- Email: [email protected]
Ultimate strength of perforated steel plates under edge shear loading
ARTICLE IN PRESS
Thin-Walled Structures 45 (2007) 301–306 www.elsevier.com/locate/tws
Ultimate strength of perforated steel plates under edge shear loading Jeom Kee Paik Department of Naval Architecture and Ocean Engineering, Pusan National University, 30 Jangjeon-Dong, Gumjeong-Gu, Busan 609-735, Republic of Korea Received 20 July 2006; received in revised form 30 January 2007; accepted 23 February 2007 Available online 27 April 2007
Abstract The aim of the present study is to investigate the ultimate strength characteristics of perforated steel plates under edge shear loading, which is a primary action type arising from cargo weight and water pressure in ships and ship-shaped offshore structures. The plates are considered to be simply supported along all (four) edges and kept straight. The cutout is circular and located at the center of the plate. A series of ANSYS nonlinear finite element analyses (FEA) are undertaken with varying the cutout size (diameter) as well as plate dimensions (plate aspect ratio and thickness). By the regression analysis of the FEA results obtained, a closed-form empirical formula for predicting the ultimate shear strength of perforated plates, which can be useful for first-cut strength estimations in reliability analyses or code calibrations, is derived. The accuracy of the ultimate strength formula developed is verified by a comparison with more refined nonlinear FEA results. r 2007 Elsevier Ltd. All rights reserved. Keywords: Ships; Ship-shaped offshore structures; Ultimate strength; Perforated plates; Elastic buckling; Elastic–plastic buckling; Edge shear loading; Empirical ultimate strength formula
1. Introduction In ships and ship-shaped offshore structures, cutouts are often located in plates to make a way of access or to lighten the structure. These perforations could reduce the ultimate strength of the plates. The cutouts are then needed to be included in the ultimate strength formulations as a parameter of influence where significant. A number of useful studies related to buckling strength of perforated plates have of course previously been undertaken in the literature. Elastic buckling strength of perforated plates was studied by many investigators [1–8]. Elastic buckling strength is often used as the basis of serviceability limit state (SLS) design for steel plates in ships and offshore structures [9–11] as well as other types of structures such as land-based structures. The previous studies related to elastic buckling of perforated steel plates are then useful for the SLS design purpose. On the other hand, it is important to realize that most moderately thick Tel.: +82 51 510 2429; fax: +82 51 512 8836.
E-mail address: [email protected]. 0263-8231/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.tws.2007.02.013
plates always involve plasticity to some degree until buckling takes place. The simple plasticity correction of elastic buckling strength is typically made to take into account the effect of plasticity using some approximate formulations such as the so-called Johnson–Ostenfeld formula. The elastic–plastic buckling strength so estimated is often termed the ‘critical’ buckling strength in maritime industry. In this regard, the elastic–plastic buckling of perforated steel plates has also been studied by many investigators [12–15]. However, it is interesting to note that the critical or elastic–plastic buckling strength of perforated steel plates obtained by the above-mentioned plasticity correction of the corresponding plate elastic buckling strength can be greater than the ultimate limit state (ULS) strength (ultimate load-carrying capacity) in some cases, typically when the plates are moderately thick and/or the size of cutout is relatively large [9–11]. This is in contrast to perfect plates, i.e., without cutout, where the ‘critical’ buckling strength estimated by the plasticity correction of the elastic buckling strength is always smaller than the plate ultimate strength at pessimistic strength side.
ARTICLE IN PRESS J.K. Paik / Thin-Walled Structures 45 (2007) 301–306
302
This means that the traditional method to estimate the critical buckling strength, i.e., by a simple plasticity correction is not valid for perforated plates. Therefore, it is required to directly consider the plate ultimate strength for ULS design purpose as long as the plates have cutouts. The ultimate strength of perforated plates or plate girders used for land-based structures has then been studied in the literature [16–18]. Since the action characteristics of plate elements used for ships and shipshaped offshore structures are quite different from those for land-based structures, similar studies are certainly necessary for perforated steel plates used for the former type of structures. The aim of the present study is to investigate the ultimate shear strength characteristics of perforated steel plates used for ships and offshore structures. A series of ANSYS [19] elastic–plastic large deflection finite element analyses (FEA) are carried out with varying the cutout size as well as plate dimensions. By regression analysis of the FEA results, an empirical closed-form formula of perforated plate ultimate strength which can be useful for first-cut strength estimations in reliability analyses or code calibrations is derived. 2. Perforated steel plates and their structural modeling For ships and ship-shaped offshore structures, cutouts are typically made in plates of ballast water tanks. In a continuous steel stiffened plate structure, a plate is surrounded by support members (stiffeners) which are typically designed so that they should not fail prior to the plate. In this regard, the plate in the present study considered to be simply supported at all (four) edges which are kept straight until the ultimate strength is reached. This boundary condition is usually adopted for plate ultimate strength design of ships and ship-shaped offshore structures [9,11]. Fig. 1 shows a rectangular plate with cutout and under edge shear loading, considered in the present study. This type of action typically arises from cargo weight and water pressure in ships and ship-shaped offshore structures and it
t ~
b 2
dc
a 2
b 2
a 2
Fig. 1. A perforated plate under edge shear loading.
is a primary load component for strength design of plate elements. Only a circular type of cutout is considered in the present study, while an elliptical or rectangular type of cutout is also used for ships and ship-shaped offshore structures. Also, it is assumed that the cutout is located at the center of the plate while it may lean toward the plate edge in some cases. The effects of cutout types other than circular type and the cutout location will be dealt with in separate papers [20–22]. As shown in Fig. 1, the plate dimensions are a b t. The Young modulus is E, the material yield stress is sY p and ffiffiffi the Poisson ratio is n. The shear yield stress is tY ¼ sY = 3. The diameter of cutout is dc. It is assumed that the plate material follows the elastic-perfectly plastic scheme, i.e., by neglecting the strain-hardening effect. The plate slenderness ratio b which represents pffiffiffiffiffiffiffiffiffiffiffiffi the degree of plate thickness is defined as b ¼ ðb=tÞ sY =E . It is to be noted that the plate slenderness ratio in ships and ship-shaped offshore structures is typically in the range 1.5–3.5 [9,11]. It is considered that the plate has an average level of initial deflection wopl which can routinely occur during welding fabrication of stiffened plate structures. For practical design purpose, the plate initial deflection is often taken as wopl ¼ 0:1b2 t in the plate buckling mode which will also be adopted for the present study. A convergence study with varying the number of finite elements was carried out to determine a relevant fine mesh model for ANSYS [19] nonlinear FEA of the plate. Fig. 2 shows a sample of the FE mesh modeling for a perforated plate with the aspect ratio of 3. 3. Ultimate strength characteristics of perforated steel plates Fig. 3 shows edge shear stress versus strain relations for a perforated plate with the aspect ratio of 3, with varying the cutout size (diameter), until and after the ultimate strength is reached. This plate is moderately thick because the plate slenderness ratio is 2.2. It is evident from Fig. 3 that the cutout significantly reduces the plate ultimate strength. For example, when the diameter of the cutout is 40% of the plate breadth, the plate ultimate strength is reduced by approximately 85% when compared with the perfect plate, i.e., without cutout. Figs. 4–6 show the ultimate strength characteristics of perforated steel plates as a function of the plate aspect ratio, the cutout size and the plate thickness. While the cutout size is a primary parameter of influence in the ultimate strength, the other two parameters, i.e., the plate aspect ratio and the plate thickness have somewhat different aspects. The ultimate shear strength of perforated plates with holding the cutout size constant tends to increase as the plate aspect ratio increases. However, the plate thickness is not a significant parameter of influence. As previously noted, the maritime industry has traditionally applied for the plasticity correction approach with
ARTICLE IN PRESS J.K. Paik / Thin-Walled Structures 45 (2007) 301–306
303
Fig. 2. A sample of the FE mesh for a plate with a centrally located circular hole, a/b ¼ 3.
1.2
dc/b=0.0
200
b
dc
dc/b=0.2 160
t ~
a dc/b=0.4
0.8
u /Y
av(MPa)
120 dc/b=0.8
FEM: 80
a/b=1 a/b=3 a/b=5
0.4
40
b×t = 800×10mm Y = 203.7 MPa wopl = 1.02t (buckling mode)
a×b×t = 2,400×800×15 mm Y = 203.7 MPa wopl = 0.12t (buckling mode) 0.0
0 0
2
4
6
8
10
av(×10−3) Fig. 3. Edge shear stress versus strain relations for a perforated plate under edge shear loading with varying the cutout size, as obtained by ANSYS nonlinear FEA; a/b ¼ 3, b ¼ 2.2.
elastic buckling strength to estimate the so-called critical (or elastic–plastic) buckling strength of steel plates, using approximate formulations. One of the most typical approximate formulations used for ships and offshore structures is the Johnson–Ostenfeld formula which is given by 8 s for sE ok2 sF ; < E scr ¼ (1) sF 1 k s : F 1 s for sE Xk2 sF ; E where scr is critical buckling strength, sE is elastic buckling strength, sF is reference yield stress depending on loading type which is taken pasffiffiffi sF ¼ sY for axial compressive loading and sF ¼ sY = 3 for edge shear loading. k1 and k2 are coefficients which are typically taken as k1 ¼ 0.25 and k2 ¼ 0.5–0.6. Fig. 7 shows the critical buckling strength tcr of perfect steel plates, i.e., without cutout under longitudinal axial compressive loading as a function of the corresponding
0.0
0.2
0.4
0.6
0.8
1.0
dc/b Fig. 4. Variation of the ultimate strength of relatively thin perforated steel plates under edge shear loads as a function of the plate aspect ratio and the cutout size; b ¼ 3.3.
elastic buckling strength tE,1, obtained by using Eq. (1) where k1 ¼ 0.25 and k2 ¼ 0.5 are used hereafter. The ultimate strength obtained by ANSYS nonlinear elastic– plastic large deflection FEA is also compared in the figure. It is evident from Fig. 7 that the so-called critical buckling strength can be a good basis of the load-carrying capacity prediction for perfect steel plates because it corresponds well with more refined nonlinear finite element solutions at somewhat pessimistic strength side. However, the above-mentioned aspect is not always the case for perforated plates. Fig. 8 shows the critical buckling strength of relatively thin perforated plates under longitudinal axial compression as obtained by Eq. (1) while a comparison with more refined nonlinear finite element solutions of the ultimate strength is made. As would be expected, it is seen from Fig. 8 that the critical buckling strength estimated from Eq. (1) is smaller than the ultimate strength although the difference is so large.
ARTICLE IN PRESS J.K. Paik / Thin-Walled Structures 45 (2007) 301–306
304
strength, i.e., specifically when the cutout size is relatively large and the plate thickness is large. This means that the traditional approach with plasticity correction of elastic buckling strength is not valid for perforated plates, and the load-carrying capacity of perforated plates must thus be dealt with on the basis of the ultimate strength rather than the so-called critical buckling strength obtained by Eq. (1).
1.2 FEM: a/b=1 a/b=3 a/b=5 0.8
u / Y
4. Empirical ultimate strength formula for perforated plates
b 0.4
dc
t ~
a b×t = 800×20 mm Y = 203.7 MPa wopl = 1.02t (buckling mode) 0.0 0.0
0.2
0.4
0.6
0.8
1.0
dc/b Fig. 5. Variation of the ultimate strength of relatively thick perforated steel under edge shear loads as a function of the plate aspect ratio and the cutout size; b ¼ 1.7.
1.2 FEM: t=10 mm t=15 mm t=20 mm
Ru = u / o
0.8
b
dc
0.4
t ~
a
A closed-form empirical formula for the ultimate shear strength of perforated plates is now derived. Although the use of such empirical formula may be limited in terms of application range, ultimate strength-based reliability analyses and code calibrations which typically require first-cut strength estimates often need closed-form formula. Based on the insights noted above, the plate ultimate reduction factor, Rtu, for edge shear may be empirically derived as a function of the cutout size and the plate aspect ratio by curve fitting based on the FE results, as follows: 2 tu dc dc ¼ C1 þ C 2 þ 1:0 Rtu , (2) tuo b b where tu, tuo are ultimate shear strengths of a plate with or without cutout, respectively, and C1, C2 are the coefficients. The coefficients C1 and C2 in Eq. (2) can be given by a2 a 0:068 0:415, (3a) C 1 ¼ 0:009 b b a2 a þ 0:309 0:787. (3b) C 2 ¼ 0:025 b b The accuracy of Eq. (2) together with Eqs. (3) is verified in Fig. 10. It is evident from Fig. 10 that Eq. (2) together with Eqs. (3) gives an excellent prediction of the ultimate shear strength for perforated steel plates when compared to more refined nonlinear finite element solutions. For completeness of Eq. (2), the ultimate strength of the perfect plate under edge shear can be given, as follows [9]: 8 > for 0o ttYE p0:5; 1:324 ttYE > > > < 3 2 tuo ¼ 0:039 tE 0:274 tE þ 0:676 tE þ 0:388 for 0:5o tE p2:0; tY tY tY tY > tY > > > 0:956 : for ttE 42:0;
b×t = 2,400×800 mm Y = 203.7 MPa wopl = 1.02t (buckling mode) 0.0
Y
0.0
0.2
0.4
0.6
0.8
1.0
dc/b Fig. 6. Variation of the ultimate strength of perforated steel plates under edge shear loads as a function of the plate thickness and the cutout size; a/ b ¼ 3.
A significant issue related to the use of Eq. (1) arises when the plate thickness of perforated plates is relatively large. Fig. 9 shows the critical buckling strength of relatively thick perforated plates under edge shear as a function of elastic buckling strength obtained by Eq. (1). It is evident from Fig. 9 that the critical buckling strength estimated from Eq. (1) is rather larger than the ultimate
(4) where tE is elastic shear buckling stress of the plate without cutout, which can be given by t 2 p2 E , (5) tE ¼ k 12ð1 n2 Þ b where k is shear buckling strength coefficient, which can be given by 8 < 4 b 2 þ 5:34 for ab X1; a k¼ : 5:34 b 2 þ 4:0 for a o1: a b
(6)
ARTICLE IN PRESS J.K. Paik / Thin-Walled Structures 45 (2007) 301–306
305
1.2 a/b = 3.0 All edges remain straight
1.1 1.0 0.9 0.8
Elastic buckling strength with plasticity correction
cr / Y
0.7 0.6 FEM:
0.5
: All edges simply supported
0.4
: Simply supported alone longitudinal edges
0.3
& clamped alone transverse edges : Clamped along longitudinal edges &simply
0.2
supported along transverse edges
0.1
: All edges clamped
0.0 0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5 4.0 E,1 / Y
4.5
5.0
5.5
6.0
6.5
7.0
Fig. 7. Critical buckling strength tcr versus elastic buckling strength tE,1 for a perfect plate under edge shear.
1.2
1.2 b
dc
t ~
FEM: a/b=1 a/b=3 a/b=5
Elastic buckling strength with plasticitycorrection
FEM : a/b=1 a/b=3 a/b=5
a Elastic bucklingstrength with plasticity correction
a/b=1
0.8
a/b=3
cr /Y
cr/Y
0.8
a/b=1 a/b=3
b
a/b=5
dc
t ~
0.4
0.4
a b×t = 800×20 mm Y = 203.7 MPa wopl = 1.02t (buckling mode)
b×t = 800×10 mm Y = 203.7 MPa wopl = 1.02t (buckling mode) 0.0
0.0 0.0
0.2
0.4
0.6
0.8
1.0
dc/b
0.0
0.2
0.4
0.6
0.8
1.0
dc/b
Fig. 8. Critical buckling strength versus ultimate strength for relatively thin perforated plates under edge shear; t ¼ 10 mm.
Fig. 9. Critical buckling strength versus ultimate strength for relatively thick perforated plates under edge shear; t ¼ 20 mm.
5. Concluding remarks
It is confirmed that the cutout significantly reduces the plate ultimate strength. It is found that the plate aspect ratio affects the ultimate strength of perforated plates to some extent when edge shear loads are predominant, while the plate thickness is not a sensitive parameter to the normalized ultimate shear strength when holding the cutout size constant. Also, the traditional approach of the plasticity correction with elastic buckling strength to
The aim of the present paper has been to investigate the ultimate strength characteristics of perforated steel plates under edge shear loads. A series of ANSYS elastic–plastic large deflection FEA were then carried out with varying the cutout size as well as plate dimensions (aspect ratio and plate thickness).
ARTICLE IN PRESS J.K. Paik / Thin-Walled Structures 45 (2007) 301–306
306
1.2 FEM: [2]
t=10 mm t=15 mm t=20 mm
[3]
0.8 Ru = u /uo
b
dc
[4]
t ~
[5]
Ru = C1 (dc / b)2 + C2 (dc/b)+1.0
[6]
C1 = −0.009 (a / b)2 − 0.068 (a/b)− 0.415 C2 = −0.025 (a / b)2 + 0.309 (a/b)−0.787
0.4
[7]
a×b×t = 2,400×800×10mm Y = 20.37 MPa wopl = 0.12t (buckling mode)
[8]
0.0 0.0
0.2
0.4
0.6
0.8
1.0
dc/b
[9] [10]
Fig. 10. Verification of the empirical ultimate shear strength formula for perforated steel plates. [11]
estimate the so-called critical buckling strength may not be valid for perforated plates, specifically when the cutout size and/or the plate thickness are relatively large. A closed-form empirical formula which can readily predict the ultimate shear strength of perforated plate as a function of cutout size as well as plate dimensions has been developed in the present study by regression analysis of FEA results. The ultimate strength formula developed will be very useful for first-cut estimation of the ultimate strength in reliability analyses or code calibrations for perforated plates under predominantly edge shear loads. Further studies are required for different types of loads or their combination loads or with different types or locations of cutout. Some contributions on the topics will be published in separate papers later on.
[12]
[13] [14]
[15]
[16]
[17] [18]
Acknowledgment The present study was undertaken at the Ship and Offshore Structural Mechanics Laboratory, Pusan National University, Korea, which is a National Research Laboratory funded by the Korea Ministry of Science and Technology (Grant no. M10600000239-06J0000-23910).
[19] [20]
[21]
References [22] [1] Shanmugam NE, Narayanan R. Elastic buckling of perforated square plates for various loading and edge conditions. In: Proceed-
ings of the international conference on finite element methods. Paper no. 103. Shanghai, China; August 1982. Sabir AB, Chow FY. Elastic buckling of flat panels containing circular and square holes. In: Proceedings of the international conference on instability and plastic collapse of steel structures. London: Granada Publishing; 1983. p. 311–21. Narayanan R, Darwish IYS. Strength of slender webs having noncentral holes. Struct Engi 1985;63B(3):57–61. Brown CJ, Yettram AL. The elastic stability of square perforated plates under combinations of bending, shear and direct load. ThinWalled Struct 1986;4(3):239–46. Brown CJ, Yettram AL, Burnett M. Stability of plates with rectangular holes. J Struct Engineer, ASCE 1987;113(5):1111–6. Shakerley TM, Brown CJ. Elastic buckling of plates with eccentrically positioned rectangular perforations. Int JMech Sci 1996; 38(8–9):825–38. Shangmugam NE, Thevendran V, Tan YH. Design formula for axially compressed perforated plates. Thin-Walled Struct 1999; 34(1):1–20. El-Sawy KM, Nazmy AS. Effect of aspect ratio on the elastic buckling of uniaxially loaded plates with eccentric holes. Thin-Walled Struct 2001;39:983–98. Paik JK, Thayamballi AK. Ultimate limit state design of steel-plated structures. Chichester, UK: Wiley; 2003. Paik JK, Thayamballi AK. Some recent developments on ultimate limit state design technology for ships and offshore structures. Ships Offshore Struct 2006;1(2):99–116. Paik JK, Thayamballi AK. Ship-shaped offshore installations: design, building, and operation. Cambridge, UK: Cambridge University Press; 2007. Azizian ZG, Roberts TM. Buckling and elasto-plastic collapse of perforated plates. In: Proceedings of the international conference on instability and plastic collapse of steel structures. London: Granada Publishing; 1983. p. 392–98. Durban D, Zuckerman Z. Elasto-plastic buckling of rectangular plates in biaxial compression/tension. Int J Mech Sci 1999;41:751–65. Betten J, Shin CH. Elasto-plastic buckling analysis of rectangular plates subjected to biaxial loads. Forschung im Ingenieurwesen 2000;65:273–8. El-Sawy KM, Nazmy AS, Martini MI. Elasto-plastic buckling of perforated plates under uniaxial compression. Thin-Walled Struct 2004;42:1083–101. Narayanan R, Rockey KC. Ultimate load capacity of plate girders with webs containing circular cut-outs. In: Proceedings of the institution of civil engineers, vol. 71 (Part 2); 1981. p. 845–62. Narayanan R, Chow FY. Ultimate capacity of uniaxially compressed perforated plates. Thin-Walled Struct 1984;2:241–64. Rockey KC, Anderson RG, Cheung YK. The behavior of square shear webs having circular holes. In: Proceedings of the international conference on Thin-Walled Structures. Swansea, London: Crosby Lockwood; 1967. p. 148–69. ANSYS, User’s manual (version 6.0), ANSYS Inc., Canonsburg, Pennsylvania, USA; 2006. Paik JK. Ultimate strength of perforated steel plates under axial compressive loading along short edges; 2006, submitted for publication. Paik JK. Ultimate strength of perforated steel plates under combined biaxial compression and edge shear loads; 2007, submitted for publication. Paik JK. Closed-form empirical formula methods for the ultimate strength prediction of perforated steel plates with various cutout shapes; 2007, submitted for publication.
Thin-Walled Structures 45 (2007) 301–306 www.elsevier.com/locate/tws
Ultimate strength of perforated steel plates under edge shear loading Jeom Kee Paik Department of Naval Architecture and Ocean Engineering, Pusan National University, 30 Jangjeon-Dong, Gumjeong-Gu, Busan 609-735, Republic of Korea Received 20 July 2006; received in revised form 30 January 2007; accepted 23 February 2007 Available online 27 April 2007
Abstract The aim of the present study is to investigate the ultimate strength characteristics of perforated steel plates under edge shear loading, which is a primary action type arising from cargo weight and water pressure in ships and ship-shaped offshore structures. The plates are considered to be simply supported along all (four) edges and kept straight. The cutout is circular and located at the center of the plate. A series of ANSYS nonlinear finite element analyses (FEA) are undertaken with varying the cutout size (diameter) as well as plate dimensions (plate aspect ratio and thickness). By the regression analysis of the FEA results obtained, a closed-form empirical formula for predicting the ultimate shear strength of perforated plates, which can be useful for first-cut strength estimations in reliability analyses or code calibrations, is derived. The accuracy of the ultimate strength formula developed is verified by a comparison with more refined nonlinear FEA results. r 2007 Elsevier Ltd. All rights reserved. Keywords: Ships; Ship-shaped offshore structures; Ultimate strength; Perforated plates; Elastic buckling; Elastic–plastic buckling; Edge shear loading; Empirical ultimate strength formula
1. Introduction In ships and ship-shaped offshore structures, cutouts are often located in plates to make a way of access or to lighten the structure. These perforations could reduce the ultimate strength of the plates. The cutouts are then needed to be included in the ultimate strength formulations as a parameter of influence where significant. A number of useful studies related to buckling strength of perforated plates have of course previously been undertaken in the literature. Elastic buckling strength of perforated plates was studied by many investigators [1–8]. Elastic buckling strength is often used as the basis of serviceability limit state (SLS) design for steel plates in ships and offshore structures [9–11] as well as other types of structures such as land-based structures. The previous studies related to elastic buckling of perforated steel plates are then useful for the SLS design purpose. On the other hand, it is important to realize that most moderately thick Tel.: +82 51 510 2429; fax: +82 51 512 8836.
E-mail address: [email protected]. 0263-8231/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.tws.2007.02.013
plates always involve plasticity to some degree until buckling takes place. The simple plasticity correction of elastic buckling strength is typically made to take into account the effect of plasticity using some approximate formulations such as the so-called Johnson–Ostenfeld formula. The elastic–plastic buckling strength so estimated is often termed the ‘critical’ buckling strength in maritime industry. In this regard, the elastic–plastic buckling of perforated steel plates has also been studied by many investigators [12–15]. However, it is interesting to note that the critical or elastic–plastic buckling strength of perforated steel plates obtained by the above-mentioned plasticity correction of the corresponding plate elastic buckling strength can be greater than the ultimate limit state (ULS) strength (ultimate load-carrying capacity) in some cases, typically when the plates are moderately thick and/or the size of cutout is relatively large [9–11]. This is in contrast to perfect plates, i.e., without cutout, where the ‘critical’ buckling strength estimated by the plasticity correction of the elastic buckling strength is always smaller than the plate ultimate strength at pessimistic strength side.
ARTICLE IN PRESS J.K. Paik / Thin-Walled Structures 45 (2007) 301–306
302
This means that the traditional method to estimate the critical buckling strength, i.e., by a simple plasticity correction is not valid for perforated plates. Therefore, it is required to directly consider the plate ultimate strength for ULS design purpose as long as the plates have cutouts. The ultimate strength of perforated plates or plate girders used for land-based structures has then been studied in the literature [16–18]. Since the action characteristics of plate elements used for ships and shipshaped offshore structures are quite different from those for land-based structures, similar studies are certainly necessary for perforated steel plates used for the former type of structures. The aim of the present study is to investigate the ultimate shear strength characteristics of perforated steel plates used for ships and offshore structures. A series of ANSYS [19] elastic–plastic large deflection finite element analyses (FEA) are carried out with varying the cutout size as well as plate dimensions. By regression analysis of the FEA results, an empirical closed-form formula of perforated plate ultimate strength which can be useful for first-cut strength estimations in reliability analyses or code calibrations is derived. 2. Perforated steel plates and their structural modeling For ships and ship-shaped offshore structures, cutouts are typically made in plates of ballast water tanks. In a continuous steel stiffened plate structure, a plate is surrounded by support members (stiffeners) which are typically designed so that they should not fail prior to the plate. In this regard, the plate in the present study considered to be simply supported at all (four) edges which are kept straight until the ultimate strength is reached. This boundary condition is usually adopted for plate ultimate strength design of ships and ship-shaped offshore structures [9,11]. Fig. 1 shows a rectangular plate with cutout and under edge shear loading, considered in the present study. This type of action typically arises from cargo weight and water pressure in ships and ship-shaped offshore structures and it
t ~
b 2
dc
a 2
b 2
a 2
Fig. 1. A perforated plate under edge shear loading.
is a primary load component for strength design of plate elements. Only a circular type of cutout is considered in the present study, while an elliptical or rectangular type of cutout is also used for ships and ship-shaped offshore structures. Also, it is assumed that the cutout is located at the center of the plate while it may lean toward the plate edge in some cases. The effects of cutout types other than circular type and the cutout location will be dealt with in separate papers [20–22]. As shown in Fig. 1, the plate dimensions are a b t. The Young modulus is E, the material yield stress is sY p and ffiffiffi the Poisson ratio is n. The shear yield stress is tY ¼ sY = 3. The diameter of cutout is dc. It is assumed that the plate material follows the elastic-perfectly plastic scheme, i.e., by neglecting the strain-hardening effect. The plate slenderness ratio b which represents pffiffiffiffiffiffiffiffiffiffiffiffi the degree of plate thickness is defined as b ¼ ðb=tÞ sY =E . It is to be noted that the plate slenderness ratio in ships and ship-shaped offshore structures is typically in the range 1.5–3.5 [9,11]. It is considered that the plate has an average level of initial deflection wopl which can routinely occur during welding fabrication of stiffened plate structures. For practical design purpose, the plate initial deflection is often taken as wopl ¼ 0:1b2 t in the plate buckling mode which will also be adopted for the present study. A convergence study with varying the number of finite elements was carried out to determine a relevant fine mesh model for ANSYS [19] nonlinear FEA of the plate. Fig. 2 shows a sample of the FE mesh modeling for a perforated plate with the aspect ratio of 3. 3. Ultimate strength characteristics of perforated steel plates Fig. 3 shows edge shear stress versus strain relations for a perforated plate with the aspect ratio of 3, with varying the cutout size (diameter), until and after the ultimate strength is reached. This plate is moderately thick because the plate slenderness ratio is 2.2. It is evident from Fig. 3 that the cutout significantly reduces the plate ultimate strength. For example, when the diameter of the cutout is 40% of the plate breadth, the plate ultimate strength is reduced by approximately 85% when compared with the perfect plate, i.e., without cutout. Figs. 4–6 show the ultimate strength characteristics of perforated steel plates as a function of the plate aspect ratio, the cutout size and the plate thickness. While the cutout size is a primary parameter of influence in the ultimate strength, the other two parameters, i.e., the plate aspect ratio and the plate thickness have somewhat different aspects. The ultimate shear strength of perforated plates with holding the cutout size constant tends to increase as the plate aspect ratio increases. However, the plate thickness is not a significant parameter of influence. As previously noted, the maritime industry has traditionally applied for the plasticity correction approach with
ARTICLE IN PRESS J.K. Paik / Thin-Walled Structures 45 (2007) 301–306
303
Fig. 2. A sample of the FE mesh for a plate with a centrally located circular hole, a/b ¼ 3.
1.2
dc/b=0.0
200
b
dc
dc/b=0.2 160
t ~
a dc/b=0.4
0.8
u /Y
av(MPa)
120 dc/b=0.8
FEM: 80
a/b=1 a/b=3 a/b=5
0.4
40
b×t = 800×10mm Y = 203.7 MPa wopl = 1.02t (buckling mode)
a×b×t = 2,400×800×15 mm Y = 203.7 MPa wopl = 0.12t (buckling mode) 0.0
0 0
2
4
6
8
10
av(×10−3) Fig. 3. Edge shear stress versus strain relations for a perforated plate under edge shear loading with varying the cutout size, as obtained by ANSYS nonlinear FEA; a/b ¼ 3, b ¼ 2.2.
elastic buckling strength to estimate the so-called critical (or elastic–plastic) buckling strength of steel plates, using approximate formulations. One of the most typical approximate formulations used for ships and offshore structures is the Johnson–Ostenfeld formula which is given by 8 s for sE ok2 sF ; < E scr ¼ (1) sF 1 k s : F 1 s for sE Xk2 sF ; E where scr is critical buckling strength, sE is elastic buckling strength, sF is reference yield stress depending on loading type which is taken pasffiffiffi sF ¼ sY for axial compressive loading and sF ¼ sY = 3 for edge shear loading. k1 and k2 are coefficients which are typically taken as k1 ¼ 0.25 and k2 ¼ 0.5–0.6. Fig. 7 shows the critical buckling strength tcr of perfect steel plates, i.e., without cutout under longitudinal axial compressive loading as a function of the corresponding
0.0
0.2
0.4
0.6
0.8
1.0
dc/b Fig. 4. Variation of the ultimate strength of relatively thin perforated steel plates under edge shear loads as a function of the plate aspect ratio and the cutout size; b ¼ 3.3.
elastic buckling strength tE,1, obtained by using Eq. (1) where k1 ¼ 0.25 and k2 ¼ 0.5 are used hereafter. The ultimate strength obtained by ANSYS nonlinear elastic– plastic large deflection FEA is also compared in the figure. It is evident from Fig. 7 that the so-called critical buckling strength can be a good basis of the load-carrying capacity prediction for perfect steel plates because it corresponds well with more refined nonlinear finite element solutions at somewhat pessimistic strength side. However, the above-mentioned aspect is not always the case for perforated plates. Fig. 8 shows the critical buckling strength of relatively thin perforated plates under longitudinal axial compression as obtained by Eq. (1) while a comparison with more refined nonlinear finite element solutions of the ultimate strength is made. As would be expected, it is seen from Fig. 8 that the critical buckling strength estimated from Eq. (1) is smaller than the ultimate strength although the difference is so large.
ARTICLE IN PRESS J.K. Paik / Thin-Walled Structures 45 (2007) 301–306
304
strength, i.e., specifically when the cutout size is relatively large and the plate thickness is large. This means that the traditional approach with plasticity correction of elastic buckling strength is not valid for perforated plates, and the load-carrying capacity of perforated plates must thus be dealt with on the basis of the ultimate strength rather than the so-called critical buckling strength obtained by Eq. (1).
1.2 FEM: a/b=1 a/b=3 a/b=5 0.8
u / Y
4. Empirical ultimate strength formula for perforated plates
b 0.4
dc
t ~
a b×t = 800×20 mm Y = 203.7 MPa wopl = 1.02t (buckling mode) 0.0 0.0
0.2
0.4
0.6
0.8
1.0
dc/b Fig. 5. Variation of the ultimate strength of relatively thick perforated steel under edge shear loads as a function of the plate aspect ratio and the cutout size; b ¼ 1.7.
1.2 FEM: t=10 mm t=15 mm t=20 mm
Ru = u / o
0.8
b
dc
0.4
t ~
a
A closed-form empirical formula for the ultimate shear strength of perforated plates is now derived. Although the use of such empirical formula may be limited in terms of application range, ultimate strength-based reliability analyses and code calibrations which typically require first-cut strength estimates often need closed-form formula. Based on the insights noted above, the plate ultimate reduction factor, Rtu, for edge shear may be empirically derived as a function of the cutout size and the plate aspect ratio by curve fitting based on the FE results, as follows: 2 tu dc dc ¼ C1 þ C 2 þ 1:0 Rtu , (2) tuo b b where tu, tuo are ultimate shear strengths of a plate with or without cutout, respectively, and C1, C2 are the coefficients. The coefficients C1 and C2 in Eq. (2) can be given by a2 a 0:068 0:415, (3a) C 1 ¼ 0:009 b b a2 a þ 0:309 0:787. (3b) C 2 ¼ 0:025 b b The accuracy of Eq. (2) together with Eqs. (3) is verified in Fig. 10. It is evident from Fig. 10 that Eq. (2) together with Eqs. (3) gives an excellent prediction of the ultimate shear strength for perforated steel plates when compared to more refined nonlinear finite element solutions. For completeness of Eq. (2), the ultimate strength of the perfect plate under edge shear can be given, as follows [9]: 8 > for 0o ttYE p0:5; 1:324 ttYE > > > < 3 2 tuo ¼ 0:039 tE 0:274 tE þ 0:676 tE þ 0:388 for 0:5o tE p2:0; tY tY tY tY > tY > > > 0:956 : for ttE 42:0;
b×t = 2,400×800 mm Y = 203.7 MPa wopl = 1.02t (buckling mode) 0.0
Y
0.0
0.2
0.4
0.6
0.8
1.0
dc/b Fig. 6. Variation of the ultimate strength of perforated steel plates under edge shear loads as a function of the plate thickness and the cutout size; a/ b ¼ 3.
A significant issue related to the use of Eq. (1) arises when the plate thickness of perforated plates is relatively large. Fig. 9 shows the critical buckling strength of relatively thick perforated plates under edge shear as a function of elastic buckling strength obtained by Eq. (1). It is evident from Fig. 9 that the critical buckling strength estimated from Eq. (1) is rather larger than the ultimate
(4) where tE is elastic shear buckling stress of the plate without cutout, which can be given by t 2 p2 E , (5) tE ¼ k 12ð1 n2 Þ b where k is shear buckling strength coefficient, which can be given by 8 < 4 b 2 þ 5:34 for ab X1; a k¼ : 5:34 b 2 þ 4:0 for a o1: a b
(6)
ARTICLE IN PRESS J.K. Paik / Thin-Walled Structures 45 (2007) 301–306
305
1.2 a/b = 3.0 All edges remain straight
1.1 1.0 0.9 0.8
Elastic buckling strength with plasticity correction
cr / Y
0.7 0.6 FEM:
0.5
: All edges simply supported
0.4
: Simply supported alone longitudinal edges
0.3
& clamped alone transverse edges : Clamped along longitudinal edges &simply
0.2
supported along transverse edges
0.1
: All edges clamped
0.0 0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5 4.0 E,1 / Y
4.5
5.0
5.5
6.0
6.5
7.0
Fig. 7. Critical buckling strength tcr versus elastic buckling strength tE,1 for a perfect plate under edge shear.
1.2
1.2 b
dc
t ~
FEM: a/b=1 a/b=3 a/b=5
Elastic buckling strength with plasticitycorrection
FEM : a/b=1 a/b=3 a/b=5
a Elastic bucklingstrength with plasticity correction
a/b=1
0.8
a/b=3
cr /Y
cr/Y
0.8
a/b=1 a/b=3
b
a/b=5
dc
t ~
0.4
0.4
a b×t = 800×20 mm Y = 203.7 MPa wopl = 1.02t (buckling mode)
b×t = 800×10 mm Y = 203.7 MPa wopl = 1.02t (buckling mode) 0.0
0.0 0.0
0.2
0.4
0.6
0.8
1.0
dc/b
0.0
0.2
0.4
0.6
0.8
1.0
dc/b
Fig. 8. Critical buckling strength versus ultimate strength for relatively thin perforated plates under edge shear; t ¼ 10 mm.
Fig. 9. Critical buckling strength versus ultimate strength for relatively thick perforated plates under edge shear; t ¼ 20 mm.
5. Concluding remarks
It is confirmed that the cutout significantly reduces the plate ultimate strength. It is found that the plate aspect ratio affects the ultimate strength of perforated plates to some extent when edge shear loads are predominant, while the plate thickness is not a sensitive parameter to the normalized ultimate shear strength when holding the cutout size constant. Also, the traditional approach of the plasticity correction with elastic buckling strength to
The aim of the present paper has been to investigate the ultimate strength characteristics of perforated steel plates under edge shear loads. A series of ANSYS elastic–plastic large deflection FEA were then carried out with varying the cutout size as well as plate dimensions (aspect ratio and plate thickness).
ARTICLE IN PRESS J.K. Paik / Thin-Walled Structures 45 (2007) 301–306
306
1.2 FEM: [2]
t=10 mm t=15 mm t=20 mm
[3]
0.8 Ru = u /uo
b
dc
[4]
t ~
[5]
Ru = C1 (dc / b)2 + C2 (dc/b)+1.0
[6]
C1 = −0.009 (a / b)2 − 0.068 (a/b)− 0.415 C2 = −0.025 (a / b)2 + 0.309 (a/b)−0.787
0.4
[7]
a×b×t = 2,400×800×10mm Y = 20.37 MPa wopl = 0.12t (buckling mode)
[8]
0.0 0.0
0.2
0.4
0.6
0.8
1.0
dc/b
[9] [10]
Fig. 10. Verification of the empirical ultimate shear strength formula for perforated steel plates. [11]
estimate the so-called critical buckling strength may not be valid for perforated plates, specifically when the cutout size and/or the plate thickness are relatively large. A closed-form empirical formula which can readily predict the ultimate shear strength of perforated plate as a function of cutout size as well as plate dimensions has been developed in the present study by regression analysis of FEA results. The ultimate strength formula developed will be very useful for first-cut estimation of the ultimate strength in reliability analyses or code calibrations for perforated plates under predominantly edge shear loads. Further studies are required for different types of loads or their combination loads or with different types or locations of cutout. Some contributions on the topics will be published in separate papers later on.
[12]
[13] [14]
[15]
[16]
[17] [18]
Acknowledgment The present study was undertaken at the Ship and Offshore Structural Mechanics Laboratory, Pusan National University, Korea, which is a National Research Laboratory funded by the Korea Ministry of Science and Technology (Grant no. M10600000239-06J0000-23910).
[19] [20]
[21]
References [22] [1] Shanmugam NE, Narayanan R. Elastic buckling of perforated square plates for various loading and edge conditions. In: Proceed-
ings of the international conference on finite element methods. Paper no. 103. Shanghai, China; August 1982. Sabir AB, Chow FY. Elastic buckling of flat panels containing circular and square holes. In: Proceedings of the international conference on instability and plastic collapse of steel structures. London: Granada Publishing; 1983. p. 311–21. Narayanan R, Darwish IYS. Strength of slender webs having noncentral holes. Struct Engi 1985;63B(3):57–61. Brown CJ, Yettram AL. The elastic stability of square perforated plates under combinations of bending, shear and direct load. ThinWalled Struct 1986;4(3):239–46. Brown CJ, Yettram AL, Burnett M. Stability of plates with rectangular holes. J Struct Engineer, ASCE 1987;113(5):1111–6. Shakerley TM, Brown CJ. Elastic buckling of plates with eccentrically positioned rectangular perforations. Int JMech Sci 1996; 38(8–9):825–38. Shangmugam NE, Thevendran V, Tan YH. Design formula for axially compressed perforated plates. Thin-Walled Struct 1999; 34(1):1–20. El-Sawy KM, Nazmy AS. Effect of aspect ratio on the elastic buckling of uniaxially loaded plates with eccentric holes. Thin-Walled Struct 2001;39:983–98. Paik JK, Thayamballi AK. Ultimate limit state design of steel-plated structures. Chichester, UK: Wiley; 2003. Paik JK, Thayamballi AK. Some recent developments on ultimate limit state design technology for ships and offshore structures. Ships Offshore Struct 2006;1(2):99–116. Paik JK, Thayamballi AK. Ship-shaped offshore installations: design, building, and operation. Cambridge, UK: Cambridge University Press; 2007. Azizian ZG, Roberts TM. Buckling and elasto-plastic collapse of perforated plates. In: Proceedings of the international conference on instability and plastic collapse of steel structures. London: Granada Publishing; 1983. p. 392–98. Durban D, Zuckerman Z. Elasto-plastic buckling of rectangular plates in biaxial compression/tension. Int J Mech Sci 1999;41:751–65. Betten J, Shin CH. Elasto-plastic buckling analysis of rectangular plates subjected to biaxial loads. Forschung im Ingenieurwesen 2000;65:273–8. El-Sawy KM, Nazmy AS, Martini MI. Elasto-plastic buckling of perforated plates under uniaxial compression. Thin-Walled Struct 2004;42:1083–101. Narayanan R, Rockey KC. Ultimate load capacity of plate girders with webs containing circular cut-outs. In: Proceedings of the institution of civil engineers, vol. 71 (Part 2); 1981. p. 845–62. Narayanan R, Chow FY. Ultimate capacity of uniaxially compressed perforated plates. Thin-Walled Struct 1984;2:241–64. Rockey KC, Anderson RG, Cheung YK. The behavior of square shear webs having circular holes. In: Proceedings of the international conference on Thin-Walled Structures. Swansea, London: Crosby Lockwood; 1967. p. 148–69. ANSYS, User’s manual (version 6.0), ANSYS Inc., Canonsburg, Pennsylvania, USA; 2006. Paik JK. Ultimate strength of perforated steel plates under axial compressive loading along short edges; 2006, submitted for publication. Paik JK. Ultimate strength of perforated steel plates under combined biaxial compression and edge shear loads; 2007, submitted for publication. Paik JK. Closed-form empirical formula methods for the ultimate strength prediction of perforated steel plates with various cutout shapes; 2007, submitted for publication.