Soilshallow foundation interaction models that are incorporated into most structural analysis programs generally lack accuracy and efficiency or neglect some aspects of foundation behavior. For instance, soilshallow foundation systems have been observed to show both small and large loops under increasing amplitude load reversals. This paper presents a practical macroelement model for soilshallow foundation system and its stability under simultaneous horizontal and vertical loads. The model comprises three spring elements: nonlinear horizontal, nonlinear rotational, and linear vertical springs. The proposed macroelement model was verified using experimental test results from largescale model foundations subjected to small and large cyclic loading cases.
Several researchers ([
In the seismic resistant design of structures, we are most interested in the strength reduction factors to account for the nonlinear behavior that might be experienced by a structure subjected to an earthquake ground motion. Few researchers [
Incorporation of SSI requires explicit modelling of soilfoundation system adequately. For instance, several models have been proposed depending on the foundation type, its embedment, and its rigidity ([
uncoupled spring model comprising three spring elements, for shallow foundations that are stiffer than the supporting soil;
a finite element formulation of linear (or nonlinear) foundation behavior using Winkler models, for shallow foundations that are less stiff than the supporting soil;
decoupled Winkler model, for shallow foundations that are flexible with respect to the supporting soil.
To address some of the abovementioned issues, macroelement formulations have been proposed. The first formulation has been developed by Nova and Montrasio [
In a completely different modelling approach, El Shamy and Zamani [
To overcome the difficulties in performing complete nonlinear simulations, Seylabi et al. [
In this paper, a new macroelement model is developed for the analysis of the nonlinear response of shallow foundations under cyclic loading. This model may easily be incorporated into available structural analysis programs such as OpenSees [
The problem being studied here is that of a shallow foundation of any shape embedded in soil and subjected to simultaneous axial and lateral forces, as shown in Figure
vertical translational elastic spring with stiffness
shear inelastic spring with preyield stiffness
rotational inelastic spring with preyield stiffness
Proposed macroelement model for soilshallow foundation system.
Two material models are considered in this study, namely, the BoucWen model [
In this study, we considered the Baber and Noori [
The nonlinear behavior of the soilshallow foundation system is modelled via the nonlinear shear and nonlinear rotational springs. A force
The constitutive relationship for
The equations describing the degradation behavior are described as follows:
For the complete implementable procedure to obtain
The vertical force
The macroelement model described in the previous subsections is used to simulate the soilshallow foundation interaction.
Initially, we assume that the shear and rotational springs are linear; then we replace them by the general forces
Undeformed and deformed configurations of the proposed macroelement model.
The above kinematic equations (see (
Considering the model subjected to the axial load
Considering the case of lateral displacementcontrolled analysis, (
The system
Previous converged solution
While (
Update
Compute
Compute the Jacobian matrix
end
where the Jacobian matrix
From the current estimation of the top lateral displacement
The above pseudocode (Global Newton Routine) along with the Local Newton Routine (that implements the BoucWenBaberNoori model of hysteresis) has been implemented numerically in MATLAB (Mathworks, Inc.).
To verify the validity of the proposed macroelement model in predicting the cyclic behavior of the soilshallow foundation system, its predictions are compared with experimental results. In the framework of the TRISEE Project (3D Site Effects and SoilFoundation Interaction in Earthquake and Vibration Risk Evaluation) a program of largescale 1 g model has been tested to investigate the response of soilshallow foundation under cyclic and dynamic loads. The experiment was carried out at ELSA (European Laboratory for Structural Assessment) in Ispra, Italy; test results are reported in many references ([
The TRISEE experiment consists of three phases; only Phases I and III are considered in the comparison because the proposed macroelement model is restricted to 2D loading conditions only. However, the model may be extended to include the 3D case.
The experimental setup consists of a square steel shallow foundation
The HD and LD specimens were loaded vertically by
The new macroelement model is used to simulate the foundation. The parameters of the numerical model are presented in Table
TRISEE: parameters of the numerical model (units: kN, m).
Phase  Shear spring  Rotational spring  Vertical spring  









HD  
I  132.2  12.5  60  58.6  10  60  120 
III  70  99  9  35  111  2  80 


LD  
I  54  3.8  47  25.4  3.8  4  65 
III  35  40.4  1  8  33.3  1  27 
Parameters of the BoucWenBaberNoori model, HD test.
Phase  Shear spring  Rotational spring  














 
I  1  0.5  0.5  1  0  −0.01  0  1  0.5  0.5  1  0  −0.01  0 
III  1  0.5  0.5  0.7  0  0  0.1  1  0.1  0.9  0.7  0  0  0.1 
Parameters of the BoucWenBaberNoori model, LD test.
Phase  Shear spring  Rotational spring  














 
I  1  0.5  0.5  1  0  −0.01  0  1  0.5  0.5  1  0  −0.01  0 
III  1  0.5  0.5  0.3  0  −0.01  0.1  1  0.33  0.67  0.2  0  −0.01  0.1 
Figures
Comparison of experimental and numerical results: HD test, Phase I.
Comparison of experimental and numerical results: LD test, Phase I.
Comparison of experimental and numerical results: HD test, Phase III.
Comparison of experimental and numerical results: LD test, Phase III.
The uplift is important for the HD Phase III test, and this can be observed from the S shaped momentrotation curve (see Figure
In the previous section, large rotations have been considered in formulating the macroelement model. However, assuming that small rotations permits the construction of the foundation stiffness matrix. In this case, (
Differentiate the equations of equilibrium (in (
in which
Differentiate the equations of kinematics (see (
Substitute
where
A practical macroelement model is presented in this paper to simulate the response of soilshallow foundation systems. The proposed model was verified against experimental test results of largescale model foundations subjected to small and large loading cycles. A summary of the main points presented in this paper is given below:
The proposed macroelement model can simulate with a good accuracy the lateral response and rocking of shallow foundations under quasistatic cyclic loadings.
The uplift could be simulated adequately using wellchosen parameters of the BoucWenBaberNoori model of hysteresis.
The soil squeezeout phenomenon observed by El Ganainy and El Naggar [
The proposed model does not take into consideration full coupling between the different springs. Nevertheless, the model represents a first step for future improvements.
The authors declare that they have no competing interests.
This work was supported by the Sustainable Construction Material and Structural Systems (SCMASS) Research Group of the University of Sharjah, UAE.