Free Vibration of Composite Thin-Walled Beams with Chordwise Asymmetric Closed Cross-Sections

In this paper, some analytical results via extended Galerkin method on free vibration characteristics of an anisotropic composite beam, which is modeled as a nonuniform thin-walled structure with a chordwise asymmetric closed cross-section and corrected the warping functions, are newly presented. For this study, nonclassical parameters such as warping restraint, transverse shear flexibility, and structural couplings induced by two special configurations, such as circumferentially uniform stiffness (CUS) and circumferentially asymmetric stiffness (CAS), are incorporated. And also, design parameters of the beam associated with preset angles, pretwist angles, taper ratios, and section ratios are additionally investigated. The results of this study could play an important role in more efficient designs of composite thin-walled beams.

However, in general, aircraft wings have an asymmetrical cross-section in the chordwise direction.Therefore, in this study, we investigated cross-sectional characteristics through the mathematical modeling of asymmetric cross-section and the correction of the warping function.In order to compensate the warping function of the asymmetric cross-section in the chordwise direction, the shape function of the crosssection contour is derived on the basis of an arbitrary pole of the cross-section and the shape function based on the pole is determined so as to satisfy the equilibrium conditions of the warping constraint.And then, the relation of the geometric cross-section is modified based on the pole of the new position and the warping function is corrected and applied to the governing equation.
In this paper, the beam modeled as a chordwise asymmetric cross-section is analyzed for free vibration characteristics numerically, and in order to investigate the effect of the asymmetry of the cross-section, a comparative study was conducted on a symmetrical cross-section model with the same chordwise length and cross-section thickness as a special case of the considered model.In the mathematical modeling, warping restraint, transverse shear flexibility, and structural couplings induced by circumferentially uniform stiffness (CUS) and circumferentially asymmetric stiffness (CAS) configurations [1,4] are considered.
In addition, the present paper examined the effects of the following parameters, such as preset angle, pretwist angle, taper ratio, and section ratio on the eigenfrequencies of the beam.Also, we compared the case where the warping function was corrected and the case where the warping function was not corrected.As a result, in this study, numerical examples were carefully analyzed and compared to investigate the effect of various design parameters on the natural frequencies that provide the basis for dynamic characteristics of beams.
The results presented as families of curves or tables, which may be used for the direct estimation of the eigenfrequencies for any given beam.

Mathematical Modeling
2.1.Basic Assumptions.The following basic assumptions have been used in this study [1,2,4,9]: (i) The cross-sections do not deform within their own planes, i.e., no in-plane deformations are allowed (ii) Transverse shear effects are incorporated.Moreover, the transverse shear strains γ xy and γ yz are uniform over the entire beam cross-section (iii) Although the rate of twist is assumed as constant in Saint-Venant torsion, it cannot be assumed to be constant along the beam axis while a warping restraint exists but it has been assumed to be a function of the axis coordinates in this paper (iv) Primary and secondary warping effects are sufficiently important to be included.The first one is related to the warping displacement of points on the midline cross-section and the second one is related to the points off the midline contour (v) In the absence of an internal pressure field, the resultant hoop stress is negligibly small as compared to the remaining stresses (vi) The beam has a nonuniform and asymmetric laminar flow airfoil profile, and in this sense, the beam is considered to be linearly tapered in both the horizontal and vertical planes (vii) Deformations are small and linear elasticity theory has been applied 2.2.Geometric Configuration of the Model.In this study, Figure 1 shows a chordwise asymmetric closed cross-section of a composite thin-walled cantilevered beam composed of 2 contours, that is, biconvex and ellipse.The pole (P m ) of the section is located at connection points of the 2 different contours on the chord line.And also, the considered beam in Figure 2 is modeled as an aircraft wing with preset angle (α 0 , so-called initial angle of attack), pretwist angle (γ 0 ), taper ratio (σ), and section ratio (ξ) for the analysis.

Governing Equations.
In principle of the theory for beams, shells and plates, and massive bodies by using the equations of three-dimensional continuum theory should be able to derive with taking advantage of the factors which serve to distinguish each type of structure.In this sense, the theory of plates and shells constitutes a two-dimensional approximation of the three-dimensional elasticity theory, while solid cross-section and thin/thick-walled beams are both one-dimensional approximations of the three-dimensional continuum theory [1,2].
Therefore, using the equations of the 3-D continuum theory, the considered thin-walled beam model was induced to 1-D equations as follows but omitted the process of the transformation in this paper; see reference [1].
where s denotes the arc length measured along the circumferential coordinate (whose origin is arbitrarily but conveniently  2 International Journal of Aerospace Engineering chosen) and n denotes the coordinate in the thickness direction.In addition, where ⋅ ds is the integral along the closed midline contour and r n s = x s l + y s m is a geometric quantity (see Figure 3), where l = cos n, x , m = sin n, x denote direction cosines.Also, F 1 s is the primary warping function and F 2 n, s is the secondary warping function.The boundary value problem, which consists of differential equations and boundary conditions, can be conveniently derived using the extended Hamilton's principle, which is as follows:

3
where δT and δV denote the kinetic energy and strain energy, respectively, and δW is the virtual work due to nonconservative forces.
As a result, the governing equations and boundary conditions are as follows but also omitted the process of the transformation in this paper; see reference [1].Table 1: Material properties and geometrical dimensions.3 International Journal of Aerospace Engineering where a ij i, j = 1, 2, … , 6 are stiffness coefficients and b j j = 1, 2, … , 18 are mass coefficients.

Material Properties and Geometrical Dimensions.
Wing structures modeled as a cantilevered thin-walled beam of a chordwise asymmetric closed cross-section made of T300/5208 carbon epoxy material are considered for numerical simulations.Their geometrical dimensions and material properties are shown in Table 1.

Correction of the Primary Warping
Functions.The warping displacement in pure twist can be obtained as the product of the warping function and the rate of twist [1,2].
where F 1 s = s 0 λ − r n ds + C, the constant C being determined by satisfying the following warping equilibrium condition: International Journal of Aerospace Engineering In equation (7), ⋅ ds denotes the integration around the whole midline contour of the cross-section.
From the above relations, equations ( 6)-( 9), the warping functions could be corrected for the model (see Figure 4) as the following procedures.
Step 1.In advance, the geometrical shape functions of the contour are derived as follows: For an ellipse contour, from x 2 /a 2 + y 2 /b 2 = 1, And also, the length and the area of the contours are , 13 where a ≥ b), Ω and β are the all area and all length of the contours, and ⋅ e and ⋅ b mean the physical quantities of the ellipse and biconvex contours, respectively.
From the above relations, we have the geometrical shape functions of the contour as follows: Step 2. And then, for correction of the warping functions, so as to satisfy equilibrium condition (7), the pole (P m ) should be translated to the new one (P n ) and also the shape functions (equation ( 14)) modified as follows:  where x c is a translation quantity along the chordwise (see Figure 4), that is, To verify the above results through an example, in case of the noncorrected primary warping functions (equation ( 14)), the function's graph can be depicted in Figure 5.In Figure 5, the sum of the area based on the x-axis is not zero.That means that the warping function should be corrected [1].Therefore, from the corrected warping functions (equations (15) and ( 16)), Figure 6 can be drawn and shows that the all area is zero.As a result, the above relations, equations (15) and (16), will be applied to governing equations and boundary conditions of the beam for analysis.

3.3.
Characteristics of the Cross-Section.For a survey of the characteristics of the chordwise asymmetric closed crosssection (Model A), using the corrected warping functions, mass coefficients b j and stiffness coefficients a ij of equations ( 4) and (5) were found.The details of the mass and stiffness coefficients could be found in reference [1], which are equations (4.1-4) for mass coefficients and Tables 4.3-1 for stiffness coefficients of chapter 4. The results of the nonzero coefficients were provided in Tables 2 and 3 for comparing between Model A and a symmetric one (Model C, see Figure 7) according to a preset angle (α 0 ) and a pretwist angle (γ 0 ) of CUS and CAS configurations for each.In the tables, "S" entries are valued elements of the symmetric section (Model C) and "A" entries are those of the asymmetric section (Model A) in addition to the results of Model C.
And then, stiffness coefficients a ij = a ji in cases of (α 0 = 0 °and γ 0 = 0 °) and (α 0 = 4 °and γ 0 = 10 °) were depicted in Figures 8 and 9 for CUS and CAS at η = 1 0, σ = 0 6, and ξ = 0 4.These figures show that a 14 , a 15 , a 24 , a 25 , a 27 , a 34 , a 35 , a 37 , a 45 , a 46 , a 56 , and a 67 have asymmetric values and the others have symmetric values at a ply angle of 90 °.International Journal of Aerospace Engineering Furthermore, Table 4 shows that some stiffness coefficients of the model have different values between α 0 = 0 °and γ 0 = 0 °and α 0 ≠ 0 °or γ 0 ≠ 0 °and some are the same or the others only appeared in case of α 0 ≠ 0 °or γ 0 ≠ 0 °.And especially, when i = j, a ii have all the same values for CUS and CAS configurations.
where ψ r j are trial functions and have to satisfy the geometrical boundary conditions at z = 0 and also, a j , b j , c j , d j , e j , and f j are arbitrary coefficients.Now, the discretization is accomplished directly in the Hamilton's principle functional and then written in a matrix form as follows: where K pq and M pq are stiffness and mass matrices, respectively, and Q r r = 1, 2, … , 6 is a force vector.
Similarly, for the survey of the characteristics of the chordwise asymmetric closed cross-section (Model A), using the corrected warping functions, the mass and stiffness matrices of equation (18) were found.The results of the nonzero coefficients were provided in Tables 5 and 6 for comparing between Model A and a symmetric one (Model C, see Figure 7) according to a preset angle (α 0 ) and a pretwist angle (γ 0 ) of CUS and CAS configurations for each.In the tables, "S" entries are valued elements of the symmetric section (Model C) and "A" entries are those of the asymmetric section (Model A) in addition to Model C.
At this point, Table 5 shows that the stiffness matrix for CAS configuration is composed of the same elements between Model A and Model C, in cases of α 0 = 0 and γ 0 = 0 and α 0 ≠ 0 or γ 0 ≠ 0. And also, mass matrices have all the same values for CUS and CAS configurations at any case.

Effects of Preset Angle and Pretwist Angle on
Eigenfrequency.In order to investigate the effects of preset and pretwist angles on eigenfrequencies calculated for Model A, both CUS and CAS configurations.The results are shown in Figure 10 and Table 7.
In Figure 10 and Table 7, preset and pretwist angles have no near effects on eigenfrequencies; nevertheless, the differences exist but very negligibly small.

Effects of Correction of the Primary Warping Functions.
Next, based on ply angles, the effects of corrected (CW) and noncorrected (NCW) warping functions on eigenfrequencies were estimated for Model A, both CUS and CAS configurations.The results are shown in Figure 11 and Table 8.
In Figure 11 and Table 8, the effects of corrected (CW) and noncorrected (NCW) warping functions on eigenfrequencies were observed to have very small differences for ply angles.However, for precise calculation, the primary warping function should be properly corrected according to the warping equilibrium condition from equations ( 6)-(9).

Effects of Geometrical Parameters.
From now on, the effects of geometrical parameters (α 0 , γ 0 , σ, and ξ) on eigenfrequencies were investigated for Model A, both CUS and CAS configurations.
(1) Preset Angle, α 0 .After calculating the eigenfrequencies via preset angles α 0 , the results are shown in Figure 12.Generally, from the results, the preset angle effects on the eigenfrequencies are very insignificant.
(2) Pretwist Angle, γ 0 .Also, after calculating the eigenfrequencies via pretwist angles γ 0 , the results are shown in Figure 13.As a result, lower-order eigenfrequencies were not almost affected by the pretwist angle but the higher modes increase the effects which is associated with the preset angles being higher.
(3) Taper Ratio, σ.Next, the eigenfrequencies via taper ratios σ of the beam were found.The results are shown in Figure 14.In this case, the taper ratio has more effect on  Table 7: Eigenfrequencies of Model A via α 0 = 0 and γ 0 = 0 for CUS and CAS.

CUS ( °)
ω 1 (rad/s) ω 2 (rad/s)  11 International Journal of Aerospace Engineering of the chordwise asymmetric closed cross-sections.However, this is an indispensable procedure for initial sizing through analytical methods in the early stages of design development, such as aircraft wings and rotor blades.
For this purpose, in this paper, a model with a chordwise asymmetric closed cross-section (Model A) of a composite thin-walled beam was selected considering the contour shape for generalization of chordwise asymmetric closed cross-sections.And then, mathematical modeling was carried out by modifying the warping functions of the asymmetric cross-section and considering the transverse shear effect, warping restraint effect, and geometrical parameters, such as preset angle (α 0 ), pretwist angle (γ 0 ), taper ratio (σ), and section ratio (ξ) of the beam.12 International Journal of Aerospace Engineering Based on the mathematical modeling with the corrected warping function, mass coefficients, stiffness coefficients, and eigenfrequencies of the beam with a chordwise asymmetric closed cross-section were investigated for the CUS and CAS configurations of Model A. In addition, the present paper examined the effects of the following parameters: preset angle, pretwist angle, taper ratio, and section ratio of the beam.The results are presented as families of curves and tables, which may be used for the direct estimation of the eigenfrequencies for any given beam.Also, a comparative study was conducted on the symmetrical cross-section model (Model C) and the symmetrical cross-section model (Model B) with same chord length and section thickness and the results were also provided.
From the above results, it is shown that the theoretical explicit analysis of the thin-walled beam with chordwise asymmetric closed cross-sections through the correction technique of the warping functions proposed in this study is possible and can be applied to the field of dynamic characteristics and aeroelasticity and so on.As a future research work, other lay-up schemes which may yield different elastic couplings might be considered.Moreover, in order to obtain more practical results, the multicell theory as well as stiffeners (spars and ribs) of the wing should be considered.

Figure 7 :
Figure 7: Model B and Model C for comparing.

Figure 8 :
Figure 8: Stiffness coefficients a ij = a ji vs. ply angle of Model A for CUS.

Figure 9 :
Figure 9: Stiffness coefficients a ij = a ji vs. ply angle of Model A for CAS.

6 Figure 15 :
Figure 15: Eigenfrequencies vs. ply angle of Model A via ξ for CUS and CAS.

Table 2 :
Mass coefficients of Model A and Model C for CUS and CAS.

Table 3 :
Stiffness coefficients of Model A and Model C.

Table 8 :
Eigenfrequencies of Model A via CW and NCW for CUS and CAS.Figure 12: Eigenfrequencies vs. ply angle of Model A via α 0 = 0 for CUS and CAS.
i Figure 13: Eigenfrequencies vs. ply angle of Model A via γ 0 = 0 for CUS and CAS.