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Nonlinear finite element method models for ultimate strength analysis of steel stiffened-plate structures under combined biaxial compression and lateral pressure actions—Part II: Stiffened panels
ARTICLE IN PRESS Thin-Walled Structures 47 (2009) 998–1007
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Nonlinear finite element method models for ultimate strength analysis of steel stiffened-plate structures under combined biaxial compression and lateral pressure actions—Part II: Stiffened panels Jeom Kee Paik , Jung Kwan Seo Department of Naval Architecture and Ocean Engineering, LRET Research Centre of Excellence, Pusan National University, Busan 609-735, Republic of Korea
a r t i c l e in f o
a b s t r a c t
Available online 9 September 2008
The present paper (Part II) is a sequel to the previous paper (Part I) [Paik JK, Seo JK. Nonlinear finite element method models for ultimate strength analysis of steel stiffened-plate structures under combined biaxial compression and lateral pressure actions—Part I: Plate elements. Thin-Walled Struct 2008, this issue, doi:10.1016/j.tws.2008.08.005.] on the application of nonlinear finite element methods for ultimate strength analysis of steel stiffened-plate structures under combined biaxial compression and lateral pressure actions. In contrast to Part I dealing with plate elements, the present paper (Part II) treats stiffened panels surrounded by strong support members such as longitudinal girders and transverse frames. In similar to Part I, some important factors of influence such as structural dimensions, initial imperfections, loading types and computational techniques in association with ultimate limit states are studied. Some useful insights in terms of nonlinear finite element method modeling are developed using ANSYS code together with the ALPS/ULSAP semi-analytical method, the latter being for the purpose of a comparison. & 2008 Published by Elsevier Ltd.
Keywords: Ultimate limit states (ULS) Ultimate strength Steel stiffened-plate structures Plate elements Stiffened panels Combined biaxial compression and lateral pressure loads Nonlinear finite element method Semi-analytical method
1. Introduction In recent years, it tends to be mandatory that limit state approaches are employed for design and strength assessment of ships and ship-shaped offshore structures [1–5]. Within the framework of limit states-based approaches, the primary task is to determine the ultimate strength. In this regard, substantial efforts have been directed toward the development of useful methodologies and guidelines to predict the limit states [6,7]. The present paper (Part II) is a sequel to the previous paper (Part I) dealing with ultimate limit states of steel plate elements surrounded by longitudinal stiffeners and transverse frames, which are the most basic structural component in ships and ship-shaped offshore installations. In contrast to Part I, the present paper is focused on ultimate limit states of stiffened panels surrounded by strong support members such as longitudinal girders and transverse frames. Some benchmark studies on the methods of ultimate strength computations for plate elements [8], stiffened panels [9] and ship’s hull girders [10] were previously reported by the authors. In the present study, some useful insights on the application of nonlinear finite element methods are developed using ANSYS Corresponding author. Tel.: +82 51 510 2429; fax: +82 51 512 8836.
E-mail address: [email protected] (J.K. Paik). 0263-8231/$ - see front matter & 2008 Published by Elsevier Ltd. doi:10.1016/j.tws.2008.08.006
code [11]. For the purpose of a comparison, the ALPS/ULSAP semianalytical method [12] is also employed. 2. Methods of ULS assessment The present paper adopts ANSYS nonlinear finite element method [11] for ultimate limit state assessment. ALPS/ULSAP semi-analytical method [12] is also used for the purpose of a comparison. 2.1. Nonlinear finite element method For ultimate limit state assessment, both geometric and material nonlinearities are taken into account, involving elastic– plastic large deflection behavior until and after the ultimate strength is reached. The elastic-perfectly plastic material model is adopted by neglecting the effect of strain-hardening. The details of structural modeling for nonlinear finite element analysis will be discussed in the later sections, while the plate-shell elements will be used for modeling of stiffened panels. 2.2. ALPS/ULSAP method ALPS/ULSAP method is based on semi-analytical approaches and provides ULS computations of plate elements and stiffened-plate
ARTICLE IN PRESS J.K. Paik, J.K. Seo / Thin-Walled Structures 47 (2009) 998–1007
999
Longitudinal girders
Stiffened panels Plating Transverse frames
Stiffeners Fig. 1. A stiffened-plate structure [6].
b
b
b
zp
t
A
N
t
A
N
hw
zp
tw
hw
t
A
N tw
hw
tf
tf
tw
zp
bf
bf
Fig. 2. Typical types of stiffeners: nomenclature: (a) flat bar, (b) angle bar and (c) Tee bar.
ns. Tra
or
flo
B
17
46
3x
8+
.8m
2x
172 mm 17 mm 8 mm 463 mm
a/2
m
17
17
a
(T)
a/2
Fig. 3. The stiffened panel studied in the present study.
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structures. Because the theoretical details of the ALPS/ULSAP method are found in [6,7], only a brief description is given herein for the ULS computations of stiffened panels. The collapse mode of a stiffened panel can be classified into the following six types, namely [6]
mode II: collapse of plating between stiffeners without failure of stiffeners,
mode III: beam-column type collapse of stiffeners with attached plating,
mode IV: local buckling of stiffener web after collapse of plating,
mode I: overall collapse mode,
mode V: lateral–torsional buckling of stiffeners after collapse of plating,
Table 1 Geometric properties of the stiffened panels considered in the present study Panel
a (mm)
B (mm)
b (mm)
ns
t (mm)
hw (mm)
tw (mm)
bf (mm)
tf (mm)
B1 B2 B3
4300 4300 4300
16,422 16,300 16,150
782 815 850
20 19 18
17.80 17.80 17.80
463 463 463
8 8 8
172 172 172
17 17 17
Note: ns ¼ number of longitudinal stiffeners.
mode VI: gross yielding of the entire panel. Analytical or semi-analytical solutions of panel ultimate strength for each of the six collapse modes noted above are derived. The real ultimate strength value of the panel is then the minimum value of ultimate strengths obtained from the six solutions. Various types of influencing parameters such as loading condition and initial imperfections as well as geometric and material properties are considered in the ULS computations.
C ns. Tra
or
flo
Lo
ngi
tud
B
ina
l ed
ge
A
a/2
z y
a/2
x a/2 a/2
A’ Fig. 4. The extent of the nonlinear finite element analysis.
σy σx
p = 0.16 MPa
σx σy
Fig. 5. A stiffened panel under combined biaxial compression and lateral pressure actions.
C’
ARTICLE IN PRESS J.K. Paik, J.K. Seo / Thin-Walled Structures 47 (2009) 998–1007
3. Structural modeling for nonlinear finite element analysis The target structure of the present study is a stiffened panel as shown in Fig. 1, where it is surrounded by longitudinal girders and transverse frames in contrast to Part I which dealt with plate elements (or plating) surrounded by longitudinal stiffeners and transverse frames. Such plate elements or stiffened panels are key strength members in deck or bottom parts of ships and shipshaped offshore installations.
3.1. Geometric and material properties The geometrical dimensions of plate panels are a (panel length), b (plate breadth or longitudinal spacing) and t (plate
1.0
ANSYS (p = 0.16 MPa), Longitudinal edges simply supported ALPS/ULSAP (p = 0.16 MPa)
0.8
0.6 .4: =0 x: σ y
0.4
σ
σyu/σYeq
0.6
C 0.2
0.0
σx, σy
σy
σx
thickness). The material properties are denoted by E (Young’s stress). The plate modulus), n (Poisson’s ratio) and sYp (yield pffiffiffiffiffiffiffiffiffiffiffiffiffi slenderness ratio is denoted by b ¼ ðb=tÞ sYp =E. Three types of stiffeners are usually relevant for building marine structures as shown in Fig. 2. zp is the distance from the plate bottom to the neutral axis of a plate–stiffener combination, i.e., single stiffener with attached plating. The material of stiffeners is often different from that of plating. For example, stiffeners are made of high-tensile steel, while plating is made of mild steel. In this regard, the yield stress of stiffeners can be defined with a different parameter of sYs, although Young’s modulus and Poisson’s ratio must be the same. A parameter sYeq is often used to represent an equivalent value of the material yield stress for both plating and stiffeners when they have different materials [6]. For a stiffened panel with identical material, sY ¼ sYeq ¼ sYp ¼ sYs. For an illustrative example, the present study adopts outer bottom stiffened-plate structures of 100,000 ton class double-hull oil tankers, where the geometry of the panel is a ¼ 4300 mm. The plate breadth or longitudinal stiffener spacing is varied at 782, 815 and 850 mm, because the number of longitudinal stiffeners with the type of Tee bar is varied at 18, 19 and 20. Therefore, the total breadth of the stiffened panel is subsequently varied at B ¼ 16,150, 16,300 and 16,422 mm. The plate thickness is t ¼ 17.8 mm. The stiffener height (hw) is 463 mm and the stiffener web thickness (tw) is 8 mm. The stiffener flange breadth (bf) is 172 mm and the stiffener flange thickness (tf) is 17 mm. Young’s modulus of the material is E ¼ 205.8 GPa, and Poisson’s ratio is n ¼ 0.3. The yield stress is sY ¼ sYeq ¼ sYp ¼ sYs ¼ 315 MPa for both plating and stiffeners. Fig. 3 shows the geometry of the stiffened panel considered in the present study. Table 1 summarizes the geometrical properties of the stiffened panels. 3.2. Initial distortions
B
A 0.0
0.2
0.4
0.4 0.6 σxu/σYeq
A→C A→B→C
0.8
1.0
It is well recognized that welding-induced initial imperfections significantly affect the ultimate strength behavior of stiffenedplate structures. Therefore, it is important to model the shape and magnitude of initial imperfections in a relevant way. In the present study, the following initial distortions are considered, although it is assumed that welding residual stress does not exist, namely:
C
0.3
Plate initial deflection (wopl) with the shape corresponding to
σy
σy/σYeq
0.2
0.1 σx 0
B
A
-0.1 -2
1001
0
2 4 6 Displacement (mm)
8
10
Fig. 6. (a) Loading path for a combination of longitudinal compression and transverse compression and (b) ultimate strength behavior with varying the loading path for a combination of longitudinal compression and transverse compression.
buckling mode due to a combination of biaxial compressive actions having the magnitude of b/200. Column-type initial distortions (woc) of stiffeners with the shape corresponding to buckling mode, due to a combination of biaxial compressive actions having the magnitude of a/1000. Sideways initial distortions (wos) of stiffeners with the shape corresponding to buckling mode due to a combination of biaxial compressive actions having the magnitude of a/1000.
For ALPS/ULSAP computations, the three factors of initial distortions, namely wopl, woc and wos are dealt with as parameters of influence, and thus they are simply defined as the data of input. For nonlinear finite element computations, however, the coordinates of nodal points in the finite element model must be reallocated in advance. Some useful procedures to assign the initial distortions of stiffened panels are as follows:
perform the buckling eigenvalue analysis for the target stiffened panel with the same finite element model, and find out the related buckling modes of plating and stiffeners, the
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Tr an
s.
flo
or
Fig. 7. Nonlinear finite element model developed for the outer bottom stiffened-plate structures.
0.5
ANSYS without lateral pressure (p = 0.0 MPa)
1.0
ANSYS with lateral pressure (p = 0.16 MPa)
0.8
0.3 σx/σYeq
σy/σYeq
ANSYS without lateral pressure (p = 0.0 MPa) ANSYS with lateral pressure (p = 0.16 MPa)
0.4
0.2 0.1
0.6 0.4 0.2 Loading ratio σx:σy = 0.79:0.21
Loading ratio σx:σy = 0.0:1.0
0.0 0.0
0.5
0.5
1.0 1.5 2.0 Strain (x10-3)
2.5
0.0
3.0
0.0
1.0
ANSYS without lateral pressure (p = 0.0 MPa)
0.8
0.3
0.6
σx/σYeq
σy/σYeq
2.0
ANSYS with lateral pressure (p = 0.16 MPa)
0.4
0.1
1.0 1.5 Strain (x10-3)
ANSYS without lateral pressure (p = 0.0 MPa)
ANSYS without lateral pressure (p = 0.16 MPa)
0.2
0.5
0.4 0.2
Loading ratio σx:σy = 0.4:0.6
0.0
Loading ratio σx:σy = 1.0:0.0
0.0 0.0
0.5
1.0 1.5 2.0 Strain (x10-3)
2.5
3.0
0.0
0.5
1.0 1.5 Strain(x10-3)
2.0
Fig. 8. The ultimate strength behavior of the bottom stiffened panel (Case B2) for various load combinations as obtained from ANSYS nonlinear FEA: (a) sx:sy ¼ 0.0:1.0, (b) sx:sy ¼ 0.4:0.6, (c) sx:sy ¼ 0.79:0.21 and (d) sx:sy ¼ 1.0:0.0.
ARTICLE IN PRESS J.K. Paik, J.K. Seo / Thin-Walled Structures 47 (2009) 998–1007
latter being necessary to consider two types, namely column (global) type and sideways; separate the buckling patterns (shapes) into three types, namely plate initial deflection, column-type initial distortions of stiffeners, and sideways initial distortions of stiffeners; amplify each of the initial distortion shapes up to the target value; superimpose the three types of amplified initial distortions altogether.
Instead of the eigenvalue analysis, an approximate approach using theoretical formulations is often utilized for each of the three initial-distortion patterns. 3.3. Boundary condition Because the boundary of stiffened panels is supported by strong members such as longitudinal girders and transverse
i ng Lo
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Longitudinal edge
e dg le ina d tu
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Longitudinal edge
e edg al n i tud
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Longitudinal edge
e edg al n i tud
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Tr an
s.
Tr an
s.
flo
or
flo o
r
Longitudinal edge Fig. 9. Deformed shapes and von Mises stress distributions of the bottom stiffened-plate structure (Case B2) at the ultimate limit state for various combinations of longitudinal and transverse compressive actions together with or without lateral pressure actions (amplification factor of 10). Simply supported at longitudinal edges, and (a) without and (b) with lateral pressure (sx:sy ¼ 1.0:0.0); (c) without and (d) with lateral pressure (sx:sy ¼ 0.79:0.21); (e) without and (f) with lateral pressure (sx:sy ¼ 0.4:0.6); (g) without and (h) with lateral pressure (sx:sy ¼ 0.0:1.0).
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Lo
Tr
an
s.
Tr flo
an
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s.
or
itu
fl o
a d in
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Longitudinal edge
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or
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Longitudinal edge
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Longitudinal edge
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Longitudinal edge Fig. 9. (Continued)
frames (or floors), the degree of rotational restraints at the panel boundary is never zero equivalent to simply supported condition, although it may neither be infinite corresponding to fixed (or clamped) condition. Therefore, it is important to model the panel edge condition in a relevant way. Nevertheless, the simply supported boundary condition (i.e., with zero rotational restraints) is often adopted in maritime industry when analytical or semi-analytical methods are applied for the practical design purpose with the benefit of mathematical simplicity [6,13]. On the other hand, refined nonlinear finite element method could resolve more accurately the issue of rotational restraints by
taking a wide extent of the finite element analysis including the support members, as shown in Fig. 4. It is seen from Fig. 4 that a half-length of the target panel in each side of the adjacent panels in the longitudinal direction has been included in the finite element model, although the extent of the target panel breadth in the transverse direction has been included. This is because by doing this the rotational restraints along transverse floors can be more realistically counted in the nonlinear finite element computations. Actually, it is desirable to expand the finite element model extent in the transverse direction to take account of the rotational restraints along longitudinal girders. However, the
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model size may become too large in this case, and most of all the effect of rotational restraints in short edges, i.e., along the longitudinal girders is supposed to be relatively small as long as longitudinal compressive actions are predominant at least. Furthermore, it is sometimes required to directly include the transverse support members (i.e., transverse floors) in the finite element mesh modeling, but for simplicity of the computations a simpler model in which no lateral deflection is allowed along the transverse support members is often adopted instead of a direct inclusion in the finite element mesh modeling of transverse support members. The following are the boundary conditions of the nonlinear finite element model extending with the two ð12 þ 1 þ 12Þ bay in the longitudinal direction, namely (see Fig. 4)
at boundaries A–C and A0 –C0 : edges along longitudinal girders are modeled to be simply supported, i.e., T[1, 1, 0], R[1, 0, 0], each edge having equal y-displacement; Table 2 Ultimate strength computations for the bottom-stiffened panel (Case B2) with varying the biaxial compressive loading ratio with or without lateral pressure actions Loading ratio, sx:sy
p (MPa)
ANSYS
ALPS/ULSAP
BC
sxu/sY
syu/sY
sxu/sY
syu/sY
1.0:0.0
0 0.16
LS LS
0.8139 0.7488
0 0
0.7920 0.7650
0 0
0.79:0.21
0 0.16
LS LS
0.7692 0.7121
0.2045 0.1893
0.6882 0.6328
0.1868 0.1718
0.4:0.6
0 0.16
LS LS
0.2289 0.2121
0.3344 0.3179
0.2410 0.1969
0.3615 0.2953
0.0:1.0
0 0.16
LS LS
0 0
0.3496 0.3319
0 0
0.3795 0.3056
Note: BC ¼ boundary condition at longitudinal edges, LS ¼ longitudinal edges simply supported.
1005
at transverse frame (floor) intersections: T[1, 1, 0] at plate nodes and T[1, 0, 1] at stiffener web nodes;
at boundaries A–A0 and C–C0 : symmetric conditions with R[1, 0, 0] at all plate nodes and stiffener nodes having equal xdisplacement for the present (illustrative) panel with an odd number of buckling half-wave, i.e., m ¼ 15, in the panel length (X) direction. For a panel with an even number of buckling half-wave, however, the straight condition (or equal xdisplacement) at plate nodes may only be applied. where T[x, y, z] indicates translational constraints and R[x, y, z] indicates rotational constraints about x-, y-, and z-coordinates. Respectively a ‘‘0’’ indicates constraint and a ‘‘1’’ indicates no constraint. 3.4. Loading condition The most vulnerable loading condition of the plate panels in outer bottom stiffened-plate structures is biaxial compression and lateral pressure actions, as shown in Fig. 5. Because the present study considers outer bottom stiffened-plate structures of a 100,000 ton class double-hull oil tanker, the magnitude (p) of lateral pressure actions is taken as p ¼ 0.16 MPa in extreme (design) condition. For nonlinear finite element computations, the lateral pressure actions are first applied, and keeping the lateral pressure actions constant a combination of biaxial compressive actions is applied. As discussed in Part I of the present study [14], the shape and magnitude of initial distortions in plate panels can be changed after application of lateral pressure actions and subsequently the ultimate strength behavior can be affected by the characteristics of lateral pressure actions. It is also considered that the panel ultimate strength behavior can be affected by the loading path or order for a combination of longitudinal compression and transverse compression. Fig. 6 shows the ultimate strength behavior with varying the loading path, i.e., (a) A-C and (b) A-B-C when the ratio of sx versus sy is 0.4 versus 0.6. In case (a), the loading ratio between sx and sy is kept constant during the ultimate strength computations, and in
1.0 ANSYS (p = 0.0 MPa), Longitudinal edges simply supported ANSYS (p = 0.16 MPa), Longitudinal edges simply supported ALPS/ULSAP (p = 0.0 MPa) ALPS/ULSAP (p = 0.16 MPa)
0.8
σyu/σYeq
0.6
=0 .4
:0
.6
0.4
= σ x:σ y
σ
x
:σ
y
0.2
: 0.79
0.21
0.0 0.0
0.2
0.4
0.6
0.8
1.0
σxu/σYeq Fig. 10. Ultimate strength interaction relationships of the panel B2 with or without lateral pressure actions.
Fig. 11. Comparison of the ultimate strength behavior for the three panel Cases B1–B3 under biaxial compression and lateral pressure actions.
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case (b) longitudinal compression (sx) is applied first and then transverse compression (sy) is applied. It is seen from Fig. 6(b) that the loading path affects the ultimate strength behavior before or after the panel reaches the ultimate limit state, but its effect is small on ultimate strength value itself. In this regard, the constant loading ratio approach is often adopted for practical purpose of the nonlinear finite element computations. This is simply due to the matter of simplicity, but the present study also adopts this approach with the focus on ULS computations.
3.5. Nonlinear finite element mesh modeling Fig. 7 shows the nonlinear finite element model for analyzing the ultimate strength behavior of the outer bottom stiffened-plate structures. Plate-shell elements are used for plating, stiffener web and stiffener flange. The number of plate-shell elements is 10 for plating between stiffeners in the plate breadth direction, 6 for stiffener web in the web height direction and 4 for stiffener flange in the flange breadth direction. The number of finite elements in
the panel length direction is allocated so that the element aspect ratio becomes almost unity.
4. Numerical computations and discussions Fig. 8 shows the ultimate strength behaviors of the stiffenedplate structures of Case B2 under different loading conditions as obtained by nonlinear finite element computations. Fig. 9 shows the deformed shapes and von Mises stress distributions of the stiffened-plate structure for Case B2 at the ultimate limit state for various combinations of longitudinal and transverse compressive actions together with or without lateral pressure actions. Table 2 summarizes the ultimate strength computations obtained by ANSYS nonlinear FEA and ALPS/ULSAP methods. Fig. 10 plots the ultimate strength interaction relations of the panel B2. It is found that the lateral pressure actions reduce the panel ultimate strength and ALPS/ULSAP solutions are in good agreement with more refined nonlinear finite element computations. Similar computations were undertaken for Cases B1 and B3, and the results were compared with Case B2. Fig. 11 compares the
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flo
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dg
e
dg
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Longitudinal edge
Lo
Tr
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s.
Tr fl o
an
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s.
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Longitudinal edge
Lo
Tr
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s.
Tr flo
or
an
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s.
itu
flo
a d in
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Longitudinal edge Fig. 12. Deformed shapes and von Mises stress distributions of the bottom stiffened-plate structure (Cases B1–B3) at the ultimate limit states under biaxial compression and lateral pressure actions (amplification factor of 10). Simply supported at longitudinal edges, and with lateral pressure (sx:sy ¼ 0.79:0.21) for (a) Case B1, (b) Case B2 and (c) Case B3.
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Table 3 Ultimate strength computations for the bottom-stiffened panel under biaxial compression and lateral pressure actions keeping sx:sy ¼ 0.79:0.21 Case
ANSYS
B1 B2 B3
ALPS/ULSAP
sxu/sY
syu /sY
sxu/sY
syu/sY
Collapse mode
0.7201 0.7121 0.6787
0.1914 0.1893 0.1804
0.6568 0.6369 0.6157
0.1746 0.1693 0.1637
IV IV IV
(3)
1.0 Bottom Stiffened Panels FEA (Transverse floor spacing = 4300mm) ALPS/ULSAP (Transverse floor spacing = 4300mm)
0.9
(4)
(5)
0.8 σxu/σY
(6)
1007
members are also included in the finite element mesh modeling, although a simpler model with zero lateral deflection along the transverse support members is often adopted instead of their direct inclusion in the finite element model as long as the transverse support members are supposed to be strong enough. The effect of loading path in association with biaxial compressive actions is negligibly small in terms of ultimate strength itself. However, it is certain that the ultimate strength behavior is significantly affected by the loading path during the load application before and after the ultimate strength is reached. Some further studies are recommended in this regard. Lateral pressure actions reduce the panel ultimate strength under predominantly compressive actions as would be expected. ALPS/ULSAP method solutions well correlate to more refined ANSYS nonlinear finite element computations of the panel ultimate strength. The insights developed from the present study will be useful for nonlinear finite element method applications to ULS assessment of steel stiffened-plate structures.
0.7 Acknowledgments
0.6
0.5 760 770 780 790 800 810 820 830 840 850 860 870 Longitudinal stiffener spacing (mm) Fig. 13. Variation of panel ultimate strength as a function of longitudinal stiffener spacing when sx:sy ¼ 0.79:0.21 with p ¼ 0.16 MPa.
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 Science and Engineering Foundation (Grant no. ROA-2006000-10239-0). The author is pleased to acknowledge the support of Lloyd’s Register Educational Trust (LRET) via the LRET Research Centre of Excellence at Pusan National University. References
ultimate strength behavior of three panels when the loading ratio is sx:sy ¼ 0.79:0.21. Fig. 12 compares the deformed shapes and von Mises stress distributions for the three panel cases. It is found that all of the three panels collapsed with the same mode, i.e., by mode IV due to local buckling of stiffener web. Table 3 summarizes the ultimate computations and Fig. 13 plots the variation of the panel ultimate strength as a function of longitudinal stiffener spacing. The panel ultimate strength tends to decrease with increase in the longitudinal stiffener spacing in this specific panel cases. 5. Concluding remarks The aim of the present study has been to develop some useful insights on nonlinear finite element method applications for ULS assessment of steel stiffened-plate structures under biaxial compression and lateral pressure actions. In contrast to Part I dealing with plate elements, the present paper (Part II) was focused on stiffened panels. Based on the results obtained from the present study, the following conclusions can be drawn: (1) It is considered that the two ð12 þ 1 þ 12Þ bay model in the panel length direction gives reasonable solutions of nonlinear finite element computations although one bay model is taken as the extent of the analysis in the panel breadth direction. (2) When the bending rigidity of transverse support members is relatively small, it may be required that the transverse support
[1] IMO. Goal-based standards. London: International Maritime Organization; 2006. [2] ISO 18072-1. Ships and marine technology—ship structures—Part 1: general requirements for their limit state assessment. Geneva: International Organization for Standardization; 2007. [3] ISO/CD 18072-2. Ships and marine technology—ship structures—Part 2: requirements for their ultimate limit state assessment. Geneva: International Organization for Standardization; 2006. [4] IACS. Common structural rules for double hull oil tankers. London: International Association of Classification Societies; 2006. [5] IACS. Common structural rules for bulk carriers. London: International Association of Classification Societies; 2006. [6] Paik JK, Thayamballi AK. Ultimate limit state design of steel-plated structures. Chichester, UK: Wiley; 2003. [7] Paik JK, Thayamballi AK. Ship-shaped offshore installations: design, building, and operation. Cambridge, UK: Cambridge University Press; 2007. [8] Paik JK, Kim BJ, Seo JK. Methods for ultimate limit state assessment of ships and ship-shaped offshore structures: Part I—unstiffened plates. Ocean Eng 2008;35:261–70. [9] Paik JK, Kim BJ, Seo JK. Methods for ultimate limit state assessment of ships and ship-shaped offshore structures: Part II stiffened panels. Ocean Eng 2008;35:271–80. [10] Paik JK, Kim BJ, Seo JK. Methods for ultimate limit state assessment of ships and ship-shaped offshore structures: Part III hull girders. Ocean Eng 2008;35: 281–6. [11] ANSYS. User’s manual (version 10.0). Houston: Swanson Analysis Systems Inc.; 2007. [12] ALPS/ULSAP, A computer program for ultimate limit state assessment for stiffened panels (version 2008.1). Ship and Offshore Structural Mechanics Laboratory, Pusan National University, Korea, 2008. [13] Paik JK, Thayamballi AK. Buckling strength of steel plating with elastically restrained edges. Thin-Walled Struct 2000;37:27–55. [14] Paik JK, Seo JK. Nonlinear finite element method models for ultimate strength analysis of steel stiffened-plate structures under combined biaxial compression and lateral pressure actions—Part I: Plate elements. Thin-Walled Struct 2008, this issue, doi:10.1016/j.tws.2008.08.005.
Contents lists available at ScienceDirect
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Nonlinear finite element method models for ultimate strength analysis of steel stiffened-plate structures under combined biaxial compression and lateral pressure actions—Part II: Stiffened panels Jeom Kee Paik , Jung Kwan Seo Department of Naval Architecture and Ocean Engineering, LRET Research Centre of Excellence, Pusan National University, Busan 609-735, Republic of Korea
a r t i c l e in f o
a b s t r a c t
Available online 9 September 2008
The present paper (Part II) is a sequel to the previous paper (Part I) [Paik JK, Seo JK. Nonlinear finite element method models for ultimate strength analysis of steel stiffened-plate structures under combined biaxial compression and lateral pressure actions—Part I: Plate elements. Thin-Walled Struct 2008, this issue, doi:10.1016/j.tws.2008.08.005.] on the application of nonlinear finite element methods for ultimate strength analysis of steel stiffened-plate structures under combined biaxial compression and lateral pressure actions. In contrast to Part I dealing with plate elements, the present paper (Part II) treats stiffened panels surrounded by strong support members such as longitudinal girders and transverse frames. In similar to Part I, some important factors of influence such as structural dimensions, initial imperfections, loading types and computational techniques in association with ultimate limit states are studied. Some useful insights in terms of nonlinear finite element method modeling are developed using ANSYS code together with the ALPS/ULSAP semi-analytical method, the latter being for the purpose of a comparison. & 2008 Published by Elsevier Ltd.
Keywords: Ultimate limit states (ULS) Ultimate strength Steel stiffened-plate structures Plate elements Stiffened panels Combined biaxial compression and lateral pressure loads Nonlinear finite element method Semi-analytical method
1. Introduction In recent years, it tends to be mandatory that limit state approaches are employed for design and strength assessment of ships and ship-shaped offshore structures [1–5]. Within the framework of limit states-based approaches, the primary task is to determine the ultimate strength. In this regard, substantial efforts have been directed toward the development of useful methodologies and guidelines to predict the limit states [6,7]. The present paper (Part II) is a sequel to the previous paper (Part I) dealing with ultimate limit states of steel plate elements surrounded by longitudinal stiffeners and transverse frames, which are the most basic structural component in ships and ship-shaped offshore installations. In contrast to Part I, the present paper is focused on ultimate limit states of stiffened panels surrounded by strong support members such as longitudinal girders and transverse frames. Some benchmark studies on the methods of ultimate strength computations for plate elements [8], stiffened panels [9] and ship’s hull girders [10] were previously reported by the authors. In the present study, some useful insights on the application of nonlinear finite element methods are developed using ANSYS Corresponding author. Tel.: +82 51 510 2429; fax: +82 51 512 8836.
E-mail address: [email protected] (J.K. Paik). 0263-8231/$ - see front matter & 2008 Published by Elsevier Ltd. doi:10.1016/j.tws.2008.08.006
code [11]. For the purpose of a comparison, the ALPS/ULSAP semianalytical method [12] is also employed. 2. Methods of ULS assessment The present paper adopts ANSYS nonlinear finite element method [11] for ultimate limit state assessment. ALPS/ULSAP semi-analytical method [12] is also used for the purpose of a comparison. 2.1. Nonlinear finite element method For ultimate limit state assessment, both geometric and material nonlinearities are taken into account, involving elastic– plastic large deflection behavior until and after the ultimate strength is reached. The elastic-perfectly plastic material model is adopted by neglecting the effect of strain-hardening. The details of structural modeling for nonlinear finite element analysis will be discussed in the later sections, while the plate-shell elements will be used for modeling of stiffened panels. 2.2. ALPS/ULSAP method ALPS/ULSAP method is based on semi-analytical approaches and provides ULS computations of plate elements and stiffened-plate
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999
Longitudinal girders
Stiffened panels Plating Transverse frames
Stiffeners Fig. 1. A stiffened-plate structure [6].
b
b
b
zp
t
A
N
t
A
N
hw
zp
tw
hw
t
A
N tw
hw
tf
tf
tw
zp
bf
bf
Fig. 2. Typical types of stiffeners: nomenclature: (a) flat bar, (b) angle bar and (c) Tee bar.
ns. Tra
or
flo
B
17
46
3x
8+
.8m
2x
172 mm 17 mm 8 mm 463 mm
a/2
m
17
17
a
(T)
a/2
Fig. 3. The stiffened panel studied in the present study.
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structures. Because the theoretical details of the ALPS/ULSAP method are found in [6,7], only a brief description is given herein for the ULS computations of stiffened panels. The collapse mode of a stiffened panel can be classified into the following six types, namely [6]
mode II: collapse of plating between stiffeners without failure of stiffeners,
mode III: beam-column type collapse of stiffeners with attached plating,
mode IV: local buckling of stiffener web after collapse of plating,
mode I: overall collapse mode,
mode V: lateral–torsional buckling of stiffeners after collapse of plating,
Table 1 Geometric properties of the stiffened panels considered in the present study Panel
a (mm)
B (mm)
b (mm)
ns
t (mm)
hw (mm)
tw (mm)
bf (mm)
tf (mm)
B1 B2 B3
4300 4300 4300
16,422 16,300 16,150
782 815 850
20 19 18
17.80 17.80 17.80
463 463 463
8 8 8
172 172 172
17 17 17
Note: ns ¼ number of longitudinal stiffeners.
mode VI: gross yielding of the entire panel. Analytical or semi-analytical solutions of panel ultimate strength for each of the six collapse modes noted above are derived. The real ultimate strength value of the panel is then the minimum value of ultimate strengths obtained from the six solutions. Various types of influencing parameters such as loading condition and initial imperfections as well as geometric and material properties are considered in the ULS computations.
C ns. Tra
or
flo
Lo
ngi
tud
B
ina
l ed
ge
A
a/2
z y
a/2
x a/2 a/2
A’ Fig. 4. The extent of the nonlinear finite element analysis.
σy σx
p = 0.16 MPa
σx σy
Fig. 5. A stiffened panel under combined biaxial compression and lateral pressure actions.
C’
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3. Structural modeling for nonlinear finite element analysis The target structure of the present study is a stiffened panel as shown in Fig. 1, where it is surrounded by longitudinal girders and transverse frames in contrast to Part I which dealt with plate elements (or plating) surrounded by longitudinal stiffeners and transverse frames. Such plate elements or stiffened panels are key strength members in deck or bottom parts of ships and shipshaped offshore installations.
3.1. Geometric and material properties The geometrical dimensions of plate panels are a (panel length), b (plate breadth or longitudinal spacing) and t (plate
1.0
ANSYS (p = 0.16 MPa), Longitudinal edges simply supported ALPS/ULSAP (p = 0.16 MPa)
0.8
0.6 .4: =0 x: σ y
0.4
σ
σyu/σYeq
0.6
C 0.2
0.0
σx, σy
σy
σx
thickness). The material properties are denoted by E (Young’s stress). The plate modulus), n (Poisson’s ratio) and sYp (yield pffiffiffiffiffiffiffiffiffiffiffiffiffi slenderness ratio is denoted by b ¼ ðb=tÞ sYp =E. Three types of stiffeners are usually relevant for building marine structures as shown in Fig. 2. zp is the distance from the plate bottom to the neutral axis of a plate–stiffener combination, i.e., single stiffener with attached plating. The material of stiffeners is often different from that of plating. For example, stiffeners are made of high-tensile steel, while plating is made of mild steel. In this regard, the yield stress of stiffeners can be defined with a different parameter of sYs, although Young’s modulus and Poisson’s ratio must be the same. A parameter sYeq is often used to represent an equivalent value of the material yield stress for both plating and stiffeners when they have different materials [6]. For a stiffened panel with identical material, sY ¼ sYeq ¼ sYp ¼ sYs. For an illustrative example, the present study adopts outer bottom stiffened-plate structures of 100,000 ton class double-hull oil tankers, where the geometry of the panel is a ¼ 4300 mm. The plate breadth or longitudinal stiffener spacing is varied at 782, 815 and 850 mm, because the number of longitudinal stiffeners with the type of Tee bar is varied at 18, 19 and 20. Therefore, the total breadth of the stiffened panel is subsequently varied at B ¼ 16,150, 16,300 and 16,422 mm. The plate thickness is t ¼ 17.8 mm. The stiffener height (hw) is 463 mm and the stiffener web thickness (tw) is 8 mm. The stiffener flange breadth (bf) is 172 mm and the stiffener flange thickness (tf) is 17 mm. Young’s modulus of the material is E ¼ 205.8 GPa, and Poisson’s ratio is n ¼ 0.3. The yield stress is sY ¼ sYeq ¼ sYp ¼ sYs ¼ 315 MPa for both plating and stiffeners. Fig. 3 shows the geometry of the stiffened panel considered in the present study. Table 1 summarizes the geometrical properties of the stiffened panels. 3.2. Initial distortions
B
A 0.0
0.2
0.4
0.4 0.6 σxu/σYeq
A→C A→B→C
0.8
1.0
It is well recognized that welding-induced initial imperfections significantly affect the ultimate strength behavior of stiffenedplate structures. Therefore, it is important to model the shape and magnitude of initial imperfections in a relevant way. In the present study, the following initial distortions are considered, although it is assumed that welding residual stress does not exist, namely:
C
0.3
Plate initial deflection (wopl) with the shape corresponding to
σy
σy/σYeq
0.2
0.1 σx 0
B
A
-0.1 -2
1001
0
2 4 6 Displacement (mm)
8
10
Fig. 6. (a) Loading path for a combination of longitudinal compression and transverse compression and (b) ultimate strength behavior with varying the loading path for a combination of longitudinal compression and transverse compression.
buckling mode due to a combination of biaxial compressive actions having the magnitude of b/200. Column-type initial distortions (woc) of stiffeners with the shape corresponding to buckling mode, due to a combination of biaxial compressive actions having the magnitude of a/1000. Sideways initial distortions (wos) of stiffeners with the shape corresponding to buckling mode due to a combination of biaxial compressive actions having the magnitude of a/1000.
For ALPS/ULSAP computations, the three factors of initial distortions, namely wopl, woc and wos are dealt with as parameters of influence, and thus they are simply defined as the data of input. For nonlinear finite element computations, however, the coordinates of nodal points in the finite element model must be reallocated in advance. Some useful procedures to assign the initial distortions of stiffened panels are as follows:
perform the buckling eigenvalue analysis for the target stiffened panel with the same finite element model, and find out the related buckling modes of plating and stiffeners, the
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Tr an
s.
flo
or
Fig. 7. Nonlinear finite element model developed for the outer bottom stiffened-plate structures.
0.5
ANSYS without lateral pressure (p = 0.0 MPa)
1.0
ANSYS with lateral pressure (p = 0.16 MPa)
0.8
0.3 σx/σYeq
σy/σYeq
ANSYS without lateral pressure (p = 0.0 MPa) ANSYS with lateral pressure (p = 0.16 MPa)
0.4
0.2 0.1
0.6 0.4 0.2 Loading ratio σx:σy = 0.79:0.21
Loading ratio σx:σy = 0.0:1.0
0.0 0.0
0.5
0.5
1.0 1.5 2.0 Strain (x10-3)
2.5
0.0
3.0
0.0
1.0
ANSYS without lateral pressure (p = 0.0 MPa)
0.8
0.3
0.6
σx/σYeq
σy/σYeq
2.0
ANSYS with lateral pressure (p = 0.16 MPa)
0.4
0.1
1.0 1.5 Strain (x10-3)
ANSYS without lateral pressure (p = 0.0 MPa)
ANSYS without lateral pressure (p = 0.16 MPa)
0.2
0.5
0.4 0.2
Loading ratio σx:σy = 0.4:0.6
0.0
Loading ratio σx:σy = 1.0:0.0
0.0 0.0
0.5
1.0 1.5 2.0 Strain (x10-3)
2.5
3.0
0.0
0.5
1.0 1.5 Strain(x10-3)
2.0
Fig. 8. The ultimate strength behavior of the bottom stiffened panel (Case B2) for various load combinations as obtained from ANSYS nonlinear FEA: (a) sx:sy ¼ 0.0:1.0, (b) sx:sy ¼ 0.4:0.6, (c) sx:sy ¼ 0.79:0.21 and (d) sx:sy ¼ 1.0:0.0.
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latter being necessary to consider two types, namely column (global) type and sideways; separate the buckling patterns (shapes) into three types, namely plate initial deflection, column-type initial distortions of stiffeners, and sideways initial distortions of stiffeners; amplify each of the initial distortion shapes up to the target value; superimpose the three types of amplified initial distortions altogether.
Instead of the eigenvalue analysis, an approximate approach using theoretical formulations is often utilized for each of the three initial-distortion patterns. 3.3. Boundary condition Because the boundary of stiffened panels is supported by strong members such as longitudinal girders and transverse
i ng Lo
Tr an
s.
1003
l ina tud
e edg
Tr an
s. flo o
flo
or
r
Longitudinal edge
e dg le ina d tu
i ng Lo
Tr an
s.
Tr an
s.
flo
or
flo o
r
Longitudinal edge
e edg al n i tud
i ng Lo
Tr an
s.
Tr an
s. flo o
flo
or
r
Longitudinal edge
e edg al n i tud
i ng Lo
Tr an
s.
Tr an
s.
flo
or
flo o
r
Longitudinal edge Fig. 9. Deformed shapes and von Mises stress distributions of the bottom stiffened-plate structure (Case B2) at the ultimate limit state for various combinations of longitudinal and transverse compressive actions together with or without lateral pressure actions (amplification factor of 10). Simply supported at longitudinal edges, and (a) without and (b) with lateral pressure (sx:sy ¼ 1.0:0.0); (c) without and (d) with lateral pressure (sx:sy ¼ 0.79:0.21); (e) without and (f) with lateral pressure (sx:sy ¼ 0.4:0.6); (g) without and (h) with lateral pressure (sx:sy ¼ 0.0:1.0).
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Lo
Tr
an
s.
Tr flo
an
ng
s.
or
itu
fl o
a d in
le
dg
e
or
Longitudinal edge
Lo
Tr
an
s.
Tr flo
an
ng
s.
or
itu
flo
d
l ina
ed
ge
or
Longitudinal edge
Lo
Tr
an
s.
Tr flo
an
ng
s.
or
itu
flo
a d in
le
dg
e
or
Longitudinal edge
Lo
Tr
an
s.
Tr flo
an
or
ng
s.
itu
flo
d
l ina
ed
ge
or
Longitudinal edge Fig. 9. (Continued)
frames (or floors), the degree of rotational restraints at the panel boundary is never zero equivalent to simply supported condition, although it may neither be infinite corresponding to fixed (or clamped) condition. Therefore, it is important to model the panel edge condition in a relevant way. Nevertheless, the simply supported boundary condition (i.e., with zero rotational restraints) is often adopted in maritime industry when analytical or semi-analytical methods are applied for the practical design purpose with the benefit of mathematical simplicity [6,13]. On the other hand, refined nonlinear finite element method could resolve more accurately the issue of rotational restraints by
taking a wide extent of the finite element analysis including the support members, as shown in Fig. 4. It is seen from Fig. 4 that a half-length of the target panel in each side of the adjacent panels in the longitudinal direction has been included in the finite element model, although the extent of the target panel breadth in the transverse direction has been included. This is because by doing this the rotational restraints along transverse floors can be more realistically counted in the nonlinear finite element computations. Actually, it is desirable to expand the finite element model extent in the transverse direction to take account of the rotational restraints along longitudinal girders. However, the
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model size may become too large in this case, and most of all the effect of rotational restraints in short edges, i.e., along the longitudinal girders is supposed to be relatively small as long as longitudinal compressive actions are predominant at least. Furthermore, it is sometimes required to directly include the transverse support members (i.e., transverse floors) in the finite element mesh modeling, but for simplicity of the computations a simpler model in which no lateral deflection is allowed along the transverse support members is often adopted instead of a direct inclusion in the finite element mesh modeling of transverse support members. The following are the boundary conditions of the nonlinear finite element model extending with the two ð12 þ 1 þ 12Þ bay in the longitudinal direction, namely (see Fig. 4)
at boundaries A–C and A0 –C0 : edges along longitudinal girders are modeled to be simply supported, i.e., T[1, 1, 0], R[1, 0, 0], each edge having equal y-displacement; Table 2 Ultimate strength computations for the bottom-stiffened panel (Case B2) with varying the biaxial compressive loading ratio with or without lateral pressure actions Loading ratio, sx:sy
p (MPa)
ANSYS
ALPS/ULSAP
BC
sxu/sY
syu/sY
sxu/sY
syu/sY
1.0:0.0
0 0.16
LS LS
0.8139 0.7488
0 0
0.7920 0.7650
0 0
0.79:0.21
0 0.16
LS LS
0.7692 0.7121
0.2045 0.1893
0.6882 0.6328
0.1868 0.1718
0.4:0.6
0 0.16
LS LS
0.2289 0.2121
0.3344 0.3179
0.2410 0.1969
0.3615 0.2953
0.0:1.0
0 0.16
LS LS
0 0
0.3496 0.3319
0 0
0.3795 0.3056
Note: BC ¼ boundary condition at longitudinal edges, LS ¼ longitudinal edges simply supported.
1005
at transverse frame (floor) intersections: T[1, 1, 0] at plate nodes and T[1, 0, 1] at stiffener web nodes;
at boundaries A–A0 and C–C0 : symmetric conditions with R[1, 0, 0] at all plate nodes and stiffener nodes having equal xdisplacement for the present (illustrative) panel with an odd number of buckling half-wave, i.e., m ¼ 15, in the panel length (X) direction. For a panel with an even number of buckling half-wave, however, the straight condition (or equal xdisplacement) at plate nodes may only be applied. where T[x, y, z] indicates translational constraints and R[x, y, z] indicates rotational constraints about x-, y-, and z-coordinates. Respectively a ‘‘0’’ indicates constraint and a ‘‘1’’ indicates no constraint. 3.4. Loading condition The most vulnerable loading condition of the plate panels in outer bottom stiffened-plate structures is biaxial compression and lateral pressure actions, as shown in Fig. 5. Because the present study considers outer bottom stiffened-plate structures of a 100,000 ton class double-hull oil tanker, the magnitude (p) of lateral pressure actions is taken as p ¼ 0.16 MPa in extreme (design) condition. For nonlinear finite element computations, the lateral pressure actions are first applied, and keeping the lateral pressure actions constant a combination of biaxial compressive actions is applied. As discussed in Part I of the present study [14], the shape and magnitude of initial distortions in plate panels can be changed after application of lateral pressure actions and subsequently the ultimate strength behavior can be affected by the characteristics of lateral pressure actions. It is also considered that the panel ultimate strength behavior can be affected by the loading path or order for a combination of longitudinal compression and transverse compression. Fig. 6 shows the ultimate strength behavior with varying the loading path, i.e., (a) A-C and (b) A-B-C when the ratio of sx versus sy is 0.4 versus 0.6. In case (a), the loading ratio between sx and sy is kept constant during the ultimate strength computations, and in
1.0 ANSYS (p = 0.0 MPa), Longitudinal edges simply supported ANSYS (p = 0.16 MPa), Longitudinal edges simply supported ALPS/ULSAP (p = 0.0 MPa) ALPS/ULSAP (p = 0.16 MPa)
0.8
σyu/σYeq
0.6
=0 .4
:0
.6
0.4
= σ x:σ y
σ
x
:σ
y
0.2
: 0.79
0.21
0.0 0.0
0.2
0.4
0.6
0.8
1.0
σxu/σYeq Fig. 10. Ultimate strength interaction relationships of the panel B2 with or without lateral pressure actions.
Fig. 11. Comparison of the ultimate strength behavior for the three panel Cases B1–B3 under biaxial compression and lateral pressure actions.
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case (b) longitudinal compression (sx) is applied first and then transverse compression (sy) is applied. It is seen from Fig. 6(b) that the loading path affects the ultimate strength behavior before or after the panel reaches the ultimate limit state, but its effect is small on ultimate strength value itself. In this regard, the constant loading ratio approach is often adopted for practical purpose of the nonlinear finite element computations. This is simply due to the matter of simplicity, but the present study also adopts this approach with the focus on ULS computations.
3.5. Nonlinear finite element mesh modeling Fig. 7 shows the nonlinear finite element model for analyzing the ultimate strength behavior of the outer bottom stiffened-plate structures. Plate-shell elements are used for plating, stiffener web and stiffener flange. The number of plate-shell elements is 10 for plating between stiffeners in the plate breadth direction, 6 for stiffener web in the web height direction and 4 for stiffener flange in the flange breadth direction. The number of finite elements in
the panel length direction is allocated so that the element aspect ratio becomes almost unity.
4. Numerical computations and discussions Fig. 8 shows the ultimate strength behaviors of the stiffenedplate structures of Case B2 under different loading conditions as obtained by nonlinear finite element computations. Fig. 9 shows the deformed shapes and von Mises stress distributions of the stiffened-plate structure for Case B2 at the ultimate limit state for various combinations of longitudinal and transverse compressive actions together with or without lateral pressure actions. Table 2 summarizes the ultimate strength computations obtained by ANSYS nonlinear FEA and ALPS/ULSAP methods. Fig. 10 plots the ultimate strength interaction relations of the panel B2. It is found that the lateral pressure actions reduce the panel ultimate strength and ALPS/ULSAP solutions are in good agreement with more refined nonlinear finite element computations. Similar computations were undertaken for Cases B1 and B3, and the results were compared with Case B2. Fig. 11 compares the
Lo
Tr
an
s.
Tr flo
an
ng
s.
or
itu
flo
a d in
le
dg
e
dg
e
dg
e
or
Longitudinal edge
Lo
Tr
an
s.
Tr fl o
an
ng
s.
or
itu
fl o
a d in
le
or
Longitudinal edge
Lo
Tr
an
s.
Tr flo
or
an
ng
s.
itu
flo
a d in
le
or
Longitudinal edge Fig. 12. Deformed shapes and von Mises stress distributions of the bottom stiffened-plate structure (Cases B1–B3) at the ultimate limit states under biaxial compression and lateral pressure actions (amplification factor of 10). Simply supported at longitudinal edges, and with lateral pressure (sx:sy ¼ 0.79:0.21) for (a) Case B1, (b) Case B2 and (c) Case B3.
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Table 3 Ultimate strength computations for the bottom-stiffened panel under biaxial compression and lateral pressure actions keeping sx:sy ¼ 0.79:0.21 Case
ANSYS
B1 B2 B3
ALPS/ULSAP
sxu/sY
syu /sY
sxu/sY
syu/sY
Collapse mode
0.7201 0.7121 0.6787
0.1914 0.1893 0.1804
0.6568 0.6369 0.6157
0.1746 0.1693 0.1637
IV IV IV
(3)
1.0 Bottom Stiffened Panels FEA (Transverse floor spacing = 4300mm) ALPS/ULSAP (Transverse floor spacing = 4300mm)
0.9
(4)
(5)
0.8 σxu/σY
(6)
1007
members are also included in the finite element mesh modeling, although a simpler model with zero lateral deflection along the transverse support members is often adopted instead of their direct inclusion in the finite element model as long as the transverse support members are supposed to be strong enough. The effect of loading path in association with biaxial compressive actions is negligibly small in terms of ultimate strength itself. However, it is certain that the ultimate strength behavior is significantly affected by the loading path during the load application before and after the ultimate strength is reached. Some further studies are recommended in this regard. Lateral pressure actions reduce the panel ultimate strength under predominantly compressive actions as would be expected. ALPS/ULSAP method solutions well correlate to more refined ANSYS nonlinear finite element computations of the panel ultimate strength. The insights developed from the present study will be useful for nonlinear finite element method applications to ULS assessment of steel stiffened-plate structures.
0.7 Acknowledgments
0.6
0.5 760 770 780 790 800 810 820 830 840 850 860 870 Longitudinal stiffener spacing (mm) Fig. 13. Variation of panel ultimate strength as a function of longitudinal stiffener spacing when sx:sy ¼ 0.79:0.21 with p ¼ 0.16 MPa.
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 Science and Engineering Foundation (Grant no. ROA-2006000-10239-0). The author is pleased to acknowledge the support of Lloyd’s Register Educational Trust (LRET) via the LRET Research Centre of Excellence at Pusan National University. References
ultimate strength behavior of three panels when the loading ratio is sx:sy ¼ 0.79:0.21. Fig. 12 compares the deformed shapes and von Mises stress distributions for the three panel cases. It is found that all of the three panels collapsed with the same mode, i.e., by mode IV due to local buckling of stiffener web. Table 3 summarizes the ultimate computations and Fig. 13 plots the variation of the panel ultimate strength as a function of longitudinal stiffener spacing. The panel ultimate strength tends to decrease with increase in the longitudinal stiffener spacing in this specific panel cases. 5. Concluding remarks The aim of the present study has been to develop some useful insights on nonlinear finite element method applications for ULS assessment of steel stiffened-plate structures under biaxial compression and lateral pressure actions. In contrast to Part I dealing with plate elements, the present paper (Part II) was focused on stiffened panels. Based on the results obtained from the present study, the following conclusions can be drawn: (1) It is considered that the two ð12 þ 1 þ 12Þ bay model in the panel length direction gives reasonable solutions of nonlinear finite element computations although one bay model is taken as the extent of the analysis in the panel breadth direction. (2) When the bending rigidity of transverse support members is relatively small, it may be required that the transverse support
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