The analysis of shallow foundations subjected to seismic loading has been an important area of research for civil engineers. This paper presents an upper-bound solution for bearing capacity of shallow strip footing considering composite failure mechanisms by the pseudodynamic approach. A recently developed hybrid symbiosis organisms search (HSOS) algorithm has been used to solve this problem. In the HSOS method, the exploration capability of SQI and the exploitation potential of SOS have been combined to increase the robustness of the algorithm. This combination can improve the searching capability of the algorithm for attaining the global optimum. Numerical analysis is also done using dynamic modules of PLAXIS-8.6v for the validation of this analytical solution. The results obtained from the present analysis using HSOS are thoroughly compared with the existing available literature and also with the other optimization techniques. The significance of the present methodology to analyze the bearing capacity is discussed, and the acceptability of HSOS technique is justified to solve such type of engineering problems.
The subject of bearing capacity is one of the important aspects of geotechnical engineering problems. Loads from buildings are transmitted to the foundation by columns or by load-bearing walls of the structures. Many researchers like Prandtl [
Nowadays, nature-based global optimization algorithms such as genetic algorithms (GA), particle swarm optimization (PSO) algorithm, and many other algorithms have been successfully applied to solve different science and engineering complex optimization problems, especially civil engineering problems such as slope stability [
Therefore, the main contributions of this paper are summarized as follows: Evaluation of pseudodynamic bearing capacity coefficient of shallow strip footing resting on A single pseudodynamic bearing capacity coefficient is presented here considering the simultaneous resistance of unit weight, surcharge, and cohesion. A recent hybrid optimization algorithm (called HSOS) is used to solve the pseudodynamic bearing capacity minimization optimization problem. PLAXIS-8.6v software is used to solve this abovementioned problem numerically for the validation of the analytical formulation. The obtained results are compared with the other results which are available in literature and the results obtained by other state-of-the-art algorithms.
The remaining part of the paper is organized as follows: Section
Let us consider a shallow strip footing of width (
Composite failure mechanism [
Elastic wedge.
Log-spiral zone.
At collapse, it is assumed that the footing and the underlying zone ABE moves in phase with each other at the same absolute velocity
Weight of the wedge ABE,
If the base of the wedge is subjected to harmonic horizontal and vertical seismic accelerations of amplitude
The mass of a thin element of the elastic wedge at depth
The total horizontal and vertical inertia forces acting within the elastic zone can be expressed as follows:
Weight of the wedge BCD,
The mass of a thin element of the elastic wedge at depth
The acceleration at any depth
The total horizontal and vertical inertia force acting within the passive Rankine zone can be expressed as follows:
Weight of the log-spiral shear zone BDE,
The log-spiral zone BDE is divided into “
Generalized slice and centre of gravity of the log-spiral zone.
Mass of strip on the
The acceleration at any depth
The total horizontal and vertical inertia force acting within this
Now, the total horizontal and vertical inertia force acting on log-spiral shear zone is expressed as
The incremental external works due to the foundation load
The incremental internal energy dissipation along the velocity discontinuities AE and CD and the radial line DE is
Equating the work expended by the external loads to the power dissipated internally for a kinematically admissible velocity field, we can get the expression of pseudodynamic ultimate bearing capacity of shallow strip footing. The classical ultimate bearing capacity equation of shallow strip footing,
After solving the above equations, the simplified form of the bearing capacity coefficients is as follows:
An attampt is made to present ‘single seismic bearing capacity coefficient’ for simultaneous resistance of unit weight, surcharge and cohesion as in a practical situation, there will be a single failure mechanism for the simultaneous resistance of unit weight, surcharge, and cohesion. So, we get
After simplification of equations, the expression of
Here,
The hybrid symbiosis organisms search (HSOS) algorithm is a recently developed hybrid optimization algorithm which is used to solve this pseudodynamic bearing capacity of shallow strip footing minimal optimization problem.
Symbiosis organisms search (SOS) algorithm is a population-based iterative global optimization algorithm for solving global optimization problems, proposed by Cheng and Prayogo [
In this section, the three-point quadratic interpolation is discussed. Considering the two organisms
The SQI is intended to enhance the entire search capability of the algorithm. Here,
In the development of heuristic global optimization algorithm, the balance of exploration and exploitation capability plays a major role [
Flowchart of the HSOS algorithm.
If an organism is going to an infeasible region, then the organism is reflected back to the feasible region using the following equation [
The algorithmic steps of hsos are given below:
The pseudodynamic bearing capacity coefficient (
Static condition.
|
2 |
|
|||
---|---|---|---|---|---|
0.25 | 0.5 | 0.75 | 1 | ||
20° | 0 | 8.349 | 11.756 | 15.087 | 18.377 |
0.25 | 11.175 | 14.488 | 17.769 | 21.029 | |
0.5 | 13.886 | 17.155 | 20.407 | 23.649 | |
30° | 0 | 30.439 | 40.168 | 49.719 | 59.177 |
0.25 | 35.903 | 45.497 | 54.975 | 64.391 | |
0.5 | 41.263 | 50.771 | 60.189 | 69.557 | |
40° | 0 | 144.24 | 178.37 | 211.78 | 244.89 |
0.25 | 157.48 | 191.17 | 224.5 | 257.5 | |
0.5 | 170.47 | 203.94 | 237.1 | 270.1 |
Pseudodynamic bearing capacity coefficient (
|
|||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|
|
2 |
|
|
|
|||||||||
|
|||||||||||||
0.25 | 0.5 | 0.75 | 1 | 0.25 | 0.5 | 0.75 | 1 | 0.25 | 0.5 | 0.75 | 1 | ||
20° | 0 | 5.881 | 8.559 | 11.172 | 13.753 | 5.882 | 8.538 | 11.128 | 13.687 | 5.878 | 8.51 | 11.079 | 13.614 |
0.25 | 8.589 | 11.17 | 13.73 | 16.277 | 8.69 | 11.247 | 13.782 | 16.303 | 8.797 | 11.323 | 13.832 | 16.329 | |
0.5 | 11.146 | 13.688 | 16.224 | 18.754 | 11.331 | 13.851 | 16.359 | 18.865 | 11.523 | 14.015 | 16.502 | 18.98 | |
|
|||||||||||||
30° | 0 | 21.98 | 29.575 | 37.021 | 44.385 | 22.107 | 29.638 | 37.028 | 44.331 | 22.223 | 29.691 | 37.07 | 44.243 |
0.25 | 26.85 | 34.306 | 41.675 | 48.996 | 27.128 | 34.529 | 41.834 | 49.081 | 27.415 | 34.739 | 41.979 | 49.169 | |
0.5 | 31.59 | 38.965 | 46.287 | 53.564 | 32.016 | 39.325 | 46.573 | 53.793 | 32.447 | 39.688 | 46.879 | 54.028 | |
|
|||||||||||||
40° | 0 | 101.41 | 127.17 | 152.55 | 177.72 | 102.32 | 127.86 | 153.02 | 177.89 | 103.17 | 128.49 | 153.41 | 178.11 |
0.25 | 112.36 | 137.9 | 163.11 | 188.16 | 113.51 | 138.82 | 163.85 | 188.69 | 114.74 | 139.83 | 164.61 | 189.16 | |
0.5 | 123.12 | 148.5 | 173.67 | 198.58 | 124.63 | 149.79 | 174.65 | 199.42 | 126.15 | 151.03 | 175.72 | 200.21 | |
|
|||||||||||||
|
|||||||||||||
|
2 |
|
|
|
|||||||||
|
|||||||||||||
0.25 | 0.5 | 0.75 | 1 | 0.25 | 0.5 | 0.75 | 1 | 0.25 | 0.5 | 0.75 | 1 | ||
|
|||||||||||||
20° | 0 | 3.627 | 5.554 | 7.437 | 9.299 | 3.495 | 5.257 | 7.019 | 8.781 | 3.45 | 5.049 | 6.644 | 8.24 |
0.25 | 6.254 | 8.093 | 9.926 | 11.754 | 6.276 | 8.01 | 9.739 | 11.464 | 6.345 | 7.94 | 9.535 | 11.131 | |
0.5 | 8.672 | 10.493 | 12.312 | 14.129 | 8.889 | 10.605 | 12.321 | 14.036 | 9.201 | 10.801 | 12.394 | 13.987 | |
|
|||||||||||||
30° | 0 | 14.856 | 20.524 | 26.077 | 31.56 | 14.451 | 19.881 | 25.193 | 30.44 | 13.88 | 19.024 | 24.045 | 29.002 |
0.25 | 19.156 | 24.697 | 30.169 | 35.606 | 19.026 | 24.314 | 29.539 | 34.734 | 18.803 | 23.785 | 28.716 | 33.621 | |
0.5 | 23.312 | 28.779 | 34.201 | 39.615 | 23.416 | 28.634 | 33.811 | 38.98 | 23.505 | 28.422 | 33.304 | 38.169 | |
|
|||||||||||||
40° | 0 | 68.468 | 87.465 | 106.13 | 124.61 | 67.336 | 85.521 | 103.4 | 121.14 | 65.74 | 82.948 | 99.846 | 116.59 |
0.25 | 77.438 | 96.227 | 114.79 | 133.22 | 76.783 | 94.766 | 112.51 | 130.11 | 75.728 | 92.708 | 109.51 | 126.1 | |
0.5 | 86.315 | 104.98 | 123.41 | 141.79 | 86.076 | 103.87 | 121.54 | 139.08 | 85.49 | 102.38 | 119.02 | 135.61 | |
|
|||||||||||||
|
|||||||||||||
|
2 |
|
|
|
|||||||||
|
|||||||||||||
0.25 | 0.5 | 0.75 | 1 | 0.25 | 0.5 | 0.75 | 1 | 0.25 | 0.5 | 0.75 | 1 | ||
|
|||||||||||||
20° | 0 | 2.843 | 4.058 | 5.268 | 6.471 | 4.291 | 5.476 | 6.642 | 7.792 | — | — | — | — |
0.25 | 5.178 | 6.389 | 7.592 | 8.794 | 9.036 | 10.181 | 11.325 | 12.468 | — | — | — | — | |
0.5 | 7.51 | 8.713 | 9.915 | 11.118 | 13.713 | 14.848 | 15.984 | 17.119 | — | — | — | — | |
|
|||||||||||||
30° | 0 | 9.3 | 13.312 | 17.24 | 21.118 | 8.356 | 11.737 | 15.116 | 18.496 | 8.15 | 10.853 | 13.557 | 16.261 |
0.25 | 13.109 | 17 | 20.856 | 24.684 | 12.336 | 15.716 | 19.085 | 22.433 | 12.366 | 15.074 | 17.783 | 20.492 | |
0.5 | 16.736 | 20.57 | 24.389 | 28.195 | 16.285 | 19.633 | 22.96 | 26.286 | 16.541 | 19.25 | 21.958 | 24.667 | |
|
|||||||||||||
40° | 0 | 44.498 | 58.106 | 71.465 | 84.714 | 40.605 | 52.714 | 64.622 | 76.375 | 35 | 45.195 | 55.199 | 65.068 |
0.25 | 51.859 | 65.283 | 78.574 | 91.7 | 48.428 | 60.359 | 72.112 | 83.809 | 43.394 | 53.371 | 63.261 | 73.021 | |
0.5 | 59 | 72.396 | 85.56 | 98.686 | 56.083 | 67.85 | 79.556 | 91.165 | 51.543 | 61.416 | 71.164 | 80.912 |
In this section, a brief parametric study and a comparative study have been presented. The effect of soil friction angle (
Figure
The variations of the bearing capacity coefficient with respect to seismic acceleration (
Figure
The variations of the bearing capacity coefficient with respect to seismic acceleration (
Figure
The variations of the bearing capacity coefficient with respect to seismic acceleration (
From Figures
The variations of the bearing capacity coefficient with respect to seismic acceleration (
A detailed comparative study of the present analysis with previous research on similar type of works with different approaches is done here. Figure
The comparison of pseudodynamic bearing capacity coefficient obtained from present analysis with previous seismic analyses with respect to different seismic accelerations (
Comparison of seismic bearing capacity coefficient (
|
Present study | Ghosh [ |
Budhu and Al-Karni [ |
Choudhury and Subba Rao [ |
Soubra [ | |||||
---|---|---|---|---|---|---|---|---|---|---|
|
|
|
|
|
|
|
|
M1 | M2 | |
0.1 | 14.43 | 14.23 | 20.39 | 20.04 | 10.21 | 9.46 | 8.4 | 7.76 | 15.6 | 18.9 |
0.2 | 8.78 | 8.65 | 9.98 | 8.82 | 3.81 | 2.86 | 2.85 | 2 | 8.9 | 10.3 |
0.3 | 5.68 | 5.67 | 3.85 | 2.35 | 1.21 | 0.56 | 0.98 | 0.29 | 4.5 | 4.9 |
The performance results, that is, pseudodynamic bearing capacity coefficients obtained by the HSOS algorithm are compared with other metaheuristic optimization algorithms. Table
Comparison of seismic bearing capacity coefficient (
|
|
DE | PSO | ABC | HS | BSA | ABSA | SOS | HSOS |
---|---|---|---|---|---|---|---|---|---|
(a) 2 |
|||||||||
20° | 0.1 | 11.54 | 11.771 | 11.255 | 11.62 | 11.25 | 11.284 | 11.248 | 11.247 |
0.2 | 8.714 | 8.524 | 8.11 | 8.73 | 8.1 | 8.51 | 8.02 | 8.01 | |
|
|||||||||
30° | 0.1 | 34.681 | 34.854 | 34.535 | 34.942 | 34.53 | 34.591 | 34.53 | 34.529 |
0.2 | 24.514 | 24.641 | 24.319 | 24.43 | 24.319 | 24.361 | 24.32 | 24.314 | |
|
|||||||||
40° | 0.1 | 138.90 | 139.12 | 138.85 | 139.54 | 138.83 | 138.99 | 138.9 | 138.82 |
0.2 | 94.768 | 95.546 | 94.78 | 94.89 | 94.77 | 94.82 | 94.77 | 94.766 | |
|
|||||||||
(b) 2 |
|||||||||
20° | 0.1 | 16.46 | 16.363 | 16.359 | 16.512 | 16.37 | 16.45 | 16.361 | 16.359 |
0.2 | 12.53 | 12.325 | 12.581 | 12.524 | 12.33 | 12.812 | 12.324 | 12.321 | |
|
|||||||||
30° | 0.1 | 46.91 | 46.579 | 46.942 | 46.76 | 46.59 | 46.751 | 46.575 | 46.573 |
0.2 | 33.931 | 33.823 | 34.15 | 33.99 | 33.83 | 33.97 | 33.816 | 33.811 | |
|
|||||||||
40° | 0.1 | 174.76 | 174.68 | 174.691 | 174.93 | 174.67 | 174.81 | 174.69 | 174.65 |
0.2 | 121.69 | 121.59 | 121.61 | 121.59 | 121.58 | 121.561 | 121.58 | 121.54 |
The numerical modeling of dynamic analysis of shallow strip footing is performed using a finite element software, PLAXIS 2D (v-8.6), which is equipped with features to deal with various aspects of complex structures and study the soil-structure interaction effect. In addition to static loads, the dynamic module of PLAXIS also provides a powerful tool for modeling the dynamic response of a soil structure during an earthquake.
A two-dimensional geometrical model is prepared that is to be composed of points, lines, and other components in the
HS small model soil parameters for PLAXIS-8.6v.
Sample |
|
|
|
|
|
|
|
|
|
|
|
---|---|---|---|---|---|---|---|---|---|---|---|
S1 | 20.9 | 0.5 | 32 | 2 | 0.2 | 1.00 |
1.00 |
3.00 |
0.5 | 1.00 |
1.00 |
S2 | 19.9 | 0.2 | 28 | 0 | 0.2 | 1.25 |
1.00 |
3.75 |
0.5 | 1.30 |
1.25 |
Finite element geometry and foundation load along with surcharge load.
During the calculation stage, three steps are adopted where, in the first step, calculations are done for plastic analysis where applied vertical load and weight of soil are activated. In the second step, calculations are made for dynamic analysis where earthquake data are incorporated as SMS file. And, in the final step, FOS is determined by the
El Salvador 2001 earthquake data.
The deformed mesh of model after calculation.
Vertical displacement contour after calculation.
Finite element model of shallow strip footing embedded in
Richards et al.’s [
Terzaghi’s [
The dynamic soil properties taken in numerical modeling [Plaxis-8.6v] are used same in the analytical formulation to validate it. Results obtained from the analytical solution and numerical modeling have been tabulated in Table
Comparison of settlements obtained from numerical and analytical analyses.
Soil samples | Depth factor ( |
Numerical solution | Analytical solution | ||||
---|---|---|---|---|---|---|---|
PLAXIS-8.6v | Richards et al. [ |
Present analysis | Terzaghi [ | ||||
FOS | Settlement (mm) |
|
Settlement (mm) |
|
Settlement (mm) | ||
Sample 1 | 0 | 1.12 | 49.57 | 0.02 | 127 | 41 | 48.97 |
0.25 | 1.95 | 42.82 | 0.14 | 18.27 | 53 | 36.01 | |
0.5 | 2.71 | 41.47 | 0.24 | 10.65 | 68 | 33.07 | |
1 | 3.19 | 40.1 | 0.28 | 9.13 | 75 | 30.61 | |
|
|||||||
Sample 2 | 0 | 1.03 | 47.45 | 0.01 | 255 | 37 | 36.6 |
0.25 | 1.59 | 40.99 | 0.1 | 25.58 | 49 | 31.08 | |
0.5 | 2.14 | 38.22 | 0.18 | 18.27 | 64 | 30.0 | |
1 | 2.61 | 34.47 | 0.25 | 10.23 | 72 | 27.33 |
Using the pseudodynamic approach, the effect of the shear wave and primary wave velocities traveling through the soil layer and the time and phase difference along with the horizontal and vertical seismic accelerations are used to evaluate the seismic bearing capacity of the shallow strip footing. A mathematical formulation is suggested for simultaneous resistance of unit weight, surcharge and cohesion using upper-bound limit analysis method. A composite failure mechanism which includes both planer and log-spiral zone is considered here to develop this mathematical model for the shallow strip footing resting on
Cohesion factor
Width of the footing
Cohesion of soil
Depth of footing below ground surface
Depth factor
Acceleration due to gravity
Shear modulus of soil
Horizontal and vertical seismic accelerations
Bearing capacity coefficients
Optimized single seismic bearing capacity coefficient
Optimized single static bearing capacity coefficient
Normalized reduction factor
Uniformly distributed column load
Surcharge loadings
Initial and final radii of the log-spiral zone (i.e., BE and BD), respectively
Time of vibration
Period of lateral shaking
Absolute and relative velocities, respectively
Primary wave velocity
Shear wave velocity
Base angles of triangular elastic zone under the foundation
Angle that makes the log-spiral part in log-spiral mechanism
Unit weight of soil medium
Lame’s constant
Poisson’s ratio of the soil medium
Angle of internal friction of the soil
Angular frequency
Symbiosis organisms search
Simple quadratic interpolation
Hybrid symbiosis organisms search.
The authors declare that there are no conflicts of interest regarding the publication of this paper.