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Bending, buckling and free vibration of laminated composite and sandwich beams: A critical review of literature
Accepted Manuscript Bending, buckling and free vibration of laminated composite and sandwich beams: A critical review of literature Atteshamuddin S. Sayyad, Yuwaraj M. Ghugal PII: DOI: Reference:
S0263-8223(16)33027-6 http://dx.doi.org/10.1016/j.compstruct.2017.03.053 COST 8374
To appear in:
Composite Structures
Received Date: Revised Date: Accepted Date:
29 December 2016 4 March 2017 15 March 2017
Please cite this article as: Sayyad, A.S., Ghugal, Y.M., Bending, buckling and free vibration of laminated composite and sandwich beams: A critical review of literature, Composite Structures (2017), doi: http://dx.doi.org/10.1016/ j.compstruct.2017.03.053
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Bending, buckling and free vibration of laminated composite and sandwich beams: A critical review of literature Atteshamuddin S. Sayyad1* and Yuwaraj M. Ghugal2 1
Department of Civil Engineering, SRES’s Sanjivani College of Engineering, Savitribai Phule Pune University, Kopargaon - 423601, Maharashtra, India. 2 Department of Applied Mechanics, Government College of Engineering, Karad - 415124, Maharashtra, India.
Abstract Laminated composite and sandwich structures are lightweight structures that can be found in many diverse applications especially civil, mechanical and aerospace engineering. The rapid increase in the industrial use of these structures has necessitated the development of new theories that suitable for the bending, buckling and vibration analysis of composite structures. Many review articles are reported in the literature on laminated composite plates and shells in the last few decades. But, in the whole variety of literature very few review articles are available exclusively on laminated composite and sandwich beams. In this article, a critical review of literature on bending, buckling and free vibration analysis of shear deformable isotropic, laminated composite and sandwich beams based on equivalent single layer theories, layerwise theories, zig-zag theories and exact elasticity solution is presented. In addition to this, literature on finite element modeling of laminated and sandwich beams based on classical and refined theories is also reviewed. Finally, displacement fields of various equivalent single layer and layerwise theories are summarized in the present study for the reference of researchers in this area. This article cites 512 references and highlights, the possible scope for the future research on laminated composite and sandwich beams. Keywords: beam theories; laminated; sandwich; bending; buckling; vibration.
*
Corresponding author, Email: [email protected], Ph. No.: (+91) 9763567881
1. Introduction Beams, columns, and rods made up of a composite material are being widely used in the civil, mechanical, aeronautical and aerospace industries due to their attractive properties such as; high strength-and stiffness-to-weight ratio and their anisotropic material property. The beam members are mainly subjected to bending whereas columns and rods are mainly subjected to axial tension and compression. The shear deformation effects are more pronounced in the laminated composite and sandwich beams subjected to transverse loads. Therefore, bending, free vibration and buckling analysis of laminated composite and sandwich beams have received widespread attention in recent years. The various analytical and numerical methods based on beam theories have developed by the researchers for analysis of shear deformable laminated composite and sandwich beams. Many review articles are reported in the literature on laminated composite plates and shells in the last few decades [1]. But, in the whole variety of literature very few review articles are available exclusively on laminated composite and sandwich beams [2-7]. Kapania and Raciti [2, 3] reviewed literature on bending, buckling, post-buckling, vibration and wave propagation analysis of laminated composite beams and plates using various displacement and stress based shear deformation theories. The mathematical formulation of some classical and higher order shear deformation theories is presented. Authors have also reviewed finite element models for the analysis of laminated beams and plates and provided suggestions for future research. Ghugal and Shimpi [4] presented a review of displacement and stress based refined theories for isotropic and anisotropic laminated composite beams. Merits and demerits of various equivalent single layer and layerwise theories are discussed. Marur [5] has thrown some light on distinct phases in the development of nonlinear vibration of beams. Gherlone [6] presented a brief review on zig-zag theories for the analysis of laminated composite and sandwich beams. It
is concluded that the equivalent single layer theories are easy to implement and computationally affordable but, in order to correctly describe the mechanical behavior of laminated structures the displacement field needs to be enriched by a through-the-thickness piecewise linear contribution i.e. zig-zag function. Hajianmaleki and Qatu [7] presented a review of research done by various researchers on the vibration analysis of composite and smart beams. The review is based on classical and higher order shear deformation theories developed for the analysis of beams and numerical methods used for solving governing differential equations. The well-known books on laminated composite and sandwich beams have published by Reddy [8], Wang et al. [9] and Carrera, et al. [10]. Therefore, in the present study an attempt is made to review the research done by various researchers on isotropic, laminated composite and sandwich beams using different analytical and numerical methods based on classical and refined beam theories.
2. Bending of shear deformable beams A great deal of research has been carried out since the last few decades to accurately assess the bending response of beams. Bending analysis of beams using two-dimensional elasticity theory is very complicated and this led to the development of refined shear deformation theories for beams which approximate the two dimensional solution into one dimensional solution with reasonable accuracy. Two classes of theories are available in the literature for the analysis of beams. In the first class of theories, the effects of shear deformation and rotary inertia are neglected whereas in the second class of theories, shear deformation and rotary inertia effects are considered.
2.1 Bending of isotropic beams The most commonly used theory for the bending analysis of beams is classical beam theory (CBT) developed by Bernoulli-Euler [11-16] almost 300 years ago. Chronological development of this theory from the first attempt by Galileo in 1638 till 1856 by Barre de Saint Venant is given by Love [17], Timoshenko [18] and Todhunter and Pearson [19]. This theory neglects the shear deformation and rotary inertia effects; therefore, this theory is typically accurate for thin beams and is less accurate for thicker beams. Rayleigh [20] improved the classical theory by allowing the effect of rotary inertia of the cross-sections of the beam. Boley [21] studied an accuracy of the CBT for the beams of variable section. The stresses and deflections of the beam are found, on the basis of the two-dimensional theory of plane stress. The Deflection and stresses are obtained in the form of series using Airy-s stress function; the first term of each series is identical with the strength-of materials solution and the others represent the necessary correction to CBT. Timoshenko [22] developed a new beam theory which was considered as a refinement of the classical beam theory. This theory introduced first-order shear effects as well as effect of rotational inertia in the kinetic energy. Therefore, this theory is also known as the first order shear deformation theory or Timoshenko beam theory (TBT). The TBT violates the zero shear stress conditions on the top and bottom surfaces of the beam. A shear correction factor is required to appropriately represent the strain energy of shear deformation. TBT also suffers from the boundary condition paradox. Many articles are published on Timoshenko beam theory and shear correction factors [23-56]. Recently, Elishakoff et al. [57] presented historical development of TBT and the research available in the literature based on this hundred years old theory which considers the effects of transverse shear deformation and rotary inertia.
However, CBT and TBT do not take into account the non-classical influences such as crosssectional warping, out-of-plane and in-plane deformations. The detail discussion on these influences is presented by Goodier [58]. Therefore, the higher order shear deformation theories are necessary to account for these non-classical influences. These higher order shear deformation theories are developed using the method of hypotheses, the method of expansion, the method of successive approximation and the mixed method. Donnell [59] and Boley and Tolins [60] have developed refined theories using the method of successive approximations for the bending of thick beams. Refined theories in which transverse shear stress is varying parabolically across the thickness of the beam and does not require problem dependent shear correction factor are called as parabolic shear deformation theories [61-80]. Irschik [81] has established an analogy between classical Bernoulli-Euler theory and refined beam theories of Levinson [68], Rehfield and Murthy [65] and by Rychtcr [74] using the principle of virtual work to increase the acceptance of higher order refined theories. The beams of rectangular cross-section with clamped and hinged boundary conditions are considered for the detail investigation. Corrections to the classical solution have been interpreted as being due to additional self-stress acting in the Bernoulli-Euler beam. Refined theories which account for the trigonometric cosine distribution of the transverse shear stress/strain through the thickness of the beam are called as trigonometric shear deformation theories [82-88]. These theories satisfy the traction free boundary conditions at top and bottom surfaces of the beam and does not require problem dependent shear correction factor. In these theories, in-plane displacement uses trigonometric function to account the effect of shear deformation and transverse displacement uses cosine function to account the effect of normal deformations. The origin of these theories can be found in Love [17], Levy [89], Kromm [90, 91], Panc [92] for bending of elastic plates and in Vlasov and Leont’ev [93] and Stein [94] on
bending of elastic beams. Refined theories containing the hyperbolic sine and cosine functions in-terms of the thickness coordinate to account for the effects of transverse shear and normal deformations in the displacement field are called as hyperbolic shear deformation theories [9092]. Beam theories deduced from such plate theories are presented by various authors [95-98]. Karama et al. [99] have developed a refined theory containing the exponential functions in-terms of the thickness coordinate in the displacement field. This theory is called as exponential shear deformation theory. It is reported that the exponential functions are richer than sine and cosine functions, because an exponential function has all even and odd power in its expansion unlike sine function, which has only odd power. The numerical results for bending and free vibration analysis of beams with different boundary conditions are presented using Levy’s solution technique. Recently, Sayyad [100, 101] presented comparison of parabolic, trigonometric, hyperbolic, and exponential shear deformation theories for the bending of isotropic beams. All these functions are accounted in the unified shear deformation theory and closed form solution is obtained using the Navier solution technique. Gao and co-authors [102-104] derived a refined theory of rectangular and curved beams by using the Papkovich-Neuber solution and Lure method without adhoc assumptions. Shi and Voyiadjis [105] and Wang and Shi [106] proposed a new sixth order shear deformation theory for the analysis of shear deformable isotropic beams with various boundary conditions. A sixth order beam theory is developed to resolve the paradoxes of TBT on the displacement boundary conditions or continuity conditions of shear flexible beams. It is also recommended that this theory can be easily used in the finite element analysis of composite beams. Carrera and coauthors [107-110] developed several refined theories for the linear static analysis of beams made of isotropic materials based on Carrera’s Unified Formulation (CUF).
Lin and Zhang [111] developed a new displacement based beam element with two degrees of freedoms per node for finite element analyses of isotropic and composite beams using Timoshenko beam theory. A layered approach is employed to describe the layered characteristic of the composite beams. Miranda et al. [112] developed a generalized beam theory considering the effect of shear deformation for a cantilever beam of various cross-sections. 2.2 Bending of laminated and sandwich beams Multilayered composite and sandwich beams are widely used in many engineering applications. While composite laminates provide higher stiffness, strength and reduce the weight over conventional metallic structures, sandwich structures can further reduce the structural weight and improve performance without sacrificing stiffness and strength. The geometry and layer numbering scheme for the general laminated composite beam is shown in Fig. 1 and 2 respectively. In this section, literature on bending of laminated composite and sandwich beams using equivalent single layer (ESL) theories, layerwise theories (LW), zig-zag (ZZ) theories and Carrera’s Unified Formulation is reviewed. This section also presents the literature on finite element modeling of laminated composite and sandwich beams and exact elasticity solutions.
Fig. 1. Laminated composite beam geometry
Fig. 2. Layer numbering system of laminated composite beam
2.2.1 Bending of laminated and sandwich beams using ESL theories Equivalent single layer (ESL) theories are derived from the method of hypotheses. In the equivalent single layer displacement based theories, one single expansion for each displacement component is used through the entire thickness of the laminate. Equivalent single layer theories are widely used for the bending analysis of simply supported isotropic, laminated composite and sandwich beams. These theories are classified as classical beam theory, first order shear deformation theory, second and third order shear deformation theories based on Taylor series expansion and higher order shear deformation theories involving parabolic, trigonometric, hyperbolic, exponential and mixed functions in the displacement field. Boay and Wee [113] studied the effect of coupling on bending, buckling and vibration of angle-ply laminated composite beams using the classical lamination beam theory. A closed form solution for clamped and simply supported beams is obtained and the results are compared with those generated using the finite element method. Chen et al. [114] presented bending of symmetrically cross-ply laminated composite beams using first order deformations (TBT) based on modified couple stress theory. The deflection and stresses of symmetric composite beams are obtained using CBT and TBT and compared with each other. Hajianmaleki and Qatu [115] applied first order shear deformation for the static analysis of composite laminated beams of rectangular and tubular
cross-section. A three-dimensional finite element model is used to verify the proposed beam model. In addition, the model is also applied to tubular cross section beams and the results were compared with experimental data and other models existing in the literature. Silverman [116] presented a parabolic shear deformation theory for the narrow doubly symmetrical three-layer rectangular beam. Valisetty [117] has developed a refined bending theory for the beam of solid circular cross section considering non-classical influences observed by Goodier. The theory is developed from the engineering stresses and the solution is provided to the problem of propagation of harmonic flexural waves in a beam of solid circular cross section. Hu et al. [118] presented experimental validation of parabolic shear deformation theory. A cantilever cross-ply laminated beam is tested experimentally where strains are measured using strain gauges and deflections are measured with dial gauges. The strain distributions in the endeffect region were found to be much closer to those predicted by the parabolic shear deformation theory. The displacement based higher order theories based on Taylor series expansion [119-128] are cumbersome and computationally more demanding, because; with each additional power of the thickness coordinates, an additional dependent unknown variable is introduced into the theory which becomes difficult to interpret physically. It has been noted that due to the higher order of terms included, the theories are not convenient to use. This observation is more or less true for many other higher order theories as well. This drawback led to the development of simple to use higher order beam theory with less number of variables. Such displacement based higher order shear deformation theories are developed and applied by Reddy [129], Khdeir and Reddy [130], Soldatos and Watson [131], Zenkour [132], Reddy et al. [133], Matsunaga [134], Liu and Soldatos [135], Ferreira et al. [136], Kroker and Becker [137], Cernescu and Romanoff [138],
El-Nady and Negm [139] and Pawar et al. [140] for the analysis of laminated composite and sandwich beams. These theories contain three or four unknown variables only and accounts for traction free conditions at the top and bottom surfaces of the beam using constitutive relations. Also these theories do not require shear correction factor. Frostig and his co-authors such as Frostig and Baruch [141-143], Frostig et al. [144], Frostig [145-148], Peled and Frostig [149], Frostig and Peled [150], Frostig and Shenhar [151], Shenhar et al. [152], Swanson [153] and Swanson and Kim [154] published series of research papers on bending of sandwich beams with soft core considering transverse flexibility of the core. Kosmatka [155] developed a new onedimensional theory to obtain section modulus of prismatic beams with arbitrary cross-section. The theory accurately accounts for the effects of shear deformation out of the cross-sectional plane and anticlastic type deformations within the cross-sectional plane. Savoia and Tullini [156] proposed a displacement based shear deformation theory for the bending of orthotropic beams. The stress distributions are presented for cantilever and clamped boundary conditions. Rand [157] carried out multilevel analysis of laminated composite beams based on a hierarchy of solution levels that enabled the prediction of wide spectrum of physical phenomena including gross and local quantities such as beam bending, extension, twist components, in-plane warping, interlaminar stresses and delamination effects. Mistou et al. [158], Naik [159], Arya et al. [160], Zenkour et al. [161, 162], Sayyad and Ghugal [163, 164], Vo and Thai [165], Sayyad et al. [166, 167], Nazargah et al. [168], Ghugal and Shikhare [169] and Nguyen and Nguyen [170] developed displacement based higher order shear deformation theories considering trigonometric functions in terms of thickness coordinate to account the effects of transverse shear and normal deformations for the bending analysis of laminated composite and sandwich beams. All these authors have used Navier’s solution
technique to obtain a closed form solution for the simply supported beams. The kinematics of the trigonometric shear deformation theories is much richer than those of the other higher order shear deformation theories, because if the trigonometric term is expanded in power series, the kinematics of higher order theories is implicitly taken into account to good deal of extent without loss of physics of the problem. Kant et al. [171] proposed a simple semi-analytical method for the stress analysis of laminated and sandwich narrow beams. The proposed model is based on the solution of a two-point boundary value problem governed by ordinary differential equations through thickness of the beam. Forth-Order Runge-Kutta-Gill algorithm is used for numerical integration. The proposed model maintains continuity of transverses stresses and displacements at the laminae interfaces. Apetre et al. [172] carried out analytical modelling of a simply supported sandwich beams using first order, third order, higher order shear deformation theories based on Fourier–Galerkin method. The numerical results are compared with the finite element analysis. The core of the one-dimensional sandwich panel is made up of functionally graded material. Karama et al. [99, 173] and Aydogdu [174] presented a new shear stress function in the form of the exponential function, to predict the mechanical behaviour of multi-layered laminated composite beams. Mechab et al. [175] presented bending analysis of short laminated composite beams subjected to mechanical and thermal loading. The formulation is based on classical and higher order shear deformation theories. The beam is subjected to a uniform load and heated at a constant temperature. It is concluded that the refined theories used gives a higher value of deflection when compared to the classical theory because of the contribution from shear deformation. Tonelli et al. [176] studied stress state of sandwich beams under bending and shear using first order shear deformation theory, zig-zag theory and higher order shear deformation theory. The numerical
results for simply supported sandwich beams subject to uniform loading obtained by these three analytical models compared with those obtained from finite element simulations. Nonlinear material response of a sandwich beam with bilinear elasto-plastic constitutive relations for the transverse normal and shear stresses at the core is studied by Salami et al. [177] based on a proposed improved high order sandwich panel theory. The face sheets are modelled using the first order shear deformation theory whereas the core is modeled using the two-dimensional elasticity. Authors also performed experimental investigations under conditions of three point bending on sandwich beams. Nonlinear bending analysis of shear deformable anisotropic laminated composite beams with various kinds of distributed loads resting on a two-parameter elastic foundation is investigated by Li and Zhao [178]. Sayyad et al. [179] applied two variable plate theory developed by Shimpi and Patel [180] for the stress analysis of laminated composite and sandwich beams. In this theory, transverse displacement consists of two parts (bending and shear) along the center line of the beam. The Navier solution technique is employed to obtain closed form solutions for simply supported laminated composite and soft core sandwich beams. The displacement fields of various reviewed equivalent single layer theories considering effects of transverse shear and normal deformations are given below in Tables 1 and 2.
Table 1 Displacement fields of equivalent single layer beam theories considering effect of transverse shear deformation ( z 0 ). Reference
Axial displacement (u)
Bernoulli [11-13], Euler [14]
u u0 z
Timoshenko [22]
u u0 z
Krishna Murty [67]
Levinson [68-71], Reddy [129]
u u0 z
Transverse displacement (w)
dw0 dx
w w0 w w0
4 z 2 dwb z 1 dx 3 h
2 dw 4 z u u0 z 0 3 h dx
w wb ws w w0
Kant and Manjunath [121]
u u0 z x z 2u*0
w w0
Kant and Manjunath [121]
u u0 z x z 3 x*
w w0
Kant and Manjunath [121]
u u0 z x z 2u*0 z 3 x*
w w0
Soldatos and Elishakoff [95]
u u0 z
dw0 1 z z cosh h sinh dx 2 h
w w0
Ghugal and Shimpi [82]
u u0 z
dw0 h z sin dx h
w w0
Karama et al. [99]
u u0 z
z 2 dw0 z exp 2 dx h
w w0
Benatta et al. [182, 183]
u u0 z
dw0 dw f z 0 dx dx
w w0
Aydogdu [174]
dw u u0 z 0 z dx
Mahi et al. [184], Sayyad and Ghugal [98, 101]
u u0 z
Shi and Voyiadjis [105]
z 5 z 3 dw 5 4z3 u u0 z 2 2 0 4 3h 4 3h dx
Vo and Thai [185], Sayyad et al. [167]
u u0 z
dwb h z dw z sin s dx h dx
w wb ws
Vo and Thai [185]
u u0 z
dwb 4 z 3 dws dx 3h2 dx
w wb ws
Ambartsumian [187]
u u0 z
dw0 z h2 z 2 dx 2 4 3
w w0
2 z / h ln
2
dw0 dw f z 0 dx dx
w w0
w w0 w w0
Table 1 Continue. Kruszewski [23], Panc [92], Reissner [188]
u u0 z
Ray [189]
u u0 z
Shimpi et al. [190] Shimpi et al. [190]
Shimpi and Patel [180], Sayyad et al. [179]
2 dw0 5 z 4 z 1 dx 4 3 h
3 dw0 1 3 z z 2 Qx dx x 2 h h dw u u0 z b dx
u u0 z
d dx
w w0
w w0 w wb ws w w0
1 5 z 2 dws dwb u u0 z z dx 4 3 h dx z2 dw0 z sech 2 z sech 1 tanh dx 4 4 2 h
w wb ws
Akavci [191]
u u0 z
Akavci [191]
u u0 z
dw0 3 z 1 h tanh z sec2 h dx 2 h 2
w w0
Mantari et al. [192]
u u0 z
2 dw0 2 z / h zm dx
w w0
Meiche et al. [193]
u u0 z
dwb h / sinh z / h z dws dx cosh / 2 1 dx
w wb ws
Xiang et al. [194]
12 u u0 zx nh
Mantari et al. [196]
u u0 z
z dw0 z m cos h m z sin e dx h h
w w0
Mantari et al. [197]
u u0 z
dw0 mh tan mz mz sec2 dx 2
w w0
Daouadji et al. [198]
u u0 z
Thai et al. [200]
n 1
dw z n x x,t 0 dx
dwb 3 1 3 z dw z 1 sech 2 h tanh s dx 2 2 2 h dx
4z3 4z3 u u0 z 2 x 2 x 3h 3h z2 dwb dw z 1 sech 2 z sech 1 tanh s dx h 4 2 4 dx
Daouadji et al. [202]
u u0 z
Grover et al. [203]
u u0 z
Mahi et al. [205]
u u0 z
Sarangan and Singh [206]
u u0 z
dw0 2r rz sinh -1 z dx h h r2 4
z3 dw0 h dw 4 2z tanh 0 2 dx 2 h dx 3cosh 1 h3
dw0 g z z dx
w w0
w w0
w wb ws w w0
w wb ws w w0
w w0 w w0
Table 2 Displacement fields of equivalent single layer beam theories considering effects of transverse shear and normal deformation ( z 0 ) . Reference
Axial displacement (u)
Kant and Manjunath [122]
u u0 z x
w w0 z z
Kant and Manjunath [122]
u u0 z x z 2u*0
w w0 z z
Kant and Manjunath [122]
u u0 z x z 3 x*
w w0 z z
Kant and Manjunath [122]
u u0 z x z 2u*0 z 3 x*
w w0 z z
Zenkour [181]
u u0 zu1
Zenkour [132]
dw dw u u0 z 0 u1 z 2 1 dx dx 2 dw dw2 z 3 1 0 u1 dx 3 dx
Maiti and Sinha [126]
u u0 z x z 2u*0 z 3 x*
Sayyad and Ghugal [163]
u u0 z
dw0 h z sin dx h
u u0 z
dwb 4 z 3 dws dx 3h2 dx
Vo et al. [186]
Pawar et al. [140]
u u0 z
Transverse displacement (w)
w w0 zw1 z 2 w2
dw0 x, t dx
w w0 zw1 z 2 w2
w w0 z z z 2 w*0 z 3 z* w w0
z cos h h
4z2 w wb ws 1 2 wz 3h w w0 f ' z
f z
Neves et al. [195]
z u u0 z u1 sin uz h
w w0 z w1 z 2 w2
Neves et al. [199]
z u u0 z u1 sinh uz h
w w0 z w1 z 2 w2
Bessaim et al. [201] Zenkour [204]
u u0 z
dwb z 1 dw z h sinh z cosh s dx h 2 dx
u u0 z
3 dw0 z 4 z 1 d h sinh cosh 2 dx h 3 h 2 dx
z 1 w wb ws cosh cosh h 2
w w0
2 1 z 4z 1 cosh 2 cosh 12 h h 2
2.2.2 Bending of laminated and sandwich beams using LW/ZZ theories The major drawback of ESL theories is double valued shear stress at the layer interface. In order to remove this defect, it is necessary to describe each composite laminate as an assembly of individual layers bonded perfectly. In layerwise (LW) theories, the axial displacement field is approximated across the thickness, layerwise or sublaminate wise with continuity at each interface, and transverse displacement is usually assumed to be independent of the thickness coordinate. Krajcinovic [207], Swift and Heller [208], Davalos et al. [209], Shimpi and Ghugal [210, 211], Arya et al. [212] and Ghugal and Shinde [213], Tahani [214], Afshin and Behrooz [215] and Aitharaju and Averill [216] developed a variationally consistent layerwise shear deformation theory for bending of laminated composite and sandwich beams. The displacement fields of various reviewed layerwise beam theories are given in Table 3. Results obtained using layerwise theories are very close to exact three-dimensional elasticity 3D results. ESL theories are easy to implement and computationally affordable but, assumptions of continuous function of inplane displacements through the laminate thickness cannot produce conspicuous kinks (zigzag) in the inplane displacements distribution typical of thick laminates as given by the exact elasticity solution. The zigzag term of the displacement field is used to model the local distortion of the cross section in each lamina of multilayered structures and is related to the continuity of transverse stresses. Therefore, in zig-zag (ZZ) theories, a piecewise cubic displacement function is superimposed over a linear displacement field and the transverse displacement is usually assumed to be independent of the thickness coordinate. Carrera [217] presented a historical review on zig-zag theories for the analysis of multilayered laminated structures and made an important conclusion that the Lekhnitskii [218] was the first to use zig-zag theory for solving an elasticity problem involving a layered beam. Arya et al. [160], Aitharaju and Averill [216], Icardi
[219-222], Di Sciuva and Icardi [223], Di Sciuva et al. [224], Kapuria et al. [225-230], Vidal and Polit [231], Cook and Tessler [232], Tessler et al. [233-235], Tessler [236] and Di Sciuva et al. [237] developed higher order zig-zag models for the analysis of laminated composite and sandwich beams.
Table 3 Displacement fields of layerwise theories considering the effects of transverse shear and normal deformations. Reference
Axial Displacement (u)
Transverse displacement (w)
Davalos et al. [209]
w w0
n
u u0 U j x,t j z j 1
Shimpi and Ghugal [210]
1
u
Shimpi and Ghugal [211]
dw0 z / h C1 C2sin dx 2 0.5 dw z / h z h 0 sin dx 2 0.5
w w0
dw0 z / h h C1 C2sin dx 2 0.5 dw z / h z h 0 h C3 +sin dx 2 0.5
w w0
u z h 2
u z h 1
u
2
w Wk x,t k z
Tahani [214],
u U k x,t k z
Afshin and Behrooz [215]
k 1, 2,.....,N 1
Aitharaju and Averill [216]
u k ub z b z 2
k 1 i 1
z zi
z z wk 1 wb wt h h
2.2.3 Bending of laminated and sandwich beams using finite element method (FEM) Since Navier solution technique is applicable to bending, buckling and free vibration analysis of simply supported beams only, many researchers have performed studies on other boundary conditions of beams using the finite element technique. Finite element method is the most widely used numerical method for the bending, buckling and vibration analysis of composite beams.
Various finite element models based on equivalent single layer, layerwise, zig-zag theories are often proposed for describing the bending behaviour of laminated composite and sandwich beams. Lee and Lee [238] developed a finite element model based on classical lamination theory for the flexural–torsional analysis of I-shaped symmetrically and unsymmetrically laminated composite beams. Numerical results are obtained for thin-walled composites under vertical and torsional loading, addressing the effects of fiber angle, and laminate stacking sequence. Lee and Kim [239], Goyal and Kapania [240], Back and Will [241], Sudhakar et al. [242], Mahieddine and Ouali [243] and Kim and Reddy [244] developed finite element models based on first order shear deformation theory for the bending problems of laminated composite and sandwich beams. Several studies have been performed on finite element modeling of laminated composite and sandwich beams based on higher order shear deformation theories [245-273]. Shimpi and Ainapure [274] extended layerwise trigonometric shear deformation theory of Shimpi and Ghugal [210] for the finite element analysis of two layered antisymmetric laminated composite beams. The finite element is free from shear locking. The numerical results are obtained for the deflection, stresses and natural frequencies of two layered anti-symmetric laminated composite beams. Aitharaju and Averill [216], Averill [275], Averill and Yip [276], Friedman and Kosmatka [277], Degiovanni et al. [278], Chakrabarti et al. [279-281], Gherlone et al. [282], Onate et al. [283], Nanda et al. [284], Iurlaro et al. [285] and Di Sciuva et al. [286] developed finite element models based on higher order zigzag theories for the analysis of laminated and sandwich beam. 2.2.4 Bending of laminated and sandwich beams using CUF According to Carrera’s Unified Formulation (CUF), the one-dimensional theories of layered beams are developed by using the following formulas: u Ft ut Fbub Fr ur F u ,
t , b, r , r 2,......., N
n Ft nt Fb nb Fr nr F n ,
t , b, r, r 2,......., N
where Ft, Fb and Fr are the base functions used for z expansion; the first two polynomials are related to the linear part of such expansions, while Fr introduces the N -1 higher order terms i.e, Fr z r . The transverse stress demand a Legendre layer-wise expansion in thickness coordinate,
whereas both Taylor and Legendre expansions could be used for the displacement unknowns. The CUF method can be easily employed to obtain the governing equations and the boundary conditions of the various theories reviewed within the framework of PVD and/or RMVT. Most of the ESL and LW models presented in the preceding section can be viewed as particular cases of the Carrera Unified Formulation (CUF) [107-110, 287-294]. Tornabene et al. [295] presented a general formulation of a two-dimensional higher order equivalent single layer theory for free vibrations of doubly curved laminated composite shells and panels with different curvatures. Governing equations for dynamic analysis of shell structures are obtained by using a general displacement field based on the CUF including the stretching and zig-zag effects. Tornabene et al. [296] presented a two-dimensional general higher order equivalent single layer (GHESL) approach, based on the Carrera unified formulation (CUF) for the static analysis of doublycurved anisotropic shells and panels using a differential quadrature method. Tornabene et al. [297] presented free vibration analysis of arbitrarily shaped laminated composite shells based on the generalized differential quadrature (GDQ) method using equivalent single layer higher order theories developed from CUF. Tornabene et al. [298] and Bacciocchi et al. [299] developed a general formulation based on various higher-order equivalent single layer theories to evaluate the natural frequencies of several doubly-curved shells with variable thickness using generalized differential quadrature method. Tornabene et al. [300] evaluated through thickness distributions of displacement and stress components of several doubly curved panels reinforced by curvilinear
fibers using equivalent single layer (ESL) and layerwise (LW) theories which is based on the Carrera Unified Formulation (CUF). Tornabene [301] presented a general formulation for a higher order layerwise theory for the free vibrations of doubly curved laminated composite shells and panels using a general displacement field based on the Carrera Unified Formulation (CUF), including the stretching effect for each layer. D’Ottavio [302] and D’Ottavio et al. [303] presented bending of laminated composite and sandwich plates based on sublaminate generalized unified formulation (S-GUF). In S-GUF, plies are grouped into several smaller units called sublaminates, each of them characterized by an independent, variable-kinematic theory. Continuity conditions between the sublaminates are enforced in strong form during the assembly procedure of the governing equations. Demasi [304] introduce a new generalization of Carrera’s unified formulation called as generalized unified formulation (GUF). The original notations of CUF are slightly modified to make the new compact notation. The GUF is applied for the bending of laminated composite plates.
2.2.5 Exact elasticity solutions for bending of laminated and sandwich beams A two-dimensional elasticity solution of beam is necessary if the beam is thick. The exact elasticity solution is always used as a basis to check the accuracy of any refined shear deformation theory. Saint-Venant [15] studied the problem of a cantilever beam of rectangular cross section possessing anisotropy of a special form and obtained the exact elasticity solution. Various such problems are discussed by Lekhnitskii [305]. Silverman [306], Hasin [307], Gerstner [308], Rao and Ghosh [309] and Cheng et al. [310] provided exact solution for shear deformable laminated composite beams using Airy’s stress functions whereas Holt and Webber [311] provided exact solution for Honeycomb sandwich beams. Dischinger [312], Cheng and Pei
[313], Herrman [314] and Iyengar and Prabhakara [315] provided two-dimensional elasticity solution for the continuous beams. Also, Pagano [316, 317] developed exact elasticity solutions for cylindrical bending of laminated composite plates. Esendemir et al. [318] obtained the deflection of a simply supported composite beam subjected to a linearly distributed load using anisotropic elasticity theory given by Lekhnitskii [305]. 3. Free vibration of shear deformable beam 3.1 Free vibration of isotropic beams Burgreen [319, 320], Krieger [321], Eringen [322] and McDonald [323] presented nonlinear vibration problems of simply supported beam. Rissone and Williams [324] studied vibrations of non-uniform cantilever beam. Srinivasan [325, 326] employed the Ritz–Galerkin technique to solve the governing nonlinear differential equation of dynamic equilibrium for free and forced vibration of a simply supported Bernoulli-Euler beam. Evensen [327] further extended it to various boundary conditions using the perturbation method. Bennett and Eisley [328] investigated the steady-state free and forced responses and stability for large amplitude motion of a beam with clamped ends using Bernoulli-Euler beam theory. A multimode analytical and numerical technique is used to obtain theoretical solutions for both response and stability. MacBain and Genin [329] studied the effect of rotational and translational support flexibility on the fundamental frequency of clamped-clamped and cantilever beams. Kim and Kim [330] obtained accurate vibration frequencies of beams with generally restrained boundary conditions using Fourier series based on Bernoulli-Euler beam theory. The suggested method is accurate for beams with not only classical boundary conditions but also non-classical boundary conditions restrained by rotational and translational springs.
The study of transverse vibration analysis of bars of uniform cross-section can be traced to the pioneering work of Timoshenko [22, 331] using the first order shear deformation theory also known as Timoshenko beam theory. Cowper [29] derived the equations of Timoshenko beam theory by integration of the equations of three-dimensional elasticity theory and obtained a new formula for the shear coefficient and also obtained the exact fundamental frequencies for a simply supported beam. Many researchers [332-361] have applied Timoshenko beam theory for the free vibration analysis of one dimensional structure such as beams and rods with different boundary condition using various analytical and numerical methods. Raville et al. [362] analyzed the problem of the natural frequencies of vibration of fixed-fixed sandwich beams by an energy approach, wherein the Lagrangian multiplier method is used to satisfy the boundary conditions of the problem. Motaghian et al. [363] studied the free vibration problem of beams resting on partial elastic foundation of the Winkler type using Bernoulli-Euler beam theory. The authors have presented an exact solution by dividing the beam into separate segments. In addition to this, an innovative mathematical approach is proposed to find the precise analytical solution of the free vibration of beams with mixed boundary conditions. Stoykov and Ribeiro [364] studied the geometrically non-linear free and forced vibrations in space of beams with non-symmetrical cross sections by using Timoshenko’s theory for bending and Saint-Venant’s theory for torsion. The p-version finite element method is developed. The warping function, which cannot be derived analytically for complex cross sections, was obtained numerically by solving the Laplace equation with Neumann boundary conditions, using the boundary element method. The integrals over the cross section were obtained numerically by Gauss integration. The stability of solutions in forced vibrations is studied by Floquet’s theory. Stephen and Levinson [365] developed a well-known second order beam theory for the free vibration analysis of beams with different
cross-sections. The theory takes into account effects of shear curvature, transverse direct stresses and rotatory inertia. Heyliger and Reddy [77] developed a finite element model of third order parabolic shear deformation theory for the dynamic behavior of rectangular beams whereas Eisenberger [366] extended the theory using the dynamic stiffness matrix to predict the exact vibration frequencies for beams with various boundary conditions. Kant and Gupta [78] developed higher order shear and normal beam theory for the free vibration analysis of beams using finite element method. Sayyad [100, 101, and 367] presented free vibration analysis of simply supported isotropic beams using various parabolic, trigonometric, hyperbolic, and exponential shear deformation theories based on Navier solution. Carrera, et al. [368-370] presented higher-order theories on the basis of the Carrera Unified formulation for the analysis of free vibration of a beam structures. 3.2 Free vibration of laminated composite and sandwich beams In this section, literature on free vibration analysis of laminated composite and sandwich beam using ESL, LW, ZZ theories and exact elasticity solutions is reviewed and presented. Also literature on finite element modeling for free vibration of composite beam using these theories is reviewed. 3.3.1 Free vibration of laminated and sandwich beams using ESL theories Emam and Nayfeh [371, 372] obtained a closed-form solution for the vibration of a beam in a buckled position using classical beam theory. The solution is obtained for the post-buckling configurations of symmetric and non-symmetric composite beams as a function of the applied axial load. The solution obtained by authors provides benchmark solution for the post-buckling analysis of composite beams. Eisenberger et al. [373] obtained exact vibrational frequencies of generally laminated beams using Timoshenko beam theory. The exact dynamic stiffness matrix
method is used to derive stiffness matrix. The analysis of un-symmetrical beams is carried out. It is observed that the proposed method deals with the general layouts and geometries of the structure and its boundary conditions. Banerjee and William [374], Banerjee [375, 376], Banerjee and Sobey [377], Howson and Zare [378], Banerjee et al. [379] and Damanpack and Khalili [380] developed an exact dynamic stiffness matrix method for the vibration analysis of composite and sandwich beams using Timoshenko and higher order beam theories. Ferreira [381] presented free vibration of laminated composite Timoshenko beams based on the multiquadric radial basis function method. Higher order shear deformation theories considering polynomial or non-polynomial shape functions for the free vibration analysis of laminated composite and sandwich beams considering the effects of transverse shear and normal deformations are developed by many researchers [40, 95, 174, 181, 382-402] using various analytical and numerical methods. Abramovich [403], Abramovich and Livshits [404] and Abramovich et al. [405] studied free vibration and buckling of symmetrically and anti-symmetrically laminated composite beams. McCarthy and Chattopadhyay [406] carried out dynamic analysis of composite box beam using higher order theory to investigate the importance of including inplane and out-of-plane warping deformations. Rao et al. [407] obtained an analytical solution for free vibration analysis of symmetric laminated composite and sandwich beams using higher order mix theory. The natural frequencies obtained by using mixed theory are compared with equivalent single layer displacement based higher order theories. The natural frequencies are obtained for first nine modes. Soldatos and Sophocleous [408] obtained frequency equations and the characteristic functions of a general three-degrees-of-freedom beam theory of Bickford [64] that describes the plane motion of shear deformable elastic beams. Aydogdu [409, 410] applied Ritz method for the free vibration
analysis of cross-ply and angle-ply laminated beams with general boundary conditions using various higher order shear deformation theories. The governing differential equations are obtained by using Hamilton’s principle and six different boundary conditions of beams are considered for detailed numerical investigations. The numerical results are obtained for different aspect ratios and lamination schemes. Jun and coauthors [411-415] developed the dynamic stiffness matrix to study vibration analysis of arbitrary lay-up laminated composite beams based on third order shear deformation theory, trigonometric shear deformation theory and hyperbolic shear deformation theory. It is well-known that the dynamic stiffness method relies on the exact solutions of the governing differential equations of motion; therefore, it can be classified as an exact method. Natural frequencies are determined for different boundary conditions. Zhen and Wanji [416] assessed several displacement-based theories for the free vibration analysis 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. An analytical solution based on the global–local higher-order theory is obtained using Navier’s solution technique. It is concluded that, increasing the order number of in-plane and transverse displacement components, the global higher-order theories can reasonably predict the natural frequencies of laminated beams with arbitrary layouts and soft-core sandwich beams. Giunta et al. [417] presented a free-vibration analysis of simply supported, cross-ply beams via several higher order as well as classical theories. Lower and higher order theories are formulated from the three-dimensional displacement field. Euler–Bernoulli’s and Timoshenko’s theories are special cases of presented displacement field. A Navier-type, closed form solution is adopted in order to derive the governing algebraic equations and natural frequencies associated to torsional, axial, shear and mixed modes are investigated. Carrera, et al. [418] presented the free vibration
response of laminated beams using various refined beam theories by expanding the unknown displacement variables over the beam section axes using Taylor type expansions, trigonometric series, exponential, hyperbolic and zig-zag functions. Prokic et al. [419] carried out the free vibration analysis of thin-walled beams with arbitrary open cross section, made of cross-ply laminates using Vlasov’s beam theory. He and Yang [420] applied higher order shear deformation of Kant and Gupta [78] for the dynamic analysis of two-layered composite beams partial interaction. Jin et al. [421] applied Reddy’s higher order shear deformation theory for vibration analysis of sandwich beams made up of laminated composite face sheets and a viscoelastic core with arbitrary lay-ups and general boundary conditions. A modified FourierRitz method is employed to derive unified formulation. The effects of ply configuration, layer number, moduli and thickness ratios on the natural frequency are illustrated. Especially the effects of restraints from different directions on two ends of the sandwich beam are deeply investigated. Finally, it is concluded that the Fourier-Ritz method yields many new and accurate results at a low computational cost for various boundary conditions including general elastic boundaries. Pagani et al. [422] and Dan et al. [423] developed higher order shear deformation theories using Carrera Unified Formulation for the free vibration analysis of composite beams with solid and thin walled cross-sections. 3.3.2 Free vibration of laminated and sandwich beams using LW/ZZ theories Xavier et al. [424] proposed a simple layerwise higher-order zig-zag model for the vibratory response of soft cored un-symmetric sandwich beams. The theory accounts for a cubic variation of the in-plane displacements in each layer, and a parabolic variation of the transverse shear stress across the thickness of the beam. The number of unknown variables of the proposed theory is the same as that given by the first order shear deformation theory. The numerical results are
obtained for simply supported boundary conditions. Shimpi and Ainapure [425] applied layerwise trigonometric shear deformation theory for the free vibration analysis of two layered antisymmetric laminated composite beams using Navier solution technique whereas Arya [426] carried out free vibration analysis of laminated composite beams using zig-zag trigonometric shear deformation theory using Navier solution technique. Tahani [214] carried out free vibration analysis of laminated composite beams using layerwise displacement theories. Kapuria et al. [226-230] carried out an assessment of zigzag theory for the free vibration of laminated composite, sandwich piezoelectric beams. Youzera et al. [427] studied the damping and forced vibrations of three-layered symmetric laminated composite beams considering normal and shear deformations using the higher-order zig-zag theories. In the first part of this study, linear and nonlinear damping parameters of laminated composite beams are obtained whereas in the second part, nonlinear forced vibration analysis is carried out for small and large vibration amplitudes. Iurlaro et al. [428] carried out the experimental assessment of the refined zigzag theory for the free vibration analysis of sandwich beams and compared the numerical results with Timoshenko beam theory. The specimens used in the experimental investigation are sandwich beams with face-sheets made of 7075 Aluminum alloy and a soft-core made of a Rohacell structural foam, largely employed in aerospace applications. Faces are bonded to the core using a very thin layer of an epoxy structural adhesive designed for both solid panels and honeycomb sandwich constructions. In order to investigate the effect of the face-to-core stiffness ratio, two foams have been considered, namely the IG31 and the WF110. 3.2.3 Free vibration of laminated and sandwich beams using finite element method (FEM) The free vibration analysis of isotropic beams based on classical and refined shear deformation theories using finite element method is presented by Kant and Gupta [78], Kapur
[361], Thomas et al. [343], Thomas and Abbas [344], Dawe [345], Reddy [346], Carrera, et al. [369], Nickel and Secor [429], Davis et al. [430] and Abbas [431]. A variety of literature is found on the finite element models based on the classical beam theory, first order/Timoshenko beam theory, higher order equivalent shear deformation theories, layerwise theories and zig-zag theories for the free vibration analysis of laminated composite and sandwich beams. de Borbon and Ambrosini [432] and Vo and Lee [433, 434] developed an one-dimensional finite element model based on classical beam theory to predict natural frequencies and corresponding vibration modes for a thin-walled composite beam. Nabi and Ganesan [435], Bhattacharya et al. [436], Chakraborty et al. [437], Goyal and Kapania [240], Kapania and Goyal [438], Jafari-Talookolaei and Ahmadian [439], Jafari-Talookolaei et al. [440], Amichi and Atalla [441], Assaf [442], Callioglu and Atlihan [443], Kahya [444], Slimani et al. [445], Ozutok and Madenci [446], Kim and Lee [447] and Stoykov and Margenov [448] developed various finite element models based on first order shear deformation theory for the vibration problems of laminated composite and sandwich beams with various boundary conditions. Ahmed [449] used the finite element displacement method to investigate the flexural vibration characteristics of curved sandwich beams. Author has developed three displacement based models in which an element having three, four and five degrees of freedoms per node are considered. Teoh and Huang [450] and Teh and Huang [451] studied free vibration of fibre reinforced composite beams. The presented models consider the effect of shear and rotary inertia. The equations of motion based on the Timoshenko beam theory are derived using Hamilton's principle. Chen and Yang [452] carried out vibration analysis of an anisotropic symmetrically laminated beam finite element including the effect of shear deformation. Kapania and Raciti [453] and Wu and Sun [454] developed a simple one-dimensional finite element
model considering 10 degrees of freedom at each node for the vibration analysis of laminated composite beams including shear deformation. A finite element model based on a third order shear deformation theory is developed by Chandrashekhara and Bangera [455] to study the free vibration characteristics of cross-ply and angle-ply laminated composite beams. The effects of number of layers, ply orientation and various boundary conditions is studied based on CBT, TBT and third order theory. The importance of Poisson effect in one-dimensional laminated beam analysis is demonstrated in this study. Maiti and Sinha [126] and Shi and Lam [456] applied third order shear deformation theory for the free vibration of laminated composite beams using finite element method. The natural frequencies and corresponding vibration modes of a cantilever sandwich beam with a soft polymer foam core are predicted by Sokolinsky et al. [457] using the higher-order theory, a two-dimensional finite element analysis, and classical sandwich theory. Subramanian [458], Marur and Kant [459], Nazargah et al. [266], Chakrabarti et al. [460] and Vo et al. [461] proposed higher order finite element models based on higher order shear deformation theories to study the vibration response of laminated composite beams. Bassiouni et al. [462] performed an experimental investigation on the dynamic behavior for laminated composite beams and validated theoretically using finite element method. The finite element model includes separate rotational degrees of freedom for each lamina but does not require additional axial or transverse degrees of freedom. The experimental tests are carried out by using a hammer test and the frequency response function is displayed on the FFT analyzer. Numerical results are compared with the experimental ones. Ramtekkar et al. [463] developed a plane-stress mixed finite element model by using Hamilton’s energy principle for the natural vibrations of laminated composite beams using third order beam theory. Continuity of the transverse shear stress has been enforced through the thickness of the laminated beam in the formulation and the
transverse stress components have been invoked as the nodal degrees of freedom by applying elasticity relations. Ganesan and Zabihollah [464, 465] investigated the free undamped vibration response of tapered composite beams using the higher order finite element model. The complete study is presented in two parts. A higher-order finite element formulation is developed in Part I [464]. The stiffness coefficients of the tapered laminated beam are determined based on classical laminated theory. In Part II [465], the developed formulation is used for the free undamped vibration analysis of various types of tapered composite beams and a parametric study is conducted. Pradeep et al. [466] studied the thermal vibration of sandwich beams with composite facings and viscoelastic core. Each composite laminate is modeled as an equivalent single layer. The formulation is a decoupled thermo-mechanical formulation. Linear temperature variation across the thickness of each layer is assumed. The effects of fiber angle, aspect ratio and core thickness on the performance of the elements are studied. Vidal and Polit [467, 468] developed a finite element model for the vibration analysis of laminated composite beams using sinusoidal shear deformation theory. A three-noded finite element is developed with a sinus distribution with layer refinement. The theory accounts the effect of transverse normal stress. Chalak et al. [469] studied free vibration response of laminated sandwich beams having a soft core using a finite element beam model based on zig-zag theory. The theory considers the cubic distribution of axial displacement whereas quadratic distribution of transverse displacement. Nanda et al. [284] presented a spectral finite element model using an efficient and accurate layerwise zigzag theory for wave propagation analysis of highly inhomogeneous laminated composite and sandwich beams. Varello and Carrera [293] and Filippi et al. [294] presented a finite element model based on refined beam theories developed from Carrera’s Unified Formulation (CUF) for the vibration analysis of laminated and sandwich beams.
3.2.4 Exact elasticity solution for the free vibration of laminated and sandwich beams Smith [470] carried out vibration analysis of multilayer orthotropic sandwich beams using a unified analysis method based on two-dimensional elasticity theory. Chen et al. [471-473] obtained elasticity solution for free vibration of laminated beams using a new approach combining the state space method and the differential quadrature method. 4. Buckling of shear deformable beams When the axial compressive force applied to the beam, keeps it in equilibrium, then the beam is said to be stable. But, if the beam does not return to its original equilibrium configuration, the beam is said to be unstable. The magnitude of the compressive force at which the beam becomes unstable is termed the critical buckling load. Therefore, the modern use of composite materials in engineering structures, especially in ships and aircraft, has made elastic instability a problem of great importance. Urgent practical requirements have given rise in recent years to extensive investigations, both theoretical and experimental, of the conditions governing the stability of such structural elements as beams, plates and shells. The first problems of elastic instability, concerning lateral buckling of compressed members, were solved about 270 years ago by Leonard Euler in 1744 and published in the appendix “Des curvis elasticis” i.e, “Elastic curves” of his book [474]. In this section a review of buckling of shear deformable beams/columns is presented being widely used in high performance structure. The in-plane buckling problem of a three-layered sandwich beams /column was studied by Hoff and Mautner [475] and Bauld [476]. Sheinman and his co-authors [477-481] studied post buckling behaviour of laminated beams based on geometrical nonlinear theory. Somers et al. [482, 483] developed an analytical model for predicting buckling and describing the post buckling behavior of the sandwich beam. Ha [484] formulated the exact stiffness matrix for the
bending and overall buckling problem of a general class of sandwich beam taking into account the nonlinear effects of axial thrust. The term ‘exact’ is used in the sense that the solution satisfies the governing differential equation, in the same way that the ‘exact’ solution obtained by conventional frame analysis satisfies the engineering beam theory. Barbero and Raftoyiannis [485] and Barbero and Tomblin [486] developed an analytical model using classical theory for Euler buckling of pultruded composite columns. Sherbourne and Kabir [487] carried out an analytical study on the lateral buckling of thin-walled open-section fibrous composite beams. The simply supported and clamped I-section beams subjected to uniformly distributed and transverse central point loads are considered for the detail numerical study. Cheng et al. [488] presented the method of continuous analysis for predicting the local delamination buckling load of the face sheet of sandwich beams. Fourier series in conjunction with the Stokes transformation is used which provides a unified solution for problems with different end conditions. Morey et al. [489] developed a simple beam theory for the buckling of symmetric angle-ply composite beams including interaction of inplane stresses. Huang and Kardomateas [490] studied buckling and initial post-buckling behavior of sandwich beams. A nonlinear beam equation including transverse shear is developed. A perturbation procedure is subsequently applied to obtain a closed-form solution for the initial post buckling behavior. Galuppi and Carfagni [491] presented buckling of simply supported three layered sandwich beams with viscoelastic core under compression load. The phenomena of glassy, rubbery and creep buckling are examined for various values of the load. 4.1 Buckling of laminated and sandwich beams using ESL theories Emam and Nayfeh [372] presented vibration and post buckling behaviour of composite beams using Euler–Bernoulli beam theory. Challamel and Girhammar [492] presented buckling of
partial composite beam–columns including shear and axial effects using Euler–Bernoulli and Timoshenko beam–column theories. Fu et al. [493] developed analytical solutions of thermal buckling and post-buckling analysis of symmetric laminated composite beams with various boundary conditions based on the Timoshenko beam theory by considering geometric nonlinearity. The governing differential equations for beams subjected to a uniform temperature rise are derived by using Hamilton’s principle. The solutions for governing equations are obtained using analytical method, shape function method and Ritz method. Aydogdu [174], Vo and Inam [397], Khdeir and Reddy [494], Khdeir [495] and Aydogdu [496, 497] developed displacement based higher order shear deformation theories for the buckling analysis of laminated composite and sandwich beams. Matsunaga [498-500] studied the buckling analysis of a simply supported thick elastic beam subjected to axial stresses using higher order shear and normal deformation theory in which axial and transverse displacements are expanded in power series. Song and Waas [501] carried out buckling analysis of stepped laminated composite beams using simple higher order beam theory (SHOT) which assumes a cubic distribution of the displacement field through the thickness. It is observed that the results from SHOT did not show significant differences to those from Timoshenko theory for a wide range of aspect ratios of the beam geometry and material properties. In addition, the results from SHOT did not necessarily show more compliant results than those from Timoshenko beam theory. A two-dimensional elasticity solution is obtained for the buckling of clamped-clamped beams. Phan et al. [502] applied extended high-order sandwich panel theory developed by Phan et al. [503] for global buckling analysis of sandwich beams. Komijani et al. [504] studied the thermal buckling behavior of two layered Timoshenko composite beams with partial interaction between the layers. The Timoshenko kinematics is considered for both layers and the shear
connection is represented by a continuous relationship between the interface shear flow and the corresponding slip. Geometrically nonlinear behavior based on the von Karman simplification of the Green strain tensor is accounted in the formulation. Giunta et al. [505] developed hierarchical theories using Carrera’s Unified Formulation for the buckling analysis of thin-walled beams with open and closed cross-section. Smyczynski and Blandzi [506] developed a mathematical model for the dynamic stability analysis of a simply supported five layered sandwich beam. The beam has two metal faces, the metal foam core and two binding layers between faces and the core. The equations of motion are obtained using Hamilton's principle. 4.2 Buckling of laminated and sandwich beams using LW/ZZ theories Zhen and Wanji [406] carried out an assessment of several displacement based zig-zag theories for the stability analysis of laminated composite and sandwich beams. Kapuria et al. [507] developed zig-zag theories for the buckling analysis of laminated composite and sandwich beams whereas Kapuria and Alam [508] developed an efficient electromechanically coupled geometrically nonlinear zigzag theory for the buckling analysis of hybrid piezoelectric beams under electro-thermo-mechanical loads. Chakrabarti et al. [509] developed a new finite element model using zig-zag theory for the buckling analysis of soft core sandwich beams. 4.3 Buckling of laminated and sandwich beams using finite element method (FEM) Goyal and Kapania [240] developed a 21 degree-of-freedom element based on the first order shear deformation theory for the buckling analysis of laminated composite beams. Kim and Lee [447] developed a finite element model to calculate the buckling loads of thin-walled Timoshenko laminated beams subjected to variable forces. Vo et al. [461] developed a finite element model based on sinusoidal shear deformation theory to study the buckling behaviour of composite beams with arbitrary lay-ups. Pradeep et al. [466] developed a finite element
formulation for the thermal buckling of sandwich beams. Wang et al. [510] presented isogeometric finite element analysis of buckling of symmetric and anti-symmetric, cross-ply and angle-ply laminated composite beams with different boundary conditions using the first order shear deformation theory. Loja et al. [511] used a higher order shear deformation theory assuming a non-linear variation of the displacement field to develop a finite-element model and applied for the linear buckling behaviour of anisotropic multi-laminated or sandwich thick and thin beams. Dafedar and Desai [512] developed a unified mixed higher order analytical formulation to evaluate the buckling behaviour of laminated composite struts. Two sets of mixed models are presented by selectively incorporating nonlinear components of Green’s strain tensor. Hu et al. [513, 514] developed various one-dimensional finite element models based on higher order shear deformation theory for the buckling analysis of sandwich beams. Silva et al. [515] developed a finite element formulation based on nonlinear generalized beam theory for the buckling and post-buckling behavior of laminated composite cylindrical panels. Gupta et al. [516] proposed a simple, elegant, and accurate closed-form expressions for predicting the post buckling behavior of composite beams with axially immovable ends using the Rayleigh–Ritz method. Ibrahim et al. [517] presented the buckling analysis of laminated composite thin walled structures by the one dimensional finite element based unified higher-order models obtained within the framework of the Carrera Unified Formulation. Chakrabarti et al. [509] developed a new finite element model using zig-zag theory for the buckling analysis of soft core sandwich beams. Kahya [518] presented buckling analysis of laminated composite and sandwich beams by the finite element method according to the first-order shear deformation theory. Leissa [519] discussed the buckling issues of anti-symmetrically laminated plates under inplane loading conditions; however, such investigations in the case of composite beams with un-
symmetric, antisymmetric, and arbitrary lay-ups are missing in the literature. Further, exact elasticity solutions for buckling of laminated beams are not available in the literature to assess the validity of various refined theories. 4.4 Exact elasticity solutions for the buckling of laminated and sandwich beams Smith [470] presented buckling analysis of multilayer orthotropic sandwich beams using twodimensional elasticity theory. Kardomateas [520, 521] obtained three-dimensional elasticity solutions for the buckling analysis of transversely isotropic rods and sandwich columns. D’Ottavio et al. [522] developed 2D elasticity solution for the global buckling of sandwich struts.
5. Concluding remarks with future direction Beam elements made up of a composite material are being widely used in the civil, mechanical, aeronautical and aerospace industries due to their attractive properties such as; high strength-and stiffness-to-weight ratio and their anisotropic material property. The shear deformation effects are more pronounced in the laminated composite and sandwich beams subjected to transverse loads. Therefore, bending, free vibration and buckling analysis of laminated composite and sandwich beams have received widespread attention in recent years. The various analytical and numerical methods have developed by the researchers for the analysis of laminated composite and sandwich beams. This article reviews the literature available for the bending, free vibration and buckling analysis of laminated composite and sandwich beams. Also, the displacement field of various displacement based equivalent single layer theories and layerwise theories is presented in this article. Based on the literature reviewed, following conclusions/suggestions are made. 1.
Analysis of laminated composite and sandwich beams is difficult by using twodimensional elasticity theory and this led to the development of refined shear deformation
theories for beams which approximate the two-dimensional elasticity solutions with reasonable accuracy. 2.
Any refinements of classical models are meaningless, in general, unless the effects of interlaminar continuous transverse shear and normal stresses are both taken into account in a multilayered beam theory. But, in a whole variety of higher order beam theories existing in the literature, no consideration is given to the effect of transverse normal deformation/strain ( z 0 ) on bending, buckling and vibration responses when these theories are applied to laminated composite and sandwich beams in view of minimizing the number of unknown variables. Therefore, refined theories which consider the effects of transverse normal deformation need more attention in future.
3.
Buckling issues of composite beams with un-symmetric, antisymmetric, and arbitrary lay-ups are missing in the literature. Further, exact elasticity solutions for buckling of laminated beams are not available in the literature to assess the validity of various refined theories. Also, issues of torsional and dynamic instabilities are needed to be resolved in future.
4.
Exact 2-D elasticity solution for the vibration analysis of laminated composite and sandwich beams is not available in the whole variety of literature. Shear mode frequencies for laminated composite and sandwich beams are rarely available in the literature. These results can be served as the benchmark solutions for the researchers in the future.
5.
Geometrically non-linear analysis of laminated composite and sandwich beams needs more attention in future.
6.
Since equivalent single layer theories are computationally simpler, those are widely used by various researchers for the bending, free vibration and buckling analysis of simply supported laminated composite beams. But, these theories are not capable enough to capture the responses of sandwich beams. Therefore, layerwise and zig-zag theories are more accurate for the sandwich beams with a soft core.
7.
A number of equivalent single layer theories can be constructed from the Carrera’s Unified Formulation (CUF). Thus, most of the displacement based equivalent single layer theories reviewed in the present study are the particular cases of CUF.
8.
It is also pointed out that the Navier solution (for simply supported boundary condition) and the finite element solution (other boundary conditions) are widely used by various researchers for the analysis of laminated and sandwich beams. However, other numerical methods such as differential quadrature method, Rayleigh-Ritz method, Galerkin method, Radial basis functions and meshfree method are not fully explored as reviewed in the literature.
9.
Bending, bucking and vibration of laminated and sandwich beams subjected to constant, linear and non-linear thermo-mechanical load needs attention in future. The influence of temperature profiles (heat conduction problem) on the thermo-mechanical response of multilayered beams considering classical and advanced theories need to be studied extensively.
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S0263-8223(16)33027-6 http://dx.doi.org/10.1016/j.compstruct.2017.03.053 COST 8374
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Composite Structures
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Please cite this article as: Sayyad, A.S., Ghugal, Y.M., Bending, buckling and free vibration of laminated composite and sandwich beams: A critical review of literature, Composite Structures (2017), doi: http://dx.doi.org/10.1016/ j.compstruct.2017.03.053
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Bending, buckling and free vibration of laminated composite and sandwich beams: A critical review of literature Atteshamuddin S. Sayyad1* and Yuwaraj M. Ghugal2 1
Department of Civil Engineering, SRES’s Sanjivani College of Engineering, Savitribai Phule Pune University, Kopargaon - 423601, Maharashtra, India. 2 Department of Applied Mechanics, Government College of Engineering, Karad - 415124, Maharashtra, India.
Abstract Laminated composite and sandwich structures are lightweight structures that can be found in many diverse applications especially civil, mechanical and aerospace engineering. The rapid increase in the industrial use of these structures has necessitated the development of new theories that suitable for the bending, buckling and vibration analysis of composite structures. Many review articles are reported in the literature on laminated composite plates and shells in the last few decades. But, in the whole variety of literature very few review articles are available exclusively on laminated composite and sandwich beams. In this article, a critical review of literature on bending, buckling and free vibration analysis of shear deformable isotropic, laminated composite and sandwich beams based on equivalent single layer theories, layerwise theories, zig-zag theories and exact elasticity solution is presented. In addition to this, literature on finite element modeling of laminated and sandwich beams based on classical and refined theories is also reviewed. Finally, displacement fields of various equivalent single layer and layerwise theories are summarized in the present study for the reference of researchers in this area. This article cites 512 references and highlights, the possible scope for the future research on laminated composite and sandwich beams. Keywords: beam theories; laminated; sandwich; bending; buckling; vibration.
*
Corresponding author, Email: [email protected], Ph. No.: (+91) 9763567881
1. Introduction Beams, columns, and rods made up of a composite material are being widely used in the civil, mechanical, aeronautical and aerospace industries due to their attractive properties such as; high strength-and stiffness-to-weight ratio and their anisotropic material property. The beam members are mainly subjected to bending whereas columns and rods are mainly subjected to axial tension and compression. The shear deformation effects are more pronounced in the laminated composite and sandwich beams subjected to transverse loads. Therefore, bending, free vibration and buckling analysis of laminated composite and sandwich beams have received widespread attention in recent years. The various analytical and numerical methods based on beam theories have developed by the researchers for analysis of shear deformable laminated composite and sandwich beams. Many review articles are reported in the literature on laminated composite plates and shells in the last few decades [1]. But, in the whole variety of literature very few review articles are available exclusively on laminated composite and sandwich beams [2-7]. Kapania and Raciti [2, 3] reviewed literature on bending, buckling, post-buckling, vibration and wave propagation analysis of laminated composite beams and plates using various displacement and stress based shear deformation theories. The mathematical formulation of some classical and higher order shear deformation theories is presented. Authors have also reviewed finite element models for the analysis of laminated beams and plates and provided suggestions for future research. Ghugal and Shimpi [4] presented a review of displacement and stress based refined theories for isotropic and anisotropic laminated composite beams. Merits and demerits of various equivalent single layer and layerwise theories are discussed. Marur [5] has thrown some light on distinct phases in the development of nonlinear vibration of beams. Gherlone [6] presented a brief review on zig-zag theories for the analysis of laminated composite and sandwich beams. It
is concluded that the equivalent single layer theories are easy to implement and computationally affordable but, in order to correctly describe the mechanical behavior of laminated structures the displacement field needs to be enriched by a through-the-thickness piecewise linear contribution i.e. zig-zag function. Hajianmaleki and Qatu [7] presented a review of research done by various researchers on the vibration analysis of composite and smart beams. The review is based on classical and higher order shear deformation theories developed for the analysis of beams and numerical methods used for solving governing differential equations. The well-known books on laminated composite and sandwich beams have published by Reddy [8], Wang et al. [9] and Carrera, et al. [10]. Therefore, in the present study an attempt is made to review the research done by various researchers on isotropic, laminated composite and sandwich beams using different analytical and numerical methods based on classical and refined beam theories.
2. Bending of shear deformable beams A great deal of research has been carried out since the last few decades to accurately assess the bending response of beams. Bending analysis of beams using two-dimensional elasticity theory is very complicated and this led to the development of refined shear deformation theories for beams which approximate the two dimensional solution into one dimensional solution with reasonable accuracy. Two classes of theories are available in the literature for the analysis of beams. In the first class of theories, the effects of shear deformation and rotary inertia are neglected whereas in the second class of theories, shear deformation and rotary inertia effects are considered.
2.1 Bending of isotropic beams The most commonly used theory for the bending analysis of beams is classical beam theory (CBT) developed by Bernoulli-Euler [11-16] almost 300 years ago. Chronological development of this theory from the first attempt by Galileo in 1638 till 1856 by Barre de Saint Venant is given by Love [17], Timoshenko [18] and Todhunter and Pearson [19]. This theory neglects the shear deformation and rotary inertia effects; therefore, this theory is typically accurate for thin beams and is less accurate for thicker beams. Rayleigh [20] improved the classical theory by allowing the effect of rotary inertia of the cross-sections of the beam. Boley [21] studied an accuracy of the CBT for the beams of variable section. The stresses and deflections of the beam are found, on the basis of the two-dimensional theory of plane stress. The Deflection and stresses are obtained in the form of series using Airy-s stress function; the first term of each series is identical with the strength-of materials solution and the others represent the necessary correction to CBT. Timoshenko [22] developed a new beam theory which was considered as a refinement of the classical beam theory. This theory introduced first-order shear effects as well as effect of rotational inertia in the kinetic energy. Therefore, this theory is also known as the first order shear deformation theory or Timoshenko beam theory (TBT). The TBT violates the zero shear stress conditions on the top and bottom surfaces of the beam. A shear correction factor is required to appropriately represent the strain energy of shear deformation. TBT also suffers from the boundary condition paradox. Many articles are published on Timoshenko beam theory and shear correction factors [23-56]. Recently, Elishakoff et al. [57] presented historical development of TBT and the research available in the literature based on this hundred years old theory which considers the effects of transverse shear deformation and rotary inertia.
However, CBT and TBT do not take into account the non-classical influences such as crosssectional warping, out-of-plane and in-plane deformations. The detail discussion on these influences is presented by Goodier [58]. Therefore, the higher order shear deformation theories are necessary to account for these non-classical influences. These higher order shear deformation theories are developed using the method of hypotheses, the method of expansion, the method of successive approximation and the mixed method. Donnell [59] and Boley and Tolins [60] have developed refined theories using the method of successive approximations for the bending of thick beams. Refined theories in which transverse shear stress is varying parabolically across the thickness of the beam and does not require problem dependent shear correction factor are called as parabolic shear deformation theories [61-80]. Irschik [81] has established an analogy between classical Bernoulli-Euler theory and refined beam theories of Levinson [68], Rehfield and Murthy [65] and by Rychtcr [74] using the principle of virtual work to increase the acceptance of higher order refined theories. The beams of rectangular cross-section with clamped and hinged boundary conditions are considered for the detail investigation. Corrections to the classical solution have been interpreted as being due to additional self-stress acting in the Bernoulli-Euler beam. Refined theories which account for the trigonometric cosine distribution of the transverse shear stress/strain through the thickness of the beam are called as trigonometric shear deformation theories [82-88]. These theories satisfy the traction free boundary conditions at top and bottom surfaces of the beam and does not require problem dependent shear correction factor. In these theories, in-plane displacement uses trigonometric function to account the effect of shear deformation and transverse displacement uses cosine function to account the effect of normal deformations. The origin of these theories can be found in Love [17], Levy [89], Kromm [90, 91], Panc [92] for bending of elastic plates and in Vlasov and Leont’ev [93] and Stein [94] on
bending of elastic beams. Refined theories containing the hyperbolic sine and cosine functions in-terms of the thickness coordinate to account for the effects of transverse shear and normal deformations in the displacement field are called as hyperbolic shear deformation theories [9092]. Beam theories deduced from such plate theories are presented by various authors [95-98]. Karama et al. [99] have developed a refined theory containing the exponential functions in-terms of the thickness coordinate in the displacement field. This theory is called as exponential shear deformation theory. It is reported that the exponential functions are richer than sine and cosine functions, because an exponential function has all even and odd power in its expansion unlike sine function, which has only odd power. The numerical results for bending and free vibration analysis of beams with different boundary conditions are presented using Levy’s solution technique. Recently, Sayyad [100, 101] presented comparison of parabolic, trigonometric, hyperbolic, and exponential shear deformation theories for the bending of isotropic beams. All these functions are accounted in the unified shear deformation theory and closed form solution is obtained using the Navier solution technique. Gao and co-authors [102-104] derived a refined theory of rectangular and curved beams by using the Papkovich-Neuber solution and Lure method without adhoc assumptions. Shi and Voyiadjis [105] and Wang and Shi [106] proposed a new sixth order shear deformation theory for the analysis of shear deformable isotropic beams with various boundary conditions. A sixth order beam theory is developed to resolve the paradoxes of TBT on the displacement boundary conditions or continuity conditions of shear flexible beams. It is also recommended that this theory can be easily used in the finite element analysis of composite beams. Carrera and coauthors [107-110] developed several refined theories for the linear static analysis of beams made of isotropic materials based on Carrera’s Unified Formulation (CUF).
Lin and Zhang [111] developed a new displacement based beam element with two degrees of freedoms per node for finite element analyses of isotropic and composite beams using Timoshenko beam theory. A layered approach is employed to describe the layered characteristic of the composite beams. Miranda et al. [112] developed a generalized beam theory considering the effect of shear deformation for a cantilever beam of various cross-sections. 2.2 Bending of laminated and sandwich beams Multilayered composite and sandwich beams are widely used in many engineering applications. While composite laminates provide higher stiffness, strength and reduce the weight over conventional metallic structures, sandwich structures can further reduce the structural weight and improve performance without sacrificing stiffness and strength. The geometry and layer numbering scheme for the general laminated composite beam is shown in Fig. 1 and 2 respectively. In this section, literature on bending of laminated composite and sandwich beams using equivalent single layer (ESL) theories, layerwise theories (LW), zig-zag (ZZ) theories and Carrera’s Unified Formulation is reviewed. This section also presents the literature on finite element modeling of laminated composite and sandwich beams and exact elasticity solutions.
Fig. 1. Laminated composite beam geometry
Fig. 2. Layer numbering system of laminated composite beam
2.2.1 Bending of laminated and sandwich beams using ESL theories Equivalent single layer (ESL) theories are derived from the method of hypotheses. In the equivalent single layer displacement based theories, one single expansion for each displacement component is used through the entire thickness of the laminate. Equivalent single layer theories are widely used for the bending analysis of simply supported isotropic, laminated composite and sandwich beams. These theories are classified as classical beam theory, first order shear deformation theory, second and third order shear deformation theories based on Taylor series expansion and higher order shear deformation theories involving parabolic, trigonometric, hyperbolic, exponential and mixed functions in the displacement field. Boay and Wee [113] studied the effect of coupling on bending, buckling and vibration of angle-ply laminated composite beams using the classical lamination beam theory. A closed form solution for clamped and simply supported beams is obtained and the results are compared with those generated using the finite element method. Chen et al. [114] presented bending of symmetrically cross-ply laminated composite beams using first order deformations (TBT) based on modified couple stress theory. The deflection and stresses of symmetric composite beams are obtained using CBT and TBT and compared with each other. Hajianmaleki and Qatu [115] applied first order shear deformation for the static analysis of composite laminated beams of rectangular and tubular
cross-section. A three-dimensional finite element model is used to verify the proposed beam model. In addition, the model is also applied to tubular cross section beams and the results were compared with experimental data and other models existing in the literature. Silverman [116] presented a parabolic shear deformation theory for the narrow doubly symmetrical three-layer rectangular beam. Valisetty [117] has developed a refined bending theory for the beam of solid circular cross section considering non-classical influences observed by Goodier. The theory is developed from the engineering stresses and the solution is provided to the problem of propagation of harmonic flexural waves in a beam of solid circular cross section. Hu et al. [118] presented experimental validation of parabolic shear deformation theory. A cantilever cross-ply laminated beam is tested experimentally where strains are measured using strain gauges and deflections are measured with dial gauges. The strain distributions in the endeffect region were found to be much closer to those predicted by the parabolic shear deformation theory. The displacement based higher order theories based on Taylor series expansion [119-128] are cumbersome and computationally more demanding, because; with each additional power of the thickness coordinates, an additional dependent unknown variable is introduced into the theory which becomes difficult to interpret physically. It has been noted that due to the higher order of terms included, the theories are not convenient to use. This observation is more or less true for many other higher order theories as well. This drawback led to the development of simple to use higher order beam theory with less number of variables. Such displacement based higher order shear deformation theories are developed and applied by Reddy [129], Khdeir and Reddy [130], Soldatos and Watson [131], Zenkour [132], Reddy et al. [133], Matsunaga [134], Liu and Soldatos [135], Ferreira et al. [136], Kroker and Becker [137], Cernescu and Romanoff [138],
El-Nady and Negm [139] and Pawar et al. [140] for the analysis of laminated composite and sandwich beams. These theories contain three or four unknown variables only and accounts for traction free conditions at the top and bottom surfaces of the beam using constitutive relations. Also these theories do not require shear correction factor. Frostig and his co-authors such as Frostig and Baruch [141-143], Frostig et al. [144], Frostig [145-148], Peled and Frostig [149], Frostig and Peled [150], Frostig and Shenhar [151], Shenhar et al. [152], Swanson [153] and Swanson and Kim [154] published series of research papers on bending of sandwich beams with soft core considering transverse flexibility of the core. Kosmatka [155] developed a new onedimensional theory to obtain section modulus of prismatic beams with arbitrary cross-section. The theory accurately accounts for the effects of shear deformation out of the cross-sectional plane and anticlastic type deformations within the cross-sectional plane. Savoia and Tullini [156] proposed a displacement based shear deformation theory for the bending of orthotropic beams. The stress distributions are presented for cantilever and clamped boundary conditions. Rand [157] carried out multilevel analysis of laminated composite beams based on a hierarchy of solution levels that enabled the prediction of wide spectrum of physical phenomena including gross and local quantities such as beam bending, extension, twist components, in-plane warping, interlaminar stresses and delamination effects. Mistou et al. [158], Naik [159], Arya et al. [160], Zenkour et al. [161, 162], Sayyad and Ghugal [163, 164], Vo and Thai [165], Sayyad et al. [166, 167], Nazargah et al. [168], Ghugal and Shikhare [169] and Nguyen and Nguyen [170] developed displacement based higher order shear deformation theories considering trigonometric functions in terms of thickness coordinate to account the effects of transverse shear and normal deformations for the bending analysis of laminated composite and sandwich beams. All these authors have used Navier’s solution
technique to obtain a closed form solution for the simply supported beams. The kinematics of the trigonometric shear deformation theories is much richer than those of the other higher order shear deformation theories, because if the trigonometric term is expanded in power series, the kinematics of higher order theories is implicitly taken into account to good deal of extent without loss of physics of the problem. Kant et al. [171] proposed a simple semi-analytical method for the stress analysis of laminated and sandwich narrow beams. The proposed model is based on the solution of a two-point boundary value problem governed by ordinary differential equations through thickness of the beam. Forth-Order Runge-Kutta-Gill algorithm is used for numerical integration. The proposed model maintains continuity of transverses stresses and displacements at the laminae interfaces. Apetre et al. [172] carried out analytical modelling of a simply supported sandwich beams using first order, third order, higher order shear deformation theories based on Fourier–Galerkin method. The numerical results are compared with the finite element analysis. The core of the one-dimensional sandwich panel is made up of functionally graded material. Karama et al. [99, 173] and Aydogdu [174] presented a new shear stress function in the form of the exponential function, to predict the mechanical behaviour of multi-layered laminated composite beams. Mechab et al. [175] presented bending analysis of short laminated composite beams subjected to mechanical and thermal loading. The formulation is based on classical and higher order shear deformation theories. The beam is subjected to a uniform load and heated at a constant temperature. It is concluded that the refined theories used gives a higher value of deflection when compared to the classical theory because of the contribution from shear deformation. Tonelli et al. [176] studied stress state of sandwich beams under bending and shear using first order shear deformation theory, zig-zag theory and higher order shear deformation theory. The numerical
results for simply supported sandwich beams subject to uniform loading obtained by these three analytical models compared with those obtained from finite element simulations. Nonlinear material response of a sandwich beam with bilinear elasto-plastic constitutive relations for the transverse normal and shear stresses at the core is studied by Salami et al. [177] based on a proposed improved high order sandwich panel theory. The face sheets are modelled using the first order shear deformation theory whereas the core is modeled using the two-dimensional elasticity. Authors also performed experimental investigations under conditions of three point bending on sandwich beams. Nonlinear bending analysis of shear deformable anisotropic laminated composite beams with various kinds of distributed loads resting on a two-parameter elastic foundation is investigated by Li and Zhao [178]. Sayyad et al. [179] applied two variable plate theory developed by Shimpi and Patel [180] for the stress analysis of laminated composite and sandwich beams. In this theory, transverse displacement consists of two parts (bending and shear) along the center line of the beam. The Navier solution technique is employed to obtain closed form solutions for simply supported laminated composite and soft core sandwich beams. The displacement fields of various reviewed equivalent single layer theories considering effects of transverse shear and normal deformations are given below in Tables 1 and 2.
Table 1 Displacement fields of equivalent single layer beam theories considering effect of transverse shear deformation ( z 0 ). Reference
Axial displacement (u)
Bernoulli [11-13], Euler [14]
u u0 z
Timoshenko [22]
u u0 z
Krishna Murty [67]
Levinson [68-71], Reddy [129]
u u0 z
Transverse displacement (w)
dw0 dx
w w0 w w0
4 z 2 dwb z 1 dx 3 h
2 dw 4 z u u0 z 0 3 h dx
w wb ws w w0
Kant and Manjunath [121]
u u0 z x z 2u*0
w w0
Kant and Manjunath [121]
u u0 z x z 3 x*
w w0
Kant and Manjunath [121]
u u0 z x z 2u*0 z 3 x*
w w0
Soldatos and Elishakoff [95]
u u0 z
dw0 1 z z cosh h sinh dx 2 h
w w0
Ghugal and Shimpi [82]
u u0 z
dw0 h z sin dx h
w w0
Karama et al. [99]
u u0 z
z 2 dw0 z exp 2 dx h
w w0
Benatta et al. [182, 183]
u u0 z
dw0 dw f z 0 dx dx
w w0
Aydogdu [174]
dw u u0 z 0 z dx
Mahi et al. [184], Sayyad and Ghugal [98, 101]
u u0 z
Shi and Voyiadjis [105]
z 5 z 3 dw 5 4z3 u u0 z 2 2 0 4 3h 4 3h dx
Vo and Thai [185], Sayyad et al. [167]
u u0 z
dwb h z dw z sin s dx h dx
w wb ws
Vo and Thai [185]
u u0 z
dwb 4 z 3 dws dx 3h2 dx
w wb ws
Ambartsumian [187]
u u0 z
dw0 z h2 z 2 dx 2 4 3
w w0
2 z / h ln
2
dw0 dw f z 0 dx dx
w w0
w w0 w w0
Table 1 Continue. Kruszewski [23], Panc [92], Reissner [188]
u u0 z
Ray [189]
u u0 z
Shimpi et al. [190] Shimpi et al. [190]
Shimpi and Patel [180], Sayyad et al. [179]
2 dw0 5 z 4 z 1 dx 4 3 h
3 dw0 1 3 z z 2 Qx dx x 2 h h dw u u0 z b dx
u u0 z
d dx
w w0
w w0 w wb ws w w0
1 5 z 2 dws dwb u u0 z z dx 4 3 h dx z2 dw0 z sech 2 z sech 1 tanh dx 4 4 2 h
w wb ws
Akavci [191]
u u0 z
Akavci [191]
u u0 z
dw0 3 z 1 h tanh z sec2 h dx 2 h 2
w w0
Mantari et al. [192]
u u0 z
2 dw0 2 z / h zm dx
w w0
Meiche et al. [193]
u u0 z
dwb h / sinh z / h z dws dx cosh / 2 1 dx
w wb ws
Xiang et al. [194]
12 u u0 zx nh
Mantari et al. [196]
u u0 z
z dw0 z m cos h m z sin e dx h h
w w0
Mantari et al. [197]
u u0 z
dw0 mh tan mz mz sec2 dx 2
w w0
Daouadji et al. [198]
u u0 z
Thai et al. [200]
n 1
dw z n x x,t 0 dx
dwb 3 1 3 z dw z 1 sech 2 h tanh s dx 2 2 2 h dx
4z3 4z3 u u0 z 2 x 2 x 3h 3h z2 dwb dw z 1 sech 2 z sech 1 tanh s dx h 4 2 4 dx
Daouadji et al. [202]
u u0 z
Grover et al. [203]
u u0 z
Mahi et al. [205]
u u0 z
Sarangan and Singh [206]
u u0 z
dw0 2r rz sinh -1 z dx h h r2 4
z3 dw0 h dw 4 2z tanh 0 2 dx 2 h dx 3cosh 1 h3
dw0 g z z dx
w w0
w w0
w wb ws w w0
w wb ws w w0
w w0 w w0
Table 2 Displacement fields of equivalent single layer beam theories considering effects of transverse shear and normal deformation ( z 0 ) . Reference
Axial displacement (u)
Kant and Manjunath [122]
u u0 z x
w w0 z z
Kant and Manjunath [122]
u u0 z x z 2u*0
w w0 z z
Kant and Manjunath [122]
u u0 z x z 3 x*
w w0 z z
Kant and Manjunath [122]
u u0 z x z 2u*0 z 3 x*
w w0 z z
Zenkour [181]
u u0 zu1
Zenkour [132]
dw dw u u0 z 0 u1 z 2 1 dx dx 2 dw dw2 z 3 1 0 u1 dx 3 dx
Maiti and Sinha [126]
u u0 z x z 2u*0 z 3 x*
Sayyad and Ghugal [163]
u u0 z
dw0 h z sin dx h
u u0 z
dwb 4 z 3 dws dx 3h2 dx
Vo et al. [186]
Pawar et al. [140]
u u0 z
Transverse displacement (w)
w w0 zw1 z 2 w2
dw0 x, t dx
w w0 zw1 z 2 w2
w w0 z z z 2 w*0 z 3 z* w w0
z cos h h
4z2 w wb ws 1 2 wz 3h w w0 f ' z
f z
Neves et al. [195]
z u u0 z u1 sin uz h
w w0 z w1 z 2 w2
Neves et al. [199]
z u u0 z u1 sinh uz h
w w0 z w1 z 2 w2
Bessaim et al. [201] Zenkour [204]
u u0 z
dwb z 1 dw z h sinh z cosh s dx h 2 dx
u u0 z
3 dw0 z 4 z 1 d h sinh cosh 2 dx h 3 h 2 dx
z 1 w wb ws cosh cosh h 2
w w0
2 1 z 4z 1 cosh 2 cosh 12 h h 2
2.2.2 Bending of laminated and sandwich beams using LW/ZZ theories The major drawback of ESL theories is double valued shear stress at the layer interface. In order to remove this defect, it is necessary to describe each composite laminate as an assembly of individual layers bonded perfectly. In layerwise (LW) theories, the axial displacement field is approximated across the thickness, layerwise or sublaminate wise with continuity at each interface, and transverse displacement is usually assumed to be independent of the thickness coordinate. Krajcinovic [207], Swift and Heller [208], Davalos et al. [209], Shimpi and Ghugal [210, 211], Arya et al. [212] and Ghugal and Shinde [213], Tahani [214], Afshin and Behrooz [215] and Aitharaju and Averill [216] developed a variationally consistent layerwise shear deformation theory for bending of laminated composite and sandwich beams. The displacement fields of various reviewed layerwise beam theories are given in Table 3. Results obtained using layerwise theories are very close to exact three-dimensional elasticity 3D results. ESL theories are easy to implement and computationally affordable but, assumptions of continuous function of inplane displacements through the laminate thickness cannot produce conspicuous kinks (zigzag) in the inplane displacements distribution typical of thick laminates as given by the exact elasticity solution. The zigzag term of the displacement field is used to model the local distortion of the cross section in each lamina of multilayered structures and is related to the continuity of transverse stresses. Therefore, in zig-zag (ZZ) theories, a piecewise cubic displacement function is superimposed over a linear displacement field and the transverse displacement is usually assumed to be independent of the thickness coordinate. Carrera [217] presented a historical review on zig-zag theories for the analysis of multilayered laminated structures and made an important conclusion that the Lekhnitskii [218] was the first to use zig-zag theory for solving an elasticity problem involving a layered beam. Arya et al. [160], Aitharaju and Averill [216], Icardi
[219-222], Di Sciuva and Icardi [223], Di Sciuva et al. [224], Kapuria et al. [225-230], Vidal and Polit [231], Cook and Tessler [232], Tessler et al. [233-235], Tessler [236] and Di Sciuva et al. [237] developed higher order zig-zag models for the analysis of laminated composite and sandwich beams.
Table 3 Displacement fields of layerwise theories considering the effects of transverse shear and normal deformations. Reference
Axial Displacement (u)
Transverse displacement (w)
Davalos et al. [209]
w w0
n
u u0 U j x,t j z j 1
Shimpi and Ghugal [210]
1
u
Shimpi and Ghugal [211]
dw0 z / h C1 C2sin dx 2 0.5 dw z / h z h 0 sin dx 2 0.5
w w0
dw0 z / h h C1 C2sin dx 2 0.5 dw z / h z h 0 h C3 +sin dx 2 0.5
w w0
u z h 2
u z h 1
u
2
w Wk x,t k z
Tahani [214],
u U k x,t k z
Afshin and Behrooz [215]
k 1, 2,.....,N 1
Aitharaju and Averill [216]
u k ub z b z 2
k 1 i 1
z zi
z z wk 1 wb wt h h
2.2.3 Bending of laminated and sandwich beams using finite element method (FEM) Since Navier solution technique is applicable to bending, buckling and free vibration analysis of simply supported beams only, many researchers have performed studies on other boundary conditions of beams using the finite element technique. Finite element method is the most widely used numerical method for the bending, buckling and vibration analysis of composite beams.
Various finite element models based on equivalent single layer, layerwise, zig-zag theories are often proposed for describing the bending behaviour of laminated composite and sandwich beams. Lee and Lee [238] developed a finite element model based on classical lamination theory for the flexural–torsional analysis of I-shaped symmetrically and unsymmetrically laminated composite beams. Numerical results are obtained for thin-walled composites under vertical and torsional loading, addressing the effects of fiber angle, and laminate stacking sequence. Lee and Kim [239], Goyal and Kapania [240], Back and Will [241], Sudhakar et al. [242], Mahieddine and Ouali [243] and Kim and Reddy [244] developed finite element models based on first order shear deformation theory for the bending problems of laminated composite and sandwich beams. Several studies have been performed on finite element modeling of laminated composite and sandwich beams based on higher order shear deformation theories [245-273]. Shimpi and Ainapure [274] extended layerwise trigonometric shear deformation theory of Shimpi and Ghugal [210] for the finite element analysis of two layered antisymmetric laminated composite beams. The finite element is free from shear locking. The numerical results are obtained for the deflection, stresses and natural frequencies of two layered anti-symmetric laminated composite beams. Aitharaju and Averill [216], Averill [275], Averill and Yip [276], Friedman and Kosmatka [277], Degiovanni et al. [278], Chakrabarti et al. [279-281], Gherlone et al. [282], Onate et al. [283], Nanda et al. [284], Iurlaro et al. [285] and Di Sciuva et al. [286] developed finite element models based on higher order zigzag theories for the analysis of laminated and sandwich beam. 2.2.4 Bending of laminated and sandwich beams using CUF According to Carrera’s Unified Formulation (CUF), the one-dimensional theories of layered beams are developed by using the following formulas: u Ft ut Fbub Fr ur F u ,
t , b, r , r 2,......., N
n Ft nt Fb nb Fr nr F n ,
t , b, r, r 2,......., N
where Ft, Fb and Fr are the base functions used for z expansion; the first two polynomials are related to the linear part of such expansions, while Fr introduces the N -1 higher order terms i.e, Fr z r . The transverse stress demand a Legendre layer-wise expansion in thickness coordinate,
whereas both Taylor and Legendre expansions could be used for the displacement unknowns. The CUF method can be easily employed to obtain the governing equations and the boundary conditions of the various theories reviewed within the framework of PVD and/or RMVT. Most of the ESL and LW models presented in the preceding section can be viewed as particular cases of the Carrera Unified Formulation (CUF) [107-110, 287-294]. Tornabene et al. [295] presented a general formulation of a two-dimensional higher order equivalent single layer theory for free vibrations of doubly curved laminated composite shells and panels with different curvatures. Governing equations for dynamic analysis of shell structures are obtained by using a general displacement field based on the CUF including the stretching and zig-zag effects. Tornabene et al. [296] presented a two-dimensional general higher order equivalent single layer (GHESL) approach, based on the Carrera unified formulation (CUF) for the static analysis of doublycurved anisotropic shells and panels using a differential quadrature method. Tornabene et al. [297] presented free vibration analysis of arbitrarily shaped laminated composite shells based on the generalized differential quadrature (GDQ) method using equivalent single layer higher order theories developed from CUF. Tornabene et al. [298] and Bacciocchi et al. [299] developed a general formulation based on various higher-order equivalent single layer theories to evaluate the natural frequencies of several doubly-curved shells with variable thickness using generalized differential quadrature method. Tornabene et al. [300] evaluated through thickness distributions of displacement and stress components of several doubly curved panels reinforced by curvilinear
fibers using equivalent single layer (ESL) and layerwise (LW) theories which is based on the Carrera Unified Formulation (CUF). Tornabene [301] presented a general formulation for a higher order layerwise theory for the free vibrations of doubly curved laminated composite shells and panels using a general displacement field based on the Carrera Unified Formulation (CUF), including the stretching effect for each layer. D’Ottavio [302] and D’Ottavio et al. [303] presented bending of laminated composite and sandwich plates based on sublaminate generalized unified formulation (S-GUF). In S-GUF, plies are grouped into several smaller units called sublaminates, each of them characterized by an independent, variable-kinematic theory. Continuity conditions between the sublaminates are enforced in strong form during the assembly procedure of the governing equations. Demasi [304] introduce a new generalization of Carrera’s unified formulation called as generalized unified formulation (GUF). The original notations of CUF are slightly modified to make the new compact notation. The GUF is applied for the bending of laminated composite plates.
2.2.5 Exact elasticity solutions for bending of laminated and sandwich beams A two-dimensional elasticity solution of beam is necessary if the beam is thick. The exact elasticity solution is always used as a basis to check the accuracy of any refined shear deformation theory. Saint-Venant [15] studied the problem of a cantilever beam of rectangular cross section possessing anisotropy of a special form and obtained the exact elasticity solution. Various such problems are discussed by Lekhnitskii [305]. Silverman [306], Hasin [307], Gerstner [308], Rao and Ghosh [309] and Cheng et al. [310] provided exact solution for shear deformable laminated composite beams using Airy’s stress functions whereas Holt and Webber [311] provided exact solution for Honeycomb sandwich beams. Dischinger [312], Cheng and Pei
[313], Herrman [314] and Iyengar and Prabhakara [315] provided two-dimensional elasticity solution for the continuous beams. Also, Pagano [316, 317] developed exact elasticity solutions for cylindrical bending of laminated composite plates. Esendemir et al. [318] obtained the deflection of a simply supported composite beam subjected to a linearly distributed load using anisotropic elasticity theory given by Lekhnitskii [305]. 3. Free vibration of shear deformable beam 3.1 Free vibration of isotropic beams Burgreen [319, 320], Krieger [321], Eringen [322] and McDonald [323] presented nonlinear vibration problems of simply supported beam. Rissone and Williams [324] studied vibrations of non-uniform cantilever beam. Srinivasan [325, 326] employed the Ritz–Galerkin technique to solve the governing nonlinear differential equation of dynamic equilibrium for free and forced vibration of a simply supported Bernoulli-Euler beam. Evensen [327] further extended it to various boundary conditions using the perturbation method. Bennett and Eisley [328] investigated the steady-state free and forced responses and stability for large amplitude motion of a beam with clamped ends using Bernoulli-Euler beam theory. A multimode analytical and numerical technique is used to obtain theoretical solutions for both response and stability. MacBain and Genin [329] studied the effect of rotational and translational support flexibility on the fundamental frequency of clamped-clamped and cantilever beams. Kim and Kim [330] obtained accurate vibration frequencies of beams with generally restrained boundary conditions using Fourier series based on Bernoulli-Euler beam theory. The suggested method is accurate for beams with not only classical boundary conditions but also non-classical boundary conditions restrained by rotational and translational springs.
The study of transverse vibration analysis of bars of uniform cross-section can be traced to the pioneering work of Timoshenko [22, 331] using the first order shear deformation theory also known as Timoshenko beam theory. Cowper [29] derived the equations of Timoshenko beam theory by integration of the equations of three-dimensional elasticity theory and obtained a new formula for the shear coefficient and also obtained the exact fundamental frequencies for a simply supported beam. Many researchers [332-361] have applied Timoshenko beam theory for the free vibration analysis of one dimensional structure such as beams and rods with different boundary condition using various analytical and numerical methods. Raville et al. [362] analyzed the problem of the natural frequencies of vibration of fixed-fixed sandwich beams by an energy approach, wherein the Lagrangian multiplier method is used to satisfy the boundary conditions of the problem. Motaghian et al. [363] studied the free vibration problem of beams resting on partial elastic foundation of the Winkler type using Bernoulli-Euler beam theory. The authors have presented an exact solution by dividing the beam into separate segments. In addition to this, an innovative mathematical approach is proposed to find the precise analytical solution of the free vibration of beams with mixed boundary conditions. Stoykov and Ribeiro [364] studied the geometrically non-linear free and forced vibrations in space of beams with non-symmetrical cross sections by using Timoshenko’s theory for bending and Saint-Venant’s theory for torsion. The p-version finite element method is developed. The warping function, which cannot be derived analytically for complex cross sections, was obtained numerically by solving the Laplace equation with Neumann boundary conditions, using the boundary element method. The integrals over the cross section were obtained numerically by Gauss integration. The stability of solutions in forced vibrations is studied by Floquet’s theory. Stephen and Levinson [365] developed a well-known second order beam theory for the free vibration analysis of beams with different
cross-sections. The theory takes into account effects of shear curvature, transverse direct stresses and rotatory inertia. Heyliger and Reddy [77] developed a finite element model of third order parabolic shear deformation theory for the dynamic behavior of rectangular beams whereas Eisenberger [366] extended the theory using the dynamic stiffness matrix to predict the exact vibration frequencies for beams with various boundary conditions. Kant and Gupta [78] developed higher order shear and normal beam theory for the free vibration analysis of beams using finite element method. Sayyad [100, 101, and 367] presented free vibration analysis of simply supported isotropic beams using various parabolic, trigonometric, hyperbolic, and exponential shear deformation theories based on Navier solution. Carrera, et al. [368-370] presented higher-order theories on the basis of the Carrera Unified formulation for the analysis of free vibration of a beam structures. 3.2 Free vibration of laminated composite and sandwich beams In this section, literature on free vibration analysis of laminated composite and sandwich beam using ESL, LW, ZZ theories and exact elasticity solutions is reviewed and presented. Also literature on finite element modeling for free vibration of composite beam using these theories is reviewed. 3.3.1 Free vibration of laminated and sandwich beams using ESL theories Emam and Nayfeh [371, 372] obtained a closed-form solution for the vibration of a beam in a buckled position using classical beam theory. The solution is obtained for the post-buckling configurations of symmetric and non-symmetric composite beams as a function of the applied axial load. The solution obtained by authors provides benchmark solution for the post-buckling analysis of composite beams. Eisenberger et al. [373] obtained exact vibrational frequencies of generally laminated beams using Timoshenko beam theory. The exact dynamic stiffness matrix
method is used to derive stiffness matrix. The analysis of un-symmetrical beams is carried out. It is observed that the proposed method deals with the general layouts and geometries of the structure and its boundary conditions. Banerjee and William [374], Banerjee [375, 376], Banerjee and Sobey [377], Howson and Zare [378], Banerjee et al. [379] and Damanpack and Khalili [380] developed an exact dynamic stiffness matrix method for the vibration analysis of composite and sandwich beams using Timoshenko and higher order beam theories. Ferreira [381] presented free vibration of laminated composite Timoshenko beams based on the multiquadric radial basis function method. Higher order shear deformation theories considering polynomial or non-polynomial shape functions for the free vibration analysis of laminated composite and sandwich beams considering the effects of transverse shear and normal deformations are developed by many researchers [40, 95, 174, 181, 382-402] using various analytical and numerical methods. Abramovich [403], Abramovich and Livshits [404] and Abramovich et al. [405] studied free vibration and buckling of symmetrically and anti-symmetrically laminated composite beams. McCarthy and Chattopadhyay [406] carried out dynamic analysis of composite box beam using higher order theory to investigate the importance of including inplane and out-of-plane warping deformations. Rao et al. [407] obtained an analytical solution for free vibration analysis of symmetric laminated composite and sandwich beams using higher order mix theory. The natural frequencies obtained by using mixed theory are compared with equivalent single layer displacement based higher order theories. The natural frequencies are obtained for first nine modes. Soldatos and Sophocleous [408] obtained frequency equations and the characteristic functions of a general three-degrees-of-freedom beam theory of Bickford [64] that describes the plane motion of shear deformable elastic beams. Aydogdu [409, 410] applied Ritz method for the free vibration
analysis of cross-ply and angle-ply laminated beams with general boundary conditions using various higher order shear deformation theories. The governing differential equations are obtained by using Hamilton’s principle and six different boundary conditions of beams are considered for detailed numerical investigations. The numerical results are obtained for different aspect ratios and lamination schemes. Jun and coauthors [411-415] developed the dynamic stiffness matrix to study vibration analysis of arbitrary lay-up laminated composite beams based on third order shear deformation theory, trigonometric shear deformation theory and hyperbolic shear deformation theory. It is well-known that the dynamic stiffness method relies on the exact solutions of the governing differential equations of motion; therefore, it can be classified as an exact method. Natural frequencies are determined for different boundary conditions. Zhen and Wanji [416] assessed several displacement-based theories for the free vibration analysis 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. An analytical solution based on the global–local higher-order theory is obtained using Navier’s solution technique. It is concluded that, increasing the order number of in-plane and transverse displacement components, the global higher-order theories can reasonably predict the natural frequencies of laminated beams with arbitrary layouts and soft-core sandwich beams. Giunta et al. [417] presented a free-vibration analysis of simply supported, cross-ply beams via several higher order as well as classical theories. Lower and higher order theories are formulated from the three-dimensional displacement field. Euler–Bernoulli’s and Timoshenko’s theories are special cases of presented displacement field. A Navier-type, closed form solution is adopted in order to derive the governing algebraic equations and natural frequencies associated to torsional, axial, shear and mixed modes are investigated. Carrera, et al. [418] presented the free vibration
response of laminated beams using various refined beam theories by expanding the unknown displacement variables over the beam section axes using Taylor type expansions, trigonometric series, exponential, hyperbolic and zig-zag functions. Prokic et al. [419] carried out the free vibration analysis of thin-walled beams with arbitrary open cross section, made of cross-ply laminates using Vlasov’s beam theory. He and Yang [420] applied higher order shear deformation of Kant and Gupta [78] for the dynamic analysis of two-layered composite beams partial interaction. Jin et al. [421] applied Reddy’s higher order shear deformation theory for vibration analysis of sandwich beams made up of laminated composite face sheets and a viscoelastic core with arbitrary lay-ups and general boundary conditions. A modified FourierRitz method is employed to derive unified formulation. The effects of ply configuration, layer number, moduli and thickness ratios on the natural frequency are illustrated. Especially the effects of restraints from different directions on two ends of the sandwich beam are deeply investigated. Finally, it is concluded that the Fourier-Ritz method yields many new and accurate results at a low computational cost for various boundary conditions including general elastic boundaries. Pagani et al. [422] and Dan et al. [423] developed higher order shear deformation theories using Carrera Unified Formulation for the free vibration analysis of composite beams with solid and thin walled cross-sections. 3.3.2 Free vibration of laminated and sandwich beams using LW/ZZ theories Xavier et al. [424] proposed a simple layerwise higher-order zig-zag model for the vibratory response of soft cored un-symmetric sandwich beams. The theory accounts for a cubic variation of the in-plane displacements in each layer, and a parabolic variation of the transverse shear stress across the thickness of the beam. The number of unknown variables of the proposed theory is the same as that given by the first order shear deformation theory. The numerical results are
obtained for simply supported boundary conditions. Shimpi and Ainapure [425] applied layerwise trigonometric shear deformation theory for the free vibration analysis of two layered antisymmetric laminated composite beams using Navier solution technique whereas Arya [426] carried out free vibration analysis of laminated composite beams using zig-zag trigonometric shear deformation theory using Navier solution technique. Tahani [214] carried out free vibration analysis of laminated composite beams using layerwise displacement theories. Kapuria et al. [226-230] carried out an assessment of zigzag theory for the free vibration of laminated composite, sandwich piezoelectric beams. Youzera et al. [427] studied the damping and forced vibrations of three-layered symmetric laminated composite beams considering normal and shear deformations using the higher-order zig-zag theories. In the first part of this study, linear and nonlinear damping parameters of laminated composite beams are obtained whereas in the second part, nonlinear forced vibration analysis is carried out for small and large vibration amplitudes. Iurlaro et al. [428] carried out the experimental assessment of the refined zigzag theory for the free vibration analysis of sandwich beams and compared the numerical results with Timoshenko beam theory. The specimens used in the experimental investigation are sandwich beams with face-sheets made of 7075 Aluminum alloy and a soft-core made of a Rohacell structural foam, largely employed in aerospace applications. Faces are bonded to the core using a very thin layer of an epoxy structural adhesive designed for both solid panels and honeycomb sandwich constructions. In order to investigate the effect of the face-to-core stiffness ratio, two foams have been considered, namely the IG31 and the WF110. 3.2.3 Free vibration of laminated and sandwich beams using finite element method (FEM) The free vibration analysis of isotropic beams based on classical and refined shear deformation theories using finite element method is presented by Kant and Gupta [78], Kapur
[361], Thomas et al. [343], Thomas and Abbas [344], Dawe [345], Reddy [346], Carrera, et al. [369], Nickel and Secor [429], Davis et al. [430] and Abbas [431]. A variety of literature is found on the finite element models based on the classical beam theory, first order/Timoshenko beam theory, higher order equivalent shear deformation theories, layerwise theories and zig-zag theories for the free vibration analysis of laminated composite and sandwich beams. de Borbon and Ambrosini [432] and Vo and Lee [433, 434] developed an one-dimensional finite element model based on classical beam theory to predict natural frequencies and corresponding vibration modes for a thin-walled composite beam. Nabi and Ganesan [435], Bhattacharya et al. [436], Chakraborty et al. [437], Goyal and Kapania [240], Kapania and Goyal [438], Jafari-Talookolaei and Ahmadian [439], Jafari-Talookolaei et al. [440], Amichi and Atalla [441], Assaf [442], Callioglu and Atlihan [443], Kahya [444], Slimani et al. [445], Ozutok and Madenci [446], Kim and Lee [447] and Stoykov and Margenov [448] developed various finite element models based on first order shear deformation theory for the vibration problems of laminated composite and sandwich beams with various boundary conditions. Ahmed [449] used the finite element displacement method to investigate the flexural vibration characteristics of curved sandwich beams. Author has developed three displacement based models in which an element having three, four and five degrees of freedoms per node are considered. Teoh and Huang [450] and Teh and Huang [451] studied free vibration of fibre reinforced composite beams. The presented models consider the effect of shear and rotary inertia. The equations of motion based on the Timoshenko beam theory are derived using Hamilton's principle. Chen and Yang [452] carried out vibration analysis of an anisotropic symmetrically laminated beam finite element including the effect of shear deformation. Kapania and Raciti [453] and Wu and Sun [454] developed a simple one-dimensional finite element
model considering 10 degrees of freedom at each node for the vibration analysis of laminated composite beams including shear deformation. A finite element model based on a third order shear deformation theory is developed by Chandrashekhara and Bangera [455] to study the free vibration characteristics of cross-ply and angle-ply laminated composite beams. The effects of number of layers, ply orientation and various boundary conditions is studied based on CBT, TBT and third order theory. The importance of Poisson effect in one-dimensional laminated beam analysis is demonstrated in this study. Maiti and Sinha [126] and Shi and Lam [456] applied third order shear deformation theory for the free vibration of laminated composite beams using finite element method. The natural frequencies and corresponding vibration modes of a cantilever sandwich beam with a soft polymer foam core are predicted by Sokolinsky et al. [457] using the higher-order theory, a two-dimensional finite element analysis, and classical sandwich theory. Subramanian [458], Marur and Kant [459], Nazargah et al. [266], Chakrabarti et al. [460] and Vo et al. [461] proposed higher order finite element models based on higher order shear deformation theories to study the vibration response of laminated composite beams. Bassiouni et al. [462] performed an experimental investigation on the dynamic behavior for laminated composite beams and validated theoretically using finite element method. The finite element model includes separate rotational degrees of freedom for each lamina but does not require additional axial or transverse degrees of freedom. The experimental tests are carried out by using a hammer test and the frequency response function is displayed on the FFT analyzer. Numerical results are compared with the experimental ones. Ramtekkar et al. [463] developed a plane-stress mixed finite element model by using Hamilton’s energy principle for the natural vibrations of laminated composite beams using third order beam theory. Continuity of the transverse shear stress has been enforced through the thickness of the laminated beam in the formulation and the
transverse stress components have been invoked as the nodal degrees of freedom by applying elasticity relations. Ganesan and Zabihollah [464, 465] investigated the free undamped vibration response of tapered composite beams using the higher order finite element model. The complete study is presented in two parts. A higher-order finite element formulation is developed in Part I [464]. The stiffness coefficients of the tapered laminated beam are determined based on classical laminated theory. In Part II [465], the developed formulation is used for the free undamped vibration analysis of various types of tapered composite beams and a parametric study is conducted. Pradeep et al. [466] studied the thermal vibration of sandwich beams with composite facings and viscoelastic core. Each composite laminate is modeled as an equivalent single layer. The formulation is a decoupled thermo-mechanical formulation. Linear temperature variation across the thickness of each layer is assumed. The effects of fiber angle, aspect ratio and core thickness on the performance of the elements are studied. Vidal and Polit [467, 468] developed a finite element model for the vibration analysis of laminated composite beams using sinusoidal shear deformation theory. A three-noded finite element is developed with a sinus distribution with layer refinement. The theory accounts the effect of transverse normal stress. Chalak et al. [469] studied free vibration response of laminated sandwich beams having a soft core using a finite element beam model based on zig-zag theory. The theory considers the cubic distribution of axial displacement whereas quadratic distribution of transverse displacement. Nanda et al. [284] presented a spectral finite element model using an efficient and accurate layerwise zigzag theory for wave propagation analysis of highly inhomogeneous laminated composite and sandwich beams. Varello and Carrera [293] and Filippi et al. [294] presented a finite element model based on refined beam theories developed from Carrera’s Unified Formulation (CUF) for the vibration analysis of laminated and sandwich beams.
3.2.4 Exact elasticity solution for the free vibration of laminated and sandwich beams Smith [470] carried out vibration analysis of multilayer orthotropic sandwich beams using a unified analysis method based on two-dimensional elasticity theory. Chen et al. [471-473] obtained elasticity solution for free vibration of laminated beams using a new approach combining the state space method and the differential quadrature method. 4. Buckling of shear deformable beams When the axial compressive force applied to the beam, keeps it in equilibrium, then the beam is said to be stable. But, if the beam does not return to its original equilibrium configuration, the beam is said to be unstable. The magnitude of the compressive force at which the beam becomes unstable is termed the critical buckling load. Therefore, the modern use of composite materials in engineering structures, especially in ships and aircraft, has made elastic instability a problem of great importance. Urgent practical requirements have given rise in recent years to extensive investigations, both theoretical and experimental, of the conditions governing the stability of such structural elements as beams, plates and shells. The first problems of elastic instability, concerning lateral buckling of compressed members, were solved about 270 years ago by Leonard Euler in 1744 and published in the appendix “Des curvis elasticis” i.e, “Elastic curves” of his book [474]. In this section a review of buckling of shear deformable beams/columns is presented being widely used in high performance structure. The in-plane buckling problem of a three-layered sandwich beams /column was studied by Hoff and Mautner [475] and Bauld [476]. Sheinman and his co-authors [477-481] studied post buckling behaviour of laminated beams based on geometrical nonlinear theory. Somers et al. [482, 483] developed an analytical model for predicting buckling and describing the post buckling behavior of the sandwich beam. Ha [484] formulated the exact stiffness matrix for the
bending and overall buckling problem of a general class of sandwich beam taking into account the nonlinear effects of axial thrust. The term ‘exact’ is used in the sense that the solution satisfies the governing differential equation, in the same way that the ‘exact’ solution obtained by conventional frame analysis satisfies the engineering beam theory. Barbero and Raftoyiannis [485] and Barbero and Tomblin [486] developed an analytical model using classical theory for Euler buckling of pultruded composite columns. Sherbourne and Kabir [487] carried out an analytical study on the lateral buckling of thin-walled open-section fibrous composite beams. The simply supported and clamped I-section beams subjected to uniformly distributed and transverse central point loads are considered for the detail numerical study. Cheng et al. [488] presented the method of continuous analysis for predicting the local delamination buckling load of the face sheet of sandwich beams. Fourier series in conjunction with the Stokes transformation is used which provides a unified solution for problems with different end conditions. Morey et al. [489] developed a simple beam theory for the buckling of symmetric angle-ply composite beams including interaction of inplane stresses. Huang and Kardomateas [490] studied buckling and initial post-buckling behavior of sandwich beams. A nonlinear beam equation including transverse shear is developed. A perturbation procedure is subsequently applied to obtain a closed-form solution for the initial post buckling behavior. Galuppi and Carfagni [491] presented buckling of simply supported three layered sandwich beams with viscoelastic core under compression load. The phenomena of glassy, rubbery and creep buckling are examined for various values of the load. 4.1 Buckling of laminated and sandwich beams using ESL theories Emam and Nayfeh [372] presented vibration and post buckling behaviour of composite beams using Euler–Bernoulli beam theory. Challamel and Girhammar [492] presented buckling of
partial composite beam–columns including shear and axial effects using Euler–Bernoulli and Timoshenko beam–column theories. Fu et al. [493] developed analytical solutions of thermal buckling and post-buckling analysis of symmetric laminated composite beams with various boundary conditions based on the Timoshenko beam theory by considering geometric nonlinearity. The governing differential equations for beams subjected to a uniform temperature rise are derived by using Hamilton’s principle. The solutions for governing equations are obtained using analytical method, shape function method and Ritz method. Aydogdu [174], Vo and Inam [397], Khdeir and Reddy [494], Khdeir [495] and Aydogdu [496, 497] developed displacement based higher order shear deformation theories for the buckling analysis of laminated composite and sandwich beams. Matsunaga [498-500] studied the buckling analysis of a simply supported thick elastic beam subjected to axial stresses using higher order shear and normal deformation theory in which axial and transverse displacements are expanded in power series. Song and Waas [501] carried out buckling analysis of stepped laminated composite beams using simple higher order beam theory (SHOT) which assumes a cubic distribution of the displacement field through the thickness. It is observed that the results from SHOT did not show significant differences to those from Timoshenko theory for a wide range of aspect ratios of the beam geometry and material properties. In addition, the results from SHOT did not necessarily show more compliant results than those from Timoshenko beam theory. A two-dimensional elasticity solution is obtained for the buckling of clamped-clamped beams. Phan et al. [502] applied extended high-order sandwich panel theory developed by Phan et al. [503] for global buckling analysis of sandwich beams. Komijani et al. [504] studied the thermal buckling behavior of two layered Timoshenko composite beams with partial interaction between the layers. The Timoshenko kinematics is considered for both layers and the shear
connection is represented by a continuous relationship between the interface shear flow and the corresponding slip. Geometrically nonlinear behavior based on the von Karman simplification of the Green strain tensor is accounted in the formulation. Giunta et al. [505] developed hierarchical theories using Carrera’s Unified Formulation for the buckling analysis of thin-walled beams with open and closed cross-section. Smyczynski and Blandzi [506] developed a mathematical model for the dynamic stability analysis of a simply supported five layered sandwich beam. The beam has two metal faces, the metal foam core and two binding layers between faces and the core. The equations of motion are obtained using Hamilton's principle. 4.2 Buckling of laminated and sandwich beams using LW/ZZ theories Zhen and Wanji [406] carried out an assessment of several displacement based zig-zag theories for the stability analysis of laminated composite and sandwich beams. Kapuria et al. [507] developed zig-zag theories for the buckling analysis of laminated composite and sandwich beams whereas Kapuria and Alam [508] developed an efficient electromechanically coupled geometrically nonlinear zigzag theory for the buckling analysis of hybrid piezoelectric beams under electro-thermo-mechanical loads. Chakrabarti et al. [509] developed a new finite element model using zig-zag theory for the buckling analysis of soft core sandwich beams. 4.3 Buckling of laminated and sandwich beams using finite element method (FEM) Goyal and Kapania [240] developed a 21 degree-of-freedom element based on the first order shear deformation theory for the buckling analysis of laminated composite beams. Kim and Lee [447] developed a finite element model to calculate the buckling loads of thin-walled Timoshenko laminated beams subjected to variable forces. Vo et al. [461] developed a finite element model based on sinusoidal shear deformation theory to study the buckling behaviour of composite beams with arbitrary lay-ups. Pradeep et al. [466] developed a finite element
formulation for the thermal buckling of sandwich beams. Wang et al. [510] presented isogeometric finite element analysis of buckling of symmetric and anti-symmetric, cross-ply and angle-ply laminated composite beams with different boundary conditions using the first order shear deformation theory. Loja et al. [511] used a higher order shear deformation theory assuming a non-linear variation of the displacement field to develop a finite-element model and applied for the linear buckling behaviour of anisotropic multi-laminated or sandwich thick and thin beams. Dafedar and Desai [512] developed a unified mixed higher order analytical formulation to evaluate the buckling behaviour of laminated composite struts. Two sets of mixed models are presented by selectively incorporating nonlinear components of Green’s strain tensor. Hu et al. [513, 514] developed various one-dimensional finite element models based on higher order shear deformation theory for the buckling analysis of sandwich beams. Silva et al. [515] developed a finite element formulation based on nonlinear generalized beam theory for the buckling and post-buckling behavior of laminated composite cylindrical panels. Gupta et al. [516] proposed a simple, elegant, and accurate closed-form expressions for predicting the post buckling behavior of composite beams with axially immovable ends using the Rayleigh–Ritz method. Ibrahim et al. [517] presented the buckling analysis of laminated composite thin walled structures by the one dimensional finite element based unified higher-order models obtained within the framework of the Carrera Unified Formulation. Chakrabarti et al. [509] developed a new finite element model using zig-zag theory for the buckling analysis of soft core sandwich beams. Kahya [518] presented buckling analysis of laminated composite and sandwich beams by the finite element method according to the first-order shear deformation theory. Leissa [519] discussed the buckling issues of anti-symmetrically laminated plates under inplane loading conditions; however, such investigations in the case of composite beams with un-
symmetric, antisymmetric, and arbitrary lay-ups are missing in the literature. Further, exact elasticity solutions for buckling of laminated beams are not available in the literature to assess the validity of various refined theories. 4.4 Exact elasticity solutions for the buckling of laminated and sandwich beams Smith [470] presented buckling analysis of multilayer orthotropic sandwich beams using twodimensional elasticity theory. Kardomateas [520, 521] obtained three-dimensional elasticity solutions for the buckling analysis of transversely isotropic rods and sandwich columns. D’Ottavio et al. [522] developed 2D elasticity solution for the global buckling of sandwich struts.
5. Concluding remarks with future direction Beam elements made up of a composite material are being widely used in the civil, mechanical, aeronautical and aerospace industries due to their attractive properties such as; high strength-and stiffness-to-weight ratio and their anisotropic material property. The shear deformation effects are more pronounced in the laminated composite and sandwich beams subjected to transverse loads. Therefore, bending, free vibration and buckling analysis of laminated composite and sandwich beams have received widespread attention in recent years. The various analytical and numerical methods have developed by the researchers for the analysis of laminated composite and sandwich beams. This article reviews the literature available for the bending, free vibration and buckling analysis of laminated composite and sandwich beams. Also, the displacement field of various displacement based equivalent single layer theories and layerwise theories is presented in this article. Based on the literature reviewed, following conclusions/suggestions are made. 1.
Analysis of laminated composite and sandwich beams is difficult by using twodimensional elasticity theory and this led to the development of refined shear deformation
theories for beams which approximate the two-dimensional elasticity solutions with reasonable accuracy. 2.
Any refinements of classical models are meaningless, in general, unless the effects of interlaminar continuous transverse shear and normal stresses are both taken into account in a multilayered beam theory. But, in a whole variety of higher order beam theories existing in the literature, no consideration is given to the effect of transverse normal deformation/strain ( z 0 ) on bending, buckling and vibration responses when these theories are applied to laminated composite and sandwich beams in view of minimizing the number of unknown variables. Therefore, refined theories which consider the effects of transverse normal deformation need more attention in future.
3.
Buckling issues of composite beams with un-symmetric, antisymmetric, and arbitrary lay-ups are missing in the literature. Further, exact elasticity solutions for buckling of laminated beams are not available in the literature to assess the validity of various refined theories. Also, issues of torsional and dynamic instabilities are needed to be resolved in future.
4.
Exact 2-D elasticity solution for the vibration analysis of laminated composite and sandwich beams is not available in the whole variety of literature. Shear mode frequencies for laminated composite and sandwich beams are rarely available in the literature. These results can be served as the benchmark solutions for the researchers in the future.
5.
Geometrically non-linear analysis of laminated composite and sandwich beams needs more attention in future.
6.
Since equivalent single layer theories are computationally simpler, those are widely used by various researchers for the bending, free vibration and buckling analysis of simply supported laminated composite beams. But, these theories are not capable enough to capture the responses of sandwich beams. Therefore, layerwise and zig-zag theories are more accurate for the sandwich beams with a soft core.
7.
A number of equivalent single layer theories can be constructed from the Carrera’s Unified Formulation (CUF). Thus, most of the displacement based equivalent single layer theories reviewed in the present study are the particular cases of CUF.
8.
It is also pointed out that the Navier solution (for simply supported boundary condition) and the finite element solution (other boundary conditions) are widely used by various researchers for the analysis of laminated and sandwich beams. However, other numerical methods such as differential quadrature method, Rayleigh-Ritz method, Galerkin method, Radial basis functions and meshfree method are not fully explored as reviewed in the literature.
9.
Bending, bucking and vibration of laminated and sandwich beams subjected to constant, linear and non-linear thermo-mechanical load needs attention in future. The influence of temperature profiles (heat conduction problem) on the thermo-mechanical response of multilayered beams considering classical and advanced theories need to be studied extensively.
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