Validation of the ULSAP Closed-Form Method for Ultimate Strength Analysis of Cross-Stiffened Panels

Samuel M. Dippold

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

Date June 20, 2005

Blacksburg, Virginia

Validation of the ULSAP Closed-Form Method for Ultimate Strength Analysis of Cross-Stiffened Panels

Samuel M. Dippold

(ABSTRACT)

This thesis presents the results of 67 ABAQUS elasto-plastic Riks ultimate strength

analyses of cross-stiffened panels. These panels cover a wide range of typical geometries.

Uniaxial compression is applied to the panels, and in some cases combined with lateral pressure.

For eight of the panels full-scale experimental results are available, and these verified the

accuracy of the ABAQUS results. The 67 ABAQUS results were then compared to the ultimate

strength predictions from the computer program ULSAP. In all but 10 cases the ULSAP predicted

strength is within 30% of the ABAQUS value, and in all but 4 cases the predicted failure mode

also agrees with that of ABAQUS. In one case the ULSAP predicted ultimate strength is 51%

below the experimental value, and so this case is studied in detail. The discrepancy is found to

be caused by the method which ULSAP uses for panels that experience overall collapse initiated

by beam-column-type failure. The beam-column method program ULTBEAM is used to predict

the ultimate strength of the 61 panels that ULSAP predicts to fail due to overall collapse of the

stiffeners and plating which may or may not be triggered by yielding of the plate-stiffener

combination at the midspan (Mode III or III-1). ULTBEAM is found to give more accurate results

than ULSAP for Mode III or III-1 failure. Future work is recommended to incorporate ULTBEAM

into ULSAP to predict the ultimate strength of panels that fail in Mode III or III-1.

Acknowledgements

I would like to thank my advisor and committee chair Dr. Owen Hughes for all of his

support and guidance over the past two years.

I would also like to thank Dr. Alan Brown and Dr. Eric Johnson for being members of my

advisory committee. In addition, I would like to thank Dr. William Hallauer for being Dr. Johnson’s

proxy.

Finally, I would also like to thank Jason Albright and Dhaval Makhecha for all of their help

with this work.

iii

Table of Contents

LIST OF FIGURES ____________________________________________________________ V

LIST OF TABLES ____________________________________________________________ VI

NOMENCLATURE ___________________________________________________________ VII

1. INTRODUCTION _________________________________________________________ 1

2. 1½ BAY ABAQUS MODEL FOR ULTIMATE STRENGTH ANALYSIS OF 3 BAY PANEL 5

2.1. ABAQUS – MODIFIED RIKS ANALYSIS ___________________________________________ 6 2.2. MATERIAL PROPERTIES________________________________________________________ 6 2.3. FINITE ELEMENTS ____________________________________________________________ 7 2.4. INITIAL IMPERFECTIONS _______________________________________________________ 8 2.5. BOUNDARY CONDITIONS ______________________________________________________ 9 2.6. GEOMETRIC PROPERTIES______________________________________________________ 10

3. SMITH PANEL ABAQUS MODELS FOR ULTIMATE STRENGTH ANALYSIS________ 12

3.1. SMITH PANEL PROPERTIES ____________________________________________________ 13 3.2. FINITE ELEMENTS ___________________________________________________________ 14 3.3. SMITH PANEL INITIAL IMPERFECTIONS ___________________________________________ 14 3.4. RESIDUAL STRESS ___________________________________________________________ 16 3.5. BOUNDARY CONDITIONS _____________________________________________________ 18

4. ANALYTICAL METHODS FOR ULTIMATE STRENGTH ANALYSIS _______________ 21

4.1. OVERALL COLLAPSE (MODE I) _________________________________________________ 21 4.2. BEAM-COLUMN COLLAPSE (MODE III)___________________________________________ 24 4.3. ULTBEAM________________________________________________________________ 25

5. COMPARISON OF ULTIMATE STRENGTH PREDICTIONS ______________________ 27

5.1. 1½ BAY MODEL RESULTS ____________________________________________________ 27 5.2. SMITH PANEL RESULTS_______________________________________________________ 31 5.3. SMITH PANEL 6 _____________________________________________________________ 33

6. CONCLUSIONS AND RECOMMENDATIONS FOR FUTURE WORK _______________ 36

6.1. CONCLUSIONS ______________________________________________________________ 36 6.2. RECOMMENDATIONS FOR FUTURE WORK _________________________________________ 36

REFERENCES ______________________________________________________________ 37

APPENDIX A _______________________________________________________________ 38

APPENDIX B _______________________________________________________________ 69

VITA ______________________________________________________________________ 78

iv

List of Figures

Figure 1.1 A three-bay stiffened panel under uniaxial compression .............................................. 1 Figure 1.2 Stiffened panel under uniaxial compression and lateral pressure ................................ 1 Figure 1.3 Cross-section of a plate-stiffener combination.............................................................. 2 Figure 1.4 Mode I - Overall collapse of a cross-stiffened panel ..................................................... 3 Figure 1.5 Mode II - Collapse due to biaxial compression ............................................................. 3 Figure 1.6 Mode III - Beam-column-type collapse.......................................................................... 3 Figure 1.7 Mode III-2 - Local S-shaped mechanism ...................................................................... 4 Figure 1.8 Mode IV - Local buckling of stiffener web ..................................................................... 4 Figure 1.9 Mode V - Stiffener tripping ............................................................................................ 4 Figure 2.1 Symmetry about midspan of middle bay in 3 bay panel ............................................... 5 Figure 2.2 Proportional loading with an unstable response (ABAQUS User's manual)................. 6 Figure 2.3 Idealized elastic-perfectly plastic stress-strain curve .................................................... 7 Figure 2.4 Finite element mesh for 1½ bay model ......................................................................... 8 Figure 2.5 Initial imperfections in the 1½ bay panel models .......................................................... 9 Figure 2.6 Transverse and longitudinal edges of a stiffened panel.............................................. 10 Figure 3.1 Finite element mesh for Smith panel 6........................................................................ 14 Figure 3.2 Initial imperfections in Smith panel models ................................................................. 15 Figure 3.3 Photo of the post collapse condition of Smith panel 6 ................................................ 16 Figure 3.4 Initial Imperfections of Smith Panel 6.......................................................................... 16 Figure 3.5 Residual stress in stiffened panels (Smith, 1975)....................................................... 17 Figure 3.6 LTF - Large Testing Frame (Smith, 1975)................................................................... 19 Figure 3.7 Transverse and longitudinal edges of a stiffened panel.............................................. 20 Figure 4.1 Membrane stress in an orthotropic plate..................................................................... 23 Figure 4.2 Plate induced failure (a) & stiffener induced failure (b) in a beam-column ................. 24 Figure 5.1 Panel vt009 at collapse ............................................................................................... 30 Figure 5.2 Subdivision of cross-section into 'fibres' (Smith, 1992)............................................... 32 Figure 5.3 Subdivision of stiffened panel into elements (Smith, 1992) ........................................ 32 Figure 5.4 Weld-induced residual stress in a stiffened panel (Smith, 1992) ................................ 32 Figure 5.5 Photo of the post collapse condition of Smith panel 6 ................................................ 35 Figure 5.6 ABAQUS results for Smith panel 6 at collapse ........................................................... 35

v

List of Tables

Table 1.1 Modes of Failure (Paik and Thayamblalli, 2003) & (Hughes, 1988) .............................. 2 Table 2.1 Material Properties for 1½ Bay Panels........................................................................... 7 Table 2.2 Elements in 1½ bay finite element model of 3 bay panel............................................... 8 Table 2.3 Boundary Conditions for 1½ Bay Models ..................................................................... 10 Table 2.4 Geometric Properties of 1½ Bay Panels ...................................................................... 11 Table 3.1 Geometric and Material Properties of the Smith Panels .............................................. 13 Table 3.2 Elements in FE Model of Smith Panel 6....................................................................... 14 Table 3.3 Residual Stress in Smith Panels .................................................................................. 18 Table 3.4 Boundary Conditions for Full Panel ABAQUS Model.................................................. 19 Table 5.1 1½ Bay Model ABAQUS and ULSAP Results ............................................................. 27 Table 5.2 Smith Panel Results ..................................................................................................... 31 Table 5.3 ULTBEAM vs. ULSAP for Smith panels not failing by Mode III or III-1 ........................ 33

vi

Nomenclature

Geometric Properties a length of one bay; spacing between adjacent transverse frames

Am amplitude of the added lateral deflection due to load

A0m amplitude of the initial deflection

Asx cross-sectional area of longitudinal stiffeners

Asy cross-sectional area of transverse frames

b spacing between adjacent longitudinal stiffeners

B breadth of stiffened panel

bf flange breadth

hw web height

M0 initial bending moment

ns number of longitudinal stiffeners

nf number of transverse frames

t plate thickness

tf flange thickness

tw web thickness

teq equivalent plate thickness

wos initial deflection of a stiffener

wosm maximum initial deflection of a stiffener

wot initial sideways tilt of a stiffener

wotm maximum initial sideways tilt of a stiffener

wopl initial deflection of the plating between longitudinal stiffeners

woplm maximum initial deflection of the plating between longitudinal stiffeners

Z section modulus

0δ initial deflection

∆ total eccentricity

η eccentricity ratio

λ column slenderness parameter

µ dead load bending term

ρx correction factor to account for variation in the true deflection pattern from the assumed

sinusoidal pattern

vii

Material Properties and Strength Parameters E Young’s modulus

Ex effective Young’s modulus in the x direction

Ey effective Young’s modulus in the y direction

p applied lateral pressure

ν Poisson’s ratio

σx applied longitudinal compressive stress

σY yield stress

σYp yield stress of plating

σYs yield stress of longitudinal stiffeners

σYf yield stress of transverse frames

σYeq equivalent yield stress of plating, longitudinal stiffeners, and transverse frames

σrc compressive residual stress

σYC yield stress of plating under compression due to residual stress

σYT yield stress of plating under tension due to residual stress

σYT equivalent yield stress for orthogonally stiffened panels

σx applied longitudinal compressive stress

viii

1. Introduction

Modern ships encounter extreme loads while performing daily tasks and must have

adequate structural strength. In order to provide the necessary strength, while minimizing cost,

most ships today are constructed using stiffened panels. Stiffened panels are generally

comprised of a plate, longitudinal stiffeners, and transverse frames. Each section between

transverse frames is referred to as a bay. For example, Figure 1.1 is a three-bay stiffened panel.

Although there are many types of stiffeners, T-shaped stiffeners and frames will be used in this

thesis and are illustrated in Figure 1.3.

Stiffened panels are designed to support axial loads as well as lateral pressure as shown

in Figure 1.1 and Figure 1.2. For the purposes of this thesis, stiffened panels will be analyzed

using uniaxial compression in the longitudinal direction and in some cases combined with lateral

pressure.

Figure 1.1 A three-bay stiffened panel under uniaxial compression

Figure 1.2 Stiffened panel under uniaxial compression and lateral pressure

1

Figure 1.3 Cross-section of a plate-stiffener combination

Stiffener and frame scantlings (geometric properties) and spacings have a great influence

on the strength of a panel as well as the way in which the panel fails. Collapse behavior of

stiffened panels can be divided into six different modes (Paik and Thayamblalli, 2003). A seventh

mode (III-2) was added from (Hughes, 1988) to help describe a mode of failure that is often

mistaken for tripping. These modes are listed in Table 1.1 and are shown in Figure 1.4–Figure

1.9.

Table 1.1 Modes of Failure (Paik and Thayamblalli, 2003) & (Hughes, 1988)

Mode Type Description

I overall collapse plating and stiffeners collapse together as a unit

II biaxial compression collapse plate-stiffener intersection yields at the corners of plating between stiffeners

III beam-column-type collapse plate-stiffener combination yields at midspan

III-1 overall collapse initiated by beam-column-type collapse

plate-stiffener combination yields at midspan and leads to plating and stiffeners collapsing together as a unit

III-2 local S-shaped mechanism a local plastic mechanism forms in the flange causing a local sideways deflection due to flange yielding

IV local buckling of stiffener web stiffened panel reaches ultimate strength immediately after stiffener web buckles locally

V Stiffener tripping stiffened panel fails immediately after lateral-torsional buckling (tripping) of stiffeners

2

Figure 1.4 Mode I - Overall collapse of a cross-stiffened panel

Figure 1.5 Mode II - Collapse due to biaxial compression

Figure 1.6 Mode III - Beam-column-type collapse

3

Figure 1.7 Mode III-2 - Local S-shaped mechanism

Figure 1.8 Mode IV - Local buckling of stiffener web

Figure 1.9 Mode V - Stiffener tripping

4

2. 1½ Bay ABAQUS Model for Ultimate Strength Analysis of 3 Bay Panel

(Ghosh, 2003) determined that in inelastic analysis a 1 bay panel model can be

misleading. Axial compression causes a panel to deflect upward and downward in alternating

bays. A bay with a downward deflection will have stiffener-induced failure while a bay with an

upward deflection will have plate-induced failure. A multi-bay panel with equal upward and

downward deflections will have stiffener-induced failure (Chen, 2003). Modeling only 1 bay can

be misleading based on the fact that the analysis may indicate plate-induced failure due to the

initial eccentricity. A 3 bay model incorporates both upward and downward deflections and is

therefore more suitable for this type of analysis. Moreover, transverse frames cannot merely be

modeled as a simply supported loaded edge as the actual boundary conditions at a frame are

between simply supported and clamped boundary conditions. This further demonstrates the

need for a 3 bay model.

Symmetry about the midspan of the middle bay of a 3 bay model can be used to

minimize the complexity of the analysis thereby reducing the time necessary to generate results.

This symmetry results in a 1½ bay model, as shown in Figure 2.1

Figure 2.1 Symmetry about midspan of middle bay in 3 bay panel

The ABAQUS analysis of the 1½ bay panel models was done following the same

procedure outlined in (Ghosh, 2003). Uniaxial compression in the direction of the longitudinal

stiffeners was applied to the model as concentrated nodal forces. A “dead load” was applied to

the model before a “live load” was applied during the Riks analysis.

5

2.1. ABAQUS – Modified Riks Analysis

The finite element analysis (FEA) program ABAQUS was used in order to perform

inelastic analyses on stiffened panels. More specifically, a static analysis was done in ABAQUS

using the modified Riks method (ABAQUS, 2002). This method is valid for cases where the load

magnitude is proportional to a single scalar parameter. This scalar, λ , is known as the load

proportionality factor. The current loading on the model, , can be determined using the

following equation:

totalP

( )00 PPPP reftotal −+= λ

Here, is the dead load and is the reference load vector. 0P refP

The modified Riks method is also capable of providing a solution for cases with unstable

responses. This type of response is shown in Figure 2.2. This capability was unnecessary in all

but one of the ABAQUS analyses evaluated for this thesis.

Figure 2.2 Proportional loading with an unstable response (ABAQUS, 2002)

2.2. Material Properties

The stiffened panels analyzed using the 1½ bay models had material properties as

shown in Table 2.1.

6

Table 2.1 Material Properties for 1½ Bay Panels

Material Mild Steel

Young’s Modulus (E) 205800 MPa

Poisson’s Ratio (ν ) 0.3

Yield Stress ( Yσ ) 352.8 MPa

The ABAQUS models incorporated an idealized elastic-perfectly plastic stress-strain

curve as shown in Figure 2.3.

Figure 2.3 Idealized elastic-perfectly plastic stress-strain curve

2.3. Finite Elements

A typical finite element mesh for a 1½ bay model is shown in Figure 2.4. The number of

elements in the plate and longitudinal stiffeners can be found in Table 2.2. These elements are

4-node S4 shell elements (ABAQUS, 2002). The model is discretized into a sufficient number of

elements to adequately represent the deformation and stress gradients, and is verified in (Chen,

2003).

7

Figure 2.4 Finite element mesh for 1½ bay model

Table 2.2 Elements in 1½ bay finite element model of 3 bay panel

Panel Part # Elements Plate 120 x 96

Stiffener Web 120 x 5

Stiffener Flange 120 x 6

2.4. Initial Imperfections

Imperfections due to fabrication are common in stiffened panels. To help make the

ABAQUS analyses as realistic as possible, initial deformations were added to the panel models.

The shapes of the initial imperfections in the panels were represented using the following

equations:

⎟⎠⎞

⎜⎝⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛=

⎟⎠⎞

⎜⎝⎛=

⎟⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛=

ax

hzww

axww

by

axww

wotmot

osmos

oplmopl

π

π

ππ

sinsin

sin

sin3sin

(1)

These equations produce panel deformations as shown in Figure 1.1. Values for woplm,

wosm, and wotm were obtained from standard values used by Det Norske Veritas (DNV, 2003).

DNV uses the following formulas to obtain maximum deflection values:

8

1000

200aww

bw

ostmosm

oplm

==

=

These values are somewhat optimistic (i.e. low compared to values in actual stiffened

panels), so values from (Chen, 2003) were also used.

Figure 2.5 Initial imperfections in the 1½ bay panel models

2.5. Boundary Conditions

Large girders or similar structures usually support the loaded edges and sides of stiffened

panels in ship structures. These girders provide resistance to displacement in the direction

normal to the plating (the z direction) and also provide some resistance to rotation of the plating

about the panel edges. Clamped boundary conditions are optimistic and allow no rotation about

the plate edges. Simple support boundary conditions are pessimistic in that there is no

restraining moment about the plate edges, which allows free rotation about the plate edges.

Simple support boundary conditions were used in these ABAQUS models because these

boundary conditions are more realistic and provide a margin of safety. These boundary

conditions can be found in Table 2.3. In this table, 1, 2, & 3 denote translational restraints in the

9

x, y, and z directions, respectively. Rotational restraints about the x, y, and z axes are denoted

by 4, 5, & 6, respectively.

Table 2.3 Boundary Conditions for 1½ Bay Models

Loaded Edge 3,4,6 Symmetric Edge 1 Sides 3,5,6 Loaded & Symmetric Edge Mid Nodes 2 Side Mid Nodes 1 Frame 3

Figure 2.6 Transverse and longitudinal edges of a stiffened panel

2.6. Geometric Properties

Table 2.4 lists the scantlings of the panels analyzed in this study.

10

Table 2.4 Geometric Properties of 1½ Bay Panels (mm)

Panel No. a b t hw tw bf tf woplm wosm

vt001 2640 900 21 90 12 100 15 0.5 6.6 vt002 2640 900 21 123 12 100 15 0.5 6.6 vt003 2640 900 21 150 12 100 15 0.5 6.6 vt004 3600 900 21 75 12 100 15 0.5 9 vt005 3600 900 21 90 12 100 15 0.5 9 vt006 3600 900 21 123 12 100 15 0.5 9 vt007 3600 900 21 150 12 100 15 0.5 9 vt008 1800 900 21 75 12 100 15 0.5 4.5 vt009 1800 900 21 90 12 100 15 0.5 4.5 vt010 1800 900 21 123 12 100 15 0.5 4.5 vt011 1800 900 21 150 12 100 15 0.5 4.5 vt012 2640 900 15 165 12 100 15 4.5 2.64 vt013 2640 900 21 150 12 100 15 4.5 2.64 vt014 2640 900 21 105 12 100 15 4.5 2.64 vt015 2640 900 21 90 12 100 15 4.5 2.64 vt016 2640 900 15 260 12 100 15 4.5 2.64 vt017 2640 900 21 90 12 100 15 4.5 0 vt018 2640 900 21 123 12 100 15 4.5 2.64 vt019 3600 900 21 75 12 100 15 4.5 3.6 vt020 3600 900 21 90 12 100 15 4.5 3.6 vt021 3600 900 21 123 12 100 15 4.5 3.6 vt022 3600 900 21 150 12 100 15 4.5 3.6 vt023 1800 900 21 48 12 100 15 4.5 1.8 vt024 1800 900 21 75 12 100 15 4.5 1.8 vt025 1800 900 21 90 12 100 15 4.5 1.8 vt026 1800 900 21 123 12 100 15 4.5 1.8 vt027 1800 900 21 150 12 100 15 4.5 1.8 vt028 2640 900 21 150 12 100 15 4.5 6.6 vt029 2640 900 21 90 12 100 15 4.5 6.6 vt030 2640 900 21 123 12 100 15 4.5 6.6

11

Geometric Properties of 1½ Bay Panels (mm)

Panel No.

a b t hw tw bf tf woplm wosm

vt031 2640 900 21 75 12 100 15 4.5 6.6 vt032 2640 900 21 60 12 100 15 4.5 6.6 vt033 2640 900 21 168 12 100 15 4.5 6.6 vt034 3600 900 21 75 12 100 15 4.5 9 vt035 3600 900 21 90 12 100 15 4.5 9 vt036 3600 900 21 123 12 100 15 4.5 9 vt037 3600 900 21 150 12 100 15 4.5 9 vt038 1800 900 21 75 12 100 15 4.5 4.5 vt039 1800 900 21 90 12 100 15 4.5 4.5 vt040 1800 900 21 123 12 100 15 4.5 4.5 vt041 1800 900 21 150 12 100 15 4.5 4.5 vt042 2640 900 21 90 12 100 15 9 2.64 vt043 2640 900 21 123 12 100 15 9 2.64 vt044 2640 900 21 150 12 100 15 9 2.64 vt045 2640 900 21 180 12 100 15 9 2.64 vt046 2640 900 21 123 12 100 15 9 6.6 vt047 2640 900 21 150 12 100 15 9 6.6 vt048 2640 900 21 90 12 100 15 9 6.6 vt049 2640 900 21 75 12 100 15 9 6.6 vt050 2640 900 21 60 12 100 15 9 6.6 vt051 2640 900 21 168 12 100 15 9 6.6 vt052 3600 900 21 75 12 100 15 9 9 vt053 3600 900 21 90 12 100 15 9 9 vt054 3600 900 21 123 12 100 15 9 9 vt055 3600 900 21 150 12 100 15 9 9 vt056 1800 900 21 75 12 100 15 9 4.5 vt057 1800 900 21 90 12 100 15 9 4.5 vt058 1800 900 21 123 12 100 15 9 4.5 vt059 1800 900 21 150 12 100 15 9 4.5

3. Smith Panel ABAQUS Models for Ultimate Strength Analysis

In 1975 C. S. Smith conducted what is widely regarded as the most comprehensive

experimental study of the compressive strength of stiffened panels (Smith, 1975). Twelve full

scale welded grillages were constructed and subjected to axial compression, and in some cases

combined axial compression and lateral loads.

The 1½ bay models used in Section 2 are insufficient to fully capture all aspects of the

collapse of the Smith panels. These models only represent 3 bay panels, not 4-5 bay panels like

the Smith panels. Instead of using a physical representation of the transverse frames, the 1½

bay models use simple support boundary conditions in their place. Full panel models were

constructed to analyze several of the Smith panels. These models varied in size between 4-5

bays and included transverse frames.

12

The modified Riks method, as described in Section 2.1, was used in the ABAQUS

analyses of the Smith panels.

3.1. Smith Panel Properties

The scantlings of the stiffened panels from the Smith tests are shown in Table 3.1. The

panels were constructed of mild steel with a Young’s modulus (E) of 205,800 MPa and a

Poisson’s ratio (ν ) of 0.3. The yield stresses in the panels varied and are also shown in Table

3.1. Some of the experiments included lateral pressure in addition to axial compression, and this

value is shown for those cases in the table as well.

Table 3.1 Geometric and Material Properties of the Smith Panels

Panel No. 1a 1b 2a 2b 3a 3b 5 6

a (mm) 1219 1219 1524 1524 1524 1524 1524 1219 b (mm) 610 610 305 305 305 305 610 610 t (mm) 8.0 7.9 7.7 7.4 6.4 6.4 6.4 6.3 woplm (mm) 0.37 0.47 0.13 1.83 1.98 4.57 0.61 3.0 σY (MPa) 252 256 266 264 255 256 252 261 p (MPa) 0 0.1 0.05 0 0.02 0 0 0

ns 4 4 9 9 9 9 4 4 hw (mm) 139.5 138.2 106.1 104.8 71.4 70.9 106.6 69.9 tw (mm) 7.2 7.1 5.4 5.4 4.5 4.7 5.3 4.6 bf (mm) 79.0 76.2 46.0 44.7 25.9 27.9 46.2 27.4 tf (mm) 14.2 14.2 9.5 9.5 6.4 6.4 9.5 6.4

wosm (mm) 1.83 1.83 3.81 1.52 4.27 2.90 1.22 12.0 Long

itudi

nal

Stif

fene

rs

σY (MPa) 258 255 273 280 232 227 235 246 nf 4 4 3 3 3 3 3 4

hw (mm) 239.3 235.7 188.7 187.5 142.0 139.7 140.0 105.0 tw (mm) 9.4 9.1 8.3 8.3 6.8 6.9 6.8 5.4 bf (mm) 125.5 127.0 102.6 102.6 79.0 79.3 77.2 46.2 tf (mm) 18.3 18.3 16.3 16.3 14.2 14.2 14.2 9.5 Tr

ansv

erse

Fr

ames

σY (MPa) 286 284 249 239 270 278 275 270

As with the 1½ bay ABAQUS models, the Smith panel models assumed an idealized

elastic-perfectly plastic stress-strain curve as shown in Figure 2.3.

13

3.2. Finite Elements

A typical finite element mesh for one of the full panel models is shown in Figure 3.1. The

number of elements in the plate, longitudinal stiffeners, and transverse frames of a typical full

panel model can be found in Table 3.2. As with the 1½ bay model panels, the model is

discretized into a sufficient number of elements to adequately represent the deformation and

stress gradients, and is verified in (Chen, 2003).

Figure 3.1 Finite element mesh for Smith panel 6

Table 3.2 Elements in FE Model of Smith Panel 6

Panel Part # Elements Plate 224 x 90

Stiffener Web 224 x 5 Stiffener Flange 224 x 4

Frame Web 90 x 8 Frame Flange 90 x 6

3.3. Smith Panel Initial Imperfections

(Smith, 1975) was very precise in measuring the imperfections in the stiffened panels

before testing. Stretched wires were used to measure the vertical imperfections of the stiffener-

14

frame intersections. The maximum values of the longitudinal stiffener and the plate deflection

(wosm and woplm, respectively) were recorded in a table. Using a similar method to the one used

for the 1½ bay models, the stiffener and plate deflections were used in Equation (1) to define the

imperfections in the panels as shown in Figure 3.2.

Figure 3.2 Initial imperfections in Smith panel models

Satisfactory results were not achieved for panel 6 using the sinusoidal pattern generated

by Equation (1), however. Although the ABAQUS ultimate strength corresponded well

numerically with the experimental ultimate strength, the ABAQUS collapse mode did not match

the experimental collapse mode (shown in Figure 3.3). Using data points that were extracted

from the actual initial imperfections of one of the middle longitudinal stiffeners (Smith, 1975), a

curve fit was done using a 6th order polynomial. This curve fit resulted in an equation used to

define the initial stiffener imperfections (wos) in the ABAQUS model as shown in Figure 3.4.

15

Figure 3.3 Photo of the post collapse condition of Smith panel 6

Figure 3.4 Initial Imperfections of Smith Panel 6

3.4. Residual Stress

High amounts of weld-induced longitudinal residual stress were found in the stiffened

panels analyzed in (Smith, 1975). These stresses were estimated under the assumption that the

transverse residual stress was negligible. Following the same technique used in (Smith, 1992),

residual stress was added to the finite element models of the stiffened panels. This method

assumes that weld-induced residual stress has a form similar to that shown in

16

Figure 3.5. The stress in the tensile zone is equal to the yield stress and has a width of

tη2 . The width of the compressive zone is thus equal to tb η2− . The stress in the

compressive zone ( rcσ ) is calculated by balancing the forces of the residual stresses along the

loaded edges of the panel as shown in Equation (2).

( )( )( )( )tbn

tn

s

sYrc η

ησσ21

21−+

+= (2)

For the purposes of this thesis, residual stress in the web of the stiffeners due to welding

was assumed to be negligible.

Figure 3.5 Residual stress in stiffened panels (Smith, 1975)

The plate of the finite element model was divided into two different sections of nodes.

One section contained the node set under compression due to residual stress, and the other

contained the set under tension. The yield stresses of these sections were then adjusted to

account for the residual stress as follows:

YYYYT

rcYYC

σσσσσσσ

2=+=−=

The residual stress effectively strengthens the area of plating near the stiffener and

weakens the area between stiffeners. This straightforward method is valid in these cases due to

the fact that the residual stress and loading are assumed to be acting along the same axis. Table

3.3 shows how yield stress was changed in the plating of the panel models due to residual stress.

17

Table 3.3 Residual Stress in Smith Panels

σY

(MPa) η σrc(MPa)

σYC(MPa)

σYT(MPa)

1a 253.3 ⎯ ⎯ 253.3 253.31b 256.4 ⎯ ⎯ 256.4 256.42a 265.6 6.4 127.4 138.2 531.22b 264.1 5.1 86.5 177.6 528.23a 254.8 6.6 97.3 157.5 509.63b 256.4 7.2 111.1 145.3 512.85 251.7 9.4 62.3 189.4 503.46 261.0 11.4 80.8 180.2 522.0

3.5. Boundary Conditions

Tests in (Smith, 1975) were carried out very meticulously on a specially built test rig.

This rig, the Large Testing Frame (LTF) shown in Figure 3.6, was designed so that the panel

edges were simply supported. Light flexural plates and tiebars restrained panel movement in the

vertical direction on the panel edges. While restraining the panel in the vertical direction, these

supports did not offer any resistance to rotation at the panel edges. The flexural plates and

tiebars allowed the panel to deform longitudinally and transversely in the plane of the panel.

In order to ensure that loads being applied to the panels were distributed evenly, and to

avoid premature failure in the end bays, reinforcement was added to the panels. This

reinforcement consisted of attaching doubler plates to the webs and flanges of the longitudinal

stiffeners as well as to the plating. The doubler plates covered 2/3 of the span of the end bays.

In order to maintain consistency between the actual Smith panels and the ABAQUS models, this

reinforcement was added to the ABAQUS models. This is important to note when looking at the

results of the ABAQUS analyses of the Smith panels in Appendix B as the longitudinal stiffeners

and plating in the end bays have a relatively small amount of stress due to the reinforcement.

18

Figure 3.6 LTF - Large Testing Frame (Smith, 1975)

In order to match the restraints in the experiments, boundary conditions shown in Table

3.4 were applied to the ABAQUS models. The transverse and longitudinal edges were restrained

in the z direction. Rotations in the axes orthogonal to the edges were restrained, while rotations

parallel to the edges were not. Nodes at the midspan of the longitudinal edges of the panel were

restrained in x direction. Nodes at the midspan of the transverse edges were restrained in the y

directions. The restraints on the mid nodes were added in order to prevent free-body motion.

Table 3.4 Boundary Conditions for Full Panel ABAQUS Model

Sides 3,4,6 Loaded Edges 3,5,6 Loaded Edge Mid Nodes 2 Side Mid Nodes 1

19

Figure 3.7 Transverse and longitudinal edges of a stiffened panel

20

4. Analytical Methods for Ultimate Strength Analysis

Finite element analysis offers precision that can only be surpassed by experimental

results. However, the time required to make an analysis with this precision of a single stiffened

panel using ABAQUS is significant. Upwards of 24,000 elements were used to construct the fine

mesh ABAQUS models. Creating ABAQUS input files with this many elements is extremely time

consuming. Analyses on panels of this size can take between 8-12 hours, even on modern

computers. The cliché “time is money” is significant in this case as many hours are spent waiting

for results of a single ABAQUS analysis.

Analytical methods provide a more time-effective (and thus cost-effective) means of

calculating the ultimate strength of stiffened panels. A group of stiffened panels can be analyzed

using a current generation computer in a fraction of a second. However, the time saved comes at

the cost of accuracy.

One such analytical method is the computer program ULSAP developed by Professor

Jeom Paik of Pusan National University, Korea (Paik, 2003). ULSAP stands for Ultimate Limit

State Assessment Program. This computer program calculates the ultimate strength of stiffened

panels using 5 different methods corresponding to the 5 modes of failure that are possible, and

also takes initial imperfections and residual stress into account. The lowest value given by these

methods is the ultimate strength of the panel, and the panel is predicted to fail in that mode.

Overall collapse and beam-column collapse are examined in Sections 4.1 & 4.2 because they

pertain directly to the topic of this thesis.

4.1. Overall Collapse (Mode I)

For overall buckling (Mode I) ULSAP uses orthotropic plate theory to calculate the

ultimate strength of a stiffened panel. Orthotropic plate theory is a method of idealizing a cross-

stiffened panel. The stiffeners and frames of a cross-stiffened panel are smeared into the plating

creating an orthotropic plate in place of the stiffened panel. Orthotropic plate theory assumes that

the stiffeners and frames in a given direction are similar, relatively numerous, and small (Paik and

Thayamballi, 2003). This ensures that the stiffeners deflect with the plating and remain stable

through collapse.

In calculating the ultimate strength of a panel that fails due to overall buckling, orthotropic

plate theory assumes that a stiffened panel is primarily subjected to an axial load xavσ (parallel to

21

the stiffeners). Lateral pressure acting on the plating (p) is considered a secondary load. Also in

this particular study transverse loads ( yavσ ) were not considered. Initial yielding will occur due to

combined compression and bending in the panel. The plating and/or stiffener flange will yield at

the point of maximum deflection. The Mises-Hencky yield criterion gives the following equation

(Paik and Thayamballi, 2003):

122

=⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟

⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛−⎟

⎟⎠

⎞⎜⎜⎝

⎛

Yeq

yb

Yeq

yb

Yeq

xb

Yeq

xb

σσ

σσ

σσ

σσ

(3)

where

( )

( )

⎟⎟⎠

⎞⎜⎜⎝

⎛+=⎟

⎠⎞

⎜⎝⎛ +=

+=+=

++=+

=

++

=++

=

+=

=

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛

−−

+−=

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛

−−

+−=

∫

atAn

EEBtAnEE

aAn

ttBAntt

aAn

BAnt

ttt

AnBtAnBt

AnBtAnBt

dyB

BamAt

EB

AAAE

BamAt

Ea

AAAEm

ffy

ssx

ffyeq

ssxeq

ffssyeqxeqeq

ff

YfffYpYy

ss

YsssYpYx

YyYxYeq

B

xxav

xyx

meqy

mmmyxyb

yyx

mxeqx

mmmxxxavxb

1,1

,

222

,

2

1

1282

1282

0

22

20

2

22

20

22

σσσ

σσσ

σσσ

σσ

ππννν

πρσ

πνπνν

πρσσ

Equation (3) is the initial yield condition of the outer surface of the orthotropic plate. One

possible failure theory is that failure of the panel occurs due to yielding in the outermost fibers of

the panel (either the plating or the stiffener flange) as a result of bending. Due to the smearing of

the stiffeners and frames into the plating to create an equivalently thick orthotropic plate, this

method is rather conservative (i.e. this method gives pessimistic results that are lower than

experimental/FEA results). The outermost fibers in the equivalent plate will not experience the

same stresses as the outer most fibers in an actual stiffened panel. Because of this, there is an

alternative method for calculating the ultimate strength of Mode I based on membrane stress.

22

The membrane stress method does not use Equation (3) to find the ultimate strength of a

stiffened panel because some of the load can be transferred to the boundaries of the panel, by a

buckling-induced redistribution of membrane stress as shown in Figure 4.1. If this redistribution

occurs, then the panel will collapse when yield occurs at the sides of the panel. At this location

there is no bending stress; only membrane stress.

Figure 4.1 Membrane stress in an orthotropic plate

In this case the following equation gives the ultimate strength:

12

minminmax

2

max =⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟

⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛−⎟

⎟⎠

⎞⎜⎜⎝

⎛

Yeq

y

Yeq

y

Yeq

x

Yeq

x

σσ

σσ

σσ

σσ

(4)

where

( )

( )

( )

( )2

02

max

20

2

max

20

22

min

20

22

max

82

828

28

2

BAAAE

BAAAEa

AAAEma

AAAEm

mmmyxy

mmmyxy

mmmxxxavx

mmmxxxavx

+=

+−=

++=

+−=

πρσ

πρσ

πρσσ

πρσσ

23

The ultimate strength for overall collapse, , is obtained from the solution to Equation

(4), the membrane stress equation.

Ixuσ

4.2. Beam-Column Collapse (Mode III)

Beam-column collapse focuses on the plate-stiffener combination, as shown in Figure

1.3. Ultimate strength in a panel with Mode III failure is said to occur when axial stress reaches

yield stress in either the plate or stiffener side of the panel. These two types of failure are known

as plate-induced failure and stiffener-induced failure, respectively. These two types of failure are

shown in Figure 4.2. Failure in a panel with simple support boundary conditions generally occurs

at the midspan, as shown in Figure 1.6.

Figure 4.2 (a) Plate induced failure & (b) stiffener induced failure in a beam-column

ULSAP uses the modified Perry-Robertson formula for beam-column collapse. This

method is a way of approximating ultimate strength based on three nondimensional parameters

from the stiffened panel. These parameters are the column slenderness parameter ( λ ),

eccentricity ratio (η ), and dead load bending term ( µ ) and are defined as:

24

( )

Yeq

Yeq

ZM

ZAE

L

σµ

δη

σρπ

λ

0

0

=

∆+=

⋅=

These terms are then input into a strength ratio equation as follows:

5.0

222111

4111

21

⎥⎦

⎤⎢⎣

⎡ −−⎟

⎠⎞

⎜⎝⎛ +

+−−⎟⎠⎞

⎜⎝⎛ +

+−=λ

µλ

ηµλ

ηµR (5)

R is the ratio of the ultimate stress to the yield stress of the panel. So the ultimate

stress can then be calculated by using the following equation:

Yult Rσσ = (6)

The Perry-Robertson formula assumes that any yielding in the flange of a stiffener will

cause failure. For stiffened panels with relatively small longitudinal stiffeners, the Perry-

Robertson formula gives pessimistic results. ULSAP assumes that yielding may travel down the

web of the stiffener before failure actually occurs. Based on this assumption, ULSAP excludes

the possibility of stiffener-induced failure and focuses only on plate-induced failure for panels with

relatively small stiffeners.

A minimum value for the ultimate strength of a stiffened panel with beam-column collapse

is set by ULSAP as a weighted average of bare plate and orthotropic plate ultimate strengths.

This is shown in Equation (7).

xeq

GOxuxeq

GBxuIII

ult tttt

++

≥σσ

σ (7)

Here, is the ultimate strength of the bare plate and is the ultimate strength of

the orthotropic plate calculated using the outer surface stress method.

GBxuσ GO

xuσ

4.3. ULTBEAM

ULTBEAM is a computer program developed by Yong Chen (Chen, 2003) in which a

beam-column approach is used to calculate the ultimate strength of stiffened panels. This

25

program uses a step-by-step numerical method to calculate the ultimate strength of stiffened

panels with uniaxial compression or combined uniaxial compression and lateral pressure.

ULTBEAM accounts for initial imperfection, but currently does not allow for residual stress. Four

cases are examined by ULTBEAM to calculate the ultimate strength. The beam-column remains

fully elastic in Case 1. Yielding occurs in the flange in Case 2, in the web in Case 3; and in the

plating in Case 4. More information about the methods used by ULTBEAM can be found in

(Chen, 2003)

26

5. Comparison of Ultimate Strength Predictions

The results of the stiffened panel ultimate strength analyses will be discussed using the

following nomenclature:

σult,EXP → Experimental ultimate strength σult,SFEA → Smith FEA ultimate strength σult,ABQ → ABAQUS ultimate strength σult,ULSAP → ULSAP ultimate strength σult,ULTBEAM → ULTBEAM ultimate strength

5.1. 1½ Bay Model Results

The results of the 1½ bay models are shown below in Table 5.1. Plots of the panels at

collapse are shown in Appendix A.

Table 5.1 1½ Bay Model ABAQUS and ULSAP Results

Panel No.

Y

ABQult

σσ ,

Y

ULSAPult

σσ ,

ABQult

ULSAPult

,

,

σσ

Y

ULTBEAMult

σσ ,

ABQult

ULTBEAMult

,

,

σσ

ABAQUS Failure Mode

ULSAP Failure Mode

vt001 0.521 0.395 0.757 0.548 1.052 III-1 III or III-1 vt002 0.675 0.526 0.780 0.684 1.013 III-1 III or III-1 vt003 0.780 0.609 0.780 0.816 1.046 III-1 III or III-1 vt004 0.515 0.226 0.439 0.513 0.995 III-1 III or III-1 vt005 0.528 0.288 0.546 0.530 1.005 III-1 III or III-1 vt006 0.585 0.427 0.730 0.601 1.027 III-1 III or III-1 vt007 0.664 0.528 0.796 0.670 1.009 III-1 III or III-1 vt008 0.592 0.425 0.717 0.628 1.061 III-1 III or III-1 vt009 0.684 0.486 0.711 0.723 1.057 III-1 III or III-1 vt010 0.828 0.593 0.716 0.910 1.098 III-1 III or III-1 vt011 0.892 0.656 0.736 0.914 1.025 III-1 III or III-1 vt012 0.709 0.872 1.230 0.849 1.198 III III or III-1 vt013 0.810 0.773 0.954 0.896 1.106 III III or III-1 vt014 0.637 0.607 0.953 0.764 1.199 III-1 III or III-1 vt015 0.545 0.520 0.954 0.694 1.273 III III or III-1 vt016 0.682 0.789 1.157 ⎯ ⎯ IV IV vt017 0.705 0.688 0.975 0.977 1.385 III III or III-1 vt018 0.715 0.690 0.965 0.827 1.157 III III or III-1 vt019 0.511 0.266 0.520 0.620 1.212 III-1 III or III-1 vt020 0.528 0.348 0.658 0.654 1.239 III-1 III or III-1

27

1½ Bay Model ABAQUS and ULSAP Results

Panel No.

Y

ABQult

σσ ,

Y

ULSAPult

σσ ,

ABQult

ULSAPult

,

,

σσ

Y

ULTBEAMult

σσ ,

ABQult

ULTBEAMult

,

,

σσ

ABAQUS Failure Mode

ULSAP Failure Mode

vt021 0.585 0.537 0.918 0.761 1.301 III-1 III or III-1 vt022 0.703 0.670 0.953 0.844 1.200 III-1 III or III-1 vt023 0.467 0.404 0.864 0.568 1.214 III-1 III or III-1 vt024 0.639 0.590 0.924 0.752 1.177 III-1 III or III-1 vt025 0.717 0.665 0.928 0.822 1.147 III-1 III or III-1 vt026 0.808 0.770 0.952 0.962 1.190 III III or III-1 vt027 0.831 0.819 0.986 0.943 1.135 III III or III-1 vt028 0.760 0.609 0.801 0.816 1.073 III III or III-1 vt029 0.531 0.395 0.743 0.548 1.033 III III or III-1 vt030 0.661 0.526 0.796 0.684 1.034 III III or III-1 vt031 0.462 0.325 0.704 0.494 1.068 III-1 III or III-1 vt032 0.435 0.254 0.584 0.445 1.023 III-1 III or III-1 vt033 0.807 0.652 0.808 0.928 1.150 III III or III-1 vt034 0.508 0.226 0.446 0.513 1.010 III-1 III or III-1 vt035 0.527 0.288 0.547 0.530 1.006 III-1 III or III-1 vt036 0.585 0.427 0.730 0.601 1.027 III-1 III or III-1 vt037 0.660 0.528 0.800 0.670 1.015 III-1 III or III-1 vt038 0.583 0.423 0.725 0.628 1.077 III-1 III or III-1 vt039 0.527 0.485 0.919 0.723 1.371 III-1 III or III-1 vt040 0.763 0.592 0.775 0.736 0.964 III III or III-1 vt041 0.796 0.655 0.823 0.914 1.148 III III or III-1 vt042 0.542 0.520 0.959 0.694 1.280 III III or III-1 vt043 0.691 0.690 0.998 0.827 1.197 III III or III-1 vt044 0.778 0.773 0.994 0.896 1.151 III III or III-1 vt045 0.793 0.829 1.046 0.900 1.135 III III or III-1 vt046 0.644 0.526 0.817 0.684 1.061 III III or III-1 vt047 0.735 0.609 0.828 0.816 1.110 III III or III-1 vt048 0.513 0.395 0.769 0.548 1.069 III III or III-1 vt049 0.457 0.325 0.712 0.494 1.080 III-1 III or III-1 vt050 0.420 0.254 0.605 0.445 1.060 III-1 III or III-1 vt051 0.767 0.652 0.850 0.928 1.210 III-1 III or III-1 vt052 0.483 0.226 0.468 0.513 1.060 III III or III-1 vt053 0.511 0.288 0.565 0.530 1.039 III-1 III or III-1 vt054 0.578 0.427 0.739 0.601 1.039 III-1 III or III-1 vt055 0.653 0.528 0.808 0.670 1.025 III1 III or III-1 vt056 0.564 0.421 0.747 0.628 1.114 III-1 III or III-1 vt057 0.627 0.483 0.771 0.723 1.154 III-1 III or III-1 vt058 0.766 0.591 0.772 0.910 1.188 III III or III-1 vt059 0.738 0.654 0.887 0.914 1.238 III-1 III or III-1

Mean: 0.799 Mean: 1.116 COV: 0.206 COV: 0.087

28

Table 5.1 shows the ABAQUS, ULSAP, and ULTBEAM results for the 1½ bay panel

models described in Section 2. These results indicate that ULSAP generally gives fairly good,

although slightly conservative, predictions for the ultimate strength of stiffened panels. Analyzing

the results further shows that ULSAP predictions for panels with medium to large stiffeners are

more accurate than predictions for panels with small stiffeners.

ULSAP does a good job predicting the mode of failure of the panels as well. All 59

panels in this series feature either beam-column-type failure or overall buckling triggered by

beam-column-type yield. ULSAP predicts all but one of these panels to fail either in a pure Mode

III, beam-column-type failure, or a combined (III-1) failure, which means overall (Mode I) collapse

triggered by a Mode III beam-column yield. Figure 5.1 shows one of the panels in this series with

combined beam-column failure and overall collapse. Yielding occurs in the web and flange of the

longitudinal stiffeners in the middle bay due to compression from the downward deflection. The

flange in the end bay is also yielding due to tension from the upward deflection. The failure of this

panel is very similar to beam-column stiffener-induced failure shown in Figure 4.2 (b). It is

important to note that because von Mises stress is being plotted, the sign of the stress cannot be

used to determine if a region is in tension or compression. Instead, deflections can be used to

establish tension and compression regions.

For the panels ULSAP predicts to fail by Mode III or III-1, ULTBEAM was also used to

predict the ultimate strength. ULTBEAM ultimate strength predictions shown in Table 5.1

correspond very well with the results given by the ABAQUS analyses. The error in the ULTBEAM

results is less than 60% of the error in the ULSAP results. Also, the coefficient of variation of the

ULTBEAM results is less than half of that of ULSAP.

29

Figure 5.1 Panel vt009 at collapse

30

5.2. Smith Panel Results

The results of the analyses of the Smith Panels are shown in Table 5.2. Plots of the panels

at collapse are shown in Appendix B.

Table 5.2 Smith Panel Results

Panel No. Y

EXPult

σσ ,

Y

SFEAult

σσ ,

EXPult

SFEAult

,

,

σσ

Y

ABQult

σσ ,

EXPult

ABQult

,

,

σσ

Y

ULSAPult

σσ ,

EXPult

ULSAPult

,

,

σσ

Y

ULTBEAMult

σσ ,

EXPult

ULTBEAMult

,

,

σσ Exp.

Collapse Mode

ABAQUS Collapse

Mode

ULSAP Collapse

Mode

1a 0.76 0.69 0.908 0.858 1.128 0.762 1.003 ⎯ ⎯ V V IV 1b 0.73 0.57 0.781 0.718 0.984 0.571 0.782 ⎯ ⎯ V V II 2a 0.91 0.81 0.890 0.960 1.054 0.816 0.897 0.923 1.014 IV V & III-2 III or III-1 2b 0.83 0.82 0.988 0.940 1.132 0.841 1.013 ⎯ ⎯ IV V & III-2 V 3a 0.69 0.63 0.913 0.755 1.094 0.589 0.854 0.763 1.105 IV III-2 III or III-1 3b 0.61 0.60 0.984 0.701 1.149 0.609 0.998 ⎯ ⎯ IV & V V & III-2 V 5 0.72 0.55 0.764 0.770 1.069 0.549 0.763 ⎯ ⎯ IV & V V & III-2 V 6 0.49 ⎯ ⎯ 0.554 1.131 0.238 0.486 0.410 0.837 I III-1 III or III-1

(Smith, 1992) is the fourth of a series of papers in the same series as (Smith, 1975). In

this paper, a computer program was developed to analyze the collapse and post-collapse

behavior of stiffened panels. This program was used to analyze the series of panels that was

constructed and tested in (Smith, 1975).

A beam-column finite element model was used to analyze the panels under the

assumption that the panels contained a large number of longitudinal stiffeners that behaved

identically. The model contained one stiffener-plate combination as shown in Figure 5.2, and

represented two adjacent bays. Each bay was assumed to be symmetric about a central plane,

so only half of the interframe span was modeled for each bay as shown in Figure 5.3. Boundary

conditions were applied to the model instead of adding physical transverse frames. These

frames were assumed to be flexurally rigid but torsionally weak and therefore simple support

boundary conditions were used. The boundary conditions, and the assumption of a center plane

of symmetry in the interframe spans, mean that this model actually represents an infinitely long

structure that repeats every two bays.

31

Figure 5.2 Subdivision of cross-section into 'fibres' (Smith, 1992)

Figure 5.3 Subdivision of stiffened panel into elements (Smith, 1992)

Residual stress was added to the FE model as shown in Figure 5.4. The yield stress of

the plating and stiffeners was modified to account for the presence of compressive and tensile

stresses due to the weld-induced residual stresses. This method is similar to the method outlined

in Section 3.4, with the addition of taking residual stress in the web of the stiffener into account.

The results from the FE analyses in (Smith, 1992) are shown in Table 5.2 as σult,SFEA.

Figure 5.4 Weld-induced residual stress in a stiffened panel (Smith, 1992)

ULSAP ultimate strength results are shown in Table 5.2. The ULSAP results have good

agreement with the experimental results for the majority of the Smith panels analyzed. The

32

prediction of failure mode for the Smith panels is acceptable, although the mode is not predicted

correctly in a few cases.

ULTBEAM results for the Smith panels predicted to fail by Mode III or III-1 failure by

ULSAP are shown in Table 5.2. The table clearly shows that ULTBEAM gives more accurate

ultimate strength predictions for Mode III or III-1 failure than ULSAP.

ULTBEAM was also used to find the ultimate strength of the Smith panels that were not

predicted to fail by Mode III or III-1 by ULSAP. These results are shown in Table 5.3. Since

ULSAP predicts a different mode of failure, it is expected that ULTBEAM will predict a higher

ultimate strength for these cases. For all but one of the panels ULTBEAM does predict a higher

ULTIMATE strength than ULSAP. For the case in which ULTBEAM is lower than ULSAP, the

values given by the two programs are virtually identical.

Table 5.3 ULTBEAM vs. ULSAP for Smith panels not failing by Mode III or III-1

Panel No. Y

ULSAPult

σσ ,

Y

ULTBEAMult

σσ ,

1a 0.762 0.760 1b 0.571 0.754 2b 0.841 0.954 3b 0.609 0.778 5 0.549 0.691

5.3. Smith Panel 6

Good agreement was reached between the experimental and ABAQUS results for Smith

panel 6. Figure 5.5 is a photo of panel 6 post collapse from the experiments of (Smith, 1975).

The ABAQUS results in Figure 5.6 show an overall failure mode with significant upward and

downward bending of the longitudinal stiffeners and transverse frames. This failure is triggered

by yielding in the flanges of the longitudinal stiffeners in the downward deflected portion of the

panel.

It is important to note that while the ABAQUS results pictured in Figure 5.6 show some

regions of high stress concentrations in the plating near the base of the longitudinal stiffeners,

these regions are not yielding. This is because residual stress was added indirectly to the model

and the color codes do not take this into account. The yield stress in the tensile zones is twice

the normal value as described in Section 3.4, so these regions are actually safe from yielding. It

is also important to note that the yield stress of the longitudinal stiffeners is lower than the yield

33

stress for the transverse frames. Because of this, the coloring in Figure 5.6 is misleading. While

the color of the longitudinal stiffeners at the sections of maximum stress does not correspond to

the highest value on the stress scale in Figure 5.6, the stress in the longitudinal stiffeners has

actually reached yield stress at these locations.

(Smith, 1992) did not analyze panel 6 because of limitations in the FEA program, which

was only able to account for interframe collapse of stiffened panels. The program could not

handle collapse modes in which the frames are involved, and as such was not able to model the

overall collapse mode of panel 6.

The ULSAP result for panel 6 is shown in Table 5.2. The value given by ULSAP for the

ultimate compressive stress is ultra conservative as it is less than half the experimental value

(0.486). However, the mode of failure given by ULSAP does match what was found in the

experimental and ABAQUS results. ULSAP predicts panel 6 to experience “III or III-1”, which

means either a pure Mode III (beam-column) failure or a Mode I (overall panel) failure triggered

by a Mode III yielding, and the latter matches the actual (experimental) failure mode.

In calculating the ultimate strength for Mode III failure, ULSAP assumes that plate-

induced failure occurs as shown in Figure 1.4. ABAQUS results show that this is not the case for

panel 6 as the maximum von Mises stress in the plating is approximately 150 MPa, which is well

below the yield stress of 261 MPa. Yielding in the flange of the stiffeners in the downward

deflected portion of the panel is what causes failure. This stiffener-induced failure is ignored by

ULSAP as previously discussed in Section 4.2.

The ULTBEAM result for panel 6 is also shown in Table 5.2. The ultimate strength

predicted by ULTBEAM has good agreement with the experimental results. Moreover the error in

the ULTBEAM ultimate strength prediction (1 – 0.837 = 0.163) is less than one third of the error of

the ULSAP prediction (1 – 0.486 = 0.514).

34

Figure 5.5 Photo of the post collapse condition of Smith panel 6

Figure 5.6 ABAQUS results for Smith panel 6 at collapse

35

6. Conclusions and Recommendations for Future Work

6.1. Conclusions

This study analyzed a total of 67 stiffened panels using the finite element analysis

program ABAQUS. The ultimate strengths of these panels were compared to results from the

computer programs ULSAP and ULTBEAM. Eight of the panels were also compared to

experimental results, as well as to results from Smith’s finite element program. For the cases in

which a mode of failure other than Mode III or III-1 is predicted, ULSAP gives satisfactory results.

For 10 of the cases in which Mode III or III-1 is predicted, ULSAP gives values of ultimate

strength with more than 30% error. The other 51 cases with Mode III or III-1 failure have

satisfactory results, but ULTBEAM gives more accurate predictions for the ultimate strength.

6.2. Recommendations for Future Work

The results from Smith panel 6 highlight a portion of the ULSAP code that could benefit

from some improvement. This panel failed by overall buckling, which was brought on by yielding

in the longitudinal stiffener flange. As noted in Section 4.2, ULSAP currently does not check for a

stiffener-induced failure, and this study has shown that this failure mode is needed. ULTBEAM

examines this failure mode explicitly, and because of this it gave a more accurate result for all of

the panels predicted to fail in Mode III of III-1. Therefore it is recommended that ULTBEAM be

incorporated into ULSAP to examine this mode of failure.

It is also necessary to correct a minor flaw in ULTBEAM. ULTBEAM currently has only

one input for yield stress of a stiffened panel. As the Smith panels show however, the yield stress

in the plating, longitudinal stiffeners, and transverse frames can differ. It is important to take this

into account in order to provide the most accurate ultimate strength predictions possible.

36

References

ABAQUS. ABAQUS/Standard user’s manual, Vol. I-III, ver. 6.1, Hibbitt, Karlsson & Sorenson Inc,

RI, 2002.

Chen, Y. Ultimate Strength Analysis of Stiffened Panels Using a Beam-Column Method, PhD.

Dissertation, Department of Aerospace and Ocean Engineering, Virginia Polytechnics

Institute and State University, Blacksburg, VA, 2003.

Det Norsk Veritas (DNV). PULS 1.5 – User’s Manual, DNV, 2003

Ghosh, B. Consequences of Simultaneous Local and Overall Buckling in Stiffened Panels,

Department of Aerospace and Ocean Engineering, Virginia Polytechnics Institute and

State University, Blacksburg, VA, 2003.

Hughes, O.F. Ship structural design, a rationally-based, computer-aided optimization approach,

SNAME, New Jersey, 1988.

Paik, J.K., Thayamballi, A.K. Ultimate Limit State Design of Steel-Plated Structures, John Wiley

& Sons, LTD, 2003.

Smith, C.S. Compressive Strength of Welded Steel Ship Grillages, RINA Transactions, Vol. 117,

1975.

Smith, C.S. Strength of Stiffened Plating Under Combined Compression and Lateral Pressure,

RINA Transactions, Vol. 134, 1992.

37

Appendix A

This appendix contains color-coded plots of the von Mises stress (MPa) in the

midthickness of the 1½ bay panel models (vt001 – vt059) at collapse. Each panel was analyzed

using ABAQUS to find the ultimate strength of the panel when subjected to uniaxial compression,

as described in Section 2. The post-processing software PATRAN was used to plot the stress

distribution.

38

vt001

vt002

39

vt003

vt004

40

vt005

vt006

41

vt007

vt008

42

vt009

vt0010

43

vt0011

vt0012

44

vt0013

vt0014

45

vt015

vt016

46

vt017

vt018

47

vt019

vt020

48

vt021

vt022

49

vt023

vt024

50

vt025

vt026

51

vt027

vt028

52

vt029

vt030

53

vt031

vt032

54

vt033

vt034

55

vt035

vt036

56

vt037

vt038

57

vt039

vt040

58

vt041

vt042

59

vt043

vt044

60

vt045

vt046

61

vt047

vt048

62

vt049

vt050

63

vt051

vt052

64

vt053

vt054

65

vt055

vt056

66

vt057

vt058

67

vt059

68

Appendix B

This appendix contains color-coded plots of the von Mises stress (MPa) in the

midthickness of the Smith panels at collapse. Each panel was analyzed using ABAQUS to find

the ultimate strength of the panel when subjected to uniaxial compression, and in some cases

lateral pressure, as described in Section 3. The post-processing software PATRAN was used to

plot the stress distribution. Photos from the Smith experiments are also shown if available.

69

Smith Panel 1a

70

Smith Panel 1b

71

Smith Panel 2a

72

Smith Panel 2b

73

Smith Panel 3a

74

Smith Panel 3b

75

Smith Panel 5

76

Smith Panel 6

77

Vita

Samuel Dippold was born in Washington, DC on September 4, 1981. He is the second

son of Vance and Sandy Dippold, and has an older brother Vance and a younger sister Laurie.

Samuel began his undergraduate studies at Virginia Tech in the fall of 1999, dual-majoring in

Ocean Engineering and Aerospace Engineering. In May of 2003, Samuel completed his

Bachelor of Science in Ocean Engineering at Virginia Tech and immediately started his graduate

work. This thesis completes his Master of Science in Ocean Engineering from Virginia Tech.

78