Modified Pseudo-Dynamic Bearing Capacity of Shallow Foundations Subjected to Inclined-Eccentric Combined Loading


 Shallow foundations are commonly subjected to simultaneous inclined and eccentric combined loadings exerted by the overlying superstructure and geo-environmental sources. The performance of such footings in seismic-prone areas is a topic of great interest in geotechnical engineering practice. In this paper, a comprehensive parametric study is conducted to evaluate the seismic bearing capacity of shallow strip foundations overlying dry and cohesionless granular soil under the action of vertical-horizontal-moment combined loading. For this purpose, a systematic combination of the lower-bound theorems of the limit analysis, the finite element method, and the nonlinear programming is implemented. The second-order cone programming (SOCP) is adopted for efficient optimization purposes so as to model the actual nonlinear form of the universal Mohr-Coulomb yield function. In addition, the equilibrium equations associated with the combined loading are incorporated into the lower bound formulations. The seismic condition is simulated by the well-established modified pseudo-dynamic approach by accounting for the significant influence of phase difference as well as the primary and shear waves propagation through applying non-uniform inertia forces along the vertical and horizontal directions, respectively. The employed formulations are rigorously validated against a majority of high-quality studies in the literature in the static combined loading condition. The results of the seismic bearing capacity are presented in the forms of spectral responses and failure envelopes for the eccentric and inclined loadings. Accordingly, the influences of non-dimensional frequency, induced seismic acceleration and material damping on the ultimate bearing capacity of eccentrically and obliquely loaded strip footings are thoroughly examined and discussed. The results show that the most critical responses of the shallow foundation are captured in the resonance condition of earthquake excitation. As the seismic intensity decreases and the damping ratio increases, the spectral vertical, horizontal and moment bearing capacities of surface footings become smoother. In addition, the failure envelopes of the shallow foundation subjected to either inclined or eccentric loading significantly shrink with the increase in the earthquake accelerations and decrease in the material damping of the underlying soil mass. The amount of changes in the size of failure envelopes in the normalized V - H and V - M spaces due to the variation of seismic intensity and material damping depends directly on the non-dimensional frequency of the earthquake excitation.


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In this section, the detailed formulations of the finite element limit analysis (FELA), simulation 126 of the eccentric and inclined loadings applied on the foundation and its incorporation into the 127 FELA formulations, and also the simulation of the modified pseudo-dynamic seismic loading are 128 thoroughly described.  sufficiently so as to accommodate potential boundary effects. The soil beneath the foundation is 153 a cohesionless granular material, whose behaviour is considered to be elastic-perfectly plastic,  (1) In these equations, the linear shape function is characterized by = ( + + ) 2 ⁄ , where 168 , and are shape function coefficients and is an area of each triangular element. The shape 169 function coefficients are defined based on the horizontal and vertical coordinates, as follow: The area of each triangular element in the finite element mesh is also estimated according to In the matrix notation, these equations can be presented as:

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In the finite element limit analysis formulations, one single coordinate may be assigned to the 187 nodes of different elements. In other words, more than one node may share the same coordinate, 188 which means that the unknown stresses are generally discontinuous along adjacent edges of the 189 triangles. In order to satisfy the continuity of stresses throughout the problem domain, the normal 190 and shear nodal stresses on the permitted discontinuities are bound to be equal on the shared 191 planes at the common edges of the adjacent triangular elements; i.e. 1 = 2 , 1 = 2 ; 192 3 = 4 , 3 = 4 (Fig. 1). In the matrix notation, the constraint of discontinuity at each edge of the horizontal ground surface can be expressed as follows: In terms of boundary condition at the contact surface between the foundation and the underlying 205 soil deposit, the magnitude of the mobilized shear stress at the soil-foundation interface should 206 not exceed the shear strength of the interface, i.e. | | ≤ − tan , where δ is the interface 207 friction angle. In the matrix form, this can be presented by the following inequality: 2.1.7. Yield criterion enforcement 210 As mentioned earlier, the soil mass beneath the shallow foundation in this study is considered to 211 be perfectly plastic, following the universal Mohr-Coulomb yield criterion. According to the 212 fundamental concepts of FELA approach, the mobilized stresses at all nodal points within the 213 soil medium should not exceed the Mohr-Coulomb yield function, defined as: In this approach, in order to convert the Mohr-Coulomb yield function into the standard form of adopting the conic quadratic (second-order) constraint ( 3 ), defined as: The matrix form of the corresponding interrelationship equation is expressed as: The three nodal auxiliary variables, i.e. 1 , 2 and 3 , are related to the nonlinear optimization 222 scheme adopted in the current study, which has been elaborated in the following section.
where is the foundation surface area, is the distance from the point of load application, is 232 the point of load application, and and τ are the normal and shear stresses along the footing 233 nodes, respectively. Eqs. (10a) and (10b) could be divided and simplified into the following 234 correlation:

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In the matrix form, the equilibrium equations at each node along the soil-foundation interface are 236 expressed as follow: Moreover, in order to allow segregation of the footing and underlying soil (which may occur due 239 to the developed tensile contact stress under eccentric loading condition), a "tension cut-off" 240 constraint must be applied to the soil-foundation interface nodes for the vertical normal stresses 241 beneath the foundation at each node being compressive (negative) only ( ≤ 0) as:

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In the preliminary formulation of FELA presented by Sloan (1988), the nonlinear form of Mohr- along the soil-footing interface, must be maximized:

Modified pseudo-dynamic seismic loading 264
In the so-called pseudo-dynamic loading, a more realistic dynamic nature of the seismic where for the horizontal seismic acceleration, we have: In addition, for the vertical seismic acceleration, we have: is the primary wave velocity, and is the material damping.

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As an iterative algorithm, the MPD equations are incorporated into the FELA formulations. In formulations integrated with the finite element limit analysis theory is provided in Fig. 2 to have 308 a vision on the current study conducted methods.   constant rate for all seismic intensities and non-dimensional frequencies (Figs. 4a, 4c, 4e). 393 Despite the vertical bearing capacity, the ultimate horizontal load the foundation can sustain 394 without reaching the failure state does not follow a constant increasing or decreasing trend.

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Conversely, for all intensities of the incident wave, the horizontal bearing capacity and its 396 corresponding fluctuation increase with the load inclination up to about 12 o , beyond which they 397 show reduction (Figs. 4b, 4d, 4f). This observation can be attributed to the very fact that the 398 horizontal limit load component is coupled with its vertical counterpart, which undergoes sharp 399 reductions with load inclination. In other words, the horizontal limit load will be a multiplication       (Figs. 10a, 10c, 10e). On the contrary, at a constant seismic intensity, the 588 normalized moment bearing capacity and its corresponding fluctuations increase with the load 589 eccentricity ratio up to about / ratio of 0.2 (to be determined more precisely in the failure 590 envelopes), and decrease thereafter (Figs. 10b, 10d, 10f). Likewise the inclination effect, the 591 developed moment bearing capacity is a synergy of both the eccentricity and the vertical limit 592 load per se; indeed, after a certain eccentricity ratio, let's say 0.2, the vertical limit load decreases 593 substantially, thus, rendering an overall decreasing moment bearing capacity. As a result, it can 594 be generally expressed that the surface footing fails to function efficiently when the load is 595 applied with an eccentricity of more than about 0.4B.