Vibration Analysis of Inclined Laminated Composite Beams under Moving Distributed Masses

The dynamic response of laminated composite beams subjected to distributed moving masses is investigated using the finite element method (FEM) based on the both first-order shear deformation theory (FSDT) and the classical beam theory (CLT). Six and ten degrees of freedom beam elements are used to discretize the CLT and FSDT equations of motion, respectively. The resulting spatially discretized beam governing equations including the effect of inertial, Coriolis, and centrifugal forces due to moving distributed mass are evaluated in time domain by applying Newmark’s scheme. The presented approach is first validated by studying its convergence behavior and comparing the results with those of existing solutions in the literature. Then, the effect of incline angle, mass, and velocity of moving body, layer orientation, load length, and inertial, Coriolis, and centrifugal forces due to the moving distributed mass and friction force between the beam and the moving distributed mass on the dynamic behavior of inclined laminated composite beams are investigated.


Introduction
The composite beams have been increasingly used over the past few decades in the field of aerospace and civil and mechanical engineering due to their excellent engineering features.They are generally used as structural components of light-weight heavy load-bearing elements because of the high strength and stiffness-to-weight ratios, the ability of being different strengths in different directions, and the nature of being tailored to satisfy design requirements of strength and stiffness in practical designs.Hence, the better understanding of vibration characteristics of these structural elements under dynamic loads is essential for their engineering design and manufacture.
The dynamic behavior of isotropic beams subjected to moving loads using the analytical and numerical methods has been investigated by some researchers; see, for examples, [1][2][3][4].In most of these works, the horizontal beams under moving loads have been considered and the effects of the Coriolis and the centrifugal forces have been neglected.In an interesting work, Wu [5] studied the vibration of an inclined isotropic beam under moving mass by considering the effects of the Coriolis and the centrifugal forces.On the other hand, little attentions have been paid to dynamic response of the laminated composite beams subjected to the moving loads.
Chandrashekhara et al. presented an exact solution for the free vibration of symmetrically laminated composite beams using first-order shear deformation and rotary inertia has been included in their analysis [6].The solution procedure is applicable to arbitrary boundary conditions.Kadivar and Mohebpour studied the dynamic response of an unsymmetric laminated composite beam subjected to moving loads [7,8].They developed a finite element method based on the third-order shear deformation theory.Law and Zhu studied the dynamics of damaged reinforced concrete bridge structures under moving vehicular loads [9]. Lee and Yhim studied the dynamics of single-and two-span continuous composite plate structures subjected to multimoving loads [10].Kiral et al. studied the dynamic behavior of the laminated composite beams subjected to a single force traveling at a constant velocity using a three-dimensional finite element model based on the classical lamination theory [11].Yang  edge cracks under an axial compressive force and a concentrated transverse load moving along the longitudinal direction [12].S ¸ims ¸ek and Kocatürk investigated free vibration characteristics and the dynamic behavior of a functionally graded simply supported Euler-Bernoulli beam under a concentrated moving harmonic load [13].Malekzadeh et al. studied the dynamic response of thick laminated annular sector plates with simply supported radial edges subjected to a radially distributed line load moving along the circumferential direction by using a three-dimensional hybrid method composed of series solution, the layerwise theory, and the differential quadrature method in conjunction with the finite difference method [14].S ¸ims ¸ek studied vibration of a functionally graded (FG) simply supported beam due to a moving mass by using Euler-Bernoulli, Timoshenko, and the third-order shear deformation beam theories [15].Khalili et al. presented a mixed method to study the dynamic behavior of functionally graded (FG) beams subjected to moving loads.The theoretical formulations are based on Euler-Bernoulli beam theory, and the governing equations of motion of the system are derived using the Lagrange equations [16].Mohebpour et al. studied the dynamic behavior of laminated composite plates traversed by a moving oscillator using the finite element method [17].Mohebpour et al. studied the dynamic response of the laminated composite beams subjected to the moving oscillator using the first-order shear deformation theory [18].Kahya studied the dynamic response of laminated composite beams under moving loads using the finite element method [19].Yas and Heshmati investigated the dynamic response of functionally graded nanocomposite beams reinforced by randomly oriented carbon nanotube under the action of moving load [20].S ¸ims ¸ek et al. studied dynamic behavior of an axially functionally graded beam under the action of a moving harmonic load [21].Mamandi and Kargarnovin investigated nonlinear vibrations analysis of an inclined pinned-pinned self-weight Timoshenko beam made of linear, homogenous, and isotropic material with constant cross section and finite length subjected to a traveling mass/force with constant velocity [22].
To the best of authors' knowledge, the dynamic behavior of the inclined laminated composite beams subjected to moving distributed mass has not been investigated so far.This motivates the authors to study the problem considered here.The aim of this study is to examine the finite element dynamic analysis of the straight and the inclined composite laminated beams subjected to moving distributed mass with constant speed based on the classical (CLBT) and the first-order (FSDT) laminated beam theories.The algorithm developed here also accounts the shear deformation and the rotary inertia for the beam and centrifugal and Coriolis forces due to moving distributed mass.In addition, the friction force between the inclined beam and the moving distributed mass is considered as a uniform distributed load.The time variable is evaluated by using Newmark's scheme [18].

Governing Equations
Consider an inclined laminated composite beam of the length , width , thickness ℎ, and incline angle  as shown in Figure 1.The coordinate system is placed at the midplane of the laminate.The thickness of all orthotropic layers is assumed to be equal.The fibers direction is indicated by the angle  measuring from positive -axes in counterclockwise direction.The displacement components of an arbitrary material point along the and -directions are denoted as (, , ) and (, , ), respectively.
In this paper, the following assumptions are adopted.
(i) Inertial effects of both the beam and the moving mass are taken into account as well as the gravity effects of the load.(ii) Friction force between the moving mass and the beam is taken into account as a uniform distributed load.(iii) The load moves at constant velocity and keeps in contact with the beam at all times.(iv) The beam is assumed to carry the whole load at the left-hand support when the motion just started.(v) The Coriolis and the centrifugal forces induced by the moving distributed mass are taken into account.(vi) Poisson effect is neglected except when it is mentioned.
The governing equations can be derived using Hamilton's principle, where  and the  denote variation of strain energy of beam and  is work done by external loads.Based on the classical beam theory, the variation of the strain energy and the kinetic energy becomes where   is the axial stress in the laminated beam and  0 is the mass density of the beam.The strain components are related to the displacement components as where  and  are the transverse and the axial displacements of a material point on the midplane in the and -directions, respectively.
The virtual work of the external loads is as follows: where  is the transverse distributed force on the surface of the laminated beam and  is the axial force.Inserting ( 2) and ( 4) into (1), the axial and transversal equations of motion of the beam are obtained as where the resultant forces and moment are defined as and the stiffness coefficients are as The axial and transversal loads on the beam due to moving distributed mass are as follows: where ) and  = V + /2.In ( 8), V is the velocity of the moving mass,   is mass,  is load length,  is acceleration of gravity, and () is Heaviside unit step function.
In a similar manner, the governing equations of the beam based on the first-order shear deformation theory can be derived in the following.For this purpose, the variation of the strain energy and the kinetic energy of the beam can be written as where   is the bending rotation of the beam cross section.The variation of strain components is as follows: = 0.
Inserting ( 9) into (1) and performing the integration by parts, one gets the following relations: where where   transverse shear is force and  is the shear correction factor.

Property Matrices of Moving Distributed Mass.
At any instant moving, distributed mass is on one or more beam elements depending on the beam element lengths; thus, the mass, stiffness, damping, and force matrices of moving, distributed mass are functions of time.Since moving distributed mass and beam are always in contact and moving mass slides on the beam surface, so they have the same transversal displacements while their axial displacements, are different.The governing equations of moving distributed mass in  and  directions are given by where   is force due to friction between the moving distributed mass and inclined beam and   are interaction forces between moving distributed mass and the beam.  have components inertials, Coriolis, and the centrifugal forces of mass, respectively.  is axial displacement of the moving distributed mass.

Free Vibration of
In the six-degree-of-freedom element (see Figure 2(b)) used in both CLT beam weak form and moving distributed mass formulations, linear Lagrange interpolation functions are used for nodal forces and displacements in  direction.For its  direction, cubic hermit shape functions are used as follows: where   ( = 1, 4) are used for nodal force and displacements in  direction.  ( = 2, 3, 5, 6) are of them in  direction.
On the other hand, FSDT-related ten-degree-of-freedom element (see Figure 2(a)) shape functions are the same as CLT ones.Shape functions used in FSDT rotation of cross section,   , are also cubic hermit shown in (15a).
Matrix form of equation of motion of distributed moving mass is as follows: where } and {} = {    } are the force and displacement vectors, respectively.For convenient of application of the finite element method, it is convenience to transform local matrices and vectors to those of global ones.Thus, according to Figure 1, the following relationships are adopted: where transformation matrix of a ten-degree-of-freedom  is defined as follows: Equation ( 16) can be rewritten in the global coordinate system as where

Equation of Motion of the Entire Vibrating System.
For a multiple-degree-of-freedom damped structural system, equation of motion is given by where [()], [()], and [()] are the instantaneous overall mass, damping, and stiffness matrices, respectively, and they are time-dependent matrices.(), Ṙ(), and R() are the displacement, velocity, and acceleration vectors, respectively, while () is the instantaneous external force vector.

Overall Property Matrices.
To take the effect of inertial and centrifugal forces of moving distributed mass into account, one must add their contribution to all property matrices of entire system.For doing this, we should add the above-mentioned moving distributed mass given by ( 16) and (19) to the inclined beam property matrices [  ] and [  ] that are mass and stiffness matrices, respectively.Thus, the instantaneous overall mass and stiffness matrices of the entire vibrating system are established by where In the last equations,  represents the total degrees of freedom of the entire vibrating system and subscripts   and   (,  = 1, 2, . . ., ) represent the numbering for the 10 degrees of freedom of the nodes of the beam elements on which the moving distributed mass applies at time .
The overall damping matrix [  ] of the inclined beam is determined by using the theory of Rayleigh damping [17] as where In ( 25), (26a), and (26b), [()] and [()] are the overall mass and stiffness matrices given by ( 22a) and (22b), respectively, and   and   are damping ratios corresponding to any two natural frequencies of the structure,   and   .
If the Coriolis force induced by the distributed moving mass is considered, one must add the contribution of the damping matrix of the moving distributed mass element, [()], to the overall damping matrix of the inclined beam itself, [  ], to establish the instantaneous overall damping matrix, [()]; that is, where except

Equivalent Nodal Forces and Overall External Force
Vector.The equivalent force vector induced by the moving distributed mass at any time  is given by where  and  are unit vectors in the local  and  directions (see Figure 3), while   and   are corresponding force components given by In the last expressions,  is the acceleration of gravity,   is the distributed moving mass, and  is the inclined angle of the beam.Besides,   is the frictional force that is assumed to be uniformly distributed between moving distributed mass and inclined beam.The force equilibrium along the incline beam in  direction requires that Thus, where  is the friction coefficient, so (31a) should be satisfied; then, a load may move along the incline beam with a constant velocity V.

Numerical Results and Discussion
To obtain results given in this paper, Newark's scheme [23] has been utilized for time derivatives of ( 21) which are to be solved for each time interval.In this section, firstly, the convergence and accuracy of the presented formulation and the validity of obtained results are investigated.Then, the numerical results of simplified problems are presented and, whenever possible, compared to available analytical and numerical results in the literature.Finally, parametric analysis is carried out to investigate the effects of the system components properties.The shear correction factor  is assumed to be equal to 5/6 in all examples.In this section, numerical results have been presented for symmetrical four-layer AS/3501-6 clamped-clamped graphite-epoxy beams (/−/−/).The results are obtained using the FSDT without Poisson effect.Table 1 shows the nondimensional fundamental frequencies ( =  2 √/(  ℎ 2 )) of four-layer symmetrical angle-ply beams for clampedclamped boundary conditions.The convergence behavior (34)

Forced Vibration of Symmetrically Laminated Straight and
Inclined Beam.In this section, numerical results for forced vibration of symmetrically laminated straight and inclined beam under moving distributed load are investigated.Numerical solutions performed the same beam properties as in [8].

Straight Beam.
In Figure 5, the variation of the midspan dynamic deflection of the beam divided by the maximum static deflection (DMF) is drawn versus speed ratio (V/V cr ).
Comparison of DMF of this graphite-epoxy (0 ∘ /90 ∘ /90 ∘ /0 ∘ ) composite beam under moving load is investigated using  CLT.As one can see, the results are in good agreement with those of [8].
(1) Influence of Layer Orientation on Transverse Deflection of Straight Beam.In Figure 6, the effect of layer orientation on the transverse deflection of straight beam based on FSDT under moving distributed mass is investigated.Load length  = 0.01 m, velocity V = 50 m/s, and damping ratio  = 0.005 are assumed.As can be expected, when the fiber angle of layers changes from 60 ∘ to 30 ∘ , the transverse deflections decrease due to increasing of the beam stiffness.(3) Influence of Mass of the Moving Distributed Load on the Transverse Deflection of the Graphite-Epoxy Composite Laminated Beam.The effect of mass of moving distributed load on the transverse deflection of beam based on the FSDT is shown in Figure 9.It can be seen that the deflections considerably change with considering the mass of moving load (without its Coriolis and centrifugal effects).In addition, time when maximum deflections occurred will be delayed when the mass of load is included.

Inclined Beam.
For numerical results in this section, beam properties are taken the same as [8] except the fact that the moving distributed mass is assumed to be 4.45 kg.
(1) Influence of the Incline Angle on the Vertical and Horizontal Deflections of Graphite-Epoxy Composite Laminated Beam.
Effect of the incline angle on vertical deflection of inclined beam is investigated in Figure 10.Load length  = 0.02 m and velocity V = 400 m/s are assumed.According to Figure 10, incline angle has no a significant effect on the vertical deflections.
Effect of incline angle of the beam on the horizontal deflection of incline beam is investigated in Figure 11.It can be seen that, with increasing the incline angle to 45 ∘ , horizontal deflection increases and maximum deflection in this angle occurs.For angles greater than 45 ∘ , the horizontal deflection decreases because the frictional force decreases.in Figure 18.Moving mass length and velocity are 0.01 m and 90 m/s, respectively.As seen in the figure, the Coriolis and centrifugal forces have no effect on the vertical deflections.Effect of friction itself is not considerable but is considerable when the mass inertia is included.beam the bigger values than the others due to its lower stiffness.
(5) Load Length and Mass Effects on Dynamic Magnification Factor (DMF).Effects of length and mass of moving distributed load on the DMF of 45 ∘ inclined composite laminated beam are shown in Figures 21 and 22, respectively.
As one can see in Figure 21, for velocities under 100 m/s and  > 0.01 m, DMF is less than 1.But, with increasing velocity to 500 m/s, DMF reaches to the maximum value (2.40).For velocities greater than 500 m/s, DMF decreases.
Figure 22 shows that effect of mass only in velocities ranged from 50 to 700 m/s is considerable.In this velocity range, increasing velocity causes increasing in DMF and vice versa.It is important to say that the length and mass of the moving distributed load do not change the velocity that at which maximum DMF occurred; that is, V = 500 m/s.

Conclusion
The dynamic analysis of the incline laminated composite beams traversed by a moving distributed mass has been investigated.Finite element method based on the first-order shear deformation and classical theories has been used to perform the equations of motion of the beam.Comparisons between the results obtained by proposed method with available results in the literature show a good agreement.It was observed that (1) the moving mass inertial effect is considerable especially for small load lengths; (2) moving distributed mass length and mass have no effect on critical velocity; (3) the friction force between moving distributed mass and beam has considerable effect on deflections; (4) maximum horizontal deflection takes place at incline angle 45 ∘ , where it has maximum friction coefficient; (5) the friction effect is considerable while considering mass of moving load but the Coriolis and centrifugal

Figure 1 :
Figure 1: (a) Geometry of the problem and (b) beam cross section.
Symmetrically AS/3501-6 Graphite-Epoxy Laminated Beam.According to Figure 1, for an arbitrary point on the inclined beam,  and  displacement components in the local  and  directions are related to  and  in global  and  directions,  =  cos  −  sin , (14a)  =  sin  +  cos .

Figure 3 :
Figure 3: An incline beam element on which resultant moving distributed mass weight    applies.

Figure 4 : 4 . 1 .
Figure 4: Time histories of transverse displacements of simply supported straight beam under moving concentrated mass.

Figure 5 :Figure 6 :
Figure 5: Influence of moving load velocity on dynamic magnification factor (DMF) of graphite-epoxy composite laminated beam.

( 2 )
Influence of the Coriolis and Centrifugal Forces of the Moving Distributed Mass on the Transverse Deflection.Figures 7 and 8 show the effect of the Coriolis ([]) and centrifugal ([]) forces induced by the moving distributed mass on the transverse deflection of beam based on the FSDT theory, respectively.It can be seen that the transverse deflections do not change considerably with considering the Coriolis and centrifugal forces.It should be noted that, with ([] ̸ = 0) or without ([] = 0) the Coriolis force effect, the centrifugal force term is included in calculations and vice versa.

Figure 20 :Figure 21 :
Figure 20: Influence of layer orientation on the horizontal deflection of the incline 45 ∘ composite laminated beam under moving distributed mass ( = 0.02 m, V = 250 m/s and  = 70.45).