ISSN : 2288-0232(Online)

**Journal of the Korean Society for Advanced Composite Structures Vol.4 No.1 pp.9-14**

DOI : https://doi.org/10.11004/kosacs.2013.4.1.009

# 복합소재 기둥 구조의 적층배열 변화에 따른 고유진동 및 모드 특성

# Natural Frequency and Mode Characteristics of Composite Pole Structures for Different Layup Sequences

### Abstract

- 2.이상열_영문.pdf1.41MB

- 1. Introduction
- 2. Basic Theory
- 3. Finite element model
- 4. Numerical examples
- 4.1 Quadrangle poles
- 4.2 Hexagon poles
- 4.3 Mode shapes
- 5. Summary and Conclusion
- 감사의 글

### 1. Introduction

With the advancement of technology in composite materials, applications of laminated composites to pole structures are advantageous because of their light weight, high specific stiffnesses and high specific strengths. However, laminated poles with a long length may result in a significant change of the dynamic characteristics during various external loading. Therefore, it is essential to study the effect of fiber angles and geometries simultaneously on the frequency of layered poles. The mechanical behaviors of a structure depend on the properties of the fibers and the matrix and in the amount and orientations of the fibers.

Numerous theories exist for laminated composite plates or structures. Many these theories were developed originally for thin shells and are based on the Kirchhoff-Love kinematic hypothesis (Sanders, 1963). However, these classical theories are inadequate to describe accurately the kinematic behavior of anisotropic composite shells. The first-order shear deformation theory (FSDT) has been extended by other investigators to deal with those problems (Kraus, 1967; Sanders, 1059; Zukas and Vinson, 1971). In general, a first order shear deformation theory can describe easily and accurately the kinematic behavior of a thin composite structure (Reddy, 2004). The composite pole types can be practically applied to wind turbine towers. Polyzois et al. (2009) studied static and dynamic characteristics of multi-cell jointed glass fiber-reinforced polymer (GFRP) wind turbine towers.

In this paper, we performed a finite element analysis based on FSDT to analyze the free vibration of laminated poles with a long length. Very little research has been studied to identify the interaction between vibrations and layup sequences for long composite pole type structures. The length-thickness ratio and layup sequence of laminated poles with different geometries could play a dominant role in determining and efficiently controlling the dynamic characteristics of the composite structure without causing a change in its material properties. Numerical examples are focused on the changes in natural frequencies for different radius-length ratios, layup sequences, shapes of the cross-section and interactions between these parameters.

### 2. Basic Theory

In the laminated structures, the stiffness properties are function of the normal coordinate. In Fig. 1 a cross-section of laminated plate composed of N layers is presented. A linear elastic properties of the anisotropic layers are characterized by the tensor of elasticity C_{ijkl}. In many applications, it can be as sumed that calculations of shell stiffness properties can be performed neglecting the differences in spa-tial and shell mid-surface metrics. In this case, the Hook’s law for each layer can be written by

**Fig. 1** Cross-section of laminated plate composed of N layers.stress resultants and couples and laminated shell stiffness characteristics

Since the present formulation is based on the natural coordinate reference frame, we introduce here an explicit transformation scheme between natural coordinates and the global coordinate system, to obtain a natural coordinate based constitutive equation. The stress tensor in the natural coordinate can be written as follows:

where is the determinant of the Jacobian matrix and is the constitutive matrix for orthotropic materials with the material angle θ.The transformation matrix in Eq. 2 is given T by Han et al. (2004). The strain energy U can be expressed by

After integration, throughout the thickness, the strain energy can be obtained in terms of shell quantities:

The Reissner’s value of 5/6 is used as the transverse shear correction factor in the FE formulation based on the first shear deformation theory.

The shell element displays resultant forces acting on a laminate which are obtained by integration of stresses through the laminate thickness. In this study, we impose the plane state on the natural constitutive equation of Eq. 2 before forming the equivalent constitutive equation. The constitutive relations of the composite T laminate are as follows:

### 3. Finite element model

In the current study, the ABAQUS program was used to develop finite element models for simulating the structural behavior of laminated composite poles. The poles tested had an average height of 5.00 m from the base. The sectional properties are shown in Fig. 2. Each ply of the laminates has equal thickness. The material properties of the plies are E1=40.0E2, G12=0.5E2, G23=0.6E2 G13=0.5E2, and Poisson’s ratio=0.25. The produced natural frequencies in this study are normalized. Fig. 3 shows finite element laminated composite models using different cross-section types in ABAQUS program. The arrangement of the fibers in a pole structure is governed by the structural requirements and by the process used to fabricate the part. The mass density is assumed for convenience to have a unit value. The boundary conditions are free-clamped supported such as a cantilever beam. Fig. 4 shows the whole finite element modelling of pole structures using ABAQUS.

**Fig. 2** Two type test models

**Fig. 3** ABAQUS modeling of laminated composites.

**Fig. 4** ABAQUS modeling of two types.

### 4. Numerical examples

#### 4.1 Quadrangle poles

Table 1~3 show non-dimensional natural frequencies of quadrangle poles for different layup sequences and length-thickness ratios (a/h). It can be observed from the tables that the use of different layup sequences (symmetry or anti-symmetry) has a negligible effect on the laminated poles, regardless of the fiber angles and length-thickness ratios. On the other hand, the induced frequency for the same layup sequence is strongly dependent on the fiber angle, especially for [0/90]s and [15/-75]_{2}.

**Table 1.** Non-dimensional natural frequencies for different layup sequences (quadrangle, a/h=5.0)

**Table 2.** Non-dimensional natural frequencies for different layup sequences (quadrangle, a/h=10.0)

**Table 3.** Non-dimensional natural frequencies for different layup sequences (quadrangle, a/h=50.0)

#### 4.2 Hexagon poles

Table 4~6 show non-dimensional natural frequencies of hexagon poles for different layup sequences and length-thickness ratios (a/h). The produced natural frequencies for hexagon poles show a similar trend with those of quadrangle poles. The highest frequency occurs for the laminate with a layup sequence of [0/90/90/0] or [0/90/0/90] because its stiffness is greater than that of the others. The differences between the layup sequences for different fiber angles depend on many geometrical parameters such as shape of the cross-section, length-to-thickness ratio, and boundary conditions.

**Table 4.** Non-dimensional natural frequencies for different layup sequences (hexagon, a/h=5.0)

**Table 5.** Non-dimensional natural frequencies for different layup sequences (hexagon, a/h=10.0)

**Table 6.** Non-dimensional natural frequencies for different layup sequences (hexagon, a/h=50.0)

#### 4.3 Mode shapes

Figs. 5~6 show the mode shapes of symmetric cross-ply laminated quadrangle and hexagon poles with free-clamped ends. Each figure represents (a) symmetric vertical bending, (b) symmetric horizontal bending, (c) torsional, and (d) antisymmetric horizontal bending modes, respectively. As shown in the figures, some peculiar and complex mode shapes are produced due to the combined effect of both shapes of the cross-section of and layup stacking sequences.

**Fig. 5** Mode shapes of quadrangle laminated composite poles (0/90/90/0).

**Fig. 6** Mode shapes of hexagon laminated composite poles (0/90/90/0).

### 5. Summary and Conclusion

A technique based on the first-order shear deformation plate theory is developed to analyze the static and free vibration behavior of pole structures made of laminated composite materials. The usage of composite for poles could be an attractive approach from the practical point of view, not only because it has high stiffness but also because we can avoid the undesirable corrosion occurred in steel structures.

The technique is then implemented for quadrangle and hexagon poles of various layup sequences and end conditions to compare the results obtained from different length-to-thickness ratios. It is observed that the use of different layup sequences make little difference regardless of fiber angles, but the produced frequencies for different fiber angles become significant for various length-to-thickness ratios.

For composite poles with the closed section, e.g., quadrangle or hexagon box, the significance could be greater, because it is not only the properties of the materials but also the sectional properties of the member that makes large contributions to the overall behavior of the structure. The results of this study may serve as a benchmark for future guidelines in designing composite pole structures. But our parametric study is only an example and more studies should be carried out for individual design cases.

### 감사의 글

본 연구는 국토해양부 건설기술연구사업 연구비 지원(과제번호: 12CCTI-B063597-01, 10MW급 강재 및 3MW급 복합 합성구조 풍력발전타워 설계기술 개발)에 의해 수행되었음.

### Reference

2.Han, S. C., Kim, K. D., Kanok-Nukulchai, W. (2004). "An element-based 9-node resultant shell element for large deformation analysis of laminated composite plates and shells." Struct. Eng. Mech., Vol. 18, No. 6, pp. 807–829.

3.Kraus, H. (1967). Thin elastic shells, Willy, New York.

4.Reddy, J. N. (2004). Mechanics of Laminated Composite Plates and Shells: Theory and Analysis, CRC Press, New York.

5.Sanders, J. L. Jr. (1959). An improved first approximation theory of thin shells, NASA TR-R24.

6.Sanders, J. L. Jr. (1963). "Nonlinear theories for thin shells Q." App. Math., Vol. 21, pp. 21–36.

7.Zukas, J. A. and Vinson, J. R. (1971). "Laminated transversely isotropic cylindrical shells." J. App. Mech., Vol. 38, pp. 400–407.