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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


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


(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


)(

)(

)()( ,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


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


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=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=Hu et al

Shell Plate Local Buckling, K=4

Nastran Tripping Mode

Nastran 1st Mode


(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


(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


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

http://www.maestromarine.com/


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)(

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