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Force Based DesignFundamentals
Ian BuckleC f CDirector Center for Civil Engineering Earthquake Research
University of Nevada, Reno
Learning OutcomesgExplain difference between elastic forces, actual forces and Modified Design Forcesactual forces and Modified Design ForcesDefine a plastic hingeE l i C it P t ti Phil hExplain Capacity Protection PhilosophyDescribe the Component Capacity/Demand R t fit M th d (M th d C)Retrofit Method (Method C)Describe how P-Δ effects are considered in d idesign
Force vs. Displacement Force Displ. RetroTopic Applicability
Elastic Response
rce FEQ
ΔEQ
eral
For FEQ
Late
Displacement - ΔΔEQ
p
Elastic Analysis & Response
Force vs. DisplacementTopic ApplicabilityForce Displ. Retrop
For moderate-large earthquakes, FEQ is a very large force (can be > structure weight) and it islarge force (can be > structure weight) and it is uneconomical to design an ordinary bridge to resist this force elasticallyInelastic behavior (damage) is therefore accepted, provided it
does not cause collapsedoes not cause collapseis ductile in nature (not brittle), and occurs in designated components (members)
Exceptions are made for special structuresExceptions are made for special structures where the higher cost of elastic (or almost elastic) behavior can be justified
Force vs. Displacement Force Displ. RetroTopic Applicability
orce
Pl ti hi
Fmechanism
Fmechanism
ater
al F
o
First-Order Rigid-Plastic Analysis
Plastic hinge
FLa
Displacement - Δ Plastic hinge
Fmechanism
First Order Rigid-Plastic analysis assumes all deformations take place at discrete regions, called plastic hinges, and that a sufficient number of plastic
First-Order Rigid Plastic Pier Response
d sc ete eg o s, ca ed p ast c ges, a d t at a su c e t u be o p ast chinges have formed to form a mechanism in the pier/bent.
Force vs. Displacement
c
Force Displ. RetroTopic Applicability
Elastic Response
rst-O
rder
gid-
Pla
stic
espo
nse
l For
ce Fir
Rig
Re
Late
ra IdealizedElasto-PlasticResponse
Displacement - Δ
Idealized Elasto-Plastic Response
Force vs. Displacement Force Displ. RetroTopic Applicability
orce FEQ
ΔEQActual Response
Fmax
ater
al F
o
2 3 4 Plastic Hinge
Fmax
La
Displacement Δ
1
Displacement - ΔActual Response Milestones
1 P d i ld P i t1 - Pseudo-yield Point2 – Maximum Plastic Deformations3 – Onset of Collapse4 - Collapsep
Force vs. Displacement Force Displ. RetroTopic Applicability
Elastic Response
FEQF
ΔEQ
Actual
l For
ce
FEQActual Response
Late
ra
Displacement - ΔΔEQ
p
Elastic vs. Actual Response
Force vs. Displacement
c
Force Displ. RetroTopic Applicability
Elastic Response
A t l Rrst-O
rder
gid-
Pla
stic
espo
nse
l For
ceActual ResponseFi
rR
igR
e
Late
ra IdealizedElasto-PlasticResponse
Displacement - Δ
Idealized Elasto-Plastic Response
Force vs. Displacement Force Displ. RetroTopic Applicability
Elastic Analysis
orce
First-OrderElastic-Plastic
ater
al F
o
Second-Order Elastic-Plastic (slender column response shown- requirements are
f )La
Displacement - Δ
in place to limit this type of behavior)
Elastic Analysis conducted using to linear methodsFirst-order elasto-plastic behavior includes reduced pier/bent stiffness and strength with progressive plastic hinge formation.
Pier Response with Higher Order Effects
Second-order elasto-plastic behavior traces formation of plastic hinges and includes geometric non-linearities (e.g. P-Δ)
Pier Response with Higher Order Effects
Force vs. DisplacementTopic ApplicabilityForce Displ. Retro
p
Member strengths:gNominal strength, SnDesign strength, Sd = φ Sn
h φ i th t th d ti f t <1 0where φ is the strength reduction factor <1.0 (see AASHTO LRFD Specifications)[Expected strength, Se = φe Sn[Expected strength, Se φe Snwhere, in absence of mill certificates, φe=1.2 for steel and 1.3 for concrete members] Over-strength, So = φo Snwhere φo is the over-strength factor >1.0 (1.7 – 2.7)
Force vs. Displacement Force Displ. RetroTopic Applicability
e D i f l f
Felastic
al F
orce Design for lesser of
Felastic or FoverstrengthFoverstrengthFmechanism
Overstrength Factor
Late
ra Force Displ. Retro1.7 1.7
Overstrength Factor
2.0
Displacement - Δ
Foverstrength = φo Fmechanism where φo = over-strength factor
Over-strength Forces
overstrength φo mechanism φo g
Force vs. Displacement Topic ApplicabilityForce Displ. Retro
cera
l For
c
μ = Δ
Late
r
Δ Δ
μ = ΔmaxΔyield
Displacement - Δ
Δyield Δmax
Displacement Ductility Factor - μ
Force vs. DisplacementTopic ApplicabilityForce Displ. Retrop
To calculate ΔmaxNonlinear time history analysis using software such as SAP2000, SEISAB-NL, ADINA, ABAQUS…Elastic analysis using either ULM SMSA MMSA THElastic analysis using either ULM, SMSA, MMSA, TH to find maximum elastic displacement ΔEQ and then assumeeither: Equal displacements, i.e. Δmax = ΔEQ
or: Equal work done during elasto-plastic response as during elastic response, and solve for Δmax
Force vs. DisplacementTopic ApplicabilityForce Displ. Retro
rce
Felastic
Then:Δ = ΔEQ
tera
l Fo
Felastic R and:
Δ R
FelasticDefine R =
Fmechanism
Fmechanism
Δmax ΔEQ
Lat
Δmax = ΔEQΔyield
μ = Δmax = RΔyield
(Displacement
mechanism
(Response Modification Factor)
Relationship between ductility and “R”
Displacement - Δ( pDuctility Factor)
p yEqual Displacement Assumption
Force vs. DisplacementTopic ApplicabilityForce Displ. Retro
rce
Felastic
Then:Δ = ΔEQ
tera
l Fo
Felastic R and:
Δ R
FelasticDefine R =
Fmechanism
Fmechanism
Δmax ΔEQ
Lat
Δmax = ΔEQΔyield
μ = Δmax = RΔyield
(Displacement
mechanism
(Response Modification Factor)
Relationship between ductility and “R”
Displacement - Δ( pDuctility Factor)
p yEqual Displacement Assumption
Force vs. DisplacementTopic ApplicabilityForce Displ. Retrop
Felastic Then:
Forc
eand:
Felastic R
FelasticDefine R =
Fmechanism F
Δmax= (R ΔEQ + Δyield) / 2
Late
ral and:
μ = Δmax ≈ (R2 + 1) / 2Δyield
(Displacement
Rmechanism
(Response Modification Factor)
Fmechanism
Displacement - ΔΔyield Δmax
Ductility Factor)ΔEQ
Relationship between ductility and “R” Equal Work Done Assumption
Force vs. DisplacementTopic ApplicabilityForce Displ. Retro
pFelastic Then:
Forc
eand:
Felastic R
FelasticDefine R =
Fmechanism F
Δmax= (R ΔEQ + Δyield) / 2
Late
ral and:
μ = Δmax ≈ (R2 + 1) / 2Δyield
(Displacement
Rmechanism
(Response Modification Factor)
Fmechanism
Displacement - ΔΔyield Δmax
Ductility Factor)ΔEQ
Relationship between ductility and “R” Equal Work Done Assumption
Force vs. Displacement Force Displ. RetroTopic Applicability
rce
μ = R = 1
ilityμ = Δmax ≈ R
Δyield
eral
For
μ = R = 2
ng D
uctiyield
Late
μ = R = 3
μ = R = 5
ncre
asin
ΔyieldΔmax
InStrength and Ductility Relationship
Displacement - Δyield
Substructure Type
Force vs. Displacement Force Displ. RetroTopic Applicability
rce
μ = R = 1
ilityμ = Δmax ≈ R
Δyield xibi
lity
ΔE
Q
eral
For
μ = R = 2
ng D
uctiyield
sing
flex
high
er Δ
Late
μ = R = 3
μ = R = 5
ncre
asin
Incr
eas
perio
d,
ΔyieldΔmax
In
NO
TE:
long
er
Strength and Ductility RelationshipDisplacement - Δ
yield
Substructure Type
Force vs. Displacement Force Displ. RetroTopic Applicability
rce
μ = R = 1
ilityμ = Δmax ≈ R
Δyield xibi
lity
ΔE
Q
eral
For
μ = R = 2
ng D
uctiyield
sing
flex
high
er Δ
Late
μ = R = 3
μ = R = 5
ncre
asin
Incr
eas
perio
d,
ΔyieldΔmax
In
NO
TE:
long
er
Strength and Ductility RelationshipDisplacement - Δ
ΔyieldSubstructure Type
Force vs. Displacement Force Displ. RetroTopic Applicability
e ct R
Felastic
F
AxialForce (Pn , Mn)
(Pn , Mn) w/ Over-strength
al F
orce
Effe
cof
R
Fmechanism = Fdesign
F
Foverstrength
ΦPn ΦMn
P Mign
Late
ra FelasticR
P,Melastic
P,M
des
Displacement - Δ MomentFelastic & P,Melastic force effect generated from Extreme Event 1 load combination
Elastic vs. Modified Design vs.Over-strength Forces
Capacity Protection Topic ApplicabilityForce Displ. Retrop y
The principle of Capacity Protection DesignThe principle of Capacity Protection Designis that yielding in one component (or member) will cap (or limit) the forces andmember) will cap (or limit) the forces and moments in an adjacent component (member)(member). i.e. reaching the elastic capacity (yield) of one member protects adjacent membersone member protects adjacent members from excessive forces
Capacity Protection Topic ApplicabilityForce Displ. Retrop y
F, Δ A B C D
Bridge Geometry
, A B C D
h1 h2
EF
2
Capacity Protection Topic ApplicabilityForce Displ. Retrop y
F, Δ F, ΔA B C D
Bending Moment Diagram
Mp
Bridge Geometry
1
2 4
, ,
VEB
A B C D
h1 h2 M
p
3EF
VFC
Two plastic hinges in BE,
2 Mp
VEB max = 2 Mp / h1
Capacity Protection Topic ApplicabilityForce Displ. Retrop y
F, Δ F, ΔA B C D
Bending Moment Diagram
Mp
Bridge Geometry
1
2 4
, ,
VEB
A B C D
h1 h2 M
p
3
EB
EF
VFC
Two plastic hinges in BE,
2 Mp
FVEB max = 2 Mp / h1
F
1
23 4
VFC
Δ
1
VEB
F = VEB + VFC
Capacity Protection Topic ApplicabilityForce Displ. Retrop y
F, Δ F, ΔA B C D
Bending Moment Diagram
Mp
Bridge Geometry
1
2 4
, ,
VEB
A B C D
h1 h2 M
p
3
EB
EF
VFCTwo plastic hinges in EB, V = 2 M / h
2 Mp
F VEB max = 2 Mp / h1F
1
23 4
VFCMaximum shear and moment in foundation at E is limited by yielding
Δ
1
VEB
F = VEB + VFC
foundation at E is limited by yielding in EB, i.e. the foundation is capacity protected by column yield
Capacity Protection Topic ApplicabilityForce Displ. Retrop y
F, Δ F, ΔA B C D
Bending Moment Diagram
Mp
Bridge Geometry
1
2 4
, ,
VEB
A B C D
h1 h2 M
p
3
EB
EF
VFCTwo plastic hinges in EB, V = 2 M / h
2 Mp
F VEB max = 2 Mp / h1F
1
23 4
VFCNOTE: For design Mp = φ Mn, but to calculate max value of V should
Δ
1
VEB
F = VEB + VFC
calculate max. value of VEB, should use the overstrength moment, i.e. Mp=φo Mn where (φo > 1)
Seismic Retrofit ProcessTopic ApplicabilityForce Displ. Retro
Three step process:Three step process:Screening and PrioritizationD t il d E l tiDetailed EvaluationRetrofit – strategies, approaches and measures
Evaluation MethodsTopic ApplicabilityForce Displ. Retro
FHWA Manual describes 5 basic methodsFHWA Manual describes 5 basic methods for evaluationMethods vary in rigor from ‘no analysis’ toMethods vary in rigor from no-analysis to ‘nonlinear dynamic time history analysis’S l ti / li ti d d thSelection / application depends on the Seismic Retrofit Category A – D for the b id ( i i h d d fbridge (seismic hazard and performance required), and its geometry (regularity).
Evaluation MethodsTopic ApplicabilityForce Displ. Retro
A and B: Default capacity methodA and B: Default capacity methodchecks capacity of components due to non-seismic loads against specified g pminima – no analysis is required (e.g. connections and support lengths)C: Capacity/demand method compares component capacities against forced d f l ti l idemands from an elastic analysis - on a component-by-component basis
Evaluation MethodsTopic ApplicabilityForce Displ. Retro
D: Capacity/spectrum method(s) useD: Capacity/spectrum method(s) use nonlinear capacity models for individual piers (or complete bridge), i.e. a pushover p ( p g ) pcurve, and displacement demands from an elastic analysis E: Nonlinear dynamic method explicitly models nonlinear behavior of components i ti hi t l i f l tin a time history analysis of complete bridge
Evaluation Method CTopic ApplicabilityForce Displ. Retro
Capacity / Demand Ratio MethodCapacity / Demand Ratio MethodForce demands are calculated by elastic methods, such as multi-modal spectral analysis method (i.e. without regard to any yielding that may occur). Elastic displacements also usedCapacities are obtained from combination ofCapacities are obtained from combination of theory (strength of materials) and engineering judgment, for each major componentj g , j pGives good results for bridges that behave elastically, or nearly so
Evaluation Method CTopic ApplicabilityForce Displ. Retro
Five step procedure (FHWA 2006):Five step procedure (FHWA 2006):1. Determine applicability of the method2 Determine capacity Q i for all components (i)2. Determine capacity, Qci for all components (i)
using theory and engineering judgement (e.g. Appendix D, FHWA Manual, 2006)pp , , )
3. Determine sum of non-seismic force and displacement demands, ΣQNSi for all NSicomponents for each load combination in LRFD design specification
Evaluation Method CTopic ApplicabilityForce Displ. Retro
Procedure continued:4. Determine seismic demand, QEQi on each
component by an elastic method of analysis
5. For each component determine capacity/demand ratio from:capacity/demand ratio from:
Qci – Σ QNSir =ri =
QEQi
Evaluation Method CTopic ApplicabilityForce Displ. Retro
If ri > 1.0 component has adequate capacity
If 0.5 < ri < 1.0 component may be acceptable i ywithout retrofitting, depending on
other deficiencies in member, if any, and f f ilconsequences of failure
If 0.5 > ri component has inadequate capacity and retrofitting is indicated
Evaluation Method CTopic ApplicabilityForce Displ. Retro
Note that for each component, several c/d ratios (ri) may need to be calculated. Example 1: Support lengths and bearingsp pp g g
Evaluation Method CTopic ApplicabilityForce Displ. Retro
Example 2: ColumnsFive (5) c/d ratios should be found for each column for the following: g
Anchorage length of longitudinal rebar, rca
Transverse confinement, rccTransverse confinement, rcc
Splice length, rcs
Shear force rrsplice
r con
finem
ent
Shear force, rcv
Bending moment, rec ranchorage
r
P-Δ Effects in Bridge Columns
Topic ApplicabilityForce Displ. Retro
ΔPPP
Bridge Columns
P
FF, Δ
PP
Fh Kpier
Kpier
Kpier h
Equilibrium in deformed state requires ΣMA = 0 i e F h + P Δ = (K i Δ) h
A
Equilibrium in deformed state requires ΣMA 0, i.e. F h + P Δ (Kpier Δ) h
Therefore F = (Kpier – P / h) Δ = K'pier Δ
P-Δ Effects in Elastic Bridge Columns
Topic ApplicabilityForce Displ. Retro
Elastic Bridge ColumnsElastic capacity curve for
Forcecolumn, slope = Kpier Elastic capacity curve
with P-Δ included, slope = K'pier
FLoss in capacity at displacement Δ, due to P-Δ effect
= PΔ / h
DisplacementΔ
P-Δ Effects in Yielding Bridge Columns
Topic ApplicabilityForce Displ. Retro
Yielding Bridge ColumnsElasto-plastic capacity curve for column; initial slope = Kpier, second slope = 0 Elasto-plastic capacity curve
Force
p pier p Elasto-plastic capacity curve with P-Δ included;initial slope = K'piersecond slope = - P/h
FLoss in capacity at displacement Δdue to P-Δ effect
= PΔ / h
DisplacementΔyield Δ
P-Δ Effects in Yielding Bridge Columns
Topic ApplicabilityForce Displ. Retro
Yielding Bridge ColumnsElasto-plastic capacity curve for column; initial slope = Kpier, second slope = 0 Elasto-plastic capacity curve
Force
p pier p Elasto-plastic capacity curve with P-Δ included;initial slope = K'piersecond slope = - P/h
FLoss in capacity at displacement Δdue to P-Δ effect
= PΔ / h
DisplacementΔyield Δ
P-Δ Effects in Yielding Bridge Columns
Topic ApplicabilityForce Displ. Retro
Yielding Bridge ColumnsAASHTO LRFD Specifications andAASHTO LRFD Specifications and FHWA Retrofit Manual require that if
PΔ / h > 0 25 Mp / hPΔ / h > 0.25 Mp / ha refined analysis must be undertaken that explicitly includes nonlinear geometricexplicitly includes nonlinear geometric effects (P-Δ)Encouraged to limit Δ such thatEncouraged to limit Δ such that
Δmax < 0.25 Mp / P
Learning OutcomesgExplain difference between elastic forces, actual forces and Modified Design Forcesactual forces and Modified Design ForcesDefine a plastic hingeE l i C it P t ti Phil hExplain Capacity Protection PhilosophyDescribe the Component Capacity/Demand R t fit M th d (M th d C)Retrofit Method (Method C)Describe how P-Δ effects are considered in d idesign
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