# Abstract

The moment-rotation response of shallow foundations is inherently nonlinear being controlled by the “elastic” stiffness at small rotations and the moment capacity of the foundation at larger rotations. Simple elastic soil structure interaction (SSI) does not model this rotational response and elastic radiation damping is very small for rocking response. The purpose of this paper is to report on shallow foundation field tests at a site in Silverdale which complement earlier tests done at a site in Albany. In these tests the foundations were subject to bi-directional loading so that the response was first measured by loading in one direction and then loading in the opposite direction. Initially, two-way slow cyclic loading was applied to the foundations with gradually increasing maximum moments until the moment capacity of the foundation was found. After this bi-directional snap-back testing was done, with the snap-back release from progressively increasing moments. The pull-back parts of these tests confirm the nature of the moment-rotation curves for the foundations and the measured curves are compared with those derived from nonlinear finite element modelling with PLAXIS 3D. The snap-back parts of the tests confirm that the damping in the first few half cycles of response is large, that the damping is amplitude dependent and decreases as the motion decays. The paper details the process for obtaining the damping values and discusses how this behaviour influences shallow foundation behaviour.

# 1 INTRODUCTION

The moment-rotation response of shallow foundations subjected to rocking is highly non-linear, a consequence of partial detachment of the footing edges and subsequent reattachment during cyclic loading. This reduction in contact area of the footing during rocking may result in reduction of actions applied to the superstructure which influences the seismic performance of the structure. Kelly (2009) explained that much of the NZ multi-storey building stock does not have adequate weight to resist the overturning moment that may occur during earthquakes and in such cases initiation of rocking can be beneficial as it limits the actions transmitted to the superstructure. Various researchers over the years have studied the possible benefits of rocking and how it may be incorporated into the design of shallow foundations under earthquake excitation. In order to achieve this it is important to estimate the amount damping that occurs during the rocking process as well as the change in stiffness that occurs with the foundation rotation.

Figure 1: CPT data (left) and water content and shear wave velocity profiles (right

Algie (2011) carried out several large scale field tests on embedded strip footings (L/B = 5 and D_{f}/B = 1) to study the rocking and dynamic response of shallow foundations on stiff clay. In addition to this he also undertook some finite element modelling using ABAQUS (Simulia 2010) to study the nonlinear moment-rotation response of the footings. This paper presents results from the snap-back experiments performed as a follow-up to Algie’s testing. One of the key features of this testing program was that separate snap-back tests were performed with moment about the shorter and longer axis. Our finite element studies showed, using PLAXIS 3D (Plaxis BV. 2012), that rectangular footings exhibit different rotational stiffness depending on the axis about which the moment is applied. This paper discusses the experimental set-up and static moment-rotation response obtained from slow-cyclic and snap-back tests. The damping values obtained for both of the test footing configurations are presented and their influence of design of rocking shallow foundations is discussed.

# experimental programme

## 2.1 Site description

The tests were carried out on a site located in Silverdale, north of Auckland. The soil properties at the site showed that the ground is comprised of stiff to very stiff residual clay with the ground water table at approximately 1m below the ground surface. Geotechnical investigation was undertaken at the test location which comprised of CPT’s and seismic dilatometer (SDMT) tests. Shear wave velocity profiles, using geophysical methods (MASW), were obtained after the foundation load tests were completed. Some of the CPT results along with the water content and wave velocity profile are shown in Figure 1. As seen from CPT data, stiff silty clay (cone resistance greater than 2 MPa) was encountered up to a depth of around 6m from the ground surface. The shear wave velocity profile reveals that the shear modulus of the soil increases with depth. The tests were performed at three different locations within the site. The ground surface was levelled at the test locations so that the footing can rest on a horizontal surface.

Figure 2: (a) Steel frame assembly for the experimental rig; (b) PFC380 channel sections with stiffeners used as footings

Figure 3: Experimental set-up assembled on site for snap-back test

Figure 4: Hysteresis moment-rotation curves from slow-cyclic tests

Figure 5: Partial detachment of the footing with the underlying soil

## 2.2 Experimental set-up for slow cyclic and snap-back tests

The structural details and dimensions of the test assembly used for the experiments are shown in Figure 2a and 2b. The test assembly consists of two frames consisting of 200UC46.2 and 100UC14.8 sections. The two frames were bolted to each other with the 6m long frame resting horizontally flat on the ground. Two PFC380 channel sections (Fig 2b) were used as footings and were bolted to the underside of the 6m frame. The entire test structure was pulled and snapped in North-south direction only but two different test footing configurations were used so that moment can be applied about the short axis (configuration 1) and long axis (configuration 2) of the footings separately. The experimental set-up assembled on site for snap-back testing with moment about the short axis is shown in Figure 3. The rotation of the footing was measured using inclinometers and LVDT’s placed at either ends of the footing. Accelerometers were installed at different heights to measure the acceleration and force time history during the snap-back tests. Draw wires were used to measure any horizontal displacement of the test structure during rocking. The test structure was pulled horizontally and both slow cyclic and snap-back tests were performed at pullback forces ranging from 5kN to 30kN.

## 2.3 Test procedure

The experimental program comprised of slow-cyclic and snap-back tests. The test structural assembly was the same for all the tests performed except that for snap-back test two different footing configurations were used. For slow cyclic tests, the test structure was pulled in horizontal direction (Figure 3) until the desired load is achieved and then slowly released. The initial pullback phase is the same for snap-back tests but instead of slow release the test structure is snapped using a quick-release mechanism, so that the structure rocks in the axis of the applied moment. The rotation-time and settlement-time history data recorded during the snap-back tests is used to estimate the magnitude of damping that occurs during rocking.

# 3 EXPERIMENTAL RESULTS

## 3.1 Slow-cyclic tests

The field experiments carried out for this study comprised of slow-cyclic and snap-back tests on two strip footings (L/B = 5). The hysteresis moment-rotation curves from the slow cyclic tests are shown in Figure 4. The tests were performed at pullback forces of 5kN, 10kN, 20kN and 30kN. For 10kN case, since the applied moment is very small (M/M_{ult} = 0.23, where, M_{ult} = ultimate moment capacity of the footing), the footing rotation is small and the moment-rotation plot is elastic. At a load of 30kN, the moment-rotation response is highly nonlinear and large plastic displacements are induced. There is also partial loss of contact between the footing and the underlying soil at this stage as seen in Figure 5. The theoretical moment-rotation curve obtained from the hyperbolic equation, modified to account for the footing shape, fits the experimental response well which confirms that nonlinear moment-rotation response of shallow foundations is hyperbolic in nature.

## 3.2 Snap-back test results

Unlike slow-cyclic tests, the snap-back tests were performed for two different footing configurations at three different locations on the site. For each configuration, snap-back releases were performed at different pull-back forces. Figure 6 shows the moment-rotation response of the footing during pull-back phase prior to the snap-back release for footing configuration 1. Also superimposed on the experimental results is the moment-rotation response from modified hyperbolic equation. From the curves it is evident that with every snap there is some degradation of rotational stiffness. The initial small strain rotational stiffness from the experimental curves was compared with theoretical stiffness predicted using Gazetas (1991) solutions. The comparison shows that the small strain rotational stiffness from the experimental curves is around 22-28% of the theoretical stiffness. The degradation of the rotational stiffness can be a result of rounding of soil surface under the footing due to footing rocking during snap-back. Pender (2015) concluded that degradation of rotational stiffness with increase in moment implied that it unlikely that ultimate moment capacity of the footing will ever be mobilized even at large PGA values. The rocking response of the footing is affected by both material and geometric (footing L/B ratio) nonlinearities.

Finite element studies carried out as part of this research showed that a rectangular footing exhibited higher rotational stiffness when moment is applied about the longer axis (configuration 2). Figure 7 has a comparison of the moment-rotation response of footings (L/B=5) obtained from pull-back phase of field experiments and that from finite element analysis. The finite element results are seen to converge with those from the field tests, which affirms that non-linear moment-rotation response of footings predicted from Plaxis3D is reliable. However, Plaxis holds the stiffness along the first part of the moment-rotation curve constant, a consequence of the way in which soil model stiffness is defined in the software.

Figure 6: Moment-rotation curves from pull-back phases for snap-back tests with moment about the short axis (configuration 1)

Figure 7: Comparison of moment-rotation curves from field experiments and PLAXIS 3D

## 3.3 Damping

One of the primary objectives of performing these snap-back tests was to estimate the amount damping that occurs during foundation rocking. The value of damping is very critical is assessing the enhanced seismic performance of rocking shallow foundations. Algie (2011), based on his snap-back experiments, concluded that the damping is significant when the footing rotation is greater than 0.6 milliradians. Pender (2014) concluded that foundation response is governed by the large damping that occurs in the first cycle of the response. Various methods are presented for estimation of damping in the past literature. Unlike dynamic loading, during snap-back test the natural frequency varies with cycles and hence, for this research, logarithmic decrement method (Chopra 2007) was used to evaluate damping. Figure 8 shows the rotation-time response obtained from snap-back tests for moment about the short axis. Damping ratio was estimated for each peak of the rotation time plot and the results are presented in Figure 9. The plots confirm that large damping occurs in the first cycle of the response, the magnitude of the damping is greater for snap-back from 30kN because the larger footing rotation at the time of snap. Thus the larger the

Figure 8: Comparison of rotation-time plots for snap-back tests with snap at 10kN and 20kN, with moment applied about short axis

Figure 9: Variation of damping ratio vs footing rotation at each impact for snap at 15kN and 30kN

footing rotation at snap release the greater is the energy dissipation in the first rocking cycle and hence the initial damping value is larger. An additional factor which is important for foundation performance is the residual rotation when the motion ceases. The results in Figure 8 show that for rotations up to at least 10 milli-radians (about half a degree) at pull-back there is negligible residual rotation. This is very critical in seismic performance of rocking foundations, because significant residual rotation of the foundation means that tall structures have an easily perceived lean. The field test results show that even modest foundation rotation at snap-back results in good damping performance with negligible residual rotation. Figure 9 shows how the damping varies during successive cycles of response. More discussion of these matters is given by Salimath (2017) and Pender et al. (2017).

# 4 CONCLUSIONS

The following conclusions are reached:

- The test results show that, for the fixed static bearing strength factor of safety at which the tests were done, the moment-rotation response of shallow foundations resting on clay is highly nonlinear and hyperbolic in nature, Figures 4, 6 & 7;
- A contributing factor to the nonlinearity of the moment-rotation curve is the reducing area of contact as the moment applied to the foundation increases, Figure 5;
- Extensive finite element modelling has confirmed this conclusion for cyclic loading from other static factors of safety;
- With the increase in moment at each snap there is degradation of rotational stiffness of the footing. The magnitude of decrease in rotational stiffness also depends on the initial static factor of safety of the footing;
- The small strain rotational stiffness obtained from experimental moment-rotation curves was around 28% of the theoretical rotational stiffness (based on the shear wave velocity of the soil);
- The foundation moment-rotation curve obtained during the pull-back phase of the field tests can be modelled quite closely with PLAXIS 3D, Figure 7;
- When footing rotation at snap-back is larger, the number of impacts in the rotation-time plot is less which implies that the energy dissipation in the first few rocking cycles is high, Figure 8;
- Damping ratio is higher for the first rocking cycle, the magnitude also depends on the footing rotation at snap-back, 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.

Chopra, A.K. (2007) *Dynamics of Structures, Theory and Application of Earthquake Engineering*, third edition. Pearson Education, Inc.

Gazetas, G. (1991) Foundation Vibrations. In: *Foundation Engineering Handbook*, H. Y. Fang, Ed. Van Nostrand, pp. 553-593.

Hakhamaneshi, M., and Kutter, B.L. (2016) Effect of footing shape and embedment on the settlement, recentering and energy dissipation of shallow footings subjected to rocking. *Journal of Geotechnical and Geoenvironmental Engineering, *Vol 12.

Pender M.J. (2014) “Integrated design of structure-foundation systems: the current situation and emerging challenges”. Keynote address: *Proceedings of NZSEE Conference*, Auckland, New Zealand.

Pender, M.J., Algie, T.B., Storie, L.B., and Salimath, R.S. (2017) One dimensional moment-rotation macro-element for performance based design of shallow foundations. Paper 332, *PBD III Earthquake Geotechnical Engineering conference, *Vancouver.

Plaxis BV. PLAXIS 3D (2012) Reference Manual. Delft, Netherlands.

Salimath, R (2017) Experimental and finite element study of moment-rotation response of shallow foundations on clay, PhD thesis, University of Auckland.

Simulia (2010) ABAQUS 6.8-EF2. Dessault Systemes.