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.(10134)
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
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