61st Annual Structural Engineering Conference
University of Kansas
Michael J. Garlich, S.E., P.E.
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It is unacceptable for bridges to
collapse at any time
Such events (and near misses)
are too common during
erection and/or demolition
The majority of engineering
effort in projects is being placed
in design rather than
construction
There is a general lack of
criteria and guidance
Stability is a complicated issue
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Bridge configuration changing
Design standards
Erection vs. final design
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Tensile and compressive stresses produced by bending
Result: lateral translation and twisting of cross-section
– Compression flange buckles & laterally translates
– Tension flange doesn’t buckle so shape must twist
Vertical Uniform Loading
Unbraced Length
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Timoshenko’s equation for elastic critical buckling moment
Mcr (k-in):
– Lb = unbraced length (in) = spacing between cross-frames or
panel points of lateral truss on comp. flange
– E = modulus of elasticity (ksi)
– G = shear modulus (ksi)
– J = torsional constant (in4)
– Cw= warping constant (in6)
– Iy = weak-axis moment of inertia (in4)
(AASHTO A6.3.3-8)
• 1st term under radical: Warping torsional stiffness
• 2nd term under radical: Saint-Venant torsional stiffness
• Applicable to doubly-symmetric and singly-symmetric shapes
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Timoshenko’s expression valid for uniform (constant)
moment, which is worst case
Cb accounts for beneficial moment gradient effect
Lateral torsional buckling moment modification
factor Cb (AISC Equation)
– Cb = 12.5Mmax/(2.5Mmax + 3MA + 4MB + 3MC)
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Moment gradient increases buckling capacity (Cb ≥ 1)
AASHTO LRFD uses different expression
– From formulation applicable to straight-line moment
diagrams
– Variable definitions to approximate non-linear moment
diagrams
– AASHTO expression in stress, not moment
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Load distribution
– Live load
– Wind
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Girder stability
Combined wind and stability
Must provide strength and stiffness
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0.005 Lb L Mu2
Mbr =
(Str)
2
n E Ieff Cb ho
βt =
2.4 L Mu2
n E Ieff Cb2
(Stif)
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Most girder lifts: single crane or spreader beam
– Lifting clamp connects to girder top flange
– Ideally lifted near girder quarter points
– Unbraced length Lb is entire girder length
 Lift points provide no lateral-torsional restraint
a
LLift
a
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Buckling behavior of girders during lifting
– Moment modification (lift adjustment) factor CbL
 Function of overhang length relative to total length L
 CbL = 2.0 for LLIFT/L ≤ 0.225
 CbL = 6.0 for 0.225 < LLIFT/L < 0.3
 CbL = 4.0 for LLIFT/L ≥ 0.3
Note: unbraced length Lb = total length L
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Primary source of stability is
girder self-weight
Crane Lifting Force
– Self-weight acts below support
point
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Primary source of instability is
lateral-torsional buckling
–
–
–
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Long unsupported lengths
Overhang region “a”
Region between pick points “LLift”
Which region controls stability?
 Depends on locations of pick
points
Girder Self Weight
6
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Horizontally curved girder segments
– Can experience rigid body rotation
during lifting
– Centroid of curved segment may have
eccentricity
 Girder will rotate until…
 Centroid is below support line
– Excessive girder rotation makes air
splices difficult
– Local rotation / deflection effects
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Geometry of horizontally curved girders
– De-stabilizing moment must be considered during erection
– Overturning moment is due to:
 Offset of center-of-gravity relative to support axis
Girder Bottom
Flange
Plumb Line
Dropped from Top
Flange- Lateral
Deformation = 10
inches
Global Buckling
(Mcr = 792 k-ft)
Buckling Between
Cross-Frames
(Mcr = 1384 k-ft)
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2 SE
π
Mb =
√ Ieff Iy (2 girders, symmetric)
Lg2
S = girder spacing
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ΦMb > 1.5 Mo
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Non-symmetric; 3- and 4-girder equations available
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Simply supported—for cantilever and continuous
modifier in development
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Evaluate global and potentially local stability
Analysis program can solve for…
– Buckling mode shape (eigenvector)
– Buckling mode value (eigenvalue)
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Analysis provides elastic buckling capacity
– Material inelasticity is not considered, but…
– Stresses during construction in elastic range anyway
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Most commercial software tools can do eigenvalue
buckling
Straight bridges
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Eigenvalue λ: factor applied to reference load
This determines critical buckling load
Pcr = λPref
– Pcr = corresponding buckling load (k)
– λ = eigenvalue
– Pref = magnitude of applied force (k)
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Target, Pcr ≥ 1.75 × selfweight
Contractor’s
preferred
erection method
 Complete 1st
splice on
ground
 Erect Span 1
with cantilever
portion
Case
Erected Girders
Case
1
5
2
6
3
7
4
8
Erected Girders
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Eigenvalue analysis of Case 1 (Span 1)
– Girder 1 erected with no intermediate bracing
 Eigenvalue on unfactored selfweight = 0.48 < 1
 Buckled shape shown; girder clearly inadequate (LTB)
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Eigenvalue analysis of Case 2 (Span 1)
– Girders 1 & 2 erected with 50% cross-frames
 Eigenvalue on unfactored selfweight = 1.57
 Buckled shape shown; case borderline (1.25<λ<1.75)
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Contractor’s erection method
(continued)
Erect Span 2 with air splice
Erect remaining cross-frames
– 50% of intermediate cross-frames
installed originally
Evaluate each case’s stability with
UT Bridge
Eigenvalue analysis using
unfactored self-weight
Case
Eigenvalue
1
0.48
2
1.57
3
1.84
4
1.96
5
1.32
6
2.36
7
3.15
8
3.55
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Rollover caused by:
– Initial girder rotation
compounded by:
 Lack of flatness of
PPC bottom flange
 Roll flexibility of
bearings…
 Leading to increased
girder rotation
Note: Figure adapted from Mast (1993)
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Girder rollover stability can be influenced by:
– Bearing slope and bearing type
– Bearing skew relative to girder centerline
– Girder imperfections (sweep)
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Rollover controls stability, not lateral-torsional buckling
– PPC girders designed to not crack under selfweight
– Relatively large Iy and J: no LTB
Courtesy of Dr. D. Jauregui
Courtesy of Dr. D. Jauregui
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FHWA developed
comprehensive manual
and training course
Provides:
– Summary of lessons learned
– Understanding and analysis
of global stability
– Design criteria for erection
– Guidance and best practices
– Design examples
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AASHTO
– 300 PLF
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ASCE 7-10
– F = .00256 kz kzt kd G Cf A V2
– Load factor in velocity
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ASCE 37 Reduction Factors
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Cf=2.2 (min)
Cf=2(1 + 0.05 s/d) ≤ 4.0
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One day, design velocity = 20 mph
Velocity modification factors (Vmod=FV)
Duration
Factor
0 – 6 weeks
0.65
6 weeks – 1 year
0.70
1 year – 2 years
0.8
2 years – 5 years
0.85
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Weather
Stop-work
Equipment issues
Load Combinations
and Load Factors
DC
CDL
CLL, CR
CW
Strength I
1.25
1.25
1.75
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Strength III
1.25
1.25
—
1.0
Strength V
1.25
1.5
—
1.0
Strength IV
1.40
1.40
1.50
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Service
1.00
1.00
1.00
1.0