NCSU

Overview of Direct

Displacement-Based Design of

Bridges

July 9, 2012

Mervyn J. Kowalsky

Professor of Structural Engineering

North Carolina State University

kowalsky@ncsu.edu

919 515 7261

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Outline

• Brief History

• DDBD Fundamentals

• SDOF Example

• MDOF Fundamentals

• MDOF Example

• Design verification

• Sources for more information

• Current and future areas of study

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Things to think about during the talk:

• Philosophical differences: DDBD,

AASHTO LRFD (Force based), and

AAHSTO Guide Spec for Seismic Design

(Displacement-based).

• Examples: How would they be handled

with current AASHTO methods?

• End Result: Does DDBD Make a

difference? (Best to try it for yourself!)

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Brief History

• 1993 “Myths and

Fallacies” paper by

Priestley.

• Continual development

from 1993 through 2007.

• Culminated in 2007 book.

• Chapter in 2013 Bridge

engineering handbook.

• Continued refinement,

adaptations, and

verifications.

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For seismic design…

• “You are the boss of the structure – tell it what

to do!” Tom Paulay

• “Strength is essential, but otherwise

unimportant.” Hardy Cross

• “Analysis should be as simple as possible, but

no simpler.” Albert Einstein

• “Always follow the principle of consistent

crudeness.” Nigel Priestley

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Within the context of

Performance Based Design:

• What should the structural strength be (i.e.

base shear force)?

• How should the strength be distributed?

• How can design be elevated by analysis?

• What should the strength of capacity

protected actions be?

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DDBD Fundamentals

• Displacement Response Spectrum (DRS)

based.

– DRS can be easily obtained from code ARS

or site specific.

• Utilizes equivalent linearization (inelastic

spectra also possible)

– Effective stiffness.

– Equivalent viscous damping

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Fundamentals

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Basic Method (SDOF) • Select target displacement, Dd

– Strain, Drift, or Ductility

• Calculate yield displacement, Dy

– Fundamental member property

• Calculate equivalent viscous damping, z

– Relationships between damping and

ductility available and easily obtained

• Calculate effective period, Teff

– From Response spectra

• Calculate effective stiffness, Keff

– Keff = 4p2m/Teff2

• Calculate design base shear force, Vb

– Vb = KeffDd

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How Are Damping Equations Obtained?

Area based hysterestic damping from above is corrected (NLTHA)

and then combined with viscous damping (i.e. 5% tangent stiffness)

to obtain expressions for equivalent viscous damping for a given

hysteretic shape, i.e. RC Column or steel beam, etc.

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Example – Single bent bridge

H=10m

d=2m fy=470MPa

Es=200GPa

W=5000kN

qd=0.035

md=4

Target Displacement:

Drift: Dd=(0.035)(10m) = 0.350 m

Ductility: Dd=mdDy

Dy=fyH2/3

fy=2.25ey/D=0.00264 1/m

Dy= 0.088 m

Dd = 4(0.088) = 0.353 m

875

mm

4 sec.

z=5%

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Example – Single bent bridge Equivalent Viscous Damping (These expressions all assume 5% tangent

stiffness proportional viscous damping and hysteretic damping):

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Example – Single bent bridge

Dc 5% = 875 mm

Tc = 4

z=5%

Obtaining Effective Period:

Dd = 350 mm

Dc 15.5% = 553 mm z=15.5%

Teff = 2.53 Period (sec)

Disp (mm)

NOTE: Dc X% = Dc 5% Rx 2 + z

Rx = 7

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Example – Single bent bridge Obtaining Effective Stiffness:

Obtaining Design Base Shear:

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Simplified Base Shear Equation for DDBD

a = 0.5 for regular conditions

a = 0.25 for velocity pulse conditions

NOTE: Damping expressed as ratio in the above equation (not %).

NOTE: Equation assumes a linear DRS to the corner point.

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•Transverse design displacement profiles

• Dual seismic load paths

• Effective system properties

•displacement, damping, mass

• Degree of fixity at column top

•Impact of abutment support conditions

•Iterative, in some cases.

Complexities for Multi-Span Bridges

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Transverse Displaced Shapes

D

D

D

D

D

(a) Symm., Free abuts. (b) Asymm., Free abuts. (c) Symm., free abuts. Rigid SS translation Rigid SS translation+rotation Flexible SS

D

D3

D4

D5

D

D

D D

D

D

D

D

D3

D3

D3

D3

D4

D4

D4

D4

D5

D5

D5

D5

(d) Symm,. Restrained abuts. (e) Internal movement joint (f) Free abuts., M.joint Flexible SS Rigid SS, Restrained abuts. Flexible SS

D

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Obtaining Displaced Shape D

ispla

cem

ent

Position along bridge Note: Stars are limit state displacements based on strain, ductility, or drift

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System Displacement and Effective Mass

From work balance between MDOF and SDOF systems:

From force equilibrium between MDOF and SDOF systems:

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Pier Damping:

System Damping:

Damping Components

System damping obtained by weighting component damping

according to work done by each component

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Force is distributed in proportion to mass and pier top displacement.

Base Shear Distribution

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Higher mode effects?

• In general, not a problem for most bridges

with regards to displaced shape.

• Possible to use “Effective modal analysis” to

define displaced shape, but takes more effort.

• Higher modes can be an issue for

superstructure bending and abutment

reactions – use dynamic amplification

factors.

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Longitudinal Design: If the bridge is straight, this is generally

straightforward, and will often dominate design requirements.

Effective damping and design displacement are the main issues.

Transverse Design: More complex, but often doesn’t govern.

Displacement shape may not be obvious at start. Design

displacement, damping, higher mode effects may need to be

considered.

DDBD OF MDOF BRIDGES

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Multi-span bridge – longitudinal direction

1. In longitudinal direction, multi-span bridge is an SDOF system.

2. Shortest pier will govern target displacement.

3. Only complexity is that damping of each pier must be weighted.

4. For bridges restrained in the transverse direction but free

longitudinally, the governing direction is longitudinal.

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Force

Displ.

C

A

B

Design Choice: Equal moment capacity, piers.

Shears inversely proportional to height

Yield curvatures of piers are equal

Design Displacement based on shortest pier.

Ductility, and hence damping of piers are different.

abutments

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12m

(39.4ft)

2.0m dia

(78.7in)

10MN 2250kips

Material props:

f’c=30MPa: f’ce=39MPa (5.7ksi)

fy=420MPa: fye=462MPa (67ksi)

fu/fy=1.35

Long.bars: 40mm (1.575in) dia.

Trans.bars:20mm @100mm (4in)

Displacement for damage-control limit state for fixed top case

= 0.326m: based on strains (concrete governs at 0.0136 over

steel at 0.06).

Design Displacement for a Footing-Supported

Column under Long. Response (Central Pier)

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Multi-span bridge – transverse direction • Estimate portion of base shear to be carried by abutments due to

superstructure bending.

• Define column and abutment target displacements.

• Define displaced shape.

• Scale shape to critical column or abutment displacement.

• Express MDOF bridge as equivalent SDOF structure

– system displacement

– system mass

– system damping

• Calculate design base shear (use simplified equation)

• Distribute base shear to each abutment and bent according to

displaced shape and mass of each bent/abutment.

• Conduct secant stiffness analysis and compare response displaced

shape to target displaced shape.

• Revise proportion of force carried by abutment, if needed

• Iterate until convergence

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40m 50m 50m 40m

16m 12m

16m

B

C

A

D

E

(Not to scale)

Transverse Design (1): Ductile Piers, restrained abutments:

Dd = 0.485m; x = 0.51; x = 0.085; VBase = 11.1MN

Transverse Design (2): Isolated piers and abutment:

Dd = 0.5m; x = 0.0, x = 0.163; VBase = 5.6MN

Transverse Design Example

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Example 10.5, p507

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Da = De = 40mm

Db = Dd = 961mm

Dc = 596mm

Therefore, limit state displacements are:

Since displacements of pier b and d are estimated

as 70% of pier c, pier c is critical and profile is:

Da = De = 40mm; Db = Dd = 417mm; Dc = 596mm

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Da = De = 40mm; Db = Dd = 417mm; Dc = 596mm (Target)

Da = De = 43mm; Db = Dd = 383mm; Dc = 572mm (actual)

NCSU Da = De = 41mm; Db = Dd = 396mm; Dc = 593mm (rev. x)

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2nd iteration

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Sample Design and Analysis

Result for MDOF Bridge

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Other verification results

• 2, 4, 6 span bridges with 9 different

support conditions.

• Each bridge designed with DDBD and

then analyzed with NLTH analysis.

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6 Span Bridge Results

X-Z X-Z X-Z

R-R PR-R R-PR

X: Longitudinal restraint

Z: Transverse restraint

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6 Span Bridge Results

X-Z X-Z X-Z

PR-PR U-R R-U

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6 Span Bridge Results

X-Z X-Z X-Z

U-U U-PR PR-U

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Sources for more information

• NCSU (every fall on campus and by distance): CE 725 covers

displacement based design of structures. Currently developing a bridge

specific course as well (CE 725 will be a pre-req).

http://engineeringonline.ncsu.edu/index.html • Courses at the Rose School in Pavia Italy: Numerous courses on DDBD

(I teach the bridge course every three years, usually in May – next 2013)

http://www.roseschool.it/

• Seminars (Past: NC, DR, Ecuador, Vancouver, NZ, Italy).

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Sources for more information

• Textbook.

• Numerous papers.

• Bridge engineering handbook, 2013, will

have a chapter on DDBD of Bridges.

• Call or email me any time.

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Current and Future Areas of Study

• DDBD of curved and irregular bridges (PhD

student Easa Khan), and arch bridges (with

Easa Khan and Dr. Tim Sullivan of Rose

School).

• Impact of load history (and path) on limit state

definitions and the relationship between strain

and displacement.

• Seismic behavior of reinforced concrete filled

pipes

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Research Methods

Analytical

• Moment curvature analysis of sections

• Fiber and FEM analysis of members

Experimental

• Material tests

• Large scale tests (30)

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Specimen Design

• 2ft Diameter

• 8ft Cantilever Length

• Single Bending

• 16 #6 Longitudinal Bars

• #3 or #4 Transverse at

Variable Spacing

• ½” Cover to Spiral

Quasi-Static Load Procedure

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Optotrak Certus HD® Position Sensor

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Experimental Tests

Tes

t M

atri

x

Specimens 1-12 Load History

Specimens 13-18 Transverse Steel

Load History

Specimens 19-24

Testing Aug - Dec

Axial Load

Aspect Ratio

Specimens 25-30 Longitudinal Steel

Axial Load

Currently 18 Tests Completed

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Quasi-Static Earthquake Loading Procedure

0 25 50 75 100 125 150-1

-0.75

-0.5

-0.25

0

0.25

0.5

0.75

1

1.25

1.5

Time (sec)

Acc

eler

atio

n (

g)

-254 -154 -54 46 146 246

-356

-256

-156

-56

44

144

244

344

-80

-60

-40

-20

0

20

40

60

80

-10 -8 -6 -4 -2 0 2 4 6 8 10

Displacement (mm)

La

tera

l F

orc

e (k

N)

La

tera

l F

orc

e (k

ips)

Displacement (in)

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Load History Characteristics

Test 9 Shown

Symmetric Three Cycle Set Monotonic – Test 1

El Centro 1940 – Test 4 Tabas 1978 – Tests 5 and 6

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Load History Characteristics

Japan 2011 – Test 12 Kobe 1995 – Test 11

Chichi 1999 – Test 10 Chile 2010 – Test 8

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Is Load History Important?

Load History as the Only Variable

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Load history and Buckling of Steel

• Characteristic compression strain

capacity:

– Impacted by boundary conditions, which

are effected by load history (i.e. large

compressive cycles which yield

transverse steel)

• Tensile Strain Demand:

– Impacted by number of reversals and

strain accumulation.

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Strain Profiles

0

200

400

600

800

1000

1200

0

5

10

15

20

25

30

35

40

45

-0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.04 0.05

Loca

tion

(m

m)

Loca

tion

(in

)

Strain

Ductility 1 +3

Ductility 1.5 +3

Ductility 2 +3

Ductility 3 +3

Ductility 4 +3

Ductility 6 +3

Ductility 8 +1

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Curvature Profiles

y = -37229x + 96y = -2499.5x + 20

R² = -0.018

y = -1978.9x + 21.575R² = 0.6693

y = -1547.3x + 25.102R² = 0.8831

y = -1429.2x + 28.909R² = 0.9692

y = -1113.4x + 32.04R² = 0.9857

y = -822.27x + 32.325R² = 0.9859

0 0.02 0.04 0.06 0.08 0.1 0.12

0

200

400

600

800

1000

1200

0

5

10

15

20

25

30

35

40

45

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04

Curvature (1/m)

Loca

tion

(m

m)

Lo

cati

on

(in

)

Curvature (1/ft)

Ductility 1 +3

Ductility 1.5 +3

Ductility 2 +3

Ductility 3 +3

Ductility 4 +3

Ductility 6 +3

Ductility 8 +1

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Plastic Hinge Method

φy

Lc

Lsp

φp

Lp

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Modified Plastic Hinge Method

φy

Lc

Lsp

φp

Lp

Elastic Flexure + Plastic Flexure + Strain Penetration + Shear

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0

1

2

3

4

5

6

7

8

9

10

0 20 40 60 80 100 120 140

Dis

pla

cem

ent

(in

)

Data Point Number

Original Plastic Hinge Method

Physical Test

Curvature Ductility Dependent Method

Comparison with Plastic Hinge Method

Input of φbase from Test Results

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L

M

M M

M1

M2

M1

M2

(a) Multi-column Pier (b) Single Column, Single Bearing

(c) Single Column, Multiple Bearing (d) Single Column, Monolithic.

F F

F F

WSS

fully fixed

at design

displace.

pinned

top moment

depends on

SS flexibility

top moment

indeterminate (2

modes)

Degree of Fixity at Pier Top