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

Authors: Arijit Saha; Apu Kumar Saha; Sima Ghosh
Journal: Advances in Civil Engineering
 (2018)

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## 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.


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## Body

## 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 (2c/γB0), depth factor (Df/B0), and horizontal and vertical seismic accelerations (kh, kv) on the normalized reduction factor (Nγe/Nγs) 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 onc-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: Section2 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 (B0) resting below the ground surface at a depth of Df 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 ≤ B0), the overburden pressure is idealized as a surcharge (q = γDf) 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.Figure 1
Composite failure mechanism [41]. (a) Collapse mechanism. (b) Velocity hodograph.
(a)
(b)Figure 2
Elastic wedge.Figure 3
Log-spiral zone.


### 2.2. Collapse Mechanism

At collapse, it is assumed that the footing and the underlying zone ABE moves in phase with each other at the same absolute velocityV1 making an angle Φ with the discontinuity line AE in order to represent the normality condition for an associated flow rule Coulomb material. Hence, there is no dissipation of energy along the soil-structure interface. Whereas the radial log-spiral shearing zone BED is bounded by a log-spiral curve ED. The equation for the curve in polar coordinates (r, θ) is r=r0eθtanϕ. The centre of this log-spiral ED is at point B, and the radius r0 is the length of the line BE, where r0=B0sinα/cosϕ and the width of the footing AB=B0. Note that, in this mechanism, we have assumed that the line AE is a tangent to the log-spiral curve at point E; hence, there is no velocity discontinuity along BE. The radial shear zone BED may be considered to be composed of a sequence of rigid triangles, as in the investigations by Chen, using the symmetrical Hill and Prandtl’s mechanisms. All the small triangles move as rigid bodies in directions which make an angle Φ with the discontinuity line ED. The velocity of each small triangle is determined by the condition that the relative velocity between the triangles in contact has the direction which makes an angle Φ to the contact surface. It has been shown that the velocity V of each triangle is V1=V0eθtanϕ. The log-spiral curve ED is assumed to be tangent to the line DE at D; hence, there is no velocity discontinuity along the line BD. Finally, the triangular wedge BCD is assumed to be rigid, moving with velocity, V2=Vβ=V1eβtanϕ. Therefore, the velocities so determined constitute a kinematically admissible velocity field. Velocity hodograph of this composite failure mechanism is shown in Figure 1(b). Having established the velocity field of the kinematically admissible failure mechanism, the incremental external work done and the incremental internal energy dissipation are calculated following the procedure as mentioned in [42].


### 2.3. Analysis of Bearing Capacity

#### 2.3.1. Elastic Wedge

Weight of the wedge ABE,(1)W=γ2B02sinαsinρcosϕ,where ρ=π/2−α−ϕ.If the base of the wedge is subjected to harmonic horizontal and vertical seismic accelerations of amplitudeahg and avg, respectively, the acceleration at any depth z and time t, below the top of the surface, can be expressed as(2)ahz,t=ahsinωt−r0sinρ−zVs,(3)avz,t=avsinωt−r0sinρ−zVp.The mass of a thin element of the elastic wedge at depthz is(4)mz=γg1tanα+1tanρr0sinρ−zdz.The total horizontal and vertical inertia forces acting within the elastic zone can be expressed as follows:(5)Qh=γkh1tanα+1tanρ∫0r0sinρr0sinρ−zsinωt−r0sinρ−zVsdz=γB02λkhsin2ρsin2α2πr0sinρcos2ϕ1tanα+1tanρcos2πtT−r0sinρλ+r0sinρ2πλsin2πtT−r0sinρλ−sin2πtT=γ2B02qh,(6)Qv=γkv1tanα+1tanρ∫0r0sinρr0sinρ−zsinωt−r0sinρ−zVpdz=γB02ηkvsin2ρsin2α2πr0sinρcos2ϕ1tanα+1tanρcos2πtT−r0sinρη+r0sinρ2πηsin2πtT−r0sinρη−sin2πtT=γ2B02qv.


#### 2.3.2. Passive Rankine Zone

Weight of the wedge BCD,(7)W1=γ2B02e2βtanϕsin2αcosβ−α+ϕcosϕsinξ.The mass of a thin element of the elastic wedge at depthz1 is(8)mz1=γgtanβ+ϕ−α+1tanβ−αr0eβtanϕcosβ+ϕ−α−zdz.The acceleration at any depthz1 and time t, below the top of the surface, can be expressed as(9)ahz1,t=ahsinωt−r0eβtanϕcosβ+ϕ−α−z1Vs,(10)avz1,t=avsinωt−r0eβtanϕcosβ+ϕ−α−z1Vp.The total horizontal and vertical inertia force acting within the passive Rankine zone can be expressed as follows:(11)Qh1=γkhtan⁡β+ϕ-α+1tan⁡β-α∫0r0eβtan⁡ϕcos⁡β+ϕ-αr0eβtan⁡ϕcos⁡β+ϕ-α-zsin⁡ωt-r0eβtan⁡ϕcos⁡β+ϕ-α-z1Vsdz1=γB02λkhe2βtan⁡ϕcos2⁡β+ϕ-αsin2⁡α2πr0eβtan⁡ϕcos⁡β+ϕ-αcos2⁡ϕtan⁡β+ϕ-α+1tan⁡β-αcos⁡2πtT-r0eβtan⁡ϕcos⁡β+ϕ-αλ+r0eβtan⁡ϕcos⁡β+ϕ-α2πλsin⁡2πtT-r0eβtan⁡ϕcos⁡β+ϕ-αλ-sin⁡2πtT=γ2B02qh1,(12)Qv1=γkvtan⁡β+ϕ-α+1tan⁡β-α∫0r0eβtan⁡ϕcos⁡β+ϕ-αr0eβtan⁡ϕcos⁡β+ϕ-α-zsin⁡ωt-r0eβtan⁡ϕcos⁡β+ϕ-α-z1Vpdz1=γB02ηkve2βtan⁡ϕcos2⁡β+ϕ-αsin2⁡α2πr0eβtan⁡ϕcos⁡β+ϕ-αcos2⁡ϕtan⁡β+ϕ-α+1tan⁡β-αcos⁡2πtT-r0eβtan⁡ϕcos⁡β+ϕ-αη+r0eβtan⁡ϕcos⁡β+ϕ-α2πηsin⁡2πtT-r0eβtan⁡ϕcos⁡β+ϕ-αη-sin⁡2πtT=γ2B02qv1.


#### 2.3.3. Log-Spiral Shear Zone

Weight of the log-spiral shear zone BDE,(13)W2=12∫0βr02dθ=r02e2βtanϕ−14tanϕ=B02sin2αe2βtanϕ−12sin2ϕ.The log-spiral zone BDE is divided into “n” number of slices which makes the angle of log-spiral center β into “n” number of dβ angles, that is, β=ndβ, as shown in Figure 4.Figure 4
Generalized slice and centre of gravity of the log-spiral zone.Mass of strip on theith slice of the log-spiral zone BDE,(14)mzi=γg2πdβ360°sin2ρ+i−0.5dβ∫0Hizidzi,where Hi=r0/2ei−1dβtanϕ+eidβtanϕsinρ+i−0.5dβ.The acceleration at any depthzi and time t of any ith slice of the log-spiral shear zone, below the top of the surface, can be expressed as(15)ahz,ti=khgsinωt−Hi−ziVs,(16)avz,ti=kvgsinωt−Hi−ziVp.The total horizontal and vertical inertia force acting within thisith slice can be expressed as follows:(17)Qhi=2πγdβkh360°sin2ρ+i−0.5dβ∫0Hizisinωt−Hi−ziVsdzi=γdβkhλB02sin2αei−1dβtanϕ+eidβtanϕ2360°∗4Hicos2ϕλHi2πsin2πtT−sin2πtT−Hiλ−cos2πtT=γ2B02qh2ei−1dβtanϕ+eidβtanϕ2,(18)Qvi=2πγdβkv360°sin2ρ+i−0.5dβ∫0Hizisinωt−Hi−ziVpdzi=γdβkvηB02sin2αei−1dβtanϕ+eidβtanϕ2360°∗4Hicos2ϕηHi2πsin2πtT−sin2πtT−Hiη−cos2πtT=γ2B02qv2ei−1dβtanϕ+eidβtanϕ2.Now, the total horizontal and vertical inertia force acting on log-spiral shear zone is expressed as(19)Qh2=∑i=1nQhi=γ2B02qh2∑i=1nei−1dβtanϕ+eidβtanϕ2,(20)Qv2=∑i=1nQvi=γ2B02qv2∑i=1nei−1dβtanϕ+eidβtanϕ2.The incremental external works due to the foundation loadP, surcharge load q, the weight of the soil wedges ABE, BCD, and BDE, and their corresponding inertial forces are(21)WP=PLB0V1sin⁡α-ϕ+khsin⁡2πtT-Hλcos⁡α-ϕ-kvsin⁡2πtT-Hηsin⁡α-ϕ,(22)WABE=γ4B02sin⁡αsin⁡2ρcos⁡ϕV1,(23)WBCD=-γ4B02e3βtan⁡ϕsin2⁡αsin⁡2β-α+ϕcos⁡ϕsin⁡ξV1,(24)WQ=-γDfB0sin⁡αe2βtan⁡ϕsin⁡ξ-khsin⁡2πtT-Hλcos⁡β-α+ϕ+1-kvsin⁡2πtT-Hηsin⁡β-α+ϕV1,(25)WBDE=-γ2∫0βr02dθVθsin⁡θ+ϕ-α=-γ2B02sin2⁡αcos2⁡ϕ1+9tan2⁡ϕ3tan⁡ϕe3βtan⁡ϕsin⁡β+ϕ-α-sin⁡ϕ-α-e3βtan⁡ϕcos⁡β+ϕ-α+cos⁡ϕ-αV1,(26)WQh=QhV1cos⁡α-ϕ,WQv=-QvV1sin⁡α-ϕ,(27)WQh1=Qh1V1eβtan⁡ϕcos⁡β-α+ϕ,WQv1=Qv1V1eβtan⁡ϕsin⁡β-α+ϕ,(28)WQh2=γ2B02qh2∑i=1nei-1dβtan⁡ϕ+eidβtan⁡ϕ2∫i-1dβidβVθcos⁡θ+ϕ-αdθ=γ2B02qh2V11+tan⁡ϕ∑i=nnei-1dβtan⁡ϕ+eidβtan⁡ϕ2eidβtan⁡ϕsin⁡idβ+ϕ-α+tan⁡ϕcos⁡idβ+ϕ-α-ei-1dβtan⁡ϕsin⁡i-1dβ+ϕ-α+tan⁡ϕcos⁡i-1dβ+ϕ-α,(29)WQv2=γ2B02qv2∑i=1nei-1dβtan⁡ϕ+eidβtan⁡ϕ2∫i-1dβidβVθsin⁡θ+ϕ-αdθ=γ2B02qv2V11+tan2⁡ϕ∑i=1nei-1dβtan⁡ϕ+eidβtan⁡ϕ2eidβtan⁡ϕ-cos⁡idβ+ϕ-α+tan⁡ϕsin⁡idβ+ϕ-α-ei-1dβtan⁡ϕ-cos⁡i-1dβ+ϕ-α+tan⁡ϕsin⁡i-1dβ+ϕ-α.The incremental internal energy dissipation along the velocity discontinuities AE and CD and the radial line DE is(30)DAE=cACV1cosϕ=cB0sinρV1,(31)DCD=cCDV2cosϕ=cB0e2βtanϕsinαcosβ−α+ϕsinβ−αV1,(32)DDE=c∫0βr0dθVθcosϕ=cB0sinαe2βtanϕ−12tanϕV1.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,(33)PL=Qult=cNc+γDfNq+0.5γB0Nγ.After solving the above equations, the simplified form of the bearing capacity coefficients is as follows:(34)Nc=sin⁡α2tan⁡ϕe2βtan⁡ϕ-1+cos⁡α-ϕ+e2βtan⁡ϕsin⁡αcos⁡β-α+ϕsin⁡β-α,(35)Nq=sin⁡αe2βtan⁡ϕsin⁡ξ-khsin⁡2πtT-Hλcos⁡β-α+ϕ+1-kvsin⁡2πtT-Hηsin⁡β-α+ϕ,(36)Nγ=sin2⁡αcos2⁡ϕ1+9tan2⁡ϕ3tan⁡ϕe3βtan⁡ϕsin⁡β+ϕ-α-sin⁡ϕ-α-e3βtan⁡ϕcos⁡β+ϕ-α+cos⁡ϕ-α-sin⁡αsin⁡2ρ2cos⁡ϕ+e3βtan⁡ϕsin2⁡αsin⁡2β-α+ϕ2cos⁡ϕsin⁡ξ-qhcos⁡α-ϕ+qvsin⁡α-ϕ-qh1eβtan⁡ϕcos⁡β+ϕ-α-qv1eβtan⁡ϕsin⁡β+ϕ-α-∑i=1nei-1dβtan⁡ϕ+eidβtan⁡ϕ21+tan2⁡ϕqh2∑i=1neidβtan⁡ϕsin⁡idβ+ϕ-α+tan⁡ϕcos⁡idβ+ϕ-α-ei-1dβtan⁡ϕsin⁡i-1dβ+ϕ-α+tan⁡ϕcos⁡i-1dβ+ϕ-α+qv2∑i=1neidβtan⁡ϕsin⁡idβ+ϕ-α+tan⁡ϕcos⁡idβ+ϕ-α-ei-1dβtan⁡ϕsin⁡i-1dβ+ϕ-α+tan⁡ϕcos⁡i-1dβ+ϕ-α.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(37)PL=γ2B0N.After simplification of equations, the expression ofN is given below.(38)N=sin2⁡αcos2⁡ϕ1+9tan2⁡ϕ3tan⁡ϕe3βtan⁡ϕsin⁡β+ϕ-α-sin⁡ϕ-α-e3βtan⁡ϕcos⁡β+ϕ-α+cos⁡ϕ-α+e3βtan⁡ϕsin2⁡αsin⁡2β-α+ϕ2cos⁡ϕsin⁡ξ-sin⁡αsin⁡2ρ2cos⁡ϕ+2DfB0sin⁡αe2βtan⁡ϕsin⁡ξ-khsin⁡2πtT-Hλcos⁡β-α+ϕ+1-kvsin⁡2πtT-Hηsin⁡β-α+ϕ-qhcos⁡α-ϕ+qvsin⁡α-ϕ-qh1eβtan⁡ϕcos⁡β+ϕ-α-qv1eβtan⁡ϕsin⁡β+ϕ-α-∑i=1nei-1dβtan⁡ϕ+eidβtan⁡ϕ21+tan2⁡ϕqh2∑i=1neidβtan⁡ϕsin⁡idβ+ϕ-α+tan⁡ϕcos⁡idβ+ϕ-α-ei-1dβtan⁡ϕsin⁡i-1dβ+ϕ-α+tan⁡ϕcos⁡i-1dβ+ϕ-α+qv2∑i=1neidβtan⁡ϕsin⁡idβ+ϕ-α+tan⁡ϕcos⁡idβ+ϕ-α-ei-1dβtan⁡ϕsin⁡i-1dβ+ϕ-α+tan⁡ϕcos⁡i-1dβ+ϕ-α+2cγB0sin⁡α2tan⁡ϕe2βtan⁡ϕ-1+cos⁡α-ϕ+e2βtan⁡ϕsin⁡αcos⁡β-α+ϕsin⁡β-α.Here,N is the single pseudodynamic bearing capacity coefficient of shallow strip footing under seismic loading condition. In this formulation, the objective function pseudodynamic bearing capacity coefficient depends on these Φ, c, α, β, t/T, kh, kv, H/λ, and H/η functions. For a particular soil and seismic condition, all other terms are constant except α, β, and t/T. So, optimization of pseudodynamic bearing capacity coefficient is done with respect to α, β, and t/T using the HSOS algorithm. The advantage of this HSOS algorithm is that it can improve the searching capability of the algorithm for attaining the global optimization. Here, the optimum value of N is represented as Nγe.Now, pseudodynamic ultimate bearing capacity,(39)Qult=0.5γB0Nγe.


## 2.1. Consideration of Model

Let us consider a shallow strip footing of width (B0) resting below the ground surface at a depth of Df 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 ≤ B0), the overburden pressure is idealized as a surcharge (q = γDf) 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.Figure 1
Composite failure mechanism [41]. (a) Collapse mechanism. (b) Velocity hodograph.
(a)
(b)Figure 2
Elastic wedge.Figure 3
Log-spiral zone.


## 2.2. Collapse Mechanism

At collapse, it is assumed that the footing and the underlying zone ABE moves in phase with each other at the same absolute velocityV1 making an angle Φ with the discontinuity line AE in order to represent the normality condition for an associated flow rule Coulomb material. Hence, there is no dissipation of energy along the soil-structure interface. Whereas the radial log-spiral shearing zone BED is bounded by a log-spiral curve ED. The equation for the curve in polar coordinates (r, θ) is r=r0eθtanϕ. The centre of this log-spiral ED is at point B, and the radius r0 is the length of the line BE, where r0=B0sinα/cosϕ and the width of the footing AB=B0. Note that, in this mechanism, we have assumed that the line AE is a tangent to the log-spiral curve at point E; hence, there is no velocity discontinuity along BE. The radial shear zone BED may be considered to be composed of a sequence of rigid triangles, as in the investigations by Chen, using the symmetrical Hill and Prandtl’s mechanisms. All the small triangles move as rigid bodies in directions which make an angle Φ with the discontinuity line ED. The velocity of each small triangle is determined by the condition that the relative velocity between the triangles in contact has the direction which makes an angle Φ to the contact surface. It has been shown that the velocity V of each triangle is V1=V0eθtanϕ. The log-spiral curve ED is assumed to be tangent to the line DE at D; hence, there is no velocity discontinuity along the line BD. Finally, the triangular wedge BCD is assumed to be rigid, moving with velocity, V2=Vβ=V1eβtanϕ. Therefore, the velocities so determined constitute a kinematically admissible velocity field. Velocity hodograph of this composite failure mechanism is shown in Figure 1(b). Having established the velocity field of the kinematically admissible failure mechanism, the incremental external work done and the incremental internal energy dissipation are calculated following the procedure as mentioned in [42].


## 2.3. Analysis of Bearing Capacity

### 2.3.1. Elastic Wedge

Weight of the wedge ABE,(1)W=γ2B02sinαsinρcosϕ,where ρ=π/2−α−ϕ.If the base of the wedge is subjected to harmonic horizontal and vertical seismic accelerations of amplitudeahg and avg, respectively, the acceleration at any depth z and time t, below the top of the surface, can be expressed as(2)ahz,t=ahsinωt−r0sinρ−zVs,(3)avz,t=avsinωt−r0sinρ−zVp.The mass of a thin element of the elastic wedge at depthz is(4)mz=γg1tanα+1tanρr0sinρ−zdz.The total horizontal and vertical inertia forces acting within the elastic zone can be expressed as follows:(5)Qh=γkh1tanα+1tanρ∫0r0sinρr0sinρ−zsinωt−r0sinρ−zVsdz=γB02λkhsin2ρsin2α2πr0sinρcos2ϕ1tanα+1tanρcos2πtT−r0sinρλ+r0sinρ2πλsin2πtT−r0sinρλ−sin2πtT=γ2B02qh,(6)Qv=γkv1tanα+1tanρ∫0r0sinρr0sinρ−zsinωt−r0sinρ−zVpdz=γB02ηkvsin2ρsin2α2πr0sinρcos2ϕ1tanα+1tanρcos2πtT−r0sinρη+r0sinρ2πηsin2πtT−r0sinρη−sin2πtT=γ2B02qv.


### 2.3.2. Passive Rankine Zone

Weight of the wedge BCD,(7)W1=γ2B02e2βtanϕsin2αcosβ−α+ϕcosϕsinξ.The mass of a thin element of the elastic wedge at depthz1 is(8)mz1=γgtanβ+ϕ−α+1tanβ−αr0eβtanϕcosβ+ϕ−α−zdz.The acceleration at any depthz1 and time t, below the top of the surface, can be expressed as(9)ahz1,t=ahsinωt−r0eβtanϕcosβ+ϕ−α−z1Vs,(10)avz1,t=avsinωt−r0eβtanϕcosβ+ϕ−α−z1Vp.The total horizontal and vertical inertia force acting within the passive Rankine zone can be expressed as follows:(11)Qh1=γkhtan⁡β+ϕ-α+1tan⁡β-α∫0r0eβtan⁡ϕcos⁡β+ϕ-αr0eβtan⁡ϕcos⁡β+ϕ-α-zsin⁡ωt-r0eβtan⁡ϕcos⁡β+ϕ-α-z1Vsdz1=γB02λkhe2βtan⁡ϕcos2⁡β+ϕ-αsin2⁡α2πr0eβtan⁡ϕcos⁡β+ϕ-αcos2⁡ϕtan⁡β+ϕ-α+1tan⁡β-αcos⁡2πtT-r0eβtan⁡ϕcos⁡β+ϕ-αλ+r0eβtan⁡ϕcos⁡β+ϕ-α2πλsin⁡2πtT-r0eβtan⁡ϕcos⁡β+ϕ-αλ-sin⁡2πtT=γ2B02qh1,(12)Qv1=γkvtan⁡β+ϕ-α+1tan⁡β-α∫0r0eβtan⁡ϕcos⁡β+ϕ-αr0eβtan⁡ϕcos⁡β+ϕ-α-zsin⁡ωt-r0eβtan⁡ϕcos⁡β+ϕ-α-z1Vpdz1=γB02ηkve2βtan⁡ϕcos2⁡β+ϕ-αsin2⁡α2πr0eβtan⁡ϕcos⁡β+ϕ-αcos2⁡ϕtan⁡β+ϕ-α+1tan⁡β-αcos⁡2πtT-r0eβtan⁡ϕcos⁡β+ϕ-αη+r0eβtan⁡ϕcos⁡β+ϕ-α2πηsin⁡2πtT-r0eβtan⁡ϕcos⁡β+ϕ-αη-sin⁡2πtT=γ2B02qv1.


### 2.3.3. Log-Spiral Shear Zone

Weight of the log-spiral shear zone BDE,(13)W2=12∫0βr02dθ=r02e2βtanϕ−14tanϕ=B02sin2αe2βtanϕ−12sin2ϕ.The log-spiral zone BDE is divided into “n” number of slices which makes the angle of log-spiral center β into “n” number of dβ angles, that is, β=ndβ, as shown in Figure 4.Figure 4
Generalized slice and centre of gravity of the log-spiral zone.Mass of strip on theith slice of the log-spiral zone BDE,(14)mzi=γg2πdβ360°sin2ρ+i−0.5dβ∫0Hizidzi,where Hi=r0/2ei−1dβtanϕ+eidβtanϕsinρ+i−0.5dβ.The acceleration at any depthzi and time t of any ith slice of the log-spiral shear zone, below the top of the surface, can be expressed as(15)ahz,ti=khgsinωt−Hi−ziVs,(16)avz,ti=kvgsinωt−Hi−ziVp.The total horizontal and vertical inertia force acting within thisith slice can be expressed as follows:(17)Qhi=2πγdβkh360°sin2ρ+i−0.5dβ∫0Hizisinωt−Hi−ziVsdzi=γdβkhλB02sin2αei−1dβtanϕ+eidβtanϕ2360°∗4Hicos2ϕλHi2πsin2πtT−sin2πtT−Hiλ−cos2πtT=γ2B02qh2ei−1dβtanϕ+eidβtanϕ2,(18)Qvi=2πγdβkv360°sin2ρ+i−0.5dβ∫0Hizisinωt−Hi−ziVpdzi=γdβkvηB02sin2αei−1dβtanϕ+eidβtanϕ2360°∗4Hicos2ϕηHi2πsin2πtT−sin2πtT−Hiη−cos2πtT=γ2B02qv2ei−1dβtanϕ+eidβtanϕ2.Now, the total horizontal and vertical inertia force acting on log-spiral shear zone is expressed as(19)Qh2=∑i=1nQhi=γ2B02qh2∑i=1nei−1dβtanϕ+eidβtanϕ2,(20)Qv2=∑i=1nQvi=γ2B02qv2∑i=1nei−1dβtanϕ+eidβtanϕ2.The incremental external works due to the foundation loadP, surcharge load q, the weight of the soil wedges ABE, BCD, and BDE, and their corresponding inertial forces are(21)WP=PLB0V1sin⁡α-ϕ+khsin⁡2πtT-Hλcos⁡α-ϕ-kvsin⁡2πtT-Hηsin⁡α-ϕ,(22)WABE=γ4B02sin⁡αsin⁡2ρcos⁡ϕV1,(23)WBCD=-γ4B02e3βtan⁡ϕsin2⁡αsin⁡2β-α+ϕcos⁡ϕsin⁡ξV1,(24)WQ=-γDfB0sin⁡αe2βtan⁡ϕsin⁡ξ-khsin⁡2πtT-Hλcos⁡β-α+ϕ+1-kvsin⁡2πtT-Hηsin⁡β-α+ϕV1,(25)WBDE=-γ2∫0βr02dθVθsin⁡θ+ϕ-α=-γ2B02sin2⁡αcos2⁡ϕ1+9tan2⁡ϕ3tan⁡ϕe3βtan⁡ϕsin⁡β+ϕ-α-sin⁡ϕ-α-e3βtan⁡ϕcos⁡β+ϕ-α+cos⁡ϕ-αV1,(26)WQh=QhV1cos⁡α-ϕ,WQv=-QvV1sin⁡α-ϕ,(27)WQh1=Qh1V1eβtan⁡ϕcos⁡β-α+ϕ,WQv1=Qv1V1eβtan⁡ϕsin⁡β-α+ϕ,(28)WQh2=γ2B02qh2∑i=1nei-1dβtan⁡ϕ+eidβtan⁡ϕ2∫i-1dβidβVθcos⁡θ+ϕ-αdθ=γ2B02qh2V11+tan⁡ϕ∑i=nnei-1dβtan⁡ϕ+eidβtan⁡ϕ2eidβtan⁡ϕsin⁡idβ+ϕ-α+tan⁡ϕcos⁡idβ+ϕ-α-ei-1dβtan⁡ϕsin⁡i-1dβ+ϕ-α+tan⁡ϕcos⁡i-1dβ+ϕ-α,(29)WQv2=γ2B02qv2∑i=1nei-1dβtan⁡ϕ+eidβtan⁡ϕ2∫i-1dβidβVθsin⁡θ+ϕ-αdθ=γ2B02qv2V11+tan2⁡ϕ∑i=1nei-1dβtan⁡ϕ+eidβtan⁡ϕ2eidβtan⁡ϕ-cos⁡idβ+ϕ-α+tan⁡ϕsin⁡idβ+ϕ-α-ei-1dβtan⁡ϕ-cos⁡i-1dβ+ϕ-α+tan⁡ϕsin⁡i-1dβ+ϕ-α.The incremental internal energy dissipation along the velocity discontinuities AE and CD and the radial line DE is(30)DAE=cACV1cosϕ=cB0sinρV1,(31)DCD=cCDV2cosϕ=cB0e2βtanϕsinαcosβ−α+ϕsinβ−αV1,(32)DDE=c∫0βr0dθVθcosϕ=cB0sinαe2βtanϕ−12tanϕV1.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,(33)PL=Qult=cNc+γDfNq+0.5γB0Nγ.After solving the above equations, the simplified form of the bearing capacity coefficients is as follows:(34)Nc=sin⁡α2tan⁡ϕe2βtan⁡ϕ-1+cos⁡α-ϕ+e2βtan⁡ϕsin⁡αcos⁡β-α+ϕsin⁡β-α,(35)Nq=sin⁡αe2βtan⁡ϕsin⁡ξ-khsin⁡2πtT-Hλcos⁡β-α+ϕ+1-kvsin⁡2πtT-Hηsin⁡β-α+ϕ,(36)Nγ=sin2⁡αcos2⁡ϕ1+9tan2⁡ϕ3tan⁡ϕe3βtan⁡ϕsin⁡β+ϕ-α-sin⁡ϕ-α-e3βtan⁡ϕcos⁡β+ϕ-α+cos⁡ϕ-α-sin⁡αsin⁡2ρ2cos⁡ϕ+e3βtan⁡ϕsin2⁡αsin⁡2β-α+ϕ2cos⁡ϕsin⁡ξ-qhcos⁡α-ϕ+qvsin⁡α-ϕ-qh1eβtan⁡ϕcos⁡β+ϕ-α-qv1eβtan⁡ϕsin⁡β+ϕ-α-∑i=1nei-1dβtan⁡ϕ+eidβtan⁡ϕ21+tan2⁡ϕqh2∑i=1neidβtan⁡ϕsin⁡idβ+ϕ-α+tan⁡ϕcos⁡idβ+ϕ-α-ei-1dβtan⁡ϕsin⁡i-1dβ+ϕ-α+tan⁡ϕcos⁡i-1dβ+ϕ-α+qv2∑i=1neidβtan⁡ϕsin⁡idβ+ϕ-α+tan⁡ϕcos⁡idβ+ϕ-α-ei-1dβtan⁡ϕsin⁡i-1dβ+ϕ-α+tan⁡ϕcos⁡i-1dβ+ϕ-α.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(37)PL=γ2B0N.After simplification of equations, the expression ofN is given below.(38)N=sin2⁡αcos2⁡ϕ1+9tan2⁡ϕ3tan⁡ϕe3βtan⁡ϕsin⁡β+ϕ-α-sin⁡ϕ-α-e3βtan⁡ϕcos⁡β+ϕ-α+cos⁡ϕ-α+e3βtan⁡ϕsin2⁡αsin⁡2β-α+ϕ2cos⁡ϕsin⁡ξ-sin⁡αsin⁡2ρ2cos⁡ϕ+2DfB0sin⁡αe2βtan⁡ϕsin⁡ξ-khsin⁡2πtT-Hλcos⁡β-α+ϕ+1-kvsin⁡2πtT-Hηsin⁡β-α+ϕ-qhcos⁡α-ϕ+qvsin⁡α-ϕ-qh1eβtan⁡ϕcos⁡β+ϕ-α-qv1eβtan⁡ϕsin⁡β+ϕ-α-∑i=1nei-1dβtan⁡ϕ+eidβtan⁡ϕ21+tan2⁡ϕqh2∑i=1neidβtan⁡ϕsin⁡idβ+ϕ-α+tan⁡ϕcos⁡idβ+ϕ-α-ei-1dβtan⁡ϕsin⁡i-1dβ+ϕ-α+tan⁡ϕcos⁡i-1dβ+ϕ-α+qv2∑i=1neidβtan⁡ϕsin⁡idβ+ϕ-α+tan⁡ϕcos⁡idβ+ϕ-α-ei-1dβtan⁡ϕsin⁡i-1dβ+ϕ-α+tan⁡ϕcos⁡i-1dβ+ϕ-α+2cγB0sin⁡α2tan⁡ϕe2βtan⁡ϕ-1+cos⁡α-ϕ+e2βtan⁡ϕsin⁡αcos⁡β-α+ϕsin⁡β-α.Here,N is the single pseudodynamic bearing capacity coefficient of shallow strip footing under seismic loading condition. In this formulation, the objective function pseudodynamic bearing capacity coefficient depends on these Φ, c, α, β, t/T, kh, kv, H/λ, and H/η functions. For a particular soil and seismic condition, all other terms are constant except α, β, and t/T. So, optimization of pseudodynamic bearing capacity coefficient is done with respect to α, β, and t/T using the HSOS algorithm. The advantage of this HSOS algorithm is that it can improve the searching capability of the algorithm for attaining the global optimization. Here, the optimum value of N is represented as Nγe.Now, pseudodynamic ultimate bearing capacity,(39)Qult=0.5γB0Nγe.


## 2.3.1. Elastic Wedge

Weight of the wedge ABE,(1)W=γ2B02sinαsinρcosϕ,where ρ=π/2−α−ϕ.If the base of the wedge is subjected to harmonic horizontal and vertical seismic accelerations of amplitudeahg and avg, respectively, the acceleration at any depth z and time t, below the top of the surface, can be expressed as(2)ahz,t=ahsinωt−r0sinρ−zVs,(3)avz,t=avsinωt−r0sinρ−zVp.The mass of a thin element of the elastic wedge at depthz is(4)mz=γg1tanα+1tanρr0sinρ−zdz.The total horizontal and vertical inertia forces acting within the elastic zone can be expressed as follows:(5)Qh=γkh1tanα+1tanρ∫0r0sinρr0sinρ−zsinωt−r0sinρ−zVsdz=γB02λkhsin2ρsin2α2πr0sinρcos2ϕ1tanα+1tanρcos2πtT−r0sinρλ+r0sinρ2πλsin2πtT−r0sinρλ−sin2πtT=γ2B02qh,(6)Qv=γkv1tanα+1tanρ∫0r0sinρr0sinρ−zsinωt−r0sinρ−zVpdz=γB02ηkvsin2ρsin2α2πr0sinρcos2ϕ1tanα+1tanρcos2πtT−r0sinρη+r0sinρ2πηsin2πtT−r0sinρη−sin2πtT=γ2B02qv.


## 2.3.2. Passive Rankine Zone

Weight of the wedge BCD,(7)W1=γ2B02e2βtanϕsin2αcosβ−α+ϕcosϕsinξ.The mass of a thin element of the elastic wedge at depthz1 is(8)mz1=γgtanβ+ϕ−α+1tanβ−αr0eβtanϕcosβ+ϕ−α−zdz.The acceleration at any depthz1 and time t, below the top of the surface, can be expressed as(9)ahz1,t=ahsinωt−r0eβtanϕcosβ+ϕ−α−z1Vs,(10)avz1,t=avsinωt−r0eβtanϕcosβ+ϕ−α−z1Vp.The total horizontal and vertical inertia force acting within the passive Rankine zone can be expressed as follows:(11)Qh1=γkhtan⁡β+ϕ-α+1tan⁡β-α∫0r0eβtan⁡ϕcos⁡β+ϕ-αr0eβtan⁡ϕcos⁡β+ϕ-α-zsin⁡ωt-r0eβtan⁡ϕcos⁡β+ϕ-α-z1Vsdz1=γB02λkhe2βtan⁡ϕcos2⁡β+ϕ-αsin2⁡α2πr0eβtan⁡ϕcos⁡β+ϕ-αcos2⁡ϕtan⁡β+ϕ-α+1tan⁡β-αcos⁡2πtT-r0eβtan⁡ϕcos⁡β+ϕ-αλ+r0eβtan⁡ϕcos⁡β+ϕ-α2πλsin⁡2πtT-r0eβtan⁡ϕcos⁡β+ϕ-αλ-sin⁡2πtT=γ2B02qh1,(12)Qv1=γkvtan⁡β+ϕ-α+1tan⁡β-α∫0r0eβtan⁡ϕcos⁡β+ϕ-αr0eβtan⁡ϕcos⁡β+ϕ-α-zsin⁡ωt-r0eβtan⁡ϕcos⁡β+ϕ-α-z1Vpdz1=γB02ηkve2βtan⁡ϕcos2⁡β+ϕ-αsin2⁡α2πr0eβtan⁡ϕcos⁡β+ϕ-αcos2⁡ϕtan⁡β+ϕ-α+1tan⁡β-αcos⁡2πtT-r0eβtan⁡ϕcos⁡β+ϕ-αη+r0eβtan⁡ϕcos⁡β+ϕ-α2πηsin⁡2πtT-r0eβtan⁡ϕcos⁡β+ϕ-αη-sin⁡2πtT=γ2B02qv1.


## 2.3.3. Log-Spiral Shear Zone

Weight of the log-spiral shear zone BDE,(13)W2=12∫0βr02dθ=r02e2βtanϕ−14tanϕ=B02sin2αe2βtanϕ−12sin2ϕ.The log-spiral zone BDE is divided into “n” number of slices which makes the angle of log-spiral center β into “n” number of dβ angles, that is, β=ndβ, as shown in Figure 4.Figure 4
Generalized slice and centre of gravity of the log-spiral zone.Mass of strip on theith slice of the log-spiral zone BDE,(14)mzi=γg2πdβ360°sin2ρ+i−0.5dβ∫0Hizidzi,where Hi=r0/2ei−1dβtanϕ+eidβtanϕsinρ+i−0.5dβ.The acceleration at any depthzi and time t of any ith slice of the log-spiral shear zone, below the top of the surface, can be expressed as(15)ahz,ti=khgsinωt−Hi−ziVs,(16)avz,ti=kvgsinωt−Hi−ziVp.The total horizontal and vertical inertia force acting within thisith slice can be expressed as follows:(17)Qhi=2πγdβkh360°sin2ρ+i−0.5dβ∫0Hizisinωt−Hi−ziVsdzi=γdβkhλB02sin2αei−1dβtanϕ+eidβtanϕ2360°∗4Hicos2ϕλHi2πsin2πtT−sin2πtT−Hiλ−cos2πtT=γ2B02qh2ei−1dβtanϕ+eidβtanϕ2,(18)Qvi=2πγdβkv360°sin2ρ+i−0.5dβ∫0Hizisinωt−Hi−ziVpdzi=γdβkvηB02sin2αei−1dβtanϕ+eidβtanϕ2360°∗4Hicos2ϕηHi2πsin2πtT−sin2πtT−Hiη−cos2πtT=γ2B02qv2ei−1dβtanϕ+eidβtanϕ2.Now, the total horizontal and vertical inertia force acting on log-spiral shear zone is expressed as(19)Qh2=∑i=1nQhi=γ2B02qh2∑i=1nei−1dβtanϕ+eidβtanϕ2,(20)Qv2=∑i=1nQvi=γ2B02qv2∑i=1nei−1dβtanϕ+eidβtanϕ2.The incremental external works due to the foundation loadP, surcharge load q, the weight of the soil wedges ABE, BCD, and BDE, and their corresponding inertial forces are(21)WP=PLB0V1sin⁡α-ϕ+khsin⁡2πtT-Hλcos⁡α-ϕ-kvsin⁡2πtT-Hηsin⁡α-ϕ,(22)WABE=γ4B02sin⁡αsin⁡2ρcos⁡ϕV1,(23)WBCD=-γ4B02e3βtan⁡ϕsin2⁡αsin⁡2β-α+ϕcos⁡ϕsin⁡ξV1,(24)WQ=-γDfB0sin⁡αe2βtan⁡ϕsin⁡ξ-khsin⁡2πtT-Hλcos⁡β-α+ϕ+1-kvsin⁡2πtT-Hηsin⁡β-α+ϕV1,(25)WBDE=-γ2∫0βr02dθVθsin⁡θ+ϕ-α=-γ2B02sin2⁡αcos2⁡ϕ1+9tan2⁡ϕ3tan⁡ϕe3βtan⁡ϕsin⁡β+ϕ-α-sin⁡ϕ-α-e3βtan⁡ϕcos⁡β+ϕ-α+cos⁡ϕ-αV1,(26)WQh=QhV1cos⁡α-ϕ,WQv=-QvV1sin⁡α-ϕ,(27)WQh1=Qh1V1eβtan⁡ϕcos⁡β-α+ϕ,WQv1=Qv1V1eβtan⁡ϕsin⁡β-α+ϕ,(28)WQh2=γ2B02qh2∑i=1nei-1dβtan⁡ϕ+eidβtan⁡ϕ2∫i-1dβidβVθcos⁡θ+ϕ-αdθ=γ2B02qh2V11+tan⁡ϕ∑i=nnei-1dβtan⁡ϕ+eidβtan⁡ϕ2eidβtan⁡ϕsin⁡idβ+ϕ-α+tan⁡ϕcos⁡idβ+ϕ-α-ei-1dβtan⁡ϕsin⁡i-1dβ+ϕ-α+tan⁡ϕcos⁡i-1dβ+ϕ-α,(29)WQv2=γ2B02qv2∑i=1nei-1dβtan⁡ϕ+eidβtan⁡ϕ2∫i-1dβidβVθsin⁡θ+ϕ-αdθ=γ2B02qv2V11+tan2⁡ϕ∑i=1nei-1dβtan⁡ϕ+eidβtan⁡ϕ2eidβtan⁡ϕ-cos⁡idβ+ϕ-α+tan⁡ϕsin⁡idβ+ϕ-α-ei-1dβtan⁡ϕ-cos⁡i-1dβ+ϕ-α+tan⁡ϕsin⁡i-1dβ+ϕ-α.The incremental internal energy dissipation along the velocity discontinuities AE and CD and the radial line DE is(30)DAE=cACV1cosϕ=cB0sinρV1,(31)DCD=cCDV2cosϕ=cB0e2βtanϕsinαcosβ−α+ϕsinβ−αV1,(32)DDE=c∫0βr0dθVθcosϕ=cB0sinαe2βtanϕ−12tanϕV1.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,(33)PL=Qult=cNc+γDfNq+0.5γB0Nγ.After solving the above equations, the simplified form of the bearing capacity coefficients is as follows:(34)Nc=sin⁡α2tan⁡ϕe2βtan⁡ϕ-1+cos⁡α-ϕ+e2βtan⁡ϕsin⁡αcos⁡β-α+ϕsin⁡β-α,(35)Nq=sin⁡αe2βtan⁡ϕsin⁡ξ-khsin⁡2πtT-Hλcos⁡β-α+ϕ+1-kvsin⁡2πtT-Hηsin⁡β-α+ϕ,(36)Nγ=sin2⁡αcos2⁡ϕ1+9tan2⁡ϕ3tan⁡ϕe3βtan⁡ϕsin⁡β+ϕ-α-sin⁡ϕ-α-e3βtan⁡ϕcos⁡β+ϕ-α+cos⁡ϕ-α-sin⁡αsin⁡2ρ2cos⁡ϕ+e3βtan⁡ϕsin2⁡αsin⁡2β-α+ϕ2cos⁡ϕsin⁡ξ-qhcos⁡α-ϕ+qvsin⁡α-ϕ-qh1eβtan⁡ϕcos⁡β+ϕ-α-qv1eβtan⁡ϕsin⁡β+ϕ-α-∑i=1nei-1dβtan⁡ϕ+eidβtan⁡ϕ21+tan2⁡ϕqh2∑i=1neidβtan⁡ϕsin⁡idβ+ϕ-α+tan⁡ϕcos⁡idβ+ϕ-α-ei-1dβtan⁡ϕsin⁡i-1dβ+ϕ-α+tan⁡ϕcos⁡i-1dβ+ϕ-α+qv2∑i=1neidβtan⁡ϕsin⁡idβ+ϕ-α+tan⁡ϕcos⁡idβ+ϕ-α-ei-1dβtan⁡ϕsin⁡i-1dβ+ϕ-α+tan⁡ϕcos⁡i-1dβ+ϕ-α.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(37)PL=γ2B0N.After simplification of equations, the expression ofN is given below.(38)N=sin2⁡αcos2⁡ϕ1+9tan2⁡ϕ3tan⁡ϕe3βtan⁡ϕsin⁡β+ϕ-α-sin⁡ϕ-α-e3βtan⁡ϕcos⁡β+ϕ-α+cos⁡ϕ-α+e3βtan⁡ϕsin2⁡αsin⁡2β-α+ϕ2cos⁡ϕsin⁡ξ-sin⁡αsin⁡2ρ2cos⁡ϕ+2DfB0sin⁡αe2βtan⁡ϕsin⁡ξ-khsin⁡2πtT-Hλcos⁡β-α+ϕ+1-kvsin⁡2πtT-Hηsin⁡β-α+ϕ-qhcos⁡α-ϕ+qvsin⁡α-ϕ-qh1eβtan⁡ϕcos⁡β+ϕ-α-qv1eβtan⁡ϕsin⁡β+ϕ-α-∑i=1nei-1dβtan⁡ϕ+eidβtan⁡ϕ21+tan2⁡ϕqh2∑i=1neidβtan⁡ϕsin⁡idβ+ϕ-α+tan⁡ϕcos⁡idβ+ϕ-α-ei-1dβtan⁡ϕsin⁡i-1dβ+ϕ-α+tan⁡ϕcos⁡i-1dβ+ϕ-α+qv2∑i=1neidβtan⁡ϕsin⁡idβ+ϕ-α+tan⁡ϕcos⁡idβ+ϕ-α-ei-1dβtan⁡ϕsin⁡i-1dβ+ϕ-α+tan⁡ϕcos⁡i-1dβ+ϕ-α+2cγB0sin⁡α2tan⁡ϕe2βtan⁡ϕ-1+cos⁡α-ϕ+e2βtan⁡ϕsin⁡αcos⁡β-α+ϕsin⁡β-α.Here,N is the single pseudodynamic bearing capacity coefficient of shallow strip footing under seismic loading condition. In this formulation, the objective function pseudodynamic bearing capacity coefficient depends on these Φ, c, α, β, t/T, kh, kv, H/λ, and H/η functions. For a particular soil and seismic condition, all other terms are constant except α, β, and t/T. So, optimization of pseudodynamic bearing capacity coefficient is done with respect to α, β, and t/T using the HSOS algorithm. The advantage of this HSOS algorithm is that it can improve the searching capability of the algorithm for attaining the global optimization. Here, the optimum value of N is represented as Nγe.Now, pseudodynamic ultimate bearing capacity,(39)Qult=0.5γB0Nγe.


## 3. The Hybrid Symbiosis Organisms Search Algorithm

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.


### 3.1. The Symbiosis Organisms Search Algorithm

Symbiosis organisms search (SOS) algorithm is a population-based iterative global optimization algorithm for solving global optimization problems, proposed by Cheng and Prayogo [33]. This algorithm is based on the basic concept of symbiotic relationships among the organisms in nature (ecosystem). Three types of symbiotic relationships are occurring in an ecosystem. These are mutualism relationship, commensalism relationship, and parasitism relationship. Mutualism relationship describes the relationship where both organisms get benefits from the interaction. Commensalism relationship is a symbiotic relationship between two different organisms where one organism gets the benefit and the other is not significantly affected. In the symbiotic parasitism relationship, one organism gets the benefit and the other is harmed, but not always killed. Based on the concept of three relationships, the SOS algorithm is executed. In the SOS algorithm, a group of organisms in an ecosystem is considered as a population size of the solution. Each organism is analogous to one solution vector, and the fitness value of each organism represents the degree of adaptation to the desired objective. Initially, a set of organisms in the ecosystem is generated randomly within the search region. The new candidate solution is generated through the biological interaction between two organisms in the ecosystem which contains the mutualism, commensalism, and parasitism phases, and the process of interaction is continued until the termination criterion is satisfied. A detailed description of SOS algorithm can be seen in [33].


### 3.2. The Simple Quadratic Interpolation (SQI) Method

In this section, the three-point quadratic interpolation is discussed. Considering the two organismsOrgj and Orgk, where Orgj=orgj1,orgj2,orgj3,…,orgjD and Orgk=orgk1,orgk2,orgk3,…,orgkD from the ecosystem, the organism Orgi is updated according to the three-point quadratic interpolation [40]. The three-point approximate minimal point for organism Orgi is determined by the following equation:(40)Org¯im=0.5Orgim2−Orgjm2fk+Orgjm2−Orgkm2fi+Orgkm2−Orgim2fjOrgim−Orglmfk+Orgjm−Orgkmfi+Orgkm−Orgimfj,where m = 1, 2, 3, …, D.The SQI is intended to enhance the entire search capability of the algorithm. Here,fi, fj, and fk are the fitness values of ith, jth, and kth organisms, respectively.


### 3.3. The Hybrid Symbiosis Organisms Search Algorithm

In the development of heuristic global optimization algorithm, the balance of exploration and exploitation capability plays a major role [43], where “Exploration is the process of visiting entirely new regions of a search space, whilst exploitation is the process of visiting those regions of a search space within the neighborhood of previously visited points” [43]. As discussed above, the SQI method may be used for the better exploration when executing the optimization process. On the other hand, Cheng and Prayoga [33] have elaborately discussed the better exploitation ability of SOS for global optimization. To balance the exploration capability of SQI and the exploitation potential of SOS, the hybrid symbiosis organisms search (HSOS) algorithm has been proposed. This hybrid method can increase the robustness as well as the searching capability of the algorithm for attaining the global optimization. By incorporating the SQI into the SOS algorithm, the HSOS algorithm is developed and the flowchart of the HSOS algorithm is shown in Figure 5. The HSOS algorithm is able to explore the new search region with the SOS algorithm and to exploit the population information with the SQI.Figure 5
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 [44]:(41)Orgi=UBi+rand0,1∗UBi−LBiifOrgi<LBi,UBi−rand0,1∗UBi−LBiifOrgi>UBi,where LBiandUBi are, respectively, the lower and upper bounds of the ith organism.The algorithmic steps of hsos are given below:Step 1. Ecosystem initialization: initialize the algorithm parameters and ecosystem organisms and evaluate the fitness value for each corresponding organism.Step 2. Main loop.Step 2.1. Mutualism phase: select one organism Orgj randomly from the ecosystem. The organism Orgi intersects with the organism Orgj and then they try to improve the survival capabilities in the ecosystem. The new organism for each of Orgi and Orgj is calculated by the following equations:(42)Orginew=Orgi+rand0,1∗Orgbest−MutualVector∗BF1,(43)Orgjnew=Orgj+rand0,1∗Orgbest−MutualVector∗BF2,where MutualVector=Orgi+Orgj/2. Here, BF1 and BF2 are called the benefit factors, the value of which be either 1 or 2. The level of benefits of the organism represents these factors, that is, whether an organism gets, respectively, partial or full benefit from the interaction. Orgbest is the best organism in the ecosystem. MutualVector represents the relationship characteristic between the organisms Orgi and Orgj.Step 2.2. Commensalism phase: between the interaction of the organisms Orgi and Orgj, the organism Orgi gets benefit by the organism Orgj and try to improve the beneficial advantage in the ecosystem to the higher degree of adaption. The new organism Orgi is determined by the following equation:(44)Orginew=Orgi+rand−1,check space1∗Orgbest−Orgj,where i ≠ j and Orgbest is the best organism in the ecosystem.Step 2.3. Parasitism phase: by duplicating randomly selected dimensions of the organism Orgi, an artificial parasite (Parasite_Vector) is created. From the ecosystem, another organism Orgj is selected randomly that is treated as a host to the Parasite_Vector. If the objective function value of the Parasite_Vector is better than the organism Orgj, it can kill the organism Orgj and adopt its position in the ecosystem. If the objective function value of Orgj is better than the Parasite_Vector, Orgj will have resistance to the parasite and the Parasite_Vector will not be able to reside in that ecosystem.Step 2.4. Simple quadratic interpolation: the two organisms Orgj and Orgk (j ≠ k) are selected randomly from the ecosystem, and then the organism Orgi is updated by quadratic interpolation passing through these three organisms which can be expressed by (40).Step 3. If the stopping criteria are not satisfied to go to Step 2, then it will proceed until the best objective function value is obtained.


## 3.1. The Symbiosis Organisms Search Algorithm

Symbiosis organisms search (SOS) algorithm is a population-based iterative global optimization algorithm for solving global optimization problems, proposed by Cheng and Prayogo [33]. This algorithm is based on the basic concept of symbiotic relationships among the organisms in nature (ecosystem). Three types of symbiotic relationships are occurring in an ecosystem. These are mutualism relationship, commensalism relationship, and parasitism relationship. Mutualism relationship describes the relationship where both organisms get benefits from the interaction. Commensalism relationship is a symbiotic relationship between two different organisms where one organism gets the benefit and the other is not significantly affected. In the symbiotic parasitism relationship, one organism gets the benefit and the other is harmed, but not always killed. Based on the concept of three relationships, the SOS algorithm is executed. In the SOS algorithm, a group of organisms in an ecosystem is considered as a population size of the solution. Each organism is analogous to one solution vector, and the fitness value of each organism represents the degree of adaptation to the desired objective. Initially, a set of organisms in the ecosystem is generated randomly within the search region. The new candidate solution is generated through the biological interaction between two organisms in the ecosystem which contains the mutualism, commensalism, and parasitism phases, and the process of interaction is continued until the termination criterion is satisfied. A detailed description of SOS algorithm can be seen in [33].


## 3.2. The Simple Quadratic Interpolation (SQI) Method

In this section, the three-point quadratic interpolation is discussed. Considering the two organismsOrgj and Orgk, where Orgj=orgj1,orgj2,orgj3,…,orgjD and Orgk=orgk1,orgk2,orgk3,…,orgkD from the ecosystem, the organism Orgi is updated according to the three-point quadratic interpolation [40]. The three-point approximate minimal point for organism Orgi is determined by the following equation:(40)Org¯im=0.5Orgim2−Orgjm2fk+Orgjm2−Orgkm2fi+Orgkm2−Orgim2fjOrgim−Orglmfk+Orgjm−Orgkmfi+Orgkm−Orgimfj,where m = 1, 2, 3, …, D.The SQI is intended to enhance the entire search capability of the algorithm. Here,fi, fj, and fk are the fitness values of ith, jth, and kth organisms, respectively.


## 3.3. The Hybrid Symbiosis Organisms Search Algorithm

In the development of heuristic global optimization algorithm, the balance of exploration and exploitation capability plays a major role [43], where “Exploration is the process of visiting entirely new regions of a search space, whilst exploitation is the process of visiting those regions of a search space within the neighborhood of previously visited points” [43]. As discussed above, the SQI method may be used for the better exploration when executing the optimization process. On the other hand, Cheng and Prayoga [33] have elaborately discussed the better exploitation ability of SOS for global optimization. To balance the exploration capability of SQI and the exploitation potential of SOS, the hybrid symbiosis organisms search (HSOS) algorithm has been proposed. This hybrid method can increase the robustness as well as the searching capability of the algorithm for attaining the global optimization. By incorporating the SQI into the SOS algorithm, the HSOS algorithm is developed and the flowchart of the HSOS algorithm is shown in Figure 5. The HSOS algorithm is able to explore the new search region with the SOS algorithm and to exploit the population information with the SQI.Figure 5
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 [44]:(41)Orgi=UBi+rand0,1∗UBi−LBiifOrgi<LBi,UBi−rand0,1∗UBi−LBiifOrgi>UBi,where LBiandUBi are, respectively, the lower and upper bounds of the ith organism.The algorithmic steps of hsos are given below:Step 1. Ecosystem initialization: initialize the algorithm parameters and ecosystem organisms and evaluate the fitness value for each corresponding organism.Step 2. Main loop.Step 2.1. Mutualism phase: select one organism Orgj randomly from the ecosystem. The organism Orgi intersects with the organism Orgj and then they try to improve the survival capabilities in the ecosystem. The new organism for each of Orgi and Orgj is calculated by the following equations:(42)Orginew=Orgi+rand0,1∗Orgbest−MutualVector∗BF1,(43)Orgjnew=Orgj+rand0,1∗Orgbest−MutualVector∗BF2,where MutualVector=Orgi+Orgj/2. Here, BF1 and BF2 are called the benefit factors, the value of which be either 1 or 2. The level of benefits of the organism represents these factors, that is, whether an organism gets, respectively, partial or full benefit from the interaction. Orgbest is the best organism in the ecosystem. MutualVector represents the relationship characteristic between the organisms Orgi and Orgj.Step 2.2. Commensalism phase: between the interaction of the organisms Orgi and Orgj, the organism Orgi gets benefit by the organism Orgj and try to improve the beneficial advantage in the ecosystem to the higher degree of adaption. The new organism Orgi is determined by the following equation:(44)Orginew=Orgi+rand−1,check space1∗Orgbest−Orgj,where i ≠ j and Orgbest is the best organism in the ecosystem.Step 2.3. Parasitism phase: by duplicating randomly selected dimensions of the organism Orgi, an artificial parasite (Parasite_Vector) is created. From the ecosystem, another organism Orgj is selected randomly that is treated as a host to the Parasite_Vector. If the objective function value of the Parasite_Vector is better than the organism Orgj, it can kill the organism Orgj and adopt its position in the ecosystem. If the objective function value of Orgj is better than the Parasite_Vector, Orgj will have resistance to the parasite and the Parasite_Vector will not be able to reside in that ecosystem.Step 2.4. Simple quadratic interpolation: the two organisms Orgj and Orgk (j ≠ k) are selected randomly from the ecosystem, and then the organism Orgi is updated by quadratic interpolation passing through these three organisms which can be expressed by (40).Step 3. If the stopping criteria are not satisfied to go to Step 2, then it will proceed until the best objective function value is obtained.


## 4. Discussion on Results Obtained by the HSOS Algorithm

The pseudodynamic bearing capacity coefficient (Nγe) has been optimized using the HSOS algorithm with respect to α, β, and t/T variables. The algorithm was performed with 1000 fitness evaluations, 30 independent runs, and 50 eco-sizes. The best result has been taken among these 30 results. This optimized single seismic bearing capacity coefficient (Nγe) is presented in Tables 1 and 2 for static and seismic conditions (kh = 0.1, 0.2, and 0.3), respectively, which can be used by the field engineers in earthquake-prone areas for the simultaneous resistance of unit weight, surcharge, and cohesion.Table 1
Static condition.
Φ
2c/γB0
D
f/B0
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
270.1Table 2
Pseudodynamic bearing capacity coefficient (Nγe) for kh = 0.1, 0.2, and 0.3.
k
h = 0.1
Φ
2c/γB0
k
v = 0
k
v = kh/2
k
v = kh
D
f/B0
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
k
h = 0.2
Φ
2c/γB0
k
v = 0
k
v = kh/2
k
v = kh
D
f/B0
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
k
h = 0.3
Φ
2c/γB0
k
v = 0
k
v = kh/2
k
v = kh
D
f/B0
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


### 4.1. Parametric Study

In this section, a brief parametric study and a comparative study have been presented. The effect of soil friction angle (Φ), depth factor (Df/B0), cohesion factor (2c/γB0), and seismic accelerations (kh and kv) on normalized reduction factor (Nγe/Nγs) is discussed. Normalized reduction factor (Nγe/Nγs) is the ratio of optimized seismic and static bearing capacity coefficient. The variations of parameters are as follows: Φ = 20°, 30°, and 40°; kh = 0.1, 0.2, and 0.3; kv = 0, kh/2, and kh; 2c/γB0 = 0, 0.25, and 0.5; and Df/B0 = 0.25, 0.75, 0.5, and 1. A detailed comparative study with other available previous research is also discussed in this section.


#### 4.1.1. Effect on Nγe/Nγs due to Variation of Φ

Figure6 shows the variations of the normalized reduction factor (Nγe/Nγs) with respect to horizontal seismic acceleration (kh) at different soil friction angles (Φ = 20°, 30°, and 40°) at 2c/γB0 = 0.25, Df = 0.5, and kv = kh/2. It is seen that the normalized reduction factor (Nγe/Nγs) increases with increase in soil friction angle (Φ). Due to an increase in Φ, the internal resistance of the soil particles will be increased, which resembles the fact that there is an increase in the seismic bearing capacity factor.Figure 6
The variations of the bearing capacity coefficient with respect to seismic acceleration (kh) at different soil friction angles (Φ = 20°, 30°, and 40°) for 2c/B0 = 0.25, Df/B0 = 0.5, and kv = kh/2.


#### 4.1.2. Effect on Nγe/Nγs due to Variation of 2c/γB0

Figure7 shows the variations of the normalized reduction factor (Nγe/Nγs) with respect to seismic acceleration (kh) at different cohesion factors (2c/γB0 = 0, 0.25, and 0.5) at Φ = 30°, Df/B0 = 0.5, and kv = kh/2. It is seen that the normalized reduction factor (Nγe/Nγs) increases with an increase in the cohesion factor (2c/γB0). Due to an increase in cohesion, the seismic bearing capacity factor will be increased as the increase in cohesion causes an increase in intermolecular attraction among the soil particle, which offers more resistance against the shearing failure of the foundation.Figure 7
The variations of the bearing capacity coefficient with respect to seismic acceleration (kh) at different cohesion factors (2c/B0 = 0, 0.25, and 0.5) for Φ = 30°, δ = Φ/2, Df/B0 = 0.5, and kv = kh/2.


#### 4.1.3. Effect on Nγe/Nγs due to Variation of Df/B0

Figure8 shows the variations of the normalized reduction factor (Nγe/Nγs) with respect to seismic acceleration (kh) for different depth factors (Df/B0 = 0.25, 0.5, and 1) at Φ = 30°, 2c/γB0 = 0.25, and kv = kh/2. It is seen that the normalized reduction factor (Nγe/Nγs) increases with an increase in the depth factor (Df/B0). Due to an increase in the depth factor (Df/B0), surcharge weight increases, which increases the passive resistance and hence increases the seismic bearing capacity factor.Figure 8
The variations of the bearing capacity coefficient with respect to seismic acceleration (kh) at different depth factors (Df/B0 = 0.25, 0.5, and 1) for Φ = 30°, δ = Φ/2, 2c/B0 = 0.25, and kv = kh/2.


#### 4.1.4. Effect on Nγe/Nγs due to Variation of Seismic Accelerations (kh and kv)

From Figures6–9, it is seen that the normalized reduction factor (Nγe/Nγs) decreases along with an increase in horizontal seismic acceleration (kh). And Figure 9 shows the variations of the normalized reduction factor (Nγe/Nγs) with respect to seismic acceleration (kh) at different vertical seismic accelerations (kv = 0, kh/2, and kh) for Φ = 30°, Df = 0.5, and 2c/γB0 = 0.25. It is seen that the normalized reduction factor (Nγe/Nγs) decreases with the increase in vertical seismic acceleration (kv) also. Due to an increase in seismic acceleration and due to the sudden movement of different waves, the disturbance in the soil particles increases, which allows more soil mass to participate in the vibration and hence decreases its resistance against bearing capacity.Figure 9
The variations of the bearing capacity coefficient with respect to seismic acceleration (kh) at different vertical seismic accelerations (kv = 0, kh/2, and kh) for Φ = 30°, 2c/B0 = 0.25, Df/B0 = 0.5, and δ = Φ/2.


#### 4.1.5. Comparison of Result

A detailed comparative study of the present analysis with previous research on similar type of works with different approaches is done here. Figure10 and Table 3 show the comparison of a pseudodynamic bearing capacity coefficient obtained from the present analysis with previous seismic analyses with respect to different seismic accelerations (kh = 0.1, 0.2, and 0.3) for Φ = 30°. It is seen that, for the lower value of seismic accelerations here in Figure 10, kh = 0.2, the values obtained from the present study are less than the values obtained from Soubra [10] (M1 and M2) [17]. But when horizontal seismic acceleration increases from 0.2, the bearing capacity coefficient also increases gradually, and at kh = 0.3, the present analysis provides greater value in comparison to all the compared methods. At kh = 0.1, around 7.5%, 24%, and 29% decrease in Nγe coefficient, and at kh = 0.2, around 2%, 15%, and 12% decrease in Nγe coefficient in comparison to that in Soubra [10] (M1 and M2) and Ghosh [17], respectively. But at kh = 0.3, it increases around 26%, 16%, and 48%, respectively, in comparison with the respective analyses.Figure 10
The comparison of pseudodynamic bearing capacity coefficient obtained from present analysis with previous seismic analyses with respect to different seismic accelerations (kh = 0.1, 0.2, and 0.3) for Φ = 30°.Table 3
Comparison of seismic bearing capacity coefficient (Nγe) for different values of kh and kv with Φ = 30°.
k
h
Present study
Ghosh [17]
Budhu and Al-Karni [7]
Choudhury and Subba Rao [13]
Soubra [10]
k
v = kh/2
k
v = kh
k
v = kh/2
k
v = kh
k
v = kh/2
k
v = kh
k
v = kh/2
k
v = kh
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
4.9The performance results, that is, pseudodynamic bearing capacity coefficients obtained by the HSOS algorithm are compared with other metaheuristic optimization algorithms. Table4 shows the performance result obtained by DE [45], PSO [46], ABC [47], HS [48], BSA [49], ABSA [50], SOS [33], and HSOS [39] algorithms at different conditions that are compared here. From this table, it is observed that the performance result, that is, pseudodynamic bearing capacity coefficient (Nγe) obtained from this HSOS algorithm is lesser than the other compared algorithms in different soil and seismic conditions. From the above investigations, it can be said that HSOS algorithm can satisfactorily be used to evaluate the seismic bearing capacity of shallow strip footing suggested here.Table 4
Comparison of seismic bearing capacity coefficient (Nγe) obtained by different standard algorithms.
Φ
k
h
DE
PSO
ABC
HS
BSA
ABSA
SOS
HSOS
(a) 2c/γB0 = 0.25, Df/B0 = 0.5, and kv = kh/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) 2c/γB0 = 0.5, Df/B0 = 0.75, and kv = kh/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


## 4.1. Parametric Study

In this section, a brief parametric study and a comparative study have been presented. The effect of soil friction angle (Φ), depth factor (Df/B0), cohesion factor (2c/γB0), and seismic accelerations (kh and kv) on normalized reduction factor (Nγe/Nγs) is discussed. Normalized reduction factor (Nγe/Nγs) is the ratio of optimized seismic and static bearing capacity coefficient. The variations of parameters are as follows: Φ = 20°, 30°, and 40°; kh = 0.1, 0.2, and 0.3; kv = 0, kh/2, and kh; 2c/γB0 = 0, 0.25, and 0.5; and Df/B0 = 0.25, 0.75, 0.5, and 1. A detailed comparative study with other available previous research is also discussed in this section.


### 4.1.1. Effect on Nγe/Nγs due to Variation of Φ

Figure6 shows the variations of the normalized reduction factor (Nγe/Nγs) with respect to horizontal seismic acceleration (kh) at different soil friction angles (Φ = 20°, 30°, and 40°) at 2c/γB0 = 0.25, Df = 0.5, and kv = kh/2. It is seen that the normalized reduction factor (Nγe/Nγs) increases with increase in soil friction angle (Φ). Due to an increase in Φ, the internal resistance of the soil particles will be increased, which resembles the fact that there is an increase in the seismic bearing capacity factor.Figure 6
The variations of the bearing capacity coefficient with respect to seismic acceleration (kh) at different soil friction angles (Φ = 20°, 30°, and 40°) for 2c/B0 = 0.25, Df/B0 = 0.5, and kv = kh/2.


### 4.1.2. Effect on Nγe/Nγs due to Variation of 2c/γB0

Figure7 shows the variations of the normalized reduction factor (Nγe/Nγs) with respect to seismic acceleration (kh) at different cohesion factors (2c/γB0 = 0, 0.25, and 0.5) at Φ = 30°, Df/B0 = 0.5, and kv = kh/2. It is seen that the normalized reduction factor (Nγe/Nγs) increases with an increase in the cohesion factor (2c/γB0). Due to an increase in cohesion, the seismic bearing capacity factor will be increased as the increase in cohesion causes an increase in intermolecular attraction among the soil particle, which offers more resistance against the shearing failure of the foundation.Figure 7
The variations of the bearing capacity coefficient with respect to seismic acceleration (kh) at different cohesion factors (2c/B0 = 0, 0.25, and 0.5) for Φ = 30°, δ = Φ/2, Df/B0 = 0.5, and kv = kh/2.


### 4.1.3. Effect on Nγe/Nγs due to Variation of Df/B0

Figure8 shows the variations of the normalized reduction factor (Nγe/Nγs) with respect to seismic acceleration (kh) for different depth factors (Df/B0 = 0.25, 0.5, and 1) at Φ = 30°, 2c/γB0 = 0.25, and kv = kh/2. It is seen that the normalized reduction factor (Nγe/Nγs) increases with an increase in the depth factor (Df/B0). Due to an increase in the depth factor (Df/B0), surcharge weight increases, which increases the passive resistance and hence increases the seismic bearing capacity factor.Figure 8
The variations of the bearing capacity coefficient with respect to seismic acceleration (kh) at different depth factors (Df/B0 = 0.25, 0.5, and 1) for Φ = 30°, δ = Φ/2, 2c/B0 = 0.25, and kv = kh/2.


### 4.1.4. Effect on Nγe/Nγs due to Variation of Seismic Accelerations (kh and kv)

From Figures6–9, it is seen that the normalized reduction factor (Nγe/Nγs) decreases along with an increase in horizontal seismic acceleration (kh). And Figure 9 shows the variations of the normalized reduction factor (Nγe/Nγs) with respect to seismic acceleration (kh) at different vertical seismic accelerations (kv = 0, kh/2, and kh) for Φ = 30°, Df = 0.5, and 2c/γB0 = 0.25. It is seen that the normalized reduction factor (Nγe/Nγs) decreases with the increase in vertical seismic acceleration (kv) also. Due to an increase in seismic acceleration and due to the sudden movement of different waves, the disturbance in the soil particles increases, which allows more soil mass to participate in the vibration and hence decreases its resistance against bearing capacity.Figure 9
The variations of the bearing capacity coefficient with respect to seismic acceleration (kh) at different vertical seismic accelerations (kv = 0, kh/2, and kh) for Φ = 30°, 2c/B0 = 0.25, Df/B0 = 0.5, and δ = Φ/2.


### 4.1.5. Comparison of Result

A detailed comparative study of the present analysis with previous research on similar type of works with different approaches is done here. Figure10 and Table 3 show the comparison of a pseudodynamic bearing capacity coefficient obtained from the present analysis with previous seismic analyses with respect to different seismic accelerations (kh = 0.1, 0.2, and 0.3) for Φ = 30°. It is seen that, for the lower value of seismic accelerations here in Figure 10, kh = 0.2, the values obtained from the present study are less than the values obtained from Soubra [10] (M1 and M2) [17]. But when horizontal seismic acceleration increases from 0.2, the bearing capacity coefficient also increases gradually, and at kh = 0.3, the present analysis provides greater value in comparison to all the compared methods. At kh = 0.1, around 7.5%, 24%, and 29% decrease in Nγe coefficient, and at kh = 0.2, around 2%, 15%, and 12% decrease in Nγe coefficient in comparison to that in Soubra [10] (M1 and M2) and Ghosh [17], respectively. But at kh = 0.3, it increases around 26%, 16%, and 48%, respectively, in comparison with the respective analyses.Figure 10
The comparison of pseudodynamic bearing capacity coefficient obtained from present analysis with previous seismic analyses with respect to different seismic accelerations (kh = 0.1, 0.2, and 0.3) for Φ = 30°.Table 3
Comparison of seismic bearing capacity coefficient (Nγe) for different values of kh and kv with Φ = 30°.
k
h
Present study
Ghosh [17]
Budhu and Al-Karni [7]
Choudhury and Subba Rao [13]
Soubra [10]
k
v = kh/2
k
v = kh
k
v = kh/2
k
v = kh
k
v = kh/2
k
v = kh
k
v = kh/2
k
v = kh
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
4.9The performance results, that is, pseudodynamic bearing capacity coefficients obtained by the HSOS algorithm are compared with other metaheuristic optimization algorithms. Table4 shows the performance result obtained by DE [45], PSO [46], ABC [47], HS [48], BSA [49], ABSA [50], SOS [33], and HSOS [39] algorithms at different conditions that are compared here. From this table, it is observed that the performance result, that is, pseudodynamic bearing capacity coefficient (Nγe) obtained from this HSOS algorithm is lesser than the other compared algorithms in different soil and seismic conditions. From the above investigations, it can be said that HSOS algorithm can satisfactorily be used to evaluate the seismic bearing capacity of shallow strip footing suggested here.Table 4
Comparison of seismic bearing capacity coefficient (Nγe) obtained by different standard algorithms.
Φ
k
h
DE
PSO
ABC
HS
BSA
ABSA
SOS
HSOS
(a) 2c/γB0 = 0.25, Df/B0 = 0.5, and kv = kh/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) 2c/γB0 = 0.5, Df/B0 = 0.75, and kv = kh/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


## 4.1.1. Effect on Nγe/Nγs due to Variation of Φ

Figure6 shows the variations of the normalized reduction factor (Nγe/Nγs) with respect to horizontal seismic acceleration (kh) at different soil friction angles (Φ = 20°, 30°, and 40°) at 2c/γB0 = 0.25, Df = 0.5, and kv = kh/2. It is seen that the normalized reduction factor (Nγe/Nγs) increases with increase in soil friction angle (Φ). Due to an increase in Φ, the internal resistance of the soil particles will be increased, which resembles the fact that there is an increase in the seismic bearing capacity factor.Figure 6
The variations of the bearing capacity coefficient with respect to seismic acceleration (kh) at different soil friction angles (Φ = 20°, 30°, and 40°) for 2c/B0 = 0.25, Df/B0 = 0.5, and kv = kh/2.


## 4.1.2. Effect on Nγe/Nγs due to Variation of 2c/γB0

Figure7 shows the variations of the normalized reduction factor (Nγe/Nγs) with respect to seismic acceleration (kh) at different cohesion factors (2c/γB0 = 0, 0.25, and 0.5) at Φ = 30°, Df/B0 = 0.5, and kv = kh/2. It is seen that the normalized reduction factor (Nγe/Nγs) increases with an increase in the cohesion factor (2c/γB0). Due to an increase in cohesion, the seismic bearing capacity factor will be increased as the increase in cohesion causes an increase in intermolecular attraction among the soil particle, which offers more resistance against the shearing failure of the foundation.Figure 7
The variations of the bearing capacity coefficient with respect to seismic acceleration (kh) at different cohesion factors (2c/B0 = 0, 0.25, and 0.5) for Φ = 30°, δ = Φ/2, Df/B0 = 0.5, and kv = kh/2.


## 4.1.3. Effect on Nγe/Nγs due to Variation of Df/B0

Figure8 shows the variations of the normalized reduction factor (Nγe/Nγs) with respect to seismic acceleration (kh) for different depth factors (Df/B0 = 0.25, 0.5, and 1) at Φ = 30°, 2c/γB0 = 0.25, and kv = kh/2. It is seen that the normalized reduction factor (Nγe/Nγs) increases with an increase in the depth factor (Df/B0). Due to an increase in the depth factor (Df/B0), surcharge weight increases, which increases the passive resistance and hence increases the seismic bearing capacity factor.Figure 8
The variations of the bearing capacity coefficient with respect to seismic acceleration (kh) at different depth factors (Df/B0 = 0.25, 0.5, and 1) for Φ = 30°, δ = Φ/2, 2c/B0 = 0.25, and kv = kh/2.


## 4.1.4. Effect on Nγe/Nγs due to Variation of Seismic Accelerations (kh and kv)

From Figures6–9, it is seen that the normalized reduction factor (Nγe/Nγs) decreases along with an increase in horizontal seismic acceleration (kh). And Figure 9 shows the variations of the normalized reduction factor (Nγe/Nγs) with respect to seismic acceleration (kh) at different vertical seismic accelerations (kv = 0, kh/2, and kh) for Φ = 30°, Df = 0.5, and 2c/γB0 = 0.25. It is seen that the normalized reduction factor (Nγe/Nγs) decreases with the increase in vertical seismic acceleration (kv) also. Due to an increase in seismic acceleration and due to the sudden movement of different waves, the disturbance in the soil particles increases, which allows more soil mass to participate in the vibration and hence decreases its resistance against bearing capacity.Figure 9
The variations of the bearing capacity coefficient with respect to seismic acceleration (kh) at different vertical seismic accelerations (kv = 0, kh/2, and kh) for Φ = 30°, 2c/B0 = 0.25, Df/B0 = 0.5, and δ = Φ/2.


## 4.1.5. Comparison of Result

A detailed comparative study of the present analysis with previous research on similar type of works with different approaches is done here. Figure10 and Table 3 show the comparison of a pseudodynamic bearing capacity coefficient obtained from the present analysis with previous seismic analyses with respect to different seismic accelerations (kh = 0.1, 0.2, and 0.3) for Φ = 30°. It is seen that, for the lower value of seismic accelerations here in Figure 10, kh = 0.2, the values obtained from the present study are less than the values obtained from Soubra [10] (M1 and M2) [17]. But when horizontal seismic acceleration increases from 0.2, the bearing capacity coefficient also increases gradually, and at kh = 0.3, the present analysis provides greater value in comparison to all the compared methods. At kh = 0.1, around 7.5%, 24%, and 29% decrease in Nγe coefficient, and at kh = 0.2, around 2%, 15%, and 12% decrease in Nγe coefficient in comparison to that in Soubra [10] (M1 and M2) and Ghosh [17], respectively. But at kh = 0.3, it increases around 26%, 16%, and 48%, respectively, in comparison with the respective analyses.Figure 10
The comparison of pseudodynamic bearing capacity coefficient obtained from present analysis with previous seismic analyses with respect to different seismic accelerations (kh = 0.1, 0.2, and 0.3) for Φ = 30°.Table 3
Comparison of seismic bearing capacity coefficient (Nγe) for different values of kh and kv with Φ = 30°.
k
h
Present study
Ghosh [17]
Budhu and Al-Karni [7]
Choudhury and Subba Rao [13]
Soubra [10]
k
v = kh/2
k
v = kh
k
v = kh/2
k
v = kh
k
v = kh/2
k
v = kh
k
v = kh/2
k
v = kh
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
4.9The performance results, that is, pseudodynamic bearing capacity coefficients obtained by the HSOS algorithm are compared with other metaheuristic optimization algorithms. Table4 shows the performance result obtained by DE [45], PSO [46], ABC [47], HS [48], BSA [49], ABSA [50], SOS [33], and HSOS [39] algorithms at different conditions that are compared here. From this table, it is observed that the performance result, that is, pseudodynamic bearing capacity coefficient (Nγe) obtained from this HSOS algorithm is lesser than the other compared algorithms in different soil and seismic conditions. From the above investigations, it can be said that HSOS algorithm can satisfactorily be used to evaluate the seismic bearing capacity of shallow strip footing suggested here.Table 4
Comparison of seismic bearing capacity coefficient (Nγe) obtained by different standard algorithms.
Φ
k
h
DE
PSO
ABC
HS
BSA
ABSA
SOS
HSOS
(a) 2c/γB0 = 0.25, Df/B0 = 0.5, and kv = kh/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) 2c/γB0 = 0.5, Df/B0 = 0.75, and kv = kh/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


## 5. Numerical Analysis

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.


### 5.1. Numerical Modeling

A two-dimensional geometrical model is prepared that is to be composed of points, lines, and other components in thex-y plane. The PLAXIS mesh generator based on the input of the geometry model automatically performs the generation of a mesh at an element level. The shallow strip footing was modeled as a plane strain, and 15 nodded triangular elements are used to simulate the foundation soil. The extension of the mesh was taken 100 m wide and 30 m depth as the earthquake forces cannot affect the vertical boundaries. Standard earthquake boundaries are applied for earthquake loading conditions using SMC files, and then the mesh is generated. Cluster refinement of the mesh is followed to obtain precise medium-sized mesh. HS small model was used to incorporate dynamic soil properties of the soil samples. Two different soil samples were used to analyze the shallow strip footing under seismic loading condition as shown in Table 5. A uniformly distributed load of 100 kN/m applying on the section of the foundation along with different surcharge loads to represent the load coming from superstructure is analyzed in this paper as shown in Figure 11. Initial stresses are generated after turning off the initial pore water pressure tool.Table 5
HS small model soil parameters for PLAXIS-8.6v.
Sample
γ (kPa)
c (kPa)
Φ
ψ
ν
E
50
ref (kPa)
E
oed
ref (kPa)
E
ur
ref (kPa)
m
G
0 (kPa)
γ
0.7
S1
20.9
0.5
32
2
0.2
1.00E + 4
1.00E + 4
3.00E + 4
0.5
1.00E + 5
1.00E − 4
S2
19.9
0.2
28
0
0.2
1.25E + 4
1.00E + 4
3.75E + 4
0.5
1.30E + 5
1.25E − 4Figure 11
Finite element geometry and foundation load along with surcharge load.


### 5.2. Calculation

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 thec-Φ reduction method. El Salvador 2001 earthquake data (moment magnitude, Mw=7.6) are given as input in the dynamic calculation as SMC file as shown in Figure 12. The vertical settlement of the foundation and the corresponding factor at safety of each condition obtained from the numerical modeling are obtained. Figures 13 and 14 show the deformed mesh and vertical displacement contour, respectively, after undergoing staged calculations.Figure 12
El Salvador 2001 earthquake data.Figure 13
The deformed mesh of model after calculation.Figure 14
Vertical displacement contour after calculation.


### 5.3. Numerical Validation

Finite element model of shallow strip footing embedded inc-Φ soil is analyzed in PLAXIS-8.6v for the validation of the analytical solution. The results obtained from this analytical analysis are compared with the numerical solutions to validate the analysis. At first, settlement of foundation is calculated analytically using two classical equations, such asRichards et al.’s [12] seismic settlement equation:(45)se=0.174V2AgkhA−4tanαAE,where V is the peak velocity for the design earthquake (m/sec), A is the acceleration coefficient for the design earthquake, and g is the acceleration due to gravity, and the value of αAE depends on Φ and critical acceleration kh∗.Terzaghi’s [2] immediate settlement equation:(46)si=qnB01−ν2EIf,where qn is the net foundation pressure, qn=Qult−γDf/FOS, ν is the Poisson ratio, E is the Young modulus of soil, and If is the influence factor for shallow strip footing. Here, Qult is the pseudodynamic ultimate bearing capacity which is obtained from (39).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 Table6. Two different types of soil models have been analyzed. Settlement of shallow foundation for the corresponding soil model is calculated using (41) and (42). Settlement values obtained from the finite element model in PLAXIS are also tabulated. It is seen that settlement obtained from analytical solution is slightly in lower side in comparison with the settlement obtained from PLAXIS-8.6v as in the analytical settlement calculation, the only initial settlement is considered. So, the formulation of pseudodynamic bearing capacity is well justified after the numerical validation.Table 6
Comparison of settlements obtained from numerical and analytical analyses.
Soil samples
Depth factor (Df/B0)
Numerical solution
Analytical solution
PLAXIS-8.6v
Richards et al. [12]
Present analysis
Terzaghi [2]
FOS
Settlement (mm)
k
h
∗
Settlement (mm)
N
γe
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


## 5.1. Numerical Modeling

A two-dimensional geometrical model is prepared that is to be composed of points, lines, and other components in thex-y plane. The PLAXIS mesh generator based on the input of the geometry model automatically performs the generation of a mesh at an element level. The shallow strip footing was modeled as a plane strain, and 15 nodded triangular elements are used to simulate the foundation soil. The extension of the mesh was taken 100 m wide and 30 m depth as the earthquake forces cannot affect the vertical boundaries. Standard earthquake boundaries are applied for earthquake loading conditions using SMC files, and then the mesh is generated. Cluster refinement of the mesh is followed to obtain precise medium-sized mesh. HS small model was used to incorporate dynamic soil properties of the soil samples. Two different soil samples were used to analyze the shallow strip footing under seismic loading condition as shown in Table 5. A uniformly distributed load of 100 kN/m applying on the section of the foundation along with different surcharge loads to represent the load coming from superstructure is analyzed in this paper as shown in Figure 11. Initial stresses are generated after turning off the initial pore water pressure tool.Table 5
HS small model soil parameters for PLAXIS-8.6v.
Sample
γ (kPa)
c (kPa)
Φ
ψ
ν
E
50
ref (kPa)
E
oed
ref (kPa)
E
ur
ref (kPa)
m
G
0 (kPa)
γ
0.7
S1
20.9
0.5
32
2
0.2
1.00E + 4
1.00E + 4
3.00E + 4
0.5
1.00E + 5
1.00E − 4
S2
19.9
0.2
28
0
0.2
1.25E + 4
1.00E + 4
3.75E + 4
0.5
1.30E + 5
1.25E − 4Figure 11
Finite element geometry and foundation load along with surcharge load.


## 5.2. Calculation

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 thec-Φ reduction method. El Salvador 2001 earthquake data (moment magnitude, Mw=7.6) are given as input in the dynamic calculation as SMC file as shown in Figure 12. The vertical settlement of the foundation and the corresponding factor at safety of each condition obtained from the numerical modeling are obtained. Figures 13 and 14 show the deformed mesh and vertical displacement contour, respectively, after undergoing staged calculations.Figure 12
El Salvador 2001 earthquake data.Figure 13
The deformed mesh of model after calculation.Figure 14
Vertical displacement contour after calculation.


## 5.3. Numerical Validation

Finite element model of shallow strip footing embedded inc-Φ soil is analyzed in PLAXIS-8.6v for the validation of the analytical solution. The results obtained from this analytical analysis are compared with the numerical solutions to validate the analysis. At first, settlement of foundation is calculated analytically using two classical equations, such asRichards et al.’s [12] seismic settlement equation:(45)se=0.174V2AgkhA−4tanαAE,where V is the peak velocity for the design earthquake (m/sec), A is the acceleration coefficient for the design earthquake, and g is the acceleration due to gravity, and the value of αAE depends on Φ and critical acceleration kh∗.Terzaghi’s [2] immediate settlement equation:(46)si=qnB01−ν2EIf,where qn is the net foundation pressure, qn=Qult−γDf/FOS, ν is the Poisson ratio, E is the Young modulus of soil, and If is the influence factor for shallow strip footing. Here, Qult is the pseudodynamic ultimate bearing capacity which is obtained from (39).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 Table6. Two different types of soil models have been analyzed. Settlement of shallow foundation for the corresponding soil model is calculated using (41) and (42). Settlement values obtained from the finite element model in PLAXIS are also tabulated. It is seen that settlement obtained from analytical solution is slightly in lower side in comparison with the settlement obtained from PLAXIS-8.6v as in the analytical settlement calculation, the only initial settlement is considered. So, the formulation of pseudodynamic bearing capacity is well justified after the numerical validation.Table 6
Comparison of settlements obtained from numerical and analytical analyses.
Soil samples
Depth factor (Df/B0)
Numerical solution
Analytical solution
PLAXIS-8.6v
Richards et al. [12]
Present analysis
Terzaghi [2]
FOS
Settlement (mm)
k
h
∗
Settlement (mm)
N
γe
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


## 6. Conclusion

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 onc-Φ soil. The HSOS algorithm is used to solve this problem. The advantage of this HSOS algorithm is that it can improve the searching capability of the algorithm for attaining the global optimum. From the comparison, the results obtained by the HSOS algorithm with other standard algorithms show the acceptability of the results in all soil and seismic conditions. Hence, using the HSOS algorithm, the coefficient of seismic bearing capacity is presented in a tabular form. Numerical modeling of shallow strip footing is also analyzed using PLAXIS-8.6v software for the validation of the analytical solution. It is observed that the results obtained from this analytical analysis are well justified with the numerical solutions. The effect of various parameters such as soil friction angle (Φ), seismic accelerations (kh and kv), cohesion factor (2c/γB0), and depth factor (Df/B0) is studied here. It is seen that the pseudodynamic bearing capacity coefficient (Nγe) increases with the increase in Φ, 2c/γB0, and Df/B0, but it decreases with the increase in horizontal and vertical seismic accelerations (kh and kv). The values obtained from the present analysis are thoroughly compared with the available pseudostatic analysis as well as pseudodynamic analysis values, and it is seen that the values obtained from the present study are comparable reasonably. Using the values as provided in Tables 1 and 2, the ultimate bearing capacity of foundation under seismic loading condition can be evaluated.



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*Source: 3729360-2018-04-05.xml*