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# 축방향 압축을 받는 폐단면리브로 보강된 복합적층판의 좌굴 해석연구

# Buckling Analysis of Laminated Composite Plates Longitudinally Stiffened with U-Shaped Ribs

### Abstract

- 1. Introduction
- 2. Analysis Modeling
- 2.1 cross-section dimensions and material properties
- 2.2 Finite element modeling
- 2.3 Local plate buckling strength
- 3. Numerical Analysis Results
- 4. Conclusions
- Acknowledgement

- 201203-05-최병호.pdf604.6KB

### 1. Introduction

The laminated composite plates have competent features for structural plates, because they are fairly lightweight and their stiffness and strength could be easily controlled to meet the design requirements by only adjusting the ply orientations and stacking sequences. Large plate structures subjected to an axial compression are stiffened by attaching longitudinal stiffeners in order to enhance the cross-sectional efficiency. the longitudinal stiffeners are classified as the open section type and the closed section type as shown in Fig. 1. Though the open-section type has been mainly used so far due to its simplicity of design and fabrication, various types of large structures stiffened with U-shaped cross-section ribs (below U-rib) are being appeared especially in the Western Europe region(Chou et al., 2006, ASSHTO LRFD 2007). Currently, there is still a lack in clear design guidelines or sufficient research about the optimum section design of U-ribs (Korean specifications, 2008; AASHTO LRFD Bridge design specifications 4th ed, 2007; Chou et al., 2006). Furthermore, technical literature related to the application and design of laminated composite plates stiffened by U-rib is not quite sufficient.

**Fig. 1. **Cross-section of stiffened plate model: (a) U-ribs; (b) open-type section ribs

Williams Stein(1976) examined the buckling stress of lonitudinally stiffened laminated panels, and Romeo (1986) conducted an experimental study to evaluate buckling strength of laminated composite plates. Heo et al.(1999) performed a series of parametric studies on post-buckling behavior of stiffened plates by open-section type stiffeners. As shown in Figure 1, it is expected that the effective width-to-thickness ratio could be reduced for the U-rib stiffened plates when compared with using the open-section ribs.

Choi and Choi (2012) presents the parametric study results from a series of the buckling analyses of isotropic stiffened plates along with the bending stiffness of U-ribs. In order to make a rational design of U-rib stiffened plates, it is required to figure out the buckling modes along with the U-rib stiffness and the minimum required stiffness for a target strength. Thus, this study conducted a series of parametric studies of the elastic buckling analyses of laminated composite plates along with the sectional dimension of the U-rib, in which the finite element modeling based on the classical laminated plate theory was applied.

The main objective of this study is to investigate the variations of the buckling modes and strengths of laminated composite plates along with U-rib sectional dimensions and stiffness. This kind of study will be helpful to provide preliminary design information about an optimum sectional design of laminated composite plates stiffened with U-ribs.

### 2. Analysis Modeling

#### 2.1 cross-section dimensions and material properties

The stiffened plate model section of the numerical parametric study is shown in Fig. 2, which have two U-ribs arranged in the spacing of w. Detailed dimensions for the U-ribs and the stiffened plate models are presented in Table 1, in which w_{1} should be the effective width of the stiffened plate models for plate buckling behavior, since w_{3} is the half size of w_{1} .

**Fig. 2. **Cross-section of U-rib stiffened plate model

**Table 1.**Sectional dimensions of U-ribs

The U-rib thickness t_{u} were varied along with the range given in Table 1 for the parametric analysis studies. Therefore, the effective aspect ratio that mainly affects the buckling behavior of the stiffened plate is expressed as below:

The material properties for the analysis models are same as Table 2, and the symmetric laminates [(0°)4]s and [(0°/90°)2]s are respectively considered for the ply orientations of the analysis model composite plates.

**Table 2.**Material properties

#### 2.2 Finite element modeling

The current numerical analysis studies were conducted by the ABAQUS finite element code and the stiffenedplates modeled by using the 4-node plane element S4R5. The w_{1} was divided into at least 12 elements as shown in Figure 3, which can obtain a sufficient stability and convergence of the numerical analysis (Park and Choi, 2011).

**Fig. 3. **Finite element mesh and boundary conditions

The axial uniform compression was loaded at the both end sides as shown in Figure 4, and the stiffened plates were simply supported along the 4-sides. The parametric bifurcation analyses were conducted along with the thickness of the U-ribs. This is intended to investigate the variation of buckling modes along with the bending stiffness of U-ribs.

**Fig. 4. **Modeling scheme

#### 2.3 Local plate buckling strength

The analysis results present the buckling stress of the laminated composite plates, which can be compared with the theoretical strength equations. As discussed in the previous section, w_{1} should be the effective width of the stiffened plate models for the elastic plate buckling. The elastic buckling equations with simply supported boundary condition is derived as Eq. (2) based on the classical laminated plate theory. When the U ribs have an adequate stiffness, it is expected that the buckling strength from the analysis results can attain the amount of Eq. (2). The minimum required stiffness values can be obtained from the comparison.

### 3. Numerical Analysis Results

The buckling eigenvalues were obtained from the parametric finite element analyses, some of which are presented in Table 3. Table 3 provides the ply thickness range of the analysis models and the variation of the buckling stress along the range. It was found from the finite element analysis results shown in Figs. 5 ~ 8 that the buckling modes could be classified into 3 types, which are the U-rib local buckling (ULB), the subpanel plate local buckling (PLB), and the global column buckling (GCB) modes.

**Table 3.**Bifurcation analysis results

The subpanel local buckling mode of the stiffened laminated composite plates is featured as the nodal lines formed along the webs of the U-ribs. It should be noted from the PLB mode that the local plate buckling mode of a half-sine curve is shown in the w₂ region under the U-ribs. In contrast to the subpanel local buckling mode, the global column buckling (GCB) mode does not show such the nodal lines formed at the U-rib locations, in which the U-ribs should be buckled like a column subjected to a axial compressive loading.

Figs. 5(a)~8(a) show the representative appearance of the ULB mode, which occurs when the ply thickness belongs to relatively lower range. The compressive strength of the stiffenened plate is extremely reduced when the ULB mode occurs. Thus, it is fairly important to recognize the minimum thickness to prevent the ULB mode. Since the minimum thickness t_{U_Limit } is influenced in a complex way by the ply thicknesses and orientations, a rational method to determine it must be considered for the future study.

**Fig. 5. **Variation of buckling modes along with U-rib thicknesses ([(0°)4]S, α =10, t_{p_ply}=3mm )

**Fig. 6. **Variation of buckling modes along with U-rib thicknesses ([(0°/90°)2]S, α =10, t_{p_ply}=1mm)

**Fig. 7. **Variation of buckling modes along with U-rib thicknesses ([(0°)4]S, α = 5, t_{p_ply}=2mm )

**Fig. 8. **Variation of buckling modes along with U-rib thicknesses ([(0°/90°)2]S, α = 3, t_{p_ply}=1mm )

Then, in a little thicker range that the ULB mode does not appear any more, first the global column buckling appears as shown in Fig. 5~6(b). If the thickness of the U-ribs is increased continuously and then the GCB mode is restrained, the PLB mode should be noted in which the local plate buckling mode of a half-sine curve appeared in the w_{2} region under the U-ribs.

A similar half-sine curve also was found within the plate inner the U-ribs, which provides an restraining effect to the GCB buckling mode. Thus, when the theoretical equation Eq. (2) is applied with the effective width w₁, the analysis results show at least 10% larger than the values from Eq. (2) that means the increment of the elastic buckling strength. This is also ascertained form Figs. 9~12 that plot the bifurcation buckling strengths along with the U-rib bending stiffness.

The elastic buckling stresses were obtained from the parametric numerical analyses which were presented in Figs. 9 ~ 12. The critical stresses are proportionally increased along with the increment of the ply thickness of the U-ribs. After reaching a threshold value, the increasing rate of the critical stress is remarkably reduced.

**Fig. 9. **Variation of buckling strength along with t_{u_ply} ([(0°)_{4}]_{s}, t_{p_ply}=2mm, α = 3)

**Fig. 10. **Variation of buckling strength along with t_{u_ply} ([(0°)_{4}]_{s}, t_{p_ply}=2mm, α = 5)

**Fig. 11. **Variation of buckling strength along with t_{u_ply} ([(0°/90°)_{2}]_{s}, t_{p_ply}=1mm, α = 5)

**Fig. 12. **Variation of buckling strength along with t_{u_ply} ([(0°/90°)_{2}]_{s}, t_{p_ply}=1mm, α=10)

Above the threshold point, the buckling mode is altered into the PLB mode. In addition, the buckling strength of the PLB mode exceeds the amount of Eq. (2). However, from this point the buckling strength is not meaningfully increased. Thus, it is found that increasing the ply thickness above the threshold point should be an inefficient way for the aspect of the buckling strength. In other words, the threshold point that initiates the PLB mode can be selected as an optimum design for the U-ribs. The ply thickness corresponding to the threshold point is presented for each case in Table 4 which is marked as t_{u_ply(req.), } since it has an important meaning of a sufficient bend stiffness to make a nodal line along the attached locations on the plates. For the model cases, the minimum required ply thickness t_{u_ply(req.) } is influenced more significantly by the laminated composite plate thickness t_{p_ply} than the effectived aspect ratio α . Since the bending stiffness of the laminated composite U-ribs be varied along with the ply orientations and sequences, additional studies are required for more various cases to provide a rational optimum design guideline for the U-ribs.

**Table 4.**Comparitive study on buckling strengths

In Table 4, the theoretical estimate "F_{r_theory} " are the calculated values from Eq. (2). It is revealed in Table 4 that, for the analysis model cases with the U-rib ply thickness of f_{u_ply(req.), } the buckling strengths could attain at least 14% to 30% larger values than Eq. (2). Thus it can be concluded that the stiffened plates by the adequate U-ribs with the minimum required ply thickness t_{u_ply(req.)} may hold an enhanced rigidity.

In Table 4, the theoretical estimate "F_{cr_theory} " are the calculated values from Eq. (2). It is revealed in Table 4 that, for the analysis model cases with the U-rib ply thickness of t_{u_ply(req} ), the buckling strengths could attain at least 14% to 30% larger values than Eq. (2). Thus it can be concluded that the stiffened plates by the adequate U-ribs with the minimum required ply thickness t_{u_ply(req.) } may hold an enhanced rigidity.

### 4. Conclusions

This study examines the varied buckling mode shapes of the laminated composite plates along with U-rib ply thickness by using the numerical analysis method. Based on the parametric numerical analysis results, it was found that the typical 3 types of buckling modes consist of: the U-rib local buckling (ULB), the subpanel plate local buckling (PLB), and the global column buckling (GCB) modes.

It should be noted from the PLB mode that the nodal lines should be formed along the attached locations of the U-ribs and the local plate buckling mode of a half-sine curve throughout the effective width w_{2} region. The ply thickness corresponding to the threshold point that initiates the PLB mode is defined as t_{u_ply(req} .), which are intensively reviewed with the design parameters.

The buckling strength of the PLB mode is similar with the amount of Eq. (2), which is the buckling strength of the simply supported plates. when reviewed in more detail, it is revealed that for the analysis cases with the U-rib ply thickness of t_{u_ply(req.)} the buckling strengths could attain at least 14% to 30% larger values than Eq. (2). Thus it can be concluded that the stiffened plates by the adequate U-ribs with the minimum required ply thickness t_{u_ply(req.)} may hold an enhanced rigidity.

Since the bending stiffness of the laminated composite U-ribs be varied along with the ply orientations and sequences, additional studies are required for more various cases to provide a rational optimum design guideline for the U-ribs.

### Acknowledgement

This study is supported by the National Research Foundation of Korea (2011-0013918). Author appreciates the financial support.

### Reference

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4.Heo, S. P., Yang, W. H., Seong, K. D., and Cho, M. R. (1999). "A study on the buckling and postbuckling behaviors of laminated composite plates and stiffened laminated composite panels by finite element method." J. of Korean Comput. Struct. Eng., Vol.12, No.4, pp.599-606.

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