# Analytical Solution of Deformations for Two-Layer Timoshenko Beams Glued by a Viscoelastic Interlayer

Authors: Zhiyi Pei; Lin Wang; Peng Wu; Jiandong Zhang; Ding Zhou
Journal: Mathematical Problems in Engineering
 (2019)

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## Abstract

An analytical solution of stresses and deformations for two-layer Timoshenko beams glued by a viscoelastic interlayer under uniform transverse load is presented. The standard linear solid model is employed to simulate the viscoelastic characteristics of the interlayer, in which the memory effect of strains is considered. The mechanical behavior of each layer is described by the first-order shear deformation theory (FSDT). By means of the principle of minimum potential energy, a group of equations for displacements and rotation angles are derived out. The final solution is obtained by conducting the Laplace transform and the inversion of Laplace transform to the equation group. Numerical comparison shows that the present solutions and the finite element results are in good agreement. It is shown that the present results are more accurate than those obtained from the Euler-Bernoulli beam theory, especially for thick beams. And the present solutions can accurately describe the variation of stresses and deformations of the beam with the time, compared with those ignoring the memory effect of strains. Finally, the effects of the geometric parameters and material properties of the interlayer on stresses and deformations of the beam are studied in detail.


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## Body

## 1. Introduction

Owing to excellent combination advantages, layered composite structures are extensively used in civil engineering. The common applications are steel-concrete composite beams [1, 2], wood-concrete composite panels [3, 4], fiber reinforced concrete [5, 6], laminated glasses [7, 8], and so on. The existing analysis indicates that the interface connection property has considerable impact on the stresses and deformations in the layered structures [9, 10]. Behr et al. [11] studied the temperature effect on behavior of laminated glass. Galuppi and Royer-Carfagni [12] studied the effective width of equivalent slab for composite beam with shear connectors. In practical engineering, the adhesives are sometimes used to connect several structural components to form a composite structure. For the viscoelastic materials, the dependence of the stress state on the strain history is intrinsic. This study focuses on the mechanical behavior of two-layer beams glued by adhesives. The adhesives possess viscoelastic characteristics in nature, and thus the mechanical properties of the layered beams are time-varying and even subjected to static loads.As is known to all, the Euler-Bernoulli (EB) beam theory assumes that the cross sections in the beam remain plane and perpendicular to the neutral axis during deformation. Based on this theory, several studies can be found for layered beams glued by viscoelastic interlayer. Zhang and Wang [13] analyzed the viscoelastic property of concrete beams reinforced by fiber reinforced polymers and gave the analytical solutions by the use of the numerical Laplace transformation method. Lei [14] studied the dynamic characteristics of damped viscoelastic nonlocal EB beams with different boundary conditions by the use of the transfer function method. Wang [15] presented the analytical solutions for bending behavior of viscoelastic beams under quasi-static loading. Bending and buckling behaviors of layered beams glued by viscoelastic interlayer were investigated by Galuppi and Royer-Carfagni [16, 17]. In their report, two kinds of analytical models for the interlayer were proposed: the fully viscoelastic model considering the memory effect of strains and the quasi-elastic model ignoring the memory effect of strains (the viscoelastic material is approximately modelled as an “aging” material). Li [18] investigated the time-varying behaviors of layered functionally graded beams with viscoelastic interlayer by using the analytical Laplace transformation.However, the transverse shear deformation is ignored in the EB beam theory, which will lead to imprecise results when the transverse shear deformations are considerable, such as deep beams. In order to consider the transverse shear deformations, the first-order shear deformation theory (FSDT) was proposed. This theory assumes that the transverse shear strain exists and is invariable along the beam thickness. We call such a beam as the Timoshenko beam. Based on the FSDT theory, Lenci and Clementi [19] studied the natural frequency of layered beams sandwiching an elastic interlayer. Murakami [20] presented a laminated Timoshenko theory which considers the interlayer slip in the beam. Wang [21] studied the vibration and damping effect in a three-layered composite annular plate with a viscoelastic interlayer based on the FSDT theory. Bardella [22] studied the reliability of FSDT model for sandwich beams. Mattei and Bardella [23] and Tessler et al. [24] developed the structural analysis for sandwich beams based on the zigzag model. Frostig et al. [25] studied the behavior of sandwich beams based on the high-order theory. The investigation of bending response for functionally graded viscoelastic sandwich beam with elastic core resting on Pasternak’s elastic foundations was presented by Zenkour [26]. Arikoglu [27] dealt with the vibration of a three-layered composite beam with a viscoelastic core by using the differential transform method. The analytical solutions of stresses and displacements of two-layer Timoshenko beams connected by viscoelastic interlayer, whose memory effect of strains was ignored, were studied by Wang et al. [28]. Aenlle and Pelayo [29, 30] studied the dynamic effective thickness of laminated-glass beams and plates. Serpilli and Lenci [31–33] developed the asymptotic expansion method to study the mechanical properties of three-layer beams and plates with soft adhesive.In the present study, we extend the study from [16] on the slender layered beams with viscoelastic interlayer to the thick layered beams with viscoelastic interlayer. The motivation is to analyze the shear effect of length-to-thickness ratio on the deformations of the layered beams. An analytical solution based on the FSDT is developed for a two-layer Timoshenko beam glued by epoxy. The memory effect of the strains of the interlayer material is considered in the analysis. A very simple model, the standard linear solid model (SLSM), is used to model the viscoelasticity of the epoxy interlayer. The deformation and rotation angle of each beam layer are described in terms of the FSDT. The viscoelastic property in the interlayer is simulated by the standard linear solid model. By means of the principle of minimum potential energy and the Laplace transformation, the solutions of deformations and rotation angles are obtained analytically.


## 2. Analytical Model

Consider a simply supported layered beam of lengthl and width b as shown in Figure 1, consisting of two external elastic layers with thicknesses h1 and h2 adhesively glued by a thin epoxy with thicknessh, subjected to transverse distributed load p(x,t).H is the distance between the geometric axes of upper and lower layers. A Cartesian coordinate system is established with origin at the bottom surface of the beam. The elastic moduli of two external layers are E1 and E2, and Poisson’s ratios are μ1 and μ2, respectively. For isotropic materials, the shear modulus in the beam layers is expressed by Gi=Ei/2(1+μi),i=1, 2. The shear modulus of the epoxy interlayer is time-dependent and denoted by G(t) [34].Figure 1
Simply supported two-layer beam glued by a viscoelastic interlayer.


### 2.1. Assumptions

The present study is carried out based on the following assumptions:(i)
The strain and displacement in the beam are small.(ii)
The interlayer is very thin with respect to the external layers. The shear stress is constant along its thickness.(iii)
The transverse displacements and rotation angles of external elastic layers are the same, but the axial displacements about the geometric axis of each layer are different.(iv)
Since the interlayer, made of viscoelastic adhesives, is relatively soft than the beam layer, the longitudinal normal stress and strain in the interlayer are negligible.


### 2.2. Basic Equations

According to the first-order shear deformation theory (FSDT), the longitudinal displacementu(x,z,t) and the transverse displacementw(x,z,t) of the layered beam are given by (1)ux,z,t=u1x,t-z-z1φx,th2+h≤z≤h2+h+h1u2x,t-z-z2φx,t0≤z≤h2,wx,z,t=wx,t,where u1(x,t) and u2(x,t) are displacements at geometric axes of upper and lower layers, respectively; φ(x,t) is the rotation angle of transverse cross-section of the layered beam; w(x,t) is the deflection of the layered beam;t denotes time. z1 and z2 are the distances from the geometric axes of upper and lower layers to the bottom surface of the beam, and z1=h1/2+h+h2, z2=h2/2. The longitudinal normal strain εxx and the transverse shear strain γxz in the beam are given by(2)εxxx,z,t=∂u1x,t∂x-z-z1∂φx,t∂xh2+h≤z≤h2+h+h1∂u2x,t∂x-z-z2∂φx,t∂x0≤z≤h2,γxzx,z,t=∂wx,t∂x-φx,t.The longitudinal normal stressσxx and the transverse shear stress τxz in the beam are further obtained by(3)σxxx,z,t=Eεxxx,z,t=E1∂u1x,t∂x-z-z1∂φx,t∂xh2+h≤z≤h2+h+h1E2∂u2x,t∂x-z-z2∂φx,t∂x0≤z≤h2(4)τxzx,z,t=Gkγxzx,z,t=G1k1∂wx,t∂x-φx,th2+h≤z≤h2+h+h1G2k2∂wx,t∂x-φx,t0≤z≤h2where k1 and k2 are, respectively, the shear coefficients in FSDT for the upper and lower beam layers. For beams with rectangular cross-section, the shear coefficients can be given as [35] (5)k=101+μ12+11μwhere μ is Poisson’s ratio of materials of the upper and lower elastic layer. We assume that the internal interlayer is very thin with respect to the external plies. This leads to constant shear stress along the thickness of the interlayer. For an elastic material, the longitudinal shear strain γ0(x,t) and shear stress τ0(x,t) in the interlayer can be expressed by the deformation and rotation angle in the upper and lower beam layers [36], as follows:(6)γ0x,t=1hu1x,t-u2x,t+φx,tH,τ0x,t=Gthu1x,t-u2x,t+φx,tHwhere H is the distance between the geometric axes of the upper and lower layers and H=h1/2+h+h2/2. G(t) is the shear modulus of the interlayer.In Cartesian coordinates, the strain energy in the beam can be written as(7)U=12∫0l∫Aσxxx,z,tεxxx,z,t+τxzx,z,tγxzx,z,t+τ0x,tγ0x,tdAdxwhere A is the cross-sectional area of the beam. The energy of the system is given by (8)Π=∫0lLdxwhere L is the Lagrange density function, which is (9)L=12R1∂u1∂x2+R2∂u2∂x2+D1+D2∂φ∂x2+B1∂w∂x-φ2+bGthu1x,t-u2x,t+φx,tH2+pxwx,tbwhere R1 and R2 are the extensional rigidities, D1 and D2 are the bending rigidities for upper and lower beam layers, and B1 is the shearing rigidity of the whole beam. These parameters can be expressed as(10)R1=E1bh1,R2=E2bh2,D1=E1bh1312,D2=E2bh2312,B1=bG1h1k1+bG2h2k2Lagrange’s equations for the beam are(11)∂∂x∂L∂u1′-∂L∂u1=0,∂∂x∂L∂u2′-∂L∂u2=0,∂∂x∂L∂φ′-∂L∂φ=0,∂∂x∂L∂w′-∂L∂w=0Equilibrium equations can be obtained by substituting (9) into (11), as follows:(12)R1u1′′x,t-bτ0x,t=0,R2u2′′x,t+bτ0x,t=0(13)D1+D2φ′′x,t+B1w′x,t-φx,t-bHτ0x,t=0(14)B1w′′x,t-φ′x,t-pxb=0Considering the memory effect of strains in the interlayer, the constitutive equation for viscoelastic interlayer can be obtained by the Boltzmann superposition principle [37], as follows:(15)τ0x,t=Gtγ0x,0+∫0tGt-ξ∂γ0x,ξ∂ξdξ=G0γ0x,t+∫0tdGt-ξdt-ξγ0x,ξdξIt can be seen that the stresses at one point not only depend on the strains at that moment, but also depend on the whole strain history. According to the Riemann-Stieltjes convolution [38, 39], (15) can be rewritten as (16)τ0x,t=γ0x,t∗dGtin which ∗ means convolution.By substituting (16) into (12)-(14), one has(17)R1u1′′x,t-bγ0x,t∗dGt=0,R2u2′′x,t+bγ0x,t∗dGt=0(18)D1+D2φ′′x,t+B1w′x,t-φx,t-bHγ0x,t∗dGt=0(19)B1w′′x,t-φ′x,t-pxb=0


## 2.1. Assumptions

The present study is carried out based on the following assumptions:(i)
The strain and displacement in the beam are small.(ii)
The interlayer is very thin with respect to the external layers. The shear stress is constant along its thickness.(iii)
The transverse displacements and rotation angles of external elastic layers are the same, but the axial displacements about the geometric axis of each layer are different.(iv)
Since the interlayer, made of viscoelastic adhesives, is relatively soft than the beam layer, the longitudinal normal stress and strain in the interlayer are negligible.


## 2.2. Basic Equations

According to the first-order shear deformation theory (FSDT), the longitudinal displacementu(x,z,t) and the transverse displacementw(x,z,t) of the layered beam are given by (1)ux,z,t=u1x,t-z-z1φx,th2+h≤z≤h2+h+h1u2x,t-z-z2φx,t0≤z≤h2,wx,z,t=wx,t,where u1(x,t) and u2(x,t) are displacements at geometric axes of upper and lower layers, respectively; φ(x,t) is the rotation angle of transverse cross-section of the layered beam; w(x,t) is the deflection of the layered beam;t denotes time. z1 and z2 are the distances from the geometric axes of upper and lower layers to the bottom surface of the beam, and z1=h1/2+h+h2, z2=h2/2. The longitudinal normal strain εxx and the transverse shear strain γxz in the beam are given by(2)εxxx,z,t=∂u1x,t∂x-z-z1∂φx,t∂xh2+h≤z≤h2+h+h1∂u2x,t∂x-z-z2∂φx,t∂x0≤z≤h2,γxzx,z,t=∂wx,t∂x-φx,t.The longitudinal normal stressσxx and the transverse shear stress τxz in the beam are further obtained by(3)σxxx,z,t=Eεxxx,z,t=E1∂u1x,t∂x-z-z1∂φx,t∂xh2+h≤z≤h2+h+h1E2∂u2x,t∂x-z-z2∂φx,t∂x0≤z≤h2(4)τxzx,z,t=Gkγxzx,z,t=G1k1∂wx,t∂x-φx,th2+h≤z≤h2+h+h1G2k2∂wx,t∂x-φx,t0≤z≤h2where k1 and k2 are, respectively, the shear coefficients in FSDT for the upper and lower beam layers. For beams with rectangular cross-section, the shear coefficients can be given as [35] (5)k=101+μ12+11μwhere μ is Poisson’s ratio of materials of the upper and lower elastic layer. We assume that the internal interlayer is very thin with respect to the external plies. This leads to constant shear stress along the thickness of the interlayer. For an elastic material, the longitudinal shear strain γ0(x,t) and shear stress τ0(x,t) in the interlayer can be expressed by the deformation and rotation angle in the upper and lower beam layers [36], as follows:(6)γ0x,t=1hu1x,t-u2x,t+φx,tH,τ0x,t=Gthu1x,t-u2x,t+φx,tHwhere H is the distance between the geometric axes of the upper and lower layers and H=h1/2+h+h2/2. G(t) is the shear modulus of the interlayer.In Cartesian coordinates, the strain energy in the beam can be written as(7)U=12∫0l∫Aσxxx,z,tεxxx,z,t+τxzx,z,tγxzx,z,t+τ0x,tγ0x,tdAdxwhere A is the cross-sectional area of the beam. The energy of the system is given by (8)Π=∫0lLdxwhere L is the Lagrange density function, which is (9)L=12R1∂u1∂x2+R2∂u2∂x2+D1+D2∂φ∂x2+B1∂w∂x-φ2+bGthu1x,t-u2x,t+φx,tH2+pxwx,tbwhere R1 and R2 are the extensional rigidities, D1 and D2 are the bending rigidities for upper and lower beam layers, and B1 is the shearing rigidity of the whole beam. These parameters can be expressed as(10)R1=E1bh1,R2=E2bh2,D1=E1bh1312,D2=E2bh2312,B1=bG1h1k1+bG2h2k2Lagrange’s equations for the beam are(11)∂∂x∂L∂u1′-∂L∂u1=0,∂∂x∂L∂u2′-∂L∂u2=0,∂∂x∂L∂φ′-∂L∂φ=0,∂∂x∂L∂w′-∂L∂w=0Equilibrium equations can be obtained by substituting (9) into (11), as follows:(12)R1u1′′x,t-bτ0x,t=0,R2u2′′x,t+bτ0x,t=0(13)D1+D2φ′′x,t+B1w′x,t-φx,t-bHτ0x,t=0(14)B1w′′x,t-φ′x,t-pxb=0Considering the memory effect of strains in the interlayer, the constitutive equation for viscoelastic interlayer can be obtained by the Boltzmann superposition principle [37], as follows:(15)τ0x,t=Gtγ0x,0+∫0tGt-ξ∂γ0x,ξ∂ξdξ=G0γ0x,t+∫0tdGt-ξdt-ξγ0x,ξdξIt can be seen that the stresses at one point not only depend on the strains at that moment, but also depend on the whole strain history. According to the Riemann-Stieltjes convolution [38, 39], (15) can be rewritten as (16)τ0x,t=γ0x,t∗dGtin which ∗ means convolution.By substituting (16) into (12)-(14), one has(17)R1u1′′x,t-bγ0x,t∗dGt=0,R2u2′′x,t+bγ0x,t∗dGt=0(18)D1+D2φ′′x,t+B1w′x,t-φx,t-bHγ0x,t∗dGt=0(19)B1w′′x,t-φ′x,t-pxb=0


## 3. Analytical Solution

### 3.1. Formulation of Solution

The boundary conditions of simply supported beams atx=0, l are given as (20)u1′0,t=u1′l,t=0,u2′0,t=u2′l,t=0,φ′0,t=φ′l,t=0,w0,t=wl,t=0Here, we expand the displacements, rotation angles, and applied loads of the beam into Fourier series, as follows:(21)u1x,t=∑j=1Majtcos⁡λjx,u2x,t=∑j=1Mbjtcos⁡λjx(22)φx,t=∑j=1Mcjtcos⁡λjx,wx,t=∑j=1Mdjtsin⁡λjx(23)px=∑j=1Mpjsin⁡λjx,pj=2l∫0lpxsin⁡λjxdxwhere aj(t), bj(t), cj(t), and dj(t) are the unknown Fourier coefficients and λj=jπ/l. By substituting (21)- (23) into (17)- (19), we obtain(24)-R1λj2ajt-bhajt-bjt+Hcjt∗dGt=0(25)-R2λj2bjt+bhajt-bjt+Hcjt∗dGt=0(26)-D1+D2λj2cjt+B1λjdjt-cjt-bHhajt-bjt+Hcjt∗dGt=0(27)B1-λj2djt+λjcjt-bpjt=0Several models, which can comprehensively describe the viscoelastic properties of materials, were developed for polymers [16, 40]. It is found that the standard linear solid (SLS) model is suitable for simulating the viscoelastic property of some adhesives such as epoxy [34]. The SLS model, as shown in Figure 2, consists of a Maxwell Model and a Hookean spring in parallel. Notations η and G0 represent the viscosity coefficient and the relaxation shear modulus andG∞ is the long-term shear modulus. By using this model, the time-dependent shear modulusG(t) in the interlayer is given by (28)Gt=G∞+G0e-t/θwhere θ=η/G0 is the relaxation time. The shear stress τ0(x,t) of the viscoelastic interlayer is further expressed by (29)τ0x,t=G∞+G0hu1x,t-u2x,t+φx,tH-G0hθ∫0te-t-ξ/θu1x,ξ-u2x,ξ+φx,ξHdξFigure 2
Standard linear solid model


### 3.2. Laplace Transformation

By conducting the Laplace transforms to (24)- (27), one has (30)-R1λj2-bhsG-sbhsG-s-bHhsG-s0bhsG-s-R2λj2-bhsG-sbHhsG-s0-bHhsG-sbHhsG-s-D1+D2λj2-B1-bH2hsG-sB1λj00B1λj-B1λj2a-jsb-jsc-jsd-js=000bp-jsin whichs is the Laplace transform variable, G-s=G∞/s+G0/s+1/θ. The solution of (30) is (31)a-js=C4bsG-sp-jsλj3C1sG-s+C2λj2+C3sG-s(32)b-js=-C5bsG-sp-jsλj3C1sG-s+C2λj2+C3sG-s(33)c-js=-bp-jsC6λj2+C7sG-sλj3C1sG-s+C2λj2+C3sG-s(34)d-js=-bp-jsB1λj2-bp-jsC6λj2+C7sG-sλj4C1sG-s+C2λj2+C3sG-swhere(35)C1=R1+R2D1+D2b,C2=R1R2D1+D2h,C3=R1R2H2b,C4=R2Hb,C5=R1Hb,C6=R1R2h,C7=R1+R2b


### 3.3. Determination of Unknowns

Making the inverse of Laplace transforms to (31)-(34) gives (36)ajt=C1j+C2je-C3jt,bjt=-C4j-C5je-C3jt,cjt=-C6j-C7je-C3jt,djt=-C8j-C9j-C10je-C3jtThe analytical expressions of coefficients in the above equations are given in the Appendix.Substituting these coefficients back to (21), (22), (3), and (29), the final expressions of the displacements, rotation angles, longitudinal normal stress of the beam, and the shear stresses in the interlayer can be obtained, respectively.


## 3.1. Formulation of Solution

The boundary conditions of simply supported beams atx=0, l are given as (20)u1′0,t=u1′l,t=0,u2′0,t=u2′l,t=0,φ′0,t=φ′l,t=0,w0,t=wl,t=0Here, we expand the displacements, rotation angles, and applied loads of the beam into Fourier series, as follows:(21)u1x,t=∑j=1Majtcos⁡λjx,u2x,t=∑j=1Mbjtcos⁡λjx(22)φx,t=∑j=1Mcjtcos⁡λjx,wx,t=∑j=1Mdjtsin⁡λjx(23)px=∑j=1Mpjsin⁡λjx,pj=2l∫0lpxsin⁡λjxdxwhere aj(t), bj(t), cj(t), and dj(t) are the unknown Fourier coefficients and λj=jπ/l. By substituting (21)- (23) into (17)- (19), we obtain(24)-R1λj2ajt-bhajt-bjt+Hcjt∗dGt=0(25)-R2λj2bjt+bhajt-bjt+Hcjt∗dGt=0(26)-D1+D2λj2cjt+B1λjdjt-cjt-bHhajt-bjt+Hcjt∗dGt=0(27)B1-λj2djt+λjcjt-bpjt=0Several models, which can comprehensively describe the viscoelastic properties of materials, were developed for polymers [16, 40]. It is found that the standard linear solid (SLS) model is suitable for simulating the viscoelastic property of some adhesives such as epoxy [34]. The SLS model, as shown in Figure 2, consists of a Maxwell Model and a Hookean spring in parallel. Notations η and G0 represent the viscosity coefficient and the relaxation shear modulus andG∞ is the long-term shear modulus. By using this model, the time-dependent shear modulusG(t) in the interlayer is given by (28)Gt=G∞+G0e-t/θwhere θ=η/G0 is the relaxation time. The shear stress τ0(x,t) of the viscoelastic interlayer is further expressed by (29)τ0x,t=G∞+G0hu1x,t-u2x,t+φx,tH-G0hθ∫0te-t-ξ/θu1x,ξ-u2x,ξ+φx,ξHdξFigure 2
Standard linear solid model


## 3.2. Laplace Transformation

By conducting the Laplace transforms to (24)- (27), one has (30)-R1λj2-bhsG-sbhsG-s-bHhsG-s0bhsG-s-R2λj2-bhsG-sbHhsG-s0-bHhsG-sbHhsG-s-D1+D2λj2-B1-bH2hsG-sB1λj00B1λj-B1λj2a-jsb-jsc-jsd-js=000bp-jsin whichs is the Laplace transform variable, G-s=G∞/s+G0/s+1/θ. The solution of (30) is (31)a-js=C4bsG-sp-jsλj3C1sG-s+C2λj2+C3sG-s(32)b-js=-C5bsG-sp-jsλj3C1sG-s+C2λj2+C3sG-s(33)c-js=-bp-jsC6λj2+C7sG-sλj3C1sG-s+C2λj2+C3sG-s(34)d-js=-bp-jsB1λj2-bp-jsC6λj2+C7sG-sλj4C1sG-s+C2λj2+C3sG-swhere(35)C1=R1+R2D1+D2b,C2=R1R2D1+D2h,C3=R1R2H2b,C4=R2Hb,C5=R1Hb,C6=R1R2h,C7=R1+R2b


## 3.3. Determination of Unknowns

Making the inverse of Laplace transforms to (31)-(34) gives (36)ajt=C1j+C2je-C3jt,bjt=-C4j-C5je-C3jt,cjt=-C6j-C7je-C3jt,djt=-C8j-C9j-C10je-C3jtThe analytical expressions of coefficients in the above equations are given in the Appendix.Substituting these coefficients back to (21), (22), (3), and (29), the final expressions of the displacements, rotation angles, longitudinal normal stress of the beam, and the shear stresses in the interlayer can be obtained, respectively.


## 4. Numerical Examples

### 4.1. Convergence Study

Consider a simply supported beam subjected to the uniform loadq=1 N/mm2. The two external layers of the beam are elastic ones and the interlayer is viscoelastic one. The parameters of the beam are h1=h2=50 mm,h=0.1 mm,b=30 mm,l=500 mm, E1=E2=70 Gpa, μ1=μ2=0.3, G0=470.529 Mpa,G∞=0.471 Mpa andθ=1 s. The terms of Fourier series are truncated up to M. The convergences of σxx, τ0,u,w at different points whent=300 s and 10000 s, respectively, are given in Figure 3. It can be seen from Figure 3 that the stresses and displacements rapidly converge to constants with the increase of M. WhenM⩾30, a satisfied accuracy has been obtained. Thus, the terms of Fourier series is fixed atM=30 in the following studies unless stated.Figure 3
The convergences ofσxx, τ0,u,w with respect to the series term M fort=300 s andt=1000 s, respectively.
(a)
t=300 s (b)
t=1000 s


### 4.2. Comparison Study

The present results (TFV) based on the Timoshenko beam theory are compared with those (EFV) based on the EB beam theory proposed by Galuppi and Royer-Carfagni [16] and those (FE) obtained from the finite element solutions by using the software ANSYS. The parameters of the beam are E1=E2=70 Gpa, G0=470.529 Mpa,G∞=0.471 Mpa,θ=1 s, μ1=μ2=0.3, h1=h2=50 mm,h=0.1 mm, andb=30mm, respectively. The load distribution is q(x)=sin⁡(πx/l) N/mm2 and the length l of the beam is variable. In the FE modelling, the viscoelastic property of the interlayer is simulated by VISCO-88 element with four nodes, which can consider the memory effect of viscoelasticity. The PLANE-183 element with four nodes is used for the external layers. Considering the thickness of the interlayer only to be 0.1 mm, the length and width of rectangular elements for interlayer are fixed at 0.5 mm and 0.1 mm, respectively. Considering the external layers are much thicker than the interlayer, the length and width of rectangular elements for external layers are fixed at 0.5 mm and 2.5 mm, respectively. Taking the beam with the length-to-thickness ratio to be 10 as example, 2000 elements for the interlayer and 80000 elements for the external layers should be used in the FE computations. Table 1 gives the comparisons of mid-span deflection wm between the TFV solutions with the EFV and FE ones for different length-to-thickness ratios l/h0, where h0 is the total thickness of the whole beam, h0=h1+h+h2. Three different time values,t=10 s, 100 s, 1000 s, are considered, respectively. It can be seen from Table 1 that there are good agreements between the TFV results and the FE results. The maximum error is less than 0.5641% for all cases. Furthermore, it is seen that the TFV results coincide with the EFV ones well for slender beams; however, the difference significantly increases in the decrease of l/h0, especially at the initial time of applying load. The maximum error of wm reaches 16.806% for l/h0=2.5 when the load is at first applied to the beam. The maximum error of wm reaches 7.075% for l/h0=2.5 when the interlayer is fully relaxed. Figure 4 gives the comparisons of wm between the TFV results with the EFV and FE ones whent=10000 s with respect to length-to-thickness ratio l/h0. It can be seen from Figure 4 that, for the small length-to-thickness ratiosl/h0, the TFV results obviously have a better accuracy than the EFV ones. Tables 2 and 3 give the comparisons of the maximum longitudinal normal stress σxx and the maximum shear stress τ0 between the TFV solutions with the EFV and FE ones for different length-to-thickness ratios l/h0, respectively. It can be seen from Tables 2 and 3 that the results from the TFV solutions are the same as those from the EFV solutions. This can be reasonably explained. For a simply supported layered beam considered here, the shear forces and moments at cross-section of the beam can be directly determined by only considering the force equilibrium. It is well known that both the Euler-Bernoulli beam theory and the Timoshenko beam theory follow the plane cross-section assumption. Therefore, the normal stress distributions on the cross-section for two different beam theories should be identical, if the moments on the cross-section are the same. Moreover, it can be easily deduced that the rotational angles and the longitudinal displacements of the simply supported layered beam are all independent of the shear coefficient, as seen in (31)-(33). Therefore, the shear stresses in the interlayer from the Timoshenko beam theory are also the same as those from the Euler-Bernoulli beam theory. Such a conclusion is valid only for statically determinate beams; however, it is invalid for statically indeterminate beams.Table 1
Comparisons of maximum deflectionwm from the present results (TFV) with the EFV and FE ones for different length-to-thickness ratios l/h0 and time values t.
l / h 0 t TFV EFV FE Error to EFV (%) Error to FE (%) 20 10 s -28.4128 -28.2849 -28.5083 0.450 0.3361 100 s -30.0366 -29.9087 -30.1347 0.426 0.3264 1000 s -39.3301 -39.2022 -39.4403 0.325 0.2802 10 10 s -1.8391 -1.8072 -1.847 1.739 0.4281 100 s -2.2298 -2.1979 -2.2378 1.434 0.3566 1000 s -3.9382 -3.9062 -3.9458 0.812 0.1934 5 10 s -0.1306 -0.1226 -0.1313 6.123 0.5641 100 s -0.2147 -0.2067 -0.2151 3.724 0.1930 1000 s -0.3745 -0.3665 -0.3748 2.134 0.0726 2.5 10 s -0.0119 -0.0099 -0.0118 16.81 0.5059 100 s -0.0241 -0.0221 -0.0239 8.270 0.7887 1000 s -0.0282 -0.0262 -0.0281 7.075 0.2485 Note: Error to EFV=(EFV-TFV)/TFV; Error to FE=(FE-TFV)/TFV.Table 2
Comparisons of maximum longitudinal normal stressσxx from the present results (TFV) with the EFV and FE ones for different length-to-thickness ratios l/h0 and time values t.
l / h 0 t TFV EFV FE Error to EFV (%) Error to FE (%) 20 10 s 243.2934 243.2934 243.9783 0.000 0.2815 100 s 247.9726 247.9726 248.6582 0.000 0.2765 1000 s 274.7517 274.7517 275.4347 0.000 0.2486 10 10 s 61.2769 61.2769 61.5935 0.000 0.5168 100 s 65.7803 65.7803 66.0864 0.000 0.4653 1000 s 85.4707 85.4707 85.7213 0.000 0.2932 5 10 s 15.7629 15.7629 15.9729 0.000 1.3325 100 s 19.6412 19.6412 19.8069 0.000 0.8435 1000 s 27.0106 27.0106 27.1155 0.000 0.3883 2.5 10 s 4.3476 4.3476 4.4776 0.000 2.9907 100 s 6.6032 6.6032 6.651 0.000 0.7242 1000 s 7.3554 7.3554 7.4064 0.000 0.6931 Note: Error to EFV=(EFV-TFV)/TFV; Error to FE=(FE-TFV)/TFV.Table 3
Comparisons of maximum shear stressτ0 in interlayer from the present results (TFV) with the EFV and FE ones for different length-to-thickness ratios l/h0 and time values t.
l / h 0 t TFV EFV FE Error to EFV (%) Error to FE (%) 20 10 s -9.5159 -9.5159 -9.5234 0.000 0.0785 100 s -9.3327 -9.3327 -9.3396 0.000 0.0736 1000 s -8.2843 -8.2843 -8.2879 0.000 0.0439 10 10 s -4.7225 -4.7225 -4.7232 0.000 0.0157 100 s -4.3698 -4.3698 -4.3701 0.000 0.0066 1000 s -2.8280 -2.8280 -2.8267 0.000 0.0447 5 10 s -2.2917 -2.2917 -2.2867 0.000 0.2204 100 s -1.6844 -1.6844 -1.6825 0.000 0.1103 1000 s -0.5302 -0.5302 -0.5307 0.000 0.0869 2.5 10 s -1.0184 -1.0184 -1.009 0.000 0.9266 100 s -0.3119 -0.3119 -0.317 0.000 1.6224 1000 s -0.0763 -0.0763 -0.077 0.000 0.8887 Note: Error to EFV=(EFV-TFV)/TFV; Error to FE=(FE-TFV)/TFV.Figure 4
Comparisons of deflectionwm of the present results (TFV) with the EFV and FE ones for different length-to-thickness ratios l/h0 whent=10000s. Error to EFV=(EFV-TFV)/TFV and Error to FE=(FE-TFV)/TFV.Furthermore, we compare the present results (TFV) with those (TFV (NME)) proposed by Wang et al. [28] and those obtained from the finite element (FE) solution by using the software ANSYS. The TFV (NME) solutions are also based on the FSDT; however, the memory effect of strains in the interlayer is neglected; i.e., the viscoelastic material is approximately modelled as an “aging” material. We consider a simply supported two-layer Timoshenko beam glued by a viscoelastic interlayer and subjected to the uniform loadq=1 N/mm2. The parameters of the beam are h1=h2=50 mm,b=30 mm,l=1000 mm, E1=E2=70 Gpa, μ1=μ2=0.3,G∞=0.5 Mpa, G0=500 Mpa,h=0.5 mm,θ=0.5 s, respectively. Figure 5 shows the comparisons of σxx, τ0,u, and w from three different solutions with respect to time t. It can be seen from Figure 5 that the present results (TFV) are in agreement with the FE ones. However, there are considerable differences between the present results (TFV) and those (TFV(NME)) without considering the memory effect of strains in the interlayer. Therefore, the results ignoring the memory effect of strains in the interlayer cannot accurately describe the varying of stresses and displacements of the beam with the time. The same responses have been obtained and recorded by Galuppi and Royer-Carfagni [16, 36] for beam with external plies modelled as EB beams.Figure 5
Comparisons ofσxx, τ0, u and w from three different solutions with respect to the time t.
(a)
σ x x at x=500 mm, z=100.5 mm (b)
τ 0 at x=0 (c)
u at x=0, z=100.5 mm (d)
w at x=500 mm


### 4.3. Parametric Study

Consider a simply supported beam loaded by the uniform loadq=1 N/mm2, which is composed of two outside elastic layers bonded by a viscoelastic interlayer. The parameters of the beam are h1=h2=50 mm,b=30 mm,l=1000 mm, E1=E2=70 Gpa, μ1=μ2=0.3, respectively.Figure6 shows the distributions of stresses and displacements at the timet=10, 100, and 1000 s, respectively, forh=0.5 mm, G0=500 Mpa,G∞=0.5 Mpa, andθ=0.5 s. It can be seen from Figure 6 that (1) the absolute values of the longitudinal normal stress σxx at the outside surfaces of the beam and the axial displacement u at the ends of the beam increase with time; (2) the transverse shear stress τxz in the external elastic layers is almost no change with the time; (3) the mid-span deflection w increases with time; (4) the maximum value of the shear stress τ0 in the interlayer appears at the ends of the beam and decreases with time.Figure 6
Distributions of stresses and displacements of the laminated beam.
(a)
σ x x at x=500 mm (b)
τ x z (c)
u at x=0 (d)
w (e)
τ 0Figure7 shows the effects of parametersh, G0,G∞, andθ on the mid-span deflection wm with respect to the time. It can be seen from Figure 7 that the mid-span deflection wm increases with time and tends to be constants for all cases. wm increases with the increase of glue thicknessh; however, it decreases with the increase of long-term shear modulusG∞. The relaxation timeθ has no effect on the values of wm at the limited timest=0 andt=∞.Figure 7
Effects of parametersh, G0, G∞, and θ on the mid-span deflection wm with respect to the time t.
(a)
G 0=500 Mpa, G∞=0.5 Mpa, θ=0.5 s (b)
h=0.5 mm, G∞=0.5 Mpa, θ=0.5 s (c)
h=0.5 mm, G0=500 Mpa, θ=0.5 s (d)
h=0.5 mm, G0=500 Mpa, G∞=0.5 MpaFurther consider the layered beam with a viscoelastic interlayer as mentioned above. The thickness of the interlayer ish=0.1 mm. The mid-span deflection wm (atx=500 mm) with respect to the time for different viscoelasticity parameters G0 andG∞ are shown in Figure 8. The monolithic limit, which corresponds to the static solution of G→∞, and the layered limit, which corresponds to the static solution ofG=0, are also plotted for references. Physically, for the layered limit the beam is composed of free-sliding plies and for the monolithic limit the beam is composed of no relatively slipping plies [16, 36]. It is seen from Figure 8 that the deflections of the beam for all cases are between the monolithic limit and the layered limit. Namely, the deflection of the beam with a viscoelastic interlayer is always larger than its monolithic limit, but smaller than its layered limit. Moreover, it can be seen from Figure 8 that G0 has the significant influence on the short-term response of deflection of the layered beam andG∞ has significant influence on the long-term response of deflection of the layered beam.Figure 8
The effect of viscoelasticity parameters of interlayer on the mid-span deflectionwm of the layered beam with a viscoelastic interlayer.


## 4.1. Convergence Study

Consider a simply supported beam subjected to the uniform loadq=1 N/mm2. The two external layers of the beam are elastic ones and the interlayer is viscoelastic one. The parameters of the beam are h1=h2=50 mm,h=0.1 mm,b=30 mm,l=500 mm, E1=E2=70 Gpa, μ1=μ2=0.3, G0=470.529 Mpa,G∞=0.471 Mpa andθ=1 s. The terms of Fourier series are truncated up to M. The convergences of σxx, τ0,u,w at different points whent=300 s and 10000 s, respectively, are given in Figure 3. It can be seen from Figure 3 that the stresses and displacements rapidly converge to constants with the increase of M. WhenM⩾30, a satisfied accuracy has been obtained. Thus, the terms of Fourier series is fixed atM=30 in the following studies unless stated.Figure 3
The convergences ofσxx, τ0,u,w with respect to the series term M fort=300 s andt=1000 s, respectively.
(a)
t=300 s (b)
t=1000 s


## 4.2. Comparison Study

The present results (TFV) based on the Timoshenko beam theory are compared with those (EFV) based on the EB beam theory proposed by Galuppi and Royer-Carfagni [16] and those (FE) obtained from the finite element solutions by using the software ANSYS. The parameters of the beam are E1=E2=70 Gpa, G0=470.529 Mpa,G∞=0.471 Mpa,θ=1 s, μ1=μ2=0.3, h1=h2=50 mm,h=0.1 mm, andb=30mm, respectively. The load distribution is q(x)=sin⁡(πx/l) N/mm2 and the length l of the beam is variable. In the FE modelling, the viscoelastic property of the interlayer is simulated by VISCO-88 element with four nodes, which can consider the memory effect of viscoelasticity. The PLANE-183 element with four nodes is used for the external layers. Considering the thickness of the interlayer only to be 0.1 mm, the length and width of rectangular elements for interlayer are fixed at 0.5 mm and 0.1 mm, respectively. Considering the external layers are much thicker than the interlayer, the length and width of rectangular elements for external layers are fixed at 0.5 mm and 2.5 mm, respectively. Taking the beam with the length-to-thickness ratio to be 10 as example, 2000 elements for the interlayer and 80000 elements for the external layers should be used in the FE computations. Table 1 gives the comparisons of mid-span deflection wm between the TFV solutions with the EFV and FE ones for different length-to-thickness ratios l/h0, where h0 is the total thickness of the whole beam, h0=h1+h+h2. Three different time values,t=10 s, 100 s, 1000 s, are considered, respectively. It can be seen from Table 1 that there are good agreements between the TFV results and the FE results. The maximum error is less than 0.5641% for all cases. Furthermore, it is seen that the TFV results coincide with the EFV ones well for slender beams; however, the difference significantly increases in the decrease of l/h0, especially at the initial time of applying load. The maximum error of wm reaches 16.806% for l/h0=2.5 when the load is at first applied to the beam. The maximum error of wm reaches 7.075% for l/h0=2.5 when the interlayer is fully relaxed. Figure 4 gives the comparisons of wm between the TFV results with the EFV and FE ones whent=10000 s with respect to length-to-thickness ratio l/h0. It can be seen from Figure 4 that, for the small length-to-thickness ratiosl/h0, the TFV results obviously have a better accuracy than the EFV ones. Tables 2 and 3 give the comparisons of the maximum longitudinal normal stress σxx and the maximum shear stress τ0 between the TFV solutions with the EFV and FE ones for different length-to-thickness ratios l/h0, respectively. It can be seen from Tables 2 and 3 that the results from the TFV solutions are the same as those from the EFV solutions. This can be reasonably explained. For a simply supported layered beam considered here, the shear forces and moments at cross-section of the beam can be directly determined by only considering the force equilibrium. It is well known that both the Euler-Bernoulli beam theory and the Timoshenko beam theory follow the plane cross-section assumption. Therefore, the normal stress distributions on the cross-section for two different beam theories should be identical, if the moments on the cross-section are the same. Moreover, it can be easily deduced that the rotational angles and the longitudinal displacements of the simply supported layered beam are all independent of the shear coefficient, as seen in (31)-(33). Therefore, the shear stresses in the interlayer from the Timoshenko beam theory are also the same as those from the Euler-Bernoulli beam theory. Such a conclusion is valid only for statically determinate beams; however, it is invalid for statically indeterminate beams.Table 1
Comparisons of maximum deflectionwm from the present results (TFV) with the EFV and FE ones for different length-to-thickness ratios l/h0 and time values t.
l / h 0 t TFV EFV FE Error to EFV (%) Error to FE (%) 20 10 s -28.4128 -28.2849 -28.5083 0.450 0.3361 100 s -30.0366 -29.9087 -30.1347 0.426 0.3264 1000 s -39.3301 -39.2022 -39.4403 0.325 0.2802 10 10 s -1.8391 -1.8072 -1.847 1.739 0.4281 100 s -2.2298 -2.1979 -2.2378 1.434 0.3566 1000 s -3.9382 -3.9062 -3.9458 0.812 0.1934 5 10 s -0.1306 -0.1226 -0.1313 6.123 0.5641 100 s -0.2147 -0.2067 -0.2151 3.724 0.1930 1000 s -0.3745 -0.3665 -0.3748 2.134 0.0726 2.5 10 s -0.0119 -0.0099 -0.0118 16.81 0.5059 100 s -0.0241 -0.0221 -0.0239 8.270 0.7887 1000 s -0.0282 -0.0262 -0.0281 7.075 0.2485 Note: Error to EFV=(EFV-TFV)/TFV; Error to FE=(FE-TFV)/TFV.Table 2
Comparisons of maximum longitudinal normal stressσxx from the present results (TFV) with the EFV and FE ones for different length-to-thickness ratios l/h0 and time values t.
l / h 0 t TFV EFV FE Error to EFV (%) Error to FE (%) 20 10 s 243.2934 243.2934 243.9783 0.000 0.2815 100 s 247.9726 247.9726 248.6582 0.000 0.2765 1000 s 274.7517 274.7517 275.4347 0.000 0.2486 10 10 s 61.2769 61.2769 61.5935 0.000 0.5168 100 s 65.7803 65.7803 66.0864 0.000 0.4653 1000 s 85.4707 85.4707 85.7213 0.000 0.2932 5 10 s 15.7629 15.7629 15.9729 0.000 1.3325 100 s 19.6412 19.6412 19.8069 0.000 0.8435 1000 s 27.0106 27.0106 27.1155 0.000 0.3883 2.5 10 s 4.3476 4.3476 4.4776 0.000 2.9907 100 s 6.6032 6.6032 6.651 0.000 0.7242 1000 s 7.3554 7.3554 7.4064 0.000 0.6931 Note: Error to EFV=(EFV-TFV)/TFV; Error to FE=(FE-TFV)/TFV.Table 3
Comparisons of maximum shear stressτ0 in interlayer from the present results (TFV) with the EFV and FE ones for different length-to-thickness ratios l/h0 and time values t.
l / h 0 t TFV EFV FE Error to EFV (%) Error to FE (%) 20 10 s -9.5159 -9.5159 -9.5234 0.000 0.0785 100 s -9.3327 -9.3327 -9.3396 0.000 0.0736 1000 s -8.2843 -8.2843 -8.2879 0.000 0.0439 10 10 s -4.7225 -4.7225 -4.7232 0.000 0.0157 100 s -4.3698 -4.3698 -4.3701 0.000 0.0066 1000 s -2.8280 -2.8280 -2.8267 0.000 0.0447 5 10 s -2.2917 -2.2917 -2.2867 0.000 0.2204 100 s -1.6844 -1.6844 -1.6825 0.000 0.1103 1000 s -0.5302 -0.5302 -0.5307 0.000 0.0869 2.5 10 s -1.0184 -1.0184 -1.009 0.000 0.9266 100 s -0.3119 -0.3119 -0.317 0.000 1.6224 1000 s -0.0763 -0.0763 -0.077 0.000 0.8887 Note: Error to EFV=(EFV-TFV)/TFV; Error to FE=(FE-TFV)/TFV.Figure 4
Comparisons of deflectionwm of the present results (TFV) with the EFV and FE ones for different length-to-thickness ratios l/h0 whent=10000s. Error to EFV=(EFV-TFV)/TFV and Error to FE=(FE-TFV)/TFV.Furthermore, we compare the present results (TFV) with those (TFV (NME)) proposed by Wang et al. [28] and those obtained from the finite element (FE) solution by using the software ANSYS. The TFV (NME) solutions are also based on the FSDT; however, the memory effect of strains in the interlayer is neglected; i.e., the viscoelastic material is approximately modelled as an “aging” material. We consider a simply supported two-layer Timoshenko beam glued by a viscoelastic interlayer and subjected to the uniform loadq=1 N/mm2. The parameters of the beam are h1=h2=50 mm,b=30 mm,l=1000 mm, E1=E2=70 Gpa, μ1=μ2=0.3,G∞=0.5 Mpa, G0=500 Mpa,h=0.5 mm,θ=0.5 s, respectively. Figure 5 shows the comparisons of σxx, τ0,u, and w from three different solutions with respect to time t. It can be seen from Figure 5 that the present results (TFV) are in agreement with the FE ones. However, there are considerable differences between the present results (TFV) and those (TFV(NME)) without considering the memory effect of strains in the interlayer. Therefore, the results ignoring the memory effect of strains in the interlayer cannot accurately describe the varying of stresses and displacements of the beam with the time. The same responses have been obtained and recorded by Galuppi and Royer-Carfagni [16, 36] for beam with external plies modelled as EB beams.Figure 5
Comparisons ofσxx, τ0, u and w from three different solutions with respect to the time t.
(a)
σ x x at x=500 mm, z=100.5 mm (b)
τ 0 at x=0 (c)
u at x=0, z=100.5 mm (d)
w at x=500 mm


## 4.3. Parametric Study

Consider a simply supported beam loaded by the uniform loadq=1 N/mm2, which is composed of two outside elastic layers bonded by a viscoelastic interlayer. The parameters of the beam are h1=h2=50 mm,b=30 mm,l=1000 mm, E1=E2=70 Gpa, μ1=μ2=0.3, respectively.Figure6 shows the distributions of stresses and displacements at the timet=10, 100, and 1000 s, respectively, forh=0.5 mm, G0=500 Mpa,G∞=0.5 Mpa, andθ=0.5 s. It can be seen from Figure 6 that (1) the absolute values of the longitudinal normal stress σxx at the outside surfaces of the beam and the axial displacement u at the ends of the beam increase with time; (2) the transverse shear stress τxz in the external elastic layers is almost no change with the time; (3) the mid-span deflection w increases with time; (4) the maximum value of the shear stress τ0 in the interlayer appears at the ends of the beam and decreases with time.Figure 6
Distributions of stresses and displacements of the laminated beam.
(a)
σ x x at x=500 mm (b)
τ x z (c)
u at x=0 (d)
w (e)
τ 0Figure7 shows the effects of parametersh, G0,G∞, andθ on the mid-span deflection wm with respect to the time. It can be seen from Figure 7 that the mid-span deflection wm increases with time and tends to be constants for all cases. wm increases with the increase of glue thicknessh; however, it decreases with the increase of long-term shear modulusG∞. The relaxation timeθ has no effect on the values of wm at the limited timest=0 andt=∞.Figure 7
Effects of parametersh, G0, G∞, and θ on the mid-span deflection wm with respect to the time t.
(a)
G 0=500 Mpa, G∞=0.5 Mpa, θ=0.5 s (b)
h=0.5 mm, G∞=0.5 Mpa, θ=0.5 s (c)
h=0.5 mm, G0=500 Mpa, θ=0.5 s (d)
h=0.5 mm, G0=500 Mpa, G∞=0.5 MpaFurther consider the layered beam with a viscoelastic interlayer as mentioned above. The thickness of the interlayer ish=0.1 mm. The mid-span deflection wm (atx=500 mm) with respect to the time for different viscoelasticity parameters G0 andG∞ are shown in Figure 8. The monolithic limit, which corresponds to the static solution of G→∞, and the layered limit, which corresponds to the static solution ofG=0, are also plotted for references. Physically, for the layered limit the beam is composed of free-sliding plies and for the monolithic limit the beam is composed of no relatively slipping plies [16, 36]. It is seen from Figure 8 that the deflections of the beam for all cases are between the monolithic limit and the layered limit. Namely, the deflection of the beam with a viscoelastic interlayer is always larger than its monolithic limit, but smaller than its layered limit. Moreover, it can be seen from Figure 8 that G0 has the significant influence on the short-term response of deflection of the layered beam andG∞ has significant influence on the long-term response of deflection of the layered beam.Figure 8
The effect of viscoelasticity parameters of interlayer on the mid-span deflectionwm of the layered beam with a viscoelastic interlayer.


## 5. Conclusion Remarks

According to the FSDT, this paper studies the stresses and deformations of simply supported two layers of Timoshenko beams glued with a viscoelastic interlayer and subjected to arbitrary loads. The viscoelasticity of the interlayer is described by the standard linear solid model and the memory effect of strains in the interlayer is considered. Taking the sinusoidal load and uniform load for example, the effects of external layer thickness and the interlayer material properties on deflection of the laminated beam are checked in detail. The results show that there are good agreements between the present results and those obtained with the Euler-Bernoulli beam theory when the length-to-thickness ratio is large; however, the errors between the two results increase in the decrease of length-to-thickness ratio. Moreover, the error of deflection from the Euler-Bernoulli beam theory at the initial time of applying load is significantly larger than that when the interlayer is fully relaxed. Therefore, the present results have a better accuracy than those in terms of the Euler-Bernoulli beam theory. The effect of interlayer material property on the deflection of the laminated beam is similar to those obtained from the Euler-Bernoulli beam theory [16, 36].



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*Source: 7620816-2019-04-18.xml*