- Email: [email protected]
Nonlinear thin shell theories for numerical buckling predictions
Thin-Walled Structures 31 (1998) 89–115
Nonlinear thin shell theories for numerical buckling predictions J.G. Teng*, T. Hong Department of Civil and Structural Engineering, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong, China
Abstract A basic building block in any numerical (geometrically) nonlinear and buckling analysis is a set of nonlinear strain–displacement relations. A number of such relations have been developed in the past for thin shells. Most of these theories were developed in the pre-computer era for analytical studies when simplicity was emphasized and terms judged to be small relative to other terms were omitted. With the availability of greatly increased computing power in recent years, accuracy rather than simplicity is given more emphasis. Additional complexity in the strain–displacement relations leads to only a small increase in computational effort, but the omission of a term which may be important in only a few complex problems is a major flaw. It is therefore necessary to re-examine classical shell theories in the context of numerical nonlinear and buckling analysis. This paper first describes a set of nonlinear strain–displacement relations for thin shells of general form developed directly from the nonlinear theory of three-dimensional solids. In this new theory, all nonlinear terms, large and small, are retained. When specialized for thin shells of revolution, this theory reduces to that previously derived by Rotter and Jumikis and others. Analytical and numerical comparisons are carried out for thin shells of revolution between Rotter and Jumikis’ theory as a special case of the present theory and other commonly used nonlinear theories. The paper concludes with comments on the suitability of the various nonlinear shell theories discussed here for use in numerical buckling analysis of complex branched shells. 1998 Elsevier Science Ltd. All rights reserved. Keywords: Shells; Buckling; Theories; Stability
* Corresponding author. 0263-8231/98/$—see front matter 1998 Elsevier Science Ltd. All rights reserved. PII: S 0 2 6 3 - 8 2 3 1 ( 9 8 ) 0 0 0 1 4 - 7
90
J.G. Teng, T. Hong / Thin-Walled Structures 31 (1998) 89–115
Nomenclature A1, A2 E H1, H2, H3 n R R1, R2 s t u, v, w u, v, w ␣1, ␣2, ␣3 ␣, , → → →
␣, , ␣,  n ⑀
lame coefficients of curvilinear coordinates on shell reference surface elastic modulus lame coefficients coordinate normal to reference surface of shells of revolution radius of circumferential coordinate principal radii of curvature meridional coordinate for shells of revolution thickness of shell displacements in curvilinear coordinates displacements of a point on shell reference surface orthogonal curvilinear coordinates orthogonal curvilinear coordinates for a general shell unit vectors corresponding to coordinates ␣,  and rotations about a tangent to coordinate lines ␣ and  rotation about the normal to reference surface strains meridional angular coordinate for shells of revolution curvatures Poisson’s ratio circumferential angular coordinate for shells of revolution
Subscripts ᐉ nᐉ ␣ 
linear nonlinear along curvilinear coordinate ␣ along curvilinear coordinate  meridional circumferential normal to reference surface
1. Introduction Many attempts have been made to develop linear and nonlinear theories for thin shells. The first complete linear theory, based to some extent on Kirchhoff’s earlier work on plates, was given by Love [1]. This theory is often referred to as Love’s first approximation. Later linear theories of thin shells have been established by Donnell [2], Reissner [3], Sanders [4], Koiter [5], Novozhilov [6], and a theory appropriate for moderately thick shells was developed by Flugge [7]. Nonlinear and stability analyses are commonly based on nonlinear strain–displacement relations and corresponding equilibrium equations. The earliest work of some generality in this regard is Marguerre’s [8] nonlinear theory of shallow shells. Donnell [2]
J.G. Teng, T. Hong / Thin-Walled Structures 31 (1998) 89–115
91
developed an approximate theory specifically for cylinders and suggested its extension for a general shell (often referred to as the Donnell–Mushtari–Vlasov theory). Novozhilov [9] derived a general large deflection theory of thin shells, but did not go into details on further simplifications nor discuss corresponding equilibrium equations. Sanders [10] and Koiter [11] independently developed a nonlinear thin shell theory and showed that previous theories could be derived from theirs with appropriate further approximations. For shell problems which may be classified as being of small strains and moderately large rotations, the nonlinear thin shell theories of Donnell [2] and Sanders [10] have been the most widely used. These classical shell theories were developed in the pre-computer era for analytical studies when simplicity was emphasized and terms judged to be small relative to other terms were omitted. Nevertheless, Sanders’ theory [10] has been considered to be sufficiently accurate in most applications and implemented in many large shells of revolution computer codes for large deflection and buckling analysis (e.g. Ref. [12]). With the availability of greatly increased computing power in recent years, accuracy rather than simplicity is given more emphasis. Additional complexity in the strain–displacement relations leads to only a small increase in computational effort, but the omission of a term which may be important in only a few complex problems is a major flaw. Furthermore, unlike classical studies which have examined only shells of simple forms, a shell buckling computer code can be applied to problems of great complexity and of a range unintended by the code writer, consequently it is difficult to ensure that apparently small terms in most problems do not become significant in some other problems. It is therefore necessary to re-examine the classical shell theories in the new context of numerical buckling analysis. Many shell theories have been presented which include both kinematic (strain–displacement) and static (equilibrium) expressions, the present study is concerned only with the kinematic relations as the equilibrium equations are generally established using the virtual work principle in numerical analysis. Rotter and Jumikis appeared to be the first to consider this issue when they developed a new nonlinear shell theory for thin shells of revolution in the early eighties. This new theory together with interesting comparisons with other available theories is documented in the PhD thesis of Jumikis [13] and in Rotter and Jumikis [14]. It was found during the present study that the same relations were also arrived at independently by Su et al. [15] and Yin et al. [16] for thin shells of revolution. Whilst this coincidence is a little surprising, the real reason being that in this set of relations, all nonlinear terms, large or small, have been retained. These relations thus constitute a complete set of nonlinear strain–displacement relations for these shells in the sense that no term has been neglected based on a judgment of its relative magnitude. For convenience, these relations are referred to as Rotter and Jumikis’ theory. All these researchers derived these relations directly for thin shells of revolution, so a corresponding nonlinear thin shell theory has not been developed for shells of general form. This paper will present the results of a recent study on linear and nonlinear shell theories in numerical buckling analysis. A new set of nonlinear strain–displacement relations for thin shells of general form developed directly from the nonlinear elastic
92
J.G. Teng, T. Hong / Thin-Walled Structures 31 (1998) 89–115
theory of three-dimensional solids will be presented first. Shells of revolution have many more applications than most other shell forms, so the theory will then be specialized for thin shells of revolution to arrive at the same relations as those derived by Rotter and Jumikis [14] and others. Analytical and numerical comparisons will be carried out between the present theory and other commonly used linear and nonlinear theories to explore their differences as well as similarities. Particular attention is paid to the effect of using a different set of nonlinear strain–displacement relations on the predicted buckling loads.
2. Strain–displacement relations for thin shells of general form 2.1. Strain components in curvilinear coordinates for three dimensional solids In the nonlinear theory of elasticity, the expressions for the strain components in an arbitrary orthogonal coordinate system are [17]:
⑀11 ⫽ e11 ⫹
1 2 (e ⫹ e221 ⫹ e231) 2 11
⑀22 ⫽ e22 ⫹
1 2 (e ⫹ e212 ⫹ e232) 2 22
⑀33 ⫽ e33 ⫹
1 2 (e ⫹ e213 ⫹ e223) 2 33
(1)
⑀12 ⫽ ⑀21 ⫽ (e12 ⫹ e21) ⫹ e11e12 ⫹ e22e21 ⫹ e31e32 ⑀13 ⫽ ⑀31 ⫽ (e13 ⫹ e31) ⫹ e11e13 ⫹ e33e31 ⫹ e21e23 ⑀23 ⫽ ⑀32 ⫽ (e23 ⫹ e32) ⫹ e22e23 ⫹ e33e32 ⫹ e12e13 where e11 ⫽
1 ∂u 1 ∂H1 1 ∂H1 ⫹ v⫹ w H1 ∂␣1 H1H2 ∂␣2 H1H3 ∂␣3
e22 ⫽
1 ∂v 1 ∂H2 1 ∂H2 ⫹ w⫹ u H2 ∂␣2 H2H3 ∂␣3 H1H2 ∂␣1
e33 ⫽
1 ∂w 1 ∂H3 1 ∂H3 ⫹ u⫹ v H3 ∂␣3 H1H3 ∂␣1 H2H3 ∂␣2
e21 ⫽
1 ∂v 1 ∂H1 ⫺ u H1 ∂␣1 H1H2 ∂␣2
e12 ⫽
1 ∂u 1 ∂H2 ⫺ v H2 ∂␣2 H1H2 ∂␣1
and
(2)
J.G. Teng, T. Hong / Thin-Walled Structures 31 (1998) 89–115
e13 ⫽
1 ∂u 1 ∂H3 ⫺ w H3 ∂␣3 H1H3 ∂␣1
e31 ⫽
1 ∂w 1 ∂H1 ⫺ u H1 ∂␣1 H1H3 ∂␣3
e32 ⫽
1 ∂w 1 ∂H2 ⫺ v H2 ∂␣2 H2H3 ∂␣3
e23 ⫽
1 ∂v 1 ∂H3 ⫺ w H3 ∂␣3 H2H3 ∂␣2
93
(3)
In Eq. (1), ⑀ii (i ⫽ 1, 2, 3) are the direct strains and ⑀ij (i, j ⫽ 1, 2, 3; i⫽j) are the shear strains. Eq. (1) Eq. (2) Eq. (3) are the nonlinear relations between the strains ⑀ and the displacements u, v and w for a three-dimensional elastic solid. In these equations, H1, H2 and H3 are the Lame coefficients of the coordinate lines ␣1, ␣2 and ␣3, respectively. These relations may be specialized for thin shells by invoking appropriate assumptions. A curvilinear coordinate system is first established, which consists of two orthogonal curvilinear coordinates ␣ and  along the lines of principal curvature, → → . Let ␣ and  be the unit vectors along ␣ and , respectand a normal coordinate → ively, and the unit vector along the normal to the shell reference surface (usually the middle surface), and u, v and w be the components of the displacement vector in the corresponding orthogonal coordinates (Fig. 1). The Lame coefficients H1, H2 and H3 are now given by H1 ⫽ A1(1 ⫹ /R1); H2 ⫽ A2(1 ⫹ /R2); H3 ⫽ 1
(4)
where A1 and A2 are the Lame coefficients of the curvilinear coordinates on the
Fig. 1.
Shell reference surface and orthogonal curvilinear coordinates.
94
J.G. Teng, T. Hong / Thin-Walled Structures 31 (1998) 89–115
reference surface, and R1 and R2 are the principal radii of curvature. Substituting Eq. (4) into Eq. (2) Eq. (3) leads to
冉 冉
冊 冊
e␣␣ ⫽
1 1 ∂A1 w 1 ∂u ⫹ v⫹ 1 ⫹ /R1 A1 ∂␣ A1A2 ∂ R1
e ⫽
1 ∂v 1 1 ∂A2 w ⫹ u⫹ 1 ⫹ /R2 A2 ∂ A1A2 ∂␣ R2
e ⫽
∂w ∂
e␣ ⫽
1 1 ∂A1 1 ∂v ⫺ u 1 ⫹ /R1 A1 ∂␣ A1A2 ∂
e␣ ⫽
1 1 ∂A2 1 ∂u ⫺ v 1 ⫹ /R2 A2 ∂ A1A2 ∂␣
e␣ ⫽
∂u ∂
e␣ ⫽
1 ∂w 1 u ⫺ 1 ⫹ /R1 A1 ∂␣ R1
e ⫽
1 v 1 ∂w ⫺ 1 ⫹ /R2 A2 ∂ R2
e ⫽
∂v ∂
and
冉 冉
(5)
冊 冊 (6)
冉 冉
冊 冊
2.2. The Love–Kirchhoff assumptions To reduce the strain–displacement relations for three-dimensional solids to those for thin shells, it is necessary to invoke the well-known Love–Kirchhoff assumptions: 1. The thickness t of the shell is negligibly small in comparison with the least radius of curvature Rmin of the reference surface. 2. The displacements are sufficiently small for quantities of the second and higher orders to be neglected compared with those of the first order. 3. The stress component normal to the reference surface is small compared with the other stress components and may be neglected in the stress–strain relations. 4. Normals of the undeformed reference surface remain straight and normal to the deformed reference surface. 5. Normals to the reference surface undergo negligible change in length during deformation.
J.G. Teng, T. Hong / Thin-Walled Structures 31 (1998) 89–115
95
The first assumption of thin shells can be written as t/Rmin¿1
(7)
Assumption 2 means that one can retain only linear terms in the expression of strains. The use of this assumption leads to a linear thin shell theory. The third assumption relates to the constitutive relations for the shell, and defines the throughthickness stress as negligible, allowing equations for plane stress conditions to be used. That is
⫽ 0
(8)
Assumption 4 is analogous to the Bernoulli–Euler hypothesis of beam theory that “plane section remains plane”. The preserved straightness of normals allows transverse shear strains to be ignored, that is
␥␣ ⫽ ␥ ⫽ 0
(9)
The last assumption implies that strains normal to the reference surface are negligible, i.e.
⑀ ⫽ 0
(10)
2.3. Strain–displacement relations for general shells By the application of the Love–Kirchhoff assumptions, the displacement components are all assumed to vary linearly across the thickness and can be represented by the following relations: u(␣,,) ⫽ u(␣,) ⫹
∂u ∂
v(␣,,) ⫽ v(␣,) ⫹
∂v ∂
w(␣,,) ⫽ w(␣,) ⫹
(11)
∂w ∂
where the quantities u, v and w represent the components of the displacement vector of a point on the reference surface. Since the linear component of the normal strain is, in general, an order of magnitude larger than the nonlinear component, the fifth Love–Kirchhoff assumption is applied only to the linear component of the normal strain as was done by Rotter and Jumikis [14]. The following equation is then obtained:
⑀l ⫽ e ⫽
∂w ⫽0 ∂
(12)
96
J.G. Teng, T. Hong / Thin-Walled Structures 31 (1998) 89–115
Application of the Love–Kirchhoff assumption which requires transverse shear strains to be ignored only to the linear component of the transverse shear strain (Eq. (1) Eq. (6)) leads to e␣ ⫹ e␣ ⫽
冉
冊
1 u ∂u 1 ∂w ⫺ ⫹ ⫽0 1 ⫹ /R1 A1 ∂␣ R1 ∂
(13)
It is easy to see that the quantity ∂u/∂ represents the rotation of a tangent to the middle surface oriented along the coordinate line ␣. This rotation is defined as ␣ and given by
␣ ⫽
冉
冊
u ∂u 1 1 ∂w ⫽ ⫺ ∂ 1 ⫹ /R1 R1 A1 ∂␣
(14)
Similarly, , which represents the rotation of a tangent to the middle surface oriented along the coordinate line  is given by
 ⫽
冉
冊
∂v 1 1 ∂w v ⫽ ⫺ ∂ 1 ⫹ /R2 R2 A2 ∂
(15)
Substituting Eq. (12) Eq. (14) Eq. (15) into Eq. (11), the three displacement components become u(␣,,) ⫽ u(␣,) ⫹ ␣ v(␣,,) ⫽ v(␣,) ⫹ 
(16)
w(␣,,) ⫽ w(␣,) By substituting Eq. (16) into Eq. (5) Eq. (6), Eq. (5) Eq. (6) may be rewritten as e␣ ⫽ 1 ⫹ ␥1 e␣ ⫽ 2 ⫹ ␥2 e␣␣ ⫽ 3 ⫹ ␥3
(17)
e ⫽ 4 ⫹ ␥4 e ⫽ 5 ⫹ ␥5 e␣ ⫽ 6 ⫹ ␥6 where
1 ⫽ ⫺ ␣ ⫽ 2 ⫽
冉
冊 冊
1 ∂w u 1 ⫺ 1 ⫹ /R1 A1 ∂␣ R1
冉
1 1 ∂A1 1 ∂v ⫺ u 1 ⫹ /R1 A1 ∂␣ A1A2 ∂
J.G. Teng, T. Hong / Thin-Walled Structures 31 (1998) 89–115
3 ⫽
冉
1 ∂u w 1 1 ∂A1 ⫹ v⫹ 1 ⫹ /R1 A1 ∂␣ A1A2 ∂ R1
4 ⫽ ⫺  ⫽
冉
1 ∂w v 1 ⫺ 1 ⫹ /R2 A2 ∂ R2
冉 冉
冊
冊
5 ⫽
1 ∂v w 1 1 ∂A2 ⫹ u⫹ 1 ⫹ /R2 A2 ∂ A1A2 ∂␣ R2
6 ⫽
1 ∂u 1 1 ∂A2 ⫺ v 1 ⫹ /R2 A2 ∂ A1A2 ∂␣
冊
97
(18)
冊
and
␥1 ⫽ ⫺
1 ∂A1 1  1 ⫹ /R1 A1 ∂ ␣
冉
冊
␥2 ⫽
1 1 ∂A1 1 ∂ ⫺  1 ⫹ /R1 A1 ∂␣ A1A2 ∂ ␣
␥3 ⫽
1 1 ∂␣ 1 ∂A1 ( ⫹  ) 1 ⫹ /R1 A1 ∂␣ A1A2 ∂ 
␥4 ⫽ ⫺
1 ∂A2 1  1 ⫹ /R2 A2 ∂ 
冉 冉
(19)
冊 冊
␥5 ⫽
1 ∂ 1 1 ∂A2 ⫹  1 ⫹ /R2 A2 ∂ A1A2 ∂␣ ␣
␥6 ⫽
1 ∂␣ 1 1 ∂A2 ⫺  1 ⫹ /R2 A2 ∂ A1A2 ∂␣ 
In terms of Eq. (1) and Eq. (17) Eq. (18) Eq. (19), the linear part of the strain components can be written as
⑀␣ᐉ ⫽ 3 ⫹ ␥3 ⑀ᐉ ⫽ 5 ⫹ ␥5
(20)
⑀␣ᐉ ⫽ (2 ⫹ 6) ⫹ (␥2 ⫹ ␥6) The above equations are the same as the strain–displacement relations of Flugge’s theory [7]. These equations are appropriate for use in the analysis of moderately thick shells. For the analysis of thin shells, for which /Ri¿1 , Eq. (18) Eq. (19) are further simplified to
1 ⫽ ⫺ ␣ ⫽
1 ∂w u ⫺ A1 ∂␣ R1
98
J.G. Teng, T. Hong / Thin-Walled Structures 31 (1998) 89–115
2 ⫽
1 ∂v 1 ∂A1 ⫺ u A1 ∂␣ A1A2 ∂
3 ⫽
w 1 ∂u 1 ∂A1 ⫹ v⫹ A1 ∂␣ A1A2 ∂ R1
4 ⫽ ⫺  ⫽
(21)
v 1 ∂w ⫺ A2 ∂ R2
5 ⫽
1 ∂v 1 ∂A2 w ⫹ u⫹ A2 ∂ A1A2 ∂␣ R2
6 ⫽
1 ∂u 1 ∂A2 ⫺ v A2 ∂ A1A2 ∂␣
and
␥1 ⫽ ⫺
1 ∂A1  A1 ∂ ␣
␥2 ⫽
1 ∂ 1 ∂A1 ⫺  A1 ∂␣ A1A2 ∂ ␣
␥3 ⫽
1 ∂␣ 1 ∂A1 ⫹  A1 ∂␣ A1A2 ∂ 
␥4 ⫽ ⫺
(22)
1 ∂A2  A2 ∂ 
␥5 ⫽
1 ∂ 1 ∂A2  ⫹ A2 ∂ A1A2 ∂␣ ␣
␥6 ⫽
1 ∂␣ 1 ∂A2 ⫺  A2 ∂ A1A2 ∂␣ 
Substituting Eq. (17) into Eq. (1), the membrane strains of general thin shells are given by
⑀␣ ⫽ 3 ⫹
1 2 ( ⫹ 22 ⫹ 23) 2 1
⑀ ⫽ 5 ⫹
1 2 ( ⫹ 25 ⫹ 26) 2 4
⑀␣ ⫽ (2 ⫹ 6) ⫹ (14 ⫹ 25 ⫹ 36) and the curvatures are given by
␣ ⫽ ␥3 ⫹ (1␥1 ⫹ 2␥2 ⫹ 3␥3)
(23)
J.G. Teng, T. Hong / Thin-Walled Structures 31 (1998) 89–115
99
 ⫽ ␥5 ⫹ (4␥4 ⫹ 5␥5 ⫹ 6␥6)
(24)
␣ ⫽ (␥2 ⫹ ␥6) ⫹ (1␥4 ⫹ 2␥5 ⫹ 3␥6 ⫹ 4␥1 ⫹ 5␥2 ⫹ 6␥3) in which 1 苲 6 and ␥1 苲 ␥6 are found in Eq. (21) Eq. (22). It is now a simple matter to write out explicitly the linear and nonlinear membrane strains separately as follows:
⑀␣␣ᐉ ⫽ 3 ⫽
w 1 ∂u 1 ∂A1 ⫹ v⫹ A1 ∂␣ A1A2 ∂ R1
⑀ᐉ ⫽ 5 ⫽
w 1 ∂v 1 ∂A2 ⫹ u⫹ A2 ∂ A1A2 ∂␣ R2
⑀␣ᐉ ⫽ 2 ⫹ 6 ⫽
冉
(25)
冊 冉
1 ∂A1 1 ∂A2 1 ∂v 1 ∂u ⫺ u ⫹ ⫺ v A1 ∂␣ A1A2 ∂ A2 ∂ A1A2 ∂␣
冊
and
⑀␣nᐉ ⫽
1 2 ( ⫹ 22 ⫹ 23) 2 1
⑀nᐉ ⫽
1 2 ( ⫹ 25 ⫹ 26) 2 4
(26)
⑀␣nᐉ ⫽ (14 ⫹ 25 ⫹ 36) Similarly, the linear and nonlinear curvature terms can also be separately written:
␣␣ᐉ ⫽ ␥3 ⫽
1 ∂␣ 1 ∂A1 ⫹  A1 ∂␣ A1A2 ∂ 
ᐉ ⫽ ␥5 ⫽
1 ∂ 1 ∂A2  ⫹ A2 ∂ A1A2 ∂␣ ␣
␣ᐉ ⫽ ␥2 ⫹ ␥6 ⫽
冉
(27)
冊 冉
冊
1 ∂ 1 ∂ ␣ 1 ∂A1 1 ∂A2 ⫺ ⫺  ⫹  A1 ∂␣ A1A2 ∂ ␣ A2 ∂ A1A2 ∂␣ 
and
␣nᐉ ⫽ (1␥1 ⫹ 2␥2 ⫹ 3␥3) nᐉ ⫽ (4␥4 ⫹ 5␥5 ⫹ 6␥6)
(28)
␣nᐉ ⫽ (1␥4 ⫹ 2␥5 ⫹ 3␥6 ⫹ 4␥1 ⫹ 5␥2 ⫹ 6␥3) The significance of nonlinear curvatures will not be discussed here as they do not appear to be important for thin shells. Readers may consult Rotter and Jumikis [14] for a brief discussion.
100
J.G. Teng, T. Hong / Thin-Walled Structures 31 (1998) 89–115
3. Comparison with other theories for general thin shells 3.1. Comparison of linear strain components with other theories Since Love [1] proposed his well known first-approximation shell theory in 1888, a number of other theories have been presented. Those commonly used include the Donnell–Mushtari–Vlasov (DMV) theory [2], Reissner’s theory [3], Novozhilov’s theory [6], Sanders’ “best” first order theory [4,5,18], Flugge’s theory [7], Rotter– Jumikis’ theory [14]. All these theories except Flugge’s theory are for thin shells only. Flugge’s theory is suitable for moderately thick shells. The present linear relations (Eq. (20)), before the thinness assumption is invoked, are the same as those of Flugge’s theory. Introducing the thinness assumption by replacing 1 ⫹ /Ri (i ⫽ 1, 2) with 1, linear strain–displacement relations for thin shells only are obtained. The present theory (Eq. (25) Eq. (26)) is then the same as Reissner’s theory. Rotter and Jumikis’ relations are developed directly for thin shells of revolution. Applying the special conditions relevant to thin shells of revolution as shown later, the present relations reduce to the linear relations of Rotter and Jumikis. Novozhilov [6] gave a set of linear strain–displacement relations slightly different from the present set (only in the twisting curvature term) due to a slightly different imposition of the thinness assumption. Strain–displacement relations of the DMV theory are the simplest set available and are appropriate only for shallow shells. The expressions for the linear membrane strains given in the theories which are appropriate for thin shells only are the same. Differences between the different linear strain–displacement relations are therefore only found in the expressions for the curvatures and twisting curvature. The differences are highlighted in Table 1. The expressions for the curvatures and twisting curvature of the DMV theory differ from those of other theories in that contributions of the in-plane displacements are neglected. The linear theory of elastic thin shells proposed by Sanders has been considered to be the “best”. The linear strain–displacement relations of Sanders’ theory have one obvious advantage over those of other theories in that all strains vanish for small rigid-body motions of the shell. For all other theories, when rigid-body motions are introduced, the twisting curvature ␣ does not vanish except in the case of a spherical shell, a flat plate or a symmetrically loaded shell of revolution, so these theories may lead to significant errors in other cases. The present strain–displacement relations before the introduction of the thinness assumption predicts that no strains result from rigid-body motions. It can thus be seen that the nonvanishing twisting curvature from rigid-body motions in many thin shell theories is directly related to the thinness assumption. In practical applications, the different thin shell theories discussed above lead to only small differences for a few cases [19]. 3.2. Comparison of nonlinear strain components with other theories Nonlinear strain terms have to be included in a nonlinear or buckling analysis, as they are the only means by which destabilizing effects are introduced into the analysis. The main task of the present study is thus to examine the differences between
1 ∂w v − R2 A2 ∂
1 ∂w v − R2 A2 ∂
1 ∂w v − R2 A2 ∂
1 ∂w v − R2 A2 ∂
1 ∂w u − R1 A1 ∂␣
1 ∂w u − R1 A1 ∂␣
1 ∂w u − R1 A1 ∂␣
1 ∂w A2 ∂
−
1 ∂w A1 ∂ ␣
1 ∂w u − R1 A1 ∂␣
−

␣
(*) 2, 6 are given by Eq. (21).
Present
Sanders
Novozhilov
Reissner
DMV
Theory
冉
n
冊
1 ∂A2v ∂A1u − 2A1A2 ∂␣ ∂
Table 1 Comparison of curvatures and twisting curvature
1 ∂A1 1 ∂␣ +  A1 ∂␣ A1A2 ∂ 
1 ∂A1 1 ∂␣ +  A1 ∂␣ A1A2 ∂ 
1 ∂A1 1 ∂␣ +  A1 ∂␣ A1A2 ∂ 
1 ∂A1 1 ∂␣ +  A1 ∂␣ A1A2 ∂ 
1 ∂A1 1 ∂␣ +  A1 ∂␣ A1A2 ∂ 
␣␣ᐉ
1 ∂A2 1 ∂ +  A2 ∂ A1A2 ∂␣ ␣
1 ∂A2 1 ∂ +  A2 ∂ A1A2 ∂␣ ␣
1 ∂A2 1 ∂ +  A2 ∂ A1A2 ∂␣ ␣
1 ∂A2 1 ∂ +  A2 ∂ A1A2 ∂␣ ␣
1 ∂A2 1 ∂ +  A2 ∂ A1A2 ∂␣ ␣
ᐉ
冉 冉 冉
1 ∂A2 1 ∂␣ −  A2 ∂ A1A2 ∂␣ 
冊 冊
1 ∂A1 1 ∂ −  + A1 ∂␣ A1A2 ∂ ␣
1 1 − R1 R2
冊
1 ∂A2 1 ∂␣ −  − A2 ∂ A1A2 ∂␣ 
1 ∂A1 1 ∂ ␣ + − A1 ∂␣ A1A2 ∂
冊 冊
1 ∂A2 1 ∂␣ −  + A2 ∂ A1A2 ∂␣ 
冊 冊
冊 冊
1 ∂A1 1 ∂ −  + A1 ∂␣ A1A2 ∂ ␣
1 ∂A2 1 ∂␣  − A2 ∂ A1A2 ∂␣ 
1 ∂A1 1 ∂  + − A1 ∂␣ A1A2 ∂ ␣
1 ∂A2 1 ∂␣ −  A2 ∂ A1A2 ∂␣ 
n
冉 冉
冊 冊
1 ∂A1 1 ∂  + − A1 ∂␣ A1A2 ∂ ␣
6 2 + (*) R1 R2
冉 冉
冉 冉
冉 冉
␣ᐉ
J.G. Teng, T. Hong / Thin-Walled Structures 31 (1998) 89–115 101
102
J.G. Teng, T. Hong / Thin-Walled Structures 31 (1998) 89–115
the different nonlinear shell theories. Such a comparison was first made by Rotter and Jumikis [14]. Because Rotter and Jumikis developed their strain–displacement relations directly for thin shells of revolution, they made their comparison of the different relations for thin shells of revolution only. The comparison presented here attempts to shed new light on the differences between the nonlinear terms of membrane strains to which the calculated buckling loads are sensitive, in addition to reinforcing some of the conclusions reached by Rotter and Jumikis. The comparison is carried out here for general shells, with a more detailed discussion for shells of revolution later in the paper. The differences between the present theory and the widely used theories of Donnell [2] and Sanders [10] for general shells are highlighted in Table 2. Sanders’ nonlinear shell theory has been the most widely used in numerical nonlinear and buckling analysis of thin-shell structures. Sanders introduced the rotation about the normal to the reference surface n into the nonlinear strain–displacement relations. It will be shown in the following that the present theory for general shells reduces Table 2 Comparison of nonlinear membrane strains Theory
Present
Sanders
Donnell
⑀␣nᐉ
1 2 ( + 22 + 23) 2 1
1 2 ( + 2n) 2 ␣
1 2  2 ␣
⑀nᐉ
1 2 ( + 25 + 26) 2 4
1 2 ( + 2n) 2 
1 2  2 
⑀␣nᐉ
14 + 25 + 36
␣
␣
1 = − ␣
1 ∂w u − A1 ∂␣ R1
1 ∂w u − A1 ∂␣ R1
2 or n
1 ∂A1 1 ∂v − u A1 ∂␣ A1A2 ∂
1 ∂A2v ∂A1u − 2A1A2 ∂␣ ∂
3
1 ∂A1 1 ∂u w + v+ A1 ∂␣ A1A2 ∂ R1
–
–
4 = − 
1 ∂w v − A2 ∂ R2
1 ∂w v − A2 ∂ R2
1 ∂w A2 ∂
5
1 ∂A2 1 ∂v w + u+ A2 ∂ A1A2 ∂␣ R2
–
–
6
1 ∂A2 1 ∂u − v A2 ∂ A1A2 ∂␣
–
–
冉
1 ∂w A1 ∂␣
冊
–
J.G. Teng, T. Hong / Thin-Walled Structures 31 (1998) 89–115
103
to that of Sanders if the assumption is adopted that linear in-plane membrane strains are much smaller than out-of-plane rotations and rotation about the normal to the reference surface. A comparison of n (Table 2) with the pseudo-rotation terms defined in Eq. (21) reveals that
n ⫽
1 ( ⫺ 6) 2 2
(29)
2 ⫽
1 ( ⫹ 6) ⫹ n 2 2
(30)
6 ⫽
1 ( ⫹ 6) ⫺ n 2 2
so
From Eq. (23), it is clear that 2 ⫹ 6 is the linear membrane shear strain term ⑀␣ᐉ. In Sanders’ theory, n is the rotation about the normal to the reference surface. For those thin shell problems which may be classified as being of small strains and moderately large rotations, the linear in-plane membrane strains may be assumed to be much smaller than the rotation about the normal to the middle surface, that is
nÀ⑀␣l ⫽ (2 ⫹ 6)
(31)
Based on Eq. (31), Eq. (30) may be rewritten as
2 ⬵ n
(32)
6 ⬵ ⫺ n Substituting Eq. (32) into Eq. (23), the membrane strains of the present theory become
⑀␣ ⫽ ⑀␣␣ᐉ ⫹ ⑀␣nᐉ ⬵ 3 ⫹
1 2 ( ⫹ 2n ⫹ 23) 2 1
⑀ ⫽ ⑀ᐉ ⫹ ⑀nᐉ ⬵ 5 ⫹
1 2 ( ⫹ 25 ⫹ ( ⫺ n)2) 2 4
(33)
⑀␣ ⫽ ⑀␣ᐉ ⫹ ⑀␣nᐉ ⫽ (2 ⫹ 6) ⫹ (14 ⫹ 25 ⫹ 36) In Eq. (33), the linear strain terms 3 and 5 are small, so the square of 3 and 5 may be neglected. Similarly, (2 ⫹ 6), representing the linear membrane shear strain, is a small term, so (25 ⫹ 36) which has the order of the square of (2 ⫹ 6) can be neglected. With these simplifications, Eq. (33) becomes
104
J.G. Teng, T. Hong / Thin-Walled Structures 31 (1998) 89–115
⑀␣ ⫽ ⑀␣␣ᐉ ⫹ ⑀␣nᐉ ⬵ 3 ⫹
1 2 ( ⫹ 2n) 2 1
⑀ ⫽ ⑀ᐉ ⫹ ⑀nᐉ ⬵ 5 ⫹
1 2 ( ⫹ ( ⫺ n)2) 2 4
(34)
⑀␣ ⫽ ⑀␣ᐉ ⫹ ⑀␣nᐉ ⬵ (B2 ⫹ 6) ⫹ 14 The nonlinear membrane strains may thus be expressed as
⑀␣nᐉ ⫽
1 2 ( ⫹ 2n) 2 1
⑀nᐉ ⫽
1 2 ( ⫹ ( ⫺ n)2) 2 4
(35)
⑀␣nᐉ ⫽ 14 The above relations are identical to Sanders’ theory (Table 2). It is therefore concluded that, for shells undergoing moderately large rotations with small strains, the present theory for general shells can be simplified to that of Sanders if the assumption is adopted that linear in-plane membrane strains are much smaller than out-of-plane rotations and rotations about the normal to the reference surface.
4. Thin shells of revolution 4.1. Strain–displacement relations for thin shells of revolution For thin shells of revolution, the coordinate system is shown in Fig. 2, where → →
s , and → n are unit vectors in the meridional, circumferential and normal directions, respectively. The Lame coefficients now become
Fig. 2.
Geometry and displacements of shells of revolution.
J.G. Teng, T. Hong / Thin-Walled Structures 31 (1998) 89–115
A1 ⫽ R1; A2 ⫽ R
105
(36)
In this coordinate system, the following relations hold ds ⫽ R1d; R ⫽ R2 sin
(37)
Substitution of Eq. (36) and 37 into Eq. (21) and 22 leads to
1 ⫽ ⫺  ⫽
∂w ∂ ∂v ⫺u ; 2 ⫽ ∂s ∂s ∂s
3 ⫽
∂u ∂ 1 ∂w ⫺ v sin ] ⫹w ; 4 ⫽ ⫺  ⫽ [ ∂s ∂s R ∂
5 ⫽
1 ∂v 1 ∂u [ ⫹ u cos ⫹ w sin ]; 6 ⫽ [ ⫺ v cos ] R ∂ R ∂
(38)
and
␥1 ⫽ ⫺ 
∂ ∂ ; ␥2 ⫽ ∂s ∂s
␥3 ⫽
∂ 1 ; ␥4 ⫽ ⫺  sin ∂s R
␥5 ⫽
1 ∂ 1 ∂ ⫹  cos ; ␥6 ⫽ ⫺  cos R ∂ R ∂
冉
冊
(39)
冉
冊
The membrane strains in shells of revolution are expressed as
⑀ ⫽ ⑀ᐉ ⫹ ⑀nᐉ ⫽ 3 ⫹ ⑀ ⫽ ⑀ᐉ ⫹ ⑀nᐉ ⫽ 5 ⫹
1 2 ( ⫹ 22 ⫹ 23) 2 1
1 2 ( ⫹ 25 ⫹ 26) 2 4
(40)
␥ ⫽ ␥ᐉ ⫹ ␥nᐉ ⫽ (2 ⫹ 6) ⫹ (14 ⫹ 25 ⫹ 36) The curvature terms in shells of revolution are given by
= ᐉ + nᐉ = ␥3 + (1␥1 + 2␥2 + 3␥3) = ᐉ + nᐉ = ␥5 + (4␥4 + 5␥5 + 6␥6)
(41)
= ᐉ + nᐉ = (␥2 + ␥6) + (1␥4 + 2␥5 + 3␥6 + 4␥1 + 5␥2 + 6␥3) The above relations (Eq. (40) Eq. (41)) are identical to those derived by Rotter and Jumikis [14], Su et al. [15] and Yin et al. [16] specifically for thin shells of
106
J.G. Teng, T. Hong / Thin-Walled Structures 31 (1998) 89–115
revolution, except for a small difference in the linear twisting curvature of Su et al. [15]. These relations will be referred to as Rotter and Jumikis’ theory in the following discussion. 4.2. Comparison of nonlinear strains with existing theories Table 3 presents a comparison similar to that of Table 2 of the following nonlinear shell theories specifically for thin shells of revolution: Donnell’s theory, Sanders’ theory, Simplified Sanders’ theory, and Rotter and Jumikis’ theory as a special case of the present theory. The simplified Sanders’ theory refers to that obtained from Sanders’ theory by omitting nonlinear strain terms associated with the rotation about the normal to the reference surface n. Donnell’s nonlinear shell theory can be obtained from the simplified Sanders’ theory if all nonlinear strain terms due to inplane displacements are ignored. Donnell’s theory, being the simplest, has been Table 3 Comparison of nonlinear strains for shells of revolution Rotter and Jumikis
Sanders
Simplified Sanders
Donnell
⑀nᐉ
1 2 ( + 22 + 23) 2 1
1 2 ( + 2n) 2
1 2  2
1 2  2
⑀nᐉ
1 2 ( + 25 + 26) 2 4
1 2 ( + 2n) 2
1 2  2
1 2  2


⑀nᐉ
14 + 25 + 36

1 = − 
∂ ∂w −u ∂s ∂s
∂w ∂ −u ∂s ∂s
2 or n
∂v ∂s
3
∂ ∂u +w ∂s ∂s
4 = − 
1 ∂w − v sin R ∂
5
1 ∂v + u cos + w sin R ∂
6
1 ∂u − v cos R ∂
冉 冉 冉
∂w ∂ −u ∂s ∂s
∂w ∂s
∂v ∂u 1 R + v cos − 2R ∂s ∂
–
–
–
–
–
冉
冊 冊
冉
冊
冉
冊
1 ∂w − v sin R ∂
1 ∂w R ∂
–
–
–
–
–
–
1 ∂w − v cos R ∂
冊
冊
J.G. Teng, T. Hong / Thin-Walled Structures 31 (1998) 89–115
107
widely used in analytical studies of shell buckling. It is generally known that Donnell’s theory is not suitable for the analysis of shells in which the buckling mode is fewer than three waves around the circumference.
5. Numerical examples Whilst the assumptions required to derive the various less complete nonlinear shell theories from the theory of Rotter and Jumikis as a special case of the present nonlinear shell theory are clear, it is of interest to examine the numerical implications of such assumptions. For many shells, the different nonlinear shell theories discussed here give closely similar results [20]. However, they may give significantly different results for a number of special but realistic problems. Numerical comparisons for two examples are presented below to explore these differences. Several versions of the NEPAS program for nonlinear buckling analysis of shells of revolution [20] have been developed using the different nonlinear shell theories discussed earlier to obtain the numerical results. 5.1. In-plane buckling of an annular plate ring Rotter and Jumikis [14] suggested that a good test problem is the in-plane buckling of an annular plate (Fig. 3). For such a plate undergoing in-plane buckling, all buckling displacements are in-plane. The annular plate used here has a width b ⫽ 0.25 m, a thickness t ⫽ 0.05 m and an inner radius R ⫽ 25 m and is subjected to a radially inward distributed load at its inner edge. The material properties used are: elastic modulus E ⫽ 2 ⫻ 105 MPa and Poisson’s ratio ⫽ 0.3. To ensure that in-plane buckling is the critical mode, the plate is restrained throughout against out-of-plane displacements. The classical ring in-plane buckling solution for loads whose direction remains unchanged during buckling predicts the following buckling load: qcr ⫽
Et 3
冉 冊 b Rc
3
(42)
Fig. 3.
Annular plate subject to radial inward load.
108
J.G. Teng, T. Hong / Thin-Walled Structures 31 (1998) 89–115
The critical number of circumferential waves for the in-plane buckling of unrestrained rings is 2, provided rigid body motions are prevented. The finite element in-plane buckling loads obtained using different shell theories are listed in Table 4 together with the classical solution. It is seen that Donnell’s theory and Sanders’ simplified theory with the rotation about the normal n omitted give no solution at all for this problem, as has been found previously by Rotter and Jumikis [14]. This is to be expected, as in these two theories the nonlinear strains are equal to zero for this problem for both axisymmetric and non-axisymmetric in-plane displacements. Sanders’ theory without simplification and Rotter and Jumikis’ theory are both able to predict the correct buckling load accurately, contrary to the conclusion reached by Rotter and Jumikis [14]. It is possible to show analytically that these two theories should lead to the same and correct answer for this problem. When specialized for the annular plate with only in-plane buckling displacements, Eq. (38) may be expressed as
1 ⫽ 4 ⫽ 0; 2 ⫽ 5 ⫽
冉
冊
∂v ∂u ; 3 ⫽ ∂s ∂s
冉
1 ∂v 1 ∂u ⫹ u ; 6 ⫽ ⫺v R ∂ R ∂
(43)
冊
using the conditions that w ⫽ 0, ⫽ 0, ∂/∂s ⫽ 0, leads to
⑀nᐉ ⫽ ⫽
1 2
1 2 1 (1 ⫹ 22 ⫹ 23) ⫽ (22 ⫹ 23) 2 2
(44)
冉 冊 冉 冊 ∂v ∂s
2
⫹
∂u ∂s
1 2
2
and
⑀nᐉ ⫽ ⫽
1 2 1 ( ⫹ 25 ⫹ 26) ⫽ (25 ⫹ 26) 2 4 2
冋冉
1 1 ∂v ⫹u 2 R ∂
冊册 冋 冉 2
⫹
1 1 ∂u ⫺v 2 R ∂
(45)
冊册
2
According to the linear strain–displacement relations, the linear membrane strains are given by Table 4 In-plane buckling loads of annular plates Shell theory
Donnell
Sanders n omitted
Sanders n retained
Rotter and Jumikis
Classical solution
qcr (N/mm)
–
–
3.317
3.316
3.284
J.G. Teng, T. Hong / Thin-Walled Structures 31 (1998) 89–115
⑀ᐉ ⫽
∂u ∂ ∂u ⫹w ⫽ ∂s ∂s ∂s
⑀ᐉ ⫽
1 ∂v 1 ∂v ⫹ w sin ⫹ u cos ⫽ ⫹u R ∂ R ∂
109
(46)
冉
冊 冉
冊
So
⑀ ⫽ ⑀ᐉ ⫹ ⑀nᐉ ⫽
冉 冊 冉 冊
∂u 1 ⫹ ∂s 2
∂v ∂s
2
⫹
⑀ ⫽ ⑀ᐉ ⫹ ⑀nᐉ ⫽
(47)
冉
冊
∂u ∂s
1 2
冉
2
冊
冉
冊
(48)
1 ∂v 1 1 ∂v 1 1 ∂u ⫹u ⫹ [ ⫹ u ]2 ⫹ [ ⫺ v ]2 R ∂ 2 R ∂ 2 R ∂
As both ∂u/∂s and 1/R(∂v/∂ ⫹ u) are small, the squares of these two linear strain terms may be neglected. Eq. (47) Eq. (48) then reduce to
冉 冊 ∂v ∂s
⑀ ⫽
∂u 1 ⫹ ∂s 2
⑀ ⫽
1 ∂v 1 1 ∂u ⫹u ⫹ [ ⫺ v ]2 R ∂ 2 R ∂
Defining
␣⫽
冉
冊
冉
冊
1 ∂u v⫺ R ∂
2
(49)
冉
冊
(50)
(51)
where ␣ represents a small rotation of a circumferential line element (Fig. 4), Eq. (44) Eq. (45) become
冉 冊 ∂v ∂s
⑀nᐉ ⫽
1 2
⑀nᐉ ⫽
1 2 ␣ 2
2
(52)
(53)
Eq. (52) and 53 are the nonlinear strain terms obtained from the present relations. Now substituting the conditions that w ⫽ 0, ⫽ 0, ∂/∂s ⫽ 0 into Sanders’ relations, the rotations are simplified to
110
J.G. Teng, T. Hong / Thin-Walled Structures 31 (1998) 89–115
Fig. 4.
 ⫽  ⫽ 0
Circumferential line element before and after deformation.
冉
1 ∂v ∂u 1 v⫺ ⫹ 2 ∂s 2R ∂
n ⫽
(54)
冊
As a result, the nonlinear strains become
⑀nᐉ ⫽ ⑀nᐉ ⫽
冉
冊
1 2 1 1 ∂v ∂u 2 1 n ⫽ [ v⫺ ] ⫹ 2 2 2 ∂s 2R ∂
(55)
Introduction of Eq. (51) into the above equation leads to
⑀nᐉ ⫽ ⑀nᐉ ⫽
冋
1 1 ∂v 1 ⫹ ␣ 2 2 ∂s 2
册
2
(56)
From Fig. 5, it can been seen that  is a small rotation of a radial line element and is related to the circumferential displacement v through
冉
v⫹
⫽
冊
∂v ds ⫺ v ∂s ∂v ⫽ ds ∂s
(57)
According to the classical analysis of circular rings [21], it may be assumed that the rotation of a radial line element is approximately equal to the rotation of a circumferential line element, that is ␣ ⫽ . With this assumption, the nonlinear membrane strain terms of Sanders’ theory become
J.G. Teng, T. Hong / Thin-Walled Structures 31 (1998) 89–115
Fig. 5.
⑀nᐉ ⫽ ⑀nᐉ ⫽
111
Normal to centroidal line before and after deformation.
冉
冊
1 1 1 ␣⫹  2 2 2
2
⬵
1 2 1 2 ␣ ⫽  2 2
(58)
It can thus be concluded that Sanders’ theory and Rotter and Jumikis’ theory should lead to closely similar results for this problem, which is in agreement with numerical predictions. 5.2. Buckling of ring-stiffened cylinders under external pressure A simply-supported stiffened cylinder under uniform external pressure is considered. The geometry of the cylinder is given in Fig. 6. The buckling loads obtained using different nonlinear strain–displacement relations are given in Table 5. The buckling loads in Table 5 are in terms of the dimensionless buckling pressure cr ⫽ pR(1 ⫺ 2)/(Et), in which p is the external pressure, E is the elastic modulus, is the Poisson’s ratio, R is the cylinder radius and t is the cylinder thickness. In
Fig. 6. Ring-stiffened cylinder under external pressure.
Sanders’ theory n omitted
6.403 (3)
6.248 (3)
Donnell theory
5.192 (3)a
5.163 (3)
6.014 (3)
6.000 (3)
Sanders’ theory n retained
Number of critical circumferential buckling waves.
a
Linear buckling Nonlinear buckling
cr × 103
General instability
Table 5 Buckling pressures of ring-stiffened cylinders
–
6.451 (3)
Subbiah and Natarajan
5.962 (3)
5.967 (3)
Rotter and Jumikis’ theory
3.239 (12)
3.745 (13)
Donnell theory
3.248 (12)
3.755 (13)
Sanders’ theory n omitted
3.243 (12)
3.749 (13)
Sanders’ theory n retained
–
3.755 (13)
Subbiah and Natarajan
Interframe buckling
3.232 (12)
3.749 (13)
Rotter and Jumikis’ theory
112 J.G. Teng, T. Hong / Thin-Walled Structures 31 (1998) 89–115
J.G. Teng, T. Hong / Thin-Walled Structures 31 (1998) 89–115
113
the numerical analyses, the following values were used: E ⫽ 2.0 ⫻ 105 MPa, ⫽ 0.3, R/t ⫽ 100. The finite element model of the shell included only the ring-stiffened cylinder with one end allowed to have only meridional rotations and the other end allowed to have both meridional rotations and axial displacements. The circular plate covers at the ends of the cylinder were not included in the finite element model, but the pressure acting on the plate was included in the form of a circumferential line load on the cylinder end which was not restrained in the axial direction. There are two possible buckling modes in such stiffened cylinders: general instability and interframe buckling. These two modes correspond to two different numbers of circumferential waves. In the global mode of instability, the ring stiffeners’ displacements are nearly all in the plane of the ring. On the other hand, these rings deform out-of-plane in the interframe mode of instability. Both linear buckling loads (prebuckling large deflection effect ignored) and nonlinear buckling loads (prebuckling large deflection effect included) are presented. The effect of prebuckling large deflections is seen to be small. It can be seen that for the interframe buckling mode, the results from the different shell theories are very close to each other as buckling displacements are out-of-plane in both the cylinder and the ring. For the general mode of instability, there are some significant differences. The predictions using the theory of Rotter and Jumikis are similar to those using Sanders’ theory, however the simplified Sanders’ theory in which the rotation about the normal is ignored predicts buckling loads which are 7% higher than those of Rotter and Jumikis’ theory. It is interesting to note that Subbiah and Natarajan’s [22] results are in very close agreement with those of the simplified Sanders’ theory, but are different from those of Rotter and Jumikis’ theory. Subbiah and Natarajan [22] indicated that Sanders’ theory was used in their analysis but did not elaborate whether the rotation about the normal was included. It seems very likely that they used the simplified Sanders theory. The unconservative predictions of the simplified Sanders’ theory are due to its inability in modelling the in-plane buckling behaviour of the annular plate web of the T-section ring stiffeners as demonstrated earlier. Compared to the theory of Rotter and Jumikis, the buckling loads predicted using Donnell’s theory are lower by about 9%. The Donnell approximations in the nonlinear strain terms lead to conservative predictions of the buckling load.
6. Conclusion A new theory for numerical nonlinear and buckling analysis of thin shells has been presented. This new theory is found to reduce to that of Rotter and Jumikis [14], Su et al. [15] and Yin et al. [16] for thin shells of revolution. The nonlinear theories of Sanders and Donnell can be deduced from the present theory by invoking a number of assumptions. Numerical comparisons have shown that the simplified Sanders’ theory with the rotation about the normal omitted should not be used in finite element codes for complex shell structures. No obvious differences in numerical results have been found between Rotter and Jumikis’ theory and Sander’s more complete theory, justifying to some extent the assumptions made for arriving at San-
114
J.G. Teng, T. Hong / Thin-Walled Structures 31 (1998) 89–115
ders’ theory from the present theory or that of Rotter and Jumikis for shells of revolution. As the use of the present theory leads to little additional computational effort in numerical analysis, it is recommended that the new theory or Rotter and Jumikis’ theory for shells of revolution be used in numerical buckling analysis for complete peace of mind that no terms, however small, have been omitted.
Acknowledgements Both authors are grateful to the Hong Kong Polytechnic University for its financial support.
References [1] Love AEH. The small free vibrations and deformation of a thin elastic shell. Philosophical Transactions of the Royal Society of London 1888;179:491–546. [2] Donnell LH. Stability of thin-walled tubes under torsion. NACA Report No. 479, 1934. [3] Reissner E. A new derivation of the equations for the deformation of elastic shells. American Journal of Math. 1941;63:177–88. [4] Sanders JL, Jr. An improved first-approximation theory for thin shells. NASA Technical Report, NASA-TR-R24, 1959. [5] Koiter WT. A consistent first approximation in the general theory of thin elastic shells. In: Proceedings of the I.U.T.A.M. Symposium on the Theory of Thin Elastic Shells. Amsterdam: North-Holland, 1960: 12–33. [6] Novozhilov VV. Thin shell theory. Groningen, The Netherlands: P. Noordhoff, Ltd, 1964. [7] Flugge W. Static und Dynamik der Schalen. Berlin: Springer Verlag, 1934. [8] Marguerre K. Zur Theorie der gekrummten Platte grosser Formanderung. In: Proceedings of the Fifth International Congress of Applied Mechanics. New York: Wiley and Sons, 1938: 93–101. [9] Novozhilov VV. Foundations of the nonlinear theory of elasticity. Rochester, NY: Graylock Press, 1953. [10] Sanders JL. Nonlinear theories for thin shells. Quarterly J. of Applied Math. 1963;21:21–36. [11] Koiter WT. On the nonlinear theory of thin elastic shells. Proceedings of the Koninklijke Nederlandse Akademie van Wetenschappen, Series B 1966;69:1–54. [12] Bushnell D. BOSOR5—program for buckling of elastic–plastic complex shells of revolution including large deflections and creep. Computers and Structures 1976;6:221–39. [13] Jumikis PT. Stability problems in silo structures, PhD thesis, School of Civil and Mining Engineering, University of Sydney, 1987. [14] Rotter JM, Jumikis PT. Nonlinear strain–displacement relations for thin shells of revolution. Res. Rep. R563, School of Civil and Mining Engineering, University of Sydney, 1988. [15] Su XY, Wu JK, Hu HC. A large displacement analysis for thin shells of revolution. In: Science in China, 1987: 400–10. [16] Yin J, Suo XZ, Combescure A. About two new efficient nonlinear shell elements. SMIRT Conference Proceedings, Lausanne, Paper No. B12/1, 1987. [17] Saada AS. Elasticity: Theory and Applications. New York: Pergamon Press, 1974. [18] Budiansky B, Sanders JL. On the “best” first-order linear shell theory. In: Progress in applied mechanics (Prager Anniversary Volume). New York, 1963: 129–40. [19] Kraus H. Thin elastic shells. New York: John Wiley and Sons, 1967. [20] Teng JG, Rotter JM. Non-symmetric bifurcation analysis of geometrically nonlinear elastic–plastic axisymmetric shells subject to combined loads including torsion. Computers and Structures 1989;32:453–77.
J.G. Teng, T. Hong / Thin-Walled Structures 31 (1998) 89–115
115
[21] Brush DO, Almroth BO. Buckling of bars, plates, and shells. New York: McGraw-Hill, 1975. [22] Subbiah J, Natarajan R. Stability analysis of ring-stiffened shells of revolution. Journal of Ship Research 1982;26:125–34.
Nonlinear thin shell theories for numerical buckling predictions J.G. Teng*, T. Hong Department of Civil and Structural Engineering, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong, China
Abstract A basic building block in any numerical (geometrically) nonlinear and buckling analysis is a set of nonlinear strain–displacement relations. A number of such relations have been developed in the past for thin shells. Most of these theories were developed in the pre-computer era for analytical studies when simplicity was emphasized and terms judged to be small relative to other terms were omitted. With the availability of greatly increased computing power in recent years, accuracy rather than simplicity is given more emphasis. Additional complexity in the strain–displacement relations leads to only a small increase in computational effort, but the omission of a term which may be important in only a few complex problems is a major flaw. It is therefore necessary to re-examine classical shell theories in the context of numerical nonlinear and buckling analysis. This paper first describes a set of nonlinear strain–displacement relations for thin shells of general form developed directly from the nonlinear theory of three-dimensional solids. In this new theory, all nonlinear terms, large and small, are retained. When specialized for thin shells of revolution, this theory reduces to that previously derived by Rotter and Jumikis and others. Analytical and numerical comparisons are carried out for thin shells of revolution between Rotter and Jumikis’ theory as a special case of the present theory and other commonly used nonlinear theories. The paper concludes with comments on the suitability of the various nonlinear shell theories discussed here for use in numerical buckling analysis of complex branched shells. 1998 Elsevier Science Ltd. All rights reserved. Keywords: Shells; Buckling; Theories; Stability
* Corresponding author. 0263-8231/98/$—see front matter 1998 Elsevier Science Ltd. All rights reserved. PII: S 0 2 6 3 - 8 2 3 1 ( 9 8 ) 0 0 0 1 4 - 7
90
J.G. Teng, T. Hong / Thin-Walled Structures 31 (1998) 89–115
Nomenclature A1, A2 E H1, H2, H3 n R R1, R2 s t u, v, w u, v, w ␣1, ␣2, ␣3 ␣, , → → →
␣, , ␣,  n ⑀
lame coefficients of curvilinear coordinates on shell reference surface elastic modulus lame coefficients coordinate normal to reference surface of shells of revolution radius of circumferential coordinate principal radii of curvature meridional coordinate for shells of revolution thickness of shell displacements in curvilinear coordinates displacements of a point on shell reference surface orthogonal curvilinear coordinates orthogonal curvilinear coordinates for a general shell unit vectors corresponding to coordinates ␣,  and rotations about a tangent to coordinate lines ␣ and  rotation about the normal to reference surface strains meridional angular coordinate for shells of revolution curvatures Poisson’s ratio circumferential angular coordinate for shells of revolution
Subscripts ᐉ nᐉ ␣ 
linear nonlinear along curvilinear coordinate ␣ along curvilinear coordinate  meridional circumferential normal to reference surface
1. Introduction Many attempts have been made to develop linear and nonlinear theories for thin shells. The first complete linear theory, based to some extent on Kirchhoff’s earlier work on plates, was given by Love [1]. This theory is often referred to as Love’s first approximation. Later linear theories of thin shells have been established by Donnell [2], Reissner [3], Sanders [4], Koiter [5], Novozhilov [6], and a theory appropriate for moderately thick shells was developed by Flugge [7]. Nonlinear and stability analyses are commonly based on nonlinear strain–displacement relations and corresponding equilibrium equations. The earliest work of some generality in this regard is Marguerre’s [8] nonlinear theory of shallow shells. Donnell [2]
J.G. Teng, T. Hong / Thin-Walled Structures 31 (1998) 89–115
91
developed an approximate theory specifically for cylinders and suggested its extension for a general shell (often referred to as the Donnell–Mushtari–Vlasov theory). Novozhilov [9] derived a general large deflection theory of thin shells, but did not go into details on further simplifications nor discuss corresponding equilibrium equations. Sanders [10] and Koiter [11] independently developed a nonlinear thin shell theory and showed that previous theories could be derived from theirs with appropriate further approximations. For shell problems which may be classified as being of small strains and moderately large rotations, the nonlinear thin shell theories of Donnell [2] and Sanders [10] have been the most widely used. These classical shell theories were developed in the pre-computer era for analytical studies when simplicity was emphasized and terms judged to be small relative to other terms were omitted. Nevertheless, Sanders’ theory [10] has been considered to be sufficiently accurate in most applications and implemented in many large shells of revolution computer codes for large deflection and buckling analysis (e.g. Ref. [12]). With the availability of greatly increased computing power in recent years, accuracy rather than simplicity is given more emphasis. Additional complexity in the strain–displacement relations leads to only a small increase in computational effort, but the omission of a term which may be important in only a few complex problems is a major flaw. Furthermore, unlike classical studies which have examined only shells of simple forms, a shell buckling computer code can be applied to problems of great complexity and of a range unintended by the code writer, consequently it is difficult to ensure that apparently small terms in most problems do not become significant in some other problems. It is therefore necessary to re-examine the classical shell theories in the new context of numerical buckling analysis. Many shell theories have been presented which include both kinematic (strain–displacement) and static (equilibrium) expressions, the present study is concerned only with the kinematic relations as the equilibrium equations are generally established using the virtual work principle in numerical analysis. Rotter and Jumikis appeared to be the first to consider this issue when they developed a new nonlinear shell theory for thin shells of revolution in the early eighties. This new theory together with interesting comparisons with other available theories is documented in the PhD thesis of Jumikis [13] and in Rotter and Jumikis [14]. It was found during the present study that the same relations were also arrived at independently by Su et al. [15] and Yin et al. [16] for thin shells of revolution. Whilst this coincidence is a little surprising, the real reason being that in this set of relations, all nonlinear terms, large or small, have been retained. These relations thus constitute a complete set of nonlinear strain–displacement relations for these shells in the sense that no term has been neglected based on a judgment of its relative magnitude. For convenience, these relations are referred to as Rotter and Jumikis’ theory. All these researchers derived these relations directly for thin shells of revolution, so a corresponding nonlinear thin shell theory has not been developed for shells of general form. This paper will present the results of a recent study on linear and nonlinear shell theories in numerical buckling analysis. A new set of nonlinear strain–displacement relations for thin shells of general form developed directly from the nonlinear elastic
92
J.G. Teng, T. Hong / Thin-Walled Structures 31 (1998) 89–115
theory of three-dimensional solids will be presented first. Shells of revolution have many more applications than most other shell forms, so the theory will then be specialized for thin shells of revolution to arrive at the same relations as those derived by Rotter and Jumikis [14] and others. Analytical and numerical comparisons will be carried out between the present theory and other commonly used linear and nonlinear theories to explore their differences as well as similarities. Particular attention is paid to the effect of using a different set of nonlinear strain–displacement relations on the predicted buckling loads.
2. Strain–displacement relations for thin shells of general form 2.1. Strain components in curvilinear coordinates for three dimensional solids In the nonlinear theory of elasticity, the expressions for the strain components in an arbitrary orthogonal coordinate system are [17]:
⑀11 ⫽ e11 ⫹
1 2 (e ⫹ e221 ⫹ e231) 2 11
⑀22 ⫽ e22 ⫹
1 2 (e ⫹ e212 ⫹ e232) 2 22
⑀33 ⫽ e33 ⫹
1 2 (e ⫹ e213 ⫹ e223) 2 33
(1)
⑀12 ⫽ ⑀21 ⫽ (e12 ⫹ e21) ⫹ e11e12 ⫹ e22e21 ⫹ e31e32 ⑀13 ⫽ ⑀31 ⫽ (e13 ⫹ e31) ⫹ e11e13 ⫹ e33e31 ⫹ e21e23 ⑀23 ⫽ ⑀32 ⫽ (e23 ⫹ e32) ⫹ e22e23 ⫹ e33e32 ⫹ e12e13 where e11 ⫽
1 ∂u 1 ∂H1 1 ∂H1 ⫹ v⫹ w H1 ∂␣1 H1H2 ∂␣2 H1H3 ∂␣3
e22 ⫽
1 ∂v 1 ∂H2 1 ∂H2 ⫹ w⫹ u H2 ∂␣2 H2H3 ∂␣3 H1H2 ∂␣1
e33 ⫽
1 ∂w 1 ∂H3 1 ∂H3 ⫹ u⫹ v H3 ∂␣3 H1H3 ∂␣1 H2H3 ∂␣2
e21 ⫽
1 ∂v 1 ∂H1 ⫺ u H1 ∂␣1 H1H2 ∂␣2
e12 ⫽
1 ∂u 1 ∂H2 ⫺ v H2 ∂␣2 H1H2 ∂␣1
and
(2)
J.G. Teng, T. Hong / Thin-Walled Structures 31 (1998) 89–115
e13 ⫽
1 ∂u 1 ∂H3 ⫺ w H3 ∂␣3 H1H3 ∂␣1
e31 ⫽
1 ∂w 1 ∂H1 ⫺ u H1 ∂␣1 H1H3 ∂␣3
e32 ⫽
1 ∂w 1 ∂H2 ⫺ v H2 ∂␣2 H2H3 ∂␣3
e23 ⫽
1 ∂v 1 ∂H3 ⫺ w H3 ∂␣3 H2H3 ∂␣2
93
(3)
In Eq. (1), ⑀ii (i ⫽ 1, 2, 3) are the direct strains and ⑀ij (i, j ⫽ 1, 2, 3; i⫽j) are the shear strains. Eq. (1) Eq. (2) Eq. (3) are the nonlinear relations between the strains ⑀ and the displacements u, v and w for a three-dimensional elastic solid. In these equations, H1, H2 and H3 are the Lame coefficients of the coordinate lines ␣1, ␣2 and ␣3, respectively. These relations may be specialized for thin shells by invoking appropriate assumptions. A curvilinear coordinate system is first established, which consists of two orthogonal curvilinear coordinates ␣ and  along the lines of principal curvature, → → . Let ␣ and  be the unit vectors along ␣ and , respectand a normal coordinate → ively, and the unit vector along the normal to the shell reference surface (usually the middle surface), and u, v and w be the components of the displacement vector in the corresponding orthogonal coordinates (Fig. 1). The Lame coefficients H1, H2 and H3 are now given by H1 ⫽ A1(1 ⫹ /R1); H2 ⫽ A2(1 ⫹ /R2); H3 ⫽ 1
(4)
where A1 and A2 are the Lame coefficients of the curvilinear coordinates on the
Fig. 1.
Shell reference surface and orthogonal curvilinear coordinates.
94
J.G. Teng, T. Hong / Thin-Walled Structures 31 (1998) 89–115
reference surface, and R1 and R2 are the principal radii of curvature. Substituting Eq. (4) into Eq. (2) Eq. (3) leads to
冉 冉
冊 冊
e␣␣ ⫽
1 1 ∂A1 w 1 ∂u ⫹ v⫹ 1 ⫹ /R1 A1 ∂␣ A1A2 ∂ R1
e ⫽
1 ∂v 1 1 ∂A2 w ⫹ u⫹ 1 ⫹ /R2 A2 ∂ A1A2 ∂␣ R2
e ⫽
∂w ∂
e␣ ⫽
1 1 ∂A1 1 ∂v ⫺ u 1 ⫹ /R1 A1 ∂␣ A1A2 ∂
e␣ ⫽
1 1 ∂A2 1 ∂u ⫺ v 1 ⫹ /R2 A2 ∂ A1A2 ∂␣
e␣ ⫽
∂u ∂
e␣ ⫽
1 ∂w 1 u ⫺ 1 ⫹ /R1 A1 ∂␣ R1
e ⫽
1 v 1 ∂w ⫺ 1 ⫹ /R2 A2 ∂ R2
e ⫽
∂v ∂
and
冉 冉
(5)
冊 冊 (6)
冉 冉
冊 冊
2.2. The Love–Kirchhoff assumptions To reduce the strain–displacement relations for three-dimensional solids to those for thin shells, it is necessary to invoke the well-known Love–Kirchhoff assumptions: 1. The thickness t of the shell is negligibly small in comparison with the least radius of curvature Rmin of the reference surface. 2. The displacements are sufficiently small for quantities of the second and higher orders to be neglected compared with those of the first order. 3. The stress component normal to the reference surface is small compared with the other stress components and may be neglected in the stress–strain relations. 4. Normals of the undeformed reference surface remain straight and normal to the deformed reference surface. 5. Normals to the reference surface undergo negligible change in length during deformation.
J.G. Teng, T. Hong / Thin-Walled Structures 31 (1998) 89–115
95
The first assumption of thin shells can be written as t/Rmin¿1
(7)
Assumption 2 means that one can retain only linear terms in the expression of strains. The use of this assumption leads to a linear thin shell theory. The third assumption relates to the constitutive relations for the shell, and defines the throughthickness stress as negligible, allowing equations for plane stress conditions to be used. That is
⫽ 0
(8)
Assumption 4 is analogous to the Bernoulli–Euler hypothesis of beam theory that “plane section remains plane”. The preserved straightness of normals allows transverse shear strains to be ignored, that is
␥␣ ⫽ ␥ ⫽ 0
(9)
The last assumption implies that strains normal to the reference surface are negligible, i.e.
⑀ ⫽ 0
(10)
2.3. Strain–displacement relations for general shells By the application of the Love–Kirchhoff assumptions, the displacement components are all assumed to vary linearly across the thickness and can be represented by the following relations: u(␣,,) ⫽ u(␣,) ⫹
∂u ∂
v(␣,,) ⫽ v(␣,) ⫹
∂v ∂
w(␣,,) ⫽ w(␣,) ⫹
(11)
∂w ∂
where the quantities u, v and w represent the components of the displacement vector of a point on the reference surface. Since the linear component of the normal strain is, in general, an order of magnitude larger than the nonlinear component, the fifth Love–Kirchhoff assumption is applied only to the linear component of the normal strain as was done by Rotter and Jumikis [14]. The following equation is then obtained:
⑀l ⫽ e ⫽
∂w ⫽0 ∂
(12)
96
J.G. Teng, T. Hong / Thin-Walled Structures 31 (1998) 89–115
Application of the Love–Kirchhoff assumption which requires transverse shear strains to be ignored only to the linear component of the transverse shear strain (Eq. (1) Eq. (6)) leads to e␣ ⫹ e␣ ⫽
冉
冊
1 u ∂u 1 ∂w ⫺ ⫹ ⫽0 1 ⫹ /R1 A1 ∂␣ R1 ∂
(13)
It is easy to see that the quantity ∂u/∂ represents the rotation of a tangent to the middle surface oriented along the coordinate line ␣. This rotation is defined as ␣ and given by
␣ ⫽
冉
冊
u ∂u 1 1 ∂w ⫽ ⫺ ∂ 1 ⫹ /R1 R1 A1 ∂␣
(14)
Similarly, , which represents the rotation of a tangent to the middle surface oriented along the coordinate line  is given by
 ⫽
冉
冊
∂v 1 1 ∂w v ⫽ ⫺ ∂ 1 ⫹ /R2 R2 A2 ∂
(15)
Substituting Eq. (12) Eq. (14) Eq. (15) into Eq. (11), the three displacement components become u(␣,,) ⫽ u(␣,) ⫹ ␣ v(␣,,) ⫽ v(␣,) ⫹ 
(16)
w(␣,,) ⫽ w(␣,) By substituting Eq. (16) into Eq. (5) Eq. (6), Eq. (5) Eq. (6) may be rewritten as e␣ ⫽ 1 ⫹ ␥1 e␣ ⫽ 2 ⫹ ␥2 e␣␣ ⫽ 3 ⫹ ␥3
(17)
e ⫽ 4 ⫹ ␥4 e ⫽ 5 ⫹ ␥5 e␣ ⫽ 6 ⫹ ␥6 where
1 ⫽ ⫺ ␣ ⫽ 2 ⫽
冉
冊 冊
1 ∂w u 1 ⫺ 1 ⫹ /R1 A1 ∂␣ R1
冉
1 1 ∂A1 1 ∂v ⫺ u 1 ⫹ /R1 A1 ∂␣ A1A2 ∂
J.G. Teng, T. Hong / Thin-Walled Structures 31 (1998) 89–115
3 ⫽
冉
1 ∂u w 1 1 ∂A1 ⫹ v⫹ 1 ⫹ /R1 A1 ∂␣ A1A2 ∂ R1
4 ⫽ ⫺  ⫽
冉
1 ∂w v 1 ⫺ 1 ⫹ /R2 A2 ∂ R2
冉 冉
冊
冊
5 ⫽
1 ∂v w 1 1 ∂A2 ⫹ u⫹ 1 ⫹ /R2 A2 ∂ A1A2 ∂␣ R2
6 ⫽
1 ∂u 1 1 ∂A2 ⫺ v 1 ⫹ /R2 A2 ∂ A1A2 ∂␣
冊
97
(18)
冊
and
␥1 ⫽ ⫺
1 ∂A1 1  1 ⫹ /R1 A1 ∂ ␣
冉
冊
␥2 ⫽
1 1 ∂A1 1 ∂ ⫺  1 ⫹ /R1 A1 ∂␣ A1A2 ∂ ␣
␥3 ⫽
1 1 ∂␣ 1 ∂A1 ( ⫹  ) 1 ⫹ /R1 A1 ∂␣ A1A2 ∂ 
␥4 ⫽ ⫺
1 ∂A2 1  1 ⫹ /R2 A2 ∂ 
冉 冉
(19)
冊 冊
␥5 ⫽
1 ∂ 1 1 ∂A2 ⫹  1 ⫹ /R2 A2 ∂ A1A2 ∂␣ ␣
␥6 ⫽
1 ∂␣ 1 1 ∂A2 ⫺  1 ⫹ /R2 A2 ∂ A1A2 ∂␣ 
In terms of Eq. (1) and Eq. (17) Eq. (18) Eq. (19), the linear part of the strain components can be written as
⑀␣ᐉ ⫽ 3 ⫹ ␥3 ⑀ᐉ ⫽ 5 ⫹ ␥5
(20)
⑀␣ᐉ ⫽ (2 ⫹ 6) ⫹ (␥2 ⫹ ␥6) The above equations are the same as the strain–displacement relations of Flugge’s theory [7]. These equations are appropriate for use in the analysis of moderately thick shells. For the analysis of thin shells, for which /Ri¿1 , Eq. (18) Eq. (19) are further simplified to
1 ⫽ ⫺ ␣ ⫽
1 ∂w u ⫺ A1 ∂␣ R1
98
J.G. Teng, T. Hong / Thin-Walled Structures 31 (1998) 89–115
2 ⫽
1 ∂v 1 ∂A1 ⫺ u A1 ∂␣ A1A2 ∂
3 ⫽
w 1 ∂u 1 ∂A1 ⫹ v⫹ A1 ∂␣ A1A2 ∂ R1
4 ⫽ ⫺  ⫽
(21)
v 1 ∂w ⫺ A2 ∂ R2
5 ⫽
1 ∂v 1 ∂A2 w ⫹ u⫹ A2 ∂ A1A2 ∂␣ R2
6 ⫽
1 ∂u 1 ∂A2 ⫺ v A2 ∂ A1A2 ∂␣
and
␥1 ⫽ ⫺
1 ∂A1  A1 ∂ ␣
␥2 ⫽
1 ∂ 1 ∂A1 ⫺  A1 ∂␣ A1A2 ∂ ␣
␥3 ⫽
1 ∂␣ 1 ∂A1 ⫹  A1 ∂␣ A1A2 ∂ 
␥4 ⫽ ⫺
(22)
1 ∂A2  A2 ∂ 
␥5 ⫽
1 ∂ 1 ∂A2  ⫹ A2 ∂ A1A2 ∂␣ ␣
␥6 ⫽
1 ∂␣ 1 ∂A2 ⫺  A2 ∂ A1A2 ∂␣ 
Substituting Eq. (17) into Eq. (1), the membrane strains of general thin shells are given by
⑀␣ ⫽ 3 ⫹
1 2 ( ⫹ 22 ⫹ 23) 2 1
⑀ ⫽ 5 ⫹
1 2 ( ⫹ 25 ⫹ 26) 2 4
⑀␣ ⫽ (2 ⫹ 6) ⫹ (14 ⫹ 25 ⫹ 36) and the curvatures are given by
␣ ⫽ ␥3 ⫹ (1␥1 ⫹ 2␥2 ⫹ 3␥3)
(23)
J.G. Teng, T. Hong / Thin-Walled Structures 31 (1998) 89–115
99
 ⫽ ␥5 ⫹ (4␥4 ⫹ 5␥5 ⫹ 6␥6)
(24)
␣ ⫽ (␥2 ⫹ ␥6) ⫹ (1␥4 ⫹ 2␥5 ⫹ 3␥6 ⫹ 4␥1 ⫹ 5␥2 ⫹ 6␥3) in which 1 苲 6 and ␥1 苲 ␥6 are found in Eq. (21) Eq. (22). It is now a simple matter to write out explicitly the linear and nonlinear membrane strains separately as follows:
⑀␣␣ᐉ ⫽ 3 ⫽
w 1 ∂u 1 ∂A1 ⫹ v⫹ A1 ∂␣ A1A2 ∂ R1
⑀ᐉ ⫽ 5 ⫽
w 1 ∂v 1 ∂A2 ⫹ u⫹ A2 ∂ A1A2 ∂␣ R2
⑀␣ᐉ ⫽ 2 ⫹ 6 ⫽
冉
(25)
冊 冉
1 ∂A1 1 ∂A2 1 ∂v 1 ∂u ⫺ u ⫹ ⫺ v A1 ∂␣ A1A2 ∂ A2 ∂ A1A2 ∂␣
冊
and
⑀␣nᐉ ⫽
1 2 ( ⫹ 22 ⫹ 23) 2 1
⑀nᐉ ⫽
1 2 ( ⫹ 25 ⫹ 26) 2 4
(26)
⑀␣nᐉ ⫽ (14 ⫹ 25 ⫹ 36) Similarly, the linear and nonlinear curvature terms can also be separately written:
␣␣ᐉ ⫽ ␥3 ⫽
1 ∂␣ 1 ∂A1 ⫹  A1 ∂␣ A1A2 ∂ 
ᐉ ⫽ ␥5 ⫽
1 ∂ 1 ∂A2  ⫹ A2 ∂ A1A2 ∂␣ ␣
␣ᐉ ⫽ ␥2 ⫹ ␥6 ⫽
冉
(27)
冊 冉
冊
1 ∂ 1 ∂ ␣ 1 ∂A1 1 ∂A2 ⫺ ⫺  ⫹  A1 ∂␣ A1A2 ∂ ␣ A2 ∂ A1A2 ∂␣ 
and
␣nᐉ ⫽ (1␥1 ⫹ 2␥2 ⫹ 3␥3) nᐉ ⫽ (4␥4 ⫹ 5␥5 ⫹ 6␥6)
(28)
␣nᐉ ⫽ (1␥4 ⫹ 2␥5 ⫹ 3␥6 ⫹ 4␥1 ⫹ 5␥2 ⫹ 6␥3) The significance of nonlinear curvatures will not be discussed here as they do not appear to be important for thin shells. Readers may consult Rotter and Jumikis [14] for a brief discussion.
100
J.G. Teng, T. Hong / Thin-Walled Structures 31 (1998) 89–115
3. Comparison with other theories for general thin shells 3.1. Comparison of linear strain components with other theories Since Love [1] proposed his well known first-approximation shell theory in 1888, a number of other theories have been presented. Those commonly used include the Donnell–Mushtari–Vlasov (DMV) theory [2], Reissner’s theory [3], Novozhilov’s theory [6], Sanders’ “best” first order theory [4,5,18], Flugge’s theory [7], Rotter– Jumikis’ theory [14]. All these theories except Flugge’s theory are for thin shells only. Flugge’s theory is suitable for moderately thick shells. The present linear relations (Eq. (20)), before the thinness assumption is invoked, are the same as those of Flugge’s theory. Introducing the thinness assumption by replacing 1 ⫹ /Ri (i ⫽ 1, 2) with 1, linear strain–displacement relations for thin shells only are obtained. The present theory (Eq. (25) Eq. (26)) is then the same as Reissner’s theory. Rotter and Jumikis’ relations are developed directly for thin shells of revolution. Applying the special conditions relevant to thin shells of revolution as shown later, the present relations reduce to the linear relations of Rotter and Jumikis. Novozhilov [6] gave a set of linear strain–displacement relations slightly different from the present set (only in the twisting curvature term) due to a slightly different imposition of the thinness assumption. Strain–displacement relations of the DMV theory are the simplest set available and are appropriate only for shallow shells. The expressions for the linear membrane strains given in the theories which are appropriate for thin shells only are the same. Differences between the different linear strain–displacement relations are therefore only found in the expressions for the curvatures and twisting curvature. The differences are highlighted in Table 1. The expressions for the curvatures and twisting curvature of the DMV theory differ from those of other theories in that contributions of the in-plane displacements are neglected. The linear theory of elastic thin shells proposed by Sanders has been considered to be the “best”. The linear strain–displacement relations of Sanders’ theory have one obvious advantage over those of other theories in that all strains vanish for small rigid-body motions of the shell. For all other theories, when rigid-body motions are introduced, the twisting curvature ␣ does not vanish except in the case of a spherical shell, a flat plate or a symmetrically loaded shell of revolution, so these theories may lead to significant errors in other cases. The present strain–displacement relations before the introduction of the thinness assumption predicts that no strains result from rigid-body motions. It can thus be seen that the nonvanishing twisting curvature from rigid-body motions in many thin shell theories is directly related to the thinness assumption. In practical applications, the different thin shell theories discussed above lead to only small differences for a few cases [19]. 3.2. Comparison of nonlinear strain components with other theories Nonlinear strain terms have to be included in a nonlinear or buckling analysis, as they are the only means by which destabilizing effects are introduced into the analysis. The main task of the present study is thus to examine the differences between
1 ∂w v − R2 A2 ∂
1 ∂w v − R2 A2 ∂
1 ∂w v − R2 A2 ∂
1 ∂w v − R2 A2 ∂
1 ∂w u − R1 A1 ∂␣
1 ∂w u − R1 A1 ∂␣
1 ∂w u − R1 A1 ∂␣
1 ∂w A2 ∂
−
1 ∂w A1 ∂ ␣
1 ∂w u − R1 A1 ∂␣
−

␣
(*) 2, 6 are given by Eq. (21).
Present
Sanders
Novozhilov
Reissner
DMV
Theory
冉
n
冊
1 ∂A2v ∂A1u − 2A1A2 ∂␣ ∂
Table 1 Comparison of curvatures and twisting curvature
1 ∂A1 1 ∂␣ +  A1 ∂␣ A1A2 ∂ 
1 ∂A1 1 ∂␣ +  A1 ∂␣ A1A2 ∂ 
1 ∂A1 1 ∂␣ +  A1 ∂␣ A1A2 ∂ 
1 ∂A1 1 ∂␣ +  A1 ∂␣ A1A2 ∂ 
1 ∂A1 1 ∂␣ +  A1 ∂␣ A1A2 ∂ 
␣␣ᐉ
1 ∂A2 1 ∂ +  A2 ∂ A1A2 ∂␣ ␣
1 ∂A2 1 ∂ +  A2 ∂ A1A2 ∂␣ ␣
1 ∂A2 1 ∂ +  A2 ∂ A1A2 ∂␣ ␣
1 ∂A2 1 ∂ +  A2 ∂ A1A2 ∂␣ ␣
1 ∂A2 1 ∂ +  A2 ∂ A1A2 ∂␣ ␣
ᐉ
冉 冉 冉
1 ∂A2 1 ∂␣ −  A2 ∂ A1A2 ∂␣ 
冊 冊
1 ∂A1 1 ∂ −  + A1 ∂␣ A1A2 ∂ ␣
1 1 − R1 R2
冊
1 ∂A2 1 ∂␣ −  − A2 ∂ A1A2 ∂␣ 
1 ∂A1 1 ∂ ␣ + − A1 ∂␣ A1A2 ∂
冊 冊
1 ∂A2 1 ∂␣ −  + A2 ∂ A1A2 ∂␣ 
冊 冊
冊 冊
1 ∂A1 1 ∂ −  + A1 ∂␣ A1A2 ∂ ␣
1 ∂A2 1 ∂␣  − A2 ∂ A1A2 ∂␣ 
1 ∂A1 1 ∂  + − A1 ∂␣ A1A2 ∂ ␣
1 ∂A2 1 ∂␣ −  A2 ∂ A1A2 ∂␣ 
n
冉 冉
冊 冊
1 ∂A1 1 ∂  + − A1 ∂␣ A1A2 ∂ ␣
6 2 + (*) R1 R2
冉 冉
冉 冉
冉 冉
␣ᐉ
J.G. Teng, T. Hong / Thin-Walled Structures 31 (1998) 89–115 101
102
J.G. Teng, T. Hong / Thin-Walled Structures 31 (1998) 89–115
the different nonlinear shell theories. Such a comparison was first made by Rotter and Jumikis [14]. Because Rotter and Jumikis developed their strain–displacement relations directly for thin shells of revolution, they made their comparison of the different relations for thin shells of revolution only. The comparison presented here attempts to shed new light on the differences between the nonlinear terms of membrane strains to which the calculated buckling loads are sensitive, in addition to reinforcing some of the conclusions reached by Rotter and Jumikis. The comparison is carried out here for general shells, with a more detailed discussion for shells of revolution later in the paper. The differences between the present theory and the widely used theories of Donnell [2] and Sanders [10] for general shells are highlighted in Table 2. Sanders’ nonlinear shell theory has been the most widely used in numerical nonlinear and buckling analysis of thin-shell structures. Sanders introduced the rotation about the normal to the reference surface n into the nonlinear strain–displacement relations. It will be shown in the following that the present theory for general shells reduces Table 2 Comparison of nonlinear membrane strains Theory
Present
Sanders
Donnell
⑀␣nᐉ
1 2 ( + 22 + 23) 2 1
1 2 ( + 2n) 2 ␣
1 2  2 ␣
⑀nᐉ
1 2 ( + 25 + 26) 2 4
1 2 ( + 2n) 2 
1 2  2 
⑀␣nᐉ
14 + 25 + 36
␣
␣
1 = − ␣
1 ∂w u − A1 ∂␣ R1
1 ∂w u − A1 ∂␣ R1
2 or n
1 ∂A1 1 ∂v − u A1 ∂␣ A1A2 ∂
1 ∂A2v ∂A1u − 2A1A2 ∂␣ ∂
3
1 ∂A1 1 ∂u w + v+ A1 ∂␣ A1A2 ∂ R1
–
–
4 = − 
1 ∂w v − A2 ∂ R2
1 ∂w v − A2 ∂ R2
1 ∂w A2 ∂
5
1 ∂A2 1 ∂v w + u+ A2 ∂ A1A2 ∂␣ R2
–
–
6
1 ∂A2 1 ∂u − v A2 ∂ A1A2 ∂␣
–
–
冉
1 ∂w A1 ∂␣
冊
–
J.G. Teng, T. Hong / Thin-Walled Structures 31 (1998) 89–115
103
to that of Sanders if the assumption is adopted that linear in-plane membrane strains are much smaller than out-of-plane rotations and rotation about the normal to the reference surface. A comparison of n (Table 2) with the pseudo-rotation terms defined in Eq. (21) reveals that
n ⫽
1 ( ⫺ 6) 2 2
(29)
2 ⫽
1 ( ⫹ 6) ⫹ n 2 2
(30)
6 ⫽
1 ( ⫹ 6) ⫺ n 2 2
so
From Eq. (23), it is clear that 2 ⫹ 6 is the linear membrane shear strain term ⑀␣ᐉ. In Sanders’ theory, n is the rotation about the normal to the reference surface. For those thin shell problems which may be classified as being of small strains and moderately large rotations, the linear in-plane membrane strains may be assumed to be much smaller than the rotation about the normal to the middle surface, that is
nÀ⑀␣l ⫽ (2 ⫹ 6)
(31)
Based on Eq. (31), Eq. (30) may be rewritten as
2 ⬵ n
(32)
6 ⬵ ⫺ n Substituting Eq. (32) into Eq. (23), the membrane strains of the present theory become
⑀␣ ⫽ ⑀␣␣ᐉ ⫹ ⑀␣nᐉ ⬵ 3 ⫹
1 2 ( ⫹ 2n ⫹ 23) 2 1
⑀ ⫽ ⑀ᐉ ⫹ ⑀nᐉ ⬵ 5 ⫹
1 2 ( ⫹ 25 ⫹ ( ⫺ n)2) 2 4
(33)
⑀␣ ⫽ ⑀␣ᐉ ⫹ ⑀␣nᐉ ⫽ (2 ⫹ 6) ⫹ (14 ⫹ 25 ⫹ 36) In Eq. (33), the linear strain terms 3 and 5 are small, so the square of 3 and 5 may be neglected. Similarly, (2 ⫹ 6), representing the linear membrane shear strain, is a small term, so (25 ⫹ 36) which has the order of the square of (2 ⫹ 6) can be neglected. With these simplifications, Eq. (33) becomes
104
J.G. Teng, T. Hong / Thin-Walled Structures 31 (1998) 89–115
⑀␣ ⫽ ⑀␣␣ᐉ ⫹ ⑀␣nᐉ ⬵ 3 ⫹
1 2 ( ⫹ 2n) 2 1
⑀ ⫽ ⑀ᐉ ⫹ ⑀nᐉ ⬵ 5 ⫹
1 2 ( ⫹ ( ⫺ n)2) 2 4
(34)
⑀␣ ⫽ ⑀␣ᐉ ⫹ ⑀␣nᐉ ⬵ (B2 ⫹ 6) ⫹ 14 The nonlinear membrane strains may thus be expressed as
⑀␣nᐉ ⫽
1 2 ( ⫹ 2n) 2 1
⑀nᐉ ⫽
1 2 ( ⫹ ( ⫺ n)2) 2 4
(35)
⑀␣nᐉ ⫽ 14 The above relations are identical to Sanders’ theory (Table 2). It is therefore concluded that, for shells undergoing moderately large rotations with small strains, the present theory for general shells can be simplified to that of Sanders if the assumption is adopted that linear in-plane membrane strains are much smaller than out-of-plane rotations and rotations about the normal to the reference surface.
4. Thin shells of revolution 4.1. Strain–displacement relations for thin shells of revolution For thin shells of revolution, the coordinate system is shown in Fig. 2, where → →
s , and → n are unit vectors in the meridional, circumferential and normal directions, respectively. The Lame coefficients now become
Fig. 2.
Geometry and displacements of shells of revolution.
J.G. Teng, T. Hong / Thin-Walled Structures 31 (1998) 89–115
A1 ⫽ R1; A2 ⫽ R
105
(36)
In this coordinate system, the following relations hold ds ⫽ R1d; R ⫽ R2 sin
(37)
Substitution of Eq. (36) and 37 into Eq. (21) and 22 leads to
1 ⫽ ⫺  ⫽
∂w ∂ ∂v ⫺u ; 2 ⫽ ∂s ∂s ∂s
3 ⫽
∂u ∂ 1 ∂w ⫺ v sin ] ⫹w ; 4 ⫽ ⫺  ⫽ [ ∂s ∂s R ∂
5 ⫽
1 ∂v 1 ∂u [ ⫹ u cos ⫹ w sin ]; 6 ⫽ [ ⫺ v cos ] R ∂ R ∂
(38)
and
␥1 ⫽ ⫺ 
∂ ∂ ; ␥2 ⫽ ∂s ∂s
␥3 ⫽
∂ 1 ; ␥4 ⫽ ⫺  sin ∂s R
␥5 ⫽
1 ∂ 1 ∂ ⫹  cos ; ␥6 ⫽ ⫺  cos R ∂ R ∂
冉
冊
(39)
冉
冊
The membrane strains in shells of revolution are expressed as
⑀ ⫽ ⑀ᐉ ⫹ ⑀nᐉ ⫽ 3 ⫹ ⑀ ⫽ ⑀ᐉ ⫹ ⑀nᐉ ⫽ 5 ⫹
1 2 ( ⫹ 22 ⫹ 23) 2 1
1 2 ( ⫹ 25 ⫹ 26) 2 4
(40)
␥ ⫽ ␥ᐉ ⫹ ␥nᐉ ⫽ (2 ⫹ 6) ⫹ (14 ⫹ 25 ⫹ 36) The curvature terms in shells of revolution are given by
= ᐉ + nᐉ = ␥3 + (1␥1 + 2␥2 + 3␥3) = ᐉ + nᐉ = ␥5 + (4␥4 + 5␥5 + 6␥6)
(41)
= ᐉ + nᐉ = (␥2 + ␥6) + (1␥4 + 2␥5 + 3␥6 + 4␥1 + 5␥2 + 6␥3) The above relations (Eq. (40) Eq. (41)) are identical to those derived by Rotter and Jumikis [14], Su et al. [15] and Yin et al. [16] specifically for thin shells of
106
J.G. Teng, T. Hong / Thin-Walled Structures 31 (1998) 89–115
revolution, except for a small difference in the linear twisting curvature of Su et al. [15]. These relations will be referred to as Rotter and Jumikis’ theory in the following discussion. 4.2. Comparison of nonlinear strains with existing theories Table 3 presents a comparison similar to that of Table 2 of the following nonlinear shell theories specifically for thin shells of revolution: Donnell’s theory, Sanders’ theory, Simplified Sanders’ theory, and Rotter and Jumikis’ theory as a special case of the present theory. The simplified Sanders’ theory refers to that obtained from Sanders’ theory by omitting nonlinear strain terms associated with the rotation about the normal to the reference surface n. Donnell’s nonlinear shell theory can be obtained from the simplified Sanders’ theory if all nonlinear strain terms due to inplane displacements are ignored. Donnell’s theory, being the simplest, has been Table 3 Comparison of nonlinear strains for shells of revolution Rotter and Jumikis
Sanders
Simplified Sanders
Donnell
⑀nᐉ
1 2 ( + 22 + 23) 2 1
1 2 ( + 2n) 2
1 2  2
1 2  2
⑀nᐉ
1 2 ( + 25 + 26) 2 4
1 2 ( + 2n) 2
1 2  2
1 2  2


⑀nᐉ
14 + 25 + 36

1 = − 
∂ ∂w −u ∂s ∂s
∂w ∂ −u ∂s ∂s
2 or n
∂v ∂s
3
∂ ∂u +w ∂s ∂s
4 = − 
1 ∂w − v sin R ∂
5
1 ∂v + u cos + w sin R ∂
6
1 ∂u − v cos R ∂
冉 冉 冉
∂w ∂ −u ∂s ∂s
∂w ∂s
∂v ∂u 1 R + v cos − 2R ∂s ∂
–
–
–
–
–
冉
冊 冊
冉
冊
冉
冊
1 ∂w − v sin R ∂
1 ∂w R ∂
–
–
–
–
–
–
1 ∂w − v cos R ∂
冊
冊
J.G. Teng, T. Hong / Thin-Walled Structures 31 (1998) 89–115
107
widely used in analytical studies of shell buckling. It is generally known that Donnell’s theory is not suitable for the analysis of shells in which the buckling mode is fewer than three waves around the circumference.
5. Numerical examples Whilst the assumptions required to derive the various less complete nonlinear shell theories from the theory of Rotter and Jumikis as a special case of the present nonlinear shell theory are clear, it is of interest to examine the numerical implications of such assumptions. For many shells, the different nonlinear shell theories discussed here give closely similar results [20]. However, they may give significantly different results for a number of special but realistic problems. Numerical comparisons for two examples are presented below to explore these differences. Several versions of the NEPAS program for nonlinear buckling analysis of shells of revolution [20] have been developed using the different nonlinear shell theories discussed earlier to obtain the numerical results. 5.1. In-plane buckling of an annular plate ring Rotter and Jumikis [14] suggested that a good test problem is the in-plane buckling of an annular plate (Fig. 3). For such a plate undergoing in-plane buckling, all buckling displacements are in-plane. The annular plate used here has a width b ⫽ 0.25 m, a thickness t ⫽ 0.05 m and an inner radius R ⫽ 25 m and is subjected to a radially inward distributed load at its inner edge. The material properties used are: elastic modulus E ⫽ 2 ⫻ 105 MPa and Poisson’s ratio ⫽ 0.3. To ensure that in-plane buckling is the critical mode, the plate is restrained throughout against out-of-plane displacements. The classical ring in-plane buckling solution for loads whose direction remains unchanged during buckling predicts the following buckling load: qcr ⫽
Et 3
冉 冊 b Rc
3
(42)
Fig. 3.
Annular plate subject to radial inward load.
108
J.G. Teng, T. Hong / Thin-Walled Structures 31 (1998) 89–115
The critical number of circumferential waves for the in-plane buckling of unrestrained rings is 2, provided rigid body motions are prevented. The finite element in-plane buckling loads obtained using different shell theories are listed in Table 4 together with the classical solution. It is seen that Donnell’s theory and Sanders’ simplified theory with the rotation about the normal n omitted give no solution at all for this problem, as has been found previously by Rotter and Jumikis [14]. This is to be expected, as in these two theories the nonlinear strains are equal to zero for this problem for both axisymmetric and non-axisymmetric in-plane displacements. Sanders’ theory without simplification and Rotter and Jumikis’ theory are both able to predict the correct buckling load accurately, contrary to the conclusion reached by Rotter and Jumikis [14]. It is possible to show analytically that these two theories should lead to the same and correct answer for this problem. When specialized for the annular plate with only in-plane buckling displacements, Eq. (38) may be expressed as
1 ⫽ 4 ⫽ 0; 2 ⫽ 5 ⫽
冉
冊
∂v ∂u ; 3 ⫽ ∂s ∂s
冉
1 ∂v 1 ∂u ⫹ u ; 6 ⫽ ⫺v R ∂ R ∂
(43)
冊
using the conditions that w ⫽ 0, ⫽ 0, ∂/∂s ⫽ 0, leads to
⑀nᐉ ⫽ ⫽
1 2
1 2 1 (1 ⫹ 22 ⫹ 23) ⫽ (22 ⫹ 23) 2 2
(44)
冉 冊 冉 冊 ∂v ∂s
2
⫹
∂u ∂s
1 2
2
and
⑀nᐉ ⫽ ⫽
1 2 1 ( ⫹ 25 ⫹ 26) ⫽ (25 ⫹ 26) 2 4 2
冋冉
1 1 ∂v ⫹u 2 R ∂
冊册 冋 冉 2
⫹
1 1 ∂u ⫺v 2 R ∂
(45)
冊册
2
According to the linear strain–displacement relations, the linear membrane strains are given by Table 4 In-plane buckling loads of annular plates Shell theory
Donnell
Sanders n omitted
Sanders n retained
Rotter and Jumikis
Classical solution
qcr (N/mm)
–
–
3.317
3.316
3.284
J.G. Teng, T. Hong / Thin-Walled Structures 31 (1998) 89–115
⑀ᐉ ⫽
∂u ∂ ∂u ⫹w ⫽ ∂s ∂s ∂s
⑀ᐉ ⫽
1 ∂v 1 ∂v ⫹ w sin ⫹ u cos ⫽ ⫹u R ∂ R ∂
109
(46)
冉
冊 冉
冊
So
⑀ ⫽ ⑀ᐉ ⫹ ⑀nᐉ ⫽
冉 冊 冉 冊
∂u 1 ⫹ ∂s 2
∂v ∂s
2
⫹
⑀ ⫽ ⑀ᐉ ⫹ ⑀nᐉ ⫽
(47)
冉
冊
∂u ∂s
1 2
冉
2
冊
冉
冊
(48)
1 ∂v 1 1 ∂v 1 1 ∂u ⫹u ⫹ [ ⫹ u ]2 ⫹ [ ⫺ v ]2 R ∂ 2 R ∂ 2 R ∂
As both ∂u/∂s and 1/R(∂v/∂ ⫹ u) are small, the squares of these two linear strain terms may be neglected. Eq. (47) Eq. (48) then reduce to
冉 冊 ∂v ∂s
⑀ ⫽
∂u 1 ⫹ ∂s 2
⑀ ⫽
1 ∂v 1 1 ∂u ⫹u ⫹ [ ⫺ v ]2 R ∂ 2 R ∂
Defining
␣⫽
冉
冊
冉
冊
1 ∂u v⫺ R ∂
2
(49)
冉
冊
(50)
(51)
where ␣ represents a small rotation of a circumferential line element (Fig. 4), Eq. (44) Eq. (45) become
冉 冊 ∂v ∂s
⑀nᐉ ⫽
1 2
⑀nᐉ ⫽
1 2 ␣ 2
2
(52)
(53)
Eq. (52) and 53 are the nonlinear strain terms obtained from the present relations. Now substituting the conditions that w ⫽ 0, ⫽ 0, ∂/∂s ⫽ 0 into Sanders’ relations, the rotations are simplified to
110
J.G. Teng, T. Hong / Thin-Walled Structures 31 (1998) 89–115
Fig. 4.
 ⫽  ⫽ 0
Circumferential line element before and after deformation.
冉
1 ∂v ∂u 1 v⫺ ⫹ 2 ∂s 2R ∂
n ⫽
(54)
冊
As a result, the nonlinear strains become
⑀nᐉ ⫽ ⑀nᐉ ⫽
冉
冊
1 2 1 1 ∂v ∂u 2 1 n ⫽ [ v⫺ ] ⫹ 2 2 2 ∂s 2R ∂
(55)
Introduction of Eq. (51) into the above equation leads to
⑀nᐉ ⫽ ⑀nᐉ ⫽
冋
1 1 ∂v 1 ⫹ ␣ 2 2 ∂s 2
册
2
(56)
From Fig. 5, it can been seen that  is a small rotation of a radial line element and is related to the circumferential displacement v through
冉
v⫹
⫽
冊
∂v ds ⫺ v ∂s ∂v ⫽ ds ∂s
(57)
According to the classical analysis of circular rings [21], it may be assumed that the rotation of a radial line element is approximately equal to the rotation of a circumferential line element, that is ␣ ⫽ . With this assumption, the nonlinear membrane strain terms of Sanders’ theory become
J.G. Teng, T. Hong / Thin-Walled Structures 31 (1998) 89–115
Fig. 5.
⑀nᐉ ⫽ ⑀nᐉ ⫽
111
Normal to centroidal line before and after deformation.
冉
冊
1 1 1 ␣⫹  2 2 2
2
⬵
1 2 1 2 ␣ ⫽  2 2
(58)
It can thus be concluded that Sanders’ theory and Rotter and Jumikis’ theory should lead to closely similar results for this problem, which is in agreement with numerical predictions. 5.2. Buckling of ring-stiffened cylinders under external pressure A simply-supported stiffened cylinder under uniform external pressure is considered. The geometry of the cylinder is given in Fig. 6. The buckling loads obtained using different nonlinear strain–displacement relations are given in Table 5. The buckling loads in Table 5 are in terms of the dimensionless buckling pressure cr ⫽ pR(1 ⫺ 2)/(Et), in which p is the external pressure, E is the elastic modulus, is the Poisson’s ratio, R is the cylinder radius and t is the cylinder thickness. In
Fig. 6. Ring-stiffened cylinder under external pressure.
Sanders’ theory n omitted
6.403 (3)
6.248 (3)
Donnell theory
5.192 (3)a
5.163 (3)
6.014 (3)
6.000 (3)
Sanders’ theory n retained
Number of critical circumferential buckling waves.
a
Linear buckling Nonlinear buckling
cr × 103
General instability
Table 5 Buckling pressures of ring-stiffened cylinders
–
6.451 (3)
Subbiah and Natarajan
5.962 (3)
5.967 (3)
Rotter and Jumikis’ theory
3.239 (12)
3.745 (13)
Donnell theory
3.248 (12)
3.755 (13)
Sanders’ theory n omitted
3.243 (12)
3.749 (13)
Sanders’ theory n retained
–
3.755 (13)
Subbiah and Natarajan
Interframe buckling
3.232 (12)
3.749 (13)
Rotter and Jumikis’ theory
112 J.G. Teng, T. Hong / Thin-Walled Structures 31 (1998) 89–115
J.G. Teng, T. Hong / Thin-Walled Structures 31 (1998) 89–115
113
the numerical analyses, the following values were used: E ⫽ 2.0 ⫻ 105 MPa, ⫽ 0.3, R/t ⫽ 100. The finite element model of the shell included only the ring-stiffened cylinder with one end allowed to have only meridional rotations and the other end allowed to have both meridional rotations and axial displacements. The circular plate covers at the ends of the cylinder were not included in the finite element model, but the pressure acting on the plate was included in the form of a circumferential line load on the cylinder end which was not restrained in the axial direction. There are two possible buckling modes in such stiffened cylinders: general instability and interframe buckling. These two modes correspond to two different numbers of circumferential waves. In the global mode of instability, the ring stiffeners’ displacements are nearly all in the plane of the ring. On the other hand, these rings deform out-of-plane in the interframe mode of instability. Both linear buckling loads (prebuckling large deflection effect ignored) and nonlinear buckling loads (prebuckling large deflection effect included) are presented. The effect of prebuckling large deflections is seen to be small. It can be seen that for the interframe buckling mode, the results from the different shell theories are very close to each other as buckling displacements are out-of-plane in both the cylinder and the ring. For the general mode of instability, there are some significant differences. The predictions using the theory of Rotter and Jumikis are similar to those using Sanders’ theory, however the simplified Sanders’ theory in which the rotation about the normal is ignored predicts buckling loads which are 7% higher than those of Rotter and Jumikis’ theory. It is interesting to note that Subbiah and Natarajan’s [22] results are in very close agreement with those of the simplified Sanders’ theory, but are different from those of Rotter and Jumikis’ theory. Subbiah and Natarajan [22] indicated that Sanders’ theory was used in their analysis but did not elaborate whether the rotation about the normal was included. It seems very likely that they used the simplified Sanders theory. The unconservative predictions of the simplified Sanders’ theory are due to its inability in modelling the in-plane buckling behaviour of the annular plate web of the T-section ring stiffeners as demonstrated earlier. Compared to the theory of Rotter and Jumikis, the buckling loads predicted using Donnell’s theory are lower by about 9%. The Donnell approximations in the nonlinear strain terms lead to conservative predictions of the buckling load.
6. Conclusion A new theory for numerical nonlinear and buckling analysis of thin shells has been presented. This new theory is found to reduce to that of Rotter and Jumikis [14], Su et al. [15] and Yin et al. [16] for thin shells of revolution. The nonlinear theories of Sanders and Donnell can be deduced from the present theory by invoking a number of assumptions. Numerical comparisons have shown that the simplified Sanders’ theory with the rotation about the normal omitted should not be used in finite element codes for complex shell structures. No obvious differences in numerical results have been found between Rotter and Jumikis’ theory and Sander’s more complete theory, justifying to some extent the assumptions made for arriving at San-
114
J.G. Teng, T. Hong / Thin-Walled Structures 31 (1998) 89–115
ders’ theory from the present theory or that of Rotter and Jumikis for shells of revolution. As the use of the present theory leads to little additional computational effort in numerical analysis, it is recommended that the new theory or Rotter and Jumikis’ theory for shells of revolution be used in numerical buckling analysis for complete peace of mind that no terms, however small, have been omitted.
Acknowledgements Both authors are grateful to the Hong Kong Polytechnic University for its financial support.
References [1] Love AEH. The small free vibrations and deformation of a thin elastic shell. Philosophical Transactions of the Royal Society of London 1888;179:491–546. [2] Donnell LH. Stability of thin-walled tubes under torsion. NACA Report No. 479, 1934. [3] Reissner E. A new derivation of the equations for the deformation of elastic shells. American Journal of Math. 1941;63:177–88. [4] Sanders JL, Jr. An improved first-approximation theory for thin shells. NASA Technical Report, NASA-TR-R24, 1959. [5] Koiter WT. A consistent first approximation in the general theory of thin elastic shells. In: Proceedings of the I.U.T.A.M. Symposium on the Theory of Thin Elastic Shells. Amsterdam: North-Holland, 1960: 12–33. [6] Novozhilov VV. Thin shell theory. Groningen, The Netherlands: P. Noordhoff, Ltd, 1964. [7] Flugge W. Static und Dynamik der Schalen. Berlin: Springer Verlag, 1934. [8] Marguerre K. Zur Theorie der gekrummten Platte grosser Formanderung. In: Proceedings of the Fifth International Congress of Applied Mechanics. New York: Wiley and Sons, 1938: 93–101. [9] Novozhilov VV. Foundations of the nonlinear theory of elasticity. Rochester, NY: Graylock Press, 1953. [10] Sanders JL. Nonlinear theories for thin shells. Quarterly J. of Applied Math. 1963;21:21–36. [11] Koiter WT. On the nonlinear theory of thin elastic shells. Proceedings of the Koninklijke Nederlandse Akademie van Wetenschappen, Series B 1966;69:1–54. [12] Bushnell D. BOSOR5—program for buckling of elastic–plastic complex shells of revolution including large deflections and creep. Computers and Structures 1976;6:221–39. [13] Jumikis PT. Stability problems in silo structures, PhD thesis, School of Civil and Mining Engineering, University of Sydney, 1987. [14] Rotter JM, Jumikis PT. Nonlinear strain–displacement relations for thin shells of revolution. Res. Rep. R563, School of Civil and Mining Engineering, University of Sydney, 1988. [15] Su XY, Wu JK, Hu HC. A large displacement analysis for thin shells of revolution. In: Science in China, 1987: 400–10. [16] Yin J, Suo XZ, Combescure A. About two new efficient nonlinear shell elements. SMIRT Conference Proceedings, Lausanne, Paper No. B12/1, 1987. [17] Saada AS. Elasticity: Theory and Applications. New York: Pergamon Press, 1974. [18] Budiansky B, Sanders JL. On the “best” first-order linear shell theory. In: Progress in applied mechanics (Prager Anniversary Volume). New York, 1963: 129–40. [19] Kraus H. Thin elastic shells. New York: John Wiley and Sons, 1967. [20] Teng JG, Rotter JM. Non-symmetric bifurcation analysis of geometrically nonlinear elastic–plastic axisymmetric shells subject to combined loads including torsion. Computers and Structures 1989;32:453–77.
J.G. Teng, T. Hong / Thin-Walled Structures 31 (1998) 89–115
115
[21] Brush DO, Almroth BO. Buckling of bars, plates, and shells. New York: McGraw-Hill, 1975. [22] Subbiah J, Natarajan R. Stability analysis of ring-stiffened shells of revolution. Journal of Ship Research 1982;26:125–34.