Local buckling of longitudinally stiffened plates with rotational stiffness of closed-section ribs

Journal of Constructional Steel Research 167 (2020) 105876

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Journal of Constructional Steel Research

Local buckling of longitudinally stiffened plates with rotational stiffness of closed-section ribs Byung H. Choi a,⁎, Arriane Nicole P. Andico a, Sang hyun Choi b a b

Department of Civil and Environmental Engineering, Hanbat National University, Daejeon 34158, Republic of Korea Korea National University of Transportation, Gyeonggi-do 16106, Republic of Korea

a r t i c l e

i n f o

Article history: Received 27 November 2018 Received in revised form 19 September 2019 Accepted 18 November 2019 Available online xxxx Keywords: Stiffened plate Closed-section rib Elastic buckling Longitudinal stiffener Plate strength Rotational stiffness

a b s t r a c t Recent studies have shown that the rotational stiffness of longitudinally installed closed-section ribs increases the local buckling strength of thin plates. Thus, in this study, the equations for the buckling strength of compressively loaded stiffened plates, which reasonably account for the partially restrained effect, were theoretically derived using the principle of minimum potential energy. Through three-dimensional refined finite element analysis performed using ABAQUS that appropriately simulates the buckling of plates stiffened with rotational rigidities along the sides, a series of parametric numerical analyses were conducted to examine the variation in buckling stresses based on the influential parameters revealed from the theoretical formulas. Further simplified and readily applicable formulas for the strength increment factor were derived from a series of rigorous regression analyses on the parametric analysis results. A comparative study of the suggested approximate solutions and the numerical analysis results was carried out to validate the proposed method. © 2019 Published by Elsevier Ltd.

1. Introduction and background Longitudinally stiffened plates are structurally economical and offer high stability even though they are thin-walled structures. Therefore, they have been widely used, and a wide range of studies based on stiffened plate buckling have been conducted. A significant amount of research [7,8,13,14] has focused on the investigation of the compressive strength or buckling behavior of stiffened plates, mainly with opensection stiffeners. As far as closed-section ribs are concerned, extensive research on the compressive strength of orthotropic plates has been conducted in Europe since the early 1970s that has focused mainly on the significance of parameters such as plate and rib geometric imperfections, width–thickness ratio, slenderness ratio, and residual stresses [6]. Until recently, the compressive strengths and post-buckling behaviors of stiffened deck plates with U-ribs have been analytically and experimentally investigated. It has recently been noticed that the local plate buckling strength of stiffened steel plates significantly increases when the rotational stiffness of the closed-section ribs exceeds a certain threshold value [1,2]. Such kinds of effects tend to be more pronounced when the closed section ribs are arranged such that such that the width of the subpanel(WS) and that of the rib(WR) are unequal; in general design practices of orthotropic deck plates, the U-rib width (WR) is equal to or similar to the subpanel width (WS). Previous studies suggest that there is no ⁎ Corresponding author. E-mail address: [email protected] (B.H. Choi).

https://doi.org/10.1016/j.jcsr.2019.105876 0143-974X/© 2019 Published by Elsevier Ltd.

reliable design guide to facilitate a reasonable specification of the restraining effect because of the rotational stiffness of closed-section stiffeners. Choi et al. [4] presented an explicit formula for the bending stiffness requirements for the closed-section longitudinal stiffeners to show that the rotational stiffness of closed-section ribs increases the local buckling strength of stiffened isotropic plates. They introduced the strength increment factor to reflect this effect. However, because the strength increment factor that was simply derived by a theoretical approximation process has a complicated form, it seems to be difficult to apply it for a practical design; it may require an implicit process. Thus, it is important to examine the strength increment factor the restraining effects of closed-section ribs along with influential design parameters because several unusual features may be expected in the buckling modes that differ from the traditional steel plates stiffened by U-ribs. Consequently, in this study, we investigate the variations in the local buckling strength of stiffened isotropic plates with a variety of arrangements and sectional dimensions of closed-section ribs. Then, for the strength increment factor, we obtain simplified formulas that rationally reflect the restraining effect of the rotational stiffness of the closedsection ribs. In this study, a comprehensive parametric numerical analysis was performed to investigate the proposed closed-form formulas for the buckling strength of a longitudinally stiffened plate along with the rotational stiffness of the closed-section ribs. First, approximate closed-form equations were theoretically derived using the energy methodology. Then, the effects of influential design parameters

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obtained from theoretical investigations were examined through a series of comparative studies with the parametric analysis results. Thus, an approximate solution that can reasonably estimate the strength increment factor was deduced by regression analysis from a large number of analyses encompassing a wide range of parameters. Subsequently, the validity and tendency of the proposed equations were verified using the numerical analysis results. 2. Theoretical formulation of buckling strength The elastic buckling stress of a simply supported plate [4,12,14], as shown in Fig. 1, is given by F cr ¼

4Dπ2 W S 2 tp

ð1Þ

The total potential energy of the stiffened plate subjected to axial compression consists of three components as shown below: Π ¼ ΠPi þ ΠRi þ ΠPa

ð2Þ

where ΠPi is the strain energy or the energy stored in the plate, ΠRi is the energy stored in the elastic spring, and ΠPa is the work done on the plate by the external forces. Fig. 2 represents a plate that is elastically restrained on both sides. Because there is an evident restraining effect due to the rotational stiffness of the closed-section ribs, the same approach can be considered for the partially restrained model as shown in Fig. 2. The boundary conditions of the partially restrained plate are ∂w w ¼ 0; ≠ 0 at x ¼ 0; L ∂x w ¼ 0;

∂w ≠ 0 at y ¼ 0; W S ∂y

w¼W

   mπx πy ψ 2πy ð1−ψÞ sin 1− cos þ sin L WS 2 WS

ð3Þ ð4Þ

ΠPi

ð5Þ

ZZ

2

2

2

4 ∂ wþ∂ w ∂x2 ∂y2

!2

8 !2 93 2 <∂2 w ∂2 w ∂ w =5 dydx −2ð1−υÞ − : ∂x2 ∂y2 ∂x∂y ;

The integration of each term gives Z

L

Z

WS

0

0

2

∂ w ∂x2

!2 dydx ¼

m4 π4 W 2 2

2L α

      7 8 8 1 − þψ −1 þ ð7Þ ψ2 8 3π 3π 2

! 2 2 ∂ w∂ w dydx ∂x2 ∂y2 0 0      2 2 4 m π W 8 8 1 −1 þ ψ 1− − ¼− ψ2 3π 3π 2 2LW s

Z

L

L

Z

Z

WS

WS

0

0

2

∂ w ∂y2

!2 dydx ¼

π4 W 2 α 2W S

2

      5 8 8 1 − þψ −1 þ ψ2 2 3π 3π 2

!2 2 ∂ w dydx ∂x∂y 0 0       m2 π 4 W 2 2 8 8 1 þψ −1 þ ¼ ψ 1− 3π 3π 2 2LW S L

kR W S kR W S þ 2Dπ

Using the shape function above, the buckling strength equation can be derived as follows: First, the energy stored in the plate or the strain

D ¼ 2

ð6Þ

Z

where ψ¼

energy stored in a deformed body, which can be obtained by determining the work the stresses acting on the body, is expressed as

Z

These boundary conditions are satisfied when a harmonic function is used for the out-of-plane buckling displacement [1,10] that is expressed as


Fig. 2. Typical buckling mode shape of plate with elastic restraints at both sides.

Z

ð8Þ

ð9Þ

WS

ð10Þ

Therefore, the energy stored in the plate is ΠPi ¼

      Dπ4 W 2 m4 7 8 8 1 − þψ −1 þ ψ2 2 2α 3 8 3π 3π 2 2W S      2 m 8 8 1 þψ −1 þ ψ2 1− þ 3π 3π 2 α       α 5 8 8 1 ψ2 − þψ −1 þ þ 2 2 3π 3π 2

ð11Þ

Second, the energy stored in the elastic spring is given by ΠRi

2 Z 1 4 L ¼ kR 2 0

!2 Z L ∂w

þ ∂y y¼0 0

!2 3

∂w

5dx ∂y y¼W S

ð12Þ

Integrating each term gives ΠRi Fig. 1. Typical buckling mode shape of plate simply supported at both sides.

" # 1 π2 W 2 α 2 ¼ kR ð1−ψÞ 2 WS

ð13Þ

B.H. Choi et al. / Journal of Constructional Steel Research 167 (2020) 105876

Finally, the work done on the plate by the external forces is as follows: ΠPa ¼ −

Ncr 2

Z

L 0

Z

WS

0

 2 ∂w dydx ∂x

3

whereτ1, τ2,and τ3 are functions of the rotational restraint stiffness and are defined as 2

ð14Þ

τ1 ¼ 86D2 π2 þ 573kR W S D þ 86kR W S 2 2

τ2 ¼ 86D2 π2 þ 229kR W S D þ 16kR W S 2

The integration yields the following equation:       Ncr m2 π 2 W 2 2 7 8 8 1 − þψ −1 þ ψ ΠPa ¼ − 8 3π 3π 2 4α

2

τ3 ¼ 172D2 π 2 þ 458kR W S D þ 43kR W S 2 ð15Þ

The total potential energy of the system can be obtained by adding the three components.      Dπ W m 8 8 1 2 7 − þ ψ −1 þ Π¼ ψ 8 3π 3π 2 2W s 2 2α 3      2 m 8 8 1 þψ −1 þ ψ2 1− þ 3π 3π 2 α ( )       α 8 8 1 1 π2 W 2 α 2 2 5 ð1−ψÞ ψ − þψ −1 þ þ kr þ 2 2 3π 3π 2 2 ws 4

2

w¼W

4

ΦðkR Þ ¼

In accordance with the concept of neutral equilibrium and the principle of stationary energy, critical loading can be determined as follows: 2



ð17Þ

 4

Now, we simplify the buckling strength equation given by Eq.(17). For m in Eq. (17), we substitute the value of mcr given by Eq. (18). This gives the buckling stress of the partially restrained plate, and is expressed as ð19Þ

where Φ(kR) is the strength increment factor that is derived using a harmonic function as the shape function and is expressed as ΦðkR Þ ¼

1 2

rffiffiffiffiffi τ1 τ3 þ τ2 4τ 2


 pffiffiffiffiffiffiffiffiffiffiffiffiffi 6 1:871 τ2 =τ 1 þ τ 3 =τ1 π2

ð22Þ

2

kR W S kR W S 2 þ τ1 ¼ 124 þ 22 D D2 2

kR W S kR W S 2 þ D D2 2

ð18Þ

4Dπ2 F cr ¼ ΦðkR Þ 2 W S tp

ð21Þ

kR W S kR W S 2 þ τ3 ¼ 102 þ 18 D D2

   931 8 71 13 1 > > > > −ψ þ ψ 2 =7 < 6 ð1−ψÞ 43 86 2 2kr W s 7         mcr ¼ α 6 þ 4 2 > >5 5 13 1 5 13 1 Dπ > ; :ψ2 −ψ þ −ψ þ > ψ2 191 86 2 191 86 2 2

"  2  3  4 # mπx y y y y þ ψ1 þ ψ2 þ ψ3 L WS WS WS WS

τ2 ¼ 24 þ 14

where the number of half-waves in the longitudinal direction corresponding to the minimum critical stress is 2

sin

where ð16Þ

F cr




where ψ1, ψ2, and ψ3 are unknown constants. Similarly, based on the shape function given by Eq. (21), the derived strength increment factor Φ(kR) becomes

     N cr m2 π2 W 2 2 7 8 8 1 − þψ −1 þ ψ − 8 3π 3π 2 4α

  9 8   13 13 1 > > > −ψ þ > ψ2 = < 2 4 2 2Dπ α 6 86 86 2 6m þm     ¼ 2 42α 3 2 5 13 1> α > m W S tp > ; :ψ2 −ψ þ > 191 86 2   93 8   71 13 1 > > > −ψ þ > ψ2 < α 43 86 2 =7 7     þ 5 13 1>5 2> > ; :ψ2 −ψ þ > 191 86 2 9 8 > > > > 2 = 2 < ð1−ψÞ 2kR α     þ 2 > 5 13 1 m W S tp > > ; :ψ2 −ψ þ > 191 86 2

On the other hand, the same boundary conditions, Eqs. (3) and (4), are also satisfied using a fourth-order polynomial function for the outof-plane buckling displacement [4,11], which is given as

ð20Þ

The approximate equations of Φ(kR) provided here have been theoretically derived. The increase in the local buckling strength of an elastically restrained plate is caused by the rotational stiffness of the closed-section ribs against the rotational displacement (ϕ), as shown in Fig. 3(a). The explicit formula for the rotational restraint stiffness kR is idealized based on the mechanical behavior depicted in Fig. 3(b) [4], and it is expressed as. kR ¼

  2EIU WT 6EIp 2− 0 0 þ WR h 2W T þ 3h

ð23Þ

where IU ¼

tu3 tp 3 Etp 3 ; Ip ¼ ;D ¼ 12 12 12ð1−ν2 Þ

3. Numerical analysis 3.1. Finite element modeling To calibrate and verify the closed-form formulas presented in the previous section, a series of parametric studies were conducted by using finite element modeling. The detailed model is shown in Fig. 4, wherein only half of the cross sections of the closed-section ribs are longitudinally installed along both the unloaded sides with the boundary conditions that corresponds to the buckling mode shape as assumed in Fig. 3. The closed-section stiffeners of the stiffened plate models were designed to possess sufficient bending stiffness to induce local plate buckling in the subpanel Ws based on the range of geometrical dimension of practical U-rib stiffened plates [6,13].

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B.H. Choi et al. / Journal of Constructional Steel Research 167 (2020) 105876

Fig. 3. Rotationally restrained plate model: (a) buckling behavior of plate stiffened by closed-section ribs with a sufficient rigidity and (b) rotational restraint stiffness attributable to stiffening by closed-section stiffener.

The dimensions of the longitudinally stiffened plates are presented in Table 1. The load distributions are applied on both sides along the longitudinal direction to apply uniform axial compression as shown in Fig. 5(a). Under the boundary condition, Fig. 5(b) shows the finiteelement mesh and the boundary condition of a prototype model. In this model, half of the U-shaped stiffeners should satisfy the symmetric boundary conditions shown in Fig. 5(b) to confer the symmetric rotational restraint effect on the subpanel during local buckling. The plates stiffened with longitudinal stiffeners were modeled using the four-node plane element S4R5 provided by the commercial finiteelement code ABAQUS 6.17. The effective subpanel was divided into at least twelve elements as suggested by previously conducted relevant studies [3,5,14], which provides sufficient stability and convergence for the numerical analysis. Variable analysis was used for the major influential parameters on the stiffened plate (WS, tp) and closed-section ribs (tu, WT, WR, h) to evaluate the increase in the local plate buckling strength due to the rotational stiffness. WS is the net spacing between the longitudinal stiffeners, WR is the lower width of the closed-section rib, WT is the upper width of the closed-section rib, tp is the thickness of the plate, tu is the thickness of the closed-section rib, and h is the height of the closed-section rib. In addition, different aspect ratios (α = 3, 5, 7) between the transverse stiffener spacing and WS were also considered in the variable analysis.

observed that half sine waves are formed along the longitudinal direction. The buckling strength of a plate stiffened with closed-section ribs significantly increases beyond 40% compared to the buckling strength with simple supports. This is attributed to the restraining effect and the rotational stiffness of the closed-section ribs, which can be observed in the buckling modes along both the sides of the subpanel in Fig. 6. From Table 2, it can also be determined that the local plate buckling stresses remain consistent in the case of similar models even with variations in α. Further, in Fig. 7, the theoretical equations for partially restrained plates derived from the energy method are compared with those calculated from the finite element analysis in terms of a percentage difference. The diagrams represent the analysis of tendencies and several design parameters and their correlations with the determined numerical analysis. Eqs. (20) and (22), exhibited a similar trend, and the difference compared with the numerical analysis was below 14%; this reveals a strong correlation with the numerical analysis. Based on the result of the comparative study, tp significantly affects the percentage difference in the buckling strength; the differences steadily increase along with the thickness of the plate (tp) as observed in Fig. 7(b). Through this comparative study, it was shown that the approximate closed-form solutions

Table 1 Model dimension of stiffened plate.

3.2. Analysis results Eigenvalue analysis was performed on the finite element models to obtain the minimum critical stress and buckling mode as shown in Fig. 6. Representative cases are listed in Table 2 that shows the comparison of the buckling stress (Fcr) of Eq. (19) using the strength increment factor derived from the suggested theoretical approaches, Eqs. (20) and (22), and Fcr derived from the finite element analysis (FEA). Finite element analysis reveals that the dominant mode shape is local plate buckling (PB) as shown in Fig. 6. Local plate buckling behavior occurs at WS, which is the net spacing between the longitudinal stiffeners. It is also

WS (mm)

WR (mm)

WT (mm)

h (mm)

WS/WR

α

250 285 324 364 405 250 285 324 364 405

162 162 162 162 162 243 243 243 243 243

112 112 112 112 112 168 168 168 168 168

105 105 105 105 105 158 158 158 158 158

1.54 1.76 2.00 2.25 2.50 1.03 1.17 1.33 1.50 1.67

3 3 3 3 3 3, 5, 7 3, 5, 7 3, 5, 7 3, 5, 7 3, 5, 7

Fig. 4. Stiffened plate model: (a) Section of plate reinforced with closed-section rib and (b) closed-section rib stiffener.

B.H. Choi et al. / Journal of Constructional Steel Research 167 (2020) 105876

5

Fig. 5. Stiffened plate model with loading and boundary condition: (a) Finite element model under an axially uniform loading state and (b) boundary condition.

typically tend to differ from the numerical analysis results based on each design parameter, and they can be modified to obtain better correlations in higher strength ranges. Furthermore, the analysis of the buckling strength variations is performed according to the coefficientρ, which is a function of the rotational stiffness kR and is given by ρ¼

kR W S D

ð24Þ

This is schematically presented in Fig. 8, where a clear correlation between the strength increment factor and the coefficient ρ can be

identified. Thus, it was found that the effect of the rotational stiffness of the closed-section ribs is dominated by ρ and is expected to be properly represented by it. Both the approximate closed-form equations (Eqs. (20) and (22)) offer high accuracy, thereby exhibiting a good correlation with the numerical analysis as shown in Fig. 8. However, it is also evident that the correlation can be improved in the lower range of the coefficient ρ. 4. Proposition of the strength factor Φ(kR) Through comparative analysis studies and the theoretical equations obtained in the previous sections, the strength increment factor is

Fig. 6. Local plate buckling (PB): (a) WS = 285 mm, (b) WS = 324 mm, (c) WS = 364 mm, and (d) WS = 405 mm.

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B.H. Choi et al. / Journal of Constructional Steel Research 167 (2020) 105876

Table 2 Results of numerical analysis (tu = 10 mm). tp

WS

7 7 8 9 10 8 9 10 11 7 8 8 9 9 10 11 8 9 9 10 11

285 324 364 364 364 405 405 405 405 285 324 364 364 405 405 405 324 364 405 405 405

WR

243 243 243 243 243 243 243 243 243 243 243 243 243 243 243 243 243 243 243 243 243

WT

168 168 168 168 168 168 168 168 168 168 168 168 168 168 168 168 168 168 168 168 168

α

3 3 3 3 3 3 3 3 3 5 5 5 5 5 5 5 7 7 7 7 7

Mode

PB PB PB PB PB PB PB PB PB PB PB PB PB PB PB PB PB PB PB PB PB

Fcr (MPa) FEA

Eq. (1)

%

Eq. (20)

%

Eq. (22)

%

673.44 537.66 544.23 665.22 796.70 445.71 546.22 655.57 774.38 688.04 682.81 549.29 673.27 550.97 662.47 783.10 683.11 672.76 552.00 661.13 779.35

447.09 345.94 357.99 453.08 559.36 289.18 365.99 451.84 546.72 447.09 451.84 357.99 453.08 365.99 451.84 546.72 451.84 453.08 365.99 451.84 546.72

50.6% 55.4% 52.0% 46.8% 42.4% 54.1% 49.2% 45.1% 41.6% 53.9% 51.1% 53.4% 48.6% 50.5% 46.6% 43.2% 51.2% 48.5% 50.8% 46.3% 42.5%

721.17 564.94 575.72 712.23 862.49 469.98 582.17 705.67 840.99 721.17 717.82 575.72 712.23 582.17 705.67 840.99 717.82 712.23 582.17 705.67 840.99

6.6% 4.8% 5.5% 6.6% 7.6% 5.2% 6.2% 7.1% 7.9% 4.6% 4.9% 4.6% 5.5% 5.4% 6.1% 6.9% 4.8% 5.5% 5.2% 6.3% 7.3%

688.12 538.64 549.44 680.88 825.92 448.23 556.02 674.97 805.51 688.12 685.65 549.44 680.88 556.02 674.97 805.51 685.65 680.88 556.02 674.97 805.51

2.1% 0.2% 0.9% 2.3% 3.5% 0.6% 1.8% 2.9% 3.9% 0.0% 0.4% 0.0% 1.1% 0.9% 1.9% 2.8% 0.4% 1.2% 0.7% 2.1% 3.2%

rationally simplified and is given by ΦðkR Þ ¼

1 2

   x1 þ x2 κ þ1 x3 þ x4 κ

ð25Þ

Because Eqs.(20) and (22) have been derived by an approximation method, they must have distinctive limitations. Thus, regression analyses were performed on 132 data cases that were collected through a series of parametric numerical analyses and then a valid equation for the strength increment factor has been obtained,

Fig. 7. Comparative study and trend analysis: (a) WS, (b) tp, (c) h, and (d) tu.

B.H. Choi et al. / Journal of Constructional Steel Research 167 (2020) 105876

7

Fig. 8. Buckling strength according to coefficient ρ: (a) WS and (b) tp.

which is more precise than the theoretically derived formula (Eq. (25)). A simplified form for regression analysis has been derived and is given as 0:87ðΦðkR Þ−0:5Þ ¼

ð3:2D þ κ Þn1 ð7:2D þ κ Þn2

ð26Þ

where κ = kRWS. A variation analysis was performed to derive n1 and n2 of the simplified formula for regression analysis as shown in Fig. 9. Consequently, nonlinear regression analysis yielded Eq. (27) for the strength increment factor that was acquired from the Minitab [9] statistical program package. 0:87ðΦðkR Þ−0:5Þ ¼

ð3:2D þ κ Þ0:98 ð7:2D þ κ Þ0:99

ð29Þ

5. Significant findings and concluding remarks

ð28Þ

The coefficient of correlation, the R − squared value, was greater than 0.99. From the regression analysis equations (Eqs. (27) and (28)), it is evident that ρ should be the actual influential parameter of Φ(kR). A diagram showing the correlation between these regression

Fig. 9. Regression data.

  3D þ kR W S ΦðkR Þ ¼ 1:2 þ 0:5 9D þ kR W S

ð27Þ

Thus, the regression equation is expected to be   3:2D þ κ 1:0 þ 0:5 ΦðkR Þrev ¼ 1:15 7:2D þ κ

equations is derived and shown in Fig. 10. From the comparative study of Fig. 10, the regression equation, Eq. (28), is found to be well suited to the finite element analysis results. However, it can be seen that there is a relatively small discrepancy at the lower range of ρ (= kRWS/D), less than 50. Because a sufficient rotational stiffness kR is inevitably adopted such that the strength increment factor Φ(kR) is appropriately applied, Eq. (28) is still applicable for attaining a reasonable estimation of the buckling strength. Eq. (29) is readily obtained by a simple adjustment of Eq. (28), and this modification produces a more accurate correlation than Eq. (28), particularly for the lower value range of ρ.

This study examined the local buckling strength of stiffened plates with closed-section ribs under uniaxial compression, which may vary along with the rotational stiffness of closed-section ribs. A closed-form formula for the strength increment factor Φ(kR) was derived by applying the energy methodology in order to precisely reflect the variations

Fig. 10. Comparative study of regression equations.

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B.H. Choi et al. / Journal of Constructional Steel Research 167 (2020) 105876

in the buckling strength due to rotational restraint effect by closedsection stiffeners. To calibrate and verify the derived closed-form formulas, a series of parametric studies were conducted using finite element analysis through the commercial software ABAQUS. The effects of the major influential parameters were quantitatively evaluated via the variable parametric analyses. The formulation of a simplified design expression for the strength increment factor was obtained based on the closed-form theoretical equation via the regression analysis of the collected data to increase the feasibility of the theoretically derived equations. Based on the regression analyses of the numerical analysis results, a simplified equation was developed to readily apply the strength increment factor. The obtained equations tend to meticulously trace the numerically analyzed results. Because they exhibited a good correlation with the threedimensional numerical analysis results, the proposed equations were validated with a sufficient value of the correlation coefficient. The findings of this study can contribute to an efficient design of structural plates by utilizing closed-section ribs as longitudinal stiffeners. In the future, a design guide can be established for the buckling strength equations that are applicable to the inelastic buckling region through experimental verification. Acknowledgement This study was supported by a National Research Foundation of Korea (NRF) grant funded by the Korean Government (MSIP) (NRF2015R1D1A1A01058201).

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