INTERNATIONAL SOCIETY FOR MECHANICS AND

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Michael J Pender University of Auckland, Auckland, New Zealand Thomas B Algie Partners in Performance, Sydney, Australia Luke B Storie & Ravindranath Salimath Tonkin & Taylor, Auckland, New Zealand

ABSTRACT Recently a number of macro-element models have been formulated for assessing the performance of shallow foundations during loading. These provide a computational tool that represents the nonlinear dynamic behavior of the in a manner much simpler than finite element modelling; consequently, they are useful for preliminary design. The basis of this paper is the 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 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.

This paper shows how the model can be applied in numerical simulation using earthquake time histories and within a pseudo-static capacity spectrum approach. The significance of the maximum displacement (foundation rotation) in relation to the damping and residual rotation at the end of the earthquake record is discussed.

1 INTRODUCTION was an appropriate method of handling soil nonlinearity. In our experience elastic soil-structure interaction (SSI) This is the intention of this paper, an extension of the modelling shows that for shallow foundations, designed in work reported by Pender et al. (2013). accordance with Load and Resistance Factor Design Field data gathered on the rocking response of (LRFD) requirements, the peak response including SSI is shallow foundations on Auckland residual was frequently only marginally different from the fixed base reported and analysed by Algie et al. (2010) and Algie response. To achieve gains from soil-structure interaction (2011) and lead to the suggestion that the shape of the nonlinear soil behaviour must be mobilised. In this context foundation moment-rotation pushover curve is hyperbolic. the term Soil-Foundation-Structure-Interaction (SFSI) has This work, and related finite element modelling by Algie recently come into vogue. (2011) and Salimath (2017), provides the basis of a Kelly (2009) made proposals for the design of shallow method for determining nonlinear moment-rotation curves foundations that rock during earthquake excitation. for shallow foundations with uplift. Also required is Rocking is understood as cyclic uplifting and reattachment information about the hysteretic damping associated with at the edges of the foundation during the course of the foundation rocking which is obtained from cyclic data earthquake. Kelly explained that low to medium-rise obtained during the free following snap-back structures on shallow foundations may not have sufficient release after static pull-back. Finally, a modification of the weight to prevent foundation rocking; in which case the substitute structure method of Shibata and Sozen (1976), designer might want to take advantage of the real benefits utilised recently by Priestley et al. (2007), is used to that follow from accepting modest amounts of rocking. obtain a single degree of freedom model (SDOF) of the One of the topics Kelly recommended for further research structure - foundation system.

With these tools, this paper proposes methods for estimating the earthquake response of shallow foundations for which uplift may occur. However, the approach does not require specific evaluation of the amount of uplift at each stage of the modelling. Rather, we hypothesise that uplift is, in part, responsible for the shape of the hyperbolic moment-rotation curve. Using the hyperbolic moment-rotation curve a shallow foundation macro-element for rotational response, is developed and applied in the paper. (A macro-element is a single computational entity that approximates the complex nonlinear soil-foundation interaction near the Figure 1. Shallow foundation snapback testing of a simple foundation.) structure loaded with and supported on shallow foundations. 2 FIELD DATA AND FOUNDATION MOMENT- ROTATION CURVES

Field experiments have been conducted at a site in Auckland with shallow foundations subject to gradual pull- back followed by cyclic response after snap-back release; more details are given by Algie et al. (2010) and Algie (2011). The set-up for the snap-back testing procedure is shown Figure 1. The pull-back load was measured with a load cell and the foundation displacement with LVDTs positioned at the front and rear of each foundation. 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 damping against the pull- Figure 2. Hyperbolic moment-rotation relation fitted to back angular displacement are presented (Algie 2011). A pull-back data. sequence of snaps from different initial loads shows how the nonlinear behaviour of the foundation develops as the applied load increases. The site used for the tests, in Albany in the northern part of Auckland, consists of a profile of stiff cohesive soil formed by in situ weathering from tertiary age sandstone and siltstone (thus it is a residual soil profile). The ends of the steel frame, shown in Figure 1, were supported on embedded shallow reinforced concrete foundations, 2.0 m in length and 0.4 m square. There were four sets of shallow foundations so the tests could be repeated at four different "sites". The steel frame structure was 2 m wide, 3.5 m high and 6 m long. Steel kentledge was strapped to the top of the frame to provide the vertical foundation load. The soil profile was investigated with CPT tests between the surface and depth of 8 m. In some of these the shear wave velocity of the soil was measured which Figure 3. Cyclic hyperbolic moment-rotation curve. indicated a reasonably consistent shear wave velocity for the materials at the site equivalent to a small strain shear there is considerable nonlinearity in the moment-rotation modulus for the soil of about 40 MPa. curves and some degradation from one snapback to the Further field testing of this type of structure was next, particularly for those tests following the snapback completed late in 2016 at Silverdale, a site further north of which applies the largest moment to the system. Auckland; the results are given by Salimath (2017). A hyperbolic curve through some of the test data is shown in Figure 2. The form of the relationship is: 3 EQUATIONS FOR MOMENT-ROTATION CURVES ϕ M = abϕ [1] Figure 2 presents the static moment-rotation curves + obtained during the application of the pullback forces to KMϕiu one of the foundation sets at the site. It is apparent that

where: M and ϕ are the foundation moment and rotation One additional aspect that needs to be included in the respectively, Kϕi and Mu are the initial rotational stiffness substitute structure model is the compliance of the soil and moment capacity of the foundation, and a and b are beneath and adjacent to the foundation. This is done by numerical values used to refine the fit of the curve to the representing the structure-foundation system as a SDOF data. Salimath (2017) found that values for a and b come structure supported on a rotational spring. Equation 3 from the shape of the foundation. The parameter a also gives the natural frequency of the substitute structure- has an effect on the amount of damping in the cyclic foundation system: hyperbolic relationship discussed below. Equation 1 provides a pushover curve for a shallow ω2 K foundation. Figure 3 shows how the pushover curve can ω=2 se = [3] substitute 2 2 be extended to give the cyclic response of a shallow Khse mehe foundation. Equation 1 acts as a backbone curve so that 1 + K whenever an unloading or reloading loop intersects the ϕ backbone curve the moment-rotation response then where: ωsubstitute and ωs are the natural frequencies follows the backbone curve (this prevents the system of the substitute structure-foundation system and the fixed developing unreasonably large foundation rotations). The base structure respectively, Ks is the stiffness of the fixed static pushover curve is bounded by the moment capacity base substitute structure, Kϕ the rotational stiffness of the of the shallow foundation and the elastic rotational foundation. Ke is the equivalent rotational stiffness of the stiffness, the same parameters control the cyclic moment- substitute structure-foundation system, and he is the rotation curve. When the direction of loading reverses the height of the substitute structure lumped mass (me). stiffness reverts to the rotational stiffness at zero moment, Kϕi. The moment capacity can be reached when the rotation is to the left or the right, so that Mu limits the moment to the right and –Mu limits it to the left. The cyclic moment rotation equation is given by:

K()() AM− M ϕ−ϕ MM=ϕi uo o + [2] o abAM− M + K ϕ−ϕ ()()uoϕ i o where: Mo and ϕo are the initial values for the current branch of the moment rotation curve, and the parameter A takes values of +1 and -1 indicating if the foundation is being loaded towards the moment capacity Mu or –Mu. The rotational stiffness is given by the slope of the current branch of the hysteretic moment-rotation loop and, as in apparent from Figures 2 and 3 and equations 1 and 2, the slope of the M-ϕ curve changes continuously with

M. Figure 4. The basis of Shibata and Sozen’s substitute 4 BUILDING-FOUNDATION SUBSTITUTE STRUCT- structure concept. URE MODEL

The aim of this section is to develop an integrated SDOF model for the rotational response of the structure- foundation system. The shape of the shallow foundation moment-rotation curve is not unlike the moment rotation curve for a reinforced concrete element. Figure 4 reproduces a diagram from the paper by Shibata and Sozen (1976) which suggests a way of incorporating the yielding of reinforced structural members into analysis of a nonlinear structure. The procedure followed in this paper is to make use of the substitute structure ideas proposed by Shibata and Sozen, but to assume that the structural elements remain elastic whilst the nonlinearity in the structure- foundation system is located at the interface between the shallow foundation and the underlying soil. Figure 5, 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 a SDOF model. Priestley et Figure 5. Implementation of the substitute structure as al. explain how the parameters for the substitute structure, presented by Priestley et al. (2007). he, me, and Ke, can be evaluated.

In equation 3 the rotational stiffnesses, Kϕ and Ke, are the Introduction. Also plotted in Figure 8 are the not constant but vary continuously with the applied accelerations of the SDOF mass calculated from the foundation moment. maximum foundation moments, clearly the amount of The algorithm presented in chapter 7 of Clough and damping of the system response is more than 5%. Penzien (1993), for calculating the incremental response The next step in the investigation of the response of of a nonlinear single degree of freedom system, was used the foundation with the hyperbolic moment-rotation curve to compute the responses discussed herein. was to scale the earthquake time history. The scale factors applied ranged between 0.01 and 4 (the results 4.1 Response of the system to the El Centro plotted in Figures 7 and 8 correspond to a scale factor of acceleration time history 1). The outputs are shown in Figure 9. Figure 9a shows that the nonlinear system period increases approximately In this section the calculated responses of a suite of linearly with the scaling factor. (The scaling factor of 0.01 structure-foundation systems to the recorded time history is used to approximate the elastic response of the from the 1940 El Centro earthquake are presented. system.) Figure 9b plots the magnitude of the normalised The buildings are 10 storeys with a basement 4 m deep. The plan dimensions of the structures ranged from 8 m to 24 m square. The natural period of all the fixed base structures is 1.0 second, a damping ratio of 3% of critical is used to specify the damping behaviour of the above ground structure. The SDOF mass is at a height of 28.5m above foundation level. As the structures are assumed to have a uniform distribution of mass with height the effective mass was taken as 0.7 of the total mass acting at 0.7 of the total height of the building. The foundation material is dense dry or with a angle of 45 degrees and a shear wave velocity of 280 m/sec. The results of the field testing shown in Figure 2 showed that the rotational stiffness of the foundation at small rotations is a fraction of that calculated from the shear wave velocity of the soil. Herein two values for the small strain rotational stiffness of the foundation are used: 100% and 30% of that calculated from shear wave velocity. The natural periods of the systems including elastic SSI effects are then 1.04 and 1.07 seconds. All the damping for the foundation is assumed to come from hysteretic energy dissipation in the soil near the foundation. Any contribution from elastic radiation damping is not considered as this is known to be small for rocking response. The earthquake time history is applied Figure 6. Substitute SDOF model and macro-element for directly beneath the basement. 10 storey buildings with 8x8 m to 24x24 m floor plans. The details of the substitute structures are shown in Figure 6. Figure 7 shows the response of the system for the 8x8m building to the El Centro time history scaled to a PGA of 0.3g (that is a scaling factor (SF) of 1). It is apparent that there is considerable hysteretic damping present but the maximum foundation rotation is modest (less than 0.003 radians (about 1/6 degrees)). Figure 8 gives the response spectrum for the calculated data shown in Figure 7 along with the response spectrum for the input motion; both spectra are for damping 5% of critical (for calculating the response spectra a damping value of 5% was used as that is the value associated with the -known El Centro motion used as input). The nonlinear system period marked on the plot, calculated from equation 3 using the secant stiffness defined by the extremes of the moment-rotation loops in Figure 7, is 1.48 seconds. Thus the nonlinear moment-rotation behaviour of the foundation gives a Figure 7. Foundation moment-rotation response when the much larger increase in system period than does linear SDOF model is subject to the El Centro time history with a elastic compliance of the soil beneath the structure; this is PGA of 0.3g (SF = 1, 8x8 m building plan). the background to the comment in the first sentence of

Figure 8. Response spectra (5% damping) for the input time history and system response (8x8m building plan). maximum foundation moment against input scale factor; a 40-fold increase in the scaling factor (0.10 to 4.0) produces only about a 6-fold increase in the maximum moment.

4.2 Effect of scaling factor on the reserve of foundation moment capacity

The static bearing strength factor of safety, under vertical Figure 9. Structure-foundation response to scaled El load only, is very large for the building foundations in Centro acceleration time histories; a) system period; b) Figure 6 as the full plan area of the foundation acts in normalised maximum foundation moment (8x8m building bearing. However, the factor of safety with respect to plan). moment is much smaller. The clearest way of appreciating this is to draw a constant vertical load section of the bearing strength surface (see Pender (2017) for details of the bearing strength surface). Figure 10 shows the moment-shear action paths for the foundation of the 8x8 building model with actions at various values of the scaling factor marked. The height to the SDOF mass sets the ratio between the moment and shear force, so all points lie along the one path. The purpose of Figure 10 is to emphasise that a large reserve of bearing strength, in terms of vertical load on the foundation, does not necessarily mean that the reserve with respect to the ultimate moment will also be large.

4.3 Damping values obtained from numerical snapback calculations

Figure 7 shows that there is considerable hysteretic damping in the system as the moment-rotation response forms reasonably large hysteresis loops; however it not Figure 10. Constant vertical load section of the shallow clear what particular value is indicated for the damping foundation bearing strength surface (BSS) showing the parameter. To elucidate the damping we did some foundation action path and the maximum moment for numerical snapback modelling. In this the foundation various scaling factors (8x8m building plan). pullback moment was applied as a ramp function up to a

Figure 11. Numerical snapback response after release from a pullback moment of 89% of Mu (8x8m building plan). desired value of moment, which was then held at a fixed value for some time to ensure that any dynamic effects induced during the ramp loading decayed, and then the moment was suddenly reduced to zero. Figure 11 has the results of one such calculation. Figure 11a gives the snapback moment-rotation response and Figure 11b gives the response in terms of foundation rotation against time. Using the logarithmic decrement approach the equivalent viscous damping ratio for the first cycle of the response is 32%. It is apparent from Figure 11a that the snapback response is dominated by the first cycle with large damping, and for subsequent cycles the damping Figure 12. Numerical snapback results (8x8m building reduces. plan): (a) damping values from snaps at various positions Figure 12a gives damping responses for several around the moment-rotation curve, (b) damping values snapback runs for the 8x8m building model. When the against foundation rotation at snap release, (c) backbone moment at the snap is a small proportion of the moment normalised secant modulus against foundation rotation. capacity the damping is small. When the moment at the snap approaches the moment capacity of the foundation The damping and secant modulus relations in Figures the logarithmic decrement damping for the first cycle of 12a and 12b are also affected by the static bearing the response is more than 30%. Figure 12b has the strength factor of safety, which is a function of the plan logarithmic decrement damping values plotted against dimensions of the foundation. The values plotted in Figure the logarithm of the foundation rotation prior to the snap 12 are for the building of 8x8m plan dimensions. release. Figure 12c has the backbone moment-rotation Damping can be reduced by setting a in equation 2 to curve in Figure 12a converted to secant moduli and a value greater than 1; for calculations with the scaling plotted against the logarithm of the foundation rotation.

Figure 13. Influence on earthquake PGA scaling factor (SF 1 to 3) on nonlinear period lengthening at maximum foundation moment. (10 storey buildings with plan dimensions: 8x8m, 12x12 16x16, 20x20 and 24x24m.) factor greater than 1 a value of 1.5 was used (the sensitivity of the response to the a value is currently being checked).

4.4 Effect of dimensionless variables on the response

Classical elastic soil-structure interaction studies propose a system of dimensionless parameters for assessing the effect of the properties of the soil around the foundation on SSI effects (for example Wolf (1985)). The most important of these are: Figure 14. Static pushover estimate of system performance (8x8 m building with EQ scale factor of 1.0). (a) trial moment-rotation points, (b) trial system periods on ω hh M SR=ee HR = e MR = e [4] the spectrum. Vaρ a3 s portion of the foundation moment capacity mobilized at where: SR is the stiffness ratio, HR the slenderness ratio peak response. As expected the period elongation is and MR the mass ratio, a is the radius of a circle having largest for the largest value of the scaling factor and, for a given scaling factor, the larger foundations have smaller the same area as the foundation and ρ is the density of period elongation. the soil. The stiffness ratio is the same for all the models 5 MODIFIED CAPACITY SPECTRUM APPROACH investigated in this paper. Both the slenderness ratio and mass ratio decrease as the plan dimensions of the An alternative to time history modelling is a method based buildings increase. The mass ratio can be thought of as on the so-called capacity spectrum method (Applied the mass of the structure in relation to the mass of a Technology Council 2005). The capacity spectrum volume of soil beneath the foundation which contributes to approach usually considers nonlinear behaviour of the the foundation stiffness. As this ratio decreases the inertia structure with a fixed base or elastic foundation, however of the foundation soil becomes more significant; herein the structure is assumed to remain elastic and the consequently the nonlinear foundation response foundation to be nonlinear. The capacity spectrum diminishes as the building plan dimension increases. method is a pseudo-static iterative approach. The Figure 13 has results for buildings with plan foundation moment-rotation curve gives the moment cap- dimensions from 8x8 to 24x24 m and with scaling factors acity of the foundation. The demand comes from the of 1, 2 and 3 applied to the El Centro earthquake record. spectrum for the earthquake motion (or the design The figure plots the difference between the nonlinear spectrum). Where these two curves intersect, the system period and the elastic SSI period against the pro-

“performance point”, indicates where the demand equals Observations after the Christchurch earthquake of the capacity. Once the soil profile details and properties February 22, 2011 indicated that a residual rotation of are known, foundation dimensions have been adopted, about 0.5 degrees (≈ 0.01 radians) is easily visible to the and the vertical load carried by the foundation evaluated, naked eye. Thus, a possible design decision would be to the steps for each iteration are as follows: limit the maximum foundation rotation during an a. Use equation (1) to develop the moment-rotation earthquake to 0.01 radians in the expectation that the relationship for the foundation. residual rotation will be much less. Reference to Figure12b shows that substantial damping is available at b. Evaluate he, me and Ke for the substitute structure rotations considerably less than 0.01 radians. This (Figure 5) following Priestley et al. (2007). answers an important criticism directed against design c. Use equation (3) to evaluate the elastic natural approaches which mobilise a significant fraction of frequency of the substitute structure-foundation foundation moment capacity. Apparently the desired system using the elastic stiffness of the foundation. objective can be achieved at modest foundation rotations, d. For the first iteration use the period derived (c) to get with consequent smaller residual rotations and no the spectral acceleration and evaluate the foundation foundation moment capacity reduction from P-∆ effects. The method discussed herein could be applied in a actions and rotation. force-based or displacement-based framework. e. Repeat step (d) using a range of foundation stiffness values less than the elastic value to obtain a range of system periods and foundation actions. 7 CONCLUSIONS f. For each iteration in (e) it is necessary to assign an appropriate damping value (Figure 12b). In this paper a single degree of freedom structure- g. Plot the results from (d) and (e) onto the foundation foundation model was developed for shallow foundations moment-rotation curve and hence determine the which are subject to vertical loads that are a small fraction actions and period where the demand crosses the of the vertical bearing strength of the foundation, thus, the response is dominated by rocking and the vertical and foundation curve. horizontal compliances are of less significance. Buildings Figure 14 gives the results of the above series of of 10 storeys with varying plan dimensions and one level calculations for the 10 storey structure with an 8x8 m floor of basement were considered. The nonlinear response of plan. Figure 14a gives the foundation moments and the system to scaled El Centro earthquake acceleration rotations for various trial values of the foundation histories was calculated. Based on the results from this rotational stiffness. Figure 14b has the El Centro work the following tentative conclusions are reached: spectrum as well as the periods for the trial foundation • The response to scaled earthquake input motions is stiffnesses. The El Centro spectrum on which these highly nonlinear (Figure 9). calculations were done is for a damping ratio of 5%. • Because of the continuous reduction in foundation Marked in the figure is a point located above the moment- rotational stiffness with increasing moment the rotation curve which, if recalculated with a damping ratio ultimate capacity of the foundation is not mobilized of 17% (about the value one would obtain from Figure 12b even at large PGA values (Figure 9b). for a foundation rotation of 0.003 radians), falls on the moment-rotation curve. These values are close to those • The foundation hysteretic damping is not a constant at the peak moment and rotation values plotted in Figure value but depends on the foundation rotation (Figure 7. 12b). The process outlined above is attractive because of its • The numerical snapback calculations show that the simplicity and “hands-on” calculation, but it does require a foundation response is dominated by the large relationship between the foundation rotation and the damping in the first cycle of response (Figure 11a). appropriate damping value (similar to that given in Figure • As the plan dimensions of the buildings and shallow 12b, at the appropriate static bearing strength factor of foundations increase, the nonlinear response is safety). reduced (Figure 13). • Static pushover analysis gives a useful indication of 6 PERFORMANCE BASED DESIGN maximum response obtained from time history calculations (Figure 14). The main challenge with this The modelling in this paper supports, from a different method is selecting the appropriate value for the perspective, what many other researchers have foundation hysteretic damping (a diagram like Figure emphasised: that nonlinear foundation behaviour has a 12b, for the appropriate static bearing strength factor beneficial effect on the earthquake response of of safety, could be used to source these). foundations for buildings and bridges. To achieve these benefits it is important to mobilise enough foundation rotation to obtain sufficient hysteretic damping. Figure 12b 8 REFERENCES shows that rotations as small as 0.002 radians give damping ratios in excess of 10%, and 0.004 radians gives Algie, T. B., Pender, M. J. and Orense, R. P. 2010. Large a damping ratio of about 20%. scale field tests of rocking foundations on an From a design point of view, the maximum foundation Auckland residual soil, In: Soil Foundation Structure rotation is important as it influences the residual rotation. Interaction (R Orense, N Chouw, and M Pender

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