Citation
Feasibility of rocking hinge bearings for bridge piers under seismic shaking

Material Information

Title:
Feasibility of rocking hinge bearings for bridge piers under seismic shaking
Creator:
Osmun, Richard L
Publication Date:
Language:
English
Physical Description:
xxv, 270 leaves : ; 28 cm

Subjects

Subjects / Keywords:
Concrete construction -- Hinges ( lcsh )
Bridges -- Foundations and piers ( lcsh )
Earthquake resistant design ( lcsh )
Bridges -- Foundations and piers ( fast )
Concrete construction -- Hinges ( fast )
Earthquake resistant design ( fast )
Genre:
bibliography ( marcgt )
theses ( marcgt )
non-fiction ( marcgt )

Notes

Bibliography:
Includes bibliographical references (leaves 267-270).
General Note:
Department of Civil Engineering
Statement of Responsibility:
by Richard L. Osmun.

Record Information

Source Institution:
University of Colorado Denver
Holding Location:
Auraria Library
Rights Management:
All applicable rights reserved by the source institution and holding location.
Resource Identifier:
61133978 ( OCLC )
ocm61133978
Classification:
LD1190.E53 1999m O75 ( lcc )

Full Text
FEASIBILITY OF ROCKING HINGE BEARINGS
FOR BRIDGE PIERS UNDER SEISMIC SHAKING
by
Richard L. Osmun
B.S.C.E. University of Wyoming, 1969
A thesis submitted to the
University of Colorado at Denver
in partial fulfillment
of the requirements for the degree of
Master of Science
Civil Engineering
1999


This thesis for the Master of Science
degree by
Richard L. Osmun
has been approved
by
Nien-Yin Chang
Date


Osmun, Richard L. (M.S., Civil Engineering)
Feasibility of Rocking Hinges for Bridge Piers Under Seismic Shaking
Thesis directed by Professor N. Y. Chang
ABSTRACT
Researchers continue to investigate ongoing problems with different
seismic connection details and high shear problems in short (squat) columns.
Many of these connection and shear problems might go away if seismic induced
loads could somehow be reduced or limited.
One way to reduce the loads is to eliminate the unknown conservatism
associated with an elastic analysis through the use of a time-history analysis.
Accordingly, a simple two column bent model with nonlinear support springs at the
top and bottom of each column was prepared as part of this thesis to compare the
most popular seismic isolation system involving lead-rubber bearings with a newly
conceived isolation system involving rocking hinge bearings. The author uses the
term rocking hinge bearings to describe a moment limiting device placed at each
end of a typical column to provide rotational stability under lateral loading through
a combination of gravity and prestress righting forces.
Rocking hinge bearings are lateral load limiting devices as are lead-rubber
bearings. However, the two devices do not limit lateral loads in the same manner.
Specifically, lead-rubber bearings limit seismic loads to the superstructure through
shear deformations in the bearing while rocking hinges limit moments which, in
turn, reduces shears to the superstructure as per V = £M/L where L is the column


length and £M is the sum of the top and bottom rocking hinge moments for a given
column. Computer modeling shows that the lead-rubber bearing is a better choice
as a lateral load reducer for bridges founded in soft soils where the foundation
materials can not support high lateral loads and the rocking hinge bearing is a
better choice for bridges founded on bedrock where the foundation materials can
support high lateral loads.
A static pushover (push) routine was developed to rank the time-history
performance of the structural alternatives. Modeling shows that the push analysis
can rank the alternatives with respect to overall ductility. However, the ductility
ranking is not necessarily the same as the dynamic ranking.
This abstract accurately represents the content of the candidate's thesis. I
recommend its publication.
IV


ACKNOWLEDGEMENT
As the saying goes, no man is an island. I believe this is especially true
when it comes to original works like this thesis. Without input from others, it is
easy to miss subtle points or bog down on esoteric issues; a little help in the right
areas makes it possible to explore more ideas. With this in mind, I wish to thank
Dr. N. Y. Chang for encouraging me to pursue this topic, and for his expert
guidance and meticulous reviews of my many draft copies. Dr. Trever Wang is
more than a member of my thesis committee; he is a friend and co-worker and has
made many valuable suggestions. The lessons learned from Dr. Mays in his
graduate courses gave me several needed tools for the computer modeling work,
and the one-on-one private conversations we had in the beginning pointed me in
the right direction during the initial phase of this project.
I must thank my family members as well. My wife has been very patient,
understanding and supportive; my daughter encouraged me when technical
problems loomed, and her help during the testing phase was appreciated. The
physical model would not have become a reality without the help of my dad; he
built the entire model to close tolerances, was a good sounding board, and helped
run the tests.
I am sincerely grateful for the support and contributions from all of you.


CONTENTS
Figures.....................................................................ix
Tables......................................................................xi
Notations..................................................................xii
Chapter
1. Introduction..........................................................1
1.1 Column Softening......................................................1
1.2 Research Objective....................................................2
1.3 Engineering Significance..............................................2
1.4 Rocking Hinge Bearings................................................3
1.5 Object-Oriented Programming...........................................5
1.6 Chapter 1 References..................................................9
2. State of the Art.....................................................10
2.1 Brief History of Seismic Specifications..............................10
2.2 Plastic Hinges.......................................................13
2.3 Seismic Isolation....................................................15
2.4 Chapter 2 References.................................................21
3. Rocking Hinge Bearings...............................................23
3.1 Concept..............................................................23
3.2 Analysis.............................................................24
3.3 Design...............................................................26
3.4 Chapter 3 References.................................................34
4. Time-History and Static Pushover Analysis............................35
vi


4.1 Quasistatic Analysis and the Pushover Model.......................35
4.2 Numerical Response Methods and the Dynamic Model..................38
4.3 Hysteresis Models.................................................42
4.4 Computer Program..................................................45
4.4.1 Header Files......................................................45
4.4.2 Spring Object File................................................46
4.4.3 Analysis File.....................................................48
4.4.4 Input Files.......................................................50
4.4.5 Program Verification..............................................51
4.5 Chapter 4 References..............................................68
5. Physical Model....................................................70
5.1 Model Description.................................................70
5.2 Test Results......................................................71
5.3 Analytical Results................................................71
6. Computer Modeling.................................................77
6.1 Description of Computer Models....................................77
6.2 Test Results......................................................79
6.3 Column Body Shears................................................84
6.4 Construction Costs................................................84
6.5 Chapter 6 References..............................................95
7. Summary, Conclusions, and Research/Testing Recommendations........96
7.1 Summary...........................................................96
7.2 Conclusions.......................................................96
7.3 Research Recommendations..........................................98
7.4 Testing Recommendations..........................................100
vii


7.5 Chapter 7 References............................................102
Appendix
A Computer Program...................................................103
B Physical Model Testing and Analysis...............................160
C Computer Model Input Calculations..................................173
D Construction Cost Estimates........................................240
E Ground Motion Input Record.........................................256
References.............................................................267
viii


FIGURES
Figure
1.1- Rocking Hinge Bearing Concepts........................................7
1.2- Rocking Hinge Bearing Details.........................................8
2.1- Confinement with Ties................................................18
2.2 - Lead-Rubber Bearing.................................................19
2.3 - Friction Pendulum Bearing...........................................20
3.1 - Geometric and Stress Block Equations.................................30
3.2 - Prestressing Equations..............................................31
3.3 - Prestressing Free-Lengths for Shorter Columns.......................32
3.4 - Pintle and Base Plate Design Procedure..............................33
4.1- Secant Stiffness Method..............................................52
4.2 - Ductility Principles, Push Plot & ARS Curves........................53
4.3 - Push Model..........................................................54
4.4 - Push Model Logic....................................................55
4.5 -Model Config. & Plane 1 Equns. of Motion.............................56
4.6 Plane 2 & 3 Equns. of Motion........................................57
4.7 - Column Element Relationships........................................58
4.8 - First Order P-A Terms...............................................59
4.9 - Pivot Spring........................................................60
4.10 - Bilinear, Rocking and Elastic Springs.............................61
4.11- Elliptical Failure Criterion........................................62
4.12- TBENT.CPP Flowchart.................................................63
4.13 Four Sample Spring Test Input Files................................64
IX


Figure
4.14- Two Sample Column Test Input Files..................................65
4.15- One Sample Push Analysis Test Input File............................66
4.16- One Sample Run Input File..........................................67
5.1 - Test Bent Elevation.................................................74
5.2 - Test Bent Details...................................................75
5.3 Test Bent Results...................................................76
6.1- Dynamic Model Results (Linear Vs. Nonlinear)........................85
6.2 - Dynamic Model Results (Plastic Hinge Config. in Hard Soils).........86
6.3 - Dynamic Model Results (Rocking Hinge Config. in Hard Soils).........87
6.4 - Dynamic Model Results (Lead-Rubber Config. in Soft Soils)...........88
6.5 - Scale Factor Performance Plots......................................89
6.6 - Column Body Nominal Shear Strength..................................90
x


TABLES
Table
6.1 -100' Span Analysis Results..........................................91
6.2 - 200' Span Analysis Results.........................................92
6.3 - Column Body Shear Performance......................................93
6.4 - Construction Cost Estimates........................................94
XI


NOTATIONS
Chapter 1
h location of load P above the supporting surface for the rocking experiment;
M " moment due to rocking action or prestressing moment to resist rocking action;
m0 = slope of the rocking moment diagram, ie, the (M/(J)) diagram;
P = applied lateral load to create rocking action;
= initial rotation at incipient rocking;
4>ot = incipient overturning angle in a pure rocking scenario.
Chapter 2
D = outer diameter of a round lead-rubber bearing;
Ps ratio of volume of transverse confining steel (spiral reinforcement) to the volume of the confined concrete core.
Chapter 3
A, = area of ultimate concrete stress block;
Av pintle shear area;
A;ps - area of individual prestressing strand;
Aps strand area in a prestressing tendon;
As = area of prestressing steel in a prestress force-rotation plot;
Asqr tendon prestressing area in a square configuration with a diagonal equal to D;;
A, = pintle bearing area;
xii


c depth of triangular stress block;
d perpendicular distance from the center of a rocking hinge bearing to its heel line;
D = column outer diameter;
D; prestressing duct outside diameter and pintle inside diameter;
Dips = diameter of individual prestressing strands;
D0 = pintle outer diameter;
e perpendicular distance from the heel line in a rocking hinge bearing to an arbitrary point within the triangular stress block;
EPs = modulus of elasticity for prestressing steel;
f = compressive stress in triangular stress block calculations;
fc- required ultimate concrete strength for a reinforced concrete column equipped with rocking hinge bearings;
fyps - prestressing strand yield stress;
fys = base plate yield strength;
F = steel ultimate breaking strength;
Fy steel yield stress at 0.2% offset;
h - column height;
L prestressing free length to create rotational ductility in a rocking hinge bearing;
m ultimate stress at the toe divided by the depth of the triangular stress block in a rocking hinge bearing;
m0 = slope of the rocking moment diagram without prestressing;
Mbp = ultimate moment in the base plate on its centerline;
hinge moment when the toe stress reaches Fy in the xiii


prestressing calculations;
hinge moment associated with the maximum possible hinge
rotation angle in the prestressing calculations;
ultimate resisting moment in a rocking hinge bearing;
required number of prestressing strands in a prestressing
tendon;
ultimate dead, live and impact reaction on a disc;
prestressing force;
prestressing force at 0;.
prestressing force at 0j;
N0 plus Nps at 0j;
ultimate bearing load;
maximum prestressing force in a prestress force-rotation
plot;
disc radius;
section modulus of the base plate on its centerline;
required plate thickness (disc or base plate as appropriate);
column shear;
horizontal shear across the pintle;
column radius in incipient collapse angle calculations,
distance perpendicular to the segment heel in stress block
calculations and the distance from the center of a disc to its
stress block reaction in prestressing calculations;
distance from the center of the base plate to the centroid of
the concrete stress block;
distance from the center of the disc to the centroid of the
triangular stress block;
xiv


y
a
ef
%
0
0C
0i
i
0!

stress block center of gravity;
a fraction of the maximum possible hinge rotation angle in
the prestressing calculations used to generate an initial
prestressing force to seat the components in a rocking hinge
bearing;
strain at the toe in a disc;
disc toe strains at yield and ultimate loading respectively;
half of the included angle in a circular segment or the hinge
rotation angle in the prestressing calculations;
incipient collapse angle;
hinge rotation angle when the toe stress reaches Fy in the
prestressing calculations;
maximum possible hinge rotation angle in the prestressing
calculations;
the maximum hinge rotation angle in the triangular stress
block; this angle describes the maximum disc crush along
the heel to toe line;
a capacity reduction factor in strength calculations.
Chapter 4
ah = horizontal seismic acceleration;
A = cross sectional area in plane of rotation for use in calculating
the radius of gyration;
actual superstructure area in the xz-plane;
gross cross sectional area of a column;
area of main column reinforcement;
Axial load ratio, ie, the applied ultimate axial load divided
As
ALR
xv


by the product of the ultimate concrete strength times the gross cross sectional area of the column;
Be - equivalent block width;
cr = rotational damping coefficient in plane of rotation;
De = equivalent block depth;
e eccentricity of axial load in a column at some location along the column shaft;
f f xX5Ay applied loads in the x and y directions for the elliptical failure criterion;
F = the force applied to a set of springs in series;
Fi = the force on one spring in a set of springs in series;
FCXFcy = member capacities in the x and y directions for the elliptical failure criterion;
g - gravitational constant;
h = column height in a column element;
h. = column one height;
h2 = column two height;
^mls ^mZ ^m3 = mass moments of inertia in plane 1,2 and 3;
Io = polar moment of inertia in plane of rotation;
Iz = area moment of inertia about the x, y and z axis in the xz, yz and xy planes respectively;
K = rotational stiffness;
ki = horizontal spring constant in the xz-plane for the connection of column one to the superstructure mass;
k2 = horizontal spring constant in the xz-plane for the connection of column two to the superstructure mass;
k3 = not used;
XVI


vertical spring constant for the connection of column one to
the superstructure mass;
vertical spring constant for the connection of column two to
the superstructure mass;
rotational spring constant in the xz-plane for the connection
of column one to the superstructure mass;
rotational spring constant in the xz-plane for the connection
of column two to the superstructure mass;
horizontal spring constant in the yz-plane for the connection
of column one to the superstructure mass;
horizontal spring constant in the yz-plane for the connection
of column two to the superstructure mass;
rotational spring constant in the yz-plane for the connection
of column one to the superstructure mass;
rotational spring constant in the yz-plane for the connection
of column two to the superstructure mass;
rotational spring constant in the yz-plane representing the
superstructure stiffness as created by the reactions at the far
bents;
rotational spring constant in the xy-plane representing the
torsional stiffness of column one;
rotational spring constant in the xy-plane representing the
torsional stiffness of column two;
secant stiffness;
the spring constant for one spring in a set of springs in
series;
rotational stiffness at the bottom of a column element due to
xvii


KpB =
Kpr =
Krb =
Krt =
Ksb
kst =
KHB, Khb =
KHT, Km =
KRB
KRT
KRBX1
KRBX2
KRBY1
KRBY2
KRTX1
KRTX2
the foundation;
rotational stiffness at the bottom of a column element due to
the plastic hinge in the column body;
rotational stiffness at the top of a column element due to the
plastic hinge in the column body;
rotational stiffness at the bottom of a column element;
rotational stiffness at the top of a column element;
rotational stiffness at the bottom of a column element due to
the support;
rotational stiffness at the top of a column element due to the
support;
horizontal spring stiffness at the bottom of a column;
horizontal spring stiffness at the top of a column;
equivalent rotational stiffness at the bottom of a column;
equivalent rotational stiffness at the top of a column;
equivalent rotational stiffness at the bottom of column one in
the xz-plane (plane of the pier bent);
equivalent rotational stiffness at the bottom of column two
in the xz-plane (plane of the pier bent);
equivalent rotational stiffness at the bottom of column one in
the yz-plane (plane of the T-firame);
equivalent rotational stiffness at the bottom of column two
in the yz-plane (plane of the T-frame);
equivalent rotational stiffness at the top of column one in the
xz-plane;
equivalent rotational stiffness at the top of column two in the
xz-plane;
XVUl


KRTY1 equivalent rotational stiffness at the top of column one in the yz-plane;
KRTY2 equivalent rotational stiffness at the top of column two in the yz-plane;
KRFB rotational stiffness at the bottom of a column due to foundation conditions;
KRPB rotational stiffness at the bottom of a column due to a plastic hinge in the column;
KRPT rotational stiffness at the top of a column due to a plastic hinge in the column;
KRSB rotational stiffness at the bottom of a column due to support conditions;
KRST rotational stiffness at the top of a column due to support conditions;
L a load which can be either a force or a moment;
Lc Le contributing superstructure length in the y direction; equivalent block length;
m = superstructure mass;
m1,m2,m3,m4 = P slopes of lines 1,2,3,4 etc. in the hysteresis diagrams; column axial load in a column element; lateral load causing Ay;
Pz superstructure gravitational reaction on the bent;
Pz. superstructure gravitational reaction on column one;
Pz2 Q superstructure gravitational reaction on column two; ultimate maximum horizontal seismic force applied to a structure;
R radius of gyration in the plane of rotation for damping
XIX


T
u
Ux> Uy,^
ULIM
WX, Wy, W7
x

y
^x> Zy>
a
P
calculations and radial distance to comer of bent in the push
analysis;
period of structure;
structure deflection due to a lateral load;
superstructure position in the x, y and z directions during a
seismic event;
ultimate displacement limit for an elastic spring;
relative superstructure position in the x, y and z directions
during a seismic event;
horizontal separation between column one and two in the xz-
plane; current horizontal displacement of the frame in the
xz-plane during a push run; displacement (linear or
rotational) in a spring plot;
the total movement for one spring in a set of springs in
series;
current horizontal displacement of the frame in the yz-plane
during a push run; load (axial, shear or moment) in a spring
plot;
ground position in the x, y and z directions during a seismic
event;
the ratio of the change in the top moment to the change in
the bottom moment in a column element; a hysteresis
parameter to locate the primary pivot points in a pivot
spring;
a hysteresis parameter to locate the pinching pivot points in
a pivot spring; a hysteresis parameter to locate the pinching
points in a rocking spring;
xx


6
A
Ac =
AMb
amt
Au =
Au, =
Au2 =
Av =
Aw =
Awx, Awx, Awx =
Awy, Awy, Awy =
Awz, Awz, Awz =
Ax
AXi
a displacement which can be either a linear or rotational
movement;
relative horizontal displacement between the top and bottom
of a column;
horizontal separation between the top and bottom of a
column in a column element;
change in moment at the bottom of a column in a column
element;
change in moment at the top of a column in a column
element;
ultimate structure deflection caused by Q; change in
superstructure horizontal position in a column element
concurrent with Az;
ultimate limit for a linear structure stiffness;
ultimate limit for a nonlinear structure stiffness;
change in horizontal reaction for a column element;
relative change between the superstructure and the earth
support, ie, Az Au;
change in the superstructure relative position, velocity and
acceleration in the x direction;
change in the superstructure relative position, velocity and
acceleration in the y direction;
change in the superstructure relative position, velocity and
acceleration in the z direction;
superstructure horizontal displacement in the xz-plane;
an incremental movement for one spring in a set of springs
in series;
xxi


Axr = superstructure horizontal displacement in the xz-plane due to the block rotation angle in said plane;
Ay superstructure horizontal displacement in the yz-plane;
Ay Az = structure deflection at yield caused by lateral load P; change in earth support position in a column element;
A0 angular change in the superstructure position as seen in the xz-plane;
A0b angular change at the bottom of a column in a column element;
A0t change in angular position of the superstructure at the top of a column element;
A0! angular change in column one's position as seen in the xz- plane during a push run;
A02 angular change in column two's position as seen in the xz- plane during a push run;
Ael3 Aeb Ae, = change in the superstructure angular position, velocity and acceleration in plane 1, ie, the xz-plane;
Ae2, Ae2, Ae2 = change in the superstructure angular position, velocity and acceleration in plane 2, ie, the yz-plane;
A03, A0 3, A 0 3 change in the superstructure angular position, velocity and
c acceleration in plane 3, ie, the xy-plane; viscous damping factor;
0 rotational motion;
0 angular velocity;
00, 0i, 0j initial, previous and current angular positions for the
xxii


superstructure in the xz-plane during a push run;
1> superstructure angular position in the xz, yz and xy-planes respectively;
P main column reinforcement ratio expressed as a percentage, ie, 100*AS/Ag.
Chapter 5
d perpendicular distance from the center of a rocking hinge bearing to its heel line;
Mu = ultimate resisting moment in a rocking hinge bearing;
P = ultimate bearing load;
Q = disc load;
T = required disc plate thickness;
P a hysteresis parameter to locate the pinching points in a rocking spring;
= strain at the toe in a disc.
Chapter 6
A - area of the circular segment;
Ae - effective shear area in the plastic hinge region;
Ah area of the bar used to make the spiral (horizontal) shear reinforcement;
Be equivalent superstructure block width for dynamic calculations;
D = overall column diameter;
D' = column core diameter after spauling;
Dcol = column outer diameter;
xxiii


De
fe
fyh
k
equivalent superstructure block depth for dynamic
calculations;
actual superstructure depth from top of deck to the soffit;
column ultimate concrete strength;
yield strength of transverse (horizontal) shear reinforcement;
a shear strength parameter within the plastic hinge region;
L
Le
Pu
PR
P-DEAD
P-L+I
MJLT
sfzddz
Sh
SF
Vc
Vd
foundation horizontal spring constant;
column height (vertical distance);
equivalent superstructure block length for dynamic
calculations;
assumed column ultimate axial load at the time plastic
hinges form;
column shear demand to capacity performance ratio;
superstructure dead load reaction at the pier;
superstructure live load reaction (without impact) at the pier
reduced for multiple lane loading as per AASHTO 3.12;
ultimate reaction at the pier due to the superstructure
including 20% of the live load reaction;
scale factor for the z direction acceleration component in a
time-history record;
vertical spacing of transverse shear reinforcement;
dynamic scale factor, ie, a multiplier used to magnify the x,
y and z time-history acceleration input in order to determine
the capacity of each model for cost evaluation purposes;
nominal concrete shear strength;
dependable design shear strength;
xxiv


nominal shear strength provided by the axial load;
nominal shear strength provided by the transverse shear
reinforcement;
ultimate applied shear load;
horizontal distance between top and bottom column axial
load reaction points;
horizontal distance to the center of gravity of the circular
segment measured from its radius point;
foundation horizontal deflection limit;
35 in the calculation of Vs and the segment angle in the
calculation of Y for determining VP;
rotational ductility;
square feet of deck area used in cost evaluations.
xxv


1.
Introduction
1.1 Column Softening
Bridge bents for medium/large span bridges (100 to 200 feet)1 usually
consist of a cap beam supported by multiple columns or a single pier wall to resist
lateral loads transversely (in the plane of the bent) and longitudinally
(perpendicular to the plane of the bent). It can be said that pier walls are always
stiffer in the transverse direction because they act as shear walls in that direction.
Likewise, it can be said that multiple column bents are generally stiffer in the
transverse direction because the frame action in the plane of the bent is usually
more rigid than the longitudinal frame action. This paper is devoted to a two
column bent and ways to make it less rigid for the purpose of ducking AASHTO
Group VH seismic loads without damaging bridge components or adversely
affecting its ability to resist AASHTO Group I live loads.
Two softening methods will be compared against a standard rigid frame
design that is capable of developing plastic hinges. Many devices have been
created over the past few years to soften seismic excitation by either isolating the
structure from ground motions or dissipating absorbed seismic energy. Included
among these devices2 is laminated lead-rubber bearings, friction pendulum
bearings, steel hysteretic dampers, hydraulic dampers and lead-extrusion dampers.
With respect to softening devices, the marketplace shows that lead-rubber bearings
have taken the lead as far as medium span bridges are concerned. However, a new
alternative, dubbed the rocking hinge bearing by the author, is of academic interest
1


and may be of practical interest for resisting large seismic loads for those bridges
founded in rock. The development of this alternative is the subject of this thesis.
1.2 Research Objective
The rocking hinge alternative was first conceived as a moment limiting
device to prevent superstructure damage; ensuing thoughts focused on structure
stability during a seismic event. Of the two thoughts, the stability concern was the
more demanding issue that began to drive the project. A time history analysis is
required to reveal the true value of rocking hinges and the following objectives
were seen as required steps to meet this need:
Prepare a simple dynamic computer model using the average
acceleration method to compare the alternatives.
Examine existing literature to find appropriate hysteresis algorithms
for modeling plastic hinges and lead-rubber bearings.
Devise a simple and inexpensive way to determine the physical
characteristics of a rocking hinge bearing in order to create a
hysteresis algorithm for such a device.
Program all of the previously mentioned hysteresis models.
Use the computer program in the first step to compare the
performance of each alternative with large and small superstructures
founded in soft and hard soils on short and tall columns.
Check to see if the rocking hinge alternative is cost effective.
1.3 Engineering Significance
The results of this study show that the rocking hinge concept is a viable
2


alternative. Specifically, this effort shows that, in all cases, plastic hinges are
average performers, lead-rubber bearings work better in soft soils and rocking
hinge bearings are the better choice in hard soils. Additionally, none of the
alternatives performed well with short (less than 10 tall) columns. Accordingly,
this research shows how column height influences the dynamic response of a
structure and suggests that lead-rubber bearings function well in soft soils because
of their shear properties whereas rocking hinges function well in hard soils that are
capable of producing significant bending moments in the hinge areas of the
longitudinal and transverse frames. In other words, this work suggests that it may
be possible to categorize isolation devices for medium/large span bridges into
shearing and flexural systems for use in soft and hard soils respectively. This work
also shows that permanent sets may appear in any of the alternatives after heavy
usage with rocking hinge bearings showing less set than lead-rubber bearings.
1.4 Rocking Hinge Bearings
A rocking hinge bearing is a bearing device that takes advantage of rocking
motions3 to resist overturning moments. A typical 12 ounce canned beverage can
be used to demonstrate the rocking motion concept. Place the can upright on a
piece of paper resting on a level table. The paper should provide enough frictional
resistance to prevent the can from sliding under a static horizontal load applied by
hand at the top of the can. Note that the horizontal force required to displace the
top of the can drops from a maximum to zero with increasing displacements until
the can tips over by gravitational forces alone. A moment is created by the applied
horizontal force. This moment is equal to the applied horizontal force times the
distance from the table top to the top of the can. A plot of this moment (M) versus
the can rotation angle () is called a rocking plot and is shown in Figure 1.1(a).
3


the can rotation angle () is called a rocking plot and is shown in Figure 1.1(a).
Incipient rotational instability occurs at the point where M = 0. Negative values of
M indicate that the direction of the applied horizontal force must be reversed in
order to prevent the can from tipping over. In this demonstration, 0 ^ (J> ^ 90. If
the gravitational force (W) and horizontal force (P) are the only forces acting on
the can, then it will be on the verge of overturning when 4) reaches a value where
A, B and C line up vertically as shown in Figure 1.1(b).
Imagine that the can had a XA" diameter hollow tube installed on its
longitudinal axis to form a passageway through the can. Furthermore, imagine
that a strong rubber band was fastened to the top of the can and passed through the
can's hollow tube and a hole in the table top so as to firmly tie the can to the table
top. Again, assume that there is enough friction between the table top and the can
to prevent the can from sliding under lateral loading. The rubber band stretches as
the can rotates under horizontal loading creating another M/a diagram, ie, the
prestressing plot as shown in Figure 1.1(a). If the rubber band were strong enough,
the superposition of rocking and prestressing diagrams could create a total diagram
with a zero or positive slope as shown in Figure 1.1(a). The positive slope would
guarantee that gravity alone would never be able topple the can under lateral
loading. The author refers to this mechanism as a rocking hinge bearing. The
practical limits for (j) in this scenario are between 0 and perhaps 20 before small
angle assumptions are violated.
The rocking hinge considered herein is shown in Figure 1.2. It consists of a
mild steel disc sandwiched between two high strength steel base plates. The base
plates are capable of cold forming the disc during rocking. A hollow pintle keeps
all three elements from rolling out of position during rocking. Rolling is caused
4


when the seismically induced lateral force is not directed toward the center of the
bearing. By design, the prestressing creates a 1% positive slope for the M/<|>
diagram as shown in Figure 1.1(a) in order to: (1) avoid instability during rocking
and, (2) to limit the moment developed during rocking. The 1% slope was selected
to avoid precision problems in the computer modeling work.
The magnitude of column moments and shears during a seismic event can
be controlled by limiting the joint moments during rocking. Accordingly, the
maximum column body and cap beam stresses are established by the limiting
moments in the rocking hinges and are therefore independent of the magnitude of
the actual seismic event. Of course, the internal rocking hinge bearing stresses are
directly proportional to the size of the seismic event and will reach failure if the
event is large enough.
On the other hand, bent horizontal translations are dependent on the size of
the event and are directly related to the ductility (inelastic movement without
failure) of the rocking-hinge bearings. The ductility of a rocking hinge bearing is
controlled by the amount of rotation the joint can develop without crushing the disc
beyond its useful limits.
1.5 Object-Oriented Programming
The time-history analysis method was used to test the rocking hinge bearing
concept. A special column element was created with 7 springs in the plane of the
bent and 7 springs in a plane perpendicular to the bent to model the members in a
two column bridge bent. Different spring types were required in order to simulate
isolation bearings, plastic hinges and rocking hinges.
5


An object-oriented language (C++) was used to program this model in order
to demonstrate the power of this paradigm to overcome the coding complications
associated with the traditional structured programming approach. Specifically,
inheritance (incorporate root properties), polymorphism (similar but different) and
overloading (providing multiple services) are powerful object-oriented tools that
allows a user to ultimately perform useful tasks like processing a heterogenous list
of springs as was done in this project. With these tools, the author
created/debugged each spring type in one module and created the dynamic model
in another module. The code in the dynamic model knows nothing about the
workings of each spring type, ie, it simply calls each spring object into action with
the same call (without regard to spring type) and utilizes the results. In effect, the
data and the tools to manipulate the data are an integral package. Accordingly, the
dynamic program is easy for an engineer (as opposed to a programmer) to
understand and to follow. The code in the dynamic model does not require a
reader to become familiar with the workings of each spring type in order to
understand the dynamic model.
Another advantage for an object-oriented language lies in its use of
memory4. While each spring object has its own data, all springs of the same type
share the same functions. This is an efficient use of memory because the data
memory requirements for each spring type is much less than the memory
requirements for the executable functions. Accordingly, it may be possible to
create a large object-oriented frame program with lots of different spring types that
runs on a PC thus promoting the non-linear time-history modeling method.
6


M
M
M
(a) Moment Curvature Plots
(b) Can Stability Similitude
Figure 1.1 Rocking Hinge Bearing Concepts
7


Typ Column Elevation
t = Cap Beam Motion
Bearing
area during
rocking
Prestress duct
-.Base Plate
X Disc
Hollow Pintle
Bearing Plan View
Detail 2
Figure 1.2 Rocking Hinge Bearing Details
8


1.6
Chapter 1 References
1. Barker, R.M., Puckett, J.A., Design of Highway Bridges Based on
AASHTO LRFD Bridge Design Specifications, John Wiley & Sons,
New York, 1997, pp. 89 & 90.
2. Priestley, Sieble, F., and Calvi, G.M., Seismic Design and Retrofit
of Bridges, John Wiley & Sons, New York, 1996, pp. 466-483.
3. Ibid., p.516.
4. Lafore, R., Object-Oriented Programming in TurboC++, Waite Group
Press, Emeryville, California, 1991, p. 230.
9


2.
State of the Art
2.1 Brief History of Seismic Specifications
Prior to 1971, the AASHO Specifications for seismic design of bridges
were based in part on the lateral force requirements for buildings as developed by
the Structural Engineering Association of California. Accordingly, California
bridges were designed for a maximum lateral force equal to 6% of the
superstructure dead load in accordance with a 12 line paragraph in the 1969
AASHO bridge code. By comparison, it took over two pages to address the wind
load requirements in that same document. Shortly after the San Fernando event,
terms like hinge restrainers, plastic hinging and concerns about minimum seat
widths, natural frequency and the loss of bond at column splices began to appear.
Obviously, the 1971 San Fernando earthquake was a major turning point in the
development of seismic design criteria for bridges, and several developments
thereafter led to the current code provisions. Some of the more important
developments were1:
In 1973, the California Department of Transportation (Caltrans) introduced
new seismic design criteria for bridges. Design parameters included the
relationship (distance) of the site to the nearest active faults, the seismic
response of the site specific soils and the dynamic response characteristics
of the bridge being investigated.
In 1975, AASHTO adopted Interim Specifications which were a slightly
10


modified version of the 1973 Caltrans provisions.
In 1979, a "Workshop on Earthquake Resistance of Highway Bridges" was
conducted by the Applied Technology Council (ATC) in San Diego. The
workshop considered the latest seismic practices, problem areas in seismic
design, and current research efforts and findings. Its objective was to
facilitate the development of new and improved seismic design standards
for highway bridges.
In 1981, the Applied Technology Council published ATC-6, "Seismic
Design Guidelines for Highway Bridges". This was a state-of-the-art
document covering recommended practices for the seismic design of
bridges.
In 1983, AASHTO adopted the ATC-6 Report as an approved alternative
guide specification for the design of highway bridges.
In 1991, AASHTO included the alternative guide specification as
Supplement A in the Standard Specifications for Seismic Design of
Highway Bridges.
The current seismic design requirements recognize the social significance
of a structure with the importance factor2. Under this provision, there are two
classifications, ie, essential bridges and other bridges. For ground line
accelerations greater than 0.29g, the importance factor causes the designer to use
the more restrictive design requirements in Seismic Performance Category D in
order to attempt to keep the structure operational during and after an event.
11


Caltrans has found increasing pressure from the public to move to a no damage
tolerance3 in order to reduce the commercial impacts of a seismic event.
Apparently California residents want more structures to be classified as Category D
facilities even though there have been more deaths from traffic accidents than from
seismic attacks.
Dramatic changes in ground motion representation have been proposed for
the design of buildings4. The return periods for maximum-magnitude earthquakes
vary significantly throughout the United States, eg, 100 years in parts of California
to 100,000 years or more in other states. Accordingly, the National Earthquake
Hazard Reduction Program (NEHRP) recommends a deterministic approach as an
alternative to the present probabilistic approach. Their deterministic approach
calls for the use of reasonable maximum levels of earthquake ground shaking for
the design of structures. In addition to saving initial construction money, this
approach would result in similar levels of seismic safety for all structures.
At least eighteen "states" contain high seismic risk areas5, ie, California,
Washington, Oregon, Nevada, Alaska, South Carolina, New York, New England,
Puerto Rico, the states bordering the New Madrid fault in Illinois, Tennessee,
Missouri, Indiana, Kentucky, Arkansas, Mississippi and Utah. These states will
benefit from a better understanding of seismic issues and technology
improvements. At the present time, however, most of these states are struggling to
simply keep their inventory of bridges up to date with respect to stream scour and
congestion relief.
12


2.2 Plastic Hinges
Along with hinge restrainers, larger seat widths and foundation
improvements, the development of plastic hinges has made it possible to keep
California bridges standing during recent seismic events. During the 1994
Northridge earthquake, no significant damage was recorded for any of the local
retrofitted bridges even though 115 of them were in areas where the ground
motions exceeded 0.25g and 36 were in areas with ground motions in excess of
0.5g6.
The development of plastic hinges began with monotonic testing of
unreinforced concrete in 19647. This work showed that the unreinforced concrete
elastic modulus degraded with each cycle due to internal microcracking. The
ensuing work in the 1980's focused on the effects of confinement reinforcement
ending with a theoretical stress-strain model by Dr. Mander8 based on the equality
of strain energy between confinement reinforcement and the confined concrete
core. Testing of this land mark model9 revealed that (1), confinement works best
on round sections because hoop tension can be employed, and (2), it is possible to
increase the ductility of square and rectangular sections by use of ties with one bar
in each comer to confine a portion of the total cross section defined by the tie
diagonals as shown in figure 2.1. By increasing ps from a minimum of 0.006 to
0.020, the ultimate axial stress can be increased by about 50% and the ductility of
the confined core can almost be doubled. Testing also showed that the strain
energy of mild and high strength rebar ties were about equal favoring the high
strength rebar because it resulted in a smaller change in the volume of the confined
concrete core.
13


Confinement for round concrete flexural members has led to an increase in
the extreme fiber strain from 0.003 to 0.02 and higher10. This flexural strain
ductility is the basis for today's plastic hinges. Caltrans research has found an
additional use for confinement reinforcement, ie, to prevent the degradation of lap
splices by limiting changes in the volume of the confined core thus reducing
relative bar slippage11.
Unfortunately, there is a down side to increasing the flexural ductility of
concrete columns; shear strength within plastic end regions is reduced with
increasing flexural ductility12. The loss in shear strength is due to the loss in
aggregate interlock across wide cracks in the plastic hinge. Recent research
involving compression field theory13 may be able to better quantify this loss in
shear strength. Until then, the approach as described in section 6.3 must be used.
There is another downside to increased column ductility that involves the
cap beam to column connection for round columns. As previously pointed out,
round columns are more efficient than rectangular sections because they employ
hoop tension to create confinement. Accordingly, they can deliver large moments
to the cap beam they support. This has posed problems on past retrofitting projects
because it was not easy or was not possible to retrofit the cap beams to match the
capacity of the columns on some bridges. Also, on new monolithic construction, it
may be necessary to overdesign portions of the superstructure framing into the cap
beam in order to take the moments delivered by the columns during a large seismic
event.
One way to mitigate this problem is to install a governor at the cap beam to
column interface. Obviously, this device would also limit the amount of plastic
14


hinging action (including the shear demands) that takes place during a seismic
event.
2.3 Seismic Isolation
Despite its attenuation potential, the acceptance and usage of seismic
isolation systems have been slow to take root. In North America, the use of
seismic isolation began in 1983 when a commercial law building in Rancho
Cucamonga, California was isolated with high damping natural rubber. Since then,
some larger bridges in Europe have been isolated with viscous damping devices14.
In the late 1980's, Japan began a program to study the application of base isolation
technology15. Part of the study focused on the development of isolators and energy
dissipators for bridges. The work involved the services of 28 private firms
consisting of material manufacturers, consultants and contractors. The
development of high energy absorbing rubber bearings received more attention
than the other alternatives which included friction, steel and viscous dampers.
Later, the Ministry of Construction sanctioned the construction of seven bridges of
various lengths to test selected alternatives.
With the installation of square lead-rubber bearings similar to the round
one shown in Figure 2.2 on a bridge at the Sierra Point Overcrossing in 1985,
Caltrans became the first U.S. agency to use seismic isolation on a bridge.
However, the official policy at that time was to use ductility rather than isolation
for the protection of bridges. This policy was not questioned until the Loma Prieta
Earthquake in 198916. With the loss of lives and service due to the collapse of the
Cypress viaduct, a new policy was created to emphasize minimal damage on
important bridges for smaller more frequent earthquakes. This policy change
15


resulted in a performance specification that allowed different types of seismic
isolation and energy dissipation devices to be bid on the same job thus opening the
door for seismic isolation.
The first test of a seismic isolated bridge came in April of 1992 when the
Eel River Bridge in Humboldt County on Highway 101 at Robinson's Ferry near the
coast line in northern California was attacked by the Cape Mendocino Earthquake
measuring 6.4 on the Richter scale. Three 200' long trusses were isolated with
lead-rubber bearings and moved 8" longitudinally and 4" transversely during the
event and then recentered themselves when the shaking stopped without causing
any damage to the piers below the isolators17.
Caltrans has been working with the University of Califomia-San Diego to
test large friction pendulum bearings18 for possible use on some of their toll bridges
such as the Benicia-Martinez and San Francisco-Oakland Bay bridges. Testing is
being done to determine if the PTFE (polytetrafluorethylene) bearing material will
melt causing slip-stick action during an XYZ time history event. See Figure 2.3.
For medium and shorter large span bridges (say 250' or less), lead-rubber
bearings appear to be one of the most economical isolation alternatives. In this
system, a lead core is press-fit into a steel laminated elastomeric bearing. The
diameter of the lead core is about 25% of the bearing's least dimension19. Lead-
rubber bearings are relatively tall devices. Their height allows them to
accommodate large horizontal motions under seismic loading without overstressing
the elastomer. The lead core provides the necessary stiffness to handle
longitudinal live and transverse wind loads without excessive translations. The
only negative aspect to this combination is the fact that the stiffness of the lead
16


core could hinder the recentering process after a seismic event if the plastic
strength of the lead core is greater than the available righting forces.
17


Confinement of Square/Rectangular Sections
Figure 2.1 Confinement with Ties
18


Plan
[-D/4 + It,
Lead core
Elastomer
Steel reinforcement
disc (t 8 thick)
Section AA
Figure 2.2 Lead-Rubber Bearing
19


Figure 2.3 Friction Pendulum Bearing
Steel base
plates
20


2.4
Chapter 2 References
1. Load and Resistance Factor Design, Participant Notebook, Publication No.
FHWA HI-95-016, January 1995, page 8-5.
2. Division I-A, Seismic Design, AASHTO Standard Specifications for
Highway Bridges, 16th Edition, Section 3.3.
3. NHI Course No. 13061 lecture by Dr. Dennis R. Mertz, University of
Delaware, Denver Colorado, August 8,1995.
4. "Changing Representation of Ground Motion in Seismic Codes", Portland
Cement Association, 5420 Old Orchard Road, Skokie, Illinois 60077-1083,
News Letter PL278.01D, pages 3 & 4.
5. "Seismic Safety", Carter & Burgess Quarterly News Letter,
4th Quarter 1996, Fort Worth, Texas, page 22.
6. Seible, F., Priestley, M.J.N., Roberts, J.E., Transportation News 194,
January-February 1998, p. 16.
7. Sinka, B.P., Gerstle, K.H., Tulin, L.G., "Stress-Strain Relations for Concrete
under Cyclic Loading", ACI Journal, Proceedings Vol. 61, No. 2, February
1964, pp. 195-212.
8. Mander, J.B., Priestley, M.J.N., Park, R., "Theoretical Stress-Strain Model
for Confined Concrete", Journal of Structural Engineering, Vol. 114, No. 8,
August 1988, pp. 1804-1826.
9. Mander, J.B., Priestley, M.J.N., Park, R., "Observed Stress-Strain Behavior
of Confined Concrete", Journal of Structural Engineering, Vol. 114, No. 8,
August 1988, pp. 1827-1849.
10. Priestley, M.J.N., Seible, F., Calvi, G.M., "Seismic Design and Retrofit of
Bridges", New York, John Wiley & Sons, 1996, p. 294.
21


11. "Memo to Designers 20-4 Attachment B, Figure B-1", California
Department of Transportation, Sacramento California, April 6,1992.
12. Priestley, M.J.N., Seible, F., Calvi, G.M., p. 331.
13. Ibid., p. 332.
14. Byron F. West (Seismic Energy Products), interview by R.L. Osmun,
June 18, 1998.
15. "Proceedings of the First U.S.-Japan Workshop on Seismic Retrofit of
Bridges", December 1990, Public Research Institute, Tsukuba Science City,
Japan, pp. 29-35.
16. Yashinsky, M., "Recent Developments in Seismic Isolation at Caltrans",
California Department of Transportation, Office of Earthquake
Engineering, Sacramento California, 1992.
17. Personal letter from Dave Rogers, Director of Bridge Sales & Marketing
for Dynamic Isolation Systems Inc., to James Siebels, Chief Engineer for
the Colorado Department of Transportation, January 31, 1997.
18. Byron F. West interview.
19. Priestley, M.J.N., Seible, F., Calvi, G.M., pp. 469-471.
22


3. Rocking Hinge Bearings
3.1 Concept
Simply put, a rocking hinge bearing is a moment limiting governor as
shown in Figure 1.2. As explained in section 1.2, each end of a column can be
fitted with a rocking hinge bearing to create flexible (soft) column support element
that is capable of ducking large horizontal loads by limiting the maximum moment
and associated shear produced at each hinge by a horizontal ground acceleration
and its associated motion. To some extent, conservation of energy requires the
column relative horizontal motion to be increased when the column moments and
shears are decreased.
The mild steel disc is cold formed by the high strength base plates during
rocking. The shape of this interface is controlled by the high strength base plates
(stronger components) which remain within the elastic range at all times. Hence,
the shape of this interface will probably be linear (as was found in the physical
model testing presented in Chapter 6). Accordingly, the shape of the bearing area
is a circular segment defined by a secant and the disc perimeter as shown in Figure
3.1(b). The size of this area depends on the magnitude of the column reaction.
The bearing stress is zero on the secant (heel) and the cold formed area starts on a
line parallel to the secant at a point where the bearing stress equals the disc yield
stress and continues to the edge of the disc (toe) where the stress is at a maximum
level.
23


The entire disc area (minus the cold formed annulus conservatively
assuming 360 of seismic damage) supports the column reaction elastically up to
and beyond incipient rocking until the disc yield stress has been reached at the
extreme fiber location (toe). The bearing area is minimized when the hinge
rotation angle has been maximized. At this point, the stress at the edge of the disc
is assumed to be equal to the ultimate stress for the disc material.
As shown in Figure 1.1(a), a supplementary prestressing force can be used
to eliminate the instability associated with a pure rocking motion. Obviously, this
additional force increases the size of the required stress block.
3.2 Analysis
The required equations to size the disc are shown in Figure 3.1(b) and the
equations for determining the required prestressing are shown in Figure 3.2. The
disc equations assume that the ultimate axial load (Pu) is known. However, Pu
includes the prestressing force (Nps) which depends on Pu. Accordingly, the
design process is iterative and is best worked out in a spreadsheet. A few
guidelines are provided herein to help streamline the iterative process assuming
that subscripts i and j refer to incipient and full rocking, respectively:
1. ) Try to make a ductile hinge with Mj only slightly greater than the
controlling service load (unfactored) moment. Rocking performance to
duck seismic loads improves when Mj is minimized. Mj is due to all loads
except seismic loads, eg, live, thermal, creep, shrinkage and wind loads.
2. ) The strong variables are disc radius (R), disc thickness (T) and the initial
24


prestressing force (a0j(Y/L)EpsAps).
3. ) Use a small initial prestressing force, say 10% of the maximum applied
prestress force, to seat the joint. This force is controlled with the a term.
4. ) Rotational ductility is directly proportional to the disc thickness. A disc
thickness of about 2" appears to be sufficient for all scenarios modeled
herein.
5. ) Moment capacity is directly proportional to the disc radius. On the
downside, the magnitued of M; and the prestressing requirements also
increase with an increase in the disc radius Additionally, the disc radius
must be large enough to prevent the loss of bearing area due to the presence
of a pintle.
6. ) The required prestressing free length (L) as shown in Figure 3.2(a) depends
on the combination of applied moment (Mu), initial prestress ratio for
seating (a) and disc thickness (T), ie, L increases with an increase in Mu, a,
and T. Of the three variables, Mu has the most influence on the required
free length.
The envisioned installation utilizes rocking hinges at the top and bottom of
each column. Accordingly, the column prestressing is common to both bearings in
shorter columns (10' or less in height). Figure 3.3 shows the relationship between
column flexibility, hinge rotations and the prestressing ffee-length for shorter
columns. As can be seen, the required prestressing area and force are independent
of the number of rotating bearings. On the other hand, the ffee-length is directly
25


proportional to the number of rotating bearings.
3.3 Design
Rules of thumb to consider for the design of a rocking hinge bearing include:
1) Depend on shear friction between the base plates and their concrete
supports to transfer column shears. Accordingly, make the diameter of the
base plates about 1" smaller than the column diameter in order to engage
the column rebar. All of the column rebar should be attached to the base
plate to create a uniform resistance even though the axial load alone may be
able to provide enough normal load to satisfy the shear friction
requirements.
2) Provide a pintle to prevent the column from rolling on the disc. Size this
member for the horizontal spring capacity less the frictional resistance on
the steel interface assuming 0.15 for the coefficient of friction of steel on
steel.
3) Double the ffee-length (L) in Figure 3.2 for shorter columns (10' or less in
height) in order to account for the rotation in the top and bottom bearings as
shown in Figure 3.3.
The high strength base plates can be fabricated from A514 steel which is a
quenched and tempered alloy rolled in heats up to 6" thick. The specifications for
this material (in ksi.) call for 90 ^ Fy ^ 100 and 100 ^ Fu< 130 depending on the
actual rolled thickness. With a Brinell hardness of 235 min. to 293 max., this steel
26


should still be machinable after it has been hardened (quenched and tempered). If
not, arrangements must be made with the steel producer to allow the fabricator to
machine (cut, drill and tap) the pieces prior to hardening.
A36 steel can be used to fabricate discs. This material is a very ductile
carbon steel which is rolled in heats up to 8" and greater in thickness. The
specifications for this material (in ksi.) call for 32 ^ Fy ^ 36 and 58 ^ Fu 80
depending on the actual rolled thickness.
Pintles can be fabricated from AISI 1040 or 1045 hot rolled rounds. This
material is a medium carbon steel which is rolled in heats up to 24" in diameter.
The specifications for 1040 steel call for yield and ultimate stresses of 42 and 76
ksi. respectively as compared to 45 and 82 ksi. respectively for 1045 steel.
All of the above suggested materials are commercially available from
domestic producers.
Figure 3.4 outlines the design procedure for sizing the pintle and base
plates. This procedure requires the disc to be sized beforehand in order to establish
Pu and Aps. With this information in hand, the pintle inner diameter, pintle outer
diameter and base plate thickness are determined in that order.
Elastic properties (instead of a plastic properties) were used to size the base
plate thickness in order to create a stiffer member to help promote a linear strain
variation within the adjacent concrete. The AASHTO Standard Specifications does
not promote the load factor design (LFD) method for sizing bearing components
even though LFD is the most appropriate method for working with extreme events.
27


Accordingly, the following limiting stresses are recommended:
4>F, = 0.5FU for the weaker steel in bearing, ie, the pintle steel.
Fv = 0.5Fy for steel in shear.
cJ)Fc = 0.85FC' for concrete in compression.
The limiting stress for steel in shear is from the Tresca yield criterion1 and
the limiting bearing stress for steel components is similar to the philosophy in the
AASHTO Standard Specifications for the allowable bearing stress on pins subject
to rotations. The limiting concrete compressive stress is for flexure because the
base plate was designed to accommodate a linear stress-strain variation for the
Whitney stress block. The capacity reduction factor () can be taken as 1.0 for
extreme events as per Caltrans and LRFD recommendations.
The maximum shear across the pintle can be taken as the horizontal earth
spring capacity minus the shear friction between the base plate-to-disc interface, ie,
the horizontal earth spring capacity minus 0.15 times the unfactored column axial
load. ASTM A53 Grade B steel pipe may be used for determining the prestressing
duct outer diameter (pintle inner diameter).
Preliminary calculations show that large prestressing tendon will be
required. Accordingly, each tendon will be assembled with a large number of 0.6"
strands. This will precipitate the design of special prestressing anchorages to
accommodate the large tendons. However, normal prestressing jacks will work
because the initial prestressing is only about 10% of the prestressing capacity.
28


The column body must be capable of supporting the superstructure reaction
plus the maximum prestressing force. Rocking hinge bearing calculations show
required ultimate concrete strengths in the 5 to 9 ksi range. Accordingly, high
performance concrete may be required for some designs. This may not be a
problem if the columns are precast in a Prestressed Concrete Institute certified
shop.
Most states require that complicated bearings be replaceable. This
provision could be satisfied by calling for split discs that are banded together
during the installation. The use of water tight covers (boots) should be considered
in damp environments and some form of cathodic protection should be provided to
protect prestressing strand placed below the water table.
29


Base plate to disc interface
y = R-cosS
dy = R-sinS-dS ignoring the minus sign
f (Fu/c)e = m(R-cos8 d)
where m = Fu/c
Pu = 2j q (f)(R'sinS)-dy
= m-K2-R 3/3)-sin 30 + d-R 2-[(sin2S)/2 0]J
M u = 2J (f)(R-sin8)(R-cos8)-dy
= (2m-R 4/4)[sin 38-cos8 + 8/2 (sin28)/4]
- 2m-R3-d(sin38)/3
Y = M U/P u
8 1 = artan(T-e f /C)
(b) Triangular Stress Block Equations
M x = (2R3/3)sin38
A = [8 (sin28)/2]R2
ft. [8 1/2-(28 48 3/3)]R2
= 283R2/3 ------ 8 Y = Mx/A = (R-sin 38)/8 3
(c) Rectangular Stress Block Equations
Figure 3.1 Geometric and Stress Block Equations
30


(a) Mechanisim
(b) Rocking
(c) Prestress
Miscelaneous Definitions
m 0 = rocking slope after lift off Eps = prestress elasticity (28E3 ksi)
N0 = ultimate D+L+l reaction N pB = prestress force
from the superstructure N u = ultimate superstructure reaction
a = a fraction of the maximum plus prestress force
prestress force, ie, [(erYj/L)EpSApS] to "seat the hinge f = = required ultimate concrete strength for the concrete column
IT = disc plus base plate thickness
fyps = prestress yield (270 ksi) i. j = yield and ultimate states respectively
Determine L
= E [YjCH-aJaj rT-ejcos(8j/2)]/L
L = E p, [Y j (1+a)Q j IT- j cos(8 j/2)]/(0.8f ypS ) increase Yj as needed to
make L > 0
Determine Ac
_______________
1-01mo-ej Ap.Ep.YjlYj-flj IT-e j cos(8 /2)]/L
A pg 1.01m0-8j-L/jEpaYj[Yj-8j IT-e j cos(8 j/2)]j
Equations for N psi and N pgj-
Npai = Ap.Ep.[Y,(aej+8l) JT-e, cos(8 ,/2)]/L
Npsj = A peEpjY; (1+)8j 5T-ejcos(8j/2)]/L
Equations for Mj, fc and N
Mj = M, m 0 (8 j 8 ,) + NrtYj
fc = Nu/(0.20ttD2/4)
Nu = N0+ N pBj
Figure 3.2 Prestressing Equations
31


length
(a) Flexible Column
(b) Rigid Body Column
p
Pj = A s [28R/(2L)]E pj = A [0R/L]Eps
(c) Prestress Force Rotation Plot
Figure 3.3 Prestressing Free-Lengths for Shorter Columns
32


o > z > o > o
I = Prestress duct O.D. = Pintle I.D.
p8 = Area of prestress tendon from
disc design
lpe = Diameter of individual strand
)DS = Area of individual prestressing
strand
sqr
I
ipa
(a) Pintle I.D.
VH = Column shear force (horizontal)
Av = Pintle shear area
A = Pintle bearing area
Fy = Pintle yield stress
Fu = Base plate ultimate stress
V h ^ 0.20P u
Av = Vh/(0.50Fy) = tt(D2-D?)A ----------- D0 = V(4Av)A + of
AX = Vh/(0.50FU) = T-D0 ------ T = Aj_/D
(b) Pintle O.D. and Base Plate Thickness for Bearing
M
s
D
A
P
T
Y
Y
bp
bp
o
e
u
c
F
ys
M
S
A
S
Y
F
bp
bp
c
y
Base plate moment
Base plate section modulus
Pintle O.D.
Concrete compression area
Ultimate axial load
Base plate thickness
C.G. concrete stress block
C.G. steel stress block from
disc design
Base plate yield strength
PJYc-YJ
(D-De)T2/6
Pu/(0.85f8)
[3Ac/(2R2)11/3
(R-sin 38)/S ^
MbpAbp------
VCSM^VKD-DoDFy]
(c) Bose Plate Thickness for Moment
Figure 3.4 Pintle and Base Plate Design Procedure
33


3.4
Chapter 3 References
1. Boresi, A.P., Schmidt, R.J., Sidebottom, O.M., "Advanced Mechanics of
Materials", New York, John Wiley & Sons, 1993, p. 131.
34


4. Time-History and Static Pushover Analysis
4.1 Quasistatic Analysis and the Pushover Model
The quasistatic analysis includes substitute structure, linear elastic and
pushover techniques for the seismic analysis of columns, bents and frames1. Linear
elastic models are important because linear elastic programs are prevalent.
The substitute structure method uses the secant stiffness approach as shown
in Figure 4.1. In this scenario, the user assumes a secant stiffness (k) on a
nonlinear structure specific load-deflection curve. The assumed stiffness is used to
(1), determine the structure deflection (u) and (2), to determine the anticipated
seismic force (Q) from an acceleration response spectrum (ARS curve) for the
foundation material. Accordingly, this approach is iterative. The user assumes a
trial stiffness and changes it until all three variables (k, u & Q) are consistent.
Caltrans uses this method for the design of hinge restrainers2.
For bents and frames, the linear elastic and pushover methods are used in
concert to determine the provided ductility for comparison with the required
ductility as shown in Figure 4.2. The required ductility is a function of the
fundamental period for the bent or frame and the supporting soils. The
fundamental period is used in conjunction with an appropriate acceleration
response spectrum (ARS curve) to determine the maximum horizontal acceleration
and associated horizontal seismic force (Q) to be applied to a linear elastic version
of the bent or frame. The horizontal seismic force (Q), the yield deflection (Ay)
35


and its associated force (P) from the linear elastic curve for the structure are
plugged into the appropriate ductility displacement equation to establish the
required ductility (Au). The bent or frame is said to have adequate seismic
ductility if the provided ductility (Au in the Push Plot) is greater than the required
ductility. In this paradigm, the entire bridge is deemed to be seismically sound if
each bent has adequate ductility.
There are two ductility displacement equations for determining the required
ductility (Au). One equation is based on the principle of equal displacements and
the other is based on the principle of equal energy. As previously explained, the
required ductility can be calculated for a given Q, Ay and P. Caltrans has
determined that the selection of the appropriate ductility displacement equation
depends on the fundamental period for the structure at hand3, ie, the equal energy
relationship is valid for those structures with a fundamental period less than 0.7
seconds and the equal displacement relationship is valid for structures with a
period of 0.7 seconds or more. Au2 in the equal energy principle is found by
equating areas 0-A-C-Au2 and O-A-B-Au!.
In the pushover method, the bent or frame is deflected (pushed) to one side
one incremental movement at a time until the frame becomes unstable and
collapses as shown in the Push Plot of Figure 4.2(b). The structure softens as
plastic hinges form and other springs break until a collapse mechanism develops
thus allowing the P-A effect to take over. The P-A effect is a recursive
phenomenon involving a series solution that produces magnified column moments
starting with an initial eccentricity in the axial load that creates additional
eccentricity (and additional column moments) that will converge (stabilize) as long
as the applied axial load is less than the Euler critical buckling load (Pcr). The
36


provided ductility at the point of collapse is Au and its value must be greater than
the required ductility.
The push frame for this project consists of a two-column bent in the xz-
plane and a T-frame in the yz-plane as shown in Figure 4.3. Each column has
seven springs in each plane:
One horizontal spring and two rotational springs at the top of the column.
The rotational springs are in series. One rotational spring can be used to
model the MAj) relationship for a support bearing (KRST) and the other
rotational spring (KRPT) can be used to model the column body plastic
hinge.
One horizontal spring and three rotational springs at the bottom of the
column. The rotational springs are in series. Two of the rotational springs
can be used to model the MAJ> relationships for the support bearing (KRSB)
and the footing (KRFB) and the third rotational spring (KRPB) can be used
to model the column body plastic hinge.
Translations produce different rotations for each column as shown in Figure
4.3(b) and (c). Each push is in a two-dimensional plane meaning that torsion is not
considered. The superstructure is assumed to be unbreakable.
Figure 4.4 shows the program logic for the push analysis used herein based
upon the geometry and variables defined in Figure 4.3. As noted, the do-while loop
continues until the structure becomes unstable, ie, until the frame no longer has
sufficient moment capacity to resist the P-A moment. The term block refers to the
superstructure mass which can be visualized as a solid block of concrete which
37


tends to rotate as it translates.
An ARS curve is an elastic response spectra created with the product of
three terms4:
"A", the anticipated maximum credible peak acceleration in bedrock.
"R", the anticipated acceleration response spectra in bedrock.
"S", the anticipated soil amplification factor for the overburden material.
Figure 4.2(c) shows the general shape of the ARS curves for soft (Bay mud) and
firm (rock like) foundations5.
The pushover method was developed as an alternative to the use of stress
reduction factors (Z or R factors) which are based on past experience as opposed to
strength estimates as described herein. The pushover routines developed in this
thesis were not intended for use to evaluate the acceptability of rocking hinges or
to compare the quasistatic methods with the time history method. Instead, the
pushover results were used to assess the ductility of a structural configuration for
comparison with the maximum achieved ductility in a time history analysis.
4.2 Numerical Response Methods and the Dynamic Model
It is not possible to use closed-form methods to analyze a multiple-degree-
of-freedom (MDOF) model that is subjected to complex excitations such as those
from a time history record. Instead, a method involving interpolation of the
excitation (for use in the Duhamel integral) or a method involving approximation
of the derivatives in the differential equations for motion must be used. Both
38


methods are applicable to linear systems; only the second method (approximation
of derivatives) is applicable to nonlinear systems.
One of the most useful methods for approximation of derivatives is the
average acceleration method'. In this method, the user must know the size of the
time step, the previous velocity and the previous acceleration in order to calculate
the change in position for a given excitation in the next time step. This method
was used to analyze the two span bridge as shown in the Model Configuration of
Figure 4.5(a). As can be seen, the superstructure is an object with mass M
supported on three bents. Horizontal restraint is provided by the center bent; all
three bents provide vertical restraint. The three view planes reveal six degrees of
freedom, ie, three translational and three rotational degrees of freedom.
The dynamic models used to develop the required motion equations for use
in the average acceleration method are presented in Figures 4.5(b) and 4.6. The
rotational and translational springs in planes 1 and 2 depend on the column element
flexural and shear properties as shown in Figure 4.7. The column element includes
the P-A effect as per the Stodola-Vianello approach starting with the first order
terms in Figure 4.8. The increased moment from these terms was included in the
column element relationships as noted in figure 4.7(b). Then, the required
constants to reflect the actual shape of the elastic curve were found by iterating on
Por for fixed-fixed and fixed-pinned columns with different stiffness and geometric
configurations. The constants from this exercise were used in the genconst(...)
function which can be found in the SPRINGS.CPP file. The comments in
genconst(...) show that the initial assumed constants were within 15% of the final
iteration results that were needed to predict Pcr with different column
configurations.
39


The equations in Figure 4.5 were solved simultaneously for the motions in
plane 1. The equations in planes 2 and 3 are coupled by springs k8 and k9 as shown
in Figure 4.6 which means that the related equations in these two planes had to be
solved simultaneously. For the column element in Figure 4.7, the moments and
shears from incremental rotations and translations can be combined by
superposition to form a set of equations that could be solved simultaneously.
The simultaneous solutions described above result in fast running in-line
code that only applies to the structural configuration shown in Figure 4.5.
Although a matrix solution would have been more elegant and versatile, it would
not have been fast running because an entirely new solution is required for each
time step and matrix code takes time to run. Structural versatility beyond what has
already been described was not important on this project. On the other hand, run
time was important because iteration was required to scale the seismic record in
order to find a system's maximum capability.
To address nonlinearity, the value of each spring constant must be
calculated based upon the known deformation at the start of a new time step. For
those springs in series, the results of this exercise can trigger a balancing exercise
if any of the spring constants have changed since the last time they were used. The
following equations can be used to balance the forces in a series of springs:
F^ZCF/lqySO/lq) (4.1)
Axi = (F-Fi)/ki (4.2)
Xi = Xi+ Axj (4.3)
The equations are executed in the indicated order. With F known from equation
40


4.1, the required incremental motion (positive or negative) to balance each spring
is determined with equation 4.2. Then, each spring's total relative movement is
determined with equation 4.3. Because this is a balancing exercise, the sum of Ax,
for the springs in series will be zero as per the derivation for equation 4.1.
The damping coefficients for the rotational terms can be determined with
equation 4.6:
Io = lx + I* + ly or Iy + \ as appropriate (4.4)
n o 3 HH II 04 (4.5)
Cr = 2 £m[E k/(mR2)]0 5 (4.6)
s II o >> (4.7)
The bridge superstructure can be converted to a solid mass that is Be wide,
De deep and Le long with the following equations:
De = [144(IJ3/U1/8 (4.8)
Be=12Ix/D3 (4.9)
Le AaLc/BeDe (4.10)
In equations 4.8 and 4.9, the inertias are determined with the actual section
geometry.
Caltrans has determined that the torsional moments of inertia for bridge
columns is only 20% effective in a seismic event because of the large cracks that
form in plastic hinge areas. Accordingly, the torsional properties for the columns
in this work were divided by five even though the selected geometry did not induce
41


torsion in the columns.
4.3 Hysteresis Models
Push programs require good spring models that properly address
nonlinearity. In addition to addressing the nonlinearity issue, the springs for time
history programs must address hysteresis in order to create realistic dynamic
models.
For reinforced concrete members, many hysteresis models have been
created starting in 19667. Among these are the Takeda, elastoplastic, bilinear,
Clough and Q-hyst models. Of these, the Takeda model was deemed to be the
most accurate and perhaps the most difficult to implement. On the other hand, the
Q-hyst model was found to be almost as accurate as the Takeda model but required
only four simple rules to implement8. An accurate even simpler model called the
Pivot Hysteresis model was developed and tested with the fiber model (a matrix of
line elements) in 1996.
The pivot model was selected for use in the preparation of this thesis as
shown in Figure 4.9. The three rules for construction of the hysteresis plot deal
with the creation of boundary and secondary lines. The three rules are:
1. For points in quadrants 1 and 3, loading and unloading in quadrant n
is directed away from or toward point Pn.
2. For points in quadrants 2 and 4, loading in quadrant n is directed
toward point PPn.
3. For points in quadrants 2 and 4, unloading in quadrant n is directed
42


away from point Pn.
Excursions beyond points PE and PJ result in failure of the column section.
Boundary lines are altered when points PD and PI are surpassed. Pinching points
PP2 and PP4 along with pivot points PI thru P4 are used to establish boundary and
secondary lines. Direction arrows show all motion possibilities. Pivot points are
tied to quadrants, ie, pivot points PI and P2 are used for creating lines in quadrants
1 and 2 while pivot points P3 and P4 are used for creating lines in quadrants 3 and
4. Where strength degradation due to splice failures etc. is not a factor, the
coordinates for pivot points PI and P2 are identical as are the coordinates for pivot
points P3 and P4. Strength degradation should not be a factor in modem columns
with proper lateral confinement. Also, none of the aforementioned hysteresis
models (including the pivot model) for reinforced concrete columns are capable of
dealing with a changing axial load due to the vertical accelerations in a time-
history analysis. Accordingly, this is a source of error for the modeling work done
herein. Although not done, the use of a larger axial load could have been
considered to develop the hysteresis parameters for an assumed maximum axial
load produced during a seismic event.
Figure 4.10 shows the geometric relationships for elastic, rocking and
bilinear springs. Again, the direction arrows show all motion possibilities for each
line. In the rocking and bilinear plots, excursions beyond points C, E and G will
result in failure of the column section. Likewise, absolute excursions beyond
ULIM will result in failure of the column section for an elastic spring.
The hysteresis algorithm for rocking hinge bearings is presented in Figure
4.10(b) and was developed with the physical model as described in Section 5.3.
43


Basically, rocking hinge hysteresis can be described with slopes m2 and m4 which
depend on the p factor to locate the pinching points D and H. The slope of m2
flattens with increasing positive rotations as does the slope of m4 as the joint
rotation becomes more negative. Obviously, the hysteresis effects increase as the
slopes of lines 2 and 4 flatten out. Independent TBENT.EXE runs showed that the
performance of rocking hinge bearings was not greatly influenced by the
magnitude of p.
One common algorithm can be used to program all four springs. First, start
at an extreme positive point on each line and unload the section until the extreme
negative failure point has been reached. Then, from an extreme negative point on
each line type, load the section until the extreme positive failure point has been
reached. The coding to accomplish these exercises results in a simple series of
nested decision statements to allow the process to stop short of the failure point as
needed.
Pivot springs were used to model the plastic hinges at the top and bottom of
each column. The program COL604N.EXE as developed by the Caltrans Special
Analysis Section was used to obtain the required plastic hinge parameters.
Rocking springs were used to model the rocking hinges at the top and bottom of
each column. The bilinear spring was used to model lead-rubber bearings with the
aid of the design parameters in Chapter 14 of the AASHTO Standard
Specifications (14th edition with 1991 interim) assuming a lead plug with a
diameter equal to 25% of the bearing's outer diameter10. The safety factor check
for the interaction of simultaneous shear strains due to rotation, axial load and
horizontal translation was ignored in order to simplify the work, ie, the primary
objective was to determine the stiffness characteristics of lead-rubber bearings as
44


opposed to their true capacity. Elastic springs were used to model all of the soil
springs with input from Chapter 13 of NAVFAC DM-7 and Caltrans field
observations11.
4.4 Computer Program
The program TBENT.EXE is an object-oriented C++ program that was
prepared for this project to compare the proposed isolation system involving
rocking hinge bearings with (1), the most popular isolation system involving lead-
rubber bearings and (2), properly confined modem columns capable of developing
plastic hinges. The C language is case sensitive and it is customary practice to
reserve the use of capital letters for the declaration of constants; functions and
variable names are always written in lower case letters. Common file extensions
include CPP for the source code, H for the header files, OBJ for compiled source
code (object files) and EXE for executable files.
TBENT was written as a multi-file program in order to clarify the coding by
grouping common functions. Specifically, spring objects were filed in
SPRING. CPP for use by the calling functions in TBENT. CPP. Two header files
were used to group needed declarations by purpose, ie, COMMON.H and
SPRINGS.H. All of the header and CPP files for TBENT can be found in
Appendix A.
4.4.1 Header Files
Both cpp files include COMMON.H and SPRINGS.H. The necessary
include directives to obtain the required library functions are found in
45


COMMON.H along with the global constants, preprocessor definitions and
prototypes for the overloaded data gathering functions that are used by both cpp
files.
The SPRINGS.H file contains all of the prototype information for the spring
classes and functions. This file makes it easy to see the I/O requirements, use of
private and public variables and the inheritance record. In general, header files
allow the user to see what a function does without the clutter associated with how
it does it. The reader must see the cpp files to find out how a function actually
works.
As can be seen in SPRINGS.H, elastic, pivot, bilinear and rocking springs
are all derived from the rather simple base class called spring. This relationship is
called inheritance and it allows a program to process a heterogenous list of springs
with the same calling function. Hence, to the TBENT.CPP file, "a spring is a
spring" and it can ask any spring to provide the spring's current stiffness and force
with the same call, ie, with getspringkp(...). By examining the list of public
functions for each of the four spring class (including the base class), the reader can
verify the presence of getspringkp(...) in each of them.
4.4.2 Spring Object File
The code for all of the prototypes and forward declarations presented in
SPRINGS.H appear in SPRINGS.CPP. As explained in section 4.3, many nested
decision statements appear in the code for getsrpingkp(...) in the pivot, bilinear and
rocking springs. Each instance of getspringkp(...) returns an integer value to the
calling function to let it know if the spring is broken. A "0" is returned if the spring
46


is not broken and a "1" is returned if the spring limit has been exceeded according
to the springs hysteresis rules. The getspringkp(...) function for a broken spring
will return a "1" and near zero values for the stiffness and force each time it is
called.
Almost half of the code in SPRINGS.CPP is devoted to the column
element. Of special interest is genconst(...) because most of the needed
information relative to the columns themselves is generated by this function.
Equations 4.1,4.2 and 4.3 were used in genconst(...) to balance the rotational mix
for springs in series (compound springs). As explained in Section 4.2, this
adjustment disturbes equilibrium conditions for the column. Accordingly, the
associated shears must be calculated with the adjusted moments in order to regain
equilibrium. This adjustment process renders the column elements useless for
tracking the current position of the superstructure block. Accordingly, this
information must be supplied by the dynamic routines. It is important to note that
the entire frame is always in static equilibrium even though the column elements
are not able to track the frames current geometric configuration. Testing with a
linear-elastic model shows that the final position of the superstructure block is
accurate to within 3% of the correct position according to the quotient of the final
position divided by the absolute value of the maximum excursion.
"Go" factors like gortx, gohbx etc. were used as broken spring detection
flags. These flags stop the program when a column becomes unstable due to the
loss of a horizontal spring or both top and bottom compound rotational springs.
For the run mode (as opposed to the push mode), genconst(...) makes interaction
checks for the combined moments, shears and deflections from the xz and yz
planes using an assumed elliptical failure criterion12 as shown in Figure 4.11. The
47


program stops when a failure is detected and informs the user as to why execution
was terminated.
4.4.3 Analysis File
Program action begins with main(...) at the end of TBENT.CPP. The
program runs in four different modes and is totally dependent on the spring file
(SPRINGS.CPP) for spring information. Calls to testspring(...) (S mode),
showtestcol(...) (C mode), runpushanal(...) (P mode) and runproblem(...) (R mode)
are made ffom within main(...) according to the mode found in the input file. In
turn, these functions call other functions to do their work. The mode information
is the first entry on the second line of the input file following a mandatory
comment line. The program checks for the proper data based on the mode it finds.
The flowchart in Figure 4.12 shows the program logic and some key functions.
Utility functions like getdatetime(...), startinput(...), postbanner(...) etc. are
located in the front part of TBENT.CPP. The function getspring(...) creates the
required springs according to a set numbering scheme, ie, 1 = elastic, 2 =
elastoplastic, 3 = pivot, 4 = bilinear and 5 = rocking. Accordingly, if getspring(...)
finds a 3, it will create a pivot spring and if it finds a 4 it will create a bilinear
spring etc. The elastoplastic spring was an early development and has been
replaced by the bilinear spring which was constructed with the logic that was
presented in section 4.3. The purpose of each of the key functions presented in
Figure 4.12 is:
getspring(.) Gets all properties for a spring from the input file and
creates a spring object.
48


getsprings(...)
getmovements(...)
testspring(...)
getcolparam(...)
getxymovements(...
genconst(...)
getpushparam(...)
runpushanal(...)
runproblem(...)
deldispl(...)
newdispl(...)
newvel(...)
Gets column springs for each column with repeated
calls to getspring(...).
Gets a list of positions to test a spring from the input
file.
Exercises the spring being tested with the list of
positions from getmovements(...).
Gets the column height (h), stiffness (ei) and
allowable superstructure drift (adrft) from the input
file.
Get a list of positions from the input file to test a
column element.
Solves for internal and external column spring forces
and deflections.
Gets the column spacing (xdist) and the
superstructure bent reaction (Pz) from the input file.
Independently pushes the frame in the x and y
directions 0.01" at a time until each frame collapses
due to instability caused by spring failures. The
applied force and frame deflection are printed out at
the point where each spring fails.
Runs a for loop to process the time-history data with
the average acceleration method.
Calculates the superstructure change in position for
each time step.
Updates the superstructure horizontal, vertical and
rotational position.
Updates the superstructure horizontal, vertical and
49


rotational velocities.
sksf(...) Gets the current stiffness and force (moment) for
each spring acting on the superstructure.
dcfdmd(...) Gets the current damping coefficients, forces and
moments acting on the superstructure.
newaccel(...) Updates the superstructure horizontal, vertical and
rotational accelerations.
4.4.4 Input Files
Sample input files appear in Figures 4.13 through 4.16. The first line in
each file is a comments line and the second line flags the intended purpose to the
program. Other details of interest include:
Figure 4.13 shows four spring test input files to demonstrate the testing of
spring types 1,3,4 and 5 as depicted in Figures 4.9 and 4.10. The third line
in each file is explained by the comments appearing after the semicolon on
each line. The x-direction movements appear on the fourth and ensuing
lines and the output is the y-axis responses for said input.
Figure 4.14 shows two column test input files to demonstrate the testing of
a fixed-fixed and a fixed-pinned column respectively. The comments
identify each spring etc. As before, the x-direction movements appear after
the column parameters and the output is the y-direction responses for said
input.
Figure 4.15 shows the input file for a push analysis. The blank line near the
50


end of the file tells signals the end of the input data to the program. The
required information for each spring is explained in the order provided in
Figures 4.9 and 4.10.
Figure 4.16 shows the input file for a dynamic run. As before, the
comments identify the required input which is followed by the time-history
record in a "time, x-acceleration, y-acceleration and z-acceleration" format.
4.4.5 Program Verification
A 0.01 second duration pulse load with a magnitude of l.Og provides a very
effective way to test for stability and compliance with the provided damping.
However, the pulse load alone can not be used to evaluate program stability when
testing a structural configuration that is on the verge of collapsing. In such cases,
the magnitude of the strength building parameters must be exercised. If the
program is stable, each strength building parameter acting alone will be able to
regain structural stability. TBENT.EXE was tested in this manner and found to be
stable.
51


Load Deflection Plot ARS Curve
Figure 4.1 Secant Stiffness Method
52


(a) Ductility Displacement Principles
(W = Structure weight)
(b) Push Plot (c) ARS Curves
Figure 4.2 Ductility Principles, Push Plot & ARS Curves
53


AX
AY.
XZ Plan
YZ Plane
(a) Motions Produced by Horizontal Displacements
I h 9 (1 -cosA m )
' 2 2f h, (1cofiA 1)
A 8 = ar3in(AX/h 1 )
A 8 2 = arsin(A X/h 2 )
A 8 = artanj[(h2(1 cosAS
h 1 (1 cosAS
(b) XZ Plane Rotation Due to AX Translation
SPRING XZ Plane YZ Plane
Col. 1 Col. 2 Col. 1 Col. 2

Horizontal Shear K| k2 kb K
Vertical Shear *4 KS K KS

Bottom Rotation KRBX1 KRBX2 KRBY1 KRBY2
Top Rotation KRTX1 KRTX2 KRTY1 KRTY2
Shear opring properties depend on KHT and KHB
KRB = (KRSB*KRPBKRFB)/(KRSB-KRPB + KRSB'KRFB + KRPB-KRFB)
KKT = (KRSr-KRFO/(KRSr + KRPT}
(d) Spring Constants
(c) XZ Plane Translation Due to
Foundation Rotation____________________
t = Location for equal vertical stiffness
Figure 4.3 Push Model
54


XZ Plane Push YZ Plane Push
INITIALIZATION: INITIALIZATION:
X = 0 AX = 0.01" R = V(h, + h2)* + X* 80 = arcos[(X/2)/R] AS = 0 = Block Rotation Y = 0 AY = 0.01" AS = 0 = Block Rotation
DO DO
Push frame to right AX (pure translation) Push frame to right AY (pure translation)
Relax block and determine final restraining forces due to associated block rotation Relax block and determine final restraining forces due to associated block rotation
Determine AXR component of AX due to block rotation produced by the vertical support springs
X = X + AX Y = Y + AY
WHILE STRUCTURE IS STABLE WHILE STRUCTURE IS STABLE
Figure 4.4 Push Model Logic
55


Span 1_

, Span 2 i
Superstructure ~1
(Mura ^
- Columns 1 4c 2
Plane 2
(YZ Plane)

* Superstructure
Block (Mara "m')
(XT Plano)
Superstructure
Block (Mass "m")
Plane 1
(XZ Plano)
(a)
Model Configuration
wx= zx ux
wz= zz- uz
ux= zx wx
uz= zz- wz
IF x = 0 yields:
m-Attx + CX-AWX+ (k,+ k2)AWx- (k,+ k2)Z-A8 = m-A2x
IFZ = 0 yields:
m-AWz+ CZ-AWZ+ (k4 + k5)AWz (k4- k5)(X/2)A8, = m-A2z
ZMy= 0 yields:
Crl-R?-AS1+ [(k,+ k2)Z2+ (k4 + k5)(X 2/4) + (k6+ k7)A81
-Ck,+ k2)Z'AWx (k4- k5)(X/2)AWz = -lm1 AS ,
(b) Plane 1 Equations of Motion
Figure 4.5 -Model Config. & Plane 1 Equns. of Motion
56



Kg

Hu.
y


K to 11 1Z
\\\W
(Span 1)
(Span 2)
C.G. Superstructure
Block (Mass "m)
WY = ZY- UY--------- UY = ZY- WY
K 12 = 3E0xiAl+ ,)a/,-2)
-i ZFy= 0 yields:
m-AWY+ CY-AlVY+ (Kb+ K9)AWy- (K8+ Kg)AQ2Z = m-A2Y
+) ZM x = 0 yields:
Cr2-Rl-AS2+ (Ka + Kg)AS2Z2+ (K10+ K,,+ K12)AS2
- (K8+ Kg)Z-AWY = -lm2A02
(a) Plane 2 Equations of Motion
Superstructure
Block (Mass m)
WY = ZY- UY---------UY = Zy WY
IFy= 0 yields:
m-AWY+ Cy-A*y+ (Kb+ KB)AWY (KB- K9)(X/2)A03 = m-A2Y
+) IM 2 = 0 yields:
C r3 'RI 'A 6 3 (K K8)(X/2)AWY
+ [(Kb+ Kg)(X2/4) + 0 (b) Plane 3 Equations of Motion
Figure 4.6 Plane 2 & 3 Equns. of Motion
57


VvVvWv
Superstructure block
*)
AST= Known Input
K FTT (K ST "K Pr)/(KST+K ft)
K RB = (K SB K PB K FB )/(K PB "K FB +K SB 'K FB SB PB )
A M H a-A M T
a = (K RB-h)/(2K RB-h6E-I) where 0 £ ot & 0.5
AMT = AST/[l/KRr+0/3-a/6)h/(E-l)]
AV = (1+a)AMT/h
(a) Pure Rotation Relationships
AW = AZ AU = known input
AC = [AV-h/3 AMB/2 + AMT-E-l/(h-K^/(E-l/h 2 P/4)
where P/4 includes first order PA effects
= AW AV(1/KHB + :/Km)
AV = 2AMB[E-l/(KRB-h2) + 1/h] Ac(2P)/(3h) AMT(2E-I)/(K rt -h2)
where (2P)/(3h) includes first order PA effects
= (AMt + AMb P-A c)/h
(b) Pure Translation Relationships
t = Joint relief
Figure 4.7 Column Element Relationships
58


e = A (Y/H) 2
<5 = [P-A/(E-l)]Jg(Y/H)2Y-dY = P-A-H 2/(4E-l)
e = [P-A/(E-l)]jgCY/H)2dY = P-A -H/(3E-I)
Figure 4.8 First Order P-A Terms


NOTES
1. Boundary (heavy) lines connect as shown.
2. Secondary (light) lines operate within the boundary lines.
3. Arrows demonstrate possible movements.
4. Subscripts identity applicable quadrants.
5. ALR = P u/Axial_Load_Capacity; p = 100Ae/Ag.
Figure 4.9 Pivot Spring
60


Y
(a) Bilinear Spring
Y
Figure 4.10 Bilinear, Rocking and Elastic Springs
61


Fy
Figure 4.11 Elliptical Failure Criterion
62


getmode(...)
Figure 4.12 TBENT.CPP Flowchart
63


Figure 4.13 Four Sample Spring Test Input Files
tbent.cpp data for testing one spring with testspringO
spring
1 2000 1.75 ;spr_type_l: ki & ulim
.5 1 2
1.5 0 -.5 -1
-2 -1.5 0
tbent.cpp data for testing one spring with testspringO
spring
3 3,9 18,14 1.6667, 0.6667 ;spr_type_3: (Fyx,Fyy), (Fux,Fuy), alpha & beta
10 4 5 13 IB
tbent.cpp data for testing one spring with testspringO
spring
4 2,4 10,5 ;spr_type_4: (Fyx,Fyy) & (Fux,Fuy)
4.2 0 4.2 10
tbent.cpp data for testing one spring with testspringO
spring
5 2,4 10,5 1
10 6 2 0 -2 -10
-10 -6 2 10 2 0
I
;spr_type_5: (Fyx,Fyy), (Fux,Fuy) & beta


Figure 4.14 Two Sample Column Test Input Files
tbent.cpp data for testing a column element with DEBUGCOLEL defined
column ;fixed- fixed column
1 1E20 1.5000 ;khtx
1 1E20 2.9900 ;khbx
1 1E20 1.0007 ;krstx
1 1E20 1.0007 ;krptx
1 1E20 1.0007 ;krsbx
1 1E20 1.0007 ;krpbx
1 1E20 1.0007 ;krfbx
1 1E20 1.5000 ;khty
1 1E20 1.9900 ;khby
1 1E20 1.0007 ;krsty
1 1E20 1.0007 ;krpty
1 1E20 1.0007 ;krsby
1 1E20 1.0007 ;krpby
1 1E20 1.0007 ;krfby
240 1EB 36 ;L, El £ drift_limit
0.00 0.00 0 0 17135, 0.10 0.10 0 0 17135, 0.10 0.10 0 0 17135, 0.10 0.10 0 0 17135
0.10 0.10 0 0 17135, 0.10 0.10 0 0 17135, 0.10 0.10 0 0 17135, 0.10 0.10 0 0 17135
0.10 0.10 0 0 17135, 0.10 0.10 0 0 17135, 0.10 0.10 0 0 17135, 0.10 0.10 0 0 17135
tbent.cpp data for testing a column element with DEBUGCOLEL defined
column ;fixed-pinned column
1 1E20 1.5000 ;khtx
1 1E20 2.9900 ;khbx
1 1.0007 jkrstx
1 1E20 1.0007 ;krptx
1 1E20 1.0007 ;krsbx
1 1E20 1.0007 ;krpbx
1 1E20 1.0007 ;krfbx
1 1E20 1.5000 ;khty
1 1E20 1.9900 ;khby
1 1.0007 ;krsty
1 1E20 1.0007 ;krpty
1 1E20 1.0007 ;krsby
1 1E20 1.0007 ;krpby
1 1E20 1.0007 ;krfby
240 1E0 36 ;L, El & drift_limit
0.00 0.00 0 0 4264, 0.10 0.10 0 0 4204, 0.10 0.10 0 0 4204, 0.10 0.10 0 0 4264
0.10 0.10 0 0 4284, 0.10 0.10 0 0 4204, 0.10 0.10 0 0 4204, 0.10 0.10 0 0 4264
0.10 0.10 0 0 4204, 0.10 0.10 0 0 4204, 0.10 0.10 0 0 4284, 0.10 0.10 0 0 42041


o
o\
a
ft
*>
Ul
O
B

C/3
69
5
T2L
S'
hd
C
09
B*

B
*5?
Ul
S'
re
te
B
O
B
c*
**a
S'
200 1 mi ch spans supported on bedrock with 661 _ x 20 H x 1% rocking-hinge
pusu 1 IE 20 1 khtx
1 1000 2 khbx
5 0.0021 54132 0.0714 57199 0.81 krstx
3 0.0019 159132 0.01351 160723 1.0 0.55 krptx
5 0.0021 54132 0.0714 57199 0.81 krsbx
3 0.0019 159132 0.01351 160723 1.0 0.55 krpbx
1 1.7E7 0.1 krfbx
1 1E20 1 khty___
1 1W0 2 khby
5 0.0021 54132 0.0714 57199 0.81 krsty
3 0.0019 159132 0.01351 160723 1.0 0.55 krpty
5 0.0021 54132 0.0714 57199 0.81 krsby
3 0.0019 159132 0.01351 160723 1.0 0.55 krpby
1 1.7E7 0.1 krfby
240 18.69E8 99 (dri ft_ Limit was 3.76) L El drif t_
1 1E20 1 khtx
1 1000 2 khbx
5 0.0021 54132 0.0714 57199 0.81 krstx
3 0.0019 159132 0.01351 160723 1.0 0.55 krptx
5 0.0021 54132 0.0714 57199 0.81 krsbx
3 0.0019 159132 0.01351 160723 1 0 0.55 krpbx
1 1, 7E7 0,1 krfbx
1 1E20 1 khty
1 1000 2 khby
5 0.0021 54132 0.0714 57199 0.81 krsty
3 0.0019 159132 0.01351 160723 1.0 0.55 krpty
5 0.0021 54132 0.0714 57199 0.81 krsby
3 0.0019 159132 0.01351 160723 1 0 0.55 krpby
1 1.7E7 0.1 krfby
240 18 69E8 99 (dri ft_ Limit was 3.76) L El drift_
1 1 7E4 2 K4
1 1.7E4 2 K5
1 164.5E6 1 K12
336 5456 xdist pz
1 3E6 1 K13
1 3E6 1 ;K14
column 1
column 2
(push, insert after kl2)
blm
14.12
blmil
141430
blmi2
1409807
blmi3 rl
1534821 100.
0.0000 0.0067 0.0016 0.0005.
0.0200 0.0046 -0.0010 0.0008,
0.0400 -0.0021 -0.0051 0.0022.
0.0600 -0.0003 -0.0039 0.0073
r2 r3
316.0 329
0.0100 0
0.0300 0
0.0500 -0
xdist zdist
7 336
0067 0,
0011 -0,
0029 -0,
52
0008
0033
0053
vdf
0
0
0
0
gc
02 386.
0009
0008
0050
pz
5456
gt qts
00 0.01
sfzddx sfzddy
12.09 12.09
sfzddz
12.09


Figure 4.16 One Sample Run Input File
200' spans supported on bedrock with 66"_ x 20'H x 1% rocking-hinge columns
run
1 1E20 1 khtx column 1
1 1000 2 khbx "
5 0.0021 54132 0.0714 57199 0.81 krstx
3 0.0019 159132 0.01351 160723 1,0 0.55 krptx "
5 0.0021 54132 0.0714 57199 0.81 krsbx "
3 0,0019 159132 0.01351 160723 1.0 0.55 krpbx "
1 1.7E7 0.1 krfbx
1 1E20 1 _ ... khty-
1 ' lO'OO 2 khby
5 0.0021 54132 0.0714 57199 0.81 krsty "
3 0.0019 159132 0.01351 160723 1.0 0.55 krpty "
5 0.0021 54132 0.0714 57199 0.81 krsby
3 0.0019 159132 0.01351 160723 1.0 0.55 krpby "
1 1.7E7 0.1 krfby
240 18.69E8 99 (drift_ limit was 3.76) L El drift_limit "
1 1E20 1 khtx column 2
1 1000 2 khbx
5 0.0021 54132 0.0714 57199 0.81 krstx "
3 0.0019 159132 0.01351 160723 1.0 0.55 krptx 11
5 0.0021 54132 0.0714 57199 0.81 krsbx "
3 0.0019 159132 0.01351 160723 1.0 0.55 krpbx "
1 1.7E7 0.1 krfbx "
1 IE 20 1 khty
1 1000 2 khby "
5 0.0021 54132 0.0714 57199 0.81 krsty
3 0.0019 159132 0.01351 160723 1.0 0.55 krpty "
5 0.0021 54132 0.0714 57199 0.81 krsby "
3 0.0019 159132 0.01351 160723 1.0 0.55 krpby "
1 1.7E7 0.1 krfby
240 18.69E8 99 (drift_ limit was 3.76) L El drift_limit "
1 1.7E4 2 K4
1 1.7E4 2 K5
1 164.5E6 1 K12
1 3E6 1 K13
1 3E6 1 K14
blm bimil blmi2 blmi3 rl
14.12 141430 1409807 1534821 100.1
0.0000 0.0067 0.0016 0.0005.
0,0200 0.0046 -0.0010 0.0008.
0.0400 -0.0021 -0.0051 0.0022.
0.0600 -0.0003 -0.0039 0.0073
r2 r3 xdist zdist
316.0 329.7 336 52
0.0100 0.0067 0.0008
0.0300 0.0011 -0.0033
0.0500 -0.0029 -0.0053
vdf gc pz qt qts
0.02 386.4 5456 00 0.01
0.0009
0.0008
0.0050
sfzddx sfzddy sfzddz
12.09 12.09 12.09
I


4.5
Chapter 4 References
1. Priestley, Seible, F., Calvi, G.M., "Seismic Design and Retrofit of
Bridges", New York, John Wiley & Sons, 1996, p. 232.
2. Bridge Design Aids 14-11, Equivalent Static Analysis of Restrainers,
California Department of Transportation, Sacramento California, October
1989.
3. Memo to Designers 20-4, Attachment A, Structural Modeling Guidelines,
California Department of Transportation, Sacramento California, April 6,
1992, p. 2.
4. "Bridge Design Specifications, Commentary for Section 3 (Loads)",
California Department of Transportation, Sacramento California, October
1989, p. 3-1.
5. Memo to Designers 20-4, Attachment A, Structural Modeling Guidelines,
California Department of Transportation, Sacramento California, October
1995, p. 19.
6. Craig, R.R., "Structural Dynamics, An Introduction to Computer Methods",
New York, John Wiley & Sons, 1981, p. 147.
7. Saiidi, M., "Hysteresis Models for Reinforced Concrete", Journal of the
Structural Division, ASCE, Vol 108, No. ST5, May 1982, pp. 1077-1087.
8. Saiidi, M., and Sozen, M.A., "Simple and Complex Models for Nonlinear
Seismic Response of Reinforced Concrete Structures", Structural Research
Series No. 465, Civil Engineering Studies, University of Illinois,
Urbana, 111., Aug., 1979.
9. Dowell, R.K., Seible, F., Wilson, E.L., "Pivot Hysteresis Model for
Reinforced Concrete Members", submitted for review to the Journal of
Structural Engineers, ASCE, 1996.
10. Priestley and Calvi, p. 470.
68


11. Memo to Designers 20-4, Attachment A, April 1992, p. 10.
12. Seyed, M., "Displacement Comparison, Capacity vs. Demand (1)",
Caltrans Special Analysis Section, October 31, 1991.
69


5. Physical Model
5.1 Model Description
A 19:1 scale model was prepared to test the rocking hinge bearing concept
with static loads as shown in Figures 5.1 and 5.2. The proportions match the
computer model with 20' tall columns as described in section 6.1. The columns act
as rigid bodies when the cap beam is swayed. The discs are made of crushable
styrofoam with an initial yield strength of about 23 psi which is 5% to 10% of the
crushing strength for the surrounding wooden members which means that the
wooden members are capable of crushing the styrofoam without deforming or
being damaged. The model was monotonically cycled four times using a
maximum lateral load of about 9 pounds. Force and displacement readings were
taken at 0.5" intervals. For each cycle, the maximum displacement was set at a
constant 1.625" which equates to about 31" in the prototype bent. As a point of
reference, the maximum observed lateral movement for the prototype bent in the
dynamic computer model was about 19".
With respect to similitude, geometrical similarities for frame members was
maintained. However, no attempt was made to scale the disc thickness and
prestressing force; enough prestressing was applied to stiffen up the frame without
crushing the discs when the frame was at rest and disc thickness was determined by
commercial availability. Accordingly, the stress-strain similarity for the
prestressing to disc stiffness and the disc to base plate stiffness relationships were
not maintained.
70


5.2 Test Results
The test results can be found in both Figure 5.3 and Appendix B. These
results show that hysteretic effects are totally related to disc crushing and became
minimal after four monotonic cycles in which the cap beam was pulled to the same
maximum displacement each time. Specifically, the final required maximum force
was equal to 90% of the initial required maximum force to deflect the cap beam
1.625" to the right. Also, a small permanent set of 0.6 (1/8" sway to the right) was
observed after the fourth cycle. The effects of disc crushing (T*e,) were added to
the equations in Figure 3.2 as a result of the physical model testing work.
Testing of the physical model shows that the initial stiffness for a rocking
hinge bearing is greater than its final stiffness after cycling one or more times as
shown in Figure 5.3(d). Additionally, the first full cycle displays the most
hysteresis which decreases with each ensuing cycle as can be once again seen in
Figure 5.3(d). Apparently, the first full cycle strain hardens the styrofoam discs.
5.3 Analytical Results
The model was analyzed as shown in Appendix B to verify the moment-
rotation concept for the triangular stress block as shown in Figure 3.1(b). The
verification focused on the measured vs. predicted force-displacement values to do
monotonic cycling in the plane of the test frame. The results show that the
triangular stress block equations in Figure 3.1(b) can be used to predict rocking
hinge moments for the initial excursion. For ensuing movements, the base plate
tends to rock on the disc heel line that was created by the previous maximum
excursion meaning that the moment in the rocking hinge bearing becomes roughly
71


the product of d times Q (or Pu) as shown in Figure 5.3(b). Accordingly, the
triangular stress block equations in Figure 3.1(b) (disc case 1) are not appropriate
for reloading situations until movement beyond the maximum previous excursion
is encountered. Disc moments for ensuing movements can be computed with
equations 5.3(a) through 5.3(c) (disc cases 2 through 4 respectively).
As pointed out in the test results, hysteretic effects are totally related to disc
crushing which causes rocking on or near the heel line with a reduced resisting
moment idealized by the produce of d times Pu. With respect to Figure 4.10(b), p
is therefore the quotient of d time Pu divided by Mu from the triangular stress block
where d and M are taken from the maximum possible excursion and Pu is the axial
load associated with at rest conditions. The "Rocking Hinge Bearing Design"
spreadsheets in Appendix C demonstrate the implementation of this design
procedure.
The lack of stress-strain similitude as discussed in section 5.1 will distort
the model results. These distortions would be intolerable in a dynamic test because
of their affect on the frame stiffness. However, these distortions have a lesser
affect on the frame in a static test aimed at establishing the basis for hysteresis in
rocking hinge bearings. Of the two distortions, the one caused by not scaling the
disc to base plate stiffness is the more important omission in a static test because it
may have an effect on the joint stiffness. This shortcoming could be corrected in
future testing by incorporating base plates made from sheet rubber with a modulus
of elasticity similar to the modulus of elasticity for the styrofoam that is used to
make the discs. Each bearing would then be a sandwich with a styrofoam disc
inserted between two rubber base plates. Because rubber has a much larger
proportional limit than styrofoam, the rubber would yield like the styrofoam but
72


would recover when the load is removed. This scenario more closely models the
prototype rocking hinge bearing because the proportional limit for the steel in the
base plates is larger than the proportional limit for the steel in the discs. The
results from testing done with rubber base plates may show that joint stiffness has
been reduced for the same relative frame displacement due to the temporary
storage of energy within the base plates; the base plates would store strain energy
during loading and release it as the load is removed.
73


Fender
washers
5/i6 0 All-
thread rod
with 1 h.
deep hole
for bicycle
cable (silver
solder cable
in hole)
Wing nut
Spring
4x4x24- Cap beam
t Bicycle
cable
40x13
round
column
2x12x4B
Base plate
$6 0x4 Eye bolt
Pull force
I*x3h0
Styrofoam
disc
5^6 0 Carriage
bolt with 1 h deep
hole for bicyle
cable (silver solder
cable in hole)
Figure 5.1 Test Bent Elevation
74


Styrofoam Disc
Figure 5.2 Test Bent Details
75