one dimensional rocking foundation macro-element for design of

one dimensional rocking foundation macro-element for design of

ONE DIMENSIONAL ROCKING FOUNDATION MACRO-ELEMENT FOR DESIGN OF SHALLOW FOUNDATIONS Michael J Pender 1, Thomas B Algie 2, Luke B Storie3, Ravindranath...

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ONE DIMENSIONAL ROCKING FOUNDATION MACRO-ELEMENT FOR DESIGN OF SHALLOW FOUNDATIONS

Michael J Pender 1, Thomas B Algie 2, Luke B Storie3, Ravindranath Salimath4

Abstract: Recently a number of macro-element models have been formulated for assessing the performance of shallow foundations during earthquake loading. These provide a computational tool that represents the nonlinear dynamic behavior of the foundation in a manner much simpler than finite element modelling; consequently, they are useful for preliminary design. The basis of this paper is the shallow foundation moment-rotation pushover curve, which is bracketed by the rotational stiffness at small deformations, determined by the small strain stiffness of the soil, and the moment capacity, which is a function of the soil shear strength and the vertical load carried by the foundation. Between these two limits there is a curved transition. The paper argues that when the vertical load carried by an embedded foundation is a small fraction of the vertical bearing strength, the moment-rotation behavior dominates the response. This means that the structurefoundation system can be reduced to a single degree of freedom (SDOF) model. The form of the shallow foundation moment-rotation curve obtained from experimental and computational modelling is approximately hyperbolic; the nonlinear shape is due in part to the nonlinear deformation of the soil beneath the foundation but also to gradual loss of contact between the underside of the foundation and the soil below. The paper proposes a generalization of the pushover curve to give a shallow foundation cyclic moment-rotation relationship. The hysteretic damping properties of the model, as a function of the foundation rotation amplitude, are demonstrated as is the relation between secant stiffness and foundation rotation. This paper shows how the model can be applied in numerical simulation of structure-foundation systems subject to earthquake time histories. The significance of the maximum displacement (foundation rotation) in relation to the damping and residual rotation at the end of the earthquake record are discussed. 1

Introduction

Background Kelly (2009) made proposals for the design of shallow foundations that rock during earthquake excitation. Rocking is understood as cyclic uplifting and reattachment at the edges of the foundation during the course of an earthquake or other cyclic loading. Kelly explained that low to medium-rise structures on shallow foundations may not have sufficient weight to prevent foundation rocking; in which case the designer might want to take advantage of the real benefits that follow from accepting modest amounts of rocking. One of the topics Kelly recommended for further research was an appropriate method of handling soil nonlinearity, which is the intention of this paper, an extension of the work reported by Pender et al. (2013). Recently a number of macro-element models have been formulated for assessing the performance of shallow foundations during earthquake loading. These provide a computational tool that represents the nonlinear dynamic behavior of the foundation in a manner much simpler than finite element modelling; consequently, they are useful 1

Department of Civil & Environmental Engineering, University of Auckland. Email: [email protected] 2 Partners in Performance, Sydney. Email: [email protected] 3 Tonkin & Taylor Ltd, Auckland. Email: [email protected] 4 Tonkin & Taylor Ltd, Auckland. Email: [email protected]

for preliminary design. The basis of this paper is the shallow foundation moment-rotation pushover curve, which is bracketed by the rotational stiffness at small deformations, determined by the small strain stiffness of the soil, and the moment capacity, which is a function of the soil shear strength and the vertical load carried by the foundation. Between these two limits there is a curved transition. The paper argues that when the vertical load carried by the foundation is a small fraction of the vertical bearing strength, moment-rotation behavior dominates the response. This means that the structure-foundation system can be reduced to a single degree of freedom model. The form of the shallow foundation moment-rotation curve obtained from experimental and computational modelling is approximately hyperbolic; the nonlinear shape is due in part to the nonlinear deformation of the soil beneath the foundation but also to gradual loss of contact between the underside of the foundation and the soil below. The paper proposes a generalization of the pushover curve to give a cyclic moment-rotation relationship for shallow foundations. The hysteretic damping properties of the model, as a function of the foundation rotation amplitude, are demonstrated as is the relation between secant stiffness and foundation rotation. Following from the above paragraph this psper explores the behaviour of a one dimensional macro-element in which there is no settlement or horizontal deformation of the foundation; the only deformation is nonlinear rotation. Modelling shallow foundations in this way reflects our observation, following the February 22, 2011 earthquake in Christchurch, of the apparently satisfactory performance of foundations for multi-storey buildings founded on gravel (Storie et al. (2015) and Storie (2017)). The foundation supports a multi-storey elastic viscously damped structure which is represented as a concentrated mass supported on an elastic viscously damped column. As there is only one mass in the system the structure-foundation system is a single degree of freedom (SDOF) model. It is intended that this will provide a simple method for foundation design sensitivity studies. Figure 1 Hyperbolic moment-rotation relation fitted to shallow foundation pull-back data (after Algie (2010). Measured shallow foundation response Field experiments have been conducted at sites north of the Auckland CBD with shallow foundations supporting a simple single storey structure subject to gradual pull-back followed by snap-back release; more details are given by Algie et al. (2010), Algie (2011), and Salimath (2017). The dynamic response of the system was recorded after each snap release. An added bonus was the static load-deflection curve obtained during the pullback phase of the test. The pullback response is in effect a static pushover curve for the foundation. Elsewhere, the relations between the rocking period and the pull-back angular displacement and damping are presented by Algie 2011). The site used for the Algie tests had a profile of stiff cohesive soil. The soil profiles were investigated with CPT tests between the surface and a depth of about 8 m. In some of these the shear wave velocity of the soil was measured which indicated a reasonably consistent shear wave velocity for the materials at the sites equivalent to a small strain shear modulus of about 40 MPa. Figure 1 shows some of the pull-back data obtained by Algie. It is apparent that there is considerable nonlinearity in the moment-rotation curves but a simple hyperbolic relationship, which asymptotes to the moment capacity of the foundation, can be fitted through the data. It is evident that some degradation occurs from one snapback to the next, particularly for those tests following the snapback which applies the largest moment to the system. Further testing of this type was completed late in 2016 at Silverdale, a site further north of Auckland, the results are given by Salimath (2017). In this case the shallow foundations were at the ground surface, whereas those for the Algie tests were embedded to a depth of 400mm. The Salimath tests, with foundation static load bearing strength factors of safety as high 10, left barely perceptible marking on the ground surface once the test rig was removed. This means that for these tests the nonlinearity in the moment-rotation curve was mostly a consequence of gradual loss of contact between the underside of the foundation and the underlying soil. Simplification at large static factors of safety An important control on foundation response under earthquake and other cyclic loading is the static bearing strength factor of safety. A common suggestion is that mobilising about one third of the vertical bearing strength of the foundation will give satisfactory static vertical load only performance (satisfactory is usually determined in relation to the settlement under working loads). However, often the actual mobilisation of bearing strength will be less, even considerably less, because of conservative decisions regarding design values for the soil parameters. Additionally, if there is a basement beneath the building, the area of the foundation may occupy the full plan area of the structure; in such cases the bearing pressure generated by the vertical load may be a very small fraction of the vertical bearing capacity of the foundation. When this occurs there is unlikely to be settlement of the foundation under earthquake actions. Likewise there will be little horizontal displacement as the lateral stiffness of the foundation will be enhanced by the lateral stiffness generated by the sidewalls of the basement. In this way the response of the foundation under earthquake will be dominated by the nonlinear rotational stiffness of the foundation.

Substitute structure The shape of the shallow foundation moment-rotation curve in Figure 1 is not unlike the moment-rotation characteristics of a reinforced concrete section. Figure 2 reproduces a diagram from the paper by Shibata and Sozen (1976). The procedure followed in this chapter is to make use of the ideas proposed by Shibata and Sozen, but to assume that the structural elements remain elastic whereas the shallow foundation is the source of nonlinearity in the structure-foundation system. Figure 3, taken from Priestley et al. (2007), shows how the approach of Shibata and Sozen can be used to reduce a multi-storey frame structure to an equivalent SDOF model; they refer to this as a substitute structure. Priestley et al. explain how the parameters for the substitute structure, h e , m e , and K e (Figure 3), can be evaluated. One additional aspect needs to be included in the substitute structure model: the compliance of the soil beneath and adjacent to the foundation. This is done by representing the structure-foundation system as a SDOF elastic structure supported on a nonlinear rotational spring, which is discussed further below and illustrated in Figure 4a. The essence of the Shibata and Sozen mechanism shown in Figure 2 is the development of hysteretic damping through ductile deformation of structural members. Priestley et al. explain that, for a given displacement of the SDOF mass, elastic soil-structure interaction reduces the ductility demand in the structure; in effect this results in a reduction of the hysteretic damping of the system. As will be explained below nonlinear foundation behaviour develops hysteretic damping and so satisfactory response of the structure-foundation system to earthquake excitation may still be feasible even though the above ground structure remains elastic. Figure 2 The basis of Shibata and Sozen’s substitute structure concept. Figure 3 Implementation of the substitute structure concept as presented by Priestley, Calvi and Kowalsky (2007). Figure 4 Single degree of freedom model of the 10 storey building with one basement level founded in dense gravel; (a) SDOF model, (b) actual structure. Performance requirements Prior to commencing the design of a foundation to resist earthquake or other cyclic loading performance criteria need to be specified. Since we are considering a one dimensional macro-element, this means specifying a permissible foundation rotation. After the Christchurch earthquake of February 2011 it was apparent that residual rotations of more than about 0.01 radian (≈ 0.5 degrees) are easily perceived by the unaided eye. So if the design is based on a maximum rotation of 0.01 radian any residual rotation will not be readily visible. This is a conservative criterion as Deng et al. (2014) observed from centrifuge data that the residual rotation decreases as the pre-earthquake bearing strength static factor of safety increases. So for typical static design shallow foundation bearing strength factors of safety the post-earthquake residual rotation can be expected to be significantly less. As will be seen below limiting the maximum foundation rotation to less than about 0.01 radians will provide benefits from nonlinear response but will still be well short of mobilising the moment capacity of the foundation, that is brief instances of bearing failure are unlikely at such modest rotations. Panagiotidou et al. (2012) found that for shallow foundations on stiff clay the onset of P-delta effects, which reduce the moment capacity of the system, occurs at rotations between 0.01 and 0.02 radians. Thus specifying a small value for the maximum design rotation has the added advantage that P-delta effects will not reduce the system capacity. 2

Equations of motion

Wolf (1985), in his book Dynamic Soil-Structure Interaction, develops the equations of motion for a single degree of freedom model in which a lumped mass is positioned at some distance above a foundation resting on an elastic medium with radiation damping. There are three components to the horizontal the displacement of the SDOF mass: the horizontal displacement of the foundation, the rotation of the foundation multiplied by the height of the mass, and the flexural displacement of the column supporting the mass, but since there is one mass in the system there is only one dynamic degree of freedom. Wolf shows how the different stiffness components contribute to the natural frequency of the structure-foundation system with the following equation:

ωe2

= 1+

ωs2 ks kh

+

k sh e 2 kϕ

(1)

where: ωe and ωs are respectively the natural frequencies of the elastic structure-foundation system and the fixed base structure, k s is the flexural stiffness of the column supporting the SDOF mass, k h is the lateral stiffness of the foundation, k ϕ the rotational stiffness of the foundation, and h e is the height above foundation level of the SDOF lumped mass (m e ). This equation shows that inclusion of foundation compliance reduces the natural frequency (lengthens the natural period) of the structure-foundation system. Equation (1) assumes that all stiffness components are elastic and gives the natural frequency of the system under steady state sinusoidal excitation, but in our case the foundation rotational stiffness is nonlinear so the natural frequency of the system under steady state excitation is variable being a function of the rotational amplitude of the foundation. In this paper we consider embedded foundations for which the lateral stiffness is large and does not have a significant effect on the system natural frequency given by the above equation. Consequently, the lateral stiffness of the foundation will not be included in the modelling below. Under earthquake excitation the response is not steady state and so the rotational stiffness of the foundation, derived from the hyperbolic moment-rotation relationship, will vary from time step to time step. This means that the equation of motion for the SDOF mass needs to be integrated incrementally. The incremental equation of motion is:

m + D∆u m + K e ( ϕ ) ∆um = g me ∆u −me ∆u

(2)

where: m e D

is the SDOF mass (tonnes), is the parameter for the equivalent viscous damping of the column supporting the SDOF mass (tonnes/sec), ϕ is the foundation rotation (radians) K e (ϕ) is the equivalent system stiffness which includes the contribution from the foundation momentrotation curve and the lateral stiffness of the column (kN/m),  ∆u,  and ∆u are respectively the horizontal acceleration increment, velocity increment, and ∆u, displacement increment of the SDOF mass,  ∆ug is the increment in input acceleration (m/sec2).

Note that the velocity dependent damping term in equation (2) comes from the elastic column supporting the SDOF mass a distance h e above the foundation level. All the foundation damping comes from hysteretic momentdeformation loops which are implicit in the foundation moment-rotation relationship discussed below. Frequency dependent elastic radiation damping is the usual energy dissipation mechanism used in elastic soil-structure interaction. However, it is well known that elastic radiation damping for foundation rotation is very small so the hysteretic action is very important during foundation rocking. The equivalent stiffness of the structure foundation system including the effect of the flexural stiffness of the column and the nonlinear rotational stiffness of the foundation is: ks K ( ϕ ) Ke ( ϕ) = 2 K ( ϕ ) + he k s

(3)

where: K(ϕ) is tangent rotational stiffness of the foundation at the current value of ϕ. A possible extension of the above approach would be to include the mass and rotational inertia of the foundation. Doing this means that the structure-foundation system now has three dynamic degrees of freedom (lateral inertial displacement of the mass at a height h e above foundation level, and lateral inertial horizontal displacement of the foundation, and inertial rotation of the foundation). Wolf (1985) states that pursuing this analysis does not produce results significantly different from those obtained with the SDOF model. Using modal analysis it is found that the two additional periods are well removed from that of the SDOF model and the first mode frequency is hardly different from that for the SDOF model. However, there is one significant factor in which the mass of the foundation block is important; the increased vertical load on the foundation enhances the moment capacity. Figure 5 Cyclic hyperbolic moment-rotation curve with the definition of the M o and ϕ o parameters.

4

Foundation hyperbolic moment-rotation relationship

Figure 1 presents the static moment-rotation curves obtained during the application of the pullback forces to one of the foundation sets at the Albany site. A hyperbolic curve is seen to fit through the data. The form of the hyperbolic relationship shown in Figure 1 is:

M=

ϕ Λ Ωϕ + Kϕ i Mu

(6)

where: M and ϕ are the current foundation moment and rotation respectively, K ϕi and M u are the initial rotational stiffness and moment capacity of the foundation respectively, and Λ and Ω are numerical parameters that may be used to refine the fit of the curve to the data (the curve fit in Figure 1 has a value of unity for both of these parameters). Salimath (2017) found that values for Λ and Ω depend on the shape of the foundation. The parameter Λ also has an effect on the amount of hysteretic damping in the cyclic hyperbolic relationship. Equation (6) provides a pushover curve for a shallow foundation, which is bounded by the moment capacity of the shallow foundation at large values of the rotation and the elastic rotational stiffness at small rotations. Equation (6) also acts as a backbone curve for the extension of hyperbolic relationship to cyclic loading given by the following equation:

M

K ϕi ( AMu − Mo )( ϕ − ϕo )

Λ ( AM − M u

o

) + ΩK (ϕ − ϕ ) ϕi

+ Mo

(7)

o

where: M o and ϕ o are the initial values for the current branch of the moment-rotation curve(defined in Figure 5), and A takes values of +1 and -1 indicating if the foundation is being loaded towards the moment capacity M u or –M u . Whenever an unloading or reloading loop intersects the backbone curve the moment-rotation response then follows the backbone curve (this prevents the system developing unreasonably large foundation rotations). When the direction of loading reverses the stiffness reverts to the rotational stiffness at zero moment, K ϕi . The moment capacity can be reached when the rotation is to the left or the right, so that M u limits the moment to the right and –M u limits it to the left. Figure 6 shows how equation 7 extends the pushover curve to give the cyclic response of a shallow foundation. The rotational stiffness of the foundation is given by the slope of the current branch of the hysteretic momentrotation loop and, as is apparent from Figure 6, the slope of the M-ϕ curve changes continuously with moment. Equations 6 and 7 are relatively simple expressions representing complex behavior. Spring bed models indicate, for a shallow foundation under a fixed vertical load and gradually increasing moment, that there will be a gradual loss of contact at the edge of the foundation. The macro-element models of Cremer et al. (2001) and Chatzigogos et al. (2009) incorporate this effect specifically, but herein it is assumed that uplift effects are handled implicitly and part of the reason for the shape of the moment-rotation curve is a consequence of the reduction of contact area. Salimath (2017) reports on an extensive series of finite element analyses, using Plaxis 3D, (Plaxis 2012) which have a rigid rectangular foundation, under constant vertical load, on a clay soil with a no-tension interface between the foundation and the underlying clay. This modelling confirms that a hyperbolic curve is a reasonable representation of the shallow foundation moment-rotation push-over curve when uplift occurs. 5

Rotation dependent foundation stiffness and hysteretic damping

Elastic soil-structure interaction modelling relies on radiation damping to take vibrational energy away from the foundation. However, for rotation of the foundation damping available from elastic radiation is small. Consequently, herein we assume that all the foundation damping is derived from the area of the hysteresis loops. Figure 6 shows that the hysteretic damping increases as the rotation amplitude increases. The apparent foundation rotational stiffness and the equivalent viscous damping ratio associated with a given hysteretic moment-rotation loop are defined in Figure 7a. Figure 7b gives the variation of the rotational stiffness with the logarithm of the rotation amplitude. Figure 7c gives the variation of the hysteretic damping against the logarithm of the rotation amplitude. When the rotation amplitude is 0.05 the hysteretic damping is 28%. Ishihara (1996) states that the maximum damping ratio for the hyperbolic model is 64%. Clearly the damping in the present model does not

approach such a value at likely rotation amplitudes, a consequence of the hysteretic loops reattaching to the backbone curve rather than crossing it. Figure 6 Steady state hysteretic shallow foundation moment-rotation loops with the Λ parameter set to 1.0. Figure 7 Hysteretic equivalent viscous damping ratio and apparent rotational stiffness as functions of foundation rotation amplitude. (a) parameter definitions; (b) foundation rotational stiffness as a function of rotation amplitude; (c) damping ratios against rotation amplitude. All the moment-rotation loops in Figure 6 are generated with the Λ parameter set to a value of 1.0. As the foundation rotation increases the slope of the line between the end points of the hysteresis loops in Figure 6 decreases; this gives the relationship, for the backbone curve, between the apparent rotational stiffness of the foundation which is plotted in Figure 7b. 6

Example calculations

In this section the response to earthquake excitation of a 10 storey building with a single level basement and a foundation with hyperbolic moment-rotation properties is presented. The input is the acceleration recorded in the 1940 El Centro earthquake (the N-S component) adjusted to a peak ground acceleration of 0.3g. The nonlinear response of the structure-foundation system was calculated to the 0.3g El Centro record scaled with factors of 0.5, 1.0, 2.0 and 3.0. The algorithm presented in chapter 7 of Clough and Penzien (1993), for calculating the incremental response of a nonlinear single degree of freedom system, was used in these computations. In addition the response of the system when the foundation behaves elastically was evaluated. All the time history and response spectra calculations discussed in this chapter were done using Mathcad 15 (PTC 2016). The earthquake acceleration histories had a time step of 0.02 seconds. The details of the 10 storey building are shown in Figure 4b, the plan dimensions are 20 m by 20 m. The damping ratio for the elastic building structure was set at 3% of the critical damping value. The building is founded on dense dry gravel with an angle of shearing resistance of 45 degrees and a shear wave velocity of 280 m/sec. The conversion from the actual structure, Figure 4b, to the SDOF model, Figure 4a, was done by assuming a uniform distribution of mass with height in the building and setting the SDOF mass to 70% of the total mass acting at 70% of the building height. Figure 8 Foundation moment-rotation loops calculated during the response to scaled El Centro ground acceleration input motions (a scale factor (SF) of 1 corresponds to an input PGA of 0.3g). The calculated foundation moment-rotation loops are plotted in Figure 8. These show increasing hysteretic damping as the scale factor applied to the input motion is increased. Figure 9 has elastic response spectra for the outputs from the scaled input motions, the spectra were obtained by processing the calculated horizontal acceleration histories of the SDOF mass. In addition there is a spectrum for the response when the foundation behaves elastically. All spectra were calculated for a damping ratio of 5% and the plotted spectral accelerations are normalized with respect to the peak input acceleration for the scaled input, so the ordinates in Figure 9 are dimensionless. It is apparent that the peak response in all cases occurs at a period close to 1 second, the natural period of the above ground part of the structure-foundation system. As the scaling of the input motion increases there is more hysteretic damping generated by the foundation as evidenced by the moment-rotation loops in Figure 8. The effect of this is that the spectral peaks in the responses of the SDOF mass are reduced as the peak input acceleration increases. This is indicative of the increasing hysteretic damping in the foundation. The damping values plotted in Figure 7c are for steady state cycling around moment-rotation loops of fixed rotation amplitude and so are not directly relevant to the damping obtained during an earthquake record. Pender et al. (2017), reporting an investigation of the performance of multi-storey buildings on shallow foundations having a hyperbolic moment-rotation curve, calculated numerically the SDOF response to snap-back release from various initial displacements. This revealed that there was considerable damping in the first half cycle of response and then a gradual reduction in damping. The spectra in Figure 9 are dominated by the response of the 10 storey elastic structure supported on the nonlinear foundation. One might have expected, with the larger values of the scale factor, some period lengthening of the peak response because of nonlinearity in the foundation. From Figure 8c we see that the peak foundation rotation during the motion when the scale factor is 2 is about 1 milli-radian; if this is the peak rotation then the rotations for the remainder of the response will be smaller. Referring to Figure 7b the equivalent stiffness for a rotation amplitude of 1 milli-radian is about 90% of the elastic value. From equation 1 the period of the structure foundation system for elastic behavior 1.04 seconds, when 0.9 of the elastic foundation rotational stiff ness is used the period is lengthened by 0.05 seconds. Given this, it is not surprising that the peaks of the response spectra for

the SDOF mass occur at periods close to the 1 second period of the above ground 10 storey elastic structure. Figure 9 shows that, despite the small foundation rotations, there is still hysteretic damping and it is this damping that reduces the peaks in the response spectra as the scale factor is increased. In other words the hysteretic damping absorbs some of the incoming energy and prevents it being passed upwards into the elastic structure. Figure 9 Response spectra, normalised with respect to the peak input acceleration, for the calculated horizontal acceleration of the SDOF mass to the scaled 1940 N-S El Centro record. The scale factors for the input ranged from 0.5 to 3.0, with the PGA for a scale factor of 1 being 0.3 g. 7

Conclusions

The motivation for the development of a one dimensional shallow foundation macro-element in this paper is threefold: first, to use the structural engineering substitute structure concept but in a different context where the nonlinearity occurs in the foundation rather than the above-ground elastic structure (Figures 2 and 3); second, to make things as simple as possible to ease sensitivity studies during the design process; third, to look specifically at situations dominated by rotational response of the foundation where settlement and sliding can be ignored. The following conclusions are reached: • • • •

8

Small foundation rotations have a significant effect on the response not because of reduced stiffness but because of hysteretic damping, Figure 8. Uplift is not considered specifically in the macro-element presented herein, but it is recognised that it is an important part of the explanation for the shape of the foundation moment-rotation curve. The rotation dependent equivalent stiffness and hysteretic damping of the model follow the expected trends, Figure 7 b and c. The hysteretic damping in the foundation shields the supported elastic structure from some of the incoming earthquake energy, Figure 9. References

Algie, T. B., Pender, M. J. and Orense, R. P. (2010). Large scale field tests of rocking foundations on an Auckland residual soil, In: Soil Foundation Structure Interaction (R Orense, N Chouw, and M Pender (eds.)), CRC Press / Balkema, The Netherlands, pp. 57- 65. Algie, T. B. (2011). Nonlinear Rotational Behaviour of Shallow Foundations on Cohesive Soil, Ph D thesis, University of Auckland. Chatzigogos, C. T., A. Pecker, and J. Salenc¸on (2009). Macroelement modeling of shallow foundations. Soil Dynamics and Earthquake Engineering 29, 765–781. Clough, R. W. & Penzien, J. (1993). Dynamics of Structures, 2nd ed., McGraw-Hill, New York. Cremer, C., A. Pecker, and L. Davenne (2001). Cyclic macro-element for soil-structure interaction: material and geometrical non-linearities. International Journal for Numerical and Analytical Methods in Geomechanics 25(13), 1257–1284. Dassault Systemes. (2010). Abaqus 6.8-EF2. Vélizy-Villacoublay, Ile-De-France, France. Deng, L., Kutter, B., and Kunnath, S. (2014). Seismic Design of Rocking Shallow Foundations: DisplacementBased Methodology. J. Bridge Eng., 19 (11) DOI: 10.1061/(ASCE)BE.1943-5592.0000616. Duncan, J. M. and Chang, C. Y. (1970). Nonlinear analysis of stress and strain in soils. Proc. ASCE, Jnl Soil Mechanics and Foundations Division, 96 (SM5), 1629-1653. Ishihara, K. (1996). Soil behaviour in earthquake geotechnics. Oxford Science Kelly, T. E. (2009). Tentative seismic design guidelines for rocking structures, Bulletin of the New Zealand Society for Earthquake Engineering. 42 (4). 239-274. Panagiotidou, A. I., Gazetas, G. and Gerolymos, N. (2012). Pushover and Seismic Response of Foundations on Stiff Clay: Analysis with P-Delta Effects. Earthquake Spectra, 28(4), 1589-1618. Pender, M. J., Algie, T. B., Storie, L. B. and Salimath, R. (2013). Rocking controlled design of shallow foundations, Proc. 2013 NZ Society for Earthquake Engineering Conference, Wellington. Pender, M. J. (2017). Bearing strength surfaces implied in conventional bearing capacity calculations. Geotechnique, 67(4), 313-424, DOI: 10.1680/geot./16-P-024.

Pender, M. J., Algie, T. B., Storie, L. B. and Salimath, R. (2017). One dimensional moment-rotation macroelement for performance based design of shallow foundations. Proceedings Performance Based Design III (PBDIII), Vancouver. Plaxis. 2012. Plaxis 3D – Version 2012, Plaxis b.v. The Netherlands. Priestley, M. J. N., Calvi, G. M. and Kowalsky, M. J. 2007. Displacement-Based Seismic Design of Structures, IUSS Press, Pavia, Italy. PTC (2015). Mathcad 15 (M045). PTC Corporation, Needham, MA, USA. Salimath, R (2017) Experimental and finite element study of moment-rotation response of shallow foundations on clay, PhD thesis, University of Auckland. Shibata, A. and Sozen, M. (1976). Substitute structure method for the seismic design in reinforced concrete, Journal of Structural Engineering, Vol. 102(1), pp. 1-18. Storie, L. B., Knappett, J. A. and Pender, M. J. (2015). Centrifuge modelling of the energy dissipation characteristics of mid-rise buildings with raft foundations on dense cohesionless soil. Proc. 6th International Conference on Earthquake Geotechnical Engineering. Christchurch, New Zealand. Storie, L. B (2017). Soil-foundation-structure interaction in the earthquake performance of multi-storey buildings on shallow foundations. PhD thesis, University of Auckland. Wolf, J. P. (1985). Dynamic Soil-Structure Interaction. Prentice Hall, New Jersey, USA.

Foundationmomentcapacity

100

Hyperboliccurve

Moment (kNm)

80 Smallstrainelasticstiffness

60

Momentrotationcurvesfor individualtests

40 20 0

1.0o

0.5o 0

10

20 30 Foundationrotation(mrad)

Figure 1

M - uncrackedsection

Moment

M - cracked section

M - for substitute (ductile) structure

Rotation Figure 2

40

he

me Shear force

F

ks

ks

Ke

Displacement Horizontal & rotational stiffness

Shallow foundation

SDOF substitute structure

Figure 3

10 floors

20m x 20m plan

SDOF mass: me Elastic column, stiffness: ks damping: D

he

Nonlinear rotational spring Basement Macro-element Figure 4

(a)

(b)

Foundation moment / Mu

Mo/Mu for 0.5 ofor

0

-0.5

-0.03

-0.02

-0.01

0.0

0.01

0.02

0.03

Foundation rotation (rad.) Figure 5

1

Moment / Mu

0.5

0

-0.5 backbone curve

-1 -40 Figure 6

-20 0 20 Foundation rotation (mrads)

40

1

a Normalised rotational stiffness

Foundation moment (kNm)

1

0.5

0

-0.5

-1 -0.03

-0.02

-0.01

0.0

0.01

0.02

b

0.8

0.6

0.4

0.2

0 -4 10

0.03

0.001

Hysteretic damping parameter

0.4

c

0.3

0.2

0.1

0 -4 10

0.001

0.1

Foundation rotation (rads)

Foundation rotation (rads)

0.01

0.1

Foundation rotation (rads) Figure 7

0.01

1

1

Foundation moment / Mu

0.3

0.5

a

0.2

0.3 0.2

0.1

0.1 0

0

-0.1 -0.1

-0.2 -0.3

-0.2

-0.4

SF = 0.5 -0.3 -0.3

-0.2

-0.1

0

0.1

0.2

SF = 1.0

-0.5 0.3

-0.4

-0.2

0

0.2

0.4

1

0.8

c

0.6

Foundation moment / Mu

b

0.4

d

0.8 0.6

0.4 0.4 0.2

0.2 0

0 backbone curve

-0.2

-0.2

-0.4 -0.4 -0.6 -0.6

-0.8

SF = 2

-1

-0.8 -2

-1

0

1

Foundation rotation (mrad) Figure 8

SF = 3

2

-4

-2

0

2

Foundation rotation (mrad)

4

Spectral acceleration / peak input acceleration

10

elastic 8

SF = 0.5 SF = 1 (PGA = 0.3g) 6

SF = 2 SF = 3

4

2

Input: El Centro NS 1940

0

0

1

2

Period (sec) Figure 9

3

4