Soil-shallow 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, soil-shallow 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 soil-shallow 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 large-scale 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 soil-foundation 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.
However, El Ganainy and El Naggar [
To address some of the above-mentioned 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
These equivalent springs represent the foundation-soil system. The macroelement model replaces the system soil-shallow foundation, thus decreasing considerably both the overall number of degrees of freedom and the computation effort required to run large models.
Proposed macroelement model for soil-shallow foundation system.
Two material models are considered in this study, namely, the Bouc-Wen model [
In this study, we considered the Baber and Noori [
The nonlinear behavior of the soil-shallow 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 soil-shallow 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 displacement-controlled 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 Bouc-Wen-Baber-Noori 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 soil-shallow foundation system, its predictions are compared with experimental results. In the framework of the TRISEE Project (3D Site Effects and Soil-Foundation Interaction in Earthquake and Vibration Risk Evaluation) a program of large-scale 1 g model has been tested to investigate the response of soil-shallow 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 Bouc-Wen-Baber-Noori 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 Bouc-Wen-Baber-Noori 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 moment-rotation 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
Then foundation stiffness matrix
A practical macroelement model is presented in this paper to simulate the response of soil-shallow foundation systems. The proposed model was verified against experimental test results of large-scale 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 well-chosen parameters of the Bouc-Wen-Baber-Noori model of hysteresis. The soil squeeze-out 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.