## 1. INTRODUCTION

Composite materials are extensively used in aerospace and civil industry engineering due to their characteristics of high strength-to-weight ratio, corrosion resistance, high energy absorption, and low thermal expansion among other construction materials. Because of their geometry and material properties, composite structures are susceptible to undergo large deformations and vulnerable to structural stability below the material strength.

Many researchers have studied the problem of flexural-torsional buckling of composite beams during recent years. Bauld and Tzeng (1984) extended Vlasov’s beam theory (1961) of isotropic material to thin-walled laminated composite beams of open cross section. To overcome the discrepancy between classical beam theory and experimental results, a number of one-dimensional finite element methods incorporating shear deformation effects have been developed to predict the response of laminated composite beams (Maddur and Chaturvedi, 2000;Back and Will, 2008;Kim, 2011). Nevertheless, many of these have been restricted to I- and C-sections.

Structural members of the T-section are usually used as bracing members or chord members of the truss and occur in the coped area of the girder. This member has a high degree of single symmetry and a relatively low torsional rigidity compared to other open cross-sections. Moreover, it is also well known that flexural deformation is coupled with torsional deformation. Therefore, this member may be subjected to flexural-torsional buckling than doubly symmetric beams. Lee (2003) performed an experimental and analytical study on the flexural-torsional buckling behavior of pultruded T-members under axial load. Back (2014) investigated flexural behavior of a laminated composite T-beam using a contour coordinate system.

A shear-flexible finite element is proposed for the flexural and buckling analyses of laminate composite T-beams. The displacement fields at the mid surface of an element are defined based on the orthogonal Cartesian coordinate system. The seven governing equations of composite beams are established. Three beam elements, namely, two-node, three-node, and four-node elements, with seven degree-of-freedom per node have been developed. The geometric stiffness of axially loaded composite beams is also developed to predict critical buckling loads. FORTRAN program is written for the implementation of the proposed laminated composite T-beam. Parametric studies are performed to investigate the influence of boundary conditions, web fiber orientation and length-to-height ratio on the maximum displacement and buckling load of composite T-beams.

## 2. VARIATIONAL FORMULATION

Fig. 1 shows two sets of coordinate systems used in the present study. One is the orthogonal Cartesian coordinate system (*x*, *y*, *z*), and the other is the contour coordinate (*x*, *n*, *s*). The contour coordinates of an arbitrary point *A* are (*r*, *q*), in which *r* and *q* are the *n*- and *s*- coordinates of A, respectively.

Based on thin-walled beam assumptions with Timoshenko’s beam theory, the longitudinal and transverse displacements of a mid-surface can be expressed as (Back and Will, 2008)

where $u\left(x\right),\hspace{0.17em}\upsilon \left(x\right),\hspace{0.17em}w\left(x\right)$, and ${\theta}_{x}\left(x\right),\hspace{0.17em}{\theta}_{y}\left(x\right),\hspace{0.17em}{\theta}_{z}\left(x\right)$ are the rigid body translations and rotations along the *x*, *y*, and *z* axes, respectively; ${\varphi}_{x}\left(x\right)$ is the change in twist about pole, and $\left({y}_{p},{z}_{p}\right)$ is the coordinates of shear center shown in Fig. 2. *ω* is the warping function.

The variation of the strain energy may be written as

where ${\in}_{x}^{o},{\kappa}_{y},{\kappa}_{z},{\kappa}_{w}$, and ${\kappa}_{xs}$ are axial strain, curvatures in the *y* and *z* direction, warping curvature about principle pole, and twisting curvature, respectively. ${N}_{x},\hspace{0.17em}{M}_{y},\hspace{0.17em}{M}_{z},{M}_{w},{V}_{y},{V}_{z},T,\hspace{0.17em}\text{and}\hspace{0.17em}{M}_{t}$ are axial force, moments about the *y* and *z* axes, bimoment, shear force in the *y* and *z* directions, two contributions to the total twisting moment.

The potential energy of a composite member subjected to a compressive axial load ${\sigma}_{x}^{o}$ is written as (Bleich, 1952)

Substituting Eq. (1) into Eqs. (2) and (3) and combining them, the weak form can be expressed as

where *I _{p}* is the polar moment of inertia about pole,

*e*is an eccentricity, and the Wagner’s coefficient

*β*is

_{z}

The stress-strain relationship for a *k*th orthotropic lamina of the flange can be written as (Back and Will, 2008).

where for the plane stress assumptions (*σ _{s}* = 0), the condensed transformed reduced stiffness coefficients are given by

Similar expressions for web may be easily obtained.

Combining Eq. (6) with the resultant forces and moment, the force-displacement relationship for a laminated composite beam can be expressed as

where *E _{ij}* is the stiffnesses of composite beams. By performing proper through-thickness integration, we explicitly express non-zero stiffness for T-section:

where a repeated index represents the sum; the index *β* takes the value 1 and 2, which denote the flange and the web, respectively. The symbols *b*_{1} and *b*_{2} represent the width of the flange and the web, respectively.

The extensional, bending-extensional coupling, and bending stiffnesses in Eq. (9) are given by

Integrating the linear part of Eq. (4) by parts and collecting each displacement component, we can write the governing equations of thin-walled composite beams as follows:

Eq. (11) is a general equilibrium equation for composite beams that takes into account warping shear deformation and material anisotropy due to various types of loadings, including axial forces, shear forces, bending moments, torques, and bimoment. You can see that the governing equations for the seven variables are fully coupled. For negligible shear deformation, the seven governing equations can be easily reduced to the four equations given by Lee and Lee (2004).

## 3. FINITE ELEMENT FORMULATION

The same shape function is adopted for all translational and rotational displacements to derive three different beam elements; linear, quadratic, and cubic elements.

where *n* and *N _{α}* are the number of nodes and the shape function of node

*α*, respectively.

The element displacement vector * d_{e}* may be written as nodal displacement vector

*:*

**d**_{α}

Substituting Eq. (12) into Eq. (4), the equilibrium equations for composite beams can be expressed as

where * f_{e}* is the element force vector and the element linear and geometric stiffness matrices

**and**

*k*_{e}**can be written in block matrix form as**

*k*_{g}

in which the expression for any block ${\kappa}_{\alpha \beta}^{e}\left(\alpha ,\hspace{0.17em}\beta =1,2,\cdots ,n\right)$ is given in Back and Will(2008).

The linear buckling problem can be written as

where *λ* is the load factor for the reference load and ${\overline{K}}_{G}$ is the initial geometric stiffness matrix at the reference load.

## 4. NUMERICAL EXAMPLES

Numerical results were made to validate the flexural analysis of laminated composite T-beams. In this study, two-node, three-node and four-node beam elements are considered. Then, apply the current model to the buckling analysis of composite beams and investigate the influence of aspect ratio and web fiber orientation on the critical buckling loads of composite T-beams.

In the following numerical analysis, the cross section shown in Fig. 2 is used: ${b}_{1}=31.47\text{mm},\hspace{0.17em}{b}_{2}=71.86\text{mm}$. The total thickness of flange and web is 3.139mm and 2.192mm, respectively. The material of the beam is made of glass epoxy with the following engineering constants: ${E}_{11}=53.78\text{GPa},\hspace{0.17em}{E}_{22}={E}_{33}=17.93\text{GPa,}$ ${G}_{12}={G}_{13}=8.96\text{GPa},\hspace{0.17em}{G}_{23}=3.45\text{GPa},$ ${\nu}_{12}={\nu}_{13}=0.25,\hspace{0.17em}{\nu}_{23}=0.34$.

### 4.1 Static Analysis

Composite beams with *L*/*h* = 10 and *L*/*h* = 20 under uniformly distributed load of 5 kN/m and 0.5 kN/m, respectively, throughout its length are performed for two types of boundary conditions: cantilever beam and simply supported beam. A total sixteen plies of the same thickness on the flange and the web are taken into account. The stacking sequence of the beam flange is unidirectional and the stem is assumed to be $\left[\theta /-\theta \right]{}_{4s}$. For two different *L*/*h* ratios, the maximum vertical displacements of simply supported and cantilever beams based on two different assumptions (*σ _{s}* = 0 and

*∈*= 0) are shown in Table 1 and Table 2, respectively. A total of 340 S9R5 shell elements are used for ABAQUS calculation, whereas two quadratic elements are used to model the beam. For both beams with

_{s}*L*/

*h*= 10 and

*L*/

*h*= 20, the present model based on plane stress assumption (

*σ*= 0) is in a very good agreement with the ABAQUS results for the entire fiber angle range considered. On the other hand, the analysis based on

_{s}*∈*= 0 assumption seems to underestimate the maximum deflection by up to 15.5% at the sequence of lamination

_{s}*θ*= 45

^{o}.

In the next example, cantilever beams with *L*/*h* = 5 and *L*/*h* = 10 under a concentrated tip load of 1 kN are considered to investigate the effects of coupling and transverse shear deformation. Flange and stem are considered four layers of unidirectional and antisymmetric angle ply laminates [*θ*, - *θ*]_{2}, respectively. The assumption of *σ _{s}* = 0 is made for every analysis. The variation in the maximum displacement of the beam with fiber angle is shown with the solutions based on classical beam theory (CBT) and first-order beam theory (FOBT) in Fig. 3.

For convenience, the non-dimensionalized maximum displacement, $\overline{\nu}=\upsilon {E}_{2}{b}_{2}^{3}/\left(P{l}^{3}\right)$, is used. According to the first-order shear deformation theory, a closed-form solution of maximum transverse deflection of laminated beams can be computed for cantilever beams under the concentrated tip load *P* as follows (Reddy, 1997):

where the first and second terms denote the deflection due to pure bending (CBT) and the deflection due to shear deformation, respectively. The variation in bending and shear components of the vertical displacement with fiber orientation is shown in Fig. 4 for the ratio of *L*/*h* = 5. As the fiber angle changes, the bending component increases faster, while the shear one does not change significantly.

It is seen from Fig. 3 that the influence of the shear deformation on the maximum deflection in this lay-up is small for *L*/*h* = 10. For lower span-to-height ratio (*L*/*h* = 5), however, the CBT solution excluding shear effect significantly underestimates the displacement over the entire range of fiber angles. Since the coupling stiffnesses ${E}_{13},{E}_{15},{E}_{27},\hspace{0.17em}\text{and}\hspace{0.17em}{E}_{35}$ do not vanish and all the other coupling stiffnesses are zero, the orthotropy solution given in Eq. (17) may not be accurate. However, due to the small coupling stiffness compared to the bending stiffness, the coupling effect due to material anisotropy becomes negligible for two different span-to-height ratios. As a result, this model closely matches with the orthotropic solution for this lay-up as shown in Fig. 3.

### 4.2 Axially Loaded Cantilever Beams

The cantilever beams with *L*/*h* = 10 and *L*/*h* = 20 are subjected to a compressive axial load at the pole. Both the flange and the web are made of sixteen plies and assume that they are symmetrically laminated about the mid-plane. Along with ABAQUS shell element results, critical buckling loads based on two different assumptions (*σ _{s}* = 0 and

*∈*= 0) are provided in Table 3 for various laminated stacking sequences. In the current finite element model, four quadratic elements are employed to represent the beam. It can be noticed that the present beam element based on plane stress assumption exhibits an excellent agreement with the ABAQUS solution for the entire fiber angle range under consideration. As expected, the longer span length reduces the critical buckling load and the lateral buckling becomes more pronounced. Columns with unidirectional fibers exhibit the highest critical buckling loads with respect to other laminations for all stacking sequences. As the ply angle

_{s}*θ*increases, the critical buckling load is significantly reduced. While the current results based on

*σ*= 0 are very consistent with the ABAQUS solution for all layups, an analysis based on plane strain assumption(

_{s}*∈*= 0) has been shown to overestimate the buckling load by up to 18.3% in the stacking sequence of

_{s}*θ*= 45

^{o}. The critical buckling mode shape for the stacking sequence of [45/-45]

_{4s}is shown in Fig. 5. Disregarding all coupling stiffness, the orthotropic closed-form solution for cantilever boundary conditions is: ${P}_{y}={\pi}^{2}{E}_{22}/4{L}^{2}$. The corresponding buckling load for the flexural mode in

*y*direction yields

*P*= 976.7 N. This bucking load, ignoring the coupling stiffness, is slightly higher than the current result.

_{y}The next example is identical to the previous example, except that the composite beam consists of four layers and the fiber angle is changed in two different ways: unidirectional fiber orientation of the flange and antisymmetric laminates [*θ*/ - *θ*]_{2} of the web (Case 1), and antisymmetric laminates [*θ*/ - *θ*]_{2} of the flange and unidirectional fiber orientation of the web (Case 2). For convenience, the following non- dimensional buckling load parameter is adopted: ${\overline{P}}_{cr}={P}_{cr}{L}^{2}{10}^{2}/\left(A{E}_{2}{b}_{2}^{2}\right)$. Buckling load parameters for cantilever beams with *L*/*h* = 10 and *L*/*h* = 20 are shown with ABAQUS results in Table 4. As seen from this table, the predictions in the present model are very consistent with the ABAQUS results. The buckling load in Case 1 is not significantly affected by the fiber angle of the web, while the buckling load decreases dramatically in Case 2 as the fiber angle in the flange changes. Columns with unidirectional fibers in the flange exhibit greater buckling loads than unidirectional fibers in the web. Also unlike the buckling load in the flexural-torsional behavior of an isotropic beam, the maximum buckling load of the anisotropic beam appears at *θ* = 0^{o}.

## 5. CONCLUSIONS

Shear-flexible finite element models have been developed for flexural and buckling analyses of thin- walled laminated T-beams by adopting an orthogonal Cartesian coordinate system. The proposed beam elements include flexural shear and warping deformation and all coupling terms due to anisotropy. The seven governing equations are derived and for negligible shear deformations, the equilibrium equations are reduced to the four equations given in the literature. One-dimensional finite element model, two-node, three-node, four-node beam elements, have been developed. The numerical examples with the proposed element showed the ability to accurately predict the displacement and buckling load of composite T-beams. The effects of span-to-height ratio and fiber orientation on the maximum displacements and critical buckling load of symmetric angle ply composite beams were investigated. The results also showed that columns with unidirectional fibers in the flange exhibited greater buckling loads than unidirectional fibers in the web. This model has been found to be adequate and efficient for the flexural and buckling problems of laminated composite T-beams.