Inelastic Analysis of Tripping Failure of Stiffened Steel Panels due … · 2020. 9. 25. ·...
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Inelastic Analysis of Tripping Failure of Stiffened Steel Panels due to Stiffener Flange Transverse Initial
Eccentricity
Scott A. Patten
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. Mayuresh Patil
Date May 8, 2006
Blacksburg, Virginia
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Inelastic Analysis of Tripping Failure of Stiffened Steel Panels due to Stiffener Flange Transverse Initial
Eccentricity
Scott A. Patten
(ABSTRACT)
This thesis studies the present methods used to predict the ultimate
tripping strength of stiffened panels under compressive axial stress. The current
methods involve the use of a bifurcation, or eigenvalue, approach to predicting
failure stress. The effects of initial transverse eccentricity of the stiffener are
ignored using such a method. Six panels were modeled and tested with
ABAQUS, a finite element software package, and the results were compared to
output from ULSAP, a closed-form ultimate strength analysis program. The
ultimate strengths predicted by ABAQUS changed with the influence of initial
deflection of the stiffener flange, while the results from ULSAP did not change.
This thesis attempts to use beam-column analysis on the imperfect stiffener
flange to predict the tripping strength. It was determined that the procedure
presented in this thesis does not accurately model the true failure mode of
stiffeners in tripping. The resulting ultimate strengths are extremely conservative
and neglect the importance of the stiffener web’s role in tripping. Future work is
recommended to expand on these findings and to incorporate the influence of the
stiffener web into a beam-column solution.
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Acknowledgments
First, I would like to thank my graduate advisor and committee chair, Dr.
Owen Hughes, for his advice and support throughout my graduate study at
Virginia Tech. For the past two years, his suggestions and responses to all of my
questions led to a rich learning experience.
I would also like to thank Dr. Alan Brown and Dr. Mayuresh Patil for
serving on my committee. I am also grateful for the valuable input I received
from Jason Albright and Feng Zhou.
Finally, I am thankful for the unending support from my family and friends.
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Table of Contents List of Figures ................................................................................................................... v List of Tables.................................................................................................................... vi Nomenclature.................................................................................................................. vii 1. Introduction................................................................................................................ 1
1.1. Stiffened Panels .................................................................................................. 1 1.2. Stiffened Panel Collapse Modes ......................................................................... 3 1.3. Smith Panel Experiments.................................................................................... 7 1.4. Previous Inelastic Tripping Analysis .................................................................. 8 1.5. Initial Eccentricity............................................................................................... 9
2. Current Methods of Tripping Analysis................................................................. 12 2.1. ABAQUS 1 ½ Bay Stiffened Panel Model....................................................... 12
2.1.1. Modified RIKS Inelastic Analysis ............................................................ 13 2.1.2. Material Properties.................................................................................... 13 2.1.3. Finite Elements ......................................................................................... 14 2.1.4. Initial Deflections...................................................................................... 14 2.1.5. Boundary Conditions ................................................................................ 15
2.2. ABAQUS Test Panel Results............................................................................ 16 2.3. ULSAP.............................................................................................................. 18 2.4. ULSAP Predictions of Panels 1 – 6 .................................................................. 18
3. Beam-Column Method........................................................................................... 21 3.1. ULTBEAM ....................................................................................................... 21
3.1.1. Assumptions.............................................................................................. 21 3.1.2. M – Φ – P Relationship............................................................................. 21 3.1.3. Step-by-Step Method ................................................................................ 22
3.2. Applying the Beam-Column Method to Tripping ............................................ 25 3.2.1. Geometry and Assumptions...................................................................... 25 3.2.2. M – Φ – P Relationship for Rectangular Cross Section ........................... 26
4. Comparison of Results .......................................................................................... 30 4.1. ABAQUS Beam Model .................................................................................... 30 4.2. Flange Scantlings .............................................................................................. 30 4.3. Validation of the Flange Beam-Column Method.............................................. 31 4.4. Revisit the 1 ½ Bay Panels ............................................................................... 33
5. Conclusions and Recommendations for Future Work...................................... 34 5.1. Conclusions....................................................................................................... 34 5.2. Recommendations for Future Work.................................................................. 34
References ...................................................................................................................... 35 Appendix .......................................................................................................................... 36 Vita.................................................................................................................................... 43
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List of Figures Figure 1.1 Cross-section of tanker hull............................................................................... 1 Figure 1.2 Ship in sagging loading condition ..................................................................... 2 Figure 1.3 Three bay stiffened panel with axial compression ............................................ 2 Figure 1.4 Three bay stiffened panel with axial compression ............................................ 3 Figure 1.5 Plate with attached T-stiffener........................................................................... 3 Figure 1.6 Mode I – Overall collapse ................................................................................. 4 Figure 1.7 Mode II – Biaxial compressive collapse ........................................................... 5 Figure 1.8 Mode III-1 – Overall collapse due to beam-column yielding ........................... 5 Figure 1.9 Mode III-2 – Local S-shaped mechanism ......................................................... 6 Figure 1.10 Mode IV – Local buckling of stiffener web .................................................... 6 Figure 1.11 Mode V – Collapse due to stiffener tripping ................................................... 6 Figure 1.12 Smith Grillage 1a; tripping.............................................................................. 7 Figure 1.13 Smith Grillage 3a; beam-column collapse ...................................................... 8 Figure 1.14 Cross-section deformation of flexible web...................................................... 9 Figure 1.15 Discrete form of the beam-column cross-section............................................ 9 Figure 1.16 Load deflection curve for eccentric columns ................................................ 10 Figure 1.17 Eccentric column........................................................................................... 11 Figure 2.1 ABAQUS 1 ½ bay model mesh ...................................................................... 12 Figure 2.2 Ideal Elastic-Perfectly Plastic Stress-Strain Plot............................................. 13 Figure 2.3 Initial deflections for 1 ½ bay ABAQUS panel .............................................. 15 Figure 2.4 Boundary conditions........................................................................................ 16 Figure 2.5 ABAQUS versus ULSAP for Panels 1 – 3; wotm = 0.5mm............................. 19 Figure 2.6 ABAQUS versus ULSAP for panels 4 – 6; wotm = 6.6mm ............................. 19 Figure 3.1 Three-bay simply supported column............................................................... 22 Figure 3.2 Free body diagram of the beam-column.......................................................... 23 Figure 3.3 Modified step-by-step procedure for three-bay beam-column........................ 24 Figure 3.4 Initial transverse eccentricity of a T-stiffener ................................................. 25 Figure 3.5 Stress and strain diagrams for stages of rectangular cross section.................. 27 Figure 4.1 Column slenderness effect on accuracy of flange beam-column method ....... 32 Figure 4.2 Flange eccentricity effect on flange beam-column ultimate strength ............. 32
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List of Tables Table 1.1 Collapse Modes................................................................................................... 4 Table 2.1 Material Properties............................................................................................ 13 Table 2.2 Panel Elements.................................................................................................. 14 Table 2.3 Boundary conditions ......................................................................................... 16 Table 2.4 Dimensions of ABAQUS test panels................................................................ 17 Table 2.5 Ultimate strength and failure mode computed by ABAQUS ........................... 17 Table 2.6 ULSAP Predictions of ultimate strength for panels 1 – 6................................. 18 Table 4.1 Flange Scantlings.............................................................................................. 31 Table 4.2 ABAQUS versus flange beam-column analysis results ................................... 31 Table 4.3 Panel 4 Flange Scantlings ................................................................................. 33 Table 4.4 Panel 4 ultimate strength calculated with the flange beam-column method .... 33
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Nomenclature Geometric Properties: A cross-sectional area of beam-column I area moment of inertia of beam-column L length of column between supports Le effective length of column a bay length between transverse frames b panel width between longitudinal girders bf breadth of stiffener flange dPH length of plastic hinge region dPP length of primary plastic region dSP length of the secondary plastic region on “upper” edge dSPD length of the secondary plastic region on “lower” edge h height of elastic region of beam-column cross-section hw height of stiffener web t plating thickness tf thickness of stiffener flange tw thickness of stiffener web w deflection w0 initial deflection wopl initial deflection of plating woplm maximum initial deflection of plating wos initial vertical deflection of the stiffener wosm maximum initial vertical deflection of the stiffener wot initial transverse deflection of the stiffener wotm maximum initial vertical deflection of the stiffener wT total deflection Φ curvature δ eccentricity of column θend slope of end bay θmid slope of middle bay λ column slenderness parameter ρ radius of gyration of column cross-section φ magnification factor
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Material and Strength Properties: E Young’s Modulus M Internal bending moment MR Internal bending moment at end of middle bay P axial load PE Euler column critical buckling load P0 dead load for RIKS analysis Pref maximum applied live load for RIKS analysis Ptotal present load for RIKS analysis Ω load scale factor for RIKS analysis ε strain εU strain on “upper” edge of the stiffener flange εL strain on the “lower” edge of the stiffener flange σx axial compressive stress σY yield stress σABQ ultimate stress calculated by ABAQUS σB-C ultimate stress calculated by flange beam-column method σULSAP ultimate stress calculated by ULSAP σU stress on “upper” edge of the stiffener flange σL stress on the “lower” edge of the stiffener flange
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1. Introduction
1.1. Stiffened Panels Modern ship hulls, commercial or military, undergo a wide spectrum of
loading conditions while at sea. The necessity of building a sufficiently strong
hull is accompanied by the desire to build vessels that are lightweight and
spacious enough to maximize cargo or payload. These conditions lead to the
use of thin-walled box-shaped hulls comprised of stiffened panels as seen in
Figure 1.1.
Figure 1.1 Cross-section of tanker hull (Hughes 1988)
The components of a stiffened panel are plating, longitudinal stiffeners,
and transverse frames. For the purpose of this study, a stiffened panel is
defined as the structure spanning multiple bay lengths longitudinally and
transversely bounded by two longitudinal girders. A bay length is defined as
the section of stiffened panel between transverse frames. In a previous
study, B. Ghosh determined that the analysis of a one-bay stiffened panel is
not sufficient. The boundary conditions at the edges of a real panel are
somewhere between simply supported and clamped; therefore, the use of a
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three-bay panel provides a central bay with mixed boundary condition
characteristics (Ghosh, 2003).
Consider a stiffened panel on the deck of a ship. Under a sagging
condition shown in Figure 1.2, the deck stiffened panels are being subjected
to a uniform compressive axial load due to hull girder bending.
Figure 1.2 Ship in sagging loading condition
Figure 1.3 and Figure 1.4 show a three-bay stiffened panel with an
applied uniform axial load, σx. For the purpose of this thesis, T-shaped
stiffeners and frames are being used. Figure 1.5 shows the cross section and
geometrical definitions of a T-stiffener and attached plating.
Figure 1.3 Three bay stiffened panel with axial compression
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Figure 1.4 Three bay stiffened panel with axial compression
Figure 1.5 Plate with attached T-stiffener
1.2. Stiffened Panel Collapse Modes A stiffened panel under compressive loads may fail in a variety of
collapse modes. The actual collapse mode is highly dependant on the
geometrical properties of the stiffeners, frames and plating. Six stiffened
panel collapse modes have been established to describe the form of failure
(Paik & Thayamballi, 2003; Hughes, 1988). Table 1.1 lists the collapse
modes, and Figure 1.6 - Figure 1.11 show diagrams of panels failing in each
mode.
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Table 1.1 Collapse Modes (Paik & Thayamballi, 2003; Hughes, 1988)
Mode Type Collapse Description
I Overall Collapse Stiffeners and plating collapse as a unit
II Biaxial Compressive Collapse
Yielding occurs at intersection of plating and stiffeners near the corners of local plating between stiffeners
III-1 Overall collapse due to beam-column yielding
Yielding of plate-stiffener at mid-span leads to beam column collapse
III-2 Local S-Shaped Mechanism
Yielding in stiffener flange causes a local sideways deflection which leads to a beam-column collapse
IV Local Buckling of Stiffener Web
Panel collapses due to local compressive buckling of stiffener web
V Stiffener Tripping Lateral-torsional buckling of the stiffener causes panel collapse
Figure 1.6 Mode I – Overall collapse
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Figure 1.7 Mode II – Biaxial compressive collapse
Figure 1.8 Mode III-1 – Overall collapse due to beam-column yielding
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Figure 1.9 Mode III-2 – Local S-shaped mechanism
Figure 1.10 Mode IV – Local buckling of stiffener web
Figure 1.11 Mode V – Collapse due to stiffener tripping
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Collapse due to stiffener tripping, mode V, is the primary focus of this
thesis. Tripping is defined as the state where the stiffeners rotate about the
line of attachment to the plating. Once the stiffeners have failed, nothing
remains to support the plate, which immediately leads to collapse of the
panel.
The geometry of the stiffener plays a large role in the tripping
resistance of the panel. In a perfect structure without initial deflection, the
web height, web thickness and the breadth of the flange are the controlling
factors. This thesis suggests that an initial sideways eccentricity also plays a
major role in the tripping resistance of a stiffened panel.
1.3. Smith Panel Experiments Dr. C. S. Smith conducted several tests on full scale welded steel
panels. A test rig was constructed to load the panels with axial compressive
stress and lateral pressure. These panels provide an example of the failure
modes discussed in this thesis. Figure 1.12 shows the collapsed shape of one
of the test panels that failed in mode V, tripping. Figure 1.13 shows the
collapsed shape of a panel that failed in mode III, beam-column collapse
(Smith, 1975).
Figure 1.12 Smith Grillage 1a; tripping (Smith, 1975)
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Figure 1.13 Smith Grillage 3a; beam-column collapse (Smith, 1975)
1.4. Previous Inelastic Tripping Analysis Dr. O. F. Hughes and Dr. M. Ma developed an inelastic stiffener
buckling model using a Rayleigh–Ritz method with four degrees of freedom.
Figure 1.14 shows the allowable deformation degrees of freedom of the
model. The elastic solution is derived from energy methods and is an efficient
eigenvalue problem. More information about the derivation of the solution
may be obtained from the authors’ elastic publication (Hughes & Ma, 1996a).
For the purpose of inelastic study, the model is broken into discrete regions
shown in Figure 1.15. The thickness of each region is determined by the
tangent modulus, and each region is treated as having an individual constant
modulus. Because plasticity acts to reduce the stiffness of a structure, the
use of a tangent modulus to reduce the stiffness is a logical solution. For the
inelastic case, an iterative method was developed to solve the energy based
technique. More information on this inelastic solution may be obtained from
the authors’ inelastic publication (Hughes & Ma, 1996b).
The eigenvalue method provides an efficient solution to the tripping
failure mode, so it was adopted by the closed-form program ULSAP. One
drawback to this method is, being an eigenvalue, or sometimes called a
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bifurcation problem, there is no allowance for any effects of initial eccentricity
of the stiffener flange.
Figure 1.14 Cross-section deformation of flexible web (Hughes & Ma, 1996)
Figure 1.15 Discrete form of the beam-column cross-section (Hughes & Ma, 1996)
1.5. Initial Eccentricity Elastic critical load analysis is a well known method for approximating
the ultimate load of structures. Equation (1) is the critical Euler buckling axial
load for a column:
10
2
2
eE L
EIP π= (1)
The effect of initial deflection is not considered in critical load analysis.
For example, Figure 1.16 shows a plot of a load deflection diagram for a
column under axial load. The ideal Euler buckling load solution is plotted
alongside the load deflection curve of a column with an initial eccentricity. A
perfect column will follow a vertical path until it reaches the Euler buckling
load, at which point the column will buckle; however, an imperfect column
with follow a curved path.
Figure 1.16 Load deflection curve for eccentric columns (Hughes 1988)
Equation (2) is the differential equation for a column under axial load:
( ) 02
2
=++ wEIP
dxwd δ (2)
As δ increases, the bending moment in the column increases. The
deflection will be magnified by the initial deflection. Consider the case of a
column with the initial deflection of a combination of sine waves as shown in
Equation 3 and Figure 1.17:
∑∞
=⎟⎠⎞
⎜⎝⎛=
1sin
nn L
xnπδδ (3)
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Figure 1.17 Eccentric column (Hughes 1988)
The effect of initial deflection of a column is proven to simply act as a
magnification factor as seen in Equations 4 – 5 (Hughes 1988):
φδ=Tw (4)
where PP
P
E
E
−=φ (5)
As initial eccentricity increases, the initial slope of the load-deflection
curve decreases. The magnification of the deflection may lead to premature
yielding, thus decreasing the ultimate strength of the member.
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2. Current Methods of Tripping Analysis
This thesis examines two current methods for determining the tripping
strength of stiffened panels: the finite element analysis program, ABAQUS,
and a closed-form ultimate strength program, ULSAP.
2.1. ABAQUS 1 ½ Bay Stiffened Panel Model The benefit of using finite element analysis is the ability to model
complex shapes with a fine mesh. A finer mesh yields more accurate results
at the cost of computation time. Not only are the calculations computationally
intense, but the generation of the model is time consuming as well. It is
possible to create and use an input file generation subroutine or computer
graphical user interface to reduce model creation time.
For the analysis performed, a 1 ½ bay ABAQUS model was created.
Symmetric loading was used, so a model of only half of the three-bay panel
was required. Figure 2.1 shows the 1 ½ bay ABAQUS model. This 1 ½ bay
ABAQUS model is verified in (Chen 2003 & Dippold 2005).
Figure 2.1 ABAQUS 1 ½ bay model mesh
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2.1.1. Modified RIKS Inelastic Analysis ABAQUS uses a modified RIKS method for inelastic analysis. The
load is proportional, which means it is applied gradually by a scale factor.
The magnitude of the load is defined in Equation (6):
( )00 PPPP reftotal −Ω+= (6)
P0 is the dead load applied before the RIKS analysis, and Pref is the
maximum applied live load. Ω is the scale factor and is determined by
ABAQUS for each load increment. With this loading method, ABAQUS
can track even the most complex load deflection paths (ABAQUS, 2002).
2.1.2. Material Properties
Table 2.1 Material Properties
Material Steel
Young’s Modulus 205800 N/mm2
Poisson’s Ratio 0.3
Yield Stress 352.8 N/mm2
The stress-strain curve is assumed to be the idealized elastic-
perfectly plastic as shown in Figure 2.2:
Figure 2.2 Ideal Elastic-Perfectly Plastic Stress-Strain Plot
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2.1.3. Finite Elements The finite element model of the panel was created using four-node
S4 shell elements (ABAQUS, 2002). The number of elements in the plate
and flange are shown in Table 2.2.
Table 2.2 Panel Elements
Plate Elements 120 x 96
Flange Elements 120 x 6
Web Elements 120 x 6
2.1.4. Initial Deflections The initial deflections of the panel were implemented in the form of
Equations (7)-(9):
⎟⎠⎞
⎜⎝⎛
⎟⎠⎞
⎜⎝⎛=
by
axww oplmopl
ππ sin3sin (7)
⎟⎠⎞
⎜⎝⎛=axww osmosπsin (8)
⎟⎠⎞
⎜⎝⎛
⎟⎠⎞
⎜⎝⎛−
=ax
axwh
zww
otmw
otmotπ
πsin
sin 222
(9)
where ≤≤ z0 ⎟⎠⎞
⎜⎝⎛−axwh otmwπ222 sin
These deflections make the panel take the form shown in Figure
2.3.
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Figure 2.3 Initial deflections for 1 ½ bay ABAQUS panel (Dippold, 2005)
2.1.5. Boundary Conditions The boundary conditions at the edges of the stiffened panel where
it meets with a transverse frame or longitudinal girder are modeled as
simply supported in ABAQUS even though the true boundary condition
lies somewhere between simply supported and clamped. A simple support
will cause a conservative prediction of ultimate strength. This assumption
is conservative because rotational restraint acts to reinforce the panel.
The frame through the panel is not modeled with elements; it is modeled
as a line constrained in the y-direction. This assumes the frame is of
adequate dimensions to prevent the plate from deflecting vertically along
this line.
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Table 2.3 and Figure 2.4 show the boundary conditions for the 1 ½
bay ABAQUS model. Translational constraints are labeled 1, 2, and 3 for
the x, y, and z directions. Rotational constraints are labeled 4, 5, and 6 for
rotation about the x, y, and z axes.
Table 2.3 Boundary conditions (Dippold, 2005)
Loaded Edge 3, 4, 6
Symmetric Edge 1
Sides 3, 5, 6
Loaded & Symmetric Edge Mid-Nodes 2
Frame 3
Figure 2.4 Boundary conditions (Dippold, 2005)
2.2. ABAQUS Test Panel Results Two sets of similar panels are analyzed. The magnitude of the
stiffener transverse initial deflection is different for each set. The height of the
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web is increasing from panel to panel within each set. The dimensions of the
panels are listed in Table 2.4.
Table 2.4 Dimensions of ABAQUS test panels; Units: mm
Table 2.5 shows the ultimate strength and mode of failure of the panels
as calculated by ABAQUS. All of the selected panels have failed in mode V,
tripping, according to ABAQUS. This is determined by analysis of the output
files presented in the appendix.
Table 2.5 Ultimate strength and failure mode computed by ABAQUS; Units: MPa
These panels show clear signs of tripping failure. The plating is
relatively flat, while the stiffeners are bending sideways. The red areas on the
figures in the appendix show yielding. The yielding is progressing across the
flange of the stiffeners.
The difference in initial deflection causes a slight change in shape of
the failed panel. There is only a noticeable difference in ultimate strength due
to the initial deflection between panel 1 and panel 4. These panels have the
smallest web height.
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2.3. ULSAP ULSAP is a closed-form ultimate strength analysis program for
stiffened panels. It calculates an ultimate strength for each mode of failure.
The mode of failure due to the lowest applied stress is chosen as the
controlling mode. ULSAP is very fast computationally and can provide an
ultimate strength prediction for a panel in seconds as opposed to the hours
ABAQUS requires.
For the calculation of mode III failure, ULSAP uses a modified Perry-
Robertson formula. This formula assumes the panel has failed at the first
sign of yielding in the stiffener. If ULSAP determines that the stiffeners are
not of sufficient scantlings to assume failure at first yield, it ignores them and
assumes failure at first yielding in the plate. For mode III failure, ULSAP does
take into consideration the initial vertical deflection of the stiffeners and plate.
For the calculation of mode V failure, ULSAP uses an eigenvalue
solution like the approach presented in (Hughes & Ma, 1996). This approach
does not take into effect the initial sideways deflection of the stiffener.
2.4. ULSAP Predictions of Panels 1 – 6 Panels 1 – 6 were analyzed by ULSAP and the results are presented
in Table 2.6. ULSAP predicts that panels 1 and 4 fail in mode III, and panels
2, 3, 5, and 6 fail in mode V. Figure 2.5 shows the ULSAP results plotted with
the ABAQUS results for panels 1 – 3. Figure 2.6 shows the ULSAP results
plotted with the ABAQUS results for panels 4 – 6.
Table 2.6 ULSAP Predictions of ultimate strength for panels 1 – 6; Units: MPa
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Figure 2.5 ABAQUS versus ULSAP for Panels 1 – 3; wotm = 0.5mm
Figure 2.6 ABAQUS versus ULSAP for panels 4 – 6; wotm = 6.6mm
The results for panels 2, 3, 5 and 6 show ULSAP finds the failure mode
as being mode V with a slightly conservative estimate under that of ABAQUS.
The results for panels 1 and 4 are more interesting. ULSAP predicts the
panels to fail in mode III. For panel 1, the initial transverse deflection is 0.5
mm and the ultimate strength calculated by ABAQUS is closer to ULSAP’s
mode III failure. The initial transverse deflection is increased to 6.6 mm for
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panel 4; the ultimate strength calculated by ABAQUS shifts closer to ULSAP’s
mode V failure.
The panel to the left of ULSAP’s mode III and mode V curve
intersection is affected by initial transverse deflection. Because ULSAP
ignores transverse eccentricity, it may not correctly predict the mode V failure
of a panel with a large stiffener transverse eccentricity and instead predict the
premature failure in mode III.
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3. Beam-Column Method
From the previous section, the need to include the effect of stiffener
transverse eccentricity in ULSAP’s tripping analysis becomes evident. The
search for a method sensitive to initial deflection begins with a successful
beam-column method, ULTBEAM.
3.1. ULTBEAM ULTBEAM, developed by Yong Chen (Chen, 2003), is a numerical
integration procedure based on the step-by-step method (Chen & Lui, 1987).
ULTBEAM uses an inelastic beam-column approach to calculate the ultimate
strength of stiffened panels. It is capable of handling beam-columns with
initial imperfections.
3.1.1. Assumptions There are three assumptions ULTBEAM uses:
1. The cross-section remains plane after bending, and remains
undeformed in the cross section plane.
2. The stress-strain relationship is elastic-perfectly plastic as seen in
Figure 2.2
3. The initial imperfection shape is a half-sine wave as seen in Figure
1.17.
3.1.2. M – Φ – P Relationship A nonlinear relationship between internal moment, M, curvature, Φ,
and axial load, P, must be derived for the cross-section of the beam-
column. The progression of plasticity is tracked by classifying the member
into different cases of progressive yielding. Each case has a different M –
Φ – P relationship. The crux of this relationship lies in the property of
curvature equaling the ratio of strain to the height of the remaining elastic
portion of the member as seen in Equation (10).
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hε
=Φ (10)
3.1.3. Step-by-Step Method The step-by-step method (Chen & Lui, 1987):
1. Divide the full length of the beam into n segments with n+1 stations.
2. Specify the deflection of the first station.
3. Specify the value of the applied load.
4. Calculate the deflection of the next station.
5. Repeat step 4 until the end of the beam is reached.
6. Iterate the applied load until the deflection at the end of the beam is
zero.
This method is only applicable to a single bay simply supported
beam-column. Yong Chen developed a modified procedure for a three-
bay beam column seen in Figure 3.1.
Figure 3.1 Three-bay simply supported column (Chen, 2003)
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Figure 3.2 Free body diagram of the beam-column (Chen, 2003)
This model allows for an internal moment, MR, to be present at the
central pinned locations as seen in Figure 3.2. This is a more realistic model,
as the simply supported single bay model is only applicable to a panel with
infinite bays. This three-bay model assumes that plasticity only occurs in the
middle bay, so this is where the step-by-step method is used. The inclusion
of the internal moment requires an additional iteration loop for the moment.
The slope at the pin support in the middle bay is predicted, and the moment is
iterated until the slope equals the predicted value. Figure 3.3 shows a
flowchart of the modified process. A more thorough description of ULTBEAM
may be found in Yong Chen’s dissertation (Chen, 2003).
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Figure 3.3 Modified step-by-step procedure for three-bay beam-column (Chen, 2003)
The process solves for the applied load that corresponds to the
specified initial deflection. It is repeated for a wide range of initial
deflections, and so ULTBEAM provides a simple way to plot a load-
deflection curve. The ultimate strength is determined from the peak load
of that curve.
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3.2. Applying the Beam-Column Method to Tripping
3.2.1. Geometry and Assumptions Consider a T-stiffener that is deflected to the side as shown in
Figure 3.4.
Figure 3.4 Initial transverse eccentricity of a T-stiffener
The plating at the base of the stiffener web does not provide
enough restraint to prevent the stiffener from rotating. The stiffener web
will remain rigid and rotate. The most resistance to tripping comes from
the breadth of the stiffener flange. It seems possible to analyze only the
flange of the stiffener as a beam-column having an initial deflection and
bending about the y-axis. This thesis will refer to the study of the flange
as a beam-column as the flange beam-column method.
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The assumptions associated with the flange beam-column analysis
are:
1. The initial deflection is small enough that the cross-section of the
flange remains a rectangle during bending.
2. Because it exists along the neutral axis of bending, the stiffener
web does not supply much bending stiffness and can be neglected.
3. The cross section remains plane after bending.
4. The stress-strain relationship is elastic-perfectly plastic.
5. The initial sideways deflection follows a half-sine wave along the
flange.
3.2.2. M – Φ – P Relationship for Rectangular Cross Section Yong Chen derived a set of M – Φ – P equations for three stages
as seen in Figure 3.5. The three stages are elastic, primary plastic, and
secondary plastic. During the primary plastic stage, yielding is present on
one edge of the flange. Yielding is present on both edges of the flange
during the secondary plastic stage. Once plasticity progresses across the
entire cross section, a plastic hinge is formed and the member is assumed
to fail because it cannot receive further loading.
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Figure 3.5 Stress and strain diagrams for stages of rectangular cross section (Chen, 2003)
28
The following are the M – Φ – P relationships for the above stages
of a rectangular cross-section (Chen, 2003):
Elastic: Since there is no yielding, the relationship is
EIM
=Φ (11)
Primary Plastic:
From equilibrium, P and M as a function of stress, σ, can be
derived.
∫=A
dAP σ
∫−
−
+=PP
f
f
db
bfPPfY dztzdtP
2
2
)(σσ (12)
∫=A
zdAM σ
∫−
−
+−
=PP
f
f
db
bf
PPfPPfY zdztz
dbdtM
2
2
)(2
σσ (13)
where:
Lf
PPf
LY bz
dbz σ
σσσ +⎟⎟
⎠
⎞⎜⎜⎝
⎛+
−−
=2
)( (14)
Solve Equations (12) – (14) simultaneously for dPP. From Equation
(10), the curvature for the primary plastic stage is:
)( PPf
LY
PPf
LY
dbEdb −−
=−−
=Φσσεε (15)
Secondary Plastic:
From equilibrium, P and M as a function of stress, σ, can be
derived.
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∫=A
dAP σ
∫−
−
+−=SP
f
fSPD
db
bd
fSPDSPfY dztzddtP2
2
)()( σσ (16)
∫=A
zdAM σ
∫−
−
+−
+−
=SP
f
fSPD
db
bd
fSPDf
SPDfYSPf
SPfY zdztzdb
dtdb
dtM2
2
)(22
σσσ (17)
where:
YSPDf
SPDSPf
Y db
zddb
z σσ
σ −⎟⎟⎠
⎞⎜⎜⎝
⎛−+
−−=
22
)( (18)
Solve Equations (16) – (18) simultaneously for dSP and dSPD. From
Equation (10), the curvature for the secondary plastic stage is:
)(22
SPDSPf
Y
SPDSPf
Y
ddbEddb −−=
−−=Φ
σε (19)
Perfect Plastic Hinge:
Once this condition is reached, the calculation stops.
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4. Comparison of Results
A MATLAB program using the flange beam-column method was
written to calculate the ultimate strength of a rectangular cross section
stiffener flange with initial transverse deformation. The results from the
beam-column analysis were compared to results from an ABAQUS finite
element beam model.
4.1. ABAQUS Beam Model A three-bay model of a stiffener flange was modeled in ABAQUS using
one-dimensional 2-node B23 beam elements (ABAQUS, 2002). Figure 3.1 is
a good representation of the model and the simply supported boundary
conditions. Each model has 54 elements with 55 nodes. The material
properties are the same as the steel from the 1 ½ bay stiffened panel models.
The initial deflection is modeled as a half-sine wave across each bay as
shown in Equation (20).
⎟⎠⎞
⎜⎝⎛=axww otmotπsin (20)
The axial load increment was proportionally applied by the RIKS method.
4.2. Flange Scantlings Column slenderness plays a large role in the behavior of beam-column
failure. Equation (21) shows the definition of the non-dimensional
slenderness parameter for columns:
EL Yσρ
λ = (21)
where 12
2fb
AI==ρ for rectangular cross sections (22)
For this study, four different flange beam-columns of varying λ were
chosen.
Table 4.1 shows the dimensions of the test flanges.
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Table 4.1 Flange Scantlings
4.3. Validation of the Flange Beam-Column Method Table 4.2 shows the ultimate strength of the test flanges across a
range of initial transverse deflections, wotm.
Table 4.2 ABAQUS versus flange beam-column analysis results
To show the effect of the column slenderness parameter on the
accuracy of the flange beam-column method, the ultimate strength calculated
by the flange beam-column method divided by the ultimate strength
calculated by the ABAQUS model was plotted in Figure 4.1 for each set of
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flanges. As the flanges become more slender, the accuracy of the flange
beam-column method improves greatly. The λ = 5 flange set was the most
accurate. The ultimate strength as calculated by the flange beam-column
method for this set is plotted versus the initial flange eccentricity in Figure 4.2.
The strength of the member is reduced as the initial flange eccentricity
increases.
Figure 4.1 Column slenderness effect on accuracy of flange beam-column method
Figure 4.2 Flange eccentricity effect on flange beam-column ultimate strength
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4.4. Revisit the 1 ½ Bay Panels The flange of the panels tested in chapter 2 have a column
slenderness parameter equal to 5.16. Table 4.3 shows the scantlings of the
flange from panel 4. The flange was tested for ultimate strength with the
flange beam-column method and the results are shown in Table 4.4.
Table 4.3 Panel 4 Flange Scantlings
Table 4.4 Panel 4 ultimate strength calculated with the flange beam-column method
The ultimate strength of the panel calculated by the 1 ½ bay ABAQUS
model from chapter 2 is 265.75 MPa as compared to the flange beam-column
calculation of 90.5 MPa. It appears to be inadequate to isolate and only
consider the flange for ultimate tripping strength predictions. The stress
prediction is far lower than the full panel stress. Based on the above numbers
and assuming a linear relationship, the flange is supplying about 30% of the
strength. The plating and the stiffener web are providing some rotational
restraint that must be accounted for.
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5. Conclusions and Recommendations for Future Work
5.1. Conclusions The necessity to consider initial imperfections in ultimate strength
calculations is supported by this research. Eigenvalue, or bifurcation,
methods provide optimistic estimates to a structural member’s ultimate
strength. The continuous tracking of a member’s load-deflection path is lost
with eigenvalue analysis.
This research attempted to introduce an analytical method for
calculating stiffened panel ultimate strength that considers the initial
transverse eccentricity of the stiffener flange. A method using a beam-
column approach attempted to decouple the lateral deflection of a T-stiffener
flange from the torsional restraint of the stiffener web. The results were not
consistent with finite element analysis results.
5.2. Recommendations for Future Work Future work with this method needs to investigate a way to capture the
coupled nature of the bending restraint of the flange to the torsional restraint
caused by the attachment of the web to the plate and still maintain the
method’s sensitivity to initial eccentricity.
35
References ABAQUS. ABAQUS/Standard user’s manual, Vol. I-III, ver. 6.1, Hibbitt, Karlsson
& Sorenson Inc, RI, 2002.
Chen, W.F., Lui, E.M. Structural Stability: Theory and Implementation, Elsevier, New York, 1987.
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.
Dippold, S. Validation of the ULSAP Closed-Form Method for Ultimate Strength Analysis of Cross-Stiffened Panels, Department of Aerospace and Ocean Engineering, Virginia Polytechnics Institute and State University, Blacksburg, VA, 2005.
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.
Hughes, O.F., Ma, M. Elastic Tripping Analysis of Asymmetrical Stiffeners, Department of Aerospace and Ocean Engineering, Virginia Polytechnics Institute and State University, Blacksburg, VA, 1996a.
Hughes, O.F., Ma, M. Inelastic Analysis of Panel Collapse by Stiffener Buckling, Department of Aerospace and Ocean Engineering, Virginia Polytechnics Institute and State University, Blacksburg, VA, 1996b.
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.
36
Appendix
This appendix contains plots of the six 1 ½ bay panels tested with ABAQUS in chapter 2. Each figure shows the panel deformation and stress at collapse. The color coded data shows von Mises Stress (MPa). The computer post-processing software, PATRAN, was used to generate these plots.
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Vita
Scott Patten was born in Richmond, Virginia on February 25, 1982. Scott
is the first son of Paul and Patricia Patten and has a younger brother Michael.
He attended Virginia Tech in the fall of 2000. Scott obtained his Bachelor of
Science in Ocean Engineering in May 2004 and began graduate research the
following summer. This thesis completes the requirements for his Master of
Science in Ocean Engineering.