Shallow Foundation Stiffness

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This chapter goes into detail on the results from the large scale field tests that were performed on residual soil on a site in Auckland. It begins by describing past experiments from the pioneering laboratory experiments done at the University of Auckland through to the latest centrifuge test at The University of California, Davis. The review highlights findings, including moment capacity equations, rotational stiffness degradation, and high levels of damping. The results section following the literature review begins with the forced-vibration tests and progresses to the snap-back tests. The chapter ends with conclusions and observations made from the experiment results. Throughout the results it is evident that data gathered supports past conclusions, mentioned above, and the benefits and detriments of rocking foundations become apparent. The presentation of results is similar for both sets of tests with a few additions specific to either forced-vibration or snap-back tests. The topics covered in the results are: Bearing capacity prior to testing Input forcing function (forced-vibration tests only) Moment-rotation – including the static and dynamic moment-rotations (snap-back tests only), the moment capacity equation, and fitting of static pull back moment-rotations to a hyperbolic curve (snap-back tests only). Rotation stiffness – the initial stiffness and the degradation throughout the testing Shear (forced-vibration tests only) Damping Frequency response Settlements Energy Analysis (for only the snap-back tests) Appendices B and D cite the critical plots for all the forced-vibration tests and the snap-back tests respectively.


Laboratory Experiments

Bartlett (Bartlett 1976) conducted 1g experiments by testing model footings (0.50m by 0.25m) on clay subjected to slow cyclic and dynamic rocking. He presented the relationship between the vertical load and overturning moment that acts on a shallow foundation, and the influences of soil properties and foundation stiffness on the amount of energy dissipated. The dynamic experiments agreed with Housner’s conclusion that foundation rocking elongates the period of a structure (Housner 1963). They also showed large rotations, which partially separated the soil from footing, induced soil yielding, resulting in a loss of foundation stiffness. Wiessing (Wiessing 1979) conducted similar 1g tests on shallow foundations (0.50m by 0.25m) sitting on sand and subjected to slow cyclic and dynamic rocking. He found that large energy dissipation, and subsequently progressive settlement, occurred during rocking cycles. Wiessing, alongside Bartlett, described a rounding of the soil that occurs due to yielding under the edges of a foundation, which reduces the stiffness of the system and causes nonlinearity in moment-rotation behaviour. The paper by Taylor et al. (1981) summarises the two research experiments by Bartlett and Wiessing. Figures 4.1 and 4.2 give moment-rotation and settlement-rotation plots for the experiments done on clay and sand respectively. The rocking foundations induced nonlinear moment-rotation behaviour even when the underlying material was elastic because of changing contact between soil and foundation. This relationship became highly nonlinear with hysteretic damping when material yielding underneath the footings occurred. Research revealed that the yielding of soil might occur beneath a foundation without any serious affect on the bearing capacity if the initial vertical factor of safety is high. They suggested that spread footings may be designed to yield during high intensity earthquakes to capitalise on the benefits of rocking foundations. Georgiadis and Butterfield (1988) performed 1g tests on model footings (0.40m by 0.05m) resting on sand and subjected the models to eccentric and inclined loads. The tests investigated relationships between the ultimate vertical, horizontal, and moment for combined loads. They helped verify the size and shape of the interaction diagrams for N-V and N-M space. They also investigated the coupling between the vertical displacement, horizontal displacement, and rotation. They found that by increasing the eccentricity of inclined load, the vertical displacements reduce while the horizontal displacement and rotation increase. Gottardi and Butterfield (1993) carried out experiments on model footings (0.50m by 0.10m) to further analyse the interaction diagrams and failure conditions in all three loading planes. They found simple expressions to interpret the experimental data, and the experiments confirmed what previous research proposed about the failure planes for M-N and V-N space. Gottardi et al. (1999) performed tests with model circular footings (0.10 m diameter) on dry dense sand. Combinations of vertical, horizontal, and moment loadings were applied to the footings. The results provided insight to bearing capacity of footings in cases other than vertical loading. He also obtained data about the hardening law and flow rule appropriate for a plasticity model, as well as an elastic response within the yield surface. Shirato et al. (2008) performed 1g large scale shake table tests and cyclic loading of shallow foundation models (0.50m by 0.50m). They tested a range of soil densities and aspect ratios of foundations, and it is to date one of the most complete experimental data sets for the nonlinear behaviour of shallow foundations. The foundations rested in a large container filled with sand with relative densities, Dr, of 60% or 80% and were subject to either cyclic lateral loading or dynamic base shaking. During the experiments they observed a change in the contact length between the foundation and soil due to soil rounding. Consequently, there was a decrease in stiffness of the foundation altering the predominant vibrating frequency. They assumed the rotation of a shallow foundation can be broken into three parts: elastic rotation, plastic rotation, and rotation from uplift, and they found the uplift component can be as dominant as the plastic component. The research also demonstrated that yielding shallow foundations dissipate energy well, and the residual displacement is dependent on the number of cycles during a test as well as the base excitation intensity (Shirato et al. 2008).

Centrifuge Experiments

Several series of centrifuge tests have been performed on foundations sitting under shear walls, many of which were at the University of California, Davis. The first set of experiments was the KRR01 tests (Rosebrook and Kutter 2001a). The model consisted of a rigid structure sitting on strip footings made from aluminium. Two shear walls connected by a rigid floor diaphragm were tested at 20g. The diaphragm was included to maintain stability during testing. Figure 4.3 shows the model used in the testing program. The footings were embedded into dry Nevada Sand with a relative density, Dr, of either 60 or 80%. In total, 35 static events, including vertical and lateral push tests, and 46 dynamic events were applied to the structure. The dynamic events included a variety of motions such as frequency step waves, sine waves, and scaled earthquake motions. The footing size, the mass, or the relative density of the sand altered the static factor of safety (FSv) in bearing throughout the testing program. The values for the static FSv ranged from 1.3 to 6.2. After these first experiments, the KRR02 test series continued on a similar track (Rosebrook and Kutter 2001b). However, differences between the two testing series include: The KRR02 models rested on the ground surface and were not embedded as the KRR01 models were. A small amount of WD40 (lubricating oil) was applied to the sand around the foundations. This was to provide some cohesion to stop sand collapsing into the footprint before and after testing. KRR02 tested footings on dry sand with a relative density, Dr, of 60%, unlike KRR01 that tested on both 60 and 80%. Only two values of Static FSv were tested; 1.6 and 4.1. The KRR03 series was the final series performed by Rosebrook and Kutter, an investigation into shear wall structures sitting on clay (Rosebrook and Kutter 2001c). The soil was made up of, in prototype scale, a 2.72m layer of dry Nevada Sand (90% + relative density), underlying a 1.70m layer of San Francisco Bay mud. A thin layer of Monterey Sand (0.26m) was applied over the clay for ease of placing the structure on the ground surface. The clay was consolidated over time using a hydraulic press until the shear strength at the surface was around 180 kPa. However this value decreased with each event, and, for the static lateral and dynamic tests, su was around 100 kPa. The soil was saturated in flight prior to testing; the container was spun, and valves opened to allow any pore water to drain out of the bottom. Once drained, the valves were closed and water was added from above. This technique was adopted for a more accurate estimate of water content. Apart from the difference in soil type, the testing was similar to the previous two. The static FSv values in KRR03 were 2.8 and 4.8. The fourth series of tests to study the behaviour of rocking shear wall footings (SSG02) went back to having strip footings resting on dry Nevada Sand with a relative density, Dr, of 80% (Gajan et al. 2003a). However, this series tested single walls and footings, unlike the previous series which tested dual walls for stability purposes. The vertical weight on the structure varied along with the lateral load heights. A range of safety factors were recreated including 3.4, 5.3, 6.8, and 9.6. All of the footings in this series rested on the ground surface. The series SSG03, which came after the SSG02 series, was altered to include foundation embedment (Gajan et al. 2003b). Five walls were tested with varying vertical load, where the static FSv’s for SSG03 were 1.1, 4.0, 6.4, 8.2, and 11.5. A summary of the above five tests series (KRR01, KRR02, KRR03, SSG02, and SSG03) can be found in Gajan et al. (2005) describing the testing program, the data processing, and outlines important test results – mostly on sand. The paper discusses the rotational stiffness degradation displayed by a rocking foundation and gives a recommended stiffness reduction for shallow footings rocking on sand, given in Equation where Kθ_f = rotational stiffness; Kθ_max = maximum (or initial) rotational stiffness; and θf = rotation (radians). This equation was accurate within ±1 standard deviation of all the centrifuge results for foundation rocking on sand, as well as with results from 1g tests performed at the University of Auckland (Wiessing 1979). The paper goes on to discuss the relationship between permanent settlement and the static vertical factor of safety, FSv. It plots the experimental failure points in the moment-vertical (Fm-Fv) plane against two theoretical failure envelopes from Cremer et al. (2001), and Houlsby and Cassidy (2002). Consequently, an equation for the moment capacity of a foundation was given: where Mmax = moment capacity; N = vertical load; L = length of foundation in shaking direction; and Lc = critical contact length – length of foundation where the bearing capacity factor of safety equals 1. For the footings resting on the ground surface, it was shown that: where FSv = vertical factor of safety in bearing. The critical contact length, Lc, for embedded footings took into account the passive, active, and sidewall effects (Gajan et al. 2005): where Pp, Pact, and Po are the passive, active, and at rest earth pressures; embedment; and k = base/side shear coefficient. D = depth of Figure 4.4 shows the concept of a critical length, this is further discussed in (Gajan and Kutter 2008b) where it uses a more generic term of Lc, regarding it as an area (Ac) rather than a length. This allows multi-direction, or shaking outside the longitudinal plane of the foundation, to occur. Gajan and Kutter (2008b) also alter the equation to account for embedment, changing Equation 4.2 to: where A = foundation area; Ac = critical contact area; Fside = side friction resistance; and Pp = passive earth pressure resistance. The paper mentions that the last two terms in Equation 4.5, the terms with Fside and Pp, count for less than 5% of the moment capacity for relatively shallow embedded footings (footing height < D). Equation 4.5 is similar to Equation 4.4, but it includes the frictional sidewall resistance and simplifies the end wall resistance from three terms (containing Pp, Po and Pact) to one term (containing just Pp). Research by Gajan & Kutter (2008a, 2009b) discuss the SSG04 test series, the rocking shear wall tests performed on sand. The load-displacement behaviour of shallow foundations depends on the moment to shear ratio (related to the aspect ratio of the wall). For large moment to shear ratios, where the height of centre of gravity is larger than the footing length, a foundation tends to rotate more than slide. As this ratio increases, so does the energy dissipated by rocking as opposed to sliding. The damping ratio (ξ) was calculated according to the following definition for estimating the amount of energy dissipated during the rocking (Kramer 1996b): The calculated energy dissipation due to rocking can have equivalent damping ratios between 20-30% (Gajan and Kutter 2008a). The capacity of a rocking foundation is dependent on the A/Ac ratio, and from this ratio well defined moment capacities can be assumed. It suggests the design of foundations should accommodate a large A/Ac ratio that has the following positives: Allowing uplift to enable mobilisation of a well defined moment capacity that is insensitive to A/Ac. Considerable amount of energy dissipation from a rocking footing, even from such high A/Ac Minimal permanent settlement.Figure 4.5 shows the test setup for the JMT02 centrifuge test series at Davis that combined frame-wall-foundation systems (Chang et al. 2006). The test series focused on an idealised two story two bay planar reinforced concrete frame with an attached shear wall. They were among the first experiments to have reasonably accurate simulations of building nonlinearity with foundation nonlinearity. Test analysis indicated the frame-wall systems have highly asymmetric hysteretic loops due to the asymmetry of the lateral force resisting system. In addition, the moment capacity of a footing is dependent on the vertical load (Equation 4.2), and therefore, a shear wall at the end of a building will behave differently to one in the middle (Chang et al. 2006).Recently, the research focus at Davis shifted to rocking bridge foundations. Ugalde et al. (2006) tested embedded bridge footings that were attached to an elastic column and then fixed to a lumped mass. The system is considered a typical ‘lollipop’ structure. Similar conclusions about the foundation moment capacity were reached in that it is heavily dependent on vertical load and on foundation length.Figures 4.6 and 4.7 show the test setups for the LJD01, and LJD02 test series (Deng et al. 2008, Deng et al. 2009). The first series, LJD01 was similar to JAU01 in that it tested ‘lollipop’ structures on shallow foundations. However the second series, LJD02, tested holistic bridge systems where foundations were attached to a dual column bent that was attached to a deck that sat on abutments. In addition, both series introduced yielding columns as well as yielding foundations. Columns were made to a required strength and the foundations were designed to be either stronger, weaker, or match that strength to get an array of different yielding mechanisms. All the columns were notched at one diameter about the foundation to create a weak point where the rotation would occur. This coincided with the distance to the centre of a plastic hinge region of a reinforced concrete column.Figure 4.8 shows moment-rotation loops from two separate tests during the LJD02 series (Deng et al. 2010). The plots on the left correspond to a smaller foundation and the ones on the right to a larger foundation. In addition, the top row shows the footing rotation where the bottom row shows the column rotation. The dotted line in the column moment-rotation is the capacity of the column recorded during a slow cyclic test. The smaller footing rotates more and dissipates more energy than the larger footing, and the opposite is happening in the column rotation plots; the larger column rotates more and is closer to capacity. Overall more rotation was observed in the larger footing due to the larger rotation of the column. The plots show that a smaller footing can protect the structure by reducing demands in the column, and also can dissipate greater energy.

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1.1 Research Motivation
1.2 Thesis Outline
Literature Review 
2.1 Shallow Foundation Stiffness
2.2 Bearing Capacity
2.3 Soil Foundation Structure Interaction2.4 Rocking Foundation History
2.5 Discussion
Experimental Configuration
3.1 Concept
3.2 Design
3.3 Site
3.4 Auckland Residual Soil Properties
3.5 Cone Penetration Tests
3.6 Laboratory Tests
3.7 Seismic Cone Penetration Tests
3.8 Wave Activated Stiffness Tests
3.9 Spectral Analysis of Surface Waves Tests
3.10 Hand Shear Vane Tests
3.11 Instrumentation
3.12 Construction
3.13 Excitation
3.14 Vertical Load
3.15 Data Acquisition
3.16 Data Processing
3.17 Test Summary
3.18 Discussion and Conclusions
Experimental Results 
4.1 Overview of Previous Research
4.2 Forced-Vibration Test Results
4.3 Snap-Back Test Results
4.4 Discussions and Conclusions
Numerical Model Development 
5.1 Past Numerical Models of Shallow Foundations
5.2 Development of the Abaqus Model
5.3 Development of the OpenSEES Model
5.4 Comparison to Experiment Pushover Curves
5.5 Moment-Rotation Equation
5.6 Discussion and Conclusions
Rocking Foundation Design Guideline 
6.1 Past Rocking Foundation Design Guides
6.2 Displacement Based Design Method
6.3 Force Based Design Method
6.4 Single Story Shear Wall Design Example
6.5 Six Story Shear Wall Design Example
6.6 Discussions and Conclusions
7.1 Main Conclusions
7.2 Concluding Remarks from Each Chapter

Nonlinear Rotational Behaviour of Shallow Foundations on Cohesive Soil

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