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.