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Topographic Amplification of Earthquakes in Puerto Rico and its Effects in Residential Construction
University of Puerto Rico at Mayagüez
Department of Civil Engineering, P.O. Box 9041, Mayagüez PR 00681 Tel (787) 265-3815, Fax (787) 833-8260
Email: [email protected] , [email protected] ,
Final Technical Report
VOLUME II
Seismic Behavior and Retrofitting of Hillside and Hilly Terrain R/C Houses Raised on Gravity Columns
FEMA-1247-DR-PR HMGP PR-0060B
Submitted to:
Lic. Melba Acosta
Governor’s Authorized Representative Commonwealth of Puerto Rico
Hazard Mitigation Office, P.O. BOX 9023434 San Juan, Puerto Rico 00902-3434
Mr. José A. Bravo
Disaster Recovery Manager Federal Emergency Management Agency
P.O. Box 70105 San Juan, Puerto Rico 00936-8105
By
Luis E. Suárez, Principal Investigator
Ricardo R. López, Co-Principal Investigator Drianfel Vásquez Torres, Graduate Student
María Elena Arroyo Caraballo, Graduate Student
Submitted: June 30,2003
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Topographic Amplification of Earthquakes in Puerto Rico and its Effects in Residential Construction
EXECUTIVE SUMMARY
The objective of this project was to study the amplification of the earthquake waves caused by topography, and to evaluate what effects should be expected on construction located in areas prone to suffer this phenomenon.
The research was divided in to two parts. The results presented in Volume I are concerned with the amplification of the seismic waves. Volume II deals with the effects on the structures, in particular residential constructions. It was found that most reinforced concrete houses built on slender columns are vulnerable to an earthquake amplified by the topography. A rehabilitation technique based on the addition of reinforced concrete walls is proposed in the recommendations in Volume II.
The research was carried out from November 2000 to May 2003. This investigation
was performed by:
Luis E. Suárez, Principal Investigator Ricardo R. López, Co-Principal Investigator Drianfel Vázquez Torres, Graduate Student María Elena Arroyo Caraballo, Graduate Student
The two volumes include the following information:
I. Volume I: Numerical Study of The Amplification of The Seismic Ground Acceleration Due to Local Topography. This investigation presents a study of the effects of local topography on the ground acceleration produced by earthquakes. The graduate student Maria Elena Arroyo Caraballo developed a Master of Science in Civil engineering thesis based on the subject of the first phase of the project.
II. Volume II: Seismic Behavior and Retrofitting of Hillside and Hilly Terrain
R/C Houses Raised on Gravity Columns. This investigation presents a study, by means of numerical simulation, of the seismic behavior of typical residences located at hills or hillsides and raised on gravity columns. The study takes into account the topographic amplification of the ground motion due to the location of the residences. The attention is focused on the seismic evaluation of the residences with typical geometric parameters obtained from a field survey carried out across Puerto Rico. Non-linear static pushovers and non-linear dynamic transient analyses are performed for the seismic vulnerability evaluation. The results of the analyses are used to select a seismic rehabilitation technique. As part of this investigation, the graduate student Drianfel Vázquez Torres submitted a dissertation in partial fulfillment of the requirements for the degree of Ph. D. in Civil engineering.
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Abstract
This investigation presents a study of the seismic behavior of typical residences
located at hills or hillsides and raised on gravity columns by means of numerical
simulation. The study takes into account the topographic amplification of the ground
motion due to the location of the residences. The attention is focused on the seismic
evaluation of these residences with typical geometric parameters obtained from a field
survey carried out across Puerto Rico. Non-linear static pushovers and non-linear
dynamic transient analyses are performed for the seismic vulnerability evaluation. The
results of the analyses are used to select a seismic rehabilitation technique.
Artificial earthquakes with and without topographic amplification were generated
for the non-linear time history analyses. A methodology was developed for the
identification of failure of the residences. Two new criteria, namely the stiffness matrix
determinant and the fundamental period of the structure are proposed and monitored
along the time history analyses.
After the evaluation, a set of tables were developed from which a rehabilitation
system can be obtained according to the size of structural elements, the clear span and the
height of columns from the first floor. Recommendations, specifications and structural
details are presented as minimum requirement for a reliable implementation of the
seismic rehabilitation system. In addition, recommendations are suggested for the future
seismic design of residences located on hills or escarpment exposed to topographic
amplifications.
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Compendio
En esta investigación se presenta un estudio del comportamiento sísmico de
residencias típicas ubicadas en laderas o colinas soportadas sobre columnas gravitatorias
por medio de simulaciones numéricas. El estudio considera la amplificación del
movimiento sísmico debido a la topografía en la localización de la residencia. El mismo
se enfoca en la evaluación estructural de este tipo de residencias cuyos parámetros
geométricos típicos fueron obtenidos de un muestreo a través de la isla de Puerto Rico.
Se utilizaron análisis basados en el régimen no lineal monotónicos y análisis no lineales
dinámicos transitorios como métodos para evaluar la vulnerabilidad de estas residencias
bajo cargas sísmicas. Con los resultados de los análisis mencionados anteriormente, se
desarrolló un sistema de rehabilitación o de mejoramiento sísmico.
Se generaron registros artificiales de terremotos con y sin la amplificación
topográfica para el desarrollo de los análisis no lineales dinámicos transitorios. Además
se desarrolló una metodología para utilizarse como indicador de colapso en las
estructuras. Se desarrollaron y monitorearon dos indicadores de falla a lo largo de la
duración de los terremotos. El primero de estos indicadores se basa en el determinante de
la matriz de rigidez y el segundo en el periodo fundamental de la estructura.
Luego de la evaluación de las estructuras, se desarrollaron una serie de tablas de
las cuales se obtiene el sistema de rehabilitación del sistema estructural de acuerdo a los
tamaños de los elementos estructurales, el largo claro de los elementos y la altura de las
columnas del primer piso. Recomendaciones, especificaciones, además de detalles
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estructurales, se muestran como requerimientos mínimos para una implementación
confiable del sistema de rehabilitación sísmica. Se sugieren recomendaciones para el
proceso de diseño futuro de este tipo de residencias ubicadas en laderas o colinas
expuestas a amplificaciones topográficas.
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Table of content
LIST OF FIGURES .........................................................................................................viii
LIST OF TABLES...........................................................................................................xxiii
CHAPTER I INTRODUCTION.....................................................................................1
1.1 Introduction....................................................................................................1
1.2 Summary of previous works ..........................................................................4
1.3 Objectives ......................................................................................................8
1.4 Summary of the Procedure.............................................................................9
1.5 Contents of this thesis ....................................................................................12
CHAPTER II FIELD SURVEY .....................................................................................15
2.1 Introduction....................................................................................................15
2.2 Field survey....................................................................................................15
2.3 Results of the Field Survey ............................................................................16
CHAPTER III VULNERABILITY EVALUATION OF TYPICAL
RESIDENCES .................................................................................................................51
3.1 Introduction....................................................................................................51
3.2 Selection of the Residences Parameters.........................................................52
3.3 Nonlinear Static Pushover Analysis...............................................................54
3.3.1 Model Generation ...........................................................................55
3.3.2 Nonlinear Static Pushover Analysis Set Up....................................55
3.3.3 Running the Static Nonlinear Pushover..........................................60
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3.3.4 Static Nonlinear Analysis Results (SAP 2000)...............................60
3.4 The Capacity Spectrum Method ....................................................................62
3.4.1 Capacity Curves (Non-linear Static Pushover) ...............................63
3.4.2 Capacity versus Demand Curves (Capacity Spectrum Method).....64
3.5 Examination of the Results ............................................................................65
CHAPTER IV SEISMIC BEHAVIOR OF CODE DESIGNED RESIDENCES .........95
4.1 Introduction....................................................................................................95
4.2 Description of the Residences........................................................................95
4.3 Seismic design of the residences....................................................................97
4.4 Evaluation of the designed residences without topographic
amplification ........................................................................................................99
4.5 Evaluation of the designed residences with the topographic
amplification ........................................................................................................100
CHAPTER V NON LINEAR DYNAMIC TRANSIENT ANALYSIS OF THE
RESIDENCES .................................................................................................................110
5.1 Introduction....................................................................................................110
5.2 Artificial Earthquake Generation...................................................................110
5.3 Other aspects for the Non-linear Dynamic Transient Analysis .....................112
5.4 Collapse Criteria or Ultimate State ................................................................113
5.4.1 Displacement or Inter-Story Drift Criteria (FCD) ..........................114
5.4.2 Ultimate Rotation Criteria (FCR) ...................................................116
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5.4.3 Element Forces Criteria (FCEM and FCES) ..................................117
5.4.4 Collapse Mechanism Criteria (FCCM)...........................................118
5.4.5 Stiffness Matrix Determinant and Structure Period Indicator
(FCK and FCT) ........................................................................................118
5.5 Failure Criteria Methodology [FC]................................................................120
5.6 Non-Linear Dynamic Transient Analyses of the Residences ........................121
5.6.1 Non-Linear Dynamic Transient Analyses of the Extreme
Residences................................................................................................123
5.6.2 Non-Linear Dynamic Transient Analyses of the Designed
Residences................................................................................................130
5.6.3 Evaluation of the FCK and FCT indicators ....................................132
CHAPTER VI SELECTION AND VERIFICATION OF THE RETROFITTING
STRATEGY.....................................................................................................................230
6.1 Introduction....................................................................................................230
6.2 Selection of the demand or target spectrum...................................................230
6.3 Selection of the Rehabilitation Strategy ........................................................232
6.4 Rehabilitation technique implementation ......................................................238
6.5 Assumptions for the development of the rehabilitation system tables ..........239
6.6 Table with Reinforced Concrete Structural Walls .........................................241
6.6.1 Retrofit tables for the weak direction..............................................242
6.6.2 Retrofit tables for the strong direction ............................................243
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CHAPTER VII LIMITATIONS, SPECIFICATIONS AND EXAMPLES ..................247
7.1 Introduction....................................................................................................247
7.2 Specifications or Recommendations..............................................................247
7.2.1 Structural Specifications or Recommendations ..............................248
7.2.2 Soil Specifications or Recomendations ..........................................249
7.2.3 Bonding Specifications or Recommendations ................................250
7.3 Limitations .....................................................................................................252
7.4 Retrofitting Examples ....................................................................................253
CHAPTER VIII SUMMARY, CONCLUSIONS AND RECOMMENDATIONS .......260
8.1 Introduction....................................................................................................260
8.2 The Field Survey............................................................................................260
8.3 Vulnerability Analysis using the Capacity Demand Spectrum......................261
8.4 Amplified spectra and earthquake records.....................................................262
8.5 Non-linear dynamic transient analyses ..........................................................262
8.6 Retrofitting system and tables........................................................................264
8.7 Recommendation for the seismic design of residence located at hills or
escarpments..........................................................................................................265
REFERENCES ................................................................................................................267
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LIST OF FIGURES
Figure 1.1: Typical hilly residences in Puerto Rico.........................................................3
Figure 2.1: Municipalities visited during the Field Survey .............................................43
Figure 2.2: Residence in Yauco ......................................................................................43
Figure 2.3: Residence in Yauco ......................................................................................44
Figure 2.4: Residence in Hormiguero.............................................................................44
Figure 2.5: Residence in Yauco ......................................................................................45
Figure 2.6: Residence in Yauco ......................................................................................45
Figure 2.7: Residence in Jayuya .....................................................................................46
Figure 2.8: Residence in Arecibo....................................................................................46
Figure 2.9: Residence in Yauco ......................................................................................47
Figure 2.10: Residence in Cabo Rojo .............................................................................47
Figure 2.11: Residence in Jayuya ...................................................................................48
Figure 2.12: Residence in Yauco....................................................................................48
Figure 2.13: Residence in Jayuya ...................................................................................49
Figure 2.14: Residence in Arecibo..................................................................................49
Figure 2.15: Residence in Yauco....................................................................................50
Figure 3.1: Preliminary systems for vulnerability analysis .............................................53
Figure 3.2: Constitutive relation for concrete hinges (based on ATC-40). .....................56
Figure 3.3: SAP 2000 Model and hinges location ...........................................................58
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Figure 3.4: Pushover curve for case SS1a .......................................................................70
Figure 3.5: Pushover curve for case SS1b .......................................................................70
Figure 3.6: Pushover curve for case SS2a .......................................................................71
Figure 3.7: Pushover curve for case SS2b .......................................................................71
Figure 3.8: Pushover curve for case SS3a .......................................................................72
Figure 3.9: Pushover curve for case S3b .........................................................................72
Figure 3.10: Pushover curve for case S1a........................................................................73
Figure 3.11: Pushover curve for case S1b .......................................................................73
Figure 3.12: Pushover curve for case S2a........................................................................74
Figure 3.13: Pushover curve for case S2b .......................................................................74
Figure 3.14: Pushover curve for case S3a........................................................................75
Figure 3.15: Pushover curve for case S3b .......................................................................75
Figure 3.16: Capacity demand curve for case SS1a ........................................................76
Figure 3.17: Capacity demand curve for case SS1b ........................................................76
Figure 3.18: Capacity demand curve for case SS2a ........................................................77
Figure 3.19: Capacity demand curve for case SS2b ........................................................77
Figure 3.20: Capacity demand curve for case SS3a ........................................................78
Figure 3.21: Capacity demand curve for case SS3b ........................................................78
Figure 3.22: Capacity demand curve for case S1a...........................................................79
Figure 3.23: Capacity demand curve for case S1b ..........................................................79
Figure 3.24: Capacity demand curve for case S2a...........................................................80
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Figure 3.25: Capacity demand curve for case S2b ..........................................................80
Figure 3.26: Capacity demand curve for case S3a...........................................................81
Figure 3.27: Capacity demand curve for case S3b ..........................................................81
Figure 3.28: Effect of soft soil on the Capacity Demand curves. ....................................82
Figure 4.1: UBC-97 spectra for Sb and Se soil types ......................................................103
Figure 4.2: Demand Spectra for Sb and Se soils in ADRS format ..................................103
Figure 4.3: Capacity Demand plot for Residence 1 (R = 8.5, Sb soil) .............................104
Figure 4.4: Capacity Demand plot for Residence 2 (R = 8.5, Se soil) .............................104
Figure 4.5: Capacity Demand plot for Residence 3 (R = 5.5, Sb soil) .............................105
Figure 4.6: Capacity Demand plot for Residence 4 (R = 5.5, Se soil) .............................105
Figure 4.7: UBC-97 Response Spectrum........................................................................106
Figure 4.8: Original and amplified response spectra for Sb soil type..............................106
Figure 4.9: Original and amplified response spectra for Se soil type ..............................107
Figure 4.10: Capacity Demand plot for Residence 1 (amplified)....................................107
Figure 4.11: Capacity Demand plot for Residence 2 (amplified)....................................108
Figure 4.12: Capacity Demand plot for Residence 3 (amplified)....................................108
Figure 4.13: Capacity Demand plot for Residence 4 (amplified)....................................109
Figure 5.1: UBC-97 Design Spectrum for Sb and Se soil type........................................139
Figure 5.2: Original and amplified response spectra for Sb soil type ..............................139
Figure 5.3: Original and amplified response spectra for Se soil type ..............................140
Figure 5.4: WinSIMQKE, a GUI developed for SIMQKE program...............................140
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Figure 5.5: Artificial earthquake for Sb soil type.............................................................141
Figure 5.6: Artificial earthquake for Sb soil type amplified............................................141
Figure 5.7: Artificial earthquake for Se soil type ............................................................142
Figure 5.8: Artificial earthquake for Se soil type amplified............................................142
Figure 5.9: Comparison of the target and actual spectrum (Sb Soil) ..............................143
Figure 5.10: Comparison of the target and actual spectrum (Se Soil Amp.)...................143
Figure 5.11: Comparison of the target and actual spectrum (Se Soil).............................144
Figure 5.12: Comparison of the target and actual spectrum (Se Soil Amp.)...................144
Figure 5.13: GUI for MOMCU program ........................................................................145
Figure 5.14: Bilinear approach used in the non-linear analyses ......................................145
Figure 5.15: LARZW post processing GUI.....................................................................146
Figure 5.16: Soft Story Collapse Mechanism for Residence SS1a..................................147
Figure 5.17: Static Nonlinear Pushover for Residence SS1a...........................................147
Figure 5.18: Pushover stiffness matrix determinant history for Residence SS1a............148
Figure 5.19: Pushover first period ratio history for Residence SS1a...............................148
Figure 5.20: Base shear vs. displacement history for Residence SS1a (EQ_SB)............149
Figure 5.21: Stiffness matrix determinant history for Residence SS1a (EQ_SB) ...........149
Figure 5.22: First period ratio for Residence SS1a (EQ_SB)..........................................150
Figure 5.23: Base shear vs. displacement history for Residence SS1a (EQ_SE) ............150
Figure 5.24: Stiffness matrix determinant history for Residence SS1a (EQ_SE) ...........151
Figure 5.25: First period ratio history for Residence SS1a (EQ_SE) ..............................151
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Figure 5.26: Base shear vs. displacement for Residence SS1a (EQ_SB_AMP) .............152
Figure 5.27: Stiffness matrix determinant history for Residence SS1a
(EQ_SB_AMP) ................................................................................................................152
Figure 5.28: First period ratio history for Residence SS1a (EQ_SB_AMP) ...................153
Figure 5.29: Soft story collapse mechanism for Residence SS1a (EQ_SB_AMP) .........153
Figure 5.30: Base shear vs. displacement history for Residence SS1a
(EQ_SE_AMP) ................................................................................................................154
Figure 5.31: Stiffness matrix determinant history for Residence SS1a
(EQ_SE_AMP) ................................................................................................................154
Figure 5.32: First period history for Residence SS1a (EQ_SE_AMP)............................155
Figure 5.33: Soft story collapse mechanism for Residence SS1a (EQ_SE_AMP) .........155
Figure 5.34: Pushover collapse mechanism for Residence SS1b ....................................156
Figure 5.35: Non-linear static pushover for Residence SS1b ..........................................156
Figure 5.36: Pushover stiffness matrix determinant history for Residence SS1b............157
Figure 5.37: Pushover first period ratio history for Residence SS1b ..............................157
Figure 5.38: Base shear vs. displacement history for Residence SS1b (EQ_SB)............158
Figure 5.39: Stiffness matrix determinant history for Residence SS1b (EQ_SB)...........158
Figure 5.40: First period ratio history for Residence SS1b (EQ_SB) .............................159
Figure 5.41: Base shear vs. displacement history for Residence SS1b (EQ_SE)............159
Figure 5.42: Stiffness matrix determinant history for Residence SS1b (EQ_SE) ...........160
Figure 5.43: First period ratio history for Residence SS1b (EQ_SE)..............................160
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Figure 5.44: Base shear vs. displacement history for Residence SS1b
(EQ_SB_AMP) ................................................................................................................161
Figure 5.45: Stiffness matrix determinant history for Residence SS1b
(EQ_SB_AMP) ................................................................................................................161
Figure 5.46: First period ratio history for Residence SS1b (EQ_SB_AMP)...................162
Figure 5.47: Collapse mechanism for Residence SS1b (EQ_SB_AMP).........................162
Figure 5.48: Base shear vs. displacement history for Residence SS1b
(EQ_SE_AMP) ................................................................................................................163
Figure 5.49: Stiffness matrix determinant history for Residence SS1b
(EQ_SE_AMP) ................................................................................................................163
Figure 5.50: First period ratio history for Residence SS1b (EQ_SE_AMP) ...................164
Figure 5.51: Soft story collapse mechanism for Residence SS1b (EQ_SE_AMP) .........164
Figure 5.52: Pushover collapse mechanism for Residence SS2a ....................................165
Figure 5.53: Non-linear static pushover for Residence SS2a ..........................................165
Figure 5.54: Pushover stiffness matrix determinant history for Residence SS2a............166
Figure 5.55: Pushover first period ratio history for Residence SS2a...............................166
Figure 5.56: Base shear vs. displacement history for Residence SS2a (EQ_SB)............167
Figure 5.57: Stiffness matrix determinant history for Residence SS2a (EQ_SB) ...........167
Figure 5.58: First period ratio history for Residence SS2a (EQ_SB)..............................168
Figure 5.59: Base shear vs. displacement history for Residence SS2a (EQ_SE) ............168
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Figure 5.60: Pushover stiffness matrix determinant history for Residence SS2a
(EQ_SE)...........................................................................................................................169
Figure 5.61: First period ratio history for Residence SS2a (EQ_SE) ..............................169
Figure 5.62: Soft story collapse mechanism for Residence SS2a (EQ_SE) ....................170
Figure 5.63: Base shear vs. displacement history for Residence SS2a
(EQ_SB_AMP) ................................................................................................................170
Figure 5.64: Stiffness matrix determinant history for Residence SS2a
(EQ_SB_AMP) ................................................................................................................171
Figure 5.65: First period ratio history for Residence SS2a (EQ_SB_AMP) ...................171
Figure 5.66: Soft story collapse mechanism for Residence SS2a (EQ_SB_AMP) .........172
Figure 5.67: Base shear vs. displacement history for Residence SS2a
(EQ_SE_AMP) ................................................................................................................172
Figure 5.68: Stiffness matrix determinant history for Residence SS2a
(EQ_SE_AMP) ................................................................................................................173
Figure 5.69: First period ratio history for Residence SS2a (EQ_SE_AMP) ...................173
Figure 5.70: Soft story collapse mechanism for Residence SS2a (EQ_SE_AMP) .........174
Figure 5.71: Pushover collapse mechanism for Residence SS2b ....................................174
Figure 5.72: Non-linear static pushover for Residence SS2b ..........................................175
Figure 5.73: Pushover stiffness matrix determinant history for Residence SS2b............175
Figure 5.74: Pushover first period ratio history for Residence SS2b ..............................176
Figure 5.75: Base shear vs. displacement history for Residence SS2b (EQ_SB)............176
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Figure 5.76: Stiffness matrix determinant history for Residence SS2b (EQ_SB)...........177
Figure 5.77: First period ratio history for Residence SS2b (EQ_SB) .............................177
Figure 5.78: Base shear vs. displacement history for Residence SS2b (EQ_SE)............178
Figure 5.79: stiffness matrix determinant history for Residence SS2b (EQ_SE)............178
Figure 5.80: First period ratio history for Residence SS2b (EQ_SE)..............................179
Figure 5.81: Soft story collapse mechanism for Residence SS2b (EQ_SE)....................179
Figure 5.82: Base shear vs. displacement history for Residence SS2b
(EQ_SB_AMP) ................................................................................................................180
Figure 5.83: Stiffness matrix determinant history for Residence SS2b
(EQ_SB_AMP) ................................................................................................................180
Figure 5.84: First period ratio history for Residence SS2b (EQ_SB_AMP)...................181
Figure 5.85: Soft story collapse mechanism for Residence SS2b (EQ_SB_AMP).........181
Figure 5.86: Base shear vs. displacement history for Residence SS2b
(EQ_SE_AMP) ................................................................................................................182
Figure 5.87: Stiffness matrix determinant history for Residence SS2b
(EQ_SE_AMP) ................................................................................................................182
Figure 5.88: First period ratio history for Residence SS2b (EQ_SE_AMP) ...................183
Figure 5.89: Soft story collapse mechanism for Residence SS2b (EQ_SE_AMP) .........183
Figure 5.90: Pushover collapse mechanism for Residence SS3a ....................................184
Figure 5.91: Non-linear static pushover for Residence SS3a ..........................................184
Figure 5.92: Pushover stiffness matrix determinant history for Residence SS3a............185
xvi
Figure 5.93: Pushover first period ratio history for Residence SS3a...............................185
Figure 5.94: Base shear vs. displacement history for Residence SS3a (EQ_SB)............186
Figure 5.95: Stiffness matrix determinant history for Residence SS3a (EQ_SB) ...........186
Figure 5.96: First period ratio history for Residence SS3a (EQ_SB)..............................187
Figure 5.97: Soft story collapse mechanism for Residence SS3a (EQ_SB)....................187
Figure 5.98: Base shear vs. displacement history for Residence SS3a (EQ_SE) ............188
Figure 5.99: Pushover stiffness matrix determinant history for Residence SS3a
(EQ_SE)...........................................................................................................................188
Figure 5.100: First period ratio history for Residence SS3a (EQ_SE) ............................189
Figure 5.101: Soft story collapse mechanism for Residence SS3a (EQ_SE) ..................189
Figure 5.102: Base shear vs. displacement history for Residence SS3a
(EQ_SB_AMP) ................................................................................................................190
Figure 5.103: Stiffness matrix determinant history for Residence SS3a
(EQ_SB_AMP) ................................................................................................................190
Figure 5.104: First period ratio history for Residence SS3a (EQ_SB_AMP) .................191
Figure 5.105: Soft story collapse mechanism for Residence SS3a (EQ_SB_AMP) .......191
Figure 5.106: Base shear vs. displacement history for Residence SS3a
(EQ_SE_AMP) ................................................................................................................192
Figure 5.107: Stiffness matrix determinant history for Residence SS3a
(EQ_SE_AMP) ................................................................................................................192
Figure 5.108: First period ratio history for Residence SS3a (EQ_SE_AMP) .................193
xvii
Figure 5.109: Soft story collapse mechanism for Residence SS3a (EQ_SE_AMP) .......193
Figure 5.110: Pushover collapse mechanism for Residence SS3b ..................................194
Figure 5.111: Non-linear static pushover for Residence SS3b ........................................194
Figure 5.112: Pushover stiffness matrix determinant history for Residence SS3b..........195
Figure 5.113: Pushover first period ratio history for Residence SS3b ............................195
Figure 5.114: Base shear vs. displacement history for Residence SS3b (EQ_SB)..........196
Figure 5.115: Stiffness matrix determinant history for Residence SS3b (EQ_SB).........196
Figure 5.116: First period ratio history for Residence SS3b (EQ_SB)............................197
Figure 5.117: Base shear vs. displacement history for Residence SS3b (EQ_SE)..........197
Figure 5.118: Stiffness matrix determinant history for Residence SS3b (EQ_SE) .........198
Figure 5.119: First period ratio history for Residence SS3b (EQ_SE)............................198
Figure 5.120: Soft story collapse mechanism for Residence SS3b (EQ_SE)..................199
Figure 5.121: Pushover collapse mechanism for Residence R1 ......................................199
Figure 5.122: Non-linear static pushover for Residence R1............................................200
Figure 5.123: Pushover stiffness matrix determinant history for Residence R1 .............200
Figure 5.124: Pushover first period ratio history for Residence R1 ................................201
Figure 5.125: Base shear vs. displacement history for Residence R1 (EQ_SB) .............201
Figure 5.126: Stiffness matrix determinant history for Residence R1 (EQ_SB).............202
Figure 5.127: First period ratio history for Residence R1 (EQ_SB) ...............................202
Figure 5.128: Maximum number of hinges formed for Residence R1 (EQ_SB) ............203
Figure 5.129: Base shear vs. displacement history for Residence R1 (EQ_SE)..............203
xviii
Figure 5.130: Stiffness matrix determinant history for Residence R1 (EQ_SE).............204
Figure 5.131: First period ratio history for Residence R1 (EQ_SE)................................204
Figure 5.132: Maximum number of hinges formed for Residence R1 (EQ_SE) ............205
Figure 5.133: Base shear vs. displacement history for Residence R1
(EQ_SB_AMP) ................................................................................................................205
Figure 5.134: Stiffness matrix determinant history for Residence R1
(EQ_SB_AMP) ................................................................................................................206
Figure 5.135: First period ratio history for Residence R1 (EQ_SB_AMP).....................206
Figure 5.136: Collapse mechanism for Residence R1 (EQ_SB_AMP) ..........................207
Figure 5.137: Base shear vs. displacement history for Residence R1
(EQ_SE_AMP) ................................................................................................................207
Figure 5.138: Stiffness matrix determinant history for Residence R1
(EQ_SE_AMP) ................................................................................................................208
Figure 5.139: First period ratio history for Residence R1 (EQ_SE_AMP).....................208
Figure 5.140: Collapse mechanism for Residence R1 (EQ_SE_AMP)...........................209
Figure 5.141: Pushover collapse mechanism for Residences R2 and R3 ........................209
Figure 5.142: Non-linear static pushover for Residences R2 and R3..............................210
Figure 5.143: Pushover stiffness matrix determinant history for Residences R2
and R3 ..............................................................................................................................210
Figure 5.144: Pushover first period ratio history for Residences R2 and R3 ..................211
xix
Figure 5.145: Base shear vs. displacement history for Residences R2 and R3
(EQ_SB)...........................................................................................................................211
Figure 5.146: Stiffness matrix determinant history for Residences R2 and R3
(EQ_SB)...........................................................................................................................212
Figure 5.147: First period ratio history for Residences R2 and R3 (EQ_SB) .................212
Figure 5.148: Maximum number of hinges formed for Residence R2 and R3
(EQ_SB)...........................................................................................................................213
Figure 5.149: Base shear vs. displacement history for Residences R2 and R3
(EQ_SE)...........................................................................................................................213
Figure 5.150: Stiffness matrix determinant history for Residences R2 and R3
(EQ_SE)...........................................................................................................................214
Figure 5.151: First period ratio history for Residences R2 and R3 (EQ_SE)..................214
Figure 5.152: Maximum number of hinges formed for Residences R2 and R3
(EQ_SE)...........................................................................................................................215
Figure 5.153: Base shear vs. displacement history for Residences R2 and R3
(EQ_SB_AMP) ................................................................................................................215
Figure 5.154: Stiffness matrix determinant history for Residences R2 and R3
(EQ_SB_AMP) ................................................................................................................216
Figure 5.155: First period ratio history for Residences R2 and R3 (EQ_SB_AMP).......216
Figure 5.156: Maximum number of hinges formed for Residence R2 and R3
(EQ_SB_AMP) ................................................................................................................217
xx
Figure 5.157: Base shear vs. displacement history for Residences R2 and R3
(EQ_SE_AMP) ................................................................................................................217
Figure 5.158: Stiffness matrix determinant history for Residences R2 and R3...............218
(EQ_SE_AMP) ................................................................................................................218
Figure 5.159: First period ratio history for Residences R2 and R3 (EQ_SE_AMP).......218
Figure 5.160: Soft story collapse mechanism for Residences R2 and R3
(EQ_SE_AMP) ................................................................................................................219
Figure 5.161: Pushover collapse mechanism for Residence R4 ......................................219
Figure 5.162: Non-linear static pushover for Residence R4............................................220
Figure 5.163: Pushover stiffness matrix determinant history for Residence R4 .............220
Figure 5.164: Pushover first period ratio history for Residence R4 ................................221
Figure 5.165: Base shear vs. displacement history for Residence R4 (EQ_SB) .............221
Figure 5.166: Stiffness matrix determinant history for Residence R4 (EQ_SB).............222
Figure 5.167: First period ratio history for Residence R4 (EQ_SB) ...............................222
Figure 5.168: Maximum number of hinges formed for Residence R4 (EQ_SB) ............223
Figure 5.169: Base shear vs. displacement history for Residence R4 (EQ_SE)..............223
Figure 5.170: Stiffness matrix determinant history for Residence R4 (EQ_SE).............224
Figure 5.171: First period ratio history for Residence R4 (EQ_SE)................................224
Figure 5.172: Maximum number of hinges formed for Residence R4 (EQ_SE) ............225
Figure 5.173: Base shear vs. displacement history for Residence R4
(EQ_SB_AMP) ................................................................................................................225
xxi
Figure 5.174: Stiffness matrix determinant history for Residence R4
(EQ_SB_AMP) ................................................................................................................226
Figure 5.175: First period ratio history for Residence R4 (EQ_SB_AMP).....................226
Figure 5.176: Maximum number of hinges formed for Residences R4
(EQ_SB_AMP) ................................................................................................................227
Figure 5.177: Base shear vs. displacement history for Residence R4
(EQ_SE_AMP) ................................................................................................................227
Figure 5.178: Stiffness matrix determinant history for Residence R4
(EQ_SE_AMP) ................................................................................................................228
Figure 5.179: First period ratio history for Residence R4 (EQ_SE_AMP).....................228
Figure 5.180: Collapse mechanism for Residence R4 (EQ_SE_AMP)...........................229
Figure 6.1: Cross section of a hard soil mountain (left) and a soft soil mountain
(right) ...............................................................................................................................232
Figure 6.2: Capacity Spectrum for Residence S1a and Sb soil type amplified
spectra ..............................................................................................................................245
Figure 6.3: Rehabilitation strategies ...............................................................................245
Figure 6.4: Residences without beams in the weak direction.........................................246
Figure 7.1: Connection between existing column and beam with structural wall ..........255
Figure 7.2: Section view of structural wall and footing...................................................256
Figure 7.3: Plan view of wall and footing........................................................................257
Figure 7.4: Schematic drawing for Example 1 ...............................................................257
xxii
Figure 7.5: Structural walls for weak direction, strong direction using one wall
and strong direction using two walls from left to right....................................................258
Figure 7.6: Rehabilitation system for Example 2 ...........................................................259
xxiii
LIST OF TABLES
Table 1.1: Conversion factors from US customary units to SI units ..............................14
Table 2.1: Municipalities and number of residences visited............................................16
Table 2.2: Summary of the residences columns height .................................................16
Table 2.3: Summary of the columns sections ................................................................17
Table 2.4: Summary of residences spans ........................................................................17
Table 2.5: Document for residence R1 ............................................................................19
Table 2.6: Document for residence R2 …………………………………………………20
Table 2.7: Document for residence R3 ............................................................................21
Table 2.9: Document for residence R5 ............................................................................23
Table 2.10: Document for residence R6 ..........................................................................24
Table 2.11: Document for residence R7 ..........................................................................25
Table 2.12: Document for residence R8 ..........................................................................26
Table 2.13: Document for residence R9 ..........................................................................27
Table 2.14: Document for residence R10 ........................................................................28
Table 2.15: Document for residence R11 ........................................................................29
Table 2.16: Document for residence R12 ........................................................................30
Table 2.17: Document for residence R13 ........................................................................31
Table 2.18: Document for residence R14 ........................................................................32
Table 2.19: Document for residence R15 ........................................................................33
xxiv
Table 2.20: Document for residence R16 ........................................................................34
Table 2.21: Document for residence R17 ........................................................................35
Table 2.22: Document for residence R18 ........................................................................36
Table 2.23: Document for residence R19 ........................................................................37
Table 2.24: Document for residence R20 ........................................................................38
Table 2.25: Document for residence R21 ........................................................................39
Table 2.26: Document for residence R22 ........................................................................40
Table 2.27: Document for residence R23 ........................................................................41
Table 2.28: Document for residence R24 ........................................................................42
Table 3.1: Parameters of the residences in the strong direction .....................................54
Table 3.2: Parameters for the residences in the weak direction ......................................54
Table 3.3: Ductilities in the strong direction. ..................................................................65
Table 3.4: Ductilities in the weak direction. ....................................................................65
Table 3.5: UBC-97 Lateral System Classification and R ................................................67
Table 3.6: Parameters to define the model S1a ...............................................................83
Table 3.7: Parameters to define the model S1b ..............................................................84
Table 3.8: Parameters to define the model SS2a ............................................................85
Table 3.9: Parameters to define the model S2b ..............................................................86
Table 3.10: Parameters to define the model S3a .............................................................87
Table 3.11: Parameters to define the model S3b ............................................................88
Table 3.12: Parameters to define the model SS1a ..........................................................89
xxv
Table 3.13: Parameters to define the model SS1b ..........................................................90
Table 3.14: Parameters to define the model SS2a ..........................................................91
Table 3.15: Parameters to define the model SS2b ..........................................................92
Table 3.16: Parameters to define the model SS3a ..........................................................93
Table 3.17: Parameters to define the model SS3b ..........................................................94
Table 4.1: Structural systems and soil types used in the residence design ......................96
Table 4.2: Final element sizes for residence 1 ................................................................98
Table 4.3: Final element sizes for residence 2 ................................................................98
Table 4.4: Final element sizes for residence 3 ................................................................98
Table 4.5: Final element sizes for residence 4 ................................................................98
Table 4.6: Seismic parameters for the residences............................................................99
Table 5.1: Peak Ground Acceleration and Duration of the Artificial Records................112
Table 5.2: Parameters for the typical residences in the strong direction .......................121
Table 5.3: Parameters for the designed residences in the strong direction.....................122
Table 5.4: Failure collapse summary for residence SS1a ..............................................127
Table 5.5: Failure collapse summary for residence SS1b ..............................................128
Table 5.6: Failure collapse summary for residence SS2a ...............................................128
Table 5.7: Failure collapse summary for residence SS2b...............................................129
Table 5.8: Failure collapse summary for residence SS3a ...............................................129
Table 5.9: Failure collapse summary for residence SS3b...............................................129
Table 5.10: Failure collapse summary for residence R1.................................................131
xxvi
Table 5.11: Failure collapse summary for residence R2 and R3 ..................................131
Table 5.12: Failure collapse summary for residence R4.................................................131
Table 5.13: FCK and FCT values for Sb earthquake.......................................................134
Table 5.14: FCK and FCT values for Se earthquake. ......................................................135
Table 5.15: FCK and FCT values for Sb amplified earthquake. .....................................135
Table 5.16: FCK and FCT values for Se amplified earthquake.......................................136
Table 5.17: Limits of FCK and FCK when FCCM was developed................................137
Table 5.18: Post Earthquake Indicator for Residences with no FCCM...........................138
Table 6.1: Retrofitting tables for the weak direction (two R/C walls) ...........................243
Table 6.2: Retrofitting tables for the strong direction (one R/C walls) ..........................244
Table 6.3: Retrofitting tables for the strong direction (two R/C walls) .........................244
Table 8.1: Seismic parameter for the residences ............................................................265
1
CHAPTER I
INTRODUCTION
1.1 Introduction
It is well known that Puerto Rico is exposed to the risk of experiencing significant
earthquakes. This is so because there are several faults around the Island (Irizarry, 1999).
At the north, we find that the Caribbean Plate collides with the North American Plate.
These two plates are constantly moving creating a left lateral strike slip fault zone. The
Anegada Trough and the Mona Canyon are two normal faults that form the eastern and
western borders of the Puerto Rico and Virgin Islands Platforms. At the south, the Great
Southern Puerto Rico Fault Zone crosses the Island from the west to the southern coast.
These faults indicate that there are source mechanisms located to the north (Puerto Rico
Trench), west (Mona Canyon), and southeast (Anegada Trough) of Puerto Rico capable
of producing earthquakes intensities in excess of VII based on the Modified Mercalli
scale (McCann,1984).
There is evidence of three major earthquakes that affected the island since the
beginning of the colonization. The first one occurred on May 2, 1787 and damage and
destruction were reported from all areas except in the south. The magnitude of this
earthquake was at least 8.0 on the Ritcher scale and the epicenter was possibly located in
the Northern Puerto Rico Trench (NEIC, 1988). The second earthquake was reported on
November 17, 1867 with an estimated magnitude of 7.3 and affected the eastern zone of
2
Puerto Rico. The last strong earthquake that significantly affected the Island occurred in
October 11, 1918 with an estimated magnitude of 7.5 on the Ritcher scale. It was
generated at the Mona Canyon, about 50 km to the northwest of the Island. The historic
records indicate that the recurrence intervals for earthquakes of intensity VII or greater
affecting any part of Puerto Rico vary between 2 and 75 years. Therefore, currently there
is a high risk that an earthquake may occur because the last one occurred almost eighty
four (84) years ago.
The problem of having strong earthquakes is increased by the topography of
Puerto Rico. From the study of damages during past earthquakes, it has been shown that
the surface topography surrounding the site of the structure can considerably amplify the
ground motions. Ridges and hills produce a scattering and diffraction of the seismic
waves causing an amplification of the ground acceleration. Evidence of this effect was
reported in Italy in the Friuli and Irpino earthquakes as well in the Chile earthquake of
1985.
Although this phenomenon has been known for several years and despite of its
importance for sites with pronounced surface irregularities, this effect is not considered
or included in the US seismic codes and thus in the design codes adopted for Puerto Rico
(UBC 1997). One can speculate about the reasons why the US codes disregard this
phenomenon. For example, it could be that in the US mainland the regions with the
conditions that make them prone to topographic amplifications are scarcely populated. It
3
could also happen that this effect has not been studied so far in sufficient detail to account
for it in a practical way in the seismic provisions.
The geography of Puerto Rico, along with the social and economic conditions that
affect the population distribution, makes many regions of the Island prone to topographic
amplifications. The problem is aggravated by the many residential structures located on
slopes and hills that are constructed with weak first stories consisting in slender columns,
as it can be shown in Figure 1.1.
Figure 1.1: Typical hilly residences in Puerto Rico
In addition, the amplification of the seismic motion can have potentially serious
consequences in a terrain that is sensitive to landslides.
4
This project will direct its attention to the seismic response of typical houses built
on the slopes or at the top of the hills and escarpments. These structures usually have
very slender columns and are not designed taken into account the seismic action. They
can be seriously damaged during a strong earthquake, due to its vulnerable structural
system, to the higher seismic loads produced by the topographic amplification, and to the
differential motions of the bottom of columns. Models of these structures will be
subjected to the seismic input considering the topographic effect as developed by Arroyo
(2001) in the first step of this project. If, as it is expected, the residences show to be
prone to suffer severe damage or even collapse, some remedial measures will be studied
to mitigate the effects of the earthquake loads.
1.2 Summary of previous works
This investigation is a complement to the investigation realized by Arroyo (2001)
“Numerical Study of the Amplification of the Seismic Ground Acceleration Due to Local
Topography”. Arroyo studied the amplification of the seismic waves that arrive to hills
or escarpment based on a peak acceleration comparison. She developed a series of
equations to relate the amplification factor to the topography as well to the location of the
structure along the hill or escarpment. She concluded from two dimensional nonlinear
analyses using the Finite Element Method that the amplification factor varies from a
range of 1 to 2.35. These factors were implemented in this investigation in a practical
way.
5
In another study developed by Athanasopoulus and Zerva (1993) amplification
factors on the seismic response of a ridge like surface using the finite elements method
was studied. They found amplifications when the base length of the ridge is two times
the incident seismic wavelength for gentle slopes and for steep slopes when these two
quantities are equal. The seismic wavelength is the product of the shear wave velocity Vs
of the soil and the dominant period Tpeak of the seismic waves. The latter is defined as the
period with the highest ordinate in the response spectrum. The amplifications factors
obtained in this investigation ranges from 1 to 3.
Bouchon (1973) studied the effects of incident in plane waves on ridges and
valleys of an elastic homogeneous half space. Bouchon concludes that the effect of
topography on surface motion appears to be important when the wavelength of the
seismic wave is of the order of the dimension of the anomaly, and it can locally be
responsible for both strong amplification and attenuation.
Sano and Pugliese (1999) studied the topographic effects in Italy during the
earthquake of September 26, 1997 that hit the Umbria-Marge region. They used a two
dimensional model, based on the indirect boundary element method, to investigate the
phenomenon and the effects of geometric parameter changes. They found that a small
variation in the slope, horizontal dimension and geometry affects the response at high
frequency and the space variability of the motion.
Other investigators (Sanchez-Sesma and Campillo (1993), Geli et al. (1998)) used
a number of different approaches to study the phenomenon such as the finite element
6
method, one dimensional wave propagation, two dimensional scalar wave propagation
using cylindrical eigenfunctions and others. A more detailed explanation about these
investigations is presented in the first part of this investigation developed by Arroyo
(2001).
This investigation focuses on the effects of those ground amplifications in
residences located in the hills or escarpment of the Puerto Rico Island. The amplification
factors obtained in the investigation of Arroyo (2001) were used to obtain amplified
earthquake record and spectrums. With these amplified spectrum and earthquake records
and nonlinear analytical tools, the behavior of these residences is studied.
The Capacity Spectrum Method was be used for the preliminary structural
evaluation and vulnerability analysis of these residences. This method is a static non-
linear analytical procedure for the assessment of the seismic performance of the structure.
It can serve as a tool for the selection and preliminary design of a retrofitting scheme that
satisfies the seismic performance objectives (Badoux, 1998). The method combines a
non-linear static (step by step) “pushover” with the Acceleration Displacement Response
Spectra (ADRS) (Bonacci (1994), Mahaney et al. (1993)). A thorough description of the
Capacity Spectrum method and its applications are given in the report ATC-40 (1996).
One of the advantages of this method is that it lends to a graphical representation
of the seismic performance of a structure. The expected displacement demand for a
given seismic ground motion is obtained by intercepting the capacity spectrum and the
seismic demand spectrum in the spectral displacement vs. spectral acceleration plane. If
7
there is no such interception, the structural failure can be directly deduced from this
graphical representation.
Peter and Badoux (1998) and Valles et al. (1996) stated that a current handicap to
the application of the method is that reliable analytical tools for pushover analyses only
seem to be available for ductile frames, but not yet for frame-wall buildings, for example.
Masonry and reinforced concrete walls are some of the rehabilitation techniques to be
tested. The programs LARZW, LARZWS/D and LARZWT were used for this purpose.
This particular program is capable of performing the nonlinear behavior of wall or frame-
wall system including the non-linear shear deformation which is one of the deficiencies
of other programs. This is one of the reasons why the program LARZW is preferred
above the other non-linear programs.
Badoux (1998) also discussed four seismic strategies in his investigation:
Strengthening and stiffening, ductility enhancement, strengthening and stiffening with
ductility enhancement and seismic demand reduction. These strategies were used in the
selection and implemented of the rehabilitation technique developed in this investigation.
The particular residences examined were treated as non-ductile structures. This is
because they are old structures, due to the economic conditions at the time of the
construction and because the seismic provisions at the same time were less rigorous than
the current provision. These structures at least had to been designed for the gravity loads
(1.4DL and 1.7 LL). Aycardi et al. (1994) and Bracci et al. (1995) said that such
structures generally possess reinforcement details that conform to the code of practice,
8
but do not conform to the modern seismic provisions. Although such structures are
designed without the earthquake load consideration, they may still posses an inherent
lateral strength capacity that may be mobilized to resist moderate earthquakes (Aycardy
et al. (1994) and Bracci et al. (1995)). The authors performed an investigation on the
post elastic behavior of companion column and of interior and exterior slab-beam-column
subassemblages constructed simultaneously with one third scale model building using the
same materials, steel, and concrete mixes. These components were tested under a quasi-
static reversed cyclic loading prior to testing the model building to be used in to predict
the non-linear behavior of the building prior to its testing on the shaking table. From
these investigations, practical drift limitation, non-ductile columns mode of failure and
other values were obtained and used in the non-linear dynamic analyses of the residences.
1.3 Objectives
The objectives of this work are to identify typical residences in hilly terrain in
Puerto Rico, to characterize their expected behavior under seismic loading, and to
develop retrofitting strategies for improving their expected seismic behavior. Another
objective is to develop recommendations for design of safe structures built in hilly
terrain. The work is limited to the superstructure and does not consider any geotechnical
failure like footing failure, earth faulting, slides or similar movements or loss of footing
because of landslides.
9
1.4 Summary of the Procedure
The procedure to accomplish the objectives of this research will be as follows:
1. Perform a field survey across the island to define the most typical parameters of
residences on hilly terrain. The parameters to be searched for are the number of
spans, the number of stories, height of stories, the size of the structural elements,
and the reinforcing steel of the structural elements. Another parameter to be
considered is the location of the residence along the hill or escarpment.
2. With the parameters acquired in the field survey, the extreme cases (i.e. the most
flexible and most rigid structural system) will be defined to perform a nonlinear
static pushover analysis. The vulnerability of the residences will be evaluated
with the nonlinear static pushover analysis and the Capacity Spectrum Method.
In addition, another case in between the two extreme cases will be evaluated with
the intention to cover practically all the structural systems. Also code designed
residences with the typical spans, column lengths and height will be designed and
later evaluated.
3. Anticipating that the previous analyses will confirm the vulnerability of the
residences, a more detailed analysis will be performed. A Nonlinear Dynamic
Transient Analysis will be carried out to identify the causes and patterns of
collapse and examine in detail the dynamic response of the residence. The
nonlinear dynamic analysis will be performed using the structures selected form
10
the field survey and the “new” residences. The program LARZWS/D will be used
for this purpose.
4. Earthquake records representative for the Island are needed for the Dynamic
Transient Analysis. Since there is no data of strong earthquakes records
registered in Puerto Rico, artificially generated earthquakes records with and
without topographic amplifications will be developed using the UBC-97 design
spectrum for Puerto Rico and for two different soil profiles. The program
SIMQKE by Venmarke (1976) will be used to generate the artificial records.
Because SIMQKE is a MSDOS based program, a Graphical User Interface (GUI)
will be developed for the SIMQKE program to facilitate the use of the program
and for the benefit of the University of Puerto Rico researchers and FEMA
projects (Vázquez, 2001).
5. A collapse characterization methodology will be developed to define the failure or
collapse of the structures. Different collapse criteria like collapse mechanism,
maximum capacity and maximum rotation of the structural elements, drift
limitations and changes in dynamic properties (i.e. periods) are going to be used
to establish the collapse criteria. In addition the stability of the structure,
characterized by the determinant of the stiffness matrix and local collapse
mechanisms will be used in the definition of the collapse criteria. Since the
typical output of a nonlinear dynamic time history is quite large, another GUI
program will be programmed. This tool will be capable of displaying on the
11
screen the hinge patterns at any time step. In addition, it will be able to animate
the motion of the structure, show the acceleration history as well to identify drift,
displacement and rotation limits. It is expected that this program will be very
useful to facilitate the interpretation and processing of the results.
6. The definition of the structural rehabilitation technique to be used, will depend on
the previously nonlinear static pushover analyses and the capacity spectrum
method. This definition relays on the relation or behavior when compared the
capacity and demand (spectrum) in the ADRS Capacity Spectrum Plot. The lack
of stiffness, ductility or both can be obtained from these plots. Therefore the
rehabilitation technique will compensate the deficiency.
7. After the collapse or failure analyses, another parametric study using various
rehabilitation techniques will be performed to obtain a simple and economical
rehabilitation system to increase the seismic capacity of the residences. Linear
elastic steel bracing systems, partial or full masonry walls and reinforced concrete
walls, columns and beams jacketing are preliminary strategies selected as
rehabilitation techniques. After the best rehabilitation technique is defined, the
next step is to design the system in order to overcome the deficiency. Another
nonlinear analysis will be performed to observe the behavior of the rehabilitated
structure. The procedure is repeated until a satisfactory “performance point” is
achieved.
12
8. The recommendations for the safe seismic design of future residences to be built
in hilly terrain in Puerto Rico and other regions with similar topography will be
obtained based on the results of the analyses of the “new” residences.
Amplification factors like acceleration, base shear and shear in floors, etc. are the
preliminary parameters to be analyzed to corroborate or correlate with the seismic
code values.
1.5 Contents of this thesis
In Chapter II of this investigation the description of the residences obtained in the
field survey is presented. Also descriptive tables, summarizing the parameters obtained
and pictures of the residences were developed in this chapter. Chapter III presents a
vulnerability analysis of the extreme cases of the residences. Starting for obtaining the
non-linear static pushover curves for the residences, up to the complete development and
evaluation using Capacity Spectrum Method using the UBC-97 design spectra is
presented.
A study of four residences designed following the UBC-97 and ACI-317-99
seismic requirement are evaluated using the Capacity Spectrum Method like the extreme
residence. In contrast of Chapter III, the demand spectrum used in Chapter IV includes
the topographic amplification as obtained in the investigation developed by Arroyo
(2001). The implementation of these amplification factors as applied in this investigation
for the UBC-97 Sb and Se soil type spectra is also presented in this chapter.
13
In Chapter V the development of artificially generated earthquake records for the
UBC-97 Sb and Se soil type spectra is presented. Furthermore, non-linear dynamic
transient analyses were developed for the extreme and designed residences to verify the
results obtained in the vulnerability analysis and to observe the behavior of residence
under the artificially generated records. In addition, a failure criteria methodology for the
evaluation of the seismic performance and monitoring of certain dynamic parameter of
the residences is presented.
The selection and implementation of a retrofitting strategy for the seismic
rehabilitation of the residences is presented in Chapter VI of this project. A series of
tables for the selection of the rehabilitation system for a particular residence is developed
and presented in this chapter. Assumptions, failure criteria and detailing of these tables
are discussed in this chapter. The rehabilitation tables were obtained by an iterative
procedure consisting in the implementation of a particular rehabilitation system and
performing a non-linear dynamic transient analysis for the UBC Sb soil type with the
topographic amplification. Finally, Chapter VII presents the conclusions of all of the
previous chapters.
Due to the topography of the Puerto Rico Island, a significant number of
residences are located at hills with the same structural system. This investigation tries to
provide a rehabilitation system and information to solve a practical and real problem.
Puerto Rico like United States is changing the US customary unit system to international
but, at this time the practical unit system used by engineers and contractor is the US
14
system. Because of this practical point of view, the investigation was developed in the
latter system of units. The author recognizes this limitation, and the following table
provides the conversion factors to SI unit system and each table in the document provides
the required conversion factors.
Table 1.1: Conversion factors from US customary units to SI units
Unit Multiply by To obtain in 25.4 mm in 0.0254 mm ft 305 mm
Length
ft 0.3048 m in2 6.4516 cm2 in2 0.00065 m2 ft2 929.03 cm2
Area
ft2 0.0929 m2 in3 16.39 cm3 in3 1.63E-05 m3 ft3 28316.85 cm3
Volume
ft3 0.02832 m3 lb 453.6 g lb 0.4536 kg
slug 14593 g Weight
slug 14.593 kg lb 1000 kip lb 4.448 N kip 4448 N
Force
kip 4.448 KN psi 6894.76 Pa ksi 6894.76 kPa
lb/ft2(psf)* 47.88 Pa Stress,
Presure k/ft2 (ksf)** 47.88 kPa
*psf = pounds per square feet **ksf = kips per square feet
15
CHAPTER II
FIELD SURVEY
2.1 Introduction
The first step in the vulnerability evaluation of the residences is to obtain
representative physical parameters of the residences to be evaluated. A field survey was
performed to obtain the dimensions of the residences and to examine their characteristic.
The parameters interested are the spans length, the height of the columns, the number of
spans, the number of stories and the steel reinforcement of the element sections. This
chapter describes the residences visited in the field survey as well as the parameters
obtained for each one.
2.2 Field survey
The field survey started by visiting different municipalities across Puerto Rico.
These municipalities are shown in Figure 2.1. A total of twenty four (24) residences
were evaluated and measured in five (5) different municipalities. A listing of the number
of residences measured is shown in Table 2.1. A document with a general description
prepared and filled out in situ, along with a schematic drawing of the residences. Each
document contains all the parameters mentioned before for every residence and other
practical information (such as the address and other descriptive data) in a tabulated and
uniform style to facilitate its interpretation. These documents are shown in Tables 2.5 to
16
2.28 which are presented at the end of this chapter. Photos of some of the residences are
shown in Figures 2.2 to 2.14.
Table 2.1: Municipalities and number of residences visited
Municipality # of ResidencesJayuya 9Cabo Rojo 6Hormigueros 1Yauco 5Arecibo 3
Total 24
2.3 Results of the Field Survey
The first parameter taken into consideration is the height of the columns. The
columns height is one of the most important parameters since it controls the slenderness
and flexibility of the structural system. A summary of the columns heights for the twenty
four residences is shown in Table 2.2 below.
Table 2.2: Summary of the residences columns height (1 ft = 305mm)
From To From To From To8 12 12 16 16 20
# of Residences
Height [ft] Over 2012 6 3 3
17
The ranges observed are related to the columns sections as shown in Table 2.3. As
expected, the predominant height range is between 8 and 12 ft because these are the
practical heights for most residences.
Table 2.3: Summary of the columns sections (1 in = 24.5 mm)
From To From To Exactly6X12 6X18 8X12 10X10 12X12
# of Residences 97 8
Section [in]
Notice that the number of residences sections does not coincide with the number of
residences height. This is because there are some residences that lie somewhere between
the ranges.
The next parameter to evaluate is the span lengths of the residences. A summary
of the span lengths is presented in Table 2.4. The table shows that there is not a
significant variation of the spans between these ranges.
Table 2.4: Summary of residences spans (1 ft = 305mm)
From To From To8 12 12 16 over 16
# of Residences 112 11
Span [ft] cc.
The other two parameters are the steel reinforcement and the number of stories. It was
found that basically the two stories is the most common case, since only one of the entire
18
stock has three stories. The steel reinforcement parameter was not found in all the
residences because the residences were old and the owners did not have the construction
drawings. However, according to the owners, in almost all the cases they used bar size
#4 and #5 using 6 of these in the beams and columns, except for the 12X12 column in
which 8 bars were used. The predominant configurations are the six #4 bars and six #5
bars for the columns and 6 #4 bars for the beams. In the following chapters Nonlinear
static and dynamic analyses will be performed in models of residences with parameters
equal to those obtained in the field survey. The specific values of the parameters used for
each analysis are specified in Chapter III for the Non-linear Static Analysis (Pushover)
and in Chapter IV and V for the nonlinear dynamic transient analysis.
19
Table 2.5: Document for residence R1
Addr
ess
Long
Dire
ctio
n
Add
ress
Age
# S
pans
Col
umns
Squa
rew
/o fi
nish
ing
City
1b
12.0
0 in
Sta
te/P
rovi
nce
2h
12.0
0 in
ZIP/
Post
Cod
e3
Stee
l8
# 5
4P
hone
Max
Hei
ght
Beam
sR
ecta
ngul
a rw
/o fi
nish
ing
Min
Hig
htb
12.0
0 in
& w
/o s
lab
Soil
Oth
er C
omm
.h
18.0
0 in
Stee
lN
/ASo
il Su
ppor
t Len
gth
Shor
t Dire
ctio
n
# Sp
ans
Col
umns
Squ
are
w/o
fini
shin
g 1
b12
.00
in2
h12
.00
in3
Stee
l8
# 5
Max
Hei
ght
Beam
sR
ecta
ngul
arw
/o fi
nish
ing
Min
Hig
htb
12.0
0 in
& w
/o s
lab
Oth
er C
omm
.h
12.0
0 in
Stee
lN
/A
R1
Des
crip
tion
of S
truct
ures
Geo
met
ryG
ener
al In
form
atio
n
BO C
aric
aboa
Car
r 144
Km
0.3
Jayu
ya
12.0
0 ft
N/A
14.0
0 ft
Puer
to R
ico
0066
4
N/A
11.5
0 ft
16.0
0 ft
13.0
0 ft
18.0
0 ft
0.00
ft
4
3
N/A
13.0
0 ft
18.0
0 ft
12.0
0 ft
Des
crip
tion
of S
truct
ures
Geo
met
ry (C
ont.)
12.0
0 ft
13.0
0 ft
1-10
11-2
0
Uni
vers
ity o
f Pue
rto
Ric
oC
ivil
Engi
neer
ing
Dep
artm
ent
21 -
30
Mor
e th
an 3
0
Shor
t D
irect
ion
Long
Dire
ctio
n
Seis
mic
Beh
avio
r in
Res
iden
ces
20
Table 2.6: Document for residence R2
Addr
ess
Long
Dire
ctio
n
Addr
ess
Age
# Sp
ans
Col
umns
Squa
rew
/o fi
nish
ing
City
1b
6.00
inSt
ate/
Prov
ince
2h
12.0
0 in
ZIP/
Post
Cod
e3
Stee
lN
/A4
Phon
eM
ax H
eigh
tBe
ams
Rec
tang
ular
w/o
fini
shin
g M
in H
ight
b6.
00 in
& w
/o s
lab
Soil
Oth
er C
omm
.h
12.0
0 in
Stee
lN
/ASo
il Su
ppor
t Len
gth
Shor
t Dire
ctio
n
# Sp
ans
Col
umns
Rec
tang
ular
w/o
fini
shin
g 1
b6.
00 in
2h
12.0
0 in
Stee
l8
# 5
Max
Hei
ght
Beam
sR
ecta
ngul
arw
/o fi
nish
ing
Min
Hig
htb
6.00
in&
w/o
sla
bO
ther
Com
m.
h12
.00
inSt
eel
N/A
R2
Des
crip
tion
of S
truct
ures
Geo
met
ryG
ener
al In
form
atio
n
Bo G
ripiñ
as R
amal
527
Km
3.5
Jayu
ya
N/A
9.75
ftPu
erto
Ric
o00
664
N/A
9.75
ft10
.00
ft
8.00
ft8.
00 ft
29.0
0 ft
2
N/A
9.75
ft
12.0
0 ft
8.00
ft
Des
crip
tion
of S
truct
ures
Geo
met
ry (C
ont.)
9.75
ft
9.75
ft
41-
10
11-2
0
Uni
vers
ity o
f Pue
rto
Ric
oC
ivil
Engi
neer
ing
Dep
artm
ent
21 -
30
Mor
e th
an 3
0
Shor
t D
irect
ion
Long
Dire
ctio
n
Seis
mic
Beh
avio
r in
Res
iden
ces
21
Table 2.7: Document for residence R3
Addr
ess
Long
Dire
ctio
n
Addr
ess
Age
# Sp
ans
Col
umns
Squ
are
w/o
fini
shin
gC
ity1
b12
.00
inSt
ate/
Prov
ince
2h
12.0
0 in
ZIP
/Pos
t Cod
e3
Stee
l8
vars
4#5
& 4#
44
Phon
eM
ax H
eigh
tB
eam
sR
ecta
ngul
arw
/o fi
nish
ing
& w
/o s
lab
Min
Hig
htb
12.0
0 in
Soil
Oth
er C
omm
.h
15.0
0 in
Stee
l8
vars
4#5
& 4#
4So
il Su
ppor
t Len
gth
Shor
t Dire
ctio
n
# S
pans
Col
umns
Rec
tang
ular
w/o
fini
shin
g1
b12
.00
in2
h12
.00
inSt
eel
8 va
rs4#
5 &
4#4
Max
Hei
ght
Beam
sR
ecta
ngul
arw
/o fi
nish
ing
Min
Hig
htb
12.0
0 in
& w
/o s
lab
Oth
er C
omm
.h
15.0
0 in
Stee
l8
vars
4#5
& 4#
4
R3
Des
crip
tion
of S
truct
ures
Geo
met
ryG
ener
al In
form
atio
n
Bo
Rio
Gra
nde
Car
r. 14
1 km
2.6
Jayu
ya
N/A
13.5
0 ft
Pue
rto R
ico
0066
4
N/A
11.0
0 ft
13.0
0 ft
14.5
0 ft
14.5
0 ft
17.0
0 ft
2
2
N/A
14.5
0 ft
0.00
ft
Des
crip
tion
of S
truct
ures
Geo
met
ry (C
ont.)
13.5
0 ft
1-10
11-2
0
Uni
vers
ity o
f Pue
rto
Ric
oC
ivil
Engi
neer
ing
Dep
artm
ent
21 -
30
Mor
e th
an 3
0
Shor
t D
irect
ion
Long
Dire
ctio
n
Seis
mic
Beh
avio
r in
Res
iden
ces
Seis
mic
Beh
avio
r in
Res
iden
ces
22
Table 2.8: Document for residence R4
R4
Addr
ess
Long
Dire
ctio
n
Addr
ess
Age
# Sp
ans
Col
umns
Squa
reC
ity1
b12
.00
inSt
ate/
Prov
ince
2h
12.0
0 in
ZIP/
Post
Cod
e3
Stee
lN
/A4
Phon
eM
ax H
eigh
tBe
ams
Rec
tang
ular
Min
Hig
htb
12.0
0 in
Soil
Oth
er C
omm
.h
15.0
0 in
Stee
lN
/ASo
il Su
ppor
t Len
gth
Shor
t Dire
ctio
n
# Sp
ans
Col
umns
Rec
tang
ular
1b
12.0
0 in
2h
12.0
0 in
Stee
lN
/A
Max
Hei
ght
Beam
sR
ecta
ngul
arM
in H
ight
b12
.00
inO
ther
Com
m.
h12
.00
inSt
eel
N/A
R4
Des
crip
tion
of S
truct
ures
Geo
met
ry (C
ont.)
15.0
0 ft
2
2
N/A
15.0
0 ft
2.75
ft
N/A
15.0
0 ft
Puer
to R
ico
0066
4
N/A
10.2
5 ft
7.00
ft
14.5
0 ft
14.5
0 ft
9.00
ft
Des
crip
tion
of S
truct
ures
Geo
met
ryG
ener
al In
form
atio
n
Bo R
io G
rand
e C
arr.
144
km 2
.6Ja
yuya
1-10
11-2
0
Uni
vers
ity o
f Pue
rto
Ric
oC
ivil
Engi
neer
ing
Dep
artm
ent
21 -
30
Mor
e th
an 3
0
Shor
Dire
ctio
nLo
ng D
irect
ion
Seis
mic
Beh
avio
r in
Res
iden
ces
Seis
mic
Beh
avio
r in
Res
iden
ces
23
Table 2.9: Document for residence R5
Addr
ess
Long
Dire
ctio
n
Addr
ess
Age
# Sp
ans
Col
umns
Squa
rew
/o fi
nish
ing
City
1b
6.00
inSt
ate/
Prov
ince
2h
14.0
0 in
ZIP/
Post
Cod
e3
Stee
lN
/A4
Phon
eM
ax H
eigh
tBe
ams
Rec
tang
ular
w/o
fini
shin
gM
in H
ight
b6.
00 in
& w
/o s
lab
Soil
Oth
er C
omm
.h
15.0
0 in
Stee
lN
/ASo
il Su
ppor
t len
gth
Shor
t Dire
ctio
n
# Sp
ans
Col
umns
Rec
tang
ular
w/o
fini
shin
g1
b6.
00 in
2h
14.0
0 in
Stee
lN
/A
Max
Hei
ght
Beam
sR
ecta
ngul
arw
/o fi
nish
ing
Min
Hig
htb
6.00
in&
w/o
sla
bO
ther
Com
m.
h15
.00
inSt
eel
N/A
R5
Des
crip
tion
of S
truct
ures
Geo
met
ryG
ener
al In
form
atio
n
Bo Z
ama
Jayu
ya
13 ft
Spa
n is
sup
porte
d on
soi
l
10.0
0 ft
Puer
to R
ico
0066
4
N/A
13.0
0 ft
12.0
0 ft
10.0
0 ft
0.00
ft
13.0
0 ft
4
2
N/A
9.75
ft
10.0
0 ft
10.0
0 ft
Des
crip
tion
of S
truct
ures
Geo
met
ry (C
ont.)
9.75
ft
10.0
0 ft
1-10
11-2
0
Uni
vers
ity o
f Pue
rto
Ric
oC
ivil
Engi
neer
ing
Dep
artm
ent
21 -
30
Mor
e th
an 3
0
Shor
t D
irect
ion
Long
Dire
ctio
n
Seis
mic
Beh
avio
r in
Res
iden
ces
Seis
mic
Beh
avio
r in
Res
iden
ces
24
Table 2.10: Document for residence R6
Addr
ess
Long
Dire
ctio
n
Addr
ess
Age
# Sp
ans
Col
umns
Squ
are
City
1b
12.0
0 in
Stat
e/Pr
ovin
ce2
h12
.00
inZI
P/P
ost C
ode
3S
teel
6 #
54
Phon
e54
.25
ftFa
xM
ax H
eigh
tBe
ams
Rec
tang
ular
Emai
lM
in H
ight
b6.
00 in
Oth
er C
omm
.h
17.0
0 in
Incl
ude
Slab
Stee
l9
# 5
3 on
top
6 on
bot
.So
il
Soil
Supp
ort L
engt
h
Shor
t Dire
ctio
n
# Sp
ans
Col
umns
Squ
are
1b
12.0
0 in
2h
12.0
0 in
Stee
l6
# 5
Max
Hei
ght
Beam
sR
ecta
ngul
arM
in H
ight
b12
.00
inO
ther
Com
m.
h17
.00
inIn
clud
e Sl
abSt
eel
9 #
53
on to
p 6
on b
ot.
R6
Des
crip
tion
of S
truct
ures
Geo
met
ryG
ener
al In
form
atio
n
Bo Z
ama
Jayu
ya
N/A
13.7
5 ft
Puer
to R
ico
0066
4
N/A
N/A
11.3
3 ft
14.6
7 ft
8.00
ft2.
00 ft
13.0
0 ft
4
N/A
2
N/A
13.5
0 ft
8.00
ft8.
00 ft
Des
crip
tion
of S
truct
ures
Geo
met
ry (C
ont.)
13.5
0 ft
13.5
0 ft
1-10
11-2
0
Uni
vers
ity o
f Pue
rto
Ric
oC
ivil
Engi
neer
ing
Dep
artm
ent
21 -
30
Mor
e th
an 3
0
Shor
t D
irect
ion
Long
Dire
ctio
n
Seis
mic
Beh
avio
r in
Res
iden
ces
Seis
mic
Beh
avio
r in
Res
iden
ces
25
Table 2.11: Document for residence R7
Addr
ess
Long
Dire
ctio
n
Addr
ess
Age
# Sp
ans
Col
umns
Squa
reC
ity1
b11
.00
inSt
ate/
Prov
ince
2h
11.0
0 in
ZIP/
Post
Cod
e3
Stee
lN
/A4
Emai
lM
ax H
eigh
tBe
ams
Rec
tang
ular
Min
Hei
ght
b12
.00
inSo
il O
ther
Com
m.
h12
.00
inSt
eel
N/A
Soil
Supp
ort L
engt
h
Shor
t Dire
ctio
n
# Sp
ans
Col
umns
Squa
re1
b11
.00
in2
h11
.00
inSt
eel
N/A
Max
Hei
ght
Beam
sR
ecta
ngul
arM
in H
ight
b12
.00
inO
ther
Com
m.
h12
.00
inSt
eel
N/A
R7
Des
crip
tion
of S
truct
ures
Geo
met
ryG
ener
al In
form
atio
n
Bo C
anal
izo
HC
2 B
OX
668
2Ja
yuya
N/A
9.75
ftPu
erto
Ric
o00
664
7.00
ft8.
00 ft
14.0
0 ft
0.00
ft
6.00
ft
3
N/A
2
N/A
9.75
ft
14.0
0 ft
14.0
0 ft
Des
crip
tion
of S
truct
ures
Geo
met
ry (C
ont.)
10.0
0 ft
1-10
11-2
0
Uni
vers
ity o
f Pue
rto
Ric
oC
ivil
Engi
neer
ing
Dep
artm
ent
21 -
30
Mor
e th
an 3
0
Shor
t D
irect
ion
Long
Dire
ctio
n
Seis
mic
Beh
avio
r in
Res
iden
ces
Seis
mic
Beh
avio
r in
Res
iden
ces
26
Table 2.12: Document for residence R8
Addr
ess
Long
Dire
ctio
n
Addr
ess
Age
# Sp
ans
Col
umns
Rec
tang
ular
w/o
fini
shin
gC
ity1
b6.
00 in
Stat
e/Pr
ovin
ce2
h14
.00
inZI
P/Po
st C
ode
3St
eel
N/A
4Ph
one
Max
Hei
ght
Beam
sR
ecta
ngul
arw
/o fi
nish
ing
Min
Hei
ght
b6.
00 in
& w
/o s
lab
Soil
Oth
er C
omm
.h
10.0
0 in
Stee
lN
/ASo
il Su
ppor
t Len
gth
Shor
t Dire
ctio
n
# Sp
ans
Col
umns
Rec
tang
ular
w/o
fini
shin
g1
b6.
00 in
2h
14.0
0 in
Stee
lN
/A
Max
Hei
ght
Beam
sR
ecta
ngul
arw
/o fi
nish
ing
Min
Hig
htb
6.00
in&
w/o
sla
bO
ther
Com
m.
h10
.00
inSt
eel
N/A
R8
Des
crip
tion
of S
truct
ures
Geo
met
ry (C
ont.)
10.5
0 ft
3
2
N/A
10.0
0 ft
13.0
0 ft
8.00
ft
N/A
9.25
ftPu
erto
Ric
o00
664
N/A
13.0
0 ft
12.7
5 ft
13.0
0 ft
0.00
ft
13.0
0 ft
Des
crip
tion
of S
truct
ures
Geo
met
ryG
ener
al In
form
atio
n
Bo C
anal
izo
Car
r. 14
0 km
11.
9Ja
yuya
1-10
11-2
0
Uni
vers
ity o
f Pue
rto
Ric
oC
ivil
Engi
neer
ing
Dep
artm
ent
21 -
30
Mor
e th
an 3
0
Shor
t D
irect
ion
Long
Dire
ctio
n
Seis
mic
Beh
avio
r in
Res
iden
ces
Seis
mic
Beh
avio
r in
Res
iden
ces
27
Table 2.13: Document for residence R9
Addr
ess
Long
Dire
ctio
n
Addr
ess
Age
# S
pans
Col
umns
Rec
tang
ular
City
1b
10.0
0 in
Stat
e/P
rovi
nce
2h
10.0
0 in
ZIP/
Post
Cod
e3
Stee
l8
# 5
4Ph
one
Max
Hei
ght
Beam
sR
ecta
ngul
arM
in H
eigh
tb
10.0
0 in
Soil
Oth
er C
omm
.h
12.0
0 in
Stee
l9
var
3 #
4 to
p 4
# 5
& 2
# 4
bot
Soi
l Sup
port
Leng
th
Shor
t Dire
ctio
n
# S
pans
Col
umns
Squ
are
1b
10.0
0 in
2h
10.0
0 in
3St
eel
N/A
Max
Hei
ght
Beam
sR
ecta
ngul
arM
in H
ight
b10
.00
inO
ther
Com
m.
h12
.00
inSt
eel
9 va
r3#
4 to
p 4#
5 &
2#4
bot
R9
11.5
0 ft
13.0
0 ft
22.0
0 ft
11.5
0 ft
Des
crip
tion
of S
truct
ures
Geo
met
ry (C
ont.)
11.5
0 ft
17.0
0 ft
11.0
0 ft
4
313
.00
ft
11.5
0 ft
Pue
rto R
ico
0066
4
N/A
N/A
11.0
0 ft
22.0
0 ft
0.00
ftN
/A
Des
crip
tion
of S
truct
ures
Geo
met
ryG
ener
al In
form
atio
n
Cam
ino
Los
Med
inas
Car
r 531
km
1Ja
yuya
1-10
11-2
0
Uni
vers
ity o
f Pue
rto
Ric
oC
ivil
Engi
neer
ing
Dep
artm
ent
21 -
30
Mor
e th
an 3
0
Shor
t D
irect
ion
Long
Dire
ctio
n
Seis
mic
Beh
avio
r in
Res
iden
ces
Seis
mic
Beh
avio
r in
Res
iden
ces
28
Table 2.14: Document for residence R10
Addr
ess
Long
Dire
ctio
n
Add
ress
Age
# S
pans
Col
umns
Rec
tang
ular
w/fi
nish
ing
City
1b
8.50
inS
tate
/Pro
vinc
e2
h16
.75"
- 18
"ZI
P/P
ost C
ode
3St
eel
N/A
4P
hone
Max
Hei
ght
Bea
ms
Rec
tang
ular
w/fi
nish
ing
w/o
sla
bM
in H
ight
b8.
50 in
Soil
Oth
er C
omm
.h
12.0
0 in
Ste
elN
/AS
oil S
uppo
rt Le
ngth
Shor
t Dire
ctio
n
# S
pans
Col
umns
Rec
tang
ular
w/fi
nish
ing
1b
8.50
in2
h16
.75"
- 18
"3
Stee
lN
/A
Max
Hei
ght
Bea
ms
Rec
tang
ular
w/fi
nish
ing
w/o
sla
bM
in H
ight
b8.
50 in
Oth
er C
omm
.h
12.0
0 in
Stee
lN
/A
R10
Des
crip
tion
of S
truct
ures
Geo
met
ryG
ener
al In
form
atio
n
Car
r 311
Km
6.7
Sec
tor C
erillo
Cab
o R
ojo
8.58
ftN
/A
10.0
8 ft
Pue
rto R
ico
11.2
5 ft
11.5
8 ft
8.58
ft
0.00
ft
3
N/A
2
N/A
11.4
2 ft
8.58
ft8.
58 ft
Des
crip
tion
of S
truct
ures
Geo
met
ry (C
ont.)
11.2
5 ft
1-10
11-2
0
Uni
vers
ity o
f Pue
rto
Ric
oC
ivil
Engi
neer
ing
Dep
artm
ent
21 -
30
Mor
e th
an 3
0
Shor
Dire
ctio
nLo
ng D
irect
ion
Seis
mic
Beh
avio
r in
Res
iden
ces
29
Table 2.15: Document for residence R11
Addr
ess
Long
Dire
ctio
n
Addr
ess
Age
# Sp
ans
Col
umns
Rec
tang
ula r
w/fi
nish
ing
City
1b
8.00
inSt
ate/
Prov
ince
2h
18.0
0 in
ZIP/
Post
Cod
e3
Stee
l6#
54
Max
Hei
ght
Beam
sR
ecta
ngul
arw
/fini
shin
g w
/o s
lab
Soil
Min
Hig
htb
8.00
inO
ther
Com
m.
h13
.00
inSo
il S
uppo
rt Le
ngth
Stee
l6
# 5
Shor
t Dire
ctio
n
# S
pans
Col
umns
Rec
tang
ular
w/fi
nish
ing
1b
8.00
in2
h18
.00
in3
Stee
l6#
5
Max
Hei
ght
Bea
ms
Rec
tang
ular
w/fi
nish
ing
w/o
sla
bM
in H
ight
b8.
00 in
Oth
er C
omm
.h
13.0
0 in
Stee
l6
# 5
R11
Des
crip
tion
of S
truct
ures
Geo
met
ryG
ener
al In
form
atio
n
Car
r 307
Km
5.6
Sec
tor G
uani
quilla
Cab
o R
ojo
8.00
ftN
/A
15.6
7 ft
Puer
to R
ico
14.7
5 ft
11.0
0 ft
9.00
ft
17.0
0 ft
3
2
N/A
11.0
0 ft
9.00
ft8.
00 ft
Des
crip
tion
of S
truct
ures
Geo
met
ry (C
ont.)
6.00
ft
1-10
11-2
0
Seis
mic
Beh
avio
r in
Res
iden
ces
Uni
vers
ity o
f Pue
rto
Ric
oC
ivil
Engi
neer
ing
Dep
artm
ent
21 -
30
Mor
e th
an 3
0
Shor
t D
irect
ion
Long
Dire
ctio
n
Seis
mic
Beh
avio
r in
Res
iden
ces
30
Table 2.16: Document for residence R12
Addr
ess
Long
Dire
ctio
n
Addr
ess
Age
# Sp
ans
Col
umns
Rec
tang
ular
w/fi
nish
ing
City
1b
8.00
inSt
ate/
Prov
ince
2h
18.0
0 in
ZIP/
Post
Cod
e3
Stee
l6#
54
Phon
eM
ax H
eigh
tBe
ams
Rec
tang
ular
w/fi
nish
ing
w/o
sla
bM
in H
ight
b8.
00 in
Soil
Oth
er C
omm
.h
13.0
0 in
Stee
l6
# 5
Soil
Supp
ort L
engt
h
Shor
t Dire
ctio
n
# Sp
ans
Col
umns
Rec
tang
ular
w/fi
nish
ing
1b
8.00
in2
h18
.00
in3
Stee
l6#
5
Max
Hei
ght
Beam
sR
ecta
ngul
arw
/fini
shin
g w
/o s
lab
Min
Hig
htb
8.00
inO
ther
Com
m.
h13
.00
inSt
eel
6 #
5
R12
Des
crip
tion
of S
truct
ures
Geo
met
ryG
ener
al In
form
atio
n
Car
r 307
Km
5.6
Sec
tor G
uani
quilla
Cab
o R
ojo
8.00
ft2n
d Fl
oor C
ontin
uity
15.6
7 ft
Puer
to R
ico
N/A
14.7
5 ft
11.0
0 ft
9.00
ft
17.0
0 ft
3
2
N/A
11.0
0 ft
9.00
ft8.
00 ft
Des
crip
tion
of S
truct
ures
Geo
met
ry (C
ont.)
6.00
ft
1-10
11-2
0
Uni
vers
ity o
f Pue
rto
Ric
oC
ivil
Engi
neer
ing
Dep
artm
ent
21 -
30
Mor
e th
an 3
0
Shor
t D
irect
ion
Long
Dire
ctio
n
Seis
mic
Beh
avio
r in
Res
iden
ces
31
Table 2.17: Document for residence R13
Addr
ess
Long
Dire
ctio
n
Addr
ess
Age
# S
pans
Col
umns
Rec
tang
ular
w/fi
nish
ing
City
1b
6.50
inSt
ate/
Prov
ince
2h
19.0
0 in
ZIP/
Post
Cod
e3
Stee
l6#
44
Phon
eM
ax H
eigh
tBe
ams
Rec
tang
ular
w/fi
nish
ing
w/o
sla
bM
in H
ight
b6.
50 in
Soil
Oth
er C
omm
.h
13.0
0 in
Stee
lN
/ASo
il Su
ppor
t Len
gth
Shor
t Dire
ctio
n
# Sp
ans
Col
umns
Rec
tang
ular
w/fi
nish
ing
1b
6.50
in2
h19
.00
in3
Stee
l6#
4
Max
Hei
ght
Beam
sR
ecta
ngul
arw
/fini
shin
g w
/o s
lab
Min
Hig
htb
6.50
inO
ther
Com
m.
h13
.00
inSt
eel
N/A
R13
Des
crip
tion
of S
truct
ures
Geo
met
ryG
ener
al In
form
atio
n
Saba
na A
lta R
amal
331
1 km
1.3
Cab
o R
ojo
9.00
ftN
/A
11.0
0 ft
Puer
to R
ico
N/A
8.75
ft8.
75 ft
9.00
ft
0.00
ft
3
2
N/A
11.0
0 ft
9.00
ft9.
00 ft
Des
crip
tion
of S
truct
ures
Geo
met
ry (C
ont.)
11.0
0 ft
1-10
11-2
0
Uni
vers
ity o
f Pue
rto
Ric
oC
ivil
Engi
neer
ing
Dep
artm
ent
21 -
30
Mor
e th
an 3
0
Shor
Dire
ctio
nLo
ng D
irect
ion
Seis
mic
Beh
avio
r in
Res
iden
ces
Unk
now
n
32
Table 2.18: Document for residence R14
.
Addr
ess
Long
Dire
ctio
n
Addr
ess
Age
# Sp
ans
Col
umns
Rec
tang
ular
w/ f
inis
hing
City
1b
6.50
inSt
ate/
Prov
ince
2h
11.0
0 in
ZIP
/Pos
t Cod
e3
Ste
elN
/A4
Phon
eM
ax H
eigh
tBe
ams
Rec
tang
ular
w/ f
inis
hing
w/o
sla
bM
in H
ight
b6.
50 in
Soil
Oth
er C
omm
.h
11.0
0 in
Stee
lN
/ASo
il Su
ppor
t Len
gth
Shor
t Dire
ctio
n
# Sp
ans
Col
umns
Rec
tang
ular
w/ f
inis
hing
1b
6.50
in2
h17
.00
in3
Stee
lN
/A
Max
Hei
ght
Beam
sR
ecta
ngul
arw
/ fin
ishi
ng w
/o s
lab
Min
Hig
htb
6.50
inO
ther
Com
m.
h11
.00
inSt
eel
N/A
R14
Des
crip
tion
of S
truct
ures
Geo
met
ryG
ener
al In
form
atio
n
Car
r. 30
9 Pa
rcel
a Sa
n R
omal
doH
orm
igue
ro
6.92
ft
11.5
0 ft
Puer
to R
ico
11.6
7 ft
11.6
7 ft
9.00
ft
0.00
ft
3
N/A
2
N/A
10.2
5 ft
9.00
ft6.
92 ft
Des
crip
tion
of S
truct
ures
Geo
met
ry (C
ont.)
9.25
ft
1-10
11-2
0
Uni
vers
ity o
f Pue
rto
Ric
oC
ivil
Engi
neer
ing
Dep
artm
ent
21 -
30
Mor
e th
an 3
0
Shor
Dire
ctio
nLo
ng D
irect
ion
Seis
mic
Beh
avio
r in
Res
iden
ces
Unk
now
n
33
Table 2.19: Document for residence R15
Addr
ess
Long
Dire
ctio
n
Addr
ess
Age
# Sp
ans
Col
umns
Rec
tang
ular
w/fi
nish
ing
City
1b
10.0
0 in
Stat
e/Pr
ovin
ce2
h11
.00
inZI
P/Po
st C
ode
3St
eel
N/A
4Ph
one
Max
Hei
ght
Beam
sR
ecta
ngul
arw
/fini
shin
g w
/o s
lab
Min
Hig
htb
7.00
inSo
il O
ther
Com
m.
h18
.00
inSt
eel
N/A
Soil
Supp
ort L
engt
h
Shor
t Dire
ctio
n
# Sp
ans
Col
umns
Rec
tang
ular
w/fi
nish
ing
1b
10.0
0 in
2h
17.0
0 in
3St
eel
N/A
Max
Hei
ght
Beam
sR
ecta
ngul
arw
/fini
shin
g w
/o s
lab
Min
Hig
htb
7.00
inO
ther
Com
m.
h18
.00
inSt
eel
N/A
R15
Des
crip
tion
of S
truct
ures
Geo
met
ry (C
ont.)
9.25
ft
3
N/A
3
N/A
10.2
5 ft
13.0
8 ft
13.0
8 ft
13.0
8 ft
Col
umns
on
2nd
Floo
r are
7in
wid
th
11.5
0 ft
Puer
to R
ico
10.5
8 ft
11.7
5 ft
10.5
8 ft
13.0
8 ft
0.00
ft
Des
crip
tion
of S
truct
ures
Geo
met
ryG
ener
al In
form
atio
n
Car
r. 10
3 km
10.
3 C
amin
o lo
s Lo
pez
Cab
o R
ojo
1-10
11-2
0
Uni
vers
ity o
f Pue
rto
Ric
oC
ivil
Engi
neer
ing
Dep
artm
ent
21 -
30
Mor
e th
an 3
0
Shor
Dire
ctio
nLo
ng D
irect
ion
Seis
mic
Beh
avio
r in
Res
iden
ces
Unk
now
n
34
Table 2.20: Document for residence R16
Addr
ess
Long
Dire
ctio
n
Addr
ess
Age
# Sp
ans
Col
umns
Rec
tang
ular
w/fi
nish
ing
City
1b
9.50
inSt
ate/
Prov
ince
2h
11.0
0 in
ZIP/
Post
Cod
e3
Stee
lN
/A4
Phon
eM
ax H
eigh
tBe
ams
Rec
tang
ular
w/fi
nish
ing
w/o
sla
bM
in H
ight
b9.
50 in
Soil
Oth
er C
omm
.h
12.5
0 in
Stee
lN
/ASo
il Su
ppor
t Len
gth
Shor
t Dire
ctio
n
# Sp
ans
Col
umns
Rec
tang
ular
w/fi
nish
ing
1b
9.50
in2
h16
.00
in3
Stee
lN
/A
Max
Hei
ght
Beam
sR
ecta
ngul
arw
/fini
shin
g w
/o s
lab
Min
Hig
htb
9.50
inO
ther
Com
m.
h12
.50
inSt
eel
N/A
R16
Des
crip
tion
of S
truct
ures
Geo
met
ry (C
ont.)
12.0
8 ft
0.00
ft
3
3
N/A
11.9
2 ft
11.0
8 ft
6.83
ft
6.83
ftC
olum
ns o
n 2n
d Fl
oor a
re 7
in w
idth
12.5
0 ft
Puer
to R
ico
N/A
11.8
3 ft
10.5
0 ft
10.5
0 ft
9.08
ft
Des
crip
tion
of S
truct
ures
Geo
met
ryG
ener
al In
form
atio
n
Car
r. 10
3 km
10.
3 C
amin
o lo
s Ló
pez
Cab
o R
ojo
1-10
11-2
0
Uni
vers
ity o
f Pue
rto
Ric
oC
ivil
Engi
neer
ing
Dep
artm
ent
21 -
30
Mor
e th
an 3
0
Shor
Dire
ctio
nLo
ng D
irect
ion
Seis
mic
Beh
avio
r in
Res
iden
ces
Unk
now
n
35
Table 2.21: Document for residence R17
Addr
ess
Long
Dire
ctio
n
Addr
ess
Age
# S
pans
Col
umns
Squ
are
w/fi
nish
ing
City
1b
12.0
0 in
Stat
e/Pr
ovin
ce2
h12
.00
inZI
P/Po
st C
ode
3St
eel
N/A
4Ph
one
Max
Hei
ght
Beam
sS
quar
ew
/fini
shin
g w
/o s
lab
Min
Hig
htb
12.0
0 in
Soil
Oth
er C
omm
.h
12.0
0 in
Stee
l4#
5So
il Su
ppor
t len
gth
Shor
t Dire
ctio
n
# Sp
ans
Col
umns
Squ
are
w/fi
nish
ing
1b
12.0
0 in
2h
12.0
0 in
3St
eel
6#5
Max
Hei
ght
Beam
sS
quar
ew
/fini
shin
g w
/o s
lab
Min
Hig
htb
12.0
0 in
Oth
er C
omm
.h
12.0
0 in
Stee
l4#
5
R17
and
two
bath
room
s, k
itche
n, e
tc.
0 to
17
ft lin
early
apr
ox.
Des
crip
tion
of S
truct
ures
form
sG
ener
al In
form
atio
n
Cal
le 3
71 k
m 3
.5 A
lmac
illo A
ltoYa
uco
10.0
0 ft
Thre
e flo
or s
yste
m, T
hree
dor
ms
10.0
0 ft
Puer
to R
ico
11.0
0 ft
11.0
0 ft
17.0
0 ft
0.00
ft
11.0
0 ft
3
Col
umns
var
ies
from
13.9
2 ft
17.0
0 ft
0.00
ft
N/A
Des
crip
tion
of S
truct
ures
form
s 10.0
0 ft
12.0
0 ft
41-
10
11-2
0
Uni
vers
ity o
f Pue
rto
Ric
oC
ivil
Engi
neer
ing
Dep
artm
ent
21 -
30
Mor
e th
an 3
0
Shor
Dire
ctio
nLo
ng D
irect
ion
Seis
mic
Beh
avio
r in
Res
iden
ces
Unk
now
n
36
Table 2.22: Document for residence R18
Addr
ess
Long
Dire
ctio
n
Addr
ess
Age
# Sp
ans
Col
umns
Squa
rew
/fini
shin
gC
ity1
b10
.00
inSt
ate/
Prov
ince
2h
10.0
0 in
ZIP/
Post
Cod
e3
Stee
lN
/A4
Phon
eM
ax H
eigh
tBe
ams
Rec
tang
ular
w/fi
nish
ing
w/o
sla
bM
in H
ight
b9.
00 in
Soil
Oth
er C
omm
.h
11.0
0 in
Stee
lN
/ASo
il Su
ppor
t len
gth
Shor
t Dire
ctio
n
# Sp
ans
Col
umns
Squa
rew
/fini
shin
g1
b10
.00
in2
h10
.00
in3
Stee
lN
/A
Max
Hei
ght
Beam
sR
ecta
ngul
arw
/fini
shin
g w
/o s
lab
Min
Hig
htb
9.00
inO
ther
Com
m.
h11
.00
inSt
eel
N/A
R18
inte
rmed
iate
bea
ms
Des
crip
tion
of S
truct
ures
Geo
met
ryG
ener
al In
form
atio
n
Cal
le 3
71 k
m 3
.4 A
lmac
illo A
ltoYa
uco
13.6
6 ft
9.83
ftPu
erto
Ric
o
11.7
5 ft
13.6
6 ft
0.00
ft
11.7
5 ft
2
Two
Span
s w
ith
10.0
0 ft
13.6
6 ft
13.6
6 ft
N/A
Des
crip
tion
of S
truct
ures
Geo
met
ry
10.0
0 ft
9.66
ft
41-
10
11-2
0
Seis
mic
Beh
avio
r in
Res
iden
ces
Uni
vers
ity o
f Pue
rto
Ric
oC
ivil
Engi
neer
ing
Dep
artm
ent
21 -
30
Mor
e th
an 3
0
Shor
Dire
ctio
nLo
ng D
irect
ion
Seis
mic
Beh
avio
r in
Res
iden
ces
Unk
now
n
37
Table 2.23: Document for residence R19
Addr
ess
Long
Dire
ctio
n
Addr
ess
Age
# Sp
ans
Col
umns
Squa
rew
/fini
shin
gC
ity1
b12
.00
inSt
ate/
Prov
ince
2h
12.0
0 in
ZIP/
Post
Cod
e3
Stee
lN
/A4
Phon
eM
ax H
eigh
tBe
ams
Squa
rew
/fini
shin
g w
/o s
lab
Min
Hig
htb
12.0
0 in
Soil
Oth
er C
omm
.h
12.0
0 in
Stee
lN
/ASo
il Su
ppor
t len
gth
Shor
t Dire
ctio
n
# Sp
ans
Col
umns
Squa
rew
/fini
shin
g1
b12
.00
in2
h12
.00
in3
Stee
l6#
5
Max
Hei
ght
Beam
sSq
uare
w/fi
nish
ing
w/o
sla
bM
in H
ight
b12
.00
inO
ther
Com
m.
h12
.00
inSt
eel
N/A
R19
Des
crip
tion
of S
truct
ures
Geo
met
ryG
ener
al In
form
atio
n
Cal
le 3
71 k
m 3
.8 A
lmac
illo A
ltoYa
uco
8.42
ftIn
term
adia
te b
eam
s at
12.
25 ft
8.50
ftPu
erto
Ric
o
9.75
ft9.
75 ft
21.0
8 ft
0.00
ft
8.00
ft3
11.9
2 ft
21.0
8 ft
21.0
8 ft
N/A
Des
crip
tion
of S
truct
ures
Geo
met
ry
8.50
ft
9.92
ft
41-
10
11-2
0
Uni
vers
ity o
f Pue
rto
Ric
oC
ivil
Engi
neer
ing
Dep
artm
ent
21 -
30
Mor
e th
an 3
0
Shor
Dire
ctio
nLo
ng D
irect
ion
Seis
mic
Beh
avio
r in
Res
iden
ces
Unk
now
n
38
Table 2.24: Document for residence R20
Addr
ess
Long
Dire
ctio
n
Addr
ess
Age
# Sp
ans
Col
umns
Squa
rew
/fini
shin
gC
ity1
b12
.00
inSt
ate/
Prov
ince
2h
12.0
0 in
ZIP
/Pos
t Cod
e3
Ste
elN
/A4
Phon
eM
ax H
eigh
tBe
ams
Squa
rew
/fini
shin
g w
/o s
lab
Min
Hig
htb
11.0
0 in
Soil
Oth
er C
omm
.h
11.0
0 in
Stee
lN
/ASo
il Su
ppor
t len
gth
Shor
t Dire
ctio
n
# Sp
ans
Col
umns
Squ
are
w/fi
nish
ing
1b
12.0
0 in
2h
12.0
0 in
3St
eel
N/A
Max
Hei
ght
Beam
sS
quar
ew
/fini
shin
g w
/o s
lab
Min
Hig
htb
11.0
0 in
Oth
er C
omm
.h
11.0
0 in
Stee
lN
/A
R20
8in
X 8i
n
Des
crip
tion
of S
truct
ures
Geo
met
ryG
ener
al In
form
atio
n
Ram
al 3
71 C
arr.
La P
lant
aA
lmac
illo A
lto,Y
auco
22.4
2 ft
Inte
rmad
iate
bea
ms
at 1
4.17
ft
11.5
0 ft
Puer
to R
ico
11.5
0 ft
22.4
2 ft
0.00
ft
11.5
0 ft
2
12.1
7 ft
22.4
2 ft
8.25
ft
N/A
Des
crip
tion
of S
truct
ures
Geo
met
ry
11.8
3 ft
31-
10
11-2
0
Uni
vers
ity o
f Pue
rto
Ric
oC
ivil
Engi
neer
ing
Dep
artm
ent
21 -
30
Mor
e th
an 3
0
Shor
Dire
ctio
nLo
ng D
irect
ion
Seis
mic
Beh
avio
r in
Res
iden
ces
Unk
now
n
39
Table 2.25: Document for residence R21
Add
ress
Long
Dire
ctio
n
Addr
ess
Age
# Sp
ans
Col
umns
Rec
tang
ular
w/fi
nish
ing
City
1b
9.00
inSt
ate/
Prov
ince
2h
12.0
0 in
ZIP/
Post
Cod
e3
Stee
lN
/A4
Phon
eM
ax H
eigh
tBe
ams
Rec
tang
ular
w/fi
nish
ing
w/o
sla
bM
in H
ight
b8.
00 in
Soil
Oth
er C
omm
.h
12.0
0 in
Stee
lN
/ASo
il Su
ppor
t len
gth
Shor
t Dire
ctio
n
# Sp
ans
Col
umns
Rec
tang
ular
w/fi
nish
ing
1b
9.00
in2
h19
.00
in3
Stee
lN
/A
Max
Hei
ght
Beam
sR
ecta
ngul
arw
/fini
shin
g w
/o s
lab
Min
Hig
htb
8.00
inO
ther
Com
m.
h12
.00
inSt
eel
N/A
R21
Des
crip
tion
of S
truct
ures
Geo
met
ryG
ener
al In
form
atio
n
Bo D
uey
Yauc
o
9.00
ft
13.0
0 ft
Puer
to R
ico
13.5
8 ft
9.00
ft
0.00
ft
15.3
3 ft
2
13.5
0 ft
9.00
ft8.
25 ft
N/A
Des
crip
tion
of S
truct
ures
Geo
met
ry
12.0
0 ft
6.50
ft
41-
10
11-2
0
Uni
vers
ity o
f Pue
rto
Ric
oC
ivil
Engi
neer
ing
Dep
artm
ent
21 -
30
Mor
e th
an 3
0
Shor
Dire
ctio
nLo
ng D
irect
ion
Seis
mic
Beh
avio
r in
Res
iden
ces
Unk
now
n
40
Table 2.26: Document for residence R22
Addr
ess
Long
Dire
ctio
n
Addr
ess
Age
# Sp
ans
Col
umns
Rec
tang
ular
w/fi
nish
ing
City
1b
8.50
inSt
ate/
Prov
ince
2h
14.0
0 in
ZIP/
Post
Cod
e3
Stee
lN
/A4
Phon
eM
ax H
eigh
tBe
ams
Rec
tang
ular
w/fi
nish
ing
w/o
sla
bM
in H
ight
b8.
50 in
Soil
Oth
er C
omm
.h
13.5
0 in
Stee
lN
/ASo
il Su
ppor
t len
gth
Shor
t Dire
ctio
n
# Sp
ans
Col
umns
Rec
tang
ular
w/fi
nish
ing
1b
8.50
in2
h14
.00
in3
Stee
lN
/A
Max
Hei
ght
Beam
sR
ecta
ngul
arw
/fini
shin
g w
/o s
lab
Min
Hig
htb
8.50
inO
ther
Com
m.
h13
.50
inSt
eel
N/A
R22
Des
crip
tion
of S
truct
ures
Geo
met
ryG
ener
al In
form
atio
n
Car
r 129
R 6
51 K
5.7
Int D
omin
guito
Sect
or M
ata
Plat
ano,
Are
cibo
9.00
ft
10.7
5 ft
Puer
to R
ico
16.5
0 ft
9.00
ft
0.00
ft
16.5
0 ft
2
11.0
0 ft
9.00
ft9.
00 ft
787-
879-
2595
Des
crip
tion
of S
truct
ures
Geo
met
ry
10.9
2 ft
10.7
5 ft
41-
10
11-2
0
Uni
vers
ity o
f Pue
rto
Ric
oC
ivil
Engi
neer
ing
Dep
artm
ent
21 -
30
Mor
e th
an 3
0
Shor
Dire
ctio
nLo
ng D
irect
ion
Seis
mic
Beh
avio
r in
Res
iden
ces
Unk
now
n
41
Table 2.27: Document for residence R23
Addr
ess
Long
Dire
ctio
n
Addr
ess
Age
# Sp
ans
Col
umns
Rec
tang
ular
w/fi
nish
ing
City
1b
8.50
inSt
ate/
Prov
ince
2h
21.7
5 in
ZIP/
Post
Cod
e3
Stee
lN
/A4
Phon
eM
ax H
eigh
tBe
ams
Rec
tang
ular
w/fi
nish
ing
w/o
sla
bM
in H
ight
b8.
50 in
Soil
Oth
er C
omm
.h
12.0
0 in
Stee
lN
/ASo
il Su
ppor
t len
gth
Shor
t Dire
ctio
n
# Sp
ans
Col
umns
Rec
tang
ular
w/fi
nish
ing
1b
8.50
in2
h21
.75
in3
Stee
lN
/A
Max
Hei
ght
Beam
sR
ecta
ngul
arw
/fini
shin
g w
/o s
lab
Min
Hig
htb
8.50
inO
ther
Com
m.
h12
.00
inSt
eel
N/A
R23
Des
crip
tion
of S
truct
ures
Geo
met
ryG
ener
al In
form
atio
n
Car
r 129
Se
ctor
Mat
a Pl
atan
o, A
reci
bo
7.83
ft
9.92
ftPu
erto
Ric
o
14.0
0 ft
7.83
ft
0.00
ft
14.5
0 ft
2
16.7
5 ft
7.83
ft7.
83 ft
Des
crip
tion
of S
truct
ures
Geo
met
ry
16.7
5 ft
31-
10
11-2
0
Uni
vers
ity o
f Pue
rto
Ric
oC
ivil
Engi
neer
ing
Dep
artm
ent
21 -
30
Mor
e th
an 3
0
Shor
Dire
ctio
nLo
ng D
irect
ion
Seis
mic
Beh
avio
r in
Res
iden
ces
Unk
now
n
42
Table 2.28: Document for residence R24
Addr
ess
Long
Dire
ctio
n
Addr
ess
Age
# Sp
ans
Col
umns
Rec
tang
ular
w/fi
nish
ing
City
1b
6.75
inSt
ate/
Prov
ince
2h
18.5
0 in
ZIP/
Post
Cod
e3
Stee
l6#
44
Phon
eM
ax H
eigh
tBe
ams
Rec
tang
ular
w/fi
nish
ing
w/o
sla
bM
in H
ight
b6.
50 in
Soil
Oth
er C
omm
.h
15.0
0 in
Stee
l6#
4So
il Su
ppor
t len
gth
Shor
t Dire
ctio
n
# Sp
ans
Col
umns
Rec
tang
ular
w/fi
nish
ing
1b
6.75
in2
h18
.50
in3
Stee
l6#
4
Max
Hei
ght
Beam
sR
ecta
ngul
arw
/fini
shin
g w
/o s
lab
Min
Hig
htb
6.50
inO
ther
Com
m.
h15
.00
inSt
eel
6#4
R24
Des
crip
tion
of S
truct
ures
Geo
met
ryG
ener
al In
form
atio
n
Car
r 130
km
10.
9 Se
ctor
Cam
po A
legr
e, H
atillo
8.50
ft
8.00
ftPu
erto
Ric
o
12.6
7 ft
8.50
ft
0.00
ft
13.5
8 ft
2
14.9
2 ft
8.50
ft8.
50 ft
Des
crip
tion
of S
truct
ures
Geo
met
ry
11.4
2 ft
14.5
0 ft
41-
10
11-2
0
Uni
vers
ity o
f Pue
rto
Ric
oC
ivil
Engi
neer
ing
Dep
artm
ent
21 -
30
Mor
e th
an 3
0
Shor
Dire
ctio
nLo
ng D
irect
ion
Seis
mic
Beh
avio
r in
Res
iden
ces
Unk
now
n
43
Figure 2.1: Municipalities visited during the Field Survey
Figure 2.2: Residence in Yauco
44
Figure 2.3: Residence in Yauco
Figure 2.4: Residence in Hormiguero
45
Figure 2.5: Residence in Yauco
Figure 2.6: Residence in Yauco
46
Figure 2.7: Residence in Jayuya
Figure 2.8: Residence in Arecibo
47
Figure 2.9: Residence in Yauco
Figure 2.10: Residence in Cabo Rojo
48
Figure 2.11: Residence in Jayuya
Figure 2.12: Residence in Yauco
49
Figure 2.13: Residence in Jayuya
Figure 2.14: Residence in Arecibo
50
Figure 2.15: Residence in Yauco
51
CHAPTER III
VULNERABILITY EVALUATION OF TYPICAL RESIDENCES
3.1 Introduction
After the completion of the field survey, an evaluation of the structural integrity
of the most critical residences is performed and described in this chapter. A Nonlinear
Static Pushover analysis is performed on the stiffest residences, the medium stiffness
residences, and the most flexible residences, in order to evaluate their vulnerability to
potential earthquake loads. A total of six residences are analyzed to examine their
behavior when they are subjected to the seismic design inputs prescribed in the current
building code used in Puerto Rico (UBC-97). The Capacity Spectrum Method is used to
study the behavior of the structures in the non-linear range. This chapter describes the
different models of the typical residences, the creation of these structural models in the
program SAP2000, the non-linear analyses and the interpretation of the results. Also a
brief explanation of the static nonlinear pushover analysis and the methodology
implemented for the vulnerability assessment is presented.
52
3.2 Selection of the Residences Parameters
It was noticed from the field survey that the story heights of the residences
inspected varied from 8 to 20 feet. Also it was noticed that the steel reinforcement varied
from bars #4 to #5 for the beams and columns. The spans varied from 8 to 16 feet and
the height of the second floor was found to be always within the 8.5 to 9 feet range.
From this information, two extreme cases of residences resistance can be
identified. The stiffest residences are those with the shorter spans, the shorter columns
and the highest steel reinforcement, whereas the most flexible ones are those with the
opposite characteristics. Based on this consideration, three types of residential
constructions are established. The stiffer residences, i.e. those with shorter columns and
spans and highest steel reinforcement, will be identified as SS1 and S1. The most
flexible residences, i.e. those with the tallest columns, longer spans and less steel
reinforcement, will be referred to as SS3 and S3. In order to obtain a better
understanding of the seismic behavior of the residences, another system was included.
This new case, which will be called SS2 and S2, corresponds to a residence which lies
somewhere in the middle of the two extremes cases SS1 and SS3.
To obtain a preliminary vulnerability analysis, structural systems consisting of
plane frames with two spans and two stories with the parameters mentioned before were
selected. A typical structural system used in the analysis is shown in Figure 3.1.
53
Figure 3.1: Preliminary systems for vulnerability analysis
For each of the three types of residences, two frames are considered. The ones in
the strong direction (see Figure 3.1) will be identified as SS1, SS2 and SS3, whereas the
frames oriented in the weak direction will be referred to as S1, S2 and S3. Therefore, this
subdivision increases the number of cases to be considered to six. Moreover, by varying
the steel reinforcement in these six prototypes, the total number of cases to be analyzed
increases to twelve. To distinguish between the frames with more and less
reinforcement, the letters “a” and “b” will be added to the names of the cases,
respectively. Tables 3.1 and 3.2 show the geometry and properties in the weak and
strong direction for all structural models considered in the study presented in this chapter.
These parameters correspond to frames with two stories and two spans, as mentioned
before.
54
Table 3.1: Parameters of the residences in the strong direction (1 in = 25.4 mm)
Table 3.2: Parameters for the residences in the weak direction (1 in = 25.4 mm)
Model Story Height [ft] Span [ft] Beams [in] Columns [in] Beams Columns1 10 9 6X17 18X6 6#4 6#52 9 9 6X17 18X6 6#4 6#51 10 9 6X17 18X6 6#4 6#42 9 9 6X17 18X6 6#4 6#41 15 12.5 8X17 16X8 6#4 6#52 9 12.5 6X17 16X6 6#4 6#51 15 12.5 8X17 16X8 6#4 6#42 9 12.5 6X17 16X8 6#4 6#41 20 16 12X17 12X12 6#5 8#52 9 16 6x17 18X6 6#5 6#51 20 16 12X17 12X12 6#4 8#42 9 16 6x17 18X6 6#4 6#4
S2b
S3a
S3b
Sizes Reinforcement
S1a
S1b
S2a
3.3 Nonlinear Static Pushover Analysis
This section contains a summary of the Static Pushover Analysis Procedure
performed as part of the vulnerability analysis. The procedure consists of three steps,
namely the Model Generation, the analysis itself (i.e. the Nonlinear Static Pushover
Model Story Height [ft] Span [ft] Beams [in] Columns [in] Beams Columns1 10 9 6X17 6X18 6#4 6#52 9 9 6X17 6X18 6#4 6#51 10 9 6X17 6X18 6#4 6#42 9 9 6X17 6X18 6#4 6#41 15 12.5 8X17 8X16 6#4 6#52 9 12.5 8X17 8X16 6#4 6#51 15 12.5 8X17 8X16 6#4 6#42 9 12.5 8X17 8X16 6#4 6#41 20 16 12X17 12X12 6#5 8#52 9 16 6x17 6X18 6#5 6#51 20 16 12X17 12X12 6#4 8#42 9 16 6x17 6X18 6#4 6#4
Sizes Reinforcement
SS1a
SS1b
SS2a
SS2b
SS3a
SS3b
55
Analysis) and the post-processing or interpretation of results. Also a brief discussion of
the Capacity Spectrum Method is presented.
3.3.1 Model Generation
The modeling and the analysis are performed using the computer program
SAP2000 Non Linear Version. The first step in the generation of the model is to define
the geometry of the residence or structure. The geometry for all of the residences
analyzed were given in Tables 3.1 and 3.2. After the geometry of the structure is
established, the next step is to assign to the beams and columns the element sections
listed in Tables 3.1 and 3.2. The boundary conditions or restraints are also defined in this
step.
3.3.2 Nonlinear Static Pushover Analysis Set Up
To perform a Nonlinear Static Pushover Analysis in SAP 2000 it is necessary to
undertake first the following steps. The static loads (i.e. Dead Load, Live Load, and
Earthquake Load) of the model must be defined. It is also necessary to assign the joint
masses to the system before computing the natural frequencies and modes of the
structure. The last step, one of the most important ones is to assign the plastic hinges
properties and their locations. All these parameters required to build each model are
shown in Tables 3.6 to 3.17 at the end of this chapter.
56
In the present study the plastic hinges properties assigned to the columns were
different from the plastic hinges properties of the beams. The plastic hinges properties
provided by SAP 2000 are typically based on the documents FEMA–273 and ATC–40.
For the columns the PMM (flexural plus axial) hinge relation was used and the Concrete
Moment M3 (flexural) hinge relation was used for the beams. The difference between
them is the inclusion of the axial load in the PMM relation. The default hinge features
for the static nonlinear analysis used by SAP 2000 are shown in Figure 3.2. Although the
program recommends that the users define the hinges properties, this does not mean that
the results obtained using the default options are less accurate than those obtained with
the user’s supplied data. These default hinges properties are based on the
recommendations of ATC 40, which were developed from experiments or from analyses
verified by experiments, in a conservative but accurate way.
Figure 3.2: Constitutive relation for concrete hinges (based on ATC-40).
57
The following list summarizes the main features of the default hinges:
The slope between B and C is taken as 10% total strain hardening for steel.
φy = 0, since it is not needed.
Points C, D and E are based on ATC-40, Table 9.6. The four conforming
transverse reinforcing rows of the table are averaged.
My is based on the reinforcement provided. Otherwise, it is based on
minimum allowable reinforcement.
The Flexural-Axial curve is taken to be the same as the Moment curve (major
moment) in conjunction with the definition of Axial–Moment interaction
curves.
The hinges were located at a distance of each end of the frame elements equal to
10% of the length shown in Figure 3.3 taken from a screen of SAP2000.
58
Figure 3.3: SAP 2000 Model and hinges location
The next step after defining and assigning the plastics hinges is to define the
pushover load case. The procedure is similar to the definition of static load cases. SAP
2000 allows the user to define the pushover in two ways: (1) pushing the structure to a
load level defined by a pattern, and (2) defining a lateral load distribution pattern applied
to each building story and increasing it gradually until the structure reaches a certain
deformation. In this study, the lateral loads applied to each story of the systems were
defined following the vertical force distribution of the static force procedure of the 1997
Uniform Building Code. The equation for calculating the lateral load distribution is:
59
1
( ) x xx t n
i ii
W hF V FW h
=
= −
∑ (3.1)
where:
Ft = 0.07TV < 0.25V = portion of the total base shear V, considered to be
concentrated at the top of the structure in addition to Fn. This force is
added when the structure’s fundamental period is more than 0.7 sec. This
is not the case for the low rise structures studied in this work.
Wx = weight associated to level x
hx = height to level x, measured from grade
Wi = weight associated to level i
hi = height to level i, measured from grade
V = total design base shear
The values of V and Ft are not important because we are interested in the relative
distribution of the lateral loads. The program will automatically push up to the capacity
60
of the structure. The calculations of the lateral loads (in percentage of V) are also shown
in Tables 3.6 to 3.17.
3.3.3 Running the Static Nonlinear Pushover
To carry out a static nonlinear pushover it is necessary first to run a static analysis
and a modal analysis. The static analysis is needed to obtain the initial stiffness matrix
and the modal analysis is needed to obtain the structure’s period and certain coefficients
and factors used in the capacity spectrum methodology. After performing these analyses,
one is ready to run the Static Pushover analysis.
SAP 2000 is also capable of including the nonlinearity due to the geometry of the
structure. This feature, known as the P-Delta effect or Frame Instability effect, was also
included in our analyses.
3.3.4 Static Nonlinear Analysis Results (SAP 2000)
Several types of outputs can be obtained from the nonlinear static pushover
analysis:
1. The Base Reaction versus Monitored Displacement can be plotted.
61
2. The tabulated values of the Base Reaction versus Monitored Displacement at
each point along the pushover curve, along with tabulations of the number of
hinges beyond certain control points on their hinge property force-
displacement curve can be viewed on the screen, printed, or saved to a file.
3. The Base Reaction versus the Monitored Displacement can be plotted in the
ADRS format (spectral acceleration versus spectral displacement) where the
vertical axis is the spectral acceleration and the horizontal axis is the spectral
displacement. The demand spectrum can be superimposed on this plot.
4. The tabulated values of the capacity spectrum (ADRS capacity and demand
curves), the effective period and the effective damping can be viewed on the
screen, printed, or saved to a file.
5. The sequence of hinge formation and the color-coded state of each hinge can
be viewed graphically, on a step-by-step basis, for each step of the pushover
case.
6. The member forces can be viewed graphically, on a step-by-step basis, for
each step of the pushover case.
62
7. The member forces and hinge results for selected members can be written to
a file in spreadsheet format for subsequent processing by a spreadsheet
program.
For this study, we are interested in obtaining the base reaction versus the
monitored displacement curve and with the demand spectrum in an ADRS format
superimposed to it (output 3 in the previous list). The program automatically calculates
the plots in the ADRS format. More information about the conversion from capacity
curves to capacity spectrum curves can be found in Chapter 6 of the ATC-49 report.
These curves allow us to observe the behavior of the structure by comparison with the
demand spectrum.
3.4 The Capacity Spectrum Method
The Capacity Spectrum Method basically is a procedure to evaluate the Nonlinear
static response of a framework based on the intersection of the capacity or pushover
curve and a reduced response spectrum. The objective is to estimate the maximum
displacement that the structure can achieve (ATC 40, 1996). The procedure requires first
to obtain the capacity of a building to resist lateral load represented by a force-
displacement curve (i.e. pushover). Then the ground response spectrum, i.e. the
earthquake demand, is obtained. The response spectrum utilized in the present study was
the UBC-97 spectrum for rock (soil profile Sb, for which Ca = 0.3 and Cv = 0.3).
63
Therefore the possible soil amplification is not considered at this stage. The
amplifications due to the local soil and topography will be considered in a following
chapter. Figure 3.3 shows the UBC-97 spectrum utilized. A graphical representation of
these graphs in an ADRS (Acceleration – Displacement Response Spectra) format
provides a clear picture of how a building responds to a ground motion.
The capacity of a structure is its ability to resist the imposed demands. The
demand is the representation of the earthquake motion. The performance of the structure
can be obtained by comparing the capacity of the structure and the demand imposed by
the ground motion. In particular, one is interested in the intersection of the curves
describing the capacity and the demand. The interception point is called the performance
point. The vulnerability of the residences will be evaluated using the performance point.
3.4.1 Capacity Curves (Non-linear Static Pushover)
The first step in the vulnerability evaluation of the residences is to obtain the
capacity curve (pushover) of the residences. As mentioned before, three cases identified
as SS1, SS2 and SS3 were evaluated. For each case, two subcases were modeled: one for
higher steel reinforcement ratio (SS1a, etc.) and other for the lower reinforcement ratio
(SS1b, etc.). These models were evaluated in the strong and weak directions. The
models that correspond to the strong direction are named with the prefix SS and those
associated with the weak direction are identified with the prefix S. For example, SS1a
64
and S1a corresponds to the stiffer case in the strong direction and weak direction,
respectively.
Figures 3.4 to 3.15 show the pushover curves for the twelve models. Note that the
same horizontal and vertical scales were used for all cases to facilitate the comparison.
3.4.2 Capacity versus Demand Curves (Capacity Spectrum Method)
As mentioned before, to obtain the performance or vulnerability of the residences,
the capacity versus demand curves plotted in an ADRS format were first calculated.
These curves are presented in Figures 3.16 to 3.27. Using this information, the next task
was the determination of the ductility demanded by the earthquake motion and the
theoretical (capacity) ductility of the overall structure obtained from the nonlinear static
pushover.
To obtain the theoretical ductility of the structure one has to determine the
yielding point of the entire structural system. The yielding point was obtained by a
bilinear approach, which is based on approximating the pushover curve by two straight
lines. The intersection of these two lines is the theoretical yielding point used (see
Figures 3.16 to 3.27). Once the yielding point is obtained, the yielding displacement can
be obtained immediately. With the yielding displacement and the ultimate displacement
obtained from the pushover curve, one can calculate the ductility of the structure.
The demand ductility for the structure was obtained using the performance point
as the ultimate displacement. The same yielding point discussed before was used to
65
calculate the demand ductility. A summary of the ductilities for the strong and weak
direction of the models is shown in the Table 3.3 and Table 3.4, respectively.
Table 3.3: Ductilities in the strong direction.
Capacity DemandSS1a 5.13 0.70SS1b 4.57 1.17SS2a 3.50 2.28SS2b 4.70 4.64SS3a 2.43 2.33SS3b 3.13 NR
Model Ductility
Table 3.4: Ductilities in the weak direction.
Capacity DemandS1a 2.18 1.11S1b 2.24 1.63S2a 2.43 2.09S2b 2.73 NRS3a 2.52 2.47S3b 3.20 NR
DuctilityModel
NR = Performance point not reached.
3.5 Examination of the Results
The parameters to establish the vulnerability condition of the residences are the
ductilities obtained from the capacity versus demand curves (Figures 3.16 to 3.27) listed
in Tables 3.3 to 3.4). Looking at the capacity demand curve of residences in the strong
direction (SS cases) displayed in Figures 3.16 to 3.21, it can be observed that almost all
66
the residences are required to withstand the ground motion in the non linear range, except
for frame SS1 which are the stiffest ones. The models in the weak direction present the
same behavior. The S1a and SS1a are the residences which exhibit the best performance
since they resist the demand in almost the linear range. However, one should have in
mind that these spectra are based on rock, i.e. the soft soil and the topographic
amplification are not included.
In terms of demand of ductility, Tables 3.3 and 3.4 show that these two cases have
a ductility of almost one. For the strong direction, the ductility demanded by the
earthquake is 0.70 which means that, in theory, the structure will be able to resist the
ground motion in the linear range. In the weak direction the demand ductility for the
same case is 1.11, meaning that the residence will almost behave in the linear range, at
least when is founded on rock.
Notice from the same tables that the demand ductility in all the other cases
exceeded a value of two (2). Also notice in Table 3.4 (corresponding to the weak
direction) that the capacities of ductility of the cases are extended from 2.18 to 3.20.
Since these residences are old and were not designed with the ductility detailing required
in today’s codes, it is expected that in reality these residences will not develop a ductility
of more than 1.75 to 2.
A discussion that can help to validate the latter statement is presented next. The
lateral resisting system of the residences can be regarded as one of three lateral systems
67
described in the UBC-97. These three structural systems are shown in Table 3.5,
including the corresponding Response Modification Factor R also defined in the UBC-97.
Table 3.5: UBC-97 Lateral System Classification and R
Lateral Force Resisting System ROrdinary Moment Resisting Frame (OMRF) 3.5Shear Walls with OMRF 6.5Masonry Walls with OMRF 4.2
The Response Modification Factor R is a function of the product of the
Overstrength factor ROD, the Ductility factor Rµ, the Damping factor Rξ and the
Redundancy factor RR. Hence, the factor R can be defined by the following equation:
R = (ROD)(Rµ)(Rξ)(RR) (3.2)
Typical values for the overstrength factor ROD, damping factor Rξ, and Redundancy
factor RR are 2, 1.1, and 1.50 respectively. Since the largest value of the Response
Modification Factor in Table 3.18 is 6.5, it can be shown by substituting these factors and
then solving Eq. 3.2 for the Ductility factor Rµ that the highest Ductility factor becomes
equal to 1.97 which is less than 2. This value supports the conclusion that these
residences would confront problems even when the structures are on rock and evidently,
when they are on a softer soil.
68
For these reasons we conclude from these preliminary analysis that the structural
integrity of almost all the residences will be compromised when subjected to strong
motion similar to the ones considered in this study (i.e. described by the response
spectrum method used). Moreover, they will confront even more problems when the soil
and topographic effects are included in the analysis. The effect of founding this stiffest
residence (S1a) on softer soil is shown in Figure 3.28. In this figure, the ground spectra
for Sb and Se soil are plotted for comparison. Notice that when the soil profile classified
as Sd in the UBC-97 is used as seismic input, the residence is not capable of resisting the
ground motion as it did in the Sb soil.
69
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0 0.5 1 1.5 2 2.5 3 3.5
Period [sec]
Pseudo Spectral Accelerations [%g]
Figure 3.3: UBC-97 Design Spectrum for Sb soil type
70
0
5
10
15
20
25
30
35
40
45
50
0 1 2 3 4 5
Displacement [in]
Base Shear [kips]
Figure 3.4: Pushover curve for case SS1a
0
5
10
15
20
25
30
35
40
45
50
0 1 2 3 4 5
Displacement [in]
Base Shear [kip]
Figure 3.5: Pushover curve for case SS1b
71
0
5
10
15
20
25
30
35
40
45
50
0 1 2 3 4 5
Displacement [in]
Base Shear [kip]
Figure 3.6: Pushover curve for case SS2a
0
5
10
15
20
25
30
35
40
45
50
0 1 2 3 4 5
Displacement [in]
Base Shear [kip]
Figure 3.7: Pushover curve for case SS2b
72
0
5
10
15
20
25
30
35
40
45
50
0 1 2 3 4 5
Displacement [in]
Base Shear [kip]
Figure 3.8: Pushover curve for case SS3a
0
5
10
15
20
25
30
35
40
45
50
0 1 2 3 4 5
Displacement [in]
Base Shear [kip]
Figure 3.9: Pushover curve for case S3b
73
0
5
10
15
20
25
30
35
40
45
50
0 1 2 3 4 5
Displacement [in]
Base Shear [kip]
Figure 3.10: Pushover curve for case S1a
0
5
10
15
20
25
30
35
40
45
50
0 1 2 3 4 5
Displacement [in]
Base Shear [kip]
Figure 3.11: Pushover curve for case S1b
74
0
5
10
15
20
25
30
35
40
45
50
0 1 2 3 4 5
Displacement [in]
Base Shear [kip]
Figure 3.12: Pushover curve for case S2a
0
5
10
15
20
25
30
35
40
45
50
0 1 2 3 4 5
Displacement [in]
Base Shear [kip]
Figure 3.13: Pushover curve for case S2b
75
0
5
10
15
20
25
30
35
40
45
50
0 1 2 3 4 5
Displacement [in]
Base Shear [kip]
Figure 3.14: Pushover curve for case S3a
0
5
10
15
20
25
30
35
40
45
50
0 1 2 3 4 5
Displacement [in]
Base Shear [kip]
Figure 3.15: Pushover curve for case S3b
76
0
0.2
0.4
0.6
0.8
1
1.2
0 1 2 3 4 5
Spectral Displacement [in]
Spectral Acceleration [%g]
Figure 3.16: Capacity demand curve for case SS1a
0
0.2
0.4
0.6
0.8
1
1.2
0 1 2 3 4 5
Spectral Displacement [in]
Spectral Acceleration [%g]
Figure 3.17: Capacity demand curve for case SS1b
77
0
0.2
0.4
0.6
0.8
1
1.2
0 1 2 3 4 5
Spectral Displacement [in]
Spectral Acceleration [%g]
Figure 3.18: Capacity demand curve for case SS2a
0
0.2
0.4
0.6
0.8
1
1.2
0 1 2 3 4 5
Spectral Displacement [in]
Spectral Acceleration [%g]
Figure 3.19: Capacity demand curve for case SS2b
78
0
0.2
0.4
0.6
0.8
1
1.2
0 1 2 3 4 5
Spectral Displacement [in]
Spectral Acceleration [%g]
Figure 3.20: Capacity demand curve for case SS3a
0
0.2
0.4
0.6
0.8
1
1.2
0 1 2 3 4 5
Spectral Displacement [in]
Spectral Acceleration [%g]
Figure 3.21: Capacity demand curve for case SS3b
79
0
0.2
0.4
0.6
0.8
1
1.2
0 1 2 3 4 5
Spectral Displacement [in]
Spectral Acceleration [%g]
Figure 3.22: Capacity demand curve for case S1a
0
0.2
0.4
0.6
0.8
1
1.2
0 1 2 3 4 5
Spectral Displacement [in]
Spectral Acceleration [%g]
Figure 3.23: Capacity demand curve for case S1b
80
0
0.2
0.4
0.6
0.8
1
1.2
0 1 2 3 4 5
Spectral Displacement [in]
Spectral Acceleration [%g]
Figure 3.24: Capacity demand curve for case S2a
0
0.2
0.4
0.6
0.8
1
1.2
0 1 2 3 4 5
Spectral Displacement [in]
Spectral Acceleration [%g]
Figure 3.25: Capacity demand curve for case S2b
81
0
0.2
0.4
0.6
0.8
1
1.2
0 1 2 3 4 5
Spectral Displacement [in]
Spectral Acceleration [%g]
Figure 3.26: Capacity demand curve for case S3a
0
0.2
0.4
0.6
0.8
1
1.2
0 1 2 3 4 5
Spectral Displacement [in]
Spectral Acceleration [%g]
Figure 3.27: Capacity demand curve for case S3b
82
0
0.2
0.4
0.6
0.8
1
1.2
0 1 2 3 4 5
Spectral Displacement [in]
Spectral Acceleration [%g]
Sb Soil Capacity Sd Soil
Figure 3.28: Effect of soft soil on the Capacity Demand curves.
83
Table 3.6: Parameters to define the model S1a (1 ft = 305mm; 1 in = 25.4 mm; 1 kip =
4,448 N; 1 k/ft2 = 46,888 Pa; 1 k-ft = 14,441 N-m)
Model S1a
Story Height [ft] Span [ft] Beams [in] Columns [in] Beams Columns1 10 9 6X17 18X6 6#4 6#52 9 9 6X17 18X6 6#4 6#5
1st Floor Dead Load Roof Dead Load
Slab thick. 5 in Slab thick. 5 inWd = 0.0625 k/ft^2 Wd = 0.0625 k/ft^2Wa = 0.04 k/ft^2 Wa = 0.02 k/ft^2Wtotal = 0.1025 k/ft^2 Wtotal = 0.0825 k/ft^2
WD = 0.9225 k/ft WD = 0.7425 k/ftLL factor 1.2 LL factor 1.2Wdtotal = 1.107 k/ft Wdtotal = 0.891 k/ft
Mass Distribution
Total Tributary Weight
1st Floor 19.926 kips Roof 16.038 kips
Distribution of Joint Masses based on Length (1st floor)
Node Length [ft] Factor Weight [k] mass [k-s^2/ft]m1 4.5 0.25 4.982 0.155m2 9 0.5 9.963 0.309m3 4.5 0.25 4.982 0.155
Total length 18
Distribution of Joint Masses based on Length (roof)
Node Length [ft] Factor Weight [k] mass [k-s^2/ft]m1 4.500 0.250 4.010 0.125m2 9.000 0.500 8.019 0.249m3 4.500 0.250 4.010 0.125
Total length 18
Lateral Load Distribution (UBC 1997)
Height [ft] W [k] Wh [k-ft] F [%]10 19.926 199.26 0.4019 16.038 304.722 0.60
35.964 503.982
Cross Section Reinforcement
84
Table 3.7: Parameters to define the model S1b (1 ft = 305mm; 1 in = 25.4 mm; 1 kip =
4,448 N; 1 k/ft2 = 46,888 Pa; 1 k-ft = 14,441 N-m)
Model S1b
Story Height [ft] Span [ft] Beams [in] Columns [in] Beams Columns1 10 9 6X17 18X6 6#4 6#42 9 9 6X17 18X6 6#4 6#4
1st Floor Dead Load Roof Dead Load
Slab thick. 5 in Slab thick. 5 inWd = 0.0625 k/ft^2 Wd = 0.0625 k/ft^2Wa = 0.04 k/ft^2 Wa = 0.02 k/ft^2Wtotal = 0.1025 k/ft^2 Wtotal = 0.0825 k/ft^2
WD = 0.9225 k/ft WD = 0.7425 k/ftLL factor 1.2 LL factor 1.2Wdtotal = 1.107 k/ft Wdtotal = 0.891 k/ft
Mass Distribution
Total Triburary Weight
1st Floor 19.926 kips Roof 16.038 kips
Distribution of Joint Masses based on Length (1st floor)
Node Length [ft] Factor Weight [k] mass [k-s^2/ft]m1 4.500 0.250 4.982 0.155m2 9.000 0.500 9.963 0.309m3 4.500 0.250 4.982 0.155
Total length 18 ft
Total Tributary Weight
Node Length [ft] Factor Weight [k] mass [k-s^2/ft]m1 4.500 0.250 4.010 0.125m2 9.000 0.500 8.019 0.249m3 4.500 0.250 4.010 0.125
Total length 18 ft
Lateral Load Distribution (UBC 1997)
Height [ft] W [k] Wh [k-ft] F [%]10 19.926 199.26 0.4019 16.038 304.722 0.60
35.964 503.982
Cross Section Reinforcement
85
Table 3.8: Parameters to define the model SS2a (1 ft = 305mm; 1 in = 25.4 mm; 1 kip =
4,448 N; 1 k/ft2 = 46,888 Pa; 1 k-ft = 14,441 N-m)
Model S2a
Story Height [ft] Span [ft] Beams [in] Columns [in] Beams Columns1 15 12.5 8X17 16X8 6#4 6#52 9 12.5 6X17 16X6 6#4 6#5
1st Floor Dead Load Roof Dead Load
Slab thick. 5.0 in Slab thick. 5.0 inWslab = 0.0625 k/ft^2 Wd = 0.0625 k/ft^2Wextra = 0.0400 k/ft^2 Wa = 0.0200 k/ft^2Wtotal = 0.1025 k/ft^2 Wtotal = 0.0825 k/ft^2
DL = 1.28125 k/ft DL = 1.03125 k/ftLL factor 1.0 LL factor 1.0Wdtotal = 1.28125 k/ft Wdtotal = 1.03125 k/ft
Mass Distribution
Total Tributary Weight
1st Floor 32.03 kips Roof 25.78 kips
Distribution of Joint Masses based on Length (1st floor)
Node Length [ft] Factor Weight [k] mass [k-s^2/ft]m1 6.25 0.25 8.008 0.249m2 12.50 0.50 16.016 0.497m3 6.25 0.25 8.008 0.249
Total length 25
Distribution of Joint Masses based on Length (roof)
Node Length [ft] Factor Weight [k] mass [k-s^2/ft]m1 6.25 0.250 6.445 0.200m2 12.50 0.500 12.891 0.400m3 6.25 0.250 6.445 0.200
Total length 25
Lateral Load Distribution (UBC 1997)
Height [ft] W [k] Wh [k-ft] F [%]15 32.03 480.47 0.4424 25.78 618.75 0.56
57.81 1099.22
Cross Section Reinforcement
86
Table 3.9: Parameters to define the model S2b (1 ft = 305mm; 1 in = 25.4 mm; 1 kip =
4,448 N; 1 k/ft2 = 46,888 Pa; 1 k-ft = 14,441 N-m)
Model S2b
Story Height [ft] Span [ft] Beams [in] Columns [in] Beams Columns1 15 12.5 8X17 16X8 6#4 6#42 9 12.5 6X17 16X8 6#4 6#4
1st Floor Dead Load Roof Dead Load
Slab thick. 5.0 in Slab thick. 5.0 inWslab = 0.0625 k/ft^2 Wd = 0.0625 k/ft^2Wextra = 0.0400 k/ft^2 Wa = 0.0200 k/ft^2Wtotal = 0.1025 k/ft^2 Wtotal = 0.0825 k/ft^2
DL = 1.28125 k/ft DL = 1.03125 k/ftLL factor 1.0 LL factor 1.0Wdtotal = 1.28125 k/ft Wdtotal = 1.03125 k/ft
Mass Distribution
Total Tributary Weight
1st Floor 32.03 kips Roof 25.78 kips
Distribution of Joint Masses based on Length (1st floor)
Node Length [ft] Factor Weight [k] mass [k-s^2/ft]m1 6.25 0.25 8.008 0.249m2 12.50 0.50 16.016 0.497m3 6.25 0.25 8.008 0.249
Total length 25
Distribution of Joint Masses based on Length (roof)
Node Length [ft] Factor Weight [k] mass [k-s^2/ft]m1 6.25 0.25 6.445 0.200m2 12.50 0.5 12.891 0.400m3 6.25 0.25 6.445 0.200
Total length 25
Lateral Load Distribution (UBC 1997)
Height [ft] W [k] Wh [k-ft] F [%]15 32.031 480.469 0.4424 25.781 618.750 0.56
57.813 1099.219
Cross Section Reinforcement
87
Table 3.10: Parameters to define the model S3a (1 ft = 305mm; 1 in = 25.4 mm; 1 kip =
4,448 N; 1 k/ft2 = 46,888 Pa; 1 k-ft = 14,441 N-m)
Model S3a
Story Height [ft] Span [ft] Beams [in] Columns [in] Beams Columns1 20 16 12X17 12X12 6#5 8#52 9 16 6x17 18X6 6#5 6#5
1st Floor Dead Load Roof Dead Load
Slab thick. 5 in Slab thick. 5 inWd = 0.0625 k/ft^2 Wd = 0.0625 k/ft^2Wa = 0.04 k/ft^2 Wa = 0.02 k/ft^2Wtotal = 0.1025 k/ft^2 Wtotal = 0.0825 k/ft^2
WD = 1.64 k/ft WD = 1.32 k/ftLL factor 1.2 LL factor 1.2Wdtotal = 1.968 k/ft Wdtotal = 1.584 k/ft
Mass Distribution
Total Tributary Weight
1st Floor 62.976 kips Roof 50.688 kips
Distribution of Joint Masses based on Length (1st floor)
Node Length [ft] Factor Weight [k] mass [k-s^2/ft]m1 8 0.25 15.744 0.489m2 16 0.5 31.488 0.978m3 8 0.25 15.744 0.489
Total length 32
Distribution of Joint Masses based on Length (roof)
Node Length [ft] Factor Weight [k] mass [k-s^2/ft]m1 8 0.25 12.672 0.39m2 16 0.5 25.344 0.79m3 8 0.25 12.672 0.39
Total length 32
Lateral Load Distribution (UBC 1997)
Height [ft] W [k] Wh [k-ft] F [%]20 62.976 1259.520 0.4629 50.688 1469.952 0.54
113.664 2729.472
Cross Section Reinforcement
88
Table 3.11: Parameters to define the model S3b (1 ft = 305mm; 1 in = 25.4 mm; 1 kip =
4,448 N; 1 k/ft2 = 46,888 Pa; 1 k-ft = 14,441 N-m)
Model S3b
Story Height [ft] Span [ft] Beams [in] Columns [in] Beams Columns1 20 16 12X17 12X12 6#4 8#42 9 16 6x17 18X6 6#4 6#4
1st Floor Dead Load Roof Dead Load
Slab thick. 5 in Slab thick. 5 inWd = 0.0625 k/ft^2 Wd = 0.0625 k/ft^2Wa = 0.04 k/ft^2 Wa = 0.02 k/ft^2Wtotal = 0.1025 k/ft^2 Wtotal = 0.0825 k/ft^2
WD = 1.64 k/ft WD = 1.32 k/ftLL factor 1 LL factor 1Wdtotal = 1.64 k/ft Wdtotal = 1.32 k/ft
Mass Distribution
Total Tributary Weight
1st Floor 52.48 kips Roof 42.24 kips
Distribution of Joint Masses based on Length (1st floor)
Node Length [ft] Factor Weight [k] mass [k-s^2/ft]m1 8 0.25 13.12 0.407m2 16 0.5 26.24 0.815m3 8 0.25 13.12 0.407
Total length 32
Distribution of Joint Masses based on Length (roof)
Node Length [ft] Factor Weight [k] mass [k-s^2/ft]m1 8 0.25 10.56 0.328m2 16 0.5 21.12 0.656m3 8 0.25 10.56 0.328
Total length 32
Lateral Load Distribution (UBC 1997)
Height [ft] W [k] Wh [k-ft] F [%]20 52.48 1049.6 0.4629 42.24 1224.96 0.54
94.72 2274.56Var Area
Cross Section Reinforcement
89
Table 3.12: Parameters to define the model SS1a (1 ft = 305mm; 1 in = 25.4 mm; 1 kip =
4,448 N; 1 k/ft2 = 46,888 Pa; 1 k-ft = 14,441 N-m)
Model SS1a
Story Height [ft] Span [ft] Beams [in] Columns [in] Beams Columns1 10 12 6X17 6X18 6#4 6#52 9 12 6X17 6X18 6#4 6#5
1st Floor Dead Load Roof Dead Load
Slab thick. 5 in Slab thick. 5 inWd = 0.0625 k/ft^2 Wd = 0.0625 k/ft^2Wa = 0.04 k/ft^2 Wa = 0.02 k/ft^2Wtotal = 0.1025 k/ft^2 Wtotal = 0.0825 k/ft^2
WD = 1.23 k/ft WD = 0.99 k/ftLL factor 1.2 LL factor 1.2Wdtotal = 1.476 k/ft Wdtotal = 1.188 k/ft
0.123 0.099Mass Distribution
Total Tributary Weight
1st Floor 26.568 kips Roof 21.384 kips
Distribution of Joint Masses based on Length (1st floor)
Node Length [ft] Factor Weight [k] mass [k-s^2/ft]m1 4.5 0.25 6.642 0.206m2 9 0.5 13.284 0.413m3 4.5 0.25 6.642 0.206
Total length 18
Distribution of Joint Masses based on Length (roof)
Node Length [ft] Factor Weight [k] mass [k-s^2/ft]m1 4.5 0.25 5.346 0.166m2 9 0.5 10.692 0.332m3 4.5 0.25 5.346 0.166
Total length 18
Lateral Load Distribution (UBC 1997)
Height [ft] W [k] Wh [k-ft] F [%]10 26.568 265.68 0.4019 21.384 406.296 0.60
47.952 671.976
Cross Section Reinforcement
90
Table 3.13: Parameters to define the model SS1b (1 ft = 305mm; 1 in = 25.4 mm; 1 kip =
4,448 N; 1 k/ft2 = 46,888 Pa; 1 k-ft = 14,441 N-m)
Model SS1b
Story Height [ft] Span [ft] Beams [in] Columns [in] Beams Columns1 10 9 6X17 6X18 6#4 6#42 9 9 6X17 6X18 6#4 6#4
1st Floor Dead Load Roof Dead Load
Slab thick. 5 in Slab thick. 5 inWd = 0.0625 k/ft^2 Wd = 0.0625 k/ft^2Wa = 0.04 k/ft^2 Wa = 0.02 k/ft^2Wtotal = 0.1025 k/ft^2 Wtotal = 0.0825 k/ft^2
WD = 0.9225 k/ft WD = 0.7425 k/ftLL factor 1.2 LL factor 1.2Wdtotal = 1.107 k/ft Wdtotal = 0.891 k/ft
Mass Distribution
Total Tributary Weight
1st Floor 19.926 kips Roof 16.038 kips
Distribution of Joint Masses based on Length (1st floor)
Node Length [ft] Factor Weight [k] mass [k-s^2/ft]m1 4.5 0.25 4.9815 0.155m2 9 0.5 9.963 0.309m3 4.5 0.25 4.9815 0.155
Total length 18
Distribution of Joint Masses based on Length (roof)
Node Length [ft] Factor Weight [k] mass [k-s^2/ft]m1 4.5 0.25 4.0095 0.125m2 9 0.5 8.019 0.249m3 4.5 0.25 4.0095 0.125
Total length 18
Lateral Load Distribution (UBC 1997)
Height [ft] W [k] Wh [k-ft] F [%]10 19.926 199.26 0.4019 16.038 304.722 0.60
35.964 503.982
Cross Section Reinforcement
91
Table 3.14: Parameters to define the model SS2a (1 ft = 305mm; 1 in = 25.4 mm; 1 kip =
4,448 N; 1 k/ft2 = 46,888 Pa; 1 k-ft = 14,441 N-m)
Model SS2a
Story Height [ft] Span [ft] Beams [in] Columns [in] Beams Columns1 15 12.5 8X17 8X16 6#4 6#52 9 12.5 8X17 8X16 6#4 6#5
1st Floor Dead Load Roof Dead Load
Slab thick. 5.0 in Slab thick. 5.0 inWslab = 0.0625 k/ft^2 Wd = 0.0625 k/ft^2Wextra = 0.0400 k/ft^2 Wa = 0.0200 k/ft^2Wtotal = 0.1025 k/ft^2 Wtotal = 0.0825 k/ft^2
DL = 1.28125 k/ft DL = 1.03125 k/ftLL factor 1.0 LL factor 1.0Wdtotal = 1.28125 k/ft Wdtotal = 1.03125 k/ft
0.106770833 0.08594Mass Distribution
Total Tributary Weight
1st Floor 32.03 kips Roof 25.78 kips
Distribution of Joint Masses based on Length (1st floor)
Node Length [ft] Factor Weight [k] mass [k-s^2/ft]m1 6.25 0.25 8.008 0.249m2 12.50 0.50 16.016 0.497m3 6.25 0.25 8.008 0.249
Total length 25
Distribution of Joint Masses based on Length (roof)
Node Length [ft] Factor Weight [k] mass [k-s^2/ft]m1 6.25 0.25 6.445 0.200m2 12.50 0.5 12.891 0.400m3 6.25 0.25 6.445 0.200
Total length 25
Lateral Load Distribution (UBC 1997)
Height [ft] W [k] Wh [k-ft] F [%]15 32.0313 480.469 0.4424 25.7813 618.750 0.56
57.8125 1099.219
Cross Section Reinforcement
92
Table 3.15: Parameters to define the model SS2b (1 ft = 305mm; 1 in = 25.4 mm; 1 kip =
4,448 N; 1 k/ft2 = 46,888 Pa; 1 k-ft = 14,441 N-m)
Model SS2b
Story Height [ft] Span [ft] Beams [in] Columns [in] Beams Columns1 15 12.5 8X17 8X16 6#4 6#42 9 12.5 8X17 8X16 6#4 6#4
1st Floor Dead Load Roof Dead Load
Slab thick. 5.0 in Slab thick. 5.0 inWslab = 0.0625 k/ft^2 Wd = 0.0625 k/ft^2Wextra = 0.0400 k/ft^2 Wa = 0.0200 k/ft^2Wtotal = 0.1025 k/ft^2 Wtotal = 0.0825 k/ft^2
DL = 1.28125 k/ft DL = 1.03125 k/ftLL factor 1.0 LL factor 1.0Wdtotal = 1.28125 k/ft Wdtotal = 1.03125 k/ft
0.106770833 0.08594Mass Distribution
Total Tributary Weight
1st Floor 32.03 kips Roof 25.78 kips
Distribution of Joint Masses based on Length (1st floor)
Node Length [ft] Factor Weight [k] mass [k-s^2/ft]m1 6.25 0.25 8.008 0.249m2 12.50 0.50 16.016 0.497m3 6.25 0.25 8.008 0.249
Total length 25
Distribution of Joint Masses based on Length (roof)
Node Length [ft] Factor Weight [k] mass [k-s^2/ft]m1 6.25 0.25 6.445 0.200m2 12.50 0.5 12.891 0.400m3 6.25 0.25 6.445 0.200
Total length 25
Lateral Load Distribution (UBC 1997)
Height [ft] W [k] Wh [k-ft] F [%]15 32.0313 480.469 0.4424 25.7813 618.750 0.56
57.8125 1099.219
Cross Section Reinforcement
93
Table 3.16: Parameters to define the model SS3a (1 ft = 305mm; 1 in = 25.4 mm; 1 kip =
4,448 N; 1 k/ft2 = 46,888 Pa; 1 k-ft = 14,441 N-m)
Model SS3a
Story Height [ft] Span [ft] Beams [in] Columns [in] Beams Columns1 20 16 12X17 12X12 6#5 8#52 9 16 6x17 6X18 6#5 6#5
1st Floor Dead Load Roof Dead Load
Slab thick. 5 in Slab thick. 5 inWd = 0.0625 k/ft^2 Wd = 0.0625 k/ft^2Wa = 0.04 k/ft^2 Wa = 0.02 k/ft^2Wtotal = 0.1025 k/ft^2 Wtotal = 0.0825 k/ft^2
WD = 1.64 k/ft WD = 1.32 k/ftLL factor 1.2 LL factor 1.2Wdtotal = 1.968 k/ft Wdtotal = 1.584 k/ft
0.164 0.132Mass Distribution
Total Tributary Weight
1st Floor 62.976 kips Roof 50.688 kips
Distribution of Joint Masses based on Length (1st floor)
Node Length [ft] Factor Weight [k] mass [k-s^2/ft]m1 8 0.25 15.744 0.489m2 16 0.5 31.488 0.978m3 8 0.25 15.744 0.489
Total length 32
Distribution of Joint Masses based on Length (roof)
Node Length [ft] Factor Weight [k] mass [k-s^2/ft]m1 8 0.25 12.672 0.394m2 16 0.5 25.344 0.787m3 8 0.25 12.672 0.394
Total length 32
Lateral Load Distribution (UBC 1997)
Height [ft] W [k] Wh [k-ft] F [%]20 62.976 1259.520 0.4629 50.688 1469.952 0.54
113.664 2729.472
Cross Section Reinforcement
94
Table 3.17: Parameters to define the model SS3b (1 ft = 305mm; 1 in = 25.4 mm; 1 kip =
4,448 N; 1 k/ft2 = 46,888 Pa; 1 k-ft = 14,441 N-m)
Model SS3b
Story Height [ft]Span [ft] Beams [in] Columns [in] Beams Columns1 20 16 12X17 12X12 6#4 8#42 9 16 6x17 6X18 6#4 6#4
1st Floor Dead Load Roof Dead Load
Slab thick. 5 in Slab thick. 5 inWd = 0.0625 k/ft^2 Wd = 0.0625 k/ft^2Wa = 0.0400 k/ft^2 Wa = 0.0200 k/ft^2Wtotal = 0.1025 k/ft^2 Wtotal = 0.0825 k/ft^2
WD = 1.640 k/ft WD = 1.3200 k/ftLL factor 1.000 LL factor 1.0000Wdtotal = 1.640 k/ft Wdtotal = 1.3200 k/ft
0.136666667 0.11Mass Distribution
Total Tributary Weight
1st Floor 52.48 kips Roof 42.24 kips
Distribution of Joint Masses based on Length (1st floor)
Node Length [ft] Factor Weight [k] mass [k-s^2/ft]m1 8 0.25 13.12 0.407m2 16 0.5 26.24 0.815m3 8 0.25 13.12 0.407
Total length 32
Distribution of Joint Masses based on Length (roof)
Node Length [ft] Factor Weight [k] mass [k-s^2/ft]m1 8 0.25 10.56 0.328m2 16 0.5 21.12 0.656m3 8 0.25 10.56 0.328
Total length 32
Lateral Load Distribution (UBC 1997)
Height [ft] W [k] Wh [k-ft] F [%]20 52.48 1049.60 0.4629 42.24 1224.96 0.54
94.72 2274.56
Cross Section Reinforcement
95
CHAPTER IV
SEISMIC BEHAVIOR OF CODE DESIGNED RESIDENCES
4.1 Introduction
From the vulnerability analysis presented in Chapter III, it was shown that the
extreme cases evaluated are not capable to resist the lateral earthquake loads or the
required spectrum. Since these residences are old and the seismic provisions at the time
of their construction were less rigorous than the current provisions for seismic design, it
is imperative to study the behavior of similar residences designed with the current
seismic zone requirement. This chapter deals with the evaluation of similar residences
designed by the author for these purposes using the most typical sizes and parameters
found in the Field Survey (Chapter II) but satisfying all seismic zone requirement of the
UBC – 97 and the ACI 318-99. The topographic amplification is included.
4.2 Description of the Residences
From the field survey it was observed that the predominant span length ranges
from 9 to 12 ft and the predominant height for the first floor extends from 8 to 12 ft. This
information was presented in Tables 2.3 and 2.4 in Chapter II of the Field Survey. A
span length of 12 ft was selected from these data for the two-span two-story structural
system. The height of the first story was taken equal to 10 ft whereas the second story
height remains equal to 9 ft, as in the residences of the Field survey.
96
The first step in the design of the residence is to identify the structural system and
the type of soil. Since the study is focused on residences which rely on columns, the
Special Moment Resisting Frame (SMRF) and the Intermediate Moment Resisting Frame
(IMRF) were selected as the structural systems. The Ordinary Moment Resisting Frame
(OMRF) is not permitted in Seismic Zones 3 and 4. The 1997 UBC code classify the soil
in six soil types ranging from Sa for the hard rock to Sf for soft soil. These two types of
soil were not used in the analysis. The Sa soil type represents a very hard rock mostly
found in the eastern cost of the United States and it is not typical in Puerto Rico. The Sf
soil type requires a site specific evaluation and moreover, it is unlikely that this soil will
be found on hills. The soil type Sb which is defined in the UBC-97 as rock and the soil
type Se defined as soft soil, were selected for the seismic design because they are
associated with the smaller and higher demand spectra, respectively. Since two structural
systems and two soil types were selected, a total of four residences were designed: two
were designed as SMRF and two as IMRF. The four cases are described in Table 4.1
Table 4.1: Structural systems and soil types used in the residence design
Residence Structural System Soil Type 1 SMRF Sb 2 SMRF Sb 3 IMRF Se 4 IMRF Se
97
4.3 Seismic design of the residences
As mentioned before, four residences were designed with the current seismic
provisions established in the UBC-97 and ACI 318-99. The design procedure can be
summarized in the following steps:
1. Calculate gravity loads (i.e. dead load, live load and building weight).
2. Obtain the soil type and structural system.
3. Calculate the design base shear.
4. Calculate the vertical distribution of the design base shear.
5. Obtain the preliminary sizes of the structural elements.
6. Create load combinations.
7. Create a model and analyze the structural system with the computer program
(SAP2000).
8. Obtain the envelope of the loads combination.
9. Obtain the design or ultimate loads of the structural elements.
10. Design the structural elements to resist the design loads.
11. Resize the element in the computer model.
12. Reanalyze the computer model with the new element sizes.
13. Return to step 8 until the designed structural elements and the analysis elements
are the same.
98
Tables 4.2 to 4.5 show the final design or ultimate loads that control the sizing of the
structural elements as well as the capacity of the final designed elements.
Table 4.2: Final element sizes for residence 1 (1 in = 25.4 mm; 1 k-ft = 14,441 N-m)
Element Mu [k-ft] φMn [k-ft] Section ReinforcementBeams 16.29 19.62 10X10 6#5
Columns 34.81 36.18 10X12 8#5
Table 4.3: Final element sizes for residence 2 (1 in = 25.4 mm; 1 k-ft = 14,441 N-m)
Element Mu [k-ft] φMn [k-ft] Section ReinforcementBeams 21.56 27.945 10X10 8#5
Columns 35.8 36.18 10X12 8#5
Table 4.4: Final element sizes for residence 3 (1 in = 25.4 mm; 1 k-ft = 14,441 N-m)
Element Mu [k-ft] φMn [k-ft] Section ReinforcementBeams 23.46 27.945 10X10 8#5
Columns 36.79 36.18 10X12 8#5
Table 4.5: Final element sizes for residence 4 (1 in = 25.4 mm; 1 k-ft = 14,441 N-m)
Element Mu [k-ft] φMn [k-ft] Section ReinforcementBeams 24.88 27.945 10X10 8#5
Columns 38.89 44.46 10X14 8#5
After the design of the residences is completed, the next step is to perform nonlinear
analyses to observe the behavior of the structures.
99
4.4 Evaluation of the designed residences without topographic amplification
After the residences are designed, vulnerability analyses similar to the analysis
developed in Chapter III were performed to observe the behavior of the code designed
residences. A nonlinear static pushover and the Capacity Demand Method were used for
the evaluation. Table 4.6 displays the structural system, the Strength Reduction Factor R,
the seismic coefficients Ca and Cv and the percent of the base shear for each residence to
better appreciate the difference between them.
Table 4.6: Seismic parameters for the residences
Residence Structural System R Soil Type Ca Cv V [%W]1 SMRF 8.5 Sb 0.3 0.3 0.0882 SMRF 8.5 Se 0.36 0.84 0.1063 IMRF 5.5 Sb 0.3 0.3 0.1364 IMRF 5.5 Se 0.36 0.84 0.164
Two different soil types were selected to obtain the maximum and minimum
demand spectrum as described in the UBC – 97. These two spectra are presented in
Figure 4.1. Figure 4.2 shows the same spectra but in the ADRS format.
The results of the pushover analysis for each of the residences are presented in
Figure 4.3 to Figure 4.6. Notice that for each case, the designed residence presents a very
good behavior since the Capacity Demand plot shows that all the residences almost resist
the demand linearly.
100
4.5 Evaluation of the designed residences with the topographic amplification
The topographic effects were studied in the first phase of the present study by
Arroyo (2000). The conclusions are presented in their report “Numerical Study of the
Amplification of the Seismic Ground Acceleration due to Local Topography”. Two
topographic irregularities were studied in this work: escarpments or embankments and
hills or ridges. For their study they varied the slope of the escarpments and also the ratio
between the length of the base and the height of the hills when subjected to a ground
motions. The El Centro and El Salvador earthquakes scaled to the same peak ground
acceleration were used as the base ground motion. From two dimensional nonlinear
analyses using the Finite Element Method, they conclude that the amplification factor
varied from 1 to 2.35. The amplification factor obtained in their investigation was based
on absolute peak ground accelerations. The maximum amplification factors were
observed on the hills when they were subjected to the El Centro earthquake.
Since the analysis developed in this investigation is based on the Capacity
Demand Method and the amplification factors are based on the peak ground acceleration,
we need to establish how these amplification factors will be applied to the correspondent
spectrum. To include these amplification factors in a practical approach in this
investigation, we need to identify the parameters used in the UBC - 97 spectra. Figure
4.7 shows the response spectrum as prescribed in the UBC–97. Notice that the response
spectrum is function of the seismic coefficients Ca and Cv. The coefficient Ca represents
the site–dependent effective peak ground acceleration at grade and the coefficient Cv
101
represents the acceleration response at a period of 1.0 sec. for a single degree of freedom
system. It was mentioned that the amplification factors obtained by Arroyo (2001) were
defined in terms of the peak ground acceleration, which means that these factors can be
applied directly to the seismic coefficient Ca. It is evident that a change in the peak
ground acceleration will cause a change in the response spectrum, and therefore the
seismic coefficient Cv will also be affected. The amplification factor corresponding to
the seismic coefficient Cv can be obtained by different ways. For example, one could
directly apply the amplification factor to artificial accelerograms that are compatible with
the original response spectrum. Then the response spectrum for these accelerograms are
computed and the new Cv is obtained from the average response spectrum as the spectral
acceleration value at T = 1.0 sec. A simpler methodology is used in this investigation and
it can be summarized as follows:
1. First the seismic coefficients Ca and Cv for a particular soil (i.e. Sb and Se) are
obtained.
2. The ratio between the coefficients Ca and Cv is then calculated.
3. Next the amplification factor from the Arroyo report is applied to the seismic
coefficient Ca and the corresponding seismic coefficient Cv using the ratio
obtained in step 2.
4. Finally, the amplified design spectrum is obtained as prescribed in the UBC-
97.
102
The original and amplified response spectra for the Sb and Se soil types for an
amplification factor of 2.35 is presented in Figures 4.7 and 4.8, respectively.
After the development of the amplified response spectra, the next step is to obtain
the Capacity Demand plots for the four residences to observe its performance. Figures
4.9 to 4.12 illustrate the performance of the residences in terms of the Capacity-Demand
plots, for the four designed residences. Notice that none of the residences are able to
withstand earthquakes described by the amplified spectra: there is no interception (or
performance) point in the Capacity Demand plots. Therefore, is essential to include the
topographic amplification effects in the seismic design provisions in order to obtain
residences that will survive under amplified motions.
The element sizes of the residences measured in the Field Survey are considerably
smaller than the sizes designed in this chapter for similar spans and heights. For the 10 ft
height and 12 ft span, the element sizes of the field survey vary from 6x12 to 6x14
inches. In the current designs, the element sizes vary from 10x10 to 10x14 inches.
Obviously this means that if the residences designed with the current seismic provisions
do not perform satisfactory when the topographic amplification is account for, the
residences of the field survey neither will perform much worse. This was shown in
Chapter III during Vulnerability Analysis of Residences. The next step task is to study
rehabilitation techniques to come up with the most simple and economical rehabilitation
system to increase the seismic capacity of the already built residences so that they can
withstand the amplified spectra.
103
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.5 1 1.5 2 2.5 3 3.5 4
Period T [sec]
Spectral Acceleration [%g]
Sb soil Se Soil
Figure 4.1: UBC-97 spectra for Sb and Se soil types
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 1 2 3 4 5 6 7 8 9 10
Spectral Displacement [in]
Spectrall Acceleration [%g]
Sb Soil Se Soil
Figure 4.2: Demand Spectra for Sb and Se soils in ADRS format
104
0
0.2
0.4
0.6
0.8
1
1.2
1.4
0 1 2 3 4 5 6 7 8 9 10
Spectral Displacement [in]
Spectrall Acceleration [%g]
Figure 4.3: Capacity Demand plot for Residence 1 (R = 8.5, Sb soil)
0
0.2
0.4
0.6
0.8
1
1.2
1.4
0 1 2 3 4 5 6 7 8 9 10
Spectral Displacement [in]
Spectral Acceleration [%g]
Figure 4.4: Capacity Demand plot for Residence 2 (R = 8.5, Se soil)
105
0
0.2
0.4
0.6
0.8
1
1.2
1.4
0 1 2 3 4 5 6 7 8 9 10
Spectral Displacement [in]
Spectral Acceleration [%g]
Figure 4.5: Capacity Demand plot for Residence 3 (R = 5.5, Sb soil)
0
0.2
0.4
0.6
0.8
1
1.2
1.4
0 1 2 3 4 5 6 7 8 9 10
Spectral Displacement [in]
Spectral Acceleration [%g]
Figure 4.6: Capacity Demand plot for Residence 4 (R = 5.5, Se soil)
106
Figure 4.7: UBC-97 Response Spectrum
0
0.5
1
1.5
2
2.5
0 0.5 1 1.5 2 2.5 3
Period T [sec]
Spec
tral
Acc
eler
atio
n [%
g]
Response Spectrum Amplified Response Spectrum
Figure 4.8: Original and amplified response spectra for Sb soil type
107
0
0.5
1
1.5
2
2.5
0 0.5 1 1.5 2 2.5 3
Period T [sec]
Spec
tral
Acc
eler
atio
n
Response Spectrum Amplified Response Spectrum
Figure 4.9: Original and amplified response spectra for Se soil type
0
0.5
1
1.5
2
2.5
0 1 2 3 4 5 6 7 8 9 10
Spectral Displacement [in]
Spectrall Acceleration [%g]
Figure 4.10: Capacity Demand plot for Residence 1 (amplified)
108
0
0.5
1
1.5
2
2.5
0 1 2 3 4 5 6 7 8 9 10
Spectral Displacement [in]
Spectral Acceleration [%g]
Figure 4.11: Capacity Demand plot for Residence 2 (amplified)
0
0.5
1
1.5
2
2.5
0 1 2 3 4 5 6 7 8 9 10
Spectral Displacement [in]
Spectral Acceleration [%g]
Figure 4.12: Capacity Demand plot for Residence 3 (amplified)
109
0
0.5
1
1.5
2
2.5
0 1 2 3 4 5 6 7 8 9 10
Spectral Displacement [in]
Spectral Acceleration [%g]
Figure 4.13: Capacity Demand plot for Residence 4 (amplified)
110
CHAPTER V
NON LINEAR DYNAMIC TRANSIENT ANALYSIS OF THE RESIDENCES
5.1 Introduction
In this chapter, analyses more refined and detailed than the Non Linear Static
Pushover are presented. The analyses performed in this chapter are known as Nonlinear
Time History Analysis or Non-linear Dynamic Transient Analysis. These analyses were
carried out using the programs LARZWS/D.
For the non-linear dynamic transient analysis, it is necessary to submit the
structures to an earthquake record in order to obtain their response. Since there is not
enough data of strong earthquake records in Puerto Rico, a set of four artificial
earthquakes were developed. This chapter presents the artificial earthquake records
developed for the non-linear dynamic transient analysis and their results. In addition, a
collapse or failure criterion was developed to evaluate the behavior or the residences.
5.2 Artificial Earthquake Generation
As presented in Chapters III and IV of this investigation, the structures (those
from the field survey, and the code-designed residences) were subjected to a non-linear
static pushover to observe their behavior. The Capacity Spectrum Method was then used
to verify their vulnerability. Recall that the Capacity Spectrum Method represents the
capacity and the demand of the structure in the ADRS format. In that chapter, the
111
residences were evaluated for the design spectra for two types of soils (UBC-97 soil types
Sb and Se) and for two amplified spectra respectively. All of the residences show fail or
collapse when compared with the amplified spectra. The methodology used to obtain the
amplified spectrum was presented in Chapter IV of this investigation. These four spectra
are shown again in Figures 5.1 to 5.3.
In order to obtain the artificial earthquake for the analyses, the program SIMQKE
(1976) was used. Because SIMQKE is a MS-DOS program and was going to be used on
numerous occasions, a Graphical User Interface (GUI) was developed to facilitate the use
of the program. The new program with the visual interactive interface was called
WinSIMQKE. A picture taken from the screen of the GUI is presented in Figure 5.4.
There are many parameters that are needed to run the program WinSIMQKE but the most
important one is the target or desired pseudo velocity spectrum. Because in practice the
seismic codes prescribe the pseudo acceleration response spectra, the program was
modified so that the required input is now the target acceleration spectrum. For this
research the target spectra are the (pseudo) acceleration spectra with and without
topographic amplification effects presented in Figures 5.1 to 5.3. The artificially
generated earthquake records obtained were compatible with these target spectra. The
artificially generated earthquake records are presented in Figures 5.5 to 5.8 for the Sb and
Se soil types and with and without the amplification of the earthquake caused by the hill.
The peak ground acceleration and the total duration of the four artificial records are
shown in Table 5.1.
112
Table 5.1: Peak Ground Acceleration and Duration of the Artificial Records
Soil Type PGA %g Duration [sec] Sb 0.36 20
Sb Amp 0.85 20 Se 0.30 20
Se Amp 0.71 20
To validate the artificially generated earthquakes, the spectrum of each artificial
earthquake was computed and compared with the target spectrum. These comparisons
are presented in Figures 5.9 to 5.12 for the Sb and Se soil types with and without the
amplification due to topographic effect. From these figures, one can observe that the
artificially generated earthquakes have spectra that are very similar to the target spectra
(i.e. the code-based spectra).
5.3 Other aspects for the Non-linear Dynamic Transient Analysis
To perform the non-linear dynamic analysis, we need to define the geometry of
the structure, the attributed mass per floor, the gravity loads on the elements, the material
properties and the moment curvature relationship for the structural elements. All of these
parameters were explained in Chapter III, except for the moment curvature relationship.
In Chapter III, the vulnerability of the residences to strong ground motions was verified
by means of a non-linear static pushover analysis carried out using SAP2000. This
program uses as default the moment-rotation relation established in the report ATC-40.
In the LARZWS/D program, the moment-curvature relationship is another parameter to
be entered. The calculation of a moment curvature relation for a given section is a quite
113
long task, especially for beam column elements with distributed longitudinal
reinforcement. In an attempt to reduce this time consuming task, the program MOMCU
developed by López (1984) was used. In view of the fact that this program is also an
MS-DOS program, another GUI was developed to reduce the time to perform this task.
A screen shot of this modified program is presented in Figure 5.13. By using this
program, a simplified moment curvature relationship can be obtained for each of the
beams and columns of the residences. A simple bilinear approach like the one shown in
Figure 5.14 was used to perform the non-linear analyses.
5.4 Collapse Criteria or Ultimate State
The collapse or ultimate state of a structure is going to be defined as the point or
situation where the structure collapses or is not capable of resisting more forces and/or
displacements. Typically, there are two criteria to define these points: the first one is
based on the capacity of the structural elements, while the second one uses the concept of
excessive displacement and inter-story drifts. The selection of either of these two
approaches depends on the judgment of the engineer and the scope of the analyses. In
this investigation both of them are used as a failure criteria. However, additional failure
or collapse indicators were implemented in our case. Furthermore, a more detailed
methodology was developed to account for other aspects not considered in neither one of
them and explained in the next section. The following failure criteria and indicators were
used in this investigation.
114
1. Displacement or Inter-Story Drift Criterion (FCD)
2. Ultimate Rotation Criterion (FCR)
3. Element Forces Criterion (FCEF)
4. Collapse Mechanism Criterion (FCCM)
5. Stiffness Matrix Determinant (FCK)
6. Structure Period Criterion (FCT)
5.4.1 Displacement or Inter-Story Drift Criteria (FCD)
The UBC-97 establishes drift limitations for the lateral resisting system. The
code imposes limits on the maximum story drift based on the period of the structure.
These limits are as follows:
T < 0.7 sec. . . . ∆M ≤ 0.025 hs
T ≥ 0.7 sec. . . . ∆M ≤ 0.020 hs
where:
T = Fundamental period of the structure
∆M = Maximum inter-story drift
hs = Story height
In these limitations it is assumed that the structure satisfies all the ductility
requirements. In our case, these residences do not satisfy the ductility detailing. Aycardi
et al. (1994) studied the behavior of structures designed only for gravity loads. They
concluded that the following detailing deficiencies may result when the ACI’s provisions
are not applied:
115
1. columns may be weaker than the adjacent beams, potentially leading to a soft
story or column sideway mechanism.
2. lap splices of column reinforcement located in potential plastic hinge zones just
above floor slab levels
3. minimal transverse reinforcement in columns for shear and confinement,
particularly in the plastic hinge zones.
4. little or no transfer shear reinforcement in beam-column joints.
5. discontinuous positive (bottom) beam flexural reinforcement in the beam-column
joint.
All of these lacks or insufficiencies of ductility were observed in the residences of the
field survey. Aycardi et al. found that for the columns they tested, the maximum strength
was achieved between 2 and 3 percent drift. Furthermore, the columns specimens were
able to sustain at least 70 percent of their maximum load capacity at least two cycles at 4
percent drift. This information suggests that the 2.5 percent of drift established by the
ACI is conservative for a ductile system, but it is considered to be reasonable for non-
ductile or gravity designed frames. In this investigation, a 2.5 percent drift was chosen to
represent one of the failure criteria that will be named FCD (which stands for Failure
Criteria due to Drift).
116
5.4.2 Ultimate Rotation Criteria (FCR)
The ultimate rotation of the elements is one important criterion when dealing with
non-ductile systems. The deficiency of minimal transverse reinforcement and little or no
transfer shear reinforcement at beam to column joints represents a critical aspect that
must be considered. In this investigation these deficiencies were considered implicitly in
the moment curvature relation of the elements. To account for the lack of shear
reinforcement at the beams, a maximum strain deformation of 0.004 was permitted. The
ACI defines the strain deformation of 0.003 as the strain where an unconfined concrete
“fails”. However, it is recalled that this strain is for design considerations, and hence it
has to be conservative. On the other hand, the European Code establishes the previous
definition of failure of concrete at a strain deformation of 0.004. Therefore, if a strain
deformation of 0.004 in/in is considered conservative for design, the same value is
regarded to be appropriate to define failure in this investigation. This failure criteria is
denoted as FCR (for Failure Criteria due to Rotations).
Also the lack of development length of the longitudinal reinforcement is
considered in the definition of the bond slip rotation. The maximum bond slip stress
permitted by ACI is 'c5 f psi. For a concrete strength of 3,000 psi, this formula gives a
bond slip stress of 274 psi. So the previous value is conservative since the value of 600
psi is typically used for the bond slip stress. The ACI equation was used in this
investigation.
117
5.4.3 Element Forces Criteria (FCEM and FCES)
As mentioned before, the element forces are other parameters used typically for a
failure criterion only when the capacity of the element (shear and moment) is exceeded.
This criterion is implemented in this investigation in two different approaches. It is
considered that failure occurs when:
1. the maximum element moment does not exceed 5 percent of the capacity used
in the moment curvature relation (FCEM).
2. the maximum element shear does not exceed the shear capacity of the
elements (FCES).
These failure criteria are referred to as the FCEM (due to Failure Criteria of Element
Moments) and FCES (for Failure Criteria due to Element Shear). The use of 5% over the
maximum moment of the elements is an approximation to express numerically an upper
bound. The solution process of solve the non-linear dynamic transient analyses
implemented in LARZW or LARWT depends on the moment curvature relation of the
element and the hysteretic diagram used (Takeda diagram). The program solves the
problem by solving step by step the equations of motion but using the stiffness of the
previous step. The differences with the “real” values are not very significant because
usually a small time integration step is used. Nevertheless, for this reason the program
can slightly overestimate the resistance and displacements. Due to this numerical issue, a
conservative value of 5 percent was chosen for the Failure Criteria of Forces. The over-
strength due to the strain hardening of the longitudinal reinforcement of the structural
118
element is considered in the program MOMCU. However, there are other sources of
“over-strength” in the structure and a 5 percent of “overstrength” was taken
conservatively.
5.4.4 Collapse Mechanism Criteria (FCCM)
This criterion is based on the global stability of the structure. The plastic hinges
formation is being monitored at each time step of the non-linear dynamic transient
analysis to verify if a local or global collapse mechanism is formed. In this research it is
assumed that a plastic hinge is formed when the element moment exceeds the yielding
moment defined in the bilinear moment curvature relation. The collapse mechanism is
defined as the combination of plastic hinges in columns or beams at a particular time that
creates a global or local instability of the structural system. For the particular residences
that are the object of the study, it is expected that a soft story or column sideways
mechanisms will be formed because they do not comply with the UBC-97 requirement of
weak beam-strong column. In this investigation this failure criterion will be named as
FCCM (Failure Criteria of Collapse Mechanism).
5.4.5 Stiffness Matrix Determinant and Structure Period Indicator (FCK and FCT)
Another way to detect instability on a structure is by obtaining the determinant of
the stiffness matrix and the natural periods of the structure. If the determinant of the
119
stiffness matrix is less than or equal to zero, this means that there is internal instability in
the structure (Cramer’s Rule). Also, the highest (fundamental) natural period of the
structure became less than or equal to zero when the latter happened because the stiffness
matrix became singular. The program LARZW was modified to calculate the
determinant of the stiffness matrix as well the natural periods at each time step to verify
the instability of the structure. This failure criterion is termed FCK and FCT, for the
Failure Criteria of Determinant of Stiffness Matrix and Failure Criteria of Periods,
respectively.
It is important to have in mind that the FCK and FCT indicators used in this
methodology represents the values at which the structure is considered to be highly
unstable due to the collapse mechanism formed and to the degradation of the structural
elements as considered in the Takeda Hysteretic Model. These values do not represent
directly a failure, they are only other approaches considered by the author as a value to
estimate the fragility or instability of the structural system.
The FCK and FCT values are compared in a step by step procedure in which the
determinant of the stiffness matrix and the first period of the structure are calculated at
each time step and compared with the value obtained from the nonlinear static pushover.
The FCK value is plotted as a percent (%K) of the current value at a particular time step
and the value stiffness obtained from the nonlinear static pushover. The FCT value is
plotted as the ratio (T/T0) of the current period of the structure and the period obtained
from the nonlinear static pushover.
120
5.5 Failure Criteria Methodology [FC]
When performing the non-linear dynamic transient analyses the calculations were
not stopped when one of the FCD, FCR, FCEM or FCES is reached. The identification
or patterns of the collapses were monitored throughout the time history. To observe the
behavior of all of these Failure Criteria including the FCCM, FCK and FCT, the
following methodology was developed.
1. The maximum drift (FCD) at failure is calculated.
2. The maximum rotation (FCR) at failure is computed.
3. The capacity of the structural elements is calculated and the FCEM and FCES
values are obtained.
4. A non-linear static pushover of the structure is performed up to the FCCM failure
criterion is obtained
5. The determinant of the Stiffness Matrix at FCCM failure is calculated, and the
FCK value is obtained at this instant.
6. The fundamental period of the structure at FCCM failure is calculated, and the
FCT value is obtained.
7. Non-linear dynamic transient analyses are performed and from the results the
Failure Criterion that controls is identified and its magnitude is assessed.
121
5.6 Non-Linear Dynamic Transient Analyses of the Residences
The non-linear dynamic transient analysis of the extreme cases presented in
Chapter III was performed with the artificially generated records as input. Also the
“new” designed residences were subjected to the same evaluation. A summary of the
geometric and physical properties of the extreme residences is presented in Table 5.2.
The same properties but for the designed residences are shown in Table 5.3. Both set of
properties are for the frames in the strong direction. Each particular residence was
evaluated independently and discussed following the FC procedure for each of the four
earthquake records.
Table 5.2: Parameters for the typical residences in the strong direction (1 ft = 305mm; 1 in =
25.4mm)
Model Story Height [ft] Span [ft] Beams [in] Columns [in] Beams Columns1 10 9 6X17 6X18 6#4 6#52 9 9 6X17 6X18 6#4 6#51 10 9 6X17 6X18 6#4 6#42 9 9 6X17 6X18 6#4 6#41 15 12.5 8X17 8X16 6#4 6#52 9 12.5 8X17 8X16 6#4 6#51 15 12.5 8X17 8X16 6#4 6#42 9 12.5 8X17 8X16 6#4 6#41 20 16 12X17 12X12 6#5 8#52 9 16 6x17 6X18 6#5 6#51 20 16 12X17 12X12 6#4 8#42 9 16 6x17 6X18 6#4 6#4
Sizes Reinforcement
SS1a
SS1b
SS2a
SS2b
SS3a
SS3b
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Table 5.3: Parameters for the designed residences in the strong direction (1 ft = 305mm; 1 in
= 25.4mm)
Model Story Height [ft] Span [ft] Beams [in] Columns [in] Beams Columns1 10 12 10x10 10x12 6#5 8#52 9 12 10x10 10x12 6#5 8#51 10 12 10x10 10x12 8#5 8#52 9 12 10x10 10x12 8#5 8#51 10 12 10x10 10x12 8#5 8#52 9 12 10x10 10x12 8#5 8#51 10 12 10x10 10x14 8#5 8#52 9 12 10x10 10x14 8#5 8#5
Reinforcement
R1
R2
R3
R4
Sizes
The first step in the FCCM procedure is to carry out a non-linear static pushover
of the residences to obtain the critical values (i.e. FCK, FCT, etc.). As mentioned before
the program LARZWS/D was used to perform the non-linear analyses. The pushover
analyses performed in Chapter III for the vulnerability analysis are not used to obtain
these parameters because there are differences in the assumptions and implementation of
the procedure in the LARZWS/D and SAP2000 programs. Therefore, to avoid any
discrepancy in the calculations it was decided to perform again the pushovers using
LARZWS/D. The lateral load distribution used in the nonlinear static pushover was
obtained using the equivalent lateral force procedure of the UBC-97 Code as presented in
Chapter III.
Since the typical output of a nonlinear dynamic time history is quite large, a
Graphic User Interface was programmed to be used as a post-processor of the results.
This tool is capable of displaying on the screen the hinge patterns at any time step. In
addition, it is able to animate the motion of the structure, show the acceleration history as
123
well to identify drift, displacement, element forces limits, rotation limits. It can also plot
the capacity versus displacement graph of the structure, plot the stiffness matrix
determinant and the first ratio period history. Basically, all the failure criteria are
processed graphically in this program. A picture of the GUI screen is presented in Figure
5.15.
5.6.1 Non-Linear Dynamic Transient Analyses of the Extreme Residences
As mentioned before the transient dynamic non-linear analyses of the residences
were performed following the FC procedure developed in section 5.5. The residence
identified as SS1a will be used as an example to describe the methodology. The
procedure is quite lengthy but it is very good for the non-linear response evaluation of
any structure.
The first step in the implementation of the FC methodology is to obtain all the
parameters that describe the structural capacity of the system. For example to obtain the
span length, the column heights, material properties, the moment curvature relationship
of the structural elements, etc. At this step one has to define the value for the FCD. As it
was discussed in Sections 5.4.1, it was set equal to 2.5%. The FCEM and FCR can be
obtained from the moment curvature relationship. The FCES was obtained using the
following equations obtained from the report ATC-40.
124
sdfA
V
dbfA
NkV
VVV
yvs
wcg
c
scn
6.0
20005.3 '
=
⎟⎟⎠
⎞⎜⎜⎝
⎛+=
+=
λ
A detailed explanation of each of the parameters is presented in ATC-40. The reason for
using this equation is that it considers the ductility of the structural element (high,
medium or low) in the parameter k and it considers the aging of the structural element
implicitly.
Once all of the previous failure criteria are obtained, the next step is to perform a
non-linear static pushover analysis. The lateral load distribution to perform the non-
linear static pushover was defined following the equivalent static load procedure of the
UBC-97. A 1 kips load increment was selected for all of the analyses to push the
structure until the FCCM criterion was reached. Practically all the extreme residences
present a soft story collapse mechanism in the first floor because they were not designed
with the strong column-weak beam philosophy. The static pushover analyses were
stopped at this particular load step and the FCK and FCT were obtained. For example,
Figure 5.16 shows the collapse mechanism for the residence SS1a. Notice the soft story
collapse mechanism formed. Figure 5.17 displays the Base shear vs. Displacement curve
for the same structure. It can be noticed from this picture that a small increment in load
produces a large increment in the displacement indicating that a possible collapse
mechanism is formed. The variation of the determinant of the stiffness matrix as a
function of load step is presented in Figures 5.18. As the applied load increase the
125
structure suffers more damage and its stiffness decreases. The decrease in stiffness is
measured by the relative value of the determinant of the stiffness matrix, as a percent of
its original value. It can be seem that the determinant decreases up to a 1.31 % of its
original value, which is precisely the FCK value. The FCT value is obtained in a similar
fashion to the FCK value. The plot in Figure 5.19 depicts the change in the structure’s
first period as the load steps are augmented. The period change is accounted for in terms
of the ratio of the instantaneous natural period at a particular step to the initial period (of
the structure after “pushing” gravity loads). Notice that at the first steps the period ratio
is one and then its starts to increase continuously up to a value of 3.58 at the last load
step. This is latter ratio is the FCT value. This is also the last value that was needed to
perform next the nonlinear dynamic transient analysis. It is emphasized that the FCT
value is an upper limit, while the FCK value is a lower limit.
With all of the parameters obtained, a table summarizing all of the failure criteria
and indicators was prepared. Table 5.4 shows all of the parameters for the residences
SS1a. At the end of the table, the “critical values” obtained previously are presented.
The next task is to perform the non-linear dynamic transient analyses using the different
earthquakes and to check at each time step all the failure criteria. It should be pointed out
that the program will not be stopped after one failure, but rather all the possible types of
failure that occur during the complete time history will be monitored.
This particular residence (SS1a) was able to withstand the earthquake on an Sb
soil: note all the OK’s in the corresponding row in the table. The variation of the base
126
shear with the displacements of this residence is displayed in Figure 5.20. It is
illustrative to compare this graph with the pushover curve in Figure 5.17. Note that the
structure did not reach the limit of the FCES, as indicated in Table 5.4. Figures 5.21 and
Figure 5.22 show, respectively, the time variation of the stiffness matrix determinant and
the ratio of the first period. It can be seen that none of these parameters exceed the limits
established in Table 5.4.
The situation is different when the earthquake for an Se soil profile is used as
input. By following the response time history, it was found that the first and only failure
criterion (FC) exceeded was the FCR (Failure Criteria of Rotation) at the columns. When
the earthquake for and Sb soil with topographic amplification is apply to the structure, the
failure sequence the same as before, i.e. the FCR is satisfied. Then the second failure
criteria exceeded was the FCEM (Failure Criteria of Element Moments), followed by the
FCT (Failure Criteria of First Period), continuing with the FCCM (Failure Criteria of
Collapse Mechanism), the FCK (Failure Criteria of Stiffness Matrix Determinant) and
finally the FCEM were developed. The sequence in which the failure criteria are reached
can be read directly from the table, by following the order of the Roman numbers. If a
Roman number is repeated, means that the two failure criteria happened at the same time.
From Figure 5.31 of the stiffness matrix determinant history one can observer that the
limiting value or 1.57 was exceeded and in Figure 5.32 of the First Period Ratio History
the limiting value of 3.57 is also exceeded.
127
Table 5.4: Failure collapse summary for residence SS1a (1 k = 4,448 N; 1 k-in = 175,118N-m)
Columns Beams Columns Beams Columns BeamsSb OK OK OK OK OK OK OK OK OK OKSe OK I OK OK OK OK OK OK OK OK
Sb Amp OK I III II V OK OK V IV IIISe Amp VIII I V II VII OK OK IV VI III
Critical Value 2.50% 0.004384 0.004302 618.471 614.712 20.37528 19.06075 Varies 1.31 3.57
Type of Failure
EQ. Record FCD FCR [rads] FCEM [k-in] FCES [k] FCCM FCK FCT
Following the procedure described before, a failure analyses was done for the residences
SS1b, SS2a, SS2b, SS3a and SS3b. The results are presented in Tables 5.5 through 5.9.
Figures 5.16 to Figure 5.120 shows the results obtained with the modified LARZW
developed program for the six residences. The residence were subjected to the four
earthquake records except foe case SS3b, where failure occurred with the unamplified
accelerograms. The results of the failure analysis are presented in graphical way in the
following order.
1. The non-linear static pushover collapse mechanism.
2. The load vs. displacement curve for the static pushover.
3. The pushover stiffness matrix determinant history.
4. The pushover first period ratio history.
5. The base shear vs. displacement history (Dynamic Transient Analysis).
6. The stiffness matrix determinant history (Dynamic Transient Analysis).
7. The first period ratio history (Dynamic Transient Analysis).
8. The collapse mechanism (if any) for the non-linear dynamic transient
analysis.
128
The correspondence between the figures and the different residence is as follows:
Residence SS1a: Figures 5.16 to 5.33
Residence SS1b: Figures 5.34 to 5.51
Residence SS2a: Figures 5.52 to 5.70
Residence SS2b: Figures 5.71 to 5.89
Residence SS3a: Figures 5.90 to 5.109
Residence SS3b: Figures 5.110 to 5.120
Table 5.5: Failure collapse summary for residence SS1b(1 k = 4,448 N; 1 k-in = 175,118N-m)
Columns Beams Columns Beams Columns BeamsSb OK OK OK OK OK OK OK OK OK OKSe OK I III IV OK OK OK OK OK II
Sb Amp OK I II III V OK OK IV III IIISe Amp V I II II IV OK OK III II II
Critical Value 2.50% 0.004251 0.004273 443.0843 362.3015 20.37528 19.06075 Varies 0.774 4.14
Type of Failure
EQ. Record FCK FCTFCD FCR [rads] FCEM [k-in] FCES [k] FCCM
Table 5.6: Failure collapse summary for residence SS2a(1 k = 4,448 N; 1 k-in = 175,118N-m)
Columns Beams Columns Beams Columns BeamsSb OK I OK OK OK OK OK OK OK IISe VII I IV IV VI OK OK V III II
Sb Amp OK I III II IV OK OK V II IISe Amp V I II II IV OK OK III II II
Critical Value 2.50% 0.006034 0.006354 550.9025 372.1515 23.66161 25.41433 Varies 1.8776 3.1168
Type of Failure
EQ. Record FCD FCR [rads] FCEM [k-in] FCES [k] FCCM FCK FCT
129
Table 5.7: Failure collapse summary for residence SS2b (1 k = 4,448 N; 1 k-in = 175,118N-m)
Columns Beams Columns Beams Columns BeamsSb OK I OK OK OK OK OK OK III IISe V I OK IV OK OK OK III II II
Sb Amp V II OK IV OK OK OK III II ISe Amp IV I V III VI OK OK III II II
Critical Value 2.50% 0.00753 0.008428 405.2601 372.1515 23.66161 25.41433 Varies 1.7243 3.17
Type of Failure
EQ. Record FCD FCR [rads] FCEM [k-in] FCES [k] FCCM FCK FCT
Table 5.8: Failure collapse summary for residence SS3a (1 k = 4,448 N; 1 k-in = 175,118N-m)
Columns Beams Columns Beams Columns BeamsSb OK II I VI OK OK OK V IV IIISe VII II I VI OK OK OK V IV III
Sb Amp VII II I VI OK OK OK V IV IIISe Amp VI II I V VII OK OK IV III III
Critical Value 2.50% 0.011459 0.007863 535.8266 572.0768 24.97615 38.12149 Varies 5.2149 2.69
Type of Failure
EQ. Record FCD FCR [rads] FCEM [k-in] FCES [k] FCCM FCK FCT
Table 5.9: Failure collapse summary for residence SS3b (1 k = 4,448 N; 1 k-in = 175,118N-m)
Columns Beams Columns Beams Columns BeamsSb OK I OK IV OK OK OK III II ISe VI I V IV VII OK OK III II II
Sb AmpSe Amp
Critical Value 2.50% 0.012059 0.007456 388.856 467.6679 24.97615 38.12149 Varies 2.03 3.05
Type of Failure
EQ. Record FCD FCR [rads] FCEM [k-in] FCES [k] FCCM FCK FCT
As it happened in the Vulnerability Analysis of Typical Residences (Chapter III) the
stiffer residence (SS1a) was capable of resist the earthquake compatible with the UBC-97
for Sb soil type. Nevertheless, as it was the case in Chapter III, none of the residences
were capable of withstanding any of the amplified earthquakes. Furthermore, for all the
130
residence the first cause of failure was due to the maximum element rotation. This was
expected due to the low ductility of the elements. Recall that the low ductility of the
elements was taken in consideration in the development of the moment curvature relation
by limiting the ultimate strain to 0.004. According to the tables, one can observe that,
even it is assumed that the FCR does not represents an imminent failure of the structural
system, all the residences show at least three more violations to the failure criteria. After
the FCR, the predominant failure criteria were those indicators developed during this
investigation, namely the FCK and FCT. They seem to be more sensitive than the
typically used inter-story drift in predicting the collapse of the structures. Also in all
cases the FCK and FCT detects the failure before the development of a FCCM failure.
One significant conclusion is that, at least for this kinds of structures (small-size, gravity-
designed structures), the FCK and FCT failure criteria are better indicator than the inter-
story drift criterion. This last indicator (FCD) was the last criterion to occur in almost all
of the analyses.
5.6.2 Non-Linear Dynamic Transient Analyses of the Designed Residences
Similarly to the extreme residences just considered, the designed residences
(Chapter IV) were analyzed for the four earthquake records and its failure studied using
the same FC procedure. The elements’ cross sections of the designed residences in the
strong direction are shown in Table 5.3. Notice that the residences R2 and R3 have the
same cross section and thus they will be considered as a single case. The results of the
131
failure analyses obtained for the code-designed residence are presented in Tables 5.10 to
5.12 using the four spectrum-compatible synthetic earthquakes.\
Table 5.10: Failure collapse summary for residence R1 (1 k = 4,448 N; 1 k-in = 175,118N-m)
Columns Beams Columns Beams Columns BeamsSb OK OK OK OK OK OK OK OK OK OKSe OK OK OK OK OK OK OK OK OK I
Sb Amp OK III II OK OK OK OK IV I IISe Amp IV III II V VI OK OK VI I I
Critical Value 2.50% 0.008429 0.011274 528.8409 297.5931 20.81346 16.43168 Varies 3.9176 2.829
Type of Failure
EQ. Record FCD FCR [rads] FCEM [k-in] FCES [k] FCCM FCK FCT
Table 5.11: Failure collapse summary for residence R2 and R3
(1 k = 4,448 N; 1 k-in = 175,118N-m)
Columns Beams Columns Beams Columns BeamsSb OK OK OK OK OK OK OK OK OK OKSe OK OK OK OK OK OK OK OK OK OK
Sb Amp IV I III II II OK OK OK V VSe Amp IV I V II III OK OK IV III IV
Critical Value 2.50% 0.016 0.0224 528.8409 385.9118 20.81346 16.43168 Varies 2.243 3.19
Type of Failure
EQ. Record FCD FCR [rads] FCEM [k-in] FCES [k] FCCM FCK FCT
Table 5.12: Failure collapse summary for residence R4 (1 k = 4,448 N; 1 k-in = 175,118N-m)
Columns Beams Columns Beams Columns BeamsSb OK OK OK OK OK OK OK OK OK OKSe OK OK OK OK OK OK OK OK OK OK
Sb Amp OK III II I II OK OK OK OK OKSe Amp V I III II II OK OK IV IV III
Critical Value 2.50% 0.011 0.022 638.9009 385.9118 25.19524 16.43168 Varies 3.828 2.73
Type of Failure
EQ. Record FCD FCR [rads] FCEM [k-in] FCES [k] FCCM FCK FCT
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From the tables one can observe that all the designed residences are capable of resist the
Sb and Se soil type earthquake. Furthermore if the element rotations do not represent an
imminent collapse, the R4 residence can resist the Sb soil amplified earthquake with
practically no damage. However, none of the designed residences are capable of resisting
the Se soil amplified earthquake. Following the same pattern used in the residences form
the field survey; Figures 5.121 to 5.180 shows the results of the non-linear static and
dynamic analyses. The graphs presented are the pushover curves, the stiffness matrix
determinant and the first period ratio pushover and time history plots, the base shear vs.
displacement plots, and the collapse mechanism (if any).
Residence R1: Figures 5.121 to 5.140
Residence R2 and R3: Figures 5.141 to 5.160
Residence R4: Figures 5.161 to 5.180
5.6.3 Evaluation of the FCK and FCT indicators
For the failure evaluation of the residences, a computer program for non-linear
dynamic analysis was used along with six different failure criteria or indicators. Also a
methodology to evaluate the vulnerability of the structures was proposed in section 5.5
(FC) of this chapter. The typical failure criteria like drift limits, maximum rotation and
maximum element capacity were used in the methodology and two new criteria were
proposed and implemented. These failure criteria or indicators, explained in section 5.4.5
are the first period ratio history (FCR) and the determinant of the stiffness matrix history
133
(FCK). The new indicators were included in this investigation because it is the author’s
opinion that there is still a need to investigate alternative and reliable indicators for the
evaluation or the interpretation of the results of the non-linear transient analyses. This
section presents an evaluation of the results of these two parameters for the results of the
non-linear dynamic transient analyses.
When performing a non-linear dynamic analysis, there are always some doubts
about the definition of collapse or failure of the structural system. For example, the drift
limits are indicators of structural damage and thus their used as a failure or collapse
indicator can be questionable. The same happens with the maximum elements force and
maximum element rotations. Although when one of these three criteria are satisfied, this
can be used as an indication of collapse, it may be argued that this is not actually a
failure, because of the structure redundancy (in case of the element rotations) or because
of there other sources of over-strength not considered in the moment curvature relations
or in the design philosophy (maximum element forces). Therefore, it was concluded that
it would be useful to have available a better, or at least an alternative indicator of
collapse. The FCK and FCT are precisely two indicators that try to: a) predict structural
instability in terms of a collapse mechanism or b) predicts structural instability due to the
structural degradation of the elements. To verify these statements a summary of the FCK
and FCT values obtained in the previous analyses is presented in Tables 5.13 to 5.16,
including whether or not a collapse mechanism (FCCM) was formed. Recall that the
FCK is the ratio in percent of the stiffness matrix determinant at a particular instant of
134
time to the original stiffness matrix determinant (i.e. at t = 0 sec. but after “pushing” the
gravity load). Similarly, the FCT value represents the ratio between the period at a
particular time and the original (i.e. at t = 0 sec. but after “pushing” the gravity load)
period. From the plots showing the variation of the stiffness matrix determinant and the
first period ratio, one can observe that when the strong motion occurs, the FCK initially
decreases whereas the FCT increases, but then the two values oscillates around some
particular value. This is the value presented in Tables 5.13 to 5.16 as the Post-
Earthquake indicator. In addition, the limits (lower limit for the FCK and upper limit for
the FCT) are presented in the tables.
Table 5.13: FCK and FCT values for Sb earthquake.
FCK % FCT FCK % FCT FCCMSS1a 22 1.52 8 1.96 NoSS1b 17 1.69 5 2.28 NoSS2a 13 1.71 2 3.13 NoSS2b 12 1.85 1 4.34 NoSS3a 19 1.77 2 5.05 YesSS3b 12 2.21 1 6.29 YesR1 26 1.16 12 1.45 NoR2 and R3 24 1.22 12 1.48 NoR4 31 1.14 15 1.39 No
Residence Post EQ Indicator Limits
135
Table 5.14: FCK and FCT values for Se earthquake.
FCK % FCT FCK % FCT FCCMSS1a 15 1.7 4 2.72 NoSS1b 9 2.06 1 4.14 NoSS2a 4 2.4 0.5 4.87 YesSS2b 3 3 0.5 6.17 YesSS3a 8 2.77 2 5.07 YesSS3b 6 4.8 0.5 6.98 YesR1 19 1.32 3 2.83 NoR2 and R3 21 1.28 4 2.34 NoR4 28 1.18 6 2.18 No
Post EQ Indicator LimitsResidence
Table 5.15: FCK and FCT values for Sb amplified earthquake.
FCK % FCT FCK % FCT FCCMSS1a 7 2.25 1 4.61 YesSS1b 4 2.59 0.5 5.68 YesSS2a 5 1.22 0.5 4.87 YesSS2b 3 2.75 0.5 6.16 YesSS3a 7 2.69 2 5.09 YesSS3b - - - - -R1 12 1.42 1 6.19 YesR2 and R3 12 1.54 1 3.44 NoR4 17.5 1.4 5 2.36 No
Post EQ Indicator LimitsResidence
136
Table 5.16: FCK and FCT values for Se amplified earthquake.
FCK % FCT FCK % FCT FCCMSS1a 4 1 2.63 4.79 YesSS1b 1 3.73 0.5 6.25 YesSS2a 1 3.5 0.5 5.23 YesSS2b 1 4.13 0.5 6.89 YesSS3a 4 3.5 0.5 5.55 YesSS3b - - - - -R1 7 2.25 0.5 6.21 YesR2 and R3 6 1.96 0.5 4.33 YesR4 7.5 1.87 2 3.39 Yes
Post EQ Indicator LimitsResidence
Two important concepts can be extracted from the observation of these tables. From the
table one can select all the residences that developed a FCCM and prepare with them
another table. The new Table 5.17 now show only all those cases in which a collapse
mechanism was formed.
137
Table 5.17: Limits of FCK and FCK when FCCM was developed.
Residence EQ FCK % FCTSS3a Sb 2 5.05SS3b Sb 1 6.29SS2a Se 0 4.87SS2b Se 0 6.17SS1a Sb Amp 1 4.61SS1b Sb Amp 0 5.68SS2a Sb Amp 0 4.87SS2b Sb Amp 0 6.16SS3a Sb Amp 2 5.09R1 Sb Amp 1 6.19SS1a Se Amp 2.63 4.79SS1b Se Amp 0 6.25SS2a Se Amp 0 5.23SS2b Se Amp 0 6.89SS3a Se Amp 0 5.55R1 Se Amp 0 6.21R2 and R3 Se Amp 0 4.33R4 Se Amp 2 3.39
From this table one can observe that if the structure has a FCT value greater than 4.33,
there is a highly potential that the structure will form a collapse mechanism (FCCM) or it
is sufficiently deteriorated to be classify as unstable. Something similar happens with the
stiffness matrix determinant: for a determinant of about less than 2.63% of the original
stiffness matrix determinant. This is an important issue because, as mentioned before, the
values cited identify a collapse mechanism, which is the only indicator that can not be
disputed as a failure.
138
From the Tables 5.13 to 5.16 one can observe that there are some residences that
can withstand some of the earthquakes (i.e., they do not form a collapse mechanism).
The residences that do not present a collapse mechanism are listed in Table 5.18 along
with the corresponding Post-Earthquake Indicators. From this table one can observe that
the first period ratio for the residences varies from approximately 1.5 to 2.0 and the
stiffness matrix determinant varies from 10 to 20 %. These are important quantities that
indicate the condition of the structure after a seismic event and they provide useful
information for forensic engineers that needs to know the condition of a structure
following an earthquake. Also these values are important for the selection of the
rehabilitation technique, because the most flexible the structure, the least “lateral load” it
needs to withstand.
Table 5.18: Post Earthquake Indicator for Residences with no FCCM.
Residence EQ FCK % FCTSS1a Sb 22 1.52SS1b Sb 17 1.69SS2a Sb 13 1.71SS2b Sb 12 1.85SS1a Se 15 1.7SS1b Se 9 2.06R2 and R3 Sb Amp 12 1.54R4 Sb Amp 17.5 1.4
139
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.5 1 1.5 2 2.5 3 3.5 4
Period T [sec]
Spectral Acceleration [%g]
Sb soil Se Soil
Figure 5.1: UBC-97 Design Spectrum for Sb and Se soil type
0
0.5
1
1.5
2
2.5
0 0.5 1 1.5 2 2.5 3
Period T [sec]
Spec
tral
Acc
eler
atio
n [%
g]
Response Spectrum Amplified Response Spectrum
Figure 5.2: Original and amplified response spectra for Sb soil type
140
0
0.5
1
1.5
2
2.5
0 0.5 1 1.5 2 2.5 3
Period T [sec]
Spec
tral
Acc
eler
atio
n
Response Spectrum Amplified Response Spectrum
Figure 5.3: Original and amplified response spectra for Se soil type
Figure 5.4: WinSIMQKE, a GUI developed for SIMQKE program
141
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
0 2 4 6 8 10 12 14 16 18 20
Time [sec]
Acceleration [%g]
Figure 5.5: Artificial earthquake for Sb soil type
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
0 2 4 6 8 10 12 14 16 18 20
Time [sec]
Acceleration [%g]
Figure 5.6: Artificial earthquake for Sb soil type amplified
142
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
0 2 4 6 8 10 12 14 16 18 20
Time [sec]
Acceleration [%g]
Figure 5.7: Artificial earthquake for Se soil type
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
0 2 4 6 8 10 12 14 16 18 20
Time [sec]
Acceleration [%g]
Figure 5.8: Artificial earthquake for Se soil type amplified
143
0
0.5
1
1.5
2
2.5
3
0 0.5 1 1.5 2 2.5 3
Period [sec]
Spectral Acceleration [%g]
Artificial Record Spectrum UBC Spectrum [Sb Soil]
Figure 5.9: Comparison of the target and actual spectrum (Sb Soil)
0
0.5
1
1.5
2
2.5
3
0 0.5 1 1.5 2 2.5 3
Period [sec]
Spectral Acceleration [%g]
Artificial Record Spectrum SbSoil Amplified Spectrum
Figure 5.10: Comparison of the target and actual spectrum (Se Soil Amp.)
144
0
0.5
1
1.5
2
2.5
3
0 0.5 1 1.5 2 2.5 3
Period [sec]
Spectral Acceleration [%g]
Artificial Record Spectrum UBC Spectrum [Se soil]
Figure 5.11: Comparison of the target and actual spectrum (Se Soil)
0
0.5
1
1.5
2
2.5
3
0 0.5 1 1.5 2 2.5 3
Period [sec]
Spectral Acceleration [%g]
Artificial Record Spectrum Se Soil Amplified Spectrum
Figure 5.12: Comparison of the target and actual spectrum (Se Soil Amp.)
145
Figure 5.13: GUI for MOMCU program
0
100
200
300
400
500
600
0 0.0002 0.0004 0.0006 0.0008 0.001 0.0012 0.0014 0.0016 0.0018
Curvature [1/in]
Moment [k-in]
6X18 L1 L2
Figure 5.14: Bilinear approach used in the non-linear analyses
146
Figure 5.15: LARZW post processing GUI
147
Figure 5.16: Soft Story Collapse Mechanism for Residence SS1a
Figure 5.17: Static Nonlinear Pushover for Residence SS1a
148
Figure 5.18: Pushover stiffness matrix determinant history for Residence SS1a
Figure 5.19: Pushover first period ratio history for Residence SS1a
149
Figure 5.20: Base shear vs. displacement history for Residence SS1a (EQ_SB)
Figure 5.21: Stiffness matrix determinant history for Residence SS1a (EQ_SB)
150
Figure 5.22: First period ratio for Residence SS1a (EQ_SB)
Figure 5.23: Base shear vs. displacement history for Residence SS1a (EQ_SE)
151
Figure 5.24: Stiffness matrix determinant history for Residence SS1a (EQ_SE)
Figure 5.25: First period ratio history for Residence SS1a (EQ_SE)
152
Figure 5.26: Base shear vs. displacement for Residence SS1a (EQ_SB_AMP)
Figure 5.27: Stiffness matrix determinant history for Residence SS1a (EQ_SB_AMP)
153
Figure 5.28: First period ratio history for Residence SS1a (EQ_SB_AMP)
Figure 5.29: Soft story collapse mechanism for Residence SS1a (EQ_SB_AMP)
154
Figure 5.30: Base shear vs. displacement history for Residence SS1a (EQ_SE_AMP)
Figure 5.31: Stiffness matrix determinant history for Residence SS1a (EQ_SE_AMP)
155
Figure 5.32: First period history for Residence SS1a (EQ_SE_AMP)
Figure 5.33: Soft story collapse mechanism for Residence SS1a (EQ_SE_AMP)
156
Figure 5.34: Pushover collapse mechanism for Residence SS1b
Figure 5.35: Non-linear static pushover for Residence SS1b
157
Figure 5.36: Pushover stiffness matrix determinant history for Residence SS1b
Figure 5.37: Pushover first period ratio history for Residence SS1b
158
Figure 5.38: Base shear vs. displacement history for Residence SS1b (EQ_SB)
Figure 5.39: Stiffness matrix determinant history for Residence SS1b (EQ_SB)
159
Figure 5.40: First period ratio history for Residence SS1b (EQ_SB)
Figure 5.41: Base shear vs. displacement history for Residence SS1b (EQ_SE)
160
Figure 5.42: Stiffness matrix determinant history for Residence SS1b (EQ_SE)
Figure 5.43: First period ratio history for Residence SS1b (EQ_SE)
161
Figure 5.44: Base shear vs. displacement history for Residence SS1b (EQ_SB_AMP)
Figure 5.45: Stiffness matrix determinant history for Residence SS1b (EQ_SB_AMP)
162
Figure 5.46: First period ratio history for Residence SS1b (EQ_SB_AMP)
Figure 5.47: Collapse mechanism for Residence SS1b (EQ_SB_AMP)
163
Figure 5.48: Base shear vs. displacement history for Residence SS1b (EQ_SE_AMP)
Figure 5.49: Stiffness matrix determinant history for Residence SS1b (EQ_SE_AMP)
164
Figure 5.50: First period ratio history for Residence SS1b (EQ_SE_AMP)
Figure 5.51: Soft story collapse mechanism for Residence SS1b (EQ_SE_AMP)
165
Figure 5.52: Pushover collapse mechanism for Residence SS2a
.
Figure 5.53: Non-linear static pushover for Residence SS2a
166
Figure 5.54: Pushover stiffness matrix determinant history for Residence SS2a
Figure 5.55: Pushover first period ratio history for Residence SS2a
167
Figure 5.56: Base shear vs. displacement history for Residence SS2a (EQ_SB)
Figure 5.57: Stiffness matrix determinant history for Residence SS2a (EQ_SB)
168
Figure 5.58: First period ratio history for Residence SS2a (EQ_SB)
Figure 5.59: Base shear vs. displacement history for Residence SS2a (EQ_SE)
169
Figure 5.60: Pushover stiffness matrix determinant history for Residence SS2a (EQ_SE)
Figure 5.61: First period ratio history for Residence SS2a (EQ_SE)
170
Figure 5.62: Soft story collapse mechanism for Residence SS2a (EQ_SE)
Figure 5.63: Base shear vs. displacement history for Residence SS2a (EQ_SB_AMP)
171
Figure 5.64: Stiffness matrix determinant history for Residence SS2a (EQ_SB_AMP)
Figure 5.65: First period ratio history for Residence SS2a (EQ_SB_AMP)
172
Figure 5.66: Soft story collapse mechanism for Residence SS2a (EQ_SB_AMP)
Figure 5.67: Base shear vs. displacement history for Residence SS2a (EQ_SE_AMP)
173
Figure 5.68: Stiffness matrix determinant history for Residence SS2a (EQ_SE_AMP)
Figure 5.69: First period ratio history for Residence SS2a (EQ_SE_AMP)
174
Figure 5.70: Soft story collapse mechanism for Residence SS2a (EQ_SE_AMP)
Figure 5.71: Pushover collapse mechanism for Residence SS2b
175
Figure 5.72: Non-linear static pushover for Residence SS2b
Figure 5.73: Pushover stiffness matrix determinant history for Residence SS2b
176
Figure 5.74: Pushover first period ratio history for Residence SS2b
Figure 5.75: Base shear vs. displacement history for Residence SS2b (EQ_SB)
177
Figure 5.76: Stiffness matrix determinant history for Residence SS2b (EQ_SB)
Figure 5.77: First period ratio history for Residence SS2b (EQ_SB)
178
Figure 5.78: Base shear vs. displacement history for Residence SS2b (EQ_SE)
Figure 5.79: stiffness matrix determinant history for Residence SS2b (EQ_SE)
179
Figure 5.80: First period ratio history for Residence SS2b (EQ_SE)
Figure 5.81: Soft story collapse mechanism for Residence SS2b (EQ_SE)
180
Figure 5.82: Base shear vs. displacement history for Residence SS2b (EQ_SB_AMP)
Figure 5.83: Stiffness matrix determinant history for Residence SS2b (EQ_SB_AMP)
181
Figure 5.84: First period ratio history for Residence SS2b (EQ_SB_AMP)
Figure 5.85: Soft story collapse mechanism for Residence SS2b (EQ_SB_AMP)
182
Figure 5.86: Base shear vs. displacement history for Residence SS2b (EQ_SE_AMP)
Figure 5.87: Stiffness matrix determinant history for Residence SS2b (EQ_SE_AMP)
183
Figure 5.88: First period ratio history for Residence SS2b (EQ_SE_AMP)
Figure 5.89: Soft story collapse mechanism for Residence SS2b (EQ_SE_AMP)
184
Figure 5.90: Pushover collapse mechanism for Residence SS3a
.
Figure 5.91: Non-linear static pushover for Residence SS3a
185
Figure 5.92: Pushover stiffness matrix determinant history for Residence SS3a
Figure 5.93: Pushover first period ratio history for Residence SS3a
186
Figure 5.94: Base shear vs. displacement history for Residence SS3a (EQ_SB)
Figure 5.95: Stiffness matrix determinant history for Residence SS3a (EQ_SB)
187
Figure 5.96: First period ratio history for Residence SS3a (EQ_SB)
Figure 5.97: Soft story collapse mechanism for Residence SS3a (EQ_SB)
188
Figure 5.98: Base shear vs. displacement history for Residence SS3a (EQ_SE)
Figure 5.99: Pushover stiffness matrix determinant history for Residence SS3a (EQ_SE)
189
Figure 5.100: First period ratio history for Residence SS3a (EQ_SE)
Figure 5.101: Soft story collapse mechanism for Residence SS3a (EQ_SE)
190
Figure 5.102: Base shear vs. displacement history for Residence SS3a (EQ_SB_AMP)
Figure 5.103: Stiffness matrix determinant history for Residence SS3a (EQ_SB_AMP)
191
Figure 5.104: First period ratio history for Residence SS3a (EQ_SB_AMP)
Figure 5.105: Soft story collapse mechanism for Residence SS3a (EQ_SB_AMP)
192
Figure 5.106: Base shear vs. displacement history for Residence SS3a (EQ_SE_AMP)
Figure 5.107: Stiffness matrix determinant history for Residence SS3a (EQ_SE_AMP)
193
Figure 5.108: First period ratio history for Residence SS3a (EQ_SE_AMP)
Figure 5.109: Soft story collapse mechanism for Residence SS3a (EQ_SE_AMP)
194
Figure 5.110: Pushover collapse mechanism for Residence SS3b
.
Figure 5.111: Non-linear static pushover for Residence SS3b
195
Figure 5.112: Pushover stiffness matrix determinant history for Residence SS3b
Figure 5.113: Pushover first period ratio history for Residence SS3b
196
Figure 5.114: Base shear vs. displacement history for Residence SS3b (EQ_SB)
Figure 5.115: Stiffness matrix determinant history for Residence SS3b (EQ_SB)
197
Figure 5.116: First period ratio history for Residence SS3b (EQ_SB)
Figure 5.117: Base shear vs. displacement history for Residence SS3b (EQ_SE)
198
Figure 5.118: Stiffness matrix determinant history for Residence SS3b (EQ_SE)
Figure 5.119: First period ratio history for Residence SS3b (EQ_SE)
199
Figure 5.120: Soft story collapse mechanism for Residence SS3b (EQ_SE)
Figure 5.121: Pushover collapse mechanism for Residence R1
200
Figure 5.122: Non-linear static pushover for Residence R1
Figure 5.123: Pushover stiffness matrix determinant history for Residence R1
201
Figure 5.124: Pushover first period ratio history for Residence R1
Figure 5.125: Base shear vs. displacement history for Residence R1 (EQ_SB)
202
Figure 5.126: Stiffness matrix determinant history for Residence R1 (EQ_SB)
Figure 5.127: First period ratio history for Residence R1 (EQ_SB)
203
Figure 5.128: Maximum number of hinges formed for Residence R1 (EQ_SB)
Figure 5.129: Base shear vs. displacement history for Residence R1 (EQ_SE)
204
Figure 5.130: Stiffness matrix determinant history for Residence R1 (EQ_SE)
Figure 5.131: First period ratio history for Residence R1 (EQ_SE)
205
Figure 5.132: Maximum number of hinges formed for Residence R1 (EQ_SE)
Figure 5.133: Base shear vs. displacement history for Residence R1 (EQ_SB_AMP)
206
Figure 5.134: Stiffness matrix determinant history for Residence R1 (EQ_SB_AMP)
Figure 5.135: First period ratio history for Residence R1 (EQ_SB_AMP)
207
Figure 5.136: Collapse mechanism for Residence R1 (EQ_SB_AMP)
Figure 5.137: Base shear vs. displacement history for Residence R1 (EQ_SE_AMP)
208
Figure 5.138: Stiffness matrix determinant history for Residence R1 (EQ_SE_AMP)
Figure 5.139: First period ratio history for Residence R1 (EQ_SE_AMP)
209
Figure 5.140: Collapse mechanism for Residence R1 (EQ_SE_AMP)
Figure 5.141: Pushover collapse mechanism for Residences R2 and R3
210
Figure 5.142: Non-linear static pushover for Residences R2 and R3
Figure 5.143: Pushover stiffness matrix determinant history for Residences R2 and R3
211
Figure 5.144: Pushover first period ratio history for Residences R2 and R3
Figure 5.145: Base shear vs. displacement history for Residences R2 and R3 (EQ_SB)
212
Figure 5.146: Stiffness matrix determinant history for Residences R2 and R3 (EQ_SB)
Figure 5.147: First period ratio history for Residences R2 and R3 (EQ_SB)
213
Figure 5.148: Maximum number of hinges formed for Residence R2 and R3 (EQ_SB)
Figure 5.149: Base shear vs. displacement history for Residences R2 and R3 (EQ_SE)
214
Figure 5.150: Stiffness matrix determinant history for Residences R2 and R3 (EQ_SE)
Figure 5.151: First period ratio history for Residences R2 and R3 (EQ_SE)
215
Figure 5.152: Maximum number of hinges formed for Residences R2 and R3 (EQ_SE)
Figure 5.153: Base shear vs. displacement history for Residences R2 and R3
(EQ_SB_AMP)
216
Figure 5.154: Stiffness matrix determinant history for Residences R2 and R3
(EQ_SB_AMP)
Figure 5.155: First period ratio history for Residences R2 and R3 (EQ_SB_AMP)
217
Figure 5.156: Maximum number of hinges formed for Residence R2 and R3
(EQ_SB_AMP)
Figure 5.157: Base shear vs. displacement history for Residences R2 and R3
(EQ_SE_AMP)
218
Figure 5.158: Stiffness matrix determinant history for Residences R2 and R3
(EQ_SE_AMP)
Figure 5.159: First period ratio history for Residences R2 and R3 (EQ_SE_AMP)
219
Figure 5.160: Soft story collapse mechanism for Residences R2 and R3 (EQ_SE_AMP)
Figure 5.161: Pushover collapse mechanism for Residence R4
220
Figure 5.162: Non-linear static pushover for Residence R4
Figure 5.163: Pushover stiffness matrix determinant history for Residence R4
221
Figure 5.164: Pushover first period ratio history for Residence R4
Figure 5.165: Base shear vs. displacement history for Residence R4 (EQ_SB)
222
Figure 5.166: Stiffness matrix determinant history for Residence R4 (EQ_SB)
Figure 5.167: First period ratio history for Residence R4 (EQ_SB)
223
Figure 5.168: Maximum number of hinges formed for Residence R4 (EQ_SB)
Figure 5.169: Base shear vs. displacement history for Residence R4 (EQ_SE)
224
Figure 5.170: Stiffness matrix determinant history for Residence R4 (EQ_SE)
Figure 5.171: First period ratio history for Residence R4 (EQ_SE)
225
Figure 5.172: Maximum number of hinges formed for Residence R4 (EQ_SE)
Figure 5.173: Base shear vs. displacement history for Residence R4 (EQ_SB_AMP)
226
Figure 5.174: Stiffness matrix determinant history for Residence R4 (EQ_SB_AMP)
Figure 5.175: First period ratio history for Residence R4 (EQ_SB_AMP)
227
Figure 5.176: Maximum number of hinges formed for Residences R4 (EQ_SB_AMP)
Figure 5.177: Base shear vs. displacement history for Residence R4 (EQ_SE_AMP)
228
Figure 5.178: Stiffness matrix determinant history for Residence R4 (EQ_SE_AMP)
Figure 5.179: First period ratio history for Residence R4 (EQ_SE_AMP)
229
Figure 5.180: Collapse mechanism for Residence R4 (EQ_SE_AMP)
230
CHAPTER VI
SELECTION AND VERIFICATION OF THE RETROFITTING STRATEGY
6.1 Introduction
This chapter deals with the selection and implementation of the retrofitting
strategy for the seismic rehabilitation of the residences. The Capacity Spectrum Method
methodology was used to achieve to the most practical and economical retrofitting
strategy. The definition of the target or demand spectrum for the rehabilitation system is
also discussed in this chapter. Then the retrofitting strategy is selected based on the
capacity spectrum, non linear dynamic transient analyses and technical and non-technical
considerations. Subsequent to the selection, a numerical simulation of its implementation
and testing is done. A series of tables was developed in which for residences with
different span lengths, column heights and elements’ cross sections, the user can
conservatively choose the retrofitting system that should be implemented.
6.2 Selection of the demand or target spectrum
To obtain a practical rehabilitation technique or strategy is necessary to know the
demand, or in other words the seismic event that the rehabilitation system needs to resist.
The vulnerability analyses performed in Chapter III and the non-linear dynamic transient
analysis in Chapter IV shows that no residence was capable of resisting ground motions
represented by the amplified response spectra or the corresponding earthquake records. It
231
is recalled that for the non-linear dynamic transient analyses these residences were
evaluated in the strong direction. Therefore, the residences will definitively collapse in
their weak direction (as shown in the vulnerability analyses in Chapter III).
To define a practical response or target spectrum it is necessary to explain some
aspect about the amplified spectrums used in this investigation. The amplified spectrum
used in this investigation was obtained following the methodology explained in section
4.5 of Chapter IV of this investigation. But this procedure is developed assuming that the
peak ground acceleration (PGA) amplification factor can be applied directly to the design
spectra (i.e. Sb or Se UBC-97 soil type design spectra). Recall that the amplification
factors used in this investigation were those obtained in a previous investigation
developed by Arroyo (2001) for the PGA of the ground records. Thus, amplification on
the peak ground acceleration is assumed to cause the same amplification on the complete
response spectrum. This is a limitation of the approach: it is not known if the
amplification for different periods should be the same as the PGA amplification factor for
zero period. View to the lack of information it is assumed that the amplification factor
should be at least equal than the PGA amplification factor proposed by Arroyo for the top
of hills or escarpments. This is deemed to be a reasonable assumption for rock or hard
soil but, from a geological point of view, the amplification factor cannot be applied
directly for a softer soil (for example the UBC-97 Se soil type).
To explain this, consider the picture in Figure 6.1 depicting a cross section of a
hard soil (rock) mountain. It is well known that mountains and continents were formed
232
by plate tectonics due to the collision of plates and some other geological phenomena.
Therefore, there is no doubt that the topographic amplification can occur on pure rocky
mountains. However, in almost all mountains with softer soils, there is below the surface
a rocky core (see Figure 6.1) that makes too conservative the results for the “pure” soft
soil mountain obtained in the previous investigation. Although, this kind of soft soil
mountains exist, it is not a common situation and thus from a practical point of view, the
amplified soft soil or Se spectrum is considered to be too conservative. Therefore the
amplified spectrum for Sb soil type was selected in this investigation as the demand
spectrum that the retrofitting system is required to withstand.
Figure 6.1: Cross section of a hard soil mountain (left) and a soft soil mountain (right)
6.3 Selection of the Rehabilitation Strategy
As mentioned before, the Capacity Spectrum Method is a procedure to evaluate a
nonlinear static response of a framework that uses the intersection of the capacity
(pushover curve) and a reduced response spectrum to estimate the maximum
displacement that the structure can achieve. The graphical representation of these graphs
233
in an ADRS (Acceleration – Displacement Response Spectra) format provides a clear
picture of how a building responds to a seismic event. This methodology can be used as
a tool for the selection of the rehabilitation technique: it is based on the understanding of
the capacity-spectrum plot itself. Figure 6.2 presents the capacity spectrum plot for a
non-satisfactory structural system (residence S1a in this case). It is known that the
intersection between the capacity and the ground spectrum is the performing point of the
structural system. When such intersection is not achieved, like in Figure 6.2 for the
amplified spectrum, it can be said that the structure is expected to fail under the event or
demand that is represented by the spectrum.
In order to use the capacity spectrum method as a tool for the selection of a
rehabilitation technique, there are five features that can be observed from the capacity
plots in the ADRS format. From the Capacity Spectrum in the ADRS format (Figure 6.3)
one can observe that:
1. A movement of the performance point in the horizontal direction to the right
represents an increase in ultimate displacements or ductility (ductility
enhancement).
2. A movement in the horizontal to the left represents a decrease in ductility or in
the ultimate displacement (ductility reduction).
3. A movement in the vertical upward direction represents an increase in capacity
(strengthening).
234
4. A movement in the vertical downward direction represents a decrease in capacity
(softening).
5. The slopes of lines radiating from the origin of the ADRS plot represent lines of
constant period T.
In theory the most economical rehabilitation strategy will depend on a combination of
technical and non-technical considerations such as (Badoux 1998):
1. The cost of the retrofit.
2. The aesthetic impact on the structure.
3. The disruption of the structure during and after the retrofitting work.
However, from the point of view of the seismic performance, the most efficient
rehabilitation system is the one that moves the capacity faster to meet the demand in the
ADRS plot (i.e. a distance perpendicular to the demand spectrum). Nevertheless, it is not
always possible to obtain this “shortcut” with the current rehabilitation techniques
available.
A rehabilitation strategy consists in the implementation of constructive measures
or the addition of structural elements or other system that decreases the deficiencies and
improves the seismic response of the structure. For example, any element or measures
that increase the ductility (ductility enhancement), or increase the capacity
(strengthening) or both, are candidates for a rehabilitation system to satisfy the demand
spectrum. Obviously, the retrofitting system will depend also on the actual condition of
the structure (low or high ductility or strength). Although it is common to equate the
235
term “retrofitting” to “strengthening and stiffening”, the term seismic retrofitting has a
wider range of possible strategies that should be considered (Badoux 1998). The most
practical rehabilitation techniques are those that provide a combination of ductility
enhancement, stiffening, strengthening, etc. The following four strategies were selected
for this investigation and they are presented in graphical form in Figure 6.3:
1. Seismic demand reduction
2. Ductility enhancement
3. Strengthening and stiffening with ductility enhancement
4. Strengthening and stiffening
A seismic demand reduction can be obtained by the installation of base isolators
that diminish the overall stiffness of the structural system causing to reduce the natural
periods but increase dramatically the ultimate displacements. Alternatively, one can
install damping devices (in the form of viscous or viscous-elastic dampers) that increase
the damping of the structure, thus decreasing the demand spectrum. From an economical
point of view, the implementation of these strategies is totally impractical, due to the
economic conditions of the owners of these residences in Puerto Rico. Moreover, these
devices are anyway quite expensive and not justifiable for typical residential
constructions anywhere.
The ductility enhancement consists in increasing the ductility of the structural
system but without significant strengthening of the structure. This enhancement can be
obtained by the implementation of column jacketing with various materials. As a result
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of the columns jacketing, the elements are capable to maintain their resistance under
higher lateral deformations. However, for the residence S1a considered in Figure 6.2, the
performance point that needs to be reached is way too far for any practical column
jacketing (approximately 2.75 times the ultimate displacement). In addition one should
recall that the residence S1a is the strongest residence of the extreme residences.
Moreover, the ultimate displacement obtained in the static non-linear pushover performed
is expected to be greater than the real value due to the assumptions of ductility in the
program SAP2000.
As it was shown in the non-linear dynamic transient analyses presented in the
previous chapter, all the extreme residences developed a soft story collapse mechanism
due to the lack of compliance whit the strong column-weak beam criterion in their
design. However, the strengthening and stiffening with ductility enhancement can
change this behavior. The implementation of steel bracing can increase the strengthening
and stiffening of the structure and a ductility enhancement can be obtained by weakening
the beams. This combination can produce a suitable rehabilitation technique that satisfies
well the demand, as shown in Figure 6.3. However, the lack of transverse reinforcement
at the joints and elements of the residences represents a critical aspect for the connection
of the steel bracings. In addition, the cost of the steel bracings and the procedure to
weaken the beams are very expensive.
As mentioned before, moving to the left and up in the ADRS plots represents an
increase in ductility and strengthening, respectively. Stiffening can be defined as a
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combination of these two. Since the initial slope of the capacity plot is the stiffness of the
structure, an increase or decrease in this property can result in a good rehabilitation
technique. Evidently, an increase in the stiffness or decrease in the period of the structure
will produce an increment in the strength of the structure. A good rehabilitation
technique that can produce stiffening and strengthening is the use of interior shear walls.
This rehabilitation system can also reduce the ultimate displacements when compared
with the original structural system. Furthermore, shear walls can also help to decrease
the maximum displacements and inter story drift, the maximum element rotation as well
as to avoid the soft story collapse mechanism. From Figure 6.3 one can observe that by
properly designing the rehabilitation system it is possible that the ultimate displacement
of the original structural system may not exceeded. In addition, this is a conservative
rehabilitation system because the residential structures are not ductile and are not capable
of developing large displacements. Therefore, the reinforced concrete structural walls
were selected as the rehabilitation system that is best suited for the deficiencies of the
residences.
The reinforced concrete structural walls were chosen as the rehabilitation
technique for the weak direction of the residences since the example in Figure 6.2
corresponds to residence S1a (stronger in the weak direction). The decision was made
because of the low ductility, stiffness and capacity that the residences show in the weak
direction and the R/C shear walls take care of all these deficiencies. The R/C structural
wall is a rehabilitation technique that increases both the stiffness and capacity of the
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structure, and it is simpler to build and connect to the existing structure than the other
alternatives. Based on practical considerations, the same rehabilitation technique was
selected for the strong direction. Since the contractors are going to built walls in the
weak direction, it is easy to use walls in the other direction because they already have the
frameworks, workers and material for the walls.
6.4 Rehabilitation technique implementation
In Chapter V of this investigation a series of Failure Criteria (drift limits FCD,
maximum element rotation FCR, maximum element forces FCEF, collapse mechanism
identification FCCM, FCK and FCT) were implemented for the evaluation of the non-
linear dynamic response of the residences. It is the author’s opinions that all of these
failure criteria are better indicator of the seismic performance of the residences than the
monotonic static non-linear pushover needed for the capacity spectrum method. Based
on the non-linear dynamic vulnerability evaluation, the same methodology as established
in section 5.5 of Chapter V with a few minor differences was implemented for the
numerical evaluation of the seismic rehabilitation system in an iterative process. The
process consists in the addition of reinforced concrete shear walls to the model, changing
in some dynamic parameters (i.e. mass), performing the non-linear dynamic transient
analyses, and verifying all the failure indicators mentioned above.
For the predominant cross sections of the residences of the field survey, a series
of tables was developed and detailed for different span lengths and column heights to
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obtain the structural R/C walls that pass all the FC. These tables were developed for both
the strong and the weak direction and for one and two R/C structural walls.
The objective of the tables is that the users can enter with their cross sectional
dimensions, span length and column length and obtain the R/C structural wall that can be
used to retrofit their residence. The rehabilitation system for each particular case was
verified case using the Non-Linear Dynamic Transient Analysis with the program
LARZWS/D and capable of resisting earthquakes described by the UBC-97 Sb soil type
with topographic amplification included. The outputs of LARZWS/D were post-
processed and evaluated by the program developed by the author for the evaluation of the
non-linear dynamic transient analyses (Vázquez, 2002).
6.5 Assumptions for the development of the rehabilitation system tables
Some assumptions were made, for the evaluation of the proposed retrofitting
system for the residences. The most important assumption is that the structural R/C wall
is the only structural system that resists the lateral earthquake loads. This was done by
attributing all the first floor mass completely to the structural wall. Therefore, the
dynamic loading will be stronger on the walls. Another assumption is that the second
floor of the residences will suffer structural and non-structural damage. This can be
expected because the addition of the structural wall can cause to move the soft story
collapse mechanism from the first floor to second floor. It was considered that this can
be permitted because there is significant redundancy in the second floor of the residences,
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due to masonry walls and other structural elements that can help to decrease the seismic
response of this floor and are not considered in the analyses. Furthermore, the collapse
mechanism criterion was not used as a failure or collapse indicator in the analyses of the
second floor.
Other assumptions made in the evaluation of the R/C walls deal with the limits of
the failure criteria in terms of the FCK and FCT limits. For these particular residences
and from the non-linear dynamic transient analyses carried out in Chapter V, it was found
that an upper limit of 4 and a lower limit of 2% for the FCT and FCK values,
respectively, are good indicator of collapse or damage. Instead of obtaining the values
from a monotonic static pushover as in Chapter V, these two values were used as
indicators of collapse for the non-linear dynamic analyses of the retrofitted structures.
Also, a collapse mechanism at the first floor will control the static pushover and this
criterion is not considered in this evaluation.
The failure criteria limit for the FCD is 2.5% and the remaining limits (i.e. the
maximum element rotation FCR, maximum element moments FCEM and maximum
element shear FCES) depend on the cross sections of the structural elements. As
established in the FC methodology of Chapter V, all of these failure criteria were verified
at each time step for each particular residence retrofitted with the corresponding R/C
wall.
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6.6 Table with Reinforced Concrete Structural Walls
The Tables 6.1 to 6.3 were developed by an iterative process consisting in adding
one or two R/C structural walls to the residences, performing the non-linear dynamic
analyses and verifying all the failure criteria. If one of the failure criteria was achieved,
the longitudinal steel reinforcement or thickness of the walls was increased and then
analyzed again. The procedure continues up to the selection of a R/C walls that pass all
the failure criteria. The retrofitting tables were developed for the weak and strong
direction following the assumptions made in section 6.5.
As mentioned before, the retrofitting tables were developed for different cross
sections, span lengths and column heights of the first floor. It was shown (Chapter II)
that the second floor of these residences is 9 ft tall for all the residences in the field
survey. The summary of the columns heights, elements cross sections, element sizes, and
span lengths were presented in Chapter II in Tables 2.26, 2.27, and 2.28, respectively.
From Table 2.26 and from the residence data presented in Tables 2.2 to 2.25, column
heights varying from 10 to 16 ft were selected for the small to medium cross sections
(6”x18” to 8”x16”) and heights from 13 to 20 ft were used for the larger cross sections
(12”x12”). Similarly, the span lengths for the small to medium sections were varied from
10 to 16 ft and from 13 to 20 ft for the larger sections.
Basically, the tables are developed for all the element cross section measured in
the field survey and they cover all the span lengths and column heights measured. One
enters the table with the column height and the span length of the residence and obtain
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the R/C structural wall that should be built to rehabilitate the residence. Table 6.1 was
developed for the weak direction of the residences and Tables 6.2 and 6.3 were developed
for the strong direction of the residences.
6.6.1 Retrofit tables for the weak direction
It was noticed during the field survey that some of these residences did not have
beams in the weak direction, aggravating the problem. Figure 6.4 presents a photograph
of one of the residences measured in the field survey that do not have beams in the weak
direction of the structural elements.
From a practical seismic design point of view, a conservative design must have at
least two lateral resisting plane, to increase the structure’s redundancy. Following this
line of thought, two R/C structural walls were implemented for the retrofitting system in
the weak direction. In other words, for the development of the tables for the weak
direction of the residences, two R/C structural walls were used as the retrofitting system
in this direction. The set of tables for the residences in the weak direction are presented
in Table 6.1
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Table 6.1: Retrofitting tables for the weak direction (two R/C walls) (1 in = 25.4 mm; 1
ft = 305 mm)
10 13 1610 6 in Wall #3 Bars @ 12 in BD 6 in Wall #3 Bars @ 12 in BD 6 in Wall #3 Bars @ 12 in BD13 6 in Wall #3 Bars @ 12 in BD 6 in Wall #3 Bars @ 12 in BD 6 in Wall #3 Bars @ 12 in BD16 6 in Wall #4 Bars @ 12 in BD 6 in Wall #4 Bars @ 12 in BD 6 in Wall #5 Bars @ 12 in BD
10 13 1612 6 in Wall #3 Bars @ 12 in BD 6 in Wall #3 Bars @ 12 in BD 6 in Wall #3 Bars @ 12 in BD14 6 in Wall #3 Bars @ 12 in BD 6 in Wall #3 Bars @ 12 in BD 6 in Wall #3 Bars @ 12 in BD16 6 in Wall #3 Bars @ 12 in BD 6 in Wall #3 Bars @ 12 in BD 6 in Wall #3 Bars @ 12 in BD
16
14 17 2012 6 in Wall #4 Bars @ 12 in BD 6 in Wall #4 Bars @ 12 in BD 6 in Wall #4 Bars @ 12 in BD14 6 in Wall #4 Bars @ 12 in BD 6 in Wall #4 Bars @ 12 in BD 6 in Wall #4 Bars @ 12 in BD16 6 in Wall #3 Bars @ 12 in BD 6 in Wall #4 Bars @ 12 in BD 6 in Wall #4 Bars @ 12 in BD
Span [ft]
Columns 12"x12" Beams 12"x17"
Height [ft]
Span [ft]
Columns 6"x18" Beams 6"x17"
Height [ft]
Span [ft]
Columns 8'x16" Beams 8"x17"
Height [ft]
6.6.2 Retrofit tables for the strong direction
Two sets of tables were developed for the strong direction of the residences. In
the first set of tables (Table 6.2) one structural wall was used as the lateral resisting
system. The other set of tables (Table 6.3) was prepared assuming that two R/C
structural walls comprise the lateral resisting system or rehabilitation system.
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Table 6.2: Retrofitting tables for the strong direction (one R/C walls) (1 in = 25.4 mm; 1
ft = 305 mm)
10 13 1610 6 in Wall #4 Bars @ 12 in BD 6 in Wall #5 Bars @ 12 in BD 6 in Wall #5 Bars @ 12 in BD13 6 in Wall #4 Bars @ 12 in BD 6 in Wall #4 Bars @ 12 in BD 6 in Wall #5 Bars @ 12 in BD16 6 in Wall #3 Bars @ 12 in BD 6 in Wall #3 Bars @ 12 in BD 6 in Wall #4 Bars @ 12 in BD
10 13 1612 6 in Wall #4 Bars @ 12 in BD 6 in Wall #4 Bars @ 12 in BD 6 in Wall #4 Bars @ 12 in BD14 6 in Wall #4 Bars @ 12 in BD 6 in Wall #4 Bars @ 12 in BD 6 in Wall #4 Bars @ 12 in BD16 6 in Wall #4 Bars @ 12 in BD 6 in Wall #4 Bars @ 12 in BD 6 in Wall #5 Bars @ 12 in BD
16
14 17 2012 6 in Wall #5 Bars @ 12 in BD 6 in Wall #5 Bars @ 12 in BD 6 in Wall #5 Bars @ 12 in BD14 6 in Wall #4 Bars @ 12 in BD 6 in Wall #5 Bars @ 12 in BD 6 in Wall #5 Bars @ 12 in BD16 6 in Wall #5 Bars @ 12 in BD 6 in Wall #5 Bars @ 12 in BD 6 in Wall #5 Bars @ 12 in BD
Span [ft]
Height [ft]Columns 6"x18" Beams 6"x17"
Span [ft]
Columns 8'x16" Beams 8"x17"
Height [ft]
Span [ft]
Columns 12"x12" Beams 12"x17"
Height [ft]
Table 6.3: Retrofitting tables for the strong direction (two R/C walls) (1 in = 25.4 mm; 1
ft = 305 mm)
10 13 1610 6 in Wall #3 Bars @ 12 in BD 6 in Wall #3 Bars @ 12 in BD 6 in Wall #3 Bars @ 12 in BD13 6 in Wall #3 Bars @ 12 in BD 6 in Wall #3 Bars @ 12 in BD 6 in Wall #3 Bars @ 12 in BD16 6 in Wall #4 Bars @ 12 in BD 6 in Wall #4 Bars @ 12 in BD 6 in Wall #5 Bars @ 12 in BD
10 13 1612 6 in Wall #3 Bars @ 12 in BD 6 in Wall #3 Bars @ 12 in BD 6 in Wall #3 Bars @ 12 in BD14 6 in Wall #3 Bars @ 12 in BD 6 in Wall #3 Bars @ 12 in BD 6 in Wall #3 Bars @ 12 in BD16 6 in Wall #3 Bars @ 12 in BD 6 in Wall #3 Bars @ 12 in BD 6 in Wall #3 Bars @ 12 in BD
16
14 17 2012 6 in Wall #4 Bars @ 12 in BD 6 in Wall #4 Bars @ 12 in BD 6 in Wall #4 Bars @ 12 in BD14 6 in Wall #4 Bars @ 12 in BD 6 in Wall #4 Bars @ 12 in BD 6 in Wall #4 Bars @ 12 in BD16 6 in Wall #3 Bars @ 12 in BD 6 in Wall #4 Bars @ 12 in BD 6 in Wall #4 Bars @ 12 in BD
Span [ft]
Columns 12"x12" Beams 12"x17"
Height [ft]
Span [ft]
Columns 6"x18" Beams 6"x17"
Height [ft]
Span [ft]
Columns 8'x16" Beams 8"x17"
Height [ft]
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0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
Spectral Displacement [in]
Spec
tral
Acc
eler
atio
n [%
g]
Sb Soil Capacity Sb Soil Amplified
Figure 6.2: Capacity Spectrum for Residence S1a and Sb soil type amplified spectra
Figure 6.3: Rehabilitation strategies
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Figure 6.4: Residences without beams in the weak direction
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CHAPTER VII
LIMITATIONS, SPECIFICATIONS AND EXAMPLES
7.1 Introduction
This chapter presents limitations and minimum specification for a safe
implementation of the retrofitting tables or structural R/C walls. Some examples
explaining the implementation of the developed retrofitting tables are also illustrated.
Two particular residences of the field survey are retrofitted as examples for these
purposes. A set of structural specifications, structural details, bonding specifications and
some geotechnical assumptions are explained in this chapter, as minimum requirement to
accomplish a safe retrofitting system. Limitations of the use of these retrofitting tables
and structural and connection details are also included in this chapter.
7.2 Specifications and Recommendations
Basically the specification for the uses of the tables can be divided in three
subdivisions: structural specifications, bonding specifications and soil specifications.
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7.2.1 Structural Specifications and Recommendations
Listed below are the minimum material properties to be specified for the
rehabilitation of the residences. The non-linear dynamic transient analyses performed for
the selection of the R/C structural walls were used these material properties.
1. A concrete compressive strength (f’c) of 3,000 psi (based on a typical concrete
compressive stress) should be used for the R/C walls and footings.
2. The minimum yield stress (Fy) for the longitudinal and transverse reinforcing
steel should not be less than 60,000 psi.
3. The development length for the vertical reinforcing steel (penetrating into the
foundation) should be not less than (See Figure 7.1 and 7.2):
a. 12 in for #3 bar with standard hook 3 in diameter of bend.
b. 16 in for #4 bar with standard hook 3 in diameter of bend.
4. Structural walls should be anchored to the existing adjacent columns and
beams by means of rebar dowels spaced at a maximum distance of 6 inches
center to center (see Figure 7.1).
5. The reinforcement dowels at the connection should have a development length
of 24 inches penetrating into the new structural wall (see Figure 7.1).
6. Epoxy or bonding specifications are given in section 7.2.3 (See Figure 7.1).
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7.2.2 Soil Specifications and Recommendations
The following properties of the soil are the minimum properties for the correct
implementation of the retrofitting system when using the two walls retrofitting tables
(Tables 6.1 and 6.3) for both directions. These specifications are for the use of the
detailed structural footing (Figures 7.2 and 7.3). The structural engineer can implement a
foundation design greater or smaller by providing calculations or analyses based on in
situ soil characteristics.
1. Soil allowable bearing pressure (qa) not less than 2000 psf.
2. Modulus of subgrade reaction of soil not less than 150 kcf.
3. Soil unit weight of 110 pounds per cubic foot.
4. The factor of safety was limited to 1.5.
When using one structural wall (Table 6.2) for the strong direction the following
limitation apply:
1. Soil allowable bearing pressure (qa) not less than 4000 psf.
2. Modulus of subgrade reaction of soil not less than 900 kcf.
3. Soil unit weight of 110 pounds per cubic foot.
4. Factor of safety diminished to 1.2.
If the problem does not comply with any of these limitations, two walls should be used or
a detailed analysis should be performed.
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7.2.3 Bonding Specifications or Recommendations
The following recommendations should be implemented for the connection
between the proposed structural walls and the existing gravity columns and beams.
1. Epoxy Resin Adhesive for Bonding Plastic Concrete to Hardened Concrete –
Class I. High Strength Epoxy Tie could be implemented also.
a. Minimum allowable tension and shear load of 4,000 pounds
b. Ultimate Tension load or bond strength of 16,000 pounds.
c. The epoxy has to be specified by the manufacturer as a cyclic loading
tested epoxy. (i.e. earthquake and wind loads)
2. The following specifications are for reinforcement bar #4 at the connection
between the existing column or beam and the wall. Use manufacturers’
specification if higher embedment length, edge distance and drill bit
diameters are provided.
a. An embedment depth of 5 inches should be entered into the existing
columns and beam at 6 in spacing (see Figure 7.1).
b. A minimum edge distance of 3 inches for the placement of the rebar
should be provided.
251
c. A drill bit of 5/8 in diameter should be used for the rebar holes at
columns and beams.
d. A tension development length of 24 inches penetrating into the
structural walls (Figure 7.1) should be provided.
3. Preparation of the surface to be repaired.
a. Remove the concrete finishing of the columns and beams using
hammers, jack hammers or chipping hammers leaving a rough
surface.
b. Before placing the fresh concrete, clean the concrete surface and the
holes of loose or foreign material and other contaminants with an oil
free compressed air. Clean the holes with a nylon brush and blow out
the remaining dust. Dust left in rebar hole can reduce the epoxy
adhesive’s holding capacity.
c. Fill the holes with the Epoxy Resin according to the specifications or
the manufacturers written recommendations.
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7.3 Limitations
The uses of the Tables 6.1 to 6.3 are limited to a maximum height of twenty (20)
feet and span length less than sixteen (16) feet. Residences with three or more stories
should not be retrofitted with the structural walls presented in these tables. Residences
with more than two spans can be retrofitted by subdividing the residences in sections
consisting in two spans or three “frames”. Then the selection of the appropriated R/C
wall for each subdivision can be obtained from the tables.
Elements cross sections considerably different to the specified should be treated
carefully. If the element cross section is between some of the sections tabulated use the
table with the smaller section. When the span or the column height is not tabulated, use
the next longer or higher span and column height tabulated.
The implementation of the R/C walls has to be verified by a professional
structural engineer. The author spent considerable time and effort for the development
and documentation of the retrofitting tables and analyses. When using the retrofitting
tables, the engineer should understand the assumptions and limitations in their
development and know that no warranty is expressed or implied by the author on the
accuracy or the reliability of the retrofitting system. The engineer or contractor is
responsible for the implementation of the retrofitting system. The author reserves all of
the rights about the implementation of this investigation and/or reference to this
investigation.
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The structural footings were designed according to the soil assumptions made in
section 7.2.2. Also the lateral load applied for the structural design was obtained as the
80% of the maximum base shear obtained in the non-linear dynamic transient analyses.
The structural table developed for the implementation of one structural wall in the strong
direction is not recommended for soft soils as established in section 7.2.2.
7.4 Retrofitting Examples
This section presents a procedure for the selection of the structural walls by
means of two examples of residences obtained in the field survey. The first residence to
be retrofitted is presented in Figure 2.4 of Chapter II. The parameters (span, height,
element sizes, etc.) of the residences are tabulated in Table 2.18 of the same chapter. The
procedure consists in the following steps:
1. Obtain or measure the element cross sections and orientation.
2. Obtain or measure the span lengths, and columns heights for both direction of
the residence. The height of the columns should be taken from the foundation
to the second floor.
3. Create a drawing summarizing the previous data. Figure 7.4 shows the
drawing sketch for this particular residence.
4. The selection of the wall for the weak direction is obtained from Table 6.1.
Recall that two walls are needed for each two span section. Enter the table
using the element sections, span and column height. See Figure 7.5.
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5. For the strong direction, two tables were developed. Table 6.2 considers only
one structural wall for each two span section and Table 6.3 is for using two
structural walls for each two span section. Similarly, enter with the element
section, span and height to obtain the desired retrofitting system.
6. It is recommended that the walls be located at the extreme of each section for
the two wall tables. The walls selected can be of interior (Table 6.2) and
exterior walls (Table 6.3). Possible locations and sizing of the retrofitting
systems are presented in Figure 7.5. Black walls are the selected location for
the structural system and gray walls are other possible locations.
The previous procedure was followed for the residence of Example two. The data for this
residences is presented in Table 2.26 and is shown in Figure 2.8. Figure 7.6 shows the
element sizes, span length, column height as well the procedure for the selection and
implementation of the retrofitting walls. For the weak direction, 6 in walls with #3 bars
@ 12 in horizontal and vertical are selected. For the strong direction, a scheme
consisting of 6 in exterior walls reinforced with #3@12 and 6in interior wall reinforced
with #4@12 was obtained.
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Figure 7.1: Connection between existing column and beam with structural wall
256
Figure 7.2: Section view of structural wall and footing
257
Figure 7.3: Plan view of wall and footing
Figure 7.4: Schematic drawing for Example 1
258
Figure 7.5: Plan view of structural walls for weak direction, strong direction using one
wall and strong direction using two walls from left to right.
259
Figure 7.6: Rehabilitation system for Example 2
260
CHAPTER VIII
SUMMARY, CONCLUSIONS AND RECOMMENDATIONS
8.1 Introduction
In the first part of this investigation a field survey was performed across the main
Island of Puerto Rico to obtain the most typical parameters of residences located on hills
or escarpments. With the parameters obtained in the field survey, a series of different
analyses (Static Pushover Analyses and Non-linear Dynamic Transient Analyses) were
performed to verify the seismic vulnerability of these residences including the
topographic effects. The effect of the local topographic effects was taken into account by
means of factors obtained in the investigation by Arroyo (2001). After the vulnerability
evaluation was performed, a rehabilitation technique consisting of R/C structural walls
was selected and tested using Non-linear Dynamic Transient Analyses. Also, sets of
tables that allow the user to choose the proper rehabilitation system for a specific
residence were prepared for different cross sections, span lengths and column heights.
Some recommendations for the design of new residences located along hill or escarpment
was also presented.
8.2 The Field Survey
A total of 24 residences were visited, measured in situ and classified.
Representative parameters like element sizes, longitudinal steel reinforcement, span
261
lengths and column heights were obtained from this field survey. The field survey
showed that the predominant span length varies from 10 to 14 ft and the predominant
columns height varies from 10 to 16 ft. In addition, the predominant element sections are
cross sections of 6”x18” and 8”x16”. With the parameters obtained in the field survey,
two extreme cases of the typical residences (stiffer and most flexible one) were defined.
An intermediate stiffness case was also added. The vulnerability of a total of 12
residences was evaluated using the Capacity Spectrum Method.
8.3 Vulnerability Analysis using the Capacity Demand Spectrum
To verify the seismic vulnerability of the extreme residences, the Capacity
Spectrum Method was applied. A total of 24 residences (12 in the weak and 12 in strong
direction) were evaluated using this methodology. Eleven of the twelve residences
presented failure or collapse when evaluated using this methodology and the UBC-97 Sb
soil type spectrum. The remaining residence also fails when evaluated for the spectrum
for soil type Se. It is necessary to point out that all the residences showed failure or
collapse when they were evaluated using the UBC-97 design spectra and there was no
consideration of the topographic amplifications. Thus, there is a real need to implement a
rehabilitation system to improve the seismic behavior of these residences.
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8.4 Amplified spectra and earthquake records
One of the main objectives of this investigation was to verify the behavior of the
typical residences on hilly terrain subjected to amplified ground motions. In the previous
investigation developed by Arroyo (2001), the amplification of seismic waves reaching
hills and escarpments was studied numerically. The amplification was defined by a
factor that is applied to the PGA. The maximum value of the amplification factors found
among the many different configurations and soils examined was 2.35. In the present
investigation this amplification factor was implemented in a simple approach (see
Chapter IV) to obtain the corresponding amplified Sb and Se spectra for soil types. After
obtaining the amplified UBC-97 Sb and Se soil type spectra, artificially generated
earthquake records were produced to perform non-linear analyses of the residences to
observe their behavior. Also a graphic user interface was developed for the SIMQKE
program that generates the artificial records (Vázquez, 2001).
8.5 Non-linear dynamic transient analyses
From the vulnerability analyses it was shown that the residences need a retrofit
system. Although, the capacity spectrum method is a widely accepted method for the
evaluation of structures, in the case of failure or collapse, more sophisticated analyses
should be used to verify the authenticity of the failure. So the more advanced Non-linear
Dynamic Transient Analyses was used to verify again the seismic vulnerability of the
residences. To do this it was necessary to develop a Failure Criteria Methodology
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(Chapter V), based on the most common and practical failure indicators but including two
new indicators proposed by the author. These two criteria or collapse indicators are the
FCK (Failure Criteria of the Stiffness Matrix Determinant) and FCT (Failure Criteria of
First Period Ratio). All the failure criteria were monitored during all the analyses of the
residences.
The FCK and FCT values were monitored for all the extreme residence as well for
the designed residences. An evaluation of these failure criteria or indicators was
performed in terms of the FCCM (Failure Criteria of Collapse Mechanism) of the
residences (Chapter V). To do this all the residences that form a collapse mechanism
(“i.e. they are not capable of resisting the event”) were tabulated and from the tables the
limiting values of 3% and 4.5 were selected for the FCK and FCT, respectively. The
limit of the FCK means that when the stiffness matrix determinant of the structure goes
below the 3% level of the original or initial stiffness matrix determinant, there is
potentially a failure mostly due to a collapse mechanism or element degradation. The
same happens when the ratio of the first period in terms of the initial period, i.e. the FCT
indicator, is greater than 4.5.
Also from the non-linear analyses, the residences that do not form a collapse
mechanism (“i.e. they are capable of resisting the event”) show a “post earthquake” FCT
value from 1.5 to 2.0, and 10 to 20% for the FCK indicator. These FCT values basically
mean that the post-earthquake fundamental period of the structure is up to 2 times the
initial period. This is an important quantity because it can be used as an indicator of the
264
status of the structures after a seismic event. Also this value can be used by forensic
engineers for the selection of a retrofitting system for a structure that survives an event,
because it is a good indicator of the structure condition. The same applies to the stiffness
matrix determinant, but this value does not represent directly a typical seismic parameter
like periods.
As shown in the Capacity Spectrum Method, all the residences studied collapse
when subjected to the earthquake records with and without topographic amplification.
The designed residences also show failure for the Se soil amplified records but the
residence identified as R4 was able to resist the amplified earthquake record for Sb soil.
Therefore, in principle there is a need of a retrofitting system for the existing residences.
For the new designed residence, these results indicate that a modified design spectrum
that takes into account the topographic amplification effects should be used to define the
seismic input. Also a GUI for the non-linear dynamic transient analyses performed by
LARW to evaluate the Failure Criteria Methodology was developed.
8.6 Retrofitting system and tables
It was shown in Chapter VI of this investigation that the most practical
rehabilitation system to take care of the deficiencies of the typical residences is the
reinforced concrete structural walls. So a series of tables was developed in terms of the
element cross section, column heights and span lengths to obtain the R/C wall that fits a
265
particular residence. The structural walls in these tables (Chapter VI) have dimensions
such that they pass the FCK and FCT criteria as well as all the other failure criteria or
indicators used in the non-linear evaluation (drift, maximum rotation, maximum element
forces, etc.).
8.7 Recommendation for the seismic design of residence located at hills or
escarpments
From the non-linear dynamic transient analyses of the residences it was shown
that residence R4 was capable of resisting the Sb soil amplified artificial earthquake.
Going back to the design of this residence, it was designed as an Intermediate Moment
Resisting Frame on a Se soil type as shown in Table 8.1.
Table 8.1: Seismic parameter for the residences
Residence Structural System R Soil Type Ca Cv V [%W]1 SMRF 8.5 Sb 0.3 0.3 0.0882 SMRF 8.5 Se 0.36 0.84 0.1063 IMRF 5.5 Sb 0.3 0.3 0.1364 IMRF 5.5 Se 0.36 0.84 0.164
The Strength Reduction Factor R for Residence 4 corresponds to 5.5, the Seismic
Coefficient Cv had a value of 0.86 and the equivalent base shear was 0.164W for this soil
and lateral resisting system. In the engineering practice, it is common to design all of
these residences assuming that the structure falls in the plateau of the design spectrum.
This assumption is reasonable since these residences do not have very large periods. At
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the plateau, the base shear is independent of the seismic coefficient Cv and only depends
on Ca and R. If we select the 0.16W (Residence 4 in Table 8.1) as the total base shear,
one can observe that this value is almost twice (1.86 times) the value for Residence 1 (for
which a special moment resisting frame with R = 8.5). Since the values of the Ca
coefficients are similar, an amplification factor of 2 to include the topographic
amplification is a reasonable value for the special moment resisting frame. So from a
practical point of view a Strength reduction factor of 4.5 (or a factor of 2 in the base shear
equation) should be used for the design of these residences regardless of the soil type.
The ductility requirement for a Special Moment Resisting Frame should be implemented
in all of the residences designs.
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