Evaluation of Members, Connections, and...
Transcript of Evaluation of Members, Connections, and...
1
Evaluation of Members, Connections, and Subsystems
Tim Ingham
Outline
Member capacity– Axial– Flexural– Ductility
Connection capacity– Riveted– Gusset plates– Net section
Displacement capacity of subsystems
Capacity demand analysis
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Axial Capacity, per AASHTO
Tension
lagshear for factor reduction section of areanet
section of area grosssteel ofstrength tensile
steel ofstrength yield where
==
==
=
=
=
UA
Af
f
UAfP
AfP
n
g
u
y
gun
gyn
Axial Capacity, per AASHTO
Compression
Ef
rKl
AfP
AfP
y
gyn
gyn
2
where
88.0 then 2.25 If
66.0 then 2.25 If
⎟⎠⎞
⎜⎝⎛=
=>
=≤
πλ
λλ
λ λ
modulus elastic thegyration of radius the
member theoflength unbraced thefactorlength effective the
where
===
=
ErlK
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Axial Capacity, Slender Cross-Section
Flanges of boxes
Flanges of I-shapes
λsλpλrRatioMember
10141
−yFtb
tb
yF65
yF52
ry FF −238
yF190
yF110
A cross-section is slender if λ>λr for one of its components. The component will buckle before reaching the yield strain of the material.
Axial Capacity, Slender Cross-Section
Follow the procedure described in Appendix B of the Load and Resistance Factor Design Specification for Structural Steel Buildings
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For Un-Stiffened Elements
leg theof thicknesstheleg theof width the
where
176 if 20000
1760.95 if 00437.0415.1
2
==
≤
⎟⎠⎞
⎜⎝⎛
=
<<−=
tb
tb
ftbf
Q
ftb
ff
tbQ
yy
s
yyys
For Stiffened Elements
element in the stress the widthactual the
widtheffective the)(with
9.641326)()(
==
=
⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜
⎝
⎛
−==
fbfb
ftbb
tfb
fbfQ
e
ea
5
Axial Capacity, Slender Cross-Section
yfQ
crcr
gyn
gyQ
n
as
ea
ffQf
AfP
AfQP
QQQ
bttfb
fQ
cr λ
λ
λλ
λ
)(66.0)(
solution iterativean requiresequation first the
88.0 then 2.25Q If
66.0 then 2.25Q Ifiscapacity n compressio nominal The
)()(
=
=>
=≤
=
=∑
∑
Example: Slender Cross-Section
Mathcad Document
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Built-Up Members
Net Section of Built-Up Member
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Splices in Built-Up Members
Splices in Built-Up Members
Gross section capacity
Net section capacity– Of splice plates– Of member– Could occur through
intermediate plates
Load transfer capacity, through rivets or bolts
If member ductility demands exceed unity, then splices should be 25% stronger than the member
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Laced Members
Laced Members
Not treated in modern codes
Tensile capacity– Use usual methodology– Don’t count the laces in gross
area
Compressive capacity– Use usual methodology
• Member stability• Section stability, based on
width-to-thickness ratios• With a modification
– Consider also the stability of the member between the points of attachment of the laces
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Compressive Capacity of Laced Members
( )
gyration of radius themember theoflength unbraced the
factorlength effective thewhere
1.1 ,40For
3001 ,40For 2
===
=′≤
+=′>
rlK
KKrKl
rKLKK
rKl
Proportions of Laced Members
Per the AASHTO Standard Specifications, the slenderness of longitudinal elements must satisfy
}32,40{min
min rKl
ra ≤
10
Ill-Proportioned Laced Member
Must consider interaction of local and global buckling
See Bazant & Cedolin, “Stability of Structures: Elastic, Inelastic, Fracture, and Damage Theories,”Oxford University Press, 1991.
}32,40{min
min rKl
ra >
Stability of Laced Members
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Actual Member
0
200
400
600
800
1000
1200
1400
1600
0.00 0.10 0.20 0.30
Time, s
Rea
ctio
n, k
ip
Full Lacing
Yield Strength
of Member
With Partial Lacing
0
200
400
600
800
1000
1200
1400
1600
0.00 0.10 0.20 0.30
Time, s
Rea
ctio
n, k
ip
Full Lacing Partial Lacing
Yield Strength
of Member
No convergence beyond this point
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Shear Strength of Laced Members
For a single plane of single lacing
Check for external shears plus 2% of compressive capacity
f( )
properties sbar' theconsidersimplicitly andbar lacing a of
capacity ecompressiv theis )(where
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221
lP
TLPTL
TV
n
n ++
=
Shear Strength of Laced Members
For a single plane of double lacing
The 0.7 factor accounts for the stabilizing effect of one lace on the other
( )
properties sbar' theconsidersimplicitly andbar lacing a of
capacity ecompressiv theis )(where
7.02 22
221
lP
TLPTLTV
n
n ++
=
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Flexural Capacity
( )
modulussection themodulus plastic the
where
whereand element, gcontrollin on the based is where
with section,-crosscompact -non aFor
with section,-crosscompact For
==
=
⎟⎟⎠
⎞⎜⎜⎝
⎛
−−
−−=
≤≤
=
<
SZ
SfM
MMMM
ZfM
yr
pr
prppn
rp
yp
p
λ
λλλλ
λλλ
λλ
Flexural Capacity
The components of the cross-sections of built-up and laced members that are subjected to tensile stress should satisfy net section requirements if the members are to reliably develop the plastic moment. Otherwise, use the elastic capacity.
Rivet hole creates a net section in the angle
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Compactness Levels
Significant; section will achieve a rotational ductility of about 8 to 10
Ductile≤ λs
Modest; section will achieve a rotational ductility of 4
Compact≤ λp
Little; yields before local buckling, but section won’t reach the plastic moment
Non-compact≤ λr
None; local buckling occurs before yield
Slender> λr
DuctilityTermWidth-to-Thick. Ratio
Damage Levels
No damage– Loads don’t exceed the nominal capacity, per code
Minimal damage– Essentially elastic performance, characterized by
• Minor inelastic response• No apparent permanent deformations• Inconsequential yielding of secondary members
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Damage Levels
Repairable damage– Can be repaired with a minimum risk of losing functionality, i.e.,
without closing the bridge, characterized by• Yield of members, although replacement should not be
necessary• Small permanent offsets, not interfering with functionality
Significant damage– Minimal risk of collapse, but may require closure to repair,
characterized by• Yield of members, possibly requiring replacement• Permanent offset of the structure
Flexural Ductility – Definitions
θθ θθ
isrotation yield the, and moment, yield the
moment plastic thewhere
from rotation, plastic thecapacityrotation the
where
yy
p
yy
pp
p
h
p
h
M
M
MM
R
θ
θθ
θθ
θθ
=
=
=
==
=
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Flexural Ductility – Definitions
urecontraflexofpoint themoment to
maximum ofpoint thefrom distance theof 10% say,
length, hinge plastic theinertia ofmoment the
modulus elastic thewhere
===
=
p
py
y
LIE
LEIM
θ
θθ θθ
Flexural Ductility – New Members
111No11.42Minimal12.14Repairable138Significantλ ≤ λrλ ≤ λpλ ≤ λsDamage Level
Controlling Width-To-Thickness Ratio
1.0and valuesbulatedbetween ta
einterpolatlinearly 0.15.0For
5.0 force axial that theAssumes
ygyg
yg
FAPFA
FAP
<<
<
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Flexural Ductility – Laced Members
Based on only a few tests of the cyclic behavior of laced members
May also be based on finite element analysis
Not to exceed values for new members, based on width-to-thickness ratios
Reduce capacity with axial force by the same rule as for new members
Check shear on laces
Check net section of components
NoMinimalRepairableSignificantDamage Level
11.31.62Ductility
Flexural Ductility – Laced Member Tests
Lee, K., & Bruneau, M., "Seismic Vulnerability Evaluation of Axially Loaded Steel Built-up Laced Members", MCEER-04-0007, Multidisciplinary Center for Earthquake Engineering Research, June, 2004
Tested specimens with various b/t and Kl/r ratios
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Flexural Ductility – Laced Member Tests
Uang, C.M. & Kleiser, M., “Cyclic Performance of Latticed Members for San Francisco-Oakland Bay Bridge,” Report No. SSRP-97/01, Division of Structural Engineering, University of California at San Diego
Tested members for seismic retrofit of the SFOBB
Flexural Ductility – Laced Member Tests
Dietrich, A. & Itani, A., “Cyclic Behavior of Laced and Perforated Steel Members on the San Francisco-Oakland Bay Bridge,”Report No. CCEER 99-9, University of Nevada, Reno, December, 1999
Tested members for seismic retrofit of the SFOBB
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Flexural Ductility – Perforated Members
Based on one test of the cyclic behavior of perforated members
May also be based on inelastic finite element analysis
Not to exceed the values for new members, based on width-to-thickness ratios
Reduce capacity with axial force by the same rule as for new members
Check net section of components
NoMinimalRepairableSignificantDamage Level
11.42.13Ductility
Flexural Ductility – Perf. Member Tests
Dietrich, A. & Itani, A., “Cyclic Behavior of Laced and Perforated Steel Members on the San Francisco-Oakland Bay Bridge,”Report No. CCEER 99-9, University of Nevada, Reno, December, 1999
Tested members for seismic retrofit of the SFOBB
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Axial Ductility – New Members
Values assume a compact section, i.e., λ ≤ λp
Values for significant damage are based on Dowrick, “Earthquake Resistant Design,” 2nd edition, John Wiley & Sons
Values are for X or Z-braces / V or K-braces
1.0 / 1.01.0 / 1.01.0 / 1.0No1.3 / 1.11.5 / 1.21.7 / 1.4Minimal1.8 / 1.12.3 / 1.52.7 / 2.1Repairable2.4 / 1.23.5 / 1.84.5 / 3.0Significant< 120< 80< 40Damage Level
ksi 36Member yf
rKl
Axial Ductility – Laced and Perf. Members
Based on– Lee, K., & Bruneau, M.– Uang, C.M. & Kleiser, M.– Dietrich, A. & Itani, A.
Not to exceed the values for new members based on controlling slenderness ratio of the member
11.31.62Laced
NoMinimalRepairableSignificantDamage Level
11.42.13Perforated
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Riveted Connections
Could also use the yield strength of rivets for higher performance and less damage
Check bearing strength per the AASHTO LRFD
rivet theof area nominal theplanesshear ofnumber the
shear singlein rivet theofstrengh ultimate the80.0
where
===
=
⋅=⋅
r
r
rrn
AmF
mAFR
φ
φφ
Bolted and Welded Connections
Bolted Connections– Design per the AASHTO LRFD– If member ductility demands exceed unity, then connections
should be 25% stronger than the members they connect
Welded Connections– Design per the AASHTO LRFD– If member ductility demands exceed unity, then connections
should be 25% stronger than the members they connect– Partial penetration welds
• Don’t use in regions subjected to inelastic deformation• Elsewhere, provide 150% of the required strength, but not less
than 75% of the member strength
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Gusset Plates
Shall be designed to Fy for combined force and moment
Use Thorton’s methodology to calculate forces and moments
Shall be designed to Fu/√3 for uniform shear and to 0.74 Fu/√3 for flexural shear
Buckling shall be investigated using a truss analogy; with K=0.65
Gusset Plates
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Gusset Plates
Gusset Plates
Unsupported edges should satisfy
Otherwise, add a stiffener
plategusset theof thicknesstheedge theoflength dunsupporte the
where
6.1
==
≤
tl
fE
tl
y
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Gusset Plates
Added stiffeners should satisfy
( )
plategusset theof thicknessthelength) edge the(1/2 plategusset of width tributarythe
axis)own its(about stiffener theof inertia ofmoment thewhere
14483.1 ,2.9max 42
===
⎟⎠⎞⎜
⎝⎛ −≥
tbI
ttbI
s
s
Gusset Plates
Should be 25% stronger than the members they connect, if member ductility demands exceed unity
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Fracture of the Net Section
possible force maximum the and force, seismic elastic the
member theofstrength yield the
),,min(sectionnet theacross ed transferrforcemember offraction the
member theofstrength tensileminimum the
member theof area gross theareanet effective the
where
2.1
max
max
==
=
===
==
≥
FF
f
FFfAD
f
AA
fAD
AA
eq
y
eqyg
u
g
e
ugg
e
α
α
Displacement Capacity of Systems
Portal Frames
Sway Frames
Support Towers
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Sway Frame
Support Towers
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Pushover Analysis
Modeling Requirements
Column– Plastic hinging element (with yield moment dependent on axial
force)– Include axial forces acting on columns
Braces– Phenomenological model– Physical model– Finite element model
• Utilizes plastic hinging elements• Requires geometrically nonlinear analysis
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Phenomenological Model
Ikeda, Mahin, & Dermitzakis
Physical Model
Ikeda & Mahin
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Finite Element Model
Force-Displacement Response
δ δ
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Force-Displacement Response
δδ
Member Versus System Ductility
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Pushover Analysis
A concentrated load at the top of the structure
A uniform load
A uniform acceleration, or mass proportional load
A concentrated displacement at the top of the structure
A displacement pattern derived from the displaced shape corresponding to the peak drift of the
A displacement pattern equal to one or more relevant mode shapes
Capacity/Demand Analysis
Demand Systemor Member Capacity Systemor Member =DC
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Force Capacity/Demand Analysis
DemandCalculatedStrengthSection Net
Demand CalculatedStrength Spliceand/or Connection
Demand CalculatedStrength eCompressiv
Demand CalculatedStrength Tensile
==
==
==
==
DNDC
DSDC
DCDC
DTDC
N
S
C
T
Displacement Capacity/Demand Analysis
Demandnt Displaceme SubsystemCapacitynt Displaceme Subsystem
DemandDuctility n DeformatioMember CapacityDuctility n DeformatioMember
=
=
DC
DC
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Threshold Values of C/D Ratio
1.01.01.0Conn. or Splice0.670.81.0Buckling1.01.01.0Fracture0.670.81.0YieldingSec., BracingMain, RedundantMain, Non-Red.Failure Mode
Member Type
Force Capacity/Demand < 1 ?
If only slightly less– A minor amount of yielding is probably acceptable– Capacity calculation is likely to be conservative– These ideas are reflected in the threshold values
If significantly less– Inelastic analysis warranted
• Get ductility demands of the member• Evaluate force redistribution to other members
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Case 1: C/DT < 1
The tensile capacity is less than the demand– Inelastic response is implied– Fracture of the net section should be avoided– Connections and splices should be 25% stronger than the
member
Retrofit it may not be required if the ratio is above the threshold value, i.e., if C/DT > γT
Case 2: C/DT < γT
The tensile capacity is less than the (threshold) demand
Retrofit should be pursued to increase the tensile strength of the member
Revaluate strengthened member– Net section– Splices and connections
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Case 3: C/DC < γC and γT < C/DT
The compressive capacity is less than the (threshold) demand
The tensile capacity exceeds the (threshold) demand
Retrofit should be pursued to increase the compressive strength of the member
Revaluate strengthened member– Net section– Splices and connections
Case 4: γC < C/DC and γT < C/DT
The compressive capacity exceeds the (threshold) demand
The tensile capacity exceeds the (threshold) demand
Evaluate connections and splices
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Case 4a: C/DS < γS
The connection or splice capacity is less than the (threshold) demand
Strengthen connection
Case 4b: γS < C/DS
The connection or splice capacity exceeds the (threshold) demand
No retrofit is required, if the tensile capacity exceeds the demand, i.e., 1 < C/DT
Otherwise (C/DT < 1), the member response is inelastic– Case 1 applies– Connections and splices should be 25% stronger than the
member
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Interaction of Axial Force and Flexure
AASHTO LRFD interaction equations
2.0 if 0.198
2.0 if 0.10.2
≥≤⎟⎟⎠
⎞⎜⎜⎝
⎛++
<≤⎟⎟⎠
⎞⎜⎜⎝
⎛++
r
u
ry
uy
rx
ux
r
u
r
u
ry
uy
rx
ux
r
u
PP
MM
MM
PP
PP
MM
MM
PP
AASHTO Inverted
5 if 0.1
89
89
5 if 0.12
2
≤≥
⎟⎟⎠
⎞⎜⎜⎝
⎛
++⎟⎟⎠
⎞⎜⎜⎝
⎛
⎟⎟⎠
⎞⎜⎜⎝
⎛
+⎟⎟⎠
⎞⎜⎜⎝
⎛
>≥
⎟⎟⎠
⎞⎜⎜⎝
⎛
++⎟⎟⎠
⎞⎜⎜⎝
⎛
⎟⎟⎠
⎞⎜⎜⎝
⎛
+⎟⎟⎠
⎞⎜⎜⎝
⎛
r
r
rxuyryux
ryrx
u
r
rxuyryux
ryrx
u
r
u
r
rxuyryux
ryrx
u
r
rxuyryux
ryrx
u
r
PP
MMMMMM
PP
MMMMMM
PP
PP
MMMMMM
PP
MMMMMM
PP
38
Capacity Demand Ratio
5 if
89
89
5 if 2
2
≤
⎟⎟⎠
⎞⎜⎜⎝
⎛
++⎟⎟⎠
⎞⎜⎜⎝
⎛
⎟⎟⎠
⎞⎜⎜⎝
⎛
+⎟⎟⎠
⎞⎜⎜⎝
⎛
=
>
⎟⎟⎠
⎞⎜⎜⎝
⎛
++⎟⎟⎠
⎞⎜⎜⎝
⎛
⎟⎟⎠
⎞⎜⎜⎝
⎛
+⎟⎟⎠
⎞⎜⎜⎝
⎛
=
r
r
rxuyryux
ryrx
u
r
rxuyryux
ryrx
u
r
u
r
rxuyryux
ryrx
u
r
rxuyryux
ryrx
u
r
PP
MMMMMM
PP
MMMMMM
PP
DC
PP
MMMMMM
PP
MMMMMM
PP
DC
Mathcad Document