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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
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