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ISSN : 2093-5145(Print)
ISSN : 2288-0232(Online)
Journal of the Korean Society for Advanced Composite Structures Vol.10 No.1 pp.1-9
DOI : https://doi.org/10.11004/kosacs.2019.10.1.001

# Parametric Study on Buckling Resistance of the Thin BFRP (Basalt Fiber Reinforced Polymer) Plate Subjected to Combined Loading

Chhorn Buntheng1, Jung WooYoung2
1Ph.D. Student, Department of Civil Engineering, Gangneung-Wonju National University, Gangneung, Korea
2Professor, Department of Civil Engineering, Gangneung-Wonju National University, Gangneung, Korea
·

본 논문에 대한 토의를 2019년 03월 31일까지 학회로 보내주시면 2019년 04월호에 토론결과를 게재하겠습니다.

Corresponding author: Jung, WooYoung Department of Engineering, Gangneung-Wonju National University, 7 Jukhun-gil, Gangnenung, Korea. Tel: +82-33-640-2421, Fax: +82-33-646-1391, E-mail: woojung@gwnu.ac.kr
November 26, 2018 December 28, 2018 January 14, 2019

# 조합하중에 대한 현무암섬유 강화 복합재료(BFRP)의 좌굴저항에 관한 매개 변수 연구

Chhorn Buntheng1, 정 우영2
1강릉원주대학교 토목공학과 박사과정
2강릉원주대학교 토목공학과 교수

## 초록

조합하중 작용 시 현무암섬유 강화 복합재료(BFRP) 플레이트의 좌굴거동에 대한 해석적 연구를 유한 요소법(FEM)을 통하여 평가하였다. 복합재료 플레이트 내에서 고려될 수 있는 경계 조건, 치수의 종횡비 및 하중 조건과 같은 다양한 매개변수 에 대한 영향성을 연구하였다. BFRP의 역학적 성질은 국내에서 제작된 시편을 이용하여 인장 및 면내 전단 실험을 통하여 구 하였다. 산정된 물성치를 토대로 고전적인 판 이론을 이용하여 대칭으로 적층된 판을 우선적으로 분석하였다. 그 결과 2축 및 전단에 대한 조합 하중의 경우, 종횡비가 0.5~1.0 일 때 좌굴하중이 빠르게 감소한다는 것을 알 수 있었다. 이와 반대로, 종횡 비가 1.0~2.5 일 때는 좌굴하중이 약간 감소하는 경향이 보였다. 또한, 기존 축 방향 하중의 평면 내 전단 하중을 조합하여 추 가할 경우 플레이트 판 내의 좌굴하중 감소가 약 4% 정도로서 큰 영향을 미치지 못함을 알 수 있었다.

National Research Foundation of Korea
2017R1A2B3008623

## 1. INTRODUCTION

The composite materials are the most popular for using in mechanical, aeronautical, civil and marine structure. Because of their high strength, small density, high stiffness and the controllability of property that is variable with the orientation and the thickness of the laminate. The behavior of structure that was constructed by composite material, mechanical behavior of rectangular laminate plate has drawn many attentions. Especially, the buckling phenomenon is the main focus which need to be considered for the competent and dependable design and secure procedure of the structural element. Due to the orthotropic characteristics of composite material, the analysis of homogeneous isotropic plate is normally easier than the composite laminate plate.

For the same material and the volume fraction of resin, the resistance of the composite laminate plate can be different due to the stacking sequence of fiber. It has several ways of fiber arrangement which found to be very popular. For cross-ply laminates and angle-ply laminate, the stacking sequence are [0/90]n and [+θ/-θ]2n, respectively. Where, 2n is the total number of laminates. Furthermore, the symmetrical or anti-symmetrical arrangement can make the stacking sequence become different (Reddy, 1997;Kollár and George, 2003). A semi-analytical approach is developed by Shufrin and Eisenberger (2008) to analyze the buckling laminated composite plates. It demonstrated that a combination of biaxial compression, tension and shear modified the stability behaviour of angle-ply laminated plates. Besides, the buckling behaviour of the plate were changed by the presence of in-plane restraints yields supplementary in-plane forces.

Utilizing the finite element method (Avci et al., 2005), studied thermal buckling of laminate composite plate with hole for both clamped and simply supported at edges. To investigate the thermal buckling temperatures the eight-node Lagrangian finite element method was used. Thermal buckling of composite laminate plates with various types of boundary conditions was considered by Thangaratnam and Ramachandran (1989). Shiau et al (2010) demonstrated that α1/α2 ratio and fiber orientation were the main factor responsible for the buckling of composite laminate plate. For simply support square angle-ply laminate plates, the stacking sequence with fiber orientated at 45o obtained the highest critical buckling temperature. Ounis et al (2014) illustrated that the increasing of E1/E2 ratio and α1/α2 ratio can make the critical buckling temperature decrease. Also, the main factor affecting the buckling laminate composite plates were fiber orientation and boundary conditions. Shaterzadeh et al (2014) found the increment of critical thermal buckling temperature because the present of hole in the center of plate. Moreover, the critical thermal buckling temperature of symmetric composite plates were smaller than those of anti-symmetric composite plates. Sim et al (2016) studied influence of stacking sequence and temperature on buckling resistance of GFRP infill panel. Simplified analysis on steel frame infill with FRP composite panel was investigated by Seo et al (2016). Buckling resistance of basalt fiber reinforced polymer infill panel subjected to elevated temperatures was studied by Sim et al (2017).

The objective of this study is to investigate the buckling resistance of basalt fiber laminate composite rectangular plates. To evaluate the buckling behavior of basalt fiber reinforced polymer (BFRP), The numerical approach was developed by using the finite element method. Also, buckling performance of thin composite plate subjected to combined loading was determined. Multiple parameters were studied, such as, boundary conditions, aspect ratio of plate dimension (a/b) and loading condition. Mechanical properties of BFRP were conducted with tensile and in-plane shear test, by using domestically fabricated sample. The classical plate theory was utilized to analyze symmetrical laminate composite plate in this present paper.

## 2. EXPERIMENTAL PROCEDURE OF BFRP LAMINATE

### 2.1 Materials

In this study, the basalt fiber was supplied by Basalt Fiber Tech Company in Australia and the epoxy resin (EPOKUKDO YD-128) was supplied by the Kukdo Chemical Company in South Korea. The specification properties of basalt fiber and epoxy resin used in this study were presented in Fig. 1. Furthermore, the mechanical properties of unidirectional basalt and epoxy such as: density, thickness, moisture content and viscosity were shown Table 1 and Table 2, respectively.

### 2.2 Sample preparation and test program

In this study, a series of experiments were performed to investigate the tensile performance (0° and 90° unidirectional) and in-plane shear performance of the basalt fiber samples. Also, the tensile test with 0° unidirectional, tensile test with 90° unidirectional and in-plane shear test samples were shown in Fig. 2(a), 2(b) and 2(c), respectively. The elastic modulus was determined from the linear portion of the stress-strain curve from both the tensile test and the in-plane shear test (Modulus of elasticity). In order to fabricate the samples, five layers of basalt fiber was combined by hand lay-up and cured about two hours in the chamber with temperature 80°C. After taking out from the chamber, water jet cutter was used to cut the sample into specific sizes. The Universal Testing Machine (model AST-MHA) with capacity 100kN was used. Also, the strain gauge was installed at the central of the sample to measure the strain of the specimens was shown in Fig. 3. Finally, five samples were tested at each level of experiment and their average values were reported as the actual result.

Tensile tests of BFRP were conducted according to ASTM (D 3039/D 3039M - 00), the tensile test samples were prepared according to the dimension shown in Table 3. The tensile test specimens were pulled with a strain rate of 0.01/min. Also, in-plane shear test of BFRP were accomplished as per ASTM (D 4255/D 4255M - 01). The specimen for this test has the following dimension: length 154mm, width 58mm, 3 holes along the length in both sides with diameter 10mm and the gap between each hole along the length were 51mm and along the width is 58mm. The in-plane shear test was followed a standard strain rate of 0.01/min.

### 2.3 Test results

Fig. 4(a) demonstrated the stress-strain curves for tensile test (0° unidirectional). It was observed that the curve increased in a linear form. Thus, when the strain increased, the stress also increased. Fig. 4(b) showed the curve of stress-strain for tensile test (90° unidirectional). Also, it was found that the characteristics of the curve is similar to bilinear. The curves showed two distinct zones, divided at the strain of 0.0035. It increased quickly in strain from 0 to 0.0035. Conversely, for strain greater than 0.0035, this rapid increment was reduced. Fig. 4(c) showed the stress-strain for in-plane shear test. It was found that this graph followed the exponential form. When strain between 0 to 0.03 the graph increases rapidly. In contrast, it increased slightly for strain greater than 0.03. The elastic modulus and shear modulus of elasticity was determined according to ASTM D 3039/D 3039M-00 and D4255/D 4255M-01, respectively.

Table 4 summarized the average results that were obtained from the graph above. Also, these results were used as input to calculate the buckling resistance.

## 3. VARIATION FORMULATION FOR STABILITY OF COMPOSITE LAMINATE PLATES

### 3.1 Theory of Composite Laminate Plates

Thin symmetrically rectangular laminated composite plates with dimensions a and b that is subjected to combined loading was studied. The geometry of plate and external force were illustrated in Fig. 5. The x-y plane parallel to the middle plane of the plate and the z-axis is perpendicular to the middle plane.

Based on the classical Kirchhoff’s plate theory, the displacement of laminate composite plate is given by:

$u ( x , y , z ) = u 0 ( x , y ) + z w 0 , x ( x , y ) υ ( x , y , z ) = υ 0 ( x , y ) + z w 0 , y ( x , y ) w ( x , y , z ) = z w 0 ( x , y )$
(1)

Where, (u, υ, w) are the displacements components along the (x, y, z) coordinate axes, respectively, and (u0, υ0, w0) are the displacement on the middle plane of the plate. The strain-displacement relations are defined in condition of the displacement as follows:

${ ∈ } = { ∈ x ∈ y ∈ x y } = { − z ∂ 2 w ∂ x 2 + 1 2 ( ∂ w ∂ x ) 2 − z ∂ 2 w ∂ y 2 + 1 2 ( ∂ w ∂ y ) 2 − 2 z ∂ 2 w ∂ x ∂ y + ∂ w ∂ x ∂ w ∂ y }$
(2)

where, є1 and є2 represent the linear and non-linear parts of the strain are written as:

$∈ 1 = { k x k y k x y } = { − z ∂ 2 w ∂ x 2 − z ∂ 2 w ∂ y 2 − 2 z ∂ 2 w ∂ x ∂ y }$
(3)

$∈ 2 = { ∈ x o ∈ y o ∈ x y o } = { 1 2 ( ∂ w ∂ x ) 2 1 2 ( ∂ w ∂ y ) 2 ∂ w ∂ x ∂ w ∂ y }$
(4)

The [A], [B] and [D] matrices are the stiffness matrices of the laminate, and [Q] is the stiffness matrix of the ply. Fig. 6

$A i j = ∑ k = 1 K ( Q ¯ i j ) k ( z k − z k − 1 ) B i j = 1 2 ∑ k = 1 K ( Q ¯ i j ) k ( z k 2 − z k − 1 2 ) D i j = 1 3 ∑ k = 1 K ( Q ¯ i j ) k ( z k 3 − z k − 1 3 ) [ Q ] = [ [ A ] [ B ] [ B ] [ D ] ]$
(5)

The total potential energy πp of the laminate plate under combined load is equal to the sum of strain energy and external force. The total potential energy is:

$τ p = U + V$
(6)

Where, U is the strain energy, V represents the external force in-plane load.

$U = 1 2 ∫ − h / 2 h / 2 [ ∫ 0 L x ∫ 0 L y ( σ x ∈ x + σ y ∈ y + τ x y γ x y ) d y d x ] d z V = 1 2 ∫ 0 L x ∫ 0 L y ( P x ∂ 2 w ∂ x 2 + P x y ∂ 2 w ∂ x ∂ y + P y ∂ 2 w ∂ y 2 ) d y d x δ U + δ V = 0$
(7)

### 3.2 Finite Element Formulations

The composite plate was modeled by using a commercial program, MATLAB, to develop a structure physical model. The whole model was divided into 10x10 meshes for all cases. There are four nodes per element with three degrees of freedom at each node. Therefore, the simplified shape function was used in this present paper to analyze the buckling load of composite plates. Fig. 7

The virtual linear strain energy

$δ U = ∭ δ ∈ 1 Q ¯ ∈ 1 d V ⇒ δ U = δ d K d T$
(8)

$δ V = ∭ δ ∈ 2 σ d V ⇒ δ V = − δ d P x K D d T$
(9)

Where,

$K = ∬ B T D B d x d y ; K D = ∬ { ∂ N i ∂ x , ∂ N i ∂ y } [ 1 P x y / P x P x y / P x P y / P x ] { ∂ N j ∂ x ∂ N j ∂ y } d x d y$

$So , δ d K d T − δ d P x K D d T = 0$
(10)

$[ K − P x K D ] { d } = 0$
(11)

Where, K is the global stiffness matrix, KD is the geometric matrix, Px is the buckling loading and N is the shape function. The vector d is the i-th buckling mode. Thus, buckling loads can be received by resolving the eigenvalue problem in Eq. (11). Fig. 8

### 3.3 Boundary conditions

Two types of boundary conditions were used, as the following:

Four edges clamped support (CCCC):

$A t x = 0 , a : w = ϕ = ψ = 0 A t y = 0 , b : w = ϕ = ψ = 0$

Four edges simply support (SSSS):

$A t x = 0 , a : w = ψ = 0 A t y = 0 , b : w = ϕ = 0$

## 4. RESULTS AND DISCUSSIONS

The thickness of composite plates h = 2mm and a/h = 100 were studied. The parameters consider in this study are the following:

• Stacking sequence: [452/-452/012/-452/452]

• Aspect ratios of plate dimension (a/b) : 0.5, 1.0, 1.5, 2.0, and 2.5

• Boundary conditions: simply support and clamped support

• Py/Px ratio: 0.2, 0.4, 0.6, 0.8 and 1.0

where, Py is applied force in y direction and Px is applied force in x direction.

Fig. 9 illustrated the relationship between non-dimensional buckling load and aspect ratio (a/b) from 0.5 to 2.5 with different ratio of (Py/Px) from 0.2 to 1.0. Moreover, the plate edges were considered simply support. Result of biaxial loading was shown in Fig. 9(a) and combined biaxial and shear loading was shown in Fig. 9(b). While the aspect ratio (a/b) between 0.5 and 1.0, it was found that the non-dimensional buckling load decreased quickly for all (Py/Px) ratio cases, which decreasing around 58%. Alternatively, when the aspect ratio was between 1.0 and 2.5, it only slightly decreased. Thus, it meant that the bigger the aspect ratio was, the smaller the percentage decreased. Furthermore, it could be seen that the smaller (Py/Px) ratio was, the bigger the non-dimensional buckling load was for both Fig. 9(a) and 9(b).

Fig. 10 showed the relationship between non-dimensional buckling load and aspect ratio (a/b) from 0.5 to 2.5 in different ratio of (Py/Px) from 0.2 to 1.0. However, the plate edges had clamped support. It was observed that Fig. 10 had the same trend as Fig. 9 for both cases (a) and (b). According to Fig. 9 and 10, it was shown that while combined biaxial and shear load were applied, the percentage decrease of non-dimensional buckling load was less than 4% when compare to applied only biaxial load. In conclusion, the in-plane shear loading had a little effect on the non-dimensional buckling load.

Fig. 11 illustrated the relationship between non-dimensional buckling load and aspect ratio (a/b) from 0.5 to 2.5 in different ratio of Py/Pxy from 0.2 to 1.0. The plate edges were supported by simply support and clamped support. Fig. 11(a) and 11(b) shown buckling resistance of laminate composite plates that subjected to combined biaxial and shear loading, respectively. For both simply support and clamped support with the aspect ratio of 0.5 to 1.0, it was found that the non-dimensional buckling load decreased rapidly. Alternatively, it slightly decreased, when the aspect ratio was between 1.0 to 2.5. Moreover, for aspect ratio equal to 0.5, it was demonstrated that the smaller Py/Pxy ratio was, the bigger the non-dimensional buckling load became. While the aspect ratio getting bigger, in case of Py/Pxy equal to 0.6, 0.8 and 1.0 the non-dimensional buckling load had similar value for both simply support and clamped support. Therefore, it was concluded that if the aspect ratio were getting bigger, the Py/Pxy ratio 0.6, 0.8 and 1.0, had very insignificant effect on non-dimensional buckling load.

## 5. CONCLUSIONS

In this study, the buckling resistance of basalt fiber laminate composite rectangular plates subjected to combined loading was presented. Various parameters, like, aspect ratio (a/b), boundary condition and loading condition were investigated. The buckling behavior of basalt fiber reinforced polymer (BFRP) was evaluated with numerical approach by using the finite element method (FEM). The mechanical properties of BFRP were obtained under tensile and in-plane shear test, by using domestically fabricated sample. The results could be summarized as follows:

• For the cases of biaxial, combined biaxial and shear loading, when the aspect ratio was 0.5 and 1.0, the non-dimensional buckling load decreased quickly. Alternatively, it decreased slightly, when aspect ratio was 1 to 2.5.

• The smaller the (Py/Px) ratio was, the bigger the non-dimensional buckling load became. This was true for both simply support and clamped support.

## ACKNOWLEDGMENT

This work was supported by the National Research Foundation [NRF] grant funded by the Korea government [MEST][No.2017R1A2B3008623].

## Figure

FRP Material Component
Sample
Strain Gauge Attach on the Samples
Experimental Results
Geometry of an n-layer Laminate
The x, y, z Laminate Coordinate System
Flow Chart of Buckling Analysis of Laminate Composite Plates
Non-dimensional Buckling Load for Simply Support with Various Ratios (Py/Px)
Non-dimensional Buckling Load for Clamped Support with Various Ratios (Py/Px)

## Table

Raw Material Properties of Unidirectional Basalt Fiber
Raw Material Properties of Resin
Tensile Specimen Geometry Recommendations
Basalt/Epoxy Properties

## Reference

1. ASTM D 3039/D 3039M - 00 (2000), “Standard Test Method for Tensile Properties of Polymer Matrix Composite Materials”, American Society for Testing and Materials (ASTM), Pennsylvania, USA.
2. ASTM D 4255/D 4255M - 01 (2001), “Standard Test Method for In-Plane Shear Properties of Polymer Matrix Composite Materials by the Rail Shear Method”, American Society for Testing and Materials (ASTM), Pennsylvania, USA.
3. Avci, A. , Kaya, S. , and Daghan, B. (2005), “ Thermal Buckling of Rectangular Laminated Plates with a Hole ”, Journal of Reinforced Plastics and Composites, Vol. 24, No. 3, pp. 259-272. doi:
4. Kollár, L. P. and George, S. S. (2003), Mechanics of Composite Structures, Cambridge University Press, 2003.
5. Ounis, H. , Tati, A. , and Benchabane, A. (2014), “ Thermal Buckling Behavior of Laminated Composite Plates: A Finite-element Study ”, Frontiers of Mechanical Engineering, Vol. 9, No. 1, pp. 41-49. doi:
6. Reddy, J. N. (1997), Mechanics of Laminated Composite Plates: Theory and Analysis, Boca Raton.
7. Seo, H. , Son, H. , Kim, S. , and Jung, W. (2016), “ Simplified Analysis on Steel Frame Infill with FRP Composite Panel ”, World Academy of Science, Engineering and Technology International Journal ofCivil, Environmental, Structure, Construction and Architectural Engineering, Vol. 10, No. 3, pp. 381-384. doi:
8. Shaterzadeh, A. R. , Abolghasemi, S. , and Rezaei, R. (2014), “ Finite Element Analysis of Thermal Buckling of Rectangular Laminated Composite Plates with Circular Cut-out ”, Journal of Thermal Stresses, Vol. 37, No. 5, pp. 604-623. doi:. 885322
9. Shiau, L. C. , Kuo, S. Y. , and Chen, C. Y. (2010), “ Thermal Buckling Behavior of Composite Laminated Plates ”, Composite Structures, Vol. 92, No. 2, pp. 508-514. doi:
10. Shufrin, I. , Rabinovitch, O. , and Eisenberger, M. (2008), “ Buckling of Laminated Plates with General Boundary Conditions under Combined Compression, Tension, and Shear—A Semi-analytical Solution ”, Thin-Walled Structures, Vol. 46, No. 7-9, pp. 925-938. doi:. 01.040
11. Shufrin, I. , Rabinovitch, O. , and Eisenberger, M. (2008), “ Buckling of Symmetrically Laminated Rectangular Plates with General Boundary Conditions—A Semi Analytical Approach ”, Composite Structures, Vol. 82, No. 4, pp. 521-531. doi:
12. Sim, V. and Jung, W. (2017), “ Buckling Ressistance of Basalt Fiber Reinforced Polymer Infill Panel Subjected to Elevated Temperature ”, World Academyof Science, Engineering and Technology International Journal of Civil, Environmental, Structure, Constructionand Architectural Engineering, Vol. 10, No. 3, pp. 401-405. doi:
13. Sim, V. , Kim, S. , Choi, J. , and Jung, W. (2016), “ Influence of Stacking Sequence and Temperature on Buckling Resistance of GFRP Infill Panel ”, World Academy of Science, Engineering and TechnologyInternational Journal of Civil, Environmental, Structure, Construction and Architectural Engineering, Vol. 10, No. 3, pp. 401-405. doi:
14. Thangaratnam, K. R. and Ramachandran, J. (1989), “ Thermal Buckling of Composite Laminated Plates ”, Computers & Structures, Vol. 32, No. 5, pp. 1117-1124. doi: