Improved Power Series Solution of Transversely Loaded Hollow Annular Membranes: Simultaneous Modification of Out-of-Plane Equilibrium Equation and Radial Geometric Equation

Abstract

The ability to accurately predict the shape of a transversely loaded hollow annular membrane is essential to the design of bending-free hollow annular shells of revolution, which requires a further improvement in the hollow annular membrane solution to meet the needs of this accurate prediction. In this paper, the large deflection problem of a transversely loaded hollow annular membrane is reformulated by simultaneously modifying the out-of-plane equilibrium equation and radial geometric equation, and a newer and more refined power series solution is derived. The reason why the classical radial geometry equation induces errors is revealed. The convergence and asymptotic behavior of the power series solution obtained is analyzed numerically. The newly derived solution is compared with the two previously derived solutions graphically, showing that the newly derived solution performs basically as well as expected. In addition, the anticipated use of the hollow and not-hollow annular membrane solutions for the design application of bending-free annular shells of revolution is discussed.

1. Introduction

Most membranes have the ability to exhibit large elastic deflections under lateral loading, which opens up the possibility for many technical applications such as blister tests [1,2,3], bulge tests [4,5,6], constrained blister tests [7,8,9,10], contact or non-contact capacitive pressure sensors [11,12,13,14], and especially deflection-based devices [15,16,17,18,19,20,21]. Essential to these technical applications is the ability to solve these problems of a large deflection of the membranes under lateral loading analytically and accurately. However, these large deflections often give rise to strong geometric nonlinearity, and these nonlinear differential equations often present analytical difficulties [22,23,24]. In the existing literature, annular membrane problems are less in evidence in comparison with circular membrane problems [25,26,27,28,29], but are often more difficult to deal with analytically due to the boundary conditions at the inner and outer edges which need to be satisfied simultaneously [30,31,32]. This paper is devoted to the further improvement in the analytical solution of a transversely loaded hollow annular membrane, aiming at providing a more accurate analytical solution for a specialized technical application of shell form-finding [33,34,35,36,37,38].

In an earlier work [31], we proposed the concept of a hollow annular membrane whose central empty region always keeps constant during transverse loading, for the determination of the shape (spatial geometry) of a hollow annular shell of revolution. Specifically, the inner edge of an annular membrane with a fixed outer edge is attached rigidly to a weightless stiff ring, to ensure that when the annular membrane is transversely loaded the central empty region within the inner edge always keeps constant and free from the action of the transverse loads during deflection, resulting in a hollow annular membrane structure. Similar to the hanging membrane method for shell designs proposed by Heinz Isler [36,37,38], the deflection shape of a transversely loaded hollow annular membrane, if inverted (flipped 180 degrees), can be used as the shape (spatial geometry) of, for instance, a concrete hollow annular shell of revolution. The hollow annular shell of revolution shaped by this method has only compression stresses, because the transversely loaded hollow annular membrane has only tension stresses. This is the so-called “shape-finding” or “form-finding” for bending-free shells [33,34,35,36,37,38]. Reducing the bending moment is very important for improving the stability of thin shells.

Before the concept of hollow annular membranes was proposed, only solid (not-hollow) annular membranes had been presented in the literature [30]. The so-called solid (not-hollow) annular membranes refer to the annular membrane structures whose outer edges are fixed and inner edges are attached rigidly to weightless stiff circular thin plates. The central region within the inner edge of a transversely loaded solid (not-hollow) annular membrane also always keeps constant during deflection, but is not empty and not free from the action of the transverse loads due to the weightless, stiff circular thin plate that is still subjected to the action of the transverse loads, and in particular is connected to the inner edge of the solid (not-hollow) annular membrane. Therefore, the difference between the hollow and solid (not-hollow) annular membranes is only that for the same outer and inner radius, the external load action withstood by a solid (not-hollow) annular membrane is greater than that withstood by its corresponding hollow annular membrane. This first suggests that under the same conditions (including outer and inner radius, transverse loads, etc.), the deflection shape of a solid (not-hollow) annular membrane is different from that of a hollow annular membrane. Secondly, it is very possible that by only adjusting the size of the loads applied, it is difficult to obtain two hollow and solid (not-hollow) annular membranes with the same geometric shape, if the other conditions are the same for the two annular membranes. This suggests that a hollow annular shell can be shaped only by a hollow annular membrane and never by a solid (not-hollow) annular membrane; otherwise, the original intention of “bending-free” will fall through due to the errors in geometric shape, and the reverse is also true. This is the reason why we proposed the concept of hollow annular membranes.

The closed-form solution of a hollow annular membrane under uniformly distributed transverse loads is derived originally by using an assumption of a small rotation angle of the membrane [31], while its second version [32] does not use the assumption of a small rotation angle of the membrane. The accuracy of the second version [32] has been greatly improved in comparison with the original version; however, as mentioned above, the shape errors have a significant influence on the shell form-finding. In other words, the accuracy of the closed-form solution of a transversely loaded hollow annular membrane determines the accuracy of the shell form-finding, and thus determines whether the bending moment will be present in the form-found hollow annular shell of revolution under service loads, which is very important for ensuring the stability of the shell. Although the finite element method is accurate and simple to obtain, it is not suitable for the shell form-finding where the structural parameters need to be adjusted arbitrarily. This is also the motivation for why this study further improves the power series solution of transversely loaded hollow annular membranes.

In this paper, two assumptions adopted in the classical radial geometric equation, which is used in the original and second versions [31,32], are further given up (see Section 3.1 for details), and a newer and more refined closed-form solution is presented. In the following section, the large deflection problem of a hollow annular membrane subjected to the action of uniformly distributed transverse loads is reformulated by simultaneously modifying the out-of-plane equilibrium equation and radial geometric equation, and is solved by using the power series method. In Section 3, some important issues are discussed, such as the reason why the classical radial geometric equation induces errors, the convergence of the power series solutions for stress and deflection obtained in Section 2, the asymptotic behavior from the obtained hollow annular membrane solution to the well-established circular membrane solution, the difference between hollow annular membrane solutions before and after improvement, and the difference in shell design between hollow and solid (not-hollow) annular membrane solutions. The concluding remarks are given in Section 4.

2. Membrane Equation and Solution

A transversely loaded hollow annular membrane structure is shown schematically in Figure 1. An initially flat annular membrane with outer radius a, inner radius b, thickness h, Poisson’s ratio v, and Young’s modulus of elasticity E, whose outer edge is rigidly fixed and inner edge is rigidly attached to a weightless stiff ring that is movable at the transverse direction perpendicular to the initially flat annular membrane, is subjected to a uniformly distributed transverse loads q, resulting in the transverse displacement (deflection) of the annular membrane. In Figure 1, the dash-dotted line represents the initially flat geometric middle plane of the annular membrane, the centroid of the initially flat geometric middle plane is coincident with the coordinate origin o of the introduced cylindrical coordinate system (r, φ, w), the polar coordinate plane (r, φ) is coincident with the plane in which the initially flat geometric middle plane is located, r is the radial coordinate variable, φ is the angle coordinate variable (not represented in Figure 1), w is the cylindrical coordinate variable (also representing the membrane deflections), and at the inner edge, the radial and circumferential displacement are rigidly constrained but the transverse displacement is not constrained.

The hollow annular membrane structures shown in Figure 1 differ from the traditional solid (not-hollow) annular membrane structures mainly because under the same inner radius b, outer radius a, thickness h, Young’s modulus of elasticity E, Poisson’s ratio v, and transverse loads q, the structural response of the former is smaller than that of the latter, because the annular membrane in the former is subjected to the action of π(a²−b²)q (since the central region within the inner edge of the former is empty and free from the action of the transverse loads q), while the annular membrane in the latter is subjected to the action of πa²q (since the inner edge of the latter is connected rigidly with a weightless, stiff circular thin plate that is still subjected to the action of the transverse loads q). This has a great influence on the out-of-plane equilibrium in the direction perpendicular to the polar plane (r, φ). The out-of-plane equilibrium equation may be established by taking a free body as shown in Figure 2.

A free body with radius r (b ≤ r ≤ a) is assumed to be taken out from the central region of the deflected annular membrane under the transverse loads q, to study its static problem of equilibrium under the joint action of the vertical (transverse) force π(r² − b²)q produced by the loads q and the vertical force 2πrσ_rhsinθ produced by the membrane force σ_rh that is acting on the boundary r, as shown in Figure 2, where σ_r denotes the radial stress and θ denotes the slope angle of the deflected annular membrane. When the free body is in equilibrium, the resultant force in the vertical direction should be equal to zero, that is,

$$2\pi r{\sigma }_{r}h\sin \theta -(\pi r^2-\pi b^2)q=0.$$

(1)

It can be found from Figure 2 that the slope at any point on the deflection curve can be expressed as tanθ = −dw/dr. Therefore, it can be recalled from trigonometric functions that

$$\sin \theta =1/\sqrt{1+1/{\tan }^2\theta }=1/\sqrt{1+1/{(-\mathrm{d}w/\mathrm{d}r)}^2}.$$

(2)

Therefore, after eliminating sinθ and tanθ from Equations (1) and (2), the out-of-plane equilibrium equation may finally be written as

$$[q^2{(r^2-b^2)}^2-4r^2{\sigma }_r^2{h}^2]{(-\frac{\mathrm{d}w}{\mathrm{d}r})}^2+q^2{(r^2-b^2)}^2=0.$$

(3)

The so-called “in-plane” refers to the plane parallel to the polar plane (r, φ), and the in-plane equilibrium equation may be established by taking out a wedge-shaped micro area element from the deflected annular membrane under the transverse loads q. If the circumferential stress is denoted by σ_t, then the in-plane equilibrium equation may be written as [31,32]

$$\frac{\mathrm{d}}{\mathrm{d}r}(r{\sigma }_r)-{\sigma }_t=0.$$

(4)

If the radial strain is denoted by e_r, the circumferential strain by e_t, and the radial displacement by u(r), then the geometric equations (the relationships between strain and displacement), which are used in [31,32] may be modified into the following forms (the detailed derivation is arranged in Section 3.1, to keep the formulation compact)

$$e_r={[{(1+\frac{\mathrm{d}u}{\mathrm{d}r})}^2+{(-\frac{\mathrm{d}w}{\mathrm{d}r})}^2]}^{1/2}-1$$

(5)

and

$$e_t=\frac{u}{r}.$$

(6)

Moreover, the physical equations (the relationships between stress and strain) are still assumed to satisfy generalized Hooke’s law and are given by [31,32]

$$e_r=\frac{1}{E}({\sigma }_r-\nu {\sigma }_t)$$

(7)

and

$$e_t=\frac{1}{E}({\sigma }_t-\nu {\sigma }_r).$$

(8)

Equations (3) through (8) are six equations for solving σ_r, σ_t, e_r, e_t, u, and w, and the solution process is as follows. Eliminating e_r and e_t from Equations (5) through (8) yields

$${[{(1+\frac{\mathrm{d}u}{\mathrm{d}r})}^2+{(-\frac{\mathrm{d}w}{\mathrm{d}r})}^2]}^{1/2}-1=\frac{1}{E}({\sigma }_r-\nu {\sigma }_t)$$

(9)

and

$$\frac{u}{r}=\frac{1}{E}({\sigma }_t-\nu {\sigma }_r).$$

(10)

From Equations (9) and (10), the radial stress σ_r and circumferential stress σ_t can be expressed as

$${\sigma }_r=\frac{E}{1-{\nu }^2}\{\nu \frac{u}{r}+{[{(1+\frac{\mathrm{d}u}{\mathrm{d}r})}^2+{(-\frac{\mathrm{d}w}{\mathrm{d}r})}^2]}^{1/2}-1\}$$

(11)

and

$${\sigma }_t=\frac{E}{1-{\nu }^2}\{\frac{u}{r}+\nu {[{(1+\frac{\mathrm{d}u}{\mathrm{d}r})}^2+{(-\frac{\mathrm{d}w}{\mathrm{d}r})}^2]}^{1/2}-\nu \}.$$

(12)

Furthermore, eliminating σ_t from Equations (4) and (10), one has

$$\frac{u}{r}=\frac{1}{E}[\frac{\mathrm{d}}{\mathrm{d}r}(r{\sigma }_r)-\nu {\sigma }_r].$$

(13)

After substituting the u from Equation (13) into Equation (11), the consistency equation may be written as:

$$\frac{\nu -1}{E}{\sigma }_r+\frac{\nu r}{E}\frac{\mathrm{d}{\sigma }_r}{\mathrm{d}r}+{\{{[1+\frac{(1-\nu )}{E}{\sigma }_r+\frac{(3-\nu )r}{E}\frac{\mathrm{d}{\sigma }_r}{\mathrm{d}r}+\frac{r^2}{E}\frac{{\mathrm{d}}^2{\sigma }_r}{\mathrm{d}{r}^2}]}^2+{(\frac{\mathrm{d}w}{\mathrm{d}r})}^2\}}^{1/2}-1=0.$$

(14)

Equations (3), (4) and (14) are three equations for solving σ_r, σ_t, and w, and the boundary conditions to determine the specific solutions of σ_r, σ_t, and w are

$$e_t=\frac{u}{r}=\frac{1}{E}[\frac{\mathrm{d}}{\mathrm{d}r}(r{\sigma }_r)-\nu {\sigma }_r]=0 \mathrm{at} r=b,$$

(15)

$$e_t=\frac{u}{r}=\frac{1}{E}[\frac{\mathrm{d}}{\mathrm{d}r}(r{\sigma }_r)-\nu {\sigma }_r]=0 \mathrm{at} r=a$$

(16)

and

$$w=0 \mathrm{at} r=a.$$

(17)

After introducing the following dimensionless variables

$$Q=\frac{aq}{hE},W=\frac{w}{a},S_r=\frac{{\sigma }_r}{E},S_t=\frac{{\sigma }_t}{E},\alpha =\frac{b}{a},x=\frac{r}{a},$$

(18)

Equations (3), (4), (13) through (17) can be transformed into the following dimensionless forms

$$[Q^2{(x^2-{\alpha }^2)}^2-4x^2{S}_r{}^2]{(-\frac{\mathrm{d}W}{\mathrm{d}x})}^2+Q^2{(x^2-{\alpha }^2)}^2=0,$$

(19)

$$\frac{\mathrm{d}(x{S}_r)}{\mathrm{d}x}-S_t=0,$$

(20)

$$\frac{u}{r}=(1-\nu )S_r+x\frac{\mathrm{d}{S}_r}{\mathrm{d}x},$$

(21)

$$(\nu -1)S_r+\nu x\frac{\mathrm{d}{S}_r}{\mathrm{d}x}+{\{{[1+(1-\nu )S_r+(3-\nu )x\frac{\mathrm{d}{S}_r}{\mathrm{d}x}+x^2\frac{{\mathrm{d}}^2{S}_r}{\mathrm{d}{x}^2}]}^2+{(\frac{\mathrm{d}W}{\mathrm{d}x})}^2\}}^{1/2}-1=0,$$

(22)

$$(1-\nu )S_r+x\frac{\mathrm{d}{S}_r}{\mathrm{d}x}=0 \mathrm{at} x=\alpha ,$$

(23)

$$(1-\nu )S_r+x\frac{\mathrm{d}{S}_r}{\mathrm{d}x}=0 \mathrm{at} x=1$$

(24)

and

$$W=0 \mathrm{at} x=1.$$

(25)

In view of the stress and displacement being finite within α ≤ x ≤ 1, after introducing β = (1 + α)/2, S_r, and W can be expanded into the power series in powers of x − β, i.e., letting

$$S_r={\displaystyle \sum _{i=0}^{\infty }{c}_i{(x-\beta )}^i}$$

(26)

and

$$W={\displaystyle \sum _{i=0}^{\infty }{d}_i{(x-\beta )}^i}.$$

(27)

Further, after introducing X = x − β, Equations (19), (22), (26) and (27) can be transformed into

$$\{Q^2{[{(X+\beta )}^2-{\alpha }^2]}^2-4{(X+\beta )}^2{S}_r{}^2\}{(-\frac{\mathrm{d}W}{\mathrm{d}X})}^2+Q^2{[{(X+\beta )}^2-{\alpha }^2]}^2=0,$$

(28)

$$\begin{array}{l}(\nu -1)S_r+\nu (X+\beta )\frac{\mathrm{d}{S}_r}{\mathrm{d}X}-1+\{[1+(1-\nu )S_r+(3-\nu )(X+\beta )\frac{\mathrm{d}{S}_r}{\mathrm{d}X}\\ +{(X+\beta )}^2\frac{{\mathrm{d}}^2{S}_r}{\mathrm{d}{X}^2}{]}^2+{(\frac{\mathrm{d}W}{\mathrm{d}X})}^2{\}}^{1/2}=0\end{array},$$

(29)

$$S_r={\displaystyle \sum _{i=0}^{\infty }{c}_i{X}^i}$$

(30)

and

$$W={\displaystyle \sum _{i=0}^{\infty }{d}_i{X}^i}.$$

(31)

By substituting Equations (30) and (31) into Equations (28) and (29) and letting all the coefficient sums of the X with like powers be equal to zero, a system of equations for determining the recurrence formulas for the power series coefficients c_i and d_i can be obtained. The resulting recurrence formulas for c_i (i = 2, 3, 4, …) and d_i (i = 1, 2, 3, …) are listed in Appendix A.

From Appendix A, it can be seen that the coefficients c_i (i = 2, 3, 4, …) and d_i (i = 1, 2, 3, …) can be expressed into the polynomials with regard to the first two coefficients c₀ and c₁, i.e., the so-called undetermined constants. The remaining one coefficient d₀ is another undetermined constant, but is a dependent one that is determined so long as the undetermined constants c₀ and c₁ are determined. By using the boundary conditions, Equations (23)–(25), the undetermined constants c₀, c₁, and d₀ can be determined as follows. Substituting Equation (26) into Equations (23) and (24) yields

$$(1-\nu ){\displaystyle \sum _{i=0}^{\infty }{c}_i{(\frac{\alpha -1}{2})}^i}+\alpha {\displaystyle \sum _{i=1}^{\infty }i{c}_i{(\frac{\alpha -1}{2})}^{i-1}}=0$$

(32)

and

$$(1-\nu ){\displaystyle \sum _{i=0}^{\infty }{c}_i{(\frac{1-\alpha }{2})}^i}+{\displaystyle \sum _{i=1}^{\infty }i{c}_i{(\frac{1-\alpha }{2})}^{i-1}}=0.$$

(33)

Equations (32) and (33) contain only c₀ and c₁, because the coefficients c_i (i = 2, 3, 4, …) can be expressed into the polynomials with regard to c₀ and c₁. Therefore, the values of c₀ and c₁ can be determined by solving Equations (32) and (33) simultaneously. Then, substituting Equation (27) into Equation (25) yields

$$d_0=-{\displaystyle \sum _{i=1}^{\infty }{d}_i{(\frac{1-\alpha }{2})}^i}.$$

(34)

Therefore, the value of d₀ can also be determined, because the values of the coefficients d_i (i = 1, 2, 3, …) can easily be determined with the known c₀ and c₁ (see Appendix A).

Finally, with the known c₀ and c₁, the coefficients c_i (i = 0, 1, 2, 3, …) and d_i (i = 0, 1, 2, 3, …) can easily be determined, and hence the expressions of σ_r and w can be determined. As for the expressions of σ_t, e_r, e_t, and u, with the known expression of σ_r, the expression of σ_t can easily be determined by Equation (4), and with the known expressions of σ_r and σ_t, the expressions of e_r, e_t and u can also be easily determined by Equations (7), (8) and (10). The problem addressed here is thus solved analytically.

3. Results and Discussions

3.1. The Reason Why the Classical Geometric Equations Induce Errors

Comparing the geometric equations used here (Equations (5) and (6) in this paper) and the classical geometric equations used in [31,32] (Equations (5) and (6) in [31,32]), it can be observed that only the radial geometric equation, Equation (5), is different, while the circumferential geometric equation, Equation (6), is the same. The reason why the classical radial geometry equation is modified in this paper is that the classical radial geometry equation can induce errors, which is analyzed as follows. The so-called geometric equations refer to the analytical relationships between displacements and strains. The radial geometry equation can be derived by taking out an infinitely small (micro) radial line element (assumed to be the straight line $\overline{AB}$ of length Δr in Figure 3) from the initially flat annular membrane. After deformation, the initially flat micro radial straight line element $\overline{AB}$ is assumed to become the micro meridional curve $\stackrel{⏜}{A^{\prime }{B}^{\prime }}$ in Figure 3.

Suppose that the radial and transverse displacements of the point A are denoted by u(r) and w(r), respectively. Then, the radial and transverse displacements of the point B may be denoted by u(r + Δr) and w(r + Δr), respectively. The u(r + Δr) and w(r + Δr) can be expanded into a Taylor series, and after ignoring the infinitely small higher order terms in Taylor expansion, they can be approximately expressed as the following forms

$$u(r+\Delta r)=u(r)+\frac{\mathrm{d}u(r)}{\mathrm{d}r}\Delta r+\frac{1}{2!}\frac{{\mathrm{d}}^{2}u(r)}{\mathrm{d}{r}^2}{(\Delta r)}^2+\dots \cong u(r)+\frac{\mathrm{d}u(r)}{\mathrm{d}r}\Delta r$$

(35)

and

$$w(r+\Delta r)=w(r)+\frac{\mathrm{d}w(r)}{\mathrm{d}r}\Delta r+\frac{1}{2!}\frac{{\mathrm{d}}^{2}w(r)}{\mathrm{d}{r}^2}{(\Delta r)}^2+\dots \cong w(r)+\frac{\mathrm{d}w(r)}{\mathrm{d}r}\Delta r.$$

(36)

Therefore, the geometric dimensions marked on Figure 3 can be obtained, where u(r) and w(r) are abbreviated as u and w, respectively.

The radial line strain from the micro radial straight line element $\overline{AB}$ to the micro meridional curve $\stackrel{⏜}{A^{\prime }{B}^{\prime }}$ (see Figure 3) may be written as, according to the definition of line strain

$$e_r=\frac{L_{\stackrel{⏜}{A^{\prime }{B}^{\prime }}}-L_{\overline{AB}}}{L_{\overline{AB}}}.$$

(37)

The classical radial geometric equation uses the assumption of $L_{\stackrel{⏜}{A^{\prime }{B}^{\prime }}}=L_{\overline{AB}}$, that is, the length change from the micro radial straight line element $\overline{AB}$ to the micro meridional curve $\stackrel{⏜}{A^{\prime }{B}^{\prime }}$ is ignored. Then, based on the assumption of $L_{\stackrel{⏜}{A^{\prime }{B}^{\prime }}}=L_{\overline{AB}}$, Equation (37) can be changed to

$$e_r=\frac{L_{\stackrel{⏜}{A^{\prime }{B}^{\prime }}}-L_{\overline{AB}}}{L_{\overline{AB}}}=\frac{(L_{\stackrel{⏜}{A^{\prime }{B}^{\prime }}}+L_{\overline{AB}})•(L_{\stackrel{⏜}{A^{\prime }{B}^{\prime }}}-L_{\overline{AB}})}{(L_{\stackrel{⏜}{A^{\prime }{B}^{\prime }}}+L_{\overline{AB}})•L_{\overline{AB}}}\approx \frac{L_{\stackrel{⏜}{A^{\prime }{B}^{\prime }}}^2-L_{\overline{AB}}^2}{(2L_{\overline{AB}})•L_{\overline{AB}}}=\frac{L_{\stackrel{⏜}{A^{\prime }{B}^{\prime }}}^2-L_{\overline{AB}}^2}{2L_{\overline{AB}}^2}.$$

(38)

Obviously, the length of the micro straight line $\overline{A^{\prime }{B}^{\prime }}$ can be used to approximately express the length of the micro curve $\stackrel{⏜}{A^{\prime }{B}^{\prime }}$ (see Figure 3)

$$L_{\stackrel{⏜}{A^{\prime }{B}^{\prime }}}\cong L_{\overline{A^{\prime }{B}^{\prime }}}=\sqrt{{(\Delta r+\frac{\mathrm{d}u}{\mathrm{d}r}\Delta r)}^2+{(-\frac{\mathrm{d}w}{\mathrm{d}r}\Delta r)}^2}.$$

(39)

Therefore, from Equations (37) and (38), the radial line strain e_r may be written as

$$e_r=\frac{L_{\stackrel{⏜}{A^{\prime }{B}^{\prime }}}-L_{\overline{AB}}}{L_{\overline{AB}}}\approx \frac{L_{\stackrel{⏜}{A^{\prime }{B}^{\prime }}}^2-L_{\overline{AB}}^2}{2L_{\overline{AB}}^2}=\frac{{(\Delta r+\frac{\mathrm{d}u}{\mathrm{d}r}\Delta r)}^2+{(-\frac{\mathrm{d}w}{\mathrm{d}r}\Delta r)}^2-{(\Delta r)}^2}{2{(\Delta r)}^2}=\frac{\mathrm{d}u}{\mathrm{d}r}+\frac{1}{2}{(\frac{\mathrm{d}u}{\mathrm{d}r})}^2+\frac{1}{2}{(-\frac{\mathrm{d}w}{\mathrm{d}r})}^2.$$

(40)

In Equation (40), the second term $\frac{1}{2}{(\frac{\mathrm{d}u}{\mathrm{d}r})}^2$ may be neglected in comparison with the first term $\frac{\mathrm{d}u}{\mathrm{d}r}$. Therefore, the radial line strain e_r may finally be written as

$$e_r=\frac{\mathrm{d}u}{\mathrm{d}r}+\frac{1}{2}{(-\frac{\mathrm{d}w}{\mathrm{d}r})}^2.$$

(41)

Equation (41) is the classical radial geometric equation used in [31,32] (see Equation (5) in [31,32]).

It can clearly be seen from the above derivation that there are two places where the classical geometric equations can induce errors. One is through ignoring the length change from the micro radial straight line element $\overline{AB}$ to the micro meridional curve $\stackrel{⏜}{A^{\prime }{B}^{\prime }}$, and the other is via neglecting the second term $\frac{1}{2}{(\frac{\mathrm{d}u}{\mathrm{d}r})}^2$ in Equation (40). In fact, if the assumption of $L_{\stackrel{⏜}{A^{\prime }{B}^{\prime }}}=L_{\overline{AB}}$ holds, then the radial line strain e_r should be equal directly to zero, so there is no need to derive Equation (38). On the other hand, it is hard to say whether $\frac{1}{2}{(\frac{\mathrm{d}u}{\mathrm{d}r})}^2$ can be neglected in comparison with $\frac{\mathrm{d}u}{\mathrm{d}r}$; they will probably be on the same order of magnitude. This is the reason why the classical radial geometry equation induces errors.

Incidentally, the radial geometry equation used in this paper is derived directly from Equations (37) and (39). Substituting Equation (39) into Equation (37) yields

$$e_r=\frac{L_{\stackrel{⏜}{A^{\prime }{B}^{\prime }}}-L_{\overline{AB}}}{L_{\overline{AB}}}\hspace{0.17em}=\frac{\sqrt{{(\Delta r+\frac{\mathrm{d}u}{\mathrm{d}r}\Delta r)}^2+{(-\frac{\mathrm{d}w}{\mathrm{d}r}\Delta r)}^2}-\Delta r}{\Delta r}=\sqrt{{(1+\frac{\mathrm{d}u}{\mathrm{d}r})}^2+{(-\frac{\mathrm{d}w}{\mathrm{d}r})}^2}-1.$$

(42)

This is the radial geometry equation used in this paper, i.e., Equation (5), where no assumptions are used here.

3.2. Convergence Analysis of Power Series Solutions

From Appendix A, it can be seen that the recursion formulas for the power series coefficients c_i and d_i are too complicated, so the convergences of the power series solutions S_r and W have to be analyzed numerically. To this end, an annular membrane with outer radius a = 20 mm, inner radius b = 5 mm, thickness h = 0.2 mm, Young’s modulus of elasticity E = 7.84 MPa, Poisson’s ratio ν = 0.47, and subjected to the actions of q = 0.0008 Mpa and 0.15 Mpa, is used as a numerical example to illustrate how to carry out a convergence analysis of the power series solutions S_r and W. Here, the relevant dimensionless quantities are α = b/a = 0.25, β = (1 + α)/2 = 0.625, and Q = aq/Eh = 0.01020408 (when q = 0.0008 Mpa) and 1.91326531 (when q = 0.15 Mpa).

To numerically carry out a convergence analysis, all the infinite power series in Equations (32)–(34) have to be truncated and replaced with their nth partial sums, i.e.,

$$(1-\nu ){\displaystyle \sum _{i=0}^n{c}_i{(\frac{\alpha -1}{2})}^i}+\alpha {\displaystyle \sum _{i=1}^{n}i{c}_i{(\frac{\alpha -1}{2})}^{i-1}}=0,$$

(43)

$$(1-\nu ){\displaystyle \sum _{i=0}^n{c}_i{(\frac{1-\alpha }{2})}^i}+{\displaystyle \sum _{i=1}^{n}i{c}_i{(\frac{1-\alpha }{2})}^{i-1}}=0$$

(44)

and

$$d_0=-{\displaystyle \sum _{i=1}^n{d}_i{(\frac{1-\alpha }{2})}^i}.$$

(45)

Then, assign values to the parameter n in Equations (43)–(45). Start with a small n value, say, n = 4. After eliminating c_i (i = 2, 3, 4) and d_i (i = 1, 2, 3, 4) in Equations (43)–(45) by using the recurrence formulas in Appendix A, only the undetermined constants c₀ and c₁ are left in Equations (43) and (44), while Equation (45) also contains only the undetermined constants c₀ and c₁ in addition to the undetermined constant d₀. Therefore, when n = 4, the values of the undetermined constants c₀ and c₁ can be determined by simultaneously solving Equations (43) and (44), and with the known c₀ and c₁, the value of the undetermined constant d₀ can easily be determined by Equation (45). Next, continue the calculation of numerical values of c₀, c₁, and d₀ with n = 5, and then with n = 6, …, until the calculated numerical values of c₀, c₁, and d₀ are sufficiently saturated, as shown in

Table 1. The calculated numerical values of c₀, c₁ and d₀ when q = 0.0008 MPa.

N	c0	c1	d0
4	0.01646666	−0.01138984	0.10267427
5	0.03249455	−0.01199211	0.04604041
6	0.01796667	−0.00813265	0.08935338
7	0.02232676	−0.00762124	0.06828286
8	0.01880637	−0.00680932	0.08100357
9	0.01998771	−0.00662006	0.07647870
10	0.01923017	−0.00651388	0.07894926
11	0.01970104	−0.00643829	0.07802335
12	0.01960104	−0.00635337	0.07870444
13	0.01955117	−0.00633490	0.07853201

and

Table 2. The calculated numerical values of c₀, c₁ and d₀ when q = 0.15 Mpa.

N	c0	c1	d0
4	0.79797871	−0.44917499	0.95536325
5	1.12954782	−0.40144873	0.34060805
6	0.88385455	−0.29134547	0.70383470
7	0.98804969	−0.26562043	0.50138603
8	0.92645754	−0.22996321	0.58732742
9	0.95209407	−0.22476126	0.54151308
10	0.94368571	−0.21505911	0.55933494
11	0.95103284	−0.21448360	0.54489530
12	0.94887115	−0.21384688	0.55189530
13	0.94926763	−0.21357301	0.55497352

. By the way, the calculation of numerical values of c₀, c₁ and d₀ is usually not easy due to the complex recursion formulas for the power series coefficients in Appendix A. Obviously, the greater the value of the parameter n in Equations (43)–(45), the greater the difficulty of the numerical calculations, that is, the greater the time consumption of labor and computer.

The degree to which the calculated numerical values of c₀, c₁, and d₀ are fully saturated are shown in Figure 4, Figure 5, Figure 6, Figure 7, Figure 8 and Figure 9, which are plotted using the data in Table 1 and Table 2. It can be seen from Figure 4, Figure 5, Figure 6, Figure 7, Figure 8 and Figure 9 that the undetermined constants c₀, c₁, and d₀ converge very well as the parameter n progressively increases. This suggests that the undetermined constants c₀, c₁, and d₀ can, for q = 0.0008 MPa, take 0.01955117, −0.00633490, and 0.07853201 (the saturated values when n = 13 in Table 1), respectively, and for q = 0.15 MPa, take 0.94926763, −0.21357301, and 0.55497352 (the saturated values when n = 13 in Table 2), respectively.

Now, we can use c₀ = 0.01955117, c₁ = −0.00633490, and d₀ = 0.07853201 to analyze the convergences of the power series solutions of stress and deflection when q = 0.0008 MPa, and use c₀ = 0.94926763, c₁ = −0.21357301, and d₀ = 0.55497352 to analyze the convergence of the power series solutions of stress and deflection when q = 0.15 MPa. The domain of the independent variable x is [α,1], i.e., [1/4,1]. Obviously, if the power series solutions converge at the two endpoints of the closed interval [1/4,1], they converge throughout the closed interval [1/4, 1]. Therefore, it is enough to analyze the convergences of the power series solutions of stress and deflection when x = α and x = 1.

Table 3. The calculated numerical values of c_i(1 − β)ⁱ and c_i(α − β)ⁱ when q = 0.0008 MPa.

i	ci(1 − β)i	ci(α − β)i
0	1.95511735 × 10−2	1.95511735 × 10−2
1	−1.37558825 × 10−2	1.37558825 × 10−2
2	4.29204384 × 10−3	4.29204384 × 10−3
3	−1.11033139 × 10−3	1.11033138 × 10−3
4	7.60061834 × 10−4	7.60061834 × 10−4
5	−5.72876516 × 10−4	5.72876515 × 10−4
6	3.88148544 × 10−4	3.88148544 × 10−4
7	−2.64333063 × 10−4	2.64333063 × 10−4
8	1.75184624 × 10−4	1.75184624 × 10−4
9	−1.15724068 × 10−4	1.15724067 × 10−4
10	7.56547054 × 10−5	7.56547054 × 10−5
11	−4.92739879 × 10−5	4.92739879 × 10−5
12	3.19117744 × 10−5	3.19117744 × 10−5
13	−2.05943388 × 10−5	2.05943387 × 10−5

and

Table 4. The calculated numerical values of d_i(1 − β)ⁱ and d_i(α − β)^I when q = 0.0008 MPa.

i	di(1 − β)i	di(α − β)i
0	7.85320100 × 10−2	7.85320100 × 10−2
1	−5.18652202 × 10−2	5.18652202 × 10−2
2	−2.51092895 × 10−2	−2.51092895 × 10−2
3	−8.95332430 × 10−3	8.95332430 × 10−3
4	−1.19523279 × 10−3	−1.19523279 × 10−3
5	−5.55755739 × 10−4	5.55755739 × 10−4
6	−8.24902204 × 10−5	−8.24902204 × 10−5
7	−3.32502705 × 10−5	3.32502705 × 10−5
8	−9.76735354 × 10−6	−9.76735354 × 10−6
9	−8.57965649 × 10−6	8.57965648 × 10−6
10	−7.80113119 × 10−6	−7.80113119 × 10−6
11	−5.55167631 × 10−6	5.55167631 × 10−6
12	−2.77842499 × 10−6	−2.77842499 × 10−6
13	−1.58075209 × 10−6	1.580752087 × 10−6

(for q = 0.0008 Mpa) and

Table 5. The calculated numerical values of c_i(1 − β)ⁱ and c_i(α − β)ⁱ when q = 0.15 MPa.

i	ci(1 − β)i	ci(α − β)i
0	9.49267634 × 10−1	9.49267634 × 10−1
1	−8.00898788 × 10−2	8.00898788 × 10−2
2	4.87742027 × 10−2	4.87742027 × 10−2
3	−1.53845039 × 10−2	1.53845039 × 10−2
4	6.33479721 × 10−3	6.33479721 × 10−3
5	−2.97981462 × 10−3	2.97981461 × 10−3
6	1.61656333 × 10−3	1.61656332 × 10−3
7	−1.50452190 × 10−4	1.50452190 × 10−4
8	7.85985875 × 10−4	7.85985875 × 10−4
9	−6.45125426 × 10−4	6.45125426 × 10−4
10	1.57480658 × 10−4	1.57480655 × 10−4
11	−9.60276002 × 10−5	9.60276002 × 10−5
12	4.81002110 × 10−5	4.81002110 × 10−5
13	−3.91803683 × 10−5	3.91803675 × 10−5

and

Table 6. The calculated numerical values of d_i(1 − β)ⁱ and d_i(α − β)ⁱ when q = 0.15 MPa.

i	di(1 − β)i	di(α − β)i
0	5.54973520 × 10−1	5.54973523 × 10−1
1	−2.33806340 × 10−1	2.33806340 × 10−1
2	−1.28213464 × 10−1	−1.28213464 × 10−1
3	−6.29533088 × 10−2	6.29533087 × 10−2
4	−2.56029287 × 10−2	−2.56029287 × 10−2
5	−9.38273495 × 10−3	9.38273495 × 10−3
6	−1.91732946 × 10−3	−1.91732946 × 10−3
7	−1.56968420 × 10−3	1.56968419 × 10−3
8	−1.34252446 × 10−3	−1.34252446 × 10−3
9	−6.19190966 × 10−4	6.19190966 × 10−4
10	−4.07197509 × 10−4	−4.07197509 × 10−4
11	−1.92578525 × 10−4	1.92578535 × 10−4
12	−9.27655317 × 10−5	−9.27655302 × 10−5
13	−8.84091866 × 10−5	8.84091878 × 10−5

(for q = 0.15 Mpa) show the calculated numerical values of the c_i(1 − β)^I, c_i(α − β)ⁱ, and d_i(1 − β)ⁱ, d_i(α − β)ⁱ in the expressions of stress and deflection solutions when x = 1 and x = α (see Equations (26) and (27)). The variations in c_i(1–β)^I, c_i(α − β)ⁱ, d_i(1 − β)ⁱ, and d_i(α − β)ⁱ with i are shown in Figure 10, Figure 11, Figure 12 and Figure 13 (for q = 0.0008 MPa) and Figure 14, Figure 15, Figure 16 and Figure 17 (for q = 0.15 MPa). It can be seen from Figure 10, Figure 11, Figure 12 and Figure 13 and Figure 14, Figure 15, Figure 16 and Figure 17 that the power series solutions of stress and deflection converge at the two endpoints of the closed interval [1/4,1] for both q = 0.0008 MPa and q = 0.15 MPa. This suggests that when the uniformly distributed transverse loads q is between 0.0008 MPa and 0.15 MPa, the power series solutions of stress and deflection are convergent and usable.

3.3. Asymptotic Behavior from Annular Membrane Solution to Circular Membrane Solution

The reliability of the hollow annular membrane solution obtained in Section 2 can be verified in the following way. Obviously, the shape of an annular membrane with inner radius b and outer radius a can gradually approach the shape of a circular membrane with radius a as the inner radius b of the annular membrane gradually approaches zero, if the membrane thickness h, Poisson’s ratio v, Young’s modulus of elasticity E, and the uniformly distributed transverse loads q are the same. This can be used to check the reliability of the hollow annular membrane solution derived in Section 2. Figure 18 and Figure 19 show the asymptotic behavior when the uniformly distributed transverse loads q take 0.0008 MPa and 0.15 MPa: as the inner radius b of the hollow annular membrane with Young’s modulus of elasticity E = 7.84 MPa, Poisson’s ratio v = 0.47, thickness h = 0.2 mm, and outer radius a = 20 mm goes from 15 mm to 10 mm, then to 5 mm, and finally to 2 mm, its deflection curves gradually approach the deflection curve of the circular membrane with Young’s modulus of elasticity E = 7.84 MPa, Poisson’s ratio v = 0.47, thickness h = 0.2 mm and radius a = 20 mm. In Figure 18 and Figure 19, “Solution 1” refers to the annular membrane results calculated by using the hollow annular membrane solution derived in Section 2 in this paper. “Solution 2” refers to the circular membrane result calculated by using the circular membrane solution derived in [39]. Therefore, the fully asymptotic behaviors shown in Figure 18 and Figure 19 suggest that the hollow annular membrane solution derived in Section 2 is, to a certain extent, reliable.

The usually so-called large deflection problems refer to the case where the maximum deflections of thin plates or membranes are greater than 1/5 of their thicknesses, and conversely, the case where the maximum deflections of thin plates or membranes are smaller than 1/5 of their thicknesses is usually called small deflection problems. The large deflection problems usually show effects of nonlinearity between loads and deflections while the small deflection problems usually show effects of linearity. Figure 20 shows the relationship between loads q and maximum deflections w_m of the hollow annular membrane with E = 7.84 MPa, v = 0.47, a = 20 mm, b = 5 mm, and h = 0.2 mm. It can be seen from the enlarged view in Figure 20 that when w_m < h/5 = 0.04 mm, the effect of linearity predominates, and when w_m > h/5 = 0.04 mm, the effect of nonlinearity gradually predominates. In fact, such a effect from linear dominance to nonlinear dominance is not only true in thin-plate or membrane problems, but also in beam problems [40].

3.4. Comparison between Hollow Annular Membrane Solutions before and after Improvement

The deflection profiles of the hollow annular membrane with Young’s modulus of elasticity E = 7.84 MPa, Poisson’s ratio v = 0.47, thickness h = 0.2 mm, outer radius a = 20 mm, inner radius b = 15 mm, 10 mm, and 5 mm, and subjected to the action of q = 0.15 MPa are calculated by using the hollow annular membrane solutions before and after improvement, and compared with each other, as shown in Figure 21, where “Solution 1” refers to the results calculated by using the new hollow annular membrane solution derived in Section 2 in this paper, and ”Solution 3” refers to the results calculated by using the earlier hollow annular membrane solution derived in [32], and “Solution 4” refers to the results calculated by using the earliest hollow annular membrane solution derived in [31].

It can be seen from Figure 21 that the differences between the deflection profiles gradually increase as the inner radius b decreases from 15 mm to 10 mm, and then to 5 mm. This suggests that the previous improvement to the out-of-plane equilibrium equation (see reference [32] for details) has played a role in making the difference between “Solution 3” and “Solution 4”, while the difference between “Solution 1” and “Solution 3” is mainly due to the current improvement to the radial geometric equation in this paper. These differences, even subtle differences, can not be ignored if the hollow annular membrane solution is used for designing the so-called bending-free shells [33,34,35,36,37,38].

3.5. Difference in Shell Design between Hollow and Solid Annular Membrane Solutions

According to the hanging membrane method for shell designs proposed by Heinz Isler [36,37,38], the shape of a transversely loaded annular membrane, after inversion (flipped 180 degrees), can be used as the shape of, for instance, a concrete annular shell of revolution. The annular shell of revolution shaped by this method has only compression stresses, because the transversely loaded annular membrane has only tension stresses. This is the so-called “shape-finding” or “form-finding” for bending-free shells. Therefore, the accuracy of an annular membrane solution has a great influence on the shell form-finding, involving whether the form-found shell really has no bending moments under service loads.

Obviously, the choice of using a hollow annular membrane solution or using a solid (not-hollow) annular membrane solution has a greater, more obvious, and more significant effect on the shell form-finding, and the larger the inner radius b of the annular membrane, the greater the effect, involving the amount or size of the effective loads acting on the annular membrane. In the solid (not-hollow) annular membrane, its inner edge is connected rigidly with a weightless stiff circular thin plate that is still under the transverse loads q, while in the hollow annular membrane, the central region within its inner edge is empty and free from the action of the transverse loads q. Therefore, under the same conditions, the mechanical response of the hollow annular membrane is smaller than that of the corresponding solid (not-hollow) annular membrane.

Figure 22 depicts the cross sections along the diameter of the inverted hollow and solid (not-hollow) annular membranes (represents the form-found annular shells), where the two annular membranes have the same Young’s modulus of elasticity E = 7.84 MPa, Poisson’s ratio v = 0.47, thickness h = 0.2 mm, outer radius a = 20 mm, inner radius b = 5 mm, maximum deflection w_m = 12 mm (it should be called the maximum height of shells, but for convenience, is still called the maximum deflection), but are subjected to the transverse loads q = 0.159373 MPa (hollow) and q = 0.125931 MPa (not-hollow), respectively. The case depicted in Figure 23 differs from the case depicted in Figure 22 only in b = 10 mm, w_m = 8 mm, q = 0.141723 MPa (hollow), and q = 0.062377 MPa (not-hollow), while the case depicted in Figure 24 differs from the case depicted in Figure 22 only in b = 15 mm, w_m = 4 mm, q = 0.232461 MPa (hollow), and q = 0.048782 MPa (not-hollow). In Figure 22, Figure 23 and Figure 24, “Solution 1” refers to the results calculated by using the new hollow annular membrane solution derived in Section 2 in this paper, and ”Solution 5” refers to the results calculated by using the solid (not-hollow) annular membrane solution derived in [41].

It can be seen from Figure 22, Figure 23 and Figure 24 that in terms of the loads required to produce the same maximum deflection w_m, the loads q applied to the hollow annular membrane is greater than the loads q applied to the solid (not-hollow) annular membrane, and the difference between the two load values increases as the inner radius b increases. This suggests that under the same conditions, the mechanical response of the hollow annular membranes is smaller than that of the corresponding solid (not-hollow) annular membranes, and the magnitude of the difference between the two depends on the magnitude of the inner radius b of the annular membranes.

On the other hand, however, the more important issue which can be observed from Figure 22, Figure 23 and Figure 24 is that for the same outer radius a, inner radius b, and maximum deflection w_m, the shape of the inverted hollow annular membrane can never be the same as the shape of the inverted solid (not-hollow) annular membrane. This reminds us that the hollow bending-free annular shells can be shaped only by the inverted transversely loaded hollow annular membrane, and never by the inverted transversely loaded solid (not-hollow) annular membrane; otherwise, the original intention of “bending-free” will fall through due to the errors in the shapes found (in other words, the bending moment will be present in the form-found annular shells under service loads). The reverse is also true. The solid (not-hollow) bending-free annular shell can only be shaped using the inverted transversely loaded solid (not-hollow) annular membrane, for example, the case of designing an annular shell of revolution with a glass roof for natural overhead light.

4. Concluding Remarks

In this paper, the large deflection problem of a transversely loaded hollow annular membrane is reformulated by simultaneously modifying out-of-plane equilibrium equation and radial geometric equation, and the newer and more refined power series solutions for stress and deflection are derived. From this study, the following conclusions can be drawn.

The classical radial geometric equation, Equation (41), is derived based on two unreasonable assumptions (see Section 3.1 for details), and therefore should be discarded and replaced by the newly derived radial geometry equation, Equations (5) or (42).

The newly derived power series solutions for stress and deflection can meet the convergence requirements and have higher accuracy than the old solutions, which further strengthens the theoretical basis for shell form-finding.

The hollow annular shells of revolution can be shaped only by the inverted transversely loaded hollow annular membranes, and never by the inverted transversely loaded solid (not-hollow) annular membranes, and the reverse is also true. Otherwise, the original intention of “bending-free” will fall through due to the errors in the found shapes.