The Effect of Concrete Footing Shape in Differential Settlement : A Seismic Design

.is paper presents the numerical results of concrete footing-soil foundation seismic interaction mechanism..e concrete footing has beenmade with two different shapes, but with the equal volume of concrete material..e concrete footing-soil foundation has been analyzed using nonlinear finite elements, with the fixed-base state. .e simulated near-fault ground motions have been applied to the concrete footing-soil foundation..e problem has been formulated based on the settlement controlled analysis..e local geotechnical conditions of all configurations have been analyzed. .e numerical analysis results indicate that the shape of a concrete footing alters seismic response, revises inertial interaction, enhances damping ratio, improves load carry capacity, modifies cyclic differential settlement, revises failure patterns, minimizes nonlinear deformation, and changes cyclic strain energy dissipation. .e novelty of this research work is the strain energy has more been dissipated with artistic concrete footing design.


Introduction
A number of the buildings have been collapsed due to improper soil-footing interaction; it has occurred when dynamic or seismic forces have been applied to them.e improvement of soil-footing seismic interaction mechanism is an art in geotechnical earthquake engineering design and needs to select appropriate footing shape to enhance the safety of the structure.
ere are many experimental, numerical, and theoretical research studies which mainly focus on differential settlement and bearing capacity of the soil when the soil has been subjected to simulated seismic loading, liquefaction, and landslide.e outcome of these research studies has been realizing failure mitigation of soil foundation and introducing different methods for improving soil foundation stability [1][2][3][4][5][6][7][8][9][10][11][12].Using research idea from different branches of science and applying them in geotechnical earthquake engineering is an acceptable research methodology.In this regard, there is an investigation on quantitative shape evaluation of graphite particles in ductile iron [13]; with attention to this research work, the effect of concrete footing shape in minimizing differential settlement of soil foundation needs to be investigated.However, the geotechnical earthquake engineering is a young field, and the concept of the seismic stress response of concrete footingsoil foundation model is a complicated mechanism and needs more investigation.In the present study, the seismic design of concrete footing with considering the configuration of concrete footing to minimize differential settlement at the base of concrete footing has been investigated.Earthquake includes differential settlement, but the shape of concrete footing in producing differential settlement has not been studied with considering seismic response at the base of a concrete footing, energy dissipation, hysteretic soil damping, strain travel paths, inertial interaction, nonlinear deformation patterns, and footing-soil seismic interaction mechanism.We hope all we have done could support the seismic design of concrete footing in considering the safety of concrete footing and soil foundation to support solving geotechnical earthquake engineering problems.

Problem Definition
In the previous studies, many investigations have been made in understanding the differential settlement of soil, when the soil has been subjected to dynamic loading [2,3,6,12].e present investigations analyzed the soil foundation-concrete footing seismic interaction mechanism and stability, with considering concrete shape.ere is no investigation on the effect of concrete footing geometry to soil foundationconcrete footing seismic interaction present in the literature, and the associated simulation indirectly even cannot be found in the literature.e idea to do this research work is to understand the effect of concrete footing shape on (i) soil foundation-concrete footing seismic interaction mechanism, (ii) the differential settlement, (iii) the nonlinear deformation, (iv) the strain paths, and (v) the failure patterns.Two types of concrete foundations have been considered and have been analyzed numerically.e concrete footing and soil foundation, simultaneously, have been subjected to seismic loading.In all configurations, single concrete footing has been analyzed.In the present study, the specific geometry of concrete footing may increase or reduce soil-footing interaction.From the theoretical concept point of view, to provide an appropriate solution for this problem, and for solving and demonstrating this elasticity problem, the ABAQUS software has been employed.e ABAQUS has been employed to solve and explain many engineering problems in the various fields through numerical simulation [12,[14][15][16][17][18]. e ABAQUS has the ability to solve nonlinear problems in high quality.To design the concrete footing with maximum safety and minimum cost, two different configurations of concrete footing have numerically been investigated.In both configurations, the same volume and grade of concrete have been used.e method of embedded concrete footing in the soil foundation is adopted, and also the interaction method for soil foundation with concrete footing in ABAQUS environment has been adopted.e footing and soil have been loaded.e problem has been formulated based on the settlement controlled analysis.e seismic response of concrete footing and soil foundation in all models has been compared.

Domain Theoretical Concept for
Present Analysis e nonlinear deformation develops due to six components of stresses applied to a body element; the six components of stresses are σ x , σ y , σ z , τ xy , τ xz , and τ yz .
Two types of distortions arise: (i) direct strain ε x , ε y , and ε yz from σ x , σ y , and σ z and (ii) angular distortions e xy , e xz , and e yz from τ xy , τ xz , and τ yz .e strains and rotation components for each tensor may be expressed in terms of their respective displacements gradients: e strain matrix is e rotation matrix is Consequently, the complete distortion of a volume element may be expressed as the sum of corresponding strains and rotations in the matrix form [19].In this study, the nonlinear finite element analysis has been applied to analyze seismic behavior of concrete footing-soil foundation interaction.Accordingly, the soil foundation and concrete footing have numerically been investigated.e six strain components will not act independently; they have a direct relationship with the displacements.Under earthquake function, due to loading, unloading, and reloading process, an element can translate, rotate, compress, or elongate.

Materials and Modeling
e concrete footing-soil foundation has been modeled using nonlinear finite elements, with the fixed-base state.
e soil foundation is 1.8 * 1.8 * 0.9 (m).e concrete footing for configuration-1 and configuration-2 are 0.6 * 0.6 * 0.4 (m) and 0.4 * 0.9 * 0.4 (m), respectively.For both models, the concrete foundation of 0.2 * 0.2 * 0.2 (m) is installed on center of concrete footing.e soil foundationconcrete footing seismic interaction has been evaluated.e concrete footing configuration is with two different shapes and equal volume.In the numerical analysis, the typical mesh has been used.e concrete footing is placed on a horizontal surface of the soil foundation; it is shown in Figures 1 and 2. e simulated near-fault ground motions, with equal magnitude, have numerically been applied to concrete footing and soil foundation.
e acceleration history of the earthquake occurred in Norcia, Italy, has been used in numerical analysis and is shown in Figures 3 and 4. From 22 to 28 seconds of the earthquake, the main seismic excitation has been observed.In comparing E, N, and Z comps, the E comp has maximum acceleration history and 2 Advances in Civil Engineering nonlinearity tolerance in Norcia Earthquake.However, due to these reasons, the E comp has been selected for numerical analysis.e mechanical properties of materials have been used in this analysis, extracted from those reported in the literature, and are shown in Table 1.e soil is often discretized with solid finite elements.Nonlinear numerical analysis has been performed based on realistic seismic data.e earthquake data have been used in numerical analysis and are reported by AMATRICE station, with 8.9 (km) distance from the epicenter of the earthquake.e Norcia Earthquake occurred with 6.2 magnitude, at the location of 42.71 N 13.17 E, and depth of 10.0 km, on 1 : 36 : 33 UTC, 24 August 2016.

Numerical Analysis, Discussion, and Verification of the Result
Enhancement geometry of a concrete footing considerably changes soil-footing seismic interaction mechanism, and this process leads to develop a new concept for the satisfactory seismic design of a concrete footing.e morphology Advances in Civil Engineering of concrete footing influences on the shear strain travel paths and seismic energy distribution.e meaningful relationships have been observed between simulated near-fault groundshaking and energy dissipation mechanism at each configuration.e characteristics of seismic waves are altered as it is facing different simulated geomorphological conditions.e seismic wave dispersion modifies damping ratio and governs nonlinear deformation patterns of soil foundation and footing-soil seismic interaction mechanism.However, the morphology of concrete footing significantly affects the amplitude of earthquake ground motions; it may be known as "geomorphological conditions effect" in concrete footing-soil foundation seismic design.
e numerical analysis results have confirmed that the geomorphological condition influence to strain energy dissipation, and this process leads to developing nonlinear deformation patterns and differential settlement with the specific shape at each configuration, and subsequently, it is understood that the geomorphological conditions are important in the distribution of earthquake damage.e flexible soil foundation area-to-ridge concrete footing area interaction is responsible for the failure mechanism of soil foundation at each configuration.However, the design of concrete footing shape at each configuration is important in the stability of concrete footing and soil foundation as well.e modified shape of concrete footing leads to change in the concrete footing center of gravity and shape of cyclic load distribution; this phenomenon results in the modification of concrete footing-soil foundation seismic interaction mechanism and seismic load response.And on the other hand, the geomorphological conditions and morphology of concrete footing are responsible for developing characteristics of strain paths, and the strain path is a factor in developing a differential settlement, failure mechanism, deformation, and bearing capacity.e seismic site response is highly variable with respect to concrete footing morphology, while the volume of used concrete is equal in both configurations, and cost effectiveness of the project is considered with seismic design of concrete footing.e geotechnical condition is another factor in ground motion behavior prediction.
e geomorphological condition affects the seismic response of an infrastructure.It can suggest beyond the theoretical seismic design; it requires to numerically simulate the influence of geomorphological conditions to predict seismic stability of the infrastructure.e near-fault ground motions change strain energy dissipation via travel path of seismic wave propagation.e hysteretic behavior of soil significantly affects the concrete footing seismic response.
is process affects concrete footing and soil foundation inertial interaction and leads to stress response of the soil foundation.Cyclic seismic load response has been developed due to concrete footing morphology, and it is shown in Figures 5 and 6. e seismic loading and concrete footing morphology are responsible for soil compaction, modify soil foundation shear strength, and alter concrete footing-soil foundation seismic interaction mechanism.e geomorphological conditions affect wave motion behavior, and it causes differential ground deformation and differential rotations of concrete footing along its base to interact with soil foundation.
e wave energy propagates in different directions and results in forming cyclic volumetric strain and produces nonlinear deformation for each configuration; it is shown in Figures 7 and 8.In configurations 1 and 2, the different ground motions response has been observed; it  4 Advances in Civil Engineering developed with respect to concrete footing shape, and this phenomenon supports in earthquake damage estimation, when the movement of seismic wave radiates along the base and sides of the concrete footing.e seismic wave radiate dislocates the concrete footing at any possible directions, and this phenomenon results in accelerating model and releasing cyclic strain energy.e hysteretic energy dissipation is scattered in all configurations, with the critical mechanism in respect to the flexible the soil foundation area-to-the ridge concrete footing area interaction; this mechanism is responsible for failure mechanism of soil foundation at each configuration.Inertia has been developed due to near-fault ground motions applied to each configuration; it has appeared in the form of base shear, moment, and torsional excitation and causes differential displacements and nonlinear deformation rotation, with a different magnitude in respect to the soil foundation flexibility and concrete footing shape.e damping ratio in configuration-2 is reduced by 15% compared to configuration-1.e shape of concrete footing governs hysteretic soil damping and inertial interaction; this process occurs based on kinematic interaction of concrete footing-soil foundation characteristics and causes concrete foundation motions in soil foundation.e nonlinear deformation significantly influences the overall concrete footing seismic behavior, especially with respect to damping and seismic degrees of damage.e stress third invariant behavior is depicted in Figures 9-12.A comparative numerical analysis of the seismic response between two differently shaped and equal volume of concrete footing under significantly invariant stresses through establishing nonlinear finite element analysis has been made.e stress third invariant at the base of concrete footing for configuration-1 is formed with higher strain energy and lower energy dissipation magnitude compared to configuration-2.
is complex mechanism influences load carry capacity of soil foundation and types of earthquake damage.
e evaluated results show that the shearing resistance at configuration-2 is improved due to lower degradation of soil by cyclic loading.e nonlinear seismic load-cyclic strain curve expresses the variation of stiffness and shear strength at the base of the concrete footing.e cyclic strain energy causes increasing nonlinear shear deformation and soil foundation failure if cyclic shear stress increases more than the shear strength of the soil foundation.Figure 9 shows the stress invariant at the base of concrete footing for configurations 1 and 2. As a result, the cyclic strain behavior would be expected to provide a more reliable prediction of differential settlement, in considering concrete footing-soil foundation seismic interaction.e nonlinear seismic load-cyclic strain relationship at both configurations shows the higher strain energy concentration is developed at the base of the configuration-1, with respect to the magnitude and shape of the seismic loading.However, the differential settlement is significantly minimized in configuration-2.e higher cyclic strain energy concentration has a direct relationship with nonlinear shear deformation and nonlinear volume change.
e failure patterns are plotted in Figure 9, in such a way that red and blue zone implies a fully plastic state.Due to the nature of cyclic loading, nonplastic deformation occurred in any possible direction.
e geometry of the failure patterns shows the plastic slice has been occupied more area in configuration-1.e plastic deformation morphology shows that the differential settlement for both models is not the same, and a higher magnitude of differential settlement occurred in configuration-1.
Figures 10 and 11 show the cyclic stress behavior of the configurations 1 and 2 in respect to the size and shape of the plastic deformation.And it describes increment of cyclic stress invariant.
e stronger response of earthquake shaking in the configuration-1 has been observed.However, it is a possibility for permanent deformation in configuration-1.
e strong strain in several parts of the configuration-1 results in permanent nonlinear shear deformations and nonlinear volume change.
e nonlinear volumetric strain often refers to ground failure.It is the possibility of ground failure in the form of lateral ground displacement, and it can contribute to the failure of the soil foundation.
Color map surface projection techniques are used in matrix analysis to simulate results of ABAQUS software, which is reported in cyclic stress invariant.According to Figure 12, by using the matrix in numerical simulation, the development of cyclic stress invariant due to seismic acceleration load response at the base of the soil foundation is illustrated.Figure 12 shows the seismic load travel paths and seismic load directions.e loading and reloading process has been depicted in different colors.e load distribution among two soil foundations is varied; configuration-2 is subjected to more distributed cyclic load compared to configuration-1; it is a result of increasing stable soilconcrete footing interaction in configuration-2.It can be concluded that the stability of configuration-2 is higher than configuration-1.
e minimum strain energy density allows the influence of the T-stress on the mixed modes I/II fracture strength [23].And the specimen geometry can strongly influence the mode-I fracture strength [24].e hysteretic energy dissipation influences the flexible soil foundation area-to-the ridge concrete footing area interaction and governs failure mechanism.e configurations 1 and 2 affect strain energy dissipation, according to their geometry.e results of the numerical analysis show good agreements with those reported in the literature.
e different types of loads can store elastic energy before damage, and this energy storage accelerates and develops damage mechanism [25][26][27][28][29][30][31][32][33].In the present numerical analysis, it has been observed that the shape of +5.380e + 00 +4.483e + 00 +3.586e + 00 +2.690e + 00 +1.793e + 00 +8.966e -01 +1.192e -07 -8.966e -01 -1.793e + 00 -2.690e + 00 -3.586e + 00 -4.483e + 00 -5.380e + 00   6 Advances in Civil Engineering concrete footing causes storage strain energy mechanism and strain energy dissipation as well.e cyclic strain energy causes deformation, with respect to the shape of the concrete footing.e soil foundation was subjected to the seismic excitation and deforms. is deformation occurs according to an internal energy mechanism and concrete footing shape. is internal energy is known as strain energy.In the present study, the strain energy has more been dissipated with artistic concrete footing design.e configuration of soil foundation-concrete footing significantly modified strain energy storage and damage patterns.e stress component distribution due to the external forces depends on the strain energy function.However, the shape of concrete footing directly affects seismic acceleration response, damping mechanism, energy dissipation, load carry capacity, differential settlement, nonlinear deformation, strain-stress travel paths, and inertial interaction.

Conclusion
e nonlinear finite elements are applied in the analysis of concrete footing-soil seismic interaction mechanism.e concrete footing is built up with two different shapes and equal volume.
e simulated near-fault ground motions have been applied to each configuration.In the present study, the following aims have been achieved: (i) It has been found that the concrete footing-soil interaction and morphology of differential settlement have been changed with respect to the shape of the concrete footing.(ii) e local geotechnical conditions have been modified ground-shaking characteristics.e anomalous damage distributions may not derive with the select appropriate shape of a concrete footing, considering local site conditions.(iii) e morphology of concrete footing affects the seismic energy travel paths, and meaningful relationships have been observed between simulated near-fault groundshaking and energy dissipation mechanism.e strain energy has more been dissipated with artistic concrete footing design.(iv) e shape of concrete footing governs hysteretic soil damping and inertial interaction; these processes have occurred based on kinematic interaction of concrete footing-soil foundation characteristics.(v) e higher strain energy concentration has been observed at the base of the configuration-1, with respect to the magnitude and shape of the seismic loading response.e differential settlement is significantly minimized in configuration-2.(vi) e cyclic strain causes plastic cyclic deformation, with respect to the shape of concrete footing and related to increment of stress.According to the numerical results, this approach supports in forecasting the seismic stability of concrete footing.Advances in Civil Engineering

e
support by the Major Projects of Natural Science Research in Jiangsu Colleges and Universities (grant no.17KJA560001), the Science and Technology Planning Project of Jiangsu Province (grant no.BY2016061-29), the Jiangsu Province Six Talent Peak High-Level Talent Project (grant no.JZ-011), and the New Wall Materials and Development of Bulk Cement Projects of Jiangsu Economic and Information Commission (grant no.2017-21) is greatly acknowledged.

Figure 12 :
Figure 12: 3D cyclic stress invariant of soil at the base of soil foundation, using matrix for numerical simulation.(a) Configuration-1.(b) Configuration-2.