Sok Kim ,Kwnghun Kim ,,Myonghol Ri ,Yongsong Pek ,Choljun Kim

a Department of Information Engineering,Chongjin Mine & Metal University,Chongjin 999091,Democratic People’s Republic of Korea

b Department of Engineering Machine,Pyongyang University of Mechanical Engineering,Pyongyang 999093,Democratic People’s Republic of Korea

c Institute of Science,Pyongyang University of Mechanical Engineering,Pyongyang 999093,Democratic People’s Republic of Korea

d Department of Resources Development Mechanical Engineering,Pyongyang University of Mechanical Engineering,Pyongyang 999093,Democratic People’s Republic of Korea

Abstract In this paper,a semi-analytical method for the forced vibration analysis of cracked laminated composite beam (CLCB) is investigated.One computational model is formulated by Timoshenko beam theory and its dynamic solution is solved using the Jacobi-Ritz method.The boundary conditions (BCs) at both ends of the CLCB are generalized by the application of artificial elastic springs,the CLCB is separated into two elements along the crack,the flexibility coefficient of fracture theory is used to model the essential continuous condition of the connective interface.All the allowable displacement functions used to analyze dynamic characteristics of CLCB are expressed by classical Jacobi orthogonal polynomials in a more general form.The accuracy of the proposed method is verified through the compare with results of the finite element method (software ABAQUS is used in this paper).On this basis,the parametric study for dynamic analysis characteristics of CLCB is performed to provide reference datum for engineers.? 2020 Shanghai Jiaotong University.Published by Elsevier B.V.This is an open access article under the CC BY-NC-ND license.(http://creativecommons.org/licenses/by-nc-nd/4.0/)

Keywords:Semi-analytical method;Cracked laminated composite beam (CLCB);Dynamic analysis;General boundary condition;Numerical analysis.

1.Introduction

As science and technology develop,aerospace structures are needed to be lighter,and high-performance structures such as high-speed turbine machinery require high strength.Based on this background,the composite laminated structures are widely used to meet these demands in structural and engineering applications,and the beam which is a basic and fundamental component of structure is contained in nearly every engineering structure.

Fig.1.The geometric dimension of CLCB

Since beam structures are usually subjected to various external forces,the forces’ repeated actions lead to their microcracks.In order to prevent accidents by predicting the destruction of structures,it is very important to fully understand the dynamic properties of CLCBs.For these backgrounds,a lot of researchers began to pay a deep attention to the research of cracked composite material beams.Till now,many scholars studied the free vibration properties and dynamic behavior of beam structures made of isotropic materials with a crack by using various solution methods and theories.[1–11]As various composite materials are widely used,recently studies on the CLCBs has been attracting the attention of many researchers.M.Krawczuk et al.[ 12,13 ]analyzed the vibration of cantilever beams with single-open crack made of unidirectional composites and graphite fiber reinforced polyimide by using the FEM (Finite Element Method).A.R.Daneshmehr et al.[14]created a computational model based on the FSDT and obtained its solutions to investigate free vibration properties of the beam with single crack made of unidirectional material subjected to mixed force of bending and torsion.K.Wang et al.[15]applied the CBT(Classical Beam Theory)to study the vibration properties of composite beam made of unidirectional fiber-reinforced material with single crack subjected to bending-torsion coupled load,and discussed the crack effects on the vibration modes.Based on Timoshenko beam theory,S.Kitipornchai et al.[16]researched nonlinear vibration of a cracked FG Timoshenko beams and used Ritz method to obtain nonlinear vibration frequencies of cracked FGM beams.Ke,LiaoLiang et al.[17]analyzed nonlinear vibration of FGM beams with an open crack by using differential quadrature method (DQM).A.R.Daneshmehr et al.,[18]based on classical elasticity theory,theoretically investigated free vibration analysis of a FGM beam with an open crack under the bending-torsion coupled loading.Kim et al.[19–21]extended the application range of orthogonal polynomials through the free vibration analysis of the cracked Timoshenko beam and CLCB using the ultraspherical polynomials and Jacobi polynomial.C.Karaagac et al.[22]investigated the vibration behavior and lateral buckling of a cantilever slender rectangular beam as research object a laminate specimen which is one of the constituents of laminated composite beams according to crack ratio and crack position.S.M.Ghoneam [23]obtained the natural frequencies of the CLCB according to change of crack locations under various classical BCs by the theoretical method,in which the displacement field was set by classical beam theory.He also experimentally verified the compatibility of his theoretical analysis method.

Although a lot of researches have investigated the vibration problems of laminated composite beams [24–34],studies on cracked laminated fiber reinforced composite beams can be found only in some literatures.Particularly,vibrations of unidirectional composite beams and FGM beam with a crack have been extensively studied,but research on dynamic analysis of CLCB is still limited.Even if laminated structure,the direction of the fiber direction of every layer was aligned in one direction,and dynamic analysis of CLCB formed by laminating layers with different fiber directions is rarely found.

Fig.2.Continuous condition model for the cracked beam

The purpose of this paper is to propose a semi-analytical method for analyzing the forced vibration of CLCB with different boundary conditions.The two types of composite specimens are symmetrically laminated about the middle plane or are alternately laminated of them.To the authors’ knowledge,in this case,there is no expression of generalized flexibility coefficient due to crack.Therefore,in this paper,one generalized expression of flexibility coefficient when two composite specimens with different fiber orientation angles are laminated symmetrically about the middle plane or are laminated alternately is presented.Then,based on it,a semi-analytical method for the forced vibration analysis of CLCB is introduced.The accuracy and robustness of the proposed method are proved compared to the existing literature and the FEM(finite element method).The several numerical results for the free vibration of CLCB with general boundary conditions that can be provided as a reference for future engineering are depicted graphically.

2.Theory

2.1. Geometric description of CLCB

Fig.1 shows the geometric dimension and Cartesian coordinate system for the dynamic analysis of CLCB.As shown in Fig.1,the CLCB has a lengthL,widthband thicknessh.

The distance from the left side of beam to the crack isLc.The deformation of CLCB is only considered inx-zplane.The one rotating elastic spring group (kθ0,kθL) and the two translating elastic spring groups (ku0,kw0and,kuL,kwL)[35–38]determine the BCs for the beam.All layers of the beam are composed of unidirectional orthotropic materials in any orientation with respect to thex-axis.The distances from the middle surface of the beam to the top and bottom surfaces ofkth layer arezk+1andzk.For the consideration,the beam is divided into two parts along the crack surface and the continuous condition model for the cracked beam is shown in Fig.2.kuc,kwc,kθcdenote the inverses of flexibility coefficients owing to the crack.

2.2. Constitutive equation

The composite material discussed in this work is supposed to be thick,so the warping effect which is very important in the thin composite beam theory is ignored.Meanwhile,since we only consider two transitional displacements along thex,zaxes and one bending displacement,the strain components of one-dimensional beam are set by Timoshenko theory based on the coordinate system in Fig.1.The strain components are written as [39]

As the beam has only the axial displacement alongx-axis direction,vertical displacement alongz-axis direction and bending displacement with respect to the middle surface,the one-dimensional constitutive equations of CLCB are

whereNxdenote the normal force resultants on the surface,Mxdenote bending moment resultants,andQxdenote the tangential force resultants,respectively.The shear correction factorκis taken as 5/6 in this paper because the cross-section of the beam is rectangular.The transformation matrixcan be obtained as the following equation.

where

in which,the stiffness coefficients of fiber reinforced materialsAij,BijandCijare expressed as

The specific data on transformation elasticof the laminated composites can be found in Refs.[ 32,37,40 ].

The total energy of the considered LCBC is written as:

whereandTiare the strain and kinetic energies of individual parts separated due to the crack,respectively.UBis the boundary energy at two ends of CLCB andUCis the interface energy on the common section between two adjacent segments.AndWeis the energy by external work.

The strain and kinetic energies of each beam part are given as follows:

where {I0,I1,I2} are the inertia coefficients,they are defined by;

The boundary energyUbis defined as

wherekψ,0(ψ=u,w,θ)andkψ,Ldenote the boundary stiffnesses at the CLCB,respectively.

And the interface energyUCis defined as [22]

where the connection spring stiffnesseskuc,kwc,kθcare defined as inverse of the flexibility coefficients in fracture mechanics.The flexibility coefficients due to expansion of crack can be found the author’s previous researches [ 20,21 ]and they are expressed as

Since the beam in which two types composite specimens are symmetrically laminated with respect to the middle plane or alternately laminated downward from the top plane will be considered.In this case,the flexibility coefficients can be rewritten as follows:

In case of laminated downward from the top plane,

In case of laminated symmetrically with respect to the middle plane,

wherekdenotes the total number of layers,k=1,2,···,and the superscripts 1,2 denotes individual layers.

The external work is presented as follows.

whereqxandqwis the external translational loads,respectively.m?denote the external moment couples in the middle surface of the considered beam.

2.3. Unified solution methodology

In this work,the discretized equations of motion for CLCB should be expressed by the allowable displacement function.It is very important to choose the reasonable allowable displacement function for ensuring a stable convergence and accuracy of the solution.

As descripted in above subsection,the theoretical model is obtained by applying segmentation technique,and therefore,the allowable displacement functions of all beam segments may be allowed to be the same.In addition,since individual part can be regarded as a free segment,this greatly reduces the constraint of boundary conditions on allowable displacement functions,and thus greatly improves the range and flexibility of allowable displacement functions selection,which only needs satisfying being linearly independent,complete.

The displacement functions of the CLCB can be flexibly selected by the penalty parameter,and the fast convergence of accurate solution can be ensured with the appropriate value of the penalty parameter.On the treatment of continuous boundary conditions,it makes the choice of the admissible function flexible to introduce the spring stiffness,which is the penalty parameter in free [40–43].Based on the author’s previous work,in this paper,all displacements including boundary and continuous conditions are selected by Jacobi polynomials,and the specific expression is as follows [ 36,40–48 ]:

The Jacoby polynomials include Chebyshev,Legendre,and Gegenbauer polynomials depending on how the values of the constantαandβare chosen.For example,the case ofα=β=?1/2 yields the Chebyshev polynomials of the first kind,and the case ofα=β=1/2 gives the Chebyshev polynomials of the second kind.Also,choosingα=β=0 gives the Legendre polynomials,and while choosingα=βgives the Gegenbauer polynomials.Therefore,it can be seen that Jacobi polynomial is superior in general and applicable aspects than other polynomials.

Fig.3.Convergence curves of first fifth frequency parameters as increase of the boundary spring stiffness.

Table1 The values of spring stiffness on some classical and elastic boundary conditions.

In Eq.(16),Um,Wm,Θmare the generalized coordinate vectors,and U (xi),W (xi),Θ(xi) are the admissible function vectors.

By substituting the allowable displacement functions defined in Eq.(16) into Eqs.(7,8,10,11),the individual energies can be obtained,the total energy of CLCB are defined by substituting these individual energies into Eq.(6).Applying the Ritz method,the equations of motion for a CLCB can be obtained as

where M and K are total mass matrix and stiffness matrix of CLCB,and A denotes the coefficient vector,and F denotes external load vector.By solving eigen problem of Eq.(17),the natural frequencies of the CLCB can be easily obtained.

3.Parameter study and discussions

3.1. Convergence of parameters

In this subsection,the convergence studies of the parameters for dynamic analysis of CLCB are presented.If the boundary spring stiffness modulus is set as extremely small or large values,the solution obtained by the mechanical model maybe mathematically not converged [44].

For determining the boundary conditions of the CLCB,the convergence study on the spring stiffness at the both boundaries of CLCB is carried out.When the value of a spring stiffness is changed from 105to1020,the other two springs’stiffnesses are fixed as1015.

A cracked laminated beam which has geometric dimensions asL=1m,b=L/15m,h=L/15m,the crack position and the crack ratio areLc=0.4m,a/h=0.1,respectively.And the fiber orientationαf iber=[ 90?0?0?90?]will be used to convergence study,and the mechanical parameters of the CLCB used in the convergence study are as follows:

The convergence curves of the first fifth frequency parametersΩalong the boundary spring stiffness values are shown in Fig.3,in which the frequency parametersΩare calculated by

From Fig.3,it can be seen that the frequency parameters are certainly increased along the boundary spring’s stiffness values in the range from107to1011,and outside this range,frequency parameters converge to a constant value.Therefore,in this study,it is assumed that the value of spring stiffness reflecting the fixed boundary condition is1014,the value of spring stiffness value reflecting the elastic condition is108.Based on the above convergence result,Table1 shows the spring stiffness settings for some boundaries including three classical BCs and three elastic BCs are presented.

In the actual application,in order to consider the efficiency of the calculation and the accuracy of the solution,the displacement functions must be selected reasonably.Thus,the convergence studies to select the appropriate number of series in the Jacobi polynomial are presented.

Table2 Convergence results for frequency parameters of CLCB with different truncation terms.

Fig.4.Percentage error of the Non-dimensional frequencies for the different Jacobi polynomials coefficients αand β.(a) F-C,(b) C-C,(c) F-F,(d) S-D

Table2 shows the convergence results of the frequency parameters for the CLCB with fully clamped boundary condition at the both ends.Form the results of Table2,it can be found that frequency parameters are stably converged in the region where the series number M of the Jacobi polynomial is greater than 10.

Base on the above results,in this research for dynamic analysis,all of the series in Jacobi polynomial are selected asM=10.

Next,the error calculation corresponding to different parameters groupα,βof the group is presented to select the Jacobi polynomials parameter.The percentage error(Ωα,β?Ωα=0,β=0)/Ωα=0,β=0of the frequency parameters according to the change of Jacobi polynomials parameterα,βare shown in Fig.4.As can be clearly seen from Fig.4,the maximum value of the percentage error of the frequency parameters does not exceed 0.35 ×10?7,from this results,it can be found that the change of the Jacobi polynomials coefficientsαandβdoes not significantly affect the dynamic analysis of CLCB.Therefore,for numerical analysis,the Jacobi polynomials coefficientsα,βare set intoα=?0.5,β=?0.5.

Table3 Comparison of the natural frequencies for LCB without crack αf iber=[ 30 ?/ 50 ?/ 30 ?/ 50 ?]

Table4 Comparison of the natural frequencies in the CLCB

3.2. Verifications of presented method

In this subsection the accuracy and reasonability of the proposed method for the vibration analysis of the CLCB will be verified in comparison with previous results and FEM.

Firstly,the natural frequencies of the LCB without crack obtained applying the current method and those of the previous results [ 33,34 ]are compared.

If the connective spring stiffness values at crack surface are assumed as infinite large,the CLCB may be modeled as beam without crack.Thus,based on the above convergence study,the connective spring stiffness valueskuc,kwc,kθcat the crack surface will be set as1014in this verification.The natural frequencies of laminated composite beam without crack under the various classical BCs are compared with previous results and the comparison results are presented in Table3.As you can see in Table3,the analysis results of the proposed method are in good agreement with the results of the existing literature results.

To confirm the reliability and accuracy of this method for the dynamic analysis of the CLCB,then,the natural frequencies taken through the proposed method are compared with the results of the finite element analysis software ABAQUS.

The natural frequencies of CLCB according to change of crack ratioa/bunder the various BCs are compared with the results of the finite element method (FEM).Here,for comparison with results of the FEM,finite element analysis software ABQUS (in element type-S4R,elements-16891) is used.The comparison results are presented in Table4.The geometric dimensions and material properties are following as:

L=1m,h=0.06m,b=0.05m,Lc=0.3m,αfiber=[90 °/90 °/90 °/90 °],ρ=1389.23kg/m3

E1=37.41Gpa,E2=13.67Gpa,μ12=0.3,G12=5.478Gpa,G13=6.03Gpa,G23=6.666Gpa

The results of Table4 show that the results obtained applying this method are good agreement with those of the finite element method.

Based on the above results,we can see that the presented method takes the capability to conduct with the dynamic analysis of the CLCB under arbitrary boundary condition.

Before investigating the forced vibration of the CLCB,it is necessary to confirm the accuracy of this method.For verification studies,material properties and geometric parameters of laminated cracked composite beam are set as following.

L=1m,h=L/20m,b=0.05m,E1=150Gpa,E2=10Gpa,μ12=0.3,G12=G13=5Gpa,G23=6Gpa,ρ=1500kg/m3

The considered CLCB is laminated in three layers.The crack parameters areLc=0.5manda/b=0.1.The fiber angles of individual layers are equal to 90 degrees.Jacobi polynomials parameters:α=0,β=0,the maximum degree of Jacobi polynomials;M=9,and the Load PointAof concentration force;A=0.4m,and measured point;B=0.2m,concentration force;=?1N,and boundary condition is Clamped-Clamped boundary.The sweep range is from 1 to 1500,and the interval is 1 Hz.Under this condition,the normal displacement response properties are presented in Fig.5,and are compared with result of FEM.

Fig.5.The comparison of normal displacement of CLCB

Fig.6.The comparison of normal displacement response of CLCB.

From Fig.5,it can be observed that the results obtained by applying proposed method are in good agreement with the results of the FEM.

The accuracy verification study on the transient response characteristics of CLCB by presented method was performed first.The verification study was conducted by comparing the transient response results with the FEM results,and the results are shown in Fig.6.The geometrical dimensions and material properties of the beams used in the verification study for the transient response characteristics were set the same as those used in the verification study of the forced vibration.The computing timetand stepΔtare selected 10msand 0.05ms,respectively.

Comparing the theoretical results with the results obtained by FEM,we can see that the theoretical results show a good agreement with those of finite element analysis software ABAQUS (Fig.6).

Based on the accuracy of the transient responding characteristics by present method,the effects of some load types on the transient responses of CLCB are presented.The geometric and material parameters are the same as in Fig.5.

3.3. Numerical examples

The beam structures,as the foundation construction,may receive external loads in engineering applications.Therefore,it is necessary to study the forced response of CLCB.For the study of forced response,it can be divided into the stability response analysis in frequency domain and the transient response analysis in time domain.

3.3.1.Steady-statevibrationanalysis

In this subsection,the steady-state problems of CLCB under different external excitation forces are investigated.In this study,two common loads:concentration force,distribution force are discussed.The diagrammatic sketch of two applied load types for CLCB is shown in Fig.7.

Fig.7 (a) is in the case of the concentrated forcefwacting on load point A of the beam and Fig.7 (b) is in the case of distribution forcefwacting on certain length section A of the beam.The displacement response will be measured at the point B.The concentration force in expressed asfw=ˉfwsin(ωt),where the amplitude of the harmonic force is taken as ˉqw=1Nandωis the frequency of the harmonic concentration force.

Fig.8 shows the displacement-frequency characteristic of CLCB under two types of load-concentration load and distribution load.The boundary conditions of three types are set as C-C,C-F,S-S.The crack ratio isa/b=0.1.Geometric and material properties are set as Fig.8 and the acting position of the force,the magnitude of the force and the measuring position are as follows.

For concentration load:acting position:A(x)=0.4m,magnitude of load;fw==?1N,Δ f=1Hz,For distribution load:acting position:A(x1,x2)=(0.3m,0.7m),magnitude of load:fw=(L?LA)=?1N,Δf=1Hz.

As shown in the Fig.8,the displacement-frequency curves are changed according to the crack location and boundary conditions.

In the case ofLc=0.2m,the displacement of CLCB with CC and C-F boundary conditions appears lowest at the primary frequency,and the displacement increases as the frequency order increases.However,in the case ofLc=0.6,the opposite characteristics are shown,and in the case ofLc=0.4,there is a little change in displacement.

Fig.9 shows the frequency-displacement response characteristic curve of CLCB according to the crack size under several boundary conditions.The geometric and material properties of CLCB were set as shown in Fig.8.The magnitude and acting point of load are also set as F ig.8.The measure point of displacement responding isB=0.2m for concentration force,B=0.4m for distribution force.

As shown in the Fig.8,the frequency-displacement response appears differently while the concentration force and the distribution force are acted according to the crack size.

Fig.7.The diagrammatic sketch of two applied load types.(a) Concentration force;(b) Distribution force.

Fig.8.The displacement-frequency characteristic of CLCB with different boundary condition under two types of load.

3.3.2.Transientrespond

As the last part of this study,the methodology mentioned previously is applied to obtain the transient responses of CLCB subjected to different four shock loads,namely rectangular pulse,triangular pulse,half-sine pulse and exponential pulse.The sketch of load time domain curve is shown in Fig.9 and individual load curves are defined as follows:

whereftis the load amplitude;τis the pulse width;tis the time variable.

Based on the verification of the transient response of the CLCB using presented method,the transient response characteristics of the vertical displacement of the structure when the boundary condition used in the forced vibration analysis and the four types of load pulses are applied in the CLCB are shown in Fig.11.In this case,the transient loadf(t) is set to the four pulse of Fig.10 and the amplitude of the rectangular pulse is taken as:ft=1N.The calculating time step is set toΔt=0.05ms,and the loading timeτand calculating timetare equal to 50ms.The transient response of the CLCB are carried out under C-C boundary condition and acting position of force and measure point of displacement responding are same as above study.The force exerted is concentration.

Fig.9.The displacement response of CLCB under different loads

Fig.10.The sketch of load time domain curve.(a) Rectangular pulse;(b) Triangular pulse;(c) Half-sine pulse;(d) Exponential pulse

Fig.11.The displacement response of CLCB under different loads

Fig.12.The displacement response of CLCB under different loads

From Fig.11,the half-sine and triangular pulses are much smaller in magnitude than the transient response of the rectangular pulse impulse.Thus,relatively slow impact loads can reduce transient response amplitudes,while strong discontinuous shock loads will increase transient response.In the case of half-sine pulses,the shape of the transient response curve can be softened and the influence of the transient response due to the impact can be reduced.In addition,the displacement is greatest when the crack position is in the middle position of the beam regardless of the impact load type.

At the end of this study,the transient response characteristics of the CLCB according to different crack sizes are investigated.

The geometric and material properties of CLCB were set as shown in Fig.11.The magnitude and acting point of load are also set as Fig.11.The measure point of displacement responding is B=0.2m and the force exerted is distribution force.It can be clearly seen from the Fig.12 that the larger the size of crack,the greater the change of displacement.

From the analysis of the pictures above,it can be seen that when the loading time is equal,the transient response caused by the rectangular pulse is the largest,while the minimum is caused by the triangular pulse.Therefore,the magnitude of the transient response of the CLCB is closely related to the form of load action.Slow loading or unloading can reduce the amplitude of transient response,while sudden loading or unloading force can increase the transient response.

4.Conclusion

In this paper,the forced vibration and transient response characteristics of CLCB under general BCs are investigated.Jacobi polynomials have been applied to the generalization of allowable displacement functions,and displacement fields are set by the Timoshenko beam theory.The cracked composite beam is laminated symmetrically or alternately around the middle plane of two types composite specimens.Kinetic and physical compatibility conditions at arbitrary boundary conditions are established by the boundary spring technique,and continuous conditions at the crack surface are modeled using the reciprocal of the compliance flexibility coefficient of fracture mechanics theory.All numerical results obtained by presented method were found to be in good agreement with the corresponding results of the existing literature and finite element analysis.In addition,several numerical results have been reported for the dynamic behavior of CLCB.The results of the current paper can be applied to reference data for future research.

Declaration of Competing Interests

The authors declare that they have no conflict of interest.

Acknowledgments

I would like to take the opportunity to express my hearted gratitude to all those who make a contribution to the completion of my article.In addition,The authors would like to thank the anonymous reviewers for their very valuable comments.