Elastic Lateral-Torsional Buckling Analysis of Permanent ... · PDF fileelement linear...
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1 Copyright © 2011 by ASME
Proceedings of the ASME 2011 30th International Conference on Ocean, Offshore and Arctic Engineering
OMAE2011 June 19-24, 2011, Rotterdam, The Netherlands
OMAE2011-49104
TRIPPING ANALYSIS AND DESIGN CONSIDERATION OF PERMANENT MEANS OF ACCESS STRUCTURE
Ming Ma
Advanced Marine Technology Center, DRS Defense Solutions, LLC
Stevensville, MD, USA
Beom-Seon Jang
Offshore Basic Engineering Team, Samsung Heavy Industries CO. LTD
Seoul, Korea
Owen F. Hughes
Aerospace and Ocean Engineering Virginia Polytechnic Institute and State
University Blacksburg, VA, USA,
ABSTRACT
An efficient Rayleigh-Ritz approach is presented for
analyzing the lateral-torsional buckling (“tripping”)
behavior of permanent means of access (PMA)
structures. Tripping failure is dangerous and often occurs
when a stiffener has a tall web plate. For ordinary
stiffeners of short web plates, tripping usually occurs
after plate local buckling and often happens in plastic
range. Since PMA structures have a wide platform for a
regular walk-through inspection, they are vulnerable to
elastic tripping failure and may take place prior to plate
local buckling. Based on an extensive study of finite
element linear buckling analysis, a strain distribution is
assumed for PMA platforms. The total potential energy
functional, with a parametric expression of different
supporting members (flat bar, T-stiffener and angle
stiffener), is formulated, and the critical tripping stress is
obtained using eigenvalue approach. The method offers
advantages over commonly used finite element analysis
because it is mesh-free and requires only five degrees of
freedom; therefore the solution process is rapid and
suitable for design space exploration. The numerical
results are in agreement with NX NASTRAN [1]
linear
buckling analysis. Design recommendations are
proposed based on extensive parametric studies.
NOMENCLATURE
Af, Aw, Am, Amf Area of flange, web, mid stiffener
web, and mid stiffener flange
a0, a1 Distance of the neutral axis of lateral
bending for the flange and the mid-
stiffener, respectively, defined in Fig.
5.
mfmwf
mfwf
tttt
bhb
,,,
,,, Defined in Fig. 5.
b Stiffener spacing
{δ} TMMTTB vv ,,,,
Dw Flexural rigidity of web
E, G Young's modulus and shear modulus
Strain due to lateral bending, defined
in Fig. 5.
f1, f2, f3, f4, f5 Shape function defined in Eq. (7)
Izf , Izw, Izm Total moment of inertia of flange,
web, and mid flat bar respect to z-
axis
Jf St. Venant torsion constant for flange
Jm St. Venant torsion constant for mid
stiffener web
Jmf St. Venant torsion constant for mid
stiffener flange
][ GK Geometric stiffness matrix
][ LK Generalized linear stiffness matrix
k Shell plate rotational spring stiffness
l Length of stiffener between
transverse supports
λ mπ/l
m Mode number = number of half
waves lengthwise
Π Total potential energy due to
bending-torsional deformation
Applied axial stress
tp Shell plate thickness
U Strain energy due to bending-
torsional deformation
V Work done due to bending-torsional
deformation
} Defined in Fig. 6
1. INTRODUCTION AND OVERVIEW
In early January 2005, The International Maritime
Organization (IMO) introduced „Permanent means of
access‟ (PMA) regulations [2]
for gaining access to holds
and ballast tanks on oil tankers and bulk carriers. The
purpose of the regulations is to provide overall and
close-up inspections and thickness measurements of the
critical hull structure parts by inspectors, classification
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society surveyors, crew and others. This can ensure that
they are free from damage such as cracks, buckling or
deformation due to corrosion, overloading, or contact
damage and that thickness diminution over the lifetime is
within established limits. In general, a PMA platform is
usually a tall web stiffener welded perpendicularly to a
side shell or longitudinal bulkhead as shown in Fig. 1. A
PMA platform web height has to be wide enough to
provide walking space for a hull inspector. Since PMA
platforms are integrated to hull structures similar to
standard longitudinal stiffeners, they are regarded as
structural strength members and are designed to
withstand the applied hull girder loads. A PMA platform
is much wider than an ordinary stiffener, so a supporting
member, usually a flat bar, is welded onto the middle of
the platform to prevent web plate local buckling and
flange plate tripping. There have been very few studies
on PMA structures because of the relatively new
regulation requirements, and the buckling behavior of
PMA structures is not well understood. Regular stiffener
buckling formulas given by classification societies [3,4]
are not usually applicable to PMA structures due to their
unique configuration. The design of a PMA structure is
often a result of satisfying the scantling requirements of
local support members described in CSR Sec.10 Pt.2 [5]
,
which may underestimate the tripping limit state of PMA
structures.
Fig. 1. A Permanent means of access structure
Stiffener tripping is more dangerous than local plate
buckling or overall buckling, and is regarded as structure
collapse because once tripping occurs the plating is left
with no stiffening and collapse follows immediately. The
few formulations that exist are mainly adaptations of
Timoshenko‟s lateral-torsional thin-walled beam theory [6]
, the main modification to that theory being an
enforced axis of rotation, instead of the natural rotational
axis of the beam cross-section. Analytical methods for
solving stiffener tripping fall into two categories:
differential equation approaches and total potential
energy approaches. Literature reviews in this area were
given in Hu et al [7]
, Hughes and Ma [8]
, and Sheikh et al [9]
. Despite successful research in this area,
the developed models cannot be directly utilized for
PMA structures because of their unique profiles.
Consequently, only general finite element (FE) analysis
can be used as a reliable method for the assessment of
the ultimate strength of PMA structures. Recently, Jang
and Ma [10]
proposed a Rayleigh-Ritz method to analyze
the lateral-torsional buckling of PMA structure. This
study is a continuation of the authors‟ previous study. It
provides parametric formulation for different types of
web plate supporting stiffener (flat bar, T-stiffener and
angle stiffener), as well as the location of the stiffener.
The numerical results are in agreement with NX
NASTRAN linear buckling analysis. Design
recommendations are proposed based on extensive
parametric studies.
2. THEORY
2.1 PMA Strain distribution assumptions
A PMA structure consists of shell plating, a tall web
plate, a flange plate and a web plate supporting stiffener,
as shown in Fig. 1. For asymmetric cross-section beam-
column assembly, it is well known that vertical bending,
sideways bending and torsion are closely coupled.
However, because of the short frame span and a
relatively large gyradius in vertical direction, the
coupling effect of Euler buckling and lateral torsional
buckling can be ignored. The assumption is further
confirmed by NX NASTRAN linear buckling analysis.
Three types of models, the “plate” model, the “pinned”
model and the “fixed” model, were constructed for an
initial finite element linear buckling study. MAESTRO [11]
was used to create the geometries, loads and
boundary conditions because of its parametric feature of
generating multiple models efficiently. Models were
then translated to a NASTRAN data file and to carry out
linear buckling analysis using NX NASTRAN. The
“plate” model consists of a PMA structure, a shell plate
and four stiffeners. The PMA structure has a web plate
of 1100 mm wide and 8 mm thick while its flange is 150
mm wide and 10 mm thick. The mid stiffener is a 250
mm wide and 13 mm thick flat bar. The smaller
stiffeners have a 250X8 web plate and a 100X10 flange
plate. The stiffener spacing is 440 mm. The “plate”
model was simply supported along the four edges of the
base plate and rotation along the edges of base plate is
allowed, as shown in Fig. 2(a). The “pinned” model
consists only of a PMA structure with the root of the
web plate simply supported, as shown in Fig. 2(b). The
“fixed” model is the same as the pinned model with the
addition of the rotational constraint along the web root,
as shown in Fig. 2(c). A unit axial pressure load is
applied at both ends of the models. The purpose of the
initial study is to assess the coupling effect of Euler
buckling and lateral torsional buckling, the effect of
plate rotational constraint, and the strain distribution of
the PMA structure.
(a) “Plate” (b)”Pinnded (c) “Fixed”
Fig. 2. Boundary conditions of three different models
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(a) “Plate”
(b) “Pinned”
(c) “Fixed”
Fig. 3. Tripping mode of three different models
The tripping mode shape is shown in Fig. 3, and the
critical tripping stresses are plotted in Fig. 4. The result
showed that critical tripping stresses were almost
identical for the three types of models in a practical
design range (l=4000 mm to l=6000 mm), indicating
that the coupling effect between Euler buckling and the
tripping, and plate rotational constraint can be ignored.
The models presented here are single bay models.
However, 3-bay models and different mesh density
models were also analyzed using NX NASTRAN. The
results were almost identical.
Fig. 4. Tripping mode of three different models
Based on plate elements‟ mid-plane stress distribution
from NX NASTRAN, the strain distribution of a PMA
structure can be assumed as the following,
Fig. 5. Strain distribution for PMA platform lateral bending
The proposed strain distribution is similar to Hughes and
Ma‟s [8]
assumption for asymmetric stiffener tripping.
Note that a PMA structure does not have a single lateral
bending neutral axis because of the flexibility of the web
plate. The neutral axis of the flange plate, a0, is very
close to the web plate. a0 and a1 can be obtained from the
force equilibrium,
0 xF i.e.
mfmf
m
mm
m
f
f
f
mwwf
f
f
tba
abtab
a
abta
b
a
ab
tathtab
a
ab
1
1
1
1
1
0
0
0
10
0
0
2
1
2
1
2
2
2
1
2
1
2
2
In the above equation, the terms associated with a0
canceled out, and a1 is given by
)(2
22
1
mfmfmmffww
mfmfmmm
tbtbtbth
tbbtba
(1)
a0 depends on the flexibility of the web plate, and may
be given as,
w
w
h
t
a
a
11
0
The flange plate strain energy associated with a0,
∫
is very small comparing to other
terms. For simplicity, it is ignored in this study, which
implies a0 is set to 0.
2.2 Displacement Field
The deformation of the cross section can be described
approximately by a displacement field that has five
degrees of freedom (vT, vM, φT, φM, φB). The cross section
deformations vT, vM, φT, φM and φB are shown in Fig. 6. vT,
and vM are the lateral displacements of the flange and the
mid stiffener with respect to the y axis, respectively. φB
and φM are the rotations at the baseline of the web
and the mid stiffener, respectively. φT is the rotation at
the tip of the web.
Fig. 6, Cross-section deformation of PMA platform
The displacement field for the stiffener web is
),(
0
)(,1
zxvv
w
xvau
w
w
xMw
(2)
for the stiffener flange it is
40
60
80
100
120
140
160
180
200
2000 3000 4000 5000 6000 7000 8000 9000
Cri
tic
al S
tre
ss
(MP
A)
Length(mm)
NX Nastran Linear Buckling Tripping Stress vs. PMA Length
Pinned
Fixed
Plate
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4 Copyright © 2011 by ASME
)(
)(
)()( ,0
xvv
xyw
xvyau
Tf
Tf
xTf
(3)
and for the mid stiffener,
)(
)(
)()( ,1
xvv
xyw
xvyau
Mm
Mm
xMm
(4)
The flexural displacement vw of the web, of which the
maximum value is vT in Fig. 6, is obtained by assuming
that the web buckles out-of-plane as a 5th order
polynomial, so that
)()()()(
)()()()()()(
54
321
zfxzfxv
zfxzfxvzfxv
MM
TTBw
(5)
In which f1, f2, f3, f4, and f5 are 5th order polynomials that
can be obtained by the following compatibility identities.
m
m
w
w
hzzwM
hzwM
hzwT
zzwB
hzzwT
v
vv
vv
v
v
)(
)(
)(
)(
)(
,
0,
,
(6)
By using Eq. (5) and Eq. (6), the polynomials can be
written as
21
212
133
12245551235
21
122
133
225545253
121412
242
54322
5
3253
54232223
4
24
543222
3
325
542322232
2
42
5432223
1
w
ww
w
www
w
ww
w
www
w
www
h
zzhzhzf
h
zzhzhzhf
h
zzhzhzf
h
zzhzhzhf
h
zzhzhzhzf
(7)
when =1/2the shape functions become, as shown in[10]
,
5432
5
432
4
5432
3
5432
2
5432
1
1640328
163216
485
2452347
412136
wwww
w
www
wwww
w
wwww
wwwww
w
h
z
h
z
h
z
h
zhf
h
z
h
z
h
zf
h
z
h
z
h
z
h
zhf
h
z
h
z
h
z
h
zf
h
z
h
z
h
z
h
z
h
zhf
(8)
If the web rotation is fully restrained at the shell plate
connection, then
)()()()()()()()( 5432 zfxzfxvzfxzfxvv MMTTw
2.3 Total Potential Energy
The derivation of the total potential energy is given in
Appendix A. The strain energy stored during buckling
can be written as
l
B
xzww
zzwxxwxzzzwzxxwx
l
xxMzw
l
xMmfm
l
xxMzm
l
xTf
l
xxTzf
dxK
dxdzvt
GvvDvDvD
dxvIEdxJJG
dxvIEdxJGdxvIEU
0
2
,2
3
,,,2
,2
0
2
,
0
2
,
0
2
,
0
2
,
0
2
,
2
1
1242
2
1
2
1
2
1
2
1
2
1
2
1
(9)
and the work done during buckling as
m
wf
A
l
xMxM
A
l
xw
A
l
xTxT
d x d Ayv
d x d Avd x d AyvV
0
2
,
2
,
0
2
,
0
2
,
2
,
)(2
1
2
1)(
2
1
(10)
Transformation of the total potential energy into the
desired stiffness expression requires the selection of the
displacement functions to describe the behavior of the
structure. It is assumed that displacements and twists
vary sinusoidally lengthwise along the member. The
buckling deformation in the lengthwise direction can be
written as
n
m
m
n
m
m
M
M
T
T
B
M
M
T
T
B
xl
xmC
v
v
v
v
11
sin)12(
sin
(11)
where mMmMmTmTmB CCvCCvC ,,,, are the
maximum amplitudes of buckling displacements, and
2m-1 is the number of half waves lengthwise.
Substitute the mode shapes (11) to (9) and (10), the total
potential energy can be written as,
G
T
L
TKKVU
2
1
2
1
(12)
Where [KL] is the linear stiffness matrix, and [KG] is the
geometric stiffness matrix. The usual stability condition
0 GL KK must be satisfied. The stiffness elements
of [KL] and [K
G] were obtained in explicit form using
Maple [12]
, a mathematical software package. The
eigenvalue solution to equation (12) is the critical load,
while the corresponding eigenvector {} describes the
buckled shape. This eigenvalue problem is 5(2m-1) x
5(2m-1) for the “plate” or “pinned” condition, and 4(2m-
1) x 4(2m-1) for the “fixed” condition. The equation can
be solved by a numerical eigenvalue routine.
3. NUMERICAL RESULTS AND COMPARISON OF FEM
3.1. PMA structure without shell plating
To validate the present method, a series of calculations
for 3 different models were carried out and the results
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were compared with those of NX NASTRAN. All
models were simply supported without plating
(“pinned”). The web platform of the PMA structure was
1100 mm wide and 8 mm thick while its flange was 150
mm wide and 10 mm thick. Model-I was supported by a
flat bar stiffener 250 mm wide and 13 mm thick,
positioned at the middle of the web plate (=0.5).
Model-II had the same scantling as Model-I with the flat
bar stiffener positioned at =0.5625. Model-III was
supported by a T-stiffener with a web height of 187.5
mm and a thickness of 13 mm while its flange had a
width of 62.5 mm and a thickness of 13 mm, as shown in
Table 1.
Table 1 “Pinned” Model Scantlings
Model -I Model-II Model-III
hw = 1100 mm tw = 8 mm
bf = 150 mm tf = 10 mm
bm = 250 mm
tm = 13 mm bmf = 0
tmf = 0
=0.5
hw = 1100 mm tw = 8 mm
bf = 150 mm
tf = 10 mm bm = 250 mm
tm = 13 mm
bmf = 0 tmf = 0
=0.5625
hw = 1100 mm tw = 8 mm
bf = 150 mm
tf = 10 mm bm = 187.5mm
tm = 13 mm
bmf = 62.5 mm tmf = 13 mm
=0.5
The results showed very good agreement between the
present study and the linear buckling finite element
analysis using NX NASTRAN, as shown in Fig. 7. Two
conclusions can be drawn from the analysis,
Critical tripping stress increases as the
supporting stiffener moves towards to the
flange plate.
Support from a flat bar stiffener is more
effective than support from a T-stiffener.
Fig. 7. Comparison to NX NASTRAN
3.2. Position of mid-stiffener
To further identify the effect of the mid-stiffener position,
a series of models with the same cross sectional profile
as Model-I were constructed. The length of one set of
models was 4500mm, and the length of the other was
6000 mm. The position of the mid flat bar stiffener had a
range of =0.5 to =0.65. The results of the present
study and NX NASTRAN are shown in Fig. 8. The plate
local buckling stress of the bottom portion of the web
plate, which indicates the lower bound and the upper
bound of the web plate local buckling, is also plotted in
Fig. 8. The elastic web plate buckling stress is given as
following,
2
2
2
112
w
ww
w
eh
tEk
(13)
For simplicity, use Kw=4 for simply supported condition,
and Kw=6.98 for clamped condition.
Fig. 8. Tripping stress and bottom web plate local buckling
stress vs. Mid-stiffener position
Figure 8 shows that the web plate local buckling may
precede tripping as the mid stiffener moves towards the
flange plate. This finding is validated by NX NASTRAN
finite element linear buckling analysis. Table 2 gives a
summary of tripping stress and local plate buckling
stress, and their corresponding modes, where the “plate”,
the “pinned” and the “fixed” model are defined in
section 2.1.
Table 2 Summary of NX NASTRAN Buckling Stresses & Modes
Length
(mm)
Boundary
Condition Tripping
Stress
(MPA)
Buckling
Stress of
1st Mode (MPA)
Tripping
Mode
6000 Pinned 0.5
0.5625 0.575
0.625
128.34
154.34 160.57
188.175
128.34
148.63 141.73
121.16
1
3 8
14
4500 Pinned 0.5 0.5625
0.575
0.625
163.51 191.09
197.90
230.95
163.51 148.88
141.84
121.43
1 9
10
14
4500 Fixed 0.5
0.5625
0.575 0.625
163.51
194.17
201.54 234.83
163.51
194.17
186.7 159.3
1
1
5 12
4500 Plate 0.5 0.5625
0.625
0.6875
161.65 189.1
226.76
278.33
161.65 189.1
162.1
134.55
1 1
11
15
3.3. Shell plate rotational constraint
A shell plate has restraint on the web plate of PMA‟s
torsion. A simple rotational spring stiffness of the shell
0
50
100
150
200
250
2500 4500 6500 8500 10500 12500 14500
Tri
pp
ing
Str
ess M
PA
Length (mm)
Tripping Stress vs. Mid-Stiffener Shape and Location
Nastran-Flatbar-alpha=0.5
Nastran-T-alpha=0.5
Nastran-Flatbar-alpha=0.5625
Fatbar-alpha=0.5
T-alpha=0.5
Flatbar-alpha=0.5625
50
100
150
200
250
300
0.4 0.45 0.5 0.55 0.6 0.65 0.7
Trip
pin
g St
ress
(M
PA
)
Tripping Stress vs. Mid-stiffener Location Nastran L=4500mm
Nastran L=6000mm
Present, m=1, L=6000mm
Present, m=1, L=4500mm
Plate Local Buckling, Pinned, K=4
Plate Local Buckling, Clamped, K=6.98
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plate on the stiffener‟s torsion can be found in Hughes
and Ma [8]
, as the following,
2
3
13
b
Etk
p
(14)
Hu et al.
[7] considered the influence of the plate‟s
buckling mode, and assumed the rotational spring
stiffness decreases when the plate is subjected to axial
compression. They proposed the following equation,
crpr
crp
crp
r
e
fck
fckk
)1(
)1(
(15)
where crp is the plate critical local buckling stress, f is
coefficient based on the plate and stiffener‟s buckling
modes, and cr is defined as,
3
3
41
1
w
pw
r
t
t
b
hc (16)
Two series of finite element models were constructed to
study the effect of shell plate rotational constraints. Both
series were the “plate” models with the same PMA
structure in section 2.1. The shell plate was supported by
two 330X8+150X10 T-stiffeners. The stiffener spacing
was 880 mm. The only difference was the shell plate
thickness; one series‟ models were 13mm thick and the
other series‟ were 16mm thick. Linear buckling analysis
was carried out using NX NASTRAN. Three types of
buckling modes; shell plate local buckling, web plate
local buckling and PMA tripping, are illustrated in Table
3.
Table 3 NX NASTRAN Buckling Modes
=0.625, Tp=13mm, b=880mm =0.625, Tp=16mm, b=880mm
1st Mode, Shell Plate Local
Buckling
cr=157.13 MPA
1st Mode, Web Plate Local
Buckling
cr=163.69 MPA
Tripping Mode
cr=230.1 MPA
Tripping Mode
cr=225.98 MPA
Two sets of NX NASTRAN results are shown in Fig. 9.
One set of results is the buckling stress of the first mode,
and the other is the tripping stress. Figure 9 also plots the
shell plate local buckling stress, computed using
equation (13) with Kw=4, and tripping stresses of this
study. Figure 9(a) shows the shell plate local buckling
that occurs prior to PMA tripping as well as web plate
local buckling when >0.5, i.e. the shell plate local
buckling modes are dominant. For 16mm shell plate
models, the PMA tripping and web local buckling occur
prior to shell plate local buckling. Shell plate rotational
constraint of equation (14) shows good agreement with
the finite element linear buckling results.
(a)
(b)
Fig. 9. Tripping stress and bottom web plate local buckling
stress vs. Mid-stiffener position
3.4 Tripping and web plate local buckling
A convergence study of m, the number of terms in the
lengthwise mode shape, was also carried out. The “plate”
model is defined in section 3.4 with the shell plate
thickness of 13 mm. The “pinned” and the “fixed” model
are defined in section 2.1. It can be seen that 5 terms
(m=5) were needed for the “plate” model, and four terms
(m=4) were needed for the “pinned” and “fixed” models,
as shown in Fig. 10. Further increasing the sinusoidal
terms does not improve the results significantly. The
results of the present study and the results obtained from
NX NASTRAN have excellent agreement for the “plate”
model. For the “Pinned” and the “Fixed” model, the
results are also in excellent agreement when tripping
precedes web plate local buckling, and agree
qualitatively when web plate local buckling precedes
tripping. It can be concluded that the PMA structure will
fail by web plate local buckling after the critical tripping
stress reaches maximum.
100
150
200
250
300
0.4 0.45 0.5 0.55 0.6 0.65 0.7
Trip
pin
g St
ress
(M
PA
)
Tp=13mm, b=880mm
Present, m=1, K=Hughes & Ma
Present,m=1, K=Hu et al
Present, m=5, K=Hughes & Ma
Present, m=5, K=Hu et al"
Shell Plate Local Buckling, K=4
Nastran 1st Mode
Nastran Tripping Mode
100
150
200
250
300
0.4 0.45 0.5 0.55 0.6 0.65 0.7
Trip
pin
g St
ress
(M
PA
)
Tp=16mm,b=880mm
Present, m=1 K=Hughes & Ma
Present, m=1, K=Hu et al
Present, m=5, K=Hughes & Ma
Present, m=5, K=Hu et al
Shell Plate Local Buckling, K=4
Nastran Tripping Mode
Nastran 1st Mode
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7 Copyright © 2011 by ASME
(a)
(b)
(c)
(d)
Fig. 10. Convergence Comparison
The result of the convergence study also indicates that
the tripping mode is the 1st buckling mode as long as the
web plate local buckling is prevented. To validate the
assumption, a small plate strip (50mm x 13mm) was
added near the center of the bottom half of the web plate
to prevent the web plate local buckling mode from
occurring first, as shown in Table 3. The “plate” model
without the plate strip is defined in section 2.1. A series
of linear buckling analyses using NX NASTRAN were
conducted. The results showed proof that tripping indeed
became the first mode and were in good agreement with
the present method, as shown in Table 3 and Fig. 11.
Fig. 11. Comparison of Present Study with NX NASTRAN 1st
Mode
Table 3 Summary of NX NASTRAN 1st Buckling Modes
Model with a plate
strip to prevent plate
local bucking
1st Mode without a
plate strip
1st Mode with a
plate strip
3.5. Scantling effect of the mid-stiffener and the
flange plate
To learn the scantling effect of the mid-stiffener and the
flange plate, the model of 16mm shell plate in section
3.4 was used. The length of the model was 4500mm. The
critical tripping stress was computed by varying the plate
thickness of the mid-stiffener or the flange plate while
the plate cross-sectional area remained a constant. The
results of the mid-stiffener are shown in Fig. 12, and the
results of the flange plate are shown in Fig. 13. The
following conclusions can be drawn from the analyses of
the prototype model,
The most effective way to increase critical
tripping stress is to move the mid-stiffener
towards to the flange plate (increasing .
Increase the width of the flange plate will
effectively increase the critical tripping stress.
The PMA tripping strength is not necessarily
improved by increasing the width of the mid-
stiffener web plate.
100
150
200
250
300
0.4 0.45 0.5 0.55 0.6 0.65 0.7
Trip
pin
g St
ress
(M
PA
)
Comparison with NX Nastran (L=4500mm, Plate)
Present, m=1
Present, m=3
Present, m=5
Present, m=6
Nastran First Mode
Nastran Tripping
Tripping Web Local Buckling
100
150
200
250
300
0.4 0.5 0.6 0.7
Trip
pin
g St
ress
(M
PA
)
Comparison with NX Nastran (L=4500mm, Fixed)
Present, m=1Present, m=3Present, m=4Present, m=5Nastran First ModeNastran Tripping
100
150
200
250
300
0.4 0.45 0.5 0.55 0.6 0.65 0.7
Trip
pin
g St
ress
(M
PA
)
Comparison with NX Nastran (L=4500mm, Pinned)
Present, m=1Present, m=3Present, m=4Present, m=5Nastran First ModeNastran Tripping
100
150
200
250
300
350
400
0.4 0.45 0.5 0.55 0.6 0.65 0.7
Trip
pin
g St
ress
(M
PA
)
Comparison with NX Nastran, L=4500mm Present, Pinned, m=1Present, Pinned, m=4Pined, Nastran 1st ModePined, Nastran TrippingPresent, Fixed, m=1Present, Fixed, m=4Fixed, Nastran 1st ModeFixed, Nastran TrippingPresent, Plate, m=1Present, Plate, m=5Plate, Nastran 1st Mode
100
150
200
250
300
0.4 0.45 0.5 0.55 0.6 0.65 0.7
Trip
pin
g St
ress
(M
PA
)
Buckling Stress of 1st Mode
With Strip, Nastran 1st Mode
Without Strip, Nastran 1st Mode
Present, Plate, m=1
Present, Plate, m=5
Plate Strip
(50 mm x 13
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8 Copyright © 2011 by ASME
(a)
(b)
Fig. 12. Tripping Stress vs. Mid-Stiffener’s Scantlings
(a)
(b)
Fig. 13. Tripping Stress vs. Flange Plate’s Scantlings
4. CONCLUSIONS AND DESIGN RECOMMENDATIONS
An energy method has been presented for analyzing the
tripping behavior of Permanent Means of Access
Structures subjected to axial compression. The results
showed very good agreement with finite element linear
buckling analysis NX NASTRAN. The method is able to
accurately predict not only the PMA tripping stress, but
also the web plate local buckling stress. A transition
point of tripping failure and web local buckling failure
can be identified using the present method. The
following observations, results and conclusions can be
drawn from the study:
Because of the wide platform of a typical PMA
structure, tripping often occurs in the elastic range
and may happen prior to plate local buckling.
To increase the PMA structure‟s tripping resistance,
the mid-stiffener should be moved towards the
flange as far as possible, provided the bottom half of
the web plate‟s local buckling is prevented.
The most effective way to increase critical tripping
stress is to position the supporting mid-stiffener
towards the flange plate.
Adding a small stiffener support at the bottom half
of the web plate, in conjunction with moving the
mid-stiffener towards to the flange, will greatly
increase the critical tripping stress and the web local
buckling stress of the PMA structure.
The fact of the critical tripping stresses being almost
identical for the “plate” model and the “pinned”
model indicated that the shell plate rotational
constraint has very little effect on the PMA‟s
tripping failure; i.e. even if the shell plate fails by
local buckling and provides no rotational support to
the PMA structure, the critical tripping stress of
PMA structure will remain the same.
To increase the critical tripping stress, it is effective
to increase the width of the flange plate. However, it
is not very effective to increase the width of the
mid-stiffener web plate.
Using flat bars to support web platforms is more
effective than using T-stiffeners.
REFERENCES
1. NX NASTRAN Version 10.1 (2009). Siemens
Product Lifecycle Management Software Inc.
2. Safety of Life at Sea (SOLAS), 2002. Regulation II-
1/3-6, Maritime Safety Committee (MSC) of
International Maritime Organization (IMO).
3. American Bureau of Shipping (ABS), 2004.
Buckling and ultimate strength assessment for
offshore structures, Houston, TX 77060, USA
4. Det Norske Veritas, 2002. Buckling strength of
plated structures, Recommended practice DNV-RP-
C201, Høvik, Norway.
5. IACS, 2006. Common structural rules for double
hull oil tankers, International Association of
Classification Societies, London.
6. Timoshenko, S. P., and Gere, J. M., Theory of
Elastic Stability. Second edition. Engineering
Societies Monographs, McGraw-Hill, NY, 1961
7. Hu, Y., Chen, B., Sun, J., 2000. Tripping of thin-
walled stiffeners in the axially compressed stiffened
panel with lateral pressure, Thin Wall Struct , 37, 1-
26
8. Hughes, O. F., Ma, M., 1996a. Elastic tripping
analysis of asymmetrical stiffeners. Comput. Struct.
60, 369-389.
100
150
200
250
300
350
5 10 15 20 25
Trip
pin
g St
ress
(M
PA
)
Mid-Stiffener Web Plate Thickness (mm)
Constant Mid-Stiffener Cross-sectional Area L=4500mm
05
06
07
100
150
200
250
300
350
100 200 300 400 500
Trip
pin
g St
ress
(M
PA
)
Mid-Stiffener Web Plate Height(mm)
Constant Mid-Stiffener Cross-sectional Area L=4500mm
05
06
07
50
100
150
200
250
300
350
400
5 10 15 20 25
Trip
pin
g St
ress
(M
PA
)
Flange Plate Thickness (mm)
Constant Flange Plate Cross-sectional Area L=4500mm
05
06
07
50
100
150
200
250
300
350
400
50 100 150 200 250
Trip
pin
g St
ress
(M
PA
)
Flange Plate Width(mm)
Constant Flange Plate Cross-sectional Area L=4500mm
05
06
07
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9 Copyright © 2011 by ASME
9. Sheikh, I.A., Grondin, G.Y., Elwi, A.E., 2002.
Stiffened steel plates under uniaxial compression J.
Constr Steel Res, 58, 2002, 1061-1080
10. Jang, B.S., Ma, M, Elastic Lateral-Torsional
Buckling Analysis of Permanent Means of Access
Structure, to be published in Ocean Engineering
11. MAESTRO 9.1. (2010). Advanced Marine
Technology Center, DRS Defense Solutions LLC,
http://www.maestromarine.com.
12. Maple 14 (2010). Maplesoft, a division of Waterloo
Maple Inc.
APPENDIX A. DERIVATION OF TOTAL POTENTIAL ENERGY
The total potential energy of a stiffened panel subjected
to edge loading is the sum of the strain energy UT and
the potential energy of the applied load, :
T TU
The strain energy for a three-dimensional isotropic
medium referred to arbitrary orthogonal coordinates may
be written
dvU yzyzxzxzxyxyzzyy
v
xxT )(2
1
After omitting xz, xz and z in accordance with the
basic approximations of thin-plate theory, the strain
energy becomes
pwf
xyxyyy
v
xxT
UUU
dvU
)(2
1
It is assumed that the strains and curvatures are
everywhere much less than unity. The finite-strain
expressions for the in-plane strain components of the
mid-surface are given by
yxyxyxxy
c
xy
yyyy
c
y
xxxx
c
x
wwvvuuvu
wvuv
wvuu
,,,,,,,,
2
,
2
,
2
,,
2
,
2
,
2
,,
5.0
5.0
(A-1)
In the plate theory of von Karman, only displacement
gradients w,x and w,y are expected to achieve significantly
large amplitudes, so of the nonlinear terms in Eq. (A-1),
only w x,
2, w y,
2 and w wx y, , are retained. However, in
the present application, gradients of u and v, in addition
to gradients of w, may become significantly large due to
in-plane rotation, especially for the flange component.
The terms u,x and v,y are of higher orders than the other
terms, therefore the second order terms involving u,x and
v,y are ignored. Hence
yxxy
c
xy
yyy
c
y
xxx
c
x
wwvu
wuv
wvu
,,,,
2
,
2
,,
2
,
2
,,
5.0
5.0
(A-2)
The derivation can be found in many books. The strains
in an arbitrary location of a plate component can be
written as
yxxyxy
yyyyy
xxxxx
wwzwvu
wuzwv
wvzwu
,,,,,
2
,
2
,,,
2
,
2
,,,
2
5.0
5.0
(A-3)
For the flange:
xyfyfxfxfyffxy
yyfyfyfyffy
xxfxfxfxffx
zwwwvu
zwwuv
zwwvu
,,,,,
,
2
,
2
,,
,
2
,
2
,,
2
5.05.0
5.05.0
By Hooke‟s law
fxyfxy
fxfyfy
fyfxfx
E
E
E
1
)(1
)(1
2
2
Since the flange acts as a beam, y is assumed to be
equal to zero, thus
and by eliminating , it is possible to write the
following stress-strain relationship:
fxyfx
fxfx
G
Eu
where for an isotropic material G=E/[2(1+v)]is the shear
modulus. Substituting the above into Uf, noting that
0 zdz , and ignoring 4th order terms,
v
yfxfxyxfxfx
v
xyfxfyf
v
xxfxf
v
yfxfxyfxfyf
v
xfxfxxfxf
v
xyx
f
dvwwwv
dvwzvuG
dvwzuE
dvwwzwvuG
dvwvzwuE
dvGEU
,,
2
,
2
,
2
,
22
,,
2
,
22
,
2
,,,,,
22
,
2
,,,
22
2)(2
1
4)(2
1
2
1
22
1
5.05.02
1
)(2
1
(A-4)
where
xfyfxy
xfx
vuG
Eu
,,
,
By the flange displacement assumption (3)
xTxyf
xxTxxf
xTxf
xxToxf
xfxTyf
w
yw
yw
vyau
vvu
,,
,,
,,
,,
,,,
)(
Also note that 12/3
2/
2/
2 tdzz
t
t
, and . Then Eq.
(A-4) becomes
fy fx
y
xy 0
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l
xTxTfx
l
xT
ff
l
xxT
f
flange
xxTo
f
dxyvtdxbt
G
dxdyyt
EdvvyaEU
))((2
1)(
32
1
)(122
1
2
1
2
,
2
,
2
,
3
2
,
3
2
,
2
Since the term l
xxT
fdxdyy
tE 2
,
3
)(122
1
is quite small
compared to other terms, it can be ignored. Thus
l
xTxTfx
l
xTf
flange
xxTo
f
dxyvt
dxdyJEdvvyaEU
))((2
1
2
1
2
1
2
,
2
,
2
,
2
,
2
where3
3
ff
f
btJ
By the mid stiffener displacement assumption (4)
xMxym
xxMxxm
xMxm
xxMxm
xmxMym
w
yw
yw
vyau
vu
,,
,,
,,
,1,
,,,
)(
dxdAyv
dxJJGdvvyaEU
mfm AA
l
xMxM
l
xMmfm
stiffenermid
xxM
m
0
2
,
2
,
2
,
2
,
2
1
)(2
1
)(2
1
2
1
where 3
3
mmm
btJ
and
3
3
mfmf
mf
btJ
By the web displacement assumption (2)
0,
,1,
,,,
zw
xxMxw
xwxzw
w
vau
wwu
wgplaneofoutwebinplaneweb
wxywxywzwz
v
wxwx
w
uuu
dvU
)(2
1
where
dxdzuEtdxdzuEt
dxdzwuwuwuEt
u
xwwxww
xwzwzwxwzwxw
w
inplaneweb
,2
,2
2
2
,,,,
2
,
2
,2
2
1
12
1
)(2
12
12
1
(A-5)
Note that Eq. (A-5) can also be derived in a same
manner as for the flange; i.e., by assuming y to be zero,
the strain energy due to in-plane deformation becomes
dxdzvaEtu xxMwinplaneweb
2
,12
1
and the strain energy due to out-of-plane deformation is
dxdzvt
GvvDvDvD
dxdzvvvvvEt
u
xzww
zzwxxwxzzzwzxxwx
xzwzzwxxwzzwxxww
planeofoutweb
,2
3
,,,2
,2
,2
,,,2
,2
2
3
1242
2
1
)1(22)1(122
1
where
)1(2
)1(12 2
3
EG
DD
EtDD
xxz
wzx
and
dxdzvvvvtu zwxwwxzzwwzxwwxwwg ,,,2
,2 2
2
1
The total strain energy can be obtained by summation of
each component‟s strain energy,
dxdAyv
dxdAvdxdAyv
udxvIEdxJJG
dxvIEdxJGdxvIE
UUUU
mfm
wf
AA
l
xMxM
A
l
xw
A
l
xTxT
planeofoutweb
l
xxMzw
l
xMmfm
l
xxMzm
l
xTf
l
xxTzf
wmf
T
0
2
,
2
,
0
2
,
0
2
,
2
,
0
2
,
0
2
,
0
2
,
0
2
,
0
2
,
)(2
1
2
1)(
2
1
2
1
2
1
2
1
2
1
2
1
where
mfmfmmmmm
A A
mzm
w
A
zw
A
ff
f
ozf
tbbatbabab
dAbadAayI
htadAaI
tbab
dAayI
m mf
w
f
2
1
2
1
2
1
3
2
1
2
1
2
1
2
1
2
0
3
2
23
)(
12)(