1

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

2

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.

iii

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.

iv

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

v

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

vi

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

vii

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

viii

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

1

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

2

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

3

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.

4

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

5

Figure 1.7 Mode II – Biaxial compressive collapse

Figure 1.8 Mode III-1 – Overall collapse due to beam-column yielding

6

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

7

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)

8

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

9

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)

11

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.

12

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

13

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

14

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.

15

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.

16

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

17

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.

18

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

19

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

20

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.

21

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).

22

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)

23

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).

24

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.

25

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.

26

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.

27

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.

29

∫=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.

30

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.

31

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

32

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

33

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.

34

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.

37

38

39

40

41

42

43

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.