NONLINEAR DYNAMIC ANALYSIS OF BRIDGE PIERS
By
CESAR FERNANDES, JR.
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
Copyright 1999
by
Cesar Femandes, Jr.
I would like to dedicate this dissertation to my parents, Cesar and Dalva, my brothers,
Magno and Marcus, and to my Fiancee, Leandra. I could not have reached such an
accomplishment without their help.
ACKNOWLEDGMENTS
I would like to thank my parents, Cesar and Dalva, my brothers, Magno and
Marcus, and all my family, for their unconditional support in all phases of my graduate
studies. I also would like to thank my fiancee, Leandra, for being so patient and
supportive during all my work.
I also would like to thank all the faculty from the Civil Engineering Department
at the University of Florida, and the members of my supervisory committee Dr. Hoit,
Dr. McVay, Dr. Fagundo, Dr. Hays, and Dr. Wilson for always having their doors open
to answer my questions.
Finally I would like to thank all the graduate students from the workstation lab,
specially, Mark and Wirat, for their support and friendship.
TABLE OF CONTENTS
page
ACKNOWLEDGMENTS ......................................... ....................... iv
LIST O F TA B L ES ......................................................................... ............................ viii
L IST O F FIG U R E S ............................................................................ ...........................x
A B STRA CT ............................... ............................................... xvi
CHAPTERS
1 IN TR O D U C TIO N ....................................................................... ............................1
B background ................................................................... ...........................................
Literature R review ........................... .... ....................................................................3
Lim stations ................................................................................ ......................... 7
O rganization................................... .................................................................... 7
2 NONLINEAR DYNAMIC ANALYSIS ........................ .......................9
Theory ................................................ ....................................... ......................... 9
Equations of Motion, Mass, and Damping Matrices ................................................... 10
Equations of Motion for Ground Motion............................. .....................14
Mass Matrices, Consistent and Lumped ................................. ......................18
Mass Matrix for the Uniform 3DBeam Element..........................................18
C onsistent............................................ .................................................18
Lum ped ................................................... ............................................ 20
Mass Matrix for the Shell Element.................................. .........................22
C onsistent............................................ ................................................. 22
Lum ped .............................................. .................................................. 25
Remarks about the mass matrix ................................. ....................26
D am ping............................................................................... ..................................27
Estimating Modal Damping Ratios............................. ... ........................29
M ass Condensation ........................................ ... ............................................. 30
TimeHistory Analysis. Direct Integration Methods................................................34
Numerical Evaluaton of Dynamic Response. Newmark's Method...........................36
Choice of tim e step At.......................... ..... ................................................ 40
N onlinear Problem s ....................................................................... ...................... 41
Analysis of the Nonlinear Response using Newmark's Method ........................41
Nonlinear Dynamic Analysis Algorithm ..............................................................47
3 DISCRETE ELEMENT MODEL AND MATERIAL HYSTERESIS .....................51
Discrete Element Derivation....................... ........... .........................51
Element Deformation Relations.......................... ....... .......................52
Integration of Stresses for Nonlinear Materials......................... ......................54
Elem ent End Forces ............................................ ............................................. 58
Elem ent Stiffness .................................................................... .........................59
Secant and Tangent Stiffness of the Discrete Element..............................................59
H ysteresis M odels................................................ ............................................ 60
M material M odels.............................................. .................................................62
Uniaxial Mild Steel Model.................................................................63
Uniaxial Monotonic Concrete Model Used in FLPIER.......................................64
Proposed Models for the Uniaxial Inelastic Cyclic Behavior of Concrete..................67
Rational M odel................................................ ...........................................67
L loading ....................................................................... ...............................68
U nloading........................................... .................................................. 72
Reloading ............................ ..... .................. ........................... 75
B ilinear M odel ............................................ ...............................................77
Strain Rate Effect.............................................. ...............................................78
Confinem ent Effect.......................... ... ..................... ........................... 79
4 M ODAL ANALYSIS .............................................................. ........................83
Natural Vibration Frequencies and Modes ................................. ...........................83
Modal and Spectral Matrices ...................... ....................................86
Normalization of Modes ....... ....................................................... .......................87
M odal Equations ..................................................................... ......................... 88
Elem ent Forces.................................................. ............................................... 92
Modal Equations for Ground Motion .................................. ........................92
Response Spectrum Analysis................................ ........................................... 95
Modal Combination Rules.............................................................96
How FLPIER handles Modal Analysis...................... ... ...............................99
5 MULTIPLE SUPPORT EXCITATION......................... .................................101
6 SOIL STRUCTURE INTERACTION ............................... .........................108
U coupled M ethod.............................................. ...........................................109
Coupled Method...................................... .................. ................ 10
Cyclic Behavior of Soil........................................................ .......................... 12
Cyclic D degradation ............................................................... .........................113
Strain R ate E ffect.................................................................. ......................... 14
Radiation D am ping............................................................... ......................... 15
vi
7 PREDICTIONS OF RESPONSE ..................................... ....................... 116
Example 1 Steel Section 1 ........................................................116
Example 2 Steel Section 2...................................................... ......... 118
Example 3 Circular Reinforced Concrete Column 1 .............................................. 121
Example 4 Circular Reinforced Concrete Column 2........................................... 125
Example 5 Rectangular Reinforced Concrete Column........................................ 128
M onotonic Tests........................ ..................... .............. ............129
C yclic Test SO ............................................... ........................................... 136
C yclic Test S ............................................. .............................................142
Cyclic Test S2.......................... .... ................... ............................ 146
Cyclic Tests S3 and S4 ..................... ....................................... 148
Cyclic Test S10.............................................................. .........................153
Example 6 Piles in Sand.................................................. ............... 159
T est SP ........................................................................... ................................159
Tests PG2 and PG3 .................. ....................................................... 162
Example 7 Mississippi Dynamic Test...................................................................166
8 CON CLU SION S .................................................................. ......................... 172
APPENDICES
A M A SS UNITS............................................. ..................................................175
B GAUSS QUADRATURE ..........................................................178
C FLPIER Manual for dynamic analsyis........................... .. ....................184
REFERENCES ....................................................................................................184
BIOGRAPHICAL SKETCH ...........................................................191
vii
LIST OF TABLES
Table page
21. Natural frequencies of a uniform cantilever beam: ConsistentMass Finite Element
and exact solution ................................................................. ......................... 21
22. Natural frequencies of a uniform cantilever beam: LumpedMass Finite Element
and Exact Solution.................................... .....................................................21
23. Recommended damping rations for structures ................... ..........................30
31. Curves coefficients ............................................................... .......................... 73
71. Results for Exam ple 2............................................................ ....................... 119
72. Design details for Example 3..........................................................121
73. Model parameters for Example 3 ................................ .........................121
74. Design details for Example 4....................... .... ................................125
75. Loading for each test .......................................................................128
76. Parametric tests, units are KN/mm2....................................................... .............131
77. Parametric tests for confinement under different strain rates.................................135
78. Parameters for test SO units are KN/mm2.......................... ........................ 139
79. Parameters for test S1 units are KN/mm2....................... .............................143
710. Parameters used in test S3 and S4...................... ....................149
712. Some soil properties for CSP1.........................................................160
713. Masses (tons) for pile group tests...................... ... ...............................162
A 1. M ass density units.............................................................................. .............. 177
B1. GaussLegendre abscissas and weights ................................. ............................182
LIST OF FIGURES
Figure page
11. Bridge pier com ponents........................................................ ...........................2
21. Tower subjected to ground motion after Chopra (1995).........................................15
22. Support motion of an Lshaped frame.
a) Lshaped frame; b) influence vector f: static displacements due to Dg=1; c)
effective load vector after Chopra (1995).................................................. 17
23. 3D B eam elem ent ............................................................... .................................19
24. True 9node rectangular element.................................................22
25. Mapping for a true rectangular 9node shell element...............................................22
26. Shell element of uniform thickness .................................................. ...................23
27. Lumped mass matrix at the nodes of true rectangular 9node shell element.
Numbers shown are fractions of the total element mass at each node.....................26
2.8. Full and condensed versions of the structure............................ ........................34
29. Average acceleration ............................................................ ......................... 37
210. Linear acceleration ............................................................. .........................37
211. Secant and Tangent approaches. After Chopra(1995)...................................... 43
212. NewtonRaphson Method............................. ..... ........................45
31. Representation of discrete element. After Hoit et al., 1996 ....................................51
32. Discrete element displacements. After Hoit et al. 1996 ..........................................54
33. Various components of total strain in the section. After Hoit et al., 1996 ................55
34. Rectangular section with integration points. After Hoit et al., 1996.......................56
35. Circular section with integration points. After Hoit et al, 1996 ..............................57
36. Secant and tangent material stiffness............................ ........................ 60
38. Elasticperfectly plastic model for mild steel......................................................64
39. FLPIER concrete points......................... ............................ ...........................65
310. Concrete strains ................................................................... ........................66
311. Concrete stresses.................................... ...................................................... 66
312. Envelop curve for concrete.......................... ...... ........................68
313. Typical compression loading........................ ...... ..........................70
314. Typical loading in tension ...................... ......................................71
315. Typical unloading in tension............................................................73
316. Typical unloading in compression............................ .........................74
317. Compression unloading with gap ....................... .. ................................74
318. Compression unloading with no gap................................. ......................75
319. Typical loading, unloading and reloading in compression........................... ...76
320. Concrete behavior with gap............................... .... ..........................76
321. Bilinear model for concrete..........................................................77
322. Stressstrain curve for concrete in FLPIER........................................................78
323. Confined and unconfined concrete models response ............................................. 79
324. Core width for different cross sections........................... ....................... 81
325. Confined concrete model...................... ....... ..........................82
41. Generalized SDF system for the nth natural mode............................................90
42. Typical response spectrum ...................... .....................................96
43. M odal analysis of pier ........................................................... ........................ 99
5.1. M multiple support m option ................................................... ........................... 102
52. 2D frame submitted to multiple support motion ................................................. 105
53. Pile subjected to multiple support excitation.....................................................106
61. a)Coupled model; b)Uncoupled model.............................. ........................11
62. Typical py curve................................................................. ...... ........................ 12
63. Cyclic soil m odel...................... ......... ..............................................................113
71. Exam ple 1 com puter m odel.....................................................................................117
72. Comparison FLPIER x reference for cyclic loading ............................................. 117
73. Exam ple 2 com puter m odel.................................................................................... 118
74. FLPIER x Hays, shear force comparison .................................. ........................119
75. FLPIER x Hays, bending moment comparison......................................... ..........120
76. Shear comparison Example 3 x FLPIER........................ .................................123
77. Moment comparison Example 3 and FLPIER.....................................................124
78. Dynamic model comparison................................ .......................124
79. Static shear comparison Example 4....................................................... 126
710. Dynamic moment comparison Example 4. Hoop spacing = 5 in........................127
711. Dynamic moment comparison Example 4. Hoop spacing = 2 in........................127
712. Computer model for Test 3..........................................................130
713. Column capacity using FLPIER.......................... ...... ....................130
714. E and E, changed........................................ ..............................................132
715.fc and f changed ............................................................... ....................... ...... 132
716. E andf changed.................................................................... .......................133
xii
717. All parameters changed .................... .........................133
718. Confinement and Strain rate effect............................. .......................135
719. Imposed displacement history for the first 90 seconds........................................137
720. Comparison FLPIER x Test, original properties..............................................138
721. Comparison FLPIER x Test, modified properties................................................138
722. Imposed tip displacement history for test.......................... ............ ..................140
723. Com prison tsOl ........................................ ..............................................140
724. Com prison ts02............................................................................................141
725. Com prison ts03 ............................................................. ................................... 141
726. Comparison ts04........................ ................. ............................................142
727. Imposed displacement in X direction..................................................................142
728. Imposed displacement in Y direction......................... .............................143
729. Com prison tsl 1........................................ ..............................................144
730. Comparison tsl c ....................................................................................144
731.Com prison ts12 ........................................... .............................................. 145
732. Com prison tsl2c ............................................................ ..........................145
733. Imposed displacements in X direction....................... ................................ 146
734. Imposed displacements in Y direction......................... ...............................147
735. Comparison test S2 x FLPIER ................................... ........................147
736. Imposed X displacement for test S3.................................................................... 149
737. Imposed Y forces for test S3 ...................................................149
738. Comparison Test S3 x FLPIER ts31 ............................. .........................150
739. Comparison Test S3 x FLPIER ts32 .............................. .........................150
xiii
740. Imposed displacement in X direction for test S4..................................................151
741. Imposed load in Y direction for test S4.......................... ............................151
742. C om prison ts4 ............................................................... ................................. 152
Fig 743. Com prison ts42....................................... ............................................ 152
744. Com prison ts43 ............................................ ............................................153
745. Imposed force in Xdirection for S10...................................................................154
746. Imposed force in the Zdirection for S10 ........................ ............ ............ 155
747. Com prison tslO0 x test ................ .................................. ......................... 155
748. Com prison ts 02 x test .................................................. ... ....................... 156
749. Com prison tsl03 x test ..................................................... ....................... 156
750. Comparison ts104 x test .........................................................157
751. Single pile in sand............................................................ .......................... 160
752. Top ring acceleration........................................................ ......................... 161
753. Top displacement comparison........................ ...... ......................161
754. Moments comparison ................. ............... ........................................162
755. 2 x 2 pile group in sand ..........................................................163
756. 3 x 3 pile group in sand ............................................................................163
757. Top of pile lateral displacement comparison, 2 x 2 group ..................................64
758. Top of pile bending moment comparison, 2 x 2 group .........................................164
759. Top of pile bending moment comparison, 3 x 3 group .......................................165
760. M ississippi test structure .................................................... .......................167
761. Load history for Mississippi test ................................ .........................168
762. Displacement history for Mississippi test.......................... ......................... 168
xiv
763. Comparison test x FLPIER with CPT soil properties..........................................169
764. Comparison test and FLPIER with SPT soil properties ......................................169
765. Comparison test and FLPIER with original soil properties.................................170
B1. (a) Trapezoidal approximation using the abscissas 1 and 1. (b) Trapezoidal
approximation using abscissas x, and x2 after Mathews, 1987............................179
Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
NONLINEAR DYNAMIC ANALYSIS OF BRIDGE PIERS
By
Cesar Femandes, Jr.
August 1999
Chairman: Marc Hoit
Major Department: Civil Engineering
Bridge piers are often subjected to lateral loading that is not neglectible
when compared to vertical loads. Such loading conditions may include wind, water and
earthquakes. In order to develop a time domain analysis for the nonlinear dynamic
response of piers and their foundations the computer program FloridaPier was
modified. FloridaPier is a nonlinear finiteelement program, developed at the
University of Florida, designed for analyzing bridge pier structures composed of pier
columns and cap supported on a pile cap and piles with nonlinear soil. The program was
developed in conjunction with the Florida Department of Transportation (FDOT)
structures division. The piers and the piles are modeled as nonlinear 3D beam discrete
elements. These elements use the true material stressstrain curves (steel and concrete)
to develop its behavior and stiffness modeling. Nonlinear dynamic capability was added
to these elements by adding to the stressstrain curves the ability to represent loading,
unloading and reloading behavior, typical of dynamic loading. In addition, a mass
matrix for this element was also implemented. The mass matrix for the piles cap was
also developed. The cap is modeled as linear shell elements. The stiffness matrix for this
type of element is considered to remain unchanged during the dynamic analysis. A
nonlinear model was also added for the soil. The py springs generated by the program
now have the capability of loading, unloading and reloading, just like the steel and
concrete. Such models can then be applied to dynamic analysis of reinforced concrete
structures subjected to seismic, impulsive, or wind loads. In addition, being the current
stateofpractice, modal analysis was also implemented. Seismic load can now be
applied to the linear structure considering the nonlinear behavior of the foundation. A
description of the analysis used to model the nonlinear dynamic behavior of bridge
piers, as well as the implemented nonlinear material behavior, is presented in this
dissertation.
CHAPTER 1
INTRODUCTION
Background
Extensive research efforts have been directed to the nonlinear response of
structures subjected to extreme load events. These extreme load events could be an
earthquake or hurricane for a building, a ship impact for a bridge, or the effects of waves
and wind action for offshore oil platforms. Traditionally, large factors of safety have
been used in such cases, resulting in overconservative design and cost ineffectiveness.
On the other hand, an unsafe design could result in catastrophic human and economic
losses. Because a more sophisticated nonlinear dynamic analysis is computationally
expensive, these structures are designed using factored static loads to account for the
dynamic effects. This procedure is acceptable for very low frequency vibrations,
however the introduction of nonlinearity, damping, and pilesoil interaction during
transient loading may significantly alter the response.
Because in recent years computers have become much faster and cheaper, it has
become possible to consider, and consequently to study, the dynamic nonlinear behavior
of structures considering many factors neglected in the past. In this dissertation, the
computer program Florida Pier (Hoit et al., 1996), which will be referred simply as
FLPIER from now on, has been modified to allow the nonlinear dynamic analysis of
bridge piers. FLPIER is a computer program based on the Finite Element Method
developed by Drs. Hoit, Mcvay, and Hays at the University of Florida for the nonlinear
static analysis of bridge piers. Nonlinear aspects of structural analysis, such as material
and geometric nonlinearity, as well as structuresoil interaction can be incorporated into
the analysis leading to more accurate results. It can model all the components of a
bridge pier and its foundation, such as pier, pier cap, piles cap, piles, and soil, as shown
in Fig.11. The pier, pier cap, and piles can be represented using discrete elements that
can incorporate the effects of material and geometric nonlinear behavior. More details
about the discrete element are found in Chapter 3. The piles cap is modeled as linear 9
node shell elements. The lateral soil resistance is modeled as nonlinear py springs,
while the axial resistance is modeled as nonlinear tz springs.
Pier Cap
Bridge Springs e
Piers
Substructure Pile Cap
Piles
Soil layers
Fig. 11. Bridge pier components
FLPIER is now used by many DOTs throughout the United States because of its
reliability and ease of use. Unlike other general Finite Element programs, like ADINA
and SAP, where modeling and analyzing can be time consuming, in FLPIER it is easy
and fast for the user to perform these tasks thanks to a user friendly interface for model
generation. The modification of soil or structure parameters in the model is not difficult
either. The results can also be seen through a graphic interface that is currently being
updated. In the modified dynamics version, resulting from this research, speed and ease
were maintained, allowing the user to easily perform the nonlinear dynamic analysis and
modify parameters in the soil or structure if necessary. Although the program is more
suitable for the analysis of bridge piers, other types of structures can also be modeled.
The new contribution for the field is the proposed concrete model. This model
was implemented in the FLPIER code to allow the nonlinear dynamic analysis of
reinforced concrete sections.
Literature Review
Over the last years different analytical models have been proposed for the
analysis of reinforced concrete structures. Models for these types of structures, which
are under primarily flexural and axial loads, can be classified as:
(i) Simple or lumped models.
(ii) Discrete models.
(iii) Fiber models.
(iv) Finite element models.
Singledegreeoffreedom (SDOF) models belong to the first class of analytical
models. In this class of models it is assumed that the structure's response to an
earthquake is dominated by its first natural frequency, allowing the system to be
represented as a SDOF system with lumped mass and stiffness properties (Crandall
(1956), Craig (1981), Paz (1985), and Chopra (1995)). A more general representation
for multidegreeof freedom systems (MDOF) is derived using the concept of shear
building. In this model the stiffness of each story is represented by nonlinear springs,
and the beams are considered to be infinitely rigid. Despite its simplicity and
satisfactory performance in predicting the maximum response, this class of models does
not provide enough data for more detailed seismic analysis. Furthermore for more
complicated frames the assumption that the beams are infinitely rigid may not be
correct.
In the second class of analytical models, the discrete models, there is a
correspondence between the analytical model and the actual structure. In such models, a
linear elastic element and a nonlinear spring represent the structural elements. The most
common case is that of a nonlinear spring attached to both ends of a linear beam
element. Atalay (1975), Clough (1966), Nakata et al. (1978), Park (1984), and Takeda
(1970), among others, have extensively used this class of models to analyze the behavior
of reinforced concrete structures. In these models a set of predefined rules defines the
hysteretic behavior of the nonlinear springs. These rules are usually obtained from
laboratory experiments with real scale specimens. It is mainly the difference between
these rules that distinguishes the models. Although these models give satisfactory
results, its main disadvantage is the fact that the nonlinear spring's rules are based on
experiments that may not correspond to the actual structural member, or type of loading,
that they are representing.
The fiber models have been used in the study of reinforced concrete (Ala
Saadeghvaziri (1997), Hajjar et al. (1998), Park et al. (1972), and Zeris and Mahin
(1991a), Zeris and Mahin (1991b)) and steel members (Baron and Venkatesan (1969),
Chen and Atsuta (1973)). These models are based on the finite element approach, and
are better suited for members and structures under complex loading histories. In these
models the crosssection is divided into segments. Each segment can then be divided in
one or more fibers. Each fiber is assumed to obey a uniaxial stressstrain relationship.
From the integration of the stresses of each fiber over the cross section, the element
forces can be calculated, and from the evaluation of the stiffness of each fiber the
overall element stiffness can be obtained. Once the element forces and stiffness are
obtained the analysis is carried out using standard Finite Element Method procedures.
Therefore only the stressstrain relationships for concrete and reinforcing steel in the
case of reinforced concrete sections, or steel, in the case of steel sections, are necessary
to describe the properties of each section of the element. This makes these models very
effective under complex loads. The main difference among all the fiber models are the
rules adopted for the uniaxial behavior of the different materials that make the cross
section. In the case of most civil engineering structures, steel and concrete, but other
materials can also be used if the stressstrain relationships are known.
In the specific case of concrete the model's backbone is the envelope curve
obtained from a monotonic test. This curve limits the concrete stresses in any loading
phase. In some models the compression envelope curve for concrete is represented by
the wellknown Hognestad parabola (Ala Saadeghvaziri (1997), Park et al. (1972)).
Another approach is to use multilinear curves to define concrete behavior in
compression (Zeris and Mahin (1991a), Zeris and Mahin (1991b)). In the case of
dynamic analysis, the unloading and reloading rules are particular for each author. In the
case of Ala Saadeghvaziri (1997), and Zeris and Mahin (1991a), Zeris and Mahin
(1991b), unloading under compressive stress has a slope equal to the initial Young's
modulus of the material. However Hajjar et al. (1998) and Park et al. (1972), have their
own more complicated expressions for the unloading curve. The reloading curves are
very specific for each model, and the reader is referred to the mentioned references for
more information. The tension strength of concrete can be neglected (Zeris and Mahin
(1991a), Zeris and Mahin (1991b)), or assumed to be equal to the concrete tensile
strength, (Ala Saadeghvaziri (1997), Park et al. (1972)) in which case its slope is
assumed to be equal to the initial slope of the compression side. The unloading and
reloading criteria are again specific for each model, and the reader is referred to the
above references for more information. Confinement, strainrate, and stiffness
degradation effects are also particular for each model.
In the case of steel, no distinction is made between steel sections and
reinforcement steel. The rules are valid for both cases. For the stressstrain curves, the
Baushinger's effect can be considered (Park et al. (1972), Baron and Venkatesan
(1969)), or ignored, in which case a bilinear or trilinear relationship (elastoplastic with
kinematic or isotropic hardening) is used (Ala Saadeghvaziri (1997), Chen and Atsuta
(1973), Zeris and Mahin (1991a), Zeris and Mahin (1991b)).
The major disadvantage of these models is that they are computationally very
expensive, but with the recent advances in computer technology, this class of models
has become more popular because of its versatility. This is the analytical model used in
this work. FLPIER already incorporates a nonlinear discrete element, which uses fiber
modeling at two points along the element's length to characterize its nonlinear behavior.
New stressstrain curves were introduced to allow the nonlinear discrete element to
perform dynamic analysis. The details can be found in Chapter 3.
The last class of analytical methods is the finite element method. In this class of
methods different elements are used to represent the structural members, such as truss
members to represent the reinforcing steel, and plane stress elements to represent the
concrete. The cracking typical of concrete represents a computational difficulty for these
models, requiring the development of more sophisticated elements. The development of
such elements is a challenge based on complex elasticity and plasticity theories. Like
fiber modeling, this class of methods is computationally expensive and time consuming.
Limitations
It is very difficult for a model to incorporate all the aspects inherent to nonlinear
dynamic analysis, and the model presented here is no exception. The first limitation is
the fact that all the theory developed in Chapter 2 for nonlinear dynamic analysis is
based on small displacement theory. The second limitation comes from the fact that the
effect of shear deformation is not included in the constitutive models. It was also not
included in the original derivation (Hoit et al., 1996). Problems of local buckling are
also outside the scope of this work.
Organization
In Chapter 2 all the theory necessary for the formulation of the problem is
presented. In Chapter 3 the derivation of the discrete element is described and the
8
adopted constitutive models for concrete and steel are presented. Then Chapter 4
discusses the actual stateofdesign procedure for modal analysis. In Chapter 5 the
concept of multiple support excitation is introduced. In Chapter 6 the soil structure
interaction and the dynamic soil behavior are explained. In Chapter 7 the response
predicted by FLPIER is compared to various literature results. Finally in Chapter 8 the
conclusions and suggestions for future work are discussed.
CHAPTER 2
NONLINEAR DYNAMIC ANALYSIS
Theory
In a static problem the frequency of the excitation applied to the structure is less
than one third of the structure's lowest natural frequency. In this case the effects of
inertia can be neglected and the problem is called quasistatic. For such problems the
static equations [K]{D} = {R} are sufficiently accurate to model the response, even
though the loads R} and displacements {D vary (slowly) with time. The static loads
{R) may result from surface loads and /or body forces.
On the other hand if the excitation frequencies are higher than noted above or if
the structure vibrates freely, the inertia effects must be considered in the analysis. The
inertia effects are accounted for by the mass matrix, written as [m] for an element and
[M] for a structure, which is a discrete representation of the continuous distribution of
mass in a structure. The effects of damping, if important, are accounted for by the
damping matrices [c] and [C].
The dynamics problems can be categorized as either wave propagation problems
or structural dynamic problems. In wave propagation problems the loading is often an
impact or an explosive blast. The excitation and the structural response are rich in high
frequencies. In such problems we are usually interested in the effects of stress waves.
Thus the time duration of analysis is usually short and is typically of the order of a wave
transversal time across a structure. A problem that is not a wave propagation problem,
but for which inertia is important, is called a structural dynamics problem. In this
category, the frequency of excitation is usually of the same order as the structure's
lowest natural frequencies of vibration.
A typical example of a wave propagation problem would be that of analyzing the
stresses in a pile when it is grounded. It is important not to exceed the allowable stresses
in order not to damage the pile. Earthquake analysis of structures is a typical structural
dynamics problem, where the inertia forces govern the response of the structure.
Problems of structural dynamics can still be subdivided into two broad
classifications. In the first one, we are interested in the natural frequencies of vibration
and the corresponding mode shapes. Usually, we want to compare natural frequencies of
the structure with frequencies of excitation. In design, it is usually desirable to assure
that these frequencies are well separated. In the second classification, we want to know
how a structure moves with time under prescribed loads, like under impacts, blasts or
wind loads, and/or motions of its supports, like in the case of an earthquake. We are
interested in the timehistory analysis. The two most popular methods of dynamic
analysis are modal methods and direct integration methods.
Equations of Motion, Mass, and Damping Matrices
The equations that govern the dynamic response of a structure will be derived by
requiring the work of external forces to be absorbed by the work of internal, inertial and
viscous forces, for any small kinematically admissible motion (i.e., any small motion
that satisfies both compatibility and essential boundary conditions). For a single
element, this work balance becomes
I{.} I{F}dV + {Su} {(Q}dS + (8 {}, (p
', S Eq. 2. 1
= ({8 {}+{8u} p{i}+{6.} c,{u}) V
Ve
where {8u} and {(e} are respectively small arbitrary displacements and their
corresponding strains, {F} are body forces, {(} are prescribed surface tractions (which
are typically nonzero over only a portion of surface Se), {P}, are concentrated loads that
act at total of n points on the element, {8u}, 'is the displacement of the point at which
load {p}i is applied, p is the mass density of the material, Cd is a materialdamping
parameter analogous to viscosity, and the volume integration is carried out over the
element volume V,.
Using usual Finite Element notation, we may write the continuous displacement
field {u}, which is a function of both space and time, and its first two time derivatives,
as
{u} =[N]{d} {u} = [N]{d} {ii} = [N]{d Eq.2.2
In Eqs. 2.2 the so called shape functions [N] are functions of space only, and the
nodal DOF {d} are functions of time only. Thus Eqs. 2.2 represent a local separation of
variables. Combination of Eqs. 2.1 and 2.2 yields
{8d})' f[B {ao}dV+ Jp[N]'[N]dVd}j + kd [N]T[N]dVd}j
ve v' V Eq. 2.3
I[N]j{F}dV [Nj]Tf{dS {p}, 0
Ve Se I
in which it has been assumed that the concentrated loads {p}j are applied only at the
nodal points locations. Since {Sd} is arbitrary, Eq. 2.3 can be written as
[m{} + [c]{d} + {r"} = {r"'} Eq. 2.4
where the element mass and damping matrices are defined as
[m]= p[N] [N]dV Eq. 2.5
[c] = cd[N]T[N]dV Eq. 2.6
and the element internal forces and external loads vector are defined as
{rin" = [B]T{a}dV Eq. 2.7
{r"'} = [N]' {F}dV + f[N]' {D}dS + ( {p} Eq. 2.8
Ve Se =1
Equation 2.4 is a system of coupled, secondorder, ordinary differential
equations in time and is called a finite element semidiscretization because although the
displacements, {d}, are discrete functions of space, they are still continuous functions of
time. Methods of dynamic analysis focus on how to solve this equation. Modal methods
focus on how to uncouple the equations, transforming the NDOF coupled system into N
uncoupled SDOF systems, each of which can be solved independently of others. More
details about this method of analysis can be found in Chapter 4. Direct integration
methods discretize Eq. 2.4 in time to obtain a sequence of algebraic equations.
Structure matrices [M], [C], and {R"'t} are constructed by standard Finite
Element Method procedures, i.e. conceptual expansion of element matrices [m], [c], and
{rn"'} to "structure size" followed by addition of overlapping coefficients, in the same
way it is done for assembling stiffness matrices in static problems. However, the exact
manner in which {R'"} is computed depends on the dynamic analysis procedure
adopted.
When Eqs. 2.5 and 2.6 are evaluated using the same shape functions [N] as used
in the displacement field interpolation (Eqs. 2.2), the results are called consistent mass
and consistent damping matrices. These matrices are symmetric. On the element level,
they are generally full, but on the structure level, they have the same sparse form as the
structure stiffness matrix. When p and Cd are nonzero, consistent matrices [m] and [c]
are positive definite. Using the mass matrix for example, the kinetic energy
S{rd}[m]{d} is positive definite for any nonzero {d}.
In typical structural analysis we are more interested in dry fiction and hysteresis
loss, than in viscous damping. It is still not well understood how the damping
mechanisms develop in structures, so from a practical standpoint Eq. 2.6 does not
correctly represent structural damping.
The internal force vector, Eq. 2.7, represents loads at nodes caused by straining
of material. Equations 2.4 and 2.7 are valid for both linear and nonlinear material
behavior; that is, in Eq. 2.7, {a} could be a nonlinear function of strain or strain rate.
For linearly elastic material behavior, {a } = [E][B]{ d} and Eq. 2.7 becomes
ri"'} =[k]{d) Eq. 2. 9
where the usual definition of the stiffness matrix holds, that is,
[k]= f[B]T[E][B]dV Eq. 2.10
ee e e
when Eq. 2.10 is used, Eq. 2.4 becomes
[m]{d} +[c]{d} +[k]{d) = {r')} Eq. 2. 11
which can be interpreted as saying that external loads are equilibrated by a combination
of inertial, damping, and elastic forces. For the assembled structure, from Eq. 2.11 we
get the equation of motion for linear systems,
[M]{D} +[C]{D) + [K]{D} = {R'} Eq. 2.12
where {R"et} corresponds to loads {R} of a static problem, but is in general a function of
time. Or, returning to Eq. 2.4, equations of the assembled structure can be written in the
alternative form
[M]{}D +[C]{b} + {Rt} = (R'} Eq. 2.13
which does not require that the material be linearly elastic and represents the equation of
motion for nonlinear systems.
Equations of Motion for Ground Motion
It is now opportune to derive the equations of motion for structural systems
subjected to ground motion. Consider the tower shown in Fig. 21, modeled as a
cantilever beam with concentrated masses at the nodes
The displacement of the ground is denoted by Dg, the total (or absolute)
displacement of the mass mj by DLj and the relative displacement between this mass and
the ground by Dj. At each instant of time these displacements are related by
D (t) = D, (t) + D, (t) Eq. 2.14
Such equations for all the n masses can be combined in vector form:
{D}'(t)= {D)(t)+ {D),(t)[1]
where [1] is a vector of order n with each element equal to unity.
n D
Rigidbody
Motion
D,
Fig. 21. Tower subjected to ground motion after Chopra (1995)
Only the relative motion [D] between the masses and the base due to structural
deformations produce elastic and damping forces (i.e. the rigidy body component of the
displacement of the structure produces no internal forces). Thus Eq. 2.13 is still valid,
however the inertia forces are related to the total acceleration {f' and from Eq. 2.15
we can write
{b}'(t) = {b}(t) + {D (t)[1]
Eq. 2.16
and substituting this value back into Eq. 2.13 we obtain
[M]{b +[C]{) + R'"') = }
Eq. 2.15
Eq. 2.17
The external force vector now becomes the effective earthquake forces vector,
and is given by
P, = [M]{i,} Eq.2.18
A generalization of the preceding formulation is useful if all the DOF's of the
system are not in the direction of the ground motion, or if the earthquake excitation is
not identical at all the structural supports (see Chapter 5 for more details). In this general
approach the total displacement of each mass is expressed as its displacement DYj due to
static application of the ground motion plus the dynamic displacement D, relative to the
quasistatic displacement:
{D}'(t)= {D) + {D}(t) Eq.2. 19
The quasistatic displacements can be expressed as {D}'(t)= {D}) (t), where
the influence vector i represents the displacements of the masses resulting from static
application of a unit ground displacement; thus Eq. 2.19 becomes
{D}'(t) = {D}(t) + {D}, (t) Eq. 2.20
The equations of motion are obtained as before, except that Eq. 2.20 is used
instead of Eq. 2.15, resulting in
[MN]{b+ [Ctiv + {R" }= [Mf )D}, Eq. 2.21
Now the effective earthquake forces are
PI j= [M]ebj
Eq. 2.22
To help illustrate the concept, consider the inverted Lshaped frame with lumped
masses subjected to horizontal ground motion shown in Fig. 22. Assuming the elements
to be axially rigid, the three DOF's are as shown. Static application ofDg= 1 results in
the displacements shown in Fig. 22. Thus f={l 1 0}T in Eq. 2.21, and Eq. 2.22
becomes
Pg (t)= [MiDg (t)
m,+m,
[0
Eq. 2.23
12=1
U 13=0
U3
SI= 1
(m2 +m3)D(t)
Stationary base
c)
Fig. 22. Support motion of an Lshaped frame.
a) Lshaped frame; b) influence vector f: static displacements due to Dg=1; c)
effective load vector after Chopra (1995)
Note that the mass corresponding to b, = 1 is m2+m3, because both masses will
undergo the same acceleration since the connecting beam is axially rigid. The effective
forces in Eq. 2.23 are shown in Fig 22. Observe that the effective forces are zero in the
vertical DOF's because the ground motion is horizontal.
Mass Matrices. Consistent and Lumped
A mass matrix is a discrete representation of a continuous distribution of mass.
A consistent element mass matrix is defined by Eqs. 2.5 that is, by
[m]= Jp[N] [N]dV. It is termed "consistent" because [N] represents the same shape
v
functions as are used in the displacement field interpolation, and to generate the element
stiffness matrix. A simpler formulation is the lumped mass matrix, which is obtained by
placing particle masses mi at nodes I of an element, such that Y m, is the total element
mass. Particle "lumps" have no rotary inertia unless rotary inertia is arbitrarily assigned,
as is sometimes done for the rotational DOF of beams and plates. A lumped mass matrix
is diagonal but a consistent mass matrix is not. The two formulations have different
merits, and various considerations enter into deciding which one, or what combination
of them, is best suited to a particular analysis procedure. The mass matrix for a 3D
uniform beam element, which is used to model the pier and piles, and for a shell
element, which is used to model the pile's cap, are developed next.
Mass Matrix for the Uniform 3DBeam Element
Consistent
The formulation found here is given by Przemieniecki (1968). As a local
coordinate system, consider the system shown in Fig.2.3. The origin is at node 1 with
the ox axis taken along the length of the beam and with the oy and oz axis as the
principal axes of the beam cross section. The matrix N for this element consists of
twelve displacements, six deflections and six rotations, that is,
Eq. 2.24
ty
dd5
d3
d,
.. dd7 d
dd 7 d10 x
Fig. 23. 3D Beam element
Using the engineering theory of bending and torsion and neglecting shear
deformations, we can easily show that the matrix N in the relationship {u}=[N]{d} is
given by Eqs. 2.25.
1
6(4 _2)
6(4 2)
(1 4 34 2)L4
(1+443 2)Lr
6( _2)
6( ~
0
( 24 + 32)Le
(24 32)L
0
132+2(3
0
(1 )L
0
(4 245 +3)L
0
3 2 24
0
0
(2 4)L
0
0
132 +243
(1 4)Lr
( +2 2 )L
0
0
0
322 3
34 2 s
LM
(2 3)L
0
NT
Eq. 2.25
The nondimensional parameters used in these equations are
x y z
SL Eq. 2.26
where L is the length of the beam. The matrix N can then be substituted into Eq. 2.5, and
performed integration over the whole volume of the element. The resulting 12 x 12
consistent mass matrix is given by
13 61
 +..
35
0 _; i
13 61
0 0 I' 4
0 0 0 0 40nc
o o o  o a 
9 6 210 13 1. 13 6
0 * 0 0 ?0 2
35210 1L? I 2f.
o o o o o
9 a ', 3 33 1 ,
T 1O S 3Q4TL 0 0 0 3514,9
0 0 0 0 0 0 0
S ~0  0 0 o 
9 131 13 L z T lw 13 6 2 2
0 a o o 4 0 0 0
,&TGZ 14 o 3a 0 o 0 0 w T+i 7.7
0 M I i 12o I 1, 21+
4o t 0 0o 0 Z 210 o G 0 o o T054+
where the matrix terms with the moments of inertia ly or I, represent rotatory inertia and
the terms with the polar moment of inertia J, represent torsional inertia of the element.
Lumped
The lumped mass matrix for the uniform beam segment of a threedimensional
frame element is simply a diagonal matrix in which the coefficients corresponding to
translator and torsional displacements are equal to onehalf of the total inertia of the
beam segment while coefficients corresponding to flexural rotations are assumed to be
zero. The diagonal lumped mass matrix for the uniform beam of distributed mass
m = pA and polar mass moment l. = pJ, of inertia per unit of length may be written
conveniently as
m= iil 1 1 / 0 0 1 1 1 Ij/m 0 o]
Eq. 2.28
Tables 21 and 22 compare the accuracy of the consistent and lumped mass
formulations using finite elements.
Table 21. Natural frequencies of a uniform cantilever beam:
Element and exact solution
ConsistentMass Finite
Number of Finite Elements, Ne
Mode 1 2 3 4 5 Exact
1 3.53273 3.51772 3.51637 3.51613 3.51606 3.51602
2 34.8069 22.2215 22.1069 22.0602 22.0455 22.0345
3 75.1571 62.4659 62.1749 61.9188 61.6972
4 218.138 140.671 122.657 122.320 120.902
5 264.743 228.137 203.020 199.860
Source: Chopra (1995).
Table 22. Natural frequencies of a uniform cantilever beam: LumpedMass Finite
Element and Exact Solution
Number of Finite Elements, Ne
Mode 1 2 3 4 5 Exact
1 2.44949 3.15623 3.34568 3.41804 3.45266 3.51602
2 16.2580 18.8859 20.0904 20.7335 22.0345
3 47.0294 53.2017 55.9529 61.6972
4 92.7302 104.436 120.902
5 153.017 199.860
Source: Chopra (1995).
Mass Matrix for the Shell Element
Because in FLPIER the pile's cap is modeled as true rectangular 9node shell
elements, we will limit the formulation to this particular type of element.
Consistent
Consider the true rectangular 9node shell element shown in Fig. 24 below.
I b b
Fig. 24. True 9node rectangular element
Now consider the following mapping:
Fig. 25. Mapping for a true rectangular 9node shell element
It is easy to verify that the shape functions N for each node are
1
N, = ( 1)(1)p' I
1
N2 4(l + 1)( 1)P11
1
N, 4(p + 1)(T + 1)LT
N4 (P 1)(W + 1)pq
1
N, =  + 1)( 1)( 1))
1
N6 = (p + 1)(pi)(i 1)(1 + 1)
1
N,  (p + 1)(p 1)(n1 + l)Tq + 1
N9 = (+1)( 1)(I + 1)(1)
Eqs. 2. 29
u 08
Centerline
Fig. 26. Shell element of uniform thickness
If now we consider for each node the six DOF's illustrated in Fig. 26, which
define the shell element, the matrix of shape functions [N], recalling again the
relationship {u}= [N]{d}, takes the form
it (
Ni...N9 zeros
N,. N,
N = NEq. 2. 30
zeros N,...N,
Ni...N,_
Note that [N]'[N] has dimensions 54 x 54, which are the correct dimensions for
the mass matrix considering the shell element illustrated in Fig. 26.
It is easily verified that for the true rectangular element illustrated in Fig. 24 the
change of coordinates from Ip and Tr to x and y can be expressed as
x = aL y= br Eq. 2.31
and recalling for convenience Eq, 2.5
[m]= Ip[N]'[N]dV Eq. 2.32
If we now consider uniform the thickness t and the mass density p over the
entire element, the differential volume dV can be written dV=tdxdy and Eq. 2.5 now
becomes
[m]= pt [N]r[N]dxdy Eq.2.33
the differencials dx and dy can obtained directly from Eq. 2.31,
dx = adit dy = bdrI Eq. 2.34
and Eq. 2.32 can be rewritten
[m] = ptab J [N]'[NJ]dld Eq. 2.35
1
The integral in Eq. 2.35 can be easily evaluated by means of Gaussian
Quadrature (the reader is referred to Appendix B for more details on this procedure),
therefore any element of the mass matrix can be obtained using Eqs. 2.30 and Eqs. 2.35
and this completes the formulation of the mass matrix for the true rectangular 9node
shell element.
Lumped
Lumping the mass for a beam element is a process that seems to be possible by
intuition and physical insight, however, for higherorder elements, like the shell
element, or elements of irregular shape, intuition can be risky. Accordingly, systematic
schemes for lumping are necessary. In FLPIER the HRZ scheme is used.
The HRZ scheme (Cook et al., 1989) is an effective method for producing a
diagonal mass matrix. It can be recommended for arbitrary elements. The idea is to use
only the diagonal elements of the consistent mass matrix, but to scale then in such a way
that the total mass of the element is preserved. Specifically, the procedural steps are as
follows (Cook et al., 1989):
1) Compute only the diagonal coefficients of the consistent mass matrix.
2) Compute the total mass of the element, m.
3) Compute a number s by adding the diagonal coefficients mi, associated with
the translational DOF (but not rotational DOF, if any) that are mutually
parallel and in the same direction.
4) Scale all the diagonal coefficients by multiplying them by the ratio m/s, thus
preserving the total mass of the element.
Following this procedure for the true rectangular 9node shell element with
uniform thickness t, results in the following lumped mass distribution
4
1 36 1
36 36
4 16 4
0o 
36 36 36
1 1
36 4 36
36
Fig. 27. Lumped mass matrix at the nodes of true rectangular 9node shell element.
Numbers shown are fractions of the total element mass at each node
Remarks about the mass matrix
Cook et al. (1989) makes important remarks about the mass matrix. The first one
is the fact that the mass matrix chosen must correctly represent the mass distribution on
the element, because the product [m]{d} or [M] {D) must yield the correct total force on
an element according to Newton's law F = ma when {d} represents a rigidbody
translational acceleration. The second remark is about the consistent and lumped mass
matrix. While the consistent mass matrix is always positive definite, the same can not be
said about the lumped mass matrix. If it contains zeros or negative entries in the
diagonal, then it is called positive semidefinite. This may or may not cause some matrix
operations to give strange results. He also suggests that a consistent mass matrix may be
more suitable for flexural problems, while lumped mass matrices usually yield natural
frequencies that are less than the exact values.
The main reason for the development of lumped mass schemes was to save
memory space in computational calculations. In the past this was a problem, but today
computers are much faster and memory has become cheap and abundant. Therefore the
use of consistent mass formulation is justified, since it avoids certain instabilities in the
matrices operations.
Damning
Damping in structures is not viscous, i.e. is not proportional to velocity; rather it
is due to mechanisms such as hysteresis in the material and slip in connections. These
mechanisms are not yet well understood. Moreover, these mechanisms are either too
difficult to incorporate into the analysis, or they make the equations computationally too
expensive. Therefore with the actual limited knowledge about damping mechanisms,
viscous damping is usually adopted in most analysis. Comparisons of theory and
experiment show that this approach is sufficiently accurate in most cases.
Damping in structures can be considered in two ways:
1)phenomenological damping methods, in which the actual physical dissipative
mechanisms such as elasticplastic hysteresis loss, structural joint friction, or
material microcracking are modeled.
2) spectral damping methods, in which viscous damping is introduced by means
of specified fractions of critical damping (Critical damping, for which the
damping ratio is = 1, marks the transition between oscillatory and
nonoscillatory response).
The first class of methods requires detailed models for the dissipative
mechanisms and almost always result in nonlinear analyses. In the second class of
methods, experimental observations of the vibratory response of structures are used to
assign a fraction of critical damping as a function of frequency, or more commonly, a
single damping fraction for the entire frequency range of a structure. The damping ratio
4 depends on the material properties at the stress level. For example, in steel piping
4 ranges from about 0.5% at low stress levels to about 5% at high stress levels. In bolted
or riveted steel structures, and in reinforced or prestressed concrete, S has the
approximate range 2% to 15% (Cook et al., 1989).
A popular spectral damping scheme, called Rayleigh or proportional damping is
used to form the damping matrix [C] as a linear combination of the stiffness and mass
matrices of the system, that is
[C]=a[K]+ 3[M] Eq. 2.36
where a and 0 are called, respectively, the stiffness and mass proportional damping
constants. Matrix [C] given by Eq. 2.36 is an orthogonal damping matrix because it
permits modes to be uncoupled by eigenvectors associated with the undamped
eingenproblem. The relationship between a, P, and the fraction of critical damping E is
given by the following equation,
S (aw + Eq. 2.37
damping constants a and P are determined by choosing the fractions of critical damping
(tj and 2) at two different frequencies (ol and 02) and solving simultaneous equations
for a and p. Thus:
a=2(22 )/(22 2)
Eq. 2.38
P = 2Bo(o(,2 co,) / (o o2) Eq. 2. 39
The damping factor a applied to the stiffness matrix [K] increases with
increasing frequency, whereas the damping factor P applied to the mass matrix [M]
increases with decreasing frequency. For structures that may have rigidbody motion, it
is important that the massproportional damping not be excessive.
Usually, the natural frequencies ot and Co2 are chosen to bound the design
spectrum. Therefore ol is taken as the lowest natural frequency of the structure, and )2
is the maximum frequency of interest in the loading or response. Cook et al. (1989)
suggests a value of 30 Hz as the upper frequency for seismic analysis, because the
spectral content of seismic design spectra are insignificant above that frequency.
Estimating Modal Damping Ratios
Because damping is still an unknown subjected the estimate of damping rations
still presents some challenge. Recommended damping values are given in Table 23 for
two levels of motion: working stress levels or stress levels no more than onehalf the
yield point, and stresses at or just below the yield point. For each stress level, a range of
damping values is given; the higher values of damping are to be used for ordinary
structures, and the lower values are for special structures to be designed more
conservatively. In addition to Table 23, recommended damping values are 3% for
unreinforced masonry structures and 7% for reinforced masonry construction. It is
important to note that the recommended damping ratios given in Table 23 can only be
used for the linearly elastic analysis of structures with classical damping. This implies
that all the material components of the structure are still behaving in their linearelastic
phase. For structures subjected to strong motions, that will lead to crushing of concrete
or yielding of steel, characterizing nonlinear material behavior, hysteretic damping must
be added to the analysis through nonlinear forcedeformation relationships.
Table 23. Recommended damping rations for structures
Stress Level Type and Condition Damping Ratio(%)
of structure
Working stress, no more Welded steel, prestressed 23
than about /2 yield point concrete, wellreinforced
concrete (only slight
cracking)
Reinforced concrete with 35
considerable cracking
Bolted and/or riveted steel, 57
wood structures with nailed
or bolted joints
At or just below yield point Welded steel, prestressed 57
concrete (without complete
loss in prestress)
Prestressed concrete with 710
no prestress left
Reinforced concrete 710
Bolted and/or riveted steel, 1015
wood structures with bolted
joints
Wood structures with nalied 1520
joints
Source: Chopra (1995).
Mass Condensation
A useful tool to decrease the number of DOF's in the static analysis of a system
without losing accuracy is static condensation. In this approach some degrees of
freedom of the structure are chosen as master degrees of freedom and the remaining
ones are called slave degrees of freedom. The choice for the master DOF is concerned
with those DOF that give a better representation of the system, i.e. in a building where
the beams are much stiffer than the columns the shear DOF's would be a good choice,
instead of the rotations. The condensed stiffness matrix obtained includes the effects of
the slave degrees of freedom, which can be recovered at any time during analysis. The
same approach can be applied to the mass matrix, however dynamic condensation
(Miller, (1981), Paz (1985)) is not exact, as will be shown later in this section, but it can
give good results if some rules are observed in the modeling. The approach described
here is given by Meirovitch (1980, pp.371372):
Let us write the equations for the potential and kinetic energy for a system in the
matrix form:
V = [D]T[K][D] Eq. 2.40
T= [D]T[M][b] Eq. 2.41
and divide the displacement vector [D] into the master displacement vector q2
and slave displacement vector qi, or
[D] =[] Eq. 2.42
Then the stiffness matrix [K] and mass matrix [M] can be partitioned
accordingly, with the result
V = K,, Eq. 2.43
= Lq2 L K2, Mj2 q2 Eq. 2.44
2L2 L 2L M22JLq2i Eq.2.4I
where K21=K2T and M21=M1T. The condition that there be no applied forces in
the direction of the slave displacements can be written symbolically in the form
aV
q=q,K,, +q2K21 Eq.2.45
aq,
where the equation implies equilibrium in the direction of the slave
displacements. Solving Eq. 2.45 for qI, we obtain
q, = K1'Kl2q2
Eq. 2.46
which can be used to eliminate q, from the problem formulation.
Equation 2.46 can be regarded as a constraint equation, so that the complete
displacement vector q can be expressed in terms of the master vector q2 in the
form
q = Pq,
Eq. 2.47
where P is a rectangular constraint matrix having the form
P [ ,]
Eq. 2.48
in which l is a unit matrix of the same order as the dimension of q2. Introducing
Eq. 2.47 into Eqs. 2.40 and 2.41, we obtain
V = q2 qK2q2
T2
where the reduced stiffness and mass matrices are simply
K, = P'KP = K22 K,2K 'K,
Eq. 2. 49
Eq. 2. 50
Eq. 2.51
M, = PMP= M, KK' M2 M,2K1'K21+KK,' M,,K' K, Eq.2.52
The matrix MI is generally known as the condensed mass matrix.
What is being sacrificed as a result of the condensation process?
To answer this question, let us consider the complete eigenvalue problem, which
can be separated into
K,,q1 + K12q2 = (M1q1 + M12q2)
K21q1 + K22q2 = (M21q, + M22q2)
Solving Eq. 2.54 for q2, we have
q, = (M2 K21)(K22 M22)q2
Eq. 2. 53
Eq. 2.54
Eq. 2.55
so that, introducing Eq. 2.55 into Eq. 2.53, we obtain
(K,, K,, K2' K2,)q = x(M,, K,, K,' M,, M2 K,'K2, +K KK,' M,, K,'K,,)q, +
L2(M,2K, 'M,,2 K,2K,'M,,K,'K,, K,, K, M,2K,2 M2, + K,2KM'M2K,2K21)q, +
Eq. 2. 56
Examining Eqs. 2.56 and 2.52 we conclude that the condensation used earlier is
ignoring second and higher order terms in X in Eq. 2.56, which can be justified if
the coefficients of X, X3, ... are significantly smaller than the coefficient of X.
For this to be true we must have the entries of M12 and M22 much smaller than
the entries of K12 and K22. Physically, this implies that the slave displacements
should be chosen from areas of high stiffness and low mass. Moreover the nodes
that carry a timevarying load should be retained as master.
FLIPIER condenses the stiffness and mass to the top of the piles. While this
procedure is exact for the stiffness, note that it is not for the mass. This is because in
pier structures the slave DOF's are located in areas of high mass concentration, like the
pile's cap. Also note that the mass of the superstructure on the pier, is usually modeled
as lumped masses at the top of the pier. In the case of earthquake loading for example
note that we have the loading function acting on slave DOF's, what is not acceptable in
this approach. Because of these limitations for mass condensation, this approach is not
recommended for the dynamic analysis of bridge piers. FLPIER was then modified to
allow a "full" analysis of the structure, where neither the stiffness nor the mass matrices
are condensed to the top of the piles. A typical full and the respective condensed version
of a structure are shown in Fig. 28.
In a typical condensed static analysis, the stiffness and loads of the
superstructure are condensed to the top of the piles. The condensed analysis is then
carried out. At the end of the analysis the superstructure's forces and displacements are
recovered and the analysis is terminated.
Structure
S Cap
Piles
Condensed masses
Condensed
Piles
Fig. 2.8. Full and condensed versions of the structure
TimeHistory Analysis. Direct Integration Methods
In direct integration methods or stepbystep methods, a finite difference
approximation is used to replace the time derivatives appearing in Eq. 2.12 or 2.13 at
various instants of time. Direct integration is an alternative to modal analysis methods.
For many structural dynamics and wave propagation problems, including those with
complicated nonlinearities, direct integration is easier to implement. In direct
integration, the approach is to write the equation of motion (2.12), at a specific instant in
time,
[M]({)} +[C]({b} +[K]{ D}, = (R"
Eq. 2. 57
where the subscript n denotes time nAt and At is the size of the time increment or time
step. The absence of time step subscripts on matrices [M], [C], and [K] in Eq. 2.57
implies linearity. For problems with material or geometric nonlinearity, [K] is a function
of displacement and therefore of time as well. Accordingly, from Eq. 2.57,
[M]j{D} +[C]{D}) + R"}) = {R"'} Eq. 2.58
{R'"',} is the internal force vector at time n At due to straining of material. It is obtained
by assembling element internal force vectors, {ri't},, given by Eq. 2.7 using {o}n. For
nonlinear problems, {R""}, is a nonlinear function of {D}n and possibly time derivatives
of {D)}, if the strain rate is an issue. For linear problems, the internal force vector is
given by the relationship {R"in,=[K]{D},. In Eq. 2.58 [M] and [C] are taken as time
independent, although for some problems these may also be nonlinear. In this work [M]
and [C] remain constant during the analysis, and the internal force vector, {R'"'}, is only
a function of the displacements {D},. The nonlinear relationship for {o}, will be
developed later.
Different methods for direct integration of Eqs. 2.57 and 2.58 can be categorized
as explicit or implicit. The first category, the explicit methods, has the form
{ D} = f({ D),,O {/(}, { D}_,,...) Eq. 2. 59
and hence permit {D}+,, to be determined in terms of completely historical information
consisting of displacements and time derivatives of displacements at time n At and
before. The main characteristic of explicit methods is the fact that the next
approximation for the displacements is based only on the known previous
approximations for the displacements and their respective derivatives. Note the use of
the equilibrium condition at time n. The Central Difference is an example of an explicit
method.
The second category, the implicit methods, has the form
{D) = f( { { D,.... )Eq.2.60
and hence computation of {D},+1 requires knowledge of the time derivatives of {D},+1,
which are unknown. The main characteristic of the implicit methods is the fact that the
next approximation for the displacements depends on unknown values of their
derivatives. Note that the equilibrium condition is used at time step n+l. Newmark's
and the WilsonTheta are examples of implicit methods.
There is vast literature about the advantages and disadvantages of each approach,
the reader is referred to Cook et al. (1989), Chopra (1995), Paz (1985), Craig (1981), or
Crandall (1956), for a more extensive discussion on these approaches. Generally
speaking under certain conditions the implicit methods are more stable than the explicit
methods. Because an implicit method was used in this research we will limit our
discussion to this class of methods.
Numerical Evaluaton of Dynamic Response. Newmark's Method
As mentioned earlier implicit methods are those in which the approximation for
the next displacements {D},,+ depends on unknown values for its time derivatives. The
main advantage of the implicit methods is the fact that most of the useful methods are
unconditionally stable and have no restriction on the time step size other than required
for accuracy.
In 1959 N. M. Newmark developed a family of timestepping methods based
assumptions for the variation of the acceleration over the time step. The first method is
37
called average acceleration and is shown in Fig. 29. Successive application of the
Trapezoidal Rule leads to Eqs. 2.61 to 2.65.
ui
ti ti+l t
At
Fig. 29. Average acceleration
ii() =0(",+i+) Eq. 2.61
At
= "u + 2 (, + ki,) Eq. 2. 63
2
u(T) = u, + iiu + (ii, + ,) Eq. 2. 64
At2
u,,I =u, + iAt + (ii, + ii) Eq. 2.65
The second method is called linear acceleration and is illustrated in Fig.210.
The Trapezoidal Rule is also used to obtain Eqs. 2.66 to 2.70.
U A
u,1
*
t t,+1 t
At
Fig. 210. Linear acceleration
ii() = ii, + (ii,, + ii ) Eq. 2. 66
At
12
u(T)= +iT + (ii, +u) Eq. 2.67
2At
At
ui1 = u + 2(i,, + +i,) Eq. 2. 68
2 3
,C2 C
u(t)= u, +th+iu,+ (u, u,) Eq. 2.69
2 6At
u,, =u, +z,At+(At)2 ~i, + ij Eq. 2.70
The Newmark's family of methods can be summarized into the following two
equations:
",+, = u, + [( )iii +yii,, ]At Eq. 2.71
u,W = u, + uAt + p ii, + pii/, At2 Eqs. 2. 72
where y and p are parameters that can be determined to obtain integration accuracy and
stability. Note that when y = V2 and P = 1/6, this method reduces to the linear
acceleration method. Newmark originally proposed as an unconditionally stable scheme,
the constant average acceleration method, in which case y = /2 and P = '. This can be
shown considering that Newmark's method is stable if (Chopra, 1995):
At 1 1
< Eq. 2. 73
T. n 5 2 P
For y = /2 and 1 = /4 this condition becomes
At
 < o Eq. 2. 74
This implies that the average acceleration method is stable for any At, no matter
how large, as mentioned before; however, it is accurate only if At is small enough. For y
= / and p =1/6, Eq. 2.73 indicates that the linear acceleration method is stable if
At
<: 0.551 Eq. 2. 75
T
For the solution of displacements, velocities and acceleration at time i+1 we
now consider the equilibrium equation also at time i+1:
Miii, + Ci,,, + Ku,1 = F+,, Eq. 2. 76
Solving Eq. 2.72 for uii, in terms of ui+ and substituting in Eq. 2.71, equations
for u,,+,and u,,,in terms of the unknown displacements u,+; only are obtained.
Substituting these equations into Eq. 2.76, a system of equations is obtained, which can
be solved to obtain u+IA:
(boM + b,C + K)u,, = F,, +M(bou, +b2i, + bii,)+C(b,u, +b4ii, +bii)Eq. 2. 77
where
b = ;b, = ;Ab = b =b 1;b4 = 1;b5 =At f2/2Eq.2.78
S Atr2 PAt t 20e 04
and finally all the quantities at time i+1 can be written as
iii+l = b (u,,, u,)+ b7u, + b8ii,
Eq. 2. 79
ui, = u, + bii, + boii, Eq. 2.80
,i = u, + (u,, u) Eq. 2.81
where
be = bo; b7 = b2; bs = b3 ; b9 = At(y) ; blo = y.At Eq. 2.82
Equation 2.77 can be now written in the condensed form:
Ku,,, = p(t) Eq. 2. 83
where the effective stiffness K is given by
K=boM+b,C+K Eq. 2.84
and the effective load vector P(t) is
p(t) = F,, +M(bou, +b26i, +b3ii,)+C(bu, +b4u, + bsii,) Eq. 2.85
which completes the formulation for linear systems.
Choice of time step At
The unconditional stability of the average acceleration method may lead some
analysts to adopt larger time steps because of economic needs, implying a solution that
may not be accurate, because unconditional stability does not mean unconditional
accuracy. Cook et al. (1989) suggests the following expression for the time step At:
At <(27t/,.)/20 0.3/o, Eq. 2.86
where (o, is the highest frequency of interest in the loading or response of the structure.
However, he suggests that in the case of convergence difficulties, the analysis should be
repeated with a smaller time step for additional assurance of a correct solution.
Nonlinear Problems
In structural analysis, a problem is nonlinear if the stiffness matrix or the load
vector depends on the displacements. The nonlinearity in structures can be classified as
material nonlinearity (associated with changes in material properties, as in plasticity) or
as geometric nonlinearity (associated with changes in configuration, as in large
deflections of a slender elastic beam). In general, for a timeindependent problem
symbolized as [K]{D}={R}, in linear analysis both [K] and {R} are regarded as
independent of {D}, whereas in nonlinear analysis {K} and/or {R} are regarded as
functions of {D}. In this dissertation we will address the nonlinearities associated with
changes in material properties.
Analysis of the Nonlinear Response using Newmark's Method
In this section the notation for SDOF systems is used to simplify the approach,
the extension to MDOF systems is done later, although the concept is exactly the same.
In the numerical evaluation of the response of a dynamic system we go from time step i,
where the equation of motion can be written
mii + cu, + (s), = p, Eq. 2.87
to time step i +1, where the dynamic equilibrium can be written
mii,, + cli,, + (fs), = p,1 Eq. 2. 88
where (fs); is the system's resisting force at time i. For a linear system (fs) = kui.
However for a nonlinear system the resisiting forcefs would depend on the prior history
of displacements and velocities. It is then necessary to consider the incremental
equilibrium equation, the difference between Eqs. 2.84 and 2.85, can be written:
mAii, + cAu, + (Afs), = Ap, Eq. 2. 89
The incremental resisting force can be written
(Afs), = (k,) Au Eq. 2. 90
where the secant stiffness (kj)sec, shown in Fig. 211, cannot be determined because ui,+
is not known. If however we make the assumption that over a small time step At the
secant stiffness (k,)se, can be replaced by the tangent stiffness (k,)an, then Eq. 2.90 can
be rewritten:
(Afs), = (ki) Au, Eq. 2. 91
The incremental dynamic equilibrium equation is now:
mAii, + cAi, + (k,), Au, = Ap, Eq. 2. 92
Equation 2.92 suggests that the analysis of nonlinear systems can be done by
simply replacing the stiffness matrix k by the tangent stiffness (k,)rto be evaluated at the
beginning of each time step. However this procedure for constant time steps At can lead
to unacceptable results for two reasons:
a) The tangent stiffness was used instead of the secant stiffness.
b) The use of a constant time step delays detection of the transitions in the force
deformation relationship.
I U, U,I, u
Fig. 211. Secant and Tangent approaches. After Chopra(1995)
These errors can be minimized by using an iterative procedure within each time
step. The idea is to guarantee dynamic equilibrium before going to the next time step.
We must then solve the equation
ku,,, = h(t) Eq. 2. 93
where now the effective stiffness k, becomes
k, = k, + bm + bc Eq. 2.94
For convenience we drop the subscript i in ki and replace it by T to emphasize
that this is the tangent stiffness, and we can rewrite Eq. 2.94 as
k, = kT + bm+b c Eq. 2. 95
The effective force P(t) is now given by
k(t) = f(t),, + m(b2z, + bii) + c(b4z, + b,u,,,) r, Eq. 2.96
where r, is the internal force at time step i.
Since we are using the tangent estimate for the stiffness, we must iterate within
each time step to guarantee dynamic equilibrium. The first step is to apply the effective
force P(t) and get the first approximation for the next displacement uki+1. Associated
with this displacement is the new tangent stiffness kkr and the true force f, which
includes the internal force rk, the inertia force/;, and the damping force/, at iteration
k. An outof balance force defined as A p(t) k+ = (t) k / is then generated. The new
effective stiffness for the system is computed and the system of equations
kAuk = APk is solved to obtain the incremental displacements Auk, which is added to
the previous displacement u,, giving a new estimate for the displacements, velocities and
accelerations at time i+1. The dynamic equilibrium is checked again with the new
values and another outof balance force is generated. The process continues until the
outofbalance force is within a specified tolerance. Figure 212 helps to understand the
process. This process is known as the NewtonRaphson method.
If the reader is familiar with numerical methods, will notice that what is defined
as the The NewtonRaphson method in some references (Cook et al. (1989), Chopra
(1995) and Bathe (1996)) and above is in fact a NewtonRaphson approach. Although
irrelevant to this discussion the author found important to define the NewtonRaphson
method from a mathematical point of view.
The NewtonRaphson method is a mathematical method for finding roots. The
method relies on the fact that the functionf(x), and its first derivative f(x), and second
derivative f '(x) are continuous near a root p. Note how rigorous the mathematical
definition for the method is, since this was not assumed in the previous application.
With this information it is possible to develop algorithms that produce sequences {xk)
that converge faster top. Note however thatf(x) and its first and second derivatives must
be continuous for the method to work. The convergence rate of the NewtonRaphson
method is quadratic. This method is also called the tangent method because the slope of
the curve is used in the formulation. The recursive formula for the method is:
T(xk,)=k f(xk ,for k = 1,2,...
f'(x )
Eq. 2. 97
and the sequence will converge top.
I U3
Fig. 212. NewtonRaphson Method
However sometimes it may be difficult if not impossible to implement Newton's
method if the first derivative is complicated. The secant method may then be used. The
secant method is the same as the Newton's method except that the first derivative is
replaced by its approximation: the slope of the line through the two previous points.
Note that when successive points get close, the method becomes unstable because
division by zero may occur. It can be shown that the convergence rate is approximately
1.62, being slower than Newton's method. The recursive formula for the secant method
is shown in Eq. 2.97. Note the clear connection between the mathematical and
engineering terms used to define both approaches. More detailed information about both
methods can be found in Mathews (1987).
T(x.,,)= x. x x ( x ) Eq. 2. 98
"f' (x) f(x,) f(x,, ) f(x,) f(x,, )
Xn X,_1
The extension to MDOF systems is immediate by replacing all the scalar
quantities by its respective vector equivalents. Note however that for a SDOF system the
tangent stiffness kr is easily obtained. Such a simple evaluation is not available if there
are MDOF. However, in practice, as it will be discussed later, the physics of the
problem allows us to calculate the tangentstiffness matrix [K,]. In a MDOF context NR
iteration involves repeated solution of the equations [Kt]i{AD}i+i={AR},+i, where the
tangent stiffness matrix [K,] and load imbalance {AR} are updated after each cycle. The
solution process seeks to reduce the load imbalance, and consequently {AD}, to zero.
The internal force in the equation of motion, Eq. 2.35, is written as
R" i ,= R"}+[K,]{AD} Eq. 2.99
where
{AD} = {D},,, {D}, Eq. 2. 100
Combining Eqs. 2.99 and 2.100 with the equations of motion, Eq. 2.35, and the
trapezoidal rule, we obtain
[K" {AD) }={IR Eq. 2. 101
where
K = 4 [M+ 2 [C]+[K,] Eq. 2.102
At2 At
and
R. = {R "}.+ {RM + [M (4 {D} + (D,) + [C{, Eq. 2.103
Note that [K,] must be predicted using {D}n (and possibly {)} if strain rates
effects are important) and must be factored at least once each time step during nonlinear
response. If [K,] is not an accurate prediction of the true tangentstiffness matrix from
time nAt to time (n + 1)At, then the solution of Eq. 2.101 for {AD} will be in error. The
error in nodal forces, that is the residual, is given by the imbalance in the equation of
motion as
{R~"= {R",,,, [M]{D ,, [C]jd,, {R'n }., Eq. 2. 104
where the internal force vector {R'"'}n,+ is computed elementby element. The process
stops when the outofbalance force vector {(R"} is smaller than a specified tolerance.
Finally, considering all the topics discussed in this section, the procedure for
nonlinear dynamic analysis can be summarized in the algorithm described next.
Nonlinear Dynamic Analysis Algorithm
Step (1). Form the initial stiffness matrix K, mass matrix M, and damping C for
the structure.
Step (2). Compute the constants for the numerical method chosen:
Wilson0 Newmark
6 3 1 6
bo= 6 ;b, 3 bo= ;b,
(OAt)2 'OA aAt2' aAt
b2 = 2b,;b = 2 b2 =l/aAt;b3 =1/2a 1
b =2;b, =6At/2 b4 =l;b5 =At(2)/2
b6 = b e /0;b, = b/0
b6 = bo;b, = b2
b, =13/0;b9 =At/2
b = b3;b9 = At(1)
6 Ab2/6 = 5At
For the Newmark's Method Average Acceleration: a = and 5 = '.
For the Newmark's Method Linear Acceleration: a = 1/6 and 6 = '.
For the Wilson0 Method: 0 = 1.4 (0 = 1.0 for Newmark's).
Step (3). Initialize iio0, and uo.
Step (4). Form the effective stiffness matrix using the initial stiffness matrix K:
K = b,M +b,C +K
Step (5). Beginning of time step loop.
Step (6). Form the effective load vector for the current time step:
F,, = F, +0(F,+, F,)+ M(b2i, +b3ii,)+C(b4u, + bii,) R,
Step (7). Solve for the displacement increments
6u = I^l@ /
Step (8). Beginning of dynamic equilibrium loop.
Step (9). Evaluate approximations for ii,u and u:
ii'l = bo0u'' b22, b3,i
ul'' = bu'0' b4 u, b,ii,
u i = u,,8u,'
Ilt+OAI
Step (10). Evaluate actual tangent stiffness K' and internal forces
R'K for all the elements in the structure.
Step (11). Evaluate new effective stiffness matrix K, based on
actual tangent stiffness K'.
Step (12). Evaluate the outofbalance dynamic forces:
s = F, +e(F, F,) [Mii +Ciu +R''
Step (13). Evaluate the ith corrected displacement increment:
Step (14). Evaluate the corrected displacement increments:
6ui =6u''i +Au'
Step (15). Check for the convergence of the iteration process:
NO
'NO T, < (tolerance)
Return to Step (8)
YES
Step (16). Return to Step (5) to process the next time step.
In this dissertation the full NewtonRaphson Method is applied. In this approach
the tangent stiffness matrix k, is updated for every iteration, in contrast with the
Modified NewtonRaphson Method in which the tangent stiffness k, is update once
every time step (it remains constant during the iterations performed within each time
step). This improves convergence, but additional computational effort is required in
forming a new tangent stiffness matrix k, and factorizing it at each iteration cycle. The
50
iterations within each time step proceed until the value of the outofbalance force
,', is smaller than a specified tolerance T. The mass matrix Mand damping matrix C
are considered to be constant throughout the analysis.
As mentioned earlier in this Chapter the correct choice of time step, which
depends on the lowest natural frequency of interest for the system, enforces stability and
accuracy to the analysis. Note however that for a nonlinear analysis, the case is that due
to yielding of steel or crushing and cracking of concrete, a lower stiffness will be
obtained during the analysis, which will result in even lower values for the lowest
natural frequency of interest. The reader should have this in mind when choosing an
adequate time step for a nonlinear dynamic analysis.
CHAPTER 3
DISCRETE ELEMENT MODEL AND MATERIAL HYSTERESIS
Discrete Element Derivation
A representation of the discreteelement used in FLPIER is shown in Fig. 31. It
was developed by Mitchell (1973) and modified by Andrade (1994). The center bar can
both twist and extent but is otherwise rigid. The center bar is connected by two universal
joints to two rigid and blocks. The universal joints permit bending at the quarter points
about the y and z axes. Discrete deformational angle changes ?1, P2, 3j and Tf4 occur
corresponding to the bending moments Mi, M2, M3 and M4, respectively. A discrete
axial shortening (6) corresponds to the axial thrust (T), and the torsional angle T's
corresponds to the torsional moments in the center bar Ms.
^, *!*t_
Universal Join (Top View) Spring
(Side View) Rigid end Block
Fig. 31. Representation of discrete element. After Hoit et al., 1996
Element Deformation Relations
In Fig. 32, wlw3 and w7w9 represent the displacements in the x, y and z
directions at the left and right ends, respectively. The displacements w4 and w1o
represent axial twists (twists about the xaxis) at the left and right ends, respectively; and
finally w5w6 and w1wu1 represent the angles at the left and right end blocks about the x
and z axes, respectively. Based on a small displacement theory we can write:
h
n = w, w, (w5 + w,1) Eq.3.1
h
s = w, w (2 + w,,) Eq. 3.2
The elongation of the center section of the element is calculated as follows:
8 = w, w, Eq. 3.3
The angle changes for the center section about the z and y axes are the defined as
s ws w2 w6 +w12
0,2 612 Eq.3.4
h h 2
s wzw w +w,
02 51 Eq. 3. 5
h h 2
The discretized vertical and horizontal angle changes at the two universal joints
are then
', = 0 w6 Eq.3.6
Y2 = w, 02 Eq. 3.7
3 = wl2 01 Eq. 3. 8
Y4 =021 WII
Eq. 3.9
and the twist in the center part of the element is defined as
5 = wI0 w4
Eq. 3. 10
Therefore, the internal deformations of the discrete element model are uniquely
defined for any combination of element end displacements. The curvature for small
displacements at the left and right universal joints about the y and z axes are defined as
follows:
At the left joint
=4, = /h
=, 2 = / h
At the right joint
T3 = /h
S= h h
The axial strain at the center of the section is given by
c 2h
Eq. 3. 11
Eq. 3.12
Eq. 3.13
Eq. 3.14
Eq. 3.15
Side View
{3 ^ '4' f.
*4 Top Vi.ew 
1 I
2 2 IS
Fig. 32. Discrete element displacements. After Hoit et al. 1996
Integration of Stresses for Nonlinear Materials
For a beam subjected to both bending and axial loads, it is assumed that the
strains vary linearly over the area of the cross section. This assumption enables the
strain components due to bending about the z and yaxes, and the axial strain to be
combined using super position. Examples of these three components are represented
separately in Fig. 33(ac) and combined in Fig. 33(d) also shown in Fig. 33(d) is a
differential force, dF,, acting on a differential area, dAi, relationship for the material
dF = a,dA,
Eq. 3.16
Finally, Fig. 33(e) represents the stressstrain curve. Then, for the left joint, the
relationship for the strain at any point in the cross section is
End View
f, 1
'12
S=E, ly, l,2Z
Then to satisfy equilibrium
M = fdFY=j a,YdA,
A A
M, = \\dF,.Z, = Jfc,Z,d4,
T = d =J IadA,
AS A
a) Strain due to b) Strain due to
zaxis bending yaxis banding
) Strssstrain relationship
9) Stressstrain relationship
Eq.3.17
Eq. 3.18
Eq.3.19
Eq. 3.20
c) Strain due to
axial thrust
d) Combined strains
Fig. 33. Various components of total strain in the section. After Hoit et al., 1996
Numerical integration of Eqs. 3.18, 3.19 and 3.20 is done using Gaussian
Quadrature. To use the method of Gaussian Quadrature, the function being integrated
must be evaluated at those points specified by the position factor. These factors are then
multiplied by the appropriate weighting factors and the products accumulated. Fig. 34
shows a square section with 25 integration points (a 5 x 5 mesh). The actual number of
defaults integration points for a square section is set at 81 (a 9 x 9 mesh). For a steel H
section the default number of points is 60. For circular sections, the section is divided
into circular sectors (12 radial divisions and five circumferencial divisions as shown in
Fig. 35), totaling 60 points. The sections are integrated at the centroids of each sector
using weighting factors of 1.0. The stress in all steel bars is evaluated at the centroid and
a weighting factor of 1 is used for each bar. When a circular void is encountered in a
square section, the force is first computed on the unvoided section and then the force
that would be acting on the voided circular area is computed and subtracted from the
force computed for the unvoided section. Circular sections with voids are divided into
sectors omitting the voided portion. This method of dividing the sections into points,
getting the strains and stresses at each point and then integrating to get forces is usually
called fiber modeling.
S
S S x X
S. i
S Concrete tegrtion Points
S Steel Rebar (xl integration)
Fig. 34. Rectangular section with integration points. After Hoit et al., 1996
 i Steelbar(Ixl Integration)
\'\
SConcrete subdivided
into sectors

Note: Integration points (lxl) for concrete are
at the geometric centrods of each sector
Fig. 35. Circular section with integration points. After Hoit et al, 1996
Even for a nonlinear material analysis, the torsional moment M is assumed to
be a linear function of the angle of twist T, and the torsional stiffness GJ, where J is
the cross section torsional constant; and G is the material shear modulus, resulting in
the following expression for the torsional moment
M = GJT / 2h Eq. 3.21
In this discrete approach the curvature is evaluated at two deformational joints
inside the element. These points are located at the quarter points from each node along
the element length. Therefore the effective position of any plastic hinge that might form
in the structure is restricted to these two locations. This should not cause any practical
limitation for most problems. However, it should be considered if trying to match a
theoretical solution with pure plastic hinges at theoretical ends of members. These
discrete joints, at which the deformations are concentrated, correspond to the integration
points along the length of the element, if a conventional finiteelement solution was
being made.
Element End Forces
The element internal forces are necessary to assemble the global internal force
vector necessary for equilibrium of the equation of motion. From equilibrium of the
center bar
V, =(M4 M)l/h Eq. 3.22
V, =(M M3)/h Eq. 3.23
and from equilibrium of the end bars
fi = T Eq. 3.24
f2 = V Eq. 3.25
3 = V2 Eq. 3.26
f4 =M Eq. 3.27
f,= M +V,2 h/2+Th/2.w5 Eq. 3.28
f =M2 +V, h/2+Th/2.w6 Eq.3.29
f7 = T Eq. 3.30
fs = V Eq. 3. 31
fA = V2 Eq. 3.32
fo = M5 Eq. 3. 33
f, =M3 +V2 h/2+T.h/2 w,, Eq. 3.34
f =M4 +V, h/2+T.h/2w12 Eq.3.35
wherefi f3 andf7 f9 are the acting end forces; andf4 f6 andfiofi2 are the acting end
moments.
Element Stiffness
As mentioned in Chapter 2 the element stiffness may change for either nonlinear
static or dynamic analysis. Therefore its is necessary to evaluate the element stiffness
for each iteration in each timestep. The procedure adopted uses the standard definition
of the stiffness matrix; for an element having n DOF the stiffness matrix is a square
matrix [K] of dimensions n x n, in which Ki is the force necessary in the ith DOF to
produce a unit deflection of thejth DOF. The stiffness computed is that obtained by one
of the two methods described below. The transformation of the discrete element
stiffness matrix to global coordinates and the assembly of the different components of
the global stiffness matrix follow standard direct stiffness procedures.
Secant and Tangent Stiffness of the Discrete Element
During the iteration process, the element stiffness matrix is reevaluated in each
new deformed position. For each iteration, the stiffness for each integration point along
the cross section within an element is stored. Then, on 12 subsequent passes, a unit
displacement is applied to each element DOF keeping all other DOF fixed and the
forces corresponding to that unit displacement are calculated over the cross section of
the element as described earlier. If the stiffness for each of the integration points is
defined by dividing the present stress by the present strain as shown in Fig. 36, then it
is called secant stiffness. On the other hand, if the stiffness for each segment is defined
by the slope of the stressstrain curve at the specific point being integrated, then it is
called the tangent stiffness as also illustrated in Fig.36. Note that if the tangent stiffness
is used, the solution of the nonlinear problems becomes NewtonRaphson, while if the
secant stiffness is used we have a secant method solution.
Stress
/ secanA
Strain
Fig. 36. Secant and tangent material stiffness
Hysteresis Models
We have shown so far how it is possible to get the element internal forces and
form the updated stiffness for each iteration in the previous paragraphs. But we have
considered only the case in which the element is subjected to loading. For dynamic
analysis usually the case is that the element will be subjected to two additional phases:
unloading and reloading. It is important to notice that even for nonlinear static analysis
these two new phases could be present due to a nonlinear redistribution of forces on the
structure that could cause some elements to be overloaded and others to be under
loaded. The curve that is used to describe this behavior is called a hysteresis. For the
inelastic analysis, a proper selection of hysteretic models for the materials is one of the
critical factors in successfully predicting the dynamic response under strong motion.
Several models have been proposed in the past for reproducing various aspects of
reinforced concrete behavior under inelastic loading reversals. In order to closely
reproduce the hysteretic behavior of various components, a highly versatile model is
required in which several significant aspects of hysteretic loops can be included, i.e.,
stiffness degradation, strength deterioration, pinching behavior and the variability of
hysteresis loop areas at different deformation levels under repeated loading reversals.
However, the model should also be as simple as possible since a large number of
inelastic spring are necessary in modeling the entire structure, and additional parameters
to describe a complicated hysteresis loop shape may sometimes require excessive
amount of information.
(d) Aq () r s0ra,
(9) Ct. (h) W (I) *
W (n) k
Fig. 37. Models for hysteresis loops proposed by some authors. After Mo, 1994
Some of the existing popular models: Clough (1966), Fukada(1969), Ayoama
(1971), Kustu (1975), Tani (1973), Takeda (1970), Park (1984), Iwan (1973),
Takayanagi (1977), Muto (1973), Atalay (1975), Nakata (1978), Blakeley (1973), and
Mo (1988) are shown in Fig.37. It appears that most of the available models are aimed
at a particular type of component, such as for use of beams, columns or shear walls only,
and therefore, fall short of the versatility required for modeling practical buildings
having a large number of different components. Most of these models were obtained by
performing curve fits to experimental data. This approach results in really good
approximations for the behavior of the specific member being studied, but lacks
versatility when applied to a real structure.
The advantage of the discrete element with fiber modeling is that the elements
general behavior will be governed by its material properties, instead of experimental
observations, making this procedure useful for studying multiple cross section
configurations under more general load histories.
Material Models
Rather than trying to develop a new element that would model this behavior
using elasticity and plasticity theories, it was thought to change the discrete element
used in FLPIER to develop such behavior. Based on various studies and experiments
(Chen (1982), Park and Paulay (1975), Mo (1994), Roufaiel and Meyer (1987), Park et
al. (1972), Agrawal et al. (1965), Sinha et al. (1964a), Sinha et al. (1964b), Kwak
(1997), Magdy and Mayer (1985), Soroushian et al. (1986), Nilsson (1979), Ozcebe
(1989), Peinzien (1960), and Tseng (1976)) about the uniaxial cyclic behavior of
concrete and steel, and the behavior of R/C members, the stressstrain curves of these
materials were modified to work for dynamic analysis. Shear deformations were not
included because it was not included in the works cited. Moreover it was also not
included in the formulation of the discrete element. By using this approach it was
possible to overcome some of the difficulties found in the various hysteresis models.
Although computationally more expensive, this fiber modeling approach gave the
modified discrete element the versatility not found in many models. A description of
these modified curves follows next.
Uniaxial Mild Steel Model
Because the cyclic behavior of steel is very dependent on the heat from which it
was produced, it was decided that rather than trying to predict the steel behavior based
on exponential curves (Agrawal et al. (1965), Shen and Dong (1997)), to use the bilinear
representation for the steel behavior. This represents a safe lower bound solution that is
adequate for most of the construction steel. So the mild steel reinforcement is assumed
to be perfectly elasticplastic (no hardening) and similar in both tension and
compression as shown in Fig. 38. The parameters needed for the mild steel bilinear
model are the modulus of elasticity E, and yield stress f. The rules for this model are as
follows (refer to Fig. 38):
Loading is represented by segment ab, the tangent stiffness is the initial
modulus of elasticity for steel E, and the stress is given by = E,.E .
Yielding: If E > e, yielding occurs and is represented by segment bc. The
tangent stiffness E, is equal to 0 and the stress a = fy, the yielding stress for steel. The
residual strain e, is given by e, = e c .
Unloading is represented by segment cd, the tangent stiffness is E, and the
stress is given by a = (E s,).E,.
64
Reloading is represented by segment ef, the tangent stiffness is E, and the
stress is given by a = (E e,).E,.
For any of these phases the secant stiffness Es, is given by
E, = Eq. 3.36
C
b c
f r
e
Fig. 38. Elasticperfectly plastic model for mild steel
Uniaxial Monotonic Concrete Model Used in FLPIER
The concrete model used in FLPIER is generated based on the values of the
concrete strength fc and modulus of elasticity of concrete Ec, input by the user. The
compression portion of the curve, which is is based on the work of Wang and Reese
(1993), is highly nonlinear and has a maximum compressive stress f which is related
to but not always equal to the compressive strength of a standard test cylinder, f'. Based
on experimental research, f, is taken to be 85% of fc, the maximum cylinder
compression stress.
The tension side of the curve is based on the tension stiffening model proposed
by Mitchell (1973). This procedure assumes an average tensile stressstrain curve for
concrete. The stress strain relation of concrete in tension is very close to linear with
cracking occurring at a small rupture stress f,, The high stresses actually experienced at
tensile cracks in the concrete will not be reproduced by the model. However the average
response over a finite length of beam will be adequately represented.
Based on the user input the program will generate the concrete curve as a series
of points connected by straight lines as shown in Fig. 39. Values in between the points
are obtained by interpolation of the extreme points in the interval. For values offc = 6
ksi and Ec = 4615 ksi the strain and stress values for the concrete stressstrain curve
generated by the program can be seen in Figs. 310 and 311 respectively.
/
/

13r2,,S
Fig. 39. FLPIER concrete points
StressShaSbn
for Concr le
Sress 0.16E03C
V~arl~o lil _I
/0
/0.6
09901
03.00E 02 0.132iEC
S0C.1650 02
'f^ fe02
33 OE03
1E03
03
IL3.n 25/ 4/1999
Fig. 310. Concrete strains
StressS!to
fr^ Conorete
Siress 0 5009
S 163
It3.; 25/ 4/1999
Fig. 311. Concrete stresses

000,I
0010
S 0004
 n.ZCc
DOne
iolve
~
I
r~r/~ 0
Proposed Models for the Uniaxial Inelastic Cyclic Behavior of Concrete
Two concrete models were implemented in FLPIER. One, called the rational
model, is based on the work of Sinha et al. (1964a), the other is a bilinear model,
similar to the mild steel model presented earlier. These two models are discussed in
more details next.
Rational Model
Figure 39 shows the default envelop of the stress strain curve supplied by the
program, which is a function of f' and E, input by the user. This curve is the backbone
of the model, because it limits the value of the stress in concrete during all phases in the
analysis. The compression portion of the concrete curve is highly nonlinear and is
defined by the Hognstead parabola up to a stress equal to 0.85fc. Beyond this point, a
straight line is adopted, connecting this maximum to another point with residual stress
of 0.20fc and strain equal to 4 c,, as shown in the Fig. 312. For strains greater than 4
,,, the stress is kept at a constant 0.20fc (residual stress) as suggested by Chen (1982).
For the tension portion the curve is assumed linear up to a stress of f, and then has a
tensionsoftening portion as shown in Fig. 312. The tensionsoftening portion attempts
to account for the untracked sections between cracks where the concrete still carries
some stress. The value of f, is based on the fixed value of e, shown in the figure and
the modulus of elasticity E,, input by the user. For English units this will give a value
of f, = 75J5 f. The rules for the rational model are described next.
0.30 f7"
fA"
Fig. 312. Envelop curve for concrete
Loading
Compression
Compression follows the curve described above, a typical loading in
compression phase is illustrated in Fig. 313. The equations that define this phase are:
When
=f =24 4{I Eq.3.37
The tangent modulus of elasticity is defined as:
If
E, = E, Eq. 3.38
If >e>e then
Ec= f  Eq.3.39
This correction was necessary because the derivative of the Hongnestead
parabola gives very high values of the tangent modulus for low values of the
compressive strain e, which caused instabilities to the model. Based on research (Chen
(1982), Park and Paulay (1975)) it was found that at about 30% off'c it is reasonable to
assume that concrete still have the initial tangent modulus, Ec,. Equation 3.39 is just the
derivative of Eq. 3.37 with respect to E.
When the strain e > co, the concrete enters a phase called softening. In this
research this phase is defined as:
If c < c < 4 e, then
0.8 f,"
0.8f:"
E, = 0 Eq. 3.41
(4e, s,)
and if a >4 E,
f= 0.20f" Eq. 3. 42
and
E, =0
Eq. 3.43
stress
(ksi) 2
2.5
0.002 0.0018
0.0012 0.001 0.0008
strain in/in
0.0002 0
Fig. 313. Typical compression loading
Tension
A typical loading in tension is shown in Fig. 314. Tension is also represented by
the envelope curve described earlier. The following equations define loading in tension
If 6 < ,
f =E Eq. 3. 44
E, = E, Eq. 3.45
If c, < e < saf then
f =fr +( E 03 ) E
Y
ii
Eq. 3. 46
S f,
Ec
6,  W
If Ef < s < en, then
( 05f, 
S0.5f,
and finally ife > 6. then
f=o
E, =0
0.6
0.3
stress
(ksi) 0.2
0.1
0.1
0.0001
strain in/in
strain in/in
Fig. 314. Typical loading in tension
Eq. 3.47
Eq. 3.48
Eq. 3.49
Eq. 3. 50
Eq. 3.51
0 0005 0 000 0.00
Unloading
Typical unloading phases in tension and compression are illustrated if Fig. 315
and Fig. 316, respectively. The expressions for compression and tension unloading are
based on the work of Sinha et al. (1964a) defined for different concrete mixes. Due to
the lack of more experimental data, it is assumed that for stronger concrete (fc > 4 ksi),
the response will be that of 4 ksi concrete. If new experiments are done for stronger
concrete these changes can be easily incorporated into the program. The family of
unloading curves is represented by second order curves. From experiments for various
concrete strengths a good fit was obtained with the formula:
J
a+H= ( X)2 Eq. 3.52
X
where H and J are experimental constants whose values for the three mixes used are
given in Table 31. For stress in units of ksi and strains in units of in/in, and X is a
particular parameter, different values of which represent different members of the
family. To determine the value of the parameter X, for a curve passing though a certain
point in the stressstrain plane oaE,, the coordinate values are substituted into Eq. 3.52
which is then solved for the required value of X, leading to the expression:
o,+H + H, 2 2)
X=E, + H T + H _V Eq. 3.53
2J 2J
The tangent modulus Ec for unloading is taken as the slope for two consecutive
points in the unloading curve:
E, Eq. 3. 54
6,, E
Table 31. Curves coefficients
fc (psi) H
3000 0.07
3750 0.09
4000 0.10
J
0.95
0.52
0.61
K
3.42
2.52
4.61
L
1.26
1.03
1.01
There are two choices for when the unloading in compression crosses the strain
axis. The first option assumes that a gap is formed in compression, and concrete will
totally unload until reloading in tension, as illustrated in Fig. 318. The second option
assumes that no gap is formed, and concrete will go straight into the tension reloading
phase from compression, as shown in Fig. 317. When unloading from tension a gap is
always formed, and concrete will not go into compression until the gap is closed, as
shown in Fig. 315.
0.6
0.5
oas : i, .. ,
04_.
stress
0,
o 2 i.. : ..
o : .... I i ,
0.1 0 Q.01
0.0001 0 0.0001
Fig. 315. Typical unloading in tension
0.0003 0.0005 0.0007 0.0009
strain (inrin)
0.5
1
stress
(ksi)
1.5
2
2.5 i
0.0006 0.0005 0.0004 0.0003 0.0002 0.0001
strain (in/in)
Fig. 316. Typical unloading in compression
1
0.5 ... .. _
(ksi)
0.5.......
1.5
2 .....
0.0006 0.0005 0004 0.0003 0.0002 0.0001
strain (in/in)
Fig. 317. Compression unloading with gap
0.2
0.4 : i ,
0.6 .....
0.8
stress 1
(ksi) 
1.2
1.6
2
0.0006 0.0005 0.0004 0.0003 0.0002 0.0001 0
strain (in/in)
Fig. 318. Compression unloading with no gap
Reloading
A typical reloading in compression is illustrated in Fig. 319. The reloading
curves in either compression or tension are represented by a family of converging
straight lines; accordingly, an expression:
a + K = Y(e + L) Eq. 3. 55
was chosen, in which K and L are experimental constants, and Y, is a parameter. The
value of Y can be found, similarly to the value of X in the previous section. By
substituting the coordinates o,,E, of a known point on the reloading curve into Eq. 3.55;
and solving for Y, which is also take as the tangent modulus, Ec, one obtains:
o, +K
Y = +Eq. 3. 56
+ L
0
1.5
2
0.006 0.0005 0 004 0.0003 0.0002 0.0001 0
strain (in/in)
Fig. 319. Typical loading, unloading and reloading in compression
When concrete goes into tension, if it goes back to compression it will reload
only when the gap resulting form the previous cycle is closed. The process can be
visualized in Fig. 320.
stress
strain
Fig. 320. Concrete behavior with gap
Bilinear Model
In another study, Agrawal et al. (1965) claimed that the response of under
reinforced concrete beams is governed by the steel. It was therefore proposed to use a
simplification of the concrete model to simplify the analysis of doubly reinforced
concrete beams. Idealized elastic plastic curves were drawn with a yield stress equal to
the nominal strengthfc value for concrete, and an elastic modulus equal to the average
stiffness of the initial portion of the actual stressstrain curve as shown in Fig. 321.
Also, the tensile strength of concrete is neglected in this model. In FLPIER the stress
strain curve for concrete is based on 13 points as shown in Fig. 322. Referring to Fig.
322, the value of the average elastic modulus is taken as
Ec ( Eq. 3. 57
and the value of the yielding strain Ey is taken as
a(2)
Y Eq. 3. 58
E,
Fig. 321. Bilinear model for concrete
1"oiue
Fig. 322. Stressstrain curve for concrete in FLPIER
Strain Rate Effect
The strength of reinforced concrete sections is very dependent on the strain rate
(Shkolnik (1996)), and degree of confinement of the section (Saatcioglu and Razvi
(1992), and Soroushian et al. (1986)). At a fast strain rate, both the modulus of elasticity
and the strength of concrete increase. The increase in strength can be as much as 17 %
for a strain rate of 0.01/sec, as reported by Park and Paulay (1975). It is very difficult to
test structural members under such conditions. So before using the results from static
tests, it is important to consider the effect of strain rate in dynamic analysis. Important
findings about the strain rate effects on reinfoced concrete members are summarized
next (Otani, (1980)):
1) High strain rates increased the initial yield resistance, but caused small
differences in either stiffness or resistance in subsequent cycles at the same
displacement amplitude.
2) Strain rate effect on the resistance diminished with increased deformation in a
strainhardening range.
3) Non substantial changes were observed in ductility and overall energy
absorption capacity.
Otani (1980) also suggests that the strain rate during an oscillation is highest at
low stress levels, and gradually decreases toward a peak strain. Cracking, crushing and
yielding will contribute to a reduction in the system's stiffness, elongating the period of
oscillation. Such damage is caused by the lower modes of vibration having long periods.
Therefore, the strain rate effect is small in earthquake analysis, and has a small effect on
the response. So, the static hysteretic behavior can be used in a nonlinear dynamic
analsyis of reinforced concrete structures.
Confinement Effect
The confinement of reinforced concrete columns is a good way to increase its
strength and ductility, as shown in Fig. 323.
Confined column
Moment
Curvatute
Fig. 323. Confined and unconfined concrete models response
Confinement basically decreases the slope of the descending branch of the
loading curve of concrete, making the confined concrete member more ductile and less
brittle. The reader is referred to Saatcioglu and Razvi (1992), and Soroushian et al.
(1986) for a good discussion on the subject.
Soroushian et al. (1986) proposed a very simplified model, that incorporates
both, strainrate and confinement effects in the compression envelop curve for concrete.
The following constitutive model (Soroushian et al. (1986)) was implemented in
FLPIER:
K 2E E Y
KK 0.002K,K, 0.002K, K,
f =8 0.002KK3 Eq. 3.59
K,K2 f[1z(e 0.002K,K,)]
2 0.002KK3
where
f= concrete compressive stress
E = concrete compressive strain
K, =1+
V 4A
p, = volumetric ratio of the hoop reinforcement to concrete core = E'" = _4A
V sh'
f ~= 28day compressive strength of concrete, adopted as 0.85fc
fih = yield strength of transverse reinforcement
0.5
3 + .002f 5 (psi)
+3 3p 0.002KK
z = f 1000 +4 0.002KK3 Eq. 3.60
(MPa)
3 + 0.29M_ 3 h'
+145 1000 p, 0.002K,K,
145f, 1000 4 sf
h'= width of concrete core measured to outside of the transverse reinforcement,
as shown in Fig. 3.24 for square, rectangular, and circular sections.
h h' '
Fig. 324. Core width for different cross sections
s = centerto center spacing of transverse reinforcement
K2 = 1.48 + 0.1601ogo0 + 0.0127(log0 g)2 Eq. 3.61
K, = 1.08 +0.1601og0io + 0.0193(logi0 d)2 Eq. 3. 62
Note: For 9 < 10s / sec, K2 = K3 = 1.0.
In this model, K2 represents the strain rate effect on the compressive strength of
the concrete, and K3 takes care of the strain rate effect on the strain at maximum stress.
It is assumed that the strain rate effect on the slope of the descending branch of the
stressstrain diagram is similar to the strain rate effect on the compressive strength of
concrete. This is supported by test results. The model representation can be seen in Fig.
325 below. There is no change in the unloading or reloading curves.
Concrete Compressive Stress (f)
KIKf'c
0.2KKf z
0.002KiK, Concrete Compressive Strain (e)
Fig. 325. Confined concrete model
The following modifications were also proposed by Soroushian et al. (1986) for
the secant and tangent stiffness of concrete when subjected to dynamic loading:
E = 1.241+ 0.111ogio + 0.127(logl0 g)2 Eq. 3.63
Ecs
E = 1.061+ 0.464 log,1 + 0.00683(logo s)2 Eq. 3.64
E,,
Where Ecd = dynamic secant modulus of elasticity, E,, = static secant modulus of
elasticity, Eld = dynamic tangent modulus of elasticity, and E,, = static tangent modulus
of elasticity. Note that when compared to the secant modulus, the tangent modulus
seems to be less influenced by the rate of straining. These changes were also
implemented in FLPIER.
CHAPTER 4
MODAL ANALYSIS
Because the most frequently used procedure for designing bridges for
earthquakes is based on modal analysis, a brief explanation of this method of analysis is
given next. It should be noted that modal analysis is an extensive subject and that what
is presented here is only an introduction of the basic concepts necessary to understand
the method. The interested reader is referred to Chopra (1995), or Paz (1985), for a more
detailed discussion on modal analysis.
Natural Vibration Frequencies and Modes
Before getting to modal analysis it is opportune to introduce the eigenvalue
problem whose solution gives the natural frequencies and vibration modes of a system.
The free vibration of an undamped system in one of its natural vibration modes can be
described mathematically by
d(t) = q, (t) Eq. 4. 1
where the deflected shape n, does not vary with time. The time variation of the
displacements is described by the simple harmonic function
q,(t) = A, cosco,t + B, sinco,t Eq. 4.2
where A, and B, are constants of integration that can be determined from the initial
conditions that initiate the motion. Combining Eqs. 4.1 and 4.2 gives

16
The external force vector now becomes the effective earthquake forces vector,
and is given by
{P'j,) = [M]{bg) Eq. 2.18
A generalization of the preceding formulation is useful if all the DOFs of the
system are not in the direction of the ground motion, or if the earthquake excitation is
not identical at all the structural supports (see Chapter 5 for more details). In this general
approach the total displacement of each mass is expressed as its displacement Lfj due to
static application of the ground motion plus the dynamic displacement Dj relative to the
quasistatic displacement:
{Â£}'(/) = {Â£>} +{DY(t) Eq. 2.19
The quasistatic displacements can be expressed as {Â£>}* (t) = t{D}g (t), where
the influence vector ( represents the displacements of the masses resulting from static
application of a unit ground displacement; thus Eq. 2.19 becomes
{/)}'(/) = {/)}(0 + 4}g(0 Eq. 2. 20
The equations of motion are obtained as before, except that Eq. 2.20 is used
instead of Eq. 2.15, resulting in
[m]{d}+ [c]{d}+ {r}=[*/]$>}, Eq. 2. 21
Now the effective earthquake forces are
\p,A=WY{ds)
Eq. 2. 22
94
The solution of q(l) can readily be obtained by comparing Eq. 4.39 to the
equation of motion for the nthmode SDF system, an SDOF system with vibration
propertiesnatural frequency
system. Equation 4.39 with Â§ = Â¡; which governs the motion of this SDOF system
subjected to ground motion iig (t), is shown here with u replaced by D to emphasize its
connection to the nth mode:
Dn + 2+o)2D = iig(t) Eq. 4. 40
Comparing Eq. 4.39 to 4.40 gives
Thus q(t) is readily available once Eq. 4.40 has been solved for D(t), utilizing
numerical time stepping methods for SDOF systems.
As mentioned earlier there are two procedures available to determine the forces
in the elements from the displacements u(t). The second of these procedures, using
equivalent static forces, is preferred in earthquake analysis because it facilitates
comparison of dynamic analysis procedures with earthquake design forces specified in
building codes. Equation 4.32 defines the static forces associated with the nthmode
response, where q(t) is given by Eq. 4.41. Putting these equations together and using s
(Eq. 4.38) leads to
/(/) = * 4(0 Eq. 4. 42
where,
4(0 = co2D(0
Eq. 4. 43
15
{D}'(0 = {o}(0 + {D}g(/)[l] Eq.2.15
where [1] is a vector of order n with each element equal to unity.
Fig. 21. Tower subjected to ground motion after Chopra (1995)
Only the relative motion [D] between the masses and the base due to structural
deformations produce elastic and damping forces (i.e. the rigidy body component of the
displacement of the structure produces no internal forces). Thus Eq. 2.13 is still valid,
however the inertia forces are related to the total acceleration {/)'} and from Eq. 2.15
we can write
{Â£}'(/) = {Â£}(/) + {Â£}f(f)[l] Eq. 2. 16
and substituting this value back into Eq. 2.13 we obtain
[Mp}+[c]{>}+{/r} = {p,#}
Eq. 2. 17
85
When the determinant is expanded, a polynomial of order N in co2 is obtained.
Eq. 4.7 is known as the characteristic equation or frequency equation. This equation has
N real and positive roots for to2 because m and k, the structural mass and stiffness
matrices are symmetric and positive definite'. The positive definite property of k is
assured for all the structures supported in a way that prevents rigidbody motion. Such is
the case for civil engineering structures of interest to us, but not for unrestrained
structures such as aircraft in flight. The positive definite property of m is also assured if
m is consistent, and can be assured when m is lumped by having positive masses in all
diagonal terms.
The N roots of Eq. 4.7 determine the N natural frequencies co (n = 1,2,...,JV) of
vibration. These roots of the characteristic equation are also known as eigenvalues,
characteristic values, or normal values. When a natural frequency co is known, Eq. 4.6
can be solved for the correspondent vector <> to within a multiplicative constant. The
eigenvalue problem does not fix the absolute amplitude of the vectors , only the shape
of the vector given by the relative values of the N displacements ifJn (j = 1,2,...,1V).
Corresponding to the N natural vibration frequencies co of an ,VDOF system, there are
N independent vectors (> which are known as natural modes of vibration, or mode
shapes of vibration. These vectors are also known as eigenvectors, characteristic
vectors, or normal modes.
' In mathematical terms a matrix A e R'" is positive definite if x Ax > 0 for all
nonzero x e R". In particular all the diagonals entries of A are positive.
170
Fig. 765. Comparison test and FLPIER with original soil properties
From the comparisons we observe that the FLPIER models are in good
agreement with the original test data, specially the SPT model. Note that the initial
period for the system is very close to the actual test in all the cases, what shows that the
estimate for the initial stiffness of the system is very good. However the same agreement
is not observed later in the response. Note that the FLPIER model is stiffer than the
actual test. This is expected since the FLPIER soil model does not include soil
degradation yet. Also note that there is a small residual displacement at the end of the
analysis, which shows that there was some damage to the piles. This is also found in the
FLPIER model, showing that the concrete model is adequate even for prestressed piles.
189
Soil Dynamic Properties
This section allows the user to specify the soil properties necessary for dynamic analysis. If not specified they are set
equal to zero.
SDYN
N=N50 T=SLR V=SWVS1,SWVS2,... F=SFDF
where
N50 is the number of cycles necessary to degrade the soil by 50%.
SLR is the rate of loading for slow cyclic loading.
SWVSi is the shear wave velocity for each soil layer.
SFDF is the fully degraded soil factor.
This section must end with a blank line.
Free Vibration Option
This section allows the user to run a free vibration problem with initial imposed parameters.
FREEV
T=DT L=NUMN Q=3 G=GF
where DT is the time step.
NUMN is the number of nodes with imposed displacement and velocity.
3 is the free vibration option
GF is the gravity factor
For each node, there must be a line with the following information
NODE D = x, y z, 0x 0y 0x xv yv, zv, 0xv, 0yv, 0zv
where
NODE is the node number where the initial conditions are applied.
x,y,z are imposed displacements in the global directions X,Y, and Z respectively.
0x 0y 0x are imposed rotations about the global axis X.Y, and Z respectively,
xv yv, zv are imposed velocities in the global directions X.Y, and Z respectively.
0xv 0yv 0zv are imposed angular velocities about the global axis X,Y, and Z respectively.
Confinement
This section allows the user to specify the parameters for the concrete confinement.
DCONF
F= FYHOOP S= HOOPS C= HOOPC D= HOOPD R= EPP
where FYHOOP is the yield stress for the hoop steel.
Note for English units use KSI, for SI units use MPA.
HOOPS is the hoop spacing.
HOOPC is the hoop core diameter.
HOOPD is the hoop bar diameter.
EPP is the strain rate for concrete.
740. Imposed displacement in X direction for test S4 151
741. Imposed load in Y direction for test S4 151
742. Comparison ts41 152
Fig 743. Comparison ts42 152
744. Comparison ts43 153
745. Imposed force in Xdirection for S10 154
746. Imposed force in the Zdirection for S10 155
747. Comparison tslOl x test 155
748. Comparison tsl02 x test 156
749. Comparison tsl03 x test 156
750. Comparison tsl04 x test 157
751. Single pile in sand 160
752. Top ring acceleration 161
753. Top displacement comparison 161
754. Moments comparison 162
755. 2x2 pile group in sand 163
756. 3x3 pile group in sand 163
757. Top of pile lateral displacement comparison, 2x2 group 164
758. Top of pile bending moment comparison, 2x2 group 164
759. Top of pile bending moment comparison, 3x3 group 165
760. Mississippi test structure 167
761. Load history for Mississippi test 168
762. Displacement history for Mississippi test 168
xiv
67
Proposed Models for the Uniaxial Inelastic Cyclic Behavior of Concrete
Two concrete models were implemented in FLPIER. One, called the rational
model, is based on the work of Sinha et al. (1964a), the other is a bilinear model,
similar to the mild steel model presented earlier. These two models are discussed in
more details next.
Rational Model
Figure 39 shows the default envelop of the stress strain curve supplied by the
program, which is a function of f'c and Ec input by the user. This curve is the backbone
of the model, because it limits the value of the stress in concrete during all phases in the
analysis. The compression portion of the concrete curve is highly nonlinear and is
defined by the Hognstead parabola up to a stress equal to 0.85/',;. Beyond this point, a
straight line is adopted, connecting this maximum to another point with residual stress
of 0.20fc and strain equal to 4 e, as shown in the Fig. 312. For strains greater than 4
e, the stress is kept at a constant 0.20/c (residual stress) as suggested by Chen (1982).
For the tension portion the curve is assumed linear up to a stress of fr and then has a
tensionsoftening portion as shown in Fig. 312. The tensionsoftening portion attempts
to account for the uncracked sections between cracks where the concrete still carries
some stress. The value of fr is based on the fixed value of er shown in the figure and
the modulus of elasticity Ea input by the user. For English units this will give a value
of fr = 7.5^/ f\ The rules for the rational model are described next.
119
Table 71. Results for Example 2
FLPIER
Hays
(tangent)
Ax(in)
Secant
Tangent
S(kips)
M(kip.in)
S(kips)
M(kip.in)
S(kips)
M(kip.in)
0.5
3.5
906.4
3.5
906.4
3.5
907
1.0
7.1
1812.7
7.1
1812.7
7.1
1810
1.5
11.0
2721.6
10.8
2717.8
10.6
2720
2.0
NC
NC
11.4
3145.5
11.5
3140
2.5
10.9
3297.8
10.9
3290
3.0
9.7
3363.2
9.7
3360
3.5
8.4
3393.6
8.4
3390
4.0
7.0
3407.3
7.0
3400
NC = No convergence achieved for the analysis.
Fig. 74. FLPIER x Hays, shear force comparison
130
approach is used. In the dynamics version the tangent stiffness approach is used, so the
softening behavior can be modeled. The moment capacity for this column, from the
static analsyis, was found to be approximately 100,000 KN.mm, at a correpondent tip
displacement of 15.4 mm. Note from Fig. 713 that both models are in very good
agreement.
k = 1500 KN/mm
Top lateral displacement (mm)
Fig. 713. Column capacity using FLPIER
6
compression (Zeris and Mahin (1991a), Zeris and Mahin (1991b)). In the case of
dynamic analysis, the unloading and reloading rules are particular for each author. In the
case of Ala Saadeghvaziri (1997), and Zeris and Mahin (1991a), Zeris and Mahin
(1991b), unloading under compressive stress has a slope equal to the initial Youngs
modulus of the material. However Hajjar et al. (1998) and Park et al. (1972), have their
own more complicated expressions for the unloading curve. The reloading curves are
very specific for each model, and the reader is referred to the mentioned references for
more information. The tension strength of concrete can be neglected (Zeris and Mahin
(1991a), Zeris and Mahin (1991b)), or assumed to be equal to the concrete tensile
strength, (Ala Saadeghvaziri (1997), Park et al. (1972)) in which case its slope is
assumed to be equal to the initial slope of the compression side. The unloading and
reloading criteria are again specific for each model, and the reader is referred to the
above references for more information. Confinement, strainrate, and stiffness
degradation effects are also particular for each model.
In the case of steel, no distinction is made between steel sections and
reinforcement steel. The rules are valid for both cases. For the stressstrain curves, the
Baushingers effect can be considered (Park et al. (1972), Baron and Venkatesan
(1969)), or ignored, in which case a bilinear or trilinear relationship (elastoplastic with
kinematic or isotropic hardening) is used (Ala Saadeghvaziri (1997), Chen and Atsuta
(1973), Zeris and Mahin (1991a), Zeris and Mahin (1991b)).
The major disadvantage of these models is that they are computationally very
expensive, but with the recent advances in computer technology, this class of models
has become more popular because of its versatility. This is the analytical model used in
176
second law, that is, a force of 1 Ibf is exerted on a mass of 1 lbm by the gravitational pull
of the earth.
An other source of confusion relies on the fact that usually there is no distinction
between the Ibf and Ibm, they are simply referred as lb. Because FLPIER needs the mass
density of the elements for dynamic analysis, we decided to include the procedure of
how to obtain this quantity for two of the most used materials in construction: concrete
and steel. Assuming first that in the English system g is given by:
g = 32.2 ft/sec2 = 386 in/sec2 A3
The unit weight of normal weight concrete is usually taken as yc = 150 Ibf/ft1
(note the units of Ibf), therefore the mass density, p, is obtained by dividing yc by the
acceleration of gravity g:
g 32.2 f ft ft1 ff
Note that the slug is defined for Ibf and ft. If the units are kip and in the
following should be done:
47 lbf.s2 ft*
124 ft4 in'
2.25x1 O'04
2.25xlO07
in'
A5
Imposed top ydisplacement (mm)
147
Top lateral displacement (mm)
Fig.735. Comparison test S2 x FLPIER
123
ran to compare the effects of including confinement in the analysis. A plot comparing
the FLPIER unconfined and confined dynamic models to the reference is shown in Fig.
78. Note that there is still some discrepancy in the predicted and test stiffness of the
column, which may be caused by the facts mentioned earlier. Although all the
approaches could predict very well the ultimate moment, only the confined model could
predict the more ductile behavior shown in Fig. 78 for the original test.
This test is important because it shows clearly the effects of confinement. Note
that for the static analysis the theoretical column strength is just slightly over the actual
test value, however the maximum displacement achieved is much less than the one in
the test. This is because no confinement was considered. This illustrates the main
characteristic of confinement, that although there is no great increase in the column
strength (Fig. 78), the column's ductility is reasonably increased, an important
characteristic that should be present in columns in seismic regions. Although the
confined model gives a response much closer to the original test, failure of the column
at about 23 in can not be predicted.
Fig. 76. Shear comparison Example 3 x FLPIER
73
Table 31. Curves coefficients
f c (psi)
H
J
K.
L
3000
0.07
0.95
3.42
1.26
3750
0.09
0.52
2.52
1.03
4000
0.10
0.61
4.61
1.01
There are two choices for when the unloading in compression crosses the strain
axis. The first option assumes that a gap is formed in compression, and concrete will
totally unload until reloading in tension, as illustrated in Fig. 318. The second option
assumes that no gap is formed, and concrete will go straight into the tension reloading
phase from compression, as shown in Fig. 317. When unloading from tension a gap is
always formed, and concrete will not go into compression until the gap is closed, as
shown in Fig. 315.
0.6
0.5
0.4
stress
(ksi) 03
0.2
0.1
o
0.1
0.0001 0 0.0001 0.0003 0.0005 0.0007 0.0009
strain (in/in)
Fig. 315. Typical unloading in tension
3
FLPIER is now used by many DOTs throughout the United States because of its
reliability and ease of use. Unlike other general Finite Element programs, like ADINA
and SAP, where modeling and analyzing can be time consuming, in FLPIER it is easy
and fast for the user to perform these tasks thanks to a user friendly interface for model
generation. The modification of soil or structure parameters in the model is not difficult
either. The results can also be seen through a graphic interface that is currently being
updated. In the modified dynamics version, resulting from this research, speed and ease
were maintained, allowing the user to easily perform the nonlinear dynamic analysis and
modify parameters in the soil or structure if necessary. Although the program is more
suitable for the analysis of bridge piers, other types of structures can also be modeled.
The new contribution for the field is the proposed concrete model. This model
was implemented in the FLPIER code to allow the nonlinear dynamic analysis of
reinforced concrete sections.
Literature Review
Over the last years different analytical models have been proposed for the
analysis of reinforced concrete structures. Models for these types of structures, which
are under primarily flexural and axial loads, can be classified as:
(i) Simple or lumped models.
(ii) Discrete models.
(iii) Fiber models.
(iv) Finite element models.
Singledegreeofffeedom (SDOF) models belong to the first class of analytical
models. In this class of models it is assumed that the structures response to an
earthquake is dominated by its first natural frequency, allowing the system to be
115
0.03 for calcareous soils. Both the rate effect and cyclic degradation models are built
into FLPIER.
Radiation Damning
Radiation damping is modeled through the use of dashpots having constants C
(ONeill et al., 1997) attached to the pile nodes, where
Ch = 2D{ys + v) Eq. 6. 4
g
where C* is for horizontal (py) resistance in units of FT/L2, D is the pile diameter, y is
the unit weight of the soil, g is the acceleration of gravity, vs is the shear wave velocity
of the soil, which would need to be estimated at a given site, and v is a velocity in
between the shear and compression wave velocities of the soil. A lower bound solution
for Ch, which is used in FLPIER, is given by taking v = vs:
Ch = 4Dvs Eq. 6. 5
g
A similar expression can be developed later for axial loading.
LD
1780
199j
.f3,
3 1262 07056 2219
UNIVERSITY OF FLORIDA
3 1262 06554 4764
Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
NONLINEAR DYNAMIC ANALYSIS OF BRIDGE PIERS
By
Cesar Fernandes, Jr.
August 1999
Chairman: Marc Hoit
Major Department: Civil Engineering
Bridge piers are often subjected to lateral loading that is not neglectible
when compared to vertical loads. Such loading conditions may include wind, water and
earthquakes. In order to develop a time domain analysis for the nonlinear dynamic
response of piers and their foundations the computer program FloridaPier was
modified. FloridaPier is a nonlinear finiteelement program, developed at the
University of Florida, designed for analyzing bridge pier structures composed of pier
columns and cap supported on a pile cap and piles with nonlinear soil. The program was
developed in conjunction with the Florida Department of Transportation (FDOT)
structures division. The piers and the piles are modeled as nonlinear 3D beam discrete
elements. These elements use the true material stressstrain curves (steel and concrete)
to develop its behavior and stiffness modeling. Nonlinear dynamic capability was added
to these elements by adding to the stressstrain curves the ability to represent loading,
unloading and reloading behavior, typical of dynamic loading. In addition, a mass
xvi
105
If the ground (or supports) accelerations Dg(t) are prescribed, the velocities
Dg(t) can be obtained by numerical integration of the accelerations Dg(t), and peÂ¡Hf) is
known from Eq. 5.11, and this completes the formulation of the governing equation (Eq.
5.5).
Equation 5.11 can be simplfied on two counts. First, if the damping matrices are
proportional to the stiffness matrices (i.e. c = aÂ¡k and cg = ajkg), the damping term is
zero because of Eq. 5.7. Second, if the mass is idealized as lumped at the DOF, the
mass matrix is diagonal, implying that mg is a null matrix and m is diagonal. With these
simplifications Eq. 5.11 can reduces to
P,g (0 = ~m(Dg(t) Eq. 5.12
For a better understanding of how the influence t matrix is formed, consider the
2D frame with the DOF illustrated in Fig. 52 below
Fig. 52. 2D frame submitted to multiple support motion
70
o
0.5
stress
(kS) 2
0.002 0.0018 00012 0001 "0.0008
strain in/in
0.0002 o
Fig. 313. Typical compression loading
Tension
A typical loading in tension is shown in Fig. 314. Tension is also represented by
the envelope curve described earlier. The following equations define loading in tension
If e < Er
f Ez
Ec = Ea
If z, < e < Es//then
/=/,+
Eq. 3. 44
Eq. 3. 45
Eq. 3. 46
173
In the case of the structures where soil was present it is clear that more studies
still need to be done when multiple piles are used, because the dynamic group effect is
not yet well understood. However the approximations given by FLP1ER gave reasonable
estimates for the displacements and forces acting in the structure from very simple
parameters.
The next steps in the research are:
1) Introduce a new steel constitutive model including strain hardening.
2) Study the behavior of full structures modeled with the nonlinear discrete
element and compare the results to the literature.
3) Study the behavior of other cross sections, such as prestressed sections,
sections with voids and sections with steel sections or pipes encased.
4) Analyze structures modeled with the nonlinear discrete element including the
soil effects and compare, if possible, the results to the literature.
5) Perform more tests with pile groups to calibrate the FLPIER model for the
pile group effects.
The approach presented will require more study to be accurate for more
complicated cases, but what is important is that in all examples the program gave a good
idea about the magnitude of the forces and displacements under a variety of extreme
loads. Note for example the case of the pile groups in sand. At first it seems very
difficult to predict the response of a pier under earthquake loads, including the soil
structure interaction, nevertheless FLPIER gave good approximations for the single and
2x2 pile group, and for the 3 x 3 group, its prediction is off by about 50%. In the case
of the Mississippi example note that FLPIER gave an excellent prediction for the
68
Fig. 312. Envelop curve for concrete
Loading
Compression
Compression follows the curve described above, a typical loading in
compression phase is illustrated in Fig. 313. The equations that define this phase are:
When E
E'7
The tangent modulus of elasticity is defined as:
If E<6e then
E' = Ed Eq. 3. 38
If E>Ee then
180
7V,/0 = f/(*,)^/(*2) = /(*.) + /(*2) B4
We shall use the method of undetermined coefficients to find the abscissas x/, xÂ¡
and the weights wi, wÂ¡ so that the formula
l/(x)dx wj(xl) + w2f(x2) B5
is exact for cubic polynomials (i.e., f(x) = a3x3 + a2x2 + olx + a0). Since four
coefficients w;, wÂ¡, xÂ¡, and xÂ¡ need to be determined in Eq. B5, we can select four
conditions to be satisfied, using the fact that integration is additive, it will suffice that
Eq. B5 is exact for the four functions _/(x)=l, x, x2, x3. The four integral conditions are
/(x) = 1: jlc& = 2 = wÂ¡ + w2. B6
i
i
fix) = x : jxofr = 0 = wÂ¡xÂ¡ + w2x2. B7
i
i
/(x) = x2 : jx2dx =  = w,x2 + w2x2 B8
i
i
/(x) = x3 : jx3dx = 0 = WjX,3 + w2x2 B9
i
Solving the system of nonlinear equations indicated above we obtain:
w, = w2 = 1 and x, = x2 = 0.5773502692. BIO
We have found the nodes and the weights that make up the two point Gauss
Legendre rule. Since the formula is exact for cubic equations, the error term will involve
the fourth derivative.
Moment (kip.in)
124
C
Fig. 77. Moment comparison Example 3 and FLPIER
Lateral top displacement (in)
Fig. 78. Dynamic model comparison
189
Saatcioglu, M., and Razvi, S. R. (1992). Strength and ductility of confined concrete.
ASCE Journal of Structural Engineering, Vol. 118, No. 6 (June), pp. 15901607.
Schnabel, P. B., Lysmer, J., and Seed, H. B. (1972). SHAKE: a computer program for
earthquake response analysis of horizontally layered site. Report No.
USB/EERC 72/12, University of California, Berkeley, Dec., p. 102.
Shen, Z., and Dong, B. (1997). An experimentalbased cumulative damage mechanics
model of steel under cyclic loading. Advances in Structural Engineering An
International Journal, Vol. 1, No. 1 (October), MultiScience Publishing Co.
Ltd., London, UK.
Shkolnik, I. E. (1996). Evaluation of dynamic strength of concrete from results of static
tests. ASCE Journal of Engineering Mechanics, Vol. 122, No. 12 (December),
pp. 11331138.
Sinha, B. P., Gerstle, K. H., and Tulin, L. G. (1964a). Stressstrain relations for
concrete under cyclic loading. Journal of American Concrete Institute, Vol. 61,
No. 2 (February), pp. 195211.
Sinha, B. P., Gerstle, K. H., and Tulin, L. G. (1964b). Response of singly reinforced
beams to cyclic loading. Journal of American Concrete Institute, Vol. 61, No. 8
(August), pp. 10211037.
Soroushian, P., Choi, K. B., Alhamad, A. (1986). Dynamic constitutive behavior of
concrete. Journal of the American Concrete Institute, Vol.83, No.2 (March
April), pp. 251259.
Takayanagi, T., and Schnobrich, W. C., Computed behavior of coupled shear walls.
Proc. of 6th WCEE, New Delhi, 1977.
Takeda, T., Sozen, M. A., and Nielsen, N. N. (1970). Reinforced concrete responses to
simulated earthquakes. ASCE Journal of Structural Division, Vol. 96, No.ST
12.
Tani, S., and Nomura, S. (1973). Response of reinforced concrete structures
characterized by skeleton curve and normalized characteristic loops to ground
motion. Proc of 5th WCEE, Rome.
Ting, J. M. (1987). Fullscale cyclic dynamic lateral pile responses. ASCE Journal of
Geotechnical Engineering, Vol. 113, No. 1 (January), New York.
Tseng, G., Stea, W, Weissman, S., Dobbs, N., and Price, P. (1976). Elastic and elasto
plastic computerized dynamic analysis of frame structures subjected to blast
overpressure. Proceedings of the National Structural Engineering Conference,
pp. 977987.
21
zero. The diagonal lumped mass matrix for the uniform beam of distributed mass
m = pA and polar mass moment Im = pJx of inertia per unit of length may be written
conveniently as
TL
m = [ 1 1 1 Is/m 0 0 1 1 1 Im/m 0 o] Eq.2.28
Tables 21 and 22 compare the accuracy of the consistent and lumped mass
formulations using finite elements.
Table 21. Natural frequencies of a uniform cantilever beam: ConsistentMass Finite
Element and exact solution
Number of Finite Elements, Ne
Mode
1
2
3
4
5
Exact
1
3.53273
3.51772
3.51637
3.51613
3.51606
3.51602
2
34.8069
22.2215
22.1069
22.0602
22.0455
22.0345
3
75.1571
62.4659
62.1749
61.9188
61.6972
4
218.138
140.671
122.657
122.320
120.902
5
264.743
228.137
203.020
199.860
Source: Chopra (1995).
Table 22. Natural frequencies of a uniform cantilever beam: LumpedMass Finite
Element and Exact Solution
Number of Finite Elements, Ne
Mode
1
2
3
4
5
Exact
1
2.44949
3.15623
3.34568
3.41804
3.45266
3.51602
2
16.2580
18.8859
20.0904
20.7335
22.0345
3
47.0294
53.2017
55.9529
61.6972
4
92.7302
104.436
120.902
5
153.017
199.860
Source: Chopra (1995).
Top of column lateral diplacement (m)
164
Fig. 757. Top of pile lateral displacement comparison, 2x2 group
Time (s)
Fig. 758. Top of pile bending moment comparison, 2x2 group
26
Following this procedure for the true rectangular 9node shell element with
uniform thickness t, results in the following lumped mass distribution
Fig. 27. Lumped mass matrix at the nodes of true rectangular 9node shell element.
Numbers shown are fractions of the total element mass at each node
Remarks about the mass matrix
Cook et al. (1989) makes important remarks about the mass matrix. The first one
is the fact that the mass matrix chosen must correctly represent the mass distribution on
the element, because the product [m\{d} or must yield the correct total force on
an element according to Newtons law F = ma when {d} represents a rigidbody
translational acceleration. The second remark is about the consistent and lumped mass
matrix. While the consistent mass matrix is always positive definite, the same can not be
said about the lumped mass matrix. If it contains zeros or negative entries in the
diagonal, then it is called positive semidefinite. This may or may not cause some matrix
operations to give strange results. He also suggests that a consistent mass matrix may be
more suitable for flexural problems, while lumped mass matrices usually yield natural
frequencies that are less than the exact values.
160
run, where only the shear wave velocity for the soil layers was changed. Because the
values for the shear wave velocity for the soil were not given in the original work,
reasonable values obtained from Richart et al. (1970) were adopted. The values vi = 68
m/s and vj = 78 m/s were adopted for the upper and lower layers respectively, because
they gave the best response. The values of the soil dampers are based on these values as
mentioned on Chapter 6. No damping was added to the structure. The acceleration
record for the top ring of the centrifuge was taken as input for the FLPIER, it can be
seen in Fig. 752. The comparison between FLPIER and the test for displacements and
moments can be seen in Figs. 753 and 754 respectively. Note that the comparison is
very good for a single pile.
Table 712. Some soil properties for CSP1
Soil Type
Nevada Sand
pmax
1.76
pmin
1.41
Unit Weight (g/cmJ)
1.52 Upper layer
1.66 Lower layer
Relative Density
3545 % Upper layer
7580% Lower layer
After Wilson et al. (1997)
5.4 m
16.8 m
] M = 50 tonnes
Loose saturated Nevada sand, t = 9.1 m
Dense saturated Nevada sand, t = 7.6 m
Fig. 751. Single pile in sand
11
\{bu}T{F}dV+ J{8u}r{}dS + {8u},r{p}(
n M Eq.2.1
= J({5e}7 {ct} + {8u}7 p{m} + {Sw}Tcd {u})iV
Ve
where {8m} and {8e} are respectively small arbitrary displacements and their
corresponding strains, {F} are body forces, {d>} are prescribed surface tractions (which
are typically nonzero over only a portion of surface Se), {/>), are concentrated loads that
act at total of n points on the element, {8m) ] is the displacement of the point at which
load {/>}, is applied, p is the mass density of the material, cj is a materialdamping
parameter analogous to viscosity, and the volume integration is carried out over the
element volume Ve.
Using usual Finite Element notation, we may write the continuous displacement
field {m}, which is a function of both space and time, and its first two time derivatives,
as
{m}=[JV]{
In Eqs. 2.2 the so called shape functions [A'] are functions of space only, and the
nodal DOF {d} are functions of time only. Thus Eqs. 2.2 represent a local separation of
variables. Combination of Eqs. 2.1 and 2.2 yields
\[B]T {a)dV + \p[N]T[N]dv{d}+ \kd[N]T[N]dv{d)
Eq. 2.3
 \[N]T{F}dV {[*]>}<Â£Â£{p},
= 0
Ve Se i1 J
in which it has been assumed that the concentrated loads {p}Â¡ are applied only at the
nodal points locations. Since {Sc/} is arbitrary, Eq. 2.3 can be written as
Top of column lateral diplacement (m)
161
3
2
_o
s
Q
<
Time (s)
Fig. 752. Top ring acceleration
0.008
Fig. 753. Top displacement comparison
128
Example 5 Rectangular Reinforced Concrete Column
In order to verify the concrete model under cyclic loading a series of tests
performed by Bousias et al. (1995) was used for comparison. This test consisted of a
series of 11 tests performed to study the behavior of reinforced concrete columns
subjected to various types of cyclic loading. The tests referred as SO, SI, S2, S3, S4 and
S10 in the original work, were used as comparison because they reflect more closely the
type of cyclic loading common in dynamic analysis. Tests SO to S4 consist of a
cantilever column subjected to imposed displacements or forces in the X and Y direction
with a constant axial load. Test S10 is also a cantilever column, but the axial force (Z
direction) is not constant during the test. These directions are indicated in Fig. 712.
Table 75 has a summary of the loading parameters and concrete strength for each test.
Table 75. Loading for each test
Test
fc (Mpa)
Axial (KN)
Loading path
SO
30.75
300
Imposed X displacement
SI
29.0
212
Imposed X displacement
Imposed Y displacement
S2
31.1
284
Imposed X displacement
Imposed Y displacement
S3
29.9
310
Imposed X displacement
Imposed Y force
S4
27.7
253
Imposed X displacement
Imposed Y force
S10
28.5
Variable
Imposed X force
Imposed Z force
The details about the specimens used in the test can be found in Gutierrez et al.
(1993). The specimens had a 0.25msquare cross section and a free length of 1.50m,
and were built in as a cantilever into a 1msquare, 0.5 m thick, heavily reinforced
foundation base. Longitudinal reinforcement consisted of eight 16mmdiameter bars,
163
Fig. 755. 2x2 pile group in sand
Fig. 756. 3x3 pile group in sand
185
NPRT = O maximum displacements and maximum forces caused by the maximum
displacements (default).
NPRT = 1 all displacements and maximum forces.
NPRT = 2 maximum displacements and all forces.
NPRT = 3 all displacements and all forces.
Note that NPRT = 2 or 3 options only allow the program to compute the element forces for the
options above (because this may take some time for a large structure), to print them out you still have to use these in
addition to the print out option. For example, if you want the pile forces for every time step use 0=4 and set P=1 under
the PRINT label. If you want only the structure forces use T=1. The maximum forces are the forces caused by the
maximum displacements, note that these can be smaller than the maximum forces for the structure. For options 2 and 3
a summary of the maximum forces and the time step when it occurred will be printed out at the end.
SMASS is the concentrated mass adopted for the soil. This mass is applied to all the translational DOF X, Y
and Z to represent the attached soil mass.
NSHM is the option for the mass matrix for the cap.
NSHM = 0 consistent mass matrix (default).
NSHM = 1 lumped mass matrix.
NBMM is the option for the mass matrix for the structure and piles.
NBMM = 0 consistent mass matrix (default).
NBMM = 1 lumped mass matrix.
NDYSOL is the option for the type of numerical solution. This option is only valid for step by step solution
NDYNS=0
NDYSOL = 0 Newmark's method average acceleration (default).
NDYSOL = 1 Newmark's method linear acceleration (default).
NDYSOL = 2 WilsonTheta method.
NMSE is the option for multiple support excitation.
NMSE = 0 standard analysis.
NMSE = 1 multiple support excitation.
NCMOD is the option for the concrete model in nonlinear analysis.
NCMOD = 0 rational model 1.
NCMOD = 1 rational model 1 with crushing and cracking option ON.
NCMOD = 2 rational model 2.
NCMOD= 3 rational model 2 with crushing and cracking option ON.
NCMOD= 4 bilinear model.
D1 ...D7 this is an option if the user wants to save a specific NODE displacement to a file for possible later
plotting or checking.
D1 is the number of NODES that will me saved (maximum = 6).
D2...D7 is the number of the NODE to be saved. For example:
P=6,1,2,3,4,5,6
D1 = 6 six NODES will be saved to the file.
1 ...6 NODES 1 to 6 will be saved to the file. They do not have to be in any specific order.
The displacements will be saved in a text file with the name: 'inputname'.DSn, the velocities in
'inputname'.VSn, and the accelerations in 'inputaname'.ASn, where n is the file number for the chosen node. In the
example above if the input file is called test.in, the displacements for node 1 are saved in test.DSI, for node 2 in
test.DS2 and so on.
For the cap, add in the same line where you input the cap properties M = RMASS.
where RMASS is the mass density for the cap material.
A typical input line for step by step analysis would look like:
DYN
Y=0 C=1 F=.05,.05,.0,.0 S=100 J=1.309E09 K=1.309E09 0=1
A typical input line for the spectrum analysis would look like:
DYN
Y=1 K=1.309E09
A typical input line for the cap would look like:
CAP
E=4400 U=0.2 T=48 M=2.267E07
This section MUST end with a blank line.
5
The fiber models have been used in the study of reinforced concrete (Ala
Saadeghvaziri (1997), Hajjar et al. (1998), Park et al. (1972), and Zeris and Mahin
(1991a), Zeris and Mahin (1991b)) and steel members (Baron and Venkatesan (1969),
Chen and Atsuta (1973)). These models are based on the finite element approach, and
are better suited for members and structures under complex loading histories. In these
models the crosssection is divided into segments. Each segment can then be divided in
one or more fibers. Each fiber is assumed to obey a uniaxial stressstrain relationship.
From the integration of the stresses of each fiber over the cross section, the element
forces can be calculated, and from the evaluation of the stiffness of each fiber the
overall element stiffness can be obtained. Once the element forces and stiffness are
obtained the analysis is carried out using standard Finite Element Method procedures.
Therefore only the stressstrain relationships for concrete and reinforcing steel in the
case of reinforced concrete sections, or steel, in the case of steel sections, are necessary
to describe the properties of each section of the element. This makes these models very
effective under complex loads. The main difference among all the fiber models are the
rules adopted for the uniaxial behavior of the different materials that make the cross
section. In the case of most civil engineering structures, steel and concrete, but other
materials can also be used if the stressstrain relationships are known.
In the specific case of concrete the models backbone is the envelope curve
obtained from a monotonic test. This curve limits the concrete stresses in any loading
phase. In some models the compression envelope curve for concrete is represented by
the wellknown Hognestad parabola (Ala Saadeghvaziri (1997), Park et al. (1972)).
Another approach is to use multilinear curves to define concrete behavior in
183
*f,.baA /a + i> ba)
\f(t)dt^^YJWllkf\^ + xNj[^y B15
Two and Three Dimensions. Multidimensional Gauss rules, called Gaussian
product rules, are formed by successive application of onedimensional Gauss rules. In
two dimensions, consider the function /faj*), the formula, neglecting the error term, is
i. l.f(x,y)dxdy = X X wnj v,, I/O*.*) B16
>=1 =1
and in three dimensions, considering the fimctionX**)^) the formula becomes
, , llf(x,y,z)dxdydz = )/(**.;)/(**.*) B17
1=1 7=1 Â£=1
It is not necessary to use the same Gauss rule in all directions, but doing so is
most common.
LIST OF TABLES
Table page
21. Natural frequencies of a uniform cantilever beam: ConsistentMass Finite Element
and exact solution 21
22. Natural frequencies of a uniform cantilever beam: LumpedMass Finite Element
and Exact Solution 21
23. Recommended damping rations for structures 30
31. Curves coefficients 73
71. Results for Example 2 119
72. Design details for Example 3 121
73. Model parameters for Example 3 121
74. Design details for Example 4 125
75. Loading for each test 128
76. Parametric tests, units are KN/mm2 131
77. Parametric tests for confinement under different strain rates 135
78. Parameters for test SO units are KN/mm2 139
79. Parameters for test SI units are KN/mm2 143
710. Parameters used in test S3 and S4 149
712. Some soil properties for CSP1 160
713. Masses (tons) for pile group tests 162
Al. Mass density units 177
viii
99
is considered, the damping matrices must not only remain constant during the process,
but also be in the classical form (i.e. proportional to the mass and stiffness matrices).
How FLPIER handles Modal Analysis
The computer program FLPIER has also been implemented with modal analysis
capabilities, referring to Fig. 43 the following iterative procedure was adopted:
Fig. 43. Modal analysis of pier
In the first cycle the earthquake is applied to the structure and the initial forces at
the base of the piers are computed. Initially the springs that represent the foundation are
considered very stiff, to simulate fixed supports. Then for each column, a vector of six
22
Mass Matrix for the Shell Element
Because in FLPIER the piles cap is modeled as true rectangular 9node shell
elements, we will limit the formulation to this particular type of element.
Consistent
Consider the true rectangular 9node shell element shown in Fig. 24 below.
Fig. 24. True 9node rectangular element
Now consider the following mapping:
1
1
8
9
6
1
5
2
1
1
>B
Fig. 25. Mapping for a true rectangular 9node shell element
It is easy to verify that the shape functions N for each node are
2
developed by Drs. Hoit, Mcvay, and Hays at the University of Florida for the nonlinear
static analysis of bridge piers. Nonlinear aspects of structural analysis, such as material
and geometric nonlinearity, as well as structuresoil interaction can be incorporated into
the analysis leading to more accurate results. It can model all the components of a
bridge pier and its foundation, such as pier, pier cap, piles cap, piles, and soil, as shown
in Fig. 11. The pier, pier cap, and piles can be represented using discrete elements that
can incorporate the effects of material and geometric nonlinear behavior. More details
about the discrete element are found in Chapter 3. The piles cap is modeled as linear 9
node shell elements. The lateral soil resistance is modeled as nonlinear py springs,
while the axial resistance is modeled as nonlinear tz springs.
Fig. 11. Bridge pier components
81
z =
0.5
3 + 0.002/;
+ p,J0.002K.K,
fc 1000
4 Vs
0.5
3 + 0.29/;
+ P,J0.002K.K,
145/ 1000
4yls 13
(psi)
(MPa)
Eq. 3. 60
h' = width of concrete core measured to outside of the transverse reinforcement,
as shown in Fig. 3.24 for square, rectangular, and circular sections.
Fig. 324. Core width for different cross sections
s = centerto center spacing of transverse reinforcement
K2 = 1.48 + 0.1601ogloE + 0.0127(log10 e)2 Eq.3.61
=1.08 + 0.1601ogloE +0.0193(log10B')2 Eq. 3. 62
Note: For e' < lO05/sec, = KÂ¡ = 1.0.
In this model, KÂ¡ represents the strain rate effect on the compressive strength of
the concrete, and KÂ¡ takes care of the strain rate effect on the strain at maximum stress.
It is assumed that the strain rate effect on the slope of the descending branch of the
stressstrain diagram is similar to the strain rate effect on the compressive strength of
concrete. This is supported by test results. The model representation can be seen in Fig.
325 below. There is no change in the unloading or reloading curves.
140
Fig. 722. Imposed tip displacement history for test
Top lateral displacement (mm)
Fig. 723. Comparison tsOl
102
The equation of dynamic equilibrium for all the DOF can now be rewritten in
partitioned matrix form:
Fig. 5.1. Multiple support motion
Observe that no external forces are applied along the superstructure DOF. In Eq.
5.1 the mass, damping, and stiffness matrices can be determined from the properties of
the structure using the traditional methods of matrix analysis, while the support motions
Dg(J), Â£>g(t) and Dg(t) must be specified. Because the data that is usually available
for an earthquake is the acceleration record Dg(t), the quantities Dg(t) and Dg(t) can
be obtained by numerical integration of Dg(t), by using, for example, the Trapezoidals
mle. It is desired to determine the displacements t in the superstructure DOF and the
support forces pg(t).
Lets first separate the displacements into two parts:
C
186
Fukada, Y. (1969). A study of the restoring characteristics of reinforced concrete
buildings. Proc. of the Kanto District Symposium of AIJ, Tokyo, Japan,
November.
Gutierrez, E., Magonette, G., and Verzeletti, G. (1993). Experimental studies of
loading rate effects on reinforced concrete columns. ASCE Journal of
Engineering Mechanics, Vol. 119, No. 5 (May), pp. 887904.
Hajjar, J. F., Schiller P. H., and Molodan A. (1998). A distributed plasticity model for
concretefilled steel tube beamcolumns with interlayer slip. Engineering
Structures, Vol. 20, No. 8, pp.663, 676, Elsevier, Amsterdam.
Hays, C. O. (1975). Nonlinear dynamic analysis of framed structures with pile
foundations. ASCE National Structural Engineering Convention, April, New
Orleans, LA.
Hoit, M. I., McVay, M., Hays, C., and Andrade, P. W. (1996). Nonlinear pile
foundation analysis using FloridaPier. ASCE Journal of Bridge Engineering,
Vol.l, No. 4 (November), pp.135142.
Iwan, W. D., (1973). A model for the dynamic analysis of deteriorating structures.,
Proc. of 5th WCEE, Rome, 1973.
Kustu, O., and Bouwkamp, J. G. (1975). Behavior of reinforced concrete deep beam
columns subassembleges under cyclic loads. UCB/EERC Report 738,
University of California, Berkeley, May.
Kwak, H. G. (1997). Nonlinear response and modeling of RC columns subjected to
varying axial load. Engineering Structures, Vol. 19, No. 6 (June), pp.417424.
Magdy, S. L., and Meyer, C. (1987). Analytical modeling of hysteretic behavior of R/C
frames. ASCE Journal of Structural Engineering, Vol. 113, No. 3 (March), pp.
429444.
Mathews, J. H. (1987). Numerical methods for computer science, engineering and
mathematics. PrenticeHall, Inc., Englewood Cliffs, NJ.
Matlock, H., Foo, S. H., and Bryant, L. L. (1978). Simulation of lateral pile behavior.
Earthquake Engineering and Soil Dynamics, July, ASCE, New York, pp. 600
619.
McVay, M. C., Shang, T., and Casper, R. (1996). Centrifuge testing of fixedhead
laterally loaded battered and plumb pile groups in sand. ASTM Geotech.
Testing Journal (March), pp. 4150.
79
1) High strain rates increased the initial yield resistance, but caused small
differences in either stiffness or resistance in subsequent cycles at the same
displacement amplitude.
2) Strain rate effect on the resistance diminished with increased deformation in a
strainhardening range.
3) Non substantial changes were observed in ductility and overall energy
absorption capacity.
Otani (1980) also suggests that the strain rate during an oscillation is highest at
low stress levels, and gradually decreases toward a peak strain. Cracking, crushing and
yielding will contribute to a reduction in the systems stiffness, elongating the period of
oscillation. Such damage is caused by the lower modes of vibration having long periods.
Therefore, the strain rate effect is small in earthquake analysis, and has a small effect on
the response. So, the static hysteretic behavior can be used in a nonlinear dynamic
analsyis of reinforced concrete structures.
Confinement Effect
The confinement of reinforced concrete columns is a good way to increase its
strength and ductility, as shown in Fig. 323.
Fig. 323. Confined and unconfined concrete models response
74
o
0.5
1
stress
(ksi)
1.5
2
2.5
0.0006 0.0005 0.0004 0.0003 0.0002 0.0001 0
strain (in/in)
Fig. 316. Typical unloading in compression
1
l I I I 1 1
0.5 I ; 
0
stress
(ksi) /
0.5 y i
1 _y_ ^^1
1.5 / I 
2 L i l J
0.0006 0.0005 0.0004 0.0003 0.0002 0.0001 0
strain (in/in)
Fig. 317. Compression unloading with gap
134
The following comments can be made about these monotonic tests:
a) Note that when the modulus of elasticity is changed there is no significant
decrease in the column strength, note however that reducing the modulus of elasticity of
steel has a more significant influence than reducing the modulus of elasticity of concrete
(Fig. 714).
b) From Fig. 715 note that by reducing by half the concrete strength decreases
the column capacity, but not in the same proportion. Also note that by reducing the steel
strength by half decreases the column strength by a larger amount than in the concrete
case.
c) From Fig. 716 note that reducing both values of the modulus of elasticity by
half makes the model unstable, which is expected. However decreasing the strengths by
half, decreases the column capacity by approximately the same proportion, as expected,
but the model did not became unstable.
d) Finally by decreasing all the parameters by half, the column strength and
stiffness is decreased by half as expected.
The file bml_5.in did not converge. This is the test where the elastic modulus of
steel and concrete were decreased by half, but the original values for stresses were
maintained. Note that decreasing drastically one of the parameters, while keeping the
other constant, may cause instabilities to the model.
In order to illustrate the strain rate effect, another set of tests was done
considering confinement at different strain rates. The strain rates were changed form 1 O
5/s (very slow) to 1/s (very fast). The concrete and steel properties were not changed,
and are those of file bml.in as described in Table 76. Table 77 helps to identify the
155
Top lateral displacement (mm)
Fig. 747. Comparison tslOl x test
86
Modal and Spectral Matrices
The N eigenvalues, N natural frequencies, and N natural modes can be assembled
compactly into matrices. Let the natural mode <> corresponding to the natural frequency
co have elements
displayed in a single square matrix, each column of which is a natural mode:
11 12 4*1 V
The matrix O is called the modal matrix for the eigenvalue problem, Eq 4.5. The
N eigenvalues co2 can be assembled into a diagonal matrix Q2, which is known as the
spectral matrix of the eigenvalue problem, Eq. 4.5:
Eq. 4. 9
Each eigenvalue and eigenvector satisfies Eq. 4.5, which can be rewritten as the
relation
n2 =
Â£ = moj2 Eq. 4. 10
By using the modal and spectral matrices, it is possible to assemble all of these
relations into a single matrix equation:
Eq. 4. 11
76
o
0.5
stress
(ksi)
1.5
2
2.5
0.0006 0.0005 0.0004 0.0003 0.0002 0.0001 0
strain (in/in)
Fig. 319. Typical loading, unloading and reloading in compression
When concrete goes into tension, if it goes back to compression it will reload
only when the gap resulting form the previous cycle is closed. The process can be
visualized in Fig. 320.
Fig. 320. Concrete behavior with gap
Moment (KN.mm)
132
Top lateral displacement (mm)
Fig. 715. fc and fy changed
39
< oo Eq. 2. 74
Tm
This implies that the average acceleration method is stable for any At, no matter
how large, as mentioned before; however, it is accurate only if At is small enough. For y
= 14 and p =1/6, Eq. 2.73 indicates that the linear acceleration method is stable if
<0.551 Eq. 2. 75
T
n
For the solution of displacements, velocities and acceleration at time j+1 we
now consider the equilibrium equation also at time i+l:
Mm+Cm+Kum=Fm Eq. 2. 76
Solving Eq. 2.72 for w,+]in terms of w,+i and substituting in Eq. 2.71, equations
for M+1and +1 in terms of the unknown displacements ,+/ only are obtained.
Substituting these equations into Eq. 2.76, a system of equations is obtained, which can
be solved to obtain m,+A/:
(b0M + btC + = Fm + M(b0u, + b1: +b,l)+C(b,ul +bl +biiii)Eq. 2. 77
where
bn ~~
pAt
A =
pAt
A = ;6, =l;h4
2 PAt 3 2p 4
li
At 2 /2 Eq. 2. 78
and finally all the quantities at time i+1 can be written as
m =b6(uMul)+b1,+bsiii
Eq. 2. 79
717. All parameters changed 133
718. Confinement and Strain rate effect 135
719. Imposed displacement history for the first 90 seconds 137
720. Comparison FLPIERx Test, original properties 138
721. Comparison FLPIER x Test, modified properties 138
722. Imposed tip displacement history for test 140
723. Comparison tsOl 140
724. Comparison ts02 141
725. Comparison ts03 141
726. Comparison ts04 142
727. Imposed displacement in X direction 142
728. Imposed displacement in Y direction 143
729. Comparison tsl 1 144
730. Comparison tsl lc 144
731 .Comparison tsl2 145
732. Comparison tsl2c 145
733. Imposed displacements in X direction 146
734. Imposed displacements in Y direction 147
735. Comparison test S2 x FLPIER 147
736. Imposed X displacement for test S3 149
737. Imposed Y forces for test S3 149
738. Comparison Test S3 x FLPIER ts31 150
739. Comparison Test S3 x FLPIER ts32
xiii
150
Moment (KN.mm) Moment (KN.mm)
156
Top lateral displacement (mm)
Fig. 748. Comparison tsl 02 x test
Top lateral displacement (mm)
Fig. 749. Comparison tsl03 x test
36
The second category, the implicit methods, has the form
{0L,=/({Â£>L,R+1{^) e260
and hence computation of {D}+1 requires knowledge of the time derivatives of {Z)}+1,
which are unknown. The main characteristic of the implicit methods is the fact that the
next approximation for the displacements depends on unknown values of their
derivatives. Note that the equilibrium condition is used at time step n+1. Newmarks
and the WilsonTheta are examples of implicit methods.
There is vast literature about the advantages and disadvantages of each approach,
the reader is referred to Cook et al. (1989), Chopra (1995), Paz (1985), Craig (1981), or
Crandall (1956), for a more extensive discussion on these approaches. Generally
speaking under certain conditions the implicit methods are more stable than the explicit
methods. Because an implicit method was used in this research we will limit our
discussion to this class of methods.
Numerical Evaluaton of Dynamic Response. Newmarks Method
As mentioned earlier implicit methods are those in which the approximation for
the next displacements {D}+Â¡ depends on unknown values for its time derivatives. The
main advantage of the implicit methods is the fact that most of the useful methods are
unconditionally stable and have no restriction on the time step size other than required
for accuracy.
In 1959 N. M. Newmark developed a family of timestepping methods based
assumptions for the variation of the acceleration over the time step. The first method is
20
The nondimensional parameters used in these equations are
Eq. 2. 26
where L is the length of the beam. The matrix N can then be substituted into Eq. 2.5, and
performed integration over the whole volume of the element. The resulting 12 x 12
consistent mass matrix is given by
_13 6/z
35
0
0
0
ILL 4
210 + IQAL
0
9 64
70 SAL2
0
0
0
131 4
42D+~\0AL
13 64,
35 "m?
0
in 4
210 104
0
9
70 5/1Z.2
0
13L ly
420_ld4L
0
105
0
0
0
13L 4
~420+104Z;
0
JLJl.
140 304
0
2J:
1Q5+"IÂ¡
0
131 4
3
13 64
420~ 104Â£
35+S4iF
0
0
0
0
0
0
0
0
0
L2 4
0
UL 4
140 30A
210 10AL
J3 6ly
35+S?
Jx
14
UL ly 2Iy
210 + 104Z. 105 + 154
0 0 0
Â£. 2/
1Q5"^Ts4
Eq. 2. 27
where the matrix terms with the moments of inertia Iy or Iz represent rotatory inertia and
the terms with the polar moment of inertia Jx represent torsional inertia of the element.
Lumped
The lumped mass matrix for the uniform beam segment of a threedimensional
frame element is simply a diagonal matrix in which the coefficients corresponding to
translatory and torsional displacements are equal to onehalf of the total inertia of the
beam segment while coefficients corresponding to flexural rotations are assumed to be
34. Rectangular section with integration points. After Hoit et al., 1996 56
35. Circular section with integration points. After Hoit et al, 1996 57
36. Secant and tangent material stiffness 60
38. Elasticperfectly plastic model for mild steel 64
39. FLPIER concrete points 65
310. Concrete strains 66
311. Concrete stresses 66
312. Envelop curve for concrete 68
313. Typical compression loading 70
314. Typical loading in tension 71
315. Typical unloading in tension 73
316. Typical unloading in compression 74
317. Compression unloading with gap 74
318. Compression unloading with no gap 75
319. Typical loading, unloading and reloading in compression 76
320. Concrete behavior with gap 76
321. Bilinear model for concrete 77
322. Stressstrain curve for concrete in FLPIER 78
323. Confined and unconfined concrete models response 79
324. Core width for different cross sections 81
325. Confined concrete model 82
41. Generalized SDF system for the nth natural mode 90
42. Typical response spectrum 96
xi
158
cut off for the maximum and minimum values for the moments (flat portion of test data
in the Figures), but the original work does not mention anything about it.
The following paragraphs summarize the observations made for the examples
presented in this section:
a) In the beginning of the modeling phase, when 16 elements were included in
the analysis, the system became unstable, because additional hinges formed within the
additional elements. It was also observed that the model is very sensitive to the size of
the time step used in the analysis. For cycles of very short duration, like in example SO,
very small time steps should be used in order to obtain a correct response.
b) Although reducing the values of the modulus of elasticity for steel and
concrete improves the model behavior, it may also become unstable for large
displacements.
c) It is clear that for larger displacement amplitudes the bilinear model is not a
good representation of the steel behavior. Note the change in the shape of the hysteresis
graphics for the larger amplitudes. This suggests that a more accurate model for the
steel, including strain hardening and possibly cyclic degradation, will result in a better
response. The introduction of a concrete gap seems to model the member stiffness
degradation very well.
d) The changes in the tangent elastic modulus suggested by Soroushian et al.
(1986), were not used since it is was found that the model is very sensitive to such
changes.
e) The gap model suggested in Chapter 3 was abandoned because it introduced
instability in the analysis.
32
where the equation implies equilibrium in the direction of the slave
displacements. Solving Eq. 2.45 for qi, we obtain
which can be used to eliminate qi from the problem formulation.
Equation 2.46 can be regarded as a constraint equation, so that the complete
displacement vector q can be expressed in terms of the master vector in the
form
q = Pq2 Eq. 2. 47
where P is a rectangular constraint matrix having the form
/
k;! k
Eq. 2. 48
in which I is a unit matrix of the same order as the dimension of qi. Introducing
Eq. 2.47 into Eqs. 2.40 and 2.41, we obtain
V =
Eq. 2. 49
r=
i
M2q2
Eq. 2. 50
where the reduced stiffness and mass matrices are simply
K, = PrKP = K22 Kl2Kn'K2t Eq. 2. 51
Mi=PtMP=M22 K2]K~' M2I MI2KU'K2I + K^K;Â¡ MuK;,'K21 Eq. 2. 52
The matrix M) is generally known as the condensed mass matrix.
What is being sacrificed as a result of the condensation process?
To answer this question, let us consider the complete eigenvalue problem, which
can be separated into
+^i2?2 =^(Mu
K2lqt +K12q2 =\(M2lq, + M22q2) Eq. 2. 54
Solving Eq. 2.54 for q2, we have
q i = (k/Vfj, 31, ){Kn XM22 )q2
Eq. 2. 55
Moment (KN.mm) Moment (KN.mm)
150
Top lateral displacement (mm)
Fig.738. Comparison Test S3 x FLP1ER ts31
Top lateral displacement (mm)
Fig.739. Comparison Test S3 x FLPIER ts32
62
Mo (1988) are shown in Fig.37. It appears that most of the available models are aimed
at a particular type of component, such as for use of beams, columns or shear walls only,
and therefore, fall short of the versatility required for modeling practical buildings
having a large number of different components. Most of these models were obtained by
performing curve fits to experimental data. This approach results in really good
approximations for the behavior of the specific member being studied, but lacks
versatility when applied to a real structure.
The advantage of the discrete element with fiber modeling is that the elements
general behavior will be governed by its material properties, instead of experimental
observations, making this procedure useful for studying multiple cross section
configurations under more general load histories.
Material Models
Rather than trying to develop a new element that would model this behavior
using elasticity and plasticity theories, it was thought to change the discrete element
used in FLPIER to develop such behavior. Based on various studies and experiments
(Chen (1982), Park and Paulay (1975), Mo (1994), Roufaiel and Meyer (1987), Park et
al. (1972), Agrawal et al. (1965), Sinha et al. (1964a), Sinha et al. (1964b), Kwak
(1997), Magdy and Mayer (1985), Soroushian et al. (1986), Nilsson (1979), Ozcebe
(1989), Peinzien (1960), and Tseng (1976)) about the uniaxial cyclic behavior of
concrete and steel, and the behavior of R/C members, the stressstrain curves of these
materials were modified to work for dynamic analysis. Shear deformations were not
included because it was not included in the works cited. Moreover it was also not
included in the formulation of the discrete element. By using this approach it was
763. Comparison test x FLPIER with CPT soil properties 169
764. Comparison test and FLPIER with SPT soil properties 169
765. Comparison test and FLPIER with original soil properties 170
Bl. (a) Trapezoidal approximation using the abscissas 1 and 1. (b) Trapezoidal
approximation using abscissas xÂ¡ and xÂ¡ after Mathews, 1987 179
xv
BIOGRAPHICAL SKETCH
Cesar Fernandes, Jr., was bom in 1968 in Rio de Janeiro, RJ, Brazil, where he
lived until he was thirteen. Then his family moved to Sao Paulo, SP, where they still
live. After attending junior and high school in Sao Paulo, he began attending the Escola
de Engenharia Maua, an engineering school, in Sao Caetano do Sul, SP, from 1985 to
1990, when he got his bachelors degree.
While working for the Ferraco Company in Sao Paulo, in 1991, the author
entered the graduate school of the Universidade de Sao Paulo from which he received
his of master of science degree in 1995. At the same time he applied for admission to
the graduate school at the University of Florida, Gainesville, FL, USA. From 1996 to
1999 he worked on his dissertation and finally in 1999 he received the degree of doctor
of philosophy from the University of Florida.
191
159
f) The bilinear model was not tested because the approximation obtained with
the rational model was very reasonable.
g) It is essential that the effects of anchorage slip be included in the nonlinear
cyclic analysis of reinforced concrete members subjected to extreme cyclic loading.
Example 6 Piles in Sand
In this example the response given by FLPIER considering the soil effects is
compared to tests conducted by Wilson et al. (1997). This was a series of dynamic
centrifuge tests to investigate soilpilestructure interaction in liquefiable sand. The
models consisted of structures supported on single piles, and 2x2 and 3x3 pile groups.
Although in the original work a variety of tests were performed, three tests were chosen
for comparison. From the series of tests referred as CSP2, a single pile with a single
column called SP, a 2 x 2 pile group with single a column called PG2, and a 3 x 3 pile
group with single column called PG3, were selected for comparison. In the original test
the models were subjected to a series of shaking events, with liquefaction occurring
during the stronger events. In this work, to avoid the effects of liquefaction not yet
included in the model, the structure is subjected only to the first event on the series, and
the results are then compared. The structure is assumed to behave linearly in all the
tests. The three tests are described next.
Test SP
System SP simulates a superstructure with a mass of 50 tonnes (1 tonne = 1000
kg) supported by a single steel pipe pile 0.67 m in diameter, 16.8 m long, and with a 19
mm wall thickness. The column height is 5.4 m. Figure 751 shows the model used in
FLPIER. The soil data for all the tests is described in Table 712. A set of five tests were
91
For given dynamic forces defined by p{t), the dynamic response of a MDOF
system can then be determined by solving Eq. 4.25 or 4.27 for the modal coordinates
q(t). Each modal equation is of the same form as the equation of motion for a SDOF
system. Thus the solution methods and results available for SDOF systems can be
adapted to obtain solutions qjf) for the modal equations. Once the modal equations
have been determined, Eq. 4.18 indicates that the contribution of the nth mode to the
displacement u(t) is
MO = 4> (')?(') Eq. 4. 28
and combining these modal contributions gives the total displacements:
(0 = >(0 = >?(0 Eq. 4. 29
n= 1 =1
Calculating all <)> in Eq. 4.6 seems to be a prohibitive computational expense for
large systems. But for many problems, the higherfrequency modes participate little in
the structural response and therefore only a small number of lowfrequency modes need
to be used. Thus only the first m equations of Eq. 4.6, where typically m N, are
solved and the transformation is approximated by
m m
u(t) = ^u(0 = yi<7(f) where m
it1 n.
This procedure is known as classical modal analysis or the classical mode
superposition method because individual (uncoupled) modal equations are solved to
determine the modal coordinates q(t) and the modal responses u(t), and the latter are
combined to obtain the total response u(t). More precisely, this method is called the
classical mode displacement superposition method because modal displacements are
171
It is expected that once soil degradation is added to the analysis, an almost perfect match
will be obtained. It is interesting to note that FLPIER gave good approximations for a
rather complicated model of nonlinear dynamic analysis. Note the presence of
prestressed battered piles and four soil layers. However, modeling was easy and straight
forward using the FLPIER program.
Moment (KN.mm) Moment (KN.mm)
138
Top lateral displacement (mm)
Fig. 720. Comparison FLPIER x Test, original properties
15 10 5 0 5 10 15 20
Top lateral displacement (mm)
Fig. 721. Comparison FLPIER x Test, modified properties
166
On the other hand a very good approximation for the moment was obtained from
FLPIER as can be seen in Fig. 758. For the 3 x 3 group such a good approximation was
not possible. FLPIER over estimated the moments by about 50 percent. Note that as the
number of piles is increased the results for the moments get worse. This is expected,
since the group influence, not included in the FLPIER model, is more predominant. It is
clear that more study is necessary in this area, but note that the values obtained for the
FLPIER approximation are perfectly good for a predesign. The advantage of using
FLPIER is that the pile and structure configurations can be easily changed, and once the
displacements are in the desirable range a more sophisticated analysis can be done. In
the future, with a larger database, the FLPIER soil model can be calibrated to give even
closer approximations for earthquake analysis.
Example 7 Mississippi Dynamic Test
The last test is one of the few real scale tests done to analyze the dynamic
response of a pile group. The actual data for the test was obtained from the East
Pascagula River test program report. The tests details can be found in Brown (1998).
Basically the tests consists of six prestressed rectangular piles subjected to a pulse load.
Four of the piles were battered. The structure is shown in Figs. 760. The load history,
applied to the piles cap as illustrated in Fig. 760, is shown in Fig. 761. The
displacement history for the piles cap is shown in Fig. 762. The test results were then
compared to the FLPIER program using different sets of soil properties. The first and
second comparison were based on CPT and SPT soil properties estimates. In the third
comparison the proposed original soil properties were used.
CHAPTER 2
NONLINEAR DYNAMIC ANALYSIS
Theory
In a static problem the frequency of the excitation applied to the structure is less
than one third of the structures lowest natural frequency. In this case the effects of
inertia can be neglected and the problem is called quasistatic. For such problems the
static equations [Â£]{ D} = {/?} are sufficiently accurate to model the response, even
though the loads {R}, and displacements {Â£>}, vary (slowly) with time. The static loads
{R} may result from surface loads and /or body forces.
On the other hand if the excitation frequencies are higher than noted above or if
the structure vibrates freely, the inertia effects must be considered in the analysis. The
inertia effects are accounted for by the mass matrix, written as [m] for an element and
[M] for a structure, which is a discrete representation of the continuous distribution of
mass in a structure. The effects of damping, if important, are accounted for by the
damping matrices [c] and [C].
The dynamics problems can be categorized as either wave propagation problems
or structural dynamic problems. In wave propagation problems the loading is often an
impact or an explosive blast. The excitation and the structural response are rich in high
frequencies. In such problems we are usually interested in the effects of stress waves.
Thus the time duration of analysis is usually short and is typically of the order of a wave
9
182
Table Bl. GaussLegendre abscissas and weights
N
Abscissas xv, *
Weights ws.k
Truncation error
E(f,N)
2
0.5773502692
+0.5773502692
1.000000000
1.000000000
/(4)M
135
3
0.7745966692
0.00000000000
0.555555556
0.888888888
/(6)(0
15,750
4
0.8611363116
0.3399810436
0.3478548451
0.6521451549
/(8)(c)
3,472,875
5
0.9061798459
0.5384693101
0.00000000000
0.2369268851
0.4786286705
0.5688888888
/'%)
1,237,732,650
6
0.9324695142
0.6612093865
0.2386191861
0.1713244924
0.3607615730
0.4679139346
/<12)(c)213[6!]4
[l2lf 13!
7
0.9491079123
0.7415311856
0.4058451514
0.00000000000
0.1294849662
0.2797053915
0.3818300505
0.4179591837
/(,4>(c)2,5[7!l4
[l4!fl5!
8
0.9602898565
0.7966664774
0.5255324099
0.1834346425
0.1012285363
0.2223810345
0.3137066459
0.3626837834
/
[l !]3! 7!
Source: Mathews (1987).
Theorem (Interval transformation for GaussLegendre rules): Let xv* and
Wjv./t be the abscissa and weights for the Appoint GaussLegendre rule on the interval 1
< x < 1. To apply the rule to integrate f(t) on a < t < b, use the change of variables
and dt=dx. B13
2
Then the integral relationship is
'a + b ba\ba
a+b ba
I = + x
2 2
f/(t)dt = \f\
2 2
dx
B14
and the GaussLegendre rule for [a,6] is
50
iterations within each time step proceed until the value of the outofbalance force
is smaller than a specified tolerance t. The mass matrix M and damping matrix C
are considered to be constant throughout the analysis.
As mentioned earlier in this Chapter the correct choice of time step, which
depends on the lowest natural frequency of interest for the system, enforces stability and
accuracy to the analysis. Note however that for a nonlinear analysis, the case is that due
to yielding of steel or crushing and cracking of concrete, a lower stiffness will be
obtained during the analysis, which will result in even lower values for the lowest
natural frequency of interest. The reader should have this in mind when choosing an
adequate time step for a nonlinear dynamic analysis.
56
must be evaluated at those points specified by the position factor. These factors are then
multiplied by the appropriate weighting factors and the products accumulated. Fig. 34
shows a square section with 25 integration points (a 5 x 5 mesh). The actual number of
defaults integration points for a square section is set at 81 (a 9 x 9 mesh). For a steel bi
section the default number of points is 60. For circular sections, the section is divided
into circular sectors (12 radial divisions and five circumferencial divisions as shown in
Fig. 35), totaling 60 points. The sections are integrated at the centroids of each sector
using weighting factors of 1.0. The stress in all steel bars is evaluated at the centroid and
a weighting factor of 1 is used for each bar. When a circular void is encountered in a
square section, the force is first computed on the unvoided section and then the force
that would be acting on the voided circular area is computed and subtracted from the
force computed for the unvoided section. Circular sections with voids are divided into
sectors omitting the voided portion. This method of dividing the sections into points,
getting the strains and stresses at each point and then integrating to get forces is usually
called fiber modeling.
s
X
X
X
X
X
X
X
X
.
r
X
X
X
X
X
X
X
x Concrete Integration Points
Steel Rebar (lxl Integration)
Fig. 34. Rectangular section with integration points. After Hoit et al., 1996
109
response of the structure is often a purely structural fundamental" mode that does not
capture the accurate response of the foundation. The primary soil response mode
approximates a freefield shear deformation in which the superstructure is displaced in a
nearrigidbody manner. The two modes may not be closely coupled and therefore
driving the design of the piles with the response of the superstructure may not accurately
estimate the actual response of the entire system. In the absence of more rational
analysis data, the uncertainties present in the simplified analysis make it difficult to
predict whether the resulting pile foundations are over or underdesigned. It is therefore
necessary to develop models that can account for the coupling between foundation and
structure.
A simplified coupled solution is to select a single pile ifom the pile group with
the superstructure being modeled as single degree of freedom system having the same
period as the fundamental period of the structure, and a mass equal to the contribution of
the superstructure to each pile. The pile is then subjected to displacement time histories
previously calculated from a dynamic site response analysis and forces and defections
are calculated. There are basically two methods for applying the displacement histories
to the piles: the Uncoupled Method and the Coupled Method. Both methods are
described next.
Uncoupled Method
This method is illustrated in Fig. 61 (a). The analysis is done in two steps:
1) The computation of the free field motions;
2) The prediction of the pilesuperstructure response.
90
Mnqn + Cnq + Knql: = Pn(t)
Eq. 4. 25
where C is given by
Eq. 4. 26
This equation governs the response of the SDOF system shown in Fig. 41.
Dividing Eq. 4.25 by M gives
Eq. 4. 27
where t,n is the damping ratio for the nth mode. The damping ratio is usually estimated
on experimental data for structures similar to the one being analyzed. Equation 4.27
governs the nth modal coordinate q(t), and the parameters M, K, C and P(t) depend
only on the nthmode <>, not on other modes. Thus we have JV uncoupled equations like
Eq. 4.25, one for each natural mode. In summary, the set on N coupled differencial
equations 4.17 in nodal displacements Uj(t) has been transformed to the set of N
uncoupled equations (Eqs. 4.25) in modal coordinates q(t).
qn(t)
Fig. 41. Generalized SDF system for the nth natural mode
165
Time (s)
Fig. 759. Top of pile bending moment comparison, 3x3 group
The comparison between FLPIER and the test is shown in Figs. 757 and 758
for the 2 x 2 group and in Fig. 759 for the 3 x 3 group (the displacement data was not
available for this configuration). For the 2x2 group FLPIER under predicted the
displacements as it can be seen in Fig. 757. Flowever when the magnitude of these
displacements is compared to the structure dimensions, we notice that they are very
small. Considering that we are trying to predict a maximum displacement of about 0.002
m for a structure that is about 30 m (from the bottom of the piles to the top of the piers),
the approximations are very good. Some error could also have been introduced by
incorrect measurement of this data.
103
Eq. 5. 2
In this equation Lf is the vector of structural displacements due to static
application of the prescribed support displacements Dg at each time instant. The two are
related through
k
kl
Eq. 5.3
where pg are the support forces necessary to statically impose displacements Dg that
vary with time; obviously, D* varies with time and therefore is known as the vector of
quasistatic displacements. Observe that pg = 0 if the structure is statically determinate
or if the support system goes rigidbody motion; for the latter condition an obvious
example is identical horizontal motion of all supports. The remainder D of the structural
displacements are known as dynamic displacements because a dynamic analysis is
necessary to evaluate them.
With the total structural displacements split into quasistatic and dynamic
displacements, Eq 5.2, we return to the first of the two partitioned Eq. 5.1.
mb' + mgDg + cD +cgDg + kD' + kgDg = 0 Eq. 5. 4
Substituting Eq. 5.2 and transferring all terms involving Dg and D' to the right
side leads to
mD + cD + kD = prjf (/)
Eq. 5. 5
106
Note that the support DOF, 4 and 5, are numbered last, according to Eqs. 5.1. By
applying unit displacements to DOF 4 and 5 we get the 4th and 5th columns of k shown
below
'24
61
61
12
12
El
6 L
&L2
21}
61
0
6 L
21}
8
0
61
12
61
0
12
0
12
0
6 L
0
12
If we make kg = [^gli > kg 2 ], where kgj and kgj are respectively the 4th and 5th
columns of k, and the influence matrix ( = [f ,,f 2] (note that f is Nx Ng), by solving
k(l: = kgJlwe get nth column of (.. This procedure can be extended to 3D analysis as
well. In this case there are six possible directions for the motion of each support,
however in most cases the horizontal movement controls the analysis. In FLPIER the
support motions are applied to the soil springs. In the case of the simplified pile shown
in Fig. 53, considering the DOF illustrated, the support stiffness kg (8 x 4) is then given
by Eq. 514.
*2 io
kÂ¡ 11
k< 12
J4H42
Fig. 53. Pile subjected to multiple support excitation
7
this work. FLPIER already incorporates a nonlinear discrete element, which uses fiber
modeling at two points along the elements length to characterize its nonlinear behavior.
New stressstrain curves were introduced to allow the nonlinear discrete element to
perform dynamic analysis. The details can be found in Chapter 3.
The last class of analytical methods is the finite element method. In this class of
methods different elements are used to represent the structural members, such as truss
members to represent the reinforcing steel, and plane stress elements to represent the
concrete. The cracking typical of concrete represents a computational difficulty for these
models, requiring the development of more sophisticated elements. The development of
such elements is a challenge based on complex elasticity and plasticity theories. Like
fiber modeling, this class of methods is computationally expensive and time consuming.
Limitations
It is very difficult for a model to incorporate all the aspects inherent to nonlinear
dynamic analysis, and the model presented here is no exception. The first limitation is
the fact that all the theory developed in Chapter 2 for nonlinear dynamic analysis is
based on small displacement theory. The second limitation comes from the fact that the
effect of shear deformation is not included in the constitutive models. It was also not
included in the original derivation (Hoit et al., 1996). Problems of local buckling are
also outside the scope of this work.
Organization
In Chapter 2 all the theory necessary for the formulation of the problem is
presented. In Chapter 3 the derivation of the discrete element is described and the
34
Structure
\
k
Condensed
structure
Piles
Piles
Fig. 2.8. Full and condensed versions of the structure
TimeHistory Analysis. Direct Integration Methods
In direct integration methods or stepbystep methods, a finite difference
approximation is used to replace the time derivatives appearing in Eq. 2.12 or 2.13 at
various instants of time. Direct integration is an alternative to modal analysis methods.
For many structural dynamics and wave propagation problems, including those with
complicated nonlinearities, direct integration is easier to implement. In direct
integration, the approach is to write the equation of motion (2.12), at a specific instant in
time,
Eq. 2. 57
where the subscript n denotes time nt\t and A/ is the size of the time increment or time
step. The absence of time step subscripts on matrices [M\, [C], and [K\ in Eq. 2.57
113
unloading path. Once the gap develops at a particular level due to lateral loading, the
axial performance of the pile is affected because the development of shearing resistance
is no longer permitted at that location, however this effect is not taken into account in
the actual dynamic version of FLPIER at this time.
p(KN/m)
Fig. 63. Cyclic soil model
Cyclic Degradation
Cyclic degradation is specified through the use of Eq. 6.1 (ONeill et al., 1997):
Eq. 6.1
In this equation X is a soil degradation parameter specified by the user, pc is the
soil resistance (e. g,,p for py curve corresponding to a given value of deflection y) for
the current cycle of loading, pp is the value of resistance corresponding to the present
vale of y on the previous cycle of loading, and p is the fully degraded value of p at the
present value of y. The analysts inputs the back bone, or first cycle, py curve and also a
fully degraded py curve. In FLPIER the initial py curve can be automatically generated
by the program based on the soil properties. The fully degraded py curve is given by a
100
forces is generated, the three forces Fx, Fy, and Fz in the x, y, and z directions, and the
three Mx, My, and Mz respective moments.
Then each of these forces is applied to the foundation, one at a time, like in a
regular static analysis. This will produce three displacements, dx, dy, and dz, in the x, y,
and z directions, and three respective rotations Oy, Qy, and 0z, at the base of each
column. These define the first column of the flexibility matrix for the foundation. After
all six forces are applied we have the six by six flexibility matrix for the foundation, one
for each column. Inverting this matrix we obtain the new stiffness for the foundation,
which becomes the foundation springs for the base of each pier for the next cycle.
The analysis is carried out until two consecutive base forces for one pier are the
same within a tolerance. To compare vectors we use norms, also called absolute values.
The 2norm2 of two consecutive base force vectors is computed, and if within a
stipulated tolerance, the analysis is terminated and the final forces are printed.
It is important to note that in this analysis the structure is always considered to
be linear, the main prerequisite for modal analysis of any type. However, the springs
generated for each cycle will reflect the characteristic nonlinear behavior of the
foundation. This is an approximate method used for bridge pier analsyis.
2 The 2norm of a vector is defined as: If = u)? then the 2norm of u is
2 = 4u'u.
23
Ai5=(^ + l)(Ml)(ril)ri
Ni = 4 (t1 + OCn i)nri
a^6 = Cm + i)(n)(n i)(n +1)
N> = 4 (f + iXn + i)nn
A^^M + lXHlXTl + l^ + l
Nt =(nl)(t + l)nii
a^8 = ^ O^Xm iXn + iXti O
w9 = (n + 1)(M l)(ri + IXri 1)
Eqs. 2. 29
Fig. 26. Shell element of uniform thickness
If now we consider for each node the six DOFs illustrated in Fig. 26, which
define the shell element, the matrix of shape functions [A], recalling again the
relationship {}= [JV]{d}, takes the form
cap lateral displacement (in) cap lateral displacement (in)
169
Fig. 763. Comparison test x FLPIER with CPT soil properties
Fig. 764. Comparison test and FLPIER with SPT soil properties
37
called average acceleration and is shown in Fig. 29. Successive application of the
Trapezoidal Rule leads to Eqs. 2.61 to 2.65.
ii
t
At
Fig. 29. Average acceleration
"(t)= 2
Eq. 2. 61
Eq. 2. 62
At
um =/ +y(w +,)
Eq. 2. 63
2
/ \ T
= UÂ¡ + W,T + (M + Â¡)
Eq. 2. 64
At2
UM =U,+ u,At + (uM + u,)
Eq. 2. 65
The second method is called linear acceleration and is illustrated in Fig.210.
The Trapezoidal Rule is also used to obtain Eqs. 2.66 to 2.70.
Fig. 210. Linear acceleration
CHAPTER 6
SOIL STRUCTURE INTERACTION
The earthquake response of bridge structures depends on the soilpilestructure
interaction during the earthquake loading. The subject has been extensively studied by
many researchers (Nogami (1997), Prakash (1997), Turner (1995), Bandoni (1996),
Reese (1974), Matlock (1978), Anderson (1972), Ting (1987), and McVay et al.
(1996)). There are two basic limitations in these studies. The first one is the fact that
most of them were done for single pile structures, making it impossible to account for
multiple pile interaction effect, characteristic of a real structure. The second one is that
because of obvious cost problems, large scale tests are rarely done. Another limitation is
the fact that the original data for most of these studies is not available.
In FLPIER, the soilstructure interaction can be accounted for by the use of
equivalent nonlinear springs. In particular, the complicated soilpilesuperstructure
interaction is captured by determining the primary structural earthquake response, and
then driving this response into the foundation system. Thus, the coupled problem of soil
structure interaction is solved in terms of structure first and foundation system second,
as mentioned in chapter 5. This procedure is considerably simpler than performing a
completely coupled solution process; however, there are some approximations and
uncertainty in the simplified procedure.
Abghari and Chai (1995) suggest that the main source of uncertainty in the
present approach can be illustrated by a simple modal response argument. The primary
108
43
Fig. 211. Secant and Tangent approaches. After Chopra(1995)
These errors can be minimized by using an iterative procedure within each time
step. The idea is to guarantee dynamic equilibrium before going to the next time step.
We must then solve the equation
kuM = pit) Eq.2.93
where now the effective stiffness k, becomes
k, = k, + b0m + btc Eq. 2. 94
For convenience we drop the subscript i in kÂ¡ and replace it by T to emphasize
that this is the tangent stiffness, and we can rewrite Eq. 2.94 as
kT = kT + b0m + bÂ¡c Eq. 2. 95
The effective force p{t) is now given by
p(0 = /(0f+i +m(b2,+b3,) + c(bil+bsuM)rl Eq.2.96
where rÂ¡ is the internal force at time step
59
Element Stiffness
As mentioned in Chapter 2 the element stiffness may change for either nonlinear
static or dynamic analysis. Therefore its is necessary to evaluate the element stiffness
for each iteration in each timestep. The procedure adopted uses the standard definition
of the stiffness matrix; for an element having n DOF the stiffness matrix is a square
matrix [X] of dimensions n x n, in which K,Â¡ is the force necessary in the z'th DOF to
produce a unit deflection of the yth DOF. The stiffness computed is that obtained by one
of the two methods described below. The transformation of the discrete element
stiffness matrix to global coordinates and the assembly of the different components of
the global stiffness matrix follow standard direct stiffness procedures.
Secant and Tangent Stiffness of the Discrete Element
During the iteration process, the element stiffness matrix is reevaluated in each
new deformed position. For each iteration, the stiffness for each integration point along
the cross section within an element is stored. Then, on 12 subsequent passes, a unit
displacement is applied to each element DOF keeping all other DOF fixed and the
forces corresponding to that unit displacement are calculated over the cross section of
the element as described earlier. If the stiffness for each of the integration points is
defined by dividing the present stress by the present strain as shown in Fig. 36, then it
is called secant stiffness. On the other hand, if the stiffness for each segment is defined
by the slope of the stressstrain curve at the specific point being integrated, then it is
called the tangent stiffness as also illustrated in Fig.36. Note that if the tangent stiffness
NONLINEAR DYNAMIC ANALYSIS OF BRIDGE PIERS
By
CESAR FERNANDES, JR.
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1999
139
The next step was to run the complete test changing the modulus of elasticity for
steel and concrete. The total duration of the test is 700 seconds. The complete imposed
tip displacement history is shown in Fig. 722. It is important to note that failure for the
column is characterized at about 15 mm. Note that in the test the column is subjected to
tip displacements of about 90 mm, what is very extreme. The analysis was done with a
time step equal to 0.01s. Table 78 helps to identify the modulus of elasticity of the
materials for each run. The confinement was also added to the analysis. Figures 723 to
726 show the comparison of each model to the test results. Note that the response given
by FLPIER using the original material properties is always stiffer than the original
response, although the column strength is very close for all cases. However when the
modulus of elasticity for steel and concrete were decreased, note that there is a much
better agreement, but the FLPIER model is still stiffer. It seems that there is an
additional stiffness degradation factor, caused by the large amplitude of the cyclic
loading, not accounted for by the analysis. Note that cyclic degradation is usually related
to a large number of low intensity loading cycles, but in this example, after a relatively
small number of cycles, the stiffness degradation is considerable. This may be
introduced in the model in future works.
Table 78. Parameters for test SO units are KN/mm2
File
tsOl
ts02
ts03
ts04
Ec
26.25
26.25
20
20
Es
200
200
90
90
Confinement
No
Yes
No
Yes
17
To help illustrate the concept, consider the inverted Lshaped frame with lumped
masses subjected to horizontal ground motion shown in Fig. 22. Assuming the elements
to be axially rigid, the three DOFs are as shown. Static application of Dg = 1 results in
the displacements shown in Fig. 22. Thus t={ 1 1 0}T in Eq. 2.21, and Eq. 2.22
becomes
Peg (0 = ~[M]*Ar (0 = ~Dt (Oi m2+mA Eq. 2. 23
l 0 J
h=l
a) b) c)
Fig. 22. Support motion of an Lshaped frame.
a) Lshaped frame; b) influence vector t: static displacements due to Dg= 1; c)
effective load vector after Chopra (1995)
Note that the mass corresponding to D2 = 1 is because both masses will
undergo the same acceleration since the connecting beam is axially rigid. The effective
forces in Eq. 2.23 are shown in Fig 22. Observe that the effective forces are zero in the
vertical DOFs because the ground motion is horizontal.
177
The same procedure can be used for steel with ys = 490 Ibf/ft3, resulting in:
p,=7.35rfO07^ A6
in
The same confusion happens in SI units because the old unit of force, the kgf (1
kgf= 9.8 N), is still used to designated the unit weight of materials, sometimes without
the distinction from the masskilogram that is represented by kg. Table Al has a
summary of these results in the English and SI units.
Table Al. Mass density units
English
Concrete
Steel
V Obf/ft3)
150
490
p (slug/ft3)
4.67
15.23
p (kip s2/in4)
2.25 x 1 O'07
7.35 x 1 O'07
SI
Concrete
Steel
y (kgf/m3)
2320
7860
p (KN s2/m4)
2.32
7.86
Bl. GaussLegendre abscissas and weights
54
Fig. 32. Discrete element displacements. After Hoit et al. 1996
Integration of Stresses for Nonlinear Materials
For a beam subjected to both bending and axial loads, it is assumed that the
strains vary linearly over the area of the cross section. This assumption enables the
strain components due to bending about the z and yaxes, and the axial strain to be
combined using super position. Examples of these three components are represented
separately in Fig. 33(ac) and combined in Fig. 33(d) also shown in Fig. 33(d) is a
differential force, dF acting on a differential area, dA,. relationship for the material
dF, = aÂ¡dAÂ¡ Eq.3.16
Finally, Fig. 33(e) represents the stressstrain curve. Then, for the left joint, the
relationship for the strain at any point in the cross section is
cap lateral displacement (in) force (tons)
168
700
600
500
400
300
200
100
0
100
1
[
i
0 0.5 1 1.5
time (s)
Fig. 761. Load history for Mississippi test
2.5
Fig. 762. Displacement history for Mississippi test
8
adopted constitutive models for concrete and steel are presented. Then Chapter 4
discusses the actual stateofdesign procedure for modal analysis. In Chapter 5 the
concept of multiple support excitation is introduced. In Chapter 6 the soil structure
interaction and the dynamic soil behavior are explained. In Chapter 7 the response
predicted by FLPIER is compared to various literature results. Finally in Chapter 8 the
conclusions and suggestions for future work are discussed.
J
18
Mass Matrices. Consistent and Lumped
A mass matrix is a discrete representation of a continuous distribution of mass.
A consistent element mass matrix is defined by Eqs. 2.5 that is, by
[m] =  p[A] \N]dV. It is termed consistent because [A] represents the same shape
v
functions as are used in the displacement field interpolation, and to generate the element
stiffness matrix. A simpler formulation is the lumped mass matrix, which is obtained by
placing particle masses mÂ¡ at nodes / of an element, such that ^ m, is the total element
mass. Particle lumps have no rotary inertia unless rotary inertia is arbitrarily assigned,
as is sometimes done for the rotational DOF of beams and plates. A lumped mass matrix
is diagonal but a consistent mass matrix is not. The two formulations have different
merits, and various considerations enter into deciding which one, or what combination
of them, is best suited to a particular analysis procedure. The mass matrix for a 3D
uniform beam element, which is used to model the pier and piles, and for a shell
element, which is used to model the piles cap, are developed next.
Mass Matrix for the Uniform 3DBeam Element
Consistent
The formulation found here is given by Przemieniecki (1968). As a local
coordinate system, consider the system shown in Fig.2.3. The origin is at node 1 with
the ox axis taken along the length of the beam and with the oy and oz axis as the
principal axes of the beam cross section. The matrix N for this element consists of
twelve displacements, six deflections and six rotations, that is,
14
[m]{}+[c]{rf}+[*]{
which can be interpreted as saying that external loads are equilibrated by a combination
of inertial, damping, and elastic forces. For the assembled structure, from Eq. 2.11 we
get the equation of motion for linear systems,
[M]{d} + [C]{d} + [Â£]{/>} = {*"'} Eq. 2.12
where {Rext} corresponds to loads {R} of a static problem, but is in general a function of
time. Or, returning to Eq. 2.4, equations of the assembled structure can be written in the
alternative form
[M]{} +[C]{d) +{/T'} = {/?'} Eq. 2.13
which does not require that the material be linearly elastic and represents the equation of
motion for nonlinear systems.
Equations of Motion for Ground Motion
It is now opportune to derive the equations of motion for structural systems
subjected to ground motion. Consider the tower shown in Fig. 21, modeled as a
cantilever beam with concentrated masses at the nodes
The displacement of the ground is denoted by Dg, the total (or absolute)
displacement of the mass m, by tÂ¡ and the relative displacement between this mass and
the ground by DÂ¡. At each instant of time these displacements are related by
D'J(t) = Dj(l) + Ds(t) Eq. 2.14
Such equations for all the n masses can be combined in vector form:
CHAPTER 1
INTRODUCTION
Background
Extensive research efforts have been directed to the nonlinear response of
structures subjected to extreme load events. These extreme load events could be an
earthquake or hurricane for a building, a ship impact for a bridge, or the effects of waves
and wind action for offshore oil platforms. Traditionally, large factors of safety have
been used in such cases, resulting in overconservative design and cost ineffectiveness.
On the other hand, an unsafe design could result in catastrophic human and economic
losses. Because a more sophisticated nonlinear dynamic analysis is computationally
expensive, these structures are designed using factored static loads to account for the
dynamic effects. This procedure is acceptable for very low frequency vibrations,
however the introduction of nonlinearity, damping, and pilesoil interaction during
transient loading may significantly alter the response.
Because in recent years computers have become much faster and cheaper, it has
become possible to consider, and consequently to study, the dynamic nonlinear behavior
of structures considering many factors neglected in the past. In this dissertation, the
computer program Florida Pier (Hoit et al., 1996), which will be referred simply as
FLPIER from now on, has been modified to allow the nonlinear dynamic analysis of
bridge piers. FLPIER is a computer program based on the Finite Element Method
1
25
The integral in Eq. 2.35 can be easily evaluated by means of Gaussian
Quadrature (the reader is referred to Appendix B for more details on this procedure),
therefore any element of the mass matrix can be obtained using Eqs. 2.30 and Eqs. 2.35
and this completes the formulation of the mass matrix for the true rectangular 9node
shell element.
Lumped
Lumping the mass for a beam element is a process that seems to be possible by
intuition and physical insight, however, for higherorder elements, like the shell
element, or elements of irtegular shape, intuition can be risky. Accordingly, systematic
schemes for lumping are necessary. In FLPIER the HRZ scheme is used.
The HRZ scheme (Cook et al., 1989) is an effective method for producing a
diagonal mass matrix. It can be recommended for arbitrary elements. The idea is to use
only the diagonal elements of the consistent mass matrix, but to scale then in such a way
that the total mass of the element is preserved. Specifically, the procedural steps are as
follows (Cook et al., 1989):
1) Compute only the diagonal coefficients of the consistent mass matrix.
2) Compute the total mass of the element, m.
3) Compute a number s by adding the diagonal coefficients m;, associated with
the translational DOF (but not rotational DOF, if any) that are mutually
parallel and in the same direction.
4) Scale all the diagonal coefficients by multiplying them by the ratio m/s, thus
preserving the total mass of the element.
40
M = w, + bgÂ¡ + btoM Eq. 2. 80
m = U; + (,Eq. 2. 81
where
&6 = o; 7 = 62; 8 = 63; 69 = At(ly); >10 = y. A/ Eq. 2.82
Equation 2.77 can be now written in the condensed form:
KuM=p(t) Eq. 2.83
where the effective stiffness K is given by
K^b^M+bfi + K Eq. 2. 84
and the effective load vector p(t ) is
pit) = Fm + M(b0u: + b1l +b3Â¡)+C(bÂ¡ul + bfiÂ¡ + bi) Eq. 2. 85
which completes the formulation for linear systems.
Choice of time step At
The unconditional stability of the average acceleration method may lead some
analysts to adopt larger time steps because of economic needs, implying a solution that
may not be accurate, because unconditional stability does not mean unconditional
accuracy. Cook et al. (1989) suggests the following expression for the time step At:
At < (2k /fl>)/20 s> 0.3/co Eq. 2. 86
where to is the highest frequency of interest in the loading or response of the structure.
However, he suggests that in the case of convergence difficulties, the analysis should be
repeated with a smaller time step for additional assurance of a correct solution.
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35
implies linearity. For problems with material or geometric nonlinearity, [K\ is a function
of displacement and therefore of time as well. Accordingly, from Eq. 2.57,
Eq. 2. 58
{R'"'} is the internal force vector at time n At due to straining of material. It is obtained
by assembling element internal force vectors, given by Eq. 2.7 using {a}. For
nonlinear problems, {R'"'} is a nonlinear function of {D} and possibly time derivatives
of {D}, if the strain rate is an issue. For linear problems, the internal force vector is
given by the relationship {Rm'}=[K\{D}. In Eq. 2.58 [M] and [C] are taken as time
independent, although for some problems these may also be nonlinear. In this work [M\
and [C] remain constant during the analysis, and the internal force vector, {/?"}, is only
a function of the displacements {D}. The nonlinear relationship for {a} will be
developed later.
Different methods for direct integration of Eqs. 2.57 and 2.58 can be categorized
as explicit or implicit. The first category, the explicit methods, has the form
Eq. 2. 59
and hence permit {D}+\ to be determined in terms of completely historical information
consisting of displacements and time derivatives of displacements at time n At and
before. The main characteristic of explicit methods is the fact that the next
approximation for the displacements is based only on the known previous
approximations for the displacements and their respective derivatives. Note the use of
the equilibrium condition at time n. The Central Difference is an example of an explicit
method.
179
(a) (b)
Fig. Bl. (a) Trapezoidal approximation using the abscissas 1 and 1. (b) Trapezoidal
approximation using abscissas x/ and xÂ¡ after Mathews, 1987
If we can use nodes xÂ¡ and xj that lie inside the interval, the line through the two
points (x/,f(x/)), (X2,fixÂ¡)) crosses the curve and the are under the line more closely
approximates the are under the curve (see Fig. Bl(b)). The equation of this line is
(x x, )[/(x,) fix,)]
y = f(x,) + mzv 2/ a B
x2x,
and the area of the trapezoid under this line is
area =
2x,
x2x.
f(x2).
B3
Notice that the trapezoidal rule is a special case of Eq. B2. When we choose x/ =
1, X2 = 1, and h = 2, then
151
Fig. 740. Imposed displacement in X direction for test S4
Fig. 741. Imposed load in Y direction for test S4
33
so that, introducing Eq. 2.55 into Eq. 2.53, we obtain
(* Kl2K^K2t)q2 = x(m22 KnK~Â¡ M2, MnKu'K2Â¡ + KÂ¡2KU' MUKU'K2l)q, +
k1(MÂ¡2K;l'MK,2Ku'MuKu'K2lKÂ¡2KM22KM2Â¡+KÂ¡2KM22KK2l)q, +
0(V)
Eq. 2. 56
Examining Eqs. 2.56 and 2.52 we conclude that the condensation used earlier is
ignoring second and higher order terms in X in Eq. 2.56, which can be justified if
the coefficients of X2, X3, ... are significantly smaller than the coefficient of X.
For this to be true we must have the entries of MÂ¡Â¡ and M22 much smaller than
the entries of KÂ¡2 and K22. Physically, this implies that the slave displacements
should be chosen from areas of high stiffness and low mass. Moreover the nodes
that carry a timevarying load should be retained as master.
FLIPIER condenses the stiffness and mass to the top of the piles. While this
procedure is exact for the stiffness, note that it is not for the mass. This is because in
pier structures the slave DOFs are located in areas of high mass concentration, like the
pile's cap. Also note that the mass of the superstructure on the pier, is usually modeled
as lumped masses at the top of the pier. In the case of earthquake loading for example
note that we have the loading function acting on slave DOFs, what is not acceptable in
this approach. Because of these limitations for mass condensation, this approach is not
recommended for the dynamic analysis of bridge piers. FLPIER was then modified to
allow a full analysis of the structure, where neither the stiffness nor the mass matrices
are condensed to the top of the piles. A typical full and the respective condensed version
of a structure are shown in Fig. 28.
In a typical condensed static analysis, the stiffness and loads of the
superstructure are condensed to the top of the piles. The condensed analysis is then
carried out. At the end of the analysis the superstructures forces and displacements are
recovered and the analysis is terminated.
Moment (KN.mm) Moment (KN.mm)
152
150000
100000
50000
0
50000
100000
40 30 20 10 0 10 20 30 40 50
Top lateral displacement (mm)
40 30 20 10 0 10 20 30 40 50
Top lateral displacement (mm)
Fig 743. Comparison ts42
118
Example 2 Steel Section 2
In Example 2 the geometric nonlinearities already incorporated in FLPIER are
included in the analysis, with the presence of an axial load. In this test the tangent
stiffness approach added to the code is compared to the current approach, the secant
stiffness, considering the pA effects. The computer model and the dimensions for the
cross section are illustrated in Fig. 73. The FLPIER results are compared to the results
using a computer program that also considers material and geometric nonlinearities
developed by Hays (1975). A summary of all the results for this test can be seen in
Table 71. The comparison between FLPIER and Hays (1975) is shown in Fig. 74 and
75.
14.585
Units are inches.
fy = S0 ksi, E, = 29000 ksi
P = 537.5 kips
k lx 10' kip/in
L =
15'= 180"
x
mm
Fig. 73. Example 2 computer model
188
Where Ti,Fi are the time and force (or acceleration) values for the point being specified.
This section MUST end with a blank line.
Point Mass
This section allows the addition of point masses to a structure.
MASS
The next line specifies the mass to be added to a node for each of the six global directions.
NS.NF.NI M=MX,MY,MZIMRX,MRY,MRZ
where NS
NF
Nl
MX
MY
MZ
MRX
MRY
MRZ
is the starting node to add the mass to.
is the final node to add the mass to.
is the increment to generate additional node numbers at between NS and NF at which to add mass.
\
I
} are the mass values for the translational and rotational X,Y,Z directions
I
I
/
This section must end with a blank line.
Point Dampers
This section allows the addition of point dampers to a structure.
DAMP
The next line specifies the dampers to be added to a node for each of the six global directions.
NS.NF.NI M=MX,MY,MZ,MRX,MRY,MRZ
where NS
NF
Nl
MX
MY
MZ
MRX
MRY
MRZ
is the starting node to add the dampers to.
is the final node to add the dampers to.
is the increment to generate additional node numbers at between NS and NF at which to add
dampers.
\
I
} are the dampers values for the translational and rotational X,Y,Z directions
I
I
/
You can not add concentrated masses or dampers to the pile's nodes.
This section must end with a blank line.
30
phase. For structures subjected to strong motions, that will lead to crushing of concrete
or yielding of steel, characterizing nonlinear material behavior, hysteretic damping must
be added to the analysis through nonlinear forcedeformation relationships.
Table 23. Recommended damping rations for structures
Stress Level
Type and Condition
of structure
Damping Ratio(%)
Working stress, no more
than about Vi yield point
Welded steel, prestressed
concrete, wellreinforced
concrete (only slight
cracking)
23
Reinforced concrete with
considerable craking
35
Bolted and/or riveted steel,
wood structures with nailed
or bolted joints
57
At or just below yield point
Welded steel, prestressed
concrete (without complete
loss in prestress)
57
Prestressed concrete with
no prestress left
710
Reinforced concrete
710
Bolted and/or riveted steel,
wood structures with bolted
joints
1015
Wood structures with nalied
joints
1520
Source: Chopra (1995).
Mass Condensation
A useful tool to decrease the number of DOFs in the static analysis of a system
without losing accuracy is static condensation. In this approach some degrees of
freedom of the structure are chosen as master degrees of freedom and the remaining
ones are called slave degrees of freedom. The choice for the master DOF is concerned
with those DOF that give a better representation of the system, i.e. in a building where
4
represented as a SDOF system with lumped mass and stiffness properties (Crandall
(1956), Craig (1981), Paz (1985), and Chopra (1995)). A more general representation
for multidegreeof freedom systems (MDOF) is derived using the concept of shear
building. In this model the stiffness of each story is represented by nonlinear springs,
and the beams are considered to be infinitely rigid. Despite its simplicity and
satisfactory performance in predicting the maximum response, this class of models does
not provide enough data for more detailed seismic analysis. Furthermore for more
complicated frames the assumption that the beams are infinitely rigid may not be
correct.
In the second class of analytical models, the discrete models, there is a
correspondence between the analytical model and the actual structure. In such models, a
linear elastic element and a nonlinear spring represent the structural elements. The most
common case is that of a nonlinear spring attached to both ends of a linear beam
element. Atalay (1975), Clough (1966), Nakata et al. (1978), Park (1984), and Takeda
(1970), among others, have extensively used this class of models to analyze the behavior
of reinforced concrete structures. In these models a set of predefined rules defines the
hysteretic behavior of the nonlinear springs. These rules are usually obtained from
laboratory experiments with real scale specimens. It is mainly the difference between
these rules that distinguishes the models. Although these models give satisfactory
results, its main disadvantage is the fact that the nonlinear springs rules are based on
experiments that may not correspond to the actual structural member, or type of loading,
that they are representing.
Moment (KN.mm)
133
Top lateral displacement (mm)
Fig. 717. All parameters changed
46
division by zero may occur. It can be shown that the convergence rate is approximately
1.62, being slower than Newtons method. The recursive formula for the secant method
is shown in Eq. 2.97. Note the clear connection between the mathematical and
engineering terms used to define both approaches. More detailed information about both
methods can be found in Mathews (1987).
7X*+1) = *
/(*)
/'(*)'
/(*.)
/(*)/(*,1)
/(*)/(Vi)
Eq. 2. 98
The extension to MDOF systems is immediate by replacing all the scalar
quantities by its respective vector equivalents. Note however that for a SDOF system the
tangent stiffness kj is easily obtained. Such a simple evaluation is not available if there
are MDOF. However, in practice, as it will be discussed later, the physics of the
problem allows us to calculate the tangentstiffness matrix [A)]. In a MDOF context NR
iteration involves repeated solution of the equations [AT,],{AD },+/={ AA},/, where the
tangent stiffness matrix [K,\ and load imbalance {AR} are updated after each cycle. The
solution process seeks to reduce the load imbalance, and consequently {AD}, to zero.
The internal force in the equation of motion, Eq. 2.35, is written as
{*in'L ={*} +MaD} Eq. 2. 99
where
{ad}={d}+1{d}
Eq. 2. 100
Combining Eqs. 2.99 and 2.100 with the equations of motion, Eq. 2.35, and the
trapezoidal rule, we obtain
129
uniformly distributed around the perimeter. A double hoop pattern of 8mmdiameter
stirrups at a 70 mm spacing was used as transversal reinforcement. The steel yielding
stress is 460 MPa. No reference is made in the original work about the values of the
modulus of elasticity for steel or concrete.
For the FLPIER model, at first 16 nonlinear discrete elements were used to
model the column, but they proved to be very time consuming for these cyclic tests. The
number of discrete elements was then reduced to four, and a good agreement could still
be obtained. All the moments from FLPIER were computed at the internal node of
element four indicated in Fig. 712, to avoid the addition of the second order moments,
and allow the results to be compared to the original test. The original tests were also
setup in a way that the second order moments were avoided. All the moments reported
are about the Y axis (Fig. 712). These are basically cyclic tests with loading modeled as
imposed tip displacements, so a lumped mass m, a damper c, and a spring k were added
to the top of the pile to minimize the dynamic effects. The computer model and the
values for the dynamic parameters can be seen in Fig. 712.
Monotonic Tests
The first step in the computer analsyis was to verify the concrete model under
monotonic load. Using the actual static version of the computer program FLPIER, the
column capacity was found. This was done by applying incremental increasing forces
until no convergence was achieved. Then with the dynamics option on, the same type of
analysis was done. The plot of moment at the base of the pile versus tip displacements
for both analysis is illustrated in Fig. 713. Note that the static version can not predict
the softening behavior of the column beyond failure because the secant stiffness
98
Zr+SZP*.V
Eq. 4. 48
to show that the first summation on the right side is identical to the SRSS combination
rule of Eq. 4.46; each term in this summation is obviously positive. The crossmodal
coefficient p, can be approximated by
(lP^)2+^P*,(l + P^) + 4fe2+^)p^ q' '
where pin =a, /co. This equation also implies that p, = p p,= 1 for i = n or for
two modes with equal frequencies and equal damping ratios. For equal modal damping
4; = = i; this equation simplifies to
84z(i + pjp
Pl" dPi2)2+4?2pi(l + PJ2
Eq. 4. 50
The topic of modal analysis has been presented in a very condensed and
simplified manner. The subject is so extensive that a whole volume could be written on
it. The interested reader should consult a structural dynamics book (Chopra (1995), Paz
(1985), Paz (1994), Craig (1981), among others) for a more detailed discussion. It is
always important to emphasize that modal analysis relies on the hypotheses that the
superposition of the responses of SDOF systems is used to get the response of a MDOF
system. This implies that the mass and stiffness matrices remain constant during the
process, in other words, a linear analysis is always performed. Furthermore if damping
148
Note that in this test it was not necessary to change the materials modulus of
elasticity for a good agreement, however FLPIER could not model the stiffness
degradation present in the model due to the amplitude of the imposed displacements. It
is interesting to note that the same stiffness degradation is present in the unloading. Also
note that the column strength predicted by FLPIER is slightly larger than the one given
by the test. This indicates that like the stiffness degradation factor mentioned before, a
strength degradation factor, based on the displacement amplitude, rather than number of
cycles, should be considered in the model. Observe, however, that as the displacements
get larger, the moments are in better agreement, what indicates that the softening model
works.
Cyclic Tests S3 and S4
These are mixed mode control tests: displacementcontrolled in the Xdirection
and forcecontrolled in the Ydirection. The imposed displacement in the Xdirection
and force in the Ydirection are shown in Fig.736 and Fig.737. The comparison
between the test and FLPIER is shown in Fig. 738 and 739. Test S4 differs form S3 in
that each constant level of the Yforce is applied first in the +Y direction and then in the
Y direction, with the repetition of three cycles of Xdisplacement. The imposed
displacement in the Xdirection and force in the Ydirection for test S4 are shown in
Fig.741 and Fig.742, respectively. The comparison is shown in Fig. 743 and 744.
The test parameters are given in table 710.
58
Element End Forces
The element internal forces are necessary to assemble the global internal force
vector necessary for equilibrium of the equation of motion. From equilibrium of the
center bar
V, =(M4 M2)/h Eq.3.22
K2=(m,M,)/h Eq.3.23
and from equilibrium of the end bars
f,=T
Eq. 3. 24
Â¡sT
II
Eq. 3. 25
i
II
Eq. 3. 26
Eq. 3. 27
fs =M, +V2 h/2 + Th/2w5
Eq. 3. 28
f6 =M2+V, h/2 + Th/2w6
Eq. 3. 29
fi = T
Eq. 3. 30
f% = ~K
Eq. 3. 31
i
Eq. 3. 32
/,o =
Eq. 3. 33
fn =M3 + V2h/2 + Th/2wu
Eq. 3. 34
fn = M, +Vt h/2 + T h/2wn
Eq. 3. 35
where/; f3 and f7f9 are the acting end forces; and /, f6 and f,n fn are the acting end
moments.
69
Eq. 3. 39
This correction was necessary because the derivative of the Hongnestead
parabola gives very high values of the tangent modulus for low values of the
compressive strain e, which caused instabilities to the model. Based on research (Chen
(1982), Park and Paulay (1975)) it was found that at about 30% off'c it is reasonable to
assume that concrete still have the initial tangent modulus, Ecl. Equation 3.39 is just the
derivative of Eq. 3.37 with respect to e.
When the strain e > e0, the concrete enters a phase called softening. In this
research this phase is defined as:
If s0< e < 4 e then
Eq. 3. 40
as/;
Eq. 3. 41
and if s > 4 e
/ = 0.20/;
Eq. 3. 42
and
E=0
Eq. 3. 43
42
of displacements and velocities. It is then necessary to consider the incremental
equilibrium equation, the difference between Eqs. 2.84 and 2.85, can be written:
mAii: + cA, + (Afs), = ApÂ¡ Eq. 2. 89
The incremental resisting force can be written
(A/v), = (k, )sec Am, Eq. 2. 90
where the secant stiffness (i,)sec, shown in Fig. 211, cannot be determined because u/+i
is not known. If however we make the assumption that over a small time step A/ the
secant stiffness (Osee, can be replaced by the tangent stiffness (kÂ¡)a, then Eq. 2.90 can
be rewritten:
(A/s), =(k,)TAu, Eq. 2. 91
The incremental dynamic equilibrium equation is now:
mAj + cAw, +(k, )T An, = Apt Eq. 2. 92
Equation 2.92 suggests that the analysis of nonlinear systems can be done by
simply replacing the stiffness matrix k by the tangent stiffness (kÂ¡)rto be evaluated at the
beginning of each time step. However this procedure for constant time steps At can lead
to unacceptable results for two reasons:
a) The tangent stiffness was used instead of the secant stiffness.
b) The use of a constant time step delays detection of the transitions in the force
deformation relationship.
Copyright 1999
by
Cesar Fernandes, Jr.
APPENDIX B
GAUSS QUADRATURE
Quadrature is the name applied to evaluating an integral numerically, rather than
analytically as is done in tables of integrals. There are many quadrature rules, such as
NewtonCotes or Simpsons rule. Here we discuss only the GaussLegendre rules,
because it is the most used in Finite Elements analysis. A brief explanation of the
method extracted from Mathews (1987) follows.
We wish to find the area under the curve
y = /(x), \<,x
What method gives the best answer if only two function evaluations are to be
made? The trapezoidal rule is a method for finding the are under the curve, it use two
function evaluations at the endpoints (1,/(1)), (1,/(1)). But if the graph y = fix) is
concave down, the error in approximation is the entire region that lies between the curve
and the line segment joining the points (see Fig. Bl(a)).
178
61
reinforced concrete behavior under inelastic loading reversals. In order to closely
reproduce the hysteretic behavior of various components, a highly versatile model is
required in which several significant aspects of hysteretic loops can be included, i.e.,
stiffness degradation, strength deterioration, pinching behavior and the variability of
hysteresis loop areas at different deformation levels under repeated loading reversals.
However, the model should also be as simple as possible since a large number of
inelastic spring are necessary in modeling the entire structure, and additional parameters
to describe a complicated hysteresis loop shape may sometimes require excessive
amount of information.
W
\
(o) Clough
(b) Fukoda
(ej Aoyorr.c
7^
rjj
Z&
(d) Kust
(} Tow
(f) lokedo
i
f
f
(g) Pork
(h) toon
(i) Toko/anogi
D
fi
]3TJ
7#"
#
(j) Mulo
W Malay
(I) Nakolo
tF
(rr>) BMtetay
3^
Fig. 37. Models for hysteresis loops proposed by some authors. After Mo, 1994
Some of the existing popular models: Clough (1966), Fukada(1969), Ayoama
(1971), Kustu (1975), Tani (1973), Takeda (1970), Park (1984), Iwan (1973),
Takayanagi (1977), Muto (1973), Atalay (1975), Nakata (1978), Blakeley (1973), and
135
strain rate tests. The comparison between the unconfined and confined models, as well
as the effect of different strain rates can be seen in Fig. 718.
Note the difference between the unconfined and the confined model and the
increase in the moment capacity as the strain rate is increased. The additional strength
provided by the increase in the strain rate should be used with caution. In the case of an
earthquake, for example, it is very probable that the structure will experiment some
cracking and yielding, what will make it less stiff, increasing the period and leading to
lower strain rates. Therefore the use of small strain rates is a safer lower bound solution.
Table 77. Parametric tests for confinement under different strain rates
File
Strain rate tl/s)
Confinement
bml.in
0
No
bml srl.in
1 x 10'5
Yes
bml sr2.in
l x ur4
Yes
bml sr3.in
1 x 10'3
Yes
bml sr4.in
1 x 10'2
Yes
bml sr5.in
1 X 10'1
Yes
bml sr.in
1
Yes
Fig. 718. Confinement and Strain rate effect
110
The free field motions are computed independently through a onedimensional
site response analysis using widely available computer codes (SHAKE. Schnabel et al.,
1972). These computed ground motions are then used as boundary conditions applied at
nodal locations, corresponding to the nonlinear "py springs", in the foundation system.
Recalling Chapter 5, it is clear that what is done is a multiple support excitation
analysis, with the support motions given by step (1). Usually in this method the soil
mass is considered to be lumped at the pile nodes. This method is the stateofpractice in
dynamic analysis of foundations.
Coupled Method
This method is illustrated in Fig. 61 (b). The analysis is done by applying the
free field motion to the base of a soil column, modeled with "soil elements". The idea is
that the earthquake occurs at the rock level, so applying the free field motion to the
bottom of the soil column makes sense. The "soil elements are connected to the pile
nodes through "py springs" and dashpots. Note that in this method only one free field
motion is applied to the structurefoundation system, so there is no necessity for
multiple support excitation. One advantage of this method is that the soil mass can be
obtained from the "soil elements" using the Finite Element Method procedure (e.g. a
consistent mass formulation). The disadvantage of this model is that the "soil element"
must be a good representation of the soil, because all the interaction behavior between
the support motion and the structurefoundation system is dependent on this element.
Although this simplified procedure is efficient for a first estimate of the behavior
of the structure when subjected to earthquake, it is somewhat impractical for design.
Note that the pile cap, pier cap, and pile group effects are not present in this type of
149
Table 710. Parameters used in test S3 and S4
File
Test
fe
Ec
fy
Es
Confinement
Time step
ts31 .in
S3
0.029
26.25
0.46
200
No
0.35
ts32.in
S3
0.029
20
0.46
90
No
0.35
ts41.in
S4
0.027
26.25
0.46
200
No
0.41
ts42.in
S4
0.027
20
0.46
90
No
0.41
ts43.in
S4
0.027
20
0.25
90
No
0.41
Fig.736. Imposed X displacement for test S3
Time (s)
Fig.737. Imposed Y forces for test S3
87
Equation 4.11 provides a compact presentation of the equations relating all
eigenvalues and eigenvectors.
The orthogonality of natural modes implies that the following square matrices
are diagonal:
K = (t>Tk<& M = 0,m0 Eq.4.12
where the diagonal elements are
Kn = ts>1nki> M = f> Eq. 4.13
Since m and k are positive definite, the diagonal elements of K and Mare
positive. They are related by
K=co2M Eq.4.14
Normalization of Modes
As mentioned earlier, the eigenvalue problem, Eq. 4.5, determines the natural
modes to only within a multiplicative factor. If the vector <> is a natural mode, any
vector proportional to (> is essentially the same natural mode because it also satisfies
Eq. 4.5. Scale factors are sometimes applied to natural modes to standardize their
elements associated with amplitudes in various DOFs. This process is called
normalization. Sometimes it is convenient to normalize each mode so that its largest
element is the unity. Other times it may be advantageous to normalize each mode so that
the element corresponding to a particular DOF, say the top floor of a multistory
building, is unity. In theoretical discussions and computer programs it is common to
normalize modes so that the M have unit values. In this case
84
d(t) = t)(A cosa>/ + 2? sino),,/) Eq. 4.3
where co and <> are unknown.
Substituting this form of d(t) in the undamped equation of motion gives
[ 0>,2m(p + kh)" ]q (/) = 0 Eq. 4. 4
This equation can be satisfied in one of the two ways. Either q(t) = 0, which
implies d(t) = 0 and there is no motion of the system (this is the socalled trivial
solution), or the natural frequencies co and modes <]> must satisfy the following
algebraic equation:
kÂ§n = a>,2m<> Eq. 4. 5
which provides a useful condition. This algebraic problem is called the matrix
eigenvalue problem (or the generalized eigenvalue problem). The stiffness and mass
matrices k and m are known; the problem is to determine the scalar
n
To indicate the formal solution to Eq. 4.5, it is rewritten as
\k co2m](> = 0 Eq. 4. 6
which can be interpreted as a set of N homogeneous algebraic equations for the N
elements <>y (j=l,2,...,/'/). This set always has the trivial solution =0, which is not
useful because it implies no motion. It has nontrivial solutions if
detk (o2to = 0
Eq. 4. 7
187
Meirovich, L. (1980). Computational methods in structural dynamics. Sijthoff &
Noordhoff, Alphen aan den Rijn, The Netherlands, Rockville, MD, USA.
Meyer, C. (1996). Design of concrete structures. PrenticeHall, Inc., Upper Saddle
River, NJ.
Miller, C. A. (1980). Dynamic reduction of structural models. Journal of the
Structural Division, Proceedings of the American Society of Civil Engineers,
Vol. 106, No. ST10 (October), pp. 20972108.
Mitchell, J, S. (1973). A nonlinear analysis of biaxially loaded beamcolumns using a
discrete element model. PhD dissertation, University of Texas at Austin, TX.
Mo, Y. L. (1988). Analysis and design of lowrise structural walls under dynamically
applied shear forces. ACI Structural Journal, Vol.83, No.2 (MarchApril), pp.
180189.
Mo, Y. L. (1994). Dynamic behavior of concrete structures. Elsevier, Amsterdam.
Monti, G., Nuti, C., and Pinto, P. E. (1996). Nonlinear response of bridges under
multisupport excitation. Journal of Structural Engineering, Vol. 122, No. 10
(Octiber), ASCE, pp. 11471159.
Muto, K., Hisada, T., Tsugawa, T., and Bessho, S. (1973). Earthquake resistant design
of a 20story reinforced concrete building. Proc. of 5th WCEE, Rome.
Nakata, S., Sproul, T., and Penzien, J. (1978). Mathematical modeling of hysteretical
behavior of concrete columns. Report No. UCB/EERC78/11, Earthquake and
Engineering Center, College of Engineering, University of California, Berkeley,
CA.
Nilsson, L. G. (1979). Impact loading on concrete structures. Department of Structure
Mechanics, Chalmers University of Technology, Goteborg, Sweden.
Nogami, T. (1997). Observation and modeling in numerical analysis and model tests in
dynamic soilstructure interaction problems. Geotechnical special publications
No. 64, ASCE, New York.
ONeill, M. W., Brown, D. A., Anderson, D. G., El Naggar, M. H., Townsend, F. C.,
Mcvay, M. C. (1997). Static and dynamic lateral loading of pile groups.
NCHRP 249, Highway Research Center, Harbert Engineering Center, Auburn
University, Auburn, AL.
Otani, S. (1980). Nonlinear dynamic analysis of reinforced concrete building
structures. Canadian Journal of Civil Engineering, Vol. 7, pp. 333344.
45
that converge faster to p. Note however that f(x) and its first and second derivatives must
be continuous for the method to work. The convergence rate of the NewtonRaphson
method is quadratic. This method is also called the tangent method because the slope of
the curve is used in the formulation. The recursive formula for the method is:
= for k = 1,2,... Eq.2.97
/(**)
and the sequence will converge to p.
Fig. 212. NewtonRaphson Method
However sometimes it may be difficult if not impossible to implement Newtons
method if the first derivative is complicated. The secant method may then be used. The
secant method is the same as the Newtons method except that the first derivative is
replaced by its approximation: the slope of the line through the two previous points.
Note that when successive points get close, the method becomes unstable because
89
XH>,?r(0 + C^r9) + 2X#,,(0fP(0 Eq. 4 20
r1 r=l rl
Because of the orthogonality relations of Eq. 4.20, all terms in each of the
summations for m and k vanish, except the r n term, reducing this equation to
(' H>)?,(0 + Zcr)?,(') + (/* )r/>(') Eq. 4. 21
r=l
or
N
Mq + X C,
r= 1
where
M =i C,=t>;c<>r A.^ /> (0 = ^(0 Eq. 4.23
Equation 4.22 exists for each = 1 to jV and the set on N equations can be
written in matrix form
Mq + Cq + Kq = P(l) Eq. 4. 24
where M is a diagonal matrix of the generalized modal masses M, K is a diagonal
matrix of the generalized modal stiffnesses K, C is a nondiagonal matrix of coefficients
Cr, and P(t) is a column vector of the generalized modal forces P(t). These N equations
in modal coordinates are coupled through the damping terms because Eq. 4.22 contains
more than one modal velocity.
The modal equations will be uncoupled if the system has classical damping
(Rayleigh is a form of classical damping). For such systems Cr= 0 if n r and Eq. 4.22
reduces to
72
Unloading
Typical unloading phases in tension and compression are illustrated if Fig. 315
and Fig. 316, respectively. The expressions for compression and tension unloading are
based on the work of Sinha et al. (1964a) defined for different concrete mixes. Due to
the lack of more experimental data, it is assumed that for stronger concrete (fc > 4 ksi),
the response will be that of 4 ksi concrete. If new experiments are done for stronger
concrete these changes can be easily incorporated into the program. The family of
unloading curves is represented by second order curves. From experiments for various
concrete strengths a good fit was obtained with the formula:
Eq. 3. 52
where H and J are experimental constants whose values for the three mixes used are
given in Table 31. For stress in units of ksi and strains in units of in/in, and X is a
particular parameter, different values of which represent different members of the
family. To determine the value of the parameter X, for a curve passing though a certain
point in the stressstrain plane ct,,e,, the coordinate values are substituted into Eq. 3.52
which is then solved for the required value of X, leading to the expression:
Eq. 3. 53
The tangent modulus Ec for unloading is taken as the slope for two consecutive
points in the unloading curve:
Eq. 3. 54
190
Turner, J. P. (1995). Performance of deep foundations under seismic loading.
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FHWASA91048, June.
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interaction at soft or liquefiable soils sites centrifuge data report for CSP2.
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Engineering, College of Engineering, University of California at Davis, Davis,
CA.
Zeris, C. A., Mahin, S. A. (1991a). Behavior of reinforced concrete structures
subjected to uniaxial excitation. ASCE Journal of Structural Engineering, Vol.
117, No. 9 (September), pp. 26402656.
Zeris, C. A., Mahin, S. A. (1991b). Behavior of reinforced concrete structures
subjected to biaxial excitation. ASCE Journal of Structural Engineering, Vol.
117, No. 9 (September), pp. 26572672.
153
150000
100000
g 50000
Z
c 0
U
a
o
s 50000
100000
40 30 20 10 0 10 20 30 40 50
Top lateral displacement (mm)
Fig. 744. Comparison ts43
Note again that by changing the values of the modulus of the elasticity of
concrete and steel improves the FLPIER model, however FLPIER again over predicted
the column capacity, which also indicates the necessity of a strength degradation factor
for the FLPIER model. In ts43 the steel strength was reduced just to illustrate the effect
of changing one of material parameters in the cyclic analysis.
Cyclic Test S10
Finally S10 is a test where the axial load (Z direction) is variable. Again the
computer model is just like the previous examples, with a mass and a damper attached at
the top of the column to minimize the dynamic effects. The material properties used for
the various models are shown in Table 7.11. The imposed force in the X direction is
shown in Fig. 745, and the imposed force in the Z direction is shown in Fig. 746.
75
o
0.2
/\
0.4
/
0.6
0.8
stress 1
(ksi)
1.2
/
1.4
/
1.6
1.8
j
2
S'
0.0006 0.0005
0.0004
0.0003
0.0002
0.0001
0
strain (in/in)
Fig. 318. Compression unloading with no gap
Reloading
A typical reloading in compression is illustrated in Fig. 319. The reloading
curves in either compression or tension are represented by a family of converging
straight lines; accordingly, an expression:
c+K = Y(e + L) Eq. 3.55
was chosen, in which K and L are experimental constants, and Y, is a parameter. The
value of Y can be found, similarly to the value of X in the previous section. By
substituting the coordinates (!,,Â£, of a known point on the reloading curve into Eq. 3.55;
and solving for Y, which is also take as the tangent modulus, Ec, one obtains:
y_t+K
E, + L
Eq. 3. 56
CHAPTER 4
MODAL ANALYSIS
Because the most frequently used procedure for designing bridges for
earthquakes is based on modal analysis, a brief explanation of this method of analysis is
given next. It should be noted that modal analysis is an extensive subject and that what
is presented here is only an introduction of the basic concepts necessary to understand
the method. The interested reader is referred to Chopra (1995), or Paz (1985), for a more
detailed discussion on modal analysis.
Natural Vibration Frequencies and Modes
Before getting to modal analysis it is opportune to introduce the eigenvalue
problem whose solution gives the natural frequencies and vibration modes of a system.
The free vibration of an undamped system in one of its natural vibration modes can be
described mathematically by
d() = q Eq.4. 1
where the deflected shape does not vary with time. The time variation of the
displacements is described by the simple harmonic function
q(t) = An costo,/ + Bn sincoi Eq.4. 2
where A and B are constants of integration that can be determined from the initial
conditions that initiate the motion. Combining Eqs. 4.1 and 4.2 gives
83
181
The general Appoint GaussLegendre rule is exact for polynomials of degree <
2N\ and the numerical integration formula is
G(f,N) = wNlf(xNt) + wN2f(xN2) +... + wNNf(xSN). Bll
The abscissas xs.k and weights wy*+ to be used have been tabulated and are
easily available; Table Bl gives the values up to eight points. Also included in the table
is the form of the error term E(fN) that corresponds to G(fJV), and it can be used to
determine the accuracy of the GaussLegendre integration formula.
The values in Table Bl in general have no easy representation, this fact makes
the method less attractive for human beings to use when hand calculations are required.
But once the values are stored in a computer it is easy to call them up when needed, the
nodes are actually roots of the Legendre polynomial and the corresponding weights
must be obtained by solving a system of equations. The general formula for N points
can then be written:
 f{x)dx = f(xNk) + E(f,N)
* = 1
B12
77
Bilinear Model
In another study, Agrawal et al. (1965) claimed that the response of under
reinforced concrete beams is governed by the steel. It was therefore proposed to use a
simplification of the concrete model to simplify the analysis of doubly reinforced
concrete beams. Idealized elastic plastic curves were drawn with a yield stress equal to
the nominal strength fc value for concrete, and an elastic modulus equal to the average
stiffness of the initial portion of the actual stressstrain curve as shown in Fig. 321.
Also, the tensile strength of concrete is neglected in this model. In FLPIER the stress
strain curve for concrete is based on 13 points as shown in Fig. 322. Referring to Fig.
322, the value of the average elastic modulus is taken as
Eq. 3. 57
and the value of the yielding strain sy is taken as
g(2)
Eq. 3. 58
fc
e
Fig. 321. Bilinear model for concrete
167
Elevation (in) / Soil Type
G
Water
216
Sand Reese
312
Soft clay below water table Matlock
520
Soft clay below water table Matlock
720
Sand Reese
1640
Plan View
30 x 30 Prestressed Concrete Piles
Concrete Cap
Fig. 760. Mississippi test structure
143
Fig.728. Imposed displacement in Y direction
The axial load is constant and is equal to 212 KN. Again a parametric study was
done to calibrate the model until the best possible agreement was obtained. Table 79
has all the parameters for the tests. The values for the concrete and steel strength are
respectively, fc = 0.029 KN/mm2, and fy = 0.46 KN/mm2. The comparison between
FLPIER and test SI is shown in Fig. 729 to Fig. 732. The moment plotted is always
about the y axis, and the top lateral displacement is always in the X direction, according
to Fig. 712.
Table 79. Parameters for test SI units are KN/mm2
File
tsll
tsllc
tsl2
tsl2c
Ec
26.25
26.25
20
20
Es
200
200
90
90
Confinement
No
Yes
No
Yes
43. Modal analysis of pier 99
5.1. Multiple support motion 102
52. 2D frame submitted to multiple support motion 105
53. Pile subjected to multiple support excitation 106
61. ajCoupled model; bjUncoupled model 111
62. Typical py curve 112
63. Cyclic soil model 113
71. Example 1 computer model 117
72. Comparison FLPIER x reference for cyclic loading 117
73. Example 2 computer model 118
74. FLPIER x Hays, shear force comparison 119
75. FLPIER x Hays, bending moment comparison 120
76. Shear comparison Example 3 x FLPIER 123
77. Moment comparison Example 3 and FLPIER 124
78. Dynamic model comparison 124
79. Static shear comparison Example 4 126
710. Dynamic moment comparison Example 4. Hoop spacing = 5 in 127
711. Dynamic moment comparison Example 4. Hoop spacing = 2 in 127
712. Computer model for Test 3 130
713. Column capacity using FLPIER 130
714. Ec and Es changed 132
715 .fc and fy changed 132
716. Â£ and/changed 133
xii
38
CO = l +^(M+,)
Eq. 2. 66
X2
Eq. 2. 67
u(T ) = M, + U T + (w,+1 + U,)
2 At
At
=/+y (+/)
Eq. 2. 68
x2 x3
u(x) = u, +II + ,y+ (,+1 ,)
Eq. 2. 69
= , +ti,A/ + (A/)2Qw +
Eq. 2. 70
The Newmarks family of methods can be summarized into the following two
equations:
"m =, + [O r K + Y,t, W 271
A/2 Eqs. 2. 72
where y and (5 are parameters that can be determined to obtain integration accuracy and
stability. Note that when y = 14 and p = 1/6, this method reduces to the linear
acceleration method. Newmark originally proposed as an unconditionally stable scheme,
the constant average acceleration method, in which case y = 14 and p = 14. This can be
shown considering that Newmarks method is stable if (Chopra, 1995):
A l 1 1
Tn 7tV2 VY2P Eq.
For y = 14 and p = 14 this condition becomes
= u, + it. At +
P K + P
12
Mp}+W{}Eq24
where the element mass and damping matrices are defined as
[m] = \p[N^[N]dV Eq.2.5
V
[c\= \cd[N^ [N]dV Eq.2.6
V
and the element internal forces and external loads vector are defined as
{rim}= Eq.2.7
Ve
{'"'} = \[N{F}dV+ + Â£{/>}, Eq.2.8
Vs Ss fa 1
Equation 2.4 is a system of coupled, secondorder, ordinary differential
equations in time and is called a finite element semidiscretization because although the
displacements, {d}, are discrete functions of space, they are still continuous functions of
time. Methods of dynamic analysis focus on how to solve this equation. Modal methods
focus on how to uncouple the equations, transforming the NDOF coupled system into N
uncoupled SDOF systems, each of which can be solved independently of others. More
details about this method of analysis can be found in Chapter 4. Direct integration
methods discretize Eq. 2.4 in time to obtain a sequence of algebraic equations.
Structure matrices [M\, [C], and {/?"} are constructed by standard Finite
Element Method procedures, i.e. conceptual expansion of element matrices [m], [c], and
{r } to structure size followed by addition of overlapping coefficients, in the same
way it is done for assembling stiffness matrices in static problems. However, the exact
41
Nonlinear Problems
In structural analysis, a problem is nonlinear if the stiffness matrix or the load
vector depends on the displacements. The nonlinearity in structures can be classified as
material nonlinearity (associated with changes in material properties, as in plasticity) or
as geometric nonlinearity (associated with changes in configuration, as in large
deflections of a slender elastic beam). In general, for a timeindependent problem
symbolized as [AT]{Z>}{/?}, in linear analysis both [A] and {R} are regarded as
independent of {D}, whereas in nonlinear analysis {A} and/or {R} are regarded as
functions of {D}. In this dissertation we will address the nonlinearities associated with
changes in material properties.
Analysis of the Nonlinear Response using Newmarks Method
In this section the notation for SDOF systems is used to simplify the approach,
the extension to MDOF systems is done later, although the concept is exactly the same.
In the numerical evaluation of the response of a dynamic system we go from time step /,
where the equation of motion can be written
mii, + cu, + (fs), = p, Eq. 2. 87
to time step Â¡+1, where the dynamic equilibrium can be written
">M + cm + (Mm = Pm Eq. 2. 88
where (fs)i is the systems resisting force at time i. For a linear system (fs), = kuÂ¡.
However for a nonlinear system the resisiting force fs, would depend on the prior history
78
Done
Point
Xvalue
Yvalue
Fig. 322. Stressstrain curve for concrete in FLPIER
Strain Rate Effect
The strength of reinforced concrete sections is very dependent on the strain rate
(Shkolnik (1996)), and degree of confinement of the section (Saatcioglu and Razvi
(1992), and Soroushian et al. (1986)). At a fast strain rate, both the modulus of elasticity
and the strength of concrete increase. The increase in strength can be as much as 17 %
for a strain rate of 0.01/sec, as reported by Park and Paulay (1975). It is very difficult to
test structural members under such conditions. So before using the results from static
tests, it is important to consider the effect of strain rate in dynamic analysis. Important
findings about the strain rate effects on reinfoced concrete members are summarized
next (Otani, (1980)):
31 essStrain
for Concrete
71
E
0.5/r
Eq. 3. 47
If Es,/< Â£ < eul then
/ =
E
and finally if e > Â£, then
/ = 0
and
Eq. 3. 48
Eq. 3. 49
Eq. 3. 50
Ec=0
Eq. 3. 51
0.6
0.5
0.4
0.3
0.2
0.1
0.1
0.0001
stress
(ksi)
\l\
i
j \
/
NJ
/
 i
/
0.0003 0.0005
strain in/in
Fig. 314. Typical loading in tension
104
where the vector of effective earthquake forces is
M') = ~[mb +mA){cÂ£>' + c,i),)[kD + kiD*) E(f 5 6
This effective force vector can be rewritten in a more useful form. The last term
drops out because of Eq. 5.3 resulting
kD+kgDg=0 Eq. 5. 7
This relation also enables us to express the quasistatic displacements If in
terms of the specified support motions Dg, and we obtain
D = (Dg Eq. 5. 8
and
t = k~'kg Eq. 5. 9
We call l the influence matrix because it describes the influence of support
displacements on the structural displacements. If all the supports undergo the same
motion the influence matrix i is unitary, and we obtain the equations derived in Chapter
2. It is clear form Eq, 5.9 that t is obtained by solving the linear system of equations:
M = kg Eq. 5.10
Substituting Eqs. 5.8 and 5.7 in Eq. 5.6 gives
P,l (0 = (mf + mg)Dg (0 (ce + cg)Dg(l)
Eq. 5. 11
185
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reinforced bridge columns by steel jacketing. Report No. SSRP 91/05,
Department of Applied Mechanics and Engineering Sciences, University of
California, San Diego, La Jolla, CA.
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EM4, (August), Proceedings of the American Society of Civil Engineers, pp.
685701.
Chopra, A. K. (1995). Dynamics of structures theory and applications to earthquake
engineering. PrenticeHall, Englewood Cliffs, NJ.
Clough, R. W. (1966). Effect of stiffness degradation on earthquake ductility
requirement. Report NO. 6614, Structural and Material Research, University of
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Cook, R. D., Malkus, D. S., and Plesha, M. E. (1989). Concepts and applications of the
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International Journal of Earthquake Engineering and Structural Dynamics, Vol.
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29
P = 22(Â§lco2i;2(Dl)/(a>22
The damping factor a applied to the stiffness matrix [K\ increases with
increasing frequency, whereas the damping factor p applied to the mass matrix [M\
increases with decreasing frequency. For structures that may have rigidbody motion, it
is important that the massproportional damping not be excessive.
Usually, the natural frequencies to i and o>2 are chosen to bound the design
spectrum. Therefore coi is taken as the lowest natural frequency of the structure, and (02
is the maximum frequency of interest in the loading or response. Cook et al. (1989)
suggests a value of 30 Hz as the upper frequency for seismic analysis, because the
spectral content of seismic design spectra are insignificant above that frequency.
Estimating Modal Damping Ratios
Because damping is still an unknown subjected the estimate of damping rations
still presents some challenge. Recommended damping values are given in Table 23 for
two levels of motion: working stress levels or stress levels no more than onehalf the
yield point, and stresses at or just below the yield point. For each stress level, a range of
damping values is given; the higher values of damping are to be used for ordinary
structures, and the lower values are for special structures to be designed more
conservatively. In addition to Table 23, recommended damping values are 3% for
unreinforced masonry structures and 7% for reinforced masonry construction. It is
important to note that the recommended damping ratios given in Table 23 can only be
used for the linearly elastic analysis of structures with classical damping. This implies
that all the material components of the structure are still behaving in their linearelastic
TABLE OF CONTENTS
page
ACKNOWLEDGMENTS iv
LIST OF TABLES viii
LIST OF FIGURES x
ABSTRACT xvi
CHAPTERS
1 INTRODUCTION 1
Background 1
Literature Review 3
Limitations 7
Organization 7
2 NONLINEAR DYNAMIC ANALYSIS 9
Theory 9
Equations of Motion, Mass, and Damping Matrices 10
Equations of Motion for Ground Motion 14
Mass Matrices, Consistent and Lumped 18
Mass Matrix for the Uniform 3DBeam Element 18
Consistent 18
Lumped 20
Mass Matrix for the Shell Element 22
Consistent 22
Lumped 25
Remarks about the mass matrix 26
Damping 27
Estimating Modal Damping Ratios 29
Mass Condensation 30
TimeHistory Analysis. Direct Integration Methods 34
Numerical Evaluaton of Dynamic Response. Newmarks Method 36
Choice of time step At 40
Nonlinear Problems 41
Analysis of the Nonlinear Response using Newmarks Method 41
v
Moment (KN.mm) Moment (KN.mm)
144
150000
100000
50000
0
50000
100000
150000
80 60 40 20 0 20 40 60 80
Top lateral displacement (mm)
Fig. 729. Comparison tsl 1
Top lateral displacement (mm)
Fig. 730. Comparison tsl lc
CHAPTER 8
CONCLUSIONS
A model for the nonlinear dynamic analysis of reinforced concrete members has
been presented. The results shown in Chapter 7 indicate that the proposed model for
concrete is in reasonable agreement with the tests results reported. The fiber modeling
adopted by FLPIER seems to be effective in modeling steel as well as circular and
square reinforced concrete sections. It is clear from the test results that anchorage slip is
an important concern when analyzing reinforced concrete members under cyclic
loading, typical during earthquakes. However it seems that it is possible to model the
anchorage slip effects by decreasing the value of the modulus of elasticity for concrete
and steel. The model can be calibrated by performing tests for different cross sections
and adopting the corrected value for the modulus of elasticity in the analysis. Note that
for all types of tests performed for the rectangular cross section reported in Chapter 7,
the same value of the corrected modulus of elasticity for steel and concrete gave better
results, showing that anchorage slip is a function of the cross section, rather than
loading.
It is important to notice that the introduced concrete gap seems to model the
stiffness degradation of the section very well. However it is clear from the hysteresis
charts of all the tests that the bilinear steel model is not adequate for larger lateral
displacements. In such cases a steel model that includes strain hardening and possibly
cyclic degradation should be used. Anchorage slip should also be included in this model.
172
112
Cyclic Behavior of Soil
The cyclic behavior for soil presented here is based on the recommendations of
O'Neill et al. (1997), and is a simplified approach to the complex dynamic behavior of
soil. Although the changes in FLPIER were limited to the lateral resistance springs (p
y), the extension to the axial resistance springs (tz) is immediate. It is also important to
note that the dynamic factor usually incorporated in the py curves for soil under
dynamic loading has not been incorporated in this model. The static py curves are used
for the dynamic analysis. A typical py curve is illustrated in Fig. 62.
Fig. 62. Typical py curve
The py behavior is illustrated in Fig. 63. Loading is described by the original p
y curve. As the pile loads is reversed at a given level, the soil reaction unloads along a
path parallel to the loading path, not along the initial loading path, which produces
hysteretic damping. When the soil reaction reaches zero, no further reaction is generated
until the displacement of the pile at the location of concern reverses, at which time
follows the loading path in the opposite direction. Then, upon load reversal in the
opposite direction, a mirror image effect is generated, except that no further soil reaction
will be generated until the defection reaches the value of deflection corresponding to the
width of the gap in the previous cycle. Then, loading occurs along the previous
154
The results for the various tests run is shown in Figs. 747 to 750.
Table 711. Material properties for example S10
File
fc
Ec
fy
Es
Conf
A/(s)
tslOl
0.028
26.25
0.46
200
No
0.1
tsl02
0.028
20
0.25
90
No
0.1
tsl03
0.028
26.25
0.35
200
Yes
0.1
tsl04
0.028
20
0.35
90
Yes
0.1
Units are KN/mm2.
Fig. 745. Imposed force in Xdirection for S10
44
Since we are using the tangent estimate for the stiffness, we must iterate within
each time step to guarantee dynamic equilibrium. The first step is to apply the effective
force p(t)and get the first approximation for the next displacement ukÂ¡+Â¡. Associated
with this displacement is the new tangent stiffness I^t and the true force /, which
includes the internal force rk, the inertia force//, and the damping force fc at iteration
k. An outof balance force defined as Ap(()w = p(J)* / is then generated. The new
effective stiffness for the system is computed and the system of equations
kk Auk = Apk is solved to obtain the incremental displacements Am*, which is added to
the previous displacement u giving a new estimate for the displacements, velocities and
accelerations at time /+1. The dynamic equilibrium is checked again with the new
values and another outof balance force is generated. The process continues until the
outofbalance force is within a specified tolerance. Figure 212 helps to understand the
process. This proccess is known as the NewtonRaphson method.
If the reader is familiar with numerical methods, will notice that what is defined
as the The NewtonRaphson method in some references (Cook et al. (1989), Chopra
(1995) and Bathe (1996)) and above is in fact a NewtonRaphson approach. Although
irrelevant to this discussion the author found important to define the NewtonRaphson
method from a mathematical point of view.
The NewtonRaphson method is a mathematical method for finding roots. The
method relies on the fact that the function f(x), and its first derivative f(x), and second
derivative f(x) are continuous near a root p. Note how rigorous the mathematical
definition for the method is, since this was not assumed in the previous application.
With this information it is possible to develop algorithms that produce sequences {xiÂ¡}
28
methods, experimental observations of the vibratory response of structures are used to
assign a fraction of critical damping as a function of frequency, or more commonly, a
single damping fraction for the entire frequency range of a structure. The damping ratio
S, depends on the material properties at the stress level. For example, in steel piping
\ ranges from about 0.5% at low stress levels to about 5% at high stress levels. In bolted
or riveted steel structures, and in reinforced or prestressed concrete, \ has the
approximate range 2% to 15% (Cook et al., 1989).
A popular spectral damping scheme, called Rayleigh or proportional damping is
used to form the damping matrix [C] as a linear combination of the stiffness and mass
matrices of the system, that is
[C] = a[Â£]+P[M]
Eq. 2. 36
where a and p are called, respectively, the stiffness and mass proportional damping
constants. Matrix [C] given by Eq. 2.36 is an orthogonal damping matrix because it
permits modes to be uncoupled by eigenvectors associated with the undamped
eingenproblem. The relationship between a, P, and the fraction of critical damping \ is
given by the following equation,
Eq. 2. 37
damping constants a and P are determined by choosing the fractions of critical damping
(iji and Â£2) at two different frequencies (or 1 and m2) and solving simultaneous equations
for a and p. Thus:
= 2(42o>2 ^co,) / (o>22 oo,2)
Eq. 2.38
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/' M Im 81,9(56,7< 2) )/25,'$
136
Cyclic Test SO
In order to verify the behavior of the concrete model under cyclic loading,
which is typical of dynamic analysis, a cyclic test was done for the first 90 seconds of
the original test. The maximum imposed displacement was slightly over 15mm, which
characterizes failure from the static tests. The tip displacement history for the first 90
seconds of the original test can be seen in Fig. 719. It is also compared to the FLPIER
history. Note that they are in perfect agreement. The first run was performed with the
original data for steel and concrete. The comparison is shown in Fig. 720. Note that the
FLPIER model is stiffer than the test. It was then necessary to adjust the values of the
elastic modulus of steel and concrete to calibrate the model. In lieu of information from
the original work, the initial adopted values for steel and concrete were respectively, E,
= 199.96 KN/mm2, and Ec = 26.25 KN/mm2, which are reasonable values for steel and
concrete (Meyer (1996)). However a good agreement between the test and FLPIER was
obtained with Es = 90 KN/mm2, and Ec = 20 KN/mm2. Three factors may have
contributed for this disagreement between the test and FLPIER:
a) The value of the modulus of elasticity for concrete was computed using the
empirical formula given by Meyer (1996), which could introduce some error.
b) The modulus of elasticity for the steel was adopted considering a typical
value for construction steel. This could also introduce some error.
c) The third, and maybe the most important, component for the error could be
anchorage slip. Alsiwat and Saatcioglu (1992), and Saatcioglu et al.(1992) reported that
anchorage slip is one of the major components of inelastic deformation in reinforced
concrete. Alsiwat and Saatcioglu (1992, pp. 1) define anchorage slip as:
19
d={dt,d2,...,d]2}
Eq. 2. 24
ds
:d
/
7
d] d4
d> //'17
z
Fig. 23. 3D Beam element
Using the engineering theory of bending and torsion and neglecting shear
deformations, we can easily show that the matrix N in the relationship {}=7V]{cf} is
given by Eqs. 2.25.
Nt
ux uy uz
15
0
0
6(552)
1 3^2 + 2^3
0
6(552K
0
1 3Â¡;2 + 2^3
0
05)5
05)ti
(145 3^2);
0
(^ + 2^243)^
( 1 + 45 3^ 2)/,ri
(;2Â¡;2+Â¡;3)
0
5
0
0
6(552)
3Â¡;2 2;3
0
6(5 52>5
0
3Â¡;2 2Â¡;3
0
55
 5^1
( 25 + 3^)15
0
(52 53)
(25 352)Â¡;
(52 53)
0
Eq. 2. 25
188
Ozcebe, G., and Saatcioglu, M. (1989). Hysteretic shear model for reinforced concrete
members. Journal of Structural Engineering, Vol. 115, No. 1 (January), 1989,
ASCE.
Park, R., Kent, D. C., and Sampson, R. A. (1972). Reinforced concrete members under
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Park, R., and Paulay T. (1975). Reinfoced concrete structures. John Wiley & Sons,
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Park, Y. J., Ang, A.HS., and Wen, Y. K. (1984). Seismic damage analysis and
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Prakash, S. (1997). Seismic Analysis and design for soilpile structure interactions.
Geotechnical special publications No. 70, ASCE, New York.
Przemieniecki, J. S. (1968). Theory of matrix structural analysis. McGrawHill, New
York.
Reese, L. C., Cox, W. R., and Koop, F. D.(1974). Analysis of laterally loaded piles in
sand. Proceedings, Offshore Technology Conference, Houston, TX.
Richart, F. E., Woods, R. D., and Hall, J. R. (1970). Vibrations of soils and
foundations. PrenticeHall, Inc., Englewood Cliffs, NJ.
Roufaiel, M. S. L., and Meyer, C. (1987). Analytical modeling of hysteretic behavior
of R/C frames. ASCE Journal of Structural Engineering, Vol. 113, No.3
(March), pp. 429443.
Saatcioglu, M., Alsiwat J. M., and Ozcebe, G. (1992). Hysteretic behavior of
anchorage slip in R/C members. ASCE Journal of Structural Engineering, Vol.
118, No. 9 (September), pp. 24392458.
CHAPTER 5
MULTIPLE SUPPORT EXCITATION
Usually we assume that all supports where the structure is connected to the
ground undergo identical prescribed motion. In this section we generalize the previous
formulation of the equation of motion to allow different possibly even multi
component prescribed motions at the various supports. Such multiplesupport
excitation may arise in several situations. First, consider the earthquake analysis of
extended structures such as the Golden Gate Bridge, in San Francisco, California. The
ground motion generated by an earthquake on the nearby San Andreas fault is expected
to vary significantly over the 6450ft length of the structure. Therefore different motions
should be prescribed at the four supports: the base of the two towers, and two ends of
the bridge. Second, consider the dynamic analysis of bridge foundations. The
earthquake occurs deep in the soil, at the rock level. As the shock wave moves up in the
soil the acceleration records at different soil depths are expected to vary. Therefore the
piles will be subjected to different accelerations along its depth during the earthquake.
In order to analyze multiple support excitation systems, we must include in the
equation of motion the effects of the support motions (Fig. 51). We first divide the
displacement vector into two parts O' and Dg ; D' includes the NDOF of the
superstructure, where the superscript I denotes that these are total displacements; and
Dg contains the Ng components of support displacements.
101
82
Fig. 325. Confined concrete model
The following modifications were also proposed by Soroushian et al. (1986) for
the secant and tangent stiffness of concrete when subjected to dynamic loading:
= 1.241 + 0.111 log,0 e + 0.127(log10 s )2 Eq. 3.63
Ecs
= 1.061 + 0.464 log, + 0.00683(loglo e)2 Eq. 3. 64
E
Where Ed = dynamic secant modulus of elasticity, Ecs = static secant modulus of
elasticity, Ed = dynamic tangent modulus of elasticity, and E = static tangent modulus
of elasticity. Note that when compared to the secant modulus, the tangent modulus
seems to be less influenced by the rate of straining. These changes were also
implemented in FLPIER.
Nonlinear Dynamic Analysis Algorithm 47
3 DISCRETE ELEMENT MODEL AND MATERIAL HYSTERESIS 51
Discrete Element Derivation 51
Element Deformation Relations 52
Integration of Stresses for Nonlinear Materials 54
Element End Forces 58
Element Stiffness 59
Secant and Tangent Stiffness of the Discrete Element 59
Hysteresis Models 60
Material Models 62
Uniaxial Mild Steel Model 63
Uniaxial Monotonic Concrete Model Used in FLPIER 64
Proposed Models for the Uniaxial Inelastic Cyclic Behavior of Concrete 67
Rational Model 67
Loading 68
Unloading 72
Reloading 75
Bilinear Model 77
Strain Rate Effect 78
Confinement Effect 79
4 MODAL ANALYSIS 83
Natural Vibration Frequencies and Modes 83
Modal and Spectral Matrices 86
Normalization of Modes 87
Modal Equations 88
Element Forces 92
Modal Equations for Ground Motion 92
Response Spectrum Analysis 95
Modal Combination Rules 96
How FLPIER handles Modal Analysis 99
5 MULTIPLE SUPPORT EXCITATION 101
6 SOIL STRUCTURE INTERACTION 108
Uncoupled Method 109
Coupled Method 110
Cyclic Behavior of Soil 112
Cyclic Degradation 113
Strain Rate Effect 114
Radiation Damping 115
vi
I certify that I have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and is fully adequate, in scope <
quality, as a dissertation for the degree of Doctor of Philosophy./
Marc I. Emit, (ZhainWan
Professor of Civil Engineering
I certify that I have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and is fully adequate, in scope and
quality, as a dissertation for the degree of Doctor of Philosophy,
Fernandy E. Fagundo
Professor of Civil Engineering
I certify that I have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and is fully adequate, in scope and
quality, as a dissertation for the degree of Doctor of Philosophy.
' H *
Clifford O. Hays, Jr.
Professor of Civil Eni :e
ing
I certify that I have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and is fully adequate, in scope and
quality, as a dissertation for the degree of Doctor of Philosophy.
fhU;
dichael C. McVay
Michael C. McVay
Professor of Civil Engineering
I certify that I have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and is fully adequate, in scope and
quality, as a dissertation for the degree of Doctor of Philosophy.
David C. Wilson
Professor of Mathematics
53
^=02,'*,, Eq.3.9
and the twist in the center part of the element is defined as
= w,o W4 Eq. 3. 10
Therefore, the internal deformations of the discrete element model are uniquely
defined for any combination of element end displacements. The curvature for small
displacements at the left and right universal joints about the y and z axes are defined as
follows:
At the left joint
Eq. 3. 11
<1>2 =*,/*
Eq. 3. 12
At the right joint
h=V3/h
Eq. 3. 13
/h
Eq. 3. 14
The axial strain at the center of the section is given by
6
Eq. 3. 15
REFERENCES
Aghari, A., and Chai, J. (1995). Modeling of soilpile super structure interaction in the
design of bridge foundations. Geotechnical Special Publication No. 51,
Performance of Deep Foundations under Seismic Loading, ASCE, John Turner
(ed.), pp. 4559.
Agrawal, G. L., Tulin, L. G., and Gerstle, K. H. (1965). Response of doubly reinforced
concrete beams to cyclic loading. Journal of the American Concrete Institute,
Vol.62, No. 7 (July), pp. 823835.
Ala Saadeghvarizi, M. (1997). Nonlinear response and modeling of RC columns
subjected to varying axial load. Engineering Structures, Vol. 19, No. 6, pp. 417
424, Elsevier, Great Britain.
Alsiwat, M., Saatcioglu, M. (1992). Reinforcement anchorage slip under monotonic
loading. ASCE Journal of Structural Engineering, Vol. 118, No. 9 (September),
pp. 24212438.
Anderson, J. M. (1972). Seismic response effects on embedded structures. Bulletin of
the Seismological Society of America, Vol. 62, No. 1 (February), pp. 177194.
Andrade, P. (1994). Materially and geometrically nonlinear analysis of laterally
loaded piles using a discrete element technique. MS report, University of
Florida, Gainesville, FL.
Atalay, M. B., and Penzien, J. (1975). The seismic behavior of critical regions of
reinforced concrete components influenced by moment, shear and axial force.
UCB/EERC Report 7519, University of California, Berkeley, December.
Ayoama, H. (1971). Analysis on a school building damaged during the TockachiOki
earthquake. Proc. of Kanto District Symposium of AIJ, Tokyo Japan, January.
Bandoni, D., and Makris, N. (1996). Nonlinear response of single piles under lateral
inertial and seismic load. Soil Dynamics and Earthquake Engineering, Vol. 15,
pp. 2943.
Baron, F., and Venkatesan, M. S. (1969). Inelastic response for arbitrary histories of
loads. Journal of Engineering Mechanics Division, Vol. 95, No. EM3 (June),
Proceedings of the American Society of Civil Engineers, pp. 763786.
184
48
Wilson0
Newmark
1 S
T,^1 =
b2 = 26,; >3 = 2
>4 = 2;6S =0A//2
d6 = 60 /0 ;&7 = fc2 /0
>8 = 1 3 / 0; 69 = A//2
6,0 =At216
b2 =l/aAf,b, = l/2a 1
64 = 1;65 = A/(2)/2
a a
b6 b0',b7 b2
bs =63;6, = A/(l8)
6,o =5Ai
For the Newmarks Method Average Acceleration: a = !4 and 8 = A.
For the Newmarks Method Linear Acceleration: a = 1/6 and 8 = Vi.
For the Wilson0 Method: 0 = 1.4 (0 = 1.0 for Newmarks).
Step (3). Initialize 0,0 and uo
Step (4). Form the effective stiffness matrix using the initial stiffness matrix K\
K = b0M + blC + K
Step (5). Beginning of time step loop.
Step (6). Form the effective load vector for the current time step:
Step (7). Solve for the displacement increments
Step (8). Beginning of dynamic equilibrium loop.
Step (9). Evaluate approximations for , and u:
= W M,
=6,8M b,,bs,
ui.e,v tt/+8w
60
is used, the solution of the nonlinear problems becomes NewtonRaphson, while if the
secant stiffness is used we have a secant method solution.
Stress
Fig. 36. Secant and tangent material stiffness
Hysteresis Models
We have shown so far how it is possible to get the element internal forces and
form the updated stiffness for each iteration in the previous paragraphs. But we have
considered only the case in which the element is subjected to loading. For dynamic
analysis usually the case is that the element will be subjected to two additional phases:
unloading and reloading. It is important to notice that even for nonlinear static analysis
these two new phases could be present due to a nonlinear redistribution of forces on the
structure that could cause some elements to be overloaded and others to be under
loaded. The curve that is used to describe this behavior is called a hysteresis. For the
inelastic analysis, a proper selection of hysteretic models for the materials is one of the
critical factors in successfully predicting the dynamic response under strong motion.
Several models have been proposed in the past for reproducing various aspects of
125
Example 4 Circular Reinforced Concrete Column 2
Another monotonic test for a circular column was presented in Chai et al. (1991).
The column test data is summarized in Table 74.
Table 74. Design details for Example 4
Diameter
24"
Height
12'
Cover to main bar
0.8"
Concrete strength /).
4.725 ksi
Modulus of elasticity Ec
3918 ksi
Longitudinal steel
26#6
Yield strengthfy
45.7 ksi
Modulus of elasticity Es (adopted)
29000 ksi
Transverse steel
#2 hoop at 5"
Yield strength fy
51 ksi
Axial force
400 kips
Following the same procedure used in Example 3 an incremental load static test
was first performed using the secant and tangent approaches. No confinement was
considered. The comparison for the shear forces can be seen in Fig. 79. Note the effect
of the second order moment as the shear force decreases after 2 in of lateral
displacement. Then a pseudodynamic test was run to verify the FLPIER predictions, for
the confined and unconfined options. The comparison can be seen in Fig. 710 and Fig.
711. Note that we now have a much closer agreement in the stiffness. This agreement
can be explained by the fact that this is a much smaller column, so the factors that
contributed to the discrepancy in the previous test, have a much smaller influence in this
test.
It is interesting to note that based on the FLPIER confinement model the column
is not adequately confined. Note that in Fig. 710 the confined response is even worse
64
Reloading is represented by segment ef the tangent stiffness is E, and the
stress is given by cr = (e s.
For any of these phases the secant stiffness EÂ¡Â¡ is given by
E
ss
CT
8
Eq. 3. 36
Fig. 38. Elasticperfectly plastic model for mild steel
Uniaxial Monotonic Concrete Model Used in FLPIER
The concrete model used in FLPIER is generated based on the values of the
concrete strength fc and modulus of elasticity of concrete Ec, input by the user. The
compression portion of the curve, which is is based on the work of Wang and Reese
(1993), is highly nonlinear and has a maximum compressive stress fc which is related
to but not always equal to the compressive strength of a standard test cylinder,^ Based
on experimental research, fc is taken to be 85% of f, the maximum cylinder
compression stress.
APPENDIX C
FLPIER MANUAL FOR DYNAMIC ANALSYIS
Stiffness
The Flpier program now supports two types of stiffness, Secant stiffness and Tangent stiffness, meaning the way the
stiffness of a point in the section is obtained in the stressstrain diagram.
STIF
S=NSTIF
where NSTIF is the type of stiffness.
NSTIF = 0 secant stiffness (default).
NSTIF = 1 tangent stiffness. This option must be used for nonlinear steel sections with a time step analysis.
For static analysis it may converge faster than the secant method.
This section MUST end with a blank line.
Type of Analysis
There are two options for the type of analysis. Condensed means that all the structure will be condensed to the top of
the piles, and then the problem will be solved. Full means that there will be no condensation for the analysis. Because
the mass condensation is not exact like the stiffness condensation, use the full option for dynamic step by step analysis
when there is a structure. For single pile analysis use the condensed option.
SCND
B=NFULL
where NFULL is the type of analysis.
NFULL = 0 condensed analysis (default).
NFULL = 1 full analysis.
This section MUST end with a blank line.
Dynamics
For dynamic analysis the flag showed below must be present in the INPUT file. The next line must have the information
described below
DYN
Y=NDYNS C=NDAMP F=ALPHA1 ,BETA1 ,ALPHA2,BETA2 S=NPMAX J=DMP K=DMS 0=NPRT M=SMASS
H=NSHM L=NBMM N=NDYSOL U=NMSE R=NCMOD P=D1,02,03,04,05,06,07
where NDYNS is the type of dynamics solution.
NDYNS =0 Step by step integration (default).
NDYNS = 1 Spectrum analysis (for the structure only).
For time step analysis you have to input all data below, for spectrum analysis you have to input only the mass
density for the structure, the rest of the data is specified under the SPECTRUM label. If there is no reference
in the input file for any of the variables, the value assigned for the variable is zero.
NDAMP is the damping option.
NDAMP =0 no damping (default).
NDAMP =1 damping.
ALPHA 1 ,BETA1 coefficients for Rayleigh damping for the structure (C=ALPHA*M+BETA*K).
ALPHA2.BETA2 coefficients for Rayleigh damping for the piles.
ALPHA3.BETA3 coefficients for Rayleigh damping for the soil.
NPMAX is the maximum number of time steps for the analysis.
DMP is the mass density for the piles.
DMS is the mass density for the structure.
NPRT is the output option for time step analysis.
184
88
Mn = rmij> = 1 rm& = I Eq. 4. 15
where / is the identity matrix, a diagonal matrix with unit values along the main
diagonal. Equation 4.15 states that the natural modes are not only orthogonal but are
normalized with respect to m. They are then called a mass orthonormal set. When the
modes are normalized in this manner, Eqs. 4.13 and 4.14 become
Kn = = = 22 Eq. 4.16
Modal Equations
The equations of motion for a linear MDOF were derived in Chapter 2 and are
repeated here for convenience in local coordinates:
mil + c + ku = p(t) Eq. 4.17
As mentioned earlier, the displacements u of an MDOF system can be expanded
in terms of modal contributions. Thus the dynamic response of a system can be
expressed as
N
(0 = Z'trir (0 = q(t) Eq. 4. 18
r=\
Using this equation, the coupled equations 4.17 in u/i) can be transformed to a
set of uncoupled equations with modal coordinates q(t) as the unknowns. Substituting
Eq. 4.18 in Eq. 4.17 gives
N N N
Zmr?r(0 + Z*+,?,(0 = P(0 Eq 4 19
r=1 r=l r=l
Premultiplying each term in this equation by Ar gives
146
Again note that a much better approximation is obtained with the modified
values of the modulus of elasticity of the materials. Also note that the column strength is
close to the test value for all parameters, however note that the more ductile column
behavior, without such significant loss in strength, is only obtained with the confined
model. Note that in Fig. 732 we almost have a perfect match.
Cyclic Test S2
In this test the deflection in the transverse direction X is held constant, while
displacementcontrolled cycles are applied in the orthogonal direction Y. Note that
displacements are imposed in both direction so another spring was added to the Y
direction at the top of the column. The imposed displacement histories in the directions
X and Y respectively are shown in Fig. 733 and Fig. 734. The axial load is 284 KN.
The total test time was about 650 seconds, the adopted time step was 0.22 seconds. The
material data is fc = 0.031KN/mm2, Ec = 26.25 KN/mm!, = 0.46KN/mm2, and E, =
200KN/mm2. Confinement was not used. A good agreement with these properties was
obtained. The comparison can be seen in Fig. 735.
Fig. 733. Imposed displacements in X direction
27
The main reason for the development of lumped mass schemes was to save
memory space in computational calculations. In the past this was a problem, but today
computers are much faster and memory has become cheap and abundant. Therefore the
use of consistent mass formulation is justified, since it avoids certain instabilities in the
matrices operations.
Damping
Damping in structures is not viscous, i.e. is not proportional to velocity; rather it
is due to mechanisms such as hysteresis in the material and slip in connections. These
mechanisms are not yet well understood. Moreover, these mechanisms are either too
difficult to incorporate into the analysis, or they make the equations computationally too
expensive. Therefore with the actual limited knowledge about damping mechanisms,
viscous damping is usually adopted in most analysis. Comparisons of theory and
experiment show that this approach is sufficiently accurate in most cases.
Damping in structures can be considered in two ways:
1) phenomenological damping methods, in which the actual physical dissipative
mechanisms such as elasticplastic hysteresis loss, structural joint friction, or
material microcracking are modeled.
2) spectral damping methods, in which viscous damping is introduced by means
of specified fractions of critical damping (Critical damping, for which the
damping ratio is \ = 1, marks the transition between oscillatory and
nonoscillatory response).
The first class of methods requires detailed models for the dissipative
mechanisms and almost always result in nonlinear analyses. In the second class of
CHAPTER 3
DISCRETE ELEMENT MODEL AND MATERIAL HYSTERESIS
Discrete Element Derivation
A representation of the discreteelement used in FLPIER is shown in Fig. 31. It
was developed by Mitchell (1973) and modified by Andrade (1994). The center bar can
both twist and extent but is otherwise rigid. The center bar is connected by two universal
joints to two rigid and blocks. The universal joints permit bending at the quarter points
about they and z axes. Discrete deformational angle changes 'P;, and '\4 occur
corresponding to the bending moments MÂ¡, M3 and AT,, respectively. A discrete
axial shortening (8) corresponds to the axial thrust (7), and the torsional angle Â¥5
corresponds to the torsional moments in the center bar MÂ¡.
Fig. 31. Representation of discrete element. After Hoit et al., 1996
51
107
k
g
*i
0
0
0
0
0
0
0
0
0
*2
0
0
0
0
0
0 0
0 0
0 0
0 0
k3 0
0 0
0 k4
0 0
Eq. 5. 14
Modal analysis has also been developed for multisupport excitation. The reader
should refer to Monti et al. (1996), Der Kiurieghian (1992), Der Kiurieghian (1995),
and Chopra (1995) for more details.
126
than the unconfined response. The conclusion is that if the column is not adequately
confined the confinement model is going to give the worse results, and therefore should
not be used. However when the hoop spacing was changed to 2 in, a much closer
agreement can be observed between the confined model and the actual test (Fig. 711).
The adopted confined model makes no distinction between hoop or spiral confinement.
Fig. 79. Static shear comparison Example 4
55
E =ECW^Z
Then to satisfy equilibrium
M, = jjdF,.r, = Ijo^dA,
A A
Mv = J \dFrZ, = \ f
A A
T=f\dFl = J fo'dA,
A i A
Eq. 3.17
Eq. 3. 18
Eq. 3. 19
Eq. 3. 20
a) Strain due to
zaxis bending
e) Stressstrain relationship
b) Strain due to c) Strain due to
yaxis bending axial thrust
Fig. 33. Various components of total strain in the section. After Hoit et al., 1996
Numerical integration of Eqs. 3.18, 3.19 and 3.20 is done using Gaussian
Quadrature. To use the method of Gaussian Quadrature, the function being integrated
Ill
analysis, making it difficult to estimate the forces to design for. Therefore the computer
program FLPIER was modified to perform a timestep analysis of the structure using the
uncoupled method. It is now possible to input, for each pile node, the acceleration
record, and the program will run the timestep analysis considering a multiple support
excitation. All piles in the group are considered to be submitted to the same acceleration
record. The result is the maximum forces that each element has been subjected to during
the dynamic analysis.
Fig. 61. a)Coupled model; b)Uncoupled model
CHAPTER 7
PREDICTIONS OF RESPONSE
This Chapter provides a comparison between the response of test columns
reported in the literature (Baron and Venkatesan (1969), Chen and Atsuta (1973), Hays
(1975), Chai et al. (1991), Bousias et al. (1995), and Wilson et al. (1997)) and the
theoretical predictions of FLPIER. The tests include different cross sections, such as
steel sections, circular and square reinforced concrete sections. When the data is
available, the monotonic and the cyclic tests are compared. Three examples considering
soil are also presented, but the structure is considered to remain linear during the
analysis. The results for the various tests are presented next.
Example 1 Steel Section 1
Although this work is mainly about the behavior of reinforced concrete
structures it is opportune to verify the behavior of a steel section to validate the adopted
bilinear steel model. A W 14 x 176 steel section that was first studied by Baron and
Venkatesan (1969), and later used by Chen and Atsuta (1973) in similar studies, is used
for comparison. The steel is ASTMA36, with yielding stressfy = 36 ksi, and Youngs
Modulus Es = 29000 ksi. The history of deformations is given. No axial load is applied.
The W section is modeled as a single steel Hpile, composed of 16 nonlinear discrete
elements, the default in FLPIER, and is fixed at the base. A mass, a damper, and a
spring were attached to the top of the pile to minimize the dynamic effects, resulting in a
'pseudodynamic' model with imposed displacements. The FLPIER model and the
116
Moment (KN.mm) Moment (KN.mm)
145
Top lateral displacement (mm)
Fig. 731.Comparison tsl2
Top lateral displacement (mm)
Fig. 732. Comparison ts!2c
49
Step (10). Evaluate actual tangent stiffness K~' and internal forces
R1'1 for all the elements in the structure.
Step (11). Evaluate new effective stiffness matrix K, based on
actual tangent stiffness K1'1.
Step (12). Evaluate the outofbalance dynamic forces:
,e.v = F/ +6)+ ]
Step (13). Evaluate the ith corrected displacement increment:
ah'= [*;']"V;^
Step (14). Evaluate the corrected displacement increments:
Sh1 =8 hm +Au'
Step (15). Check for the convergence of the iteration process:
NO
Return to Step (8)
YES
Step (16). Return to Step (5) to process the next time step.
In this dissertation the full NewtonRaphson Method is applied. In this approach
the tangent stiffness matrix kr is updated for every iteration, in contrast with the
Modified NewtonRaphson Method in which the tangent stiffness kT is update once
every time step (it remains constant during the iterations performed within each time
step). This improves convergence, but additional computational effort is required in
forming a new tangent stiffness matrix kr and factorizing it at each iteration cycle. The
137
Anchorage slips occurs when the critical section of a member for flexure is
located near the adjoining member. Formation of a flexural crack at the interface
of two members strains the reinforcement crossing the crack. Widening of the
crack may produce inelastic strains in the reinforcement. This results in
penetration of yielding into the adjoining member, giving rise to significant
extension of reinforcement. Additional rigid body deformation may occur due to
slippage of reinforcement. The combined effect of reinforcement extension and
slip in the adjoining member may be referred to as anchorage slip.
Anchorage slip will result in member end rotations that are not accounted for in
flexural analysis, resulting in a member that is 'softer' than initially predicted. In the
FLPIER proposed concrete model this effect is not considered, but it seems, based on
the results presented in this work, that this effect may be considered by decreasing the
modulus of elasticity of the materials, making the member 'softer'. Alsiwat and
Saatcioglu (1992), and Saatcioglu et al. (1992) developed models for anchorage slip,
and the reader is referred to their work for more information on this topic. The strain
rate adopted for all the confined tests was 1005l/sec (very slow).
Fig. 719. Imposed displacement history for the first 90 seconds
57
Fig. 35. Circular section with integration points. After Hoit et al, 1996
Even for a nonlinear material analysis, the torsional moment M is assumed to
be a linear function of the angle of twist 'V, and the torsional stiffness GJ, where J is
the cross section torsional constant; and G is the material shear modulus, resulting in
the following expression for the torsional moment
M = GSVI2h Eq. 3.21
In this discrete approach the curvature is evaluated at two deformational joints
inside the element. These points are located at the quarter points from each node along
the element length. Therefore the effective position of any plastic hinge that might form
in the structure is restricted to these two locations. This should not cause any practical
limitation for most problems. However, it should be considered if trying to match a
theoretical solution with pure plastic hinges at theoretical ends of members. These
discrete joints, at which the deformations are concentrated, correspond to the integration
points along the length of the element, if a conventional finiteelement solution was
being made.
52
Element Deformation Relations
In Fig. 32, wiws and w7wg represent the displacements in the x, y and z
directions at the left and right ends, respectively. The displacements W4 and wÂ¡o
represent axial twists (twists about the xaxis) at the left and right ends, respectively; and
finally wjwg and wuwu represent the angles at the left and right end blocks about the x
and z axes, respectively. Based on a small displacement theory we can write:
/I = w3w9(w5 + wn)
Eq. 3. 1
h
S=wsw2(w6+wn)
Eq. 3. 2
The elongation of the center section of the element is calculated as follows:
8 = w7 w.
Eq. 3. 3
The angle changes for the center section about the z and y axes are the defined as
wsw2 w6+wn
1 h h 2
Eq. 3. 4
WS+WM
h ~ 2
Eq. 3. 5
The discretized vertical and horizontal angle changes at the two universal joints
are then
*1 =0,".
Eq. 3. 6
11
JS
1
CD
Eq. 3. 7
CD
1
II
&
Eq. 3. 8
LIST OF FIGURES
Figure page
11. Bridge pier components 2
21. Tower subjected to ground motion after Chopra (1995) 15
22. Support motion of an Lshaped frame.
a) Lshaped frame; b) influence vector l: static displacements due to Dg= 1; c)
effective load vector after Chopra (1995) 17
23. 3D Beam element 19
24. True 9node rectangular element 22
25. Mapping for a true rectangular 9node shell element 22
26. Shell element of uniform thickness 23
27. Lumped mass matrix at the nodes of true rectangular 9node shell element.
Numbers shown are fractions of the total element mass at each node 26
2.8. Full and condensed versions of the structure 34
29. Average acceleration 37
210. Linear acceleration 37
211. Secant and Tangent approaches. After Chopra(1995) 43
212. NewtonRaphson Method 45
31. Representation of discrete element. After Hoit et al., 1996 51
32. Discrete element displacements. After Hoit et al. 1996 54
33. Various components of total strain in the section. After Hoit et al., 1996 55
x
APPENDIX A
MASS UNITS
Problems in structural dynamics are based on Newton's second law, that is,
Force = (Mass)(Acceleration) A1
and these quantities, plus other directly related to them, must be expressed in a
consistent system of units. At the present time engineering practice is in the process of
conversion from English engineering units (United States Customary System) to the
International System (SI). In English engineering units dimensional homogeneity in Eq.
A1 is obtained when the force is given in Ibf, the mass in slugs, and the acceleration in
ft/sec2. This is the English ftlbfsec system of units. The slug is a derived unit, and form
Eq. Al:
1 slug = 1 lbfsec2/ft A2
The units of force and mass usually lead to confusion because of the use of the
term weight as a quantity to mean either force or mass. On the one hand, when one
speaks of an object's "weight", it is usually the mass, that is, quantity of matter, that is
referred to. On the other hand, in scientific and technological usage term "weight of a
body" has usually meant the force which applied to the body, would give an acceleration
equal to the local acceleration of free fall. This is the "weight that would be measured by
a spring scale. If the mass of a body is given in pounds (lb,) it must be divided by the
acceleration of gravity g = 32.2 ft/sec2 to obtain the mass as referred to in Newton's
175
66
Fig. 310. Concrete strains
Stresssit oin
for Concrete
Stress
* 0.5809
1 2.4
/ 3.435
/ 4.163
4.335 /
/ 4.684
^^Tdb996
1.457
38
lUJn 25/ 4/1999
Done
Point
Yvalue
Vvoiue
Fig. 311. Concrete stresses
matrix for this element was also implemented. The mass matrix for the piles cap was
also developed. The cap is modeled as linear shell elements. The stiffness matrix for this
type of element is considered to remain unchanged during the dynamic analysis. A
nonlinear model was also added for the soil. The py springs generated by the program
now have the capability of loading, unloading and reloading, just like the steel and
concrete. Such models can then be applied to dynamic analysis of reinforced concrete
structures subjected to seismic, impulsive, or wind loads. In addition, being the current
stateofpractice, modal analysis was also implemented. Seismic load can now he
applied to the linear structure considering the nonlinear behavior of the foundation. A
description of the analysis used to model the nonlinear dynamic behavior of bridge
piers, as well as the implemented nonlinear material behavior, is presented in this
dissertation.
xvii
187
Dynamic Step by Step Integration
Any structure may be analyzed using the step by step integration method of dynamic analysis. This method uses the
Newmark method and Raleigh damping to solve for dynamic response resulting from time varying loading. The time
varying loading can be applied as a single load function applied at many nodes or different time functions applied at
specific DOF. The STEP program will perform this analysis. The distributed and concentrated loads applied to the
structure will be applied as a CONSTANT load throughout the analysis. The applied load function can be a force or a
ground acceleration.
TRANSIENT
T=T1 L=L1 P=P1 Q=Q1 G=GF A=A0 B=A1
where T1
L1
P1
Q1
GF
is the time step increment for integration, default is 0.01s.
is the number of time varying load functions to be specified. The functions are applied at the
specified nodes and DOF.
is the maximum number of time points specified for any load function. If three functions are
specified, P1 is the maximum number of points used to specify any of the three functions.
is the flag for the type of load function applied. If Q1 =0 then the load is a force. If Q1=1, then the
load applied is a ground acceleration.
is the gravity factor to multiply times the acceleration or load record input below.
AO, A1 are global damping factors, implying that only one damping factor will be applied to structure, piles
and soil.
Note: you must choose one damping factor, either here or under the DYN header, only for the soil.
Both can not be chosen.
LOADING FUNCTION DEFINITIONS (L1 Sets of lines)
If you choose the multiple support excitation option, you have to create one set of loading functions for each pile node.
Including the first line.
The next lines specify the load function and its point of application. There should be L1 sets of lines.
FIRST LINES: Node and DOF application specification.
There can be as many of these lines as required to specify all loaded nodes and DOF for this load function. If the L=
portion is NOT specified, ALL active DOF will be loaded.
If Multiple Support Analysis user supplies the acceleration record for each pile node, 1 to 16. All piles are assumed to be
subjected to the same acceleration record.
N=NF,NL,NI F=L1 ,L2,L3,L4,L5,L6
where NF
NL
Nl
Li
is the first node in a generation sequence for which the DOF specification is used.
is the last node in a generation sequence for which the DOF specification is used.
is the increment for generating node numbers between NF and NL for which the DOF specification
is used. NL and Nl can be left blank if no generation is desired.
For Multiple Support Excitation these have to be piles nodes limited to 1 to 16.
is the state at which the ith DOF can have, either loaded or NO load. Therefore Li can have ONLY
the two following values;
Li = L is for loaded.
Li = N is for NO load.
The next lines specify the loading function values. The lines MUST contain FOUR pairs of numbers each. The number
of lines is dictated by the maximum number of points used to specify the function (P1). The points do NOT need to be at
even spacing.
T1.F1 T2.F2 T3.F3 T4.F4
65
The tension side of the curve is based on the tension stiffening model proposed
by Mitchell (1973). This procedure assumes an average tensile stressstrain curve for
concrete. The stress strain relation of concrete in tension is very close to linear with
cracking occurring at a small rupture stress f The high stresses actually experienced at
tensile cracks in the concrete will not be reproduced by the model. However the average
response over a finite length of beam will be adequately represented.
Based on the user input the program will generate the concrete curve as a series
of points connected by straight lines as shown in Fig. 39. Values in between the points
are obtained by interpolation of the extreme points in the interval. For values of fc = 6
ksi and Ec = 4615 ksi the strain and stress values for the concrete stressstrain curve
generated by the program can be seen in Figs. 310 and 311 respectively.
Fig. 39. FLPIER concrete points
174
response, even with complications such as nonlinear prestressed battered piles. In this
manner engineers can have a feeling of what to expect from a structure under a certain
earthquake or dynamic loading.
The nonlinear dynamic analysis of structures is a very complex subjected, and
the author with this work tried to show a fairly simple approximation to the problem that
showed good results for the types of structures studied.
92
superposed. For brevity we usually refer to this procedure as modal analysis. This
analysis method is restricted to linear systems with classical damping. Linearity of the
system is implicit in using the principle of superposition, Eq. 4.29. Damping must be of
the classical form in order to obtain modal equations that are uncoupled, a central
feature of modal analysis.
Element Forces
The element forces can be obtained by two procedures. In the first procedure the
//Â¡mode contribution r(!) to an element force r(t) is determined from nodal
displacements u(t) using element stiffness properties. Then the element force
considering contributions of all modes is
N
= Eq. 4.31
H=1
In the second procedure, the equivalent static forces associated with the nth
mode response are defined using the relationship/,(/) = knun(t), and using Eq.4.16 gives
/ (0 = (0 Eq. 4. 32
Static analysis of the structure subjected to these external forces at each time
instant gives the element force r(t), then the total force r(t) is given by Eq. 4.31.
Modal Equations for Ground Motion
The equations of motion governing the response of an MDF system to
earthquake induced ground motion are repeated:
m + c + ku = (/)
Eq. 4. 33
10
transversal time across a structure. A problem that is not a wave propagation problem,
but for which inertia is important, is called a structural dynamics problem. In this
category, the frequency of excitation is usually of the same order as the structures
lowest natural frequencies of vibration.
A typical example of a wave propagation problem would be that of analyzing the
stresses in a pile when it is grounded. It is important not to exceed the allowable stresses
in order not to damage the pile. Earthquake analysis of structures is a typical structural
dynamics problem, where the inertia forces govern the response of the structure.
Problems of structural dynamics can still be subdivided into two broad
classifications. In the first one, we are interested in the natural frequencies of vibration
and the corresponding mode shapes. Usually, we want to compare natural frequencies of
the structure with frequencies of excitation. In design, it is usually desirable to assure
that these frequencies are well separated. In the second classification, we want to know
how a structure moves with time under prescribed loads, like under impacts, blasts or
wind loads, and/or motions of its supports, like in the case of an earthquake. We are
interested in the timehistory analysis. The two most popular methods of dynamic
analysis are modal methods and direct integration methods.
Equations of Motion. Mass, and Damping Matrices
The equations that govern the dynamic response of a structure will be derived by
requiring the work of external forces to be absorbed by the work of internal, inertial and
viscous forces, for any small kinematically admissible motion (i.e., any small motion
that satisfies both compatibility and essential boundary conditions). For a single
element, this work balance becomes
47
MaZ>}={^}+, Eq.2.101
where
[^]=^[A/]+[c]+[^] Eq.2.102
and
W* L = {* L, {* L + {>L + H ] + [c]{} Eq. 2. 103
Note that [Â£,] must be predicted using {D} (and possibly {0}^ if strain rates
effects are important) and must be factored at least once each time step during nonlinear
response. If [T,] is not an accurate prediction of the true tangentstiffness matrix from
time nAt to time (n + l)Ai, then the solution of Eq. 2.101 for {AD} will be in error. The
error in nodal forces, that is the residual, is given by the imbalance in the equation of
motion as
{r}= Eq. 2. 104
where the internal force vector {R"")+i is computed elementby element. The process
stops when the outofbalance force vector Â¡R"r} is smaller than a specified tolerance.
Finally, considering all the topics discussed in this section, the procedure for
nonlinear dynamic analysis can be summarized in the algorithm described next.
Nonlinear Dynamic Analysis Algorithm
Step (1). Form the initial stiffness matrix K, mass matrix M, and damping C for
the structure.
Step (2). Compute the constants for the numerical method chosen:
162
Test
O 5 10 15 20
Time (s)
Fig. 754. Moments comparison
Tests PG2 and PG3
Tests PG2 and PG3 consist of a single column supported by a 2 x 2 (PG2) and a
3x3 (PG3) pile group respectively. The pile and soil properties are identical to those in
the single pile test described previously. The structure dimensions (in meters) and setup
are shown in Figs. 755 and 756. The masses for the cap, column and top mass are
given in Table 713. The period of the superstructure considering it is fixed at the base
of the column is 0.5 s. The prototype model in FLPIER had its pier stiffness adjusted to
have this same period. Because the structure is considered to be linear this was done
simply by changing the values of the modulus of elasticity E, and moment of inertia I,
for the linear 3D beam elements that model the pier.
Table 713. Masses (tons) for pile group tests
PG
Cap mass
Column mass
Ton mass
2x2
132
174
233
3x3
329
193
468
7 PREDICTIONS OF RESPONSE
116
Example 1 Steel Section 1 116
Example 2 Steel Section 2 118
Example 3 Circular Reinforced Concrete Column 1 121
Example 4 Circular Reinforced Concrete Column 2 125
Example 5 Rectangular Reinforced Concrete Column 128
Monotonic Tests 129
Cyclic Test SO 136
Cyclic Test SI 142
Cyclic Test S2 146
Cyclic Tests S3 and S4 148
Cyclic Test S10 153
Example 6 Piles in Sand 159
Test SP 159
Tests PG2 and PG3 162
Example 7 Mississippi Dynamic Test 166
8 CONCLUSIONS 172
APPENDICES
A MASS UNITS 175
B GAUSS QUADRATURE 178
C FLPIER Manual for dynamic analsyis 184
REFERENCES 184
BIOGRAPHICAL SKETCH 191
vii
63
possible to overcome some of the difficulties found in the various hysteresis models.
Although computationally more expensive, this fiber modeling approach gave the
modified discrete element the versatility not found in many models. A description of
these modified curves follows next.
Uniaxial Mild Steel Model
Because the cyclic behavior of steel is very dependent on the heat from which it
was produced, it was decided that rather than trying to predict the steel behavior based
on exponential curves (Agrawal et al. (1965), Shen and Dong (1997)), to use the bilinear
representation for the steel behavior. This represents a safe lower bound solution that is
adequate for most of the construction steel. So the mild steel reinforcement is assumed
to be perfectly elasticplastic (no hardening) and similar in both tension and
compression as shown in Fig. 38. The parameters needed for the mild steel bilinear
model are the modulus of elasticity Es and yield stress fi>. The mies for this model are as
follows (refer to Fig. 38):
Loading is represented by segment ab, the tangent stiffness is the initial
modulus of elasticity for steel Et and the stress is given by a = Et.s .
Yielding'. If e > ey yielding occurs and is represented by segment bc. The
tangent stiffness Es is equal to 0 and the stress a = f, the yielding stress for steel. The
residual strain Â£r is given by er =e Â£ .
Unloading is represented by segment cd, the tangent stiffness is Es and the
stress is given by a = (s s,).Â£,.
This dissertation was submitted to the Graduate Faculty of the College of
Engineering and to the Graduate School and was accepted as partial fulfillment of the
requirements for the degree of Doctor of Philosophy.
August 1999
M. J. Ohanian
Dean, College of Engineering
Winfred M. Phillips
Dean, Graduate School
186
Spectrum Analysis
The results of an EIGEN solution can be used to perform a spectrum analysis. This procedure uses the mode
superposition method to combine the individual eigen vectors into a single response, based on the excitation given by a
response spectrum. Response spectrums are usually given for earthquake loading. This procedure combines the
individual modes response for a spectrum acting in each of the three global directions, X,Y and Z. The modal responses
are combined using the Complete Quadratic Combination (CQC) procedure, the directions are combined using Square
Root Sum of the Squares (SRSS). SPECT allows only a single input response spectrum with different scale factors for
that spectrum in each of the three directions (X, Y, and Z). The program will allow either an input spectrum or an input
acceleration record and generate the spectrum values internally.
SPECTRUM
For input Spectrum use the following lines:
S=SP D=DX,DY,DZ N=NV
where SP
DX
DY
DZ
NV
is the number of spectrum points used to define the response spectrum curve. The points are given
in pairs (Circular frequency (rad / s), Value) and are assumed linear between values.
is the scale factor to apply to the input spectrum for use in the X direction. The response spectrum
values are scaled by this factor when used for the X direction.
is the scale factor to apply to the input spectrum for use in the Y direction. The response spectrum
values are scaled by this factor when used for the Y direction.
is the scale factor to apply to the input spectrum for use in the Z direction. The response spectrum
values are scaled by this factor when used for the Z direction.
is the number of eigenvectors to use for the responses. The default is the total number of vectors
solved for from EIGEN.
SPECTRUM APPLIED DIRECTIONS (Repeat for as many lines an necessary)
A spectrum analysis also needs damping ratios for each mode used in the analysis. The damping ratio $si$ is the
percentage of damping for the mode in question. These values must be specified if a spectrum analysis is to be
performed. The next lines specifies the damping ratios to use for each mode. As many lines as required can be used.
NF.NL.NI S=S1
where NF is the first mode in a generation sequence for which the damping ratio is used.
NL is the last mode in a generation sequence for which the damping ratio is used.
Nl is the increment for generating mode for which the damping ratio is used. Modes between NF and
NL will also use the specified ratio.
NL and Nl can be left blank if no generation is desired.
S1 is the damping ration value to be used (as a decimal).
SPECTRUM DEFINITION LINES (Repeat for as many lines an necessary)
The next lines specify the spectrum function values. The lines MUST contain FOUR pairs of numbers each. The
number of lines is dictated by the number of points used to specify the function (SP). The points do NOT need to be at
even spacing.
F1.A1 F2.A2 F3.A3 F4.A4
Where Fi.Ai are the frequency and acceleration values for the point being specified.
NOTE: the last point in the spectrum has to be 0,0.
122
The first step was to perform an incremental static analysis using the secant and
the tangent methods. The static test uses the actual uniaxial material model presented in
Chapter 3. The dynamic test uses the proposed concrete model also presented in Chapter
3. The comparison between FLP1ER and the test is shown in Fig. 76 and 77 for the
shears and moments respectively. From Fig. 7.6 to 7.7 note that although the column
strength values are very close, the slope of the curve is slightly different. Three factors
may have contributed to that:
a) Because the original work did not provide the values for the modulus of
elasticity for steel and concrete, they had to be adopted. For the concrete the empirical
value given by the formula (Meyer, 1996):
Ec = 57000Jf\ Eq. 7. 1
was adopted. For the steel 29,000 ksi was adopted as the modulus of elasticity, which is
typical for steel.
b) The use of a discrete curve, defined by straight lines segments, instead of a
continuous curve (parabola) for the concrete model may also have introduced some
error.
c) A phenomena called anchorageslip, which is explained in more details later,
may be present. Basically this phenomena causes additional displacements usually not
accounted for by the analysis, making the column behave softer than predicted.
Note that all these facts have direct influence on the column stiffness, explaining
the difference between the FLPIER model and the test results.
A 'pseudodynamic' test, following the same idea presented in Example 1, but
with an adjustment in the mass and damper because of the change in stiffness, was then
93
where
P,jÂ¡() = m\t(t) Eq.4.34
The displacement u of an NDOF system can be expressed as
= !>(') = Eq.4.35
=1 n=l
The spatial distribution of the effective earthquake forces pej(f) is defined by s =
mi this force distribution can be expanded as a summation of modal inertia force
distributions s:
N
mi = ^ Tmn Eq. 4.36
n1
where
T=~J = f>i M = Eq. 4.37
the factor T is usually called the modal participation factor, implying it is a measure of
the degree to which the nth mode participates in the response. The contribution of the
nth mode to the excitation vector mi is
s = Eq. 4. 38
which is independent of how the modes are normalized.
Equation 4.27 is then specialized for earthquake motion by replacing p(t) by
Pejft) to obtain
9. +2^,w9 +co lq = Tiig(t)
Eq. 4. 39
24
N
NtN,
Eq. 2. 30
Note that [JV][iV] has dimensions 54 x 54, which are the correct dimensions for
the mass matrix considering the shell element illustrated in Fig. 26.
It is easily verified that for the true rectangular element illustrated in Fig. 24 the
change of coordinates from (i and t to x and y can be expressed as
x = ap y = br\ Eq. 2.31
and recalling for convenience Eq, 2.5
[m]= \p[N[N]dV Eq. 2. 32
V
If we now consider uniform the thickness t and the mass density p over the
entire element, the differential volume dV can be written dV=tdxdy and Eq. 2.5 now
becomes
[m] = pi J[jV] [N]dxdy Eq. 2. 33
v
the differencials dx and dy can obtained directly from Eq. 2.31,
dx = adp dy = bdr\ Eq. 2.34
and Eq. 2.32 can be rewritten
i i
[m] = ptab  J[ N]7 [ N]dpdr\ Eq. 2. 35
ii
95
The resultant equivalent static forces f(t) are the product of two quantities: (1)
the nthmode contribution s to the spatial distribution mi of pejÂ¡(J) and (2) the pseudo
acceleration response of the nthmode SDOF system to g(t).
Response Spectrum Analysis
The type of modal analysis that has been presented is called response history
analysis (RHA), because the response depends on analyzing N uncoupled SDOF
equations of motion on time by any procedure used for SDOF systems. Flowever
structural design is usually based on the peak values of forces and deformations over the
duration of the earthquakeinduced response. Therefore another type of modal analysis
was developed, which is called response spectrum analysis (RSA).
This method is based on the fact that the exact value of the peak response of an
MDOF system in its nth natural mode can be obtained from the earthquake response
spectrum. A spectrum is a plot of the period T or natural frequency to versus the
maximum (or peak) response, which could be acceleration, velocity or displacement for
a SDOF system subjected to a specific earthquake with a specific damping ratio. Such a
plot is shown in Fig. 42. Therefore the peak value of A(t) is available from the pseudo
acceleration response spectrum as its ordinate A(Tn,Â£,) corresponding to the natural
period T and damping ration i;; for brevity, A(TÂ£) will be denoted as A. Therefore
the peak response is
rm = r*A,
Eq. 4. 44
80
Confinement basically decreases the slope of the descending branch of the
loading curve of concrete, making the confined concrete member more ductile and less
brittle. The reader is referred to Saatcioglu and Razvi (1992), and Soroushian et al.
(1986) for a good discussion on the subject.
Soroushian et al. (1986) proposed a very simplified model, that incorporates
both, strainrate and confinement effects in the compression envelop curve for concrete.
The following constitutive model (Soroushian et al. (1986)) was implemented in
FLPIER:
/ =
e < 0.002^^3
KxK2fc[\z{z0.002^*3)]
>0.2K,Kj:
e>0.002X,K}
Eq. 3. 59
where
/= concrete compressive stress
s = concrete compressive strain
ps = volumetric ratio of the hoop reinforcement to concrete core =
fc = 28day compressive strength of concrete, adopted as 0.85./c
fi, = yield strength of transverse reinforcement
117
dynamic parameters are shown in Fig. 71. A comparison between the actual test and the
FLPIER model is shown in Fig. 72. Based on the fact that just a few points are reported
in the original work, the comparison is very good.
m = 1 kip s2/in
a
k = 10000 kip/in
c = 200 kip s/in
L
t
1\W
Fig. 71. Example 1 computer model
Curvature x 1000 (rads)
Fig. 72. Comparison FLPIER x reference for cyclic loading
120
Fig. 75. FLPIER x Hays, bending moment comparison
The following observations can be made from this example:
a) As expected the tangent and the secant approaches give the same response
until a point where the secant approach does not converge (between 1.5 and 2.0 in).
Beyond this point FLPIER is in good agreement with the results given by Hays (1975).
This indicates that the new tangent approach introduced in the FLPIER model seems to
work, and in this particular example is more stable than the secant approach.
b) Second, the pA (or second order) moment effect is also modeled, and is clear
in Fig. 74. Note that the shear force starts to decrease as the second order moment
dominates the response. Although this is not a dynamic test, its importance relies on the
fact that it tests the tangent stiffness approach used in the dynamic analysis.
96
All response quantities r(t) associated with a particular mode, say the nth mode,
reach their peak values at the same time instant as A(t) reaches its peak.
Fig. 42. Typical response spectrum Us (S,ven)
Modal Combination Rules
How can we combine the peak modal responses rm (n = \,2,...,N) to determine
the peak value r0= max I r(t)  of the total response? It will not be possible to determine
the exact value of r from rm because, in general, the modal responses r(t) attain their
peaks at different time instants and the combined response r(t) attains its peak at yet a
different instant.
Approximations must be introduced in combining the peak modal responses rno
determine from the earthquake response spectrum because no information is available
when these peak modal values occur. The assumption that all modal peaks occur at the
13
manner in which {/?'} is computed depends on the dynamic analysis procedure
adopted.
When Eqs. 2.5 and 2.6 are evaluated using the same shape functions [;V] as used
in the displacement field interpolation (Eqs. 2.2), the results are called consistent mass
and consistent damping matrices. These matrices are symmetric. On the element level,
they are generally full, but on the structure level, they have the same sparse form as the
structure stiffness matrix. When p and cÂ¡Â¡ are nonzero, consistent matrices [m] and [c]
are positive definite. Using the mass matrix for example, the kinetic energy
y {} [m]^} is positive definite for any nonzero {cf}.
In typical structural analysis we are more interested in dry fiction and hysteresis
loss, than in viscous damping. It is still not well understood how the damping
mechanisms develop in structures, so from a practical standpoint Eq. 2.6 does not
correctly represent structural damping.
The internal force vector, Eq. 2.7, represents loads at nodes caused by straining
of material. Equations 2.4 and 2.7 are valid for both linear and nonlinear material
behavior; that is, in Eq. 2.7, {a} could be a nonlinear function of strain or strain rate.
For linearly elastic material behavior, {a} = [it][S]{cf} and Eq. 2.7 becomes
{}[*]{<*} Eq. 2. 9
where the usual definition of the stiffness matrix holds, that is,
M= \[B]T[E\B\dV
re
when Eq. 2.10 is used, Eq. 2.4 becomes
Eq. 2. 10
97
same time and their algebraic sign is ignored provides an upper bound to the peak value
of the total response:
H
ra < XkJ Eq. 4. 45
n=i
This upperbound value is usually too conservative. Therefore, this absolute sum
(ABSSUM) modal combination rule, is not popular in structural design applications.
The squarerootofsumofsquares (SRSS) rule for modal combination is
r0
Eq. 4. 46
The peak response in each mode is squared, the squared modal peaks are
summed, and the square root of the sum provides an estimate of the peak total response.
This modal combination rule provides excellent response estimates for structures with
wellseparated frequencies.
The complete quadratic combination (CQC) rule for modal combination is
applied to a wider class of structures at is overcomes the limitations of the SRSS rule.
According to the CQC rule,
/ n s y/2
= Zip// Eq. 4. 47
\/=l =1 /
Each of the N2 terms on the right side of this equation is the product of the peak
response in the ith and nth modes and the correlation coefficient p, for these two
modes; p, varies between 0 and 1 and p, = 1 for i = n. Thus Eq. 4.47 can be rewritten
as
Moment (KN.mm) Moment (KN.mm)
141
150000
100000
50000
0
50000
100000
150000
Test
FLPIER
100 80 60 40 20 0 20 40 60
Top lateral displacement (mm)
Fig. 724. Comparison ts02
80 100
100 80 60 40 20 0 20 40 60 80 100
Top lateral displacement (mm)
Fig. 725. Comparison ts03
Moment (kip. in)
127
Lateral top displacement (in)
Fig. 710. Dynamic moment comparison Example 4. Hoop spacing = 5 in
Lateral top displacement (in)
Fig. 711. Dynamic moment comparison Example 4. Hoop spacing = 2 in
ACKNOWLEDGMENTS
I would like to thank my parents, Cesar and Dalva, my brothers, Magno and
Marcus, and all my family, for their unconditional support in all phases of my graduate
studies. I also would like to thank my fiance, Leandra, for being so patient and
supportive during all my work.
I also would like to thank all the faculty from the Civil Engineering Department
at the University of Florida, and the members of my supervisory committee Dr. Hoit,
Dr. McVay, Dr. Fagundo, Dr. Flays, and Dr. Wilson for always having their doors open
to answer my questions.
Finally I would like to thank all the graduate students from the workstation lab,
specially, Mark and Wirat, for their support and friendship.
IV
157
Top lateral displacement (mm)
Fig. 750. Comparison tsl04 x test
Note again that a better response is obtained with the corrected values for the
modulus of elasticity of the materials. In this test the steel strength was also changed to
illustrate its effect in the cyclic response of the column. Note from Fig. 748 that the
column strength is greatly reduced when the steel strength is reduced. In this test the
best fit was obtained using a steel strengthfy = 0.35 KN/mm2 and the confined model,
as seen in Fig. 750. This discrepancy in the steel strength may indicate that the steel
used for this column may be weaker than the specified value. It may also indicate that
varying the axial force may also add some strength degradation to the column, not
accounted for by the actual FLPIER model. The data for the original set seems to have a
31
the beams are much stiffer than the columns the shear DOFs would be a good choice,
instead of the rotations. The condensed stiffness matrix obtained includes the effects of
the slave degrees of freedom, which can be recovered at any time during analysis. The
same approach can be applied to the mass matrix, however dynamic condensation
(Miller, (1981), Paz (1985)) is not exact, as will be shown later in this section, but it can
give good results if some rules are observed in the modeling. The approach described
here is given by Meirovitch (1980, pp.371372):
Let us write the equations for the potential and kinetic energy for a system in the
matrix form:
V = \[Df[K\[D] Eq.2.40
T = \[D]T{M}[b] Eq.2.41
and divide the displacement vector [D into the master displacement vector qÂ¡
and slave displacement vector qÂ¡, or
[D] =
9,
.92.
Eq. 2.42
Then the stiffness matrix [A!] and mass matrix [M\ can be partitioned
accordingly, with the result
Eq. 2. 43
1
9,
T
Mn
[9,1
2
.92.
m2,
m21_
1.92.
Eq. 2. 44
where K.2i=KÂ¡2 and \2i=MÂ¡2 The condition that there be no applied forces in
the direction of the slave displacements can be written symbolically in the form
SV
Qq 9,^11+92^21
Eq. 2. 45
100000
150000
100 80 60 40 20 0 20 40 60 80 100
Top lateral displacement (mm)
Fig. 726. Comparison ts04
Cyclic Test SI
In this test displacement cycles in pairs of linearly increasing amplitude are
alternately applied in the two transverse directions X and Y. The displacement history
for each direction can be seen in Fig. 727 and Fig. 728.
Fig. 727. Imposed displacement in X direction
131
A parametric study on the influence of changing the parameters of steel and
concrete was also performed for the monotonic case. Each of the parameters, modulus
of elasticity of concrete Ec, compressive strength of concrete fc, modulus of elasticity of
steel Es, and yield stress of steel fy, was changed by a factor of 0.5 for comparison.
Table 76 helps to identify the tests and the changed parameters. The input files are
identified by the .in extension. The extension .DAT identifies the output data file for the
plots. Confinement was not considered in these tests. The comparison between the
various parametric tests and the model with the original parameters is shown in Figs. 7
14 to Fig. 717.
Table 76. Parametric tests, units are KN/mm2
File
Ec
fc
Es
fy
Confinement
bml.in
26.25
0.031
200
0.46
No
bmll.in
13
0.031
200
0.46
No
bml 2.in
26.25
0.031
100
0.46
No
bml 3.in
26.25
0.015
200
0.46
No
bml 4.in
26.25
0.031
200
0.23
No
bml 5.in
13
0.031
100
0.46
No
bml 6.in
26.25
0.015
200
0.23
No
bml 7.in
13
0.015
100
0.23
No
121
Example 3 Circular Reinforced Concrete Column 1
The third example analyzed by FLPIER was the fullscale flexure column
presented in Chai et al. (1991). The reference mentions that this column represented the
current (1991) ductile design for bridge columns. Table 72 summarizes the parameters
for the test column which was subjected to a constant axial compression force of 1000
kips and a lateral cyclic displacement of increasing amplitudes until failure of the
column. Unfortunately the data for the cyclic test was not available for comparison. The
computer model is identical to the one shown in Fig. 73, except for the cross section
and length L, now 30 ft. The different parameters for the tests are shown in Table 73.
All tests have the same material properties.
Table 72. Design details for Example 3
Diameter
60"
Height
30'
Cover to main bar
4"
Concrete strength/c
5.2 ksi
Modulus of elasticity Ec
4110 ksi
Longitudinal steel
25#14
Yield strength fy
6B.9 ksi
Modulus of elasticity Es (adopted)
29000 ksi
Transverse steel
#5 spiral at 3.5"
Yield strength fy
71.5 ksi
Axial force
1000 kips
Table 73. Model parameters for Example 3
File
Stiffness
Analysis type
Confinement
Convergence
circ2.in
Secant
Static
No
OK
circ21.in
Tangent
Static
No
OK
circ22.in
Tangent
Dynamic
No
NO
circ23.in
Tangent
Dynamic
Yes
OK
114
degradation factor, given by the user, multiplied by the initial py curve. At this time the
user can not supply the fully degraded py curve. In a seismic event or for an extreme
event that involves impact loading, some cyclic degradation may occur, but the fully
degraded value may never be reached. A typical value for the soil degradation
parameter, X, is given in Eq. 62.
Eq. 6. 2
Where NÂ¡o is the number of cycles that would be necessary to degrade the soil
by 50 percent. This would be a sitespecific parameter that wold have to be obtained by
appropriate laboratory tests or from cyclic lateral pile testing.
Strain Rate Effect
Usually when soil is subjected to dynamic loading, cyclic degradation occurs
simultaneously with an increase in the apparent soil (axial or lateral) resistance caused
by rapid rates of loading that occur in the most extreme events. Once the cyclic
degradation has been accounted for, the rate effect is computed from (ONeill et al.,
1997):
Eq. 6.3
where pÂ¡ is the instantaneous soil resistance considering both cyclic and loading rate
effects, pc is the resistance considering only cyclic loading, tr is the actual rate of loading
in Hz or unit of distance per second, ts is the corresponding rate of loading appropriate
for standard slow cyclic loading (typically 0.01 to 0.1 Hz) and F2 is a soil factor, which
can be taken as 0.01 0.03 for sand, 0.02 0.07 for silts, 0.02 0.12 for clays, and 0.01 
I would like to dedicate this dissertation to my parents, Cesar and Dalva, my brothers,
Magno and Marcus, and to my Fiance, Leandra. I could not have reached such an
accomplishment without their help.
