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Interlaminar stresses in antisymmetric angle-ply cylindrical shell panels Amir K. Miri a,⇑, Asghar Nosier b a b

Mechanical Engineering Department, McGill University, 817 Sherbrooke Street West, Montreal, QC, Canada H3A 2K6 Department of Mechanical Engineering, Sharif University of Technology, P.O. Box 11365-9567, Azadi Avenue, Tehran, Iran

a r t i c l e

i n f o

Article history: Available online 6 September 2010 Keywords: Antisymmetric angle-ply composite shells Interlaminar stress Free-edge effect Layerwise theory Elasticity solution

a b s t r a c t Layerwise theory of Reddy is utilized for investigating free-edge effects in antisymmetric angle-ply laminated shell panels under uniform axial extension. Following some physical arguments, governing displacement ﬁeld is divided into local and global parts. The former is discretized through the shell thickness by a zig-zag interpolation function while the latter is calculated by a ﬁrst-order shear deformation theory. Local equilibrium equations are then solved through a state space approach. Accuracy of the proposed technical solution is subsequently veriﬁed by a novel analytical elasticity solution. For this end, the problem is analytically solved for speciﬁc boundary conditions along the edges. The numerical results show excellent agreement between two theories for various composite shell panels. Ó 2010 Elsevier Ltd. All rights reserved.

1. Introduction Composite shell structures are widely used in the modern engineering. In all applications accurate design and inclusive analysis are important to guarantee the safety. However, free-edge stresses in laminated structures may result in delamination for a loading lower than what is predicted by classical theories. In a very narrow region near to free edges, localized high transverse stresses occur due to mismatch in mechanical properties of neighboring layers. This phenomenon has been investigated through a variety of analytical and numerical solutions which examine behavior of the transverse stresses close to the edges of and within the laminated composite structures. A comparative survey is made by Kant and Swaminathan [1] on different methods used for the estimation of interlaminar stresses in laminated composite plates and shells. In a pioneer work, Li et al. [2] investigated basic characteristics of the interlaminar stresses in a double-layered circular cylindrical shell with simply supported ends and subjected to a uniform pressure. A nonlinear thick composite shell element was then developed by Hinrichsen and Palazotto [3] to impose a cubic spline function on the thickness deformation. Plate-type element formulation was also extended to the analysis of moderately thick laminated composite shell structures [4]. Out-of-plane thermomechanical stresses were analytically calculated in [5] for a simply supported cross-ply cylindrical shell with different ending conditions. Incorporating nonlinear variations of the tangential displacement, Kant and Menon [6,7] applied a C0 ﬁnite element space discretization in a composite cylindrical shell. A penalty-based ⇑ Corresponding author. Tel.: +1 5143982118; fax: +1 5143984476. E-mail address: [email protected] (A.K. Miri). 0263-8223/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.compstruct.2010.08.038

numerical procedure was done by Fraternali and Reddy [8] to cover the stretching of the transverse normal in a spherical shell. In order to get a better prediction of structural response, a third-order shell theory based on Reddy’s parabolic shear strain distribution was presented by Huang [9]. Similarly, the complete stress ﬁeld was computed in orthotropic cylindrical shells [10,11], where an effective piece-wise linear function is used. Basar and Ding [12] presented the theoretical fundamentals for a layerwise theory which includes transverse shear and transverse normal strains. It provides a compromise between the continuum theory and equivalent single-layer theories. A comparative study on the interlaminar stresses was done in [13] for cross-ply parabolic and hyperbolic caps with using ﬁrst- and higher-order shear deformation theories. A post-process method based on a higher-order theory was applied to estimate the interlaminar stresses in a simply supported cylindrical shell [14]. Superimposing a cubic global displacement ﬁeld on a zig-zag linearly varying ﬁeld, an efﬁcient higher-order shell theory was introduced in [15] for cylindrical bending of simply supported symmetric laminated shells. Rao and Ganesan [16] have also studied the interlaminar stresses in clamped cross-ply spherical shells by employing three-dimensional models. They used a semi-analytical approach by means of eight-node quadratic quadrilateral and three-node isoparametric curved elements. Accuracy of layer reduction techniques on analysis of the interlaminar shear stresses in laminated cylindrical shells was studied through a combination of the conventional single-layer and multiple-layer shell theories [17]. An analytical solution based on a reﬁned asymptotic formulation was developed by Wu and Chi [18] for doubly curved laminated shells under sinusoidal distributed loading and with various edge conditions. Brank and Carrera [19] presented some aspects of newly reﬁned analysis for multilayered composite plates

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and shells, based on a piece-wise linear variation of the displacement ﬁeld through the thickness which is applicable in layerwise analysis. Fox [20] solved the equilibrium equations for throughthickness stresses in an equivalent single-layer shell theory. He assumed a slowly varying shell curvature and a small ratio of thickness to radius of curvature. Elasticity solution with trigonometric series was applied for assessing the interlaminar stresses in a cross-ply cylindrical shell [21]. Thermo-mechanical behavior of simply supported cross-ply composite and sandwich doubly curved laminated cylindrical and spherical shell panels were investigated in [22]. Later, Hossain [23] used an improved ﬁnite element model for the stress analysis of anisotropic and laminated doubly curved moderately thick composite shells and shell panels. A ﬁnite element model, which reduces the number of free parameters for each layer, was proposed to obtain interlaminar stress distribution in singly curved laminated structures [24]. Through-thickness distribution of the transverse normal stress was added to the formulation of ﬁrst- and higher-order shear deformable shell elements in [25]. Coupled mechanical, thermal, and electric responses of smart composite thick shells were derived from a zig-zag theory [26]. A global-local higher-order model was also proposed in [27] for determining through-thickness stress distributions in laminated shells under cylindrical bending. Recently, Roque and Ferreira [28] used third-order shear deformation theory of Reddy, along with a multiquadric-function based mesh-less collocation method, to analyze the deformation of plates and shells. They also utilized an optimization technique to obtain the shape parameters which are applied in a cross-ply shell under sinusoidal loads. Furthermore, some studies were made on the dynamic behavior of the interlaminar stresses in composite shells (e.g., [29,30]). Finally, the present authors [31] have calculated interlaminar hygrothermal stresses in general cross-ply and antisymmetric angle-ply shell panels by the layerwise theory of Reddy (LWT). The above review shows that less attention has been addressed toward the analysis of the interlaminar stresses in composite shell panels under mechanical extension. The present work is devoted to analytically study the interlaminar stresses within one of the most desirable lay-ups, antisymmetric angle-ply circular cylindrical shell panels, with free edges. The LWT is employed to calculate the three-dimensional stress distributions within the aforementioned shell panels. The displacement ﬁeld within LWT exhibits only a C0-continuity through the laminate thickness and, thereby, allows for the possibility of continuous out-of-plane stresses [32]. An equivalent single-layer theory (ESL) is also utilized for the determination of layer-independent global part of the displacement ﬁeld. An exact analytical elasticity solution is then proposed for assessing the accuracy of the LWT solution. Moreover, the numerical results illustrate the effectiveness of ﬁrst-order shear deformation theory (FSDST) in predicting the global constant parameter appearing in the displacement ﬁeld as well as the accuracy of LWT in determining the transverse stress ﬁeld. Finally, boundary-layer stress distributions are considered in various composite shell panels.

2. Governing equations Consider a composite laminated cylindrical shell panel, with mean radius R and uniform thickness H, subjected to a uniform axial strain as shown in Fig. 1. The circular cylindrical coordinates (x, h, r) are placed at the middle surface of the shell panel so that x and r are the axial and radial coordinates, respectively. It must be noted that an antisymmetric angle-ply laminated panel consists of an even number of equal-thickness orthotropic laminae at +b and b ﬁber orientations with respect to the x-axis [33], as depicted in Fig. 1.

2.1. Displacement ﬁeld Strain–displacement relations of elasticity within the kth layer of the shell panel are [34] ðkÞ

ðkÞ

ðkÞ

ðkÞ

@u1 u @u 1 @u2 ðkÞ ; eh ¼ þ 3 ; erðkÞ ¼ 3 r @h @x r @r ðkÞ ðkÞ ðkÞ ðkÞ ðkÞ @u2 u2 @u @u 1 @u3 3 ¼ þ ; cðkÞ þ 1 ; xr ¼ r @h @r r @x @r ðkÞ ðkÞ @u2 1 @u1 ¼ þ r @h @x

eðkÞ x ¼ cðkÞ hr cðkÞ xh

ðkÞ

ðkÞ

ð1Þ

ðkÞ

where u1 ; u2 , and u3 represent the displacement components in the x, h, and r directions, respectively, of a material point located at (x, h, r) in the kth layer of the shell panel. It is assumed here that sensibly far from the endings, the strain components are independent of the axial coordinate x. Systematic integration of (1), along with satisfying the interfacial continuities, yield the most general form of the displacement ﬁeld ðkÞ

u1 ðx; h; rÞ ¼ xrðC 4 sin h þ C 5 cos hÞ þ C 6 x þ uðkÞ ðh; rÞ ðkÞ

u2 ðx; h; rÞ ¼ xðC 1 cos h C 2 sin h C 3 rÞ 1 x2 ðC 4 cos h C 5 sin hÞ þ v ðkÞ ðh; rÞ 2 ðkÞ u3 ðx; h; rÞ ¼ xðC 1 sin h þ C 2 cos hÞ 1 x2 ðC 4 sin h þ C 5 cos hÞ þ wðkÞ ðh; rÞ 2

ð2Þ

It can be proved that C1 represents a rigid body rotation (i.e., C1 = 0); also C2 and C4 must vanish due to the problem symmetries [31]. In addition, u(k) and v(k)(w(k)) are odd (even) functions of h. Hence the reduced displacement components for the kth layer of the antisymmetric angle-ply cylindrical shell panel are ðkÞ

u1 ðx; h; rÞ ¼ C 5 xr cos h þ C 6 x þ uðkÞ ðh; rÞ 1 ðkÞ u2 ðx; h; rÞ ¼ C 3 xr þ C 5 x2 sin h þ v ðkÞ ðh; rÞ 2 1 ðkÞ 2 u3 ðx; h; rÞ ¼ C 5 x cos h þ wðkÞ ðh; rÞ 2

ð3aÞ ð3bÞ ð3cÞ

Two physically distinct terms exist in the above relations; the terms containing Ci’s which relate to the global deformation and those corresponding to the local deformation. The global terms can be recognized by some physical statements. From relations (3) it is concluded that ðkÞ

u1 ðx ¼ a; h ¼ 0; rÞ ¼ C 5 ar C 6 a

ð4aÞ

ðkÞ u2 ðx

ð4bÞ

¼ a; h ¼ 0; rÞ ¼ C 3 ar

From (4a), it is clear that C5a and C5a stand for inﬁnitesimal rotations of the lines AB and DE in Fig. 1 about the lines cc and c’c’, respectively. Let suppose that the elongation is being applied with two ﬁxtures, which are attached at the ends of the shell panel. Hence, the constant C5 is indeed canceled out and the constant C6 merely represent the uniform axial extension (i.e., C6 e). In addition, the constants C3a and C3a correspond to inﬁnitesimal rotations of the lines AB and DE about the common line d’d’ (see Fig. 1). The terms involving constant parameters C3, C5, and C6 describe the global deformation of the shell panel, whereas the functions u(k)(h, r), v(k)(h, r), and w(k)(h, r) are related to the local deformation of individual layers within the composite laminated shell panel. It should be added that abandoning AB and DE results in more effort for analyzing the problem. Indeed C5 and C6 must be simultaneously calculated by reading the displacement at two independent points (e.g., above and below surfaces). For the sake of convenience, however, the constant C5 is neglected in the subsequent formulation.

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α γ β α

Fig. 1. The geometry of an antisymmetric angle-ply laminated circular cylindrical shell panel under extension.

2.2. Equilibrium equations The governing local equilibrium equations for the kth layer of the lengthy shell panel, in the absence of body forces, are presented as [34] ðkÞ ðkÞ @ ðkÞ @ rxh @ 2 ðkÞ @r @ ðkÞ ðkÞ r rxr þ r rhr þ r h ¼ r rr rh ¼ @r @r @r @h @h ðkÞ @r þ hr ¼ 0 @h

ð5Þ

ðkÞ ij

ðkÞ ðkÞ ðkÞ ðkÞ 1 ðkÞ ðkÞ ðkÞ 1 ðkÞ ðkÞ 1 ðkÞ C 13 u1;xr þ C 36 þ C 45 u1;hr þ C 13 C 12 u C 26 2 u1;h r r 1;x r ðkÞ ðkÞ ðkÞ ðkÞ 1 ðkÞ ðkÞ ðkÞ 1 ðkÞ u þ C 36 C 26 u þ C 36 u2;xr þ C 23 þ C 44 r 2;hr r 2;x 1 1 ðkÞ ðkÞ ðkÞ ðkÞ ðkÞ ðkÞ ðkÞ ðkÞ C 22 þ C 44 2 u2;h þ C 33 u3;rr þ C 33 C 22 u r r 3;r ðkÞ 1 ðkÞ ðkÞ 1 ðkÞ ðkÞ 1 ðkÞ þ C 45 u3;xh þ C 44 2 u3;hh C 22 2 u3 ¼ 0 r r r

ð7Þ

where r represent the stress tensor which is a function of (h, r). Obtaining the stress components from three-dimensional Hook’s law of orthotropic materials [33]

with ½CðkÞ denoting the off-axis stiffness matrix. Free-edge boundðkÞ ðkÞ ðkÞ ary conditions are rh ¼ rhr ¼ rxh ¼ 0 (at h = ±h) which may be expanded as

8 rx > > > > > > rh > > > > < rr

C 12 u1;x þ C 26

9ðkÞ > > > > > > > > > > =

> rhr > > > > > > > > > > > rxr > > > > > > > > : ;

rxh

2

C 11

6 6 C 12 6 6 6 C 13 ¼6 6 6 0 6 6 0 4 C 16

C 12

C 13

0

0

C 22

C 23

0

0

C 23

C 33

0

0

0

0

C 44

C 45

0

0

C 45

C 55

C 26

C 36

0

0

9ðkÞ 3ðkÞ 8 ex > > > > > > 7 > > > > eh > > C 26 7 > > > 7 > > > > 7 < = er > C 36 7 7 7 >c > 0 7 > hr > > > > 7 > > > > 7 > cxr > > 0 5 > > > > > > : ; c C 66 xh C 16

ð6Þ

and applying relations (1) yield the displacement-based local equilibrium equations ðkÞ ðkÞ

ðkÞ

C 55 u1;rr þ C 55 ðkÞ

þ C 66 þ

ðkÞ C 55

ðkÞ ðkÞ

1 ðkÞ ðkÞ 1 ðkÞ ðkÞ 1 ðkÞ ðkÞ ðkÞ u þ C 16 u1;xh þ C 66 2 u1;hh þ C 45 u2;rr r 1;r r r

1 ðkÞ ðkÞ 1 ðkÞ ðkÞ ðkÞ ðkÞ ðkÞ 1 ðkÞ u þ C 26 2 u2;hh þ C 55 u2;xr þ ðC 36 þ C 45 Þ u3;hr r 2;xh r r 1 ðkÞ ðkÞ 1 ðkÞ u þ C 26 2 u3;h ¼ 0 r 3;x r ðkÞ

C 45 u1;rr þ C 45

2 ðkÞ ðkÞ 1 ðkÞ ðkÞ 1 ðkÞ ðkÞ ðkÞ ðkÞ 1 ðkÞ u þ C 12 u1;xh þ C 26 2 u1;hh þ C 44 u2;rr þ C 26 u2;xh r 1;r r r r

1 ðkÞ ðkÞ 1 ðkÞ ðkÞ 1 ðkÞ ðkÞ ðkÞ u C 44 2 u2 þ C 22 2 u2;hh þ C 45 u3;xr r 2;r r r ðkÞ ðkÞ 1 ðkÞ ðkÞ 2 ðkÞ ðkÞ ðkÞ 1 ðkÞ þ C 23 þ C 44 u þ C 45 u3;x þ C 22 þ C 44 2 u3;h ¼ 0 r 3;hr r r ðkÞ

þ C 44

1 ðkÞ ðkÞ ðkÞ ðkÞ 1 ðkÞ ðkÞ ðkÞ ðkÞ 1 ðkÞ u þ C 26 u2;x þ C 22 u2;h þ C 23 u3;r þ C 22 u3 ¼ 0 r 1;h r r

ðkÞ ðkÞ

ðkÞ

ðkÞ ðkÞ

ðkÞ ðkÞ

ðkÞ ðkÞ

ðkÞ

ðkÞ

C 45 u1;r þ C 44 u2;r C 44 C 16 u1;x þ C 66

1 ðkÞ ðkÞ ðkÞ ðkÞ 1 ðkÞ u þ C 45 u3;x þ C 44 u3;h ¼ 0 r 2 r

1 ðkÞ ðkÞ ðkÞ ðkÞ 1 ðkÞ ðkÞ ðkÞ ðkÞ 1 ðkÞ u þ C 66 u2;x þ C 26 u2;h þ C 26 u3;r þ C 36 u3 ¼ 0 r 1;h r r ð8Þ

Unfortunately, there is no exact analytical solution for this kind of boundary-value problems. A technical approach such as layerwise theory may be implemented to resolve this difﬁculty. The layerwise theory of Reddy is subsequently employed to ﬁnd an exact solution. 3. Technical solution 3.1. Layerwise theory of reddy (LWT) Reddy’s layerwise theory (LWT) is used to calculate the solution of Eqs. (7) which are subjected to boundary conditions (8). In LWT the displacement components of a generic point in the laminated shell panel are conveniently given as (see [32,35])

u1 ðx; h; zÞ ¼ uk ðx; hÞUk ðzÞ u2 ðx; h; zÞ ¼ v k ðx; hÞUk ðzÞ

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A.K. Miri, A. Nosier / Composite Structures 93 (2011) 419–429

u3 ðx; h; zÞ ¼ wk ðx; hÞUk ðzÞ

ð9Þ

Mkxh ¼ Mkh ¼ Rkh ¼ 0 at h ¼ h

ð15Þ

with k, here and in what follows, being a dummy index implying summation of terms from k = 1 to k = Nm + 1 while Nm represent the number of mathematical (numerical) layers (see Fig. 2). In (9) u1, u2, and u3 denote the displacement components in the x, h, and z directions, respectively. Similarly, uk(x, h), vk(x, h), and wk(x, h) are the displacements of the points initially located on the kth surface of the shell panel in the x, h, and z directions, respectively. Furthermore, the functions U0k s in (9) are the global Lagrangian interpolation polynomials associated with the kth surface [31]. Based on the reduced displacement ﬁeld in (3), the LWT displacement ﬁeld in (9) is rewritten here

The generalized stress and moment resultants in LWT are expressed in terms of the displacement functions upon substitution of (11) into (6) and the subsequent results into (14)

kj kj 0 jk kj kj 0 ðNkz ; M kh ; Mkxh Þ ¼ Bjk 36 ; D26 ; D66 U j =R þ B23 ; D22 ; D26 V j =R kj kj jk kj kj þ Akj 33 ; B23 ; B36 W j þ B23 ; D22 ; D26 W j =R Ak36 ; Bk26 ; Bk66 RC 3 Bk36 ; Dk26 ; Dk66 C 3 Ak13 ; Bk12 ; Bk16 e

ð16aÞ

u1 ðx; h; zÞ ¼ e x þ U k ðhÞUk ðzÞ u2 ðx; h; zÞ ¼ C 3 xðR þ zÞ þ V k ðhÞUk ðzÞ

kj kj kj kj kj Q kx ; Q kh ; Rkh ¼ Akj 55 ; A45 ; B45 U j þ A45 ; A44 ; B44 V j jk kj jk jk kj 0 Bjk 45 ; B44 ; D44 V j =R þ B45 ; B44 ; D44 W j =R

ð16bÞ

u3 ðx; h; zÞ ¼ W k ðhÞUk ðzÞ;

k ¼ 1; 2; . . . ; Nm þ 1

ð10Þ

Substitution of (10) into the strain–displacement relations (1) yields the following results

ex ¼ e ; eh ¼ ðV 0k þ W k ÞUk =R; ez ¼ W k U0k ; cxz ¼ U k U0k chz ¼ W 0k V k Uk =R þ V k U0k ; cxh ¼ C 3 ðR þ zÞ þ U 0k Uk =R

ð11Þ

A prime indicates an ordinary differentiation with respect to an appropriate variable. The equilibrium equations are then obtained by employing (11) in the principle of minimum total potential energy [36]. The results are, in general, 3(Nm + 1) local equilibrium equations corresponding to 3(Nm + 1) unknown functions Uk, Vk, and Wk and a global equilibrium equation associated with the parameter C3 (if it is treated as an unknown constant) as

( k ) ( k ) ( k ) 1 dM xh 1 dM h 1 dRh k k k k k RQ x ¼ þ Rh RQ h ¼ Mh RN z ¼ 0 R dh R dh R dh

ð12Þ and

Z

þho

Z

þh=2

ðR þ zÞrxh R dz dh ¼ 0

dC 3 : ho

ð13Þ

h=2

The generalized stress and moment resultants appearing in (12) are deﬁned as

Nkz ; Q kx ; Q kh ¼

Z

þh=2 h=2

Z M kh ; Mkxh ; Rkh ¼

where the laminate rigidities in (19) are deﬁned as [31] Nm Z X kj kj Akj pq ; Bpq ; Dpq ¼ i¼1

Akpq ; Bkpq ; Bkpq ; Dkpq ¼

0 0 0 C ðiÞ pq Uk Uj ; Uk Uj ; Uk Uj dz

ziþ1 zi

Nm X i¼1

Z zi

ziþ1

0 0 C ðiÞ pq Uk ; Uk ; Uk z; Uk z dz

ð17Þ

Next, the local displacement equilibrium equations within LWT are simply obtained by substituting (19) into Eq. (9)

kj jk 2 2 dU k : Dkj U 00j Akj V 00j Akj 55 U j þ D26 =R 45 B45 =R V j 66 =R þ

h

i jk kj 2 0 Bkj 36 B45 =R þ D26 =R W j ¼ 0

2 kj dV k : Dkj U 00j Akj 45 B45 =R U j 26 =R h i kj jk kj kj 2 2 Akj V 00j 44 B44 þ B44 =R þ D44 =R V j þ D22 =R h i jk kj kj 2 0 þ Bkj 23 B44 =R þ ðD22 þ D44 Þ=R W j ¼ 0 h i jk kj 2 0 Bkj 45 B36 =R D26 =R U j h i jk kj kj kj 2 0 2 þ Bkj W 00j 44 B23 =R ðD22 þ D44 Þ=R V j þ D44 =R h i kj jk kj 2 Akj 33 þ B23 þ B23 =R þ D22 =R W j ¼ RAk36 þ Bk26 þ Bk36 þ Dk26 =R C 3 þ Ak13 þ Bk12 =R e

dW k : ðrz ; rxz ; rhz ÞU0k dz

þh=2

ðrh ; rxh ; rhz ÞUk dz

ð14Þ

h=2

Also at the edges the following traction-free boundary conditions must be satisﬁed

ð18Þ

The global equilibrium equation of the shell panel is also expressed as

Z

þho

dC 3 : ho

h Bj66 U 0j þ Bj26 V 0j þ RAj36 þ Bj26 þ Bj36 W j

i R2 A66 þ 2RB66 þ D66 C 3 þ RA16 þ B16 e dh ¼ 0

Fig. 2. Local Lagrangian linear interpolation functions.

ð19Þ

Due to the computational efforts, a simpler theory can be utilized to calculate C3. In equivalent single-layer theories, the displacements are assumed to be continuous functions over the thickness and the laminated shell panel is characterized as an equivalent homogenous layer. Regarding the balance between accurate representation and simplicity of formulation, the ﬁrstorder shear shell deformation theory (FSDST) is the most attractive approach due to its simplicity and low computational cost. Employing a similar approach to what is proposed in [31], the constant C3 can be derived as

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C3 ¼

A12 B26 A22 B16 e B26 B26 A22 D66

ð20Þ

where the introduced rigidities are

2N X Apq ; Bpq ; Dpq ¼

Z

i¼1

ziþ1

zi

ð21Þ

with ½Q being as the reduced plane-stress stiffness matrix of the shell panel [33].

Since Eqs. (18) represent 3(Nm + 1) coupled ordinary differential equations with constant coefﬁcients, one may introduce the following state space variables (see [37,38])

fg1 g ¼ fUg;

fn1 g ¼ fU 0 g ¼ fg01 g;

fg2 g ¼ fVg;

fn2 g ¼ fV 0 g ¼ fg02 g;

fg3 g ¼ fW g ¼

ð22Þ

fn3 g ¼ fWg

where, for example, fg1 gT ¼ ðU 1 ; U 2 ; U 3 ; . . . ; U Nm þ1 Þ with {g2}T, . . ., and {n3}T being deﬁned similarly. Employing (22) in (18) yield two systems of coupled ﬁrst-order ordinary differential equations

fn0 g ¼ ½Afgg;

C 26

fg0 g ¼ ½Bfng þ fP1 gC 3 þ fP2 ge

ð23Þ

where {g}T = ({g1}T, {g2}T, {g3}T), and {n}T = ({n1}T, {n2}T, {n3}T). The coefﬁcient matrices [A] and [B] and the vectors {P1} and {P2} appearing in Eqs. (23) are stated in Appendix A. Due to identical conditions at h = ho and h = ho, it is clear that the variables {n} and {g} are even and odd functions of h, respectively. Keeping this in mind, it can be readily shown that the general solutions of Eqs. (23) are

fng ¼ ½U½coshðkhÞfK 1 g ½B1 fP 1 gC 3 ½B1 fP2 ge 1

fgg ¼ ½B½U½K ½sinhðkhÞfK 1 g

ð24Þ

where the diagonal matrices [cosh(kh)] and [sinh(kh)] are deﬁned to be

½sinhðkhÞ ¼ diagðsinhðk1 hÞ; sinhðk2 hÞ; . . . ; sinhðk3ðNm þ1Þ hÞÞ

~ ðkÞ ðhÞ uðkÞ ðh; rÞ ¼ U ðkÞ ðh; rÞ þ r u ðkÞ ðkÞ v ðh; rÞ ¼ V ðh; rÞ þ r v~ ðkÞ ðhÞ ~ ðkÞ ðhÞ wðkÞ ðh; rÞ ¼ W ðkÞ ðh; rÞ þ r w

Imposing (28) on (7), along with (27) and taking into account that C3 = C5 = 0, yield two decoupled sets of equations. The ﬁrst set of equations, with its associated non-homogenous boundary conditions, has the following hyperbolic solutions ðkÞ ðkÞ ~ðkÞ ¼ A1ðkÞ sinhðcðkÞ u 1 hÞ þ A2 sinhðc2 hÞ ðkÞ ðkÞ ðkÞ ðkÞ ðkÞ ðkÞ ~ ðkÞ ¼ BðkÞ w 1 A1 coshðc1 hÞ þ B2 A2 coshðc2 hÞ þ A3 ðkÞ ðkÞ ðkÞ ðkÞ ðkÞ e ðkÞ ðkÞ e ðkÞ v~ ðkÞ ¼ dðkÞ 1 A1 sinhðc1 hÞ þ d2 A2 sinhðc2 hÞ C 5 A3 h þ C 6 e h

ð29Þ The unknown constants in (29) are listed in Appendix B. The second set of equations can be obtained from the displacement ﬁeld (28) and Eqs. (7) ðkÞ ðkÞ 1 ðkÞ ðkÞ 1 ðkÞ ðkÞ ðkÞ ðkÞ 1 ðkÞ U þ C 66 2 U ;hh þ C 45 V ;rr C 55 U ðkÞ þ C 26 2 V ;hh ;rr þ C 55 r ;r r r ðkÞ ðkÞ 1 ðkÞ ðkÞ 1 ðkÞ W þ C 26 2 W ;h ¼ 0 þ C 36 þ C 45 r ;hr r

2 ðkÞ ðkÞ 1 ðkÞ ðkÞ ðkÞ ðkÞ 1 ðkÞ 1 þ C 44 V ;rðkÞ C 44 2 V ðkÞ U þ C 26 2 U ;hh þ C 44 V ;rr r ;r r r r ðkÞ 1 ðkÞ ðkÞ ðkÞ 1 ðkÞ ðkÞ ðkÞ 1 ðkÞ W þ C 22 þ C 44 2 W ;h ¼ 0 þ C 22 2 V ;hh þ C 23 þ C 44 r r ;hr r

ðkÞ

ð25Þ

k2i ’s

In Eqs. (24) and (25) denote the eigenvalues of the matrix [A][B]([C]). The [U] and [K] are the associated modal matrix and diagonal eigenvalue matrix of [C], respectively. In addition, the vector {K1} contains the 3(Nm + 1) unknown integration constants which, on the other hand, are found by imposing the boundary conditions in (15) at only one edge (in terms of C3 and e). Finally, it is to be noted that repeated zeros may exist among the eigenvalues of the matrix [C]. For this reason, artiﬁcial small terms akjUj, akjVj, and akjWj are added to the right-hand sides of Eqs. (17), so that the eigenvalues of the matrix [C] will all be distinct. Here, akjis deﬁned as (see [35] for more details on this subject):

Z

þh=2

Uk Uj dz

ð28Þ

ðkÞ

C 45 U ðkÞ ;rr þ C 45

½coshðkhÞ ¼ diagðcoshðk1 hÞ; coshðk2 hÞ; . . . ; coshðk3ðNm þ1Þ hÞÞ

akj ¼ a

ð27Þ

To solve (7), with conditions in (27), non-homogenous terms in the boundary conditions must be isolated. Hence the unknown displacement ﬁeld is decomposed as

3.2. State space approach

fn03 g;

1 ðkÞ ðkÞ 1 ðkÞ ðkÞ ðkÞ u þ C 22 v ;h þ C 23 wðkÞ ;r ¼ C 12 e r ;h r ðkÞ 1 ðkÞ ðkÞ 1 ðkÞ ðkÞ ðkÞ C 66 u;h þ C 26 v ;h þ C 26 wðkÞ ;r ¼ C 16 e r r wðkÞ ¼ 0 at h ¼ h ðkÞ

ðiÞ Q pq ð1; z; z2 Þdz p; q ¼ 1; 2; 6

ðkÞ

0

the line d’d’ in Fig. 1 (i.e., C3 = C5 0), however, may lead to an exact analytical solution. Now, the boundary conditions are deﬁned ðkÞ ðkÞ ðkÞ to be rh ¼ u3 ¼ rxh ¼ 0 which lead to the following expressions

ð26Þ

h=2

where a is a relatively small number in comparison with the rigidities Akj pq ðpq ¼ 33; 44; 55Þ. 4. Elasticity solution 4.1. Governing equations To evaluate the accuracy of the proposed technical solution, an elasticity solution is presented for a speciﬁc set of edge conditions. As it is mentioned before, Eq. (7) subjected to boundary conditions (8) have no analytical solution. Simply-support conditions at the edges of the shell panel, while preventing the global twisting about

ðkÞ ðkÞ 1 ðkÞ ðkÞ 1 ðkÞ ðkÞ ðkÞ 1 ðkÞ U ;hr C 26 2 U ;h þ C 23 þ C 44 V C 36 þ C 45 r r r ;hr 1 ðkÞ ðkÞ ðkÞ ðkÞ ðkÞ ðkÞ 1 ðkÞ W ðkÞ C 22 þ C 44 2 V ;h þ C 33 W ;rr þ C 33 C 22 r r ;r ðkÞ 1 ðkÞ ðkÞ 1 þ C 44 2 W ;hh C 22 2 W ðkÞ ¼ 0 r r

ð30Þ

which are subjected to homogenous boundary conditions

1 ðkÞ ðkÞ 1 ðkÞ ðkÞ U þ C 22 V ;h þ C 23 W ðkÞ ;r ¼ 0 r ;h r ðkÞ 1 ðkÞ ðkÞ 1 ðkÞ ðkÞ C 66 U ;h þ C 26 V ;h þ C 36 W ðkÞ ;r ¼ 0 r r W ðkÞ ¼ 0 at h ¼ h ðkÞ

C 26

ð31Þ

4.2. Fourier series expansions Fourier series representations are used here to ﬁnd the solution of Eqs. (30) with the boundary conditions in (31). Eqs. (31) are already satisﬁed if the unknown displacement functions be deﬁned as

U ðkÞ ¼ W

ðkÞ

1 X

¼

U ðkÞ n ðrÞ sinðan hÞ;

n¼0 1 X n¼0

V ðkÞ ¼

1 X

V nðkÞ ðrÞ sinðan hÞ;

n¼0

W ðkÞ n ðrÞ cosð n hÞ

a

ð32Þ

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where an = (2n + 1)p/2h. Substituting of (32) into (30) yields the below ordinary differential equations 2 2 1 ðkÞ0 ðkÞ a ðkÞ ðkÞ a U C 66 2n U nðkÞ þ C 45 V nðkÞ00 C 26 2n V nðkÞ r n r r ðkÞ ðkÞ an ðkÞ an ðkÞ C 36 þ C 45 W ðkÞ0 n C 26 2 W n ¼ 0 r r

ðkÞ

ðkÞ

C 55 U nðkÞ00 þ C 55

~ ðkÞ ðhÞ ¼ u ~ ðkÞ ðhÞ ¼ w

a2n r2

ðkÞ

W ðkÞ n C 22

1 ðkÞ W ¼0 r2 n

kkn V ðkÞ ; n ¼ Bkn r

ðkÞ

ðkÞ u3 ðx; h; rÞ

ð34Þ

which upon substitution into (33), the following algebraic equations are obtained

2

ekn 11 6 kn 4 e21 ekn 31

9 8 9 38 ekn 13 > < Akn > = > <0> = 7 B ¼ 0 ekn 5 kn 23 > > > : ; : > ; C kn 0 ekn 33

ekn 12 ekn 22 ekn 32

ðkÞ

ð35aÞ

ðkÞ

ðkÞ

ðkÞ

ð35bÞ

To have a nontrivial solution, the determinant of the coefﬁcient matrix in (35) must vanish which will yield a sixth-order polynomial. The six roots of this polynomial are optimistically distinct and, in general, complex. Thus, the solutions (34) are rewritten as 6 X

Akni r kkni ;

V ðkÞ ðh; rÞ ¼

i¼1

W nðkÞ ðrÞ ¼

6 X

Bkni Akni r kkni ;

i¼1

6 X

C kni Akni rkkni

ð36Þ

i¼1

in which Bkni and C kni being obtained by

(

Bkni C kni

)

" ¼

ekni 12

ekni 13

ekni 22

ekni 23

#(

ekni 11 ekni 21

1 6 X X

¼

1 6 X X n¼0

i¼1

1 X

6 X

n¼0

i¼1

! Akni r

kkni

þr

^ ðkÞ u n

sinðan hÞ

i¼1

! Bkni Akni rkkni þ r v^ nðkÞ sinðan hÞ ! C kni Akni r

kkni

þr

^ ðkÞ w n

cosðan hÞ

ð39Þ

The unknown constants in (39), Akni, are found by imposing the traction-free boundary conditions at the top and bottom surfaces of the laminated shell panel, the displacement continuity conditions at the interfaces, and the stress equilibrium conditions at the interfaces. Moreover, Fourier Cosine expansion must be applied for e to be able to apply in the computations

e ¼

1 X

e bn cosðan hÞ

ð40Þ

where bn = 4(1)n/p(2n + 1).

In this section, numerical results are presented and discussed for various antisymmetric angle-ply shell panels under extension. For the sake of convenience, all physical laminae are assumed to have equal thickness (h = 1 mm) and are modeled as being made up of p numerical layers within LWT (Nm = 2N p). In all the subsequent calculations, unless mentioned, p is set equal to ‘‘10” (see also the convergence study in [37]). The non-dimensional width to thickness ratio 2b/H and mean radius to thickness ratio R/H are used here to represent the geometrical properties. The composite shell panels are also assumed to be made of graphite/epoxy T300/5208 [33] with on-axis mechanical properties given as

ðkÞ

2 2 ekn 33 ¼ C 33 kkn C 22 ðkkn þ 1Þ an C 44

U nðkÞ ðrÞ ¼

ð38Þ

5. Numerical results and discussions

2 2 2 ekn ekn 11 ¼ C 55 kkn an C 66 ; 12 ¼ C 45 kkn ðkkn 1Þ an C 26 ; ðkÞ ðkÞ ðkÞ ekn 13 ¼ an C 36 þ C 45 kkn an C 26 ; ðkÞ 2 2 ðkÞ ekn 21 ¼ C 45 kkn þ 1 an C 26 ; ðkÞ 2 2 ðkÞ ekn 22 ¼ C 44 kkn 1 an C 22 ; ðkÞ ðkÞ ðkÞ ðkÞ ekn 23 ¼ an C 23 þ C 44 kkn an C 22 þ C 44 ; ðkÞ ðkÞ ðkÞ ekn 31 ¼ an C 36 þ C 45 kkn an C 26 ; ðkÞ ðkÞ ðkÞ ðkÞ ekn 32 ¼ an C 23 þ C 44 kkn an C 22 þ C 44 ; ðkÞ

v^ nðkÞ sinðan hÞ;

n¼0

where ðkÞ

1 X

^ nðkÞ cosðan hÞ w

¼ e x þ

u2 ðx; h; rÞ ¼

ð33Þ

W nðkÞ ¼ C kn rkkn

v~ ðkÞ ðhÞ ¼

n¼0

n¼0

which have a solution of the form

U nðkÞ ¼ Akn r kkn ;

n¼0 1 X

ðkÞ u1 ðx; h; rÞ

ðkÞ ðkÞ an ðkÞ0 ðkÞ an ðkÞ ðkÞ an ðkÞ0 C 36 þ C 45 U n C 26 2 U nðkÞ þ C 23 þ C 44 V r r r n ðkÞ ðkÞ an ðkÞ ðkÞ ðkÞ 1 C 22 þ C 44 2 V nðkÞ þ C 33 W nðkÞ00 þ ðC 33 C 22 Þ W nðkÞ0 r r ðkÞ

^ ðkÞ u n sinðan hÞ;

Finally, the displacement components within the kth layer of the shell panel are

ðkÞ

C 45 U nðkÞ00 þ C 45

C 44

1 X

n¼0

2 2 ðkÞ0 ðkÞ a ðkÞ ðkÞ 1 U C 26 2n U nðkÞ þ C 44 V nðkÞ00 þ C 44 V ðkÞ0 r n r n r 2 ðkÞ 1 ðkÞ an ðkÞ ðkÞ ðkÞ an C 44 2 V ðkÞ W ðkÞ0 n C 22 2 V n C 23 þ C 44 n r r r a n ðkÞ ðkÞ C 22 þ C 44 2 W ðkÞ n ¼ 0 r

ðkÞ

compatible form of solution. Backing to (29), Fourier series for ~ ðkÞ ; v~ ðkÞ , and w ~ ðkÞ are reduced to u

) ð37Þ

here ekni pq are similar to those of (35) with replacing kkn by kkni. All the required displacement components are already identiﬁed. Unknown constants should be obtained through imposing the associated boundary and continuity conditions which needs a

E1 ¼ 132 GPa; E2 ¼ E3 ¼ 10:8 GPa; G12 ¼ G13 ¼ 5:65 GPa G23 ¼ 3:38 GPa;

m12 ¼ m13 ¼ 0:24; m23 ¼ 0:59

ð41Þ

5.1. Comparison of FSDST with LWT in predicting C3 The global constant parameter appearing in the displacement ﬁeld (10) is calculated by FSDST in a closed-form (see Eq. (20)). As it is discussed earlier, this parameter may also, with more computational effort, be calculated by LWT (see Eq. (19)). In order to assess the accuracy and effectiveness of FSDST in predicting this constant parameter, numerical results of C3H/e for various 2b/H and R/H ratios are generated and presented in Table 1. It is observed that very close agreements exist between the two theories and the slight differences diminish as the shell panels become thinner. It should be reminded that the LWT results become more accurate as the number of numerical (mathematical) layer p within each lamina is increased. Finally, it is indicated here that increasing 2b/H ratio decreases the boundary-layer effect and this makes more similarity between the two theories.

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A.K. Miri, A. Nosier / Composite Structures 93 (2011) 419–429

5.2. Comparison of LWT solution with elasticity solution

5.3. Free-edge stress distributions

The interlaminar stress components can be determined by employing (10) and (11) into the Hooke law (6). It is observed, however, these stresses are not continuous at intersections of physical layers and they also have insufﬁcient accuracy. Throughthe-thickness integration of local equilibrium Eq. (5) hopefully improves the estimations (see also the integrations in [31]). The accuracy of the stresses computed by this way is evaluated with analytical solution presented in (39). It is reminded here that the speciﬁc boundary conditions used in the elasticity solution, as stated in (27), corresponds to a supposed simply-support condition. Consequently, the following boundary conditions are imposed in the LWT formulation

Backing to the free-edge conditions (15), the effect of boundarylayer region in the shell panels will be discussed. Distribution of interlaminar stresses rz/e and rxz/e at the ﬁrst and second interfaces of six-layered [30°/45°/90°2/45°/30°] shell panel, are shown in Fig. 5. It is observed that stress magnitudes in a distance ( h) near to the free edge increase rapidly and get the maximum values at h = h, while being insigniﬁcant in the interior region of the panel. For satisfying the local equilibrium state, sign of the normal stress always change near the free edge. In the ﬁrst interface, rz after a negative value varies to a tensional state that may lead to delamination. In the second interface, however, following two sign variations it comes to compression which guarantees the system lay-up. The shear stress rxz, which is considerably greater than rhz, uniformly increases to a noticeable value. The second interface

M kxh ¼ M kh ¼ W k ¼ 0 at h ¼ h

ð42Þ

Also C3 = 0 should be applied in (10). In this particular case, the interlaminar normal stress along the free-edge can be better estiðkÞ mated by directly utilizing the Hooke law. The expression of rz obtained from (6), with taking into account Wk(±h) = 0, yield

ðkÞ ðkÞ 0 ðkÞ 0 rðkÞ z h¼h ¼ C 13 e þ C 23 V j ðh ÞUj ðzÞ=R þ C 36 U j ðh ÞUj ðzÞ=R

where 0 Dkj 26 U j

U 0j

þ

and

0 Dkj 22 V j

V 0j ’s

5

ð43Þ

can be obtained by applying (42) in (19)

¼ RBk12 e ;

0 kj 0 k Dkj 66 U j þ D26 V j ¼ RB16 e

z 0

ð44Þ

Distribution of interlaminar stress components, calculated by LWT and elasticity solutions, along 20°/75° interface of [20°/ 75°/75°/20°] shell panel with R/H = 50 and 2b/H = 5 are illustrated in Fig. 3. It must be noted that the convergence of Fourier series expansions were systematically checked in a simple trail and error manner, by increasing the truncation constants in (39). It was found that 250 coefﬁcients assure convergence for boundary-layer stresses in different lay-ups. Also, interlaminar normal stress distributions along the middle surface of [b°/b°] shell panel, with the same geometry ratios, for different values of b are presented in Fig. 4. Good agreements exist between LWT and elasticity results within these panels. In the subsequent results R/H = 30 and 2b/H = 10 are set in our calculations.

z -5

-10 Elasticity LW T

-15

xz -20 0

0.2

0.4

0.6

0.8

1.0

Fig. 3. Distribution of interlaminar stresses along the 20°/75° interface of [20°/ 75°/75°/20°] panel under a uniform axial strain. Table 1 Numerical values of C3H/e for three antisymmetric angle-ply laminated shell panels under axial strain e according to FSDST and LWT. Stacking sequence

R/H

Theory

2b/H = 5

2b/H = 10

[20°/ 20°]

10

FSDST LWT

p=1 p=4 p=8

5.2402708 5.2293949 5.1937902 5.1956329

5.2402708 5.2317129 5.2196562 5.2206797

p=1 p=4 p=8

5.2402708 5.2292066 5.1922423 5.1939467

5.2402708 5.2314105 5.2187248 5.2196732

p=1 p=4 p=8

4.8205443 4.8147118 4.8142053 4.8147877

4.8205443 4.8170197 4.8175699 4.8178615

p=1 p=4 p=8

4.8205443 4.8140434 4.8140478 4.8146568

4.8205443 4.8154028 4.8161584 4.8164610

p=1 p=4

0 3.74182E34 2.25768E32

0 9.59474E35 4.52298E32

50

FSDST LWT

10

Elasticity LW T

8

= 80

6

= 45

4 [15°/70°/70°/15°]

10

FSDST LWT

2

= 15 50

[0°/90°/0°2/90°/0°]

10

FSDST LWT

FSDST LWT

0 0

0.2

0.4

0.6

0.8

1.0

Fig. 4. Interlaminar normal stress rz distribution at the middle interface of [b°/b°] panel under a uniform axial strain for various b.

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A.K. Miri, A. Nosier / Composite Structures 93 (2011) 419–429

2 15

, z /h = -1.0 , z /h = -2.0 , z /h = -1.0 xz , z /h = -2.0 xz z z

σ / ε (GPa)

10

1

5

= 1.00 = 0.99 = 0.98 = 0.96

0

0 -1

-5

0

0.2

0.4

0.6

0.8

-2

1.0

Fig. 5. Distribution of interlaminar stresses rz and rxz along the 30°/45° and 45°/90° interfaces of [30°/45°/90°2/45°/30°] panel under a uniform axial strain.

-4

0

4

8

Fig. 7. Through-the-thickness distribution of interlaminar shear stress rxz at and near the free-edge of [(45°/45°)2] panel under a uniform axial strain.

2

1.0

= 1.00 = 0.99 = 0.98 = 0.97

p = 20 p = 10 p=5 p=2

0.5

z /h

1

z /h

-8

0

-1

0.0

-0.5

-2 -4

-2

0

2

4

6

-1.0

-2

-1

1

2

Fig. 6. Through-the-thickness distribution of interlaminar normal stress rz at and near the free-edge of [20°/60°/60°/20°] panel under a uniform axial strain.

Fig. 8. Through-the-thickness distribution of interlaminar shear stress rhz at the free-edge of [30°/30°] panel under a uniform axial strain as a function of p.

curve corresponds to more deviation of ﬁber (angular-) orientations and hence bears bigger interlaminar shear stress. Throughthe-thickness distribution of interlaminar normal stress rz/e at and near the free edge of [20°/60°/60°/20°] shell panel is demonstrated in Fig. 6. The signiﬁcantly high gradient of stress is obvious in the vicinity of the free surface and specially its magnitudes in junctions of the physical layers. Having even behavior with respect to thickness coordinate, the normal stress maximizes at the middle surface (i.e., z/h = 0). Also, thickness variation of interlaminar shear stress rxz/e in boundary-layer region of [(45°/45°)2] shell panel is presented in Fig. 7. The difference between points at the physical interfaces and the points along individual layers markedly prove the effect of material discontinuity on transverse stresses. The remaining component of interlaminar stresses is expected to be vanished along the traction-free surface. However, as shown in Fig. 8 for a two-layered [30°/30°] shell panel, it never entirely disappear. Even increasing the numerical layers used in

LWT leads to non-vanishing stress, especially near the physical interface. This may come from nature of the present formulation and particularly the weak form of the boundary condition, where the generalized stress resultant is forced to vanish (see Eq. (15)). Commonly the out-of-plane stresses calculated by LWT at exactly free edge are of little credit. Next, the single-layer shell panel is considered for free-edge effect. Boundary-layer region is generated mainly by equilibrium state of edge elements as well as the difference in physical properties of adjacent layers. The second reason is undoubtedly canceled in a single lamina. Two-layered [(0°)2] and [(90°)2] panels are chosen here because the formulation is based on 2N-layered shell panels. Distribution of normal stress rz/e along the middle surface of these panels for traction-free boundary conditions (15) and simply-support boundary conditions (42), for better comparison, are shown in Fig. 9. The order of free-edge stress in single-layered shell panels is considerably slighter than the previous cases. However, it is observed that for hypothetical

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A.K. Miri, A. Nosier / Composite Structures 93 (2011) 419–429

10

Simply-support , [02], LWT

40

Case I Case II

Simply-support , [02], Elasticity

8

Simply-support , [902], LWT

xz

30

Simply-support , [902], Elasticity

6

5(103) Traction-free , [02], LWT

20

5(103) Traction-free , [90 ], LWT

4

2

10 2 0

z

0 0

0.2

0.4

0.6

0.8

1.0

Fig. 9. Variations of interlaminar normal stress rz along the middle interface of [0°2] and [90°2] panels under a uniform axial strain.

0

z

-5

Case I Case II

- 15

0

0.2

0.4

xz

0.6

0.8

15

30

45

60

75

90

Fig. 11. Inﬂuence of C3 on the interlaminar stresses rz and rxz at intersection of the free-edge and middle surface of the [b°/b°] panel as a function of b.

creases (increase) from Case I to Case II. Blocking the twisting reduces in-plane strains and consequently in-plane stress magnitudes and as a result, that may be seen in equilibrium relations, contributions of out-of-plane shear stresses promote. Higher magnitudes of transverse shear stresses need smaller normal stress for satisfying the equilibrium conditions. For comprehensive study the effect of C3, interlaminar stresses rz/e and rxz/e at intersection of free-edge and middle surface of [b°/b°] shell panel are displayed in Fig. 11 versus b. Two extreme values of b correspond to cross-ply shell panel (i.e., C3 = 0) that show the conformity of both results. In the range 15° < b < 30°, however, notable differences are easily revealed that can be applied in engineering design. At the end, it should be noted that the effect of geometry parameters as well as various lay-ups on interlaminar (thermal) stress distributions are fully considered by the present authors in [31] and is disregarded in this paper.

z

- 10

-10 0

1.0

Fig. 10. Inﬂuence of C3 on the distributions of interlaminar stresses along the 30°/ 60° interface of the [0°/30°/60°/60° /30°/0°] panel under a uniform axial strain.

conditions (42) the order of stress is comparable with other laminated panels. It may be concluded that non-zero rhz along the edge is so important that replace other effects. 5.4. Effect of C3on free-edge stresses The effect of terms involving the global constant C3 is now considered by introducing two subsequent cases in our code: – Case I: C3 is computed by Eq. (19). – Case II: The global twisting about the line d’d’ in Fig. 1 is prevented (i.e., C3 0). Distribution of interlaminar stresses rz/e, rhz/e, and rxz/e along 30°/60° interface of [0°/30°/60°/60°/30°/0°] shell panel are displayed in Fig. 10. Conserving behavior of the stress variation, the absolute values of Case II with respect to Case I differ in the boundary-layer region. In fact, normal stress (shear stresses) de-

6. Concluding remarks A technical displacement-based approach is presented for studying the free-edge effects in long antisymmetric angle-ply laminated circular cylindrical shell panels under uniform axial extension. The most general form of the displacement ﬁeld in a laminated cylindrical shell is obtained from the displacement– strain relations. After making some physical arguments, a reduced form of the displacement ﬁeld is identiﬁed for antisymmetric angle-ply shell panels under extension. Reddy’s layerwise shell theory (LWT) is then employed to numerically determine the displacement components. First-order shear deformation shell theory (FSDST) is also used to obtain a closed-form expression for the terms indicating the global deformation. The equilibrium equations of LWT are subsequently solved through a state space approach. Moreover, the accuracy of boundary-layer stresses are investigated by generating and comparing analytical elasticity and LWT solutions for antisymmetric shell panels with speciﬁc edge conditions. The numerical results reveal very close agreements between LWT and elasticity theory. In the numerical study, various composite shell panels are considered to illustrate the interlaminar stresses. It is observed that preventing the global twisting of the aforementioned panels may increase (decrease) the localized transverse shear (normal) stresses.

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ðkÞ ðkÞ ðkÞ ðkÞ ðkÞ ðkÞ ðkÞ ðkÞ ðkÞ ðkÞ ðkÞ ðkÞ a1 b3 a3 b1 ðcðkÞ Þ4 þ a2 b3 þ a1 b4 a3 b2 a4 b1 ðcðkÞ Þ2

Appendix A The matrices [A] and [B] and vectors {P1} and {P2} appearing in (23) are given as:

2

½a11 ½a12 ½a13

3

6 7 ½A ¼ 4 ½a21 ½a22 ½a23 5; ½0 ½0 8 9 f0g > > < = fP 1 g ¼ f0g ; > : > ; fp1 g

2

½I

½0

6 ½B ¼ 4 ½0

½I

8 9 > < f0g > = fP2 g ¼ f0g > : > ; fp2 g

½0

ðkÞ ðkÞ

ðkÞ a1 ðkÞ a4

7 ½0 5; ½b31 ½b32 ½b33 ðA-1Þ

ðB-3Þ

The new constants used in (B-2) and (B-3) are found to be

3

½I

ðkÞ ðkÞ

þ a2 b4 a4 b2 ¼ 0

e ; ¼ C 66 C 26 C 3 ðkÞ

ðkÞ

26

36

ðkÞ

ðkÞ

45

ðkÞ ðkÞ ðkÞ e ðkÞ b1 ¼ C 26 C 22 C 3 ; ðkÞ b4

ðkÞ

a2 ¼ C 55 ; ðkÞ ðkÞ ðkÞ ðkÞ e ðkÞ ¼C þC þC C C

¼

ðkÞ C 22

þ

ðkÞ 2C 44

þ

26

e ; a3 ¼ C 26 C 4 ðkÞ

ðkÞ

5

ðkÞ

ðkÞ ðkÞ e ðkÞ b3 ¼ C 22 C 4 ;

ðkÞ

b2 ¼ 2C 45 ; ðkÞ e ðkÞ C C

ðkÞ C 23

ðkÞ

ðB-4Þ

5

22

where [0] and [I] are (N + 1) (N + 1) zero and identity matrices, respectively, and {0} is a zero vector with (N + 1) rows. Moreover

e ðkÞ (i=1,..,6) in above relations ½C denote the stiffness matrix. Also, C i are deﬁned as

½a11 ¼ ½d1 1 ½d2 ; ½a12 ¼ ½d1 1 ½d3 ; ½a13 ¼ ½d1 1 ½d4

e ðkÞ ¼ C 16 C 26 C 22 C 26 ; C 1 ðkÞ ðkÞ ðkÞ ðkÞ C 26 C 26 C 22 C 66

ðkÞ

½a21 ¼ ½d6 1 ½d5 ; ½a22 ¼ ½d6 1 ½d7 ; ½a23 ¼ ½d6 1 ½d8

ðkÞ

1 g ¼ R½D44 1 ðR2 fA36 g þ RfB26 g þ RfB36 g þ fD26 gÞ; fp

ðkÞ

ðA-2Þ

where

½d1 ¼ ð1=RÞ½D66 ð1=RÞ½D26 ½D22 1 ½D26 ½d3 ¼ R½A45 ½B45 T ½D26 ½D22 1 ðR½A44 ½B44 ½B44 T þ ð1=RÞ½D44 Þ ½D26 ½D22 1 ½a ½d4 ¼ ½B36 þ ½B45 T ð1=RÞ½D26 þ ½D26 ½D22 1 ð½B23 ½B44 T þ ð1=RÞ½D22 þ ð1=RÞ½D44 Þ ½d5 ¼ ½D26 ½D66 1 ðR½A55 Þ R½A45 þ ½B45 þ ½D26 ½D66 1 ½a ½d6 ¼ ð1=RÞ½D26 ½D66 1 ½D26 ð1=RÞ½D22 ½d7 ¼ ð1=RÞ½D26 ½D66 1 ðR½A45 ½B45 T Þ R½A44 þ ½B44 þ ½B44 T ð1=RÞ½D44 ½a ½d8 ¼ ½D26 ½D66 1 ð½B36 ½B45 T þ ð1=RÞ½D26 Þ þ ½B23 ½B44 T þ ð1=RÞð½D22 þ ½D44 Þ ½d9 ¼ ½B36 T ½B45 þ ð1=RÞ½D26 ½d10 ¼ ½B23 T ½B44 þ ð1=RÞð½D22 þ ½D44 Þ ½d11 ¼ R½A33 þ ½B23 þ ½B23 T þ ð1=RÞ½D22 þ ½a

ðA-3Þ

Appendix B ðkÞ

The coefﬁcients Ai ’s in (29) are the solution of the below system

9 38 8 ðkÞ 9 ðkÞ ðkÞ e > 0 coshðc1 h ÞA1 > C > > > > 1 < = < = 7 ðkÞ ðkÞ ðkÞ 7 ðkÞ ðkÞ e e e e ¼ coshð c h ÞA C 5 5 þ C C 2 2 2 6 > > > > : ; > > : ðkÞ ; 0 A3 1

c1ðkÞ cðkÞ 2

6 ðkÞ 6d 4 1 ðkÞ B1

ðkÞ

d2

ðkÞ

B2

ðB-1Þ where coefﬁcients

and

ðkÞ di ’s

are deﬁned as

a1 ðci Þ2 þ a2 ; 3 ðkÞ ðkÞ ðkÞ ðkÞ a2 ci þ a4 ci 2 e ðkÞ cðkÞ C e ðkÞ BðkÞ cðkÞ C e ðkÞ BðkÞ ¼ C 5 3 4 i i i i ðkÞ

ðkÞ

ðkÞ Bi

ðkÞ

ðkÞ

Bi ¼ ðkÞ

di

ðkÞ

ðkÞ

ðkÞ

ðkÞ

ðkÞ

ðkÞ

ðkÞ

e ðkÞ ¼ C 23 2C 22 ; C 5 ðkÞ ðkÞ C 23 C 22

ðkÞ

ðkÞ

ðkÞ

ðkÞ

e ðkÞ ¼ C 12 C 66 C 16 C 26 ; C 2 ðkÞ ðkÞ ðkÞ ðkÞ C 66 C 22 C 26 C 26 ðkÞ C 44 ðkÞ ðkÞ C 23 C 22 ðkÞ C 13 ðkÞ C 22

e ðkÞ ¼ C 4 ðkÞ

e ðkÞ ¼ C 12 C 6 ðkÞ C 23

; ðB-5Þ

References

½d2 ¼ R½A55 ½D26 ½D22 1 ðR½A45 ½B45 Þ þ ½a

2

ðkÞ

e ðkÞ ¼ C 36 þ C 45 C 26 ; C 3 ðkÞ ðkÞ C 23 C 22

½b31 ¼ R½D44 1 ½d9 ; ½b32 ¼ R½D44 1 ½d10 ; ½b33 ¼ R½D44 1 ½d11 2 g ¼ R½D44 1 ðRfA13 g þ fB12 gÞ fp

ðkÞ

ði ¼ 1; 2Þ

ðB-2Þ

and also c1 and c2 are two distinct roots of the following equation

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