Nonlinear dynamic analysis of bridge piers

MISSING IMAGE

Material Information

Title:
Nonlinear dynamic analysis of bridge piers
Physical Description:
xvii, 191 leaves : ill. ; 29 cm.
Language:
English
Creator:
Fernandes, Cesar, 1968-
Publication Date:

Subjects

Subjects / Keywords:
Bridges -- Florida   ( lcsh )
Bridges -- Foundations and piers -- Computer simulation   ( lcsh )
Concrete bridges -- Foundations and piers -- Testing   ( lcsh )
Civil Engineering thesis, Ph. D   ( lcsh )
Dissertations, Academic -- Civil Engineering -- UF   ( lcsh )
Genre:
bibliography   ( marcgt )
non-fiction   ( marcgt )

Notes

Thesis:
Thesis (Ph. D.)--University of Florida, 1999.
Bibliography:
Includes bibliographical references (leaves 184-190).
Statement of Responsibility:
by Cesar Fernandes, Jr.
General Note:
Typescript.
General Note:
Vita.

Record Information

Source Institution:
University of Florida
Rights Management:
All applicable rights reserved by the source institution and holding location.
Resource Identifier:
aleph - 030473989
oclc - 43380210
System ID:
AA00012993:00001


This item is only available as the following downloads:


Full Text










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 3D-Beam 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
Time-History 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

2-1. Natural frequencies of a uniform cantilever beam: Consistent-Mass Finite Element

and exact solution ................................................................. ......................... 21

2-2. Natural frequencies of a uniform cantilever beam: Lumped-Mass Finite Element

and Exact Solution.................................... .....................................................21

2-3. Recommended damping rations for structures ................... ..........................30

3-1. Curves coefficients ............................................................... .......................... 73

7-1. Results for Exam ple 2............................................................ ....................... 119

7-2. Design details for Example 3..........................................................121

7-3. Model parameters for Example 3 ................................ .........................121

7-4. Design details for Example 4....................... .... ................................125

7-5. Loading for each test .......................................................................128

7-6. Parametric tests, units are KN/mm2....................................................... .............131

7-7. Parametric tests for confinement under different strain rates.................................135

7-8. Parameters for test SO- units are KN/mm2.......................... ........................ 139

7-9. Parameters for test S1- units are KN/mm2....................... .............................143

7-10. Parameters used in test S3 and S4...................... ....................149

7-12. Some soil properties for CSP1.........................................................160

7-13. Masses (tons) for pile group tests...................... ... ...............................162

A -1. M ass density units.............................................................................. .............. 177








B-1. Gauss-Legendre abscissas and weights ................................. ............................182














LIST OF FIGURES


Figure page

1-1. Bridge pier com ponents........................................................ ...........................2

2-1. Tower subjected to ground motion after Chopra (1995).........................................15

2-2. Support motion of an L-shaped frame.

a) L-shaped frame; b) influence vector f: static displacements due to Dg=1; c)

effective load vector after Chopra (1995).................................................. 17

2-3. 3D B eam elem ent ............................................................... .................................19

2-4. True 9-node rectangular element.................................................22

2-5. Mapping for a true rectangular 9-node shell element...............................................22

2-6. Shell element of uniform thickness .................................................. ...................23

2-7. Lumped mass matrix at the nodes of true rectangular 9-node 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

2-9. Average acceleration ............................................................ ......................... 37

2-10. Linear acceleration ............................................................. .........................37

2-11. Secant and Tangent approaches. After Chopra(1995)...................................... 43

2-12. Newton-Raphson Method............................. ..... ........................45

3-1. Representation of discrete element. After Hoit et al., 1996 ....................................51

3-2. Discrete element displacements. After Hoit et al. 1996 ..........................................54

3-3. Various components of total strain in the section. After Hoit et al., 1996 ................55








3-4. Rectangular section with integration points. After Hoit et al., 1996.......................56

3-5. Circular section with integration points. After Hoit et al, 1996 ..............................57

3-6. Secant and tangent material stiffness............................ ........................ 60

3-8. Elastic-perfectly plastic model for mild steel......................................................64

3-9. FLPIER concrete points......................... ............................ ...........................65

3-10. Concrete strains ................................................................... ........................66

3-11. Concrete stresses.................................... ...................................................... 66

3-12. Envelop curve for concrete.......................... ...... ........................68

3-13. Typical compression loading........................ ...... ..........................70

3-14. Typical loading in tension ...................... ......................................71

3-15. Typical unloading in tension............................................................73

3-16. Typical unloading in compression............................ .........................74

3-17. Compression unloading with gap ....................... .. ................................74

3-18. Compression unloading with no gap................................. ......................75

3-19. Typical loading, unloading and reloading in compression........................... ...76

3-20. Concrete behavior with gap............................... .... ..........................76

3-21. Bilinear model for concrete..........................................................77

3-22. Stress-strain curve for concrete in FLPIER........................................................78

3-23. Confined and unconfined concrete models response ............................................. 79

3-24. Core width for different cross sections........................... ....................... 81

3-25. Confined concrete model...................... ....... ..........................82

4-1. Generalized SDF system for the nth natural mode............................................90

4-2. Typical response spectrum ...................... .....................................96








4-3. M odal analysis of pier ........................................................... ........................ 99

5.1. M multiple support m option ................................................... ........................... 102

5-2. 2D frame submitted to multiple support motion ................................................. 105

5-3. Pile subjected to multiple support excitation.....................................................106

6-1. a)Coupled model; b)Uncoupled model.............................. ........................11

6-2. Typical p-y curve................................................................. ...... ........................ 12

6-3. Cyclic soil m odel...................... ......... ..............................................................113

7-1. Exam ple 1 com puter m odel.....................................................................................117

7-2. Comparison FLPIER x reference for cyclic loading ............................................. 117

7-3. Exam ple 2 com puter m odel.................................................................................... 118

7-4. FLPIER x Hays, shear force comparison .................................. ........................119

7-5. FLPIER x Hays, bending moment comparison......................................... ..........120

7-6. Shear comparison Example 3 x FLPIER........................ .................................123

7-7. Moment comparison Example 3 and FLPIER.....................................................124

7-8. Dynamic model comparison................................ .......................124

7-9. Static shear comparison Example 4....................................................... 126

7-10. Dynamic moment comparison Example 4. Hoop spacing = 5 in........................127

7-11. Dynamic moment comparison Example 4. Hoop spacing = 2 in........................127

7-12. Computer model for Test 3..........................................................130

7-13. Column capacity using FLPIER.......................... ...... ....................130

7-14. E and E, changed........................................ ..............................................132

7-15.fc and f changed ............................................................... ....................... ...... 132

7-16. E andf changed.................................................................... .......................133




xii








7-17. All parameters changed .................... .........................133

7-18. Confinement and Strain rate effect............................. .......................135

7-19. Imposed displacement history for the first 90 seconds........................................137

7-20. Comparison FLPIER x Test, original properties..............................................138

7-21. Comparison FLPIER x Test, modified properties................................................138

7-22. Imposed tip displacement history for test.......................... ............ ..................140

7-23. Com prison tsOl ........................................ ..............................................140

7-24. Com prison ts02............................................................................................141

7-25. Com prison ts03 ............................................................. ................................... 141

7-26. Comparison ts04........................ ................. ............................................142

7-27. Imposed displacement in X direction..................................................................142

7-28. Imposed displacement in Y direction......................... .............................143

7-29. Com prison tsl 1........................................ ..............................................144

7-30. Comparison tsl c ....................................................................................144

7-31.Com prison ts12 ........................................... .............................................. 145

7-32. Com prison tsl2c ............................................................ ..........................145

7-33. Imposed displacements in X direction....................... ................................ 146

7-34. Imposed displacements in Y direction......................... ...............................147

7-35. Comparison test S2 x FLPIER ................................... ........................147

7-36. Imposed X displacement for test S3.................................................................... 149

7-37. Imposed Y forces for test S3 ...................................................149

7-38. Comparison Test S3 x FLPIER ts31 ............................. .........................150

7-39. Comparison Test S3 x FLPIER ts32 .............................. .........................150




xiii









7-40. Imposed displacement in X direction for test S4..................................................151

7-41. Imposed load in Y direction for test S4.......................... ............................151

7-42. C om prison ts4 ............................................................... ................................. 152

Fig 7-43. Com prison ts42....................................... ............................................ 152

7-44. Com prison ts43 ............................................ ............................................153

7-45. Imposed force in X-direction for S10...................................................................154

7-46. Imposed force in the Z-direction for S10 ........................ ............ ............ 155

7-47. Com prison tslO0 x test ................ .................................. ......................... 155

7-48. Com prison ts 02 x test .................................................. ... ....................... 156

7-49. Com prison tsl03 x test ..................................................... ....................... 156

7-50. Comparison ts104 x test .........................................................157

7-51. Single pile in sand............................................................ .......................... 160

7-52. Top ring acceleration........................................................ ......................... 161

7-53. Top displacement comparison........................ ...... ......................161

7-54. Moments comparison ................. ............... ........................................162

7-55. 2 x 2 pile group in sand ..........................................................163

7-56. 3 x 3 pile group in sand ............................................................................163

7-57. Top of pile lateral displacement comparison, 2 x 2 group ..................................64

7-58. Top of pile bending moment comparison, 2 x 2 group .........................................164

7-59. Top of pile bending moment comparison, 3 x 3 group .......................................165

7-60. M ississippi test structure .................................................... .......................167

7-61. Load history for Mississippi test ................................ .........................168

7-62. Displacement history for Mississippi test.......................... ......................... 168




xiv








7-63. Comparison test x FLPIER with CPT soil properties..........................................169

7-64. Comparison test and FLPIER with SPT soil properties ......................................169

7-65. Comparison test and FLPIER with original soil properties.................................170

B-1. (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 Florida-Pier was

modified. Florida-Pier is a nonlinear finite-element 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 stress-strain curves (steel and concrete)

to develop its behavior and stiffness modeling. Nonlinear dynamic capability was added

to these elements by adding to the stress-strain 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 p-y 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

state-of-practice, 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 over-conservative 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 non-linearity, damping, and pile-soil 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 non-linearity, as well as structure-soil 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.1-1. 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 p-y springs,

while the axial resistance is modeled as nonlinear t-z springs.



Pier Cap
Bridge Springs ----e
Piers

Substructure-- Pile Cap




Piles




Soil layers


Fig. 1-1. 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.

Single-degree-of-freedom (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 multi-degree-of 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

(1991-a), Zeris and Mahin (1991-b)) 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 cross-section is divided into segments. Each segment can then be divided in

one or more fibers. Each fiber is assumed to obey a uniaxial stress-strain 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 stress-strain 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 stress-strain 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 well-known 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 (1991-a), Zeris and Mahin (1991-b)). 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 (1991-a), Zeris and Mahin

(1991-b), 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

(1991-a), Zeris and Mahin (1991-b)), 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, strain-rate, 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 stress-strain curves, the

Baushinger's effect can be considered (Park et al. (1972), Baron and Venkatesan

(1969)), or ignored, in which case a bilinear or tri-linear relationship (elastoplastic with

kinematic or isotropic hardening) is used (Ala Saadeghvaziri (1997), Chen and Atsuta

(1973), Zeris and Mahin (1991-a), Zeris and Mahin (1991-b)).

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 stress-strain 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 state-of-design 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 time-history 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 material-damping

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, second-order, 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. 2-1, 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
Rigid-body
Motion


D,

Fig. 2-1. 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

quasi-static displacement:

{D}'(t)= {D) + {D}(t) Eq.2. 19

The quasi-static 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 L-shaped frame with lumped

masses subjected to horizontal ground motion shown in Fig. 2-2. 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. 2-2. 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. 2-2. Support motion of an L-shaped frame.
a) L-shaped 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 2-2. 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 3D-Beam 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. 2-3. 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+44-3 2)Lr


6(- _2)
6( -~
0
(- 24 + 32)Le
(24- 32)L


0
1-32+2(3
0
-(1 )L
0
(4 -245 +3)L
0
3 2 24
0

0
(2 -4)L


0
0
1-32 +243
-(1 4)Lr
(- +2- 2 )L
0
0
0
32-2 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 1-4 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 2-10 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 three-dimensional

frame element is simply a diagonal matrix in which the coefficients corresponding to

translator and torsional displacements are equal to one-half 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 2-1 and 2-2 compare the accuracy of the consistent and lumped mass

formulations using finite elements.


Table 2-1. Natural frequencies of a uniform cantilever beam:
Element and exact solution


Consistent-Mass 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 2-2. Natural frequencies of a uniform cantilever beam: Lumped-Mass 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 9-node shell

elements, we will limit the formulation to this particular type of element.

Consistent

Consider the true rectangular 9-node shell element shown in Fig. 2-4 below.


I b b


Fig. 2-4. True 9-node rectangular element


Now consider the following mapping:


Fig. 2-5. Mapping for a true rectangular 9-node 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. 2-6. Shell element of uniform thickness


If now we consider for each node the six DOF's illustrated in Fig. 2-6, 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. 2-6.

It is easily verified that for the true rectangular element illustrated in Fig. 2-4 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 9-node

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 higher-order 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 9-node 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. 2-7. Lumped mass matrix at the nodes of true rectangular 9-node 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 rigid-body

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 semi-definite. 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 elastic-plastic 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 rigid-body motion, it

is important that the mass-proportional 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 2-3 for

two levels of motion: working stress levels or stress levels no more than one-half 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 2-3, 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 2-3 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 linear-elastic








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 force-deformation relationships.


Table 2-3. Recommended damping rations for structures
Stress Level Type and Condition Damping Ratio(%)
of structure
Working stress, no more Welded steel, prestressed 2-3
than about /2 yield point concrete, well-reinforced
concrete (only slight
cracking)
Reinforced concrete with 3-5
considerable cracking
Bolted and/or riveted steel, 5-7
wood structures with nailed
or bolted joints
At or just below yield point Welded steel, prestressed 5-7
concrete (without complete
loss in prestress)
Prestressed concrete with 7-10
no prestress left
Reinforced concrete 7-10
Bolted and/or riveted steel, 10-15
wood structures with bolted
joints
Wood structures with nalied 15-20
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.371-372):

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

T-2


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 time-varying 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. 2-8.

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



Time-History Analysis. Direct Integration Methods


In direct integration methods or step-by-step 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 Wilson-Theta 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 time-stepping 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. 2-9. Successive application of the

Trapezoidal Rule leads to Eqs. 2.61 to 2.65.

ui





ti ti+l t

At
Fig. 2-9. 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.2-10.

The Trapezoidal Rule is also used to obtain Eqs. 2.66 to 2.70.

U A

u,1



--*
t t,+1 t

At
Fig. 2-10. 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 f-2/2Eq.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 time-independent 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. 2-11, 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. 2-11. 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 out-of 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 out-of balance force is generated. The process continues until the

out-of-balance force is within a specified tolerance. Figure 2-12 helps to understand the

process. This process is known as the Newton-Raphson method.

If the reader is familiar with numerical methods, will notice that what is defined

as the The Newton-Raphson method in some references (Cook et al. (1989), Chopra

(1995) and Bathe (1996)) and above is in fact a Newton-Raphson approach. Although

irrelevant to this discussion the author found important to define the Newton-Raphson

method from a mathematical point of view.

The Newton-Raphson 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 Newton-Raphson

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. 2-12. Newton-Raphson 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 tangent-stiffness matrix [K,]. In a MDOF context N-R

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 tangent-stiffness 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 element-by element. The process

stops when the out-of-balance 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:








Wilson-0 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, =1-3/0;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 Wilson-0 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 out-of-balance 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 Newton-Raphson Method is applied. In this approach

the tangent stiffness matrix k, is updated for every iteration, in contrast with the

Modified Newton-Raphson 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 out-of-balance 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 discrete-element used in FLPIER is shown in Fig. 3-1. 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. 3-1. Representation of discrete element. After Hoit et al., 1996








Element Deformation Relations


In Fig. 3-2, wl-w3 and w7-w9 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 x-axis) at the left and right ends, respectively; and

finally w5-w6 and w1-wu1 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 wz-w 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. 3-2. 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 y-axes, and the axial strain to be

combined using super position. Examples of these three components are represented

separately in Fig. 3-3(a-c) and combined in Fig. 3-3(d) also shown in Fig. 3-3(d) is a

differential force, dF,, acting on a differential area, dAi, relationship for the material


dF = a,dA,


Eq. 3.16


Finally, Fig. 3-3(e) represents the stress-strain 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
z-axis bending y-axis banding


) Strss-strain relationship
9) Stress-strain relationship


Eq.3.17





Eq. 3.18



Eq.3.19



Eq. 3.20


c) Strain due to
axial thrust


d) Combined strains


Fig. 3-3. 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. 3-4

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. 3-5), 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. 3-4. 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. 3-5. 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 finite-element 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+T-h/2.w5 Eq. 3.28

f =M2 +V, -h/2+T-h/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/2-w12 Eq.3.35

wherefi -f3 andf7 -f9 are the acting end forces; andf4 -f6 andfio-fi2 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 time-step. 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. 3-6, then it

is called secant stiffness. On the other hand, if the stiffness for each segment is defined

by the slope of the stress-strain curve at the specific point being integrated, then it is

called the tangent stiffness as also illustrated in Fig.3-6. Note that if the tangent stiffness








is used, the solution of the nonlinear problems becomes Newton-Raphson, while if the

secant stiffness is used we have a secant method solution.

Stress







/ secanA



Strain


Fig. 3-6. 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 over-loaded 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. 3-7. 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.3-7. 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. (1964-a), Sinha et al. (1964-b), 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 stress-strain 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 elastic-plastic (no hardening) and similar in both tension and

compression as shown in Fig. 3-8. 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. 3-8):

Loading is represented by segment a-b, 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 b-c. 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 c-d, the tangent stiffness is E, and the

stress is given by a = (E -s,).E,.





64


Reloading is represented by segment e-f, 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. 3-8. Elastic-perfectly 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 non-linear 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 stress-strain 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. 3-9. 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 stress-strain curve

generated by the program can be seen in Figs. 3-10 and 3-11 respectively.





/
/-



--

13r2,,S


Fig. 3-9. FLPIER concrete points











Stress-ShaSbn
for Concr le

Sress 0.16E-03C
V~arl~o lil _I


/-0
/-0.6


-09901

03.00E- 02 -0.132iE-C
S-0C.1650 02
'-f^ fe02


33 OE-03


1E-03


-03


IL3.n 25/ 4/1999


Fig. 3-10. Concrete strains





Stress-S!to
fr^ Conorete

Siress 0 5009













S -163





It3.; 25/ 4/1999


Fig. 3-11. 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. (1964-a), 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 3-9 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. 3-12. 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

tension-softening portion as shown in Fig. 3-12. The tension-softening 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. 3-12. Envelop curve for concrete


Loading


Compression

Compression follows the curve described above, a typical loading in

compression phase is illustrated in Fig. 3-13. 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. 3-13. Typical compression loading



Tension

A typical loading in tension is shown in Fig. 3-14. 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. 3-14. 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. 3-15

and Fig. 3-16, respectively. The expressions for compression and tension unloading are

based on the work of Sinha et al. (1964-a) 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 3-1. 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 stress-strain 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 3-1. 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. 3-18. 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. 3-17. 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. 3-15.

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. 3-15. 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. 3-16. 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. 3-17. 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. 3-18. Compression unloading with no gap



Reloading


A typical reloading in compression is illustrated in Fig. 3-19. 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. 3-19. 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. 3-20.


stress


strain


Fig. 3-20. 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 stress-strain curve as shown in Fig. 3-21.

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. 3-22. Referring to Fig.

3-22, 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. 3-21. Bilinear model for concrete

























1"oiue


Fig. 3-22. Stress-strain 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

strain-hardening 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. 3-23.

Confined column
Moment


Curvatute
Fig. 3-23. 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, strain-rate 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[1-z(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 ~= 28-day 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. 3-24. Core width for different cross sections


s = center-to 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 < 10-s / 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

stress-strain 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.

3-25 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. 3-25. 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




Full Text
xml version 1.0 encoding UTF-8
REPORT xmlns http:www.fcla.edudlsmddaitss xmlns:xsi http:www.w3.org2001XMLSchema-instance xsi:schemaLocation http:www.fcla.edudlsmddaitssdaitssReport.xsd
INGEST IEID EV79Y7PH8_ARQZD9 INGEST_TIME 2013-01-22T13:17:33Z PACKAGE AA00012993_00001
AGREEMENT_INFO ACCOUNT UF PROJECT UFDC
FILES