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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, 2009
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, 9th ed., 1965 adopted for
the first time the minimum moment of inertia of the longitudinal stiffener:

Is   t 3w
where


  0.07k 3 n4 for n  1
  0.125k 3 for n  1
with 2  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:


                       Is  8 t w  3


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, 17th 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



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


Bleich (1952) and Timoshenko
and Gere (1961) introduced
Barbré’s study (published in
German) to English speaking
world using the following model:
         Symmetric and Antisymmetric
            Buckling Mode Shapes


t               a               t


             w
                     Stiffener
     O   ·                            x 2w

             w



 y
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



                         Symmetric      Antisymmetric
                  30.6


                  30.2
      Fcr (ksi)



                  29.8


                  29.4


                   29
                     580          630         680          730
                                                     4
                            Moment of Inertia, I s (in )
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, Fcr (ksi)


                                                                                                  Fcr,       Fcr,
                w           t                    R      Is, Eq.(1)     Is, used        Fcr,
n       a                             w/t                                                        FEM,       FEM,
              (in.)       (in.)                 (ft)       (in4)         (in4)     AASHTO
                                                                                                D=w/1000   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 in4 = 0.416106 mm4; 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




I s  0.3a       2
                     nt w 3



Where   a  aspect ratio  a / w
        n  number of stiffeners
            Plate Buckling Coefficient


    6



k
    4




    2
        0       1   2   2     6   3   4
                        a/w
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

        k E  t 
            2              2

Fcr            2     
      12  1     w 

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

            60                                         0-1
                                         A A SHTO Eq.(1 34)
                                         B ifurcatio n A nalysis
                50                       No nlinear A nalysis (W/1000)
                                         No nlinear A nalysis (W/100)
            40                           SSRC Type P arabo la
    Fcr (ksi)




            30


            20


                10


                0
                     0   30   60          90              120            150
                                   w/t
Longitudinal stiffener arrangement, AASHTO




          4 Eq. Spa.     5 Eq. Spa.    4 Eq. Spa.
             9’-0”         12’-0”          9’-0”




Longitudinal stiffener arrangement, Proposed




            2 Eq. Spa.    3 Eq. Spa.   2 Eq. Spa.
               9’-0”        12’-0”         9’-0”
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 in2, tf = 0.57 in
Is = 53.5+7.35(8.995-2.12)2 = 400 in4

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

9.15t2 = 7.35, t = 0.9 in, d=7.35/0.9 = 8.17 in
Is = 0.9(8.17)3/3 =164 in4
                   Quick Comparison

Pla t e Beh a vior T h eor y
The limiting value of the slenderness ratio assuming
the residual stress of 0.3Fy is
            4 E
               2
                                        b
Fcr                           0.7 Fy   54.73  43.2
      12 1     b / t 
               2            2
                                        t

Fcr  0.005  43.2   50  40.67 ksi
                     2
Column Behavior Theory
           E  0.38 E 
be  1.92t   1      
           f  b/t f 
                29000       0.38 29000 
 1.92  1.25          1                49.3
                  40  54 /1.25 40 
           49.3
Qa  Q           0.913
            54
                           4
I s is computed as 458 in
                                                2
The area of the effective section is 142.3 in
r  458 /142.3  1.794 in
KL 1 10 12                  E  2
             66.9, Fe                64 ksi
                           KL / r 
                                     2
 r   1.794
              QFy
                    
Fcr  Q  0.658  Fy  33.9 ksi
                 Fe
                   
                   
40.67  33.9
               100  19.97%
    33.9
For transverse stiffeners at 20 ft, WT12  38
                                                    2
is needed. The effective section becomes 146.2 in
                                                4
and corresponding I s is computed as 1010 in
 r  1010 /146.2  2.63 in
KL 1 20  12                   E2
              91.25, Fe                34.37 ksi
                             KL / r 
                                       2
 r    2.63
              QFy
                   
Fcr  Q  0.658  Fy  26.18 ksi
                Fe
                  
                  
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 recognized by the
plate behavior theory.
 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.
Longitudinal stiffener arrangement, AASHTO




          4 Eq. Spa.     5 Eq. Spa.    4 Eq. Spa.
             9’-0”         12’-0”          9’-0”




Longitudinal stiffener arrangement, Proposed




            2 Eq. Spa.    3 Eq. Spa.   2 Eq. Spa.
               9’-0”        12’-0”         9’-0”
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




I s  0.3a       2
                     nt w 3



Where   a  aspect ratio  a / w
        n  number of stiffeners
END

				
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