Overview of Direct
Displacement-Based Design of
Bridges
July 9, 2012
Mervyn J. Kowalsky
Professor of Structural Engineering
North Carolina State University
[email protected]
919 515 7261
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Outline
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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
d=2m
875
mm
z=5%
H=10m
4 sec.
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
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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
Obtaining Effective Period:
Disp (mm)
Dc 5% = 875 mm
z=5%
Dc 15.5% = 553 mm
z=15.5%
Dd = 350 mm
Teff = 2.53 Tc = 4
NOTE: Dc X% = Dc 5% Rx
Rx =
7
2+z
Period (sec)
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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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Complexities for Multi-Span Bridges
•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.
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Transverse Displaced Shapes
D
D
D
D
D
D
D3
D
D3
D4
D
D5
D
(a) Symm., Free abuts.
Rigid SS translation
D4
D5
(b) Asymm., Free abuts.
Rigid SS translation+rotation
D
D
D
D
D
D3
D
D3
D4
D3
D4
D5
(c) Symm., free abuts.
Flexible SS
D4
D5
(d) Symm,. Restrained abuts. (e) Internal movement joint
Flexible SS
Rigid SS, Restrained abuts.
D5
(f) Free abuts., M.joint
Flexible SS
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Displacement
Obtaining Displaced Shape
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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Damping Components
System damping obtained by weighting component damping
according to work done by each component
Pier Damping:
System Damping:
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Base Shear Distribution
Force is distributed in proportion to mass and pier top displacement.
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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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DDBD OF MDOF BRIDGES
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.
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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
C
A
B
abutments
Displ.
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. NCSU
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Design Displacement for a Footing-Supported
Column under Long. Response (Central Pier)
10MN 2250kips
Material props:
12m
(39.4ft)
2.0m dia
(78.7in)
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).
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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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Transverse Design Example
40m
50m
A
16m
B
50m
40m
12m
16m
C
(Not to scale)
E
D
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
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Example 10.5, p507
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Therefore, limit state displacements are:
Da = De = 40mm
Db = Dd = 961mm
Dc = 596mm
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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(Work balance)
(Force equil.)
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Da = De = 40mm; Db = Dd = 417mm; Dc = 596mm (Target)
Da = De = 43mm; Db = Dd = 383mm; Dc = 572mm (actual)
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Da = De = 41mm; Db = Dd = 396mm; Dc = 593mm (rev. x) NCSU
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
R-R
X-Z
PR-R
X-Z
R-PR
X: Longitudinal restraint
Z: Transverse restraint
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6 Span Bridge Results
X-Z
PR-PR
X-Z
U-R
X-Z
R-U
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6 Span Bridge Results
X-Z
U-U
X-Z
U-PR
X-Z
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
Specimens 1-12
Load History
Test Matrix
Transverse Steel
Specimens 13-18
Currently 18 Tests Completed
Specimens 19-24
Testing Aug - Dec
Load History
Axial Load
Aspect Ratio
Longitudinal Steel
Specimens 25-30
Axial Load
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Quasi-Static Earthquake Loading Procedure
1.5
1.25
1
0.5
0.25
0
-0.25
-0.5
-0.75
Displacement (mm)
0
25
50
75
100
125
-254
80
150
Time (sec)
-154
-54
46
146
246
344
60
244
40
144
20
44
0
-56
-20
-156
-40
-256
-60
-80
-356
-10
-8
-6
-4
-2
0
2
Displacement (in)
4
6
8
10
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Lateral Force (kN)
-1
Lateral Force (kips)
Acceleration (g)
0.75
Load History Characteristics
Monotonic – Test 1
El Centro 1940 – Test 4
Test 9 Shown
Symmetric Three Cycle Set
Tabas 1978 – Tests 5 and 6
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Load History Characteristics
Japan 2011 – Test 12
Chichi 1999 – Test 10
Kobe 1995 – Test 11
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
1200
45
Ductility 1 +3
40
Ductility 1.5 +3
1000
Ductility 2 +3
Ductility 3 +3
Ductility 4 +3
30
800
Ductility 6 +3
25
Ductility 8 +1
600
20
Location (mm)
Location (in)
35
400
15
10
200
5
0
-0.03
-0.02
-0.01
0
0.01
0.02
0.03
0.04
0
0.05
Strain
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Curvature Profiles
Curvature (1/m)
0
0.02
0.04
0.06
0.08
0.1
0.12
1200
Ductility 1 +3
y = -37229x + 96
Ductility 1.5 +3
40
Ductility 2 +3
Location (in)
35
Ductility 3 +3
30
Ductility 4 +3
25
Ductility 6 +3
Ductility 8 +1
20
15
y = -2499.5x + 20
R² = -0.018
y = -1978.9x + 21.575
R² = 0.6693
y = -1547.3x + 25.102
R² = 0.8831
y = -1429.2x + 28.909
R² = 0.9692
y = -1113.4x + 32.04
R² = 0.9857
y = -822.27x + 32.325
R² = 0.9859
1000
800
600
Location (mm)
45
400
10
200
5
0
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0
0.04
Curvature (1/ft)
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Plastic Hinge Method
Lc
φy
Lp
φp
Lsp
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Modified Plastic Hinge Method
Elastic Flexure + Plastic Flexure + Strain Penetration + Shear
Lc
Lp
Lsp
φy
φp
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Comparison with Plastic Hinge Method
10
9
Original Plastic Hinge Method
8
Physical Test
Displacement (in)
7
Curvature Ductility Dependent Method
6
5
4
3
2
Input of φbase from Test Results
1
0
0
20
40
60
80
100
120
140
Data Point Number
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Degree of Fixity at Pier Top
F
F
M
fully fixed
at design
displace.
pinned
M
M
(a) Multi-column Pier
WSS
(b) Single Column, Single Bearing
F
M1
F
M1
L
top moment
depends on
SS flexibility
top moment
indeterminate (2
modes)
M2
(c) Single Column, Multiple Bearing
M2
(d) Single Column, Monolithic.
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