E)$ |

$x$ | Dimensionless radial coordinate $(r/a)$ |

${b}_{i}$ | Coefficients of the power-series for ${S}_{r}$ |

${c}_{i}$ | Coefficients of the power-series for $W$ |

## Appendix A

## Appendix B

## References

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**Figure 7.**Distribution of ${b}_{i}$ with i when n = 26: (

**a**) $i=0,2,4,6,\dots ,26$ and (

**b**) $i=2,4,6,\dots ,26$.

**Figure 8.**Distribution of ${c}_{i}$ with i when n = 26: (

**a**) $i=0,2,4,6,\dots ,26$ and (

**b**) $i=2,4,6,\dots ,26$.

Loads q [MPa] | Solution 1 | Solution 2 | Solution 3 | Solution 4 | |
---|---|---|---|---|---|

${w}_{m}$ [mm] | 0.0001 | 0.47756 | 0.47743 | 0.47718 | 1.31183 |

0.003 | 3.69050 | 3.43552 | 3.34403 | 4.07615 | |

0.01 | 5.94663 | 5.49420 | 5.27261 | 6.05366 | |

${\sigma}_{m}$ [MPa] | 0.0001 | 0.10536 | 0.10534 | 0.10534 | 0.04202 |

0.003 | 0.44479 | 0.39752 | 0.36977 | 0.33981 | |

0.01 | 0.99162 | 0.86134 | 0.80763 | 0.78009 |

n | b_{0} | b_{2} | b_{4} | b_{6} |

4 | 1.20311 × 10^{−1} | −1.65094 × 10^{−2} | −3.36329 × 10^{−3} | ― |

6 | 1.23788 × 10^{−1} | −1.55680 × 10^{−2} | −2.95482 × 10^{−3} | −9.30349 × 10^{−4} |

8 | 1.25201 × 10^{−1} | −1.52080 × 10^{−2} | −2.80646 × 10^{−3} | −8.58923 × 10^{−4} |

10 | 1.25861 × 10^{−1} | −1.50439 × 10^{−2} | −2.74022 × 10^{−3} | −8.27699 × 10^{−4} |

12 | 1.26198 × 10^{−1} | −1.49612 × 10^{−2} | −2.70718 × 10^{−3} | −8.12280 × 10^{−4} |

14 | 1.26380 × 10^{−1} | −1.49168 × 10^{−2} | −2.68953 × 10^{−3} | −8.04083 × 10^{−4} |

16 | 1.26483 × 10^{−1} | −1.48918 × 10^{−2} | −2.67964 × 10^{−3} | −7.99504 × 10^{−4} |

18 | 1.26541 × 10^{−1} | −1.48776 × 10^{−2} | −2.67400 × 10^{−3} | −7.96896 × 10^{−4} |

20 | 1.26579 × 10^{−1} | −1.48686 × 10^{−2} | −2.67044 × 10^{−3} | −7.95252 × 10^{−4} |

22 | 1.26600 × 10^{−1} | −1.48635 × 10^{−2} | −2.66844 × 10^{−3} | −7.94329 × 10^{−4} |

24 | 1.26607 × 10^{−1} | −1.48616 × 10^{−2} | −2.66770 × 10^{−3} | −7.93987 × 10^{−4} |

26 | 1.26609 × 10^{−1} | −1.48612 × 10^{−2} | −2.66754 × 10^{−3} | −7.93915 × 10^{−4} |

n | b_{8} | b_{10} | b_{12} | b_{14} |

8 | −3.27991 × 10^{−4} | ― | ― | ― |

10 | −3.11944 × 10^{−4} | −1.33536 × 10^{−4} | ― | ― |

12 | −3.04098 × 10^{−4} | −1.29313 × 10^{−4} | −5.98144 × 10^{−5} | ― |

14 | −2.99948 × 10^{−4} | −1.27091 × 10^{−4} | −5.85763 × 10^{−5} | −2.86800 × 10^{−5} |

16 | −2.97637 × 10^{−4} | −1.25857 × 10^{−4} | −5.78906 × 10^{−5} | −2.82872 × 10^{−5} |

18 | −2.96323 × 10^{−4} | −1.25157 × 10^{−4} | −5.75019 × 10^{−5} | −2.80649 × 10^{−5} |

20 | −2.95495 × 10^{−4} | −1.24716 × 10^{−4} | −5.72576 × 10^{−5} | −2.79253 × 10^{−5} |

22 | −2.95030 × 10^{−4} | −1.24469 × 10^{−4} | −5.71206 × 10^{−5} | −2.78471 × 10^{−5} |

24 | −2.94858 × 10^{−4} | −1.24377 × 10^{−4} | −5.70698 × 10^{−5} | −2.78181 × 10^{−5} |

26 | −2.94822 × 10^{−4} | −1.24358 × 10^{−4} | −5.70592 × 10^{−5} | −2.78121 × 10^{−5} |

n | b_{16} | b_{18} | b_{20} | b_{22} |

16 | −1.44729 × 10^{−5} | ― | ― | ― |

18 | −1.43427 × 10^{−5} | −7.60266 × 10^{−6} | ― | ― |

20 | −1.42610 × 10^{−5} | −7.55386 × 10^{−6} | −4.12285 × 10^{−6} | ― |

22 | −1.42152 × 10^{−5} | −7.52654 × 10^{−6} | −4.10626 × 10^{−6} | −2.29738 × 10^{−6} |

24 | −1.41983 × 10^{−5} | −7.51642 × 10^{−6} | −4.10013 × 10^{−6} | −2.29360 × 10^{−6} |

26 | −1.41947 × 10^{−5} | −7.51431 × 10^{−6} | −4.09884 × 10^{−6} | −2.29281 × 10^{−6} |

n | b_{24} | b_{26} | ||

24 | −1.31103 × 10^{−6} | ― | ― | ― |

26 | −1.31054 × 10^{−6} | −7.63614 × 10^{−7} | ― | ― |

n | c_{0} | c_{2} | c_{4} | c_{6} |

4 | 3.01849 × 10^{−1} | −2.65045 × 10^{−1} | −3.68042 × 10^{−2} | ― |

6 | 3.00567 × 10^{−1} | −2.575995 × 10^{−1} | −3.32919 × 10^{−2} | −9.97580 × 10^{−3} |

8 | 2.99505 × 10^{−1} | −2.546935 × 10^{−1} | −3.19904 × 10^{−2} | −9.31291 × 10^{−3} |

10 | 2.98554 × 10^{−1} | −2.53357 × 10^{−1} | −3.14045 × 10^{−2} | −9.02080 × 10^{−3} |

12 | 2.97949 × 10^{−1} | −2.52681 × 10^{−1} | −3.11111 × 10^{−2} | −8.87599 × 10^{−3} |

14 | 2.97569 × 10^{−1} | −2.52317 × 10^{−1} | −3.09540 × 10^{−2} | −8.79885 × 10^{−3} |

16 | 2.97332 × 10^{−1} | −2.52112 × 10^{−1} | −3.08659 × 10^{−2} | −8.75572 × 10^{−3} |

18 | 2.97207 × 10^{−1} | −2.51995 × 10^{−1} | −3.08156 × 10^{−2} | −8.73114 × 10^{−3} |

20 | 2.97173 × 10^{−1} | −2.51921 × 10^{−1} | −3.07839 × 10^{−2} | −8.71564 × 10^{−3} |

22 | 2.97171 × 10^{−1} | −2.51879 × 10^{−1} | −3.07661 × 10^{−2} | −8.70693 × 10^{−3} |

24 | 2.97171 × 10^{−1} | −2.51864 × 10^{−1} | −3.07594 × 10^{−2} | −8.70370 × 10^{−3} |

26 | 2.97171 × 10^{−1} | −2.51861 × 10^{−1} | −3.07581 × 10^{−2} | −8.70303 × 10^{−3} |

n | c_{8} | c_{10} | c_{12} | c_{14} |

8 | −3.47991 × 10^{−3} | ― | ― | ― |

10 | −3.35337 × 10^{−3} | −1.41877 × 10^{−3} | ― | ― |

12 | −3.27724 × 10^{−3} | −1.37720 × 10^{−3} | −6.26249 × 10^{−4} | ― |

14 | −3.23690 × 10^{−3} | −1.35529 × 10^{−3} | −6.14040 × 10^{−4} | −2.93518 × 10^{−4} |

16 | −3.21441 × 10^{−3} | −1.34310 × 10^{−3} | −6.07272 × 10^{−4} | −2.89687 × 10^{−4} |

18 | −3.20161 × 10^{−3} | −1.33618 × 10^{−3} | −6.03433 × 10^{−4} | −2.87518 × 10^{−4} |

20 | −3.19355 × 10^{−3} | −1.33183 × 10^{−3} | −6.01020 × 10^{−4} | −2.86156 × 10^{−4} |

22 | −3.18902 × 10^{−3} | −1.32938 × 10^{−3} | −5.99666 × 10^{−4} | −2.85392 × 10^{−4} |

24 | −3.18734 × 10^{−3} | −1.32848 × 10^{−3} | −5.99164 × 10^{−4} | −2.85109 × 10^{−4} |

26 | −3.18699 × 10^{−3} | −1.32829 × 10^{−3} | −5.99059 × 10^{−4} | −2.85050 × 10^{−4} |

n | c_{16} | c_{18} | c_{20} | c_{22} |

16 | −1.43691 × 10^{−4} | ― | ― | ― |

18 | −1.42448 × 10^{−4} | −7.27170 × 10^{−5} | ― | ― |

20 | −1.41668 × 10^{−4} | −7.22652 × 10^{−5} | −3.77500 × 10^{−5} | ― |

22 | −1.41231 × 10^{−4} | −7.20122 × 10^{−5} | −3.76022 × 10^{−5} | −2.00212 × 10^{−5} |

24 | −1.41069 × 10^{−4} | −7.19185 × 10^{−5} | −3.75475 × 10^{−5} | −1.99890 × 10^{−5} |

26 | −1.41035 × 10^{−4} | −7.18989 × 10^{−5} | −3.75361 × 10^{−5} | −1.99822 × 10^{−5} |

n | c_{24} | c_{26} | ||

24 | −1.08172 × 10^{−5} | ― | ― | ― |

26 | −1.08132 × 10^{−5} | −5.93435 × 10^{−6} | ― | ― |

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**MDPI and ACS Style**

Li, X.; Sun, J.-Y.; Zhao, Z.-H.; He, X.-T.
Large Deflection Analysis of Axially Symmetric Deformation of Prestressed Circular Membranes under Uniform Lateral Loads. *Symmetry* **2020**, *12*, 1343.
https://doi.org/10.3390/sym12081343

**AMA Style**

Li X, Sun J-Y, Zhao Z-H, He X-T.
Large Deflection Analysis of Axially Symmetric Deformation of Prestressed Circular Membranes under Uniform Lateral Loads. *Symmetry*. 2020; 12(8):1343.
https://doi.org/10.3390/sym12081343

**Chicago/Turabian Style**

Li, Xue, Jun-Yi Sun, Zhi-Hang Zhao, and Xiao-Ting He.
2020. "Large Deflection Analysis of Axially Symmetric Deformation of Prestressed Circular Membranes under Uniform Lateral Loads" *Symmetry* 12, no. 8: 1343.
https://doi.org/10.3390/sym12081343