large-scale experiments on nonlinear behavior of shallow ... - J-Stage

SOILS AND FOUNDATIONS Japanese Geotechnical Society

Vol. 48, No. 5, 673–692, Oct. 2008

LARGE-SCALE EXPERIMENTS ON NONLINEAR BEHAVIOR OF SHALLOW FOUNDATIONS SUBJECTED TO STRONG EARTHQUAKES MASAHIRO SHIRATOi), TETSUYA KOUNOii), RYUICHI ASAIiii), SHOICHI NAKATANIiv), JIRO FUKUIv) and ROBERTO PAOLUCCIvi) ABSTRACT We conducted a series of 1G large-scale shake table tests and cyclic eccentric loading tests of a shallow foundation model. The experimental parameters were the diŠerence in loading methods (i.e., dynamic and static), input seismic motions (i.e., intensity and number of cycles), soil densities (i.e., dense and medium dense), and the ratio of horizontal and overturning moment loads. The experimental data set contains the accelerations and displacements of the soil and foundation as well as the distributions of normal and shear reaction forces at the foundation base. The experimental results provide crucial data to model the coupling eŠect among vertical, horizontal, and overturning loads, the accumulation of irreversible displacement, and the foundation uplift, and so is one of the most complete benchmark data sets for the development and validation of numerical models for the nonlinear response of shallow foundations to strong earthquakes. Key words: cyclic loading, (macro-element), seismic response, shake table test, shallow foundation (IGC: E3/E8/E14) ing judgments and the idea that shallow foundations have a su‹cient safety margin, even against strong earthquakes. We have observed no harmful damage to highway bridge shallow foundations such as settlement or inclination. However, our understanding of the actual behavior of shallow foundations during strong earthquakes is not su‹cient to model the foundations as macroscopic structural elements of the total structure in the seismic design calculation. Irreversible displacement may be caused by the cyclic large force to the soil beneath the footing. The behavior of a shallow foundation subjected to a combination of vertical load (V ), horizontal load (H ), and moment (M ) has been investigated previously. However, the investigation of the macroscopic shallow foundation behavior during earthquakes is still at an early stage. Although, recently, some experimental ˆndings have been reported (e.g., Haya and Nishimura, 1998; Maugeri et al., 2000; Negro et al., 2000; Faccioli et al., 2001; Gajan et al., 2005), available experimental data sets for developing and validating numerical modeling under strong seismic loads are still limited. The present paper reports the results of large-scale 1-G tests of model pier footings on sand that were subjected

INTRODUCTION Shallow foundations are considered to have larger safety margins than pile foundations with respect to strong earthquakes, because design norms require that shallow foundations rest directly on sturdy bearing layers. The Speciˆcations for Highway Bridges (Japan Road Association, 2002), the highway bridge design norm in Japan, requires further empirical regulations for normal and small-to-mid scale earthquake designs. For example, the maximum soil reaction stress intensity and the maximum degree of partial uplift are limited to prevent shallow foundations from an excessive settlement. Such additional safety margins are attributed to the prevention of bearing failure, overturning, and excessive settlement and inclination, during even strong earthquakes. In addition, seismic loads on shallow foundations are considered to be reduced during strong earthquakes. With the foundation resting on a sturdy soil, rockinguplift behavior can occur. A progressive reduction in the contact area of the soil-footing interface can be expected, resulting in a nonlinear relationship between the moment and rotation of the footing. Recent case histories also back up empirical engineeri)

ii) iii) iv)

v) vi)

Senior Researcher, Center for Advanced Engineering Structural Assessment and Research, Public Works Research Institute, Ibaraki, Japan (shirato＠pwri.go.jp). Researcher, ditto. Engineer, Obayashi Cooperation, Japan (formerly, Visiting Researcher, Public Works Research Institute, Japan). Chief Researcher for Management System and Substructures, Center for Advanced Engineering Structural Assessment and Research, Public Works Research Institute, Japan. Advanced Construction Technology Center, Tokyo, Japan. Associate Professor, Dipartimento di Ingegneria Strutturale, Politecnico di Milano, Milan, Italy. The manuscript for this paper was received for review on October 22, 2007; approved on July 24, 2008. Written discussions on this paper should be submitted before May 1, 2009 to the Japanese Geotechnical Society, 4-38-2, Sengoku, Bunkyo-ku, Tokyo 112-0011, Japan. Upon request the closing date may be extended one month. 673

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to shake table loading, cyclic combined loading, and centered vertical loading. Although in-depth experimental reports have been published by the Public Works Research Institute (PWRI), Tsukuba, Japan, with CDROMs that include the recorded raw digital data of the experimental results (Fukui et al., 2007a; Fukui et al., 2007b), the present study focuses on the macroscopic nonlinear hysteresis of shallow foundations. We ˆrst introduce the testing methods used herein and then present the major ˆndings relevant to the development of a numerical model capable of accounting for the nonlinear foundation response to earthquake loading. Note that, based on the ˆndings of the present study, several approaches to establish a numerical simulation model of shallow foundations are also investigated in companion papers (Paolucci et al., 2008; Shirato et al., 2008). THEORETICAL DESCRIPTION OF THE MACROSCOPIC BEHAVIOR OF A SHALLOW FOUNDATION SUBJECTED TO COMBINED LOADING Before describing the experimental setup and analyzing the results, we summarize some theoretical concepts that will be helpful for discussing the experimental ˆndings. The nonlinear load-displacement behavior of a shallow foundation under inclined/eccentric monotonic loading has been successfully explained theoretically by the macro-element approach (Nova and Montrasio, 1991; Gottardi and Butterˆeld, 1995; and others). The macro-element approach was originally based on the load-settlement curve for concentric monotonic loading. Load-settlement curves follow a work hardening until the load reaches the ultimate bearing capacity. Such a rule can be expressed as:

V＝F(v)

(1)

where V is the vertical force, v is the settlement, and F is a work hardening function. Then, this relationship is extended toward monotonic combined loading. As for the ultimate bearing capacity of shallow foundations to combined loading, a failure locus in the V-H-M space has been established (Nova and Montrasio, 1991; Butterˆeld and Gottardi, 1994; Houlsby and Martin, 1993). An example of the failure locus of a footing on sand, fcr, as shown in Fig. 1, can be represented by Nova and Montasio's equation (1991):

Ø H » ＋Ø MB » －V Ø －VV » ＝ 2

fcr＝

2b

2

2

m

c

1

0

(2)

m

where Vm is the bearing capacity for centered vertical loading, while m, c, and b are constants that deˆne the shape of the failure locus. When an isotropic hardening is assumed, the yield locus expands within the failure locus in the V-H-M space as the monotonic loading progresses. Ultimately it touches the failure locus, fcr. The yield locus, f, can be deˆned as follows:

Fig. 1. Typical failure locus for footing under combined load in the VH-M space

f＝

Ø » Ø » Ø H m

2

＋

M 2 V －V 2 1－ Vc cB

»＝ 2b

0

(3)

where Vc is the V-intercept and speciˆes the size of the yield locus, considering an increasing load combination of V, H, and M and the progressive expansion of the yield locus. A generalized displacement is also introduced as:

veq＝ v 2＋au2＋b(Bu )2

(4)

where v is settlement, u is sliding, u is rotation, and a and b are non-dimensional parameters that translate the sliding and rotation into an equivalent settlement. Using the size of the yield locus, Vc, and the generalized displacement, veq, Eq. (1) can be extended to a work hardening rule for combined loading:

Vc＝F(veq)

(5)

Finally, the footing response can be obtained within the context of work hardening plasticity using a suitable plastic ‰ow rule. When the current load combination lies inside the yield locus, the load-displacement response should be elastic. As a result, a shallow foundation can be modeled with a macro-element that describes the relationship between an incremental load set (V_ , H _ , M/B _ ) and the corresponding incremental displacement set ( ·v, ·u, Bu_ ). The numerical simulation for a structural system using a macro-element is considerably more straightforward than, for example, other sophisticated ˆnite element approaches, and is considerably easier to implement in the computation of dynamic soil-structure interactions (Paolucci, 1997; Cremer et al., 2001; Okamura and Matsuo, 2002;, di Prisco et al., 2002). However, as indicated above, the macro-element approach was originally developed for monotonic loading and there is little data

SEISMIC BEHAVIOR OF SHALLOW FOUNDATIONS

demonstrating the applicability of the macro-element theory to earthquake-related loading conditions. SHAKE TABLE EXPERIMENTAL PROGRAM The shake table experiments were conducted at the Large-scale Shake Table Facility at PWRI, Tsukuba, Japan.

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in which the unit of conˆning stress, s?c, is kN/m2. A total of 33 accelerometers were embedded in the sand deposit. A model pier footing was located at the center of the sand deposit surface. Figure 2 shows a schematic diagram of the model pier footing. The model was comprised of three structural components: a top steel rack, a short steel I-beam column, and a footing. Figure 2 also shows the mass, mG, the structural moment of inertia about the center of gravity, JG, and the height from the

Test Apparatus Photograph 1 shows the model set-up. The shake table was 8 m×8 m, as viewed from the top. A laminar shear box having internal dimensions of 4 m in length, 4 m in width, and 2.1 m in depth was placed on the table. The laminar shear box was constructed of ten layered frames. A dry dense Toyoura sand deposit was constructed to a height of 2 m in the laminar shear box and was compacted in layers so that a satisfactory homogeneous soil condition was obtained. The soil relative density, Dr, was 80 z, and the mass density, r, was 1.60×103 kg/m3. CD triaxial compression tests revealed that the internal friction angle, q, was 42.19 . Undrained cyclic triaxial compression laboratory tests were conducted with three diŠerent conˆnement stress conditions. The linear regression result for the observed elastic moduli, E0, at a strain level of 10－4 is: E0＝34.92×s?c 0.4436×103 kN/m2

Photo 1.

(6)

Fig. 2. Model pier footing used in the shake table experiments (as viewed from the front and top)

Experimental setup for the shake table experiments

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Photo 2.

Arrangement of load cells (N＝North side, S＝South side)

footing base to the center of gravity, hG, for each structural component. The total weight of the model pier footing was 8,385 N. The total height was 0.753 m. The height of the center of gravity was 0.420 m from the base of the footing. The weight of the steel rack at the top was 5,227 N, including the steel plates on the rack. A short steel column having an I cross-section and end diaphragms connected the steel rack to the footing. The short steel column was markedly stiŠer than the soilfoundation system and so can be considered rigid. The footing shape was a 0.5 m square block of 0.25 m in depth. A total of 11 bi-directional load cells were attached along the shaking direction, working as the base unit of the footing, so that the distribution of normal and shear reactions to the base of the footing was captured. Photograph 2 shows the arrangement and dimensions of the load cells. The load cells were made of steel. The long side of the load cell had the same length as the side of the foundation, and the short side of the load cell was aligned in the direction of the excitation. Sandpaper was attached to the contact surface where the load cells came into contact with the soil, so that the boundary condition was rough. Holes and strain gauges were arranged in each load cell. The strain gauges measured the distortion of the load cell, so that the normal and shear forces were estimated. The calibration processes of the load cells were also reported in PWRI reports (Fukui et al., 2007a; Fukui et al., 2007b). Note that 25 accelerometers were attached to capture the horizontal and vertical accelerations of the model pier footing. The results of a monotonic centered vertical loading

experiment, which is described later herein, revealed that the static safety factor in terms of the bearing capacity with respect to the model weight (or dead load) is 29, which is considerably larger than the required safety factors in the Speciˆcations for Highway Bridges. The Speciˆcations for Highway Bridges requested a safety factor of greater than 3 for normal design, considering both dead and live loads, and a safety factor of greater than 2 for small-to-mid-scale earthquake design, considering dead loads as the vertical load and the eŠect of the earthquake-induced inclination and eccentricity of the load on the bearing capacity. However, as mentioned in the INTRODUCTION, the Speciˆcations for Highway Bridges also require the limitations of the degree of partial uplift and the intensity of subgrade reaction stress for both normal and small-to-mid-scale earthquake design. We examined several previous design results of highway bridge shallow foundations and found that the ratios of the bearing capacities to the dead loads tended to be in the range of 9 to 24, much larger factors of safety than the required minimum values. The ratio of the height of the center of gravity of the model pier footing to the footing length was 0.84 and is included among typical ratios in previous design case histories of highway bridge pier footings, which range from 0.5–2.0 (Fukui et al., 1999). Therefore, the model pier footing had mechanical properties similar to those of typical design cases. The shake table was rocked only in the North-South horizontal direction. Figure 3 shows the earthquake records that excited the shake table. We chose two diŠerent earthquake records obtained on sturdy ground. A

SEISMIC BEHAVIOR OF SHALLOW FOUNDATIONS

Fig. 3.

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Input motions

motion recorded at Shichiho Bridge, Hokkaido, Japan, during the 1993 Hokkaido Nansei-Oki Earthquake (MW ＝7.8) and the N-S component recorded at JMA (Japan Metrological Agency) Kobe during the 1995 Hyogo-ken Nanbu (Kobe) Earthquake (MW＝6.9) were adopted. The Shichiho Bridge motion is classiˆed as a Type I earthquake motion, as deˆned by the Speciˆcations for Highway Bridges. The Kobe motion is very representative of the Type II earthquake motion as deˆned by the Speciˆcations for Highway Bridges. The characteristics of inelastic behavior of both soils and structural members during an earthquake are greatly aŠected by both intensity and duration (or the number of cycles) of the earthquake, and, therefore, two types of ground motion are considered as design motions in the Speciˆcations for Highway Bridges. Type I earthquake motions are ground motions that are associated with the interplate-type earthquake having a magnitude of approximately 8 and are generated at plate boundaries in the ocean. Type II earthquake motions are ground motions that are associated with the inland-strike-type earthquakes having magnitudes of approximately 7 that are caused by faults located at short distances from bridge sites. Type I motions contain a larger number of cycles than Type II motions. The shake table was also excited by a sweep wave to check the basic vibration properties at diŠerent stages, which had an acceleration level of 0.50 m/s2 with the frequency level gradually increasing from 1 to 20 Hz. The excitation continued for 35 seconds. As shown in Fig. 4(a), the transfer function from the horizontal acceleration at the shake table to that at the soil surface indicates that the ˆrst-mode characteristic horizontal vibration fre-

Fig. 4. Transfer functions of vibration (a) X-direction acceleration at the soil surface versus X-direction acceleration of the shake table; (b) X-direction acceleration at the top of the model-pier-footing versus X-direction acceleration of the shake table; (c) Z-direction acceleration at an edge of the footing versus X-direction acceleration of the soil surface

quency of the soil deposit was approximately 22 Hz. As shown in Fig. 4(b), the transfer function from the horizontal acceleration at the soil surface to that at the top of the model pier footing shows that the characteristic sway vibration frequency of the model pier footing was approximately 11 Hz, in which the term of transfer function is used as the power spectrum ratio. As shown in Fig. 4(c), the transfer function from the horizontal acceleration of the soil surface to the vertical acceleration at an edge of the footing reveals that the characteristic rock-

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ing vibration frequency was approximately 17.5 Hz. Figure 5 shows the horizontal acceleration distributions in the model pier footing versus height when the accelerations at the top of the model pier footing were maximum and minimum during the sweep wave excitation, indicating that the rotation center was located somewhere near 100 mm above the base center of the footing.

Fig. 5. Accelerometer positions in the model pier footing and acceleration distributions versus depth at which the maximum and minimum accelerations were observed. Solids lines are linear ˆtting lines for corresponding acceleration distributions

Table 1.

The shake table motions were captured with laser displacement transducers and accelerometers.

Test Series We examined two test series, Case S1 and Case S2, where S denotes the shake table experiment. The sand deposit was separately prepared for each case. Each case had several excitation phases with diŠerent input motions. The test series are listed in Table 1. Originally, Case S1 was planned for Case S1–2, i.e., the Type I seismic motion case, and Case S2 was planned for Case S2–2, i.e., the Type II seismic motion case. We planned the excitation cases that followed Case S1–3 in Case S1 as a rehearsal for the Case S2–2 experiment, applying the Kobe motion several times with diŠerent experimental conditions. Therefore, as noted in Table 1 and Photo 3, we did not treat the model setup processes carefully, and the soil conditions underneath the footing had uncertainties compared to other excitation cases. Although the data was originally going to be dumped, the test excitation phases exhibited unique behavior, and we reconsidered these data as formal excitation cases S1–4 and S1–5 and analyzed the recorded data. Before the Case S1–4 experiment, the model pier footing was uplifted, the soil surface was leveled, and the model was replaced at the original position, with an embedment depth of 50 mm (20z of the footing depth and 10z of the footing length). Before the Case S1–5 experiment, the pier footing model was just moved 1 m to the west without any treatment of the soil surface with no embedment. In Case S2–2, the Kobe motion was applied but the acceleration amplitude was 80z of the original record in the time domain. Priority was given to the bearing capacity failure, while overturning failure as a rigid block on a ‰oor was out of the scope in this project. Accordingly, we were worried that the original motion was too strong to observe the foundation response until the end of the excitation. Although a 10 mm embedment (4z of the footing depth and 2z of the footing length) was included as a fail-safe, we assumed that the embedment eŠect on the overall foundation behavior could be disregarded.

Shake table experimental program

Case identiˆer

Earthquake motions and notes on the preparation

Planned maximum acceleration

Observed maximum acceleration on table

Embedded depth

S1–1 S1–2 S1–3 S1–4

Sweep wave Shichiho wave Sweep wave (Remove the model, recompact the soil surface, and settle the model again)ªKobe wave (Remove the model and settle it at a diŠerent position 1 m west from the original position) ªKobe waveª(Remove the model and soil deposit)

50 gal 386 gal 50 gal 812 gal

112 gal 601 gal 106 gal 712 gal

0 mm 0 mm 0 mm 50 mm

812 gal

726 gal

0 mm

Sweep wave Weakened Kobe wave Sweep wave

50 gal 650 gal 50 gal

110 gal 557 gal 113 gal

10 mm 10 mm 10 mm

S1–5

S2–1 S2–2 S2–3

SEISMIC BEHAVIOR OF SHALLOW FOUNDATIONS

Photo 3.

679

State at the beginning of Case S1–5 excitation. The soil deformation in Case S1–4 remained at the center of the soil surface

Note that earthquake motions were not perfectly reproduced at the shake table, as summarized in Table 1, because of the huge dimensions of the system.

Data Processing As shown in Fig. 6, the measured displacements and forces of the footing are expressed in terms of the displacements and resultant forces at the base center of the footing. The direction of the sliding, u, in Fig. 6 corresponds to the North direction in Photos 1 and 2 and Fig. 2. The horizontal displacement of the model pier footing relative to the soil deposit surface displacement was taken and will be shown later herein. The settlement will be shown later using the absolute displacement, because the measured settlement of the soil deposit surface at the end of each case was negligible. In the following results, accelerations were obtained with accelerometers. Displacements of the model pier footing and soil deposit surface were basically captured via an image analysis of digital Video Cassette Recorder (VCR) records. The VCR image analysis appeared to be able to more accurately capture the long-period part of the displacement, including permanent settlement and rotation. The double-integration of acceleration together with a proper high-pass ˆltering process can also provide displacement records. The displacement records obtained by the VCR image analysis and the double-integration of acceleration agreed very well in terms of the phase changes and amplitudes in the main excitation part, but the double-integration was inferior in terms of the reliable recovery of the long-period part because of the unavoidable high-pass ˆltering process. The resultant forces at the base center point of the footing were obtained with bi-directional load cells. The vertical load, V, and the horizontal loads, H, were estimated as the summations of the vertical forces and shear forces measured at the load cells. The moment, M, was estimated by taking the summation of the products of the vertical force and the distance between the load cell and the base center point of the footing for all load cells. Ac-

Fig. 6.

Sign convention and notation of load and displacement

cordingly, the P-Delta eŠect, the additional moment caused by the weight of the structure moving through a lateral displacement, is automatically included in the estimated moment, M. CYCLIC LOADING EXPERIMENT PROGRAM The experiments were conducted at the Foundation Engineering Laboratory at the PWRI, Tsukuba, Japan.

Test Apparatus Figure 7 shows a schematic diagram and a photograph of the model setup. In a deep test pit having internal dimensions of 4 m in length and 4 m in width, a dry sand deposit was constructed of Toyoura sand to a height of 2 m with an average relative soil density of Dr＝80z (soil density r＝1.60×103 kg/m3) or 60z ( r＝1.54×103 kg/ m3). The deposit was compacted in layers so that homogeneous soil conditions were achieved. At a relative density Dr of 60z, CD triaxial compression tests revealed that the internal friction angle, q, was 39.99and undrained cyclic triaxial compression laboratory tests with three diŠerent conˆnement stress conditions determined a small strain elastic modulus, E0, at a strain level of 10－4. The linear regression result for the observed elastic moduli, E0, is: E0＝24.68×s?c 0.4776×103 kN/m2

(7)

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Fig. 7.

Experimental setup for the cyclic loading experiment (as viewed from the front and top)

in which the unit of conˆning stress, s?c, is kN/m2. Figure 8 shows a schematic diagram of the model pier footing. The combined V-H-M loadings were achieved by controlling the weight of the model pier footing and then gradually applying a horizontal displacement at a ˆxed height via a computer-controlled servo actuator. Displacement modes and bearing capacities were investigated for diŠerent V-H-M combinations, soil densities, and cyclic loading patterns. The model pier footing was basically the same as that used in the shake table experiment. There were two diŠerences: 1) two cases of pier height were examined, and 2) a universal joint was attached to the top of the steel rack. Figure 8 also shows the mass, moment of inertia, and height of the center of gravity from the footing base in terms of each structural component: a steel rack, an Ibeam column, and a footing. The height of the loading point was varied using two columns of diŠerent height. The column used in the shake table experiment and another taller column having the same I cross-section were employed. We refer to these columns as the short column and the tall column, respectively. The non-dimensional ratios of applied moment, M, to horizontal force, H, at the base center point of the

footing, M/H/B, became 1.8 and 2.6 (＝1:1.44) for the short column and the tall column, respectively. The universal joint was attached to the model pier foundation to connect with the actuator system. At the universal joint, the model was free to rotate and move up and down. Because of the existence of the universal joint and the use of diŠerent pier heights, the mass of the model pier footing was diŠerent from that used in the shake table experiment. In terms of the model pier footing with the tall column, the initial static factors of safety for dead loads were 28 for a relative density of 80z and 14 for a relative density of 60z. The displacement of the model was measured using linear variable displacement transducers (LVDTs) and laser displacement transducers (LDTs). A load cell attached to the actuator measured the load on the actuator. Slow monotonic lateral loading, one-sided cyclic lateral loading, and two types of reversed cyclic lateral loading in the N-S direction were involved. The measured average rates of loading were 7.4 mm/s when the sign of incremental displacement was positive and 9.7 mm/s when the sign of incremental displacement was negative. Figure 9 shows the prescribed displacement histories, d, where d0

SEISMIC BEHAVIOR OF SHALLOW FOUNDATIONS

is the reference displacement. We speciˆed the loading patterns, referring to a draft guideline proposed by a joint research by the PWRI and the Federal Highway Administration (FHWA) of the United States for cyclic loading experiments on bridge columns (Unjoh et al., 2006). The Type I loading pattern was associated with

Type I seismic motions, whereas the Type II loading pattern was associated with Type II seismic motions. The guideline does not suggest on how to determine the reference displacement d0 in terms of an experiment to examine soil-foundation interactions. In the present experiment, d0 was assumed based on the results of the preceding monotonic lateral load case in which the same experimental conditions, except for loading patterns, were used. The displacement level at which the maximum moment appeared was taken as the reference displacement d0. Finally, the amplitude of the speciˆc displacement was gradually increased from 0.125d0 to 0.25d0 to 0.5d0, followed by i×d0. Some cycles having smaller displacement amplitudes than 1d0 were applied prior to the 1d0-cycle in order to investigate the foundation behavior during small-to-mid scale earthquakes. A total of n cycles were repeated at each displacement amplitude. For one-sided cyclic loading, during the loading phase, displacement was controlled and increased up to the speciˆed displace-

Fig. 8. Model pier footings used in cyclic loading experiment (as viewed from the front and top)

Table 2.

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Fig. 9.

Applied cyclic loading histories

Cyclic loading experimental program

Case identiˆer

Loading pattern

Relative soil density

Lever arm length, L*

C1 C2 C3

Monotonic loading One-sided cyclic loading with Type II pattern Fully-reversed cyclic loading with Type II pattern

80z 80z 80z

1300 mm (Tall column) 1300 mm (Tall column) 1300 mm (Tall column)

C4

80z

900 mm (Short column)

C5 C6

Monotonic loading, followed by one-sided cyclic loading on the reversed-side Fully-reversed cyclic loading with Type I pattern Fully-reversed cyclic loading with Type II pattern

80z 80z

900 mm (Short column) 900 mm (Short column)

C7 C8 C9

Monotonic loading Fully-reversed cyclic loading with Type I pattern Fully-reversed cyclic loading with Type II pattern

60z 60z 60z

900 mm (Short column) 900 mm (Short column) 900 mm (Short column)

* The length from the base of the footing to the loading point

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ment level. During unloading phase the load was then decreased to zero.

TEST RESULTS

Test Series Test series are tabulated in Table 2, where ``C'' denotes the cyclic loading experiment. There were a total of nine test cases, Cases C1 to C3, Cases C4 to C6, and Cases C7 to C9 belonging to the same categories. When the soil density was changed from case to case, the soil deposit was fully removed and reconstructed. Other than that, only the upper 1 m of soil was removed and reconstructed. In Case C4, ˆrst, monotonic loading was applied, and the experiment was continued using the onesided cyclic loading with the Type II pattern on the reversed side. The values of d0 were determined as 11 mm for Cases C2 and C3, 16 mm for Cases C5 and C6, and 18 mm for Cases C8 and C9.

Vertical Push Figure 10 shows the results for the centered vertical loading experiments. The measured maximum loads were 244.8 kN for a relative density of 80z and 128.8 kN for a relative density of 60z. Photograph 4 shows the observed slip line after the dense sand case experiment. The slip line appearing on the soil surface spread in every direction. Figure 11 shows enlarged views of the partial unloading-reloading processes. A least mean square approximation for the unloading paths provided the values of vertical spring constants of 89,179 kN/m2 for a relative density of 80z and 76,207 kN/m2 for a relative density of 60z. A theoretical vertical spring constant that is based on elasticity and corresponds to a vibration frequency of zero is given as follows (Gazetas, 1991):

Data Processing The measured forces and displacements of the footing are expressed in terms of the displacements and resultant forces at the base center of the footing, as shown in Fig. 6. However, the direction of sliding, u, in Fig. 6 coincides with the south direction in Figs. 7 and 8 in the cyclic loading experiment. Data processing was basically the same as in the shake table experiment, except that the displacement of the model was obtained with LVDTs and LDTs.

Kv＝4.54 G(B/2)/(1－n)

(8)

where Kv is the spring constant, G is the shear modulus of soil, B is the footing length, and n is the Poisson ratio. Using Eq. (8) and assuming n＝0.3, the approximate values of the shear moduli are obtained as 55,000 kN/m2 for a relative density of 80z and 47,000 kN/m2 for a rel-

VERTICAL LOADING EXPERIMENT PROGRAM For each soil condition (Dr＝80z and 60z), a centered vertical push experiment was also conducted. The experiments were conducted in the test pit at the Foundation Engineering Laboratory, which was also used in the cyclic loading experiment. The test pit had internal dimensions of 4 m in length, 4 m in width, and 2 m in depth. A stiŠ steel box ˆlled with concrete and having a square base of 0.5 m was used as the footing. The entire footing base was covered with pieces of the same sandpaper. A manually operated hydraulic jack was used. At an early loading stage, a series of partial unloading and reloading was applied to estimate the elastic (unloading) rigidity of the soil. Data processing was diŠerent from the other experimental programs. Four LVDTs were set to measure the settlements at the corners of the footing top. The settlement of the footing was analyzed as the average measured displacements. A load cell was attached to the jack. The vertical load at the footing base was obtained as the sum of the measured load with the load cell and the weight of the footing. The details of the experiment are included in the PWRI report of the cyclic loading experiment (Fukui et al., 2007a).

Fig. 10.

Results of the vertical loading experiment

Photo 4. Deformation of the soil surface after the concentric vertical loading experiment on dense sand

SEISMIC BEHAVIOR OF SHALLOW FOUNDATIONS

ative density of 60z.

Overall Behavior in the Shake Table Experiment and Cyclic Loading Experiment A failure locus in the V-H-M space can be expressed with Eq. (2). Next we attempt to compare the observed loads and the theoretical failure criterion. Although Nova and Montrasio recommended b＝0.95, for simplicity, we hereinafter will round this value to b＝1. m is the tangent on the H-V envelope at V＝0, while c is the tan-

Fig. 11. Partial unloading-reloading curve results of the vertical loading experiment

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gent on the ( M/B )-V envelope at V＝0. Hereinafter, the value of m is assumed to be tan q. Figure 12 plots the trajectory of the observed forces in the H-M plane and the projection of the theoretical failure locus, Eq. (2), at V＝V0 for diŠerent values of c, where V0 is the weight of the model pier footing in each case. Figure 13 shows the trajectory of the observed forces in the V-M plane and the projection of the theoretical failure locus. The values of H at which the maximum moment appeared were sought in Cases S1–2, C2, and C8, and they were used in the plots of the projections of Eq. (2). Figures 12(a) and 13(a) are associated with shake table loading of Cases S1–2, S1–4, S1–5, and S2–2. Figures 12(b) and 13(b) are associated with cyclic loading and dense sand cases, Cases C2, C3, C5, and C6. Figures 12(c) and 13(c) are associated with cyclic loading and medium dense sand cases, Cases C8 and C9. In addition, Figs. 12(b) and 13(b) contain the results of both tall and short column cases. Nova and Montrasio (1991) pointed out that the value of c is conceived to be in the range of 0.3–0.5, so that the experimental data agrees with the theoretical failure envelop under the assumptions of c＝0.45 and 0.5 for the dense and medium dense sands. Figures 12 and 13 indicate that the diŠerence in the failure locus between the shake table loading and cyclic loading was indiscernible, when the soil density was identical. Therefore, the theory is capable of accounting for the bearing capacity in both the shake table loading condition and the cyclic loading

Fig. 12.

Projection of observed forces and theoretical failure locus in H-M plane

Fig. 13.

Projection of observed forces and theoretical failure locus in M-V plane

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Photo 5.

States of the footing after excitations (Captured from VCR records)

condition, even though the theory is basically based on the results of a monotonic loading experiment. Figure 13 also indicates that the vertical force, V, varied during the shake table experiment, whereas it did not change during the cyclic loading experiment. This is because the base center of the footing rapidly moved up and down because of the rocking motion, as will be examined later. Photograph 5 shows the model pier footing states after the excitations of Case S1–2, Case S1–4, Case S1–5, and Case S2–2. In terms of Cases S1–2, S1–4, and S2–2, the model pier footing was subjected to permanent residual displacements, although the loads reached the theoretical failure locus of soil resistance during the excitations, as shown in Figs. 12 and 13. The important point is that, since earthquake loads varied with time, the fact that the loads reached the ultimate state did not necessarily indicate failure. This reinforces the need for reliable methods to check the permanent displacement of shallow foundations during earthquakes. The footing was toppled only in Case S1–5. The model tilted signiˆcantly with large pulses and was toppled by a subsequent pulse. Although a similar response was also observed in Case S1–4, the side resistance to the footing resulting from a larger embedment depth may have prevented the footing from toppling. However, in terms of Case S1–5, as noted in Table 1 and shown in Photo. 3, the model was just replaced at a position 1 m west of the original position immediately after Case S1–4, where the soil surface was probably loose and disturbed during the several preceding large excita-

tion phases. Therefore, we assumed that Case S1–5 was the most vulnerable. Although Cases S1–4 and S2–2 were both subjected to the Kobe excitation, the residual rotation of the model in Case S2–2 was markedly smaller than that in Case S1–4. In Case S2–2, the amplitude of the input motion was reduced by 80z from the original earthquake record, which explains why the residual displacement in Case S2–2 was smaller. Therefore, the accumulation of the displacement was aŠected by the acceleration intensity. However, the eŠect of the number of cycles contained in earthquake motions should be critical as well. Figure 14 shows the time histories of the model responses in Cases S1–2 and S2–2. The vertical scales of the axes for each item are the same between Case S1–2 and Case S2–2, whereas the horizontal scale is not. From top to bottom, ag is the horizontal acceleration at the ground surface, a1 is the horizontal acceleration at the center of gravity of the top steel rack, v, u, and u are the settlement, sliding, and rotation at the base center of the footing, respectively. v and u are normalized with the footing length, B. For reference, the normalized size of the yield surface of Eq. (3), rc＝Vc/Vm, is also calculated from Eq. (44) of the paper presented by Nova and Montrasio (1991) with observed combined loads:

rc＝

where

j h 2＋ m 2 1－ j2

Ø

»

1/(2b)

(9)

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Fig. 15. Rotation-settlement (M-v) curve and the evolution of residual settlement, vr, during the repeated cycles at the 2d0-cycle, 4d0-cycle and 6d0-cycle: Case C5

Fig. 14. Time histories of acceleration at the ground surface, ag, acceleration at the center of the top steel rack, a1, and settlement, sliding, and rotation of the footing, v, u, and u, respectively: Cases S1–2 and S2–2

j＝

V H M , h＝ , and m＝ mV m cBVm Vm

(10)

and c is assumed to be 0.45. We calculate the temporal size of the yield surface, rc, from Eq. (9) with instantaneous observed loads, V(t), H(t), and M(t), and set the size as rt. If the temporal size, rt, exceeds the largest value of rc ever experienced in the footing, rc is replaced by rt. Note that, if rc is larger than 1, we set rc to be 1. We also plot the time history of rt in Fig. 14, which shows that the footing may behave elastically when rt is smaller than rc, i.e., when fº0 in Eq. (3), and may become plastic when rt is equal to or larger than rc, i.e., when fÆ0 in Eq. (3), at each time. The irreversible vertical settlement at the end of the excitation of Case S1–2 was considerably larger than that of Case S2–2, although the recorded maximum accelerations on the ground level, ag, and at the top steel rack, a1,

of Case S1–2 were similar to those of Case S2–2. The time histories of rc and rt indicate that Case S1–2 underwent plastic deformation more times than Case S2–2. While the base center of the footing moved up and down, as seen in the time histories of v/B, the settlement gradually accumulated. Although the predominant displacement mode of the footing was rocking, the rocking motion induced the irreversible accumulation of settlement, v, as well as rotation, u. Note that the residual rotation at the end of the excitation in Case S2–2 was almost zero, that is, 0.001 rad. As seen above, the residual displacement depended not only on the amplitude of applied base acceleration but also on the number of cycles contained in the base acceleration. We also conˆrmed this tendency via the results of the cyclic loading experiments. A typical moment-settlement curve is shown in Fig. 15. The curve was observed in Case C5. Several repeated cycles were applied at each speciˆed loading amplitude. Note that only the model weight was applied as the vertical force in the cyclic loading experiments. The base center of the footing moved up and down alternatively, and the gradual accumulation of the irreversible settlement increased, where the irreversible settlement is deˆned as the settlement when M became zero on the way from the negative peak moment to the following positive peak moment. Figure 15 also shows that the irreversible settlement accumulated even during the repeated cycles at 2d0 and 4d0 amplitudes in Case C5. The rate of the accumulation of irreversible settlement during the cycles did not change as the cycles repeated at each speciˆed displacement amplitude. Figure 16 shows the load trajectories in the M-V plane during the ˆrst and sixth cycles at the 2d0-cycle in Case C5, where the theoretical curves are calculated with the maximum horizontal load during the Case C5 experiment. Since the combined loads appeared to reach the theoretical failure surface, we deduced that the irreversible settlement dur-

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Fig. 16.

SHIRATO ET AL.

Projection of observed forces and theoretical failure locus in the M-V plane during the ˆrst and sixth cycles at the 2d0-cycle: Case C5

Fig. 17. Relationships between residual displacement and given drift angle amplitude: Cases C3 and C6

Fig. 18. Relationships between residual displacement and given drift angle amplitude: Cases C6 and C9

ing the repeated loading cycles could be explained by a typical plasticity theory. Figure 17 compares the evolution of the irreversible settlement and rotation, at the end of the ˆnal cycle for each given drift angle amplitude, d/L, for Cases C3 and C6. The moment/vertical force ratios were diŠerent between Cases C3 and C6. Case C3 used the tall column, while Case C6 used the short column. Other than that, the experimental conditions were identical. The irreversible displacement is deˆned herein as the displacement when M equals zero in the transition from the i-th d-cycle to the (i＋1)-th d-cycle. The drift angle is deˆned as the ratio of the displacement at the point of loading, d, to the lever arm length, L, d/L, where the sliding of the footing was disregarded when estimating the drift angle amplitude because it was su‹ciently small compared to d. Both cases showed a similar tendency in the accumulation of the irreversible displacement, but the accumulation in Case C6, a short column case, gradually became larger than that in Case C3, a tall column case, as the given drift angle increased. Therefore, we recognize that there is a coupling eŠect of V-H-M on the diŠerence in the evolution of irreversible displacements, v, u, and u. Another reason for this diŠerence may be that the rotation angle at 1d of the short column case was larger than that of the tall column case. Figure 18 shows the relationships between the irreversible displacements and rotations as a function of the given

drift angle amplitude for Cases C6 and C9. Case C6 was conducted on dense sand and Case C9 was conducted on medium-dense sand. The other conditions were identical. The accumulation of the irreversible displacement in the medium-dense sand case (Case C9) was faster than that in the dense sand case (Case C6).

Uplift Behavior Figure 19 shows typical examples of the distributions of the reaction forces normal to the load cells at the foundation base in Cases S1–2, S2–2, and C4, along with the corresponding moment-rotation curves. The normal reaction force distributions and moment-rotation curves were extracted from time windows around which the maximum rotations appeared during the excitations. Based on the results of Case C4 in Fig. 19(c), the partial uplift initiated at point b and the corresponding seismic coe‹cient level (＝moment/height of the center of gravity/weight) was approximately 0.2. Note that the initial imperfection of the leveling and setting of the soil surface and footing caused a small moment at the beginning. Fukui et al. (1999) reported that the lateral seismic intensity coe‹cients at which the partial uplift initiated in typical design cases of highway bridge shallow foundations ranged between 0.07 and 0.25. With increasing rotations, one-part of the footing was uplifted while the opposite part remained in contact with the underlying soil (points d, g, j, s, and v in Fig. 19).

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Fig. 19.

687

Distributions of normal forces to the footing base and corresponding moment-rotation curves: Cases S1–2 (a), S2–2 (b), and C4 (c)

Simultaneously, at points v and s, the normal force distributions formed a sort of a square shape. From this shape,

we can ˆgure out that the upper limit moment was mobilized as a result of the soil becoming plastic. Therefore,

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688

we consider that the signiˆcant yield of the soil resistance initiated when the moment attained the ultimate value. However, during the unloading phases (points t and w) and the subsequent reloading phases (points u and x), the eŠective length of the footing did not increase signiˆcantly. This implies permanent reductions in the footing-soil contact area because of the plastic deformation of the soil underneath the footing edge. Figure 20 shows the time evolution of the area of the footing that met with the underlying soil. In Cases S1–2 and S2–2, the abscissa shows the sequential time during the experiment, and the vertical axis corresponds to the footing length. In Fig. 20, at each time, the black position indicates the contact area (or the eŠective footing length). The eŠective footing length gradually decreased as the sequential time elapsed, which was attributed to the evolution of the plastic deformation of the soil underneath both footing edges. Similar behavior was also ob-

Fig. 20. Time evolution of the contact area of the footing base with the underlying soil: Cases S1–2 and S2–2

Photo 6.

served in the cyclic loading experiment. The alternative foundation rocking rounded the soil surface shape beneath the footing. Photographs 3 and 6 show a typical example. We infer that the model pier footing was supported by sort of a soil-arching mechanism after the excitation or cyclic loading. The characteristics of nonlinear footing response changed with the reduction in the eŠective footing length. As shown in Fig. 19(c), the stiŠness during the unloading phase from point d was markedly smaller than the initial rigidity of the virgin loading from point a. The stiŠness degradation also caused the change in the predominant vibration frequency of the soil-foundation system. Figure 21 shows an enlarged view of the time histories in Case S2–2, of the acceleration at the soil surface, ag, the acceleration at the center of gravity of the

Fig. 21. Time histories of acceleration at the ground surface, ag, acceleration at the center of the top steel rack, a1, and normalized vertical force, V/V0, normalized horizontal force, H/V0, and normalized moment, 2M/(V0B), at the base center of the footing: Case S2–2

Deformation of the soil surface after a cyclic loading experiment (Case C6, Type II loading, dense sand)

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Fig. 22. Decomposition of total displacement into elastic, plastic, and uplift components Fig. 24.

Fig. 23.

Fig. 25.

u-u el, u-u pl, and u-u up relationships: Case C9

top steel rack, a1, the normalized vertical force at the base center of the footing, V/V0, the normalized horizontal force at the base center of the footing, H/V0, and the normalized moment at the base center of the footing, M/V0/ ( B/2). V0 is the weight of the model pier footing and B is the footing length. For reference, the size of the yield surface, rc, as well as rt are also plotted. Rather ‰at peak regions were observed in the time histories of a1, H and M. A ‰at peak region of M appeared around a sequential time of 7 seconds, because the combined loads reached an ultimate state as indicated by the time histories of rc and rt, in which rc＝1 is equivalent to f＝fcr and rtÆrc is equivalent to fÆ0 in Eq. (3). Then noticeable elongation of the vibration frequency seemed to initiate after a sequential time of 10 seconds, although rt was considerably smaller than rc, i.e., fº0 in Eq. (3). This infers that the uplift-rocking behavior gradually prevailed, instead of the plastic soil deformation, and that the dynamic impedance decreased after a sequential time of 10 seconds. We will herein attempt to estimate how much the uplift contributed to the total footing rocking motion. As illustrated in Fig. 22, we assume that the displacement, u, can be decomposed into the elastic component, u el, plastic component, u pl, and uplift component, u up:

u＝u el＋u pl＋u up

(11)

As a typical example, from the moment M-rotation u

Moment-rotation curves: Cases C6 and C9

Moment-rotation curves: Cases S2–2 and C4

relationship of Case C9 (cyclic loading and medium sand), we tried to estimate the evolution of each component in terms of u. At any peak rotation point on the positive side, we assume that the elastic component, u el, can be estimated by the following equation:

u el＝M/Kr, Kr＝3.6( B/2)3 G/(1－n)

(12)

The plastic component, u pl, is estimated as the irreversible rotation at M＝0, which is the rotation fully unloaded from the corresponding peak point, u. Finally, the uplift component, u up, is derived by subtracting the elastic and plastic components, u el and u pl, from the total rotation at the corresponding peak point. Figure 23 shows the derived u-u el, u-u pl, and u-u up relationships. The uplift component contributed to the total rotation as much as the plastic component and they were predominant at every total rotation level, while the contribution of the elastic component was always relatively smaller than that of the plastic and uplift components.

Hysteresis Loops Typical moment-rotation curves observed in the cyclic loading experiments are shown in Fig. 24. They were observed in Cases C6 and C9. The soil was dense and medium dense, respectively. The irreversible rotation at M＝0 in terms of each rotation amplitude in Case C9 was larger than that in Case C6, as shown in Fig. 18, so that the loop areas in the medium dense soil condition (Case C9) ap-

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Fig. 27. Fig. 26.

Rotation-settlement plots: Case C4

Rotation-settlement plots: Cases S1–2, S2–2, and C6

peared to be somewhat larger than those in the dense soil condition (Case C6) at the same rotation amplitudes. The moment-rotation relationship is likely to be modeled with a conventional peak-oriented hysteresis rule when the peak moment and rotation are considered in numerical simulations. Figure 25 superimposes the moment-rotation curves in Case S2–2 and Case C4, which were subjected to shake table loading and monotonic loading, respectively. In

both cases, Dr was 80z and the short column was attached. In Fig. 25, the negative moment and rotation relationship of Case C4 is shown after reversing the signs of the moment and rotation relationship recorded on the positive side. The backbone curve of Case S2–2 and the monotonic loading curve of Case C4 have similar shapes. This indicates that backbone curves for any loading conditions can be characterized with the monotonic loading curve. Figure 26 shows the rotation-settlement relationships in Cases S1–2, S2–2, and Case C6 (shake table loading

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CONCLUDING REMARKS

Fig. 28.

Horizontal force-sliding curve: Case S2–2

with Type I seismic motion, shake table loading with Type II seismic motion, and fully-reversed cyclic loading with Type II loading pattern, respectively). The uplift behavior and accumulation of the settlement of the footing were eminently observed. Since similar V-shaped loops were observed in all cases, we conceive that there is a coupling between the rotational component and the vertical component in the foundation behavior during earthquakes. Figure 27 shows the moment-rotation relationship, the moment-settlement relationship, and the rotation-settlement relationship at the base center of the footing in Case C4. This was the case that included a one-sided cyclic loading process in the applied loading history. Again, the rotation, u, and the settlement, v, are coupled, and this eŠect should be considered when making a numerical model. The numbers with arrows in the graphs (1, 2, 3, 4, and 5) depict the sequential order. In the one-sided cyclic loading paths, the curve tended to turn towards the previous unloading point. This may not be the behavior considered by a typical peak-oriented rule. Typical peakoriented rules are described with a function of the absolute peak point, so that reloading curves always go toward a point on the backbone curve and are characterized with the absolute peak values. Therefore, the characteristics of hysteresis loops can be described with an origin-oriented and peak-oriented rule, but should also be a function of both positive-side peak rotation point and negative side-peak rotation point. In addition, in the moment-settlement curve in Fig. 27(b), there is a threshold moment value at which the vertical displacement became negative, i.e., the base center of the footing was uplifted. This value was approximately ±1.6 kN･m. Figure 28 shows the horizontal force-sliding curve in Case S2–2 (shake table loading). The ultimate horizontal loads were observed and the sliding displacement abruptly increased when the horizontal load reached the ultimate value.

The results of large-scale shake table experiments and cyclic loading experiments of model pier footings on sand were reported. The experiments provided extensive data on the behavior of shallow foundations subjected to earthquakes, including cyclic and dynamic loading. The ˆndings are not only instructive to structural engineers, but they also provide a comprehensive benchmark for the development of numerical models. The most relevant conclusions are summarized as follows: (1) The residual displacement is dependent on the number of loading cycles during an excitation as well as the base excitation intensity. (2) The backbone curves of load-displacement loops can be modeled based on the load-displacement curves for monotonic loading. (3) The coupling eŠect among vertical displacement, horizontal displacement, and rotation must be considered. (4) The uplift signiˆcantly aŠects the foundation behavior such as the shape of hysteresis loop, the degradation in the rotational stiŠness, and the elongation of the vibration property. When we assume that the displacement can be decomposed into an elastic component, a plastic component, and an uplift component, the uplift component became as predominant as the plastic component. (5) The hysteresis rule should be a function of both positive and negative moments or both positive and negative rotations, and an origin and peak-oriented hysteresis rule is likely to be relevant in terms of the peak response. ACKNOWLEDGMENT This research was conducted as part of a ˆve-year joint research agreement between the Public Works Research Institute, Tsukuba, Japan, and Politecnico di Milano, Milan, Italy 2003–2007, on seismic design methods for bridge foundations. This study was supported in part by the Executive Program of Cooperation in the Fields of Science and Technology between the governments of Italy and Japan 2002–2006, Project No. 13B2. REFERENCES 1) Butterˆeld, R. and Gottardi, G. (1994): A three-dimensional failure envelope for shallow foundations on sand, G áeotechnique, 44(1), 181–184. 2) Cremer, C., Pecker, A. and Davenne, L. (2001): Cyclic macro-element for soil-structure interaction: material and geometrical nonlinearities, International Journal for Numerical and Analytical methods in Geomechanics, 25, 1257–1284. 3) di Prisco, C., Nova, R. and Sibilia, A. (2002): Analysis of soilstructure interaction of towers under cyclic loading, Proc. NUMOG 8, Rome (eds. by Pande and Pietruszczak), Swets & Zeitlinger, Lisse, 637–642. 4) Faccioli, E., Paolucci, R. and Vivero, G. (2001): Investigation of seismic soil-footing interaction by large scale cyclic tests and analytical models, Special Presentation Lecture SPL–05, 4th Int. Conf.

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