F.G.CANALES,J.L.MANTARI,b,*

aFaculty of Mechanical Engineering,Universidad de Ingenieríay Tecnología(UTEC),Lima 15063,Peru

bDepartment of Mechanical Engineering,University of New Mexico,Albuquerque 87131,USA

Boundary discontinuous Fourier analysis of thick beams with clamped and simply supported edges via CUF

F.G.CANALESa,J.L.MANTARIa,b,*

aFaculty of Mechanical Engineering,Universidad de Ingenieríay Tecnología(UTEC),Lima 15063,Peru

bDepartment of Mechanical Engineering,University of New Mexico,Albuquerque 87131,USA

This paper presents an analytical solution for static analysis of thick rectangular beams with different boundary conditions.Carrera’s Unified Formulation(CUF)is used in order to consider shear deformation the ories of arbitrary order.The novelty of the present work is that a boundary discontinuous Fourier approach is used to consider clamped boundary conditions in the analytical solution,unlike Navier-type solutions which are restricted to simply supported beams.Governing equations are obtained by employing the principle of virtual work.The numerical accuracy of results is ascertained by studying the convergence of the solution and comparing the results to those of a 3D finite element solution.Beams subjected to bending due to a uniform pressure load and subjected to torsion due to opposite linear forces are considered.Overall,accurate results close to those of 3D finite element solutions are obtained,which can be used to validate finite element results or other approximate methods.

1.Introduction

1D the ories are widely used to analyze behaviors of slender bodies in a computationally efficient manner.For this reason,many beam models have been developed.The most wellknown beam the ory is the classical or Euler-Bernoulli beam the ory,which yields reasonably good results for slender beams.However,this model does not take into account shear deformations in a beam.The Timoshenko beam the ory is an improvement over the classical the ory that considers a uniform shear distribution across the thickness of a beam.However,this the ory requires a shear correction factor to correct the strain energy of deformation.Discussion of shear coefficients has been presented in Refs.1–4

A large amount of Higher-order Shear Deformation Theories(HSDTs)have been developed in order to consider a nonuniform shear distribution in a beam’s cross-section.HSDTs with polynomial distributions of shear deformation across the thickness are common due to the ir simplicity,and some have been presented in Refs.5–13Theories containing trigonometric functions in thickness coordinates are also common.A trigonometric shear deformation the ory has been presented by Dahake and Ghugal.14Many polynomial and trigonometric deformation the ories have been developed for analysis of laminated beams,as presented in Refs.15–20Firstorder shear deformation the ories are popular due to the ir computational efficiency,and some have been given in Refs.21–24

In order to analyze the ories with arbitrary order in a systematic manner,a unified formulation known as Carrera’s Unified Formulation(CUF)has been developed in Ref.25This formulation has been applied to solve multifield problems,as presented in Refs.26–28Carrera and Giunta29used the 1DCUF model to analyze 1D problems with complex crosssections,and furthe r development has been presented by Carrera et al.30–32The capability of the se models to obtain quasi-3D solutions has been exploited to develop accurate static33,free vibration34,35,and buckling analysis36of composite beams.

Analytical solutions for bending of simply supported beams are obtained by using a Fourier series in Navier-type solutions.Other boundary conditions such as clamped conditions can be considered in a finite element formulation or by using the Ritz method,but accurate analytical solutions for the se boundary conditions are a fairly scarce topic in the literature.Since finite element formulations or variational methods obtain approximate results,exact analytical solutions are required as a benchmark in order to assess the validity of the results.The present work intends to provide such analytical solutions for clamped boundary conditions.

A generalization of the Fourier series method known as the boundary discontinuous Fourier method can take into account clamped boundary conditions.This method was developed by Chaudhuri in Refs.37,38Discontinuities are introduced in order to satisfy boundary constraints.This solution methodology has been applied for static and free vibration analysis of cylindrical panels,39,40doubly-curved panels,41–48and plates.49–54Since the rate of convergence of a Fourier series is slower in the presence of discontinuities,a mixed Fourier solution has also been developed in Refs.55,56in order to produce accelerated convergence.Oktem and Chaudhuri have applied the boundary discontinuous Fourier method for analysis of plates57–59and shells60–63using HSDTs.

In this paper,an analytical solution for static analysis of thick beams with Clamped-Clamped(C-C)and Clamped-Simple(C-S)boundary conditions is obtained.A general approach to obtain such an analytical solution using a unified formulation is currently unavailable in the literature,since the other option commonly used for static analysis of beams is a Navier-type solution,which can only consider simply supported edges.Theories of arbitrary order are considered in a systematic manner by using CUF.The principle of virtual work is used to obtain governing equations.The convergence of the solution is analyzed and 3D finite element solutions are obtained in order to assess the validity of results.Good results agreements with 3D finite element solutions are obtained.The results can be used as a benchmark for comparison with approximate solution methods.

2.Analytical modeling

A beam of length L,width b,and total thickness h is considered in the present analysis.The rectangular Cartesian coordinate system used in the present work is shown in Fig.1.The beam occupies the following region:-b/2≤x≤b/2,0≤y≤L,-h/2≤z≤h/2.

2.1.Elastic stress-strain relations

A general displacement vector is introduced:

The cross-sectional plane of the beam is denoted by Ω.The stress and strain components are grouped as

where σijand εijare the components of the stress and strain vectors,respectively.Subscript ‘p” stands for terms lying on planes orthogonal to the cross-section,while subscript ‘n”stands for terms lying on the cross-section.Considering small amplitude displacements,the strain-displacement relations are

The linear differential operators Dp,DnΩ,and Dnyare given by

Fig.1 Coordinate frame of beam model.

The stress components are given by constitutive laws:

where σ is the stress vector,ε is the strain vector andis the constitutive matrix.Eq.(5)can be split by using Eq.(2):

In the case of an isotropic material,the matrices,,andare given by

2.2.Displacement field

The displacement field is expressed within the framework of CUF:

where Fτare the functions of coordinates x and z on the crosssection,Mstands for the number of terms used in the expansion,uτis the vector of the generalized displacements,and the repeated subscript ‘τ” indicates summation.A Taylor-type expansion is used to determine the functions Fτ,consisting of a MacLaurin series that uses the 2D polynomials xizjas a base.Table 1 presents Mand Fτas functions of the expansion orderN.

For example,the displacement field of the second-order(N=2)Taylor-type expansion model can be expressed as

Table 1 MacLaurin’s polynomials.

Classical beam the ories can be obtained as a special case of the generalized formulation.For example,the Timoshenko beam the ory is obtained in two steps:(a)a first-order displacement field is considered:

and(b)the displacements uxand uzmust be constant in the cross-section:

2.3.Principle of virtual work

The static version of the principle of virtual work is applied:

where δ stands for the virtual variation operator,Lintstands for the strain energy,and Lextis the external work.Substituting Eqs.(3),(4),and(6)in Eq.(12),the following expression is obtained:

Substituting Eqs.(4),(7),and(8)in Eq.(13)and integrating by parts results in the following:

where Kτsis the stiffness matrix and Πτsis the matrix of the natural boundary conditions.The components of Kτsare provided as

where a cross-sectional moment parameter has been used,and a generic term is defined as

The suffix after the comma denotes the derivatives.The components of Πτsare provided as follows:

Letting Pτ= ［PxτPyτPzτ］Tdefine a vector of the generalized forces,the natural boundary conditions can be obtained by substituting Eq.(17)in Eq.(14):

2.4.Boundary discontinuous solution

Geometric boundary conditions for simply supported beams,in terms of the displacement variables given in Eq.(8),are expressed as

Geometricboundaryconditionsforclamped-clamped beams,in terms of the displacement variables given in Eq.(8),are additional constraints to those given in Eq.(19),which are given as

The displacement variables are assumed as follows:

where m is the wave number of the trigonometric term and p is the number of trigonometric terms of the series.The coefficient αmis given by

The total number of unknown Fourier coefficients introduced in Eq.(21)is M（3p+1）.The assumed solution satisfies the simply supported geometric boundary conditions given in Eq.(19).However,the clamped support boundary condition,given in Eq.(20),is not satisfied.In order to obtain an analytical solution for clamped beams,the boundary discontinuous method is used.The details of the procedure are given in Refs.37,38

The boundary discontinuous method introduces boundary Fourier coefficients arising from discontinuities of a solution at the edges y=0，L.The displacement variable uyτas given by Eq.(21b)does not satisfy the boundary condition for clamped supports given in Eq.(20).Therefore,it is forced to vanish at the se edges.The partial derivative uyτ，yis seen to vanish at the edges,thus violating the complementary boundary constraint or boundary discontinuities at the se edges;see Refs.37,38For furthe r differentiation,uyτ，yyis expanded in a Fourier series in order to satisfy the complementary boundary constraint.It is important to note that the derivative of the Fourier series of a given function is not necessarily the same as the Fourier series of the derivative of the function when this function has discontinuities.

The Fourier series of the derivative uyτ，yis given by

where Uyτm，yis the Fourier term associated with the Fourier series of the function uyτ，y.Integrating Eq.(23b)by parts and using the vanishing boundary conditions given in Eq.(20)obtains:

The Fourier term Uyτmof the Fourier series of the function uyτis recognized:

Thus,the first derivative can be obtained through term-byterm differentiation.However,it will be demonstrated that the second derivative has a different form.The Fourier series of the second derivative uyτ，yyis given by

where Uyτm，yyis the Fourier term of the function uyτ，yyand aτis a Fourier coefficient.Integrating Eq.(26b)by parts obtains:

Note that the function uyτ，yis not necessarily zero at the edges since discontinuities are introduced aty=0，L(Eq.(21b)is not valid at the edges).Substituting Eqs.(23b)and(25)in Eq.(27),the Fourier term is obtained as

Substituting Eq.(28)in Eq.(26a)the following expression is obtained:

where the Fourier coefficients aτand bτintroduce 2M new unknowns,resulting in a total of M（3p+3）unknowns.These coefficients are given by

and γm，ψmare defined as

2.5.Virtual work of load

The external work of a load q0applied on the surface z=h/2 is given by

Substituting Eq.(8)in Eq.(32)obtains:

The load is expressed using a Fourier series:

where Qmis a Fourier coefficient associated with the Fourier series of the load.Othe r types of loads can be analyzed in a similar manner.29

2.6.Governing equations

Substituting Eqs.(21a)–(21c)and the ir appropriate partial derivatives in Eq.(14),in conjunction with Eqs.(15),(29),(33),and(34),the following expressions are obtained for the case of a beam subjected to a distributed load on the surface z=h/2:

Equating the coefficients of the trigonometric functions of Eqs.(35)and(36)to zero yields M（3p+1）linear algebraic equations.Additional equations are supplied by the geometric boundary conditions related to vanishing of the displacement variables uyτat the edges y=0，L:

These equations can be expressed in a more convenient form58:

This step generates 2M additional equations,resulting in a total of M（3p+3）linear algebraic equations with as many unknowns.

2.7.Extension to other boundary conditions

In addition to the clamped-clamped boundary condition,the clamped-simple boundary condition can also be analyzed by the present method.A clamped support is considered at y=0 and a simple support at y=L.Since the geometric boundary condition given in Eq.(20b)has been relaxed,the function uyτis no longer forced to vanish at y=L,and Eq.(37b)is no longer required.This step reduces the available equations in M compared to the clamped-clamped case.In addition,discontinuities are no longer introduced at y=L,and thus uyτ，yis equal to zero at this point.By substituting uyτ，y（L）=0 in Eqs.(30a),(30b),the following relation is obtained:

This step eliminates M unknowns,and thus the system remains determinate.

3.Numerical results and discussion

The present development has been programmed in MATLAB,and numerical examples are given in the present section.An isotropic square beam,i.e.,with b=h,is considered in the numerical examples.The displacements,stresses,and geometric parameters are expressed in the following non-dimensional forms:

where E is the modulus of elasticity.The Poisson’s ratio is considered to be ν=0.3.The boundary conditions of the beam are indicated by letters C(Clamped support)and S(Simple support).

3.1.Convergence study

A study of the convergence is performed first in order to assess the stability of the results.Fig.1 shows the geometry of the beam considered.Fig.2 shows the locations of evaluation points of the studied displacements and stresses.

Fig.2 Location of evaluation points of transverse displacement,axial stress,axial displacement,transverse stress,and shear stress.

Fig.3 Convergence of transverse displacementat(0,L/2,0)and axial stressat(0,L/2,h/2)of a C-C square beam with L/h=10 subjected to a uniform load.

The beam is subjected to a uniform load q0on the surface z=h/2.Fig.3 show the transverse displacement and axial stress of a C-C square beam with L/h=10 as the number of terms in the Fourier series is increased for various expansion orders N.While Navier-type solutions converge quickly and 25 terms are usually sufficient20,a Fourier series converges slowly if the function has discontinuities.Due to this,the boundary discontinuous method has a slower convergence compared to that of a Navier solution.

Table 2 presents numerical results for the displacements and stresses as the number of terms in the Fourier series is increased,considering N=4.The transverse stressˉσzzis seen to have a slower convergence.At least 4 significant figures can be expected from the numerical results when the number of terms is m=15000,except for the transverse stressˉσzz,and this number of terms is used in the remainder of the manuscript.

3.2.Bending loads

In order to validate the results,a 3D finite element solution of static analysis of the beam has been obtained using ANSYS general purpose program.The 20-noded Solid186 element was used to model the beam,and the mesh was constructed using equally sized cubic elements.In the tables,the 3D finite element solution is denoted by ANS3Dxxx,where the subscript indicates the number of elements in the beam axis.For example,for a beam with L/h=5,the notation ANS3D200stands for a mesh with 200 elements in the beam axis and 40 elements in each axis of the cross-section,resulting in a mesh of 200×40×40.The error between the results from the present model and the 3D finite element solution(FEM)is defined as

where the most refined finite element mesh solution(i.e.the ANS3Dxxxmodel with the largest numerical subscript)is used for calculation of the error

The beam considered and the points of evaluation are the same as those in the previous section,as shown in Figs.1 and 2.Table 3 presents the values of displacements and stresses at specified points of C-C and C-S square beams with L/h=10,as obtained by the present model,and the corresponding results obtained by the 3D FEM.In order to correctly predict the shear stressˉσyz,an expansion order of at least N=3 is required.Very close agreements can be obtained for all the displacements and stresses using an expansion order of N=6 or higher.For higher expansion orders(N≥5),ahigher reported error is obtained for the transverse stressˉσzz;however,for N=7,the higher reported error is around 0.05%and within the margin of error of the 3D FEM solution.Similar trends are observed between the results for C-C and C-S beams.

Table 2 Convergence of transverse displacement,axial stress,transverse normal stress,axial displacement,and shear stressfor a C-C isotropic square beam with L/h=10 subjected to a uniform load.

Table 2 Convergence of transverse displacement,axial stress,transverse normal stress,axial displacement,and shear stressfor a C-C isotropic square beam with L/h=10 subjected to a uniform load.

Number of termsˉuzˉσyyˉσzzˉuyˉσyz 400 3.4342 25.406 1.0436 0.46820 3.5096 600 3.4375 25.386 1.0314 0.46853 3.5096 1000 3.4402 25.370 1.0206 0.46879 3.5096 1500 3.4415 25.361 1.0150 0.46892 3.5096 2500 3.4426 25.355 1.0105 0.46902 3.5096 4000 3.4431 25.351 1.0079 0.46908 3.5096 6000 3.4435 25.349 1.0065 0.46911 3.5096 10000 3.4437 25.347 1.0053 0.46914 3.5096 15000 3.4439 25.347 1.0048 0.46915 3.5096

Table 3 Comparison of transverse displacementˉuz,axial stressˉσyy,transverse normal stressˉσzz,axial displacementˉuy,and shear stressˉσyzfor C-C and C-S isotropic square beams with L/h=10 subjected to a uniform load.

Fig.4 show the distributions of the transverse displacementˉuzacross the thickness of C-C and C-S beams with L/h=10,respectively.As the expansion order increases,the results converge to those of the 3D finite element solution in an increasingly slow manner.

Fig.4 Distribution of transverse displacementˉuzat(0,L/2,z)through thickness of C-C and C-S square beams subjected to uniform load with L/h=10.

Fig.5 shows the distribution of the transverse normal stressˉσzzacross the thickness of a C-C beam with L/h=10.An overshoot in the maximum stress is observed for an expansion order of N=3.However,this is corrected by using a higher expansion order.It can be seen that the distributions of the transverse stress across the thickness obtained by using expansion orders of N=4 and N=5 are in close agreements with that of the 3D finite element solution.

Fig.6 shows the distribution of the shear stressˉσyzacross the thickness of a C-C beam with L/h=10.Higher expansion orders are required in order to obtain zero shear stress on the top and bottom surfaces of the beam.

Tables 4 and 5 present the values of displacements and stresses at specified points of C-C and C-S square beams with L/h=5 and 2,respectively.For the critical case of a thick beam with L/h=2,using an expansion order of N=7,the higher reported error is 0.23%.

Fig.7 shows the distributions of the transverse displacementˉuzacross the thickness of C-C and C-S beams with L/h=5,respectively.Results obtained from higher expansion orders gradually converge to that of the 3D FEM solution,similar to the case with L/h=10.

Fig.5 Distribution of transverse normal stressˉσzzat(0,L/2,z)through thickness of C-C square beam subjected to uniform load with L/h=10.

Fig.6 Distribution of transverse shear stressat(0,L/4,z)through thickness of C-C square beam subjected to uniform load with L/h=10.

Fig.8 shows the distribution of the transverse displacementacross the thickness of a C-C beam with L/h=2.It can be observed that the thickness stretching effect,i.e.,variation of the transverse displacement across the thickness,is more pronounced for beams with lower aspect ratios.

3.3.Torsional loads

A beam subjected to two linear loads of equal magnitude q0and opposite directions is considered,as shown in Fig.9.Classic beam models are unable to detect displacement and stresses for this loading case.Table 6 presents results of the shear stressevaluated at Point A with coordinates(0,0,h/2).It can be observed that higher expansion orders are required in order to correctly predict the shear stress.These results can be used as a benchmark for future finite element works.

Table 4 Comparison of transverse displacement,axial stress,transverse normal stress,axial displacement,and shear stressfor C-C and C-S isotropic square beams with L/h=5 subjected to uniform load.

Table 4 Comparison of transverse displacement,axial stress,transverse normal stress,axial displacement,and shear stressfor C-C and C-S isotropic square beams with L/h=5 subjected to uniform load.

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Table 5 Comparison of the transverse displacementˉuz,axial stress,transverse normal stress,axial displacement,and shear stressfor C-C and C-S isotropic square beams with L/h=2 subjected to a uniform load.

Table 5 Comparison of the transverse displacementˉuz,axial stress,transverse normal stress,axial displacement,and shear stressfor C-C and C-S isotropic square beams with L/h=2 subjected to a uniform load.

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Fig.7 Distribution of transverse displacementˉuzat(0,L/2,z)through thickness of C-C and C-S square beams subjected to uniform load with L/h=5.

Fig.8 Distribution of transverse displacementˉuzat(0,L/2,z)through thickness of C-C square beam subjected to uniform load with L/h=2.

Fig.9 Geometry of beam subject to torsion loads.

Table 6 Comparison of the shear stressˉσxyfor C-C isotropic square beam with L/h=2 subjected to torsional loads.

4.Conclusions

This paper presents an analytical solution for bending and torsion of a thick rectangular beam with clamped or simple supports.The boundary discontinuous Fourier approach is used in conjunction with shear deformation the ories of arbitrary order via Carrera’s Unified Formulation(CUF).The important conclusions that emerge from this paper can be summarized as follows:

(1)Accurate analytical results of the stresses and displacements for clamped beams can be obtained by the present model with a low computational effort.

(2)At least an expansion order of N=3 is required in order to correctly predict the maximum shear stress σyzdue to bending loads.

(3)An overshoot of the maximum transverse normal stress σzzis observed for an expansion order of N=3 when bending loads are considered.However,this is corrected by using higher expansion orders.

(4)The boundary discontinuous method requires a higher computational effort compared to that of Navier-type solutions,but it is much lower than that required for a 3D finite element solution.

Acknowledgment

This paper is dedicated to Professor Reaz Chaudhurifor his outstanding contribution to computational mechanics.

1.Kaneko T.On Timoshenko’s correction for shear in vibrating beams.J Phys D:Appl Phys 1975;8(16):1927–36.

2.Hutchinson JR.Transverse vibrations of beams,exact versus approximate solutions.J Appl Mech 1981;48(4):923–8.

3.Hutchinson JR,Zillmer SD.On the transverse vibration of beams with rectangular cross-section.J Appl Mech 1986;53(1):39–44.

4.Rychter Z.On the shear coefficient in beam bending.Mech Res Commun 1987;14(5–6):379–85.

5.Essenburg F.On the significance of the inclusion of transverse normal strain in problems involving beams with surface constraints.J Appl Mech 1975;42(1):127–32.

6.Levinson M.A new rectangular beam the ory.J Sound Vibrat 1981;74(1):81–7.

7.Rychter Z.On the accuracy of a beam the ory.Mech Res Commun 1987;14(2):99–105.

8.Rychter Z.A simple and accurate beam the ory.Acta Mech 1988;75(1):57–62.

9.Petrolito J.Stiffness analysis of beams using a higher-order beam the ory.Comput Struct 1995;55(1):33–9.

10.Murthy AVK.Towards a consistent beam the ory.AIAA J 1984;22(6):811–6.

11.Bhimaraddi A,Chandrashekhara K.Ochigherorder beam the ory.J Aerospace Eng 1993;6(4):408–13.

12.Heyliger PR,Reddy JN.A higher order beam finite element for bending and vibration problems.J Sound Vibrat 1988;126(2):309–26.

13.Kant T,Gupta A.A finite element model for a higher-order shear deformable beam.J Sound Vibrat 1988;125(2):193–202.

14.Dahake AG,Ghugal YM.A trigonometric shear deformation the ory for flexure of thick beam.Procedia Eng 2013;51:1–7.

15.Khdeir AA,Reddy JN.An exact solution for the bending of thin and thick cross-ply laminated beams.Compos Struct 1997;37(2):195–203.

16.Sayyad AS,Ghugal YM.Effect of transverse shear and transverse normal strain on bending analysis of cross-ply laminated beams.Int J Appl Math Mech 2011;7(12):85–118.

17.Shimpi RP,Ghugal YM.A new layerwise trigonometric shear deformation the ory for two-layered cross-ply beams.Compos Sci Technol 2001;61(9):1271–83.

18.Aydogdu M.A new shear deformation the ory for laminated composite plates.Compos Struct 2009;89(1):94–101.

19.Arya H,Shimpi RP,Naik NK.A zigzag model for laminated composite beams.Compos Struct 2002;56(1):21–4.

20.Mantari JL,Canales FG.A unified quasi-3D HSDT for the bending analysis of laminated beams.Aerospace Sci Technol 2016;54:267–75.

21.Senjanovic′I,Vladimir N,Hadzˇic′N,Tomic′M.New first order shear deformation beam the ory with in-plane shear influence.Eng Struct 2016;110:169–83.

22.Senjanovic′I,Vladimir N.Physical insight into Timoshenko beam the ory and its modification with extension.Struct Eng Mech 2013;48(4):519–45.

23.Senjanovic′I,Rudan S,Vladimir N.Influence of shear on the torsion of thin-walled girders.Trans FAMENA 2009;33(2):35–50.

24.Senjanovic′I,Tomasˇevic′S,Vladimir N.An advanced the ory of thin-walled girders with application to ship vibrations.Mar Struct 2009;22(3):387–437.

25.Carrera E.Theories and finite elements for multilayered plates and shells:A unified compact formulation with numerical assessment and benchmarking.Arch Comput Methods Eng 2003;10(3):215–96.

26.Carrera E.Transverse normal strain effects on the rmal stress analysis of homogeneous and layered plates.AIAA J 2005;43(10):2232–42.

27.Carrera E,Boscolo M,Robaldo A.Hierarchic multilayered plate elements for coupled multifield problems of piezoelectric adaptive structures.Formulation and numerical assessment.Arch Comput Methods Eng 2007;14(4):383–430.

28.Carrera E,Brischetto S,Nali P.Variational statements and computational models for multifield problems and multilayered structures.Mech Adv Mater Struct 2008;15(3–4):192–8.

29.Carrera E,Giunta G.Refined beam the ories based on a unified formulation.Int J Appl Mech 2010;2(1):117–43.

30.Carrera E,Giunta G,Nali P,Petrolo M.Refined beam elements with arbitrary cross-section geometries.Comput Struct 2010;88(5–6):283–93.

31.Carrera E,Petrolo M.Refined beam elements with only displacement variables and plate/shell capabilities.Meccanica 2012;47(3):537–56.

32.Carrera E,Filippi M,Zappino E.Laminated beam analysis by polynomial,trigonometric,exponential and zig-zag the ories.Eur J Mech A/Solids 2013;41:58–69.

33.Catapano A,Giunta G,Belouettar S,Carrera E.Static analysis of laminated beams via a unified formulation.Compos Struct 2011;94(1):75–83.

34.Giunta G,Biscani F,Belouettar S,Ferreira AJM,Carrera E.Free vibration analysis of composite beams via refined the ories.Compos Part B:Eng 2013;44(1):540–52.

35.Filippi M,Pagani A,Petrolo M,Colonna G,Carrera E.Static and free vibration analysis of laminated beams by refined the ory based on Chebyshev polynomials.Compos Struct 2015;132:1248–59.

36.Ibrahim SM,Carrera E,Petrolo M,Zappino E.Buckling of composite thin walled beams by refined the ory.Compos Struct 2012;94(2):563–70.

37.Chaudhuri RA.On boundary-discontinuous double Fourier series solution to a system of completely coupled P.D.E’.s.Int J Eng Sci 1989;27(9):1005–22.

38.Chaudhuri RA.On the roles of complementary and admissible boundary constraints in Fourier solutions to the boundary value problems of completely coupled RTH order PDEs.J Sound Vibrat 2002;251(2):261–313.

39.Chaudhuri RA,Abu-Arja KR.Static analysis of moderately-thick finite antisymmetric angle-ply cylindrical panels and shells.Int J Solids Struct 1991;28(1):1–15.

40.Kabir HRH,Chaudhuri RA.A direct Fourier approach for the analysis of thin finite-dimensional cylindrical panels.Comput Struct 1993;46(2):279–87.

41.Chaudhuri RA,Kabir HRH.On analytical solutions to boundaryvalue problems of doubly-curved moderately-thick orthotropic shells.Int J Eng Sci 1989;27(11):1325–36.

42.Kabir HRH,Chaudhuri RA.Free vibration of shear-flexible antisymmetric angle-ply doubly curved panels.Int J Solids Struct 1991;28(1):17–32.

43.Chaudhuri RA,Kabir HRH.A boundary-continuous-displacement based Fourier analysis of laminated doubly-curved panels using classical shallow shell the ories.Int J Eng Sci 1992;30(11):1647–64.

44.Chaudhuri RA,Kabir HRH.Sensitivity of the response of moderately thick cross-ply doubly-curved panels to lamination and boundary constraint—I.Theory.Int J Solids Struct 1993;30(2):263–72.

45.Chaudhuri RA,Kabir HRH.Sensitivity of the response of moderately thick cross-ply doubly-curved panels to lamination and boundary constraint—II.Application.Int J Solids Struct 1993;30(2):273–86.

46.Chaudhuri RA,Kabir HRH.Boundary-discontinuous Fourier analysis of doubly-curved panels using classical shallow shell the ories.Int J Eng Sci 1993;31(11):1551–64.

47.Chaudhuri RA,Kabir HRH.Static and dynamic Fourier analysis offinite cross-ply doubly curved panels using classical shallow shell the ories.Compos Struct 1994;28(1):73–91.

48.Kabir V,Chaudhuri RA.On Gibbs-phenomenon-free Fourier solution for finite shear-flexible laminated clamped curved panel.Int J Eng Sci 1994;32(3):501–20.

49.Chaudhuri RA,Kabir HRH.A boundary discontinuous Fourier solution for clamped transversely isotropic(pyrolytic graphite)Mindlin plates.Int J Solids Struct 1993;30(2):287–97.

50.Kabir HRH,Chaudhuri RA.A generalized Navier’s approach for solution of clamped moderately-thick cross-ply plates.Comput Struct 1991;17(4):351–66.

51.Chaudhuri RA,Kabir HRH.Vibration of clamped moderately thick general cross-ply plates using a generalized Navier approach.Compos Struct 1993;24(4):311–21.

52.Chaudhuri RA.Effect of boundary constraint on the frequency response of moderately thick flat laminated panels.Compos Eng 1994;4(4):417–28.

53.Kabir HRH.Free vibration of clamped,moderately thick,arbitrarily laminated plates using a generalized Navier’s approach.J Sound Vibrat 1994;171(3):397–410.

54.Chaudhuri RA,Balaraman K,Kunukkasseril VX.A combined the oretical and experimental investigation on free vibration of thin symmetrically laminated anisotropic plates.ComposStruct 2005;67(1):85–97.

55.Chaudhuri RA,Kabir HRH.Effect of boundary constraint on the frequency response of moderately thick doubly curved cross-ply panels using mixed Fourier solution functions.J Sound Vibrat 2005;283(1–2):263–93.

56.KabirHRH,Al-KhaleefiAM,ChaudhuriRA.Frequency response of a moderately thick antisymmetric cross-ply cylindrical panel using mixed type of Fourier solution functions.J Sound Vibrat 2003;259(4):809–28.

57.Oktem AS,Chaudhuri RA.Fourier solution to a thick cross-ply Levy type clamped plate problem.Compos Struct 2007;79(4):481–92.

58.Oktem AS,Chaudhuri RA.Boundary discontinuous Fourier analysis of thick cross-ply clamped plates.Compos Struct 2007;82(4):539–48.

59.Oktem AS,Chaudhuri RA.Effect ofinplane boundary constraints on the response of thick general(unsymmetric)cross-ply plates.Compos Struct 2008;83(1):1–12.

60.Oktem AS,Chaudhuri RA.Sensitivity of the response of thick cross-ply doubly curved panels to edge clamping.Compos Struct 2009;87(4):293–306.

61.Oktem AS,Chaudhuri RA.Fourier analysis of thick cross-ply Levy type clamped doubly-curved panels.Compos Struct 2007;80(4):489–503.

62.Oktem AS,Chaudhuri RA.Levy type Fourier analysis of thick cross-ply doubly curved panels.Compos Struct 2007;80(4):475–88.

63.Oktem AS,Chaudhuri RA.Higher-order the ory based boundarydiscontinuous Fourier analysis of simply supported thick cross-ply doubly curved panels.Compos Struct 2009;89(3):448–58.

64.Reddy JN.Mechanics of laminated composite plates:Theory and analysis.2nd ed.Boca Raton:CRC Press;2004.

30 June 2016;revised 28 October 2016;accepted 24 January 2017

Available online 11 July 2017

Analytical solution;

Beam;

Clamped;

Fourier;

Unified formulation

*Corresponding author.

E-mail address:jmantari@utec.edu.pe(J.L.MANTARI).

Peer review under responsibility of Editorial Committee of CJA.

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http://dx.doi.org/10.1016/j.cja.2017.06.014

1000-9361?2017 Chinese Society of Aeronautics and Astronautics.Production and hosting by Elsevier Ltd.

This is an open access article under the CC BY-NC-ND license(http://creativecommons.org/licenses/by-nc-nd/4.0/).

?2017 Chinese Society of Aeronautics and Astronautics.Production and hosting by Elsevier Ltd.This is an open access article under theCCBY-NC-NDlicense(http://creativecommons.org/licenses/by-nc-nd/4.0/).