displacement based seismic design of rc wall buildings accounting for nonlinear...

8/12/2019 Displacement Based Seismic Design of RC Wall buildings accounting for Nonlinear Soil-Structure-Interaction effects

1/134

Displacement-Based Seismic Design of RC

Wall Buildings accounting for Nonlinear Soil-

Structure Interaction

A Dissertation Submitted in Partial Fulfilment of the Requirements

for the Master Degree in

Earthquake Engineering

By

Dimitrios Sotiriadis

Supervisor(s): Dr. Timothy Sullivan

Dr. Antonio Araujo Correia

February, 2014

Istituto Universitario di Studi Superiori di Pavia

Universit degli Studi di Pavia


2/134

Abstract

i

The dissertation entitled Displacement-Based Seismic Design of RC Wall buildings

accounting for Nonlinear Soil-Structure Interaction effects, by Dimitrios Sotiriadis, has been

approved in partial fulfilment of the requirements for the Master Degree in Earthquake

Engineering.

Timothy Sullivan

Antonio Araujo Correia


3/134

Abstract

ii

ABSTRACT

The role of Soil-Structure Interaction (SSI) in the seismic response of structures has been

investigated during the last decade by many researchers proving the inadequacy of the current

seismic building codes in accounting for such effects. In general, SSI effects are believed to

act in favour of the seismic response of the structures and, in most cases, are neglected.

However, a number of real earthquake incidents along with experimental evidence, has shown

that SSI may lead to detrimental effects that may cause unexpected damage or collapse. Thus,

in order to be sure that a proper structural design has been made for a structure, properevaluation of SSI effects is necessary especially when the foundation soil and the structure

itself has properties that promote such effects.

As Performance Based Seismic design of structures becomes more and more common and it

is adopted in design, it is more than apparent that a proper evaluation of the deformations

coming from the foundation itself is necessary. Towards this direction, much experimental

and analytical work has been conducted in order to evaluate the performance of foundation

systems under seismic excitation including the energy dissipation characteristics, the stiffness

and strength degradation etc. At the same time, efficient numerical tools, called macro-

elements, which require low computational cost, have been developed and demonstrated

remarkable accuracy in representing the foundation response under static cyclic or shake table

excitation.

In this study, a shallow foundation macro-element is implemented in order to derive empirical

curves of stiffness degradation and equivalent damping in order to include SSI effects directly

into a Displacement Based Design process and obtain the optimal design of both


4/134

Abstract

iii

superstructure and foundation at the same time. The design curves are implemented in the

Seismic design of RC core wall buildings with different heights and the design outcome is

compared with the fixed based design approach.

Keywords: Soil Structure Interaction, shallow foundations, reinforced concrete, walls, seismic design.


5/134

Index

iv

TABLE OF CONTENTS

PageABSTRACT ........................................................................................................................................... iiTABLE OF CONTENTS....................................................................................................................... ivLIST OF FIGURES ................................................................................................................................ 1LIST OF TABLES .................................................................................................................................. 61 INTRODUCTION ............................................................................................................................. 1

1.1 Research Objectives ................................................................................................................... 11.2 Organization of the report .......................................................................................................... 2

2 LITERATURE OVERVIEW ON SFSI ............................................................................................. 42.1 General Description ................................................................................................................... 42.2 Kinematic Interaction................................................................................................................. 42.3 Inertial Interaction ...................................................................................................................... 52.4 Beneficial and Detrimental effects of SFSI ............................................................................... 72.5 Linear SFSI .............................................................................................................................. 112.6 Nonlinear SFSI......................................................................................................................... 122.7 Modeling of Nonlinear SFSI .................................................................................................... 13

2.7.1 Modeling of the Near-Field sub-domain ........................................................................ 152.8 Direct Displacement Based Design and SFSI ....................................................................... 182.9 Soil Structure Interaction Eurocode 8 ................................................................................... 23


6/134


7/134

Index

vi

6 CONCLUSIONS ........................................................................................................................... 1046.1 Stiffness degradation and damping curves of shallow foundations ....................................... 1046.2 Direct Displacement Based Design accounting for nonlinear SFSI ...................................... 1056.3 Future research ....................................................................................................................... 106

REFERENCES ................................................................................................................................... 107APPENDIX A ...................................................................................................................................... 1APPENDIX B ...................................................................................................................................... 1


8/134

Index

1

LIST OF FIGURES

Figure 2.1: Schematic description of SFSI. ................................................................................7Figure 2.2: Comparison between code and real earthquake response spectra with high spectral

acceleration ordinates in long periods (5% damped spectra). .............................................9Figure 2.3: Conceptual sub-domains for nonlinear SFSI modelling. .......................................14Figure 2.4: Generic SFSI modelling in the case of macro-elements. .......................................18Figure 2.5: Schematic image of ultimate loads surface used in most of the SFSI macromodels

(QN=normalized vertical force, QV=normalized horizontal force, QM=normalized

moment). ...........................................................................................................................18Figure 2.6: Variation of shallow foundation rotational secant stiffness and damping ratio for

dense and medium-dense sands, as a function of foundation rotation (Paolucci et al.,

2012). ................................................................................................................................20Figure 2.7: Empirical curves for secant stiffness degradation and corresponding increase of

damping ratio [Paolucci et al., 2012]. ...............................................................................21Figure 2.8:Flow chart summarizing the DDBD+NLSFSI design procedure by Paolucci et al.

[2012]. ...............................................................................................................................22Figure 3.1: Determination of for a SDOF rocking system under earthquake record CC1. ...28Figure 3.2: Determination of for SDOF rocking systems under earthquake record CC3. ....28


9/134

Index

2

Figure 3.3: Determination of for SDOF rocking system under earthquake record CC5. ......29Figure 3.4: Determination of for SDOF rocking system under earthquake record CC10. ....29Figure 3.5: Response of SDOF rocking system under earthquake CC5. ..................................30Figure 3.6: Response of SDOF rocking system under earthquake CC10. ................................30Figure 3.7: Displacement reduction factor for 5% tangent-stiffness based elastic damping and

r=0.05. ...............................................................................................................................31Figure 3.8: Displacement reduction factor as a function of system's effective period (5%

tangent-stiffness based elastic damping, r=0.05). .............................................................31Figure 3.9: Average values of as a function of effective period and ductility (5% tangent-

stiffness based damping, r=0.05). .....................................................................................32Figure 3.10: Displacement reduction factor vs ductility for the CC earthquake record set (5%

initial-stiffness based elastic damping, r=0.05). ...............................................................33Figure 3.11: Displacement reduction factor vs. Ductility (5% secant-stiffness based elastic

damping, r=0.05). ..............................................................................................................33Figure 3.12: Displacement reduction factor as a function of system's effective period (5%

secant-stiffness based elastic damping, r=0.05). ...............................................................34Figure 3.13: Comparison between the DRF variation for different elastic damping models. ..34Figure 3.14: Comparison between the proposed uplift model regarding (a) moment-rotation

and (b) moment-heave diagram. .......................................................................................36Figure 3.15: Effect of Safety Factor (SF) on the rotation of uplift initiation (B=4m). .............37Figure 3.16: Effetc of Safety Factor (SF) on the stiffness degradation of a 4x4m footing. .....37 Figure 3.17: Effect of foundation width on the stiffness degradation of footings with SF=2. .37Figure 3.18: Stiffness degradation curves for He/B=1. .............................................................42


10/134

Index

3

Figure 3.19: Stiffness degradation curves for He/B=2. .............................................................43Figure 3.20: Stiffness degradation curves for He/B=3. .............................................................43Figure 3.21: Stiffness degradation curves for He/B=4. .............................................................44Figure 3.22: Stiffness degradation curves with respect to foundation rotation for a 7.5x7.5m

footing. ..............................................................................................................................45Figure 3.23: Comparison between the current curves and Paolucci's curves. ..........................46Figure 3.24: Calculation of hysteretic damping. .......................................................................47Figure 3.25: EVD results for He/B3.0 ....................................................................................48Figure 3.26: EVD results for He/B=2.0 ....................................................................................48Figure 3.27: EVD results for He/B=1.0. ...................................................................................49Figure 3.28: Comparison between EVD for different "Dmg" values. ......................................50

Figure 3.29: Comparison between Paolucci EVD curves and current methodology results(He/B=4). ..........................................................................................................................51

Figure 3.30: Comparison between Paolucci EVD curves and current methodology results

(He/B=4). ..........................................................................................................................51Figure 3.31: Comparison between Paolucci EVD curves and current methodology results

(He/B=4). ..........................................................................................................................52Figure 3.32: EVD curves fitted to the analysis results for He/B=1.0. ......................................53Figure 3.33: EVD curves fitted to the analysis results for He/B=2.0. ......................................54Figure 3.34: EVD curves fitted to the analysis results for He/B3.0. ......................................54Figure 4.1: Typical plan view of case study buildings. ............................................................57


11/134

Index

4

Figure 4.2: Elastic acceleration and displacement response spectra of design earthquake (5%

damping). ..........................................................................................................................60

Figure 4.3: Base Shear comparison between FBD and DDBD. ...............................................70Figure 4.4: Base moment comparison between FBD and DDBD. ...........................................70Figure 4.5: Simplified Capacity Design Envelopes for Cantilever Walls. ...............................71Figure 4.6: Ductility component of Concrete Shear - Resisting Mechanism. ..........................74Figure 4.7: Design chart for KAE for =40 coming from log spiral analysis. ........................76Figure 4.8: Design chart for KPEcoming from log spiral analysis for non-cohesive soil.........76Figure 4.9: Variation of foundation safety factors with wall aspect ratio. ...............................77Figure 4.10: Schematic flow chart of iterative design procedure (from Sullivan et al. [2010]).

...........................................................................................................................................79Figure 4.11: Inelastic DRF curves for different elastic damping modeling. ............................87Figure 5.1: Force - displacement response of study case buildings. .........................................94Figure 5.2: Displacement spectra of the earthquake records chosen and the design earthquake.

...........................................................................................................................................96Figure 5.3: Comparison between NLTHA and Design for the 6 storey building. ....................97Figure 5.4: Comparison between NLTHA and Design for 8 storey building. ..........................97Figure 5.5: Comparison between NLTHA and Design for the 12 storey building. ..................98Figure 5.6: Moment - rotation curves of footings. ....................................................................99Figure 5.7: Comparison between NLTHA and design displacement envelopes. ...................100Figure 5.8: Comparison between NTHA and DDBD analysis moment envelopes. ...............101


12/134

Index

5

Figure 5.9: Comparison between NLTHA and DDBD analysis shear force envelopes. ........101


13/134

Index

6

LIST OF TABLES

Table 2.1: Values of parameters for the derivation of the empirical curves. ............................21Table 3.1: Foundation Macroelement parameters. ...................................................................39Table 3.2: Internal damping ratio and soil stiffness degradation due to ground shaking as per

Eurocode 8-5. ....................................................................................................................55Table 4.1: Characteristics of case study buildings. ...................................................................59Table 4.2: Parameters of design earthquake response spectra. .................................................60Table 4.3: DDBD results for fixed base individual walls. ........................................................67Table 4.4: FBD results for individual walls of case study buildings. .......................................69Table 4.5: Design outcome of case study buildings. ................................................................73Table 4.6: Soil properties used for the foundation design. .......................................................77Table 4.7: Foundation design results for the case study buildings. ..........................................78Table 4.8: Superstructure and foundation design of case study buildings accounting for

nonlinear SFSI. .................................................................................................................81Table 4.9: Comparison between the fixed base and the NLSFI design approaches. ................82Table 4.10: Reinforcement detailing of the wall buildings. ......................................................83


14/134

Index

7

Table 4.11: Design outcome of the 12-storey wall for the two approaches(DDBD_NLE_:

Iinitial stiffness DRF curve, DDBD_NLE_SS: secant stiffness DRF curve). ..................90

Table 4.12: Rocking walls reiforcement design. ......................................................................91Table 5.1: Comparison between NLTHA and Design. ...........................................................102


15/134

Chapter1. Introduction

1

1 INTRODUCTION1.1 Research ObjectivesThe principal objective of this research is to derive a new displacement-based seismic design

method of reinforced concrete wall buildings accounting for nonlinear effects of Soil

Foundation Structure interaction. Those nonlinear effects mainly refer to yielding of

foundation subsoil (material nonlinearity) and foundation uplift (geometric nonlinearity), that

is detachment between one portion of the foundation and the supporting soil.

The research is focused on buildings whose lateral load resisting system comprises of

reinforced concrete structural walls and are supported on shallow foundation lying on sand.

The choice of studying such systems comes from the argument that they are more possible topromote soil structure interaction effects during strong ground motion. Thus, the effort of

taking into account soil structure interaction effects in such systems is necessary both for

safety and economical design reasons.

During the last decades, many researchers have highlighted the importance of soil structure

interaction for the safety of various structural systems (mainly bridge piers) and some of them

have even claimed that allowing the foundation subsoil to fail could function as a safety

valve of the superstructure. This behaviour scheme has been claimed to be efficient especially

for retrofit of existing structures where the foundation has been damaged during past

earthquakes and any rehabilitation effort would be too expensive and difficult. However, quite

recently, the idea of foundation failure has been introduced in the design of new structures

and a large number of investigations on that field have been reported.


16/134


2

A lot of important findings have been published as a result of the research undergone during

the past years. These findings have contributed on the development of efficient macro

element models whose accuracy compares with the one obtained with complex finite element

models but keeping the computational cost relatively low. The development of efficient and

accurately macro element models allows researchers to conduct parametric analyses such as

the one reported herein.

1.2 Organization of the reportFollowing the research objectives stated in Chapter 1, the organization of the report is

presented. The report was organized in such a way so that the reader can understand the basic

principles and effects of soil structure interaction on the response of structures during an

earthquake and clearly realize the effort that is made herein.

In Chapter 2, an overview on the Soil Foundation Structure Interaction (SFSI) is presented

so that its basic principles and effects are introduced along with significant research

developments that have been made on that field. Moreover, a reference is made on how

modern Seismic design building codes (Eurocode 8) address this issue and some criticism is

made.

In Chapter 3, the actual contribution of this research project is described that is the derivation

of stiffness degradation and damping curves of shallow foundations as a function of the

foundation rotation. Two cases are considered; one where foundation uplift is allowed but the

subsoil remains essentially elastic and one where both uplift and soil yielding are allowed.

The curves were constructed in such a way that they can be easily implemented in a

displacement based design framework.

In Chapter 4, the case study buildings are described and designed using the tools derived in

Chapter 3. Furthermore, for comparison reasons, the case study buildings are designed

assuming a fixed based. Except for the superstructure, the foundation is designed in both

cases so that a further comparison is made.

In Chapter 5, the design verification of the case study buildings both for the fixed-base case

and the SFSI case is presented. Two software programs have been used; Ruaumoko and


17/134


3

SeismoStruct, and important modelling aspects of the RC walls behaviour are addressed such

as inelastic shear deformations.

At last, in Chapter 6, the conclusions along with the uncertainties that exist in this research arepresented and any possible future research that can be done on that field is proposed.


18/134

Chapter 2. Literature Overview on SFSI

4

2 LITERATURE OVERVIEW ON SFSI2.1 General DescriptionThe term Soil - Foundation Structure Interaction (called SFSI from now on) comes from the

fact that during earthquake shaking the foundation, the surrounding soil and the superstructure

interact and one affects the motion and the response of the others. More specifically, the soil

deforms under the influence of incident seismic waves (shear waves, dilatational P-waves,

surface waves etc.) and carries with it the foundation and the superstructure. In turn, due to

the induced motion of the superstructure, inertial forces are developed on it. This results into

dynamic stresses at the foundation which are transmitted to the supporting soil. Thus,

additional deformations, coming from the inertial forces on the superstructure, are induced in

the soil while additional waves emanate from the soil-foundation interface. In response,

foundation and superstructure exhibit further dynamic displacements which produce further

inertial forces and so on.

In order to simplify the description of the aforementioned simultaneously occurring

phenomena, SFSI is separated into two successive phenomena referred to as kinematic

interaction and inertial interaction. The complete concept of SFSI is described by the

superposition of these two interaction effects.

2.2 Kinematic InteractionThe kinematic interaction refers to the effects of the incident seismic waves on the system

which consists of the foundation and the surrounding soil, setting any mass of the

superstructure equal to zero. The main effect of the kinematic interaction is the development

of the foundation motion which is generally different than the free field motion that is the

ground surface motion if there was no foundation or structure. Usually, the foundation motion


19/134


5

is smaller than the free field motion due to the difference in stiffness between the soil and

foundation. In addition, the foundation motion generally contains a rotational component.

This component is more significant for deep foundations while for surface foundations, it is

negligible. Also, the importance of the rotational component depends on the rigidity of the

foundation itself and as a matter of fact the more flexible the foundation the more significant

the rotational component is [Mylonakis et al, 2006].

From the above, one can conclude that the kinematic interaction effects become more

important as the embedment depth of the foundation increases. In the case of a surface

foundation, where the embedment depth is zero, which is subjected to vertically propagating

S-waves, there is no kinematic interaction effect at all. In Mylonakis et al. [2006],a state-of-

the art description and treatment of SFSI is presented. As far as kinematic interaction is

concerned, a number of cases of different foundation configurations and incident waves are

presented along with the relative importance of the kinematic interaction effects. Moreover,

the transfer functions (i.e. the functions that multiply the ground surface free field motion in

order to obtain the foundation input motion) coming from analytical studies are referred for

every case. In practice, especially for shallow foundations with small embedment depth and

for noncritical structures, the kinematic interaction effects are ignored leading to slightly

conservative designs, as explained in Mylonakis et al. [2006].

2.3 Inertial InteractionThe inertial interaction refers to the response of the complete soil foundation structure

system against the inertial forces which are developed due to the foundation input motion

which was defined in the previous paragraph and is a product of the kinematic interaction.

Usually, for surface or embedded foundations, inertial interaction is conveniently performed

in two steps.

The first step consists of computing the foundation dynamic impedances (springs and dashpot

coefficients) associated with each mode of vibration. The values of the dynamic impedance

coefficients depend on the soil and foundation characteristics. In the usual case where the

foundation is considered as perfectly rigid, there are six modes of vibration (3 translational

and 3 rotational) and the soil profiles properties mostly define the dynamic impedance


20/134


6

coefficients while the foundations contribution is limited to its geometrical properties. The

general form of an impedance function in a random mode of vibration is:

The first term is termed as dynamic stiffness and reflects the stiffness and inertia of thesupporting soil. The term is defined as a dashpot coefficient and reflects the radiation andthe material damping generated in the system during the earthquake shaking. Both terms

defined are frequency dependent, however, in order to be easy to implement in computer

codes, they are typically treated as frequency independent. The most usual values of these

terms are the ones that correspond to zero frequency (static components of impedance

functions). Nevertheless, as mentioned in Pecker and Pender [2000], simple rheological

models can easily be used to represent the frequency dependence of the impedance functions.

In the case of embedded foundations, the horizontal forces along the principal axes induce

rotational oscillations and thus, cross-coupling impedance terms exist between the

translational and the rotational degrees of freedom. However, in the case of shallow

foundations, with small embedment depth, these cross-coupling terms are quite insignificant

and, usually, are neglected [Mylonakis et al., 2006]. The factors which mostly affect the

dynamic impedance functions are:

The foundation shape and dimensions (strip, rectangular, circular etc). The soil profile properties (uniform or multi-layer deposit, depth of rock layer etc). The embedment (surface foundations, embedded foundations, pile foundation).

In Mylonakis et al.[2006],a number of analytical, approximate expressions and charts for the

calculation of foundation impedances for the most common cases (with respect to the factors

stated above) met in practice are presented and discussed.

The second step of the inertial interaction analysis includes the determination of the response

of the structure and the foundation supported on the springs and dashpots, which represent the

foundation impedance functions, and subjected to the accelerations coming from the

kinematic interaction. A schematic description of the SFSI is shown in figure 2.1.


21/134


7

Figure 2.1: Schematic description of SFSI (after Mylonakis et al [2006]).

2.4 Beneficial and Detrimental effects of SFSISoil Foundation Structure interaction has traditionally been considered to be beneficial for

the seismic response of the structures [Mylonakis et al. 2000]. This is due to the period

lengthening coming from the flexibility of the foundation in comparison with a fixed-based

structure. Also, additional seismic energy is dissipated through wave radiation and hysteretic

behaviour of the soil foundation system and, therefore, the damping ratio of a flexibly

supported structure is increased with respect to a fixed based structure. The combination of

the period lengthening and increased damping leads into lower values of spectral acceleration

in a code based acceleration response spectrum which, in turn, results into lower inertial

forces on the structure and more economical design. Moreover, it was observed that when

SFSI effects are included in the analysis the ductility demands on the structure decrease and

this fact comes to strengthen the belief of the beneficial role of SFSI [Mylonakis et al. 2000].


22/134


8

However, this belief is significantly influenced by oversimplifications made in the

representation of the seismic demand in building codes. In particular, these

oversimplifications are related to a) the smooth response spectra used for design (especially

for the portion of monotonically decreasing spectral acceleration with increasing period after

the plateau of constant acceleration), b) the behaviour factors used to derive design forces and

c) the calculation of the foundation impedances assuming homogeneous half - space

conditions for the soil, an assumption which overestimates the damping of structures on real

soil profiles [Mylonakis et al. 2000].

On the other hand, it has been shown that at sites with soft soil conditions and deep, relatively

uniform layers, the highest spectral acceleration ordinates occur in long periods, as shown in

figure 2.2. Also, SFSI is highly affected by dynamic phenomena such as resonance and

forward directivity effects. This is more evident in earthquake records on soft soil deposits

(high spectral ordinates in long periods) and records including high velocity pulses, especially

in the fault normal direction. Furthermore, as mentioned in Mylonakisand Gazetas [2000],

analytical studies performed in the early 90s showed that, in soft soil sites, an increase in

structural period may increase the imposed ductility demand. In addition to this, it should be

always in mind that the displacement spectral ordinates are always increasing with increasing

period and thus, P effects become more significant. Also, in a displacement - based design

framework, period lengthening leads to larger demands on the system and consequently, SFSI

effects may not be considered beneficial, in contrast to the traditional design methods (i.e.

Force Based design). All the aforementioned lead to the conclusion that period lengthening is

not always beneficial and a careful assessment of both the seismic input and soil conditions is

essential. In Mylonakis and Gazetas [2000], a number of real earthquakes where SFSI had a

detrimental effect on structures are referred. Such examples include 10 to 12 storey

buildings founded on soft clay in Mexico earthquake (1985) and a section of the Hanshin

Expressways Route 3 highway in Kobe earthquake (1995).

Numerical analyses as well as experimental studies have been conducted on structures where

SFSI may be important in order to highlight the main effects of soil structure interaction.

Most of the structures studied were bridge piers, subjected to real earthquake records in the

transverse direction of the bridge. Also, some shear wall or cantilever wall buildings were


23/134


9

studied. These two categories of structures may be prone to SFSI effects due to their

flexibility.

Figure 2.2: Comparison between code and real earthquake response spectra with high spectralacceleration ordinates in long periods (5% damped spectra).

Mylonakis and Gazetas [2000]performed a numerical investigation of an elastoplastic bridge

pier which had the same properties of a pier of the Hanshin Expressways Route 3 highway

and was subjected to the nearby Kobe Fukiai earthquake record. The analysis they conducted

was based on traditional design methods, assigning appropriate behaviour factors, and on the

response spectra of the actual record. They concluded that when SFSI is included in the

analysis of a bridge pier supported on deformable soil, there may be an increase in theductility demand with respect to the fixed-base pier. In Cremer et al. [2002]a SFSI macro-

element model was developed and as a numerical application, a bridge pier, belonging to the

real railway bridge Viaduct de lArc in North of Marseille, was analyzed under the Aigion

earthquake (Greece 1996, Ms=6.1). The analysis included both linear and nonlinear SFSI as

well as a case with no SFSI effects at all. Linear SFSI was shown to mainly affect the

structural period of the structure by dragging it to higher values depending on the flexibility

of the soil. The higher the flexibility the longer the period is and, for that earthquake, the


24/134


10

lower the value of the overturning moment at the base of the pier. For nonlinear SFSI, it was

observed that the more important the nonlinearities, the more reduced the base shear and the

overturning moment and the larger the displacements. Also, it was observed that accounting

for SFSI effects reduces the floor accelerations which is an important aspect of the design of

equipment fixed on the deck of the bridge.

Mylonakis et al. [2006], in a state-of-the- art paper on dealing with SFSI, performed a

parametric analysis of a bridge pier where different soil moduli were applied. Both harmonic

steady-state and time history analyses were performed; the former to investigate the salient

features of the dynamic behaviour of the system and the latter to obtain prediction of the

response to actual motions. Two earthquake records were considered: an artificial

accelerogram fitted to the NEHRP-94 with PGA=0.4g and the Pacoima downstream motion

recorded on soft rock outcrop during the Northridge 1994 earthquake. The main conclusions

regarding the response of the bridge pier can be summarized in the following:

An increase in radiation damping (which depends on supporting soil profile) wouldlead to a significant reduction of the piers response if the period of the pier is below

the cut-off frequency of the soil profile. As the flexibility of the supporting soil profile

increases, the radiation damping increases because waves emitted by the foundation

penetrate in the soil half-space.

If shifting of the piers period due to SFSI leads to resonance between soil andstructure, it may also lead to increase in piers response depending on the frequency

content of the earthquake.

Nonlinear effects such as soil plastic deformations, uplift and pore water pressure leadto an increase of the structural period that may cause de-resonance or resonance and

thus, progressive collapse of the structure.

Pecker et al. [2010]performed Incremental Dynamic Analysis of an elastoplastic bridge pier,

using a set of 30 earthquake records. Linear and nonlinear SFSI effects were included in the

analysis by implementing a SFSI macro-element model. It was found that the overall

behaviour of the pier when linear SFSI effects are considered is not so different than the

fixed-base pier. However, for large Cumulative Average Velocity (CAV) values, linear SFSI

was shown to be unfavorable for the response of the pier due to higher ductility demands. On


25/134


11

the other hand, nonlinear SFSI leads to essentially lower ductility demands than the other two

cases even for large CAV values due to soil yielding which protects the superstructure.

However, permanent settlements and rotations with high variability are an issue when

nonlinear SFSI effects are accounted for. Many other examples for bridge piers and RC wall

buildings exist in the literature which can confirm more or less the aforementioned remarks.

2.5 Linear SFSIWith the significant development of computer technology, there does not seem to be any

rational reason for neglecting soil structure interaction. A multi step approach in

evaluating SFSI effects consists of first, estimating the foundation input motion according to

the kinematic interaction analysis and then, performing the inertial interaction analysis

according to the steps described earlier. This approach can be limited to the last step if a) the

system (soil, foundation, structure) remains linear, b) kinematic interaction is neglected and c)

dynamic foundation impedance functions are available.

As mentioned earlier, kinematic interaction can be conservatively neglected for shallow

foundations with small embedment depth. Also, dynamic foundation impedance functions are

available from analytical studies conducted earlier [Gazetas, 1991]. On the other hand, it is

extremely rare that the whole system can respond elastically linearly during an earthquake.

Although the superposition theorem is exact for completely linear systems, it can be applied

to moderately non linear systems too. This can be achieved by selecting reduced values of soil

properties (such as shear modulus and shear strength) which are compatible with the expected

free field strains induced by the propagating seismic waves. This approximation is the basis of

the Equivalent Linear Method proposed by Idriss and Seed in 1968, and implies that all the

soil nonlinearities come from the passage of the seismic waves and that any additional

nonlinearity developed around the edges of the footing are negligible. Thus, this kind of

equivalent linear analysis is valid only in cases when the footing or the mat foundation does

not exhibit significant rotation, uplift or sliding during the earthquake shaking which cause

the soil to reach its bearing capacity and develop significant plastic deformations. These types

of nonlinearities are a consequence of the soil foundation structure interaction.


26/134


12

2.6 Nonlinear SFSIAs mentioned in the previous paragraph, the only nonlinearities which are accounted for in

the equivalent linear method are the ones associated with the passage of the seismic waves

and the strain levels that they impose in the soil. Nonlinearities associated with the soil

structure interaction are ignored. This kind of nonlinearities is divided into geometrical and

material nonlinearities.

Geometrical nonlinearities are mainly related to the foundation uplift (for shallow foundation)

and the formation of gap between the soil and the piles (for deep foundations). Those

nonlinearities follow a reversible procedure and result into negligible permanent

deformations. Their main effect is the isolation of the system which results into reduced

forces transmitted by the foundation to the soil and therefore decreased seismic demand.

Material nonlinearities arise from soil yielding around the edges of shallow foundations and

along the shaft of piles. Their nature is irreversible as they induce permanent deformations

into the system. Their main effect, except for the additional increase of the structural period,

is also related to soil yielding which acts as a safety valve for the foundation and the

superstructure as well as to the significant energy dissipation which tend to limit the seismic

demands on the structure.

Geometrical and material nonlinearities are strongly coupled and a lot of theoretical,

analytical and experimental research effort has been made during the last decades to throw

some light on the nonlinear behaviour of soil foundation systems under slow cyclic loading

or real earthquake records. The most significant findings of recent experimental and analytical

research on shallow foundations are briefly presented below:

The dynamic response of a soil foundation system is driven by the rocking mode ofvibration as it is much more flexible than swaying and uplifting.

Load displacement behaviour of shallow foundations depends on the appliedmoment to shear ratio at the base of the footing.

The horizontal translation of footings results into insignificant permanentdeformations.

Reduction in effective contact areas between the footings base and side walls and thesurrounding soil, during earthquake shaking and after significant yielding, leads to


27/134


13

additional period elongation of the system as the rotational stiffness (mainly) of the

foundation decreases substantially.

Significant yielding of the soil foundation system initiates when the ultimate loadsare attained (when the bearing capacity is reached) and the foundation keeps in

contact with the underlying soil only in the proximity of the foundation edges,

revealing significant uplift. However, during unloading and subsequent reloading

phases, the contact areas do not increase remarkably implying permanent reduction of

the contact area. Following successive strong load pulses in both directions of loading,

the soil profile under the foundation is rounded and exhibits substantial accumulated

seismic settlement.

Significant bearing capacity degradation does not occur during loading (even in slowcyclic tests) even when the soil below the footing reaches its ultimate capacity and

yields.

As the ratio between the area of the footing (A) and the contact area (Ac) increases,less energy dissipation due to soil yielding as well as permanent settlement is observed

while uplift becomes more significant.

Substantial energy is dissipated due to rocking of footing and energy dissipationincreases with increasing foundation rotation.

Higher earthquake intensity leads to more energy dissipation, cyclic rotation and upliftand also, tilting towards one side of footing is likely to be observed.

Foundation stiffness degradation and energy dissipation increase depend on the initialsafety factor against vertical concentric loads (Nmax/N, where Nmax is the bearing

capacity of the footing under vertical concentric load). This ratio also affects the

amplitude of permanent settlements.

2.7 Modeling of Nonlinear SFSIAn engineering approximation of the aspects that nonlinear SFSI includes can be reached by

subdividing the supporting soil into two sub-domains:

- A far-field domain which extends a sufficient distance from the foundation such thatthe soil structure interaction nonlinearities to be negligible. Nonlinearities in that

domain are only governed by the propagation of seismic waves.


28/134


14

- A near-field domain, in the proximity of the foundation where all geometrical andmaterial nonlinearities due to soil structure interaction are concentrated.

The boundary between these domains is not known with precision, however in practice, its

location is not necessary. The concept described above can be easily implemented if the

degrees of freedom of the foundation are considered uncoupled (which is a valid assumption

of shallow foundations Mylonakis et al [2006]). The far-field domain is modeled with the

equivalent linear impedance functions whereas the near-field domain is lumped into a

nonlinear element which reproduces adequately the geometrical and material nonlinearities.

Damping in the far-field domain arises only from radiation damping, which is viscous-type,

while in the near-field domain, damping comes from the hysteretic behaviour of the soil

foundation system. An image of the aforementioned concept is presented in figure 2.3.

Some interesting remarks on the definition of dynamic impedance functions as well as on

radiation damping can be found in Mylonakis et al. [2006]and inWolf, J.,P. and Song, C.,

[2002]. In these papers, dynamic impedance functions for different shallow foundation

configurations can be found as well as the effects of different soil profiles with or without

shallow rock layer on the impedance functions and on the radiation damping criterion. These

remarks may prove extremely helpful for the representation of the far-field sub-domain.

Figure 2.3: Conceptual sub-domains for nonlinear SFSI modelling.

As for the modeling of the near-field sub-domain, all the nonlinearities associated with soil

structure interaction should be captured and more specifically:


29/134


15

Foundation uplifting. Soil yielding. Coupling between uplifting and soil yielding with all the inherent consequences

(permanent settlements, stiffness degradation, maximum and residual rotations and

displacements).

Sliding (although not very important). Corresponding energy dissipation in each vibration mode.

Since modeling of the nonlinearities in the near-field domain are extremely important in order

to take into account SFSI effects and even take advantage of them in design, the next

paragraph is devoted in presenting a summary of numerical models that have been developed

during the last two decades.

2.7.1 Modeling of the Near-Field sub-domainThe beneficial role of SFSI, under certain circumstances, in design has been recognized for

many years. In the past, during large earthquakes, it has been observed that instantaneous

mobilization of the bearing capacity of the soil beneath the foundation, as well as nonlinear

behaviour of the soil foundation system, occurred. Thus, since cases where the soil

foundation system behaves nonlinearly, especially when the occurring earthquake exceeds thedesign seismic demands, are common, one could argue that SFSI effects should be taken into

account and even be exploited to obtain a more economical design of the foundation and the

superstructure.

However, in order to be able to account for the SFSI effects, proper and reliable tools,

to model them, are necessary. The most rigorous methods used to study soil structure

interaction are based on the finite element technique. Nevertheless, their use in this context is

very complex, competence demanding and time consuming. Therefore, the derivation of more

efficient tools, combining precision in modeling, simplicity in use and minor computational

effort is essential. Two types of alternative models have been developed to represent the

nonlinear response of soil foundation systems: models based on Beam on Nonlinear

Winkler Foundation (BNWF) and the so-called Macro-element models. Both of them are

described in the following with more emphasis being placed on the concept of macro-

elements since they have been proven superior to the BNWFs.


30/134


16

The concept of BNWF is based on obtaining global models for shallow foundations by

considering uncoupled Winkler springs attached at the soil-foundation interface that are

characterized by an elastoplastic contact breaking law. As mentioned in Harden et al.

[2009], Q z components are used as vertical springs to model the vertical load settlement

response and to provide moment capacity calculations whereas p y components are used to

simulate the passive resistance of soil against the footing movement. At last, T z

components are necessary to model sliding at the soil foundation interface. The elastic part

of the aforementioned components represents the elastic far-field response while the plastic

part is used to simulate the near-field response that includes the nonlinearities associated with

the soil-structure interaction. The main advantage of such formulations is that they permit

derivation of global system response by integration of the local spring response, which can be

achieved analytically. On the other hand, they are subject to a certain limitations associated

with the Winkler decoupling hypothesis as they are unable to describe the coupling between

the vertical and rotational degrees of freedom which has been shown to exist. Moreover, the

difficulty of parameters calibration is another significant drawback of those formulations.

In the case of macro-elements, footing and soil are considered as a single element and

a six degrees of freedom (3D case) or a three degrees of freedom (2D case) model is

formulated describing the resultant force displacement behaviour of a point (usually the

center) of the footing in the vertical, horizontal and rotational directions. The evolution of

macro-element models for shallow foundations seems to follow somehow parallel

developments with the theory of plasticity. Roscoe and Schofield [1956, 1957] are the first to

have suggested a treatment of the non-linear behaviour of shallow foundations based on the

theory of plasticity. The stress and deformation tensors are replaced by the resultant force and

corresponding displacement vectors with respect to which, a suitably chosen elastoplastic law

is formulated. Nova and Montrasio [1991] have exploited this idea in formulating a model for

strip footings on sand under monotonic loading with an isotropic-hardening elastoplastic law.

The basic idea of this model is first to represent the bearing capacity of the foundation under

combined loading as a surface in the space of the resultant vertical and horizontal force and

moment acting on the foundation following the reasoning initiated by Salenon [1972] and

also by Butterfield [1980]. Then this ultimate surface is identified as the yield surface in the

plasticity model regardless the mechanisms governing its origin and is allowed to evolve


31/134


17

according to a suitably chosen hardening law. In parallel, the displacements of the footing are

predicted by introducing an experimentally calibrated flow rule, which turns out to be non-

associated. This two-step modeling procedure has been followed in a number of subsequent

works for different soil conditions (clays, loose and dense sands) and different foundation

geometries (strip, rectangular and circular) leading to very accurate approximations of the

ultimate surface (yield surface), the hardening rule and the flow rule. Other works aimed at

extending the applicability of such models to cyclic loading. Pedretti [1998] and subsequently

Di Prisco et al. [2003] retained the isotropic hardening rule of the Nova and Montrasio model

for the case of virgin loading and introduced a hypoplastic bounding surface formulation for

the cases of unloading/reloading. This allowed obtaining a continuous plastic response for the

footing all through the loading history. In parallel, Paolucci [1997] initiated the use of macro-

element models for earthquake engineering applications. An original modeling approach was

proposed in the works of Crmer et al. [2002] in which, two distinct non-linear mechanisms

(soil plasticization and footing uplift) are formulated independently, whereas the global

footing response is obtained through their coupling. The model was developed for strip

footings on cohesive soils under seismic loading. Uplift was described by a geometric model

and soil plasticization by a kinematic and isotropic hardening plasticity model following

Prvost [1978]. Recently, Grange et al. [2008] modified the plasticity model of Crmer et al.

[2002] for application to circular footings and three-dimensional loading. A model with

coupled uplift and soil plasticity has also been presented by Paolucci et al. [2008]. Even more

recently, Chatzigogos et al. [2011] proposed a macro-element model where, except for

rocking and uplifting, nonlinear sliding mechanism at the soil-foundation interface was

included. This model was further improved to be implemented in seismic analysis in Figini et

al. [2012] where a new mapping rule was introduced to better fit the loading path under

seismic loading and also, uplift plasticity coupling, described through the concept ofeffective footing width, was implemented. A significant asset of this last work is that a unique

set of parameters for dense sand soil conditions was introduced that fits reasonably well

results from independent large scale laboratory tests, both cyclic and dynamic.


32/134


33/134


19

Current seismic design codes approaches forbid damage occurring on the foundation and rely

on the superstructure to deform inelastically so that the seismic energy is dissipated. However,

as shown earlier, nonlinear foundation response is almost unavoidable in many practical

cases, since overturning moments may get temporarily greater than the foundation static

bearing capacity. In this case the nonlinear foundation response may lead to lower seismic

demands on the superstructure. The idea to exploit non-linear energy dissipation at the soil-

foundation interface is becoming more and more attractive and has already led to some

outstanding examples of seismic design of foundations allowed to uplift during earthquakes,

such as in the case of the Rion-Antirion suspension bridge in Greece [Pecker, 2006], as well

as to increase experimental evidence of its benefits on the superstructure behaviour [Ugalde et

al., 2007; Deng et al., 2011]. Therefore, controlled foundation uplift and/or controlled plastic

response of the soil-foundation system are expected to become soon a rational and

economically efficient earthquake protection solution [Anastasopoulos et al., 2010], and to

lead in the next future to new performance-based design approaches including non-linear soil-

foundation-structure interaction as a key element.

In Priestley et al.[2007] the direct design procedure accounting for the soil-structure

interaction was first proposed and consisted the base for other approaches. Within this

perspective, Paolucci et al. [2012], proposed a new procedure that aimed to explicitly

introduce nonlinear Soil Foundation Structure Interaction in DDBD. The procedure is

based on the use of empirical curves, quantifying the foundation stiffness degradation and the

corresponding increase of damping ratio, as a function of foundation rotation. To simplify the

procedure, only the rotational degree of freedom is considered which is quite reasonable, as

the seismic response of shallow foundations is dominated by the rocking behaviour. The study

was based on large scale experimental tests of a shallow foundation under cyclic loading

supported on medium-dense and dense sand, which were carried out at the Joint Research

Center in Ispra, Italy and at the Public Works Research Institute in Tsukuba, Japan. In figure

2.6, a set of such experimental results along with numerical simulations performed using the

macro-element model developed by di Prisco et al. [2003],are presented.


34/134


20

Figure 2.6: Variation of shallow foundation rotational secant stiffness and damping ratio for dense and

medium-dense sands, as a function of foundation rotation (Paolucci et al., 2012).

By using the aforementioned macro-element model and after calibration with theexperimental results, empirical equations for both KF/KF0() and () as a function of the

static safety factor (Nmax/N) and the relative density of the sand were derived to be used in the

DDBD. The curves obtained by this work are given in figure 2.7. The equations that give

these curves are:

(2.2)

For dense sands a saturation value F,max=0.25 and for medium-dense sands F,max=0.37 is

assigned. Regardless of the relative density of sand the lower threshold of damping ratio was

F,min=0.036 which represents the radiation damping. The values of all the other parameters

are given in table 2.1.


35/134


21

Figure 2.7: Empirical curves for secant stiffness degradation and corresponding increase of damping ratio

[Paolucci et al., 2012].

Table 2.1: Values of parameters for the derivation of the empirical curves.

Due to the limitations of the numerical model for large values of foundation rotation (e.g. no

uplift is considered), those empirical curves may not be reliable if the reduction of the secant

stiffness falls below a threshold value of 0.15.


36/134


22

Using the aforementioned empirical curves, an iterative design procedure is applied where

both the performance of the superstructure and the foundation are design input data. The

foundation should be designed preliminarily so that an initial geometry and safety factor is

known. The procedure is iterated until convergence on the performance of the superstructure

and the foundation occurs and then, the obtained performance is checked with the limits

defined by the designer. By this method an optimal integrated design of both the foundation

and the superstructure can be achieved.

Figure 2.8:Flow chart summarizing the DDBD+NLSFSI design procedure by Paolucci et al. [2012].

The procedure denoted as DDBD+NLSSI was applied in the design of a bridge pier and the

design was verified against nonlinear time history analyses including spectrum compatible

real accelerograms. It was observed that there was a reasonable agreement between the design


37/134


23

and the average values from the nonlinear time history analysis. However, a discrepancy

exists in terms of maximum rotation max, since the computed values (from NLTH) were

larger than those expected according to the design procedure. This leads to an overestimation

of the ductility demand on the superstructure as the contribution of the foundation rotation to

the overall system ductility demand is underestimated. This discrepancy is due to the fact that

no uplift was considered in the macro-element model which was used to obtain the empirical

curves which were implemented in the design procedure.

Furthermore Sullivan et al. [2010] applied a similar DDBD+NLSSI procedure for reinforced

concrete wall structures supported on shallow foundations using the empirical curves derived

by Paolucci et al. [2007].Implementation of this design procedure in buildings with different

building heights concluded that: i) the required design strengths tend to be higher when

accounting for SFSI and ii) DDBD method results in significantly lower required design

strengths compared to Force Based design methods for 6 and 8 storey buildings, suggesting

that shallow foundation solutions may be more feasible for medium rise buildings when

DDBD is used. In addition to this, it was pointed out that another means of accounting axial-

shear-moment interaction at the base of the foundations should be explored and calibrated (to

NLTH) equivalent viscous damping curves for shallow foundations should be developed.

2.9 Soil Structure Interaction Eurocode 8In most of building codes, the structure response and foundation loads are determined

neglecting SFSI and assuming a fixed base. This is due to the traditional belied that SFSI is

favorable and neglecting it results in slightly conservative design outcome. However, the

significance of the SFSI was highlighted in the previous sections and it was clear that SFSI

should be taken into account especially in a performance based design framework.

Eurocode 8 is one of the codes where the significance of SFSI was recognized and some

regulations were included. More specifically, in chapter 6 of part 5 of Eurocode 8, it is stated:

The effects of dynamic soil-structure interaction shall be taken into account in the case of:

structures where P- effects play a significant role;

structures with massive or deep seated foundations;


38/134


24

slender tall structures;

structures supported on very soft soils, with average shear wave velocity less than 100 m/s.

For pile foundations, paragraph 5.4.2 of EC8-5 treats soil-structure interaction which is

always taken into account and gives attention to the kinematic interaction which is important

for that kind of foundations.

Annex D of EC8-5 includes a brief description of SFSI and its effects on structures and states

that for most of structures (except for the cases stated above) SFSI is beneficial.

As for the seismic bearing capacity of shallow foundations, which is important in identifying

soil yielding, EC8-5 states: "The stability against seismic bearing capacity failure taking into

account load inclination and eccentricity arising from the inertia forces of the structure as well

as the possible effects of the inertia forces in the supporting soil itself can be checked with the

general expression and criteria provided in annex F. The rise of pore water pressure under

cyclic loading should be considered either in the form of undrained strength or as pore

pressure in effective stress analysis. For important structures, non linear soil behavior should

be considered in determining possible permanent deformation during earthquakes."

More specifically, the code requires that the actions on the foundation should be less than the

resistance, that is

(2.4)Sd includes the normal force Nsd, the shear force Vsd, the overturning moment Msd and the

inertial forces F developed in the soil which arise from site response analysis and kinematic

interaction. According to EC8-5, the stability of the footing against seismic bearing capacityfailure can be checked with the following expression relating the soil strength, the design

actions (Nsd, Vsd, Msd) at the foundation level and the inertial forces in the soil:


39/134


25

(2.5)

This expression comes from upper bound solutions derived from limit equilibrium methods.

This expression is combined with safety factors which are used to deal with the twofold

uncertainty arising from the fact that upper bound solution are used and that limited number

of kinematic mechanism were considered. The formula presented is applicable for both

cohesive and frictional soils. The normalized inertial forces in the soil are computed as

(2.6)

(2.7)

where is the ground acceleration, cu is the undrained shear strength and is the friction

angle.


40/134

Chapter 3. Derivation of Stiffness Degradation and Damping curves for Shallow Foundations

26

3 DERIVATION OF STIFFNESS DEGRADATION ANDDAMPING CURVES FOR SHALLOW FOUNDATIONS

In this chapter, the procedures followed to derive the stiffness degradation and damping

curves for shallow foundation systems are described. At first, only rocking systems which are

supported on elastic soil are considered. Then, foundation systems which lie on nonlinear

sand soil are considered accounting also for geometric nonlinearities (uplift).

Since a large number of analyses is required, especially for the derivation of the damping

curves, a set of 40 earthquake records are used in order to provide equivalent damping values

calibrated through Nonlinear Time History Analysis (NLTHA). This set of accelerograms is

subdivided into four subsets named as CA, CC, LA and LC. Each one of the subsets

comprises of 10 earthquake records. Sets CA and CC have relatively linearly increasing

spectral displacements up to a period of 4 seconds whereas sets LA and LC have relatively

linearly increasing displacement spectra up to a period of 8 seconds. The second letter in the

subset name stands for the type of soil on which the accelerograms was recorded. Thus letter

A stands for soil type A according to Eurocode 8, that is rock whereas letter C stands for soil

type C.

3.1 Rocking Systems lying on Elastic SoilIn this section, the behaviour of generic rocking elastic systems is investigated. The aim of

this attempt is to derive tools, such as graphs presenting the equivalent damping ratio with

ductility, which can be used in a displacement-based design procedure of such systems. In

relation with the subject of this thesis, structural systems supported on rocking foundations

are considered where the soil underneath remains essentially elastic. Thus, no dissipation of


41/134


42/134


28

The maximum displacements obtained through the analysis were then compared to the elastic

displacements coming from the displacement response spectrum of the corresponding

earthquake record for the effective period that was observed in the analysis (Ke=Fmax/max,

Te=2 (m/Ke)0.5). In figures 3.1-3.4 the procedure described is demonstrated along with

specific force displacement curves of rocking systems.

Figure 3.1: Determination of for a SDOF rocking system under earthquake record CC1.

Figure 3.2: Determination of for SDOF rocking systems under earthquake record CC3.

-2 -1 0 1 2-1000

-500

0

500

1000

displacement (m)

F

orce

(kN)

orce- sp acemen curve or = .

Nonlinear elastic system

Equivalent linear system

0 2 4 6 80

20

40

60

80

100

Period (sec)

Spectral

displacement(cm)

Te=7.87

sec

-1 -0.5 0 0.5 1-1500

-1000

-500

0

500

1000

1500

displacement (m)

Force

(kN)

SDOF Force-displacement curve for =0.45



0 2 4 6 80

20

40

60

80

100

120

Period (sec)

Spectraldisplacement(cm)

Te=3.18 sec


43/134


29

Figure 3.3: Determination of for SDOF rocking system under earthquake record CC5.

Figure 3.4: Determination of for SDOF rocking system under earthquake record CC10.

In figures 3.5 and 3.6, the response of one SDOF rocking system under earthquakes CC5 and

CC10 respectively are shown. Under the former earthquake, the system exhibits a

displacement reduction factor greater than one, thus its maximum displacement exceeds the

one obtained from equivalent linear analysis. The effect of this phenomenon is shown in the

displacement time history of the system where it is shown that the high amplitude oscillations

occur for a long time period after the earthquake excitation is over (the record duration is 40

seconds) as a consequence of very small value of equivalent damping. Under the latter

-1.5 -1 -0.5 0 0.5 1 1.5-1000

-500

0

500

1000

displacement (m)

Force(kN)




0 2 4 6 80

20

40

60

80

100

Period (sec)

Displacement(cm

)

Te=5.62

sec

-1 -0.5 0 0.5 1-1500

-1000

-500

0

500

1000

1500

displacement (m)

Force(kN)




0 2 4 6 80

20

40

60

80

100

Period (sec)

Displac

ement(cm)

Te=4.43 sec


44/134


30

earthquake, the picture is reversed as the system response is quickly damped after the

achievement of the maximum displacement indication a high value of the equivalent damping

ratio.

Figure 3.5: Response of SDOF rocking system under earthquake CC5.

Figure 3.6: Response of SDOF rocking system under earthquake CC10.

The results obtained after all the analyses were performed are shown in figures 3.7-3.9. In

figure 3.7, all the analyses results are shown along with average and average standard

deviation curves. In almost every ductility level, the coefficient of variance is about 30%

0 10 20 30 40 50 60 70 80 90-1.5

-1

-0.5

0

0.5

1

1.5

Time (sec)

Displacement(m)

Displacement time history of SDOF rocking system with =1.81

0 10 20 30 40 50 60-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

time (sec)

Displacement(m)

Displacement time history of SDOF rocking system with =0.87


45/134


31

indicating significant scattering. The average curve shows that the DRF increases with

ductility demand up to a ductility level of 4 beyond which it remains approximately constant

at a value of 1.55. Figure 3.8 is presented to demonstrate the wide range of effective periods

whereas figure 3.9 shows that for effective periods larger than 2.5 seconds DRF is relatively

independent of Te.

Figure 3.7: Displacement reduction factor for 5% tangent-stiffness based elastic damping and r=0.05.

Figure 3.8: Displacement reduction factor as a function of system's effective period (5% tangent-stiffness

based elastic damping, r=0.05).

0.000

0.00

1.000

1.00

2.000

2.00

.000

.00

.000

.00

0 2 10

, , ,

+

0.000

0.00

1.000

1.00

2.000

2.00

.000

.00

.000

.00

0 1 2


46/134


32

Figure 3.9: Average values of as a function of effective period and ductility (5% tangent-stiffness based

damping, r=0.05).

At this point it should be noted that the values of obtained were surprisingly high as they

werent expected to be that larger than unity. In fact, the use of tangent stiffness-based elastic

damping is in part responsible for this picture. If a system experiences too many post-yield

loading cycles, it oscillates with decreased damping ratio giving a similar response to the one

shown in figure 3.5. On the other hand, if the earthquake record exhibits a few significant

acceleration peaks which cause the system to exceed the yielding point and after that, the

system behaves mostly in the pre-yield branch, then the system response is damped giving

similar response to the one shown in figure 3.6.

To illustrate the significance of the elastic damping model, another set of analyses was

performed using just the earthquake records of the CC set (10 earthquake records) and

applying an initial stiffness-based elastic damping model with the same damping ratio as

before (5%). The corresponding results, with respect to the factor as a function of ductility

demand, are shown in figure 3.10.

Referring to figure 3.10, it is apparent that using an initial stiffness elastic damping model

results in obtaining decreasing values of DRF with increasing ductility demand. This fact

clearly proves the importance of the elastic damping model.

0.000

0.00

1.000

1.00

2.000

2.00

.000

0 2


47/134


33

A third set of analyses was performed by using the secant stiffness to update the tangent

stiffness damping matrix in every step of NLTHA. All the earthquake record sets were chosen

to perform those analyses. The results are given in figures 3.11-3.13.

Figure 3.10: Displacement reduction factor vs ductility for the CC earthquake record set (5% initial-

stiffness based elastic damping, r=0.05).

Figure 3.11: Displacement reduction factor vs. Ductility (5% secant-stiffness based elastic damping,

r=0.05).

0.000

0.200

0.00

0.00

0.00

1.000

1.200

1.00

1.00

1.00

2.000

0 2 10

0.000

0.00

1.000

1.00

2.000

2.00

.000

0 2 10

, , ,


48/134


34

Figure 3.12: Displacement reduction factor as a function of system's effective period (5% secant-stiffness

based elastic damping, r=0.05).

Figure 3.13: Comparison between the DRF variation for different elastic damping models.

As it was expected, the secant stiffness damping model, results into DRF values between the

tangent and initial stiffness based modelling of the elastic damping. It can be observed that the

DRF remains essentially constant and equal to unity for all the ductility demand levels. This

outcome implies that, in average, the work done by a bi-linear rocking system up to the

maximum displacement is the same with the work done by an effective linear system.

0.000

0.00

1.000

1.00

2.000

2.00

.000

0 1 2

0.000

0.00

1.000

1.00

2.000

2.00

0 2 10


49/134


35

3.1.2 Stiffness Degradation of Rocking FoundationsIn order to derive the stiffness degradation curve against the foundation rotation due to

geometric nonlinearities, an uplift model is necessary. In this study, the foundation uplift

model described in Chatzigogos et al. [2011] is adopted. In the development of this model, the

soil is represented by a linear elastic half-space and a rough soil-foundation interface with

zero tensile strength is assumed. Thus, the footing is not allowed to slide but it is allowed to

be vertically detached from the soil surface. Foundation uplift modifies the soil-footing

contact area leading to decrease in the foundation static impedances. It should be noted that

the uplift response is fully reversible (as long as the foundation does not tilt) and no hysteretic

energy is dissipated. Before uplift initiation the static impedances (stiffness and dashpots)

remain constant and equal to the values initially determined. After uplift initiation the static

impedances decrease. he comparison of this uplift model with more sophisticated finite

element formulation is given in figure

The expressions that describe the uplift model are given below along with explanation of the

variables.

In the previous expressions, =4 for strip foundations and 6 for circular foundations whereas

is equal to 2 and 3 respectively. Also, Nmax is the bearing capacity of the footing under

vertical load only, KMM,0is the initial rotational stiffness of the footing and QNis the ratio of

the current vertical load to Nmax. After some mathematical manipulation, the following

expressions have been derived:


50/134


36

In equation (3.7) Kf/Kf,0is the ratio of the effective rotational stiffness to the initial one, SF is

the Static Safety Factor which is defined as the ratio of Nmax to the current vertical load

(alternatively SF=1/QN), is the Poisson ratio of the soil, G is the Shear Modulus of the soil,

is the unit weight of the soil, B is the width of the footing and N is the bearing capacity factor

which depends on the soil friction angle. In this study, mainly square footings are considered,

so the values that correspond to equivalent circular footings were assigned to the parameters

and .

The above formula is very helpful to understand the main parameters that affect the rocking

behaviour of a shallow footing on elastic soil. One can observe that the width of footing

affects the uplift initiation proving that uplift is a geometry-related phenomenon. In figures

3.11 the effect of the Safety factor on the rotation of uplift initiation is presented whereas in

figures 3.12 and 3.12 the effect of the Safety Factor and foundation width on the stiffness

degradation is shown. The soil properties used are characteristic of sand soils.

Figure 3.14: Comparison between the proposed uplift model regarding (a) moment-rotation and (b)

moment-heave diagram.


51/134


37

Figure 3.15: Effect of Safety Factor (SF) on the rotation of uplift initiation (B=4m).

Figure 3.16: Effetc of Safety Factor (SF) on the stiffness degradation of a 4x4m footing.

Figure 3.17: Effect of foundation width on the stiffness degradation of footings with SF=2.

0

0.000

0.001

0.001

0.002

0 20 0 0 0 100 120

0

0.1

0.2

0.

0.

0.

0.

0.

0.

0.

1

0.00001 0.0001 0.001 0.01

0

0.1

0.2

0.

0.

0.

0.

0.

0.

0.

1

0.00001 0.0001 0.001 0.01


52/134


38

3.2 Foundation systems lying on inelastic soilIn this section, the derivation of stiffness degradation and equivalent viscous damping curves

of foundation systems accounting for the nonlinear nature of the soil are presented. More

specifically, in addition to the uplift mechanism, which was considered in the previous

sections, the soil plasticity mechanism is accounted for as well.

The derivation of these curves is performed through the use of a very recently developed SFSI

macro-element model which was created by

displacement based seismic design of rc wall buildings accounting for nonlinear...

Documents