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Elastic buckling analysis of longitudinally stiffened plates with flat-bar stiffeners
Ocean Engineering 58 (2012) 48–59
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Elastic buckling analysis of longitudinally stiffened plates with flat-bar stiffeners Ahmad Rahbar-Ranji n Department of Ocean Engineering, AmirKabir University of Technology, Hafez Ave., Tehran 15914, Iran
a r t i c l e i n f o
a b s t r a c t
Article history: Received 2 October 2011 Accepted 24 September 2012
The aim of present work is to investigate the characteristics of interactions of different buckling modes of flat-bar stiffened panels. Literature-based and rules-based expressions for assessing buckling strength of stiffened plates are studied. Energy method is employed for the analyses of selected buckling modes of stiffened plates including plate buckling, torsional buckling, web buckling and interactions of them when either plate or stiffener or both are under compression. Results are compared with numerical solutions using finite element method and available expressions to identify the applicability and accuracy of selected expressions for certain conditions. It is found that some of the given expressions for buckling analyses of stiffened plates have limited applicability. & 2012 Elsevier Ltd. All rights reserved.
Keywords: Plate buckling Stiffened panel Torsional buckling Web buckling Plate torsional rigidity
1. Introduction Longitudinally stiffened plates are fundamental structural components in ship structure even in ships with small length. Compressive longitudinal stresses produced by hull girder bending moment impose buckling of longitudinal stiffeners between two adjacent transverse girders. Depending on dimensions of stiffeners and attached plate, different modes of buckling from local buckling of plate or stiffeners to overall buckling of panel could be occurred. Stiffener buckling itself is divided to local buckling of web or flange, beam-column type flexural buckling and torsional buckling. Flexural buckling and torsional buckling of stiffeners interact when asymmetric stiffeners are used. Interactions of plate buckling, torsional buckling and web buckling occur regardless of stiffener types (Fig. 1). Many research papers were dedicated to buckling analysis of stiffened plates. Yao et al. (1997) have performed a series of elasto-plastic large deflection analyses for stiffened panels with flat-bar (FB) stiffeners. Van der Neut (1983) has studied interactions of web, plate, flexural and torsional buckling of Z-stiffeners by strip theory. Inclusive coupled buckling analysis of stiffened plates is possible by using energy method. Fujikubo and Yao (1999) have to study the restraining effect of the stiffeners web/ flange on the edges of the plate between stiffeners. Hughes and Ma (1996) have studied interaction of plate, web, beam-column type flexural buckling, and torsional buckling of stiffeners by this method. Byklum and Amdahl (2002) have used this method for
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coupled buckling and post-buckling analysis of stiffened plates applying large deflection theory. Danielson and Wilmer (2004) have used this method for coupled buckling analysis of stiffened plate with bulb plate stiffeners. In lieu of detailed analysis, classification society rules (American Bureau of shipping, 2007; Bureau Veritas, 2002; Det Norske Veritas, 2009; Germanischer LIoyd’s, 2004; LIoyd’s Register of Shipping, 2002) proposes simple equation for the evaluation of each modes of buckling of stiffened panels independently. In some cases, these equations are improved by introducing a rotational spring representing adjacent element. Calculation of stiffness of these springs was the subject of some researches. Paik and Thayamballi (2000) and Hughes et al. (2004) have given some expressions for the calculation of torsional rigidity of stiffeners in plate buckling analysis. Paik et al. (1998) have given expression for the calculation of rotational stiffness of attached plate and upper flange in web buckling analysis of stiffeners. Faulkner et al. (1973) have given expression for calculation of spring constant of attached plate in torsional buckling analysis of stiffeners. Det Norske Veritas (2009), Zheng and Hu (2005), and Hughes (1983) have given expressions for the calculation of spring constant of attached plate in the interaction of web and torsional buckling and interaction of web, plate and torsional buckling. The validity of results by using these expressions is sometimes unclear. Continuous stiffened plate should be modeled to represent the actual interactions of buckling modes. The main aim of the present work is to investigate coupled buckling of FB stiffened plate thoroughly and to study the accuracy and applicability of literature-based and rule-based expressions. Given equation by Fujikubo and Yao (1999) based on energy method for buckling analysis of stiffened plate is modified to consider selected
A. Rahbar-Ranji / Ocean Engineering 58 (2012) 48–59
49
Fig. 1. Description of coordinate system and buckling deformation in interaction of plate buckling (W), web and torsional buckling (V). Fig. 3. Torsional buckling (V1) and web buckling modes (V2 þ V3) in flat-bar stiffener.
V 2 ¼ Z3 W 2 V 3 ¼ Z4 W 1 þ Z5 W 2 where
Fig. 2. Plate buckling modes in stiffened plate, W1 simple supported condition and W2 clamped condition.
buckling modes in the range from plate buckling, web buckling, tripping to interactions of them. Critical Euler stresses are calculated and finite element method (FEM) is employed to confirm the accuracy of the results.
Z1 ¼
k2 k6 k6 k1 k4
Z2 ¼
k1 k3 k5 k1 k4 k6
Z3 ¼ k3 Z4 ¼
k2 k4 k1 k4 k6
Z5 ¼
k3 k5 k6 k1 k4
2. Basic equation
where Equilibrium method or energy approach can be used for buckling analysis of structures. In energy method a shape function with some unknown coefficients for buckled deformation is assumed and total potential energy during buckling is calculated. Using the principle of minimum potential energy, critical Euler stress is determined. The more precisely the buckled deformation is assumed, the more accurately the critical Euler stress is evaluated. Fujikubo and Yao (1999) have used this method to calculate critical Euler stress for coupled mode of torsional, web and plate buckling in stiffened plates. They assumed plate buckling mode shape between stiffeners as mpx py W 2 mpx 2py wðx,yÞ ¼ W 1 sin sin þ 1cos ð1Þ sin a b a b 2 where a, and b are length and width of plate panel, respectively, and m is half wave numbers in longitudinal direction (Fig. 1). In Eq. (1) first sentence represents buckling mode with simple supported condition, and second one with clamped condition (Fig. 2). Buckling mode shape of stiffener’s web is assumed as follows: z mpx mpx pz mp x pz vðx,zÞ ¼ V 1 sin þ V 3 sin þ V 2 sin 1cos sin hw a a 2hw a 2hw
ð2Þ where hw is the web height. First term represents rigid body rotation of stiffener, referred as tripping or torsional buckling, second and third terms represent web buckling mode (Fig. 3). Stiffener flange is assumed as a beam with flexural and torsional rigidity firmly connected to the web. Rigid connection assumptions of plate to web and web to flange reduces five coefficients in Eqs. (1) and (2) to two as follows: V 1 ¼ Z1 W 1 þ Z2 W 2
k1 ¼ k2 ¼
p 2
phw b 2
k3 ¼
Dp 16hw Dw b2
k4 ¼ nDw
mp2 a
mp2 a ( ) mp2 p 2 k6 ¼ Dw n þ a 2hw k5 ¼ nDw
Using principle of minimum potential energy, critical Euler stress could be found from the following equation:
k1 s2x þ k2 sx sy þ k3 s2y k4 sx k5 sy þ k6 ¼ 0
ð3Þ
where sx and sy are applied compressive stresses in longitudinal and transverse directions, respectively, and
k1 ¼ g4 g6 g25 k2 ¼ g4 g9 þ g6 g7 2g5 g8 k3 ¼ g7 g9 g28 k4 ¼ g1 g6 þ g3 g4 2g2 g5 k5 ¼ g1 g9 þ g3 g7 2g2 g8
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A. Rahbar-Ranji / Ocean Engineering 58 (2012) 48–59
k6 ¼ g1 g3 g22 where g1–g9 are given in the appendix.
k1 ¼ k2 ¼ k3 ¼ k5 ¼ 0 k4 ¼ g4 k6 ¼ g1
3. Interaction of selected buckling modes in stiffened plate Eq. (3) has derived for coupled buckling analysis of stiffened plate when all buckling deformations are considered. Literaturebased expressions are proposed for interaction of some of buckling modes. To determine the accuracy and applicability of these expressions, Eq. (3) is modified to consider interaction of selected buckling modes as follows.
With attached plate:
Z1 ¼ Z2 ¼ Z3 ¼ Z5 ¼ 0 Z4 ¼
k2 k1
3.1. Tripping of FB stiffener with rigid web assumption Without attached plate:
3.2.3. Web buckling of FB stiffener, V2 and V3 mode shape
Z1 ¼ 1
Without attached plate:
Z2 ¼ Z3 ¼ Z4 ¼ Z5 ¼ 0
Z1 ¼ Z2 ¼ Z4 ¼ 0
k1 ¼ k2 ¼ k3 ¼ k5 ¼ 0
Z3 ¼ 1
k4 ¼ g4
Z5 ¼
k6 ¼ g1 With attached plate:
Z1 ¼ k2
k5 k6
k1 ¼ k2 ¼ k3 ¼ k5 ¼ 0 k4 ¼ g6 k6 ¼ g3
Z2 ¼ Z3 ¼ Z4 ¼ Z5 ¼ 0 With attached plate:
Z1 ¼ Z2 ¼ 0 3.2. Web buckling of FB stiffener
Z3 ¼ k3
To study web buckling of stiffener, different buckling deformations with and without attached plate are investigated as follows.
Z4 ¼
3.2.1. Web buckling of FB stiffener, V2 mode shape Without attached plate
Z1 ¼ Z2 ¼ Z4 ¼ Z5 ¼ 0 Z3 ¼ 1 k1 ¼ k2 ¼ k3 ¼ k5 ¼ 0
k2 k1
Z5 ¼ 0 Z6 ¼
k2 k6 k1 k3 k5
k1 ¼ k2 ¼ k3 ¼ k5 ¼ 0 k4 ¼ Z26 g6 þ 2Z6 g5 þ g4 k6 ¼ Z26 g3 þ 2Z6 g2 þ g1
k4 ¼ g6 k6 ¼ g3 With attached plate:
Z1 ¼ Z2 ¼ Z4 ¼ Z5 ¼ 0 Z3 ¼ k3
3.3. Interaction of tripping and web buckling of FB stiffener To study interaction of web buckling and tripping, different mode shapes with and without attached plate are investigated as follows 3.3.1. Interaction of tripping and web buckling, V1 and V2 mode shapes
3.2.2. Web buckling of FB stiffener, V3 mode shape
Without attached plate:
Without attached plate:
Z1 ¼ Z4 ¼ Z5 ¼ 0
Z1 ¼ Z2 ¼ Z3 ¼ Z5 ¼ 0
Z2 ¼ 1
Z4 ¼ 1
Z3 ¼ 1
A. Rahbar-Ranji / Ocean Engineering 58 (2012) 48–59
k1 ¼ k2 ¼ k3 ¼ k5 ¼ 0 k4 ¼ g6 k6 ¼ g3 With attached plate:
Z1 ¼ k2 Z2 ¼ 0
3.4. Different mode shapes of plate buckling Hughes and Ma (1996) have proposed two different mode shapes for plate buckling between stiffeners in transverse direction, namely S-shape (Fig. 4) and U-shape (Fig. 5). Eq. (1) corresponds to U-shape mode in which one half waves in transverse direction between two adjacent stiffeners are considered. For S-shape, plate buckling mode should be assumed as follows: wðx,yÞ ¼ W 1 sin
Z3 ¼ k3
mpx 2py W 2 mpx 4py sin þ 1cos sin a b a b 2
ð4Þ
Total potential energy at the instance of bifurcation can be expressed as follows:
Z4 ¼ Z5 ¼ 0 Z6 ¼
51
Y
k2 k4 k3 k5
¼ UW
ð5Þ
k4 ¼ Z26 g6 þ 2Z6 g5 þ g4
where U is the strain energy and W is the work done by applied compressive load. These quantities can be divided into following components:
k6 ¼ Z26 g3 þ 2Z6 g2 þ g1
U ¼ U Po þ U wo
ð6Þ
W ¼ W Po þW wo
ð7Þ
k1 ¼ k2 ¼ k3 ¼ k5 ¼ 0
3.3.2. Interaction of tripping and web buckling of FB stiffener, V1 and V3 mode shapes Without attached plate:
Z1 ¼ 1 Z2 ¼ Z3 ¼ Z5 ¼ 0 Z4 ¼
k4 k6
where UPo and WPo are strain energy and work component in the attached plate due to out-of-plane deformation, and Uwo and Wwo are strain energy and work component in the web due to out-of-plane deformation. Strain energy stored in the attached plate due to out-ofplane deformation is calculated from classical plate theory as follows: DP U Po ¼ 2
Z
b=2
b=2
8 !2 93 !2 < 2 = 2 2 2 2 4 @ w þ @ w 2ð1vÞ @ w @ w @ w 5dx dy : @x2 @y2 @x @y ; @x2 @y2 a=2
Z
a=2
2
ð8Þ
k4 ¼ g4 k6 ¼ g1 With attached plate:
Z1 ¼
k2 k6 k6 k1 k4
Z2 ¼ Z3 ¼ Z5 ¼ 0 Z4 ¼
k2 k4 k1 k4 k6 Fig. 4. S-shape mode of plate buckling between adjacent stiffeners.
3.3.3. Interaction of tripping and web buckling of FB stiffener, V1, V2 and V3 mode shapes Without attached plate:
Z1 ¼ 1 Z2 ¼ 0 Z3 ¼ 1 Z4 ¼ Z5 ¼
k4 k6
With attached plate: In this case, parameters are as quoted in Section2.
Fig. 5. U-shape mode of plate buckling between adjacent stiffeners.
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A. Rahbar-Ranji / Ocean Engineering 58 (2012) 48–59
where DP is the bending rigidity of attached plate and is defined as follows: DP ¼
Et3 P 12 1n2
where E is Young’s modulus, n is Poisson’s ratio and tP is the attached plate thickness. Component of work due to out-of-plane deformation in attached plate is calculated as follows: Z a=2 Z b=2 2 1 @w t P dy dx ð9Þ W Po ¼ sE 2 @x a=2 b=2 where sE is the applied compressive stress at the time of bifurcation, called Euler stress. Strain energy stored in the web due to out-of-plane deformation is calculated from classical plate theory as follows: U wo ¼
Dw 2
Z
hw 0
Z
2
2 2 4 @ vþ@ v 2 2 @x @z a=2 a=2
!2 2ð1vÞ
8 !2 93 <@2 v @2 v @2 v =5 dx dz : @x2 @z2 @x@z ;
ð10Þ
where Dw is the bending rigidity of web plate and is defined as follows: Dw ¼
Et 3 w 12 1n2
where tw is the web thickness. Component of work due to out-ofplane deformation in web is calculated as follows: Z a=2 Z hw 2 1 @v t w dz dx ð11Þ W wo ¼ sE 2 @x a=2 0 After substituting Eqs. (4) and (2) into Eqs. (8)–(9) and (10)–(11), respectively, and manipulating them yield to the following changes in the expressions of k k2 ¼
2phw b
k3 ¼
DP 32hw Dw b2
2
4. Different loading conditions in buckling analysis of stiffened plate Although in ship structures, stiffeners and attached plate are almost under the same axial compressive stresses, however, to study the accuracy of literature-based expressions, it is necessary to consider the cases of unloaded attached plate/stiffener as well. It is worth to mention that, these cases cannot be checked by FEM, since result of eigenvalue analysis expressed as a load factor. If one concentrate load applies on one node only, or some concentrate loads apply on some nodes the results would be the same, since corresponding load factors are changed accordingly.
5. Investigated cases and discussions In this study, material is considered as mild steel with E¼ 206 GPa and v¼0.3, a length of 3200 mm has been considered for stiffener, spacing between them is taken as 600 mm and thickness of attached plate is assumed equal to web thickness. To demonstrate the accuracy of this method, a series of FEM eigenvalue analyses are performed. The extent of FE model should be appropriate to capture all buckling modes and also to reduce the time of analysis. Modeling of the whole panel although satisfies the best former objective, however, increases the time of analysis. Fujikubo and Yao (1999) have suggested the triple-span double-bay model with symmetry boundary conditions on longitudinal edges and periodical continuous conditions on transverse edges where vertical displacement at locations of transverse girders is prevented. Periodical continuous conditions means that, at the same y coordinates along the transverse edges corresponding nodal dispalcements are the same. Hughes and Ma (1996) have proposed two models, onespan double-bay model and one-span triple-bay model with simple supported conditions on transverse edges and symmetry conditions on longitudinal edges. All these models are examined and compared and it was found that all models yield the same results. Therefore, one-span double-bay model which is simple to generate and is consistent with assumed boundary conditions in Eq. (1) is used in this work. The computer code ANSYS (version 5.6) has been used for this analysis. Web and attached plate are modeled using four-node quadrilateral shell element (SHELL63) with six degrees of freedom per node. To enforce tripping about junction of web to attached plate and prevent flexural buckling, displacement in y and z directions at the baseline of web are restrained. A uniformly distributed normal stress is applied over one end while holding the other end. Fig. 6 shows FE model with applied load and boundary conditions. 5.1. Tripping of stiffener with rigid web assumption without attached plate Euler stress for tripping of thin-walled open section beams about center of torsion is calculated from following equation (Timoshenko and Gere, 1961): 2 EIW pL2 þ GJ sTE ¼ ð12Þ I0 where IW, J and I0 are sectorial moment of inertia, St. Venant’s moment of inertia, and polar moment of inertia about center of torsion, respectively, L is the length of beam and G is the shear modulus. The position of center of torsion depends on boundary
4.1. Stiffener is loaded In this case, the work done by the applied load corresponds to Euler stresses on the web of stiffener, and the first sentence in the expressions of g4, g5, and g6 in the appendix should be neglected. 4.2. Plate is loaded To determine adequate boundary conditions for plate buckling analysis, it is necessary to model continuous stiffened plate with unloaded stiffeners. In this case, the first sentence in the expressions of g4, g5, and g6 in appendix should be considered.
Fig. 6. FE model with applied load and boundary conditions.
A. Rahbar-Ranji / Ocean Engineering 58 (2012) 48–59
conditions of beam. In stiffened panels, the center of torsion is located at junction point of stiffener to attached plate. Above cited parameters for FB about this point are calculated as follows (American Bureau of shipping, 2007; Bureau Veritas, 2002; Det Norske Veritas (2009); Germanischer LIoyd’s, 2004; LIoyd’s Register of Shipping, 2002): IW ¼
3 t 3W hW
36
Table 1 Tripping Euler stress (MPa), rigid web assumption without attached plate. Flat-bar (mm)
Eq. (3)
Eq. (12)
FEM
50 5 65 6 75 7 90 8 100 9 110 11 120 12 140 13 160 14 180 15 200 16 220 16 240 17 260 18 280 19 300 20 320 21 340 22 360 24 380 26 400 28 420 30 440 30 460 32
792.76 675.76 691.08 627.19 643.24 794.51 794.93 686.24 610.17 554.30 511.73 423.73 402.78 385.64 371.39 359.41 349.24 340.23 362.61 383.21 402.49 420.60 384.69 402.04
793.51 676.37 691.69 627.71 643.75 795.10 795.48 686.64 610.46 554.49 511.82 423.73 402.71 385.49 371.16 359.11 348.86 340.07 362.02 382.47 401.59 419.53 383.58 400.75
790.29 674.07 689.50 625.87 641.94 729.95 793.40 684.96 609.07 553.32 510.84 423.01 402.11 385.84 370.79 358.84 348.70 340.01 362.07 382.65 401.92 420.03 384.18 401.52
Fig. 7. FE model and buckling deformation for tripping mode of buckling.
53
3
I0 ¼
J¼
t W hW 3
hw t 3w 3
Table 1 shows Euler tripping stress for different FB calculated by Eqs. (3) and (12) and compared with FEM. As can be seen both equations yield the same results and very good agreements with FEM are observed. Fig. 7 shows FE model and buckling deformation for this mode of buckling. 5.2. Web buckling Given equations for Euler buckling analysis of stiffener’s web for FB is as follows (Bureau Veritas (2002)): 2 t sWE ¼ 0:78E W ð13Þ hW Eq. (13) is based on buckling of a rectangular plate with three edges simply supported and one edge free. In this equation torsional rigidity of attached plate is ignored. Paik et al. (1998) have proposed following equation to take into account this rigidity: 2 t sWE ¼ 0:9kw E W ð14Þ hW where kw depends on ratio of attached plate to web rotational stiffness. Figs. 8 and 9 show Euler stresses for web buckling of FB stiffeners without attached plate, and with unloaded attached plate, respectively. Following conclusions can be made: (1) As can be seen from Fig. 8, results of Eq. (3) with V2 and V3 mode shapes have the same tendency as Eq. (13). Results of Eq. (3) with V2 þ V3 mode shape are unreliable and results of
Fig. 9. Euler stress for web buckling of FB stiffeners with attached plate.
Fig. 8. Euler stress for web buckling of FB stiffeners without attached plate.
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A. Rahbar-Ranji / Ocean Engineering 58 (2012) 48–59
V2 mode shape are higher than V3 mode shape, since, clamped assumption of V2 mode shape at the base line increase buckling strength. Results of Eq. (13) are about 20% lower than Eq. (3) with V3 mode shape, since, this mode shape is not consistent with free edge assumption at the upper edge of web. It can be concluded that rule-based expression is applicable for web buckling analysis of FB stiffener when attached plate is ignored. (2) Comparing Figs. 8 and 9, it can be concluded that torsional rigidity of attached plate would increase web buckling strength of stiffeners. Eq. (13) which ignores this rigidity underestimates Euler stress of web buckling. In contrast to previous case, results of Eq. (3) with V2 mode shape are lower than V3 and V2 þV3 mode shapes, and results of Eqs. (13) and (14) both are lower than Eq. (3) with V2 mode shape. It is worth to mention that this case cannot be checked by FEM, since it is not possible to have web buckling without tripping.
Table 2 Euler stress for the interaction of tripping and web buckling without attached plate. Flat-bar (mm)
50 5 65 6 75 7 90 8 100 9 110 11 120 12 140 13 160 14 180 15 200 16 220 16 240 17 260 18 280 19 300 20 320 21 340 22 360 24 380 26 400 28 420 30 440 30 460 32
Eq. (3)
5.3. Interaction of tripping of stiffener and web buckling without attached plate Table 2 shows Euler stress for interaction of tripping and web buckling. Comparing Tables 1 and 2, it reveals that differences between FE results are very low, and differences between results of Eq.(3) with V1 þV2 þ V3 mode shape are about 3%. The reason for this behavior is that, web buckling strength is very higher than tripping strength. Therefore, web buckling has no influence on Euler stress for tripping when attached plate is ignored. It also can be concluded that when web buckling is permitted, V1 þV3 mode shape yields more accurate results than V1 þV2 þV3.
5.4. Plate buckling with unloaded stiffener Rule-based equation (American Bureau of shipping, 2007; Bureau Veritas (2002); Det Norske Veritas (2009); Germanischer LIoyd’s, 2004; LIoyd’s Register of Shipping, 2002) for buckling analysis of plate panel between stiffeners is based on buckling of a rectangular plate with four edges simple supported (SSSS) as follows:
FEM
V1 þ V2 þ V3
V1 þV3
792.46 675.32 690.49 626.41 642.27 793.05 793.19 684.20 607.81 551.59 508.63 420.63 399.28 381.70 367.00 354.54 343.86 334.62 355.57 374.93 392.88 409.56 373.63 389.43
792.84 675.85 691.20 627.34 643.44 794.81 795.27 686.66 610.66 554.87 512.37 424.37 403.51 386.45 372.30 360.42 350.35 341.75 364.07 384.93 404.49 422.90 387.00 404.68
sPE ¼ 4 789.95 673.83 689.27 625.67 641.73 792.68 793.12 684.70 608.81 553.06 510.57 422.75 401.84 384.71 370.48 358.51 348.34 339.63 361.63 382.14 404.34 419.37 383.53 400.80
p 2 DP 2
b tP
ð15Þ
Paik and Thayamballi (2000) have shown that torsional rigidity of longitudinal stiffeners on unloaded edges would create rotational restrictions on plate edges. They have proposed
Fig. 11. Euler stress for tripping of FB stiffeners with rigd web assumption and unloaded attached plate.
Fig. 10. Euler stress for plate buckling with unloaded longitudinal stiffeners.
A. Rahbar-Ranji / Ocean Engineering 58 (2012) 48–59
where Cr is the web bending coefficient and is defined as follows:
following equation for plate Euler stress analysis: 2
sPE ¼ kx1
p DP
ð16Þ
2
b tP
0 r zL o2 2 r zL o 20
ð17Þ
20 r zL
where zL is defined as follows:
zL ¼
GJ x bDP
ð18Þ
where Jx is the St. Venant torsional constant of stiffener and G is the shear modulus. Hughes et al. (2004) have corrected Eq. (16) as follows:
sPE ¼ kCr
p2 DP
ð19Þ
2
b tP
where kCr is defined as follows: 8 3 2 > > < 0:396zCr 1:974zCr þ 3:565zCr þ 4:0 0:881 kCr ¼ 6:951 zCr 0:4 > > : 7:025
tP tw
ð22Þ
hw b
Fig. 10 shows plate Euler stress of stiffened plate with unloaded longitudinal FB stiffeners. It can be concluded that for small ratios of stiffener area to plate area, results of all equations are very close. As this ratio increases, the effect of web rigidity is more apparent and clamped mode (W2) becomes important. In this case, rule-based expression underestimates Euler stress, and Paik and Thayamballi (2000) expression overestimates it. For buckling analysis of plate panel in stiffened plate when compressive stress on FB stiffeners is ignored, Hughes et al. (2004) expression is applicable.
5.5. Tripping of stiffeners with rigid web assumption and unloaded attached plate
ð20Þ
ð21Þ
where m is the number of half waves, and depends on K as
0 r zCr o 2 2 r zCr o20
1 3
Tripping Euler stress of stiffener with unloaded attached plate can be calculated from following formulae (American Bureau of shipping, 2007; Bureau Veritas (2002); Det Norske Veritas, 2009; Germanischer LIoyd’s, 2004; LIoyd’s Register of Shipping, 2002): 2 EIW pL2 m2 þ mK2 þ GJ sTE ¼ ð23Þ I0
20 r zCr
where
zCr ¼ C r zL
Cr ¼ 1 þ 3:6
where kx1 is defined as follows: 8 3 2 > > < 0:396zL 1:974zL þ 3:565zL þ 4:0 0:881 kx1 ¼ 6:951 zL 0:4 > > : 7:025
55
Fig. 12. Euler stress for the interaction of tripping and plate buckling with rigid web assumption.
Fig. 13. Euler stress for the interaction of tripping and web buckling with unloaded attached plate.
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A. Rahbar-Ranji / Ocean Engineering 58 (2012) 48–59
case expression for spring constant of attached plate is modified as follows: Det Norske Veritas (2009) and Zheng and Hu (2005)
follows: 2
2
2
m ðm1Þ r K rm ðm þ1Þ
2
where K is defined as kf L4 K¼ 4 p EIW
kfe ¼ kf ð24Þ
where kf is the spring stiffness of attached plate, and is defined as Et3 P kf ¼ 3b 1n2
1þ
ð25Þ
Fig. 11 shows tripping Euler stress calculated by Eqs. (3) and (23). As can be seen, when torsional rigidity of attached plate is considered, the tripping strength of stiffeners is increased. For small ratios of stiffener area to plate area, tripping strength increases significantly due to high torsional rigidity of plate, which Eq. (23) fails to predict it. Therefore, rule-based expression is not applicable for tripping analysis of FB stiffeners with rigid web assumption and unloaded attached plate, especially for FBs with low ratio of stiffener area to plate area. 5.6. Interaction of tripping and plate buckling with rigid web assumption When attached plate is under compression, Euler stress for tripping would be reduced, and rotational stiffness of attached plate would be lower than Eq. (25). Faulkner et al. (1973) were the first who modified the expression of attached plate spring constant due to compressive stress applied on plate as follows: 8 < k 1 sa sa o sPE f sPE kfe ¼ ð26Þ : 0 sPE o sa where sa is the applied compressive stress and sPE is the plate Euler stress for rectangular plate with four edges simple supported (Eq. (15)). Fig. 12 shows Euler stress for interaction of tripping and plate buckling of stiffened plate calculated by using Eqs. (3) and (26) and compared with tripping Euler stress without attached plate, FEM, and Eq. (15). As can be seen, when the ratio of stiffener area to plate area is less than about 0.333, plate buckling is dominant mode and results of Eq. (26) are exactly the same as tripping Euler stress of bare stiffener, while results of Eq. (3) are the same as plate Euler stress with four edges simple supported (SSSS). Therefore, when plate buckling is dominant mode of buckling, tripping has no influence on buckling strength of stiffened plate. In this case, Euler stress reduces to plate Euler stress with four edges simple supported and clamped condition (W2) is less important. When the ratio of stiffener area to plate area is bigger than about 0.333, tripping is dominant mode and both Eqs. (3) and (26) yield Euler stress somewhere in between tripping Euler stress and plate buckling Euler stress. However, results of Eq. (26) are about 10% lower than Eq. (3). Therefore, Faulkner et al. (1973) expression is applicable for the calculation of tripping with unloaded attached plate and rigid web assumption, when tripping is dominant mode and it underestimates Euler stress about 10%. It is worth to mention that very good agreements between results of Eq. (3) and FEM can be observed regardless of dominant mode of buckling.
1 4 h t 3 w
3
ð27Þ
P
tw
b
Hughes (1983), kfe ¼ kf
1 m2 1 þ 3 a2 1 þ0:4 hbw ttwP
ð28Þ
where a is the plate aspect ratio and m is half wave numbers of tripping of stiffener. Fig. 13 shows Euler stress for interaction of tripping and web buckling. By comparing results of Eq. (3) with and without web buckling, it can be concluded that when the ratio of stiffener area to plate area is less than about 0.333, web buckling reduce Euler stress up to 10%. For higher ratio of stiffener area to plate area, web buckling has no influence on tripping. Eq. (28) has better agreements with Eq. (3) and both have the same tendency. Eq. (27) underestimates Euler stress for this mode of buckling. Therefore, for interaction analysis of tripping and web buckling, Hughes (1983) expression can be used.
5.8. Interaction of tripping, web buckling and plate buckling In this case following expressions for spring constant of attached plate are given. Det Norske Veritas (2009): kfe ¼ kf 1þ
Ca 3
4 hw 3 b
tP tw
ð29Þ Ca
where Ca is the stress factor and is defined as follows: sa a C a ¼ 1
sPE
ð30Þ
where a in general is equal to 2, and for flat-bars is equal to 1 and Ca need not to be taken less than zero. Hughes (1983): kfe ¼ kf
Ca 3 1 þ0:4 hbw ttwP
ð31Þ
where Ca is stress factor and defined as follows which should not to be taken less than zero: 2 sa m C a ¼ 1 2 1 ð32Þ 2
sPE
a
Zheng and Hu (2005): 8 2 > sa > > < kf C r 1f sEP kfe ¼ 2 2 > sa > > : kf C r 1f sPE
sa o sPE ð33Þ
sa Z sPE
where f is a parameter which depends on the ratio of number of half waves of plate buckling (m0) to number of half waves of stiffener tripping (m), and Cr is web factor and is defined as follows: 1
5.7. Interaction of tripping and web buckling with unloaded attached plate
Cr ¼
When bending of web is considered, interaction of web buckling and tripping would reduce tripping Euler stress. In this
Danielson and Wilmer (2004) have proposed the following equation for coupled buckling analysis of tripping, web and plate
1þ
4 hw 3 b
3
ð34Þ
tP tw
A. Rahbar-Ranji / Ocean Engineering 58 (2012) 48–59
buckling in stiffened plates with FB stiffeners: Ghw t3w a 2 Dp b mb þ 3 a þ mb scr ¼ 3 3 tp b t w hw 3 p2 þ
57
ð35Þ
Fig. 14 shows Euler stress for interaction of tripping, web and plate buckling. As can be seen, for the ratio of stiffener area to plate area less than about 0.333 which plate buckling is dominant buckling mode (Fig. 12), results of Eq. (3) are close to plate Euler stress with four edges simple supported. In another word, for small FB stiffeners when both plate and stiffener are under compression, interaction of plate buckling and tripping makes clamp condition of plate to be less important and tripping and web buckling have no influence on Euler stress, which neither of Eqs. (29), (31), (33) and (35) could predict this. When the ratio of stiffener area to plate area is bigger than 0.333, Hughes (1983) expression has better agreement with Eq. (3). Fig. 15 shows results of Eqs. (3) and (31) which are compared with FEM. As can be seen, Eq. (3) and FEM have good agreements and Hughes (1983) expression yields higher values when the ratio of stiffener area to plate area is higher than 0.57. It can be concluded that Hughes (1983) expression can be used for coupled buckling analysis, when plate buckling is not dominant mode of buckling and ratio of stiffener area to plate area is less than 0.57. Fig. 16 shows buckling deformation of stiffened plate in interaction of tripping, web and plate buckling.
Fig. 16. Buckling deformation of stiffened plate in interaction of tripping, web and plate buckling.
5.9. S-shape mode of plate buckling To investigate the effect of plate buckling mode shape on interaction of tripping, web and plate buckling, Euler stress for Fig. 17. Euler stress for the interaction of tripping, web and plate buckling, S-shaped and U-shape modes of plate buckling.
S-shape mode of plate buckling is calculated by using Eq. (3) and compared with U-shape and FEM (Fig. 17). As can be seen, for ratio of stiffener area to plate area less than about 0.4, S-shape has higher buckling strength. For these stiffeners, results of U-shape have very good agreement with FEM. For ratio of stiffener area to plate area bigger than 0.4, U-shape has higher buckling strength, and for these stiffeners, Sshape has better agreement with FEM. Therefore, in contrast to Hughes and Ma (1996), for buckling analysis of FB stiffened plate, S-shape mode of plate buckling is dominant mode when the ratio of stiffener area to plate area is bigger than 0.4.
Fig. 14. Euler stress for the interaction of tripping, web and plate buckling.
6. Conclusions In design rules, tripping, web, and plate buckling of stiffened plate are treated separately and interaction of them are ignored. Only in Det Norske Veritas (2009) the interaction of plate and web buckling with tripping is considered by modification of expression of attached plate rotational stiffness in tripping analysis of stiffeners. In the literature, the influence of adjacent elements is considered as a rotational spring. Fujikubo and Yao (1999) expression for coupled mode of buckling of stiffened panel is modified to assess literature-based and rules-based expressions for elastic buckling strength of FB stiffened plates and to identify the applicability of selected expressions for certain conditions. Critical Euler stresses for different buckling modes are calculated and compared with corresponding expressions and FEM. It is found that:
Fig. 15. Euler stress for the interaction of tripping, web and plate buckling compared with FEM.
1. Fujikubo and Yao (1999) expression yields accurate results for tripping analysis of FB stiffeners.
58
A. Rahbar-Ranji / Ocean Engineering 58 (2012) 48–59
2. Bureau Veritas (2002) expression for web buckling analysis of FB stiffener is applicable when attached plate is ignored. 3. Web buckling has no influence on Euler stress for tripping when attached plate is ignored. 4. For plate buckling when stiffener is not loaded, simple supported assumption is correct for small ratio of stiffener area to plate area. For larger ratio of stiffener area to plate area, Hughes et al. (2004) expression is applicable. 5. Rule-based expression is not applicable for tripping analysis of FB stiffeners with rigid web assumption and unloaded attached plate, especially for FBs with low ratio of stiffener area to plate area. 6. Faulkner et al. (1973) expression is applicable for the calculation of tripping with unloaded attached plate and rigid web assumption, when tripping is dominant mode and it underestimates Euler stress about 10%. 7. For interaction analysis of tripping and web buckling, Hughes (1983) expression can be used. When ratio of stiffener area to plate area is less than about 0.333, web buckling reduce Euler stress up to 10%. For higher ratio of stiffener area to plate area, web buckling has no influence on tripping. 8. For small FB stiffeners when both plate and stiffener are under compression, interaction of plate buckling and tripping makes clamp condition of plate to be less important and tripping and web buckling have no influence on Euler stress. Neither Det Norske Veritas (2009); Hughes (1983); Zheng and Hu (2005) nor Danielson and Wilmer (2004) could predict this. 9. In contrast to Hughes and Ma (1996), for buckling analysis of FB stiffened plate, S-shape mode of plate buckling is dominant mode when ratio of stiffener area to plate area is bigger than 0.4 Finally, to assess true Euler stress in stiffened plate, it is necessary to consider interaction of all buckling modes and corresponding design codes should be revised to take into account this interaction properly.
a2 ¼
Dw m4 p4 hw D ap3 D nm2 p2 ð1nÞDw m2 p2 c1 þ w 3 c4 þ w c7 þ c10 ah ahw 2a3 w 4hw
a3 ¼
2 2Dp p3 ab m2 1 þ 2 3 a2 b
a4 ¼
Dw m4 p4 hw D ap3 D nm2 p2 ð1nÞDw m2 p2 c2 þ w 3 c5 þ w c8 þ c11 3 2ahw 2ahw 4a 8hw
a5 ¼
Dp p4 ab 3m4 2 m2 þ þ 4 2 2 8a4 b a2 b
a6 ¼
Dw m4 p4 hw D ap3 D nm2 p2 ð1nÞDw m2 p2 c3 þ w 3 c6 þ w c9 þ c12 3 ahw ahw 2a 4hw
c1 ¼ c2 ¼
c3 ¼ c4 ¼ c5 ¼ c6 ¼ c7 ¼
Appendix
c8 ¼
g1 ¼ a1 þ a2 g2 ¼ a3 þ a4
c9 ¼
Z21 3
2
m p g4 ¼ 2a
g5 ¼ g6 ¼
Z22 3
p 16
p 8
þ
2
m2 p 4a
2
m2 p 2a
tp b þ t w hw c1 2
16t p b þ t w hw c2 3p 3t p b þ t w hw c3 8
p2
Z1 Z4
3 4 2 Z25 4 8 8 2 Z3 þ þ 1 þ 2 Z2 Z3 þ 2 Z2 Z5 þ Z3 Z5 2 p 2 p p p p
Z24 Z3 Z4
Z4 Z5
p 16
p2 8
p2 4
4
p
Z23 þ
16
Z25
Z3 Z5 4
Z24 þ Z1 Z4
Z4 Z5 þ 1
p2 8
p
c10 ¼ Z21 þ
8
þ
2
2 4 8 2 8 Z1 Z2 þ Z4 Z5 þ 1 þ 2 Z1 Z3 þ Z3 Z4 þ 2 Z1 Z5 þ Z2 Z4 3 p p p p
g3 ¼ a5 þ a6 2
Z24
þ
Z23 þ
2
p2 8
p2 8
2
p2 8
Z25 þ 1
p 2
Z2 Z3 þ Z2 Z5
Z24 þ 2Z1 Z4
c11 ¼ 2Z1 Z2 þ c12 ¼ Z22 þ
p Z1 Z3 þ Z1 Z5 þ Z2 Z4
p2 4
p Z4 Z5 þ2Z1 Z3 þ Z3 Z4 þ2 Z1 Z5 þ Z2 Z4
Z23 þ
2
p2 8
p Z25 þ2Z2 Z3 þ Z3 Z5 þ 2Z2 Z5 2
References
g7 ¼ g8 ¼ g9 ¼ a1 ¼
p2 tp a 4b 2pt p a 3b
p2 tp a 4b 2 Dp p4 ab m2 1 þ 2 4 a2 b
American Bureau of Shipping, 2007. Buckling Strength of Longitudinal Strength Members, Part 3, Chapter 2, App. 4. Bureau Veritas, 2002. Rules and Regulations for the Classification of Ships, Part B, Section 4. Byklum, E., Amdahl, J., 2002. A simplified method for elastic large deflection analysis of plates and stiffened panels due to local buckling. Thin-walled Struct. 42 (5), 701–717. Danielson, D.A., Wilmer, A., 2004. Buckling of stiffened plates with bulb flat flanges. Int. J. Solids Struct. 41, 6407–6427. Det Norske Veritas, 2009. Rules for Classification of Ships, Part 3, Chapter 1, Section 13, Buckling Control. Hovik, Norway. Faulkner, D., Adamchak, J.C., Synder, G.J., Vetter, M.R., 1973. Synthesis of welded gillage to withstand compression and normal loads. Comput. Struct. 3, 221–246.
A. Rahbar-Ranji / Ocean Engineering 58 (2012) 48–59
Fujikubo, M., Yao, T., 1999. Elastic local buckling strength of stiffened plate considering plate/stiffener interaction and welding residual stress. Mar. Struct. 12, 543–564. Germanischer LIoyd’s, 2004. Rules and Guidelines, Part 1, Section 3F. Proof of Buckling Strength. Hughes, O.F., 1983. Ship Structural Design. Wiley, New York. Hughes, O.F., Ma, M., 1996. Elastic tripping analysis of asymmetrical stiffeners. Comput. Struct. 60 (3), 369–389. Hughes, O.F., Ghosh, B., Chen, Y., 2004. Improved prediction of simultaneous local and overall buckling of stiffened panels. Thin-Walled Struct. 42, 827–850. LIoyd’s Register of Shipping, 2002. Rules and Regulations for the Classification of Ships, Part 3, Chapter 4, Section 7.
59
Paik, J.K., Thayamballi, A.K., Lee, W.H., 1998. A numerical investigation of tripping. Mar. Struct. 11, 159–183. Paik, J.K., Thayamballi, A.K., 2000. Buckling strength of steel plating with elastically restrained edges. Thin-Walled Struct. 37, 27–55. Timoshenko, S.P., Gere, J.M., 1961. Theory of Elastic Stability, second ed. McGraw-Hill, New York, Engineering Societies Monograph. Van der Neut, A., 1983. Overall Buckling of Z-Stiffened Panels in Compression. Delft University. Yao, T., Fujikubo, M., Yanagihara, D., 1997. Buckling/plastic collapse behaviour and strength of stiffened plates under thrust. Int. J. Offshore Polar Eng. 7 (4), 285–292. Zheng, G., Hu, Y., 2005. Tripping of thin-walled stiffeners in the axially compressed stiffened panel with lateral pressure and end moments. Thin-Walled Struct. 42, 789–799.
Contents lists available at SciVerse ScienceDirect
Ocean Engineering journal homepage: www.elsevier.com/locate/oceaneng
Elastic buckling analysis of longitudinally stiffened plates with flat-bar stiffeners Ahmad Rahbar-Ranji n Department of Ocean Engineering, AmirKabir University of Technology, Hafez Ave., Tehran 15914, Iran
a r t i c l e i n f o
a b s t r a c t
Article history: Received 2 October 2011 Accepted 24 September 2012
The aim of present work is to investigate the characteristics of interactions of different buckling modes of flat-bar stiffened panels. Literature-based and rules-based expressions for assessing buckling strength of stiffened plates are studied. Energy method is employed for the analyses of selected buckling modes of stiffened plates including plate buckling, torsional buckling, web buckling and interactions of them when either plate or stiffener or both are under compression. Results are compared with numerical solutions using finite element method and available expressions to identify the applicability and accuracy of selected expressions for certain conditions. It is found that some of the given expressions for buckling analyses of stiffened plates have limited applicability. & 2012 Elsevier Ltd. All rights reserved.
Keywords: Plate buckling Stiffened panel Torsional buckling Web buckling Plate torsional rigidity
1. Introduction Longitudinally stiffened plates are fundamental structural components in ship structure even in ships with small length. Compressive longitudinal stresses produced by hull girder bending moment impose buckling of longitudinal stiffeners between two adjacent transverse girders. Depending on dimensions of stiffeners and attached plate, different modes of buckling from local buckling of plate or stiffeners to overall buckling of panel could be occurred. Stiffener buckling itself is divided to local buckling of web or flange, beam-column type flexural buckling and torsional buckling. Flexural buckling and torsional buckling of stiffeners interact when asymmetric stiffeners are used. Interactions of plate buckling, torsional buckling and web buckling occur regardless of stiffener types (Fig. 1). Many research papers were dedicated to buckling analysis of stiffened plates. Yao et al. (1997) have performed a series of elasto-plastic large deflection analyses for stiffened panels with flat-bar (FB) stiffeners. Van der Neut (1983) has studied interactions of web, plate, flexural and torsional buckling of Z-stiffeners by strip theory. Inclusive coupled buckling analysis of stiffened plates is possible by using energy method. Fujikubo and Yao (1999) have to study the restraining effect of the stiffeners web/ flange on the edges of the plate between stiffeners. Hughes and Ma (1996) have studied interaction of plate, web, beam-column type flexural buckling, and torsional buckling of stiffeners by this method. Byklum and Amdahl (2002) have used this method for
n
Tel.: þ98 21 64543114; fax þ98 21 66412495. E-mail address: [email protected]
0029-8018/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.oceaneng.2012.09.018
coupled buckling and post-buckling analysis of stiffened plates applying large deflection theory. Danielson and Wilmer (2004) have used this method for coupled buckling analysis of stiffened plate with bulb plate stiffeners. In lieu of detailed analysis, classification society rules (American Bureau of shipping, 2007; Bureau Veritas, 2002; Det Norske Veritas, 2009; Germanischer LIoyd’s, 2004; LIoyd’s Register of Shipping, 2002) proposes simple equation for the evaluation of each modes of buckling of stiffened panels independently. In some cases, these equations are improved by introducing a rotational spring representing adjacent element. Calculation of stiffness of these springs was the subject of some researches. Paik and Thayamballi (2000) and Hughes et al. (2004) have given some expressions for the calculation of torsional rigidity of stiffeners in plate buckling analysis. Paik et al. (1998) have given expression for the calculation of rotational stiffness of attached plate and upper flange in web buckling analysis of stiffeners. Faulkner et al. (1973) have given expression for calculation of spring constant of attached plate in torsional buckling analysis of stiffeners. Det Norske Veritas (2009), Zheng and Hu (2005), and Hughes (1983) have given expressions for the calculation of spring constant of attached plate in the interaction of web and torsional buckling and interaction of web, plate and torsional buckling. The validity of results by using these expressions is sometimes unclear. Continuous stiffened plate should be modeled to represent the actual interactions of buckling modes. The main aim of the present work is to investigate coupled buckling of FB stiffened plate thoroughly and to study the accuracy and applicability of literature-based and rule-based expressions. Given equation by Fujikubo and Yao (1999) based on energy method for buckling analysis of stiffened plate is modified to consider selected
A. Rahbar-Ranji / Ocean Engineering 58 (2012) 48–59
49
Fig. 1. Description of coordinate system and buckling deformation in interaction of plate buckling (W), web and torsional buckling (V). Fig. 3. Torsional buckling (V1) and web buckling modes (V2 þ V3) in flat-bar stiffener.
V 2 ¼ Z3 W 2 V 3 ¼ Z4 W 1 þ Z5 W 2 where
Fig. 2. Plate buckling modes in stiffened plate, W1 simple supported condition and W2 clamped condition.
buckling modes in the range from plate buckling, web buckling, tripping to interactions of them. Critical Euler stresses are calculated and finite element method (FEM) is employed to confirm the accuracy of the results.
Z1 ¼
k2 k6 k6 k1 k4
Z2 ¼
k1 k3 k5 k1 k4 k6
Z3 ¼ k3 Z4 ¼
k2 k4 k1 k4 k6
Z5 ¼
k3 k5 k6 k1 k4
2. Basic equation
where Equilibrium method or energy approach can be used for buckling analysis of structures. In energy method a shape function with some unknown coefficients for buckled deformation is assumed and total potential energy during buckling is calculated. Using the principle of minimum potential energy, critical Euler stress is determined. The more precisely the buckled deformation is assumed, the more accurately the critical Euler stress is evaluated. Fujikubo and Yao (1999) have used this method to calculate critical Euler stress for coupled mode of torsional, web and plate buckling in stiffened plates. They assumed plate buckling mode shape between stiffeners as mpx py W 2 mpx 2py wðx,yÞ ¼ W 1 sin sin þ 1cos ð1Þ sin a b a b 2 where a, and b are length and width of plate panel, respectively, and m is half wave numbers in longitudinal direction (Fig. 1). In Eq. (1) first sentence represents buckling mode with simple supported condition, and second one with clamped condition (Fig. 2). Buckling mode shape of stiffener’s web is assumed as follows: z mpx mpx pz mp x pz vðx,zÞ ¼ V 1 sin þ V 3 sin þ V 2 sin 1cos sin hw a a 2hw a 2hw
ð2Þ where hw is the web height. First term represents rigid body rotation of stiffener, referred as tripping or torsional buckling, second and third terms represent web buckling mode (Fig. 3). Stiffener flange is assumed as a beam with flexural and torsional rigidity firmly connected to the web. Rigid connection assumptions of plate to web and web to flange reduces five coefficients in Eqs. (1) and (2) to two as follows: V 1 ¼ Z1 W 1 þ Z2 W 2
k1 ¼ k2 ¼
p 2
phw b 2
k3 ¼
Dp 16hw Dw b2
k4 ¼ nDw
mp2 a
mp2 a ( ) mp2 p 2 k6 ¼ Dw n þ a 2hw k5 ¼ nDw
Using principle of minimum potential energy, critical Euler stress could be found from the following equation:
k1 s2x þ k2 sx sy þ k3 s2y k4 sx k5 sy þ k6 ¼ 0
ð3Þ
where sx and sy are applied compressive stresses in longitudinal and transverse directions, respectively, and
k1 ¼ g4 g6 g25 k2 ¼ g4 g9 þ g6 g7 2g5 g8 k3 ¼ g7 g9 g28 k4 ¼ g1 g6 þ g3 g4 2g2 g5 k5 ¼ g1 g9 þ g3 g7 2g2 g8
50
A. Rahbar-Ranji / Ocean Engineering 58 (2012) 48–59
k6 ¼ g1 g3 g22 where g1–g9 are given in the appendix.
k1 ¼ k2 ¼ k3 ¼ k5 ¼ 0 k4 ¼ g4 k6 ¼ g1
3. Interaction of selected buckling modes in stiffened plate Eq. (3) has derived for coupled buckling analysis of stiffened plate when all buckling deformations are considered. Literaturebased expressions are proposed for interaction of some of buckling modes. To determine the accuracy and applicability of these expressions, Eq. (3) is modified to consider interaction of selected buckling modes as follows.
With attached plate:
Z1 ¼ Z2 ¼ Z3 ¼ Z5 ¼ 0 Z4 ¼
k2 k1
3.1. Tripping of FB stiffener with rigid web assumption Without attached plate:
3.2.3. Web buckling of FB stiffener, V2 and V3 mode shape
Z1 ¼ 1
Without attached plate:
Z2 ¼ Z3 ¼ Z4 ¼ Z5 ¼ 0
Z1 ¼ Z2 ¼ Z4 ¼ 0
k1 ¼ k2 ¼ k3 ¼ k5 ¼ 0
Z3 ¼ 1
k4 ¼ g4
Z5 ¼
k6 ¼ g1 With attached plate:
Z1 ¼ k2
k5 k6
k1 ¼ k2 ¼ k3 ¼ k5 ¼ 0 k4 ¼ g6 k6 ¼ g3
Z2 ¼ Z3 ¼ Z4 ¼ Z5 ¼ 0 With attached plate:
Z1 ¼ Z2 ¼ 0 3.2. Web buckling of FB stiffener
Z3 ¼ k3
To study web buckling of stiffener, different buckling deformations with and without attached plate are investigated as follows.
Z4 ¼
3.2.1. Web buckling of FB stiffener, V2 mode shape Without attached plate
Z1 ¼ Z2 ¼ Z4 ¼ Z5 ¼ 0 Z3 ¼ 1 k1 ¼ k2 ¼ k3 ¼ k5 ¼ 0
k2 k1
Z5 ¼ 0 Z6 ¼
k2 k6 k1 k3 k5
k1 ¼ k2 ¼ k3 ¼ k5 ¼ 0 k4 ¼ Z26 g6 þ 2Z6 g5 þ g4 k6 ¼ Z26 g3 þ 2Z6 g2 þ g1
k4 ¼ g6 k6 ¼ g3 With attached plate:
Z1 ¼ Z2 ¼ Z4 ¼ Z5 ¼ 0 Z3 ¼ k3
3.3. Interaction of tripping and web buckling of FB stiffener To study interaction of web buckling and tripping, different mode shapes with and without attached plate are investigated as follows 3.3.1. Interaction of tripping and web buckling, V1 and V2 mode shapes
3.2.2. Web buckling of FB stiffener, V3 mode shape
Without attached plate:
Without attached plate:
Z1 ¼ Z4 ¼ Z5 ¼ 0
Z1 ¼ Z2 ¼ Z3 ¼ Z5 ¼ 0
Z2 ¼ 1
Z4 ¼ 1
Z3 ¼ 1
A. Rahbar-Ranji / Ocean Engineering 58 (2012) 48–59
k1 ¼ k2 ¼ k3 ¼ k5 ¼ 0 k4 ¼ g6 k6 ¼ g3 With attached plate:
Z1 ¼ k2 Z2 ¼ 0
3.4. Different mode shapes of plate buckling Hughes and Ma (1996) have proposed two different mode shapes for plate buckling between stiffeners in transverse direction, namely S-shape (Fig. 4) and U-shape (Fig. 5). Eq. (1) corresponds to U-shape mode in which one half waves in transverse direction between two adjacent stiffeners are considered. For S-shape, plate buckling mode should be assumed as follows: wðx,yÞ ¼ W 1 sin
Z3 ¼ k3
mpx 2py W 2 mpx 4py sin þ 1cos sin a b a b 2
ð4Þ
Total potential energy at the instance of bifurcation can be expressed as follows:
Z4 ¼ Z5 ¼ 0 Z6 ¼
51
Y
k2 k4 k3 k5
¼ UW
ð5Þ
k4 ¼ Z26 g6 þ 2Z6 g5 þ g4
where U is the strain energy and W is the work done by applied compressive load. These quantities can be divided into following components:
k6 ¼ Z26 g3 þ 2Z6 g2 þ g1
U ¼ U Po þ U wo
ð6Þ
W ¼ W Po þW wo
ð7Þ
k1 ¼ k2 ¼ k3 ¼ k5 ¼ 0
3.3.2. Interaction of tripping and web buckling of FB stiffener, V1 and V3 mode shapes Without attached plate:
Z1 ¼ 1 Z2 ¼ Z3 ¼ Z5 ¼ 0 Z4 ¼
k4 k6
where UPo and WPo are strain energy and work component in the attached plate due to out-of-plane deformation, and Uwo and Wwo are strain energy and work component in the web due to out-of-plane deformation. Strain energy stored in the attached plate due to out-ofplane deformation is calculated from classical plate theory as follows: DP U Po ¼ 2
Z
b=2
b=2
8 !2 93 !2 < 2 = 2 2 2 2 4 @ w þ @ w 2ð1vÞ @ w @ w @ w 5dx dy : @x2 @y2 @x @y ; @x2 @y2 a=2
Z
a=2
2
ð8Þ
k4 ¼ g4 k6 ¼ g1 With attached plate:
Z1 ¼
k2 k6 k6 k1 k4
Z2 ¼ Z3 ¼ Z5 ¼ 0 Z4 ¼
k2 k4 k1 k4 k6 Fig. 4. S-shape mode of plate buckling between adjacent stiffeners.
3.3.3. Interaction of tripping and web buckling of FB stiffener, V1, V2 and V3 mode shapes Without attached plate:
Z1 ¼ 1 Z2 ¼ 0 Z3 ¼ 1 Z4 ¼ Z5 ¼
k4 k6
With attached plate: In this case, parameters are as quoted in Section2.
Fig. 5. U-shape mode of plate buckling between adjacent stiffeners.
52
A. Rahbar-Ranji / Ocean Engineering 58 (2012) 48–59
where DP is the bending rigidity of attached plate and is defined as follows: DP ¼
Et3 P 12 1n2
where E is Young’s modulus, n is Poisson’s ratio and tP is the attached plate thickness. Component of work due to out-of-plane deformation in attached plate is calculated as follows: Z a=2 Z b=2 2 1 @w t P dy dx ð9Þ W Po ¼ sE 2 @x a=2 b=2 where sE is the applied compressive stress at the time of bifurcation, called Euler stress. Strain energy stored in the web due to out-of-plane deformation is calculated from classical plate theory as follows: U wo ¼
Dw 2
Z
hw 0
Z
2
2 2 4 @ vþ@ v 2 2 @x @z a=2 a=2
!2 2ð1vÞ
8 !2 93 <@2 v @2 v @2 v =5 dx dz : @x2 @z2 @x@z ;
ð10Þ
where Dw is the bending rigidity of web plate and is defined as follows: Dw ¼
Et 3 w 12 1n2
where tw is the web thickness. Component of work due to out-ofplane deformation in web is calculated as follows: Z a=2 Z hw 2 1 @v t w dz dx ð11Þ W wo ¼ sE 2 @x a=2 0 After substituting Eqs. (4) and (2) into Eqs. (8)–(9) and (10)–(11), respectively, and manipulating them yield to the following changes in the expressions of k k2 ¼
2phw b
k3 ¼
DP 32hw Dw b2
2
4. Different loading conditions in buckling analysis of stiffened plate Although in ship structures, stiffeners and attached plate are almost under the same axial compressive stresses, however, to study the accuracy of literature-based expressions, it is necessary to consider the cases of unloaded attached plate/stiffener as well. It is worth to mention that, these cases cannot be checked by FEM, since result of eigenvalue analysis expressed as a load factor. If one concentrate load applies on one node only, or some concentrate loads apply on some nodes the results would be the same, since corresponding load factors are changed accordingly.
5. Investigated cases and discussions In this study, material is considered as mild steel with E¼ 206 GPa and v¼0.3, a length of 3200 mm has been considered for stiffener, spacing between them is taken as 600 mm and thickness of attached plate is assumed equal to web thickness. To demonstrate the accuracy of this method, a series of FEM eigenvalue analyses are performed. The extent of FE model should be appropriate to capture all buckling modes and also to reduce the time of analysis. Modeling of the whole panel although satisfies the best former objective, however, increases the time of analysis. Fujikubo and Yao (1999) have suggested the triple-span double-bay model with symmetry boundary conditions on longitudinal edges and periodical continuous conditions on transverse edges where vertical displacement at locations of transverse girders is prevented. Periodical continuous conditions means that, at the same y coordinates along the transverse edges corresponding nodal dispalcements are the same. Hughes and Ma (1996) have proposed two models, onespan double-bay model and one-span triple-bay model with simple supported conditions on transverse edges and symmetry conditions on longitudinal edges. All these models are examined and compared and it was found that all models yield the same results. Therefore, one-span double-bay model which is simple to generate and is consistent with assumed boundary conditions in Eq. (1) is used in this work. The computer code ANSYS (version 5.6) has been used for this analysis. Web and attached plate are modeled using four-node quadrilateral shell element (SHELL63) with six degrees of freedom per node. To enforce tripping about junction of web to attached plate and prevent flexural buckling, displacement in y and z directions at the baseline of web are restrained. A uniformly distributed normal stress is applied over one end while holding the other end. Fig. 6 shows FE model with applied load and boundary conditions. 5.1. Tripping of stiffener with rigid web assumption without attached plate Euler stress for tripping of thin-walled open section beams about center of torsion is calculated from following equation (Timoshenko and Gere, 1961): 2 EIW pL2 þ GJ sTE ¼ ð12Þ I0 where IW, J and I0 are sectorial moment of inertia, St. Venant’s moment of inertia, and polar moment of inertia about center of torsion, respectively, L is the length of beam and G is the shear modulus. The position of center of torsion depends on boundary
4.1. Stiffener is loaded In this case, the work done by the applied load corresponds to Euler stresses on the web of stiffener, and the first sentence in the expressions of g4, g5, and g6 in the appendix should be neglected. 4.2. Plate is loaded To determine adequate boundary conditions for plate buckling analysis, it is necessary to model continuous stiffened plate with unloaded stiffeners. In this case, the first sentence in the expressions of g4, g5, and g6 in appendix should be considered.
Fig. 6. FE model with applied load and boundary conditions.
A. Rahbar-Ranji / Ocean Engineering 58 (2012) 48–59
conditions of beam. In stiffened panels, the center of torsion is located at junction point of stiffener to attached plate. Above cited parameters for FB about this point are calculated as follows (American Bureau of shipping, 2007; Bureau Veritas, 2002; Det Norske Veritas (2009); Germanischer LIoyd’s, 2004; LIoyd’s Register of Shipping, 2002): IW ¼
3 t 3W hW
36
Table 1 Tripping Euler stress (MPa), rigid web assumption without attached plate. Flat-bar (mm)
Eq. (3)
Eq. (12)
FEM
50 5 65 6 75 7 90 8 100 9 110 11 120 12 140 13 160 14 180 15 200 16 220 16 240 17 260 18 280 19 300 20 320 21 340 22 360 24 380 26 400 28 420 30 440 30 460 32
792.76 675.76 691.08 627.19 643.24 794.51 794.93 686.24 610.17 554.30 511.73 423.73 402.78 385.64 371.39 359.41 349.24 340.23 362.61 383.21 402.49 420.60 384.69 402.04
793.51 676.37 691.69 627.71 643.75 795.10 795.48 686.64 610.46 554.49 511.82 423.73 402.71 385.49 371.16 359.11 348.86 340.07 362.02 382.47 401.59 419.53 383.58 400.75
790.29 674.07 689.50 625.87 641.94 729.95 793.40 684.96 609.07 553.32 510.84 423.01 402.11 385.84 370.79 358.84 348.70 340.01 362.07 382.65 401.92 420.03 384.18 401.52
Fig. 7. FE model and buckling deformation for tripping mode of buckling.
53
3
I0 ¼
J¼
t W hW 3
hw t 3w 3
Table 1 shows Euler tripping stress for different FB calculated by Eqs. (3) and (12) and compared with FEM. As can be seen both equations yield the same results and very good agreements with FEM are observed. Fig. 7 shows FE model and buckling deformation for this mode of buckling. 5.2. Web buckling Given equations for Euler buckling analysis of stiffener’s web for FB is as follows (Bureau Veritas (2002)): 2 t sWE ¼ 0:78E W ð13Þ hW Eq. (13) is based on buckling of a rectangular plate with three edges simply supported and one edge free. In this equation torsional rigidity of attached plate is ignored. Paik et al. (1998) have proposed following equation to take into account this rigidity: 2 t sWE ¼ 0:9kw E W ð14Þ hW where kw depends on ratio of attached plate to web rotational stiffness. Figs. 8 and 9 show Euler stresses for web buckling of FB stiffeners without attached plate, and with unloaded attached plate, respectively. Following conclusions can be made: (1) As can be seen from Fig. 8, results of Eq. (3) with V2 and V3 mode shapes have the same tendency as Eq. (13). Results of Eq. (3) with V2 þ V3 mode shape are unreliable and results of
Fig. 9. Euler stress for web buckling of FB stiffeners with attached plate.
Fig. 8. Euler stress for web buckling of FB stiffeners without attached plate.
54
A. Rahbar-Ranji / Ocean Engineering 58 (2012) 48–59
V2 mode shape are higher than V3 mode shape, since, clamped assumption of V2 mode shape at the base line increase buckling strength. Results of Eq. (13) are about 20% lower than Eq. (3) with V3 mode shape, since, this mode shape is not consistent with free edge assumption at the upper edge of web. It can be concluded that rule-based expression is applicable for web buckling analysis of FB stiffener when attached plate is ignored. (2) Comparing Figs. 8 and 9, it can be concluded that torsional rigidity of attached plate would increase web buckling strength of stiffeners. Eq. (13) which ignores this rigidity underestimates Euler stress of web buckling. In contrast to previous case, results of Eq. (3) with V2 mode shape are lower than V3 and V2 þV3 mode shapes, and results of Eqs. (13) and (14) both are lower than Eq. (3) with V2 mode shape. It is worth to mention that this case cannot be checked by FEM, since it is not possible to have web buckling without tripping.
Table 2 Euler stress for the interaction of tripping and web buckling without attached plate. Flat-bar (mm)
50 5 65 6 75 7 90 8 100 9 110 11 120 12 140 13 160 14 180 15 200 16 220 16 240 17 260 18 280 19 300 20 320 21 340 22 360 24 380 26 400 28 420 30 440 30 460 32
Eq. (3)
5.3. Interaction of tripping of stiffener and web buckling without attached plate Table 2 shows Euler stress for interaction of tripping and web buckling. Comparing Tables 1 and 2, it reveals that differences between FE results are very low, and differences between results of Eq.(3) with V1 þV2 þ V3 mode shape are about 3%. The reason for this behavior is that, web buckling strength is very higher than tripping strength. Therefore, web buckling has no influence on Euler stress for tripping when attached plate is ignored. It also can be concluded that when web buckling is permitted, V1 þV3 mode shape yields more accurate results than V1 þV2 þV3.
5.4. Plate buckling with unloaded stiffener Rule-based equation (American Bureau of shipping, 2007; Bureau Veritas (2002); Det Norske Veritas (2009); Germanischer LIoyd’s, 2004; LIoyd’s Register of Shipping, 2002) for buckling analysis of plate panel between stiffeners is based on buckling of a rectangular plate with four edges simple supported (SSSS) as follows:
FEM
V1 þ V2 þ V3
V1 þV3
792.46 675.32 690.49 626.41 642.27 793.05 793.19 684.20 607.81 551.59 508.63 420.63 399.28 381.70 367.00 354.54 343.86 334.62 355.57 374.93 392.88 409.56 373.63 389.43
792.84 675.85 691.20 627.34 643.44 794.81 795.27 686.66 610.66 554.87 512.37 424.37 403.51 386.45 372.30 360.42 350.35 341.75 364.07 384.93 404.49 422.90 387.00 404.68
sPE ¼ 4 789.95 673.83 689.27 625.67 641.73 792.68 793.12 684.70 608.81 553.06 510.57 422.75 401.84 384.71 370.48 358.51 348.34 339.63 361.63 382.14 404.34 419.37 383.53 400.80
p 2 DP 2
b tP
ð15Þ
Paik and Thayamballi (2000) have shown that torsional rigidity of longitudinal stiffeners on unloaded edges would create rotational restrictions on plate edges. They have proposed
Fig. 11. Euler stress for tripping of FB stiffeners with rigd web assumption and unloaded attached plate.
Fig. 10. Euler stress for plate buckling with unloaded longitudinal stiffeners.
A. Rahbar-Ranji / Ocean Engineering 58 (2012) 48–59
where Cr is the web bending coefficient and is defined as follows:
following equation for plate Euler stress analysis: 2
sPE ¼ kx1
p DP
ð16Þ
2
b tP
0 r zL o2 2 r zL o 20
ð17Þ
20 r zL
where zL is defined as follows:
zL ¼
GJ x bDP
ð18Þ
where Jx is the St. Venant torsional constant of stiffener and G is the shear modulus. Hughes et al. (2004) have corrected Eq. (16) as follows:
sPE ¼ kCr
p2 DP
ð19Þ
2
b tP
where kCr is defined as follows: 8 3 2 > > < 0:396zCr 1:974zCr þ 3:565zCr þ 4:0 0:881 kCr ¼ 6:951 zCr 0:4 > > : 7:025
tP tw
ð22Þ
hw b
Fig. 10 shows plate Euler stress of stiffened plate with unloaded longitudinal FB stiffeners. It can be concluded that for small ratios of stiffener area to plate area, results of all equations are very close. As this ratio increases, the effect of web rigidity is more apparent and clamped mode (W2) becomes important. In this case, rule-based expression underestimates Euler stress, and Paik and Thayamballi (2000) expression overestimates it. For buckling analysis of plate panel in stiffened plate when compressive stress on FB stiffeners is ignored, Hughes et al. (2004) expression is applicable.
5.5. Tripping of stiffeners with rigid web assumption and unloaded attached plate
ð20Þ
ð21Þ
where m is the number of half waves, and depends on K as
0 r zCr o 2 2 r zCr o20
1 3
Tripping Euler stress of stiffener with unloaded attached plate can be calculated from following formulae (American Bureau of shipping, 2007; Bureau Veritas (2002); Det Norske Veritas, 2009; Germanischer LIoyd’s, 2004; LIoyd’s Register of Shipping, 2002): 2 EIW pL2 m2 þ mK2 þ GJ sTE ¼ ð23Þ I0
20 r zCr
where
zCr ¼ C r zL
Cr ¼ 1 þ 3:6
where kx1 is defined as follows: 8 3 2 > > < 0:396zL 1:974zL þ 3:565zL þ 4:0 0:881 kx1 ¼ 6:951 zL 0:4 > > : 7:025
55
Fig. 12. Euler stress for the interaction of tripping and plate buckling with rigid web assumption.
Fig. 13. Euler stress for the interaction of tripping and web buckling with unloaded attached plate.
56
A. Rahbar-Ranji / Ocean Engineering 58 (2012) 48–59
case expression for spring constant of attached plate is modified as follows: Det Norske Veritas (2009) and Zheng and Hu (2005)
follows: 2
2
2
m ðm1Þ r K rm ðm þ1Þ
2
where K is defined as kf L4 K¼ 4 p EIW
kfe ¼ kf ð24Þ
where kf is the spring stiffness of attached plate, and is defined as Et3 P kf ¼ 3b 1n2
1þ
ð25Þ
Fig. 11 shows tripping Euler stress calculated by Eqs. (3) and (23). As can be seen, when torsional rigidity of attached plate is considered, the tripping strength of stiffeners is increased. For small ratios of stiffener area to plate area, tripping strength increases significantly due to high torsional rigidity of plate, which Eq. (23) fails to predict it. Therefore, rule-based expression is not applicable for tripping analysis of FB stiffeners with rigid web assumption and unloaded attached plate, especially for FBs with low ratio of stiffener area to plate area. 5.6. Interaction of tripping and plate buckling with rigid web assumption When attached plate is under compression, Euler stress for tripping would be reduced, and rotational stiffness of attached plate would be lower than Eq. (25). Faulkner et al. (1973) were the first who modified the expression of attached plate spring constant due to compressive stress applied on plate as follows: 8 < k 1 sa sa o sPE f sPE kfe ¼ ð26Þ : 0 sPE o sa where sa is the applied compressive stress and sPE is the plate Euler stress for rectangular plate with four edges simple supported (Eq. (15)). Fig. 12 shows Euler stress for interaction of tripping and plate buckling of stiffened plate calculated by using Eqs. (3) and (26) and compared with tripping Euler stress without attached plate, FEM, and Eq. (15). As can be seen, when the ratio of stiffener area to plate area is less than about 0.333, plate buckling is dominant mode and results of Eq. (26) are exactly the same as tripping Euler stress of bare stiffener, while results of Eq. (3) are the same as plate Euler stress with four edges simple supported (SSSS). Therefore, when plate buckling is dominant mode of buckling, tripping has no influence on buckling strength of stiffened plate. In this case, Euler stress reduces to plate Euler stress with four edges simple supported and clamped condition (W2) is less important. When the ratio of stiffener area to plate area is bigger than about 0.333, tripping is dominant mode and both Eqs. (3) and (26) yield Euler stress somewhere in between tripping Euler stress and plate buckling Euler stress. However, results of Eq. (26) are about 10% lower than Eq. (3). Therefore, Faulkner et al. (1973) expression is applicable for the calculation of tripping with unloaded attached plate and rigid web assumption, when tripping is dominant mode and it underestimates Euler stress about 10%. It is worth to mention that very good agreements between results of Eq. (3) and FEM can be observed regardless of dominant mode of buckling.
1 4 h t 3 w
3
ð27Þ
P
tw
b
Hughes (1983), kfe ¼ kf
1 m2 1 þ 3 a2 1 þ0:4 hbw ttwP
ð28Þ
where a is the plate aspect ratio and m is half wave numbers of tripping of stiffener. Fig. 13 shows Euler stress for interaction of tripping and web buckling. By comparing results of Eq. (3) with and without web buckling, it can be concluded that when the ratio of stiffener area to plate area is less than about 0.333, web buckling reduce Euler stress up to 10%. For higher ratio of stiffener area to plate area, web buckling has no influence on tripping. Eq. (28) has better agreements with Eq. (3) and both have the same tendency. Eq. (27) underestimates Euler stress for this mode of buckling. Therefore, for interaction analysis of tripping and web buckling, Hughes (1983) expression can be used.
5.8. Interaction of tripping, web buckling and plate buckling In this case following expressions for spring constant of attached plate are given. Det Norske Veritas (2009): kfe ¼ kf 1þ
Ca 3
4 hw 3 b
tP tw
ð29Þ Ca
where Ca is the stress factor and is defined as follows: sa a C a ¼ 1
sPE
ð30Þ
where a in general is equal to 2, and for flat-bars is equal to 1 and Ca need not to be taken less than zero. Hughes (1983): kfe ¼ kf
Ca 3 1 þ0:4 hbw ttwP
ð31Þ
where Ca is stress factor and defined as follows which should not to be taken less than zero: 2 sa m C a ¼ 1 2 1 ð32Þ 2
sPE
a
Zheng and Hu (2005): 8 2 > sa > > < kf C r 1f sEP kfe ¼ 2 2 > sa > > : kf C r 1f sPE
sa o sPE ð33Þ
sa Z sPE
where f is a parameter which depends on the ratio of number of half waves of plate buckling (m0) to number of half waves of stiffener tripping (m), and Cr is web factor and is defined as follows: 1
5.7. Interaction of tripping and web buckling with unloaded attached plate
Cr ¼
When bending of web is considered, interaction of web buckling and tripping would reduce tripping Euler stress. In this
Danielson and Wilmer (2004) have proposed the following equation for coupled buckling analysis of tripping, web and plate
1þ
4 hw 3 b
3
ð34Þ
tP tw
A. Rahbar-Ranji / Ocean Engineering 58 (2012) 48–59
buckling in stiffened plates with FB stiffeners: Ghw t3w a 2 Dp b mb þ 3 a þ mb scr ¼ 3 3 tp b t w hw 3 p2 þ
57
ð35Þ
Fig. 14 shows Euler stress for interaction of tripping, web and plate buckling. As can be seen, for the ratio of stiffener area to plate area less than about 0.333 which plate buckling is dominant buckling mode (Fig. 12), results of Eq. (3) are close to plate Euler stress with four edges simple supported. In another word, for small FB stiffeners when both plate and stiffener are under compression, interaction of plate buckling and tripping makes clamp condition of plate to be less important and tripping and web buckling have no influence on Euler stress, which neither of Eqs. (29), (31), (33) and (35) could predict this. When the ratio of stiffener area to plate area is bigger than 0.333, Hughes (1983) expression has better agreement with Eq. (3). Fig. 15 shows results of Eqs. (3) and (31) which are compared with FEM. As can be seen, Eq. (3) and FEM have good agreements and Hughes (1983) expression yields higher values when the ratio of stiffener area to plate area is higher than 0.57. It can be concluded that Hughes (1983) expression can be used for coupled buckling analysis, when plate buckling is not dominant mode of buckling and ratio of stiffener area to plate area is less than 0.57. Fig. 16 shows buckling deformation of stiffened plate in interaction of tripping, web and plate buckling.
Fig. 16. Buckling deformation of stiffened plate in interaction of tripping, web and plate buckling.
5.9. S-shape mode of plate buckling To investigate the effect of plate buckling mode shape on interaction of tripping, web and plate buckling, Euler stress for Fig. 17. Euler stress for the interaction of tripping, web and plate buckling, S-shaped and U-shape modes of plate buckling.
S-shape mode of plate buckling is calculated by using Eq. (3) and compared with U-shape and FEM (Fig. 17). As can be seen, for ratio of stiffener area to plate area less than about 0.4, S-shape has higher buckling strength. For these stiffeners, results of U-shape have very good agreement with FEM. For ratio of stiffener area to plate area bigger than 0.4, U-shape has higher buckling strength, and for these stiffeners, Sshape has better agreement with FEM. Therefore, in contrast to Hughes and Ma (1996), for buckling analysis of FB stiffened plate, S-shape mode of plate buckling is dominant mode when the ratio of stiffener area to plate area is bigger than 0.4.
Fig. 14. Euler stress for the interaction of tripping, web and plate buckling.
6. Conclusions In design rules, tripping, web, and plate buckling of stiffened plate are treated separately and interaction of them are ignored. Only in Det Norske Veritas (2009) the interaction of plate and web buckling with tripping is considered by modification of expression of attached plate rotational stiffness in tripping analysis of stiffeners. In the literature, the influence of adjacent elements is considered as a rotational spring. Fujikubo and Yao (1999) expression for coupled mode of buckling of stiffened panel is modified to assess literature-based and rules-based expressions for elastic buckling strength of FB stiffened plates and to identify the applicability of selected expressions for certain conditions. Critical Euler stresses for different buckling modes are calculated and compared with corresponding expressions and FEM. It is found that:
Fig. 15. Euler stress for the interaction of tripping, web and plate buckling compared with FEM.
1. Fujikubo and Yao (1999) expression yields accurate results for tripping analysis of FB stiffeners.
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A. Rahbar-Ranji / Ocean Engineering 58 (2012) 48–59
2. Bureau Veritas (2002) expression for web buckling analysis of FB stiffener is applicable when attached plate is ignored. 3. Web buckling has no influence on Euler stress for tripping when attached plate is ignored. 4. For plate buckling when stiffener is not loaded, simple supported assumption is correct for small ratio of stiffener area to plate area. For larger ratio of stiffener area to plate area, Hughes et al. (2004) expression is applicable. 5. Rule-based expression is not applicable for tripping analysis of FB stiffeners with rigid web assumption and unloaded attached plate, especially for FBs with low ratio of stiffener area to plate area. 6. Faulkner et al. (1973) expression is applicable for the calculation of tripping with unloaded attached plate and rigid web assumption, when tripping is dominant mode and it underestimates Euler stress about 10%. 7. For interaction analysis of tripping and web buckling, Hughes (1983) expression can be used. When ratio of stiffener area to plate area is less than about 0.333, web buckling reduce Euler stress up to 10%. For higher ratio of stiffener area to plate area, web buckling has no influence on tripping. 8. For small FB stiffeners when both plate and stiffener are under compression, interaction of plate buckling and tripping makes clamp condition of plate to be less important and tripping and web buckling have no influence on Euler stress. Neither Det Norske Veritas (2009); Hughes (1983); Zheng and Hu (2005) nor Danielson and Wilmer (2004) could predict this. 9. In contrast to Hughes and Ma (1996), for buckling analysis of FB stiffened plate, S-shape mode of plate buckling is dominant mode when ratio of stiffener area to plate area is bigger than 0.4 Finally, to assess true Euler stress in stiffened plate, it is necessary to consider interaction of all buckling modes and corresponding design codes should be revised to take into account this interaction properly.
a2 ¼
Dw m4 p4 hw D ap3 D nm2 p2 ð1nÞDw m2 p2 c1 þ w 3 c4 þ w c7 þ c10 ah ahw 2a3 w 4hw
a3 ¼
2 2Dp p3 ab m2 1 þ 2 3 a2 b
a4 ¼
Dw m4 p4 hw D ap3 D nm2 p2 ð1nÞDw m2 p2 c2 þ w 3 c5 þ w c8 þ c11 3 2ahw 2ahw 4a 8hw
a5 ¼
Dp p4 ab 3m4 2 m2 þ þ 4 2 2 8a4 b a2 b
a6 ¼
Dw m4 p4 hw D ap3 D nm2 p2 ð1nÞDw m2 p2 c3 þ w 3 c6 þ w c9 þ c12 3 ahw ahw 2a 4hw
c1 ¼ c2 ¼
c3 ¼ c4 ¼ c5 ¼ c6 ¼ c7 ¼
Appendix
c8 ¼
g1 ¼ a1 þ a2 g2 ¼ a3 þ a4
c9 ¼
Z21 3
2
m p g4 ¼ 2a
g5 ¼ g6 ¼
Z22 3
p 16
p 8
þ
2
m2 p 4a
2
m2 p 2a
tp b þ t w hw c1 2
16t p b þ t w hw c2 3p 3t p b þ t w hw c3 8
p2
Z1 Z4
3 4 2 Z25 4 8 8 2 Z3 þ þ 1 þ 2 Z2 Z3 þ 2 Z2 Z5 þ Z3 Z5 2 p 2 p p p p
Z24 Z3 Z4
Z4 Z5
p 16
p2 8
p2 4
4
p
Z23 þ
16
Z25
Z3 Z5 4
Z24 þ Z1 Z4
Z4 Z5 þ 1
p2 8
p
c10 ¼ Z21 þ
8
þ
2
2 4 8 2 8 Z1 Z2 þ Z4 Z5 þ 1 þ 2 Z1 Z3 þ Z3 Z4 þ 2 Z1 Z5 þ Z2 Z4 3 p p p p
g3 ¼ a5 þ a6 2
Z24
þ
Z23 þ
2
p2 8
p2 8
2
p2 8
Z25 þ 1
p 2
Z2 Z3 þ Z2 Z5
Z24 þ 2Z1 Z4
c11 ¼ 2Z1 Z2 þ c12 ¼ Z22 þ
p Z1 Z3 þ Z1 Z5 þ Z2 Z4
p2 4
p Z4 Z5 þ2Z1 Z3 þ Z3 Z4 þ2 Z1 Z5 þ Z2 Z4
Z23 þ
2
p2 8
p Z25 þ2Z2 Z3 þ Z3 Z5 þ 2Z2 Z5 2
References
g7 ¼ g8 ¼ g9 ¼ a1 ¼
p2 tp a 4b 2pt p a 3b
p2 tp a 4b 2 Dp p4 ab m2 1 þ 2 4 a2 b
American Bureau of Shipping, 2007. Buckling Strength of Longitudinal Strength Members, Part 3, Chapter 2, App. 4. Bureau Veritas, 2002. Rules and Regulations for the Classification of Ships, Part B, Section 4. Byklum, E., Amdahl, J., 2002. A simplified method for elastic large deflection analysis of plates and stiffened panels due to local buckling. Thin-walled Struct. 42 (5), 701–717. Danielson, D.A., Wilmer, A., 2004. Buckling of stiffened plates with bulb flat flanges. Int. J. Solids Struct. 41, 6407–6427. Det Norske Veritas, 2009. Rules for Classification of Ships, Part 3, Chapter 1, Section 13, Buckling Control. Hovik, Norway. Faulkner, D., Adamchak, J.C., Synder, G.J., Vetter, M.R., 1973. Synthesis of welded gillage to withstand compression and normal loads. Comput. Struct. 3, 221–246.
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Fujikubo, M., Yao, T., 1999. Elastic local buckling strength of stiffened plate considering plate/stiffener interaction and welding residual stress. Mar. Struct. 12, 543–564. Germanischer LIoyd’s, 2004. Rules and Guidelines, Part 1, Section 3F. Proof of Buckling Strength. Hughes, O.F., 1983. Ship Structural Design. Wiley, New York. Hughes, O.F., Ma, M., 1996. Elastic tripping analysis of asymmetrical stiffeners. Comput. Struct. 60 (3), 369–389. Hughes, O.F., Ghosh, B., Chen, Y., 2004. Improved prediction of simultaneous local and overall buckling of stiffened panels. Thin-Walled Struct. 42, 827–850. LIoyd’s Register of Shipping, 2002. Rules and Regulations for the Classification of Ships, Part 3, Chapter 4, Section 7.
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Paik, J.K., Thayamballi, A.K., Lee, W.H., 1998. A numerical investigation of tripping. Mar. Struct. 11, 159–183. Paik, J.K., Thayamballi, A.K., 2000. Buckling strength of steel plating with elastically restrained edges. Thin-Walled Struct. 37, 27–55. Timoshenko, S.P., Gere, J.M., 1961. Theory of Elastic Stability, second ed. McGraw-Hill, New York, Engineering Societies Monograph. Van der Neut, A., 1983. Overall Buckling of Z-Stiffened Panels in Compression. Delft University. Yao, T., Fujikubo, M., Yanagihara, D., 1997. Buckling/plastic collapse behaviour and strength of stiffened plates under thrust. Int. J. Offshore Polar Eng. 7 (4), 285–292. Zheng, G., Hu, Y., 2005. Tripping of thin-walled stiffeners in the axially compressed stiffened panel with lateral pressure and end moments. Thin-Walled Struct. 42, 789–799.