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 2
πrσrhsinθ produced by the membrane force
σrh that is acting on the boundary
r, as shown in
Figure 2, where
σr 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,
It can be found from
Figure 2 that the slope at any point on the deflection curve can be expressed as tan
θ = −d
w/d
r. Therefore, it can be recalled from trigonometric functions that
Therefore, after eliminating sin
θ and tan
θ from Equations (1) and (2), the out-of-plane equilibrium equation may finally be written as
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
σt, then the in-plane equilibrium equation may be written as [
31,
32]
If the radial strain is denoted by
er, the circumferential strain by
et, 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)
and
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]
and
Equations (3) through (8) are six equations for solving
σr,
σt,
er,
et,
u, and
w, and the solution process is as follows. Eliminating
er and
et from Equations (5) through (8) yields
and
From Equations (9) and (10), the radial stress
σr and circumferential stress
σt can be expressed as
and
Furthermore, eliminating
σt from Equations (4) and (10), one has
After substituting the
u from Equation (13) into Equation (11), the consistency equation may be written as:
Equations (3), (4) and (14) are three equations for solving
σr,
σt, and
w, and the boundary conditions to determine the specific solutions of
σr,
σt, and
w are
and
After introducing the following dimensionless variables
Equations (3), (4), (13) through (17) can be transformed into the following dimensionless forms
and
In view of the stress and displacement being finite within
α ≤
x ≤ 1, after introducing
β = (1 +
α)/2,
Sr, and
W can be expanded into the power series in powers of
x −
β, i.e., letting
and
Further, after introducing
X =
x −
β, Equations (19), (22), (26) and (27) can be transformed into
and
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
ci and
di can be obtained. The resulting recurrence formulas for
ci (
i = 2, 3, 4, …) and
di (
i = 1, 2, 3, …) are listed in
Appendix A.
From
Appendix A, it can be seen that the coefficients
ci (
i = 2, 3, 4, …) and
di (
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
and
Equations (32) and (33) contain only
c0 and
c1, because the coefficients
ci (
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
Therefore, the value of
d0 can also be determined, because the values of the coefficients
di (
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 ci (i = 0, 1, 2, 3, …) and di (i = 0, 1, 2, 3, …) can easily be determined, and hence the expressions of σr and w can be determined. As for the expressions of σt, er, et, and u, with the known expression of σr, the expression of σt can easily be determined by Equation (4), and with the known expressions of σr and σt, the expressions of er, et and u can also be easily determined by Equations (7), (8) and (10). The problem addressed here is thus solved analytically.