Vibration of laminated sandwich beams having soft core

Abstract

Free vibration response of laminated sandwich beams having a soft core is studied by using a recently developed C₀ finite element beam model. The model has been developed based on higher order zigzag theory where the in-plane displacement variation is considered to be cubic for both the face sheets and the core. The transverse displacement is assumed to be quadratic within the core while it remains constant in the faces beyond the core. The model satisfies the condition of transverse shear stress continuity at the layer interfaces and the zero transverse shear stress condition at the top and bottom of the beam. The nodal field variables are chosen in an efficient manner to overcome the problem of continuity requirement of the derivatives of transverse displacements. Numerical examples on free vibration covering different features of laminated composite and sandwich beams are presented. Many new results are also presented which should be useful for future research.

Keywords

Finite element free vibration sandwich beam soft core zigzag theory

1. Introduction

Laminated composite materials are popular as a structural material due to their superior properties such as high strength/stiffness to weight ratio and greater resistance to environmental degradation over conventional metallic materials. A typical laminated structure of great use in various weight sensitive engineering applicationssuch as marine, aerospace and civil engineering constructions is the popular sandwich construction. Sandwich is a laminated structure with low strength core and high strength face sheets in the form of composite laminates. As the sandwich structure possesses high internal damping, it has better energy absorption capacity under dynamic conditions. Laminated composite (e.g., GFRP, CFRP etc.) structures are weak in shear due to their low shear modulus compared to extensional rigidity. Thus the effect of shear deformation is quite significant and it becomes more complex in the case of sandwich construction, as the material property variation is very large between the core and the face layers.

For accurate analysis of composite laminates, a number of models have been proposed based on the single layer theory, layer-wise theory and zigzag theory. The single layer theory is also known as Reissner-Mindlin's plate theory or first order shear deformation theory (FSDT), where the deformation of plate is expressed in terms of unknown parameters of the reference plane usually taken as the middle plane of thestructure. The transverse shear strain is assumed to be uniform over the entire plate thickness (i.e., the transverse shear stress is constant). Nabi and Ganesan (1994) have developed a two node finite element (FE) beam model based on FSDT considering bi-axial bending, extensional and torsional effects as well as including shear deformation for the free vibration analysis of composite beam with arbitrary boundary conditions. Yildirim (2000) has formulated a vibration problem of laminated beams based on the stiffness matrix approachusing FSDT and investigated the effect of the longitudinal to transverse moduli ratio on natural frequencies. For free vibration and wave propagation problems a theory was presented by Chakraborty et al. (2002) based on FSDT, which shows that the coupling between axial shear and flexure has a pronounced effect on the axial and shear wave speeds, where the bending speeds are marginally affected. Goyal and Kapania (2007) developed a five node beam FE model of twenty four degrees-of-freedom (d.o.f.), based on FSDT for static as well as dynamic analysis.

This theory (FSDT) requires a shear correction factor to predict the actual parabolic variation of shear stress. To overcome the shortcoming of FSDT, higher order shear deformation theories (HSDT) are developed. In HSDT, the higher order variation of in-plane displacement through the thickness is considered to represent the actual warping of the plate cross-section so it (HSDT) becomes free from the shear correction factor. The warping of the cross section also allows higher order variation of transverse shear stresses/strains across the depth (Reddy, 1984). The theories developed for vibration analysis by Chandrashekhara and Bangera (1992, 1993), Heylinger and Reddy (1988), Kant and Gupta (1988), Khedir and Reddy (1994), Marur and Kant (1997) and many others fall under this category. Shi and Lam (1999) presented an FE formulation based on higher order beam theory (HOBT) to signify the effect of higher order element and coupling mass matrices on the frequencies of high mode flexural vibration of beam.

An analytical solution for a free vibration problem based on HSDT was presented by Kant and Swaminathan (2001), where the solutions were obtained in closed form using Navier technique. Matsunaga (2001) developed one dimensional global higher order theory, in which the fundamental equations were derived based on the power series expansions of continuous displacement components to analyze vibration and buckling problems. Karama et.al (2003) presented a multilayered laminated composite model by introducing an exponential function as a shear stress function where an FE based software package Abaqus was used to check the efficiency of the model. Murthy et al. (2005) presented an FE model based on third order shear deformation theory (i.e., HSDT) for the analysis of asymmetrically staked laminated beams with different boundary conditions. For the vibration analysis of the beam, four shear deformation theories were presented by Aydogdu (2005) with different boundary conditions. By choosing appropriate shape functions, the theories developed are named as: parabolic shear deformation beam theory, hyperbolic shear deformation beam theory, first order shear deformation beam theory and exponential shear deformation beam theory. Based on HSDT, Subramanian (2006) had presented two theories using a two node C₁ continuous beam FE model with eight d.o.f. per node for dynamic analysis of symmetrical laminated composite beams. The variation of the transverse shear stress across the depth was assumed as non-parabolic in the first theory while it was parabolic in the second. The in-plane displacement and the out of plane displacement were assumed to be quintic and quartic respectively. The displacement fields also satisfied the stress free conditions at the top and bottom surfaces of the beam. Rahmani et al. (2009) investigated the vibration analysis of a sandwich beam with a flexible core material based on higher order sandwich panel plate theory. A model based on higher order theory has been proposed by Arvin et al. (2010) for the free and forced vibration analysis of a sandwich beam by considering transverse displacement as independent for both the faces and linear through the depth of the beam core. The effects of fiber angle, geometrical changes and stiffness variation between faces and core have been investigated.

At the layer interfaces HSDT shows discontinuity in the transverse shear stress distribution, while variations of the corresponding transverse shear strain are continuous across the thickness. This is mainly due to the different values of shear rigidity at the adjacent layers. But the actual behavior of a composite laminate is exactly opposite i.e., the transverse strain may be discontinuous but the corresponding shear stress must be continuous at the layer interface (Sheikh and Chakrabarti, 2003).

The above disparity leads to the development of layer-wise theories, which started with discrete layer theories. In discrete layer theories, the unknown displacement components are taken at all the layer interfaces including the top and bottom surfaces of the structure. Discrete layer theories proposed by Ambarstsumian (1974), Lu and Liu (1992), Robbins and Reddy (1996), Srinivas (1973), Toledeno and Murakami (1987), and many others assume a unique displacement field in each layer and displacement continuity across the layers. In these theories, the number of unknowns increases directly with the increase in the number of layers due to which huge computational involvement was required.

The above problem has been subsequently solved in zigzag theories (ZZT), where the unknowns at different interfaces are defined in terms of those at the reference plane. The number of unknowns are made independent of the number of layers by introducing the transverse shear stress continuity condition at the layer interfaces of the laminate and the in-plane displacements have piece-wise variation across the plate thickness in this theory (ZZT). Averill (1994) developed a C₀ FE based on first order ZZT by overcoming the C₁ continuity requirement using the concepts of independent interpolations and penalty functions. Hermitian functions were used by Di Sciuva (1985, 1993) to approximate the transverse displacement in the formulations. Carrera (1996) used two different fields along thelaminate thickness direction for displacement and transverse shear stress respectively for his formulation. Averill and Yip (1996a) developed a C₀ FE based on cubic ZZT, using interdependent interpolations for transverse displacement and rotations and penalty function concepts.

In some improved version of these theories, the condition of zero transverse shear stresses at the plate/beam top and bottom was also satisfied. The theories developed by Cho and Averill (1997), Cho and Parmerter (1993), Di Sciuva (1987), Kapuria et al. (2003), Lee et al. (1990), Murakami (1986), and many others fall under this category. Kapuria et al. (2003) developed 1D ZZT for the analysis of simply supported beams. The effect of the laminate layup and the thickness to span ratio was investigated. Carrera (2003) presented a historical review on the zigzag theories used for the analysis of the multilayered laminate plates and shells, in which the three basic theories have been discussed namely: Lekhnitskii Multilayered Theory (LMT), Ambartsumian Multilayered Theory (AMT) and Reissner Multilayered Theory (RMT). Tahani (2007) presented the analytical solution for laminated beams by using two theories based on layer wise displacement fields. Zigzag models for laminated composite beams were developed by using trigonometric terms to represent the linear displacement field, transverse shear strains and stresses (Shimpi and Ainapure, 2001; Shimpi and Ghugal, 2001). These theories (Zigzag) provide a very accurate approximation of the structural behavior even for lower span to thickness ratio. However, the ZZT has a problem in its FE implementation as it requires C₁ continuity of the transverse displacement at the nodes.

To combine the benefits of the discrete layer wise and higher order zigzag theories (HOZT), Averill and Yip (1996b), Icardi (2001, 2003), and many other authors developed theories which are known as sub-laminated models. Cho and Averill (2000) presented an improved sub-laminate model with first order zigzag approximation of displacement within each sub-laminate, which contains an eight node C₀ FE having five displacement d.o.f. in each node for each sub-laminate. Aitharaju and Averill (1999) developed a new C₀ FE based on a quadratic zigzag layer-wise theory. For eliminating the shear locking phenomenon, the shear strain field is also made field consistent.The transverse normal stress was assumed to be constant through the thickness of the laminate. The new FE was applied to model the beam as a combination of different sub-laminates.

A C₀ plate model based on enhanced first order theory (EFSDT) was presented by Kim and Cho (2005, 2006), where it was shown that the displacements, in-plain strains and stresses can be approximated to those of the three dimensional theory or higher order theory, in the least square sense. Recently the authors (Kim and Cho, 2007), have developed a C₀ model using the EFSDT based on a mixed variational theorem, which also satisfy the lateral conditions at the top and bottom surfaces of the plate. For an accurate analysis of the stresses, Bambole and Desai (2007), Rao et al. (2001) and many others assumed stress components as d.o.f. at interface nodes along with displacement d.o.f., which are called a mixed FE approach. Ramtekkar and Desai, (2002) have developed a six node plane stress mixed beam FE model for the free vibration analysis of the composite laminated beams. Along with the mode shapes, the variation of displacements and stresses through the thickness of the beam are also shown under the various modes of vibration.

Recently, Zhen and Wanji (2008) presented global theories with higher order shear deformation and ZZTs satisfying the continuity of transverse shear stresses at interfaces for analyzing the global response sandwich laminated beams. The effects of the order number of higher order shear deformation as well as inter-laminar continuity of shear stress on the global response of laminated beams and soft core sandwich beams are studied.

In the case of a laminated sandwich structure having a soft core the transverse deformation is very significant as there is an abrupt change in the values of transverse shear rigidity and thickness of the face sheet and core. As such the variation of transverse deflection across the depth of a sandwich structure having a soft core has to be calculated with sufficient accuracy. Givil et al. (2007) has presented the dynamic model based on the higher order sandwich panel theory to study the behavior of a soft core sandwich panel under dynamic loading. Recently, Vidal and Polit (2010) presented five different sinus models based on ZZT for the free vibration analysis of laminated sandwich beams. A three node FE model with 24 d.o.f. was developed for the analysis. The results obtained were verified with the results generated by using Ansys software package.

In a laminated sandwich structure, unknown transverse displacement fields across the depth, in addition to that in the reference plane, are essential to represent the variation of transverse deflection. This can be done by using sub-laminate plate theories but the number of unknowns will increase with an increase in the number of sub-laminates. On the other hand, introduction of additional unknowns in the transverse displacement fields invites additional C₁ continuity requirements in its FE implementation by using the ZZT as mentioned earlier. However, the application of a C₁ continuous FE is not encouraged in a practical analysis.

From the above, it can be concluded that though many works have been reported on the dynamic analysis of laminated beams very few were on a soft core sandwich beam using ZZT as compared to a laminated composite beam. For the static and buckling analysis of sandwich plates with soft compressible core, Pandit et al. (2008a,b) proposed a HOZT. Recently Pandit et.al (2010) have presented an improved HOZT for vibration of soft core sandwich plates with random material properties. The core is modeled as a 3-D elastic continuum. To overcome the problem of C₁ continuity, the authors (Pandit et al., 2008a,b, 2010) have used separate field variables to define the derivatives of transverse displacements in C₀ FE formulation for the implementation of the theory. Due to this, some constrains have been imposed which are enforced variationally through a penalty approach. However, choosing a suitable value for the penalty stiffness multiplier is a well known problem in the FE method.

In this paper, free vibration response of a laminated sandwich beam having a soft compressible core has been investigated using an improved C₀ beam FE model recently developed by the authors (Chakrabarti et al., 2011) based on HOZT. In this FE model, the in-plane displacement field is assumed as a combination of a linear zigzag function with different slopes at each layer and a cubically varying function over the entire thickness. The out of plane displacement is considered to be quadratic within the core and constant in the face sheets. The model satisfies the transverse shear stress continuity conditions at the layer interfaces and the conditions of zero transverse shear stress at the top and bottom of the beam. The isoparametric quadratic beam element has three nodes with seven displacement d.o.f. (i.e., in-plane displacements and transverse displacements at the reference mid surface, at the top and at the bottom of the beam along with rotational d.o.f. only at the reference mid surface) at each node. The displacement fields are chosen in such a manner that there is no need to impose any penalty stiffness in the formulation. The element may also be matched quite conveniently with other C₀ elements. A two dimensional higher order plane FE model can also solve the sandwich laminated beam problem. However, in that

case one should use a refined theory such as ZZT to accurately predict the in-plane deformations and transverse shear deformations. Also the core deformability in a soft core sandwich structure should also be modeled properly by taking additional terms in the transverse displacement component. In the proposed beam model developed by the authors, all those effects are considered very efficiently by using a one dimensional beam model only and the results are observed to be quite accurate compared to the three dimensional elasticity solutions. Thus the proposed beam model is more efficient since it uses lesser nodal unknowns and hence lesser computational time to predict results as accurately as any two dimensional or three dimensional models.

It is observed that there is a scarcity of suitable results in the published literature on dynamic response of laminated sandwich beams having soft cores. So in this paper, the present FE beam model is applied to solve free vibration problems of laminated sandwich beams having a soft core for different geometry and boundary conditions. The vibration results obtained for different problems indicate that the present FE model is capable of analyzing laminated sandwich structures having a soft compressible core very accurately with a lesser number of d.o.f. per node when compared with two/three dimensional models.

2. Mathematical formulations

The in-plane displacement field (Figure 1) is chosen as follows:

(1)

where, u₀ denotes the in-plane displacement of any point on the mid surface, θ_x is the rotation of the normal to the middle plane about the z-axis, n_u and n_l are number of upper and lower layers respectively, β_x and η_x are the higher order unknown,

α_{xu}^{i}

and

α_{xl}^{i}

are the slopes of i-th layer corresponding to upper and lower layers respectively and

H (z - z_{i}^{u})

and

H (- z + z_{i}^{l})

are the unit step functions.

Figure 1.

General lamination scheme and displacement configuration.

The transverse displacement is assumed to vary quadratically through the core thickness and constant over the face sheets (as shown in Figure 2) and it may be expressed as,

(2)

where w_u, w₀ and w_l are the values of the transverse displacement at the top layer, middle layer and bottom layer of the core, respectively, and l₁, l₂ and l₃ are Lagrangian interpolation functions in the thickness co-ordinate.

Figure 2.

Variation of transverse displacement (w) through the thickness of laminated sandwich beam.

The stress-strain relationship based on a plane strain condition of an orthotropic layer/lamina (say k-th layer) having any fiber orientation with respect to structural axes system (x-y) may be expressed as

(3)

where

{\bar{σ}}

{\bar{ɛ}}

and

[\bar{Q_{K}}]

are the stress vector, the strain vector and the transformed rigidity matrix of k-th lamina, respectively.

Utilizing the conditions of zero transverse shear stress at the top and bottom surfaces of the beam and imposing the conditions of the transverse shear stress continuity atthe interfaces between the layers along with the conditions, u = u_u at the top and u = u_l at the bottom of the beam, β_x, η_x, $α_{xu}^{i}$ , $α_{xl}^{i}$ , $(\partial w_{u} / \partial x)$ and $(\partial w_{l} / \partial x)$ may be expressed in terms of the displacement u₀_,θ_x,u_uand u_l as

{B} = [A] {α}

(4)

where,

{\begin{matrix} {B} = {β_{x} η_{x} α_{xu}^{1} α_{xu}^{2} \dots α_{xu}^{nu - 1} α_{xl}^{1} α_{xl}^{2} \dots α_{xl}^{nl - 1} \\ (\partial w_{u} / \partial x) (\partial w_{l} / \partial x) \end{matrix}}}^{T}, {α} = {u_{0} θ_{x} u_{u} u_{l}}^{T}

and the elements of [A] are dependent on material properties. It is to be noted that the last two entries of the vector {B} help to define the derivatives of transverse displacement at the top and bottom faces of the beam in terms of the displacements u₀_, θ_x, u_u and u_l to overcome the problem of C₁ continuity as mentioned before.

Using the above equations, the in-plane displacement field as given in equation (1) may be expressed as

U = b_{1} u_{0} + b_{2} θ_{x} + b_{3} u_{u} + b_{4} u_{l}

(5)

Where, the coefficients b_i's are function of thickness coordinate unit step functions and material properties.

The generalized displacement vector {δ} for the present beam model can now be written with the help of equations (2) and (5) as

{δ} = {u_{0} w_{0} θ_{x} u_{u} w_{u} u_{l} w_{l}}^{T}

(6)

Using linear strain-displacement relation and equations (1) to (4), the strain field may be expressed in terms of unknowns (for the structural deformation) as

{\bar{ɛ}} = [\frac{\partial U}{\partial x} \frac{\partial W}{\partial z} \frac{\partial U}{\partial z} + \frac{\partial W}{\partial x}] or {\bar{ɛ}} = [H] {ɛ}

(7)

where,

and the elements of [H] are functions of z and unit step functions. The values of l₁, l₂, l₃, b_i's and the elements of [H] are used as reported in the appendix (A,B and C) by Chakrabarti et al.(2011).

In the present problem, a recently developed three node FE (Chakrabarti et al., 2011) with seven d.o.f. $(u_{0}, w_{0}, θ_{x}, u_{u}, w_{u}, u_{l} and w_{l})$ per node is employed. Using the FE method the generalized displacement vector {δ} at any point may be expressed as

{δ} = \sum_{i = 1}^{n} N_{i} δ_{i}

(8)

where,

{δ} = {u_{0} w_{0} θ_{x} u_{u} w_{u} u_{l} w_{l}}^{T}

as defined earlier, δ_i isthe displacement vector corresponding to node i, N_i is the shape function associated with the node i and n is the number of nodes per element, which is three in the present study.

With the help of equation (8), the strain vector ${ɛ}$ that appeared in equation (7) may be expressed in terms of freedom as

{ɛ} = {B} {δ}

(9)

where {B} is the strain-displacement matrix in the Cartesian coordinate system.

Using Hamilton's principle, the equation of motion is given as

(10)

where the kinetic energy (T)

T = \frac{1}{2} ∫ρ ({\overset{\cdot}{u}}^{2} + {\overset{\cdot}{w}}^{2}) dv

(11)

\overset{\cdot}{u}

and

\overset{\cdot}{w}

being time derivatives of u and w, ρ being the density of the material and the potential energy

(12)

In the above equation, the work done by external forces is neglected and the damping is not considered. The Hamilton's principle leads to the equilibrium equation of a system

[M] {\partial^{2} \bar{δ} / \partial t^{2}} + [K] {\bar{δ}} = 0

(13)

where [M], [K], {δ} and

{\partial^{2} \bar{δ} / \partial t^{2}}

are, respectively, theglobal mass matrix, stiffness matrix, nodal variablevector and acceleration vector of the system. Consequently, the frequencyλ can be evaluated by using following equation,

[K] {\bar{δ}} = λ^{2} [M] {\bar{δ}}

(14)

Now the element stiffness matrix can be derived following by the technique of energy minimization inusual manner as derived by Chakrabarti et al. (2011).

In the similar manner the consistent mass matrix can be derived as that of stiffness matrix. The acceleration at any point within the beam may be expressed as

{\partial^{2} \bar{δ} / \partial t^{2}} = {\begin{matrix} \partial^{2} \bar{u} / \partial t^{2} \partial^{2} \bar{w} / \partial t^{2} \end{matrix}} = - λ^{2} {\bar{} u \bar{w}} = - λ^{2} [Δ] {δ}

(15)

where matrix [Δ] of order 2 × 7 contains z terms and some constant quantities as that of [H].

(16)

where ρ_i is the mass density of the i-th layer and the matrix [L] is

[L] = \sum_{i = 1}^{n_{u} + n_{1}} {{∫ρ}_{i} [Δ]}^{T} [Δ] dz

(17)

A numerical code is developed to implement the abovementioned operations involved in the proposed FE model to calculate displacements and vibration frequencies in laminated sandwich beams. The skyline technique has been used to store the global stiffness matrix in a single array and Gaussian decomposition scheme is adopted for the static solution, while simultaneous iteration technique of Corr and Jennings (1976) is used in free vibration analysis.

3. Numerical results

In this section, the accuracy and applicability of the recently developed FE model based on HOZT are demonstrated by solving a number of numerical examples on free vibration analysis of laminated sandwich beams with a wide range of features. The results obtained are compared with published results in some cases and finally many new results are also generated. The following different boundary conditions are used

Simply supported boundary condition: The d.o.f. u₀, w₀, w_u and w_l are restrained while θ_x, u_u and u_l are unrestrained in one boundary. In the other boundary, the d.o.f. w₀, w_u and w_l are restrained while u₀, θ_x, u_u and u_l are unrestrained.

Clamped boundary condition: All the nodal d.o.f. at the boundary are fully restrained.

Free boundary condition: All the nodal d.o.f. at the boundary are unrestrained.

The non-dimensional quantities used, to show different results are as follows

Natural   frequency, \bar{λ} = \frac{λ l^{2}}{h} \sqrt{/ ρ E_{Tf} E_{Tf}}

where, ρ and E_Tf are the density and transverse modulus of elasticity of face layer respectively.

3.1. Three Layered Laminated Sandwich Beam (0°/90°/0°)

For the convergence studies of fundamental natural frequency, a three layer beam is analyzed taking mesh divisions 2, 5, 10, 16, 20. The thickness of all layers is considered to be same. The material properties used in this example are as shown in Table 1 (Example 1). Taking different values of thickness ratio (l/h), the results obtained by using the present FE model are presented in Table 2. It may be observed from Table 2 that the frequencies are converged at mesh division 20. Amesh division of 20 is taken for all subsequent analysis to get accurate results. The present results corresponding to arbitrary boundary conditions are compared withthose published; FSDT (Yildirim, 2000), TSDT (Khedir and Reddy, 1994), HSDT (Murthy et al., 2005) and Sinus Model (Vidal and Polit, 2010). It may be observed in Table 2 that the present results are in good agreement to the other results with a fewer number of elemental d.o.f.

Table 1.

Material properties for different sandwich beams

Example	Layer/sheet	Material properties				Mass density
Example	Layer/sheet	E₁(psi)	E₃(psi)	G₁₃(psi)	υ ₁₃	(ρ) (Ib s²in⁻⁴)
1	Face	40.0E06	1.0E06	0.6E06	0.25	100
	Core	1.0E06	1.0E06	0.5E06	0.25	100
2	Face: 1^*	19E06	1.0E06	447.845E03	0.32	1000
	Core	32.022	4.0E05	79.055E03	3.0E-5	70
3	Face: 1	19E06	1.5E06	0.9E06	0.22	1627
	Face: 2	1.5E06	1.5E06	1.0E06	0.49	1627
	Core	1.0E03	1.0E03	0.5E03	0.00	67

Face number increases from the outer layers towards inside.

Table 2.

Non-dimensional natural frequencies of a sandwich beam for different boundary conditions and thickness ratio

l/h	Reference	H-H	C-H	C-C	C-F
5	Present (2)¹	9.2738	9.9884	11.0223	4.1731
	Present (5)	9.2101	9.8717	10.8378	4.1616
	Present (10)	9.2082	9.8679	10.8318	4.1612
	Present (16)	9.2081	9.8676	10.8314	4.1611
	Present (20)	9.2081	9.8676	10.8314	4.1611
	Yildirim (2000)	–	9.6500	10.4300	4.1300
	Khedir and Reddy (1994)	9.2080	10.2390	11.6030	4.2310
	Murthy et al. (2005)	9.2070	10.2380	11.6020	4.2300
	Aydogdu (2005)	9.2070	10.2360	11.6370	4.2340
	Vidal and Polit (2010)	9.2010	10.0000	11.1000	4.1890
10	Present	13.6117	16.3914	19.2545	5.4754
	Khedir and Reddy (1994)	13.6110	16.6000	19.7190	5.4910
	Murthy et al. (2005)	13.6140	16.5990	19.7120	5.4950
20	Present	16.3360	22.7797	29.4593	6.0640
	Aydogdu (2005)	16.3370	22.9070	29.9260	6.0700

Quantities within the parentheses indicate mesh size.

3.2. Three layer laminated sandwich beam

A three layer simply supported sandwich beam is analyzed in this example using the present FE model. The core has a thickness of 0.8 h while it is 0.1 h each for the two laminated faces, where h (=1 inch) is the overall thickness of the beam. The material properties for the face sheets and core are given in Table 1 (Example 2). In Table 3, the non dimensional natural frequencies for the first five modes are reported for different thickness ratio (l/h). Results reported by Kapuria et.al. (2004) and Vidal and Polit (2010) based on the sinus model and zigzag model respectively along with the results from ANSYS software package (Vidal and Polit, 2010) are considered for the comparison. For the first three modes of l/h = 2 frequencies are identical. From Table 3, it can be concluded that the present model performance is good with fewer d.o.f. as compared to other models. As in the case of ANSYS the 2D model has 1275 d.o.f. while the present one has 287 d.o.f.

Table 3.

Non-dimensional natural frequencies of a three layered sandwich beam for different thickness ratio

l/h	Reference	1	2	3	4	5
2	Present	3.53	5.34	7.58	12.01	16.61
	Vidal and Polit (2010)	3.53	5.34	7.60	15.00	16.45
	Ansys	3.53	5.34	7.58	12.09	16.53
5	Present	7.83	17.29	26.92	33.40	36.96
	Vidal and Polit (2010)	7.83	17.31	26.97	33.37	37.06
	Ansys(2010)	7.82	17.28	26.93	33.40	37.01
	Kapuria et al. (2004)	7.82	17.27	26.90	–	36.93
10	Present	12.26	31.33	50.28	69.18	88.28
	Vidal and Polit (2010)	12.26	31.33	50.31	69.26	88.51
	Ansys (2010)	12.23	31.33	50.31	69.26	88.51
	Kapuria et al.(2004)	12.23	31.29	50.21	68.09	88.18
20	Present	15.41	49.04	87.06	163.43	201.32
	Vidal and Polit (2010)	15.41	49.04	87.07	163.69	201.94
	Ansys (2010)	15.38	48.98	87.01	163.58	201.64
	Kapuria et al. (2004)	15.38	48.94	86.90	163.12	200.87

3.3. Multilayered sandwich beam

A multilayered laminated sandwich beam (0°/90°/0°/90°/C/90°/0°/90°/0°) is analyzed in this example, which is a new problem. The ratio i.e. thickness of core to thickness of face sheets (t_c/t_f₎ is considered to be five. Material properties are as shown in Table 1 (Example 3). The values of the first six non-dimensional natural frequencies are presented in Table 4 for a different thickness ratio (l/h) ranging from 10 to 100 for different boundary conditions. To check the applicability of present FE model to this problem of a multilayered sandwich beam, the present results are compared with those obtained by using the FE software package Abaqus (Ver.6.8) in some cases. The mode shapes forthe first three free vibration modes of a beam witha thickness ratio (l/h) equal to 10 are shown in Figure 3 with different geometry boundary conditions i.e., Hinge-Hinge (H-H), Clamp-Clamp (C-C), Clamp-Hinge (C-H) and Clamp-Free (C-F).

Figure 3.

First three modes shapes (I, II & III) for laminated sandwich beams for l/h = 10. (a) Hinged-Hinged boundary; (b) Hinged-Clamped boundary; (c) Clamped-Clamped boundary.

Table 4.

Non-dimensional natural frequencies of a thick laminated sandwich beam for different boundary conditions and thickness ratio (t_c / t _f = 5)

l/h	B.C	1	2	3	4	5	6
10	H-H	1.3546	3.2554	5.9945	9.3489	9.6996	13.2680
	C-H	1.5562	3.6987	6.7046	10.6813	13.2738	13.6325
	C-C	1.8122	4.2098	7.4868	11.7332	13.2957	13.7531
20	H-H	2.5095	5.4205	8.8716	13.0789	18.1953	24.3366
	C-H	2.7011	5.8123	9.5033	13.9863	19.4111	25.8923
	C-C	2.9508	6.2516	10.2024	14.9737	20.7198	27.5536
50	H-H	5.4602	12.2784	19.1503	26.3310	33.9956	42.2995
	C-H	5.8897	12.7242	19.7836	2.1991	35.1390	43.7651
	C-C	6.4992	13.1711	20.4658	28.1191	36.3587	45.3273
100	H-H	8.1820	21.8417	35.5432	49.1378	62.8272	76.7822
	H-H^*	8.1319	21.7500	35.3045	48.5490	61.5991	74.5369
	C-H	9.8567	22.8734	36.5336	50.3094	64.2686	78.5534
	C-H^*	9.6644	22.3083	35.5163	48.6462	61.6514	78.7690
	C-C	11.5939	23.8053	37.5656	51.5127	65.7766	80.4076
	C-C^*	11.2319	22.7514	35.7481	48.7309	61.7087	79.7690

Results obtained using Abaqus (Ver.6.8).

The through thickness variations of the non-dimensional displacements and non-dimensional stresses (in-plane and transverse) at different locations of a multi-layered sandwich beam are shown in the Figures 4 to 7 for bending mode, shear mode, core deformable mode and soft-core mode respectively. The variations are presented for the first, fifth and ninth bending modes. It is observed from the variation of the in-plane normal stress that the moment is effectively resisted by the face layers. The variations of transverse normal and transverse shear stresses show continuity at the interface of the layers with different material properties. The variation of in-plane displacement is as expected in a refined higher order model. For the first shear mode, the non-dimensional in-plane displacement, transverse normal stress and the transverse shear stress variations are plotted. The variations of transverse normal stress and the transverse shear stress show a very high stress gradient at the interface between the face sheet and core and the shear stress is effectively resisted by the core only, while the in-plane displacement shows a linear variation in the core.

Figure 4.

Variation of across the depth of a sandwich beam in bending mode for l/h = 10; (a) In-plane normal stress ( ${\bar{σ}}_{xx}$ ); (b) Transverse normal stress ( ${\bar{σ}}_{xz}$ ); (c) Transverse shear stress ( ${\bar{τ}}_{xz}$ ) and (d) In-plane displacement ( $\bar{u}$ ).

Figure 5.

Variation across the depth of a sandwich beam in shear mode for l/h = 10; (a) Transverse normal stress ( ${\bar{σ}}_{xz}$ ); (b) Transverse shear stress ( ${\bar{τ}}_{xz}$ ) and In-plane displacement ( $\bar{u}$ ).

Figure 6.

Variation across the depth in of a sandwich beam in thickness mode or core deformable mode for l/h = 10; (a) In-plane normal stress ( ${\bar{σ}}_{xx}$ ); (b) Transverse normal stress ( ${\bar{σ}}_{xz}$ ); (c) Transverse shear stress ( ${\bar{τ}}_{xz}$ ) and (d) In-plane displacement ( $\bar{u}$ ).

Figure 7.

Variation across the depth in of a sandwich beam in soft-core mode for l/h = 10; (a) In-plane normal stress ( ${\bar{σ}}_{xx}$ ); (b) Transverse normal stress ( ${\bar{σ}}_{xz}$ ); (c) Transverse shear stress ( ${\bar{τ}}_{xz}$ ) and (d) In-plane displacement ( $\bar{u}$ ).

For core deformation mode or thickness mode, the variations of non-dimensional in-plane displacement, in-plane normal stress, transverse normal stress and the transverse shear stress through the thickness are presented for first, fifth and eighth thickness modes. The in-plane normal stress and transverse normal stress variation is symmetric about the mid plane. The shear stress variation shows maximum value at the interface of face layer and core. A jump is observed for in-plane displacement at interface of face sheet and core. A soft core mode is also reported and for first, third andfifth soft core modes, the variations of non dimensional in-plane normal stress, transverse normal stress, transverse shear stress and in-plane displacement are plotted.

The in-plane normal stress and transverse normal stress shows the resistance of moment is given by only the face layers but the variation is different than the variation shown for bending mode. The transverse shear stress variation shows that the negative shear stress induced in face sheets and the resistance is observed by core only. The in-plane displacement variation is as expected in the first mode but it shows very steep gradient in the remaining two modes. All these results are presented for the first time.

4. Conclusion

Free vibration analysis of laminated sandwich beam having a soft core has been carried out using a recently developed efficient C₀ FE beam model based on HOZT.The FE model has been developed by considering the in-plane displacement variation to be cubic for both the face sheets and the core and the transverse displacement to be quadratic within the core while it is constant in the faces beyond the core. The proposed model satisfies the condition of transverse shear stress continuity at the layer interfaces and the zero transverse shear stress condition at the top and bottom of the beam. There is no need to use penalty functions in the C₀ formulation as used by many previous researchers. This is due to the appropriate selection of nodal field variables to overcome the problem of continuity requirements of the derivatives of transverse displacements. Numerical examples on free vibration covering different features oflaminated composites and sandwich beams are presented to illustrate the accuracy and applicability of the present model. Many new interesting results related to core deformability (such as mode shapes and stress patterns) have also been presented in the submitted manuscript which should be a benchmark for future research.

Footnotes

Funding

This research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors.

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