phd thesis summary
TRANSCRIPT
Universitatea Tehnică de Construcții București
TECHNICAL UNIVERSITY OF CIVIL ENGINEERING BUCHAREST
Faculty of Civil, Industrial and Agricultural Buildings
Ph.D. THESIS Summary
Contributions towards improving design procedures for reinforced concrete shear
walls
Ph.D. Student
Eugen MORARIU, Eng. Ph.D. Coordinator
Tudor POSTELNICU, Ph. D. Prof. Eng.
BUCHAREST
2012
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ACKNOWLEDGEMENTS
First and foremost, I would like to express my full gratitude to my doctorate coordinator, Prof. Tudor Postelnicu, for the support, help and continuous guidance. I also want to thank him for the patience shown to bring the thesis to the present format.
For his time, scientific discussions and valuable ideas he gave me, I would like to show my sincere appreciation to As. Prof. Dan Zamfirescu. Also, I want to thank for his help and support in obtaining the research period at the University of Ljubljana, Slovenia.
I am highly indebted to Prof. Matej Fischinger from University of Ljbubljana for his guidance and constant supervision, as well as for the research period at IKPIR. This period was a step forward in my professional development providing me a direct link to international research.
I would like to express my special gratitude and thanks to Prof. Radu Pascu for the attention and interest shown towards my research and for his much welcomed interventions.
My thanks and appreciations also go to my friend Ionut Damian for the endless discussions and for his support in completing the thesis.
In addition, I would like to thank Prof. Radu Perovici and As. Prof. Daniel Dan for their willingness, able guidance and useful suggestions, which helped me in completing my research.
Finally, I would like to thank my wife for her understanding and patience, and also my parents and brother for their constant support.
KEYWORDS Inelastic shear force, shear force amplification, shear force distribution, isolated
walls, coupled walls, dual structures.
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CONTENT
1. INTRODUCTION................................................................................................................................. 18
1.1 Evolution of knowledge in seismic design of buildings with reinforced concrete walls ...........................................18
1.2 Thesis content ................................................................................................................................................................20
2. KNOWLEDGE LEVEL ON SEISMIC DESIGN OF STRUCTURES WITH REINFORCED CONCRETE WALLS ............................................................................................................................................... 22
2.1 Introduction .....................................................................................................................................................................22
2.2 Objectives .......................................................................................................................................................................22
2.3 Study organization ..........................................................................................................................................................23 2.3.1 Design codes considered ........................................................................................................................................23 2.3.2 Presentation of the analyzed structure .....................................................................................................................23
2.4 Comparative analysis of the provisions of the design codes considered ..................................................................24 2.4.1 Design resistances ..................................................................................................................................................24 2.4.2 Gravitational load in seismic design combination .....................................................................................................25 2.4.3 The structure of the equation for seismic force evaluation .......................................................................................26 2.4.4 Design seismic load combinations ...........................................................................................................................28 2.4.5 Evaluation of the design values of the sectional efforts (strength requirements) ......................................................28 2.4.6 Evaluation of the design values of the bending moment and shear resistatance ......................................................30
2.5 Overall structural safety level provided by the application of the codes considered ................................................32
2.6 Conclusions ....................................................................................................................................................................34
3. STUDY FOR THE CALIBRATION OF THE DESIGN SHEAR FORCES VALUES OF REINFORCED CONCRETE WALLS .......................................................................................................................... 36
3.1 Introduction .....................................................................................................................................................................36
3.2 Main objectives ...............................................................................................................................................................37
3.3 The general approach .....................................................................................................................................................37 3.3.1 Shear forces associated with “capacity design”, dynamic amplification of shear forces in the elastic range ............37 3.3.2 Dynamic amplification of shear forces in the inelastic range ....................................................................................39 3.3.3 Proposed equation ..................................................................................................................................................43 3.3.4 Comparinson of the proposed approach with codes of interest ................................................................................46
3.4 Parametric study objectives...........................................................................................................................................51
3.5 Parametric study presentation .......................................................................................................................................51
3.6 Study parameters............................................................................................................................................................51
3.7 Lateral force wall design ................................................................................................................................................53
3.8 Walls modeling ...............................................................................................................................................................55 3.8.1 Elastic modeling ......................................................................................................................................................55 3.8.2 Nonlinear modeling ..................................................................................................................................................56
3.9 Seismic imput and damping ...........................................................................................................................................71
3.10 Parametric study results ................................................................................................................................................75 3.10.1 Elastic analysis results.............................................................................................................................................75 3.10.2 Cheking and correction of the proposed equation for shear force evaluation at the walls base ................................84 3.10.3 Shear force distribution over walls height .................................................................................................................98
3.11 Conclusions .................................................................................................................................................................. 107
4. STUDY FOR THE CALIBRATION OF THE DESIGN SHEAR FORCES VALUES OF WALLS THAT ARE PART OF DUAL STRUCTURES ............................................................................................. 109
4.1 Study presentation ....................................................................................................................................................... 109
4.2 Study objectives ........................................................................................................................................................... 110
4.3 Particular problems ...................................................................................................................................................... 110
4.4 Study parameters.......................................................................................................................................................... 113
4.5 Lateral force structure design ...................................................................................................................................... 116
4.6 Structures modeling ..................................................................................................................................................... 116 4.6.1 Elastic modeling .................................................................................................................................................... 116 4.6.2 Nonlinear modeling ................................................................................................................................................ 117
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4.7 Seismic imput and damping ......................................................................................................................................... 121
4.8 Parametric study results .............................................................................................................................................. 122 4.8.1 Elastic analysis results........................................................................................................................................... 122 4.8.2 Cheking and correction of the proposed equation for shear force evaluation at the walls base .............................. 133 4.8.3 Cheking and correction of the proposed distribution of the shear forces over the walls height ............................... 143
4.9 Conclusions .................................................................................................................................................................. 147
5. STUDY FOR THE CALIBRATION OF THE DESIGN SHEAR FORCES VALUES OF COUPLED WALLS ............................................................................................................................................. 148
5.1 Study presentation ....................................................................................................................................................... 148
5.2 Study objectives ........................................................................................................................................................... 148
5.3 Particular problems ...................................................................................................................................................... 149
5.4 Study parameters.......................................................................................................................................................... 151
5.5 Lateral force couple systems design........................................................................................................................... 153
5.6 Coupled systems modeling.......................................................................................................................................... 154 5.6.1 Elastic modeling .................................................................................................................................................... 154 5.6.2 Nonlinear modeling ................................................................................................................................................ 154
5.7 Seismic imput and damping ......................................................................................................................................... 156
5.8 Parametric study results .............................................................................................................................................. 156 5.8.1 Elastic analysis results........................................................................................................................................... 156 5.8.2 Cheking and correction of the proposed equation for evaluation of the design shear force at the base of coupled walls
systems .............................................................................................................................................................. 166 5.8.3 Cheking of the base shear force proposed distribution methods between the two piers ......................................... 177
5.9 Conclusions .................................................................................................................................................................. 183
6. DESING RECOMMENDATIONS ...................................................................................................... 185
6.1 Design shear force values at the walls base ............................................................................................................... 185 6.1.1 Proposed equation ................................................................................................................................................ 185 6.1.2 Comparison of the proposed equation with national codes .................................................................................... 187
6.2 Shear force distribution ................................................................................................................................................ 191 6.2.1 Over the walls height ............................................................................................................................................. 191 6.2.2 At the base of coupled walls .................................................................................................................................. 192
7. CONCLUSIONS................................................................................................................................ 193
7.1 Thesis presentation ...................................................................................................................................................... 193
7.2 Chapters conclusions .................................................................................................................................................. 195 7.2.1 Chapter 2 .............................................................................................................................................................. 195 7.2.2 Chapter 3 .............................................................................................................................................................. 196 7.2.3 Chapter 4 .............................................................................................................................................................. 197 7.2.4 Chapter 5 .............................................................................................................................................................. 198
7.3 Final conclusions.......................................................................................................................................................... 199
7.4 Personale contributions ............................................................................................................................................... 200
7.5 Future research ............................................................................................................................................................. 201
8. REFERENCES.................................................................................................................................. 202
9. BIBLIOGRAPHY............................................................................................................................... 205
ANNEX A – DETAILED COMPARATIVE STUDY PRESENTED IN CHAPTER 2 ................................. 210
ANNEX B – 2D ANALYSIS PROGRAM OF ISOLATED CONCRETE WALLS (ELECTRONIC FORMAT)
ANNEX C – 2D ANALYSIS PROGRAM OF DUAL STRUCTURES (ELECTRONIC FORMAT)
ANEXA D – 2D ANALYSIS PROGRAM OF COUPLED WALLS SYSTEMS (ELECTRONIC FORMAT)
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1. THESIS PRESENTATION
The main objective of the study is to calibrate the design values of shear forces, in isolated or coupled reinforced concrete walls of multistory buildings, in which these elements are, alone or in combination with reinforced concrete frames (dual systems), the lateral seismic structure. The goal is to fully cover the application range of reinforced concrete walls in engineering practice.
In order to identify different approaches to the design of reinforced concrete walls, a comparison of the provisions for the design of structures with walls of the most important design codes worldwide was conducted in Chapter 2. A comprehensive comparative study of the results of applying the provisions of European codes, Romanian ones, those in the U.S. and New Zealand, countries where seismic engineering is the most advanced, was developed. The chapter ends with conclusions in order to clarify the structural safety level obtained by applying national codes in relation to the safety level obtained by applying other codes, at the end of Chapter 2 proposals are made for the amending of certain provisions of the Romanian code currently under review.
The calibration of the values of the design shear forces of reinforced concrete walls is addressed in sequence from simple to complex, from isolated wall to coupled walls and those that are part of dual structures.
Following this idea chapter three develops the evaluation of the design shear forces for isolated walls.
The chapter starts by analyzing the methodology for evaluating the design shear forces in walls following the principles of capacity design, whereby the design shear force is associated to the global yielding mechanism. Although statically correct, this method has the major disadvantage of neglecting the dynamic effects of the seismic response that develops in the concrete walls.
To avoid this shortcoming in current design that is based on the force method, the evaluation of the design lateral force takes in to account the dynamic effects in the elastic range of the structural response. This is done in a simplified manner in the equivalent lateral force method and in an analytical manner in the response spectra method. These are obtained from elastic static analysis of the structures using a reduction factor q, which has the same value for all vibration modes of a structure.
However studies have shown, that the level of shear forces that develops in reinforce concrete wall during seismic actions is greater than that evaluated according to the principles of capacity design, due to nonlinear dynamic amplification phenomena.
An important step in evaluating the shear forces of reinforced concrete isolated walls was made by Keintzel in the 90s [14] by introducing modal forces limit method and also a simplified application methodology, currently implemented in EC8-1 [3]. In short, the method considers that only first mode response is reduced by base yielding in the case of isolated walls, while the contribution of higher modes to the base shear force is elastic and not reduced by the behavior factor.
Based on this method and also on its improvements made by Rejec et al. [15] an equation is proposed for evaluating the design shear forces of reinforced concrete cantilever walls, equation 3.14. The results of its application are compared in turn with those obtained by the application of national, as well as European codes, both for equivalent lateral force method and response spectrum method.
At the end of Chapter 3 a parametric study, based on nonlinear dynamic analysis of a set of 54 cantilever walls with different section dimensions, different heights and different
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longitudinal reinforcement ratios, is made with the main purpose to determine whether the proposed equation (equation 3.14) is appropriate when using a distributed plasticity model, which takes into account cracking and nonlinear behavior over the wall height. Also the parametric study aimed to propose a variation rule of the design shear forces over the isolated walls height. For analysis three forms of dynamic elastic amplification response
spectrums with increasing corner periods �� where proposed, for each of the spectrums 14 spectrum compatible accelerograms where used.
a b c Fig. 3.28 Elastic acceleration spectra with 5% damping for the 14 compatible accelerograms with: (a)
spectrum of EC8-1 [3] Type 1 Soil type C with corner period �� = 0.6�, (b) spectrum of P100-1 [1] with
corner period �� = 1.6� and (c) spectrum with constant amplification
Fig. 3.10 Values of the fundamental vibration period of the proposed cantilever walls
0
1
2
3
4
5
6
7
8
0 1 2 3 4
Sse
[m/s
2]
T [s]
0
1
2
3
4
5
6
7
8
0 1 2 3 4
Sse
[m/s
2]
T [s]
0
1
2
3
4
5
6
7
8
0 1 2 3 4
Sse
[m/s
2]
T [s]
1
2
3
4
5
6
7
8
9
10
11
12
13
14
Media
Țintă
0
1
2
3
4
5
1 2 3 4 5 6 7 8 9
T [
s]
Wall No.
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a b c
Fig. 3.27 The scheme used to model the behavior of isolated walls: (a) physical model, (b) mathematical model, (c) integration section, reinforced concrete fibers and stress-strain relationships for concrete and
reinforcement
Isolated walls structures are rare in engineering practice, walls are often found associated with reinforced concrete frames. It therefore seems necessary to investigate the influence that frames have on the behavior of reinforced concrete walls, especially on wall shear forces and on the distribution of the shear forces over their height. Limited studies were found on this subject.
Chapter 4 develops a parametric study based on nonlinear dynamic analysis of four plane structural configurations starting from a isolated wall, around which frames are added to the left and respectively right, with an increasing number of bays. By varying the height and the longitudinal reinforcement of the walls a total of 72 plane structures were obtained. For analysis the same accelerograms were used as for the analysis of isolated walls.
Fig. 4.3 Values of fundamental periods of vibration for the proposed dual structures
0.00
0.50
1.00
1.50
2.00
2.50
3.00
0-0 1-1 2-2 3-3
T [
s]
Number of frame bays
N8
N12
N16
8
Fig. 4.2 Schematic presentation of the proposed dual structures
Fig. 4.6 Dual structures modeling scheme: (a) mathematical model, (b) hysteretic behavior rule for lumped plasticity beams hinges, fiber model with concrete and reinforcement fibers for columns and walls, including
stress-strain relationships for concrete and reinforcing
Structural walls always meet in buildings also functional role as partitioning or envelope walls. In this case the need of wall with openings, with a functional or technical role, is frequent. If the openings are arranged in order over the vertical a couple wall configuration is obtained.
In Chapter 5, in order to investigate the dynamic amplification of shear forces for coupled walls and to identify the influence of coupling upon them, a parametric study
9
based on nonlinear dynamic analysis is made. The research also had in mind to establish a relevant shear force distribution rule between the walls of a coupled system, due to their differentiated cracking.
The structure on which the study is made in composed of two identical walls, with the section of one of the previous study isolated wall, coupled with a coupling beam at each level. By varying the longitudinal reinforcement of walls and coupling beams and the height a total of 108 coupled wall systems resulted. If the modeling assumptions regarding the coupling beams stiffness (constant expressed only depending on the characteristics of the concrete section, independent of the reinforcement, and variable, depending on the level of reinforcement, and thus the degree of coupling) is added a total of 216 coupled wall systems are obtained. For analysis the same accelerograms were used as for the analysis of isolated walls.
Fig. 5.3 Values of coupling intensity (equation 5.1) for the 216 analyzed systems with constant and
variable period vs. the fundamental vibration period
Nonlinear modeling for both static and dynamic analysis was carried out in OpenSees program [22], for all parametric studies performed interface software’s in Matlab were made, which compiles the analysis models, which are subsequently run in OpenSees program [22].
For modeling nonlinear behavior of walls and columns distributed plasticity fiber type bar elements were used with formulation in force as Forced-Based Beam-Column Element. One element was used over a story height, each with 5 integration sections.
Beams were modeled using lumped plasticity hinges with Takeda [35] hysteretic behavior rule as uniaxialMaterial Hysteretic as part of length 0 zeroLength element. Elastic zones were modeled with elastic beam elements elasticBeamColumn.
Chapter 6 summarizes the results of Chapters 3, 4 and 5 in the form of a general equation for evaluating the design shear force at the base of concrete walls, either isolated, coupled walls or those that are part o dual structures. This includes an additional safety factor equal to 1.15. Also the proposed distributions for shear force variation over the walls height are presented and also the base distribution of the shear forces between the walls of coupled systems.
0.00
0.20
0.40
0.60
0.80
0 1 2 3 4
CI
T [s]
Const.
Var.
10
Fig. 5.1 Schematic presentation of the Fig. 5.2 Coupled walls systems modeling scheme: (a) mathematical proposed coupled walls systems model, (b) hysteretic behavior rule for lumped plasticity beams
hinges, fiber model with concrete and reinforcement fibers for walls, including stress-strain relationships for concrete and reinforcing
Finally, Chapter 7 provides a brief overview of the thesis, the chapters conclusions are resumed, the final conclusions, personal contributions and future research directions are presented.
6. DESIGN RECOMMENDATIONS
6.1 Design shear force values at the walls base
6.1.1 Proposed equation
All relationships suggested until this chapter are based on efforts obtained from static elastic analysis of structures under the lateral forces associated with the first mode of vibration ����,�, thus they are independent of the method used to evaluating the design
lateral forces.
Because not all structural analysis programs provide the first vibration mode forces and that the vast majority of design codes use for design the lateral forces obtained either through equivalent lateral force method or response spectrum method a general equation 6.1 is proposed. This allows total flexibility regarding the method used for evaluating de design efforts.
The equation can be used for all types of walls: isolated, couple, and for those that are part of dual structures and it was obtained from the equation 5.7, modified to take into account the specific aspects of each type of walls structures.
It can be seen that the dynamic amplification factor � has been increased with an additional safety factor of 1.15. This is necessary for two reasons. On the one hand, the results given by the equation 5.7 were calibrated to the average. In Table 6.1 it can be observed that the mean values minus one standard deviation of the report values between design shear forces obtained by applying equation 5.7 and the mean values of the
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maximum recorded dynamic shear forces are slightly under conservative for all the analyzed structures. On the other hand, these results correspond to mean values of the dynamic analysis results characterized by certain variability, although the variation coefficient value is only 10%.
Tab. 6.1 Mean values minus one standard deviation of the ratio between the design shear forces ��� obtained by applying the equation 5.7 and mean values of the maximum forces obtained from nonlinear
dynamic analyses with mean materials strength ����.
Walls �� ������ ��.⁄ � − Additional safety
coeficient g
Isolated 0.6 0.84 1.19
1.6 0.88 1.14
Dual structures 0.6 0.87 1.15
1.6 0.86 1.16
Coupled 0.6 0.92 1.09
1.6 0.81 1.23
��� = ��� ∙ �
= ��� ∙ 1.15 ∙ ����� ���� ∙���� �
; 1��� + ������ � �� ∙ ������ ∙ �1 −min���; 0.7 − ������ (6.1)
Where: ��� design shear force at the wall base; ���, ��� shear force and bending moment at the base obtained from static elastic
analysis of the structures under design shear forces associated to:
i) first vibration mode: ��� = ���, ��� = ���, ii) equivalent lateral force method: ��� = � �⁄ ∙ ���,���
��� = � �⁄ ∙���,���
iii) response spectrum method: ��� = �����
����∙�����
�������∙ ���,��
��� = ���,�� ��� overstrength factor that considers the various sources of overstrength; � behavior factor (reduction factor) used in the force method to assess the ability of the structure to dissipate energy; ��� design capable bending moment at the wall base; �� mass participation ratio for mode i, calculated as the ratio of the effective mass associated mode i and total mass m �� reduction factor of the second vibration mode contribution,
= 1 + 0.125(� − 2��) ≥ 1; ���� maximum dynamic amplification factor of the elastic response; ���� dynamic amplification factor to the elastic response corresponding to the vibration period of the fundamental mode;
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T vibration period of the fundamental mode; �� coupling intensity equal to 0 for isolated walls and walls of dual structures, while for coupled walls:
=���,������,������
;
���,��� design capable bending moment taken through indirect effect; ���,������ design capable bending moment of the couple wall system.
6.1.2 Comparison of the proposed equation with nation codes
In order to compare the results provided by the proposed equation with those of national codes (described in Chapter 3.3.4), the ratio between the results obtained with the proposed relationship and the product of ��� ∙ � ∙ ���� , defined by the relation 3.16, was evaluated thus yielding the values of the dynamic amplification of the shear forces in the
sense the Romanian Code P100-1 [16], � !!. In order to make the comparison certain terms of equation 5.7 need to be fixed.
Thus, for isolated walls: � = 0.63, ��� �⁄ �� = 0.1 and �� = 0, for walls that are part of dual structures the values considered are those for the analyzed dual structures with 12 levels
and 6 frame bays: � = 0.68, ��� �⁄ �� = 0.06 and �� = 0; for couple walls systems the values are those for the analyzed couple systems 12 levels and medium degree of
coupling with constant period: � = 0.67, ��� �⁄ �� = 0.09 and �� = 0.35.
The value of the behavior factor q for isolated walls and those from dual structures was 4.6, while for coupled walls was 6.25, according P100-1 [16].
Ratio was calculated for four values of flexural wall overstrength � =������
from 1 to
4, with respect to the design bending moments.
Figures 6.1 and 6.2 present the values the ratio between design shear forces obtained by applying the equation 6.1 and the product of the ��� ∙ � ∙ ���� , defined by the
relation 3.16, where ���� is the design shear force obtained from elastic static analysis of the structure under design forces, determined by equivalent lateral force method (Fig. 6.1) and response spectra method (Fig. 6.2).
It is noted that until the corner period ��, the dynamic amplification coefficient as given by P100-1 [16], does not change with the variation of the period, but varies with
structural overstrength �, with the method used for determining the design lateral forces, with the modal participation mass ratios. For the selected cases the value of these coefficients are presented in Table 6.2.
It also noted that before the corner period ��, the influence of frames in dual structures although present, is relatively low in intensity, as their presence affects especially the contribution of higher modes the shear force, on this range the first two modes contribution to the shear force are comparable. The same comment applies to couple walls.
After the corner ��, there is a rapid increase of the dynamic amplification with period increases and a decrease with overstrength increasing, due change of the fundamental mode contribution.
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a
b
c
Fig. 6.1 Values of the ratio between design shear forces obtained by applying equation 6.1 and the product
� � ∙ � ∙ ���! , defined by the equation 3.16, where ���! is the design shear force at the base obtained by
applying equivalent lateral force method vs. fundamental vibration period for corner period �� = 0.6� (Left)
and corner period �� = 1.6� (right): (a) isolated walls, (b) walls from dual structures, (c) coupled walls
1.20
0
1
2
3
4
0 1 2 3 4
εP100
T [s]
ω =4
ω =3
ω =2ω =1
1.20
0
1
2
3
4
0 1 2 3 4
εP100
T [s]
ω =4ω =3
ω =2
ω =1
1.20
0
1
2
3
4
0 1 2 3 4
εP100
T [s]
ω =4
ω =3
ω =2
ω =1
1.20
0
1
2
3
4
0 1 2 3 4
εP100
T [s]
ω =4ω =3
ω =2
ω =1
1.20
0
1
2
3
4
0 1 2 3 4
εP100
T [s]
ω =4
ω =3
ω =2ω =1
1.20
0
1
2
3
4
0 1 2 3 4
εP100
T [s]
ω =4
ω =3ω =2
ω =1
14
a
b
c
Fig. 6.2 Values of the ratio between design shear forces obtained by applying equation 6.1 and the product � � ∙ � ∙ ���! ,, defined by the equation 3.16, where ���! is the design shear force at the base obtained by
applying response spectrum method vs. fundamental vibration period for corner period �� = 0.6� (Left) and
corner period �� = 1.6� (right): (a) isolated walls, (b) walls from dual structures, (c) coupled walls.
1.20
0
1
2
3
4
0 1 2 3 4
εP100
T [s]
ω =4ω =3
ω =2
ω =1
1.20
0
1
2
3
4
0 1 2 3 4
εP100
T [s]
ω =4ω =3
ω =2
ω =1
1.20
0
1
2
3
4
0 1 2 3 4
εP100
T [s]
ω =4ω =3
ω =2
ω =1
1.20
0
1
2
3
4
0 1 2 3 4
εP100
T [s]
ω =4ω =3
ω =2
ω =1
1.20
0
1
2
3
4
0 1 2 3 4
εP100
T [s]
ω =4ω =3ω =2
ω =1
1.20
0
1
2
3
4
0 1 2 3 4
εP100
T [s]
ω =4ω =3
ω =2
ω =1
15
Fig. 6.3 Values the ration between design shear forces obtained by applying the equation 6.1 and the
product � � ∙ � ∙ ���! , defined by the relation 3.16, where ���! is the design shear force at the base obtained by
applying equivalent lateral force method, � = 1 for isolated walls, couple walls and those of dual vs period
for corner period �� = 0.6� (Left) and corner period �� = 1.6� (Right)
Fig. 6.4 Values the ration between design shear forces obtained by applying the equation 6.1 and the
product � � ∙ � ∙ ���! , defined by the relation 3.16, where ���! is the design shear force at the base obtained
by applying equivalent lateral force method, for the case � = 1 for isolated walls, couple walls and those of
dual vs. period for corner period �� = 0.6� (Left) and corner period �� = 1.6� (Right)
Figures 6.3 and 6.4 respectively, compare the values of the ratio between the design shear forces obtained by applying the equation 5.7 and the product γ"# ∙ ω ∙ V$#� for
the case ω = 1, for isolated walls, coupled and those from dual structures. It is found that the influence of both frame and coupling increases after the corner ��, due to the increasing contribution of higher modes to the shear force. However the reduction of the dynamic amplification does not exceed 15%, neither for shear forces obtained through equivalent lateral force method, nor through response spectrum method. Instead, the reduction of the shear forces associated to the first vibration mode V$#,� is greater, about
22% for walls that are part of dual structures and 16% for couple walls.
If the values of amplification factors in the sense of the Romanian Code P100-1
[16], � !!, presented in Table 6.2, are examined an approximate value of amplification factor suitable for all types of walls would be � !! = 1.3. This value corresponds to an overstrength ω ≈ 1.5, common for structures with walls designed in Romania.
1.20
0
1
2
3
4
0 1 2 3 4
εP100
T [s]
DualCoupledIsolaed
1.20
0
1
2
3
4
0 1 2 3 4
εP100
T [s]
Dual
Coupled
Isolaed
1.20
0
1
2
3
4
0 1 2 3 4
εP100
T [s]
Dual
Coupled
Isolated
1.20
0
1
2
3
4
0 1 2 3 4
εP100
T [s]
Dual
Coupled
Isolated
16
Tab. 6.2 Values of the ratio between design shear forces obtained by applying the equation 6.1 and the product � � ∙ � ∙ ���! , defined by the relation 3.16, where ���! is the design shear force at the base obtained by
applying equivalent lateral force method and response spectrum method for: isolated walls, coupled walls
and walls of dual structures with fundamental period less than the corner period ��
Walls � �" �#�#��$
�� Method
Dynamic amplification factor
�%"&&
ω=1 ω=2 ω=3 ω=4
Isolated 4.6 0.63 0.1 0
Equivalent
lateral force 1.55 1.26 1.20 1.18
Response
spectrum 1.72 1.28 1.18 1.14
Dual
structures 4.6 0.68 0.06 0
Equivalent
lateral force 1.44 1.23 1.19 1.17
Response
spectrum 1.53 1.23 1.17 1.15
Coupled 6.25 0.67 0.09 0.35
Equivalent
lateral force 1.47 1.24 1.19 1.17
Response
spectrum 1.56 1.23 1.16 1.14
6.2 Shear force distribution
6.2.1 Over walls height
Based on the parametric studies results two distributions of the shear forces over walls height were proposed, one for isolated walls and one for coupled walls and those that are part of dual structures. These are shown in Figure 6.5.
Fig. 6.5 Distribution of the design shear force ��� normalized at the base over the walls height for: isolated, coupled walls and those that are part of dual structures
6.2.2 At the base of coupled walls
The study realized suggests applying equation 6.2 for the distribution of the total shear force between the piers of a couple system. The equation involves distributing the shear force associated to the fundamental mode according to each pier capable bending moment at the base, while the shear force associated to the second mode should be distributed evenly between the coupled walls.
Equation 6.2 resulted from adaptation of equation 5.5, valid for two couple walls, to the case with n walls.
0.5, 0.5
0.4, 1
0.6, 0.5
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1
H
V
Izolați
Duale,
Cuplați
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���,� = & (��,�∑(��,�
�(*+�)
�'
( ���,������ (6.2)
Where: ���,� design shear force at the base of pier i; ���,� design capable bending moment at the base of pier i, associated to the global
plastification mechanism; ∑���,� sum of piers capable bending moments at the base, associated to the global
plastification mechanism; � dynamic amplification factor of shear forces according to relation 6.1, evaluated at the couple system level; number of coupled piers; ���,������ global design shear force evaluated according to relation 6.1. at the couple
system level.
7. CONCLUSIONS
7.3 Final conclusions
Based on research conducted it can be said that for structures with fundamental period smaller than the corner period of the design spectrum dynamic shear forces of reinforced concrete walls are slightly higher than those associated with the global plastification mechanism, if the design is done the according to the force method and considering the various sources of overstrength. Moreover, the dynamic amplification of the shear forces is constant with period for this range, decreasing with structural overstrength increase. This makes possible the calibration of factors for evaluating the dynamic amplifications of the design shear forces. It is however noted that the contribution of higher modes is relatively high and is comparable to the contribution of the first vibration mode.
Instead, beyond the corner period the dynamic amplification of the shear forces starts to be significant, increasing with period increase over the corner period and with structural overstrength decrease. The dependence on period makes it impossible to calibrate factors for evaluating the dynamic amplification of the design forces, is thus recommended that relation 6.1 is used for this range, valid for: isolate walls, couple walls and those that are part of dual structures.
Reinforced concrete frames presence reduces the dynamic amplification of the shear forces, this effect is due on the one hand to changes in the vibration characteristics of structures, which lowers the second mode modal mass and increase first mode one, on the other hand to the increase of frame contribution to the base shear force in the inelastic response compared to the elastic one. Of these two influences, only the first was considered directly and integrate in equation 6.1. Frame influence on wall shear forces increases for structures with fundamental periods large than the corner one.
Regarding the influence of coupling it was found that this leads also to decrease of dynamic amplification of shear forces, having minimum values for medium coupling intensities. A parabolic variation of coupling influence on shear forces was found with coupling increasing from minimum to maximum.
Given that dynamic amplification phenomena of shear forces reduces towards the half height of walls, as confirmed by research conducted, a simplified distribution of shear forces over isolated walls height was proposed, characterized by a rapid reduction of
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shear forces up to half height, followed by a slow decrease to the top. This distribution was corrected for walls that are part of dual structure, whereas the presence of frames has a smoothing effect on the distribution of shear forces over the walls height.
Finally, it can be said that national codes formulas for evaluating the design shear forces provide results close to the seismic response of structures at the base for fundamental periods lower than corner period of the design spectrum. The presence of frames and couplings reduces the dynamic amplification of the design shear forces of walls. It is appreciated that this phenomenon should be introduced in a relevant form in design codes.
Instead the distribution of the shear forces over the walls height given in the national codes is conservative over the whole height, it is recommended to use a simplified distributions characterized by a rapid reduction of shear forces up to half height, followed by a slow decrease to the top.
7.4 Personal contributions
A first contribution of the thesis is the correction of Rejec's [15] equation to maintain consistency with modal limit forces method, proposed Keintzel [14]. Equation 3.14 resulted, used to determine the design shear forces of isolated concrete walls.
Based on the parametric study of isolated walls, the proposed equation was further
corrected by introducing a reduction factor of second vibration mode contribution ��, which takes into account the effect of cracking and nonlinear incursion over the walls height.
Another contribution of the thesis is the proposal, for the first time, to consider frame influence on the dynamic shear force amplification of reinforced concrete walls of dual structures and the influence of coupling, as a function o coupling intensity.
Shear force distribution over the walls height were proposed for isolated, coupled and walls that are part of dual structures, which were calibrated based on the results of parametric studies. It is considered that the proposed distributions are superior to those used in current design practice.
For coupled walls a method of distribution of shear force between the piers at the base was also proposed which leads to a lower and more realistic redistribution of shear forces between piers than that obtained using the bending moment capacity of the piers. This distribution was confirmed by the results of parametric studies.
To develop the research 3 interface software in Matlab were done, which compiles the analysis models, that are run in OpenSees program. The software’s can run modal, elastic (equivalent lateral forces and response spectra), static and dynamic nonlinear analyses on isolated walls with a symmetrical section, on plane structures with a wall and frames to the left and right and on couple wall systems with two piers. All structures can have n levels and any level high. Also, the results are post processed automatically under the form of a synthesized file of results and also as various graphic representations. These software’s are attached to the thesis.
7.5 Future research
For future it appears interesting analyzing the influence that frames have on dynamic amplification coupled wall systems.
Also of interest is the possibility of introducing frame contribution to the walls shear forces at the base in the nonlinear response range and the study of shear forces in structures with walls of different sizes and different axial forces.
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