Static deflection of fully coupled composite Timoshenko beams: An exact analytical solution

Journal Pre-proof Static deflection of fully coupled composite Timoshenko beams: An exact analytical solution Olga Doeva, Pedram Khaneh Masjedi, Paul M. Weaver

PII: DOI: Reference:

S0997-7538(19)30654-0 https://doi.org/10.1016/j.euromechsol.2020.103975 EJMSOL 103975

To appear in:

European Journal of Mechanics / A Solids

Received date : 13 August 2019 Revised date : 8 February 2020 Accepted date : 11 February 2020 Please cite this article as: O. Doeva, P.K. Masjedi and P.M. Weaver, Static deflection of fully coupled composite Timoshenko beams: An exact analytical solution. European Journal of Mechanics / A Solids (2020), doi: https://doi.org/10.1016/j.euromechsol.2020.103975. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2020 Published by Elsevier Masson SAS.

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Static Deflection of Fully Coupled Composite Timoshenko Beams: An Exact Analytical Solution

a Bernal

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Olga Doevaa , Pedram Khaneh Masjedia,∗, Paul M. Weavera

Institute, School of Engineering, University of Limerick, Limerick, Ireland

Abstract

The purpose of this paper is to present the exact analytical solution for the static deflection analysis of fully coupled composite Timoshenko beams. The system of governing equations and the boundary conditions are derived from variational principles. Using the method of direct integration, the exact analytical solution of the static deflection of a Timoshenko beam is obtained by solving this system of differential equations in terms of transverse displacements and cross-sectional rotations. Static deflection analyses of Timoshenko beams, subject to various boundary conditions and uniformly distributed and tip loads, are performed and the results are compared to those obtained from classical Euler-Bernoulli theory by using different values of length-to-thickness ratio. In addition, it is shown that for the case of a cantilevered composite beam subject to tip loads, the proposed exact analytical solution is equivalent to the exact solution from the intrinsic formulation. The Chebyshev collocation method is also employed to validate the obtained exact analytical solution. In the proposed formulation the stiffness properties of the composite beam are expressed by engineering constants, therefore is not limited by the cross-sectional shape of the beam, type of material

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and thus can be utilised for engineering applications and design purposes. The exact analytical solution can also be used as a benchmark for validating results obtained from various numerical methods. Keywords: Timoshenko beam, composite beam, exact solution, analytical solution, static deflection

1. Introduction

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Composite beams are widely used in many fields of engineering such as aerospace, marine, civil, and biomedical. This fact stimulates considerable interest in structural analysis of composite beams. Beams can be defined as structural elements where their length is significantly larger compared to their width and thickness. Classical Euler-Bernoulli Beam Theory (CBT) describes the structural behaviour of slender beams, assuming that the cross section of the beam perpendicular to the neutral axis remains perpendicular to the neutral axis after deformation. This assumption implies a zero through-thickness shear ∗ Corresponding

author: phone:+353 834603273 Email addresses: [email protected] (Olga Doeva), [email protected] (Pedram Khaneh Masjedi), [email protected] (Paul M. Weaver) Preprint submitted to Elsevier

February 8, 2020

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strain. However, the effect of transverse shear strains becomes more significant with shorter/thicker beams, or when higher natural frequencies of the beam are required. To address this issue, a new beam theory was

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presented by Timoshenko [1, 2], and further extended in [3]. This theory, referred to as Timoshenko Beam Theory (TBT), introduced first-order shear and rotational inertia effects as refined effects in CBT. Static behaviour of Timoshenko beams was extensively investigated by the research community. Gordaninejad and Bert [4] presented an analytical solution for a straight sandwich beam considered as TBT with thick skins taking into account transverse shear deformation in the facings as well as stretching and bending action in the core. Hinged-hinged and clamped-free boundary conditions were assumed. Assuming that the bending rigidity of a beam is second-order differentiable with respect to the coordinate variable, Lee and Kuo [5] derived the exact static solution of the deflection of non-uniform Timoshenko beams. To illustrate the analysis, the static deflection of a simply supported beam with constant depth and linearly varying width subject to uniformly distributed and concentrated loads was determined. Wang [6] obtained the static bending solution of homogeneous single-span Timoshenko beams with general loading and boundary conditions in terms of the bending solution of Euler-Bernoulli beams. Romano [7] presented the closed form solutions for the bending of Timoshenko beams with linearly and parabolically varying depth and linearly varying width along beam’s length subjected to polynomial, exponential and sinusoidal loading conditions. Results were validated for simply supported and clamped beams. Using Green’s functions, Wang et al. [8] derived the exact solutions for deflection, buckling and vibration of Timoshenko beams resting on Winkler, Pasternak, Vlasov and generalised elastic foundations. The Boundary Element Method was applied by Antes [9] for the linear static analysis of Timoshenko beams. Using the method of weighted residuals to

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derive an integral equation description for the bending moments and the shear forces, he derived the exact solutions for a clamped-simply supported beam under transverse loading and an inclined pin-joint beam with only one hinge. Hutchinson [10] developed a three-dimensional series solution for a simply supported Timoshenko beam and investigated the behaviour of shear coefficient for a rectangular Timoshenko beam. Jeleni´c and Papa [11] used linked interpolation functions of arbitrary order to obtain the exact solutions for linear analysis of Timoshenko beams subject to loads expressed by polynomials.

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Significant effort was made by the research community to investigate the static response of functionally graded (FG) beams. Considering the rotary inertia and shear deformation, Li [12] introduced a new unified approach for the static and dynamic analysis of FG beams. He presented the deflection and stress distribution results for cantilevered FG beams with material properties expressed by arbitrary functions along the beam thickness. Using the inverse Laplace transform, Adamek and Valeˇs [13] introduced an analytical solution for the response of a heterogeneous simply supported Timoshenko beam of rectangular cross-section subject to an arbitrary transverse dynamic load. Analytical results for the case of a three-layered laminated beam and functionally graded beam with properties varying in the thickness direction according to an arbitrary even function were presented. Using logarithmic functions, Pydah and Batra [14] derived shear deformation theory 2

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for thick circular beams and provided an analytical solution for the static deformations of the bi-directional functionally graded thick circular cantilevered beams with the material properties varying according to

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exponential and power laws, respectively, in the tangential and the thickness directions. Pei et al. [15] analytically solved, compared and discussed the cases of pure shearing, pure bending and pure tension for Timoshenko FG beams. Employing the physical neutral surface concept and Timoshenko beam theory, Phuong et al. [16] obtained Navier-type analytical solutions for bending analysis of functionally graded beams having porosities. Expanding the transverse load in Fourier series, they obtained exact formulae for the static deflections of FG beams made of aluminium and alumina subject to simply supported boundary conditions. Material properties were assumed to vary continuously through-the-thickness of beam according to a power law relationship. Two types of porosity distributions, even and uneven, were considered. A large amount of papers available in the literature deals with the analysis of laminated composite beams. By using the state-space concept simultaneously with the Jordan canonical form, Khdeir and Reddy [17, 18] developed exact closed-form solutions for static analyses of laminated composite beams. Solutions were presented for rectangular symmetric and antisymmetric cross-ply beams with arbitrary boundary conditions subjected to mechanical and thermal loadings respectively. Based on classical and higher order shear deformation theories, Mechab et al. [19] introduced a closed form solution for the bending analysis of short unidirectional laminated composite beams subject to mechanical and thermal loads. Applying the Generalised First Strain Gradient Theory, Sidhardh and Ray [20] developed the exact solutions for the static bending response of laminated composite beams. The efficiency of the suggested approach was demonstrated for simply supported isotropic and orthotropic beams subject to a sinusoidal distributed mechanical load.

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Based on Zaki-Moumni three-dimensional thermomechanical model and Timoshenko beam theory, Viet et al. [21] derived an analytical model describing the bending response of superelastic shape memory alloy (SMA) laminated composite beam. They assumed that the beam configuration is formed by two SMA layers bonded to one elastic core layer. The model was validated for a cantilevered beam subject to a concentrated force at the free tip. Later, applying the same model for shape memory alloys combined with Timoshenko beam theory, Van Viet et al. [22] proposed an analytical model for laminated composite beams consisting

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of multiple alternating superelastic shape memory alloy and elastic layers. The model was validated for the case of laminated composite cantilever beam subjected to a concentrated load at the tip. Employing a quasi-3D theory, Nguyen et al. [23] obtained the Ritz-based solutions for the static and dynamic response of laminated composite beams. A combinations of polynomial and exponential functions were used as shape functions. The presented model was verified for the symmetric and unsymmetric cross-ply beams with rectangular cross section subject to a uniformly distributed load. Simply supported, clamped and cantilevered boundary conditions were assumed.

As described, a large number of Timoshenko beam models has been developed to predict the structural behaviour of composite beams (for more references see [24] and [25]). However, the proposed solutions 3

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are restricted by layup, cross-sectional shape, boundary conditions, or special types of loading. While the aforementioned restrictions were addressed by Doeva et al. [26] and Masjedi and Weaver [27] in the

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context of Euler-Bernoulli theory, to the best of authors’ knowledge, there is no exact solution for the fully coupled composite Timoshenko beams in the literature without these restrictions. Exact analytical solutions are expressed symbolically, in closed form mathematical expressions and allow evaluating the role and importance of each contribution of the solution. Additionally, exact analytical solutions introduce considerable computational efficiency by bypassing the need for numerical integration and allow avoiding stability and convergence problems that can arise in numerical solutions such as finite element method. In the present study, the limitations are addressed and an exact analytical solution for the static deflection of fully coupled shear deformable composite beams without these limitations is introduced. For that purpose, a mathematical formulation with the following characteristics is proposed: (1) the entries of the stiffness matrix are expressed in terms of the engineering constants to keep the formulation as general as possible; (2) material is linear elastic; (3) displacements, strains and rotations are small; (4) shear deformations are taken into account; (5) the shape of cross section is arbitrary and remains unchanged during the deformation. Despite the fact that the focus of this paper is to present an exact analytical solution for the static response of fully coupled composite beams, the Chebyshev Collocation Method (CCM), which has been shown to be relatively efficient and accurate in beam problems (Masjedi and Ovesy [28, 29], Masjedi and Maheri [30] and Masjedi et al. [31]), for the purpose of verification is also applied.

The rest of the paper is organised as follows: in Section 2, governing equations and boundary conditions are established to derive a general solution of fully coupled composite Timoshenko beam; in Section 3, explicit

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expressions for deflections u, v, w and rotations θx , θy , θz are derived; in Section 4 a brief description of the application of Chebyshev Collocation Method is given; in Section 5, several benchmark test problems are considered and the exact results obtained from TBT are verified by those obtained from the CCM and compared with the exact results obtained from CBT. The effect of the length-to-thickness ratio on the displacement response of the beams is also highlighted. Finally, conclusions are presented in Section 6.

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2. Governing Equations 2.1. Beam kinematics

Consider a straight composite beam. Let the beam length be denoted by ` measured along the x coordinate axis while the coordinates y and z define the cross-sectional planes, as shown in Fig. (1).

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z

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y

x

`

Figure 1: Composite beam

Displacement of a generic point on the cross-section is given by a vector U = U (x, y, z) with components:

Ux = u (x) + zθy (x) − yθz (x) ,

(1a)

Uy = v (x) − zθx (x) ,

(1b)

Uz = w (x) + yθx (x) ,

(1c)

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where u, v and w denote displacements of the beam reference line in x, y and z directions and θx , θy and θz represent the rotations of beam cross-section about x, y and z, respectively. Now, with the help of Eqns. (1), the strain measures of the beam are expressed as:

∂Ux = u0 + zθy0 − yθz0 , ∂x ∂Uy ∂Ux xy = + = (v 0 − θz ) − zθx0 , ∂y ∂x ∂Ux ∂Uz xz = + = (w0 + θy ) + yθx0 , ∂z ∂x

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xx =

(2a) (2b) (2c)

where ()0 denotes the derivative with respect to x. 2.2. Principle of virtual work

Internal work of the beam can be expressed as: Z

V

Wint dV =

Z

(σxx xx + σxy xy + σxz xz ) dV.

V

5

(3)

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Substituting strain measures xx , xy and xz , given by Eqns. (2), into Eqn. (3), the following can be obtain:

Wint dV =

V

Z

V


σxx u0 + z σxx θy0 − y σxx θz0 + σxy (v 0 − θz ) + σxz (w0 + θy ) + (y σxz − z σxy ) ϕ0 dV. (4)

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Z

Defining beam internal forces and moments as:

Z

Fx =

σxx dA,

(5a)

σxy dA,

(5b)

σxz dA,

(5c)

(y σxz − z σxy ) dA, Z My = z σxx dA, ZA Mz = − y σxx dA,

(5d)

ZA

Fy =

ZA

Fz =

Mx =

A

Z

A

(5e) (5f)

A

Eqn. (4) can be written as: Z

`

Wint dx =

0

Z

0

`


Fx u0 + Fy (v 0 − θz ) + Fz (w0 + θy ) + Mx θx0 + My θy0 + Mz θz0 dx.

(6)

The principle of virtual work is presented by the following expression:  δWint − δWext dx = 0,

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Z ` 0

(7)

where δWint and δWext are the variations of internal and external works respectively. Using Eqn. (6), variation of internal work of the beam can be written as: Z

`

δWint =

`

T

δε N dx =

0

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0

Z

Z

`

δεT Sεdx.

(8)

0

Here the vector of strains and curvatures ε and vector of internal forces and moments N can be expressed as:

ε = [x

γxy

γxz

κx

κy

κz ]T = [u0

v 0 − θz

N = [Fx

Fy

w 0 + θy Fz

Mx

θx0

θy0 My

θz0 ]T ,

(9a)

Mz ]T .

(9b)

It is noted that herein a linear relation is assumed between internal forces and moments N and strains and curvatures ε, i.e. N = Sε, which is a well-accepted approach by many researchers. Various methodologies 6

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have been proposed in the literature to obtain a linear constitutive relation for composite beams, e.g. see Giavotto et al. [32], Cesnik and Hodges [33], Hill and Weaver [34], Kim et al. [35], Morandini et al. [36] and Pai

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[37]. According to Berdichevskii [38], the 3D problem of a beam can split into a ”2D cross-sectional analysis” governing the cross-sectional elastic constants and a ”1D beam” analysis. Based on this premise, current work assumes that the 2D cross-sectional analysis and 1D beam problem are decoupled, thus, the validity of exact closed-form solutions which are presented for a 1D beam problem in this paper is independent of the methodology used to obtain the stiffness matrix and these solutions are expressed for the most general case in which the stiffness matrix is fully populated. The stiffness matrix S has the following form: 

A

S= BT

B

D

where matrices A, B and D can be written as: 

EA

GAy

SSy Sz SEF

SSy F SSz F ST F

EIy

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  A = SSy E  SSz E  S  ET  B = SSy T  SSz T  GJ   D = ST F  ST L

SSy E

SF L



,

SSz E



  SSy Sz  ,  GAz  SEL   SSy L  ,  SSz L  ST L   SF L  ,  EIz

(10)

(11)

(12)

(13)

where EA is extensional stiffness, GAy is shear stiffness in y direction, GAz is shear stiffness in z direction, GJ is twist stiffness, EIy is out-of-plane bending stiffness, EIz is in-plane bending stiffness, Sij is coupling between i and j, wherein sub-indices i and j have the following meaning: Sy is shear in y direction, Sz is shear in z direction, T is torsion, F is flap-wise bending, L is lag-wise bending, E is extension (or axial

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elongation). In order to keep the formulation as general as possible, it is assumed that the stiffness matrix is fully populated meaning that all possible coupling terms are considered. Coupling terms between different degrees of freedom can occur due to material properties such as anisotropy, beam cross-section geometry e.g. offset between centroid and beam reference line or a combination of these two, which can result in a fully populated stiffness matrix for complex realistic cross-sections (real life example with fully populated stiffness matrix can be found in Hodges and Yu [39]). Variation of external work can be represented by:

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Z

`

δWext dx =

0

Z

`

δU QT dx,

(14)

0

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where the vector of displacements and rotations U and vector of external forces and moments Q are written as:

U = [u

Q = [fx

v

fy

w

θx

fz

θy

mx

θ z ]T ,

my

mz ]T ,

(15a) (15b)

where fx is distributed axial force, fy and fz are distributed shear forces in y and z directions respectively, mx is distributed torque, my and mz are distributed bending moments.

From Eqn. (8) the following definitions of the internal forces and moments can be obtained:

(16a)

Fy = SSy E u0 + GAy (v 0 − θz ) + SSy Sz (w0 + θy ) + SSy T θx0 + SSy F θy0 + SSy L θz0 ,

(16b)

Fz = SSz E u0 + SSy Sz (v 0 − θz ) + GAz (w0 + θy ) + SSz T θx0 + SSz F θy0 + SSz L θz0 ,

(16c)

Mx = SET u0 + SSy T (v 0 − θz ) + SSz T (w0 + θy ) + GJθx0 + ST F θy0 + ST L θz0 ,

(17a)

My = SEF u0 + SSy F (v 0 − θz ) + SSz F (w0 + θy ) + ST F θx0 + EIy θy0 + SF L θz0 ,

(17b)

Mz = SEL u0 + SSy L (v 0 − θz ) + SSz L (w0 + θy ) + ST L θx0 + SF L θy0 + EIz θz0 .

(17c)

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Fx = EAu0 + SSy E (v 0 − θz ) + SSz E (w0 + θy ) + SET θx0 + SEF θy0 + SEL θz0 ,

Using Eqns. (7), (8) and (14), and definitions (16) and (17), the following system of governing equations can be obtained:

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Fx0 + fx = 0, Fy0 + fy = 0, Fz0 + fz = 0,

Mx0 + mx = 0, My0 − Fz + my = 0, Mz0 + Fy + mz = 0.

At x = 0 and x = `, boundary conditions can be written as follows: 8

(18)

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or

v=0

or

w=0 θx = 0 θy = 0 θz = 0

Fx = fbx , Fy = fby ,

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u=0

Fz = fbz ,

or

(19)

Mx = m b x,

or

My = m b y,

or

Mz = m b z,

or

where fbx is axial tip force, fby and fbz are tip shear forces, m b x is tip torque moment, m b y and m b z are tip bending moments.

Defining vectors

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and matrix

  f  x   f =  fy  ,   fz

  θ  x   θ =  θy  ,   θz   M  x   M =  My  ,   Mz   fb  x   fb = fby  ,   fbz

  u     u = v  ,   w   F  x   F = Fy  ,   Fz   m  x   m = my  ,   mz

 0   e = 0  0

0 0 1

0



  −1 ,  0

  m b  x   c = m m b ,  y m bz

(20)

(21)

(22)

(23)

governing equations (18) and boundary conditions (19) can be written in a compact matrix form as follows:

F 0 + f = 0,

(24a)

M 0 + eF + m = 0,

(24b)

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(25a)

θ = 0,

(25b)

F = fb,

(25c)

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where 0 is 3 × 1 vector of zeros.

u = 0,

c, M =m

(25d)

Note that using definition of matrices (11) - (13), internal forces (16) and internal moments (17) can also be written in matrix form in the following manner:

F = Aγ + Bκ,

(26a)

M = B T γ + Dκ,

(26b)

where the vector of curvatures κ and vector of linear strains γ can be written respectively as:

3. Exact Analytical Solution

κ = θ0 ,

(27a)

γ = u0 + eθ.

(27b)

matrix

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Before the procedure of obtaining the exact analytical solution of Eqns. (24) is presented, consider the



KA

K= KB T

KB KD



,

(28)

such that K = S −1 , i.e. K is an inverse of matrix S. Entries of K can be determined using the Block

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Matrix Inversion Formula ([40]):

KA = A−1 + A−1 B(D − B T A−1 B)−1 B T A−1 ,

(29a)

KB = −A−1 B(D − B T A−1 B)−1 ,

(29b)

KB T = −(D − B T A−1 B)−1 B T A−1 ,

(29c)

KD = (D − B T A−1 B)−1 .

(29d)

The procedure of obtaining the exact solution for Eqns. (24) is given below. 10

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First, Eqn. (24a) should be integrated to obtain the expression of the internal forces in the following

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form:

F = −xf + C 1 .

(30)

Next, substituting Eqn. (30) into Eqn. (24b), rearranging and integrating it, the expression for internal moments can be derived:

M=

1 2 x ef − x(eC 1 + m) + C 2 . 2

(31)

To obtain the expression for the vector of linear strains, Eqn. (26a) can be rearranged in a following manner:

γ = A−1 (F − Bκ).

(32)

Substituting Eqn. (32) in Eqn. (26b) and using definition (29d), the expression for κ can be obtained: κ = KD (M − B T A−1 F ).

(33)

Substituting Eqn. (27a) and the expressions of internal forces F and moments M given by Eqns. (30)

θ0 =

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and (31), into Eqn. (33), and using definitions (29), the following expression can be derived: 1 2 x KD ef − x(KB T f + KD (eC 1 + m)) + (KB T C 1 + KD C 2 ). 2

(34)

Integrating Eqn. (34), the expression for the vector of rotations can be obtained: 1 3 1 x KD ef − x2 (KB T f + KD (eC 1 + m)) + x(KB T C 1 + KD C 2 ) + C 3 . 6 2

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θ=

(35)

Substituting Eqn. (27b) and the expressions of internal forces F , curvatures κ and rotations θ, given by Eqns. (30), (33), (35) respectively, into Eqn. (32), rearranging and integrating it, the expression of the vector of the displacements can be written as follows:

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1 4 1 x eKD ef + x3 (KB ef + eKB T f + eKD (eC 1 + m)) 24 6 1 − x2 (KA f + KB (eC 1 + m) + eKB T C 1 + eKD C 2 ) 2

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u=−

+ x(KA C 1 + KB C 2 − eC 3 ) + C 4 , (36)

Eqns. (36) and (35) are the general exact closed-form solutions for the static deflection of a fully coupled composite Timoshenko beam under the action of uniformly distributed and tip loads, unrestricted by the type of boundary conditions. These general solutions have four vectors of unknown integration constants C i , i = 1, 2, 3, 4, which can be determined once the boundary conditions are given.

As an illustrative example, the expressions of the unknowns for a cantilevered composite Timoshenko beam under tip loads are provided below. To obtain the expressions for the vectors C i , i = 1, 2, 3, 4, for this case, boundary conditions (25a) and (25b) should be applied at x = 0 and boundary conditions (25c) and (25d) at x = `. Thus, using Eqns. (29), the vectors C i , i = 1, 2, 3, 4, can be written in the following form:

C 1 = fb,

(37b)

C 3 = C 4 = 0.

(37c)

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c + `efb, C2 = m

(37a)

Substituting coefficients (37) into Eqns. (36) and (35), exact solutions of the deflection of the cantilevered composite Timoshenko beam under tip loads can be obtained:    1 3 1  x eKD fb − x2 (KB e + eKB T )fb + eKD (c m + `efb) + x KA fb + KB (c m + `efb) , 6 2

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u=

  1 θ = − x2 KD efb + x KB T fb + KD (c m + `efb) . 2

(38)

(39)

It is worth mentioning that solutions (38) and (39), which are displacement based (DB), are identical to the exact solutions for the linear static deflection of cantilevered composite beams under the action of tip loads based on the force-based intrinsic formulation (IF), presented by Hodges in [41].

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4. Chebyshev Collocation Method The Chebyshev collocation method (CCM) is employed for the purpose of verification of the solutions for

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Eqns. (18) obtained from the proposed model. The Chebyshev polynomials are adopted as trial functions for the discretisation of the unknown variables u, v, w, θx , θy and θz , and the Chebyshev points are employed as collocation points. The Chebyshev polynomials Tn (x) of the first kind in x and of degree n are defined as ([42]):

Tn (x) = cos(nθ)

when x = cos θ,

−1 ≤ x ≤ +1,

n = 0, 1, . . .

(40)

The first few Chebyshev polynomials of the first kind are: T0 (x) = 1,

T1 (x) = x,

T2 (x) = 2x2 − 1,

T3 (x) = 4x3 − 3x,

(41)

T4 (x) = 8x4 − 8x2 + 1,

T5 (x) = 16x5 − 20x3 + 5x, .. .

Replacing the independent variable x in Eqn.(41) by:

2 b+a x− , b−a b−a

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(42)

the Chebyshev polynomials can be shifted from the interval −1 ≤ x ≤ +1 to any arbitrary interval a ≤ x ≤ b. The Chebyshev points in the interval −1 ≤ x ≤ +1 are given as: xi = cos



 2i − 1 π , 2N

i = 1, 2, . . . , N + 1,

(43)

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where N is the highest degree of the Chebyshev polynomials used as the trial functions. For an arbitrary interval a ≤ x ≤ b the transformed Chebyshev points are: 1 1 xi = (a + b) − (b − a) cos 2 2



 2i − 1 π , 2N

i = 1, 2, . . . , N + 1.

Using the Chebyshev polynomials, unknown variables are discretised as:

13

(44)

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u=

N X

ai Ti (x),

(45a)

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i=0

v=

N X

bi Ti (x),

(45b)

ci Ti (x),

(45c)

di Ti (x),

(45d)

ei Ti (x),

(45e)

fi Ti (x).

(45f)

i=0

w=

N X i=0

θx =

N X i=0

θy =

N X i=0

θz =

N X i=0

To apply the collocation method, Eqns. (45) are substituted into Eqns. (18) and the residuals at the Chebyshev points are set to be equal to zero. The system of 6 × (N + 1) linear equations is solved for unknown coefficients ai , bi , ci , di , ei and fi , where i = 0, 1, 2, ..., N . It is noted that for the current problem, using N = 6 gives converged results and all numerical results from the Chebyshev collocation method in the following sections are obtained on this premise.

5. Numerical Results

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In this section examples of composite Timoshenko beam under the action of uniformly distributed and tip loads are presented for clamped-free and simply supported boundary conditions. A laminated fibrereinforced beam with a rectangular cross section is considered for all numerical samples. The material and

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geometric properties of a beam are listed in Table 1.

14

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Table 1: Beam properties

0.01

Number of Layers

48

Ply Thickness

0.000125

m

Beam Thickness (h)

0.006

m

E11

135.64

GP a

10.14

GP a

5.86

GP a

E22 G12 ν12

m

lP repro of

Width (b)

0.29

Beam Cross Section Analysis Software (BECAS) ([43]), which is a finite element based code, was employed to calculate the stiffness constants. For obtaining the stiffness matrix entries, 589 rectangular four node elements were used in BECAS. For the purpose of benchmarking and in order to examine the effects of different coupling terms on the deflection of composite Timoshenko beam, symmetric, anti-symmetric, cross-ply and unsymmetric stacking sequences were considered. Their stiffness properties are listed in Table (2). To study the difference in the results obtained from the Timoshenko and the Euler-Bernoulli beam

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rna

theories, four length-to-thickness ratios were taken into consideration.

15

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Table 2: Stiffness matrix for different stacking sequences

[4524 ]s

[4524 / − 4524 ]

 1062968.944    0    0    −652.718    0  0

 4379210.004    0    0    0    5651.859  0

0

0

0

0



0

0

0

247076.915

0

0

0

4.311

−1.435

0

−1.435

3.105

0

0

0

  0    0    0    0   7.312



0

−652.718

0

948804.572

0

0

843.785

0

261043.269

0

0

0

0

4.817

0

843.785

0

0

3.662

0

22.739

0

0

0

0

0

5651.859

0

249765.521

0

142.342

0

0

0

248073.688

0

0

0

142.342

0

2.699

0

0

0

0

0

13.132

0

0

0

0

0

36.472

 1768173.009 576737.274    576737.274 579373.238    0 0    796.523 339.418    −1437.350 −697.861  0 0

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[6024 /3024 ]

S

0

rna

[024 /9024 ]

 1137235.581 392797.921    392797.921 589485.359    0 0    0 0    0 0  0 0

lP repro of

Stacking Sequence

16

0

796.523

−1437.350

0

339.418

−697.861

243228.650

0

0

0

4.341

−2.128

0

−2.128

4.889

−9.796

0

0

0

     22.739   0    0   8.095 0



0

            



     −9.796   0    0   9.424 0

Journal Pre-proof

5.1. Cantilevered composite beam under tip load A cantilevered laminated composite beam subject to a concentrated tip load is considered in this subsec-

lP repro of

tion. Numerical results for the exact tip deflection obtained from the present and Hodges [41] solutions, and from the Chebyshev collocation method based on TBT, are provided in Tables (3) - (4) for different stacking sequences. These results are compared with those obtained from the exact solution based on Euler-Bernoulli theory (see Appendix A), and the difference between two theories is calculated as follows: TBT Exact − CBT Exact × 100% Difference = TBT Exact

(46)

Four beam lengths ` are considered, namely 0.03 m, 0.06 m, 0.12 m and 0.3 m, which give four lengthto-thickness ratios (`/h): 5, 10, 20 and 50 respectively. Tip load fbz = 10 N is considered for symmetric,

unsymmetric, and cross-ply layups, and tip moment m b x = 2.5 N.m is considered for anti-symmetric case.

In all of the following tables displacements are expressed in metres and rotations are measured in radians. Graphs for `/h = 10 are presented and deflections with zero value are not shown.

As shown in Tables (3) - (6) and Figs. (2)-(3), for all types of stacking sequence studied in this subsection, the displacement-based and intrinsic formulation give identical results and agree closely with results obtained by solving the corresponding problems using CCM. However, as expected the results obtained from TBT are different from those obtained from CBT. In the case of the symmetric layup there is no difference between the results obtained from TBT and CBT for twist θx . This could be due to the fact that there is no twist-transverse shear coupling term. Whereas, for the out-of-plane displacement w the difference between the two theories increases from 0.04% to 3.42% while the length-to-thickness ratio decreases from 50 to 5.

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As there are no shear coupling terms, shear effects do not appear in the anti-symmetric stacking sequence case, therefore there is no difference in the results obtained from the two theories which is clearly shown in Table (4). As shown in Table (5), for the case of the cross-ply layup there is no difference between results obtained from TBT and CBT for the axial displacement u, but there is an increasing difference from 0.08% to 7.27% for the out-of-plane displacement w when the length-to-thickness ratio decreases from 50 to 5. This trend can be explained by the fact that no coupling exists between shear and the axial displacement

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u or out-of-plane bending w. As observed in Table (6), in the case of the unsymmetric layup the difference between results obtained from TBT and CBT for the displacement in axial direction u remains constant and significant (29.3%) for all length-to-thickness ratios. The difference between the two theories for θx is also constant, though small (0.48%), irrespective of the value of the length-to-thickness ratio. In the case of outof-plane displacement w the difference between results obtained from the two theories increases from 3.26% to 7.12% when the length-to-thickness ratio decreases from 50 down to 5. It is also worth mentioning that in Euler-Bernoulli theory there are no coupling terms for in-plane displacement v while in Timoshenko theory in-plane displacement v is coupled with shear deformation, therefore the value for this type of displacement 17

Journal Pre-proof

is 0 according to CBT and non-zero according to TBT.

TBT Exact (DB)

lP repro of

Table 3: Tip deflection of symmetric cantilevered beam under tip load

Exact (IF)

CCM (DB)

CBT Exact

Difference (%)

`/h = 5

w

0.0000354735

0.0000354735

0.0000354735

0.0000342593

3.42

θx

-0.0005703039

-0.0005703039

-0.0005703039

-0.0005703039

0.00

θy

-0.0017129659

-0.0017129659

-0.0017129659

–

–

`/h = 10

w

0.0002765029

0.0002765029

0.0002765029

0.0002740745

0.88

θx

-0.0022812158

-0.0022812158

-0.0022812158

-0.0022812158

0.00

θy

-0.0068518637

-0.0068518637

-0.0068518637

–

–

`/h = 20

w

0.0021974531

0.0021974531

0.0021974532

0.0021925963

0.22

θx

-0.0091248631

-0.0091248631

-0.0091248631

-0.0091248631

0.00

θy

-0.0274074548

-0.0274074548

-0.0274074548

–

–

0.0342714604

θx

-0.0570303945

θy

-0.1712965923

0.0342714604

0.0342714604

0.0342593185

0.04

-0.0570303945

-0.0570303945

-0.0570303945

0.00

-0.1712965923

-0.1712965923

. –

–

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w

rna

`/h = 50

18

Journal Pre-proof

Table 4: Tip deflection of anti-symmetric cantilevered beam under tip load

Exact (DB)

CBT Exact

Difference (%)

lP repro of

TBT Exact (IF)

CCM (DB)

`/h = 5

u

0.0000104288

0.0000104288

0.0000104288

0.0000104288

0.00

θx

0.0169835531

0.0169835531

0.0169835531

0.0169835531

0.00

`/h = 10

u

0.0000208576

0.0000208576

0.0000208576

0.0000208576

0.00

θx

0.0339671063

0.0339671063

0.0339671063

0.0339671063

0.00

`/h = 20

u

0.0000417151

0.0000417151

0.0000417151

0.0000417151

0.00

θx

0.0679342125

0.0679342125

0.0679342125

0.0679342125

0.00

`/h = 50

0.0001042878

0.0001042878

0.0001042878

0.0001042878

0.00

θx

0.1698355313

0.1698355313

0.1698355313

0.1698355313

0.00

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rna

u

19

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Table 5: Tip deflection of cross-ply cantilevered beam under tip load

Exact (DB)

CBT Exact

Difference (%)

lP repro of

TBT Exact (IF)

CCM (DB)

`/h = 5

u

0.0000009949

0.0000009949

0.0000009949

0.0000009949

0.00

w

0.0000166268

0.0000166268

0.0000166268

0.0000154175

7.27

θy

-0.0007708739

-0.0007708739

-0.0007708739

–

–

`/h = 10

u

0.0000039796

0.0000039796

0.0000039796

0.0000039796

0.00

w

0.0001257585

0.0001257585

0.0001257585

0.0001233398

1.92

θy

-0.0030834955

-0.0030834955

-0.0030834955

–

–

`/h = 20

u

0.0000159183

0.0000159183

0.0000159183

0.0000159183

0.00

w

0.0009915558

0.0009915558

0.0009915558

0.0000159184

0.48

θy

-0.0123339818

-0.0123339818

-0.0123339818

–

–

`/h = 50

w

0.0154295705

θy

-0.0770873865

0.0000994899

0.0000994899

0.0000994899

0.00

0.0154295705

0.0154295705

0.0154174773

0.08

-0.0770873865

-0.0770873865

–

–

rna

0.0000994899

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u

20

Journal Pre-proof

Table 6: Tip deflection of unsymmetric cantilevered beam under tip load

Exact (DB)

CBT Exact

Difference (%)

lP repro of

TBT Exact (IF)

CCM (DB) `/h = 5

u

-0.0000007086

-0.0000007086

-0.0000007086

-0.0000009162

29.30

v

-0.0000007314

-0.0000007314

-0.0000007314

0

100

w

0.0000305678

0.0000305678

0.0000305678

0.0000283898

7.12

θx

-0.0005302954

-0.0005302954

-0.0005302954

-0.0005277456

0.48

θy

-0.0014667154

-0.0014667154

-0.0014667154

–

–

θz

0.0000012822

0.0000012822

0.0000012822

–

–

`/h = 10

u

-0.0000028342

-0.0000028342

-0.0000028342

-0.0000036647

29.30

v

-0.0000029258

-0.0000029258

-0.0000029258

0

100

w

0.0002371414

0.0002371414

0.0002371414

0.0002271185

4.23

θx

-0.0021211816

-0.0021211816

-0.0021211816

-0.0021109823

0.48

θy

-0.0058668616

-0.0058668616

-0.0058668616

–

–

θz

0.0000025646

0.0000025645

0.00000256446

–

–

-0.0000113369

v

-0.0000117031

w

0.0018823296

θx

-0.0084847264

θy

-0.0234674466

θz

0.0000051289

-0.0000113369

-0.0000113369

-0.00001465869

29.30

-0.0000117031

-0.0000117031

0

100

0.0018823296

0.0018823296

0.0018169484

3.47

-0.0084847264

-0.0084847264

-0.0084439294

0.48

-0.0234674466

-0.0234674466

–

–

0.0000051289

0.0000051289

–

–

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u

rna

`/h = 20

`/h = 50

u

-0.0000708556

-0.0000708556

-0.0000708556

-0.0000916168

29.30

v

-0.0000731441

-0.0000731441

-0.0000731441

0

100

w

0.0293466428

0.0293466428

0.0293466428

0.0283898180

3.26

θx

-0.0530295402

-0.0530295402

-0.0530295402

-0.0527745585

0.48

θy

-0.1466715411

-0.1466715411

21 -0.1466715411

–

–

θz

0.0000128223

0.0000128223

0.0000128223

–

–

Journal Pre-proof

10-4

10-5 2

2.5

lP repro of

1.8 1.6

2

1.4 1.2

1.5

1

0.8

1

0.6 0.4

0.5

0.2

0

0

0

0.01

0.02

0.03

0.04

0.05

0.06

0

(a) Symmetric layup, fbz = 10 N/m

10-4

5

0.01

0.02

0.03

0.04

0.05

0.06

(b) Anti-symmetric layup, m b x = 2.5 N

10-4

4

4.5

3

4 3.5

2

3

1

2 1.5 1 0.5 0 0.01

0.02

0.03

0.04

0.05

0

-1

-2

-3

0.06

0

Jou

0

rna

2.5

(c) Cross-ply layup, fbz = 10 N/m

0.01

0.02

0.03

0.04

0.05

(d) Unsymmetric layup, fbz = 10 N/m

Figure 2: Displacement of a cantilevered beam under the action of tip load for different stacking sequences

22

0.06

Journal Pre-proof

10-3

0.035

5

0

lP repro of

0.03

0.025

-5

0.02

-10

0.015

-15

0.01

-20

0.005

-25

0

0

0.01

0.02

0.03

0.04

0.05

0.06

0

(a) Symmetric layup, fbz = 10 N/m

10-3

0

0.01

0.02

0.03

0.04

0.05

0.06

(b) Anti-symmetric layup, m b x = 2.5 N

0.04

0.035

-0.5

0.03

0.025

-1

0.02

-1.5

-2

-2.5

-3 0.01

0.02

0.03

0.04

0.05

0.01 0.005 0 -0.005 -0.01 0

0.06

Jou

0

rna

0.015

(c) Cross-ply layup, fbz = 10 N/m

0.01

0.02

0.03

0.04

0.05

0.06

(d) Unsymmetric layup, fbz = 10 N/m

Figure 3: Rotation of a cantilevered beam under the action of tip load for different stacking sequences

5.2. Cantilevered composite beam under distributed load In this subsection a cantilevered composite beam under the action of uniformly distributed load fz = 50 N/m is considered for symmetric, unsymmetric, and cross-ply stacking sequences, while for anti-symmetric case uniformly distributed moment mx = 50 N is assumed. 23

Journal Pre-proof

Tables (7) - (10) and Figs. (4) - (5) demonstrate good agreement between results obtained from the proposed exact analytical solution and CCM for the deflections of the composite Timoshenko beam for

lP repro of

different stacking sequences. It is observed that as in the previous example, there is a difference between the values obtained from TBT and CBT. It is observed that the difference in the results obtained from the two theories for axial displacement u, in-plane displacement v, and twist θx is exactly the same as in the case of cantilevered beam under tip load. However while the qualitative behaviour of out-of-plane displacement w is following a similar trend as in the previous example, just a marginal increase in values can be observed. Table 7: Tip deflection of symmetric cantilevered beam under uniformly distributed load

TBT Exact (DB)

CBT Exact

Difference (%)

0.0000020182

0.0000019271

4.51

-0.0000285152

-0.0000285152

0.00

-0.0000856483

–

–

0.0000311976

0.0000308334

1.17

-0.0002281216

-0.0002281216

0.00

-0.0006851864

–

–

0.0004933342

0.29

CCM (DB)

`/h = 5

w

0.0000020182

θx

-0.0000285152

θy

-0.0000856483

`/h = 10

w

0.0000311976

θx

-0.0002281216

θy

-0.0006851864

`/h = 20

0.0004947912

0.0004947912

θx

-0.0018249726

-0.0018249726

-0.0018249726

0.00

θy

-0.0054814909

-0.0054814909

–

–

rna

w

`/h = 50

0.0192799731

0.0192799731

0.0192708666

0.05

θx

-0.0285151973

-0.0285151973

-0.0285151973

0.00

θy

-0.0856482961

-0.0856482961

–

–

Jou

w

24

Journal Pre-proof

Table 8: Tip deflection of anti-symmetric cantilevered beam under uniformly distributed load

TBT

CBT Exact

CCM (DB)

Difference (%)

lP repro of

Exact (DB)

`/h = 5

u

0.0000031286

θx

0.0050950659

0.0000031286

0.0000031286

0.00

0.0050950659

0.0050950659

0.00

`/h = 10

u

0.0000125145

θx

0.0203802638

0.0000125145

0.0000125145

0.00

0.0203802638

0.0203802638

0.00

`/h = 20

u

0.0000500581

θx

0.0815210550

0.0000500581

0.0000500581

0.00

0.0815210550

0.0815210550

0.00

`/h = 50

0.0003128634

θx

0.5095065939

0.0003128634

0.0003128634

0.00

0.5095065939

0.5095065939

0.00

Jou

rna

u

25

Journal Pre-proof

Table 9: Tip deflection of cross-ply cantilevered beam under uniformly distributed load

TBT

CBT Exact

CCM (DB)

Difference (%)

lP repro of

Exact (DB)

`/h = 5

. u

0.0000000497

w

0.0000009579

θy

-0.0000385437

0.0000000497

0.0000000497

0.00

0.0000009579

0.0000008672

9.47

-0.0000385437

–

–

0.0000003979

0.0000003979

0.00

0.0000142385

0.0000138757

2.55

-0.0003083495

–

–

0.0000031837

0.0000031837

0.00

0.0002234629

0.0002220117

0.65

-0.0024667964

–

–

0.0000497449

0.0000497449

0.00

0.0086814009

0.0086723309

0.10

-0.0385436932

–

–

`/h = 10

u

0.0000003979

w

0.0000142385

θy

-0.0003083495

`/h = 20

u

0.0000031837

w

0.0002234629

θy

-0.0024667964

`/h = 50

0.0000497449

w

0.0086814009

θy

-0.0385436932

Jou

rna

u

26

Journal Pre-proof

Table 10: Tip deflection of unsymmetric cantilevered beam under uniformly distributed load

TBT

CBT Exact

CCM (DB)

Difference (%)

lP repro of

Exact (DB)

`/h = 5

u

-0.0000000354

v

-0.0000000356

w

0.0000017426

θx

-0.0000265148

θy

-0.0000733358

θz

0.0000000962

-0.0000000354

-0.0000000458

29.30

-0.0000000356

0

100

0.0000017426

0.0000015969

8.36

-0.0000265148

-0.0000263873

0.48

-0.0000733358

–

–

0.0000000962

–

–

-0.0000002834

-0.0000003667

29.30

-0.0000002849

0

100

0.0000267709

0.0000255508

4.56

-0.0002121182

-0.0002110982

0.48

-0.0005866862

–

–

0.0000003847

–

–

-0.0000022674

-0.0000029317

29.30

-0.0000022791

0

100

0.0004238942

0.0004088134

3.56

`/h = 10

u

-0.0000002834

v

-0.0000002849

w

0.0000267709

θx

-0.0002121182

θy

-0.0005866862

θz

0.0000003847

`/h = 20

-0.0000022674

v

-0.0000022791

w

0.0004238942

θx

-0.0016969453

-0.0016969453

-0.0016887859

0.48

θy

-0.0046934893

-0.0046934893

–

–

θz

0.0000015387

0.0000015387

–

–

-0.0000354278

-0.0000458084

29.30

rna

u

`/h = 50

-0.0000354278

Jou

u v

-0.0000356104

-0.0000356104

0

100

w

0.0165097993

0.0165097993

0.0159692726

3.27

θx

-0.0265147701

-0.0265147701

-0.0263872793

0.48

θy

-0.0733357706

-0.0733357706

–

–

θz

0.0000096167

0.0000096167

–

–

27

Journal Pre-proof

10-5

10-5 1.2

lP repro of

3

1

2.5

0.8

2

0.6

1.5

0.4

1

0.2

0.5

0

0

0

0.01

0.02

0.03

0.04

0.05

0.06

0

(a) Symmetric layup, fz = 50 N/m 10-5

1.5

0.01

0.02

0.03

0.04

0.05

0.06

(b) Anti-symmetric layup, mx = 50 N

10-5

3

2

1

1

0 0.01

0.02

0.03

0.04

0.05

-1

-2

-3

0.06

0

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0

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0.5

0

(c) Cross-ply layup, fz = 50 N/m

0.01

0.02

0.03

0.04

0.05

0.06

(d) Unsymmetric layup, fz = 50 N/m

Figure 4: Displacement of a cantilevered beam under the action of uniformly distributed load for different stacking sequences

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10-4

1

0.02 0

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0.018 0.016

-1

0.014

-2

0.012

-3

0.01

0.008

-4

0.006

-5

0.004

-6

0.002

-7

0

0

0.01

0.02

0.03

0.04

0.05

0.06

0

(a) Symmetric layup, fz = 50 N/m 10-4

0

0.01

0.02

0.03

0.04

0.05

0.06

(b) Anti-symmetric layup, mx = 50 N

10-4

8

6

4

-1

2

-3 0.01

0.02

0.03

0.04

0.05

-2

-4

-6

0.06

0

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0

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-2

0

(c) Cross-ply layup, fz = 50 N/m

0.01

0.02

0.03

0.04

0.05

0.06

(d) Unsymmetric layup, fz = 50 N/m

Figure 5: Rotation of a cantilevered beam under the action of uniformly distributed load for different stacking sequences

5.3. Simply supported composite beam under distributed load Simply supported composite beam with three types of stacking sequences, namely symmetric, unsymmetric, and cross-ply, loaded with a uniformly distributed force fz = 500 N/m and simply supported composite beam with anti-symmetric layup subject to uniformly distributed moment mx = 50 N are considered in this 29

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subsection. Tables (11) - (14) and Figures (6) and (7) display the maximum deflections for four types of stacking

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sequences of simply supported composite beam. Again, in all cases an excellent agreement is observed between the exact results and those of the CCM. Numerical values in Table (11) demonstrate that there is no difference in results obtained using TBT and CBT for θx in the case of the symmetric layup. However, there is a difference between the two approaches for the out-of-plane displacement w, and it is especially significant for shorter beams, varying from 0.13% when `/h = 50 up to 11.45% when `/h = 5. As in previous examples, in the case of the anti-symmetric stacking sequence there is no difference for the values obtained from Timoshenko and Euler-Bernoulli theories for both axial displacement u and twist θx . Table (13) shows that in the case of the cross-ply layup there is no difference in results obtained for axial displacement u by applying the two theories while the difference for the out-of-plane displacement w increases significantly with decreasing length-to-thickness ratio from 0.45% when `/h = 50 up to 31.12% when `/h = 5. As can be observed from Table (14), for the case of unsymmetric stacking sequence the differences between the results obtained based on the two approaches for the axial and in-plane displacements u and v, and twist θx remain constant for all length-to-thickness ratios and are exactly the same as in the previous two examples. The difference in results for the out-of-plane displacement w, obtained by applying TBT and CBT, exhibits the same qualitative behaviour as the cantilevered beam, but is more pronounced with values varying from 13.87% when `/h = 50 to 26.07% when `/h = 5. It also should be noted that in the current example the loading and boundary conditions are symmetric and in the case of symmetric, cross-ply and unsymmetric stacking sequences, the maximum displacement for w occurs at the middle of the beam (i.e. at x = `/2).

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point.

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However, due to the anisotropy, the maximum deformations in u, v, θx , θy , and θz do not occur at the same

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Table 11: Tip deflection of symmetric simply supported beam under uniformly distributed load

TBT

CBT Exact

CCM (DB)

Difference (%)

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Exact (DB)

`/h = 5

w

0.0000019879

θx

0.0000137194

θy

0.0001820243

0.0000019879

0.0000017602

11.45

0.0000137194

0.0000137194

0.00

0.0001820243

–

–

0.0000290739

0.0000281633

3.13

0.0001097551

0.0001097551

0.00

0.0014561943

–

–

0.0004542550

0.0004506124

0.80

0.0008780404

0.0008780404

0.00

0.0116495543

–

–

0.0176248149

0.0176020487

0.13

0.0137193807

0.0137193807

0.00

0.1820242855

–

–

`/h = 10

w

0.0000290739

θx

0.0001097551

θy

0.0014561943

`/h = 20

w

0.0004542550

θx

0.0008780404

θy

0.0116495543

`/h = 50

0.0176248149

θx

0.0137193807

θy

0.1820242856

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w

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Table 12: Tip deflection of anti-symmetric simply supported beam under uniformly distributed load

TBT

CBT Exact

CCM (DB)

Difference (%)

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Exact (DB)

`/h = 5

u

0.0000007822

θx

0.0012737665

0.0000007822

0.0000007822

0.00

0.0012737665

0.0012737665

0.00

`/h = 10

u

0.0000031286

θx

0.0050950659

0.0000031286

0.0000031286

0.00

0.0050950659

0.0050950659

0.00

`/h = 20

u

0.0000125145

θx

0.0203802638

0.0000125145

0.0000125145

0.00

0.0203802638

0.0203802638

0.00

`/h = 50

0.0000782158

θx

0.1273766485

0.0000782158

0.0000782158

0.00

0.1273766485

0.1273766485

0.00

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u

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Table 13: Tip deflection of cross-ply simply supported beam under uniformly distributed load

TBT

CBT Exact

CCM (DB)

Difference (%)

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Exact (DB)

`/h = 5

u

0.0000000239

w

0.0000007287

θy

0.0000481483

0.0000000239

0.0000000239

0.00

0.0000007287

0.0000005019

31.12

0.0000481483

–

–

0.0000001915

0.0000001915

0.00

0.0000089379

0.0000080309

10.15

0.0003851863

–

–

0.0000015317

0.0000015317

0.00

0.0001321229

0.0001284949

2.75

0.0030814900

–

–

0.0000239335

0.0000239335

0.00

0.0050420105

0.0050193358

0.45

0.0481482815

–

–

`/h = 10

u

0.0000001915

w

0.0000089379

θy

0.0003851863

`/h = 20

u

0.0000015317

w

0.0001321229

θy

0.0030814900

`/h = 50

0.0000239335

w

0.0050420105

θy

0.0481482815

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u

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Table 14: Tip deflection of unsymmetric simply supported beam under uniformly distributed load

TBT

CBT Exact

CCM (DB)

Difference (%)

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Exact (DB)

`/h = 5

u

0.0000000170

v

0.0000000185

w

0.0000016171

θx

0.0000127569

θy

0.0001407863

θz

0.0000113175

0.0000000170

0.0000000220

29.30

0.0000000185

0

100

0.0000016171

0.0000011956

26.07

0.0000127569

0.0000126956

0.48

0.0001407863

–

–

0.0000113175

–

–

0.0000001364

0.0000001763

29.30

0.0000001482

0

100

0.0000230984

0.0000191296

17.18

0.0001020554

0.0001015647

0.48

0.0011262903

–

–

0.0000452700

–

–

0.0000010909

0.0000014105

29.30

0.0000011854

0

100

0.0003584741

0.0003060735

14.62

`/h = 10

u

0.0000001364

v

0.0000001482

w

0.0000230984

θx

0.0001020554

θy

0.0011262903

θz

0.0000452700

`/h = 20

0.0000010909

v

0.0000011854

w

0.0003584741

θx

0.0008164432

0.0008164432

0.0008125175

0.48

θy

0.0090103221

0.0090103221

–

–

θz

0.0001810801

0.0001810801

–

–

0.0000170452

0.0000220396

29.30

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u

`/h = 50

0.0000170452

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u v

0.0000185211

0.0000185211

0

100

w

0.0138814744

0.0138814744

0.0119559967

13.87

θx

0.0127569247

0.0127569247

0.0126955857

0.48

θy

0.1407862822

0.1407862822

–

–

θz

0.0011317504

0.0011317504

–

–

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10-5

10-6 3.5

3

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3 2.5

2.5

2

2

1.5

1.5

1

1

0.5

0.5

0

0

0

0.01

0.02

0.03

0.04

0.05

0.06

0

(a) Symmetric layup, fz = 500 N/m 10-6

0.01

0.02

0.03

0.04

0.05

0.06

(b) Anti-symmetric layup, mx = 50 N

10-5

3

10

2.5

8

2

1.5

6

1

2

0

-2 0.01

0.02

0.03

0.04

0.05

0 -0.5 -1 -1.5

0.06

0

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0

0.5

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4

(c) Cross-ply layup, fz = 500 N/m

0.01

0.02

0.03

0.04

0.05

0.06

(d) Unsymmetric layup, fz = 500 N/m

Figure 6: Displacement of a beam simply supported at both ends under the action of uniformly distributed load for different stacking sequences

35

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10-3

2

10-3

6

1.5

1

lP repro of

5

4

0.5

3

0

2

-0.5

1

-1

-1.5

0

0

0.01

0.02

0.03

0.04

0.05

0.06

0

(a) Symmetric layup, fz = 500 N/m 10-4

4

0.02

0.03

0.04

0.05

0.06

(b) Anti-symmetric layup, mx = 50 N

10-3

1.5

3

0.01

1

2

0.5

1 0 -1 -2 -3 -4 0.01

0.02

0.03

0.04

0.05

-0.5

-1

0.06

0

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0

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0

(c) Cross-ply layup, fz = 500 N/m

0.01

0.02

0.03

0.04

0.05

0.06

(d) Unsymmetric layup, fz = 500 N/m

Figure 7: Rotation of a beam simply supported at both ends under the action of uniformly distributed load for different stacking sequences

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6. Conclusions In this paper the exact analytical solution for the static deflection analysis of fully coupled composite

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Timoshenko beams under uniformly distributed and tip loads is presented. The governing equations and the boundary conditions are obtained from a variational principle and expressed in a compact matrix form. The exact solution is derived by the method of direct integration. It is also mentioned that for the case of a cantilevered composite beam subject to uniform tip loads, the proposed exact analytical solution is equivalent to the exact solution from the intrinsic formulation. The application of the proposed model is illustrated on a composite Timoshenko beam subject to clamped-free and simply supported boundary conditions and uniformly distributed and tip loads. In addition, obtained results are compared with those obtained from the exact solution based on classical Euler-Bernoulli theory. The effect of the length-tothickness ratio on the deflection response of the beams is highlighted, and the fact that the Euler-Bernoulli model is insufficient at explaining the behaviour of thick beams is emphasised. Moreover, presented results are validated by applying the high-accuracy Chebyshev collocation method. In the proposed formulation, the entries of the stiffness matrix are expressed by engineering constants which makes the model applicable to various kinds of materials, shapes of cross section and types of boundary conditions thus it can be used for engineering applications and design purposes. The exact analytical solution can also serve as a benchmark for validating results obtained from various analytical and numerical methods as well as for convergence studies.

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References

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[5] S.-Y. Lee, Y. Kuo, Static analysis of nonuniform Timoshenko beams, Computers & structures 46 (5) (1993) 813–820. [6] C. M. Wang, Timoshenko beam-bending solutions in terms of Euler-Bernoulli solutions, Journal of engineering mechanics 121 (6) (1995) 763–765.

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[11] G. Jeleni´ c, E. Papa, Exact solution of 3D Timoshenko beam problem using linked interpolation of arbitrary order, Archive of applied mechanics 81 (2) (2009) 171–183. [12] X.-F. Li, A unified approach for analyzing static and dynamic behaviors of functionally graded Timoshenko and Euler–

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Bernoulli beams, Journal of Sound and vibration 318 (4-5) (2008) 1210–1229. [13] V. Adamek, F. Valeˇs, Analytical solution for a heterogeneous Timoshenko beam subjected to an arbitrary dynamic transverse load, European Journal of Mechanics-A/Solids 49 (2015) 373–381.

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[15] Y. Pei, P. Geng, L. Li, A modified uncoupled lower-order theory for FG beams, Archive of Applied Mechanics (2018) 1–14. [16] N. T. B. Phuong, T. M. Tu, H. T. Phuong, N. Van Long, Bending analysis of functionally graded beam with porosities resting on elastic foundation based on neutral surface position, Journal of Science and Technology in Civil Engineering (STCE)-NUCE 13 (1) (2019) 33–45.

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Appendix A. Exact solution of fully coupled composite Euler-Bernoulli beam

The exact solution for the static deflection of a fully coupled composite Euler-Bernoulli beam under

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uniformly distributed loads can be obtained as (Doeva et al. [26]):     fz w 1 1 1  = KD   x4 + C 1 x3 + C 2 x2 + C 3 x + C 4 , 24 6 2 fy v

      u fz 3 1 −1 f 1 1   = − KB   x + A BC 1 x2 − A−1  x  x2 + C 5 x + C 6 , 6 2 2 θx fy mx

(A.1)

(A.2)

where C i , i = 1, 2, . . . , 6, are the vectors of unknown coefficients to be determined. The definitions of KB

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and KD are given in Eqn. (29) and:





A=

EA

SET

SET

GJ

B=

SEF

−SEL

SF T

−SLT





D=

,



,

EIy

−SF L

−SF L

EIz

39

(A.3a)



.

(A.3b)

(A.3c)

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 A fully coupled displacement-based composite Timoshenko beam model is presented  Exact analytical solution is obtained using direct integration technique  The Chebyshev collocation method is alternatively applied as a numerical solution

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Declaration of interests

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☒ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests:

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Olga Doeva: Methodology, Software, Writing- Original draft preparation, Writing- Reviewing and Editing. Pedram Khaneh Masjedi: Conceptualisation, Methodology, Software, WritingReviewing and Editing. Paul M. Weaver: Supervision, Writing- Reviewing and Editing.