Dissertation2005 Beyer

8/20/2019 Dissertation2005 Beyer

1/118

I.U.S.S.

Istituto Universitariodi Studi Superiori di Pavia Università degli Studi

di Pavia

EUROPEAN SCHOOL OF ADVANCED STUDIES IN

REDUCTION OF SEISMIC RISK

ROSE SCHOOL

Design and Analysis of Walls Coupled by Floor

Diaphragms

A Dissertation Submitted in PartialFulfilment of the Requirements for the Master Degree in

EARTHQUAKE ENGINEERING

By

Katrin Beyer

Supervisors: Prof. M.J. N. PRIESTLEY, Prof. G. M. CALVI, Dr. R. PINHO

October, 2005


2/118

The dissertation entitled “Design and Analysis of Walls Coupled by Floor

Diaphragms”, by Katrin Beyer, has been approved as partial fulfilment of the

requirements for the Master Degree in Earthquake Engineering.

M.J.N. Priestley

G.M. Calvi

R. Pinho


3/118


4/118


5/118ii

ACKNOWLEDGEMENTS

To Prof. Priestley, Prof. Calvi and Dr. Pinho I express my grateful thanks for supervising and encouragingthis work. Their helpful comments and suggestions significantly contributed to improve the work and myunderstanding of the subject. I would also like to thank Prof. Restrepo who initiated the project during thecourse “Seismic Design of RC Structures” which he taught in September 2004 at the Roseschool.

My thanks also go to my friends with whom I started the Master Program: Christos Baltas, for company

through highs and lows of all courses and the little spare time left, Angelos Floros, for his cheerfulcharacter which guarantees high spirits whenever he is around, and Agustin Castro, whose perfectGerman language skills allowed me frequent conversations in the language which remains most familiar tome. I would like to extend my thanks to all students at the Roseschool for their friendship and companyduring my time in Pavia.

Finally, I would like to give thanks to my mum and dad and my sisters, Anne and Judith. Although wecurrently spend little more than a few days a year together I am sincerely thankful for their support andthe strength they have given me throughout my studies.


6/118iii


7/118iv

CONTENT

ABSTRACT I

ACKNOWLEDGEMENTS II

CONTENT IV

LIST OF FIGURES VII

LIST OF TABLES XI

NOTATION XII

1 INTRODUCTION 1

2 STRUCTURAL SYSTEM 2

3 DESIGN OF COUPLED WALLS 4

3.1 DDBD Method for Coupled Walls 4

3.2 Literature Review on Base Shear Demand of Coupled Walls 10

3.3 Design Shear and Bending Moments of Coupled Walls 15

3.3.1 Different proposals for the dynamic amplification factors and shear forcedistribution over height 15

3.3.2 Design base shears when treated as de-coupled walls 18

3.3.3 Design base shears when treated as coupled walls as suggested by Rutenberg

[25] 19

3.3.4 Summary of Design Shear Forces and Design Moments 21

3.4 Flexible Foundation 23

3.4.1 Dimensioning of Foundation 23

3.4.2 Dynamic Properties of Soil Foundation 24

3.4.3 Flexibility of Foundation Structure 26

3.4.4 Foundation Flexibility in DDBD 28

3.5 Summary of Chapter 3 29

4 ANALYSIS OF COUPLED WALL SYSTEMS 30

4.1 General Modelling Assumptions 30


8/118 v

4.2 Description of the Numerical Model of the coupled wall system 31

4.3 Pushover Analysis of De-coupled and Coupled Walls 31

4.3.1 Different Approaches of Pushover Analysis 31

4.3.2 Pushover Analysis of De-coupled Walls 33

4.3.3 Pushover Analysis of Walls coupled with rigid floors 34

4.4 Non-linear Time-History Analysis 38

4.4.1 Time-History Inputs 38

4.4.2 Results from Time History Analysis of coupled Walls 39

4.4.3 Comparison of Time History Results of De-coupled Walls to Coupled Walls 43

4.4.4 Time History Analysis of the Coupled Wall System with flexible foundation 46


5 BASE SHEAR FORCES IN COUPLED WALLS AND THEIR SENSITIVITY TOMODELLING ASSUMPTIONS 50

5.1 Rigid and Flexible Lateral Connection of Coupled Walls 51

5.1.1 In-plane Flexibility of Slab spanning between the two Walls 51

5.1.2 Pushover Analysis of Walls coupled with flexible floor diaphragm 52

5.1.3 Non-linear time history Analysis of Walls coupled with flexible floor diaphragm56

5.2 Shear Stiffness of Structural Walls 60

5.2.1 Shear Stiffness and Capacity of Wall Sections subjected to high Shear ForceLevels 60

5.2.2 Comparison of Pushover Analyses for different Assumptions on ShearStiffness of Wall Sections 64

5.2.3 Comparison of Non-linear time history analysis for different Assumptions onShear Stiffness of Wall Sections 70

5.2.4 Model with Shear Flexibility and Floor Diaphragm Flexibility 73

5.3 Lumped Plasticity Model vs. Fibre Element Model 75

5.3.1 Lumped Plasticity Model in Ruaumoko 75

5.3.2 Fibre Element Model 76

5.3.3 Pushover Results for the Fibre Element Model and the Lumped PlasticityModel 78

5.3.4 Non-linear time-history analysis 82


6 SUMMARY AND CONCLUSIONS 92

6.1 Summary of Design 92

6.2 Summary of Sensitivity Studies 93

6.3 Conclusions and Recommendations for Design and Analysis of Walls coupled by

Floor Diaphragms 96

7 REFERENCES 99


9/118


10/118 vii

LIST OF FIGURES

Figure 2.1 8-storey coupled wall structure: Elevation and plan view....................................................................3

Figure 3.1 Two storey system of coupled walls: (a) properties and loading, (b) floor forces and deflectedshapes after formation of a plastic hinge at the base of the long wall [28]...............................................11

Figure 3.2 Bi-linear hysteresis rule and modified Takeda hysteresis rule .......................... .................................13

Figure 3.3 Cyclic static analysis for rigidly coupled walls which hystretic behaviour is modelled with themodififed Takeday rule: Base shear, base moment and top displacement (thin lines in base shear plot:maximum base shear forces predicted by equation (3.21)) .............................. .................................. .........14

Figure 3.4 Cyclic static analysis for rigidly coupled walls which hystretic behaviour is modelled with the bi-linear rule: Base shear, base moment and top displacement (thin lines in base shear plot: maximumbase shear forces predicted by equation (3.21)) ................................. .................................. .........................15

Figure 3.5 Force-controlled pushover analyses with resultant forces at h=5.1m (Intensity based method)and h=10.5m (Traditional method).................................................................................................................21

Figure 3.6 Design bending moment envelopes......................................................................................................22

Figure 3.7 Design shear forces when walls are designed as-coupled walls as suggested by Rutenberg(“Coupled”) and as de-coupled walls (“De-coupled”).................................................................................22

Figure 3.8 Coupled wall structure with flexible foundation........................... .................................. ....................23

Figure 3.9 Scheme of analytical model with flexible foundation.........................................................................28

Figure 4.1 Adaptive pushover analysis of single walls: Base shear – top displacement and base moment –top Displacement ............................ .................................. .................................. ............................. ..................33

Figure 4.2 Adaptive pushover analysis of rigidly coupled walls: Base moment – top displacement and baseshear – top displacement relationship with the instants of formation of a plastic hinge.......................35

Figure 4.3 Adaptive pushover analysis of rigidly coupled walls: Shear forces in walls at formation of plastichinge in Wall 1 and Wall 2................................................................................................................................35

Figure 4.4 Adaptive pushover analysis for rigidly coupled walls: Floor diaphragm forces at formation ofplastic hinge in Wall 1 and Wall 2....................................................................................................................36

Figure 4.5 Base shear force as a function of the height of the resultant lateral force: absolute andnormalised to the maximum base shear of a de-coupled cantilever wall ............................ ......................37

Figure 4.6 Sketch of coupled wall system after yielding of long wall with the lateral force resultant atheight h and the largest storey force at the first level...................................................................................38

Figure 4.7 Displacement spectra for 5% damping.................................................................................................39

Figure 4.8 Rigidly coupled walls: Displacement envelope from time history analysis (thin black lines:results from time history analyses, thick blue line: average of five analyses, thick grey line: designprofile)..................................................................................................................................................................41


11/118 viii

Figure 4.9 Rigidly coupled walls: Drift envelope from time history analysis (thin black lines: results fromtime history analyses, thick blue line: average of five analyses, thick grey line: maximum design drift)...............................................................................................................................................................................41

Figure 4.10 Rigidly coupled walls: Shear force and bending moment envelope from time history analysis(thin black lines: results from time history analyses, thick blue line: average of five analyses, thick red

and green lines: design envelopes according to different methods, thick grey line: flexural capacityassigned in analysis)............................................................................................................................................42

Figure 4.11 Rigidly coupled walls: Storey Force Envelope from time history analysis (thin black lines:results from time history analyses, thick blue line: average of five analyses)................. ...........................43

Figure 4.12 Comparison of coupled and de-coupled walls: Average displacement envelopes from timehistory analysis....................................................................................................................................................44

Figure 4.13 Comparison of coupled and de-coupled walls: Average drift envelope from time historyanalysis .......................... ................................. ............................. ................................... .................................. ....44

Figure 4.14 Comparison of coupled and de-coupled walls: Average shear force and bending momentenvelope from time history analysis................................................................................................................45

Figure 4.15 Rigidly coupled walls with flexible foundation: Displacement envelope from time historyanalysis (thin black lines: results from time history analyses of flexible foundation model, thick greenand blue lines: average of results of five analyses, thick grey line: design displacement profile) ..........47

Figure 4.16 Rigidly coupled walls with flexible foundation: Drift envelope from time history analysis (thinblack lines: results from time history analyses of flexible foundation model, thick green and bluelines: average of results of five analyses, thick grey line: design displacement profile for flexiblefoundation)..........................................................................................................................................................47

Figure 4.17 Rigidly coupled walls with flexible foundation: Shear force and bending moment envelopefrom time history analysis (thin black lines: results from time history analyses of flexible foundationmodel, thick blue and green lines: average of results of five analyses, thick grey lines: flexural capacityassigned in analysis)............................................................................................................................................48

Figure 4.18 Rigidly coupled walls with flexible foundation: Storey force envelope from time historyanalysis (thin black lines: results from time history analyses of flexible foundation model, thick lines:average of results of five analyses)...................................................................................................................48

Figure 5.1 Flexible floor model (finally masses were moved back onto walls)........................... ......................52

Figure 5.2 Adaptive pushover analysis of flexibly coupled walls: Wall base shear force – top displacement...............................................................................................................................................................................53

Figure 5.3 Adaptive pushover analysis of flexibly coupled walls: System base shear force – topdisplacement........................................................................................................................................................54

Figure 5.4 Adaptive pushover analysis of flexibly coupled walls: Base moment – top displacement ...........54

Figure 5.5 Adaptive pushover analysis of flexibly coupled walls: Storey forces ............................. ..................55

Figure 5.6 Adaptive pushover analysis of flexibly coupled walls: Displacement and drift profile for topdisplacement =0.30m .......................... .................................. ................................... ............................. ............55

Figure 5.7 Force-controlled pushover analysis of flexibly coupled walls: Wall base shear force – topdisplacement........................................................................................................................................................56

Figure 5.8 Rigidly and flexibly coupled walls: Displacement envelope from time history analysis (thin blacklines: results from time history analyses with flexible coupling, thick green and blue lines: average ofresults of five analyses, thick grey line: design displacement profile) ............................ ............................58

Figure 5.9 Rigidly and flexibly coupled walls: Drift envelope from time history analysis (thin black lines:results from time history analyses with flexible coupling, thick green and blue lines: average of resultsof five analyses, thick grey line: maximum design drift) ............................. ................................... ..............58


12/118ix

Figure 5.10 Rigidly and flexibly coupled walls: Shear force and bending moment envelope from timehistory analysis (thin black lines: results from time history analyses with flexible coupling, thick greenand blue lines: average of results of five analyses, thick grey line: flexural capacity assigned in analysis)...............................................................................................................................................................................59

Figure 5.11 Rigidly and flexibly coupled walls: Storey force envelope from time history analysis (thin black

lines: results from time history analyses with flexible coupling, thick green and blue lines: average ofresults of five analyses)......................................................................................................................................59

Figure 5.12 Variation of shear stiffness with displacement..................................................................................60

Figure 5.13 Schematics of the equivalent truss model using Williot principles ([10] after [11]).....................61

Figure 5.14 Shear stiffness after shear cracking: Dependence of shear stiffness on the angle of thecompression strut towards the wall axis.........................................................................................................62

Figure 5.15 Adaptive pushover analysis of coupled walls with different assumptions about shear stiffness: Wall base ahear force – top displacement................ ............................... ................................... ....................66

Figure 5.16 Adaptive pushover analysis of coupled walls with different assumptions about shear stiffness:

System base ahear force – top displacement ............................ ............................. ................................... .....66 Figure 5.17 Adaptive pushover analysis of coupled walls with different assumptions about shear stiffness:

Base moment – top displacement of Wall 1..................................................................................................67

Figure 5.18 Adaptive pushover analysis of coupled walls with different assumptions about shear stiffness:Storey forces........................................................................................................................................................67

Figure 5.19 Adaptive pushover analysis of coupled walls with different assumptions about shear stiffness:Displacement and drift profile for top displacement =0.30m....................................................................68

Figure 5.20 Force-controlled pushover analysis of coupled walls with different assumptions about shearstiffness: Wall base ahear force – top displacement ............................... .................................. ....................70

Figure 5.21 Rigidly coupled walls with different assumptions about shear stiffness: Displacement envelopefrom time history analysis (thin black lines: results from time history analyses with tri-linear shearmodel, thick green and blue lines: average of results of five analyses, thick grey line: designdisplacement) ......................... .................................. ............................. .................................. ............................71

Figure 5.22 Rigidly coupled walls with different assumptions about shear stiffness: Drift envelope fromtime history analysis (thin black lines: results from time history analyses with tri-linear shear model,thick green and blue lines: average of results of five analyses, thick grey line: maximum design drift)...............................................................................................................................................................................72

Figure 5.23 Rigidly coupled walls with different assumptions about shear stiffness: Shear force andbending moment envelope from time history analysis (thin black lines: results from time historyanalyses with tri-linear shear model, thick green and blue lines: average of results of five analyses,

thick grey line: flexural capacity assigned in analysis)...................................................................................72

Figure 5.24 Rigidly coupled walls with different assumptions about shear stiffness: Storey force envelopefrom time history analysis (thin black lines: results from time history analyses with tri-linear shearmodel, thick green and blue lines: average of results of five analyses)......................................................73

Figure 5.25 Force-controlled pushover analysis of coupled walls with shear flexibility and floor diaphragmflexibility: Wall base shear force – top displacement....................................................................................74

Figure 5.26 Non-linear time history analysis of coupled walls with shear flexibility and floor diaphragmflexibility: Shear force and bending moment envelope from time history analysis (thin black lines:results from time history analyses with tri-linear shear model and flexible coupling, thick green andblue lines: average of results of five analyses, thick grey line: flexural capacity assigned in analysis)...74

Figure 5.27 Adaptive pushover analysis of de-coupled walls: Base shear – top displacement and basemoment – top displacement for fibre and element model (coloured line: fibre model, grey line:element model) ........................... .................................. .................................. ............................. .......................77


13/118x

Figure 5.28 Cyclic displacement-controlled pushover analysis of de-coupled walls: Base shear – topdisplacement and base moment – top displacement for fibre and element model (fibre elementmodel: coloured line, lumped plasticity model: grey line)............................................................................78

Figure 5.29 Adaptive pushover analysis of coupled walls: Base shear – top displacement and base moment – top displacement for fibre and lumped plasticity model (coloured thick lines: fibre model of

coupled walls, coloured thin lines: fibre model of de-coupled walls, grey lines: element model).........80

Figure 5.30 Adaptive pushover analysis of coupled walls: System base shear – top displacement for fibreand lumped plasticity model.............................................................................................................................80

Figure 5.31 Adaptive pushover analysis of coupled walls: Base shear – top displacement and base moment – top displacement for fibre element model for de-coupled and rigidly and flexibly coupled walls(solid thick lines: rigidly coupled walls, dashed thick lines: flexibly coupled walls, solid thin lines: de-coupled wall )......................................................................................................................................................81

Figure 5.32 Force-controlled pushover analysis of coupled walls for fibre and lumped plasticity model: Wall base ahear force – top displacement................ ............................... ................................... ....................82

Figure 5.33 Time-history analysis of coupled walls subjected to Accelerogram 5: Comparison of topdisplacement response of fibre element model and lumped plasticity model with no and 5% effectivedamping ............................. .................................. ............................. .................................. .................................85

Figure 5.34 Time-history analysis of coupled walls subjected to Accelerogram 5: Comparison of base shearresponses of fibre element model and lumped plasticity model with no and 5% effective damping ..85

Figure 5.35 Time-history analysis of coupled walls subjected to Accelerogram 5: Comparison of basemoment responses of fibre element model and lumped plasticity model with no and 5% effectivedamping ............................. .................................. ............................. .................................. .................................86

Figure 5.36 Fibre and lumped plasticity model of rigidly coupled walls: Displacement envelope from timehistory analysis (thin black lines: results from time history analyses with fibre model, thick red, greenand blue lines: average of results of five analyses, thick grey line: design displacement) .......................86

Figure 5.37 Fibre and lumped plasticity model of rigidly coupled walls: Drift envelope from time historyanalysis (thin black lines: results from time history analyses with fibre model, thick red, green andblue lines: average of results of five analyses, thick grey line: maximum design drift) .......................... .87

Figure 5.38 Fibre and lumped plasticity model of rigidly coupled walls: Shear force and bending momentenvelope from time history analysis (thin black lines: results from time history analyses with fibremodel, thick red, green and blue lines: average of results of five analyses, thick grey line: flexuralcapacity assigned in analysis) ............................. ............................. ................................... ............................. ..87


14/118xi

LIST OF TABLES

Table 3.1 Properties of Wall 1, Wall 2 and coupled wall system............................. ................................... ...........9

Table 3.2 Properties of Wall 1, Wall 2 and coupled wall system, continued......................... ..............................9

Table 3.3 Design base shear forces for the “Traditional Method” and the “Intensity Based Method” when walls are designed as de-coupled walls.................................. ............................. .................................. ...........19

Table 3.4 Design base shear forces for the “Traditional Method” and the “Intensity Based Method” when walls are designed as-coupled walls as suggested by Rutenberg........................... .................................. ....20

Table 4.1 Wall section properties.................................................. .................................. ............................. .............31

Table 4.2 Comparison analysis values of de-coupled walls to design values................ ................................. ....33

Table 4.3 Rigidly coupled walls: Comparison of peak base shear forces of pushover and time-historyanalyses.................................................................................................................................................................46

Table 5.1 Rigidly and flexibly coupled walls: Comparison of peak base shear forces of pushover and time-history analyses ........................... .................................. .................................. ............................. .......................57

Table 5.2 Shear Stiffness and Capacity of the Structural Walls at different stages......................... ..................64

Table 5.3 Effect of shear deformation on displacement at top and first level (displacements are at instantof formation of plastic hinges, i.e for Wall 1 at formation of plastic hinge in Wall 1 and for Wall 2 atformation of plastic hinge in Wall 2)...............................................................................................................65

Table 5.4 Rigidly coupled walls with different assumptions about shear stiffness: Comparison of peak baseshear forces of pushover and time-history analyses (Pushover results are peak values for topdisplacements ≤0.35m) ............................ .................................. ................................... ............................. .......71

Table 5.5 Fibre element model: Properties of walls ............................. ................................... ............................. .76

Table 5.6 Yielding sequence in fibre model of de-coupled walls ............................ .................................. ..........77

Table 5.7 Yielding sequence in fibre model of coupled walls ............................. .................................. ...............79

Table 5.8 Fibre and element model of coupled walls: Comparison of peak base shear forces of pushoverand time-history analyses (Pushover results are peak values for top displacements ≤0.475m)...........84

Table 5.9 System base shear demand, dynamic effective height and dynamic amplification factor obtainedfrom non-linear time-history analysis of different models (average results from set of fiveaccelerograms) ............................ .................................. .................................. ............................. .......................91


15/118xii

NOTATION

c A Gross concrete section of slab (effective width)

gross A Gross area of wall

i A Energy absorbed per cycle

s A Shear area of wall

s A Reinforcement area

B Half width of the foundation ( ≈10m)

b D Bar diameter of main longitudinal reinforcement

c E Concrete elastic modulus

1 EI Cracked wall stiffness

s E Reinforcement elastic modulus

jiF , Lateral force on wall at storey

G Shear modulus reduced for non-linear behaviour of soil

maxG Maximum shear modulus of soil

d M Design base moment

n M Nominal base moment

N Axial force on wall section

base N Axial compression load at base of wall

ξ ,d S Spectral displacement at corner period for effective damping ratio ξ

cT Corner period of displacement spectrum


16/118


17/118xiv

F r Post-elastic stiffness ratio of the base shear – effective displacement relationship for wall

s Vertical spacing of horizontal wall reinforcement

iv Base shear ratio of one wall

id ,∆ Design displacement at effective height of wall

sd ,∆ Design displacement of the coupled wall system

sd ,∆ System design displacement at the effective height

yeff ,∆ Yield displacement at the effective height of a wall

yi,∆ Displacement of a single wall at storey

d j ,∆ Design displacement of a wall at storey

y j ,∆ Yield displacement at storey of a single wall

js,∆ Displacement of coupled wall system at storey

i y,∆ Yield displacement of the de-coupled wall

s y,∆ Yield displacement of the coupled wall system

α Inclination of compression struts, angle measure towards member axis

β Viscous elastic damping ratio of soil

ys,ε Yield strain of main longitudinal reinforcement

oφ Overstrength factor

yφ Yield curvature of a single wall

i,∆ µ Displacement ductility of wall

υ Poisson’s ratio of soil

cυ Concrete Poisson’s ratio

θ Rotation at wall base


18/118xv

d θ Design drift

ρ Reinforcement ratio

v ρ Web reinforcement ratio, wvvv bs A / = ρ

ω Dynamic amplification factor

ieff ,ξ Effective damping ratio of wall

seff ,ξ Effective damping ratio of the coupled wall system

iel,ξ Elastic viscous damping of wall

ihyst ,ξ Hysteretic damping of wall


19/118


20/118

Chapter 2: Structural System

2

2 STRUCTURAL SYSTEM

An 8-storey building was chosen as an example which is used throughout this study. The lateral loadbearing capacity of the structure is provided by in total 8 structural walls at the perimeter of the slab. In -direction 2 long and 2 shorter walls contribute to the capacity of the system; in -direction the capacity isprovided by 4 walls with identical properties. Out-of plane stiffness of the structural walls and stiffness

provided by non-structural elements or gravity columns is neglected. Only the behaviour when loaded inthe -direction will be considered in this study. Since the structure is symmetric regarding the -axis onlyhalf of the floor area with one long and one short wall is considered in the following. The centre lines ofthe walls in -direction are at a distance of 10m. The structure is symmetric regarding the -axis and onlyone long and one short wall will be modelled.

It is assumed that the centre of mass, the centre of stiffness and the centre of strength all coincide withthe geometric centre of the slab. The storey height is 3m, the total floor plan area is 25mx20m=500m², thecontributing horizontal floor mass per storey is estimated as 400t (i.e. 200t between one long and oneshort wall).

Figure 2.1 shows a sketch of the coupled wall structure. The structural walls are assumed to have the

following typical material:- Concrete: 'c f = 30 MPa; c E = 25.7 GPa, cυ =0.25

- Reinforcement (longitudinal and transverse): y f = 450 MPa; s E = 200GPa


21/118

Chapter 2: Structural System

3

Figure 2.1 8-storey coupled wall structure: Elevation and plan view

hn

lw1 lw2

columnscarrying gravity

load only

x y

hn

lw1 lw2

hn

lw1 lw2

columnscarrying gravity

load only

x y


22/118

Chapter 3: Design of Coupled Walls

4

3 DESIGN OF COUPLED WALLS

In this chapter the design of the coupled wall structure is described and the flexibility of an example for afoundation structure is investigated. For the design of the walls a displacement based design procedure asdescribed in [18] was adopted. The design procedure is extended to coupled walls as described in [19]. Theunderlying assumptions of this method are summarised and discussed in section 3.1.

The design spectrum is also the same as described in [18], i.e. an EUROCODE 8 Spectrum for 0.4g PGAand soil class B where the limit of the constant velocity range was extended from 2 to 4s. The designspectrum is plotted in Figure 4.7 (section 4.4.1). For 5% viscous damping the spectral displacement at thecorner period is 0.60m. The walls were designed for a maximum storey drift of 2%. For the analysedstructure the drift limit proved to be more critical than the rotational capacity at the wall bases.

In section 3.2 the effect of coupling of walls of different lengths is described and a brief overview over the work by Rutenberg and his co-workers on this topic is given. Different methods to derive design baseshear forces for the individual walls are outlined in section 3.3.

3.1 DDBD METHOD FOR COUPLED WALLS

In addition to the design assumptions of a single wall the design of a coupled wall system requiresassumptions about:

- The displacement profile of the coupled wall system

- The effective damping of the coupled wall system

The base moment distribution between the two walls is the designer’s choice.

The design profile of coupled walls is expected to be a combination of the displacement profile of the de-

coupled walls. In [19] it is suggested that the displacement profile of the coupled wall system could beapproximated as the profile of the longer wall. The longer wall is likely to be stronger and is henceexpected to control the deformation of the coupled wall system.

Three points should be considered if this recommendation is followed:

- In many cases the design storey drift rather than the rotational capacity at the wall base willgovern the design. In this case, the design displacement of the single short wall will be smallerthan the design displacement of the single long wall. Assuming the profile of the coupled wallsystem is a combination of the profiles of the two single walls approximating the coupled wallprofile by the profile of the long wall will yield in a slightly unconservative design.

- If not only two but several walls are coupled together it might not be reasonable to assume that

the longest wall solely controls the displacement.- If the lateral connection between the walls is to a certain degree flexible the profiles of the two

walls will not be exactly the same.


23/118


5

A possible approach to account for these points is to weight the profiles of the single walls by the baseshear ratios when computing the system displacement profile:

jii js v ,, ∆=∆ ∑ (3.1)

js,∆ displacement of coupled wall system at storey

iv base shear ratio of one wall

ji,∆ displacement of a single wall at storey

The profiles of the single walls should have similar effective displacements. The design displacement andthe effective mass of the coupled wall system are then obtained as

sd

js j

eff

js j

js j

sd

mm

m

m

,

,

,

2

,

, ,

∆

∆=

∆

∆=∆

∑

∑

∑ (3.2)

sd ,∆ design displacement of the coupled wall system

jm storey mass

eff m effective mass of coupled wall system

The yield displacement of the coupled wall system is not directly required for the design. Based on a bi-linear approximation of the force-displacement relationship of coupled walls it can be estimated as a

function of the yield displacements ∆ of the de-coupled walls and the base shear ratios carried by the

walls:

1

,

−

∆=∆ ∑

yi

i

s y

v (3.3)

s y,∆ yield displacement of the coupled wall system

i y,∆ yield displacement of the de-coupled wall i

The system displacement ductility is obtained as usual as the ratio of design displacement and yielddisplacement.

The yield displacement of a single wall is based on a linear curvature distribution:

−=∆

n

j

j y y jh

hh

31

2

1 2, φ (3.4)

y j ,∆ yield displacement at storey of a single wall

yφ yield curvature of a single wall

jh height of storey

nh total height of structure


24/118


6

The effective height is determined from the design profile. This is different to the estimate of yield

displacement yeff ,∆ at the effective height used by Priestley and Amaris in [18]:

( )

2

, 7.03

1n y yeff hφ =∆ (3.5)

Priestley and Amaris already noted that this estimate of yield displacement is unrealistically low and hencethe effective damping, which is a function of the displacement ductility of the wall element, tends to beoverestimated. The yield curvature of the wall sections is estimated as in [18]:

w

sy

yl

ε φ

2= (3.6)

syε yield strain of main longitudinal reinforcement

wl wall length

The maximum design displacement of a wall on a rigid foundation is determined by the design drift d θ

[18]:

−

−+

−=∆

22

1

31

2

1 2,

p

jn yd

n

j

j yd j

lhh

h

hh φ θ φ (3.7)

d j ,∆ design displacement of a wall at storey

d

θ design drift

pl plastic hinge length at wall base

The first term represents the deformation of the elastic wall, the second the rotation of a rigid body aboutthe idealised plastic hinge location at half the plastic hinge length above the wall base.

The effective damping of the coupled wall system is a function of the total energy absorbed during onecycle and the kinetic energy stored in the system. The effective damping of the system can be expressed interms of the effective damping ratios of the walls.

The effective damping ratio ieff ,ξ of a SDOF system is defined as

%1002 ,

,

id i

i

ieff V

A

∆=

π ξ (3.8)

i A energy absorbed per cycle

iV base shear corresponding to a SDOF system (i.e., eff basei h M V / = )

id ,∆ design displacement at effective height of wall

The effective damping ratio of the coupled wall system seff ,ξ can hence be written as

sd

id iieff

sd s

iseff v

V

A

,

,,

,,

2 ∆

∆=

∆= ∑

∑ξ

π ξ (3.9)


25/118


7

Assuming that the design displacement of a wall is equal to the system design displacement this yield inthe equation given in [19]:

∑= ieff iseff v ,, ξ ξ (3.10)

The effective damping of a structural wall at the design displacement results from two sources: the“elastic” viscous damping and the hysteretic damping. The term representing the hysteretic damping in

the following formula is exact for the modified Takeda model with parameters α =0.5 and β =0.0 (as usedin this study, section 4.2):

+

−−+=+=

∆

∆

iF

i

F

ihyst ielieff r r

Max ,,

,,,

11

1,005.0* µ

µ π γ ξ ξ ξ (3.11)

ieff ,ξ effective damping of wall

iel,ξ elastic viscous damping of wall

ihyst ,ξ hysteretic damping of wall

F r post-elastic stiffness ratio of the base shear – effective displacementrelationship for wall

i,∆ µ displacement ductility of wall

In [22] Priestley and Grant reason that the elastic viscous damping term is best modelled by tangentstiffness proportional damping. In most analyses some elastic viscous damping is introduced with the

intention to account for energy dissipation by the following mechanisms:

- Hysteretic damping when the structural element is still elastic

- Foundation damping

- Non-structural damping

For all three sources a case for tangent stiffness proportional damping can be made [22]: Hystereticdamping of structural elements in the inelastic range is modelled by the hysteresis rule and hence noadditional damping should be added during the inelastic response. Foundation damping is proportional tothe load increments induced by the structure onto the soil and therefore diminishes when the system haspassed its yield point. Stiffness and strength of non-structural elements tend to degrade rapidly withincreasing drift demand. Moreover, the contribution of non-structural elements to the overall lateralstrength is small, commonly less than 5%. This does not include masonry infills which should be modelledby separate elements which account for the stiffness and damping characteristics of masonry infill panels.

DDBD is based on the secant stiffness and not on tangent stiffness. Hence, the elastic damping termneeds to be reduced as proposed by Priestley and Grant [22] to account for the difference in tangent andsecant stiffness at the design point. The reduction factor for the elastic damping term depends on the

displacement ductility ∆ µ and the post-elastic stiffness ratio F r :

( )( )F r −−−= ∆ 11095.01 µ γ (3.12)

The effective period eff T and base shear demand d V are determined in the established way of the DDBD

approach:


26/118


8

2

,

,

, 2,

∆=

∆=

eff

sd eff d

d

sd

ceff T

mV S

T T π

ξ

(3.13)

cT corner period of displacement spectrum

sd ,∆ system design displacement at effective height

ξ ,d S spectral displacement at corner period for effective damping ratio ξ

eff m effective mass

The reduction of spectral displacement with effective damping can be calculated using the followingrelationship:

seff

d d S S ,

%5,,2

7ξ

ξ +

= (3.14)

%5,d S spectral displacement at corner period for damping ratio of 5%

The base shear is distributed between the walls according to the chosen base shear ratios 1v and 2v and

over the height in proportion to the effective storey mass:

∑ ∆∆

=dj j

dj j

d i jim

mV vF , (3.15)

jiF , lateral force on wall at storey

For a storey mass of 200t a coupled wall system with walls of different lengths is designed (see section 2). To meet the design requirement of a maximum drift of 2% the wall lengths have been chosen as 6.0m and4.0m.

A principal design parameter is the distribution of base shear between the two walls. In this study the

design base shear was distributed between the two walls proportionally to2

wl as recommended in [15].

Distributing the base shear proportional to the square of the wall lengths yields in similar reinforcementratios for the two walls (assuming similar axial load ratios). With wall lengths of 6.0m and 4.0m the baseshear ratios are taken as:

3.0

7.0

,

,2

2

,

,1

1

==

==

bases

base

bases

base

V

V v

V

V v

(3.16)


27/118


9

Table 3.1 Properties of Wall 1, Wall 2 and coupled wall system

Wallwb

[m]

wl

[m]

wn lh /

[-]

b D

[mm]

pl

[m]

eff h

[m]

eff y,∆

[m]

eff d ,∆

[m]

∆ µ

[-]

φ µ

[-]

Wall 1 0.20 6.0 4.0 20 1.92 (17.8) 0.089 0.269 3.02 9.78

Wall 2 0.20 4.0 6.0 20 1.52 (17.9) 0.134 0.269 2.02 6.52

Coupled Walls 17.8 0.099 0.269 2.72

Table 3.2 Properties of Wall 1, Wall 2 and coupled wall system, continued

WallF

r

[-]eff ξ

[%]

eff T [s]

bd V V / [-]

d V [kN]

d M [kNm]

nV [kN]

n M [kNm]

base N [kN]

Wall 1 0.01 16.3 0.70 1080 19200 991 17600 900

Wall 2 0.01 12.9 0.30 462 8210 438 7780 750

Coupled Walls

15.3 2.82 1.00 1540 27400 1430 25400

wb wall width

wl wall length

b D bar diameter of main longitudinal reinforcement

φ µ curvature ductility

d V design base shear (at design, i.e. ultimate displacement)

bV design system base shear

d M design base moment

nV nominal base shear (at design, i.e. ultimate displacement)

n M nominal base moment

base N axial compression load at base of wall

For comparison: assuming the long wall imposes the displacement on the short wall would have lead to atotal nominal base shear value of 1500kN compared to 1540kN which resulted from the design above.

This is a difference of –2.6% which can be regarded upon as negligible. Hence, if only two walls arecoupled it does not seem significant if the system displacement is approximated by the displacement ofthe long wall or by the weighted sum of the profiles of the two single walls.


28/118


10

To simplify the study no section analysis was carried out but the post-yield stiffness ratio r of themoment curvature relationship was assumed as 0.01. Based on the concepts for a simple cantilever thepost-yield stiffness ratio of the force displacement relationship can be derived:

Elastic Stiffness:

yeff eff

n

y

n

elh

M V k

,∆=

∆= (3.17)

Post-elastic stiffness:

2

11

*

*

***

**

eff peff peff eff

plhl

EI r

hld H

EI r d

d H

dM

d

dF k =

Φ

Φ=

∆=

∆= (3.18)

r post-elastic stiffness ratio of the moment-curvature relationship at the wall

base

Post-elastic stiffness ratio:

yeff p

y

neff p

y

el

pl

F hl

r

M hl

EI r

k

k r

φ **

*

**

** 1 ∆=

∆== (3.19)

1 EI cracked wall stiffness

For the long wall a post-elastic stiffness ratio F r of 0.035 results from the above equations. For the short

wall the ratio is slightly larger (0.044). Both values lie within the commonly expected range of 0.02-0.05.

The nominal moment was obtained from the design moment based on the curvature ductility and thepost-elastic stiffness ratio of the moment-curvature relationship:

( )n pn p hlhl / 5.01 / 3

11

−

−+= ∆

µ µ φ

(3.20)

( )11 −+=

φ µ r

M M d n (3.21)

3.2 LITERATURE REVIEW ON BASE SHEAR DEMAND OF COUPLED WALLS

In 1999 Rutenberg and Leibovich [26] published pushover analysis results of coupled walls with differentlength similar to those discussed in section 4.3.3. They noted that coupling of walls of different length willlead “to a redistribution of lateral forces among walls” and that the shear forces transmitted through thefloor diaphragms might be “quite large” [26]. Paulay discusses the pushover analyses presented in thispaper [13] and notes that it is “astonishing that immediately after general yielding, when the element withthe greatest resistance lost nearly all its strength, the other elements suddenly develop resistance severaltimes greater than the strength that has been assigned to them and which they are not expected to be ableto develop” [13]. He also states that “the unsteady response of the elements immediately after onset ofyielding (…) is incompatible with reality”.


29/118


11

Rutenberg and Leibovich [27] derived analytical expressions for coupling forces between two walls ofunequal length where one wall is considerably stiffer than the other and the floor diaphragms are infinitelyrigid. The two storey case is shown in Figure 3.1. The deflected shape of the structure represents the stateafter formation of the plastic hinge at the base of the long wall and before formation of the plastic hinge

at the base of the short wall. The storey forces are required to reach compatibility of any additionaldisplacement of the long and the short wall after formation of the plastic hinge at the base of the long wall. If the walls would be de-coupled:

- The long wall would rotate as a rigid body about the plastic hinge at the base.

- The elastic deformation of the short wall would continue to increase.

The in-plane rigid floor diaphragms require that the displacements of the long and the short wall at thestorey levels are equal. The resulting storey forces for a point load at the top and infinite rigidity of thelong wall are shown in Figure 3.1. At this stage an increment of H ∆ of the external load at the top floorreduces the shear force at the base of the long wall by 1.5 H ∆ and increases the base shear force of theshort wall by 2.5 H ∆ . The increase in base shear force in the shorter wall is caused by redistribution of the

moment increase in the lower storey from the long to the short wall after the long wall has yielded [28]. Inthe upper storey the long wall is still elastic and is able to carry part of the incremental moment. Themoment transfer in the lower storey is accomplished “by means of force couples through the floordiaphragm” [28]. This is the explanation for unsteady response of the base shear forces after formation ofthe plastic hinge at the base of the long wall and the increased base shear demand on the short wall asobserved from the pushover analysis.

Figure 3.1 Two storey system of coupled walls: (a) properties and loading, (b) floor forces and deflected

shapes after formation of a plastic hinge at the base of the long wall [28]

Rutenberg suggested a design procedure on the basis of a cyclic static analysis to assess the base sheardemand in structural walls coupled by floor diaphragms. The procedure comprises the following steps:

- The system base shear demand sboV ,ωφ is determined for a substitute structure with the

combined properties of the structural walls (combined stiffness and moment capacities). Thedynamic amplification factor ω is determined on the properties of this substitute structure.

eff

in

sbh

M V

∑=

,

, (3.22)

eff h effective height as determined in Table 3.1


30/118


12

in M , nominal moment capacity of wall

- From the system base shear demand including dynamic amplification the minimum effective

height min,eff h is determined as

ω

eff

eff

hh =min, (3.23)

- A cyclic static analysis with the resultant force at the height min,eff h is carried out and a maximum

base shear of sboV ,ωφ . The base shear demand on the walls obtained from the pushover analysis

are the design shear forces.

Rutenberg [28] also carried out inelastic time-history analyses of a 10-storey building which had beenpreviously studied by Paulay and Restrepo [15]. The structure comprises four walls of different length.

Rutenberg showed that the formula given in equation (3.24) accurately predicts the maximum base sheardemand on a wall in a coupled system if elastic-plastic material behaviour is assumed.

Rutenberg [28] suggests that it is not necessary to carry out a cyclic static analysis but that the peak baseshear forces can be estimated from a uni-directional pushover analysis (again with the resultant force

sbV ,ω at the height min,eff h ) as follows:

ityiyi V V V −= 2max, (3.24)

jiV , base shear at yield of wall

jit V , base shear at wall when all walls have yielded

Although Rutenberg [28] did not explicitly state this it is believed that the above formula is based onelasto-plastic behaviour of the plastic hinge and is not applicable to general hysteresis rules. The modified

Takeda rule with parameters α =0.5 and β =0.0 seems better suited to describe the behaviour of reinforcedconcrete under seismic loading [17]. The two hysteresis rules are plotted in Figure 3.2. Both hysteresisrules have bi-linear back bone curves but the stiffnesses during the unloading and reloading phases aredifferent. After yielding of the short wall the reloading stiffness of the long wall will be smaller than itsinitial stiffness. Hence, the shear force demand on the long wall is less for the Takeda hysteresis rule thanthe bi-linear hysteresis rule. The maximum base shear demand on the short wall is the same for bothhysteresis rule because it might occur in the first loading phase. Figure 3.3 and Figure 3.4 show the resultsof cyclic static analysis for the modified Takeda hysteresis rule and the bi-linear hysteresis rule respectively.

In the first cycle the walls remains elastic, in the second and third both walls yield at the base. The thingreen and blue lines show the base shear demand for Wall 1 and 2 as predicted by Rutenberg’s formula.

Two years before Rutenberg published the design procedure for coupled walls [28] Rutenberg andLeibovich [27] had carried out pushover analyses of an 8-storey building which had been previouslystudied by Priestley and Kowalsky [20]. In the longitudinal direction of the building the stiffness isprovided by three members, one wall of 6m length and two walls of 3m length. Rutenberg and Leibovichstudied the model designing the members for two different strength distributions [27]. In the first model

the strength was distributed proportionally to2

wl , in the second model proportionally to3

wl . For the

model where strength is assigned proportionally to2

wl the results show that after yielding of the long wall

the long wall looses almost its entire strength and the two short walls carry the entire base shear. The

pushover results for the model where strength is distributed according to3

wl the base shear-displacementplot suggests that the long and the short walls yield simultaneously; as a consequence there are nocompatibility forces between the walls and the pushover results are identical to those of de-coupled walls.


31/118


32/118


14

Figure 3.3 Cyclic static analysis for rigidly coupled walls which hystretic behaviour is modelled with the

modififed Takeday rule: Base shear, base moment and top displacement (thin lines in base shear plot:

maximum base shear forces predicted by equation (3.24))


33/118


34/118


16

10 / 9.0 nTM +=ω where n≤6

8.130 / 3.1 ≤+= nTM ω where n>6(3.25)

•

“Intensity Based Method”: dynamic amplification factor based on effective period anddisplacement ductility [21]

)(1 T o

IM BΦ

+= ∆ µ

ω

with 15.1)5.0(*4.0067.0067.0 )( ≤−+=≤ eff T T B

(3.26)

The dynamic amplification factor of the “intensity based method” was derived from casestudies of single cantilevers [21].

The base shear is computed as

eff

nod

h

M V ω φ = (3.27)

oφ overstrength factor

ϖ dynamic amplification factor

eff h static effective height of the system (i.e. not considering higher modeeffects)

n M nominal base moment

The base shear was distributed over the height of the wall proportionally to the product of storey massand design displacement (in fact only proportionally to design displacement since the storey mass isconstant over the height of the building).

The design bending moment profile is different for the “traditional method” and the “intensity basedmethod”. The “traditional method” suggests a linear design moment profile over the height of the wall while the “intensity based method” proposes a bi-linear moment profile with 75% of the base moment atmidheight (Figure 3.6).

For the eight-storey structure considered in this project the dynamic amplification factor according to the

“traditional method” is TM ω =1.70. The dynamic amplification factor of the “Intensity based method” is

computed on the basis of the effective period and the displacement ductility of the system (Table 3.2)

assuming for the intensity based method an average overstrength factor of oφ = 1.08:

50.3995.008.1

72.211

995.0)5.082.2(*4.0067.0

)(

)(

=+=Φ

+=

=−+=

∆T

o

IM

T

B

B

µ ω

The dynamic amplification factor derived according to the “intensity based method” is hence 3.50.

A further approach for the derivation of the dynamic amplification was suggested by Rutenberg [28]. This

approach however is not based on approximate equations but involves non-linear time-history analysis ofa structural wall. Rutenberg [28] suggested that the amplification factor could be computed by lumping theproperties of the two walls into one super-wall and carrying out non-linear time-history analysis with


35/118


17

spectrum compatible records. The properties of the super-wall for the example of the coupled wallstructure considered here are

Table 3.3 Properties of the “super-wall” combining the stiffnesses and capacities of the two coupled walls

EI

[kNm²]

n M

[kNm]

r

[-]

Super-wall 3.04*107 25400 0.01

The dynamic amplification factor is then computed by solving equation (3.27) for ω :

max

max

M

H V eff =ω (3.28)

maxV maximum average base shear of “super-wall” from time-history analysis with spectrum compatible records

max M maximum average base moment of “super-wall” from time-history

analysis with spectrum compatible records ( max M includes increase in

moment capacity due to post-yield stiffness, i.e. on M M φ =max )

The analysis was carried out as described in chapter 4. The average of the maximum base shears from theanalyses was 5015 kN, the average of the maximum base moments 27400kNm. The dynamic amplificationfactor according to Rutenberg’s method was hence

26.327400

8.175020

max

max=

⋅==

M

H V eff SW ω

This is fairly similar to the “intensity based” dynamic amplification factor. The maximum base shear in thecoupled wall system where the walls are modelled as two walls rather than one “super-wall” is howeverconsiderably lower. From time-history analysis of the coupled wall system with properties as described insection 4.4 the system base shear was computed as the sum of the base shear forces of the long and short wall, i.e. the base shear forces of the long and short wall were added in the time domain. The average of

the maximum system base shear forces of the five time histories analyses was 4070kN, the averagemaximum system overturning moment was 26800kNm and the dynamic amplification factor for thesystem base shear force hence only

70.226800

8.174070

max

max=

⋅==

M

H V eff CW ω

Four different dynamic amplification factors were derived. The first two, the dynamic amplification

factors according to the traditional method TM ω and the intensity-based approach IM ω , were derived

from approximate equations which are suitable for design. The other two were derived from non-linear

THA of a “super-wall” ( SW ω ) combining the stiffnesses and capacities of the two coupled walls into one

and from non-linear THA of the actual coupled wall system ( CW ω )

. The latter is the “true” amplificationfactor of the coupled wall system. The resulting dynamic amplification factors are summarized in Table3.4. The dynamic amplification factor derived from the intensity-based method is very similar to the factor


36/118


18

derived from THA of the “super-wall” showing that the intensity-based method is indeed very suitablefor deriving amplification factors for single walls. For coupled walls the dynamic amplification factors aresmaller than for the equivalent “super-wall”. The dynamic amplification factor derived according to thetraditional method significantly underestimates the true amplification factor of the coupled wall system

and would hence lead to an unsafe design base shear force of the coupled wall system.

Table 3.4 Dynamic amplification factors for the base shear of the example 8-storey coupled wall system

derived from different approaches

“Traditional Method”TM ω =1.70

“Intensity based method” IM ω =3.50

THA with “super-wall”SW ω =3.26

THA with coupled wall systemCW ω =2.70

3.3.2 Design base shears when treated as de-coupled walls

In design approaches as suggested by Paulay and Restrepo [15] the coupling effect between walls ofdifferent lengths is not considered when determining the base shear demand on the walls. Once the basemoment is distributed between the coupled walls, the wall determined which controls the displacement

capacity of the system the walls are treated effectively as de-coupled walls. The dynamic amplificationfactor is however determined from the system properties, i.e. the traditional approach yields in a dynamic

amplification factor of TM ω =1.70 and the intensity-based approach in a dynamic amplification factor of

IM ω =3.50 (see Table 3.4). In the following the design base shears of the two walls are computed as for

de-coupled walls, i.e. as single cantilever walls according to equation (3.27). The resulting dynamic baseshear forces for the two different proposed equations for dynamic amplification factors are summarised in Table 3.5. The design shear force and bending moment profiles are shown in Figure 3.6 and Figure 3.7. The differences between the base shear forces yielding from the two approaches reflect the difference ofthe two dynamic amplification factors which differed by more than a factor of two.


37/118


19

Table 3.5 Design base shear forces for the “Traditional Method” and the “Intensity Based Method” when

walls are designed as de-coupled walls

Wall 1 Wall 2

Tradit.Method

IntensityBased

Method

Tradit.Method

IntensityBased

Method

Overstrength Factor oφ 1.09 1.09 1.06 1.06

Dyn. Ampl. Factor ω 1.70 3.50 1.70 3.50

Total Factor ω φ o 1.85 3.82 1.80 3.82

Design Base Shear 1840kN 3780kN 789kN 1670kN

3.3.3 Design base shears when treated as coupled walls as suggested by Rutenberg [28]

Rutenberg’s design base shear forces are based on the base shear demand on a substitute structure withthe combined properties of the walls, i.e. the properties of the entire system. The displacement ductility ofthe system is 2.72 (Table 3.1). The dynamic amplification assuming for the intensity based method an

average overstrength factor of oφ = 1.08 are hence (see Table 3.4):

Traditional method: ω =1.70Intensity based method: ω =3.50

The system base shear is determined from equation (3.22) as 1420kN. Hence, the system base shearsincluding the dynamic amplification factor and the overstrength factor are

Traditional method: sbo V ,ω φ =2610kN

Intensity based method: sbo V ,ω φ =5370kN

The average maximum system base shear obtained from the five time history analyses was 4070kN (seesection 3.3.1). The minimum effective height for the different dynamic amplification factors according toequation (3.23) are summarized for the different dynamic amplification factors in Table 3.6.


38/118


20

Table 3.6 Minimum effective height for the example 8-storey coupled wall system derived from different

approaches

Dynamic

AmlificationFactor

Minimum effective height

“Traditional Method”TM ω =1.70 min,eff h =17.8m/1.70=10.5m

“Intensity based method” IM ω =3.50 min,eff h =17.8m/3.50=5.1m

THA with “super-wall”SW ω =3.26 min,eff h =17.8m/3.26=5.5m

THA with coupled wall systemCW ω =2.70 min,eff h =17.8m/2.70=6.6m

Force-controlled pushover analysis was carried out with resultant forces at 10.5m and 5.1m height. Themodel used for this analysis is the lumped plasticity model described in section 4.2 (rigid floor diaphragms,

rigid foundation, shear area grosss A A 8.0= ). The base shear – top displacement relationships are plotted in

Figure 3.5. The top displacement at ultimate displacement is approximately 0.390m. The design base shear

forces for Wall 1 and Wall 2 based on the method suggested by Rutenberg for TM ω and IM ω are

summarised in Table 3.5. They are the base shear forces obtained from the pushover analyses when thesystem base shears are 2610kN and 5370kN respectively.

Table 3.7 Design base shear forces for the “Traditional Method” and the “Intensity Based Method” when

walls are designed as coupled walls as suggested by Rutenberg

Wall 1 Wall 2

Tradit.Method

IntensityBased

Method

Tradit.Method

IntensityBased

Method

Design Base Shear 1610kN 3400kN 1300kN 1970kN


39/118


21

Figure 3.5 Force-controlled pushover analyses with resultant forces at h=5.1m (Intensity based method)

and h=10.5m (Traditional method)

3.3.4 Summary of Design Shear Forces and Design Moments

The design bending moment envelopes are different for the traditional method and the intensity basedmethod. However, they are independent whether the walls are designed as de-coupled or coupled walls.

The design bending moment envelopes are plotted in Figure 3.6.

The shear force envelopes are plotted in Figure 3.7. For each wall there are four envelopes which resultfrom the combination of traditional and intensity based method and of design as de-coupled walls andcoupled walls. Designing the walls as coupled walls increases the design shear forces for the short wall anddecreases the design shear forces for the long wall with respect to those of de-coupled walls which aredesigned as single cantilever walls. The difference between design shear forces for one wall hence amountsfrom two different sources:

- Coupling forces between the long and short wall: Coupling forces were not considered when the walls were designed as de-coupled, single cantilever walls but they were considered when thedesign base shear force was determined by the pushover method proposed by Rutenberg [28] (seesection 3.3). The design shear force envelopes resulting from Rutenberg’s design approach wereplotted as dashed lines, the design shear forces envelopes resulting from the design approach when the walls were considered as de-coupled were plotted as solid lines.


40/118


22

- Dynamic amplification factor: The dynamic amplification factor determined according to the

traditional approach was TM ω =1.70 (green lines) while the dynamic amplification factor

according to the intensity based approach was IM ω =3.50 (red lines).

Figure 3.7 shows that the design shear forces for one wall differ greatly depending on the approachchosen to derive the base shear forces. The shear force profile over the height was assumed to be thesame for all cases. It is debatable whether this is a sensible choice since the coupling forces certainly alterthe profile of shear force demand on one wall (see section 4.3.3).

Figure 3.6 Design bending moment envelopes according to the traditional and the intensity-based design

approach

Figure 3.7 Design shear forces when walls are designed as-coupled walls as suggested by Rutenberg

(dashed lines) and as de-coupled walls (solid lines)


41/118


42/118


24

point out that combinations of actions below or on the surface are however only potentially safe since thefoundation capacity was derived from the upper bound estimate of the plasticity theorem.

The soil for this example is a dense sand of several tens of meters of thickness which is classified as soiltype B according to Eurocode 8 [4]. It is assumed that the shear wave velocity of the soil is 500m/s, the

unit weight 20kN/m³ and the angle of shearing resistance in terms of effective stresses 'φ =30o. Thematerial safety factor M γ for the angle of shearing resistance is 1.25 [5]. Only accounting for the weight of

the gliding soil body and the overlaying soil mass above the foundation level the ultimate bearing capacity

at foundation level is M γ σ / max =3780kPa. Assuming that the horizontal effective mass of the

superstructure is a good approximation for its entire mass the vertical soil stress due to weight offoundation and superstructure is 90kPa (for a constant stress distribution over the foundation area).Hence, under static loading the bearing capacity of the foundation system is more than sufficient.

Apart from the vertical load due to weight of foundation and superstructure soil supporting thefoundation has to accommodate horizontal stresses due to horizontal loading and vertical stresses due tothe overturning moment. When computing the overturning moment at foundation level the additional

moment due to the eccentricity of wall base and foundation level has to be accounted for. From thedesign values at the wall bases (section 3.2) the horizontal load and overturning moment at the foundationlevel are obtained as

( )

kNmV h M M

kN V

f f walld f

f

898303501054820

11670141544202

, =+=+=

=+⋅=

∑

Assuming rigid behaviour of the foundation the shear stress demand at foundation level is 23.3kPa and

the maximum stresses of a linear stress distribution due to the moment demand )max( M σ =±43kPa.

Hence, uplift of the foundation is not expected to occur. For an average axial stress of 90kPa the shear

stress capacity of the soil is M γ τ / max =42kPa. Hence, all demand values are well below the corresponding

capacity values. Therefore, interaction of N V and M is not checked and permanent displacements arenot expected.

3.4.2 Dynamic Properties of Soil Foundation

The dynamic properties of soil are best described by its shear modulus , its Poisson’s ratio υ and its viscous elastic damping ratio β . EC 8 [4] proposes to reduce the shear modulus with increasingacceleration level to account for non-linear soil behaviour. Reduction factors for are only given foraccelerations smaller than 0.3g while the peak ground acceleration in this design project is 0.4g and the

modification factor has been extrapolated to 0.3. The dynamic properties of the soil are hence:

kPavGG s5622

max 1053.1101.53.050081.9 / 203.03.03.0 ⋅=⋅⋅=⋅⋅=⋅== ρ

3.0=v

%15= β

Under dynamic loading the soil will deform and absorb energy. Only the lateral component and therocking component of the foundation motion is considered in the following. A schematic drawing of themodel which includes foundation flexibility is shown in Figure 3.9. The spring and dashpot elements act inhorizontal and vertical direction. The soil-structure interaction problem is treated as proposed by Gazetaz[6] and assumes that there is no interaction with adjacent foundations.

Gazetas derived some approximate formulae for the dynamic stiffness and the radiation dashpotcoefficients. The formulae applied here are derived for arbitrarily shaped partially or fully embedded


43/118


25

foundations in a homogenous halfspace. The viscous elastic material damping is added to the radiationdamping.

The static stiffnesses of an embedded foundation are estimated as the static stiffness of a foundation onthe ground surface times a factor accounting for the embedding depth. The surface stiffnesses are

approximated by the theoretical solutions for a square foundation on an elastic halfspace since thehorizontal foundation dimensions are almost equal:

rad kNmGB

K

mkN GB

K

sufracery

surface x

/ 109.71

6.3

/ 101.82

9

83

,

6

,

⋅=−

=

⋅=−

=

υ

υ (3.29)

G shear modulus reduced for non-linear behaviour of soil

B half width of the foundation ( ≈10m)

υ Poisson’s ratio

It is assumed that there is a good contact between soil and foundation wall from 1m below the groundsurface onwards. Embedding increases the lateral stiffness by a factor of 1.47 and the rotational stiffnessby a factor of 1.30.

The ratio of dynamic to static stiffness of the soil depends on the loading frequency. Direct displacementbased design addresses the maximum displacement demand. The loading is hence characterised by theeffective period of the structure which – in a first step – is approximated as the effective period of a

system with rigid foundation. The dimensionless loading frequency is defined as

sc

Ba

ω =0 (3.30)

ω frequency of excitation (here: eff T =2.82s =2.23s-1 )

B length of the foundation perpendicular to the axis of the rocking motion

sc shear velocity of the soil medium at the surface reduced to account fornon-linearity of the soil

The dimensionless frequency is very low ( =0.08) and the dynamic stiffness of the foundation can beapproximated by its static stiffness. Foundation flexibility would reduce further and the ratio ofdynamic to static stiffness would shift even closer to unity.

As outlined above soil damping is constitute of two kinds: material damping and radiation damping. Forrelative large amplitude and long period vibration it is expected that material damping will be the maincontributor to the energy absorbtion by the soil. Nevertheless, radiation damping associated with lateraland rotational motion is estimated to confirm this assumption.

Embedding a foundation increases the radiation damping due to additional energy loss at the foundation wall faces [6]. Good contact between soil and wall faces is required. For a foundation system on theground surface the radiation damping for rocking motion with long periods is almost zero and the

radiation damping for rocking motion of an embedded foundation results solely from energy loss over the wall faces. The effect of embedding on radiation damping associated with lateral foundation motion is lesssignificant. In this study embedding increases the radiation damping coefficient for lateral motion by


44/118


26

about 46%. The radiation damping coefficients for lateral and rotational motion for the embeddedfoundation are:

rad kNmsC

mkNsC

rad rot

rad y

/ 100.4

/ 107.2

6,

5

,

⋅=

⋅=

As expected the contribution of radiation damping to the overall damping is negligible. The final dampingcoefficients are almost solely governed by the material damping of the soil:

rad kNmsK C C

mkNsK C C

eff rot rad rot rot

eff yrad y y

/ 101.1 / 2

/ 107.1 / 2

8

,

6

,

⋅=+=

⋅=+=

ω β

ω β

3.4.3

Flexibility of Foundation Structure

The foundation structure is a reinforced concrete caisson which – compared to the superstructure - isrelatively stiff. However, there will be some deformation associated with inducing the wall base momentsand shear forces into the caisson. This caisson wall flexibility is modelled by elongating the wall axis to themidheight of the caisson and assigning to it the elastic properties of the wall base section (Figure 3.9). Inthe following the additional displacement and rotation at the wall bases due to deformation of the caisson walls is estimated. The estimate is only very approximative and shear deformations are not accounted for. The rotation and displacement at wall base level due to flexural deformation of the caisson walls can beexpressed with the wall base properties:

+=

+=

ω φ φ θ

ω φ φ

eff

o y

eff o y

h

L L

h

L

Lu

2

11

3

2

12

1 2

(3.31)

oφ flexural overstrength

ω dynamic amplification factor for shear forces

yφ yield curvature of the wall base

L length of the elastic element modelling the caisson wall flexibility (here

estimated as half the caisson height, i.e.1.5m)

eff h effective height of the wall

If the walls would be uncoupled the following additional displacement due to caisson wall flexibility wouldbe expected:


45/118


46/118


28

Figure 3.9 Scheme of analytical model with flexible foundation

3.4.4 Foundation Flexibility in DDBD

It was outlined in the beginning of section 3.4 that two principal sources of foundation flexibility areconsidered:

- Soil flexibility

- Flexibility of the foundation structure

Both sources result in additional elastic deformation, leading to lengthening of the effective period whichcan be easily incorporated in the direct displacement based design approach. Soil flexibility is alsoassociated with additional dissipated energy due to material and radiation damping (section 3.4.2). Theenergy dissipated by soil and structure are additive. The approach to incorporate soil flexibility is describedin detail in [19]. Additional deformation due to elastic deformation of the foundation structure increases ingeneral displacement and drift of the superstructure. The way the additional deformation can be estimated was shown in section 3.4.3.

In this study the main source of foundation flexibility is associated with local elastic deformation of thecaisson walls due to inducing wall base moments and shears into the caisson. The expected totaldisplacement at the effective height is 0.30m and the expected maximum drift at the top of the structure2.16% (section 3.4.3).

The effective period increases from 2.82s to 3.00s. The system displacement ductility reduces from

s∆ µ =2.72 (rigid foundation, section 3.2) to

31.210.003.0

30.0=

+=∆s µ

Due to the smaller displacement ductility the effective damping reduces from 15.3% to 13.8%. The new

base shear demand on the system (not including dynamic amplification) is d V =1360kN. The new intensity

based dynamic amplification factor for the coupled wall system is 3.55 compared to 3.50. Hence, on thebasis of Rutenberg’s design approach for coupled wall systems the new system base shear is 4840kNcompared to 5370kN for a system with rigid foundation. The difference is about 10%.

Plastic hinge

at wall base

Elastic wall section

Elastic foundation

beam

Plastic hinge

at wall base

Elastic wall section

Elastic foundation

beam


47/118


29

Hence, if the system would be redesigned accounting for the flexibility of the foundation the capacity ofthe wall bases could be reduced. It was stated before that in this study the walls are not redesigned butonly reanalysed with the foundation flexibility included in the model. The yield moments at the wall basesact like a “fuse” hence the shear force and bending moment demand on the wall structures is expected to

be similar for a rigid and a flexible foundation.

3.5 SUMMARY OF CHAPTER 3

In this chapter the design of the coupled wall system according to DDBD was described and differentapproaches of determining the base shear demand on the structural walls were discussed. The main pointsmade were:

- The design of coupled wall systems using the DDBD method is almost identical to the design ofa single wall. It only requires additional assumptions on the displacement profile and the effectivedamping of the coupled wall system.

-

Determining the base shear demands on the walls calls for particular attention. Rutenberg [28]explained the mechanism between base shear distribution in coupled walls of different length andproposed a design method which accounted for the coupling effect. Base shear forces obtained with this method were computed for two different assumptions for the dynamic amplificationfactor and compared to base shear forces which would be obtained if the two walls would beconsidered as decoupled. For the long wall the resulting base shear forces differed by a factor ofabout 2.3, for the short wall the resulting base shear forces differed even by a factor of 2.5.

- The effect of foundation flexibility on the design was studied in section 3.4. Shallow foundation was not a feasible solution and hence the structure was founded on a fairly rigid caisson. The only veritable source of additional flexibility for the considered design were local deformation of the

caisson walls were the structural walls connected to the caisson. This causes a lengthening of theeffective period by about 6% and a reduction in system base shear demand by about 10%.


48/118

Chapter 4: Analysis of Coupled Wall Systems

30

4 ANALYSIS OF COUPLED WALL SYSTEMS

In this chapter the assumptions behind an element model are described (section 4.1) and a detaileddescription of the model is given (section 4.2). The model was analysed using the code “Ruaumoko2D”developed by Carr [3]. Pushover analyses of the coupled wall structure are compared to those of the de-coupled walls (section 4.3). The chapter concludes with the results obtained from non-linear time-history

analysis o

Dissertation2005 Beyer

Documents

Transcript of Dissertation2005 Beyer