Power series solution of circular membrane under uniformly distributed loads: investigation into Hencky transformation

Sun, Jun‐Yi; Yang, Rong; He, Xiao‐Ting; Gao, Xiao‐Wei; Zheng, Zhao

doi:10.12989/sem.2013.45.5.631

Cited by 20 publications

(50 citation statements)

References 16 publications

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“…Compared with our early works about Hencky problem , it may easily be seen that all the expressions obtained here for displacements, strains and stresses have the same form as those in Hencky solution. But the boundary conditions, under which Eqs.…”

Section: Resultsmentioning

confidence: 50%

“…, where θ is the slope angle. According to our earlier work , the equations of equilibrium including out‐plane and in‐plane may be written as

h σ_{r} \frac{d w}{d r} = - \frac{1}{2} r q

and

\frac{d}{d r} (r h σ_{r}) - h σ_{t} = 0 .

…”

Section: Membrane Equation and Its Power Series Solutionmentioning

confidence: 99%

“…According to our earlier work , the power series solution of Eq. may be written as

Z (x) = c^{- 4 / 3} (c x) f (c x),

where c is an undetermined integral constant and

f (x)

\begin{matrix} right f (x) & = & left 1 - \frac{1}{2} x - \frac{1}{6} x^{2} - \frac{13}{144} x^{3} - \frac{17}{288} x^{4} - \frac{37}{864} x^{5} - \frac{1205}{36288} x^{6} - \frac{219241}{8128512} x^{7} \\ left - \frac{6634069}{292626432} x^{8} - \frac{51523763}{2633637888} x^{9} - \frac{998796305}{57940033536} x^{10} - \frac{118156790413}{7648084426752} x^{11} - \dots \dots . \end{matrix}

…”

Section: Membrane Equation and Its Power Series Solutionmentioning

confidence: 99%

See 2 more Smart Citations

Closed‐form solution of elastic circular membrane with initial stress under uniformly‐distributed loads: Extended Hencky solution

Sun

Lian

et al. 2015

Z Angew Math Mech

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The well‐known Hencky solution is only applicable to the problem of deformation of the elastic circular membrane without initial stress under transverse uniformly‐distributed loads. The problem considered here is a more general case: an initial tensile or compressive stress has been present in the initially flat circular membrane before the membrane is subjected to the transverse loads. The closed‐form solution of the considered problem was presented and all the expressions obtained here for displacements, strains and stresses have the same form as those in the well‐known Hencky solution. The initial stress plays an important role in the determination of numerical value of the integral constant controlling membrane equation. The solution obtained here can be regressed into the well‐known Hencky solution when the initial stress is equal to zero, and it is therefore called extended Hencky solution.

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Section: Resultsmentioning

confidence: 50%

“…, where θ is the slope angle. According to our earlier work , the equations of equilibrium including out‐plane and in‐plane may be written as

h σ_{r} \frac{d w}{d r} = - \frac{1}{2} r q

and

\frac{d}{d r} (r h σ_{r}) - h σ_{t} = 0 .

…”

Section: Membrane Equation and Its Power Series Solutionmentioning

confidence: 99%

“…According to our earlier work , the power series solution of Eq. may be written as

Z (x) = c^{- 4 / 3} (c x) f (c x),

where c is an undetermined integral constant and

f (x)

\begin{matrix} right f (x) & = & left 1 - \frac{1}{2} x - \frac{1}{6} x^{2} - \frac{13}{144} x^{3} - \frac{17}{288} x^{4} - \frac{37}{864} x^{5} - \frac{1205}{36288} x^{6} - \frac{219241}{8128512} x^{7} \\ left - \frac{6634069}{292626432} x^{8} - \frac{51523763}{2633637888} x^{9} - \frac{998796305}{57940033536} x^{10} - \frac{118156790413}{7648084426752} x^{11} - \dots \dots . \end{matrix}

…”

Section: Membrane Equation and Its Power Series Solutionmentioning

confidence: 99%

See 1 more Smart Citation

Closed‐form solution of elastic circular membrane with initial stress under uniformly‐distributed loads: Extended Hencky solution

Sun

Lian

et al. 2015

Z Angew Math Mech

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“…For verifying the reliability of the closed-form solution presented here, we made a comparison between the solution presented here and the well-known Hencky solution [12,17], based on the numerical example of a polyethylene terephthalate thin-film (considered in [23]) with = 10 mm, ℎ = 0.003 mm, = 0.8 mm, = 4.65 GPa, and V = 0.34. When the contact radius is equal to 0.0001 mm, the action, the applied loads , is about 2.69 KPa.…”

Section: Resultsmentioning

confidence: 99%

“…The continuous condition at = and the boundary conditions at = will be applied during the process of solution. As for the case prior to the contact between the deflected membrane and the frictionless rigid plate, that is, the well-known Hencky problem, it has been dealt with and its solution may be found in [12,17].…”

Section: Membrane Equation and Its Solutionmentioning

confidence: 99%

Closed-Form Solution of a Peripherally Fixed Circular Membrane under Uniformly Distributed Transverse Loads and Deflection Restrictions

Wang

2018

Mathematical Problems in Engineering

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The problem of axisymmetric deformation of a peripherally fixed and uniformly loaded circular membrane under deflection restrictions (by a frictionless horizontal rigid plate) was analytically solved, where the assumption of constant membrane stress adopted in the existing work was given up, and a closed-form solution of this problem was presented for the first time. The numerical analysis shows that the closed-form solution presented here has higher calculation accuracy than the existing approximate solution.

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Closed-form solution of well-known Hencky problem without small-rotation-angle assumption

Lian

Sun

Yang

et al. 2016

Z. Angew. Math. Mech.

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In this paper, the well-known Hencky problem, the large deflection problem of axisymmetric deformation of uniformly loaded circular membranes, was resolved, where the small-rotation-angle assumption usually adopted in membrane problems was given up. The presented closed-form solution has a higher accuracy than well-known Hencky solution, a better understanding of the non-linear behavior of the considered problem could thus be reached. The presented numerical example shows that, the important integral constant controlling differential equations should change along with the increase of the applied transverse loads, but in well-known Hencky solution it becomes a constant due to the adopted small-rotation-angle assumption, resulting in the calculation error to increase, especially when the applied transverse loads is relatively large the well-known Hencky solution will no longer apply.

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Power series solution of circular membrane under uniformly distributed loads: investigation into Hencky transformation

Cited by 20 publications

References 16 publications

Closed‐form solution of elastic circular membrane with initial stress under uniformly‐distributed loads: Extended Hencky solution

Closed‐form solution of elastic circular membrane with initial stress under uniformly‐distributed loads: Extended Hencky solution

Closed-Form Solution of a Peripherally Fixed Circular Membrane under Uniformly Distributed Transverse Loads and Deflection Restrictions

Closed-form solution of well-known Hencky problem without small-rotation-angle assumption

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