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Unresolved issues in damping and their impact on bridge
displacement demand estimation
AASHTO T-3 Meeting
June 24, 2019 – Montgomery, Alabama
Mervyn J. Kowalsky – Professor of Structural Engineering, NC State
Outline
•Why do we care about displacement demand estimation?
•Summary of Direct Displacement-Based Seismic Design
• Importance of viscous damping on displacement demand estimation
NCHRP 12-106 Team
•Tom Murphy (Modjeski and Masters)
•Lee Marsh (Berger-ABAM)
• Ian Buckle (Independent Consultant)
•Mervyn Kowalsky (NC State University)
• Jose Restrepo (Advanced Analysis and Design, LLC)
•Don Anderson (CH2M Hill)
Proposed Guidelines for Performance-Based Seismic Bridge Design
DISCLAIMER: This investigation is sponsored by TRB under the NCHRP Program.
Data reported is work in progress. The contents of this presentation has not been reviewed by the project panel
or NCHRP, nor do they constitute a standard, specification, or regulation.
Some Features of 12-106
• Goal: “Provide designers with a methodology for selecting the appropriate level of performance for a given bridge, and implementing a design process that will reasonably result in achieving that level of performance under the design ground motions.”
• Intended as an adjunct to the Seismic Guide Specification.
• First national level implementation of Performance-Based Seismic Design of Bridges.
• Deterministic flavor
• Strain-based limit state definitions (as opposed to ductility)
• Direct displacement-based option for displacement demand estimation
Direct Displacement Based Design1. Choose Limit State Displacement (strain, rotation, or ductility based)
2. Calculate yield displacement
0 0.002 0.004 0.006
Curvature (1/m)
0
10000
20000
30000
40000
Mo
men
t (k
Nm
)
Nu/f'cAg = 0
Nu/f'cAg = 0.1
Nu/f'cAg = 0.2
Nu/f'cAg = 0.3
Nu/f'cAg = 0.4
(a) Reinforcement Ratio = 1%
0 0.002 0.004 0.006
Curvature (1/m)
0
10000
20000
30000
40000
50000
Mo
men
t (k
Nm
)
Nu/f'cAg = 0
Nu/f'cAg = 0.1
Nu/f'cAg = 0.2
Nu/f'cAg = 0.3
Nu/f'cAg = 0.4
(b) Reinforcement Ratio = 3%
0 2 4 6Displacement Ductility
0
0.05
0.1
0.15
0.2
0.25D
am
pin
g (
%)
0 1 2 3 4 5Period (seconds)
0
0.1
0.2
0.3
0.4
0.5
Dis
pla
cem
en
t (m
)
5%
10%
15%
20%
30%
d
Te
Elasto-Plastic
Steel Frame
Concrete Frame
Hybrid Prestress
(c) Equivalent damping vs. ductility (d) Design Displacement Spectra
Concrete Bridge
3. Calculate displacement ductility, damping, and effective period
4. Calculate effective stiffness and required strength
me Fu
F Fn rKi
He Ki Ke
y d
Fu = Ked
FBD DDBD
Ke=4p2me/Te2
For Multi-span bridges:
Dis
pla
cem
ent
Position along bridge
Note: Stars are limit state displacements based on
strain, ductility, or drift
Establish an equivalent SDOF system:
From work balance between MDOF and SDOF systems:
From force equilibrium between MDOF and SDOF systems:
Sample 6 Span Bridge Results
0.0
0.1
0.2
0.3
0.4
0.5
0.00 0.25 0.50 0.75 1.00
Deck
Dis
pla
cem
en
t (m
)
Normalize length
EQ1 to EQ7
EQ Avg
DDBD_ST
0.0
0.1
0.2
0.3
0.4
0.5
0.00 0.25 0.50 0.75 1.00
Deck
Dis
pla
cem
en
t (m
)
Normalize length
0.2
0.3
0.4
0.5
0.6
0.7
0.00 0.25 0.50 0.75 1.00
Deck
Dis
pla
cem
en
t (m
)
Normalize length
0.2
0.3
0.4
0.5
0.6
0.7
0.00 0.25 0.50 0.75 1.00
Deck
Dis
pla
cem
en
t (m
)
Normalize length
DDBD to obtain Displacement DemandF
1
2
z
m
Sd
T
5%
15%
20%
1.Guess
2.Calculate Keff
3.Calculate Teff
𝑇𝑒𝑓𝑓 = 2𝜋𝑀
𝐾𝑒𝑓𝑓
4. Calculate ductility, m
5. Calculate damping, z.
6. Calculate new
7. Return to step 1,
iterate to converge
5
6
𝜇∆ =∆
∆𝑦
Seismic Guide Spec Displacement Demand Options
•Procedure 1: Equivalent Static Analysis• Uniform Load Method
• Single Mode Method
•Procedure 2: Elastic Dynamic Analysis
•Procedure 3: Non-linear Time History Analysis
Choice of above a function of bridge characteristics and seismic demands
Uniform load method
∆𝑑= 𝑉𝑠,𝑚𝑎𝑥
𝑝𝑒𝑝𝑜𝑅𝐷𝑅𝑑
Evaluating Displacement Demand
Why the differences?Hypothesis: Viscous damping model choices account for these differences
F
gmakcm
mkmc 22
SDOF MODEL
kinitial
Initial Stiffness damping: Constant Coefficient c, k=kinitial
Tangent Stiffness damping: Damping coefficient varies in proportion to instantaneous stiffness: ck/kinitial
Note: Peak displacement decreases as c increases
MOST ANALYSES USE CONSTANT COEFFICIENT DAMPING
What are the choices?
Reasons for Elastic Viscous Damping
Should the damping coefficient remain constant?
Absolutely Not!
Does the damping model matter?
-0.1 0 0.1Displacement (m)
-2000
-1000
0
1000
2000
Sti
ffn
ess
Forc
e (
kN
)
-0.1 0 0.1Displacement (m)
-2000
-1000
0
1000
2000
Dam
pin
g F
orc
e (
kN
)
-0.1 0 0.1Displacement (m)
-2000
-1000
0
1000
2000
Sti
ffn
ess
Forc
e (
kN
)
-0.1 0 0.1Displacement (m)
-2000
-1000
0
1000
2000
Dam
pin
g F
orc
e (k
N)
(a) Analysis with Initial Stiffness Damping
(b) Analysis with Tangent Stiffness Damping
SDOF, T=0.5 sec
FORCED SINUSOIDAL INPUT OF T=1.0sec
TAKEDA HYSTERESIS
Initial-Stiffness Damping Ad=0.83Ah
Tangent-Stiffness Damping Ad=0.15Ah
0 4 8 12 16 20Time (seconds)
-0.1
-0.05
0
0.05
0.1
0.15
Dis
pla
cem
en
t (m
)
Tangent Stiffness
Initial Stiffness
RESPONSE OF SDOF MODEL, T=0.5sec
TO 1.5*EL CENTRO 1940 NS
Impact on results – computational SDOF
35 m 40 m
14 m 14 m 14 m
1 2 3 4 5
35 m40 m
K0KS
NON VISCOUS
MODAL
M+Ki
0
500
1000
1500
2000
2500
3000
3500
4000
4500
5000
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50
Fo
rce (
kN
)
Displacement (m)
Impact on Range of Outcomes
Experimental Evidence ?
0 4 8 122 6 10Time (sec)
-0.1
0
0.1
-0.05
0.05
0.15
-0.15
Dis
pla
cem
en
t (m
)
Experiment
Tangent Stiffness
Initial Stiffness
Displacement Equivalence Rules
Displacement RatiosConsider Displacement Equivalence Rules (i.e. equal displacement approximation):
What does the mean for Displacement Demand Estimation?
•SGS Procedure 1 and 2: Based on displacement equivalence rules which were established with initial stiffness damping. Displacements will be underestimated!
•SGS Procedure 3: Need to choose correct damping model.
•DDBA: Provides accurate displacements
Research Needs in this Area
•Compare numerous techniques for evaluating displacement demands.
•Quantify impact of damping models for both:• Displacement equivalence rules
• Non-linear time history analysis
•Recommendations for both
•These are addressed in the RNS titled: “Estimating inelastic displacement demands for bridges under seismic forces”
Acknowledgements
•Nigel Priestley: Nigel brought attention to the damping question long before it became a popular topic of research.
•Current PhD Student Diego Martinez: He is studying impacts of damping on MDOF response of bridges.
•Thanks to Rich and the T3 Committee for the invitation to visit with you today.