LONGITUDINAL STIFFENERS ONLONGITUDINAL STIFFENERS ON

COMPRESSION PANELSCOMPRESSION PANELS

Chai H. Yoo, Ph.D., P.E., F. ASCEChai H. Yoo, Ph.D., P.E., F. ASCEProfessor EmeritusProfessor EmeritusDepartment of Civil EngineeringDepartment of Civil EngineeringAuburn UniversityAuburn University

CIVL 7690CIVL 7690

July 14, 200July 14, 20099

HistoryHistory

● ● The most efficient structural form is The most efficient structural form is trusstruss

with regard to its weight-to-strength ratio with regard to its weight-to-strength ratio provided that all other conditions are equal.provided that all other conditions are equal.

Old section of NY Metro Subway system,Old section of NY Metro Subway system,

Tower crane post and arms,Tower crane post and arms,

Space station,Space station,

New Orleans Super dome, etc.New Orleans Super dome, etc.

Brooklyn Bridge, New YorkBrooklyn Bridge, New York Designed by Roebling, Opened in 1883Designed by Roebling, Opened in 1883

George Washington Bridge, New YorkGeorge Washington Bridge, New YorkDesigned by Amman, opened in 1931Designed by Amman, opened in 1931

Auburn University Auburn University Highway Bridges, Past, Present, and FutureHighway Bridges, Past, Present, and Future

HistoryHistory

● ● For containment type structuresFor containment type structures

maintaining two or more separate pressuremaintaining two or more separate pressure

or temperature zones, continuous barriers,or temperature zones, continuous barriers,

membranes, plates and shells, aremembranes, plates and shells, are

required.required.

Aircraft fuselageAircraft fuselage, ,

Dome roofDome roof,,

SubmarinesSubmarines, etc., etc.

HistoryHistory

● ● When the loads (both transverse andWhen the loads (both transverse and longitudinal) arelongitudinal) are

smallsmall→→membrane, i.e., placardmembrane, i.e., placard mediummedium→→platesplates

heavyheavy→→stiffened platesstiffened plates topic of discussiontopic of discussion

BACKGROUNDBACKGROUND

AASHO Standard Specifications for Highway Bridges, 9AASHO Standard Specifications for Highway Bridges, 9 thth ed., 1965 ed., 1965 adopted for the first time the minimum moment of inertia of the adopted for the first time the minimum moment of inertia of the longitudinal stiffener:longitudinal stiffener:

where where

3sI t w

3 40.07k n for n 1

30.125k for n 1

2with k 4

There was no further stipulation as to the correct value for k.

BACKGROUNDBACKGROUND

For composite box girder compression flanges For composite box girder compression flanges stiffened longitudinally and transversely, AASHTO stiffened longitudinally and transversely, AASHTO requites the minimum moment of inertia of the requites the minimum moment of inertia of the longitudinal stiffener:longitudinal stiffener:

3sI 8 t w

It is of interest to note that the absence of a length parameter of the longitudinal stiffener in both AASHTO equations. A longitudinal stiffener attached to the compression flange is essentially a compression member.

BACKGROUNDBACKGROUND

It was found that an old bridge, It was found that an old bridge, (curved box girder approach spans (curved box girder approach spans to the to the Fort Duquesne BridgeFort Duquesne Bridge in in Pittsburg) designed and built before Pittsburg) designed and built before the enactment of the AASHTO the enactment of the AASHTO criteria on longitudinal stiffeners, did criteria on longitudinal stiffeners, did not rate well for modern-day traffic, not rate well for modern-day traffic, despite having served for many despite having served for many years.years.

BACKGROUNDBACKGROUND

Despite the practicing engineers’ Despite the practicing engineers’ intuitive realization of the intuitive realization of the unreasonableness of the equations, unreasonableness of the equations, they are still in force in both AASHTO they are still in force in both AASHTO Standard SpecificationsStandard Specifications for Highway for Highway Bridges, 17Bridges, 17thth ed. (2002) and AASHTO ed. (2002) and AASHTO LRFD Bridge Design SpecificationsLRFD Bridge Design Specifications, 4th , 4th ed. (2007) with a limitation imposed on ed. (2007) with a limitation imposed on the number of longitudinal stiffeners not the number of longitudinal stiffeners not to exceed “two.”to exceed “two.”

BACKGROUNDBACKGROUND

In a relatively short period of time, there In a relatively short period of time, there were a series of tragic collapses occurred were a series of tragic collapses occurred during the erection of the bridgesduring the erection of the bridges

Danube in 1969Danube in 1969

Milford Haven Bridge in Wales in 1970Milford Haven Bridge in Wales in 1970

West Gate Bridge in Australia in 1970West Gate Bridge in Australia in 1970

Koblenz Bridge in Germany in 1971Koblenz Bridge in Germany in 1971

BACKGROUNDBACKGROUND

These tragic collapses drew an urgent attention to steel box These tragic collapses drew an urgent attention to steel box girder bridge design and construction. Some of the girder bridge design and construction. Some of the researchers, primarily in the U.K., responded to the urgency researchers, primarily in the U.K., responded to the urgency include:include:

ChatterjeeChatterjeeDowlingDowlingDwightDwightHorneHorneLittleLittleMerrisonMerrisonNarayanaNarayana

BACKGROUNDBACKGROUND

Although there were a few variations Although there were a few variations tried, such astried, such as

Effective Width MethodEffective Width MethodEffective Length MethodEffective Length Method

these researchers were mainly these researchers were mainly interested in “Column Behavior” of the interested in “Column Behavior” of the stiffened compression flanges.stiffened compression flanges.

BACKGROUNDBACKGROUND

BarbrBarbréé studied the strength studied the strength of longitudinally stiffened of longitudinally stiffened compression flanges and compression flanges and published extensive results in published extensive results in 19371937..

BACKGROUNDBACKGROUND

BleichBleich (1952) and (1952) and TimoshenkoTimoshenko and and GereGere (1961) introduced (1961) introduced BarbrBarbréé’s’s study (published in study (published in German) to English speaking German) to English speaking world using the following world using the following model:model:

y

xO

a

2wStiffener

t t

w

w

·

Symmetric and Antisymmetric Symmetric and Antisymmetric Buckling Mode ShapesBuckling Mode Shapes

Consider the load carrying mechanics of a plateConsider the load carrying mechanics of a plate

element subjected to a element subjected to a transverse loadingtransverse loading

● ● Very thin plates depend on the Very thin plates depend on the membrane actionmembrane action as as

that in placards and airplane fuselagesthat in placards and airplane fuselages

● ● Ordinary plates depend primarily on the Ordinary plates depend primarily on the bending bending actionaction

● ● Very thick plates depend on Very thick plates depend on bending and shearbending and shear

actionaction

Our discussions herein are limited to the case of ordinary Our discussions herein are limited to the case of ordinary plateplate

Elements (no membrane action, no shear deformation)Elements (no membrane action, no shear deformation)

BACKGROUNDBACKGROUND

It was known from the early days It was known from the early days that stiffened plates with weak that stiffened plates with weak stiffeners buckle in a symmetric stiffeners buckle in a symmetric mode while those with strong mode while those with strong stiffeners buckle in an stiffeners buckle in an antisymmetric mode. The exact antisymmetric mode. The exact threshold value of the minimum threshold value of the minimum moment of inertia of the moment of inertia of the stiffener, however, was stiffener, however, was unknown.unknown.

Symmetric or antisymmetric Symmetric or antisymmetric buckling is somewhat buckling is somewhat confusing. It appears to be confusing. It appears to be just the remnant of just the remnant of terminology used by Bleich. It terminology used by Bleich. It is obvious thatis obvious that

symmetric bucklingsymmetric buckling implies implies column behaviorcolumn behavior and and

antisymmetric bucklingantisymmetric buckling implies implies plate behaviorplate behavior

It appears to be the case, at It appears to be the case, at least in the earlier days, that least in the earlier days, that the the column behaviorcolumn behavior theory theory was dominant in Europe, was dominant in Europe, Australia, and Japan while in Australia, and Japan while in North America, particularly, in North America, particularly, in the U.S., a modified the U.S., a modified plate plate behaviorbehavior theory prevailed. theory prevailed.

Japanese design of rectangular box sections of a horizontally curved continuous girder

In the In the column behaviorcolumn behavior theory, theory, the strength of a stiffened the strength of a stiffened plate is determined by plate is determined by summing the summing the column strengthcolumn strength of each of each individual longitudinal individual longitudinal stiffenerstiffener, with an effective , with an effective width of the plate to be part of width of the plate to be part of the cross section, between the the cross section, between the adjacent transverse stiffeners.adjacent transverse stiffeners.

It should be noted that in It should be noted that in symmetric buckling (column symmetric buckling (column behavior), the behavior), the stiffener bendsstiffener bends along with the plate whereas along with the plate whereas in antisymmetric buckling in antisymmetric buckling (plate behavior), the stiffener (plate behavior), the stiffener remains straightremains straight although it is although it is subjected to subjected to torsional rotationtorsional rotation..

Symmetric Mode

Antisymmetric Mode

Hence, it became intuitively Hence, it became intuitively evident that in order to evident that in order to ensure antisymmetric ensure antisymmetric buckling, the stiffener must buckling, the stiffener must be sufficiently strong.be sufficiently strong.

A careful analysis of data from A careful analysis of data from a series of finite element a series of finite element analyses made it possible to analyses made it possible to determine numerically the determine numerically the threshold value of the minimum threshold value of the minimum required moment of inertia of a required moment of inertia of a longitudinal stiffener to ensure longitudinal stiffener to ensure antisymmetric buckling.antisymmetric buckling.

Critical Stress vs Longitudinal Stiffener Size

29

29.4

29.8

30.2

30.6

580 630 680 730

Moment of Inertia, I s (in4)

Fcr

(ks

i)

Symmetric Antisymmetric

Selected example data are Selected example data are shown in the table. During shown in the table. During the course of this study, well the course of this study, well over 1,000 models have been over 1,000 models have been analyzed.analyzed.

Comparison of Ultimate Stress, Fcr (ksi)

n w(in.)

t(in.)

w/tR

(ft)Is, Eq.(1)

(in4)

Is, used

(in4)Fcr,

AASHTO

Fcr,

FEM,=w/1000

Fcr,

FEM,=w/100

3 3 120 1.50 80.0 800 1894 1902 16.4 23.6 19.1

2 3 60 0.94 64.0 200 189 189 25.6 30.0 27.3

1 3 60 1.13 53.3 200 231 233 35.6 37.3 31.8

3 5 30 0.75 40.0 200 164 165 46.2 46.7 38.4

1 5 30 1.25 24.0 300 439 442 50.0 50.0 45.6

1 5 30 1.88 16.0 200 1483 1510 50.0 50.0 49.8

(Note: 1 in. = 25.4 mm; 1 ft = 0.305 m; 1 in4 = 0.416106 mm4; 1 ksi = 6.895 MPa)

Jaques HeymanJaques Heyman, Professor emeritus, University , Professor emeritus, University of Cambridge, wrote in 1999 that there had of Cambridge, wrote in 1999 that there had been no new breakthrough since been no new breakthrough since Hardy CrossHardy Cross published published Moment DistributionMoment Distribution method in 1931. method in 1931.

I disagree.I disagree.

The most significant revolution in modern era is The most significant revolution in modern era is Finite ElementFinite Element method. Although the vague method. Although the vague notion of the method was there since the time notion of the method was there since the time of Rayleigh and Ritz, the finite element method of Rayleigh and Ritz, the finite element method we are familiar with today was not available we are familiar with today was not available until in the late 1980s encompassing the until in the late 1980s encompassing the material and geometric nonlinear incremental material and geometric nonlinear incremental analysis incorporating the updated and/or total analysis incorporating the updated and/or total LagrangianLagrangian formulation. formulation.

Despite the glitter, Despite the glitter, Finite ElementFinite Element method is method is not a design guide.not a design guide.

Daily practicing design engineers need Daily practicing design engineers need design guide in the form of design guide in the form of chartscharts, , tablestables and/or and/or regression formulasregression formulas synthesizing and quantifying vast synthesizing and quantifying vast analytical data afforded from the finite analytical data afforded from the finite element method.element method.

There exist golden opportunities for There exist golden opportunities for engineering researchers to do just those engineering researchers to do just those contributions.contributions.

REGRESSION EQUATIONREGRESSION EQUATION

2 3sI 0.3 n t w

Where aspect ratio a / w

n number of stiffeners

a / w0 1 2 3 4

k

2

4

6

2 6

Plate Buckling Coefficient

It was decided from the It was decided from the beginning of our study that beginning of our study that we wanted to make sure that we wanted to make sure that our stiffened compression our stiffened compression flanges would buckle in an flanges would buckle in an antisymmetric mode.antisymmetric mode.

In the elastic buckling range In the elastic buckling range of the width-to-thickness of the width-to-thickness ratio, the critical stress of ratio, the critical stress of the plate isthe plate is

22

cr 2

k E tF

w12 1

with k 4

AASHTO divides the sub-panel AASHTO divides the sub-panel between longitudinal stiffeners between longitudinal stiffeners or the web into three zones by or the web into three zones by the width-to-thickness ratio:the width-to-thickness ratio:

yield zone = compactyield zone = compacttransition zone = noncompacttransition zone = noncompactelastic buckling zone = slenderelastic buckling zone = slender

The regression equation for The regression equation for the minimum required the minimum required moment of inertia of the moment of inertia of the longitudinal stiffener works longitudinal stiffener works equally well for the sub-equally well for the sub-panels in all three zones.panels in all three zones.

It also works for It also works for horizontally horizontally curved box girderscurved box girders..

Critical stress vs width-to-thickness Critical stress vs width-to-thickness ratioratio

0

10

20

30

40

50

60

0 30 60 90 120 150w/t

F cr

(ksi

)

AASHTO Eq.(10-134)

Bifurcation Analysis

Nonlinear Analysis (W/1000)

Nonlinear Analysis (W/100)

SSRC Type P arabola

4 Eq. Spa. 5 Eq. Spa. 4 Eq. Spa.

9’-0” 9’-0”12’-0”

Longitudinal stiffener arrangement, AASHTO

2 Eq. Spa. 3 Eq. Spa. 2 Eq. Spa.

9’-0” 9’-0”12’-0”

Longitudinal stiffener arrangement, Proposed

Japanese design of rectangular box sections of a horizontally curved continuous girder

Stiffened Compression Stiffened Compression Panel (Japanese Panel (Japanese

Practice)Practice)

Tee shapes are stronger than Tee shapes are stronger than rectanglesrectangles

Consider the moment of inertia about the axis Consider the moment of inertia about the axis parallel to the flange and at the base of the parallel to the flange and at the base of the stiffener.stiffener.

TeeTee, WT9x25: A = 7.35 in, WT9x25: A = 7.35 in22, t, tff = 0.57 in = 0.57 in

IIss = 53.5+7.35(8.995-2.12) = 53.5+7.35(8.995-2.12)22 = = 400 in400 in44

RectangleRectangle, d/t = 0.38(E/Fy), d/t = 0.38(E/Fy)1/21/2 = 9.15 with Fy = 50 = 9.15 with Fy = 50 ksi for compact section:ksi for compact section:

9.15t9.15t22 = 7.35, t = 0.9 in, d=7.35/0.9 = 8.17 in = 7.35, t = 0.9 in, d=7.35/0.9 = 8.17 in

IIss = 0.9(8.17) = 0.9(8.17)3/3/3 =3 =164 in164 in44

Quick ComparisonQuick Comparison

2

22

2

The limiting value of the slenderness ratio assuming

the residual s

Pla

tress of 0.3 is

40.7 54.73 43.2

12

t e Beh a vior T h eor

1 /

0.005 43.2 50 40. 7

y

6

y

cr y

cr

F

E bF F

tb t

F ksi

4

0.381.92 1

/

29000 0.38 290001.92 1.25 1 49.3

40 54 /1.25

Column Behavior The

40

49.30.913

54

is com

o

puted as 458 in

ry

e

a

s

E Eb t

f b t f

Q Q

I

2

2

2

The area of the effective section is 142.3 in

458 /142.3 1.794 in

1 10 1266.9, 64 ksi

1.794 /

0.658 33.9 ksi

40.67 33.9100 19.97%

33.9

y

e

e

QF

Fcr y

r

KL EF

r KL r

F Q F

2

4

2

2

For transverse stiffeners at 20 ft, WT12 38

is needed. The effective section becomes 146.2 in

and corresponding is computed as 1010 in

1010 /146.2 2.63 in

1 20 1291.25, 34.37

2.63 /

s

e

I

r

KL EF

r KL r

ksi

0.658 26.18 ksi

40.67 26.18100 55.34%

26.18

A spacing of 20 ft is more reasonable in this case.

Hence, a 55% extra strength is recognized by th

plate behavior th

e

eory.

y

e

QF

Fcr yF Q F

Stiffened Compression Stiffened Compression Panel (Japanese Panel (Japanese

Practice)Practice)

Concluding RemarksConcluding Remarks

• The AASHTO critical stress equation appears The AASHTO critical stress equation appears to be to be unconservativeunconservative in the transition zone in the transition zone with AWS acceptable out-of-flatness with AWS acceptable out-of-flatness tolerances.tolerances.

• Residual stresses Residual stresses significantly reducesignificantly reduce the the critical stresses of slender plates.critical stresses of slender plates.

• Recognition of the postbuckling reserve Recognition of the postbuckling reserve strength in slender plates remains strength in slender plates remains debatable with regard to the adverse effect debatable with regard to the adverse effect of of large deflectionlarge deflection..

• The regression equation derived appears The regression equation derived appears now to be ready to now to be ready to replacereplace two AASHTO two AASHTO equations without any limitations imposed.equations without any limitations imposed.

Concluding Remarks -continuedConcluding Remarks -continued

• It has been proved that the It has been proved that the plate plate behavior theory behavior theory yields a more yields a more economical design than that by the economical design than that by the column behavior theorycolumn behavior theory..

• In the numerical example examined, it In the numerical example examined, it is is 20%-50%20%-50% more economical. more economical.

4 Eq. Spa. 5 Eq. Spa. 4 Eq. Spa.

9’-0” 9’-0”12’-0”

Longitudinal stiffener arrangement, AASHTO

2 Eq. Spa. 3 Eq. Spa. 2 Eq. Spa.

9’-0” 9’-0”12’-0”

Longitudinal stiffener arrangement, Proposed

Symmetric ModeColumn Behavior TheoryGlobal Buckling

Antisymmetric ModePlate Behavior TheoryLocal Buckling

J. Structural Engineering, ASCE, Vol. 127, J. Structural Engineering, ASCE, Vol. 127,

No. 6, June 2001, pp. 705-711No. 6, June 2001, pp. 705-711

J. Engineering Mechanics, ASCE, Vol. 131, J. Engineering Mechanics, ASCE, Vol. 131, No.2, February 2005, pp. 167-176No.2, February 2005, pp. 167-176

Engineering Structures, Elsevier, Vol. Engineering Structures, Elsevier, Vol. 29(9),29(9),

September 2007, pp. 2087-2096September 2007, pp. 2087-2096

Engineering Structures, ElsevierEngineering Structures, Elsevier, Vol. , Vol. 31(5),31(5),

May 2009, pp. 1141-1153May 2009, pp. 1141-1153

REGRESSION EQUATIONREGRESSION EQUATION

2 3sI 0.3 n t w

Where aspect ratio a / w

n number of stiffeners

ENDEND