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Unformatted text preview: LONGITUDINAL STIFFENERS ON
COMPRESSION PANELS Chai H. Yoo, Ph.D., P.E., F. ASCE
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 weighttostrength 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
Highway 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 ≥ φ t3w where φ = 0.07k 3 n 4 for n > 1 φ = 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 modernday 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
a σt
w
O · σt Stiffener 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 30.6 Symmetric Antisymmetric Fcr (ksi) 30.2
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)
α n w
(in.) t
(in.) w/t R
(ft) Is, Eq.(1)
(in4) Is, used
(in4) Fcr,
AASHTO Fcr,
FEM,
∆ =w/100
0 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 = 0.416× 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 = 0.3α
Where 2 3 nt w α = aspect ratio = a / w
n = number of stiffeners Plate Buckling Coefficient 6 k 4 2
0 1 2 2 a/w 6 3 4 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 widthtothickness ratio, the critical stress of the plate is kπ E t Fcr = 2
12 1 − µ w 2 ( ) with k = 4 2 AASHTO divides the subpanel between longitudinal stiffeners or the web into three zones by the widthtothickness 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 widthtothickness ratio 6
0
A
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a 3
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6
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9
0
1
2
0
1
5
0 w/t Longitudinal stiffener arrangement, AASHTO 4 Eq. Spa.
9’0” 5 Eq. Spa.
12’0” 4 Eq. Spa.
9’0” Longitudinal stiffener arrangement, Proposed 2 Eq. Spa.
9’0” 3 Eq. Spa.
12’0” 2 Eq. Spa.
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 in22 f = 0.57 in
, t
Is = 53.5+7.35(8.9952.12) = 400 in 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
4π E
2 b
Fcr =
= 0.7 Fy ⇒ = 54.73 > 43.2
2
2
t
12 ( 1 − µ ) ( b / 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 The area of the effective section is 142.3 in 2 r = 458 /142.3 = 1.794 in
KL 1× 10 × 12
πE
=
= 66.9, Fe =
= 64 ksi
2
r
1.794
( KL / r )
2 Fe
Fcr = Q 0.658 Fy = 33.9 ksi 40.67 − 33.9
× 100 = 19.97%
33.9
QFy For transverse stiffeners at 20 ft, WT12 × 38
is needed. The effective section becomes 146.2 in
and corresponding I s is computed as 1010 in 2 4 ⇒ r = 1010 /146.2 = 2.63 in
KL 1× 20 × 12
πE
=
= 91.25, Fe =
= 34.37 ksi
2
r
2.63
( KL / r )
2 QFy Fe
Fcr = Q 0.658 Fy = 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 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 outofflatness 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.
9’0” 5 Eq. Spa.
12’0” 4 Eq. Spa.
9’0” Longitudinal stiffener arrangement, Proposed 2 Eq. Spa.
9’0” 3 Eq. Spa.
12’0” 2 Eq. Spa.
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. 705711 J. Engineering Mechanics, ASCE, Vol. 131, No.2, February 2005, pp. 167176 Engineering Structures, Elsevier, Vol. 29(9),
September 2007, pp. 20872096 Engineering Structures, Elsevier, Vol. 31(5),
May 2009, pp. 11411153 REGRESSION EQUATION I s = 0.3α
Where 2 3 nt w α = aspect ratio = a / w
n = number of stiffeners END ...
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This note was uploaded on 09/24/2011 for the course CIVL 7690 taught by Professor Staff during the Summer '10 term at Auburn University.
 Summer '10
 Staff
 Finite Element Method

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