An assessment of several displacement-based theories for the vibration and stability analysis of laminated composite and sandwich beams

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Composite Structures 84 (2008) 337–349 www.elsevier.com/locate/compstruct

An assessment of several displacement-based theories for the vibration and stability analysis of laminated composite and sandwich beams Wu Zhen *, Chen Wanji Department of Aeronautics and Astronautics, Shenyang Institute of Aeronautical Engineering, Shenyang 110034, China Available online 10 October 2007

Abstract Several displacement-based theories are assessed by analyzing the free vibration and the buckling behaviors of laminated beams with arbitrary layouts as well as soft-core sandwich beams. The equations governing the dynamic response of laminated structures are derived by using Hamilton’s principle. However, equations of equilibrium for buckling problems are given by employing the principle of virtual displacements. Moreover, using Navier’s technique and solving the eigenvalue equations, analytical solutions based on the global–local higher-order theory used in this paper are first presented in present study. At the same time, the effect of the order number of higherorder shear deformation as well as interlaminar continuity of transverse shear stress on the global response of both laminated beams and soft-core sandwiches has been also studied. Numerical results show that by increasing the order number of in-plane and transverse displacement components, the global higher-order theories can reasonably predict the natural frequencies and the critical loads of laminated beams with arbitrary layouts and soft-core sandwich beams whereas these global higher-order theories are still less accurate compared to the global–local higher-order theory and the zig-zag theory used herein. Ó 2007 Elsevier Ltd. All rights reserved. Keywords: Displacement-based theories; Free vibration and stability; Laminated composite beams; Soft-core sandwiches; Analytical solutions

1. Introduction With increasing applications of fiber-reinforced composite laminates in various engineering structures due to their high specific strength and high specific stiffness, various displacement-based theories for laminated structures have been developed. As far as the development of displacement theories for laminated composite structures is concerned, two approaches are usually adopted in the published literature. One approach neglects the continuity conditions of transverse shear stresses whereas another approach can a priori satisfy continuity of transverse shear stresses at interfaces. In the first approach, the first-order theories widely used assume a constant transverse shear strain across *

Corresponding author. E-mail address: [email protected] (W. Zhen).

0263-8223/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.compstruct.2007.10.005

the thickness direction. Therefore, the shear correction factor is generally used to adjust the transverse shear stiffness for the dynamic and stability problems of laminates [1–4]. To avoid the use of shear correction factor, the global higher-order theories have been extensively developed. Firstly, Helinger and Reddy [5] have used the third order theory to analyze bending and vibration of laminated beams. Subsequently, Marur and Kant [6] used both second-order theory and third-order theory neglecting transverse normal strain to predict natural frequencies of laminated composite and sandwich beams. Further, the third-order theory considering transverse normal strain as well as other higher-order theories is extended to analyze the free vibration of laminated composite and sandwich plates [7–9]. In addition, buckling problems of skew laminated composite and sandwich panels have been also studied by using the third-order theories [10]. By increasing the order number of in-plane and transverse

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displacement components, a higher-order theory named as HSDT-98 herein, in which the in-plane displacement field consists of nineth-order polynomial in global thickness coordinate z whereas the transverse deflection is represented by an 8th-order polynomial of global coordinate z, is proposed to analyze dynamic and buckling problems of multilayered composite beams [11]. Moreover, this higher-order theory in conjunction with other global higher-order theories has been developed to analyze vibration and stability of laminated composite and sandwich plates and shells [12–15]. In addition to the global higher order theories above mentioned, other higher-order theories can be found in following articles [16–19]. Numerical results showed that the global higher-order theories can predict the satisfactory global response for general laminates (e.g., natural frequencies and buckling stresses). However, global higher-order theories can not account for continuity of transverse stress at interfaces, so these theories lose capability to accurately predict the variation of transverse shear stresses through the thickness directly from constitutive equations. Moreover, the global third order theories have severe limitations in the prediction of global response of thick laminations with arbitrary layouts [20]. In particular, the recent researches [20,21] have shown that the global third order theories considerably overestimate natural frequencies of the soft-core sandwich plates. To overcome the limitations of global higher theories, Kapuria et al. [22] have used zig-zag theory satisfying the interlaminar continuity of transverse shear stresses to predict the dynamic and buckling response of laminated beam with arbitrary layouts. Numerical comparison with exact solutions showed that the third order theory as well as the first-order theory obviously overestimates the dynamic response of laminated beams with arbitrary thickness and materials at each ply whereas the zig-zag theory can reasonably predict the natural frequencies of ‘‘special” structures. In addition, Dafedar et al. [23] used the mixed theory to predict buckling of laminated composite and sandwich plates, and a conclusion has been drawn that the global third order theories fail to accurately predict the critical loads of soft-core sandwich plates. By analyzing the stability of laminated composite and sandwich struts, Dafedar and Desai [24] have also drawn the same conclusion. Recently, authors [25,26] have used the finite element methods based on global–local higher-order theory to predict the natural frequencies and critical loads of soft-core sandwich plates. Numerical results showed that global–local higher-order theory proposed by Li and Liu [27], which a priori satisfies the continuity of transverse shear stresses at interfaces, has capability to accurately analyze vibration and stability of soft-core sandwiches. Previous investigations showed that the global thirdorder theories grossly overestimate the natural frequencies and critical loads of soft-core sandwiches. To the authors knowledge, however, a study is not available in which the higher-order theories with higher-order shear

deformation such as the previously mentioned HSDT-98 and the zig-zag theories [22,28–30] are used to analyze the vibration and stability of soft-core sandwich beams. In view of this situation, this paper will attempt to verify, with the help of numerical examples, whether both the global theories with higher-order shear deformation and the zig-zag theories satisfying continuity of transverse shear stresses at interfaces can accurately predict the global response of soft-core sandwich beams. In other words, this paper also studies the effect of the order number of higher-order shear deformation as well as interlaminar continuity of transverse shear stress on the global response of soft-core sandwich beams. To this end, several displacement-based theories have been considered herein. Moreover, analytical solutions for free vibration and stability of laminated composite and sandwich beams based on global–local higher-order theory are first reported in this paper. 2. Fundamental equations of laminated beams 2.1. Displacement theories For convenient comparison of various theories, the present work done only considers the area of laminated composite and sandwich beams. Various displacement-based theories for laminated beams, which used in present investigations, will be simply described herein. 2.1.1. Global–local higher-order theory (GLHT) The global–local higher-order theory proposed by Li and Liu [27] a priori satisfies displacements and transverse shear stresses continuity conditions at interfaces. Displacement fields of the global–local higher-order theory are simply given by uk ¼ u0 þ Uk1 ðzÞu11 þ Uk2 ðzÞu1 þ Uk3 ðzÞu2 þ Uk4 ðzÞu3 þ Uk5 w0;x wk ¼ w0 ð1Þ Uki

where are the function of material constants and thickness of laminates. Expression of Uki can be found in references [27]. Therefore, the total number of the unknown variables is only six which is independent of the number of layers in any multilayered beams. 2.1.2. Zig-zag theory (ZZT) satisfying continuity of transverse shear stresses at interfaces The zig-zag theories for general lamination configurations have been proposed by superimposing a cubic varying displacement field on a zig-zag linear displacement [28–30]. Herein, the zig-zag theory proposed by Cho and Parmerter [28] is only considered. By using transverse shear stress continuity conditions at interfaces as well as transverses shear free surface conditions, finial displacement field of zig-zag theory proposed by Cho and Parmerter [28] can be rewritten as follows

W. Zhen, C. Wanji / Composite Structures 84 (2008) 337–349

uk ¼ u0 þ Wk1 ðzÞu3 þ Wk2 w0;x k

w ¼ w0

ð2Þ

where, expression of Wki can be found in Appendix. The total number of the unknown variables in zig-zag theory is only 3. 2.1.3. Global higher-order shear deformation theories (HSDT) To approximate the three-dimensional elasticity problem to a two-dimensional plate problem, in principle, the displacement fields can be expanded as a Taylor’s series in term of the thickness coordinate. By expanding the displacement components in term of thickness coordinate, Matsunaga [11] has proposed a higher-order shear deformation theory HSDT-98. The in-plane displacement field of HSDT-98 consists of nineth-order polynomial in global thickness coordinate z whereas the transverse deflection is represented by an eighth-order polynomial of global coordinate z. Displacement fields of HSDT-98 can be detailedly given by u¼

9 X

u i zi

i¼0

w¼

8 X

ð3Þ wi z

HSDT-76 [11] u i zi

i¼0

w¼

6 X

ð4Þ wi zi

i¼0

This model includes 15 unknown variables. HSDT-54 [11] u¼

5 X

4 X

ð5Þ wi zi

This model includes 11 unknown variables. HSDT-33 [7] u¼

ð8Þ

A shear correction factor of 5/6 is adopted in computed results using the first-order theory. Moreover, this theory only comprises 3 unknown variables.

ui z

3 X

ð6Þ wi zi

i¼0

This model includes 8 unknown variables. HSDT-Reddy [31]

The analytical formulations using global–local higherorder theory are only detailedly given in this paper, and the results obtained from other models can be presented by using the same procedure. By substituting the displacement components of global–local higher-order theory into the strain–displacement equations, the following equations can be obtained by ou0 ou1 ou1 ou2 ou3 o2 w0 þ Uk1 1 þ Uk2 þ Uk3 þ Uk4 þ Uk5 2 ox ox ox ox ox ox   oUk1 1 oUk2 oUk3 oUk4 oUk5 ow0 u þ u1 þ u2 þ u3 þ 1 þ cxz ¼ oz 1 oz oz oz oz ox ex ¼

2.3. Constitutive relations



i

i¼0

w¼

u ¼ u0 þ zu1 w ¼ w0

By incorporating the plane stress condition (rz = 0) through the thick direction, the constitutive equation for the kth ply in its material-axis system may be written as

i¼0

3 X

2.1.4. First-order shear deformation theories (FSDT) The first-order shear deformation theory [32], commonly known as Mindlin plate theory, predicts constant transverse shear stress through the thickness of plate. First-order theory can be written as

ð9Þ

u i zi

i¼0

w¼

Displacement fields in Reddy’s theory only consist of 3 unknown variables.

2.2. Strain–displacement relations

This model includes 19 unknown variables. In addition to above model, the following higher-order shear deformation theories already published in previous literature are also employed for comparison. They are

7 X

In addition, Reddy [31] also proposed a higher-order shear deformation theory which can satisfy the transverse shear stress free boundary conditions. Herein, Reddy’s theory is named as HSDT-Reddy in the present work.   ow 4z3 þ z  2 cx u ¼ u0  z ox ð7Þ 3h w ¼ w0

i

i¼0

u¼

339

r1 s13

k

 ¼

c11  c213 =c33

0

0

c44

k 

e1 c13

k

ð10Þ

In the above equations, (r1, s13) are the stresses, (e1, c13) are the linear strain components and the cij are the corresponding elastic constant of the kth lamina. Using coordinate transformation, the constitutive equation of a lamina in a common structural axis system may be expressed as follows:

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rk ¼ Qk ek

ð11Þ

where, rk ¼ f rx sxz g; ek ¼ f ex cxz g; Qk is the transformed material constants for the kth layer.

h 2

orx dz ¼  ox

h2

þ

2.4. Governing equations

3 X

Uklþ1

l¼1

2.4.1. Equations of dynamic equilibrium for free vibration To derive the equations of dynamic equilibrium, Hamilton’s principle is used herein. Hamilton’s principle for present problems can be expressed in analytical form from time t1 to t2 as follows: Z t2 Z _ u_ þ wd _ wÞdV _ rx dex þ sxz dcxz  qðud dt ¼ 0 ð12Þ t1

Z du0 :

V

where q is the mass density of material; the over dot represents partial differentiation with respect to time. By employing Eqs. (1), (9), (11) and (12) and integrating by parts, the following equations of motion can be given by: Z h2 Z h2 orx du0 : dz ¼ q h2 ox h2 ! 3 X w0 k 1 k k o€ Ulþ1 €  € u0 þ U 1 € u1 þ ul þ U5 dz ox l¼1  Z h2  orx oUk1  sxz dz Uk1 du11 : ox oz h2 ! Z h2 3 X w0 k k 1 k k o€ U1 q € u0 þ U1 € Ulþ1 € u1 þ ul þ U 5 ¼ dz ox h2 l¼1  Z h Z h 2 2 oUkiþ1 k orx  sxz dz ¼ Uiþ1 Ukiþ1 q dui : ox oz h2 h2 ! 3 X o€ w 0 Uklþ1 € u11 þ ul þ Uk5  € u0 þ Uk1 € dz ox l¼1    Z h Z h 2 2 o2 rx oUk osxz Uk5 2  1 þ 5 q dw0 : dz ¼ ox oz ox h2 h2 " ! # 3 2 €0 u0 u11 X o€ ul k o€ k o€ k ko w € 0 dz þ U1 þ þ U5 2  w Ulþ1  U5 ox ox ox ox l¼1 ð13Þ in which i = 1–3. 2.4.2. Equations of equilibrium for buckling problems The principle of virtual displacements is used to derive the equations of equilibrium herein. The principle of virtual displacements for present problems can be written as follows: Z rx dex þ sxz dcxz þ rx0 ðu;x du;x þ w;x dw;x ÞdV ¼ 0 ð14Þ V

where rx0 is the axial stress. By employing Eqs. (1), (9), (11) and (14) and integrating by parts, the following equations can be given by

Z

h 2

rx0 h2

2 1 o2 u0 k o u1 þ U 1 ox2 ox2 !

3 o2 u l k o w0 þ U dz 5 ox2 ox3

 Z h Z h 2 2 oUk1 o2 u0 k orx  sxz dz ¼  : U1 rx0 Uk1 ox oz ox2 h2 h2 ! 3 2 1 3 X o2 ul k o u1 k k o w0 þU1 2 þ Ulþ1 2 þ U5 3 dz ox ox ox l¼1  Z h2  Z h2 oUkiþ1 k orx  sxz dz ¼  Uiþ1 rx0 Ukiþ1 dui : ox oz h2 h2 ! 3 2 1 2 3 X o2 u0 o u o u o w l 0  þ Uk1 21 þ Uklþ1 2 þ Uk5 3 dz ox2 ox ox ox l¼1    Z h2  Z h2 o2 r x oUk osxz dw0 : Uk5 2  1 þ 5 rx0 dz ¼  ox oz ox h2 h2 " ! 3 2 2 1 3 X o2 ul k o u0 k o u1 k k o w0  U5 þ U1 2 þ Ulþ1 2 þ U5 3 ox2 ox ox ox l¼1 # o2 w0  2 dz ox du11

ð15Þ in which i = 1–3. 3. Analytical solutions For the simply-supported beam, the following boundary conditions at edges x = 0 and x = L are expressed as ou11 ¼0 ox oui ¼0 ox w0 ¼ 0

ð16Þ

where i = 0–3. Following the Navier solution procedure, the displacement variables satisfying the above boundary conditions may be written as follows: 1 X u0 ¼ u0m cos axeixt m¼1 1 X

u11 ¼

u11m cos axeixt

m¼1

ui ¼

1 X

w0 ¼

ð17Þ uim cos axe

m¼1 1 X

ixt

w0m sin axeixt

m¼1

where i = 1–3, a = mp/L, L is the length of beam and x is the natural frequency.

W. Zhen, C. Wanji / Composite Structures 84 (2008) 337–349

Substituting Eq. (17) into Eq. (13) and colleting the coefficients, the following eigenvalue problem for dynamic problems can be given by ðK  x2 MÞU ¼ 0

ð18Þ

where U is the generalized displacement vector which can be written as U ¼ f u0m

u11m

u1m

u2m

u3m

w0m g

where, the matrix K represents the stiffness matrix, and the matrix M is the mass matrix. For buckling problems, the displacement variables satisfying the boundary conditions in Eq. (16) may be expressed as 1 X u0 ¼ u0m cos ax u11 ¼

u11m cos ax

m¼1

ui ¼

1 X

w0 ¼

ð19Þ uim cos ax

m¼1 1 X

w0m sin ax

m¼1

in which, i = 1–3, a = mp/L, L is the length of beam. Substituting Eq. (19) into Eq. (15) and colleting the coefficients, the buckling problems may be written as the following eigenvalue problem: ðK  kKG ÞU ¼ 0

where, subscript 1 signifies the direction parallel to the fibers, and subscripts 2 and 3 denote the transverse direction. In this case, L and h are the length and the thickness of laminated beam, respectively. The material properties are assumed to be the same for all layers, and the thickness of each layer is identified. The natural frequencies are normalized as X = xL2(q/E1)/h, where q, which is taken to be uniform in the thickness direction, denotes the mass density of material. An analytical result has been selected for comparison with the present study, namely HSDT-98* [11]:

m¼1 1 X

ð20Þ

in which, the matrix K still represents the stiffness matrix, and the matrix KG denotes the geometric stiffness matrix. 4. Numerical examples

341

Analytical solutions based on higher-order theory HSDT-98 are given by Matsunaga [11].

This example is chosen to mainly verify the availability of present analytical approach. Table 1 presents comparison of high frequencies calculated from different theories. Thereinto, acronyms have been introduced to represent the different analysis in Table. For example, GLHT in Table denotes the analytical results calculated from global–local higher-order theory (GLHT). Signification of other acronyms can be obtained by using the same methods. It should be indicated that numerical integration along thickness direction is adopted to compute the stiffness matrix K and the mass matrix M in the present study. Therefore, some difference between HSDT-98* presented by Matsunaga [11] and present results HSDT-98 can be seen in Table 1. Due to unobvious difference, this example can still prove that present analytical method is valid. This example also shows again that global higher-order theory even first-order theory can accurately predict natural frequencies of general laminated beams.

To objectively assess various displacement-based theories, several typical vibration and buckling problems are consider in this section.

Example 4.2. Free vibration analysis of a simply-supported five-ply beam (0°/0°/0°/0°/0°) with arbitrary layouts.

Example 4.1. Free vibration analysis of a simply-supported symmetric laminated beam (0°/90°/90°/0°).

This laminated beam has ply of thickness 0.1h/0.25h/ 0.15h/0.2h/0.3h of materials 1/2/3/1/3 [22].

The following material properties [11] are adopted: E1 ¼ 145 GPa; E2 ¼ 9:6 GPa; G13 ¼ G12 ; G23 ¼ 3:4 GPa;

E3 ¼ E2 ;

Material 1:

G12 ¼ 4:1 GPa;

E1 ¼ E2 ¼ E3 ¼ 6:9 GPa;G1 2 ¼ G1 3 ¼ G2 3 ¼ 1:38 GPa;

m12 ¼ m13 ¼ v23 ¼ 0:3

v1 2 ¼ v1 3 ¼ v2 3 ¼ 0:25

Table 1 Comparison of fundamental frequencies of symmetric laminated beam (0°/90°/90°/0°, L/h = 10) Modes

HSDT-98* [11]

HSDT-98

GLHT

ZZT

HSDT-Reddy

FSDT

1 2 3 4 5

2.3093 6.9756 12.0339 17.1006 22.1492

2.3146 6.9858 12.0459 17.1134 22.1627

2.3136 6.9833 12.0601 17.1925 22.3902

2.3158 7.0012 12.1129 17.3053 22.5985

2.3147 6.9893 12.0702 17.2036 22.4016

2.3155 6.9774 11.9805 16.9244 21.7842

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W. Zhen, C. Wanji / Composite Structures 84 (2008) 337–349

imum % errors of GLHT and ZZT are 9.54 and 22.43, respectively. However, the maximum % error of Reddy’s theory (HSDT-Reddy) is at least seven times the error in GLHT. Figs. 1–3 present the clear comparison of the % error of various theories. In Table 2, it can be also seen that the maximum errors for HSDT-33, HSDT-54, HSDT-76 and HSDT-98 are 27.79, 10.83, 8.33 and 6.70, respectively.

Material 2: E1 ¼ 224:25 GPa; E2 ¼ E3 ¼ 6:9 GPa; G1 2 ¼ G1 3 ¼ 56:58 GPa; G23 ¼ 1:38 GPa; v12 ¼ v13 ¼ v23 ¼ 0:25 Material 3: E1 ¼ 172:5 GPa; E2 ¼ E3 ¼ 6:9 GPa; G1 2 ¼ G1 3 ¼ 3:45 GPa;G23 ¼ 1:38 GPa; v12 ¼ v13 ¼ v23 ¼ 0:25

70

Exact [22]:

Exact two-dimensional elasticity solution for free vibration of laminated beams is given by Kapuria et al. [22].

60

GLHT ZZT HSDT-98 HSDT-76 HSDT-54 HSDT-33 HSDT-Reddy

50

% Error

The mass density q = 1578 kg/m3 is taken to be uniform in the thickness direction. pffiffiffiffiffiffiffiffiffiffiThe natural frequencies are normalized as X ¼ xL2 q=E0 =h, thereinto E0 = 6.9 GPa. In this example, an exact solution is chosen for comparison, namely

40 30 20 10

In this example, a laminated composite beam with different thickness and materials at each ply is analyzed. The normalized natural frequencies obtained from different theories are compared with exact solutions [22] in Table 2, and the % errors of various theories relative to exact solutions [22] are also given in bracket. It can be found that the max-

0

1

2

3

4

Modes Fig. 1. Comparison of % errors of various theories relative to exact solutions for five-layer beam with arbitrary layouts (0°/0°/0°/0°/0°, L/ h = 5).

Table 2 Comparison of frequencies as well as % error of various theories relative to exact solutions for a five-layer (0°/0°/0°/0°/0°) beam with different ply thickness and material property at each ply Modes (m) 1

2

3

4

L/h

Exact [22]

5

7.2551

10

10.152

20

11.924

5

18.837

10

29.020

20

40.606

5

32.769

10

50.832

20

76.577

5

47.602

10

75.349

20

116.08

GLHT

ZZT

HSDT-98

HSDT-76

HSDT-54

HSDT-33

HSDT-Reddy

7.4686 (2.94) 10.3694 (2.14) 12.0327 (0.91)

7.2876 (0.45) 10.1828 (0.30) 11.9510 (0.23)

7.5005 (3.38) 10.3415 (1.87) 12.0153 (0.77)

7.6620 (5.61) 10.4781 (3.21) 12.0724 (1.24)

7.9849 (10.06) 10.7376 (5.77) 12.1764 (2.12)

9.0526 (24.78) 11.4181 (12.47) 12.4189 (4.15)

10.7705 (48.45) 12.1924 (20.10) 12.6548 (6.13)

19.2328 (2.10) 29.8747 (2.95) 41.4776 (2.14)

19.4374 (3.19) 29.1505 (0.45) 40.7315 (0.30)

19.8072 (5.15) 30.0022 (3.38) 41.3661 (1.87)

20.1804 (7.13) 30.6483 (5.61) 41.9124 (3.22)

20.8615 (10.75) 31.9398 (10.06) 42.9505 (5.77)

23.8353 (26.53) 36.2106 (24.77) 45.6726 (12.48)

31.7068 (68.32) 43.0823 (48.46) 48.7696 (20.10)

33.9995 (3.76) 52.0734 (2.44) 78.7483 (2.84)

36.2256 (10.55) 51.4644 (1.24) 76.8460 (0.35)

34.9223 (6.57) 53.0128 (4.29) 78.6932 (2.76)

35.4850 (8.29) 54.1330 (6.49) 80.1914 (4.72)

36.3173 (10.83) 56.3359 (10.83) 83.1366 (8.56)

39.8386 (21.57) 64.9560 (27.79) 91.8751 (19.97)

54.1257 (65.17) 83.215 (63.71) 103.771 (35.51)

52.1422 (9.54) 76.9313 (2.10) 119.499 (2.95)

58.2790 (22.43) 77.7496 (3.19) 116.602 (0.45)

50.7929 (6.70) 79.2289 (5.15) 120.008 (3.38)

51.5703 (8.33) 80.7217 (7.13) 122.593 (5.61)

52.6974 (10.70) 83.4460 (10.75) 127.759 (10.06)

55.7109 (17.03) 95.3414 (26.53) 144.842 (24.78)

76.7002 (61.13) 126.827 (68.32) 172.329 (48.46)

The number in bracket is the % errors of various theories relative to exact solutions.

W. Zhen, C. Wanji / Composite Structures 84 (2008) 337–349

Example 4.3. Natural frequency of a non-symmetric crossply (90°/0°/90°/0°) laminated composite beam with simplysupported boundary is predicted.

70 60

GLHT ZZT HSDT-98 HSDT-76 HSDT-54 HSDT-33 HSDT-Reddy

% Error

50 40

The following orthotropic material properties [22] have been used E1 ¼ 181 GPa; E2 ¼ 10:3 GPa; E3 ¼ E2 ; G12 ¼ G13 ¼ 7:17 GPa; G23 ¼ 2:87 GPa;

30

v12 ¼ v13 ¼ 0:25; 20 10 0

1

2

3

4

Modes Fig. 2. Comparison of % errors of various theories relative to exact solutions for five-layer beam with arbitrary layouts (0°/0°/0°/0°/0°, L/ h = 10).

50 GLHT ZZT HSDT-98 HSDT-76 HSDT-54 HSDT-33 HSDT-Reddy

45 40

% Error

35 30

343

v23 ¼ 0:33

The material properties are assumed to be the same for all layers. The thickness of each layer is identified and the mass density q = 1578 kg/m3 is also taken to be uniform in the thickness direction. pffiffiffiffiffiffiffiffiffiffiThe natural frequencies are normalized as X ¼ xL2 q=E2 =h. To further assess the range of applicability of various theories, free vibration of a non-symmetric laminated beam is analyzed herein. Firstly, the normalized natural frequencies obtained from different theories as well as the % errors of various theories relative to exact solutions [22] in bracket are presented in Table 3. It is seen that the maximum % error in ZZT is 18.51, which is even higher than the maximum % error in Reddy’s theory. However, the maximum % error in GLHT is only 2.81. To clearly compare the % errors in different theories, the part results in Table 3 are plotted in Fig. 4. Numerical results show that GLHT is very suitable for predicting the natural frequencies of non-symmetric laminated beams.

25

Example 4.4. Free vibration analysis of a simply-supported five-layer (0°/90°/core/0°/90°) soft-core sandwich beam.

20 15 10

The following material properties are adopted [21]:

5 0

Face sheets : E1 ¼ 131 GPa;E2 ¼ E3 ¼ 10:34 GPa; 1

2

3

4

Modes Fig. 3. Comparison of % errors of various theories relative to exact solutions for five-layer beam with arbitrary layouts (0°/0°/0°/0°/0°, L/ h = 20).

Therefore, a conclusion can be drawn that by increasing the order number of the in-plane and the transverse displacement components, the maximum % error of global higher-order theories is gradually reduced. For free vibration of laminated beam with arbitrary thickness and material at each ply, the maximum error of HSDT-76 is close to that of GLHT. HSDT-76 consists of 15 unknown variables whereas total number of unknowns in GLHT is only 6. Therefore, compared to higher-order theories (HSDT), GLHT is efficient. In fact, the zig-zag theory (ZZT) is more efficient than GLHT whereas the maximum % error in ZZT is much two times that of GLHT. Considering the accuracy and the computational efficiency, GLHT is more suitable for prediction of high frequencies of laminated beams with arbitrary thickness and materials at each ply.

G1 2 ¼ G2 3 ¼ 6:895 GPa; G13 ¼ 6:205 GPa; v12 ¼ v13 ¼ 0:22;

v23 ¼ 0:49;

q ¼ 1627 kg=m3 ;

Core ðisotropicÞ : E1 ¼ E2 ¼ E3 ¼ 6:89  103 GPa; G1 2 ¼ G13 ¼ G23 ¼ 3:45  103 GPa v12 ¼ v13 ¼ v23 ¼ 0;

q ¼ 97 kg=m3 :

In this case, tc and tf denote the thickness of core and thickness of face sheet, respectively; the natural frequencies 1=2 are normalized as X ¼ xL2 ðq=E2 Þf =h. Rao et al. [21] have presented the exact solutions for free vibration of the soft-core sandwich plate, and a conclusion has been drawn that global higher-order theories as well as first-order theory very much overestimate the natural frequencies of the soft-core sandwich plate. It should be indicated that the third order theory is only used in their work. This example is used to verify whether the global higherorder theory can accurately predict the dynamic response of soft-core sandwich structures by increasing the order number of in-plane and transverse displacement components. As far as authors know, however, no exact solutions for soft-core sandwich beams as used in this paper have

344

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Table 3 Comparison of frequencies as well as % error of various theories relative to exact solutions for a non-symmetric four-layer beam (90°/0°/90°/0°) Modes (m) 1

2

3

4

5

6

L/h

Exact [22]

5

5.7015

10

7.1588

5

14.990

10

22.806

5

24.927

10

40.940

5

35.259

10

59.958

5

45.762

10

79.581

5

56.263

10

99.709

GLHT

ZZT

HSDT-98

HSDT-76

HSDT-54

HSDT-33

HSDT-Reddy

5.6996 (0.03) 7.1649 (0.09)

5.7622 (1.06) 7.1999 (0.57)

5.7705 (1.21) 7.2024 (0.61)

5.8105 (1.91) 7.2240 (0.91)

5.8759 (3.06) 7.2599 (1.41)

6.0473 (6.06) 7.3451 (2.60)

6.1753 (8.31) 7.4008 (3.38)

14.987 (0.02) 22.798 (0.04) 25.005 (0.31) 40.913 (0.07)

15.221 (1.54) 23.049 (1.07) 25.702 (3.11) 41.469 (1.29)

15.237 (1.65) 23.082 (1.21) 25.342 (1.66) 41.567 (1.53)

15.368 (2.52) 23.242 (1.91) 25.525 (2.40) 41.923 (2.40)

15.554 (3.76) 23.504 (3.06) 25.718 (3.17) 42.475 (3.75)

16.156 (7.78) 24.189 (6.06) 26.499 (6.31) 44.081 (7.67)

16.790 (12.00) 24.701 (8.31) 27.807 (11.55) 45.489 (11.11)

35.577 (0.90) 59.951 (0.01)

37.591 (6.61) 60.884 (1.54)

35.825 (1.60) 60.949 (1.65)

36.037 (2.21) 61.475 (2.53)

36.186 (2.63) 62.217 (3.77)

36.919 (4.71) 64.624 (7.78)

39.151 (11.03) 67.160 (12.01)

46.559 (1.74) 79.675 (0.12)

51.200 (11.88) 81.254 (2.1)

46.472 (1.55) 80.917 (1.68)

46.710 (2.07) 81.566 (2.49)

46.828 (2.33) 82.369 (3.50)

47.434 (3.65) 85.282 (7.16)

51.050 (11.55) 89.087 (11.94)

57.849 (2.81) 100.02 (0.31)

66.676 (18.51) 102.81 (3.11)

57.114 (1.51) 101.36 (1.66)

57.379 (1.98) 102.10 (2.40)

57.508 (2.21) 102.87 (3.17)

58.014 (3.11) 105.99 (6.30)

63.690 (13.13) 111.23 (11.55)

The number in bracket is the % errors of various theories relative to exact solutions.

20

15

% Error

Table 4 Comparison of high frequencies for soft-core sandwich beam (0°/90°/core/ 0°/90°, tc/tf = 10)

GLHT ZZT HSDT-98 HSDT-76 HSDT-54 HSDT-33 HSDT-Reddy

L/h

Sources

Modes 1

2

3

4

5

4

GLHT ZZT HSDT-98 HSDT-76 HSDT-54 HSDT-33 HSDT-Reddy FSDT

0.6384 0.6161 0.6547 0.6918 0.6936 1.4958 2.2499 6.3043

1.8001 1.7720 1.8242 1.8818 1.8854 3.2770 4.6821 14.9183

3.6311 3.6214 3.6609 3.7345 3.7428 5.4978 7.2938 23.2726

6.1333 6.1864 6.0928 6.2250 6.2839 7.7359 10.1769 31.4858

9.2762 9.4647 8.3652 8.7226 9.4760 10.1821 13.3961 39.6303

10

GLHT ZZT HSDT-98 HSDT-76 HSDT-54 HSDT-33 HSDT-Reddy FSDT

1.3467 1.2822 1.3956 1.4983 1.5018 3.4686 5.0127 9.1807

2.9936 2.8748 3.0814 3.2759 3.2843 7.3556 11.0757 28.6423

5.1264 4.9707 5.2395 5.5045 5.5187 11.4028 17.0514 50.2777

7.8609 7.6859 7.9926 8.3108 8.3299 15.7484 23.0898 71.9161

11.2509 11.0754 11.4016 11.7615 11.7838 20.4817 29.2632 93.2398

10

5

0

1

2

3

4

5

6

Modes Fig. 4. Comparison of % errors of various theories relative to exact solutions for non-symmetric four-layer beam (90°/0°/90°/0°, L/h = 5).

been published. However, authors [25] have presented a finite element method based on the global–local higherorder theory (GLHT) proposed by Li and Liu [27] to predict dynamic response of soft-core sandwich plates, and good results are obtained. Therefore, results from GLHT are used to serve as reference of those calculated from other theories in present study. Firstly, high frequencies calculated from different theories are presented in Table 4. To clearly compare various

theories, the results in Table 4 have been plotted in Figs. 5 and 6. Numerical results show that with increasing of the order number of in-plane and transverse displacement components, natural frequencies computed from global higher order theories are gradually close to the results obtained from GLHT and ZZT. In particular, difference of natural frequencies between HSDT-54 and HSDT-33

W. Zhen, C. Wanji / Composite Structures 84 (2008) 337–349

is very obvious. Thereinto, results obtained from HSDT-54 are in good agreement with those obtained from GLHT, ZZT, HSDT-98 and HSDT-76. Subsequently, Table 5 presents comparison of fundamental frequencies of simplysupported soft-core sandwich beams for varying length-tothickness ratios (L/h). Moreover, these results are plotted in Fig. 7. By numerical comparison, the same conclusion can be drawn that natural frequencies predicted from HSDT-54 agree well with those obtained from GLHT whereas HSDT-33, HSDT-Reddy and FSDT obviously overestimate natural frequencies of soft-core sandwich beam.

40 GLHT ZZT HSDT-98 HSDT-76 HSDT-54 HSDT-33 HSDT-Reddy FSDT

35

Frequency

30 25 20 15 10 5 0

1

345

2

3

4

5

Example 4.5. Buckling analysis of simply-supported crossply laminated beams subjected to axial compression. The following material properties [11] are chosen:

Modes Fig. 5. Comparison of high frequencies for various theories (0°/90°/core/ 0°/90°, tc/tf = 10, L/h = 4).

E2 ¼ E3 ¼ 106 psi; E1 ¼ 25E2 ; G12 ¼ G13 ¼ 0:5E2 ; G23 ¼ 0:2E2 ; v12 ¼ v13 ¼ v23 ¼ 0:25

11 100

70 60 50

9 8

Frequency

80

Frequency

10

GLHT ZZT HSD T-98 HSD T-76 HSD T-54 HSD T-33 HSDT-Reddy FSDT

90

7 6

GLHT ZZT HSDT-98 HSDT-76 HSDT-54 HSDT-33 HSDT-Reddy FSDT

5 4

40

3

30

2

20

1 10

0 0

1

2

3

4

2

4

10

20

30

5

40

50

60

70

80

90

100

L/h

Modes Fig. 6. Comparison of high frequencies for various theories (0°/90°/core/ 0°/90°, tc/tf = 10, L/h = 10).

Fig. 7. Comparison of fundamental frequencies obtained from various theories for varying length-to-thickness ratios (L/h), (0°/90°/core/0°/90°, tc/tf = 10).

Table 5 Comparison of fundamental frequencies for soft-core sandwich beam (0°/90°/core/0°/90°, tc/tf = 10) L/h

GLHT

ZZT

HSDT-98

HSDT-76

HSDT-54

HSDT-33

HSDT-Reddy

FSDT

2 4 10 20 30 40 50 60 70 80 90 100

0.4500 0.6384 1.3467 2.5588 3.6830 4.6800 5.5389 6.2648 6.8710 7.3742 7.7911 8.1369

0.4430 0.6161 1.2822 2.4347 3.5141 4.4824 5.3272 6.0503 6.6616 7.1747 7.6042 7.9637

0.4560 0.6547 1.3956 2.6533 3.8107 4.8280 5.6960 6.4223 7.0233 7.5180 7.9249 8.2602

0.4704 0.6918 1.4983 2.8470 4.0685 5.1220 6.0028 6.7254 7.3123 7.7875 8.1728 8.4864

0.4713 0.6936 1.5018 2.8531 4.0764 5.1308 6.0119 6.7343 7.3208 7.7954 8.1801 8.4931

0.8192 1.4958 3.4686 5.9900 7.5503 8.4765 9.0393 9.3965 9.6336 9.7974 9.9147 10.0013

1.1705 2.2499 5.0127 7.6885 8.8861 9.4603 9.7670 9.9466 10.0599 10.1356 10.1885 10.2268

3.7295 6.3043 9.1807 10.0440 10.2343 10.3037 10.3364 10.3543 10.3651 10.3721 10.3770 10.3804

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W. Zhen, C. Wanji / Composite Structures 84 (2008) 337–349

The critical buckling loads are normalized as K = 12L2k/(p2E2h2). To further verify the present analytical approach, buckling response of symmetric and non-symmetric laminated composite beams is predicted in this example. In Table 6, the present results obtained from various theories are compared with analytical solution given by Matsunaga [11], namely HSDT-98*. Thereinto, numerical integration along thickness direction is used to calculate the stiffness matrix K and the geometric stiffness matrix KG herein. It can be observed that the present results HSDT-98 agree well with analytical solutions presented by Matsunaga [11] (HSDT98*), which shows that present analytical method for buckling problems is valid. Example 4.6. Buckling analysis of simply-supported softcore sandwich beams subjected to axial compression. The following material properties [24] are used:

Isotropic aluminum face sheets : E ¼ 70 GPa; v ¼ 0:3; Orthotropic core : E1 ¼ 1  108 GPa;

E2 ¼ 1:09 GPa;

G12 ¼ 0:266 GPa; v12 ¼ 1  105 : The critical buckling stresses are normalized as K = kL2/ E2fh3. Moreover, an analytical solution obtained from the mixed layerwise theory is chosen for comparison, namely D& D(LW) [24]:

An analytical results obtained from the mixed layerwise theories for stability of laminated beams is given by Dafedar and Desai [24].

The present results for various values of tc/tf and L/h are compared with analytical results obtained from the mixed layerwise theories [24] in Table 7. It is well known that mixed layerwise theory (LW) [24] is accurate enough to

Table 6 Comparison of buckling stresses of laminated beams (L/h = 5) Number of layers

HSDT-98a [11]

HSDT-98

GLHT

ZZT

HSDT-Reddy

FSDT

3 4 5 6

5.9721 3.9626 5.7471 4.5375

5.9772 3.9679 5.7521 4.5431

5.8486 3.8518 5.5781 4.2894

5.9331 3.9751 5.6033 4.3117

6.3547 4.7800 6.4703 5.0635

7.1188 4.8616 6.4816 5.0856

a

Analytical solutions based on higher-order theory HSDT-98 are given by Matsunaga [11].

Table 7 Comparison of critical loads as well as % error of various theories relative to analytical results computed from mixed layerwise theory for soft-core sandwich beam tc/tf

L/h

5

25

50

D D (LW) [24]

GLHT

ZZT

HSDT-98

HSDT-76

HSDT-54

HSDT-33

HSDTReddy

FSDT

2

0.006222

5

0.01432

10

0.041084

50

0.34319

0.006719 (7.988) 0.01484 (3.631) 0.04182 (1.792) 0.3648 (6.297)

0.006794 (9.193) 0.01486 (3.771) 0.04182 (1.792) 0.3648 (6.297)

0.007216 (15.98) 0.01864 (30.17) 0.05535 (34.72) 0.4076 (18.77)

0.007358 (18.26) 0.01875 (30.94) 0.05571 (35.60) 0.4086 (19.06)

0.008945 (43.76) 0.02692 (87.99) 0.08263 (101.1) 0.4588 (33.69)

0.03851 (518.9) 0.1643 (1047) 0.3507 (753.62) 0.5601 (63.20)

0.06434 (934.1) 0.2486 (1636) 0.4313 (949.8) 0.5668 (65.16)

0.2169 (3386) 0.4499 (3042) 0.5367 (1206) 0.5728 (66.91)

0.001601 (4.647) 0.009142 (1.225) 0.03168 (1.878) 0.1558 (8.307)

0.001601 (4.647) 0.009143 (1.236) 0.03168 (1.878) 0.1558 (8.307)

0.001777 (16.15) 0.01017 (12.61) 0.03477 (11.82) 0.1586 (10.25)

0.001810 (18.31) 0.01037 (14.82) 0.03538 (13.78) 0.1591 (10.60)

0.002058 (34.52) 0.01188 (31.54) 0.03977 (27.89) 0.1624 (12.89)

0.002320 (51.64) 0.01322 (46.38) 0.04343 (39.66) 0.1647 (14.49)

0.003343 (118.5) 0.01889 (109.2) 0.05785 (86.04) 0.1711 (18.94)

0.06187 (3944) 0.1395 (1444) 0.1717 (452.2) 0.1857 (29.09)

0.001510 (4.723) 0.008692 (1.5978) 0.02756 (2.982) 0.09072 (8.999)

0.001510 (4.723) 0.008692 (1.5978) 0.02756 (2.982) 0.09072 (8.999)

0.001628 (12.91) 0.009333 (9.0903) 0.02916 (8.961) 0.09142 (9.840)

0.001709 (18.52) 0.009777 (14.28) 0.03024 (12.99) 0.09186 (10.37)

0.001803 (25.04) 0.01029 (20.276) 0.03146 (17.56) 0.09229 (10.89)

0.001820 (26.22) 0.01041 (21.679) 0.03174 (18.60) 0.09249 (11.13)

0.001958 (35.79) 0.01113 (30.095) 0.03339 (24.77) 0.09285 (11.56)

0.03281 (2175) 0.07471 (773.26) 0.09229 (244.8) 0.09997 (20.11)

2

0.0015299

5

0.0090314

10

0.031096

50

0.14385

2

0.0014419

5

0.0085553

10

0.026762

50

0.083230

The number in bracket is the % errors of various theories relative to analytical results D& D (LW) obtained from the mixed layerwise theory.

W. Zhen, C. Wanji / Composite Structures 84 (2008) 337–349

predict buckling response, so the percentage errors of various theories relative to the results of mixed layerwise theory have been given in bracket of Table 7. To clearly compare the % error of various theories, the part results in Table 7 are plotted in Figs. 8 and 9. Numerical results show that the maximum percentage errors in GLHT and ZZT are less than 10 whereas the maximum % errors in HSDT-33 as well as HSDT-Reddy are more than 1000. In addition, the maximum % error of FSDT is even close to 4000. Therefore, we can draw a conclusion that the FSDT as well as the third-order theories very much overestimate the critical loads of the soft-core sandwich beams. By increasing the order number of in-plane and transverse displacement components, the maximum % errors of global higher-order theories are gradually reduced. For example, the maximum % error of HSDT-54 is 101.1 whereas the 4000 3500 GLHT ZZT HSDT-98 HSDT-76 HSDT-54 HSDT-33 HSDT-Reddy FSDT

3000

% Error

2500 2000 1500 1000 500 0

5

25

50

tc/tf Fig. 8. Comparison of % errors of various theories relative to analytical results obtained from mixed layerwise theory for soft-core sandwich beams (L/h = 2).

1200

GLHT ZZT HSDT-98 HSDT-76 HSDT-54 HSDT-33 HSDT-Reddy FSDT

1000

% Error

800 600

400

200

0

5

25

347

maximum % error of HSDT-98 is only 34.72. Therefore, if global higher-order theory is considered to predict the critical loads of soft-core sandwich beams, HSDT-98 should be at least employed. 5. Conclusions Natural frequencies and critical loads of the simply-supported laminated beams and soft-core sandwiches have been predicted by using various displacement-based theories, and these results are compared with those previously published. By numerical comparison, the following conclusions can be drawn: 1. Global–local higher-order theory (GLHT) as well as zig-zag theory, used in this paper, can predict the satisfactory natural frequencies of laminated beams with arbitrary layouts and soft-core sandwich beams. However, for prediction of high frequencies in non-symmetric beams (L/h = 5), the zig-zag theory is less accurate compared to the global–local higher-order theory. In addition, by increasing the order number of in-plane and transverse displacement components, the global higher-order theories (e.g., HSDT-98, HSDT-76 and HSDT-54) can yield greatly improved results over the available third order theories (HSDT-33 and HSDTReddy) as well as the first-order theory (FSDT) for dynamic problems of laminated beams with arbitrary layouts. Thereinto, both the third order theories and the first-order theory very much overestimate natural frequencies of so special structures. 2. Both the global–local higher-order theory (GLHT) and the zig-zag theory (ZZT), used herein, are very suitable for prediction of critical loads for soft-core sandwich beams whereas the third order theories (HSDT-33 and HSDT-Reddy) as well as first-order theory (FSDT) considerably overestimate the critical loads of soft-core sandwich beams. Although the buckling loads obtained from HSDT-98 are more accurate compared to those computed from the third order theories as well as firstorder theory, HSDT-98 is still less accurate than GLHT and ZZT. 3. To accurately predict the dynamic and the buckling response of laminated beams with arbitrary layouts and soft-core sandwiches, the displacement-based theories a priori satisfying continuity of transverse shear stresses should be adopted. By considering the accuracy and efficient, the global–local higher-order theory (GLHT) used herein is proposed to predict natural frequencies of critical loads of arbitrarily laminated beams and soft-core sandwich beams.

50

tc/tf Fig. 9. Comparison of % errors of various theories relative to analytical results obtained from mixed layerwise theory for soft-core sandwich beams (L/h = 10).

Appendix Initial displacement fields of zig-zag theory proposed by Cho and Parmerter may be written by

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W. Zhen, C. Wanji / Composite Structures 84 (2008) 337–349

uk ðx; y; zÞ ¼ u0 ðx; yÞ þ zu1 ðx; yÞ þ z2 u2 ðx; yÞ þ z3 u3 ðx; yÞ þ

k1 X

S ix ðz  zi ÞH ðz  zi Þ;

i¼1

wk ðx; y; zÞ ¼ w0 ðx; yÞ in which H(z  zi) is the Heaviside unit step function. Using linear strain–displacement relations and threedimensional constitutive equations for cross-ply laminates, the transverse shear stresses for the kth layer can be written as follows: skxz ðx; y; zÞ ¼ Q44k ekxz ðx; y; zÞ; where ekxz ðzÞ ¼

k1 X ow0 þ u1 þ 2zu2 þ 3z2 u3 þ S ix H ðz  zi Þ: ox i¼1

Based on lower transverse shear free condition, the following equation can be obtained by: u1 ¼ 

ow0 : ox

By imposing the continuity conditions of transverse shear stresses at interfaces, the following equation can be given by: ! k1 X S kx ¼ ak 2zk u2 þ 3z2k u3 þ S ix ; i¼1

where ak ¼

Qk44 Qkþ1 44 Qkþ1 44

.

Further, the unknown variable S kx can be rewritten as S kx ¼ F k1 u2 þ F k2 u3 ; where, the coefficients for k = 1 can be easily given by F 11 ¼ 2z1 a1 ;

F 12 ¼ 3z21 a1 :

The coefficients for k > 1 can be calculated from the following recursive equations: ! ! k1 k1 X X k i k i 2 F 1 ¼ ak F 1 þ 2zk ; F 2 ¼ ak F 2 þ 3zk : i¼1

i¼1

Using the upper transverse shear free condition, following equation can be given by u 2 ¼ A1 u 3 ;

Pn1 Pn1 where, A1 ¼ ð3h2 þ i¼1 F i2 Þ=ð2h þ i¼1 F i1 Þ. Finally, the coefficients Wki can be given by Wk1 ¼ z3 þ z2 A1 þ

k1 X

Gi1 z 

i¼1

k1 X

Gi1 zi ;

Wk2 ¼ z;

i¼1

where, Gk1 ¼ F k1 A1 þ F k2 . References [1] Khatua TP, Cheung YK. Bending and vibration of multilayer sandwich beams and plates. Int J Number Meth Eng 1973;6:11–24.

[2] Chandrashekhara K, Krisnamurthy K, Roy S. Free vibration of composite beams including rotary inertia and shear deformation. Compos Struct 1990;14:269–79. [3] Liew KM, Huang YQ. Bending and buckling of thick symmetric rectangular laminates using the moving least-squares differential quadrature method. Int J Mech Sci 2003;45:95–114. [4] Huang YQ, Li QS. Bending and buckling analysis of antisymmetric laminates using the moving least square differential quadrature method. Comput Method Appl Mech Eng 2004;193: 3471–92. [5] Helinger PR, Reddy JN. A higher-order beam finite element for bending and vibration problems. J Sound Vib 1988;126: 309–26. [6] Marur SR, Kant T. Free vibration analysis of fiber-reinforced composite beams using higher-order theories and finite element modeling. J Sound Vib 1996;194:337–51. [7] Kant T, Swaminathan K. Analytical solutions for free vibration of laminated composite and sandwich plates based on a higher-order refined theory. Compos Struct 2001;53:73–85. [8] Kant T, Swaminathan K. Free vibration of isotropic, orthotropic, and multilayer plates based on higher-order refined theories. J Sound Vib 2001;241:319–27. [9] Swaminathan K, Patil SS. Analytical solutions using a higher-order refined computational model with 12 degrees of freedom for the free vibration analysis of antisymmetric angle-ply plates, Compos Struct 2007 [on line]. [10] Babu CS, Kant T. Two shear deformable finite element models for buckling analysis of skew fibre-reinforced composite and sandwich panels. Compos Struct 1999;46:115–24. [11] Matsunaga H. Vibration and buckling of multilayered composite beams according to higher-order deformation theories. J Sound Vib 2001;246:47–62. [12] Matsunaga H. Vibration and stability of cross-ply laminated composite plates according to a global higher-order plate theory. Compos Struct 2000;48:231–44. [13] Matsunaga H. Vibration and stability of angle-ply laminated composite plates subjected to in-plane stresses. Int J Mech Sci 2001;43:1925–44. [14] Matsunaga H. Thermal buckling of cross-ply laminated composite and sandwich plates according to a global higher-order deformation theory. Compos Struct 2005;68:439–54. [15] Matsunaga H. Vibration and stability of cross-ply laminated composite shallow shells subjected to in-plane stresses. Compos Struct 2007;78:377–91. [16] Kant T, Marur SR, Rao GS. Analytical solution to the dynamic analysis of laminated beams using higher-order refined theory. Compos Struct 1998;40:1–9. [17] Fares ME, Zenkour AM. Buckling and free vibration of nonhomogeneous composite cross-ply laminated plates with various plate theories. Compos Struct 1999;44:279–87. [18] Subramanian P. Dynamic analysis of laminated composite beams using higher-order theories and finite elements. Compos Struct 2006;73:342–53. [19] Frederiksen PS. Single-layer plate theories applied to the flexural vibration of completely free thick laminates. J Sound Vib 1995;186:743–59. [20] Rao MK, Desai YM. Analytical solutions for vibrations of laminated and sandwich plates using mixed theory. Compos Struct 2004;63: 361–73. [21] Rao MK, Scherbatiuk K, Desai YM, Shah AH. Natural vibrations of laminated and sandwich plates. J Eng Mech 2004;130: 1268–78. [22] Kapuria S, Dumir PC, Jain NK. Assessment of zigzag theory for static loading, buckling, free and forced response of composite and sandwich beams. Compos Struct 2004;64:317–27. [23] Dafedar JB, Desai YM, Mufti AA. Stability of sandwich plates by mixed, higher-order analytical formulation. Int J Solid Struct 2003;40:4501–17.

W. Zhen, C. Wanji / Composite Structures 84 (2008) 337–349 [24] Dafedar JB, Desai YM. Stability of composite and sandwich struts by mixed formulation. J Eng Mech 2004;130:762–70. [25] Wu Zhen, Chen Wanji. Free vibration of laminated composite and sandwich plates using global–local higher-order theory. J Sound Vib 2006;298:333–49. [26] Wu Zhen, Chen Wanji. Thermomechanical buckling of laminated composite and sandwich plates using global–local higher-order theory. Int J Mech Sci 2007;49:712–21. [27] Li XY, Liu D. Generalized laminate theories based on double superposition hypothesis. Int J Numer Method Eng 1997;40: 1197–212.

349

[28] Cho M, Parmerter R. Efficient higher-order composite plate theory for general lamination configurations. AIAA J 1993;31:1299–306. [29] Di Sciuva M. A refined transverse shear deformation theory for multilayered anisotropic plates. Atti Accademia Scienze Torino 1984;118:279–95. [30] Di Sciuva M. Multilayered anisotropic plate models with continuous interlaminar stresses. Compos Struct 1992;22:149–67. [31] Reddy JN. A simple higher-order theory for laminated composite plates. J Appl Mech 1984;51:745–52. [32] Whitney JM, Pagano NJ. Shear deformation in heterogeneous anisotropic plates. J Appl Mech 1970;37:1031–6.