Advances in Civil Engineering

Volume 2018, Article ID 3729360, 18 pages

https://doi.org/10.1155/2018/3729360

## Pseudodynamic Bearing Capacity Analysis of Shallow Strip Footing Using the Advanced Optimization Technique “Hybrid Symbiosis Organisms Search Algorithm” with Numerical Validation

^{1}Department of Civil Engineering, National Institute of Technology Agartala, Barjala, Jirania, 799046 West Tripura, India^{2}Department of Mathematics, National Institute of Technology Agartala, Barjala, Jirania, 799046 West Tripura, India

Correspondence should be addressed to Arijit Saha; moc.liamg@02tijiraahas

Received 26 July 2017; Revised 25 September 2017; Accepted 9 October 2017; Published 5 April 2018

Academic Editor: Moacir Kripka

Copyright © 2018 Arijit Saha et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

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.

#### 1. Introduction

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 [1], Terzaghi [2], Meyerhof [3, 4], Vesic [5, 6], and many more have investigated the mechanisms of bearing capacity of foundation under a static loading condition. Due to seismic loading, foundations may experience a reduction in bearing capacity and an increase in settlement. Two sources of loading must be taken into consideration, initial loading due to lateral forces imposed on superstructure and kinematic loading due to ground movements developed during the earthquake. The pioneering works in determining the seismic bearing capacity of shallow strip footings were done by Budhu and Al-Karni [7], Dormieux and Pecker [8], Soubra [9–11], Richards et al. [12], Choudhury and Subha Rao [13], Kumar and Ghosh [14], and many others using pseudostatic approach with the help of different solution techniques such as method of slices, limit equilibrium, method of stress characteristics, and upper bound limit analysis. Apart from these analytical researchers, Shafiee and Jahanandish [15] and Chakraborty and Kumar [16] used finite element method to estimate the seismic bearing capacity of strip footings on the soil using PLAXIS-2D considering the pseudostatic approach. Since, in the pseudostatic method, the dynamic loading induced by the earthquake is considered as time-independent, which ultimately assumes that the magnitude and phase of acceleration are uniform through the soil layer, pseudodynamic analysis is developed where the effects of both shear and primary waves are considered along with the period of lateral shaking. Ghosh [17] and Saha and Ghosh [18] evaluated pseudodynamic bearing capacity using limit analysis method and limit equilibrium method, respectively, considering the linear failure surface. In the earlier analyses, the resistance of unit weight, surcharge, and cohesion is considered separately. Therefore, if the solution was done for shallow strip footing resting on *c-Φ* soil, there will be three separate coefficients: one for unit weight, another for surcharge, and the other for cohesion. But in a practical situation, there will be a single failure mechanism for the simultaneous resistance of unit weight, surcharge, and cohesion. Thus, an attempt is made to present a single seismic bearing capacity coefficient for the simultaneous resistance of unit weight, surcharge, and cohesion. Here, in this paper, the pseudodynamic bearing capacity of shallow strip footing considering composite failure mechanism resting on *c-Φ* soil is solved using the upper-bound limit analysis method. A relative ease in solving geometrically complex multidimensional problem renders limit analysis, attractive as an alternative to numerical codes. The kinematic method of limit analysis hinges on constructing a velocity field that is admissible for a rigid-perfect plastic material obeying the associative flow rule.

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 [19, 20, 21–28], retaining walls [29–31], and structural design [32]. Cheng and Prayogo [33] introduced a new nature-based optimization technique, called symbiotic organisms search (SOS) algorithm. This technique is based on the interactive relationship among the organism in the ecosystem. It has no algorithm-specific control parameters. The SOS algorithm has been successfully applied to solve different engineering optimization problems [34–38]. Recently, Nama et al. [39] proposed a hybrid algorithm called hybrid symbiotic organisms search (HSOS) algorithm, which is the combination of SOS algorithm and simple quadratic interpolation method [40]. Here, in this paper, HSOS algorithm is used to optimise the pseudodynamic bearing capacity of shallow strip footing considering upper bound limit analysis method. Mathematically, the problem can be represented as a nonlinear hard optimization problem, which can be solved by the HSOS algorithm which is found to be a more satisfactory optimum solution and can be used for designing the shallow strip footing. In the HSOS algorithm, failure surface angle (*α*, *β*) and *t*/*T* are considered as the search variables. So, it can be applied to obtain optimal solutions in the different fields of science and engineering. Numerical analysis is also done using dynamic module of PLAXIS-8.6v software to validate this analytical solution. Results are presented in tabular form including comparison with other available analyses. Effects of a wide range of variation of parameters like soil friction angle (*Φ*), cohesion factor (2*c*/*γB*_{0}), depth factor (*D*_{f}/*B*_{0}), and horizontal and vertical seismic accelerations (*k*_{h}, *k*_{}) on the normalized reduction factor (*N*_{γe}/*N*_{γs}) have been studied.

Therefore, the main contributions of this paper are summarized as follows:(i)Evaluation of pseudodynamic bearing capacity coefficient of shallow strip footing resting on *c-Φ* soil considering composite failure surface using upper bound limit analysis method.(ii)A single pseudodynamic bearing capacity coefficient is presented here considering the simultaneous resistance of unit weight, surcharge, and cohesion.(iii)A recent hybrid optimization algorithm (called HSOS) is used to solve the pseudodynamic bearing capacity minimization optimization problem.(iv)PLAXIS-8.6v software is used to solve this abovementioned problem numerically for the validation of the analytical formulation.(v)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 2 discusses the formulation of the real-world geotechnical earthquake engineering optimization problem such as the pseudodynamic bearing capacity of a shallow foundation. The overview of the optimization algorithm HSOS is presented in Section 3. Section 4 presents discussions of the results obtained by the HSOS algorithm to show the efficiency and accuracy of this hybrid algorithm for solving this engineering optimization problem. Numerical analysis of shallow strip footing using the dynamic module of PLAXIS-8.6v software and the validation of analytical formulation are discussed in Section 5, and finally, Section 6 presents the conclusion and the summary of the outcome of the paper.

#### 2. Formulation of Pseudodynamic Bearing Capacity Coefficient

##### 2.1. Consideration of Model

Let us consider a shallow strip footing of width (*B*_{0}) resting below the ground surface at a depth of *D*_{f} over which a load (*P*) of column acts. The homogeneous soil of effective unit weight *γ* has Mohr–Coulomb characteristic *c-Φ* and can be considered as a rigid plastic body. For shallow foundation (*D* _{f} ≤ *B*_{0}), the overburden pressure is idealized as a surcharge (*q* = *γD*_{f}) which acts over the length of BC. The classical two-dimensional slip line field obtained by Prandtl [1] is the traditional failure mechanism which has three regions such as active zone, passive zone, and logarithmic radial-fan transition zone. In this composite failure mechanism, half of the failure is assumed to occur along the surface AEDC, which is composed of a triangular elastic zone ABE, triangular passive Rankine zone BDC, and in between them a log spiral radial shear zone BDE shown in Figure 1(a) [41]. It is a composite mechanism that is defined by the angular parameters *α* and *β* in which the log-spiral slip surface ED is a tangent to lines AE and DC at E and D, respectively. Figures 2 and 3 show the detailed free body diagram of the elastic zone ABE and composite passive Rankine zone and the log-spiral shear zone BEDC, respectively.