Consequences of Simultaneous Local and Overall Buckling in Stiffened Panels

Biswarup Ghosh

Thesis submitted to the Faculty of the Virginia Polytechnic Institute and State University

in partial fulfillment of the requirements for the degree of

Master of Science in

Ocean Engineering

Dr. Owen F. Hughes, Chair Dr. Alan J. Brown

Dr. Eric R. Johnson

April 18, 2003 Blacksburg, Virginia

Keywords: Plate Buckling, Panel Buckling, Interactive Buckling, Panel Ultimate Strength

Copyright 2003, Biswarup Ghosh

ii

Consequences of Simultaneous Local and Overall Buckling in Stiffened Panels

Biswarup Ghosh

(ABSTRACT)

In this thesis improved expressions for elastic local plate buckling and overall

panel buckling of uniaxially compressed T-stiffened 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 identifies “crossover” panels – those 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, 2003) had produced a new

step-by-step beam-column method for predicting stiffener-induced compressive collapse

of stiffened 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 stiffened panels – the 55 crossover panels and 52

others. The beam-column 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. Collapse

patterns for the crossover panels are studied and classified from von Mises stress

distribution at collapse. The collapse mechanism and load-deflection diagrams suggest

stable inelastic post collapse behavior for most panels and an abrupt drop in load carrying

capacity in only nine of the 55.

iii

Acknowledgements

First and foremost, I thank my advisor and committee chair Dr. Owen Hughes for

suggesting this topic and for his constant guidance, advice and support.

I thank Dr. Alan Brown and Dr. Eric Johnson for reviewing this work and for

their valuable comments and suggestions.

I thank Dr. Yong Chen for all his help with this work.

Finally, I thank my wife Anindita, my parents, and my family for their love,

support and encouragement.

iv

Table of Contents

List of Figures....................................................................................................... vi

List of Tables ...................................................................................................... viii

Nomenclature ....................................................................................................... ix

1. Introduction....................................................................................................... 1

1.1. Literature Review......................................................................................... 6

1.2. Obtaining a Range of Crossover Panels using ABAQUS ........................... 8

1.3. Improved Expressions for Elastic Buckling and Crossover Prediction ....... 9

1.4. Ultimate Strength Analysis of Crossover Panels ......................................... 9

1.5. Collapse Patterns for Crossover Panels ..................................................... 10

2. FE Model for Eigenvalue Buckling Analysis................................................ 12

2.1. Material Properties..................................................................................... 13

2.2. Finite Elements .......................................................................................... 13

2.3. Boundary Conditions ................................................................................. 14

2.4. Scantlings................................................................................................... 18

3. Elastic Buckling Stresses ................................................................................ 21

3.1. Local Plate Buckling Stress ....................................................................... 21

3.2. Overall Panel Buckling Stress ................................................................... 27

4. The Crossover Parameter COγ ...................................................................... 30

4.1. Klitchieff Equation for COγ ....................................................................... 30

4.2. Improved Equation for COγ ....................................................................... 31

4.3. Crossover Parameter Comparisons ............................................................ 31

v

5. FE Model for Ultimate Strength Analysis .................................................... 35

5.1. Modified RIKS Method for Inelastic Analysis.......................................... 36

5.2. Material Properties..................................................................................... 37

5.3. Finite Elements .......................................................................................... 38

5.4. Imperfections ............................................................................................. 38

5.5. Boundary Conditions ................................................................................. 39

6. Analytical Methods for Ultimate Strength Analysis.................................... 41

6.1. Orthotropic Plate Method – Outer Surface Stress...................................... 41

6.2. Orthotropic Plate Method – Membrane Stress........................................... 42

6.3. Beam-column Method for Stiffened Panels............................................... 47

7. Comparison of Ultimate Strength Predictions ............................................. 52

7.1. Orthotropic Plate Method – Outer Surface stress ...................................... 53

7.2. Orthotropic Plate Method – Membrane Stress........................................... 55

8. Collapse Mechanisms and Post-collapse for Crossover Panels .................. 57

8.1. Collapse Modes.......................................................................................... 57

8.2. Stress – Axial Deflection Curves for Crossover Panels............................. 61

9. Conclusions and Recommendations for Future Work................................ 64

9.1. Conclusions................................................................................................ 64

9.2. Recommendations for Future Work........................................................... 64

References............................................................................................................ 65

Appendix.............................................................................................................. 67

Vita ..................................................................................................................... 123

vi

List of Figures

Fig. 1.1. A stiffened panel under uniaxial compression ......................................... 1

Fig. 1.2. Cross-section of a single plate-stiffener combination .............................. 2

Fig. 1.3. Overall buckling of the plating and stiffeners as a unit............................ 3

Fig. 1.4. Buckling due to predominantly transverse compression.......................... 3

Fig. 1.5. Beam-column type plate induced buckling .............................................. 3

Fig. 1.6. Local buckling of the stiffener web.......................................................... 4

Fig. 1.7. Flexural-torsional buckling (tripping) of the stiffeners............................ 4

Fig. 1.8. Simplified design space for optimum stiffened panel design................... 5

Fig. 2.1. A one-bay 3-stiffener model for eigenvalue analysis............................. 13

Fig. 2.2. Local plate-buckling mode of a 3-stiffener crossover panel .................. 15

Fig. 2.3. Overall buckling mode of a 3-stiffener crossover panel ........................ 15

Fig. 2.4. Edge-buckling in a 5-stiffener model..................................................... 16

Fig. 2.5. A 5-stiffener crossover panel with edge-stiffeners................................. 17

Fig. 2.6. Local plate-buckling mode of a 5-stiffener panel with edge-stiffeners.. 17

Fig. 2.7. Overall buckling mode of a 5-stiffener panel with edge-stiffeners........ 18

Fig. 3.1. Buckling coefficient kCr from FEA and from eq. 3.6 ............................. 26

Fig. 3.2. Transverse shear in an axially loaded column........................................ 27

Fig. 3.3. Shear stress distribution in lightly loaded panel P61 ............................. 28

Fig. 4.1. Klitchieff’s crossover predictions compared to FE results .................... 34

Fig. 4.2. New expressions’ crossover predictions compared to FE results .......... 34

Fig. 5.1. FE model of a 3-bay panel ..................................................................... 36

Fig. 5.2. Idealized elastic-perfectly plastic stress-strain curve ............................. 37

vii

Fig. 5.3. Overall buckling shape from an eigenvalue analysis ............................. 39

Fig. 6.1. Membrane stress distribution under xσ ................................................. 42

Fig. 6.2. Yield locations at plate longitudinal edges............................................. 43

Fig. 6.3. A 3-span simply supported beam-column.............................................. 47

Fig. 6.4. Free body diagram of the 3-span beam-column..................................... 47

Fig. 6.5. Step-by-step procedure for a 3-span beam-column................................ 49

Fig. 7.1. Correlation of ultimate strength predictions with FEA.......................... 53

Fig. 7.2. Ultimate strength using FEA and orthotropic surface stress.................. 54

Fig. 7.3. Percent error in orthotropic surface stress method relative to FEA ....... 55

Fig. 7.4. Percent error in orthotropic membrane stress method relative to FEA.. 56

Fig. 8.1. Collapse stress in P52 (Group I), FEAult ,σ = 263.3 MPa ....................... 59

Fig. 8.2. Collapse stress in P98 (Group II), FEAult ,σ = 330.3 MPa...................... 60

Fig. 8.3. Collapse stress in P60 (Group III), FEAult ,σ = 272.8 MPa .................... 60

Fig. 8.4. Collapse stress in P78 (Group IV), FEAult ,σ = 337.3 MPa .................... 61

Fig. 8.5. Inelastic stress – end shortening curves for crossover panels ................ 62

viii

List of Tables

Table 2.1a. Geometric properties of crossover panels with 3 stiffeners .............. 19

Table 2.1b. Geometric properties of crossover panels with 5 stiffeners.............. 20

Table 3.1. Comparison of elastic buckling stresses (sheet 1 of 2) ....................... 24

Table 3.1. Comparison of elastic buckling stresses (sheet 2 of 2) ....................... 25

Table 4.1. Comparison of crossover parameter COγ (sheet 1 of 2)...................... 32

Table 4.1. Comparison of crossover parameter COγ (sheet 2 of 2)...................... 33

Table 6.1. Comparison of ultimate strength results (sheet 1 of 2) ....................... 45

Table 6.1. Comparison of ultimate strength results (sheet 2 of 2) ....................... 46

Table 7.1. Comparison of analytical ultimate strength predictions with FEA..... 52

Table 8.1. Collapse mechanisms in crossover panels .......................................... 58

ix

Nomenclature Geometric Properties

a length of one-bay, spacing between two adjacent transverse frames

Af sectional area of stiffener flange

Ap sectional area of plate in between adjacent stiffeners ( )tb=

As sectional area of a single longitudinal stiffener

AT sectional area of a single longitudinal stiffener plus effective plating

Aw sectional area of stiffener web

b spacing between two adjacent longitudinal stiffeners

B breadth of stiffened panel

bf breadth of stiffener flange

hw height of stiffener web

Ix, Iy moment of inertia of a single stiffener with attached plating

ns number of longitudinal stiffeners in a stiffened panel

t thickness of plate

tf thickness of stiffener flange

tw thickness of stiffener web

u1 axial shortening of bay

w0 maximum initial deflection of a longitudinal stiffener ( )a0025.0=

Π panel aspect ratio ( )Ba /=

b plate slenderness ratio

=

Etb Yσ

g ratio of flexural rigidity of plate-stiffener combination to flexural rigidity of

plating

=

DbEI x

l slenderness ratio of stiffener with attached plating

=

Ea Yσ

πρ

r radius of gyration of longitudinal stiffener with attached plating

=

T

x

AI

x

Material Properties and Strength Parameters

D flexural rigidity of isotropic plate

−

=)1(12 2

3

νtE

Dx flexural rigidity of orthotropic plate in x-direction

=

bIE x

Dy bending rigidity of orthotropic plate in y-direction

=

aIE y

E Young’s modulus

G shear modulus

+

=)1(2 ν

E

H torsional rigidity of orthotropic plate

+=

bJG

tG x3

61

Jx torsional rigidity of a longitudinal stiffener for continuous stiffening

+= )(

61 33

ffww tbth

P0 virtual aspect ratio of orthotropic plate

=

4/1

x

y

DD

Ba

ζ ratio of torsional rigidity of stiffener and bending rigidity of attached plating

=

bDJG x

h torsional stiffness parameter of orthotropic plate

=

yx DDH

n Poisson’s ratio

sE Euler column buckling stress

slocal elastic local plate buckling stress

sov ,panel elastic overall panel buckling stress

sx applied longitudinal compressive stress

sY yield stress

1

1. Introduction

In ships a common portion of structure is a multi-bay longitudinally stiffened

panel supported by transverse cross frames. If there are two cross frames, it is a three-bay

panel as shown in Fig. 1.1.

Fig. 1.1. A stiffened panel under uniaxial compression

The strength of a bare plate simply supported at the edges is strongly affected by

its width to length ratio. As this ratio increases the strength diminishes rapidly and

therefore it is structurally advantageous to subdivide the width by welding stiffeners on to

the plate. Thus, by judicious addition of a small proportion of structural weight the

strength of the panel is greatly increased. The cross section of a single plate-stiffener

combination is shown in Fig. 1.2.

z σx

a

transverse cross frames

b

y

x

σx

B

2

Fig. 1.2. Cross-section of a single plate-stiffener combination

Based on (Paik and Thayamballi, 2002), the buckling modes of a stiffened panel

can be artificially subdivided into the following categories:

• Mode I: Overall buckling of the plating and stiffeners as a unit

• Mode II: Buckling due to predominantly transverse compression

• Mode III: Beam-column buckling of the stiffeners

• Mode IV: Local buckling of the stiffener web

• Mode V: flexural-torsional buckling or “tripping” of the stiffeners

These modes as shown in Figs. 1.3 to 1.7 are neither mutually exclusive nor

independent. However, having stiffeners with good proportions can prevent the last two

buckling modes cited above. Some local bending of the stiffener web could still interact

with the other modes in otherwise practical panel dimensions.

fb

tf

t

b

N A

d tw hw

3

Fig. 1.3. Overall buckling of the plating and stiffeners as a unit

Fig. 1.4. Buckling due to predominantly transverse compression

Fig. 1.5. Beam-column type plate induced buckling

4

Fig. 1.6. Local buckling of the stiffener web

Fig. 1.7. Flexural-torsional buckling (tripping) of the stiffeners

Figure 1.8 shows a simplified design space with only two design variables, plate

thickness and height of the stiffener web. The axis normal to the page is the weight of the

stiffened panel and the contours are those of constant weight per unit width. The figure

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 (Tvergaard, 1973). This can be explained by arguing that

if the plate thickness is increased by taking material from the stiffeners and adding it to

the plate, the critical stress for local plate buckling would increase and the critical stress

for overall panel buckling would decrease for most practical combinations of other

5

parameters. Thus, the panel with the highest critical bifurcation buckling stress would

have simultaneous buckling modes.

Fig. 1.8. Simplified design space for optimum stiffened panel design

It is useful to have a structural parameter by which one can determine which

mode of buckling would occur first in a given panel. (Bleich, 1952) introduced the

following flexural rigidity ratio which is commonly used for this purpose:

bDIE x=γ

Overall buckling

optimum design point

t

hw

local plate buckling

optimum hw

optimum t

6

1.1. Literature Review

(Cox and Riddell, 1949) used the strain energy method to find a closed form

solution for the “crossover” value COγ , which is the value of γ 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 stiffeners necessary to prevent overall buckling

from occurring before local buckling of the plating between the stiffeners. Their analysis

was for panels with one, two or three longitudinal stiffeners but could be extended to

four, five or more stiffeners. In their analysis they ignored the torsional stiffness of the

stiffeners due to complications introduced in interpreting the results.

Based on Timoshenko’s system of equations to determine the critical compressive

force of a longitudinally stiffened panel, (Klitchieff, 1951) arrived at a general solution

for COγ which is valid for any number of stiffeners. Klitchieff also did not take into

account the effects of torsional rigidity of the stiffeners.

(Tvergaard and Needleman, 1975) used a combined Raleigh Ritz-finite 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. (Soares and Gordo, 1997)

identified this imperfection-sensitivity with steep load shedding characteristics of the

panel causing a “violent collapse”.

Recently, (Grondin et.al, 2002) investigated the stability of steel plates stiffened

with T – stiffeners subjected to uniaxial compression using a single-stiffener half-bay

finite element model. They did a parametric study with an extensive range of

dimensionless parameters and identified simultaneous buckling in some of their panels.

They found these panels suffered 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”.

7

The elastic critical bifurcation buckling stresses, however, are artificial for most

ship panels wherein the buckling, of whichever type, is inelastic. Because these elastic

stresses typically exceed the material yield stress, they are not the actual collapse stress,

but merely a parameter that represents the panel characteristics. Therefore, for ultimate

strength analyses the yield stress has to be taken into account, and the word “buckling”

should more appropriately be replaced with “collapse” in the modes cited earlier. For the

prediction of ultimate strength of stiffened panels, two approaches have been used with

different theoretical philosophies: large deflection orthotropic plate theory and the beam-

column approach.

Orthotropic Approach

The orthotropic plate approach assumes that the bending rigidity of the stiffeners

in both directions is “smeared” into the plating. This approach is satisfactory when the

stiffeners are numerous, uniform and closely spaced in each direction. The panel collapse

strength could be determined using the Mises-Hencky yield criterion either at the

midthickness or at the outer surface of the orthotropic plate. The major advantage of the

orthotropic plate method over the beam-column approach is that it accounts for the plate

membrane stress pattern, which is ignored by the latter. However, (Paik and

Thayamaballi, 2002) found that as the size of the stiffeners increased, the orthotropic-

predicted ultimate strength (based on midthickness stress) increased at about twice the

actual rate, as obtained from nonlinear finite element analyses. This is because the

stiffeners in fact play a distinct role in the collapse mechanism and cannot be

approximated by smeared rigidities. These authors presented an equation for ultimate

strength, which is a weighted average of the orthotropic plate (midthickness) value and

the value for the panel if the stiffeners are removed. (Paik, 2001) has implemented these

orthotropic-based methods in the computer program ULSAP (ULtimate Strength

Analysis of Panels).

8

Beam-column Approach

In the inelastic beam-column methods for geometrically imperfect members, the

load-deflection response of the member is traced from the start of loading to the collapse

load. Numerical or approximate techniques (Chen and Atsuta, 1976, 1977) such as the

Raleigh-Ritz method, Galerkin method, Newmark method and the step-by-step method

have been used. (Chen, 2003) modified the original step-by-step method (Chen and Lui,

1987) for application to a continuous beam-column and developed a simple correction

factor in terms of the product lP0 2 to give a very good estimate of the ultimate strength

of a stiffened panel under uniaxial compression. The concept of effective breadth was

used to allow for the effect of local plate buckling. The method has been implemented in

a computer program called ULTBEAM.

Due to the difficulty of experimental investigation of collapse of stiffened panels

there is very limited data available for comparison purposes. Therefore the preferred tool

now for calculating the ultimate strength is nonlinear finite element (FE) analysis and this

is used to evaluate the two approximate approaches.

1.2. Obtaining a Range of Crossover Panels using ABAQUS

In Chapter 2 the ABAQUS finite element program is used to model one-bay

stiffened panels for eigenvalue buckling analysis. For each panel the stiffener web height

was adjusted iteratively until the local and overall buckling modes coincided. Altogether

55 crossover panels that cover a wide range of typical panel geometries were obtained

and their structural dimensions are tabulated in Tables 2.1a and 2.1b.

9

1.3. Improved Expressions for Elastic Buckling and Crossover Prediction

In Chapter 3 improved expressions for local and overall buckling of one-bay

stiffened panels under uniaxial compression are developed. For local or plate buckling an

improved expression for the decrease in rotational restraint of the plating by the stiffeners

due to bending of the stiffener web is presented. The overall buckling expression

considers a modified Euler buckling formula derived in (Timoshenko, 1936) that allows

for the added deflection of an ideal column due to transverse shear. For columns of

ordinary cross section the effect is negligible, but this study shows that for typical

stiffened panels the effect is significant.

Chapter 4 examines “crossover” panels – i.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 overall buckling, this study obtains an

improved expression for the rigidity ratio COγ that uniqely identifies a crossover panel.

The accuracy of these three new expressions is demonstrated by the ABAQUS elastic

eigenvalue results for 55 crossover panels. 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

(Klitchieff, 1951) normalized by ABAQUS has a mean error of 0.739 with a large scatter

(COV = 14.8 %).

1.4. Ultimate Strength Analysis of Crossover Panels

In Chapter 5 the elasto-plastic FE analysis of the 55 crossover panels is carried

out on 3-bay multi-stiffener models using ABAQUS. It has been verified (Chen, 2003)

that a 3-bay model can be adopted as a generic representation of a multi-bay structure

10

without significant loss of accuracy and is also used by Classification societies such as

DnV in similar research.

In Chapter 6 the theory behind two classes of closed form methods for predicting

panel ultimate strength is briefly presented. The first class of methods is based on elastic

large-deflection orthotropic plate theory, saying that collapse occurs at first yield.

Recently (Chen, 2003) presented a contrasting (almost opposite) approach, based on an

improved step-by-step beam-column method.

Chapter 7 presents the ultimate strength results of 55 crossover panels and shows

that orthotropic theory based methods cannot handle stiffener-induced failure of the

crossover panels. Two possible reasons are that orthotropic theory (1) does not allow for

two simultaneous and different buckling modes and (2) does not consider the stiffener

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 %. The study shows that the improved beam-

column method is unaffected by crossover proportions and gives good results for

stiffener-induced failure: for the 55 panels the beam-column prediction normalized by

ABAQUS is 1.028 with a low scatter (COV = 2.9 %).

1.5. Collapse Patterns for Crossover Panels

The inelastic collapse behavior of a stiffened panel is extremely complex with

progressive yielding occurring through various depths of the cross-section, often

approaching or resembling a plastic hinge. Because of the variety in panel geometry there

is a corresponding variety in the pattern of plasticity at collapse. The basic pattern of

collapse however is the same for crossover and non-crossover panels.

In Chapter 8 the panels were classified into four groups based on the von Mises

stress distribution at the midthickness of the section at collapse. There are no indications

that any of the crossover panels are weaker than a non-crossover panel in terms of

11

ultimate strength. The occurrence of plasticity converts the sudden elastic bifurcation into

a smooth soft-peaked load-deflection curve, and in all but nine of the 55 panels it

prevented a steep post-buckling load-deflection curve. A common and unique property

among the nine that could explain this could not be found. However, the crossover

formula remains useful because it provides a rough estimate of the minimum size of

stiffener needed to prevent overall buckling from preceding plate buckling.

In Chapter 9 conclusions on prediction and ultimate strength characteristics of

crossover panels are drawn. Some recommendations for future work for ultimate strength

predictions of stiffened panels are made.

The Appendix contains the von Mises stress distribution at the midthickness of

the section at collapse and the load-deflection curves for the 55 crossover panels.

12

2. FE Model for Eigenvalue Buckling Analysis

For eigenvalue buckling analysis, it was found that a one-bay panel with

appropriate boundary conditions that simulate the support of the bay at transverse frames

gave the same results as a 3-bay panel. So, for this part of the analyses, a series of one-

bay panels with 3 or 5 equally spaced longitudinal T-stiffeners was modeled and

analyzed using ABAQUS. A 3-stiffener model is shown in Fig. 2.1. The stiffened panel is

discretized into sufficient number of elements to allow for free development of the

buckling modes. The use of four-node shell elements S4 allows for finite rotations and

membrane strains. Uniaxial compressive load is applied on both the left and right hand

sides of the model using the *CLOAD option. An incremental loading pattern QN is

defined in the *BUCKLE step, which is scaled by the load multipliers ωi found in the

eigenvalue problem:

0)( 0 =+ ∆M

iNM

iNM vKK ω

where

NMK 0 is the stiffness matrix corresponding to the base state,

NMK∆ is the load stiffness matrix due to the incremental loading pattern QN,

iω are the eigenvalues,

Miv are the eigenvectors (buckling mode shapes),

M and N are degrees of freedom of the whole model, and

i refers to the ith buckling mode.

The critical buckling mode is then Ni Qω . Since our models were carefully

adjusted to have crossover proportions, usually the first two modes were of interest.

13

Fig. 2.1. A one-bay 3-stiffener model for eigenvalue analysis

2.1. Material Properties

Material: steel

Young’s modulus: 205800 N/mm2

Poisson’s ratio: 0.3

Yield stress: 352.8 MPa.

2.2. Finite Elements

No. of elements in plate: 80 x 24 per stiffener (for 3 stiffener panels)

80 x 16 per stiffener (for 5 stiffener panels)

No. of elements in web: 80 x 8 per stiffener

No. of elements in flange: 80 x 6 per stiffener

14

2.3. Boundary Conditions

Let u, v, and w be the translations along x, y and z-axes respectively of Fig. 2.1.

• the mid-node of both longitudinal (unloaded) plate edges have u constrained to be

zero and the mid-node of both the loaded plate edges have v constrained to be

zero, to prevent rigid body motion.

• the longitudinal (unloaded) edges are simply supported.

• the transverse (loaded) edges are simply supported. In addition, they have

rotational restraint about the z-axis. The web nodes are constrained to have equal

v displacement which prevents sideways bending of the web at the frames. These

together simulate the support of the panel at transverse frames or bulkheads.

With other dimensions unchanged, the web height of the stiffeners was carefully

adjusted to get a crossover value of local and overall buckling stresses (typically within 1

or 2 %, maximum 5 %). Fig. 2.2 shows the local plate-buckling mode of the crossover

panel shown in Fig. 2.1 and Fig. 2.3 shows the overall buckling mode of 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 first 25 panels were 3-

stiffener models and the other 30 panels were 5-stiffener models.

15

Fig. 2.2. Local plate-buckling mode of a 3-stiffener crossover panel

Fig. 2.3. Overall buckling mode of a 3-stiffener crossover panel

16

For the panels with 5 stiffeners (and some of the 3 stiffener ones indicated with an

‘ * ‘ in Table 2.1a), 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 illustrated in

Fig. 2.4.

Fig. 2.4. Edge-buckling in a 5-stiffener model

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 effect and prevent “edge buckling”,

additional stiffeners were modeled along the longitudinal edges of the panel as shown in

Fig. 2.5.

17

Fig. 2.5. A 5-stiffener crossover panel with edge-stiffeners

This resulted in realistic uniform local plate buckling in-between the stiffeners as shown

in Fig. 2.6. The overall buckling mode of this panel is shown in Fig. 2.7.

Fig. 2.6. Local plate-buckling mode of a 5-stiffener panel with edge-stiffeners

18

Fig. 2.7. Overall buckling mode of a 5-stiffener panel with edge-stiffeners

2.4. Scantlings

Tables 2.1a and 2.1b list the scantlings of the crossover panels with 3 stiffeners

and 5 stiffeners respectively. In this study, the range of panels is grouped in terms of β.

All the panels are within practical proportions from a design point of view. As shown in

Fig. 1.1 the width B is 3600 mm for all panels.

Since this study is part of ongoing research at Virginia Tech on ultimate strength

of stiffened panels, the data presented in this thesis is a subset of a larger database of 107

panels presented in Table 1.1 of (Chen, 2003). To maintain consistency between this

thesis and (Chen, 2003), the panel numbers in the first column of Tables 2.1a and 2.1b

(and subsequent tables of data corresponding to a panel from these tables) are kept the

same. The panels are numbered as P50 to P107, excluding P57, P66 and P75 which were

not crossover panels.

19

Table 2.1a. Geometric properties of crossover panels with 3 stiffeners

Panel No.

a

b

t

wh

wt

fb

ft

ba

p

s

AA

β

P50 1800 900 21 50 20 200 30 2.00 0.37 1.77

P51 1800 900 21 84 12 100 15 2.00 0.13 1.77

P52 1800 900 21 50 10 200 30 2.00 0.34 1.77

* P53 1800 900 16 36 20 200 30 2.00 0.47 2.33

P54 1800 900 16 56 12 100 15 2.00 0.15 2.33

P55 1800 900 16 81 5 60 10 2.00 0.07 2.33

P56 1800 900 16 31 10 200 30 2.00 0.44 2.33

P58 1800 900 10 28 12 100 15 2.00 0.20 3.73

P59 1800 900 10 41 5 60 10 2.00 0.09 3.73

P60 2640 900 21 80 20 200 30 2.93 0.40 1.77

P61 2640 900 21 123 12 100 15 2.93 0.16 1.77

P62 2640 900 21 75 10 200 30 2.93 0.36 1.77

* P63 2640 900 16 58 20 200 30 2.93 0.50 2.33

P64 2640 900 16 84 12 100 15 2.93 0.17 2.33

* P65 2640 900 16 53 10 200 30 2.93 0.45 2.33

P67 2640 900 10 45 12 100 15 2.93 0.23 3.73

P68 2640 900 10 62 5 60 10 2.93 0.10 3.73

P69 3600 900 21 112 20 200 30 4.00 0.44 1.77

P70 3600 900 21 166 12 100 15 4.00 0.18 1.77

P71 3600 900 21 106 10 200 30 4.00 0.37 1.77

* P72 3600 900 16 83 20 200 30 4.00 0.53 2.33

P73 3600 900 16 120 12 100 15 4.00 0.20 2.33

P74 3600 900 16 76 10 200 30 4.00 0.47 2.33

P76 3600 900 10 65 12 100 15 4.00 0.25 3.73

P77 3600 900 10 86 5 60 10 4.00 0.11 3.73

* indicates that edge stiffeners were added

20

Table 2.1b. Geometric properties of crossover panels with 5 stiffeners

Panel No.

a

b

t

wh

wt

fb

ft

ba

p

s

AA

β

P78 1800 600 21 84 20 200 30 3.00 0.61 1.18 P79 1800 600 21 116 12 100 15 3.00 0.23 1.18 P80 1800 600 21 93 10 160 20 3.00 0.33 1.18 P81 1800 600 21 77 10 200 30 3.00 0.54 1.18 P82 1800 600 16 60 20 200 30 3.00 0.75 1.55 P83 1800 600 16 82 12 100 15 3.00 0.26 1.55 P84 1800 600 16 54 10 200 30 3.00 0.68 1.55 P85 1800 600 10 31 20 200 30 3.00 1.10 2.48 P86 1800 600 10 45 12 100 15 3.00 0.34 2.48 P87 1800 600 10 56 5 60 10 3.00 0.15 2.48 P88 2640 600 21 126 20 200 30 4.40 0.68 1.18 P89 2640 600 21 168 12 100 15 4.40 0.28 1.18 P90 2640 600 21 136 10 160 20 4.40 0.36 1.18 P91 2640 600 21 116 10 200 30 4.40 0.57 1.18 P92 2640 600 16 93 20 200 30 4.40 0.82 1.55 P93 2640 600 16 120 12 100 15 4.40 0.31 1.55 P94 2640 600 16 82 10 200 30 4.40 0.71 1.55 P95 2640 600 10 52 20 200 30 4.40 1.17 2.48 P96 2640 600 10 68 12 100 15 4.40 0.39 2.48 P97 2640 600 10 84 5 60 10 4.40 0.17 2.48 P98 3600 600 21 174 20 200 30 6.00 0.75 1.18 P99 3600 600 21 223 12 100 15 6.00 0.33 1.18 P100 3600 600 21 185 10 160 20 6.00 0.40 1.18 P101 3600 600 21 159 10 200 30 6.00 0.60 1.18 P102 3600 600 16 131 20 200 30 6.00 0.90 1.55 P103 3600 600 16 164 12 100 15 6.00 0.36 1.55 P104 3600 600 16 133 10 160 20 6.00 0.47 1.55 P105 3600 600 16 115 10 200 30 6.00 0.74 1.55 P106 3600 600 10 76 20 200 30 6.00 1.25 2.48 P107 3600 600 10 95 12 100 15 6.00 0.44 2.48

21

3. Elastic Buckling Stresses

3.1. Local Plate Buckling Stress

The equation for buckling of a simply supported bare plate was derived by

(Bryan, 1891). In terms of a buckling coefficient k Bryan’s equation is:

ktbD

local 2

2πσ = (3.1)

The expression for the buckling coefficient 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 stiffeners is simply

supported on the loaded edges and is elastically restrained by the stiffeners along the

longitudinal edges. (Paik and Thayamballi, 2000) obtained an exact solution for the

elastic buckling coefficient that allows for the rotational restraint given to the plating by

the stiffeners. The authors also presented a set of more convenient and sufficiently

accurate approximate expressions obtained by curve fitting as follows:

≥

<≤−

−

<≤+−+

=

20025.7

2024.0

881.0951.6

20565.3974.1396.04 23

ζ

ζζ

ζζζζ

for

for

for

k (3.2)

in which bDJG x=ζ is a non-dimensional parameter involving the St.Venant torsional

stiffness xJ of the stiffener.

Equation (3.2) is based on the assumption that the stiffeners remain straight until

the plating in between them buckled. But if the stiffener web is slender (either tall or thin)

then there will be bending of the stiffener web and the stiffeners will not provide the full

22

theoretical rotational restraint along their line of attachment. (Paik and Thayamballi,

2000) proposed a correction factor CL to the original torsional rigidity as follows:

ζζ LL C=

However, in this study it was found that the expression for CL gave a value of 1.0

for 2<ζ , which is a range within which many practical panels lie. Therefore an

alternative correction factor rC for web bending is proposed as follows:

ζζ rCr C= (3.3)

where

bd

tt

C

w

r 3

6.31

1

+

= (3.4)

This expression is adapted from (Sharp, 1966). 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 stiffeners but also for possible web bending in the stiffeners:

Crlocal ktbD

2

2πσ = (3.5)

where

≥

<≤−

−

<≤+−+

=

20025.7

2024.0

881.0951.6

20565.3974.1396.04 23

Cr

CrCr

CrCrCrCr

Cr

for

for

for

k

ζ

ζζ

ζζζζ

(3.6)

23

In Table 3.1, the ABAQUS eigenvalues corresponding to local and overall

buckling modes are recorded as one critical buckling stress value under the column

σbkl,FEA. The local buckling stress calculated using eq. (3.5) is normalized by σbkl,FEA and

the mean for the 55 panels presented in this thesis is 0.965 with a COV of 6 %. This

verifies the accuracy of eqs. (3.4) - (3.6).

24

Table 3.1. Comparison of elastic buckling stresses (sheet 1 of 2)

Panel No. FEAbkl ,σ localσ FEAbkl

local

,σσ

panelov,σ FEAbkl

panelov

,

,

σσ

P50 494 511 1.035 545 1.104 P51 428 410 0.957 451 1.054 P52 445 446 1.002 468 1.051 P53 367 350 0.954 392 1.067 P54 257 245 0.954 275 1.069 P55 239 235 0.985 228 0.953 P56 276 319 1.155 293 1.063 P58 109 111 1.019 125 1.145 P59 96 93 0.973 99 1.031 P60 502 506 1.009 534 1.064 P61 432 409 0.947 440 1.018 P62 442 438 0.992 446 1.008 P63 365 350 0.958 365 1.001 P64 262 245 0.934 264 1.008 P65 308 309 1.005 304 0.986 P67 112 112 0.997 121 1.081 P68 97 93 0.960 97 1.003 P69 501 502 1.002 507 1.011 P70 432 409 0.946 433 1.002 P71 440 432 0.982 440 1.000 P72 369 349 0.947 345 0.935 P73 266 244 0.919 277 1.043 P74 299 301 1.008 293 0.980 P76 115 112 0.977 121 1.055 P77 97 93 0.958 98 1.010

25

Table 3.1. Comparison of elastic buckling stresses (sheet 2 of 2)

Panel No. FEAbkl ,σ localσ FEAbkl

local

,σσ

panelov,σ FEAbkl

panelov

,

,

σσ

P78 1248 1187 0.951 1281 1.026 P79 1012 922 0.911 1035 1.023 P80 1005 930 0.926 1015 1.010 P81 1017 990 0.974 983 0.967 P82 813 818 1.006 843 1.037 P83 653 555 0.850 668 1.024 P84 666 715 0.979 658 1.023 P85 359 351 1.074 367 0.988 P86 308 266 0.863 316 1.025 P87 229 210 0.917 229 0.998 P88 1229 1167 0.949 1213 0.987 P89 1001 921 0.920 1011 1.010 P90 989 926 0.937 984 0.995 P91 1000 972 0.972 989 0.989 P92 804 815 1.014 798 0.992 P93 641 553 0.863 636 0.992 P94 642 689 1.074 631 0.983 P95 358 352 0.982 348 0.972 P96 311 267 0.859 295 0.947 P97 230 209 0.910 228 0.993 P98 1197 1149 0.960 1155 0.965 P99 984 920 0.935 970 0.986 P100 975 923 0.947 959 0.983 P101 988 960 0.972 963 0.975 P102 791 813 1.027 762 0.963 P103 631 552 0.875 624 0.989 P104 617 567 0.920 609 0.986 P105 627 667 1.063 613 0.977 P106 356 352 0.988 332 0.933 P107 308 268 0.871 286 0.928

Mean 0.965 1.007 COV 0.060 0.042

26

In Fig. 3.1 the buckling coefficient kCr calculated for the ABAQUS critical

buckling stress value is plotted together with the approximate expression eq. (3.6).

Fig. 3.1. Buckling coefficient kCr from FEA and from eq. 3.6

Crk

5 10 15 20

4.5

5

5.5

6

6.5

7

Crζ

27

3.2. Overall Panel Buckling Stress

The Euler buckling stress for a column is:

2

2

=

ρ

πσa

EE (3.7)

As shown by (Timoshenko, 1936) in a column under axial compressive load there

is some transverse shear Q due to the slope; that is )(xwPQ ′= as in Fig. 3.2.

Fig. 3.2. Transverse shear in an axially loaded column

The resulting shear strain causes an additional deflection, and the effect is to reduce the

overall (Euler) buckling stress by the factor )/( ETww AGAGA σ+ . For an ordinary

column the effect is negligible but for a stiffener-column the web area Aw is a small

fraction of the total area AT and the factor can be significant. In this study it ranged

between 0.71 for panel P52 and 0.98 for panel P76.

Fig. 3.3 corresponds to panel P61 with 15/ =wT AA . This figure illustrates the

presence of a significant amount of shear stress in the stiffener webs even when the panel

is fully elastic under the applied load. Assuming the stiffener to deflect in a cosine half

wave, the accompanying hand calculation gives a maximum shear stress of 35.8 MPa at

the stiffener ends. The hand calculation provides a rough estimate of the maximum shear

stress in the web but would not match exactly with the stress gradient shown in the color

)(xw

P

dxQ

dxxQQ

∂∂

+

y

x

28

plot. This is because in the finite element analysis the plating has absorbed some of the

shear flow.

Fig. 3.3. Shear stress distribution in lightly loaded panel P61

Therefore, the corrected overall buckling stress for a stiffener-column is given by:

+

=−ETw

wEcsov AGA

GAσ

σσ , (3.8)

A stiffened panel will undergo overall buckling if the stiffeners are relatively

small. From small deflection orthotropic theory, the elastic overall buckling stress is:

orthx

orthov kta

D2

2

,π

σ = (3.9)

where the buckling coefficient is

40

2021 Π+Π+= ηorthk (3.10)

Shear stress (XY) at midthickness MPax 1.186=σ

axCoswxw π

max)( =

mmw 77.10max =

MPaAQAQ

xdxwd

axSin

aw

xdxwd

w

Tx

ax

8.35/0128.0

0128.02640

77.10)(

)(

maxmax

max

2/

max

==××=

=×=

−=

−=

τσ

π

ππ

a

29

where 4/1

0

=Π

x

y

DD

Ba is the panel virtual aspect ratio, and

DDH

DDH

xyx

==η is the orthotropic torsional stiffness parameter.

If Π0 is small, the stiffeners become independent and the stiffener-column

buckling eq. (3.8) would give good results. There are several ways in which Π0 can be

small:

• short or wide bay (small Ba )

• heavy stiffeners (large xD )

• thin plating (small D )

• very close spacing of stiffeners (large DDx )

For cases when 0Π is not small, we convert the stiffener-column buckling eq.

(3.8) into a panel buckling equation by applying the orthotropic buckling coefficient korth

given by eq. (3.10). That is,

21 40

20,, Π+Π+

+

== − ησ

σσσETw

wEorthcsovpanelov AGA

GAk (3.11)

In eq. (3.11), Eσ 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 3.1, the overall panel buckling stress calculated using eq.

(3.11) is normalized by σbkl,FEA and the mean for the 55 panels is 1.007 with a COV of

4.2 % which verifies the accuracy of this analytical expression.

30

4. The Crossover Parameter COγ

The structural parameter DbEI x=γ is the ratio of the flexural rigidity of a plate-

stiffener combination to the flexural rigidity of the plating.

4.1. Klitchieff Equation for COγ

By transformation of a system of equations established by (Timoshenko, 1936) to

determine the critical compressive load of a longitudinally stiffened panel (Klitchieff,

1951) derived an expression for COγ assuming the plating in between the stiffeners to

have buckled in one half-sine wave in the transverse direction and ignoring the rotational

restraint given by the stiffeners. His expression is:

2,

4( 1) SCO K s

AanBC Bt

λγ λπ

= + + (4.1)

where

22 4

=

baλ

and

1 1sin sinh1 1/ /1 1 1 1cos cos cosh cos

/ 1 / 1s s

a b a bC

a b n a b n

λ λπ π

λ λ π λ λ ππ π

− +

= − +− − + +

− −+ +

31

4.2. Improved Equation for COγ

Since a crossover panel undergoes simultaneous local and overall buckling, we

can obtain an expression for COγ by equating eqs. (3.5) and (3.11):

21 40

202

2

Π+Π+

+

= ησ

σπ

ETw

wECr AGA

GAk

tbD (4.2)

Substituting 2

2

=

ρ

πσa

EE and

T

x

AI

=2ρ in eq. (4.2)

21 40

202

2

2

2

Π+Π+

+

= ησ

ππ

ETw

w

T

xCr AGA

GAAaEI

ktbD

from which

Π+Π+

+

==21 4

02

0

3

2

,

ησ

γ

ETw

w

CrTxnewCO

AGAGA

ktb

AabDIE

(4.3)

where Crk , 0Π and η have been defined earlier.

4.3. Crossover Parameter Comparisons

In Table 4.1 the crossover value of COγ obtained from the eigenvalue results for

the 55 crossover panels is tabulated under the column FEACO,γ . Compared to it are the

predictions using eqs. (4.1) and (4.3). The Klitchieff expression normalized by FEACO,γ

has a mean of 0.739 with a high scatter (COV = 14.8 %) and the normalized new

expression has a mean of 0.956 associated with a COV of 7.3 %.

32

Table 4.1. Comparison of crossover parameter COγ (sheet 1 of 2)

Panel No.

FEACO,γ KCO,γ FEACO

KCO

,

,

γγ

newCO,γ FEACO

newCO

,

,

γγ

P50 35.12 22.58 0.643 32.50 0.926 P51 22.47 18.78 0.836 20.15 0.897 P52 34.57 22.15 0.641 32.31 0.935 P53 44.60 24.12 0.541 39.19 0.879 P54 23.03 19.07 0.828 20.35 0.884 P55 18.14 17.77 0.979 18.83 1.038 P56 36.92 23.66 0.641 41.28 1.118 P58 26.97 19.92 0.738 23.88 0.885 P59 19.50 18.08 0.927 18.35 0.941 P60 69.83 48.15 0.690 65.82 0.943 P61 45.92 39.73 0.865 42.49 0.925 P62 61.74 46.61 0.755 60.50 0.980 P63 85.10 51.43 0.604 81.17 0.954 P64 46.88 40.31 0.860 43.26 0.923 P65 73.60 49.92 0.678 75.23 1.022 P67 56.01 42.12 0.752 51.56 0.920 P68 40.62 37.79 0.930 38.84 0.956 P69 120.85 90.15 0.746 119.72 0.991 P70 83.65 74.07 0.886 78.75 0.941 P71 106.72 86.16 0.807 104.54 0.980 P72 147.74 96.29 0.652 149.64 1.013 P73 92.23 75.31 0.817 80.82 0.876 P74 125.70 92.29 0.734 129.70 1.032 P76 105.18 78.46 0.746 97.27 0.925 P77 75.71 69.57 0.919 71.76 0.948

33

Table 4.1. Comparison of crossover parameter COγ (sheet 2 of 2)

Panel No.

FEACO,γ KCO,γ FEACO

KCO

,

,

γγ

newCO,γ FEACO

newCO

,

,

γγ

P78 100.22 58.57 0.584 91.24 0.910 P79 58.32 44.89 0.770 50.98 0.874 P80 68.53 48.43 0.707 61.31 0.895 P81 85.73 55.97 0.653 86.65 1.011 P82 116.54 63.63 0.546 112.44 0.965 P83 63.34 45.94 0.725 51.61 0.815 P84 98.64 61.15 0.528 110.25 0.953 P85 144.46 76.35 0.620 137.69 1.118 P86 77.64 48.87 0.629 64.72 0.834 P87 48.66 41.91 0.861 44.45 0.914 P88 195.96 129.65 0.662 187.52 0.957 P89 121.14 98.89 0.816 109.18 0.901 P90 133.62 105.31 0.788 124.40 0.931 P91 165.09 121.29 0.735 161.64 0.979 P92 231.35 140.69 0.608 236.96 1.024 P93 129.47 101.00 0.780 111.55 0.862 P94 184.73 132.30 0.716 205.39 1.112 P95 291.22 168.15 0.577 294.29 1.011 P96 157.36 107.17 0.681 142.09 0.903 P97 103.97 90.45 0.870 94.97 0.913 P98 347.81 250.55 0.720 345.91 0.995 P99 218.23 189.93 0.870 206.07 0.944 P100 236.74 199.92 0.844 226.97 0.959 P101 283.85 228.95 0.807 282.90 0.997 P102 415.25 271.50 0.654 445.03 1.072 P103 241.07 194.22 0.806 211.87 0.879 P104 262.01 210.15 0.802 242.87 0.927 P105 321.76 249.45 0.775 353.95 1.100 P106 524.44 322.68 0.615 556.80 1.062 P107 291.10 205.56 0.706 272.79 0.937

Mean 0.739 0.956 COV 0.148 0.073

34

Figures 4.1 and 4.2 graphically demonstrate this improved prediction. In these

figures the x and y-axes were chosen to be As / Ap and a / b respectively because these

parameters were observed to have a significant influence over COγ . There are other

parameters that influence the crossover panel dimensions but they do not have such a

noticeable influence.

Fig. 4.1. Klitchieff’s crossover predictions compared to FE results

Fig. 4.2. New expressions’ crossover predictions compared to FE results

a / b

FEACO,γ

COγ

ps AA /

KCO,γ

a / b

FEACO,γ

COγ

ps AA /

newCO,γ

35

5. FE Model for Ultimate Strength Analysis

For inelastic analysis the panels being modeled should be appropriate to capture

all the mechanisms that lead to collapse of the structure. Subjected to longitudinal

compression, an interframe bay would deflect in an upward or downward half-sine wave

(which are the plate-induced and stiffener-induced modes respectively), while the next

bay would deflect in the opposite sense. (Chen, 2003) demonstrated using 107 FE models

that a multi-bay structure with unbiased (equal upward and downward) initial

eccentricities is weaker in the stiffener induced buckling mode and that failure of this bay

would lead to the collapse of the structure. In other words, if the initial eccentricity is the

same in the upward and downward directions, collapse of a multi-bay panel is always

caused by a stiffener-induced failure. Therefore, a one-bay model as used by (Grondin,

2002) and some other researchers which could undergo either plate induced or stiffener

induced collapse depending on the initial eccentricity could be misleading, and drawing

conclusions for a multi-bay structure would be inappropriate. Moreover, the boundary

conditions at a transverse frame are intermediate between simply supported and clamped,

and cannot be accurately modeled as a simply supported loaded edge. Therefore, for

inelastic analysis a 3-bay model is most appropriate, which can be represented as a

symmetric 1½ bay model as shown in Fig. 5.1. Also, due to inclusion of the inelastic

properties which involve complex collapse mechanisms, it was found that edge stiffening

of the panels was not necessary.

In our ABAQUS models, four-node S4 shell elements were used and a fine mesh

generated to adequately represent the deformations and stress gradients. Uniaxial

compressive load was applied to the right hand side of the model only as concentrated

nodal forces using the *CLOAD option. The loads were applied in two portions – one

portion as a ‘dead load’ in a previous step, and a ‘live load’ in the current RIKS step.

36

Fig. 5.1. FE model of a 3-bay panel

5.1. Modified RIKS Method for Inelastic Analysis

The modified RIKS method as implemented in ABAQUS assumes proportional

loading, which means that the load magnitude varies with a single scalar parameter. The

algorithm works well even for complex unstable problems. The current load magnitude,

totalP , is defined by )( 00 PPPP reftotal −Ω+= , where 0P is the “dead load” from the

previous load history, refP is the reference load vector defined in the current RIKS step,

and Ω is the “load proportionality factor”. Ω is found as part of the solution.

The basic algorithm uses the Newton’s method. It uses only a 1 % extrapolation

of the strain increment. The essence of this method is that it finds a single equilibrium

path in a space defined by the nodal variables and the loading parameter, and

simultaneously solves for displacements and loads. Since the load magnitude is an

z σx

a

line of symmetry

transverse cross frame

b

y

x

1½ bay symmetric model (dark shaded)

mmB 3600=

37

additional unknown variable, ABAQUS uses the “arc length” along the static equilibrium

path to measure the progress of the solution. An initial increment in arc length, inl∆ ,

along the static equilibrium path is given on the data line of the *STATIC option. The

initial load proportionality factor, in∆Ω , is equal to this initial increment in arc length,

but is adjusted if the increment fails to converge. Thereafter, the value of Ω is calculated

automatically. The increment size is limited by moving a given distance (determined by

ABAQUS/Standard’s convergence rate dependent automatic incrementation scheme)

along a tangent line to the current solution point. It then searches for equilibrium in the

plane that passes through the point thus obtained and that is orthogonal to the same

tangent line. Since the loading magnitude is part of the solution, a method needs to be

specified for completion of the RIKS step. This can be either a maximum value of Ω , a

maximum displacement value at a specified degree of freedom, or the maximum number

of increments specified in the *STEP option. In our analyses a sufficient number of

increments were specified to get the post collapse response.

5.2. Material Properties

These are the same as in the eigenvalue analyses. In addition, in these analyses the

idealized ‘elastic-perfectly plastic’ stress-strain curve as shown in Fig. 5.2 was adopted.

Fig. 5.2. Idealized elastic-perfectly plastic stress-strain curve

stress

Yσ

strain

38

5.3. Finite Elements

No. of elements in plate: 120 x 24 per stiffener (for 3 stiffener panels)

120 x 16 per stiffener (for 5 stiffener panels)

No. of elements in web: 120 x 8 per stiffener

No. of elements in flange: 120 x 6 per stiffener

5.4. Imperfections

In order to analyze the inelastic buckling situation the problem has to be posed as

a continuous response problem instead of a bifurcation problem by introducing an initial

imperfection for the stiffeners and the plating. This is achieved by using the

*IMPERFECTION option, and 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 deflection in the full bay and a downward deflection in the half bay,

which is shown in Fig. 5.3. The scaling factor for the initial imperfection of the stiffeners

is w0 = 0.0025 a, where a is the length of one bay. Since there will always be some local

subpanel deflection (more or less, depending on the size of stiffener and the size of

subpanel) in an overall buckling mode shape, the initial deflection of plating is

automatically included once the scaling factor is applied.

39

Fig. 5.3. Overall buckling shape from an eigenvalue analysis

5.5. 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. 5.3. Let a “1” denote no

constraint.

• 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 3-bay model, has symmetric boundary conditions. This is simulated with

T [0, 1, 1] and R [1, 0, 1].

40

• 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].

Additionally, the stiffener web nodes should also be constrained to have equal y

displacement at the frame. However, since all the 107 panels that were modeled

were well proportioned and uniaxially compressed only, this boundary condition

was found redundant.

41

6. Analytical Methods for Ultimate Strength Analysis

6.1. Orthotropic Plate Method – Outer Surface Stress

The governing differential equations for large deflection orthotropic plate theory

are the equilibrium equation and the compatibility equation (Troitsky, 1976). Considering

idealized initial imperfections, boundary conditions and load application (Paik and

Thayamballi, 2003) solved the governing differential equations. The amplitude of the

added lateral deflection function Am was first solved for. With increase in the lateral

deflection of the orthotropic plate, there is local yield due to action of bending. Using the

Mises-Henckey yield criterion first yield on the outer surface of the orthotropic plate,

σorth,surface , occurs at the value of σxav that satisfies the following equation:

122

=

+

−

Y

yb

Y

yb

Y

xb

Y

xb

σσ

σσ

σσ

σσ

(6.1)

where

+

−+

−+

−=22

2

22

1)(

28)2(

BamAAt

Ea

AAAEmy

yx

ommeqx

ommmxxxavxb

πνπνν

πρσσ ,

+

−+

−+

−=22

2

2

1)(

28)2(

BamAAt

EB

AAAEx

yx

ommeqy

ommmyxyb

ππννν

πρσ ,

∫=B

oxxav dy

Bσσ 1 ,

=eqt equivalent plate thickness 2 2 2

s s

xeq s s

n At tt t n AB tB

+ ++= = = + ,

42

xρ is a correction factor to account for the variation in the true deflection pattern from

the assumed sinusoidal pattern,

omA is the amplitude of the buckling mode initial lateral deflection,

mA is the amplitude of the added lateral deflection due to load,

m is the half wave number in buckling,

xE and yE are the orthotropic equivalent values of Young’s modulus, and

xν and yν are the orthotropic equivalent values of Poisson’s ratio.

6.2. Orthotropic Plate Method – Membrane Stress

Solving the governing differential equations for large deflection orthotropic plate

theory, (Paik and Thayamballi, 2003) obtained the membrane stress distribution at

midthickness of the orthotropic plate under predominantly longitudinal compressive

loads (Fig. 6.1).

Fig. 6.1. Membrane stress distribution under xσ

a

B

43

The maximum and minimum membrane stresses in the x and y directions are:

+=

+−=

++=

+−=

2

2

min

2

2

max

2

22

min

2

22

max

8)2(

8)2(

8)2(

8)2(

BAAAE

BAAAE

aAAAEm

aAAAEm

ommmyxy

ommmyxy

ommmxxxavx

ommmxxxavx

πρσ

πρσ

πρσσ

πρσσ

(6.2)

where ∫=B

oxxav dy

Bσσ 1 ,

xρ , omA , mA , m, xE and yE have been defined in Section 6.1.

These researchers found that collapse of the panel may not always be associated

with first 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

as shown in Fig. 6.2. Using the von Mises yield criterion the ultimate stress, σorth,mem , is

the value of xavσ that satisfies eq. (6.2) and the following equation:

12

minminmax2

max =

+

−

Y

y

Y

y

Y

x

Y

x

σσ

σσ

σσ

σσ

(6.3)

Fig. 6.2. Yield locations at plate longitudinal edges

C : compressive T : tensile

xavσ

: yield

44

In other words the ultimate stress σorth,mem is the value of the applied x-stress xavσ

when the membrane midthickness stresses cause yield at the midlength of the

longitudinal edges.

The above theory has been implemented in the computer program ULSAP

(ULtimate Strength Analysis of Panels) (Paik, 2001) which has been used to calculate the

stresses tabulated under sorth,surface and σorth,mem in Table 6.1. ULSAP however is not

restricted to orthotropic theory and provides independent ultimate strength algorithms for

all five of the failure modes listed in the introduction.

45

Table 6.1. Comparison of ultimate strength results (sheet 1 of 2)

Panel No.

λ Y

FEAult

σσ ,

Y

surfaceorth

σσ ,

Y

memorth

σσ ,

Y

ULTBEAM

σσ

FEAult

surfaceorth

,

,

σσ

FEAult

memorth

,

,

σσ

FEAult

ULTBEAM

,σσ Collapse

mode

P50 0.737 0.78 0.83 1.00 0.81 1.066 1.288 1.049 C1a1

P51 0.838 0.68 0.83 1.00 0.68 1.216 1.470 1.007 C3b1

P52 0.736 0.75 0.83 1.00 0.79 1.114 1.340 1.056 C1a1

P53 0.889 0.64 0.83 1.00 0.68 1.293 1.558 1.058 B1a1

P54 1.095 0.50 0.76 0.99 0.50 1.504 1.978 1.002 C1b1

P55 1.190 0.40 0.69 0.90 0.42 1.710 2.234 1.047 C1b1

P56 0.967 0.58 0.81 1.00 0.61 1.417 1.738 1.062 C1a1

P58 1.656 0.30 0.53 0.63 0.31 1.737 2.071 1.014 C2b2

P59 1.853 0.25 0.46 0.54 0.25 1.802 2.130 0.986 C1c1

P60 0.776 0.77 0.87 1.00 0.80 1.130 1.293 1.039 C3a1

P61 0.869 0.68 0.87 1.00 0.68 1.290 1.474 1.007 C3b1

P62 0.812 0.72 0.87 1.00 0.76 1.204 1.380 1.043 C1a1

P63 0.953 0.64 0.87 1.00 0.68 1.353 1.557 1.062 C1a1

P64 1.137 0.52 0.80 1.00 0.52 1.550 1.919 1.008 C3b1

P65 1.010 0.59 0.86 1.00 0.64 1.449 1.684 1.075 B1a1

P67 1.702 0.33 0.59 0.62 0.33 1.796 1.911 1.017 C2b2

P68 1.893 0.28 0.52 0.53 0.27 1.874 1.898 0.991 C1b1

P69 0.814 0.78 0.90 1.00 0.81 1.165 1.290 1.051 C3a3

P70 0.889 0.70 0.91 1.00 0.71 1.297 1.422 1.010 C3b1

P71 0.847 0.74 0.90 1.00 0.78 1.221 1.350 1.056 C3a1

P72 0.998 0.66 0.90 1.00 0.70 1.367 1.520 1.066 C3a3

P73 1.120 0.57 0.88 1.00 0.57 1.537 1.749 1.006 C3b1

P74 1.060 0.62 0.89 1.00 0.66 1.450 1.623 1.075 C3a1

P76 1.712 0.36 0.69 0.67 0.37 1.913 1.874 1.020 C2a1

P77 1.902 0.32 0.63 0.56 0.32 1.944 1.725 0.986 C1b2

46

Table 6.1. Comparison of ultimate strength results (sheet 2 of 2)

Panel No.

λ Y

FEAult

σσ ,

Y

surfaceorth

σσ ,

Y

memorth

σσ ,

Y

ULTBEAM

σσ

FEAult

surfaceorth

,

,

σσ

FEAult

memorth

,

,

σσ

FEAult

ULTBEAM

,σσ Collapse

mode

P78 0.473 0.96 0.82 1.00 0.96 0.853 1.046 1.000 C4a4 P79 0.542 0.94 0.87 1.00 0.98 0.927 1.067 1.042 C4a4 P80 0.520 0.94 0.86 1.00 0.96 0.916 1.069 1.028 C4a4 P81 0.500 0.94 0.83 1.00 0.95 0.881 1.068 1.017 C3a4 P82 0.600 0.90 0.83 1.00 0.91 0.919 1.109 1.010 C1a4 P83 0.691 0.82 0.88 1.00 0.83 1.064 1.213 1.006 C3a1 P84 0.640 0.88 0.83 1.00 0.88 0.950 1.139 1.006 C1a1 P85 0.946 0.67 0.82 1.00 0.69 1.224 1.488 1.029 B1a1 P86 1.030 0.57 0.84 1.00 0.55 1.464 1.739 0.958 B1a1 P87 1.203 0.43 0.73 0.89 0.42 1.696 2.081 0.981 C1a1 P88 0.506 0.95 0.86 1.00 0.97 0.912 1.058 1.024 C1a4 P89 0.563 0.93 0.91 1.00 0.97 0.978 1.080 1.046 C4a4 P90 0.553 0.92 0.90 1.00 0.95 0.971 1.083 1.026 C4a4 P91 0.534 0.92 0.87 1.00 0.96 0.944 1.081 1.043 C3a4 P92 0.637 0.89 0.87 1.00 0.90 0.980 1.125 1.016 C1a4 P93 0.722 0.81 0.91 1.00 0.82 1.128 1.239 1.010 C3a3 P94 0.691 0.84 0.88 1.00 0.88 1.041 1.187 1.042 C1a1 P95 0.993 0.67 0.86 1.00 0.69 1.280 1.484 1.017 B1a3 P96 1.079 0.56 0.88 1.00 0.55 1.554 1.771 0.983 B1a1 P97 1.220 0.45 0.79 0.94 0.44 1.743 2.061 0.974 C3b1 P98 0.530 0.94 0.89 1.00 0.99 0.949 1.068 1.052 C1a4 P99 0.583 0.91 0.93 1.00 0.99 1.020 1.100 1.091 C3a3 P100 0.574 0.91 0.92 1.00 0.97 1.008 1.094 1.062 C4a4 P101 0.561 0.92 0.90 1.00 0.99 0.985 1.092 1.077 C3a3 P102 0.662 0.88 0.90 1.00 0.92 1.019 1.138 1.045 C1a3 P103 0.736 0.81 0.93 1.00 0.83 1.146 1.228 1.022 C3a3 P104 0.734 0.82 0.93 1.00 0.86 1.129 1.220 1.046 C3a3 P105 0.722 0.83 0.90 1.00 0.89 1.085 1.199 1.064 C1a1 P106 1.028 0.66 0.89 1.00 0.69 1.338 1.505 1.031 B1a3 P107 1.103 0.59 0.91 1.00 0.57 1.554 1.700 0.976 C3a3

Mean 1.274 1.455 1.028 COV 0.241 0.238 0.029

47

A - A

P

0w

P

a a a

A

A

Middle span (half) Stiffener induced failure

End span remains elastic

P P

MR

qend = qmid P

MRqmid

P M

1w

nwx

x

z

6.3. Beam-column Method for Stiffened Panels

Newmark’s method (Newmark, 1943) and the Numerical step-by-step procedure

(Chen and Lui, 1987) have been used to predict the ultimate strength of a pinned-pinned

beam column when the axial compressive load is increased in steps. The Newmark

method can only trace the load-deflection curve of the structure from the start of loading

to the peak ultimate strength. Since for crossover panels we are also interested in the

descending branch of the curve, the Numerical step-by-step procedure should be used.

(Chen, 2003) developed a modified step-by-step procedure for a three-span

simply supported beam column (Fig. 6.3). The free body diagram of the continuous

beam-column is shown in Fig. 6.4.

Fig. 6.3. A 3-span simply supported beam-column

Fig. 6.4. Free body diagram of the 3-span beam-column

48

Based on the following assumptions:

• the cross section remains plane after bending and remains undeformed in the cross

sectional plane,

• the stress-strain relationship is idealized elastic-perfectly plastic (Fig. 7.2),

• the initial imperfection of the beam-column follows a half sine wave,

the moment (M) – curvature (Ф) – thrust (P) relationships are derived for the following

cases:

• cross section of middle span remains elastic

• formation of primary / secondary plastic hinge

• formation of perfect plastic hinge.

The cross section of a beam-column extracted from one of our stiffened panels is

not symmetric about its neutral axis (Fig. 1.2). So when the member deflects, there is a

reaction bending moment MR at the intermediate support. The procedure therefore starts

with an initial guess of this unknown bending moment MR. The algorithm is shown in the

flowchart in Fig. 6.5.

49

Fig. 6.5. Step-by-step procedure for a 3-span beam-column

The end bay shown in Fig. 6.4 is assumed to remain elastic and the analytical load

deflection equation is given by:

axw

ax

kakxkx

EIkMxw R πφ sin)1

tansin(cos)( 02 +−+−= (6.4)

where

EPP

−=

1

1φ ,

?

Nown = 0

Set w1

?

Set initial P

Guess initial MR

Step-by-step procedure P M Ф wi+1 wn

Adjust MR

Yes

Yes Adjust P

'nw = qend No

Start

Print P

50

EIPk = , and

TEE AP σ= .

The first derivative of w(x) gives the slope at the connection between the middle

bay and end bay

aw

akak

EIkM

w Rend

πφθ 0

2

1tan

)0(' +

+−== (6.5)

At this connection the two boundary conditions for the middle bay are:

2'0 axat

ww

end

=

==

θ (6.6)

With these boundary conditions the procedure presented in Fig. 6.5 was

implemented to find the axial force P corresponding to a specified deflection w1.

In order to apply the beam-column method to a stiffened panel, (Chen, 2003)

applied the concept of effective breadth based on (Faulkner, 1975) to account for local

plate buckling effect as follows:

−=≤

=≥=

− µ

ββµ

σσ

µ

1

2

,

12,1

,1

bb

bb

e

epanelov

local

(6.7)

where localσ and panelov,σ are given by eq. (3.5) and eq. (3.11) respectively.

The beam-column ultimate strength is then corrected by a factor R obtained by

curve-fitting the finite element values of ultimate strength for 107 stiffened panels, which

include the 55 crossover panels presented in Tables 2.1a and 2.1b. The resulting

expression for the factor R is:

40

220 8566.17585.20.1 Π+Π+= λλR (6.8)

51

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 ULTBEAMσ in Table 6.1.

For panels with very small stiffeners, 5.2/ <ww th , it was found that the depth of

yield in the stiffener web could not be accurately ascertained. For such small stiffeners,

the panel strength will be slightly higher than the bare plate ultimate strength. Therefore,

for 5.2/ <ww th , ULTBEAM uses the following formula to predict the ultimate

strength:

2

)5.2/( 5.2/

)(

−+= =

wwplbarethULTBEAMplbareULTBEAM

thww

σσσσ (6.9)

52

7. Comparison of Ultimate Strength Predictions

Table 7.1 summarizes the results of Table 6.1 giving statistical comparisons of the

ultimate strength predictions of orthotropic theory based on outer surface stress, that

based on membrane stress and the beam-column method (as implemented in ULTBEAM)

compared to the finite element results (ABAQUS).

Table 7.1. Comparison of analytical ultimate strength predictions with FEA

Method Mean COV

Orthotropic surface stress

1.274 24.1 %

Orthotropic membrane stress

1.455 23.8 %

Beam-column method for stiffened panels (ULTBEAM)

1.028 2.9 %

The correlation of the three methods with the FE solutions is plotted in Fig. 7.1.

The orthotropic outer surface stress based results are optimistic for most cases. The

orthotropic membrane stress approach gives a collapse stress value that is nearly equal to

the material yield stress. This method is too optimistic and therefore not recommended

particularly for crossover panels. The beam-column method is unaffected by the

crossover proportions. We now consider what might be the reasons for the above errors

in the orthotropic based methods.

53

Fig. 7.1. Correlation of ultimate strength predictions with FEA

7.1. Orthotropic Plate Method – Outer Surface stress

Firstly, of its very nature, orthotropic plate theory is elastic, and does not consider

the growth and spread of plasticity. Secondly, orthotropic plate theory is based on a

regular buckled pattern of m x n half waves. If the stiffeners are small it will correctly

predict overall buckling, with m and n being small. If the stiffeners are large it will

correctly predict local plate buckling, with m being roughly B / (ns + 1) where ns is the

number of stiffeners. But in a crossover panel these two buckling modes are occurring

Y

FEAult

σσ ,

Y

ult

σσ

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Orthotropic outer surface stressOrthotropic membrane stressBeam-column method (ULTBEAM)Series4

54

together and orthotropic plate theory does not allow for two simultaneous elastic buckling

modes. Fig. 7.2 shows the normalized ultimate strength value from FEA and the

orthotropic strength based on surface stress plotted against λ, which is the slenderness

ratio of the stiffener with attached plating. As expected, the FEA ultimate strength

decreases sharply with λ, whereas the orthotropic surface stress based results remain

nearly unchanged.

In Fig. 7.2, Panel nos. P58, P59, P67, P68, P76 and P77 have been excluded. For

these panels the plate slenderness parameter β is 3.73, as seen in Table 2.1a. This is

unusually slender and permits the stiffeners to behave independently, which by itself is

sufficient reason for the orthotropic plate approach to have less accuracy.

Fig. 7.2. Ultimate strength using FEA and orthotropic surface stress

Ea Yσ

πρλ =

Y

ult

σσ

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.00 0.25 0.50 0.75 1.00 1.25 1.50

FEA

Orthotropic outersurface stress

55

Fig. 7.3 follows from Fig. 7.2 and plots the percent error in the orthotropic surface

stress results compared to FEA against λ. Whereas one would expect that the accuracy of

orthotropic plate theory would improve as the stiffener size decreases (larger λ) but here

it is the opposite. Again, this may be because orthotropic plate theory does not allow for

two simultaneous elastic buckling modes.

Fig. 7.3. Percent error in orthotropic surface stress method relative to FEA

7.2. Orthotropic Plate Method – Membrane Stress

The membrane stress based prediction is also orthotropic in nature. Orthotropic

plate theory cannot distinguish between plate-induced and stiffener-induced failure.

-20

0

20

40

60

80

100

0.00 0.25 0.50 0.75 1.00 1.25 1.50

% e

rror w

ith F

EA

Orthotropic outersurface stress

Ea Yσ

πρλ =

56

Because of the specified initial imperfection (Fig. 5.3) all the panels in this study

underwent stiffener-induced failure in which bending stress in the stiffener is a key

factor. In stiffener-induced failure the maximum bending stress is proportional to the

inverse of the section modulus flangex zIZ /= , where flangez is the distance from the

neutral axis for longitudinal bending to the stiffener flange. So the large overestimate of

ultimate strength in using the membrane approach could be due to neglecting the bending

stress. To investigate this Fig. 7.4 plots the percent error in the orthotropic membrane

stress based results for all crossover panels excluding the 6 panels with β = 3.73 versus

the ratio b2t / Z. A pronounced correlation is observed from the figure.

Fig. 7.4. Percent error in orthotropic membrane stress method relative to FEA

Ztb2

0

20

40

60

80

100

120

140

160

180

200

0 20 40 60 80 100 120 140 160 180 200

% e

rror w

ith F

EA

Orthotropic membrane stressSeries2

57

8. Collapse Mechanisms and Post-collapse for Crossover Panels

8.1. Collapse Modes

The von Mises stress distribution at midthickness can be portrayed in color-coded

plots using the post-processing software MSC Patran. In the last column of Table 6.1

labeled “Collapse Mode”, the collapse mechanisms at the ultimate load carrying capacity

of each panel are identified from these plots using the following nomenclature:

A : Stiffeners elastic in middle bay

B : Web partially plastic in middle bay

C : Approximate plastic hinge in middle bay

1 : Plate elastic in middle bay

2 : Plate corners yielded in middle bay

3 : Plate mid-longitudinal edges yielded in middle bay

4 : Plate gross yield in middle bay

a : Stiffeners elastic in end bay

b : Web partially plastic in end bay

c : Approximate plastic hinge in end bay

1 : Plate elastic in end bay

2 : Plate corners yielded in end bay

3 : Plate mid-longitudinal edges yielded in end bay

4 : Plate gross yield in end bay

58

As observed in Table 6.1 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, 2003). In Table 8.1, we present a broader classification of the 14

different collapse mechanisms occurring in the 55 crossover panels in 4 groups.

Table 8.1. Collapse mechanisms in crossover panels

END BAY

PLATE ELASTIC

END BAY

PLATE YIELDED

MID BAY PLATE

ELASTIC

Group I

C1a1 (9) C1c1 (1) B1a1 (5) C1b1 (3) ------ 18

Group II

C1a3 (1) B1a3 (2) C1a4 (4) ------ 7

MID BAY PLATE

YIELDED

Group III

C2a1 (1) C3a1 (4) C3b1 (6) ------ 11

Group IV

C2b2 (3) C3b3 (8) C3a4 (2) C4a4 (6) ------- 19

From Table 8.1, we see that 18 panels in Group I reaches their ultimate load with

yield in the stiffeners only, while the plate midthickness in both middle and end bay is

still elastic. This further illustrates that the orthotropic membrane stress prediction of

ultimate strength can be optimistic. Note that while the pattern and extent of plasticity in

the plating varies widely, “stiffener yield through web” is a common factor in all 55

cases. For the stiffeners in the middle bay the yield zone reached the full depth of the web

(“approximate plastic hinge”; first letter C) in 48 cases, and in the other 7 cases the yield

zone extended some distance into the web. The consistent presence of stiffener web yield

and the inconsistent presence of plate yield suggests that for stiffener-induced failure

“stiffener web yield” (say through 2/3 of the web height to be conservative) is a better

59

criterion for panel collapse than the “initial yield of plating” criteria that is used by the

orthotropic-based methods.

Figs. 8.1 to 8.4 show the von Mises stress distribution at midthickness at the

maximum load carrying capacity of one panel from each group.

Fig. 8.1. Collapse stress in P52 (Group I), FEAult ,σ = 263.3 MPa

von Mises stress at midthickness (MPa)

Stiffener yield through web

Plate elastic in both bays

60

Fig. 8.2. Collapse stress in P98 (Group II), FEAult ,σ = 330.3 MPa

Fig. 8.3. Collapse stress in P60 (Group III), FEAult ,σ = 272.8 MPa

von Mises stress at midthickness (MPa)

Stiffener yield through web

Plate yield in end bay

Plate elastic in mid bay

von Mises stress at midthickness (MPa)

Stiffener yield through web

Plate elastic in end bay

Plate yield in mid bay (ref Fig. 6.2)

61

Fig. 8.4. Collapse stress in P78 (Group IV), FEAult ,σ = 337.3 MPa

8.2. Stress – Axial Deflection 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 stress-end shortening curve was then drawn for every panel.

In Fig. 8.5, we present the curves for the 4 panels which are a good representation of

what we have seen for all the crossover panels in this study.

Stiffener yield through web

Plate yield in both bays

von Mises stress at midthickness (MPa)

62

Fig. 8.5. Inelastic stress – end shortening curves for crossover panels

All the crossover panels in this study collapsed due to a stiffener-induced failure

of the middle bay. The first loss of stiffness as shown in Fig. 8.5 is caused by progressive

yield through the stiffener 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 stiffener 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 (Fig. 6.2) or the four corners of the bay. In some

panels yielding in the stiffeners caused by shear was observed in the end bay. Although

this facilitated overall panel collapse, it is not considered to be a major cause.

0

0.2

0.4

0.6

0.8

1

0 0.00025 0.0005 0.00075 0.001 0.00125 0.0015)3(/1 au

P 52

P 60

P 98 P 78

First loss of stiffness Ultimate strength (Collapse)

Y

FEAx

σσ ,

63

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. 8.5, three out of the four panels have a stable post collapse

behavior. This was seen in 46 out of the 55 panels. The remaining 9 panels (P52, P56,

P62, P63, P69, P71, P104, P105, P106) suffered from a steep drop in load carrying

capacity similar to P52 shown in the figure.

64

9. Conclusions and Recommendations for Future Work

9.1. Conclusions

In this study, 55 stiffened panels with proportions suitable for use in ship design

which had simultaneous local and overall elastic buckling stresses were modeled and

analyzed using the finite element software ABAQUS. Modified expressions for elastic

local plate buckling and overall panel buckling expressions were derived and compared

to elastic FE bifurcation buckling results. An improved expression for prediction of

crossover proportions was derived and compared to the bifurcation results. Inelastic

RIKS analysis for ultimate collapse stress and post collapse behavior using ABAQUS

was carried out on these panels. Ultimate stress was also calculated using orthotropic-

based methods and a modified beam-column method for stiffened panels and compared

to the FE results. It was found that for panels having crossover proportions, orthotropic-

based methods are unsatisfactory and the beam-column method is most suitable for

ultimate stress prediction. Collapse patterns were studied and classified from von Mises

stress distribution at collapse and were not found to be unusual. Load-deflection diagrams

showed stable inelastic post collapse behavior for most panels and an abrupt drop in load

carrying capacity in only nine of the 55.

9.2. Recommendations for Future Work

The beam-column method has been found to be unaffected by the crossover

criteria and is therefore reliable for ultimate strength prediction of any stiffened panel

under uniaxial compression. It has also been shown by (Chen, 2003) to be accurate for

panels under transverse compression only. The method may therefore be extended to

evaluate the ultimate strength of a panel under biaxial compression, with lateral pressure.

65

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Klitchieff, J.M. (1951). On the stability of plates reinforced by longitudinal ribs,

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Newmark, N.M. (1943). Numerical procedure for computing deflections,

moments and buckling loads, Transaction, ASCE, Vol.108, pp.1161.

Paik, J.K., Thayamballi, A.K. (2000). Buckling strength of steel plating with

elastically restraining edges, Thin-Walled Structures, Vol.37, pp.27-55.

Paik, J.K. (2001). ULSAP user’s manual, Proteus Engineering, MD.

Paik, J.K., Thayamballi, A.K. (2003). Ultimate Limit State Design of Steel Plated

Structures, John Wiley & Sons.

Sheikh, I.A., Grondin, G.Y. and Elwi, A.E. (2002). Stiffened steel plates under

uniaxial compression, J. of Constructional Steel Research, Vol.58, pp.1061-1080.

Timoshenko, S. (1936). Theory of elastic stability, McGraw Hill, New York.

Troitsky, M.S. (1976). Stiffened panels : bending, stability and vibrations,

Elsevier, Amsterdam.

Tvergaard, V. (1973). Influence of post-buckling behavior on optimum design of

stiffened panels, Int. J. Solids Structures, Vol.9, pp.1519.

Tvergaard, V., Needleman, A. (1975). Buckling of eccentrically stiffened elastic-

plastic panels on two simple supports or multiply supported, Int. J. Solids Structures,

Vol.11, pp.647-663.

67

Appendix

This Appendix contains color-coded von Mises stress distribution gradients in the

midthickness of the cross-section of the crossover panels P50 to P107 (excluding P57,

P66 and P75) at collapse. All the panels were subjected to uniaxial compression and the

finite element program ABAQUS was used to obtain the ultimate strength as detailed in

Chapter 5. The stress distribution was plotted using the post processing software MSC

PATRAN. For each panel, the stress-end shortening curve as obtained from ABAQUS is

also plotted.

68

Fig. A1. Collapse stress in P50, 78.273, =FEAultσ MPa

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.0002 0.0004 0.0006 0.0008 0.001 0.0012 0.0014 0.0016 0.0018 0.002

end shortening / panel length

appl

ied

stre

ss /

Yiel

d st

ress

von Mises stress at midthickness (MPa)

Plate elastic in both bays

Stiffener yield through web

69

Fig. A2. Collapse stress in P51, 05.240, =FEAultσ MPa

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.0002 0.0004 0.0006 0.0008 0.001 0.0012 0.0014 0.0016 0.0018 0.002

end shortening / panel length

appl

ied

stre

ss /

Yiel

d st

ress

von Mises stress at midthickness (MPa)

Plate elastic in end bay

Plate yield in midbay (ref Fig. 6.2)

Stiffener yield through web

70

Fig. A3. Collapse stress in P52, 3.263, =FEAultσ MPa

von Mises stress at midthickness (MPa)

Plate elastic in both bays

Stiffener yield through web

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.0002 0.0004 0.0006 0.0008 0.001 0.0012 0.0014 0.0016 0.0018 0.002

end shortening / panel length

appl

ied

Stre

ss /

Yiel

d st

ress

Steep post-collapse load shedding characteristic observed.

71

Fig. A4. Collapse stress in P53, 38.226, =FEAultσ MPa

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.0002 0.0004 0.0006 0.0008 0.001 0.0012 0.0014 0.0016 0.0018 0.002

end shortening / panel length

appl

ied

stre

ss /

Yiel

d st

ress

von Mises stress at midthickness (MPa)

Plate elastic in both bays

Stiffener web partially plastic

72

Fig. A5. Collapse stress in P54, 15.177, =FEAultσ MPa

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.0002 0.0004 0.0006 0.0008 0.001 0.0012 0.0014 0.0016 0.0018 0.002

end shortening / panel length

appl

ied

stre

ss /

Yiel

d st

ress

von Mises stress at midthickness (MPa)

Plate elastic in both bays

Stiffener yield through web

73

Fig. A6. Collapse stress in P55, 8.142, =FEAultσ MPa

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.0002 0.0004 0.0006 0.0008 0.001 0.0012 0.0014 0.0016 0.0018 0.002

end shortening / panel length

appl

ied

stre

ss /

Yiel

d st

ress

von Mises stress at midthickness (MPa)

Plate elastic in both bays

Stiffener yield through web

74

Fig. A7. Collapse stress in P56, 9.202, =FEAultσ MPa

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.0002 0.0004 0.0006 0.0008 0.001 0.0012 0.0014 0.0016 0.0018 0.002

end shortening / panel length

appl

ied

stre

ss /

Yiel

d st

ress

Steep post-collapse load shedding characteristic observed.

von Mises stress at midthickness (MPa)

Plate elastic in both bays

Stiffener yield through web

75

Fig. A8. Collapse stress in P58, 48.107, =FEAultσ MPa

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.0002 0.0004 0.0006 0.0008 0.001 0.0012 0.0014 0.0016 0.0018 0.002

end shortening / panel length

appl

ied

stre

ss /

Yiel

d st

ress

von Mises stress at midthickness (MPa)

Plate corners yield in both bays

Stiffener yield through web

76

Fig. A9. Collapse stress in P59, 1.89, =FEAultσ MPa

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.0002 0.0004 0.0006 0.0008 0.001 0.0012 0.0014 0.0016 0.0018 0.002

end shortening / panel length

appl

ied

stre

ss /

Yiel

d st

ress

von Mises stress at midthickness (MPa)

Plate elastic in both bays

Stiffener yield through web

77

Fig. A10. Collapse stress in P60, 83.272, =FEAultσ MPa

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.0002 0.0004 0.0006 0.0008 0.001 0.0012 0.0014 0.0016 0.0018 0.002

end shortening / panel length

appl

ied

stre

ss /

Yiel

d st

ress

Plate elastic in end bay

Stiffener yield through web

Plate yield in mid bay (ref Fig. 6.2)

von Mises stress at midthickness (MPa)

78

Fig. A11. Collapse stress in P61, 3.239, =FEAultσ MPa

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.0002 0.0004 0.0006 0.0008 0.001 0.0012 0.0014 0.0016 0.0018 0.002

end shortening / panel length

appl

ied

stre

ss /

Yiel

d st

ress

Plate elastic in end bay

Plate yield in mid bay (ref Fig. 6.2)

Stiffener yield through web

von Mises stress at midthickness (MPa)

79

Fig. A12. Collapse stress in P62, 7.255, =FEAultσ MPa

Plate elastic in both bays

Stiffener yield through web

von Mises stress at midthickness (MPa)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.0002 0.0004 0.0006 0.0008 0.001 0.0012 0.0014 0.0016 0.0018 0.002

end shortening / panel length

appl

ied

stre

ss /

Yiel

d st

ress

Steep post-collapse load shedding characteristic observed.

80

Fig. A13. Collapse stress in P63, 63.226, =FEAultσ MPa

Plate elastic in both bays

Stiffener yield through web

von Mises stress at midthickness (MPa)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.0002 0.0004 0.0006 0.0008 0.001 0.0012 0.0014 0.0016 0.0018 0.002

end shortening / panel length

appl

ied

stre

ss /

Yiel

d st

ress

Steep post-collapse load shedding characteristic observed.

81

Fig. A14. Collapse stress in P64, 08.183, =FEAultσ MPa

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.0002 0.0004 0.0006 0.0008 0.001 0.0012 0.0014 0.0016 0.0018 0.002

end shortening / panel length

appl

ied

stre

ss /

Yiel

d st

ress

von Mises stress at midthickness (MPa)

Plate elastic in end bay

Plate yield in mid bay (ref Fig. 6.2) Stiffener yield

through web

82

Fig. A15. Collapse stress in P65, 43.209, =FEAultσ MPa

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.0002 0.0004 0.0006 0.0008 0.001 0.0012 0.0014 0.0016 0.0018 0.002

end shortening / panel length

appl

ied

stre

ss /

Yiel

d st

ress

von Mises stress at midthickness (MPa)

Plate elastic in both bays

Web partially plastic

83

Fig. A16. Collapse stress in P67, 23.115, =FEAultσ MPa

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.0002 0.0004 0.0006 0.0008 0.001 0.0012 0.0014 0.0016 0.0018 0.002

end shortening / panel length

appl

ied

stre

ss /

Yiel

d st

ress

von Mises stress at midthickness (MPa)

Plate corners yield in both bays

Stiffener yield through web

84

Fig. A17. Collapse stress in P68, 75.97, =FEAultσ MPa

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.0002 0.0004 0.0006 0.0008 0.001 0.0012 0.0014 0.0016 0.0018 0.002

end shortening / panel length

appl

ied

stre

ss /

Yiel

d st

ress

von Mises stress at midthickness (MPa)

Plate elastic in both bays

Stiffener yield through web

85

Fig. A18. Collapse stress in P69, 45.273, =FEAultσ MPa

Plate yield in endbay (ref Fig. 6.2)

Plate yield in midbay (ref Fig. 6.2)Stiffener yield

through web

von Mises stress at midthickness (MPa)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.0002 0.0004 0.0006 0.0008 0.001 0.0012 0.0014 0.0016 0.0018 0.002

end shortening / panel length

appl

ied

stre

ss /

Yiel

d st

ress

Steep post-collapse load shedding characteristic observed.

86

Fig. A19. Collapse stress in P70, 58.246, =FEAultσ MPa

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.0002 0.0004 0.0006 0.0008 0.001 0.0012 0.0014 0.0016 0.0018 0.002

end shortening / panel length

appl

ied

stre

ss /

Yiel

d st

ress

von Mises stress at midthickness (MPa)

Plate yield in mid bay (ref Fig. 6.2) Stiffener yield

through web

Plate elastic in end bay

87

Fig. A20. Collapse stress in P71, 35.261, =FEAultσ MPa

von Mises stress at midthickness (MPa)

Plate elastic in end bay

Plate yield in mid bay (ref Fig. 6.2) Stiffener yield

through web

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.0002 0.0004 0.0006 0.0008 0.001 0.0012 0.0014 0.0016 0.0018 0.002

end shortening / panel length

appl

ied

stre

ss /

Yiel

d st

ress

Steep post-collapse load shedding characteristic observed.

88

Fig. A21. Collapse stress in P72, 10.232, =FEAultσ MPa

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.0002 0.0004 0.0006 0.0008 0.001 0.0012 0.0014 0.0016 0.0018 0.002

end shortening / panel length

appl

ied

stre

ss /

Yiel

d st

ress

Plate yield in end bay (ref Fig. 6.2)

Plate yield in mid bay (ref Fig. 6.2) Stiffener yield

through web

von Mises stress at midthickness (MPa)

89

Fig. A22. Collapse stress in P73, 55.201, =FEAultσ MPa

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.0002 0.0004 0.0006 0.0008 0.001 0.0012 0.0014 0.0016 0.0018 0.002

end shortening / panel length

appl

ied

stre

ss /

Yiel

d st

ress

Plate elastic in end bay

Plate yield in mid bay (ref Fig. 6.2) Stiffener yield

through web

von Mises stress at midthickness (MPa)

90

Fig. A23. Collapse stress in P74, 33.217, =FEAultσ MPa

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.0002 0.0004 0.0006 0.0008 0.001 0.0012 0.0014 0.0016 0.0018 0.002

end shortening / panel length

appl

ied

stre

ss /

Yiel

d st

ress

von Mises stress at midthickness (MPa)

Plate elastic in both bays

Plate yield in mid bay (ref Fig. 6.2)

Stiffener yield through web

91

Fig. A24. Collapse stress in P76, 6.126, =FEAultσ MPa

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.0002 0.0004 0.0006 0.0008 0.001 0.0012 0.0014 0.0016 0.0018 0.002

end shortening / panel length

appl

ied

stre

ss /

Yiel

d st

ress

Plate elastic in end bay

Plate corners yield in mid bay

Stiffener yield through web

von Mises stress at midthickness (MPa)

92

Fig. A25. Collapse stress in P77, 7.113, =FEAultσ MPa

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.0002 0.0004 0.0006 0.0008 0.001 0.0012 0.0014 0.0016 0.0018 0.002

end shortening / panel length

appl

ied

stre

ss /

Yiel

d st

ress

von Mises stress at midthickness (MPa)

Plate corners yield in both bays

Stiffener yield through web

93

Fig. A26. Collapse stress in P78, 25.337, =FEAultσ MPa

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.0002 0.0004 0.0006 0.0008 0.001 0.0012 0.0014 0.0016 0.0018 0.002

end shortening / panel length

appl

ied

stre

ss /

Yiel

d st

ress

Plate gross yield in both bays

Stiffener yield through web

von Mises stress at midthickness (MPa)

94

Fig. A27. Collapse stress in P79, 5.330, =FEAultσ MPa

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.0002 0.0004 0.0006 0.0008 0.001 0.0012 0.0014 0.0016 0.0018 0.002

end shortening / panel length

appl

ied

stre

ss /

Yiel

d st

ress

von Mises stress at midthickness (MPa)

Plate gross yield in both bays

Stiffener yield through web

95

Fig. A28. Collapse stress in P80, 330, =FEAultσ MPa

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.0002 0.0004 0.0006 0.0008 0.001 0.0012 0.0014 0.0016 0.0018 0.002

end shortening / panel length

appl

ied

stre

ss /

Yiel

d st

ress

Plate gross yield in both bays

Stiffener yield through web

von Mises stress at midthickness (MPa)

96

Fig. A29. Collapse stress in P81, 25.330, =FEAultσ MPa

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.0002 0.0004 0.0006 0.0008 0.001 0.0012 0.0014 0.0016 0.0018 0.002

end shortening / panel length

appl

ied

stre

ss /

Yiel

d st

ress

Plate gross yield in end bay

Stiffener yield through web

Plate yield in mid bay (ref Fig. 6.2)

von Mises stress at midthickness (MPa)

97

Fig. A30. Collapse stress in P82, 318, =FEAultσ MPa

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.0002 0.0004 0.0006 0.0008 0.001 0.0012 0.0014 0.0016 0.0018 0.002

end shortening / panel length

appl

ied

stre

ss /

Yiel

d st

ress

Plate gross yield in end bay

Plate elastic in mid bay

Stiffener yield through web

von Mises stress at midthickness (MPa)

98

Fig. A31. Collapse stress in P83, 7.290, =FEAultσ MPa

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.0002 0.0004 0.0006 0.0008 0.001 0.0012 0.0014 0.0016 0.0018 0.002

end shortening / panel length

appl

ied

stre

ss /

Yiel

d st

ress

Plate elastic in end bay

Stiffener yield through web

Plate yield in mid bay (ref Fig. 6.2)

von Mises stress at midthickness (MPa)

99

Fig. A32. Collapse stress in P84, 8.309, =FEAultσ MPa

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.0002 0.0004 0.0006 0.0008 0.001 0.0012 0.0014 0.0016 0.0018 0.002

end shortening / panel length

appl

ied

stre

ss /

Yiel

d st

ress

Plate elastic in both bays

Stiffener yield through web

von Mises stress at midthickness (MPa)

100

Fig. A33. Collapse stress in P85, 9.236, =FEAultσ MPa

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.0002 0.0004 0.0006 0.0008 0.001 0.0012 0.0014 0.0016 0.0018 0.002

end shortening / panel length

appl

ied

stre

ss /

Yiel

d st

ress

The load-shedding portion of the curve could not be obtained because of numerical instability in the analysis after reaching the ultimate load.

Web partially plastic

Plate elastic in both bays

von Mises stress at midthickness (MPa)

101

Fig. A34. Collapse stress in P86, 53.202, =FEAultσ MPa

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.0002 0.0004 0.0006 0.0008 0.001 0.0012 0.0014 0.0016 0.0018 0.002

end shortening / panel length

appl

ied

stre

ss /

Yiel

d st

ress

Plate elastic in both bays

Web partially plastic

von Mises stress at midthickness (MPa)

102

Fig. A35. Collapse stress in P87, 63.151, =FEAultσ MPa

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.0002 0.0004 0.0006 0.0008 0.001 0.0012 0.0014 0.0016 0.0018 0.002

end shortening / panel length

appl

ied

stre

ss /

Yiel

d st

ress

Plate elastc in both bays

Stiffener yield through web

von Mises stress at midthickness (MPa)

103

Fig. A36. Collapse stress in P88, 5.333, =FEAultσ MPa

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.0002 0.0004 0.0006 0.0008 0.001 0.0012 0.0014 0.0016 0.0018 0.002

end shortening / panel length

appl

ied

stre

ss /

Yiel

d st

ress

Plate gross yield in end bay

Plate elastic in mid bay

Stiffener yield through web

von Mises stress at midthickness (MPa)

104

Fig. A37. Collapse stress in P89, 5.326, =FEAultσ MPa

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.0002 0.0004 0.0006 0.0008 0.001 0.0012 0.0014 0.0016 0.0018 0.002

end shortening / panel length

appl

ied

stre

ss /

Yiel

d st

ress

Plate gross yield in both bays

von Mises stress at midthickness (MPa)

Stiffener yield through web

105

Fig. A38. Collapse stress in P90, 75.325, =FEAultσ MPa

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.0002 0.0004 0.0006 0.0008 0.001 0.0012 0.0014 0.0016 0.0018 0.002

end shortening / panel length

appl

ied

stre

ss /

Yiel

d st

ress

Plate gross yield in both bays

Stiffener yield through web

von Mises stress at midthickness (MPa)

106

Fig. A39. Collapse stress in P91, 25.326, =FEAultσ MPa

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.0002 0.0004 0.0006 0.0008 0.001 0.0012 0.0014 0.0016 0.0018 0.002

end shortening / panel length

appl

ied

stre

ss /

Yiel

d st

ress

Plate gross yield in end bay

Plate yield in mid bay (ref Fig. 6.2) Stiffener yield

through web

von Mises stress at midthickness (MPa)

107

Fig. A40. Collapse stress in P92, 5.313, =FEAultσ MPa

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.0002 0.0004 0.0006 0.0008 0.001 0.0012 0.0014 0.0016 0.0018 0.002

end shortening / panel length

appl

ied

stre

ss /

Yiel

d st

ress

Plate elastic in mid bay Plate gross yield

in end bay

Stiffener yield through web

von Mises stress at midthickness (MPa)

108

Fig. A41. Collapse stress in P93, 83.284, =FEAultσ MPa

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 0.0002 0.0004 0.0006 0.0008 0.001 0.0012 0.0014 0.0016 0.0018 0.002

end shortening / panel length

appl

ied

stre

ss /

Yiel

d st

ress

Plate yield in end bay (ref Fig. 6.2)

Plate yield in mid bay (ref Fig. 6.2) Stiffener yield

through web

von Mises stress at midthickness (MPa)

109

Fig. A42. Collapse stress in P94, 25.297, =FEAultσ MPa

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.0002 0.0004 0.0006 0.0008 0.001 0.0012 0.0014 0.0016 0.0018 0.002

end shortening / panel length

appl

ied

stre

ss /

Yiel

d st

ress

Plate elastic in both bays

Stiffener yield through web

von Mises stress at midthickness (MPa)

110

Fig. A43. Collapse stress in P95, 65.237, =FEAultσ MPa

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.0002 0.0004 0.0006 0.0008 0.001 0.0012 0.0014 0.0016 0.0018 0.002

end shortening / panel length

appl

ied

stre

ss /

Yiel

d st

ress

The load-shedding portion of the curve could not be obtained because of numerical instability in the analysis after reaching the ultimate load.

Plate yield in end bay (ref Fig. 6.2)

Plate elastic in mid bay

Web partially plastic

von Mises stress at midthickness (MPa)

111

Fig. A44. Collapse stress in P96, 03.199, =FEAultσ MPa

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.0002 0.0004 0.0006 0.0008 0.001 0.0012 0.0014 0.0016 0.0018 0.002

end shortening / panel length

appl

ied

stre

ss /

Yiel

d st

ress

Plate elastic in both bays

Web partially plastic

von Mises stress at midthickness (MPa)

112

Fig. A45. Collapse stress in P97, 4.160, =FEAultσ MPa

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.0002 0.0004 0.0006 0.0008 0.001 0.0012 0.0014 0.0016 0.0018 0.002

end shortening / panel length

appl

ied

stre

ss /

Yiel

d st

ress

Plate elastic in end bay

Plate yield in mid bay (ref Fig. 6.2)

Stiffener yield through web

von Mises stress at midthickness (MPa)

113

Fig. A46. Collapse stress in P98, 25.330, =FEAultσ MPa

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.0002 0.0004 0.0006 0.0008 0.001 0.0012 0.0014 0.0016 0.0018 0.002

end shortening / panel length

appl

ied

stre

ss /

Yiel

d st

ress

Plate gross yield in end bay

Plate elastic in mid bay

Stiffener yield through web

von Mises stress at midthickness (MPa)

114

Fig. A47. Collapse stress in P99, 75.320, =FEAultσ MPa

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.0002 0.0004 0.0006 0.0008 0.001 0.0012 0.0014 0.0016 0.0018 0.002

end shortening / panel length

appl

ied

stre

ss /

Yiel

d st

ress

von Mises stress at midthickness (MPa)

Plate gross yield in end bay

Plate yield in mid bay (ref Fig. 6.2)

Stiffener yield through web

115

Fig. A48. Collapse stress in P100, 5.322, =FEAultσ MPa

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.0002 0.0004 0.0006 0.0008 0.001 0.0012 0.0014 0.0016 0.0018 0.002

end shortening / panel length

appl

ied

stre

ss /

Yiel

d st

ress

von Mises stress at midthickness (MPa)

Plate gross yield in both bays

Stiffener yield through web

116

Fig. A49. Collapse stress in P101, 0.323, =FEAultσ MPa

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.0002 0.0004 0.0006 0.0008 0.001 0.0012 0.0014 0.0016 0.0018 0.002

end shortening / panel length

appl

ied

stre

ss /

Yiel

d st

ress

Plate yield in end bay (ref Fig. 6.2)

Plate yield in mid bay (ref Fig. 6.2) Stiffener yield

through web

von Mises stress at midthickness (MPa)

117

Fig. A50. Collapse stress in P102, 0.310, =FEAultσ MPa

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.0002 0.0004 0.0006 0.0008 0.001 0.0012 0.0014 0.0016 0.0018 0.002

end shortening / panel length

appl

ied

stre

ss /

Yiel

d st

ress

Plate yield in end bay (ref Fig. 6.2)

Plate elastic in mid bay

Stiffener yield through web

von Mises stress at midthickness (MPa)

118

Fig. A51. Collapse stress in P103, 3.287, =FEAultσ MPa

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.0002 0.0004 0.0006 0.0008 0.001 0.0012 0.0014 0.0016 0.0018 0.002

end shortening / panel length

appl

ied

stre

ss /

Yiel

d st

ress

Plate yield in end bay (ref Fig. 6.2)

Plate yield in mid bay (ref Fig. 6.2) Stiffener yield

through web

von Mises stress at midthickness (MPa)

119

Fig. A52. Collapse stress in P104, 05.289, =FEAultσ MPa

Plate yield in end bay (ref Fig. 6.2)

Plate yield in mid bay (ref Fig. 6.2) Stiffener yield

through web

von Mises stress at midthickness (MPa)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.0002 0.0004 0.0006 0.0008 0.001 0.0012 0.0014 0.0016 0.0018 0.002

end shortening / panel length

appl

ied

stre

ss /

Yiel

d st

ress

Steep post-collapse load shedding characteristic observed.

120

Fig. A53. Collapse stress in P105, 1.294, =FEAultσ MPa

Plate elastic in both bays

Stiffener yield through web

von Mises stress at midthickness (MPa)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.0002 0.0004 0.0006 0.0008 0.001 0.0012 0.0014 0.0016 0.0018 0.002

end shortening / panel length

appl

ied

stre

ss /

Yiel

d st

ress

Steep post-collapse load shedding characteristic observed.

121

Fig. A54. Collapse stress in P106, 43.234, =FEAultσ MPa

Plate yield in end bay (ref Fig. 6.2)

Plate elastic in mid bay

Stiffener yield through web

von Mises stress at midthickness (MPa)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.0002 0.0004 0.0006 0.0008 0.001 0.0012 0.0014 0.0016 0.0018 0.002

end shortening / panel length

appl

ied

stre

ss /

Yiel

d st

ress

Steep post-collapse load shedding characteristic observed.

122

Fig. A55. Collapse stress in P107, 4.207, =FEAultσ MPa

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.0002 0.0004 0.0006 0.0008 0.001 0.0012 0.0014 0.0016 0.0018 0.002end shortening / panel length

appl

ied

stre

ss /

Yiel

d st

ress

The load-shedding portion of the curve could not be obtained because of numerical instability in the analysis after reaching the ultimate load.

Plate yield in end bay (ref Fig. 6.2)

Plate yield in mid bay (ref Fig. 6.2) Stiffener yield

through web

von Mises stress at midthickness (MPa)

123

Vita

Biswarup Ghosh, the son of Bimalendu and Basanti Ghosh, was born in Calcutta,

India on May 27, 1974. He graduated with a Bachelor’s degree in Marine Engineering

from the Marine Engineering and Research Institute, Calcutta in 1996. He was employed

with Seaarland Shipping Management GmbH. as a sea-going engineer from 1996 to

2001. Biswarup Ghosh came to Virginia Tech in Fall 2001 to begin his study towards a

Master of Science degree with the Department of Aerospace and Ocean Engineering.

This thesis completes his MS degree in Ocean Engineering from Virginia Tech.