A Novel Approach for Lateral Buckling Assessment of Double Tapered Thin-Walled Laminated Composite I-Beams

Document Type : Research Paper

Author

Department of Civil Engineering, Faculty of Engineering, University of Kashan, Kashan, Iran

Abstract

The purpose of this paper is to introduce a simple and novel method for discussing the lateral-torsional stability of thin-walled symmetric balanced laminated beams with varying I-section. Based on the classic lamination theory and Vlasov‘s model, the total potential energy for the flexural displacements and the twist angle is established. The variational formulation is then constructed only in terms of the angle of twist using an auxiliary function. The buckling loads are finally determined by applying the Ritz method. To demonstrate the accuracy of the proposed formulation, the analytical solutions for a sample case of tapered I-beam are compared with results obtained from ANSYS's shell element. Moreover, this new procedure is very efficient in reducing the computational effort. Eventually, based on a selected load, the influences of some parameters such as the tapering ratios, transverse load position, and fiber orientation on lateral stability resistance of composite tapered I-beams under simply supported end conditions are discussed in detail. The results show that the lateral buckling resistance of composite beam with tapered I-section decreases significantly as the fiber angle in both flanges is rotated off-axis. Also, the maximum lateral buckling load for simply supported web and flanges tapered beam under uniformly distributed load is obtained by placing fibersin the web and  in both flanges.

Keywords


A Novel Approach for Lateral Buckling Assessment of Double Tapered Thin-Walled Laminated Composite I-Beams
M. Soltani*
Department of Civil Engineering, Faculty of Engineering, University of Kashan, Kashan, Iran.


KEYWORDS        ABSTRACT
Lateral buckling
Laminated composites
Tapered I-beam
Classical lamination theory
Ritz method        The purpose of this paper is to introduce a simple and novel method for discussing the lateral-torsional stability of thin-walled symmetric balanced laminated beams with varying I-section. Based on the classic lamination theory and Vlasov‘s model, the total potential energy for the flexural displacements and the twist angle is established. The variational formulation is then constructed only in terms of the angle of twist using an auxiliary function. The buckling loads are finally determined by applying the Ritz method. To demonstrate the accuracy of the proposed formulation, the analytical solutions for a sample case of tapered I-beam are compared with results obtained from ANSYS's shell element. Moreover, this new procedure is very efficient in reducing the computational effort. Eventually, based on a selected load, the influences of some parameters such as the tapering ratios, transverse load position, and fiber orientation on lateral stability resistance of composite tapered I-beams under simply supported end conditions are discussed in detail. The results show that the lateral buckling resistance of composite beam with tapered I-section decreases significantly as the fiber angle in both flanges is rotated off-axis. Also, the maximum lateral buckling load for simply supported web and flanges tapered beam under uniformly distributed load is obtained by placing fibers±45^(∘ )in the web and 0^∘ in both flanges.
 
Introduction
The use of thin-walled structural components in the most innovative engineering fields, including aircraft wings, helicopters, turbine blades, steel frames, and decks of the bridge, has become increasingly common throughout the years. Although thin-walled open cross-sections have some outstanding features, such as the high value of stiffness-to-weight and strength-to-weight ratios, they have some drawbacks, such as poor torsion rigidity and low out-of-plane bending resistance. As a result of these disadvantages, a laterally unbraced thin-walled beam subjected to bending around its strong axis may suddenly buckle in a flexural–torsional mode. This situation hence results in a lower stability strength. Moreover, in recent years, the application of thin-walled beams with variable cross-section has been extensively come into vogue in many advanced industries such as aeronautical and mechanical installations due to the importance of having an optimum distribution of weight and strength. With the development of fabrication processes, specifically pultrusion, thin-walled structural components made up of fiber-reinforced composite materials in aeronautical and mechanical installations have become increasingly common throughout the years. The main reason for this increase is the desirable feature of composites, such as high fatigue resistance, durability, corrosion resistance, and high stiffness-to-weight and strength-to-weight ratios. A review of the literature shows that different researchers have conducted several investigations to study the mechanical responses of thin-walled structural components made up of homogenous and/or composite materials. In the following, a short description of a few of them is presented.
Rajasekaran and Nalinaa [1] assessed the vibrational characteristics and buckling behavior of non-prismatic composite spatial members with generic thin-walled sections via the finite element method within the context of non-linear strain displacement relationship. Based on a geometrically non-linear theory and the assumptions of large displacements and rotations, Machado and Cortinez [2-5] studied the free vibrational response and buckling behavior of composite beams with doubly-symmetric thin-walled open cross-section loaded by arbitrary external forces. Using the finite element methodology, the flexural-torsional coupled free vibrational behavior and buckling problem of thin-walled composite beams were precisely investigated by Vo and Lee [6], considering the impacts of axial load on the vibration characteristics. Based on linear fracture mechanics and the Castigliano theory, the influence of edge crack ratio and position on free vibration responses and lateral buckling strength of laminated composite slender beam was studied by Karaagac et al. [7]. To estimate the buckling resistance of simply supported thin-walled structural members made of Fiber Reinforced Polymer (FRP) loaded by axial and uniformly transverse forces, Ascione et al. [8] developed a mechanical model based on the assumptions of small strains and moderate rotations. In order to exhaustively examine the static and dynamic responses of beams made from FG piezoelectric materials, an innovative and improved three-noded beam element was formulated by Lezgy-Nazargah et al. [9]. Moreover, the elastic behavior of functionally graded piezomagnetic cylinders was studied in Refs. [10-13] under thermal, longitudinally non-uniform pressure, magnetic and mechanical loads. Using updated Lagrangian formulation, Lanc et al. [14] analyzed the lateral buckling and post-buckling behavior of functionally graded materials (FGMs), thin-walled beams with mono-symmetric I and channel sections for various boundary conditions based on Vlasov theory for thin-walled cross-sections and Euler–Bernoulli beam assumptions. Within the framework of finite stain theory, Mohandes and Ghasemi [15, 16] investigated nonlinear free vibrational characteristics of laminated beams through different shear deformation theories. Based on Ritz and Galerkin's methods, Saoula et al. [17] studied the stability resistance of laterally unrestrained simply supported thin-walled box beam elements subjected to combined bending and axial forces. Ghasemi and Mohandes [18-20] assessed the mechanical response including interlaminar normal and shear stresses of laminated composite size-dependent beam in the thermal and/or subjected to transverse load, according to finite strain assumption and a modified couple stress theory. To facilitate and increase the speed of mathematical computations to perform the elasto-plastic analysis of thin-walled beams, an innovative finite element formulation was suggested by Lezgy-Nazargah [21] based on the theory of generalized layered global-local beam (GLGB). Moreover, Nguyen et al. [22, 23] proposed a new finite element solution for computing lateral-torsional critical loads of FGM thin-walled beams with singly symmetric open sections. The precise free vibrational behavior of a doubly curved piezoelectric nanoshell resting on Pasternak’s foundation was completely studied by Arefi [24] according to the first-order shear deformation theory. In another study, Ahmadi and Rasheed [25] employed the generalized semi-analytical technique to analyze the lateral-torsional buckling of anisotropic laminated beams with the rectangular thin-walled cross-section under simply supported end supports based on the classical laminated plate theory. Within the context of first-order shear deformation theory and using a semi-analytical solution methodology, the mechanical response of thin-walled laminated beams with constant open and/or closed cross-sections was assessed by Wackerfuß and Kroker [26]. In addition, Arefi and Zenkor [27] applied the higher-order sinusoidal shear deformation beam theory to extract the governing equations of sandwich microbeams with two piezoelectric face-layers and a homogeneous core loaded by a transverse force along with an electrical one. Using Navier’s solution, Ghasemi and Meskini [28] assessed the free vibrational response of simply supported porous laminated rotating circular cylindrical shells within the framework of Love’s shell theory. The lateral-torsional stability analysis of tapered thin-walled beam with arbitrary open cross-section under different boundary conditions was completely investigated by Soltani et al. [29-33] using different numerical methodologies. Ghasemi et al. [34-37] analyzed fiber-metal laminate (FML) cylindrical shells under different boundary conditions. For further numerical techniques-based investigations on the static and dynamic analyses, the reader is referred to [38-45] for composite beams subjected to different loading cases and end conditions.
Due to the application of composite structural members with thin-walled cross-sections in the design of sensitive and modern structures such as aircraft wings, helicopters, and turbine blades, it is necessary to study the problem of sandwich laminated thin-walled beams with varying cross-section. For this reason, the main aim of the present work is to analyze the lateral stability of laminated doubly-symmetric tapered I-beams with symmetrical lay-up for all section walls by presenting an innovative analytical solution. Within the classic lamination theory framework and Vlasov’s model without considering the shear deformation, the total potential energy for the vertical and lateral deformations and the twist angle are determined. Note that bending–twisting coupling is accounted for in our formulation. Based on the method proposed by Soltani et al. [31-33] and using an auxiliary function, the variational statement is then obtained only in terms of the twist angle. The acquired formulation can be applied for estimating the lateral-buckling load of composite I-beam under different boundary conditions. However, simply supported beam with free warping at both supports is contemplated here. The trigonometric function, which satisfies the simply supported beam end conditions, is thus used to acquire the analytical solutions through the Ritz method. The superiority of the proposed approach is to simplify and drastically decrease the essential computational efforts to calculate the lateral buckling load of symmetrically laminated thin-walled beams with varying cross-sections. To check the accuracy and the efficiency of the proposed methodology, our results are compared with numerical ones from the ANSYS code, and a good agreement is observed. For measuring the effects of web and flanges non-uniformity ratios, load height position, and fiber angle on lateral-torsional stability of simply supported laminated composite web and/or flanges tapered I-beams under uniformly distributed load, an exhaustive numerical example is finally presented. 
Derivation of formulation
Geometrical description of double tapered I-beam
In the current study, the linear lateral-torsional buckling analysis is conducted for a laminated composite I-beam with a tapered web and flanges. The right-hand Cartesian coordinate system, with x as the initial longitudinal axis measured from the left end of the beam, the y-axis in the lateral direction, and the z-axis along the vertical direction, are considered as indicated in Fig. 1. The origin of these axes (O) is located at the centroid of the doubly-symmetric I-section. The symmetrically laminated I-beam is initially subjected to a laterally distributed load qz in z-direction along with a line (PP'). The arbitrarily distributed force is thus applied on point P located on the section contour with the eccentricity zP(x) (Fig. 1). 
In this study, Vlasov's model for non-uniform torsion is applied for the description of the displacement field of a point on the section contour. Based on this classical theory, only slender and long beams are considered and there are no shear deformations in the mean surface of the section. It is further admitted that the cross-section does not change shape during deflection. This means that the cross-section is rigid in its own plane and consequently no distortional deformations occur. From these assumptions, the displacement fields for an arbitrary point on the beam can be expressed as follows [46]:
U(x,y,z)=u_0 (x)-y dv(x)/dx
-z dw(x)/dx-ω(y,z)  dθ(x)/dx           (1)
V(x,y,z)=v(x)-zθ(x)    
W(x,y,z)=w(x)+yθ(x)    
In these equations, U is the axial displacement and displacement components V and W represent lateral and vertical displacements (in direction y and z). Note that the axial displacement U is extracted from the nullity of shear deformations in the main surface. The term ω(y,z) signifies a cross-section variable that is called the warping function, which can be defined based on Vlasov’s torsion theory. u0, v, and w are the 3D displacement components of the reference point O. Ɵ is twisting angle. 
Total potential energy for a composite beam with tapered I-section
The equilibrium equations for beam are derived from the potential energy condition given by: 
δΠ=δ(U_l+U_0-W_e )=0    (2)
δ illustrates a virtual variation in the last formulation. U_l represents the elastic strain energy, U_0 expresses the strain energy due to effects of the initial stresses and We is the work of the applied loads. According to the applied load, δΠ is reduced to the following terms:
 

Fig. 1. Geometrical scheme for doubly symmetric I-section beam subjected to an arbitrarily distributed external load

δΠ=∫_0^L▒∫_A▒(σ_xx δε_xx^l+σ_xy δγ_xy^l+σ_xz δγ_xz^l )dAdx
+∫_0^L▒∫_A▒(σ_xx^0 δε_xx^*+σ_xy^0 δγ_xy^*+σ_xz^0 δγ_xz^* )dAdx
-∫_0^L▒〖q_z δw_p dx〗=0    (3)
in which, L and A express the beam length and the cross-sectional area, respectively. (δε_xx^l,δγ_xz^l,δγ_xy^l) and (δε_xx^*,δγ_xz^*,δγ_xy^*) are the variation of the linear and the non-linear parts of the strain tensor, respectively. σ_ijdenotes the Piola–Kirchhoff stress tensor, and σ_ij^0 is the initial stress, also called the pre-buckling stress. Based on the assumption of Green’s strain-tensor, the linear and the non-linear parts of strain-displacement relations and their first variation are [29, 31]:
ε_xx^l=u_0^'-yv^″-zw^″-ωθ^″
δε_xx^l=δu_0^'-yδv^″-zδw^″-ωδθ^″    (4)
ε_xx^*=1/2 (v'^2+w'^2+r^2 θ'^2 )
+yw^' θ^'-zv^' θ^'
δε_xx^*=v^' δv^'+w^' δw^'+r^2 θ^' δθ^' "
" "  "+yθ'δw'+yw'δθ'-zθ'δv'-zv'δθ'    
with r^2=y^2+z^2    
γ_xz^l=2ε_xz^l=(y-∂ω/∂z) θ^'
δγ_xz^l=(y-∂ω/∂z)δθ^'    
γ_xz^*=-(v_^'+θ^' z)θ
δγ_xz^l=-θδv^'-v^' δθ
             -zθδθ^'-zθ^' δθ    
γ_xy^l=2ε_xy^l=-(z+∂ω/∂y) θ^'
δγ_xy^l=-(z+∂ω/∂y)δθ^'    
γ_xy^*=(w_^'+θ^' y)θ
δγ_xy^*=θδw^'+w^' δθ+yθδθ^'+yθ^' δθ    
In this study, the applied external loads in the z-direction cause the pre-buckling shear force (V_z^0) and the pre-buckling bending moment (M_y^0) loading. Both these forces induce the pre-buckling axial and shear stresses as 
σ_xx^0=-(M_y^0)/I_y  z,σ_xz^0=(V_z^0)/A=-〖M_y^0〗^'/A,
 σ_xy^0=0"  "     (5)
where σ_xx^0and (σ_xz^0,σ_xy^0) are the pre-buckling bending stress and shear stress often called the initial stresses. In Eq. (3), w_P is the vertical displacement of point P. According to kinematics used in Asgarian et al. [29] and by adopting the quadratic approximation, the vertical displacement of the point P and its first variation are as:
w_P=w-z_P  θ^2/2→ δw_P=δw-z_P θδθ    (6)
In this equation, zP is used to imply the eccentricity of the applied loads from the centroid of the cross-section. Substituting equations (4) to (6) into relation (3), the expression of the virtual potential energy can be carried out as:
δΠ=∫_0^L▒∫_A▒〖σ_xx (█(&δu_0^'-yδv^″-zδw^″@&-ωδθ^″ ))dAdx〗    
+∫_0^L▒∫_A▒〖τ_xy (-(z+∂ω/∂y)δθ^' )dAdx〗    
+∫_0^L▒∫_A▒〖τ_xz ((y-∂ω/∂z)δθ^' )dAdx〗    (7)
+∫_0^L▒∫_A▒〖(-(M_y^0)/I_y  z)(█(v^' δv^'+w^' δw^'@+r^2 θ^' δθ^'@+yθ'δw'+yw'δθ'@-zθ'δv'-zv'δθ^' ))dAdx〗    
+∫_0^L▒∫_A▒〖(-〖M_y^0〗^'/A)(█(-θδv^'-v^' δθ@-zθδθ^'-zθ^' δθ))dAdx〗    
-∫_0^L▒(q_z δw-M_t θδθ)dx=0    
in which, M_t=q_z z_P denotes the second order torsion moments due to load eccentricity. The variation of strain energy can be formulated in terms of section forces acting on cross-sectional contour of the elastic member in the buckled configuration. The section stress resultants are presented by the following expressions: 
N=∫_A▒〖σ_xx dA〗    (8a)
M_y=∫_A▒〖σ_xx zdA〗    (8b)
M_z=-∫_A▒〖σ_xx ydA〗    (8c)
B_ω=-∫_A▒〖σ_xx ωdA〗    (8d)
M_sv=∫_A▒(τ_xz (y-∂ω/∂z)-τ_xy (z+∂ω/∂y))dA    (8e)
where N is the axial force applied at end member. My and Mz denote the bending moments about major and minor axes, respectively. B is the bi-moment. Msv is the St-Venant torsion moment. In this stage, by integrating Eq. (7) over the cross-section area of the beam and using relations (8a)–(8e), the final form of the variation of total potential energy ) is acquired as:
δΠ=∫_L▒(█(Nδu_0^'+M_z δv^″@-M_y δw^″+B_ω δθ^″ ))dx
+∫_0^L▒(M_sv δθ^' )  dx
+∫_0^L▒(M_y^0 (θ'δv'+v'δθ'))dx
+∫_0^L▒(〖M_y^0〗^' (θδv^'+v^' δθ))dx
-∫_0^L▒(q_z δw-q_z z_P θδθ)dx=0    (9)
or 
δΠ=
∫_L▒(█(Nδu_0^'+M_z δv^″@-M_y δw^″+B_ω δθ^″+M_sv δθ^' ))dx
+∫_0^L▒(-M_y^0 v^″ δθ-M_y^0 θδv^″ )  dx
-∫_0^L▒(q_z δw-q_z z_P θδθ)dx=0    (10)
The present model is applied in the case of balanced and symmetrical lay-ups of the web and both flanges. In the context of classical laminated plate theory and substitution Eq. (4) into Eq. (8), the stress resultants of symmetrically balanced laminates are derived in terms of displacement components as [47]
N=(EA)_com u_0^'    (11a)
M_z=(EI_z )_com v^″    (11b)
M_y=-(EI_y )_com w^″    (11c)
B_ω=(EI_ω )_com θ^″    (11d)
M_sv=(GJ)_com θ^'    (11e)
where (EA)_comdenotes axial rigidity. (EI_y )_comand (EI_z )_com represent the flexural rigidities of the y- and z-axes, respectively. (EI_ω )_comand (GJ)_comare, respectively, warping and torsional rigidities of composite thin-walled beams with doubly symmetric I-section, defined by [47]:
(EA)_com=2bA_11^f+dA_11^w    (12a)
(EI_y )_com=2bD_11^f+d^2/2 bA_11^f
+d^3/12 A_11^w    (12b)
(EI_z )_com=b^3/6 A_11^f+dD_11^w    (12c)
(EI_ω )_com=(d^2/4 A_11^f+D_11^f)b^3/6+d^3/12 D_11^w    (12d)
(GJ)_com=4(2bD_66^f+dD_66^w)    (12e)
That indexes f and w refer to the web and the flange of the beam cross-section, respectively. A_11^f,A_11^wand D_11^f,D_11^w,D_66^f,D_66^w are the matrices of extensional and bending stiffness of both flanges and web, respectively, which are calculated as
(A_ij^f,D_ij^f)=∫▒〖Q_ij^f (1,z^2)dz〗    (13)
(A_ij^w,D_ij^w)=∫▒〖Q_ij^w (1,y^2)dy〗    
where Q_ij^fand Q_ij^ware the transformed reduced stiffness related to the flanges and web, respectively. Since the breadth of the flanges and the height of the web are assumed to vary linearly along the length of the beam, the above stiffness terms Eq. (12) are not constant.
Substituting Eq. (11) into (10), one gets the variation of the total potential is then a function of the virtual displacements δu_0,δv,δwand δθ, and of their derivatives as
δΠ=
∫_L▒(█((EA)_com u_0^' δu_0^'+(EI_z )_com v^″ δv^″@+(EI_y )_com w^″ δw^″@+(EI_ω )_com θ^″ δθ^″+(GJ)_com θ^' δθ^' ))dx    
+∫_0^L▒(-M_y^0 v^″ δθ-M_y^0 θδv^″ )  dx    (14)
-∫_0^L▒(q_z δw-q_z z_P θδθ)dx=0    
or equivalently
∫_L▒((EA)_com u_0^' δu_0^' )dx=0    (15a)
∫_L▒〖((EI_y )_com w^″ δw^″-q_z δw)dx=0〗    (15b)
∫_L▒〖((EI_z )_com v^″-M_y^0 θ)δv^″ dx=0〗    (15c)
∫_L▒(█(&(EI_ω )_com θ^″ δθ^″+(GJ)_com θ^' δθ^'@&-M_y^0 v^″ δθ-z_P q_z θδθ))dx
=0    (15d)
Based on the straightforward methodology presented by Soltani et al. [29-33], Eq. (15c) can be rewritten in the following form for any acceptable lateral buckled configuration:
v^″=(M_y^0)/((EI_z )_com ) θ    (16)
whose substitution in Eq. (15d) enables its redefinition in an uncoupled form just dependent on the twist angle θ, independently from the lateral displacement v, i.e.
∫_L▒(█(&(EI_ω )_com θ^″ δθ^″+(GJ)_com θ^' δθ^'@&-〖M_y^0〗^2/((EI_z )_com ) θδθ-z_P q_z θδθ))dx=0    (17)
One of the most convenient methodologies to precisely estimate the lateral-torsional stability limit state of different types of continuous structural elements is the Rayleigh-Ritz method. Additionally, the main advantage of the present approach is that the Central Processing Unit (CPU) requires less time to acquire the solution with excellent precision [30, 31]. In the following, the Rayleigh-Ritz method is thus employed to obtain buckling load of non-prismatic columns. Based on the assumptions of this classical technique, it is essential to replace buckled shape of the elastic beam with appropriate deformation shapes of the element after lateral buckling satisfying both geometrical and natural boundary conditions of the system. Since the resulting weak form of lateral equilibrium equations is only in terms of the twisting angle, the approximate buckled shape of the beam must satisfy all the essential boundary conditions for torsional rotation.
In the case of cantilevers, the left end (fixed one) of the beam is prevented from freely warping (θ(0)=θ^' (0)=0), while, the right end of the I-section is free to warp (θ^″ (L)=0). Thus, the first displacement mode in torsion can be approximated by trigonometric functions as [48, 49]
θ(x)=θ_1 (1-cos⁡(πx/2L) )    (18)
+θ_2 (1-cos⁡(3πx/2L) )    
+θ_3 (1-cos⁡(5πx/2L))    
For simply supported beams, the twist angle equals zero at both ends (θ(0)=θ(L)=0). Further, the ends of the I-shape beam are free to warp (θ^″ (0)=θ^″ (L)=0). Therefore, the torsional rotation mode shape becomes [48, 49]:
θ(x)=θ_1  sin⁡(πx/L)    (19)
For fixed-fixed cases, both end supports are prevented from freely warping. This means that the twist angle and the rate of twist at the fixed support are null (θ(0)=θ(L)=θ^' (0)=θ^' (L)=0). The expression for the angle of twist can be approximated as [48, 49]
θ(x)=θ_1 (1-cos⁡(2πx/L))    (20)
+θ_2 (1-cos⁡(3πx/L))    
+θ_3 (1-cos⁡(5πx/L))    
In the above equations, θ_i (i=1,2,3) are the Ritz coefficients. It is important to note that the final variational formulation (Eq. (17)) is applicable for lateral stability analysis of composited tapered I-beam under various end conditions [29, 30, 33, 48], but, in the next section, only simply supported beam is considered for implementation of the Ritz method.
 Lateral buckling analysis
To demonstrate the application of Eq. (17), simply supported beam subjected to uniformly distributed load is considered, as shown in Fig. 2. For this loading condition, the bending moment distribution through the x-axis is given by:
M_y^0=q_z  L^2/2(x/L-x^2/L^2 )    (21)
The substitution of the expression of the internal bending moment, the corresponding buckled shape function of the simply supported beam Eq. (19), and its derivatives into Eq. (17) yields.
∫_L▒((1/4 〖q_z〗^2  ((Lx-x^2 )^2)/((EI_z )_com )+z_P q_z)(sin⁡(  πx/L))^2 )dx    (22)
    
=(π/L )^2 ∫_L▒(█(&(π/L )^2 (EI_ω )_com (sin⁡(  πx/L))^2@&+(GJ)_com (cos⁡(  πx/L))^2 ))dx    
Again remind that in the present study, the height of the web and the width of both flanges are varying linearly such as
d(x)=d_0 (1+α)(x/L)+d_0    0≤α,β
(23)
b(x)=b_0 (1+β)(x/L)+b_0        
The termsβandαare the flanges and web tapering ratios, respectively, which are defined as β=b_L/b_0-1 and α=d_L/d_0-1. The subscripts 0 and L indicate dimensions at x=0 and x=L. Note that the prismatic cross-sections case is achieved by equating these two parameters (βandα) to zero. 
 

Fig. 2. Simply supported beam with uniformly 
distributed load.
The three stiffness quantities required for lateral-torsional stability of double tapered I-beam subjected to transverse loading are finally determined by substitution Eq. (23) into Eq. (12).
(EI_z )_com=1/6 (b_L (1+β)(x/L)+b_L ) A_11^f
+(d_L (1+α)(x/L)+d_L ) D_11^w    (24a)
(EI_ω )_com=1/6 (b_L (1+β)(x/L)+b_L )^3 D_11^f
+1/24 (d_L (1+α)(x/L)+d_L )^2⨯
(b_L (1+β)(x/L)+b_L )^3 A_11^f
+1/12 (d_L (1+α)(x/L)+d_L )^3 D_11^w    (24b)
(GJ)_com=8(b_L (1+β)(x/L)+b_L ) D_66^f
+4(d_L (1+α)(x/L)+d_L ) D_66^w    (24c)
In this study, it is supposed that uniformly transverse load can be applied at three different positions: the top flange, the centroid (shear center), and the bottom flange and, therefore
z_P (x)=z_P0 (1+α)(x/L)+z_P0    (25)
z_P0=-d_0/2,0,d_0/2    
By inserting the equation presented above Eq.s (24 and 25) into Eq. (22) and after appropriate integrations over the beam’s length, as well as some calculations, the critical values of qz can be obtained.
 Numerical Example
In the preceding section, an analytical methodology has been formulated to calculate the lateral-torsional buckling loads of thin-walled laminated composite beam with varying I-section. In this section, a comprehensive example is conducted to show the effects of significant parameters such as fiber angle orientation, loading position, and non-uniformity ratios (β,α) on the lateral buckling capacity of multi-layered composite tapered I-beam. To that end, a simply supported laminated double-tapered I-beam with symmetric lamination with a span of 8m subjected to uniformly distributed load is considered. At the left end section, both flanges are assumed to be 100mm wide (bL), and the web of the I-shape is 200mm deep (dL).  All section walls (flanges and web) are assumed to be laminated symmetrically concerning its mid-plane and made of 16 plies, each 0.25mm thick (total thickness: tw=tf=4 mm). All the layers are made of glass/epoxy (S2) with the following elastic properties [2-5]: 
E_1=48.3(GPa),〖       E〗_2=19.8(GPa), 
G_12=8.96(GPa),〖     G〗_13=8.96(GPa),
 G_23=6.19(GPa),〖    ν〗_12=0.27,
ν_13=0.27,〖                  ν〗_23=0.6.
where directions parallel and perpendicular to fibers are presented by subscripts ‘1’ and ‘2’, respectively. The main features of this type of glass epoxy composite are its high tensile strength, high resistance to damage, and improved impact resistance. Therefore, S-glass is widely used in the aerospace and building industries.
To have a better understanding of the numerical outcomes, the evaluated lateral-torsional buckling load is presented in the non-dimensional form as 
q_nor=(q_cr L^3)/(E_y t_w d_L^3 )     (26)
The current section is divided into two different subsections: the first one for verification of the formulation proposed herein, and the second one is for studying the influence of the above-mentioned factors on the linear lateral buckling behavior of the considered member. 
 Verification 
The absence of numerical studies on the thin-walled laminated composite beams with varying cross-sections, herein, and the accuracy of the predicted results based on the present formulation are checked with those acquired via SHELL281 of ANSYS code [50]. To this end, the lowest values of the non-dimensional lateral buckling parameter (q_nor) of the contemplated beam with variable thin-walled I-section for two different loading positions and various values of tapering ratios (β=α=0, 0.2, and 0.5) are evaluated and depicted in Table 1. In this section, six different stacking sequences are considered for the web and flanges. Fig. 3 schematically shows two different lay-up arrangements of both flanges of the I-section. Additionally, the relative errors  associated with the current approach are given by the following expression:
Δ=|q_nor^Ritz-q_nor^ANSYS |/(q_nor^ANSYS )×100    (27)
One observes a good agreement between the present analytical methodology and ANSYS simulations. The error between the Ritz method and ANSYS is below 10%.
 

 

Fig. 3: Ply stack of top and bottom flanges,
 (a) [0/90]_4s, (b) [(0_3/90)_2 ]_s.

 
 
Fig. 4. Variation of the lateral buckling parameter with respect to fiber angle change in both flanges for different tapering ratios, (a) load applied at the top flange, (b) load applied at the centroid.

 
Table 1. Comparative evaluation of the lateral buckling response for a simply supported composite web and
 flanges tapered beam under distributed load.
Tapering parameter    Lay-up    Loading position
        Top flange    Centroid
    Flanges    web    Present    ANSYS    Δ(%)    Present    ANSYS    Δ(%)
β_f=β_w=0    [0]_16    [0]_16    0.069    0.068    2.361    0.093    0.090    2.623
    [0/90]_4s    [±45]_8    0.064    0.060    5.742    0.081    0.077    5.223
    [(0_3/90)_2 ]_s    [±45]_8    0.071    0.067    5.675    0.091    0.087    5.163
    [0/90]_4s    [0/90]_4s    0.056    0.055    1.974    0.073    0.071    2.186
    [0/90]_4s    [(0/±45/90)_2 ]_s    0.060    0.058    2.939    0.077    0.075    2.953
    [(0_3/90)_2 ]_s    [(0/±45/90)_2 ]_s    0.067    0.065    2.990    0.087    0.084    3.015
β_f=β_w=0.2    [0]_16    [0]_16    0.089    0.086    3.610    0.122    0.117    4.478
    [0/90]_4s    [±45]_8    0.079    0.074    6.413    0.103    0.097    6.548
    [(0_3/90)_2 ]_s    [±45]_8    0.089    0.083    6.373    0.118    0.110    6.520
    [0/90]_4s    [0/90]_4s    0.071    0.069    3.058    0.095    0.091    3.886
    [0/90]_4s    [(0/±45/90)_2 ]_s    0.075    0.072    3.889    0.099    0.095    4.539
    [(0_3/90)_2 ]_s    [(0/±45/90)_2 ]_s    0.084    0.081    3.994    0.113    0.108    4.654
β_f=β_w=0.5    [0]_16    [0]_16    0.129    0.120    8.153    0.183    0.168    8.904
    [0/90]_4s    [±45]_8    0.110    0.100    9.301    0.148    0.135    9.569
    [(0_3/90)_2 ]_s    [±45]_8    0.125    0.114    9.594    0.171    0.156    9.829
    [0/90]_4s    [0/90]_4s    0.100    0.094    6.844    0.138    0.128    7.684
    [0/90]_4s    [(0/±45/90)_2 ]_s    0.105    0.098    7.323    0.143    0.132    8.047
    [(0_3/90)_2 ]_s    [(0/±45/90)_2 ]_s    0.120    0.111    7.788    0.166    0.153    8.454

 
Parametric Study
In this section, to assess the effect of fiber angle orientation, load height parameter, and web and flanges non-uniformity ratios, three cases are considered. It is necessary to note that the laminated non-uniform I-beam with equal web height and flanges width tapering ratios (β=α) under a uniform loading distribution applied on the top flange and the shear center is studied. 
The first case is that the web and flanges plates are made of 16 plies with unidirectional [0]_16lay-up for the web, while the top and bottom flanges are assumed to have symmetrical [±θ]_4Slay-ups. In this case, Fig. 4 exhibits the effect of the non-uniformity ratio (β=α) on the variation of the lateral buckling load parameters of laminated composite tapered I-beam considering the fiber angle () of its flanges. It is seen that the lateral stability capacity decreases monotonically with an increase in fiber angle (), where the effect of fiber angle change increases when  varies between 20 and 60. In the first lateral buckling mode, the laminated thin-walled beam becomes weaker and more unstable as the angle of orientation increases. The maximum lateral stability strength is thus obtained with unidirectional [0]_16lay-up for both flanges. In the case of a prismatic member, these results are confirmed in [47].
The next section is the same as before, except that the top and bottom flanges are considered unidirectional, [0]_16whereas the web laminate is assumed to have symmetric angle-ply laminations [±θ]_4S. Now, to study the influence of fiber angle orientation in the web of doubly-symmetric I-section, the variation of lateral buckling parameters for a simply supported laminated tapered beam with various tapering ratios versus fiber angle change is presented in Fig. 5. As the fiber angle is rotated off axis, the lateral buckling capacity is maximized at  = 45° and then sharply minimized at  = 90°. The higher lateral stability strength for prismatic and tapered I-beam is thus obtained by aligning the fiber orientation in the web around 45°. Similar behavior can also be observed for the two different loading positions.
The third case is that all section walls, including flanges and the web, are sixteen-layered symmetric angle-ply laminate [±θ]_4Swith equal thickness. For this lay-up, Fig. 6 displays the variation of the lowest lateral buckling parameter of simply supported composite I-beam considering the change in fiber orientation in the flanges and web for six different tapering parameters (β=α = 0, 0.2, 0.4, 0.6, 0.8, and 1). Fig. 5 shows that the lateral stability decreased steadily with increasing fiber angle. In addition, as shown in Fig. 6, the magnitude of critical parameters relating to the first lateral buckling mode decreases sharply for 20≤θ≤60 whereas, the lateral buckling resistance decreases slightly and reaches the minimum magnitude forθ>60. Similar trends in the results are also observed.
According to the illustrations, it is found out that for any value of fiber angle orientation, the stability of prismatic beam (β=α=0) and double tapered one with β=α=1 is the minimum and maximum, respectively. Hence, the lateral buckling parameter increases significantly with an increase in web and/or flange non-uniformity ratios ( and ) due to the enhancement of all geometrical characteristics of cross-section and, consequently, flexural stiffness and torsional rigidity of the elastic member. 
 

 

Fig. 5. Variation of the lateral buckling parameter with respect to fiber angle change in the web for different tapering ratios, (a) load applied at the top flange, (b) load applied at the centroid.

 

 

Fig. 6. Variation of the lateral buckling parameter with respect to fiber angle change in the web and both flanges for different tapering ratios, (a) load applied at the top flange, (b) load applied at the centroid.
Based on the results presented in Figs. 4-6, it can be concluded that the optimum fiber angles for achieving the highest lateral buckling resistance of simply supported web and flanges tapered beam under distributed load are ±45^∘ in the web and 0^∘in both flanges. This statement is reasonable since the flexural stiffness (EI_z )_comand the warping rigidity(EI_ω )_com, which are expressed in terms of unidirectional stiffness A11 and D11 (Eq. (24)), achieved their maximum magnitude by aligning the fibers at 0^∘ in both flanges and web. As these two stiffness quantities ((EI_z )_com and (EI_ω )_com) are mostly controlled by the fiber angle on the top and bottom flanges, hence the fiber orientation should be placed at 0^∘in both flanges to improve the lateral stability strength. Based on Eq. (24), the laminate torsional stiffness (GJ)_comis presented in terms of twisting stiffness in both flanges D_66^f and the webD_66^w, which are often maximum for fiber orientation around45^∘. Therefore, the linear lateral-torsional capacity becomes higher by placing the web fiber angle at ±45^∘.
In the following sections, the optimal stacking sequence is adopted. In order to investigate the influence of high load parameter (zP) on the lateral stability behavior, the variations of the lateral buckling load parameters (q_nor) of the thin-walled laminated beam with varying I-section versus tapering ratios (varying from 0 to 1) is plotted in Fig. 7 for the three loading positions.
The magnitude of the non-dimensional lateral-torsional buckling parameter (q_nor) for various combinations of web height and flange width tapering ratios, with different loading positions are listed in Table 2.
Fig. 6 and Table 2 show that the uniformly transverse load position has a significant effect on the stability strength of composite beams with varying doubly-symmetric I-section, especially for larger tapering ratios. Regarding these load cases, the lateral buckling strength will become best when the distributed load location is on the bottom flange due to the reduction of the rotation of the I-section from its origin, and the lower values are obtained when the load is applied on the top flange position.
 

Fig. 7. Variation of the lateral buckling parameter versus the non-uniformity ratio for three different transverse loading positions

Table 2: Lateral buckling parameter for laminated tapered I-beam with different tapering ratios and loading position (top and bottom flanges: [0]_16, web: [±45]_4s)
Loading position    β    Web tapering ratio (α)
        0    0.3    0.6    0.9
Top flange    0    0.078    0.083    0.088    0.093
    0.3    0.102    0.110    0.118    0.127
    0.6    0.132    0.144    0.157    0.170
    0.9    0.168    0.186    0.204    0.223
Centroid    0    0.101    0.109    0.118    0.127
    0.3    0.137    0.150    0.163    0.176
    0.6    0.181    0.200    0.219    0.239
    0.9    0.233    0.260    0.288    0.317
Bottom flange    0    0.131    0.144    0.157    0.169
    0.3    0.182    0.199    0.217    0.234
    0.6    0.241    0.264    0.287    0.310
    0.9    0.308    0.338    0.368    0.397
Moreover, the effect of the rate of flanges width tapering parameter () is seen to be higher than the effect of the web non-uniformity ratio (). The reason is that the lateral-torsional buckling phenomenon occurs concerning the minor axis moment of inertia.
Conclusions
In this paper, the lateral stability of a tapered thin-walled balanced laminated beam with an I-section is investigated by presenting an innovative analytical technique. It is assumed that all section walls (the web and both flanges) are laminated symmetrically concerning its mid-plane. Considering the bending–twisting coupling, the total potential energy is determined based on the assumptions of Vlasov’s model and the classic lamentation theory. The resulting formulation is thus obtained in terms of the vertical and lateral deformations and the twist angle. By presenting an auxiliary function, the variational statement is established only in terms of the twist angle. The Ritz method is finally employed to estimate the lateral buckling load. It is believed that the methodology proposed herein facilitates lateral stability analysis of symmetrically laminated thin-walled beams with varying cross-sections. Therefore, this new procedure is very efficient in reducing computational effort and also saving computing time. After verification, the impact of web and/or flanges tapering ratios, fiber angle, and transverse loading position on lateral-torsional stability of simply supported composite tapered I-beam is exhaustively surveyed. According to the numerical outcomes, it is concluded that the mentioned parameters play significant roles in the stability strength of laminated tapered I-beam. For all transverse loading positions, it was found that the lateral buckling parameter of composite beam with tapered I-section decreases as the fiber orientation in both flanges is rotated off axis, whereas the lateral-torsional buckling resistance increases as the web and/or flanges tapering ratios increase. It is also illustrated that the effect of the flange tapering parameter (β) on the buckling capacity is higher than that of the web one (α).
Additionally, it is observed that the buckling capacity of a simply supported laminated beam with doubly-symmetric I-section will become best when the uniformly distributed load is applied on the bottom flange. Also, it can be interpreted that the effect of fiber angle change in both flanges on lateral buckling strength is significant. Finally, it can be concluded that the maximum lateral buckling load for simply supported web and flanges tapered beam under uniformly distributed load is obtained by placing fibers at ±45^∘ in the web and 0^∘in both flanges.
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[1]   Rajasekaran, S. and Nalinaa, K., 2005. Stability and vibration analysis of non-prismatic thin-walled composite spatial members of generic section. International Journal of Structural Stability and Dynamics, 5(04), pp.489-520.
[2]   Machado, S.P. and Cortínez, V.H., 2005. Non-linear model for stability of thin-walled composite beams with shear deformation. Thin-Walled Structures, 43(10), pp.1615-1645.
[3]   Machado, S.P. and Cortínez, V.H., 2007. Free vibration of thin-walled composite beams with static initial stresses and deformations. Engineering Structures, 29(3), pp.372-382.
[4]   Machado, S.P., 2007. Geometrically non-linear approximations on stability and free vibration of composite beams. Engineering structures, 29(12), pp.3567-3578.
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