LONGITUDINAL STIFFENERS ON COMPRESSION PANELS

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LONGITUDINAL STIFFENERS ON

COMPRESSION PANELS



Chai H. Yoo, Ph.D., P.E., F. ASCE

Professor Emeritus

Department of Civil Engineering

Auburn University



CIVL 7690



July 14, 200
9




History



The most efficient structural form is
truss


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



Old section of NY Metro Subway system,


Tower crane post and arms,


Space station,


New Orleans Super dome, etc.


Brooklyn Bridge, New York


Designed by Roebling, Opened in 1883

George Washington Bridge, New York

Designed by Amman, opened in 1931


Auburn University
Highway Bridges, Past, Present, and Future

History



For containment type structures


maintaining two or more separate pressure


or temperature zones, continuous barriers,


membranes, plates and shells, are


required.



Aircraft fuselage
,


Dome roof
,


Submarines
, etc.

History



When the loads (both transverse and


longitudinal) are



small

membrane, i.e., placard




medium

plates



heavy

stiffened plates


topic of discussion






BACKGROUND


AASHO Standard Specifications for Highway Bridges, 9
th

ed., 1965 adopted for
the first time the minimum moment of inertia of the longitudinal stiffener:










where

3
s
I t w



3 4
0.07k n for n 1

 

3
0.125k for n 1

 
2
with k 4
 
There was no further stipulation as to the correct value for k.

BACKGROUND


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




3
s
I 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.

BACKGROUND


It was found that an old bridge,
(curved box girder approach spans to
the
Fort Duquesne Bridge

in Pittsburg)
designed and built before the
enactment of the AASHTO criteria on
longitudinal stiffeners, did not rate
well for modern
-
day traffic, despite
having served for many years.

BACKGROUND


Despite the practicing engineers’ intuitive
realization of the unreasonableness of the
equations, they are still in force in both
AASHTO
Standard Specifications

for
Highway Bridges, 17
th

ed. (2002) and
AASHTO
LRFD Bridge Design Specifications
,
4th ed. (2007) with a limitation imposed on
the number of longitudinal stiffeners not to
exceed “two.”

BACKGROUND


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




Danube in 1969



Milford Haven Bridge in Wales in 1970



West Gate Bridge in Australia in 1970



Koblenz Bridge in Germany in 1971

BACKGROUND


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




Chatterjee



Dowling



Dwight



Horne



Little



Merrison



Narayana

BACKGROUND


Although there were a few variations tried,
such as




Effective Width Method



Effective Length Method



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

BACKGROUND




Barbr
é

studied the strength of
longitudinally stiffened
compression flanges and
published extensive results in
1937
.

BACKGROUND




Bleich

(1952) and
Timoshenko

and
Gere

(1961) introduced
Barbr
é
’s

study (published in
German) to English speaking
world using the following model:

y
x
O
a
2w
Stiffener

t

t
w
w

Symmetric and Antisymmetric
Buckling Mode Shapes

Consider the load carrying mechanics of a plate

element subjected to a
transverse loading




Very thin plates depend on the
membrane action

as


that in placards and airplane fuselages




Ordinary plates depend primarily on the
bending action




Very thick plates depend on
bending and shear


action


Our discussions herein are limited to the case of ordinary plate

Elements (no membrane action, no shear deformation)

BACKGROUND


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


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



symmetric buckling

implies
column behavior

and


antisymmetric buckling

implies
plate behavior


It appears to be the case, at least
in the earlier days, that the
column behavior

theory was
dominant in Europe, Australia,
and Japan while in North America,
particularly, in the U.S., a
modified
plate behavior

theory
prevailed.

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


In the
column behavior

theory,
the strength of a stiffened plate is
determined by summing the
column strength

of each
individual longitudinal stiffener
,
with an effective width of the
plate to be part of the cross
section, between the adjacent
transverse stiffeners.


It should be noted that in
symmetric buckling (column
behavior), the
stiffener bends

along with the plate whereas in
antisymmetric buckling (plate
behavior), the stiffener
remains
straight

although it is subjected
to
torsional rotation
.

Symmetric Mode

Antisymmetric Mode



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


A careful analysis of data from a
series of finite element analyses
made it possible to determine
numerically the threshold value of
the minimum required moment of
inertia of a longitudinal stiffener
to ensure 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
(in
4
)
F
cr
(ksi)
Symmetric Antisymmetric


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

Comparison of Ultimate Stress,
F
cr

(ksi)


n

a

w

(in.)

t

(in.)

w/t

R

(ft)

I
s
,

Eq.(1)

(in
4
)

I
s
,

used

(in
4
)

F
cr
,

AASHTO

F
cr
,

FEM,

D
=w
/1000

F
cr
,

FEM,

D
=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 in
4

= 0.416

10
6

mm
4
; 1 ksi = 6.895 MPa)




Jaques Heyman
, Professor emeritus, University of
Cambridge, wrote in 1999 that there had been no
new breakthrough since
Hardy Cross

published
Moment Distribution

method in 1931.



I disagree.



The most significant revolution in modern era is
Finite Element

method. Although the vague notion
of the method was there since the time of Rayleigh
and Ritz, the finite element method we are familiar
with today was not available until in the late 1980s
encompassing the material and geometric nonlinear
incremental analysis incorporating the updated
and/or total
Lagrangian

formulation.



Despite the glitter,
Finite Element

method is
not a design guide.



Daily practicing design engineers need
design guide in the form of
charts
,
tables

and/or
regression formulas

synthesizing and
quantifying vast analytical data afforded
from the finite element method.



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


REGRESSION EQUATION

2 3
s
I 0.3 n t w
a

Where


 
aspect ratio a/w
a


n number of stiffeners
a / w
0
1
2
3
4
k
2
4
6
2
6
Plate Buckling Coefficient


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


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




 

 

 
2
2
cr
2
k E t
F
w
12 1


with


k 4

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



yield zone = compact


transition zone = noncompact


elastic buckling zone = slender


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



It also works for
horizontally
curved box girders
.

Critical stress vs width
-
to
-
thickness ratio

0
10
20
30
40
50
60
0
30
60
90
120
150
w/t
F
cr
(ksi)
AASHTO Eq.(10-134)
Bifurcation Analysis
Nonlinear Analysis (W/1000)
Nonlinear Analysis (W/100)
SSRC Type Parabola

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
Panel (Japanese Practice)

Tee shapes are stronger than rectangles


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



Tee
, WT9x25: A = 7.35 in
2
, t
f

= 0.57 in


I
s

= 53.5+7.35(8.995
-
2.12)
2

=
400 in
4



Rectangle
, d/t = 0.38(E/Fy)
1/2

= 9.15 with Fy = 50 ksi
for compact section:




9.15t
2

= 7.35, t = 0.9 in, d=7.35/0.9 = 8.17 in


I
s

= 0.9(8.17)
3/
3 =
164 in
4



Quick Comparison









2
2
2
2
The limiting value of the slenderness ra
tio assuming
the residual s
Pla
tress of 0.3 is
4
0.7 54.73 43.2
12
t e Beha vi or Theor
1/
0.005 43.2 50 40.7
y
6

y
cr y
cr
F
E b
F F
t
b t
F ksi


    

   


4
0.38
1.92 1
/
29000 0.38 29000
1.92 1.25 1 49.3
40 54/1.25
Column Behavior The
40
49.3
0.913
54
is com
o
puted as 458 in
ry
e
a
s
E E
b 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 12
66.9, 64 ksi
1.794
/
0.658 33.9 ksi
40.67 33.9
100 19.97%
33.9
y
e
e
QF
F
cr y
r
KL E
F
r
KL r
F Q F

 
 
   
 
 
 
 
 

 




2
4
2
2
For transverse stiffeners at 20 ft, WT1
2 38
is needed. The effective section become
s 146.2 in
and corresponding is computed as 1010 i
n
1010/146.2 2.63 in
1 20 12
91.25, 34.37
2.63
/
s
e
I
r
KL E
F
r
KL r


  
 
   
ksi


0.658 26.18 ksi
40.67 26.18
100 55.34%
26.18
A spacing of 20 ft is more reasonable in
this case.
Hence, a 55% extra strength is recognize
d by th
plate behavior th
e
eory
.
y
e
QF
F
cr y
F Q F
 
 
 
 
 

 
Stiffened Compression
Panel (Japanese Practice)

Concluding Remarks


The AASHTO critical stress equation appears to
be
unconservative

in the transition zone with
AWS acceptable out
-
of
-
flatness tolerances.


Residual stresses
significantly reduce

the
critical stresses of slender plates.


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


The regression equation derived appears now
to be ready to
replace

two AASHTO equations
without any limitations imposed.


Concluding Remarks
-
continued


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



In the numerical example examined, it is
20%
-
50%

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 Mode

Column Behavior Theory

Global Buckling

Antisymmetric Mode

Plate Behavior Theory

Local Buckling

J. Structural Engineering, ASCE, Vol. 127,

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

J. Engineering Mechanics, ASCE, Vol. 131,

No.2, February 2005, pp. 167
-
176

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

September 2007, pp. 2087
-
2096

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

May 2009, pp. 1141
-
1153

REGRESSION EQUATION

2 3
s
I 0.3 n t w
a

Where


 
aspect ratio a/w
a


n number of stiffeners
END