Inelastic Behaviour of Structures under Variable Loads

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Vertical cracks may be identified within the mm wide central span, which is subjected to pure bending constant moment while their inclination evidently changes in the two zones of combined moment shear loading between the load points and supports. Unfortunately, the available laboratory equipment did not allow testing with axial load to measure the effect of the pier weight on the section response. As expected, the weight of the pier, which is present in the centrifuge model tests, results in considerable increase of the section moment capacity, although at the cost of reduced ductility.

It is important to note that the USC-RC-predicted moment capacity for the actual axial load is in good agreement with the maximum moment load recorded in the centrifuge tests. Only the column, where structural failure was expected to take place, was modeled using the RC material, whereas the deck and the foundation were made of steel and aluminum, respectively. Care was taken in the design and setup of the deck-to-column and column-to-foundation joints to achieve full fixity. Yet, soil structure interaction is expected to drastically increase this value especially in the case of the rocking pier.

Experimental setup: a photo of the pier model assembly mass, RC column, and foundation and attached accelerometers indicating characteristic dimensions and b schematic cross section of the model within the laminar box showing instrumentation. All dimensions are in millimeters model scale. Similitude is an important consideration in physical modeling using reduced scale models that are intended to capture the response of field-scale prototypes. Centrifuge modeling is particularly useful for the investigation of soil—foundation—structure interaction problems where realistic simulation of the stress dependent soil behavior plays a key role.

Thanks to the enhanced gravity field in a geotechnical centrifuge, a 1: n scale model will have the same effective stress acting at homologous points in the model soil and the full-scale prototype soil. A set of scaling laws have been developed to achieve similitude in centrifuge modeling, as detailed by 48 , Dynamic centrifuge model testing was carried out in the 3.

The soil model was prepared within an equivalent shear beam ESB container with flexible walls, described in It was instrumented using two vertical arrays of five ADXL78 MEMS accelerometers, one array buried under the foundation centerline and the other at a distance large enough to record free field response. The motion of the pier was recorded using identical accelerometers attached to the foundation and the deck. Vertical displacements of two foundation corners, recorded by Linear Variable Differential Transformers LVDTs , were used to calculate settlement and rotation in the direction of shaking.

Another pair of LVDTs recorded the horizontal displacements of the deck center of mass. Taking advantage of the effectiveness of this device in simulating desired motions 51 , an ensemble of records from historic earthquakes were utilized as base excitation. The motions were band pass filtered between 0. Shaking table acceleration time histories used in the different earthquake scenarios, compared with target records from earthquakes of different magnitudes in Greece, Italy, USA, and Japan.

By contrast, the Rinaldi and Takatori motions, recorded during the M s 6. Characterized by near fault directivity effects, these latter motions have increased spectral ordinates within a particularly broad band of periods especially in the range of particular interest, i.

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Having considerable duration and number of cycles, the record of the M s 6. Central to the issue of seismic performance evaluation is the recognition that damage in a component is cumulative, and the level of damage depends not only on the maximum deformation but also on the history of deformations. In practice, structures may, over their lifespan, be subjected to several motions of varying intensities. Thus, attempting to account for seismic history, the performance assessment undertaken here involves a succession of motions considering two different seismic history scenarios.

In Earthquake Scenario A , the ground motion order was such that the intensity roughly increased throughout the test, as if a number of weak or medium intensity motions were to precede a major catastrophic event, either in a sense of foreshocks or as independent events taking place in a relatively short period. Earthquake Scenario B explores the case where a very strong earthquake is followed by weaker shaking events or aftershocks. It should be noted that in some cases, when scenarios A or B did not lead to collapse, additional shaking was applied using the Takatori record until ultimate or practical failure was reached.

In this section, the seismic performance of the two pier—foundation systems during seismic scenarios A and B is evaluated and systematically compared. Focusing on damage accumulation due to successive earthquakes and the importance of shaking history, time histories of recorded illustrative demand parameters, such as deck acceleration and displacement, as well as moment-rotation and shear force-displacement hysteretic response loops are subsequently presented at prototype scale. As anticipated, the rocking pier experiences invariably lower acceleration than the conventional pier.

This advantage becomes more significant as excitation intensity increases and the ultimate capacities are mobilized. Hence, upon shaking with the L'Aquila record, the maximum transient demand experienced by the rocking pier is half the demand on the conventional pier. Acceleration time history sequence recorded during Earthquake Scenario A at a the deck of the rocking pier; b the deck of the conventional pier; and c the model base. The rocking isolation effect can be easily quantified with regard to the moment capacity of the rocking foundation, being therefore potentially useful for the estimation of seismic demand in design.

Yet, this value coincides with the measured peak mass acceleration only in the very first shaking event using the Aegion record. Thereupon, overstrength effects, associated with soil densification during shaking, lead to some considerable increase in this value, which yet remains substantially lower than the peak demand on the conventional pier.

On the basis of the aforementioned discussion, it may be deduced that even the relatively low intensity Aegion excitation is sufficient to momentarily mobilize the capacity of the rocking footing and induce its rocking—uplifting response. Evidently, as anticipated, in the case of the rocking pier, rotational movement prevails throughout the entire shaking sequence. The opposite is the case with the conventional pier, where deck drift is almost exclusively associated with deformation and failure of the RC column.

More importantly, the rocking pier demonstrates a crucial advantage over conventional design. Not only does it experience significantly lower drift in the first shaking event Aegion but it also retains an increasingly favorable performance during the following earthquakes of greater intensity and duration. In particular, having experienced the first two excitations, it is practically unaffected by the sustained shaking of the L'Aquila record, resulting in a small total drift of 0.

Note that in this test, failure is assumed to take place when the deck mass hits the horizontal LVDT, which prevents further movement in the direction of shaking. Inelastic action is not only unavoidable but also essential in providing the required energy absorption to enable survival of the structure under intense seismic shaking. The two design alternatives, both relying on inelastic response, differ only in the component where the inelastic deformation is directed to.

Subsequent loading with the stronger Lefkada motion causes an important number of excursions into the nonlinear regime, accumulating substantial permanent deformation at the column base. However, having experienced this damage, the pier column appears unable to sustain further excitation with the equally strong L'Aquila motion, which exhausts its ductility capacity causing rapid deterioration of strength after a couple of cycles and eventually collapse.

On the other hand, the column of the rocking pier responds, as expected, practically within the linear-elastic regime throughout the entire sequence. Verifying its design, the conventional foundation responds linear elastically with increased rotational stiffness, in comparison with the monotonic backbone curve, owing to densification of the underlying soil.

By contrast, the rocking foundation presents a broad moment—rotation hysteresis receiving comparatively larger rotational demand. Although the collapse of the conventional pier column after excitation with the L'Aquila record ended Test 2, Test 1 was continued by applying two additional very strong motions, the Rinaldi and the Takatori records. Response of the rocking pier subjected to successive base excitation with the Rinaldi and the Takatori motions, after having survived shaking with the three preceding lower magnitude motions of Earthquake Scenario A, in terms of a deck drift and b foundation settlement time histories.

Remarkably, despite having been subjected to a sequence of three earthquakes with intensity equivalent to, or exceeding, its design earthquake and having suffered considerable foundation deformation, the rocking pier survives the excess demands imposed by the Rinaldi motion. The pier eventually failed again by hitting the horizontal LVDT during excitation with the last, extremely strong excitation using the Takatori record.

Photos of the bridge models after Tests 1 and 2 Scenario A : a failure of the conventional pier after the first three medium-strong intensity motions and b the damage at the pier base compared with c the rocking pier subjected to the same motions plus two additional very strong motions, focusing on d foundation uplift. The response of nonlinear systems strongly depends on the exact loading history. Hence, it was decided to further study the response of the two pier designs under an alternative earthquake sequence in order to generalize the previously made observations.

This loading scenario differs from Earthquake Scenario A in that the very intense Rinaldi record is applied first, whereas the weaker Aegion and Lefkada records, subsequently imposed on the models, may be perceived as smaller aftershocks. Acceleration time history sequence recorded during shaking with Earthquake Scenario B at a deck of rocking pier smaller foundation ; b deck of conventional pier larger foundation ; and c excitation. Again, in agreement with results from Tests 1 and 2, it may be seen that for the rocking pier, deck deflections are mainly because of the foundation rotation, whereas column deflection plays a minor role.

The opposite is the case for the conventional pier. Despite being considerably distressed by the first very strong motion, the rocking pier demonstrates a remarkably stable response during shaking with the following three smaller excitations. Because both piers survived Earthquake Scenario B without collapse, testing was continued by applying the deleterious Takatori motion.

In fact, the second shaking with the Takatori excitation was required to induce failure of the rocking pier. In contrast, the conventional pier failed just after the application of the first Takatori motion. Again, the damage pattern hinge length, crack formation, and compressive spalling implies realistic modeling of the actual response of RC elements. Drift response of a the rocking pier and b the conventional pier to base excitation with two successive Takatori records following the shaking sequence of Earthquake Scenario B.

Photos of the conventional pier after Test 6 indicating the damage induced by shaking with Earthquake Scenario B followed by one additional very strong motion the Takatori record : a view of the entire pier model b in-plane and c out-of-plane views of the column—foundation joint. This experimental campaign has provided proof of the concept of deliberately designing for foundation nonlinearity to render RC structures safe under intense seismic excitation. Emphasis is placed on the response of RC bridge piers designed in accordance with modern seismic codes and hence having well-confined cross sections.

In contrast to previous studies, these piers are modeled at a highly reduced scale using a recently developed scale model reinforced concrete, which captures the behavior of prototype RC sections with a high level of fidelity. The seismic performance of a moderately tall pier supported on a square footing on top of a layer of medium-dense sand was studied through a series of dynamic centrifuge experiments. Two design alternatives are considered: i conventional capacity design, in which the foundation is as usual overdesigned, guiding plastic hinging into the superstructure and ii rocking isolation design, in which the foundation is deliberately underdesigned to promote uplifting and soil yielding, guiding plastic deformation below the ground.

The performance of the two design alternatives is evaluated and compared with the emphasis on the resistance of potential plastic hinge zones to cumulative damage due to successive shaking protocols. The testing sequence involved two different shaking scenarios, representing particularly destructive earthquake motions either preceded or followed by earthquakes closer to those against which the pier was designed. On the basis of the presented tests, the following conclusions can be drawn:.

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In both shaking scenarios, the rocking-isolated pier survived the excessive demands imposed by particularly strong shaking sequences, which caused catastrophic structural failure of the conventionally designed pier. Naturally, owing to its significantly lower FS V , the rocking foundation is prone to suffering increased settlements in comparison with the overdesigned foundations involved in conventional capacity design. The effect of such settlements on performance of the bridge could not be measured in the presented tests, where the continuity of the deck and the interaction between consecutive piers have not been taken into account.

Moreover, this drawback may be remediated through soil improvement measures. The authors gratefully acknowledge the support of the manager of the UoD centrifuge facility, Dr Andrew Brennan, and the invaluable contribution of the technical staff within the Department of Civil Engineering, especially Mark Trusswell and Colin Stark, who assisted in conducting the experiments. National Center for Biotechnology Information , U. Earthq Eng Struct Dyn. Published online Jul 7. Author information Article notes Copyright and License information Disclaimer.

This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited. Abstract Experimental proof is provided of an unconventional seismic design concept, which is based on deliberately underdesigning shallow foundations to promote intense rocking oscillations and thereby to dramatically improve the seismic resilience of structures.

Keywords: centrifuge modeling, seismic performance, soil—structure interaction, rocking isolation, capacity design, concrete failure. Background and Objectives Capacity design, which forms the cornerstone of modern seismic design, aims at controlling seismic damage by strategically directing inelastic deformation to structural components, which are less important to the overall system stability 1. Supporting evidence for this new approach has been provided by the following theoretical and empirical findings: Several theoretical and numerical studies on the rocking response of rigid blocks and elastic single-degree-of-freedom SDOF oscillators [e.

Open in a separate window. Figure 1. Table 1 RC column load calculations and design assessment with respect to the Eurocode. RC, reinforced concrete. Table 2 Foundation design: summary of actions and factors of safety FS. Experimental Methods The experimental program was carried out at the UoD and involved three different parts. Figure 2. Figure 3. Figure 4. Centrifuge modeling Similitude is an important consideration in physical modeling using reduced scale models that are intended to capture the response of field-scale prototypes.

Figure 5. Figure 6. Table 3 Testing program: sequence of seismic excitations. Experimental Results In this section, the seismic performance of the two pier—foundation systems during seismic scenarios A and B is evaluated and systematically compared. Figure 7. Figure 8. Figure 9. Figure Response to Earthquake Scenario B The response of nonlinear systems strongly depends on the exact loading history. Concluding Remarks This experimental campaign has provided proof of the concept of deliberately designing for foundation nonlinearity to render RC structures safe under intense seismic excitation.

References 1. Seismic design of reinforced concrete and masonry buildings. New York: Wiley; Villaverde R. Methods to assess the seismic collapse capacity of building structures: state of the art. Ground motions versus geotechnical and structural damage in the Christchurch February earthquake. Seismological Research Letters. Housner GW. The behavior of inverted pendulum structures during earthquakes.

Bulletin of the Seismological Society of America. Dynamic centrifuge model testing was carried out in the 3. The soil model was prepared within an equivalent shear beam ESB container with flexible walls, described in It was instrumented using two vertical arrays of five ADXL78 MEMS accelerometers, one array buried under the foundation centerline and the other at a distance large enough to record free field response. The motion of the pier was recorded using identical accelerometers attached to the foundation and the deck.

Vertical displacements of two foundation corners, recorded by Linear Variable Differential Transformers LVDTs , were used to calculate settlement and rotation in the direction of shaking. Another pair of LVDTs recorded the horizontal displacements of the deck center of mass.

Taking advantage of the effectiveness of this device in simulating desired motions 51 , an ensemble of records from historic earthquakes were utilized as base excitation. The motions were band pass filtered between 0. Shaking table acceleration time histories used in the different earthquake scenarios, compared with target records from earthquakes of different magnitudes in Greece, Italy, USA, and Japan.

By contrast, the Rinaldi and Takatori motions, recorded during the M s 6. Characterized by near fault directivity effects, these latter motions have increased spectral ordinates within a particularly broad band of periods especially in the range of particular interest, i. Having considerable duration and number of cycles, the record of the M s 6. Central to the issue of seismic performance evaluation is the recognition that damage in a component is cumulative, and the level of damage depends not only on the maximum deformation but also on the history of deformations.

In practice, structures may, over their lifespan, be subjected to several motions of varying intensities. Thus, attempting to account for seismic history, the performance assessment undertaken here involves a succession of motions considering two different seismic history scenarios. In Earthquake Scenario A , the ground motion order was such that the intensity roughly increased throughout the test, as if a number of weak or medium intensity motions were to precede a major catastrophic event, either in a sense of foreshocks or as independent events taking place in a relatively short period.

Earthquake Scenario B explores the case where a very strong earthquake is followed by weaker shaking events or aftershocks. It should be noted that in some cases, when scenarios A or B did not lead to collapse, additional shaking was applied using the Takatori record until ultimate or practical failure was reached. In this section, the seismic performance of the two pier—foundation systems during seismic scenarios A and B is evaluated and systematically compared.

Focusing on damage accumulation due to successive earthquakes and the importance of shaking history, time histories of recorded illustrative demand parameters, such as deck acceleration and displacement, as well as moment-rotation and shear force-displacement hysteretic response loops are subsequently presented at prototype scale. As anticipated, the rocking pier experiences invariably lower acceleration than the conventional pier.

This advantage becomes more significant as excitation intensity increases and the ultimate capacities are mobilized. Hence, upon shaking with the L'Aquila record, the maximum transient demand experienced by the rocking pier is half the demand on the conventional pier.

Acceleration time history sequence recorded during Earthquake Scenario A at a the deck of the rocking pier; b the deck of the conventional pier; and c the model base.

The rocking isolation effect can be easily quantified with regard to the moment capacity of the rocking foundation, being therefore potentially useful for the estimation of seismic demand in design. Yet, this value coincides with the measured peak mass acceleration only in the very first shaking event using the Aegion record. Thereupon, overstrength effects, associated with soil densification during shaking, lead to some considerable increase in this value, which yet remains substantially lower than the peak demand on the conventional pier.

On the basis of the aforementioned discussion, it may be deduced that even the relatively low intensity Aegion excitation is sufficient to momentarily mobilize the capacity of the rocking footing and induce its rocking—uplifting response. Evidently, as anticipated, in the case of the rocking pier, rotational movement prevails throughout the entire shaking sequence.

The opposite is the case with the conventional pier, where deck drift is almost exclusively associated with deformation and failure of the RC column. More importantly, the rocking pier demonstrates a crucial advantage over conventional design. Not only does it experience significantly lower drift in the first shaking event Aegion but it also retains an increasingly favorable performance during the following earthquakes of greater intensity and duration.

In particular, having experienced the first two excitations, it is practically unaffected by the sustained shaking of the L'Aquila record, resulting in a small total drift of 0. Note that in this test, failure is assumed to take place when the deck mass hits the horizontal LVDT, which prevents further movement in the direction of shaking. Inelastic action is not only unavoidable but also essential in providing the required energy absorption to enable survival of the structure under intense seismic shaking.

The two design alternatives, both relying on inelastic response, differ only in the component where the inelastic deformation is directed to. Subsequent loading with the stronger Lefkada motion causes an important number of excursions into the nonlinear regime, accumulating substantial permanent deformation at the column base.

However, having experienced this damage, the pier column appears unable to sustain further excitation with the equally strong L'Aquila motion, which exhausts its ductility capacity causing rapid deterioration of strength after a couple of cycles and eventually collapse. On the other hand, the column of the rocking pier responds, as expected, practically within the linear-elastic regime throughout the entire sequence. Verifying its design, the conventional foundation responds linear elastically with increased rotational stiffness, in comparison with the monotonic backbone curve, owing to densification of the underlying soil.

By contrast, the rocking foundation presents a broad moment—rotation hysteresis receiving comparatively larger rotational demand. Although the collapse of the conventional pier column after excitation with the L'Aquila record ended Test 2, Test 1 was continued by applying two additional very strong motions, the Rinaldi and the Takatori records.

Response of the rocking pier subjected to successive base excitation with the Rinaldi and the Takatori motions, after having survived shaking with the three preceding lower magnitude motions of Earthquake Scenario A, in terms of a deck drift and b foundation settlement time histories. Remarkably, despite having been subjected to a sequence of three earthquakes with intensity equivalent to, or exceeding, its design earthquake and having suffered considerable foundation deformation, the rocking pier survives the excess demands imposed by the Rinaldi motion.

The pier eventually failed again by hitting the horizontal LVDT during excitation with the last, extremely strong excitation using the Takatori record. Photos of the bridge models after Tests 1 and 2 Scenario A : a failure of the conventional pier after the first three medium-strong intensity motions and b the damage at the pier base compared with c the rocking pier subjected to the same motions plus two additional very strong motions, focusing on d foundation uplift. The response of nonlinear systems strongly depends on the exact loading history.

Hence, it was decided to further study the response of the two pier designs under an alternative earthquake sequence in order to generalize the previously made observations. This loading scenario differs from Earthquake Scenario A in that the very intense Rinaldi record is applied first, whereas the weaker Aegion and Lefkada records, subsequently imposed on the models, may be perceived as smaller aftershocks. Acceleration time history sequence recorded during shaking with Earthquake Scenario B at a deck of rocking pier smaller foundation ; b deck of conventional pier larger foundation ; and c excitation.

Again, in agreement with results from Tests 1 and 2, it may be seen that for the rocking pier, deck deflections are mainly because of the foundation rotation, whereas column deflection plays a minor role. The opposite is the case for the conventional pier.

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Despite being considerably distressed by the first very strong motion, the rocking pier demonstrates a remarkably stable response during shaking with the following three smaller excitations. Because both piers survived Earthquake Scenario B without collapse, testing was continued by applying the deleterious Takatori motion. In fact, the second shaking with the Takatori excitation was required to induce failure of the rocking pier. In contrast, the conventional pier failed just after the application of the first Takatori motion. Again, the damage pattern hinge length, crack formation, and compressive spalling implies realistic modeling of the actual response of RC elements.

Drift response of a the rocking pier and b the conventional pier to base excitation with two successive Takatori records following the shaking sequence of Earthquake Scenario B. Photos of the conventional pier after Test 6 indicating the damage induced by shaking with Earthquake Scenario B followed by one additional very strong motion the Takatori record : a view of the entire pier model b in-plane and c out-of-plane views of the column—foundation joint.

This experimental campaign has provided proof of the concept of deliberately designing for foundation nonlinearity to render RC structures safe under intense seismic excitation. Emphasis is placed on the response of RC bridge piers designed in accordance with modern seismic codes and hence having well-confined cross sections. In contrast to previous studies, these piers are modeled at a highly reduced scale using a recently developed scale model reinforced concrete, which captures the behavior of prototype RC sections with a high level of fidelity.

The seismic performance of a moderately tall pier supported on a square footing on top of a layer of medium-dense sand was studied through a series of dynamic centrifuge experiments. Two design alternatives are considered: i conventional capacity design, in which the foundation is as usual overdesigned, guiding plastic hinging into the superstructure and ii rocking isolation design, in which the foundation is deliberately underdesigned to promote uplifting and soil yielding, guiding plastic deformation below the ground.

The performance of the two design alternatives is evaluated and compared with the emphasis on the resistance of potential plastic hinge zones to cumulative damage due to successive shaking protocols. The testing sequence involved two different shaking scenarios, representing particularly destructive earthquake motions either preceded or followed by earthquakes closer to those against which the pier was designed. On the basis of the presented tests, the following conclusions can be drawn:. In both shaking scenarios, the rocking-isolated pier survived the excessive demands imposed by particularly strong shaking sequences, which caused catastrophic structural failure of the conventionally designed pier.

Naturally, owing to its significantly lower FS V , the rocking foundation is prone to suffering increased settlements in comparison with the overdesigned foundations involved in conventional capacity design. The effect of such settlements on performance of the bridge could not be measured in the presented tests, where the continuity of the deck and the interaction between consecutive piers have not been taken into account. Moreover, this drawback may be remediated through soil improvement measures.

The authors gratefully acknowledge the support of the manager of the UoD centrifuge facility, Dr Andrew Brennan, and the invaluable contribution of the technical staff within the Department of Civil Engineering, especially Mark Trusswell and Colin Stark, who assisted in conducting the experiments. National Center for Biotechnology Information , U. Earthq Eng Struct Dyn. Published online Jul 7.

Author information Article notes Copyright and License information Disclaimer. This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited. Abstract Experimental proof is provided of an unconventional seismic design concept, which is based on deliberately underdesigning shallow foundations to promote intense rocking oscillations and thereby to dramatically improve the seismic resilience of structures.

Keywords: centrifuge modeling, seismic performance, soil—structure interaction, rocking isolation, capacity design, concrete failure. Background and Objectives Capacity design, which forms the cornerstone of modern seismic design, aims at controlling seismic damage by strategically directing inelastic deformation to structural components, which are less important to the overall system stability 1.

Supporting evidence for this new approach has been provided by the following theoretical and empirical findings: Several theoretical and numerical studies on the rocking response of rigid blocks and elastic single-degree-of-freedom SDOF oscillators [e. Open in a separate window. Figure 1. Table 1 RC column load calculations and design assessment with respect to the Eurocode. RC, reinforced concrete. Table 2 Foundation design: summary of actions and factors of safety FS.

Experimental Methods The experimental program was carried out at the UoD and involved three different parts. Figure 2. Figure 3. Figure 4.

Centrifuge modeling Similitude is an important consideration in physical modeling using reduced scale models that are intended to capture the response of field-scale prototypes. Figure 5. Figure 6. Table 3 Testing program: sequence of seismic excitations. Experimental Results In this section, the seismic performance of the two pier—foundation systems during seismic scenarios A and B is evaluated and systematically compared.

Figure 7. Figure 8. Figure 9. Figure Response to Earthquake Scenario B The response of nonlinear systems strongly depends on the exact loading history. Concluding Remarks This experimental campaign has provided proof of the concept of deliberately designing for foundation nonlinearity to render RC structures safe under intense seismic excitation. References 1. Seismic design of reinforced concrete and masonry buildings.

New York: Wiley; Villaverde R. Methods to assess the seismic collapse capacity of building structures: state of the art. Ground motions versus geotechnical and structural damage in the Christchurch February earthquake. Seismological Research Letters. Housner GW.

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The behavior of inverted pendulum structures during earthquakes. Bulletin of the Seismological Society of America. Meek JW. Effect of foundation tipping on dynamic response. Simplified earthquake analysis of structures with foundation uplift. Paolucci R. Simplified evaluation of earthquake induced permanent displacements of shallow foundations. Journal of Earthquake Engineering. Rocking response of rigid blocks to earthquakes. Earthquake Engineering and Structural Dynamics. Shenton HW. Criteria for initiation of slide, rock, and slide-rock rigid-body modes. Makris N, Roussos YS.

Rocking response of rigid blocks under near-source ground motions. Zhang J, Makris N.