2648
2648
2648
www.elsevier.com/locate/tws
Abstract
In this paper, improved expressions for elastic local plate buckling and overall panel
buckling of uniaxially compressed T-stiened panels are developed and validated with 55
ABAQUS eigenvalue buckling analyses of a wide range of typical panel geometries. These
two expressions are equated to derive a new expression for the rigidity ratio (EIx/Db)CO that
uniquely identies crossover panelsthose for which local and overall buckling stresses
are the same. The new expression for (EIx/Db)CO is also validated using the 55 FE models.
Earlier work by Chen (Ultimate strength analysis of stiened panels using a beam-column
method. PhD Dissertation, Department of Aerospace and Ocean Engineering, Virginia Polytechnic Institute and State University, Blacksburg, VA, 2003) had produced a new stepby-step beam-column method for predicting stiener-induced compressive collapse of stiened
panels. An alternative approach is to use orthotropic plate theory. As part of the validation
of the new beam-column method, ABAQUS elasto-plastic Riks ultimate strength analyses
were made for 107 stiened panelsthe 55 crossover panels and 52 others. The beamcolumn and orthotropic approaches were also used. A surprising result was that the orthotropic
approach has a large error for crossover panels whereas the beam-column method does
not. Some possible reasons for this are suggested.
# 2004 Elsevier Ltd. All rights reserved.
Keywords: Plate buckling; Panel buckling; Interactive buckling; Panel ultimate strength
0263-8231/$ - see front matter # 2004 Elsevier Ltd. All rights reserved.
doi:10.1016/j.tws.2004.01.003
828
Nomenclature
Geometric properties
a
length of one-bay, spacing between two adjacent transverse frames
Af
sectional area of stiener ange
Ap
sectional area of plate in between adjacent stieners (=bt)
As
sectional area of a single longitudinal stiener
AT
sectional area of a single longitudinal stiener plus eective plating
Aw
sectional area of stiener web
b
spacing between two adjacent longitudinal stieners
B
breadth of stiened panel
bf
breadth of stiener ange
hw
height of stiener web
Ix, Iy moment of inertia of a single stiener with attached plating
ns
number of longitudinal stieners in a stiened panel
t
thickness of plate
tf
thickness of stiener ange
tw
thickness of stiener web
u1
axial shortening of bay
w0
maximum initial deection of a longitudinal stiener (=0.0025a)
P
panel aspect ratio (=a/B)
p
b
plate slenderness parameter ( b=t rY =E )
c
ratio of exural rigidity of platestiener combination to exural
rigidity of plating ( EIx =Db)
p
k
slenderness ratio of stiener with attached plating ( a=pq rY =E )
q
radius
of gyration of longitudinal stiener with attached plating
p
( Ix =AT )
Material properties and strength parameters
D
exural rigidity of isotropic plate ( Et3 =121 m2 )
Dx
exural rigidity of orthotropic plate in x-direction ( EIx =b)
Dy
bending rigidity of orthotropic plate in y-direction ( EIy =a)
E
Youngs modulus
G
shear modulus ( E=21 m)
H
torsional rigidity of orthotropic plate ( 1=6Gt3 GJx =b)
Jx
torsional rigidity of a longitudinal stiener for continuous stiening
( 1=6hw t3w bf t3f )
P0
virtual aspect ratio of orthotropic plate ( a=BDy =Dx 1=4 )
f
ratio of torsional rigidity of stiener and bending rigidity of attached
plating ( GJx =Db)
p
g
torsional stiness parameter of orthotropic plate ( H= Dx Dy )
m
Poissons ratio
rE
Euler column buckling stress
rlocal
rov,panel
rx
rY
829
1. Introduction
In ships, a common portion of structure is a multi-bay longitudinally stiened
panel supported by transverse cross-frames. If there are two cross-frames, it is a
three-bay panel as shown in Fig. 1. The cross-section of a single platestiener
combination is shown in Fig. 2.
Based on Paik and Thayamballi [13], the buckling modes of a stiened panel can
be articially subdivided into the following categories:
.
.
.
.
.
Mode
Mode
Mode
Mode
Mode
These modes are neither mutually exclusive nor independent. However, having
stieners with good proportions can prevent the last two buckling modes cited
above. Some local bending of the stiener web could still interact with the other
modes in otherwise practical panel dimensions. For a stiened panel subjected to
830
uniaxial compression only, overall buckling and local plate buckling are usually
two distinct modes. However, there are specic geometric dimensions when these
two modes occur together and interact very closely. Fig. 3 shows a simplied
design space with only two design variables, plate thickness and height of the stiener web. The axis normal to the page is the weight of the stiened panel, and the
contours are those of constant weight per unit width. The gure shows the constraints against local plate buckling and overall panel buckling, and it is evident
that the optimum design would be at the junction of these two constraints.
Such an optimum panel would have the highest bifurcation buckling stress in its
class of panels of equal weight per unit width [18]. It is useful to have a structural
831
parameter by which one can determine which mode of buckling would occur rst
in a given panel. Bleich [1] introduced the following exural rigidity ratio which is
commonly used for this purpose:
c
EIx
Db
Cox and Riddell [5] used the strain energy method to nd a closed form solution
for the crossover value cCO, which is the value of c at which the elastic local and
overall buckling stresses are equal. This crossover value is an important threshold
value because it is the minimum size of stieners necessary to prevent overall
buckling from occurring before local buckling of the plating between the stieners.
Their analysis was for panels with one, two or three longitudinal stieners but
could be extended to four, ve or more stieners. In their analysis, they ignored the
torsional stiness of the stieners due to complications introduced in interpreting
the results. Based on Timoshenkos system of equations to determine the critical
compressive force of a longitudinally stiened panel, Klitchie [9] arrived at a general solution for cCO which is valid for any number of stieners. Klitchie also did
not take into account the eects of torsional rigidity of the stieners.
Tvergaard and Needleman [19] used a combined Raleigh Ritz-nite element
method to study the bifurcation behavior and initial post-bifurcation behavior of
perfect panels compressed into the plastic range. Their studies revealed considerable imperfection sensitivity both for panels that bifurcate in the plastic range and
for panels with a yield stress a little above the elastic bifurcation stress. Guedes
Soares and Gordo [7] identied this imperfection sensitivity with steep load
shedding characteristics of the panel causing a violent collapse. Recently, Grondin
et al. [15] investigated the stability of steel plates stiened with T-stieners subjected to uniaxial compression using a single-stiener half-bay nite element model.
They did a parametric study with an extensive range of dimensionless parameters
and identied simultaneous buckling in some of their panels. They found these
panels suered an abrupt drop in load carrying capacity in the post-buckling range
and attributed this to interaction buckling referring to this behavior as the
interaction failure mode.
This paper has ve parts. The rst two parts (Sections 3 and 4) develop
improved expressions for the elastic bifurcation buckling stress of one-bay stiened
panels under uniaxial compression for local and overall buckling. For local or
plate buckling, it presents an improved expression for the decrease in rotational
restraint of the plating by the stieners due to bending of the stiener web. For
overall buckling, it considers a modied Euler buckling formula derived by
Timoshenko [16] that allows for the added deection of an ideal column due to
transverse shear. For columns of ordinary cross-section, the eect is negligible, but
the paper shows that for typical stiened panels, the eect is signicant.
The third part (Sections 5 and 6) examines crossover panelsi.e. panels whose
proportions are such that the elastic local and overall bifurcation buckling stresses
are equal. Bifurcation theory predicts that crossover panels have a steep post-buckling load shedding curve. By equating the improved expressions for local and
832
overall buckling, the paper obtains an improved expression for the rigidity ratio cCO
that uniquely identies a crossover panel. The accuracy of these three new expressions is demonstrated by performing a ne mesh ABAQUS elastic eigenvalue
analysis of 55 one-bay panels that cover a wide range of typical panel geometries.
For each panel, the stiener web height was adjusted iteratively until the local and
overall buckling modes coincided. The mean value of the local buckling stress normalized by the ABAQUS eigenvalue is 0.965 with a COV of 6%. The mean value
of the overall buckling stress normalized by the ABAQUS result is 1.007 with a
COV of 4.2%. The new crossover expression normalized by the ABAQUS crossover value has a mean of 0.956 associated with a COV of 7.3%, whereas the customary expression due to Klitchie [9] normalized by ABAQUS has a mean error
of 0.739 with a large scatter (COV 14:8%).
The fourth part (Sections 710) presents the results of elasto-plastic nite
element (ABAQUS) ultimate strength analysis of 55 crossover panels (but now
three bays in length) and the predictions of two classes of closed form methods for
predicting panel ultimate strength. The rst class of methods is based on elastic
large deection orthotropic plate theory, saying that collapse occurs at rst yield.
This part of the paper shows that such methods cannot handle stiener-induced
failure of crossover panels. Two possible reasons are that orthotropic theory (1)
does not allow for two simultaneous and dierent buckling modes and (2) does not
consider the stiener web height, but only an equivalent thickness. For the 55
panels, two representative orthotropic methods normalized by the ABAQUS
ultimate strength have means of 1.274 and 1.455 with COV being around 24%.
Recently, Chen [4] presented a contrasting (almost opposite) approach, based on
an improved step-by-step beam-column method. The paper shows that this method
is unaected by crossover proportions and gives good results for stiener-induced
failure: for the 55 panels, the beam-column prediction normalized by ABAQUS is
1.028 with a low scatter (COV 2:9%).
Because of the variety in panel geometry, there is a corresponding variety in the
pattern of plasticity at collapse. Based on these patterns, the panels were classied
into four groups in the fth part of the paper. The occurrence of plasticity converts
the sudden elastic bifurcation into a smooth soft-peaked loaddeection curve, and
in all but nine of the 55 panels, it prevented a steep post-buckling loaddeection
curve. The authors were unable to nd any common and unique property among
the nine that could explain this. However, the crossover formula remains useful
because it provides a rough estimate of the minimum size of stiener needed to
prevent overall buckling from preceding plate buckling.
833
834
the same panel. This adjustment procedure was performed to get 55 crossover
panels covering a wide range of practical panel dimensions used in ship design. The
rst 25 panels were three-stiener models and the other 30 panels were ve-stiener
models.
For the panels with ve stieners (and some of the three stiener ones indicated
with an in Table 1(a)), it was found that the plate buckled at the longitudinal
edges only with a low stress value, while the rest of the panel remained unbuckled.
This is because the two edge subpanels are weaker than the others. In reality, the
longitudinal edges would have other longitudinal structure that would provide
some rotational restraint to the plating. To simulate this eect and prevent edge
buckling, additional stieners were modeled along the longitudinal edges of
the panel which resulted in realistic uniform local plate buckling in between the
stieners.
835
Table 1
Geometric properties of crossover panels with three stieners and ve stieners
t
hw
tw
bf
tf
a/b
As/Ap
21
21
21
16
16
16
16
10
10
21
21
21
16
16
16
10
10
21
21
21
16
16
16
10
10
50
84
50
36
56
81
31
28
41
80
123
75
58
84
53
45
62
112
166
106
83
120
76
65
86
20
12
10
20
12
5
10
12
5
20
12
10
20
12
10
12
5
20
12
10
20
12
10
12
5
200
100
200
200
100
60
200
100
60
200
100
200
200
100
200
100
60
200
100
200
200
100
200
100
60
30
15
30
30
15
10
30
15
10
30
15
30
30
15
30
15
10
30
15
30
30
15
30
15
10
2.00
2.00
2.00
2.00
2.00
2.00
2.00
2.00
2.00
2.93
2.93
2.93
2.93
2.93
2.93
2.93
2.93
4.00
4.00
4.00
4.00
4.00
4.00
4.00
4.00
0.37
0.13
0.34
0.47
0.15
0.07
0.44
0.20
0.09
0.40
0.16
0.36
0.50
0.17
0.45
0.23
0.10
0.44
0.18
0.37
0.53
0.20
0.47
0.25
0.11
1.77
1.77
1.77
2.33
2.33
2.33
2.33
3.73
3.73
1.77
1.77
1.77
2.33
2.33
2.33
3.73
3.73
1.77
1.77
1.77
2.33
2.33
2.33
3.73
3.73
(b) Five
P78
P79
P80
P81
P82
P83
P84
P85
P86
P87
P88
P89
P90
P91
P92
P93
P94
P95
P96
P97
21
21
21
21
16
16
16
10
10
10
21
21
21
21
16
16
16
10
10
10
84
116
93
77
60
82
54
31
45
56
126
168
136
116
93
120
82
52
68
84
20
12
10
10
20
12
10
20
12
5
20
12
10
10
20
12
10
20
12
5
200
100
160
200
200
100
200
200
100
60
200
100
160
200
200
100
200
200
100
60
30
15
20
30
30
15
30
30
15
10
30
15
20
30
30
15
30
30
15
10
3.00
3.00
3.00
3.00
3.00
3.00
3.00
3.00
3.00
3.00
4.40
4.40
4.40
4.40
4.40
4.40
4.40
4.40
4.40
4.40
0.61
0.23
0.33
0.54
0.75
0.26
0.68
1.10
0.34
0.15
0.68
0.28
0.36
0.57
0.82
0.31
0.71
1.17
0.39
0.17
(continued on
Panel
no.
stieners
1800
1800
1800
1800
1800
1800
1800
1800
1800
1800
2640
2640
2640
2640
2640
2640
2640
2640
2640
2640
600
600
600
600
600
600
600
600
600
600
600
600
600
600
600
600
600
600
600
600
1.18
1.18
1.18
1.18
1.55
1.55
1.55
2.48
2.48
2.48
1.18
1.18
1.18
1.18
1.55
1.55
1.55
2.48
2.48
2.48
next page)
836
Table 1 (continued )
Panel
no.
hw
tw
bf
tf
a/b
As/Ap
P98
P99
P100
P101
P102
P103
P104
P105
P106
P107
3600
3600
3600
3600
3600
3600
3600
3600
3600
3600
600
600
600
600
600
600
600
600
600
600
21
21
21
21
16
16
16
16
10
10
174
223
185
159
131
164
133
115
76
95
20
12
10
10
20
12
10
10
20
12
200
100
160
200
200
100
160
200
200
100
30
15
20
30
30
15
20
30
30
15
6.00
6.00
6.00
6.00
6.00
6.00
6.00
6.00
6.00
6.00
0.75
0.33
0.40
0.60
0.90
0.36
0.47
0.74
1.25
0.44
1.18
1.18
1.18
1.18
1.55
1.55
1.55
1.55
2.48
2.48
2.3. Scantlings
Table 1(a),(b) lists the scantlings of the crossover panels with three stieners and
ve stieners, respectively. In this study, the range of panels is grouped in terms of
b. All the panels are within practical proportions from a design point of view. As
shown in Fig. 1, the width B is 3600 mm for all panels.
Since this study is part of ongoing research at Virginia Tech on ultimate strength
of stiened panels, the data presented in this paper are a subset of a larger database of 107 panels presented in Table 1.1 of Chen [4]. To maintain consistency
between this paper and Chen [4], the panel numbers in the rst column of Table 1
(and subsequent tables of data corresponding to a panel from these tables) are kept
the same. The panels are numbered as P50P107, excluding P57, P66 and P75
which were not crossover panels.
p2 D
k
b2 t
The expression for the buckling coecient k depends on the type of boundary
support, and for long simply supported plates, it is usually assumed that k 4. In
our one-bay panels under consideration, the bare plating in between the stieners is
simply supported on the loaded edges and is elastically restrained by the stieners
along the longitudinal edges. Paik and Thayamballi [11] obtained an exact solution
for the elastic buckling coecient that allows for the rotational restraint given to
the plating by the stieners. They also presented a set of more convenient and su-
837
for 0 f < 2
for 2 f < 20
for f 20
where
Cr
1
3
t
d
1 3:6
tw b
This expression is adapted from Sharp [14]. The factor 3.6 in the denominator is
the value that gave the best agreement with the ABAQUS eigenvalue solutions for
the 55 crossover panels.
We now have an expression for local plate buckling which allows not only for
rotational restraint by the stieners but also for possible web bending in the stieners:
rlocal
p2 D
kCr
b2 t
where
kCr
8
3
2
>
>
< 4 0:396fCr 1:974fCr 3:565fCr
0:881
6:951
>
f
>
Cr 0:4
:
7:025
for fCr 20
In Table 2, the ABAQUS eigenvalues corresponding to local and overall buckling modes are recorded as one critical buckling stress value under the column
rbkl,FEA. The local buckling stress calculated using Eq. (5) is normalized by
rbkl,FEA and the mean for the 55 panels presented in this paper is 0.965 with a
COV of 6%. This veries the accuracy of Eqs. (4)(6).
838
Table 2
Comparison of elastic buckling stresses
Panel no.
rbkl,FEA
rlocal
rlocal/rbld,FEA
rov,panel
rov,panel/rbld,FEA
P50
P51
P52
P53
P54
P55
P56
P58
P59
P60
P61
P62
P63
P64
P65
P67
P68
P69
P70
P71
P72
P73
P74
P76
P77
P78
P79
P80
P81
P82
P83
P84
P85
P86
P87
P88
P89
P90
P91
P92
P93
P94
P95
P96
P97
P98
P99
P100
P101
494
428
445
367
257
239
276
109
96
502
432
442
365
262
308
112
97
501
432
440
369
266
299
115
97
1248
1012
1005
1017
813
653
666
359
308
229
1229
1001
989
1000
804
641
642
358
311
230
1197
984
975
988
511
410
446
350
245
235
319
111
93
506
409
438
350
245
309
112
93
502
409
432
349
244
301
112
93
1187
922
930
990
818
555
715
351
266
210
1167
921
926
972
815
553
689
352
267
209
1149
920
923
960
1.035
0.957
1.002
0.954
0.954
0.985
1.155
1.019
0.973
1.009
0.947
0.992
0.958
0.934
1.005
0.997
0.960
1.002
0.946
0.982
0.947
0.919
1.008
0.977
0.958
0.951
0.911
0.926
0.974
1.006
0.850
0.979
1.074
0.863
0.917
0.949
0.920
0.937
0.972
1.014
0.863
1.074
0.982
0.859
0.910
0.960
0.935
0.947
0.972
545
451
468
392
275
228
293
125
99
534
440
446
365
264
304
121
97
507
433
440
345
277
293
121
98
1281
1035
1015
983
843
668
658
367
316
229
1213
1011
984
989
798
636
631
348
295
228
1155
970
959
963
1.104
1.054
1.051
1.067
1.069
0.953
1.063
1.145
1.031
1.064
1.018
1.008
1.001
1.008
0.986
1.081
1.003
1.011
1.002
1.000
0.935
1.043
0.980
1.055
1.010
1.026
1.023
1.010
0.967
1.037
1.024
1.023
0.988
1.025
0.998
0.987
1.010
0.995
0.989
0.992
0.992
0.983
0.972
0.947
0.993
0.965
0.986
0.983
0.975
839
Table 2 (continued )
Panel no.
P102
P103
P104
P105
P106
P107
rbkl,FEA
791
631
617
627
356
308
rlocal
rlocal/rbld,FEA
813
552
567
667
352
268
1.027
0.875
0.920
1.063
0.988
0.871
Mean
COV
0.965
0.060
rov,panel
762
624
609
613
332
286
rov,panel/rbld,FEA
0.963
0.989
0.986
0.977
0.933
0.928
1.007
0.042
p2 Dx
korth
a2 t
10
p
where P0 a=BDy =Dx 1=4 is the panel virtual aspect ratio, and g H= Dx Dy
p
H= Dx D is the orthotropic torsional stiness parameter.
If P0 is small, the stieners become independent and the stiener-column buckling
Eq. (8) would give good results. There are several ways in which P0 can be small:
. short or wide bay (small a/B)
. heavy stieners (large Dx)
840
Aw G
rov;panel rov;sc korth rE
11
1 2gP20 P40
Aw G AT rE
In Eq. (11), rE is the Euler column buckling stress, the term in parenthesis
accounts for the transverse shear force, and the term in braces accounts for the
panel geometric properties. In Table 2, the overall panel buckling stress calculated
using Eq. (11) is normalized by rbkl,FEA and the mean for the 55 panels is 1.007
with a COV of 4.2% which veries the accuracy of this analytical expression.
5. The crossover parameter cCO
5.1. Klitchie equation for cCO
By transformation of a system of equations established by Timoshenko [16] to
determine the critical compressive load of a longitudinally stiened panel, Klitchie [9]
derived an expression for cCO assuming the plating in between the stieners to
have buckled in one half-sine wave in the transverse direction and ignoring the
rotational restraint given by the stieners. His expression is:
4ak
2 AS
k
cCO;K ns 1
12
pBC
Bt
where
k2 4
a 2
b
and
p
p
k1
k1
p
p
sin
sinh
1
1
a=b
a=b
p
p
C p
p
k1
k1
p
p
k1
k1
p cos
cosh
cos
p cos
ns 1
a=b
ns 1
a=b
5.2. Improved equation for cCO
Since a crossover panel undergoes simultaneous local and overall buckling, we
can obtain an expression for cCO by equating Eqs. (5) and (11):
p2 D
Aw G
k
r
1 2gP20 P40
13
Cr
E
2
b t
Aw G AT rE
EIx a2 AT 6
3 6
Db
bt 4
841
7
k
7
Cr
5
Aw G
2
4
1 2gP0 P0
Aw G AT rE
14
842
Table 3
Comparison of crossover parameter cCO
Panel no.
cCO,FEA
cCO,K
cCO,K/cCO,FEA cCO,new
cCO,new/cCO,FEA
P50
P51
P52
P53
P54
P55
P56
P58
P59
P60
P61
P62
P63
P64
P65
P67
P68
P69
P70
P71
P72
P73
P74
P76
P77
P78
P79
P80
P81
P82
P83
P84
P85
P86
P87
P88
P89
P90
P91
P92
P93
P94
P95
P96
P97
P98
P99
P100
P101
35.12
22.47
34.57
44.60
23.03
18.14
36.92
26.97
19.50
69.83
45.92
61.74
85.10
46.88
73.60
56.01
40.62
120.85
83.65
106.72
147.74
92.23
125.70
105.18
75.71
100.22
58.32
68.53
85.73
116.54
63.34
98.64
144.46
77.64
48.66
195.96
121.14
133.62
165.09
231.35
129.47
184.73
291.22
157.36
103.97
347.81
218.23
236.74
283.85
22.58
18.78
22.15
24.12
19.07
17.77
23.66
19.92
18.08
48.15
39.73
46.61
51.43
40.31
49.92
42.12
37.79
90.15
74.07
86.16
96.29
75.31
92.29
78.46
69.57
58.57
44.89
48.43
55.97
63.63
45.94
61.15
76.35
48.87
41.91
129.65
98.89
105.31
121.29
140.69
101.00
132.30
168.15
107.17
90.45
250.55
189.93
199.92
228.95
0.643
0.836
0.641
0.541
0.828
0.979
0.641
0.738
0.927
0.690
0.865
0.755
0.604
0.860
0.678
0.752
0.930
0.746
0.886
0.807
0.652
0.817
0.734
0.746
0.919
0.584
0.770
0.707
0.653
0.546
0.725
0.528
0.620
0.629
0.861
0.662
0.816
0.788
0.735
0.608
0.780
0.716
0.577
0.681
0.870
0.720
0.870
0.844
0.807
0.926
0.897
0.935
0.879
0.884
1.038
1.118
0.885
0.941
0.943
0.925
0.980
0.954
0.923
1.022
0.920
0.956
0.991
0.941
0.980
1.013
0.876
1.032
0.925
0.948
0.910
0.874
0.895
1.011
0.965
0.815
0.953
1.118
0.834
0.914
0.957
0.901
0.931
0.979
1.024
0.862
1.112
1.011
0.903
0.913
0.995
0.944
0.959
0.997
32.50
20.15
32.31
39.19
20.35
18.83
41.28
23.88
18.35
65.82
42.49
60.50
81.17
43.26
75.23
51.56
38.84
119.72
78.75
104.54
149.64
80.82
129.70
97.27
71.76
91.24
50.98
61.31
86.65
112.44
51.61
110.25
137.69
64.72
44.45
187.52
109.18
124.40
161.64
236.96
111.55
205.39
294.29
142.09
94.97
345.91
206.07
226.97
282.90
843
Table 3 (continued )
Panel no.
cCO,FEA
cCO,K
cCO,K/cCO,FEA cCO,new
cCO,new/cCO,FEA
P102
P103
P104
P105
P106
P107
415.25
241.07
262.01
321.76
524.44
291.10
271.50
194.22
210.15
249.45
322.68
205.56
0.654
0.806
0.802
0.775
0.615
0.706
1.072
0.879
0.927
1.100
1.062
0.937
Mean
COV
0.739
0.148
445.03
211.87
242.87
353.95
556.80
272.79
0.956
0.073
7.1. Imperfections
The imperfection pattern is obtained from an overall buckling mode shape of an
eigenvalue buckling analysis. The selected mode shape has an upward half wave
deection in the full bay and a downward deection in the half bay, which is
shown in Fig. 7. The scaling factor for the initial imperfection of the stieners is
w0 0:0025a, where a is the length of one-bay. Since there will always be some
local subpanel deection (more or less, depending on the size of stiener and the
size of subpanel) in an overall buckling mode shape, the initial deection of plating
is automatically included once the scaling factor is applied.
7.2. Boundary conditions
Let a 0 on T[x, y, z] denote translation constraints and on R[x, y, z] denote
rotational constraints about the x-, y- and z-coordinates in Fig. 7. Let a 1
denote no constraint.
844
. the mid-width node in each of the two transverse edges has T [1, 0, 1] to prevent
rigid body motion in the y-direction.
. the longitudinal edges are simply supported with T [1, 1, 0] and R[1, 0, 0], with
all the nodes along each edge having equal y-displacement.
. the transverse edge on the left hand side, which is the midlength of the mid-bay
of the full three-bay model, has symmetric boundary conditions. This is simulated with T [0, 1, 1] and R[1, 0, 1].
. the transverse edge on the right hand side, which is the loaded edge, is simply
supported with T [1, 1, 0] and R[0, 1, 0]. Only the plate nodes have equal
x-displacements.
. the transverse cross-frame is not modeled, but is simulated with T [1, 1, 0].
8. Orthotropic plate theory for ultimate strength prediction
8.1. Orthotropic plate methodouter surface stress
The governing dierential equations for large deection orthotropic plate theory
are the equilibrium equation and the compatibility equation [17]. Considering idealized initial imperfections, boundary conditions and load application, Paik and
Thayamballi [13] solved the governing dierential equations. The amplitude of the
added lateral deection function was rst solved for. With increase in the lateral
deection of the orthotropic plate, there is local yield due to combined membrane
and bending stress. Collapse is assumed to occur when the Hencky-von Mises
stress on the outer surface of the orthotropic plate rorth,surface reaches rY.
8.2. Orthotropic plate methodmembrane stress
Solving the governing dierential equations for large deection orthotropic plate
theory, Paik and Thayamballi [13] obtained the membrane stress distribution at
midthickness of the orthotropic plate under predominantly longitudinal compressive loads. They found that collapse of the panel may not always be associated with
rst yield on the outer surface of the orthotropic plate. As long as it is possible to
redistribute the applied loads to the straight plate boundaries by membrane action,
collapse does not occur. Collapse occurs when the most stressed boundary locations yield. The corresponding value of the applied load is designated rorth,mem.
The above theory has been implemented in the computer program ULSAP (ultimate strength analysis of panels) [12] which has been used to calculate the stresses
tabulated under rorth,surface and rorth,mem in Table 4. ULSAP however is not restricted to orthotropic theory and provides independent ultimate strength algorithms
for all ve of the failure modes listed in Introduction.
9. Beam-column method for ultimate strength prediction
Newmarks method [10] and the numerical step-by-step procedure [3] have been
used to predict the ultimate strength of a pinnedpinned beam column when the
845
Table 4
Comparison of ultimate strength results
Panel
no.
P50
P51
P52
P53
P54
P55
P56
P58
P59
P60
P61
P62
P63
P64
P65
P67
P68
P69
P70
P71
P72
P73
P74
P76
P77
P78
P79
P80
P81
P82
P83
P84
P85
P86
P87
P88
P89
P90
P91
P92
P93
P94
P95
P96
P97
P98
P99
0.737
0.838
0.736
0.889
1.095
1.190
0.967
1.656
1.853
0.776
0.869
0.812
0.953
1.137
1.010
1.702
1.893
0.814
0.889
0.847
0.998
1.120
1.060
1.712
1.902
0.473
0.542
0.520
0.500
0.600
0.691
0.640
0.946
1.030
1.203
0.506
0.563
0.553
0.534
0.637
0.722
0.691
0.993
1.079
1.220
0.530
0.583
rult,
rorth,
FEA/rY
surface/rY mem/rY
0.78
0.68
0.75
0.64
0.50
0.40
0.58
0.30
0.25
0.77
0.68
0.72
0.64
0.52
0.59
0.33
0.28
0.78
0.70
0.74
0.66
0.57
0.62
0.36
0.32
0.96
0.94
0.94
0.94
0.90
0.82
0.88
0.67
0.57
0.43
0.95
0.93
0.92
0.92
0.89
0.81
0.84
0.67
0.56
0.45
0.94
0.91
0.83
0.83
0.83
0.83
0.76
0.69
0.81
0.53
0.46
0.87
0.87
0.87
0.87
0.80
0.86
0.59
0.52
0.90
0.91
0.90
0.90
0.88
0.89
0.69
0.63
0.82
0.87
0.86
0.83
0.83
0.88
0.83
0.82
0.84
0.73
0.86
0.91
0.90
0.87
0.87
0.91
0.88
0.86
0.88
0.79
0.89
0.93
0.81
0.68
0.79
0.68
0.50
0.42
0.61
0.31
0.25
0.80
0.68
0.76
0.68
0.52
0.64
0.33
0.27
0.81
0.71
0.78
0.70
0.57
0.66
0.37
0.32
0.96
0.98
0.96
0.95
0.91
0.83
0.88
0.69
0.55
0.42
0.97
0.97
0.95
0.96
0.90
0.82
0.88
0.69
0.55
0.44
0.99
0.99
rorth,
1.00
1.00
1.00
1.00
0.99
0.90
1.00
0.63
0.54
1.00
1.00
1.00
1.00
1.00
1.00
0.62
0.53
1.00
1.00
1.00
1.00
1.00
1.00
0.67
0.56
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
0.89
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
0.94
1.00
1.00
1.066
1.216
1.114
1.293
1.504
1.710
1.417
1.737
1.802
1.130
1.290
1.204
1.353
1.550
1.449
1.796
1.874
1.165
1.297
1.221
1.367
1.537
1.450
1.913
1.944
0.853
0.927
0.916
0.881
0.919
1.064
0.950
1.224
1.464
1.696
0.912
0.978
0.971
0.944
0.980
1.128
1.041
1.280
1.554
1.743
0.949
1.020
1.288
1.470
1.340
1.558
1.978
2.234
1.738
2.071
2.130
1.293
1.474
1.380
1.557
1.919
1.684
1.911
1.898
1.290
1.422
1.350
1.520
1.749
1.623
1.874
1.725
1.046
1.067
1.069
1.068
1.109
1.213
1.139
1.488
1.739
2.081
1.058
1.080
1.083
1.081
1.125
1.239
1.187
1.484
1.771
2.061
1.068
1.100
1.049
C1a1
1.007
C3b1
1.056
C1a1
1.058
B1a1
1.002
C1b1
1.047
C1b1
1.062
C1a1
1.014
C2b2
0.986
C1c1
1.039
C3a1
1.007
C3b1
1.043
C1a1
1.062
C1a1
1.008
C3b1
1.075
B1a1
1.017
C2b2
0.991
C1b1
1.051
C3a3
1.010
C3b1
1.056
C3a1
1.066
C3a3
1.006
C3b1
1.075
C3a1
1.020
C2a1
0.986
C1b2
1.000
C4a4
1.042
C4a4
1.028
C4a4
1.017
C3a4
1.010
C1a4
1.006
C3a1
1.006
C1a1
1.029
B1a1
0.958
B1a1
0.981
C1a1
1.024
C1a4
1.046
C4a4
1.026
C4a4
1.043
C3a4
1.016
C1a4
1.010
C3a3
1.042
C1a1
1.017
B1a3
0.983
B1a1
0.974
C3b1
1.052
C1a4
1.091
C3a3
(continued on next page)
846
Table 4 (continued )
Panel
no.
rorth,
rorth,
rult,
FEA/rY surface/rY mem/rY
P100
P101
P102
P103
P104
P105
P106
P107
0.574
0.561
0.662
0.736
0.734
0.722
1.028
1.103
0.91
0.92
0.88
0.81
0.82
0.83
0.66
0.59
0.97
0.99
0.92
0.83
0.86
0.89
0.69
0.57
1.008
0.985
1.019
1.146
1.129
1.085
1.338
1.554
1.094
1.092
1.138
1.228
1.220
1.199
1.505
1.700
1.062
1.077
1.045
1.022
1.046
1.064
1.031
0.976
Mean
COV
1.274
0.241
1.455
0.238
1.028
0.029
0.92
0.90
0.90
0.93
0.93
0.90
0.89
0.91
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
C4a4
C3a3
C1a3
C3a3
C3a3
C1a1
B1a3
C3a3
axial compressive load is increased in steps. Chen [4] developed a modied step-bystep procedure for a three-span simply supported beam column (Fig. 8). In order
to apply the beam-column method to a stiened panel, it is necessary to account
for local plate buckling, which is usually done by means of an eective breadth
be [6,8]. Section 4.8.1 of Paik and Thayamballi [13] gives an analytical solution for
be using large deection orthotropic plate theory and taking into account the eect
of initial plate imperfection, wopl, which for average welding is given by
wopl 0:1b2 t
15
16
For panels with very small stieners, hw =tw < 2:5, it was found that the depth of
yield in the stiener web could not be accurately ascertained. For such small stif-
847
feners, the panel strength will be slightly higher than the bare plate ultimate
strength. Therefore, for hw =tw < 2:5, ULTBEAM uses the following formula to
predict the ultimate strength:
hw =tw 2
rULTBEAM rbare pl rULTBEAM hw =tw 2:5 rbare pl
17
2:5
This procedure has been implemented in the computer program ULTBEAM.
For the crossover panels in this study, the ultimate strength is also calculated using
ULTBEAM and the results are tabulated under rULTBEAM in Table 4.
Table 5
Comparison of ultimate strength predictions by dierent methods with FEA
Method
Mean
COV (%)
1.274
1.455
1.028
24.1
23.8
2.9
848
plating. As expected, the FEA ultimate strength decreases sharply with k, whereas
the orthotropic surface stress based results remain nearly unchanged.
In Fig. 10, Panel nos. P58, P59, P67, P68, P76 and P77 have been excluded. For
these panels, the plate slenderness parameter b is 3.73, as seen in Table 1(a). This is
unusually slender and permits the stieners to behave independently, which by
itself is sucient reason for the orthotropic plate approach to have less accuracy.
Fig. 11 follows from Fig. 10 and plots the percentage error in the orthotropic
surface stress results compared to FEA against k. Whereas one would expect that
the accuracy of orthotropic plate theory would improve as the stiener size decreases (larger k), but here it is the opposite. Again, this may be because orthotropic
plate theory does not allow for two simultaneous elastic buckling modes.
10.2. Orthotropic plate methodmembrane stress
The membrane stress based prediction is also orthotropic in nature. Orthotropic
plate theory cannot distinguish between plate-induced and stiener-induced failure.
Because of the specied initial imperfection (Fig. 7), all the panels in this study
underwent stiener-induced failure in which bending stress in the stiener is a key
849
Fig. 10. Ultimate strength using FEA and orthotropic surface stress for crossover panels.
850
Fig. 11. Percentage error in orthotropic surface stress method relative to FEA.
In the last column of Table 4 labeled collapse mode, the collapse mechanisms
at the ultimate load carrying capacity of each panel are identied from these plots
using the following nomenclature:
A: Stieners elastic in middle bay
B: Web partially plastic in middle bay
C: Approximate plastic hinge in middle bay
1: Plate
2: Plate
3: Plate
4: Plate
851
Fig. 12. Percentage error in orthotropic membrane stress method relative to FEA.
1:
2:
3:
4:
Plate
Plate
Plate
Plate
As observed in Table 4, the collapse mechanisms are complex and varied. However, they are not unusual and are similar to those observed in non-crossover
panels, as presented by Chen [4]. In Table 6, we present a broader classication of
the 14 dierent collapse mechanisms occurring in the 55 crossover panels in four
groups.
From Table 6, we see that 18 panels in Group I reaches their ultimate load with
yield in the stieners only, while the plate midthickness in both middle and end
bays is still elastic. This further illustrates that the orthotropic membrane stress
852
853
854
Table 6
Collapse mechanisms in crossover panels
Group I
C1a1 (9)
C1c1 (1)
B1a1 (5)
C1b1 (3)
Group II
C1a3 (1)
B1a3 (2)
C1a4 (4)
18
Group III
C2a1 (1)
C3a1 (4)
C3b1 (6)
Group IV
C2b2 (3)
C3b3 (8)
C3a4 (2)
C4a4 (6)
11
19
prediction of ultimate strength can be optimistic. Note that while the pattern and
extent of plasticity in the plating varies widely, stiener yield through web is a
common factor in all 55 cases. For the stieners in the middle bay, the yield zone
reached the full depth of the web (approximate plastic hinge; rst letter C) in 48
cases, and in the other seven cases, the yield zone extended some distance into the
web. The consistent presence of stiener web yield and the inconsistent presence
of plate yield suggest that for stiener-induced failure stiener web yield
(say through 2/3 of the web height to be conservative) is a better criterion for
panel collapse than the initial yield of plating criteria that is used by the
orthotropic-based methods.
Fig. 1316 show the von Mises stress distribution at midthickness at the
maximum load carrying capacity of one panel from each group.
11.2. Stressaxial deection curves for crossover panels
As part of the RIKS analysis using ABAQUS, the axial deformation or end
shortening of the panel u1 was measured at a loaded edge plate node at every load
increment. The normalized stressend shortening curve was then drawn for every
panel. In Fig. 17, we present the curves for the four panels which are a good representation of what we have seen for all the crossover panels in this study.
All the crossover panels in this study collapsed due to a stiener-induced failure
of the middle bay. The rst loss of stiness as shown in Fig. 17 is caused by progressive yield through the stiener web at the most stressed location which is at the
midlength of the middle bay. Collapse occurs with the formation of an approximate plastic hinge at that location, the depth of yield depending on the stiener
proportions, with or without yield in the plate in one or more bays. Yield locations
in the plate were either at the midlength of the longitudinal edges or the four cor-
855
ners of the bay. In some panels yielding in the stieners caused by shear was
observed in the end bay. Although this facilitated overall panel collapse, it is not
considered to be a major cause.
The post-collapse behavior is associated with gradual spread of plasticity in the
panel, and to obtain equilibrium ABAQUS reduced the applied load in subsequent
increments. As shown in Fig. 17, three out of the four panels have a stable postcollapse behavior. This was seen in 46 out of the 55 panels. The remaining nine
panels (P52, P56, P62, P63, P69, P71, P104, P105, and P106) suered from a steep
drop in load carrying capacity similar to P52 shown in the gure.
12. Conclusions
In this study, 55 stiened panels with proportions suitable for use in ship design
which had simultaneous local and overall elastic buckling stresses were modeled
and analyzed using the nite element software, ABAQUS. Modied expressions
for elastic local plate buckling and overall panel buckling expressions were derived
856