Seismic bearing capacity of shallow strip footing embedded in slope resting on two-layered soil

As shown in Fig. 3, wedge AMKJ is a known active wedge that is posited at the top layer, giving pressure to the passive wedge. The weight of the wedge WA=2B0−h1 cot αA12h1γ1, {W_A} = {{2{B_0} - {h_1}\,\cot \,{\alpha _{A1}}} \over 2}{h_1}{\gamma _1}, with the length of JK being equal to (1) (B0−h1 cot αA1) \left( {{B_0} - {h_1}\,\cot \,{\alpha _{A1}}} \right)

Total load acting on the foundation is given by (2) P1=pLB0 {P_1} = {p_L}{B_0}

Total cohesive force (C₁) on the slip lines AJ and MK is calculated as (3) C1(MK)=c1h1,C1(AJ)=c1AJ=c1h1 cos ecαA1C2(JK)=c2JK=c2(B0−h1 cot αA1) \matrix{ {{C_{1\left( {MK} \right)}} = {c_1}{h_1},{C_{1\left( {AJ} \right)}} = {c_1}AJ = {c_1}{h_1}\,\cos \,ec{\alpha _{A1}}} \hfill \cr {{C_{2\left( {JK} \right)}} = {c_2}JK = {c_2}\left( {{B_0} - {h_1}\,\cot \,{\alpha _{A1}}} \right)} \hfill \cr }

The intensity of load at layer thickness h₁ is expressed as (depicted in Fig. 7) (4) pL1=pLB0(B0+h1) {p_{L1}} = {{{p_L}{B_0}} \over {\left( {{B_0} + {h_1}} \right)}}

Conceding to limit equilibrium conditions, the authors can write (5) ∑V=0⇒C1(AJ) sin αA1+C1(MK)+RA1 cos(αA1−ϕ1)++ PA1 sin δ1+(pL1+γ1h1)JK(1−kv)−−(P1+WA)(1−kv)=0 \matrix{ {\sum {V = 0} } \hfill \cr { \Rightarrow {C_{1\left( {AJ} \right)}}\,\sin \,{\alpha _{A1}} + {C_{1\left( {MK} \right)}} + {R_{A1}}\,\cos \left( {{\alpha _{A1}} - {\phi _1}} \right) + } \hfill \cr { + {P_{A1}}\,\sin \,{\delta _1} + \left( {{p_{L1}} + {\gamma _1}{h_1}} \right)JK\left( {1 - {k_v}} \right) - } \hfill \cr { - \left( {{P_1} + {W_A}} \right)\left( {1 - {k_v}} \right) = 0} \hfill \cr } (6) ∑H=0⇒−C1(AJ)cos αA1+RA1 sin(αA1−ϕ1)−PA1 cos(δ1)−−{(pL1+γ1h1)JKkh tan ϕ2+c2(JK)JK}++(P1+WA)kh=0 \matrix{ {\sum {H = 0} } \hfill \cr { \Rightarrow - {C_{1\left( {AJ} \right)}}\cos \,{\alpha _{A1}} + {R_{A1}}\,\sin \left( {{\alpha _{A1}} - {\phi _1}} \right) - {P_{A1}}\,\cos \left( {{\delta _1}} \right) - } \hfill \cr { - \left\{ {\left( {{p_{L1}} + {\gamma _1}{h_1}} \right)JK{k_h}\,\tan \,{\phi _2} + {c_{2\left( {JK} \right)}}JK} \right\} + } \hfill \cr { + \left( {{P_1} + {W_A}} \right){k_h} = 0} \hfill \cr }

After solving Equations 5 and 6 and modifying both, the active pressure can be obtained as given below: (7) PA1=pLB0{(1−kv)sin(αA1−ϕ1)+kh cos(αA1−ϕ1)cos(αA1−ϕ1−δ1)}−{pLB0(B0+h1)(B0−h1 cot αA1)}{(1−kv)sin(αA1−ϕ1)+kh tan ϕ2 cos(αA1−ϕ1)cos(αA1−ϕ1−δ1)}++ 2B0−h1 cot αA12h1γ1{(1−kv)sin(αA1−ϕ1)+kh cos(αA1−ϕ1)cos(αA1−ϕ1−δ1)}−−γ1h1(B0−h1 cotαA1){(1−kv)sin(αA1−ϕ1)+kh tan ϕ2 cos(αA1−ϕ1)cos(αA1−ϕ1−δ1)}−−2c1h1sin(αA1−ϕ1)cos(αA1−ϕ1−δ1)−c2B0cos(αA1−ϕ1)cos(αA1−ϕ1−δ1)−c1h1 cot αA1cos(αA1−ϕ1)cos(αA1−ϕ1−δ1)+−c2h1 cot αA1cos(αA1−ϕ1)cos(αA1−ϕ1−δ1) \matrix{ {{P_{A1}} = {p_L}{B_0}\left\{ {{{\left( {1 - {k_v}} \right)\sin \left( {{\alpha _{A1}} - {\phi _1}} \right) + {k_h}\,\cos \left( {{\alpha _{A1}} - {\phi _1}} \right)} \over {\cos \left( {{\alpha _{A1}} - {\phi _1} - {\delta _1}} \right)}}} \right\}} \hfill \cr { - \left\{ {{{{p_L}{B_0}} \over {\left( {{B_0} + {h_1}} \right)}}\left( {{B_0} - {h_1}\,\cot \,{\alpha _{A1}}} \right)} \right\}} \hfill \cr {\left\{ {{{\left( {1 - {k_v}} \right)\sin \left( {{\alpha _{A1}} - {\phi _1}} \right) + {k_h}\,\tan \,{\phi _2}\,\cos \left( {{\alpha _{A1}} - {\phi _1}} \right)} \over {\cos \left( {{\alpha _{A1}} - {\phi _1} - {\delta _1}} \right)}}} \right\} + } \hfill \cr { + {{2{B_0} - {h_1}\,\cot \,{\alpha _{A1}}} \over 2}{h_1}{\gamma _1}} \hfill \cr {\left\{ {{{\left( {1 - {k_v}} \right)\sin \left( {{\alpha _{A1}} - {\phi _1}} \right) + {k_h}\,\cos \left( {{\alpha _{A1}} - {\phi _1}} \right)} \over {\cos \left( {{\alpha _{A1}} - {\phi _1} - {\delta _1}} \right)}}} \right\} - } \hfill \cr { - {\gamma _1}{h_1}\left( {{B_0} - {h_1}\,\cot {\alpha _{A1}}} \right)} \hfill \cr {\left\{ {{{\left( {1 - {k_v}} \right)\sin \left( {{\alpha _{A1}} - {\phi _1}} \right) + {k_h}\,\tan \,{\phi _2}\,\cos \left( {{\alpha _{A1}} - {\phi _1}} \right)} \over {\cos \left( {{\alpha _{A1}} - {\phi _1} - {\delta _1}} \right)}}} \right\} - } \hfill \cr { - 2{c_1}{h_1}{{\sin \left( {{\alpha _{A1}} - {\phi _1}} \right)} \over {\cos \left( {{\alpha _{A1}} - {\phi _1} - {\delta _1}} \right)}} - {c_2}{B_0}{{\cos \left( {{\alpha _{A1}} - {\phi _1}} \right)} \over {\cos \left( {{\alpha _{A1}} - {\phi _1} - {\delta _1}} \right)}}} \hfill \cr { - {c_1}{h_1}\,\cot \,{\alpha _{A1}}{{\cos \left( {{\alpha _{A1}} - {\phi _1}} \right)} \over {\cos \left( {{\alpha _{A1}} - {\phi _1} - {\delta _1}} \right)}} + } \hfill \cr { - {c_2}{h_1}\,\cot \,{\alpha _{A1}}{{\cos \left( {{\alpha _{A1}} - {\phi _1}} \right)} \over {\cos \left( {{\alpha _{A1}} - {\phi _1} - {\delta _1}} \right)}}} \hfill \cr }

From Fig. 4, active pressure distributing from the top layer to the wedge JKE.

Figure 4

The weight of the wedge: (8) WB=12(B0−h1 cot αA1)h2γ2 {W_B} = {1 \over 2}\left( {{B_0} - {h_1}\,\cot \,{\alpha _{A1}}} \right){h_2}{\gamma _2}

Base shear at the interface between the two layers is given as: (9) (pL1+γ1h1)kh tan φ2+c2and(pL1+γ1h1)kh tan φ1+c1 \matrix{ {\left( {{p_{L1}} + {\gamma _1}{h_1}} \right){k_h}\,\,\tan \,{\varphi _2} + {c_2}} \hfill \cr {{\rm{and}}} \hfill \cr {\left( {{p_{L1}} + {\gamma _1}{h_1}} \right){k_h}\,\,\tan \,{\varphi _1} + {c_1}} \hfill \cr }

Total cohesive force at the slip lines JE and KE is expressed as: (10) C2(KE)=c2KE=c2h2,C2(JE)=c2KE=c2h2 cos ecαA2andC1(JK)=c1JK=c1(B0−h1 cot αA1) \matrix{ {{C_{2\left( {KE} \right)}} = {c_2}KE = {c_2}{h_2},} \hfill \cr {{C_{2\left( {JE} \right)}} = {c_2}KE = {c_2}{h_2}\,\cos \,ec{\alpha _{A2}}} \hfill \cr {{\rm{and}}} \hfill \cr {{C_{1\left( {JK} \right)}} = {c_1}JK = {c_1}\left( {{B_0} - {h_1}\,\cot \,{\alpha _{A1}}} \right)} \hfill \cr }

Conceding to limit equilibrium conditions, (11) ∑V=0⇒C2(JE)sin αA2+C2(KE)++ RA2 cos(αA2−ϕ2)+PA2sin δ2−−(pL1+γ1h1)JK(1−kv)−WB(1−kv)=0 \matrix{ {\sum {V = 0} } \hfill \cr { \Rightarrow {C_{2(JE)}}\sin \,{\alpha _{A2}} + {C_{2\left( {KE} \right)}} + } \hfill \cr { + {R_{A2}}\,\cos \left( {{\alpha _{A2}} - {\phi _2}} \right) + {P_{A2}}\sin \,{\delta _2} - } \hfill \cr { - \left( {{p_{L1}} + {\gamma _1}{h_1}} \right)JK\left( {1 - {k_v}} \right) - {W_B}\left( {1 - {k_v}} \right) = 0} \hfill \cr } (12) ∑H=0⇒C1(JK)−C2(JE) cos αA2++ RA2 sin(αA2−ϕ2)−PA2 cos δ2++ (pL1+γ1h1)JKh tan ϕ1+WBkh=0 \matrix{ {\sum {H = 0} } \hfill \cr { \Rightarrow {C_{1\left( {JK} \right)}} - {C_{2\left( {JE} \right)}}\,\cos \,{\alpha _{A2}} + } \hfill \cr { + {R_{A2}}\,\sin \left( {{\alpha _{A2}} - {\phi _2}} \right) - {P_{A2}}\,\cos \,{\delta _2} + } \hfill \cr { + \left( {{p_{L1}} + {\gamma _1}{h_1}} \right)J{K_h}\,\tan \,{\phi _1} + {W_B}{k_h} = 0} \hfill \cr }

Solving Equations 11 and 12 and simplifying them, we obtain (13) PA2=f1(pL1,c2,ϕ2,γ2,αA2) {P_{A2}} = {f_1}\left( {{p_{L1,}}{c_2},{\phi _2},{\gamma _2},{\alpha _{A2}}} \right)

The details of equations of P_A2 are given in Appendix I.

Hence, total active pressure from both the layers is given by (14) PA=PA1+PA2 {P_A} = {P_{A1}} + {P_{A2}}

Due to active pressure generated in the top layer, the passive zone gives resistance to the active pressure. The weight of the passive wedge, as depicted in Fig. 5, is given as (15) Wc=[h1 cot αp1+2h2 cot αp22h1−−14(h1 cot αp1+h2 cot αp2−Dftan i+B02)2tan αp1]γ1 \matrix{ {{W_c} = \left[ {{{{h_1}\,\cot \,{\alpha _{p1}} + 2{h_2}\,\cot \,{\alpha _{p2}}} \over 2}} \right.{h_1} - } \hfill \cr {\left. { - {1 \over 4}{{\left( {{h_1}\,\cot \,{\alpha _{p1}} + {h_2}\,\cot \,{\alpha _{p2}} - {{{D_f}} \over {\tan \,i}} + {{{B_0}} \over 2}} \right)}^2}\tan \,{\alpha _{p1}}} \right]{\gamma _1}} \hfill \cr }

Figure 5

Extra loadingacting on the foundation is expressed as surcharge load (16) q1=γ1(YM/2)=γ112tan i(Dftan i−B02) \matrix{ {{q_1}} \hfill & { = {\gamma _1}\left( {YM/2} \right)} \hfill \cr {} \hfill & { = {\gamma _1}{1 \over 2}\tan \,i\left( {{{{D_f}} \over {\tan \,i}} - {{{B_0}} \over 2}} \right)} \hfill \cr }

Total surcharge load in the top layer of the passive zone is given as (17) Q1=q1MR=γ12tan i(Dftan i−B02)2 \matrix{ {{Q_1}} \hfill & { = {q_1}MR} \hfill \cr {} \hfill & { = {{{\gamma _1}} \over 2}\tan \,i{{\left( {{{{D_f}} \over {\tan \,i}} - {{{B_0}} \over 2}} \right)}^2}} \hfill \cr }

Total cohesive force in the slip lines KG and GD of the passive wedge is given as (18) C1(KG)=c1KG−c1h2 cot αp2,C1(MK)=c1MK=c1h1C1(GD)=c1GD=c1h1 cos ecαp1−−c1{12(h1 cot αp1+h2 cot αp2−Dftan i+B02)sec αp1} \matrix{ {{C_{1\left( {KG} \right)}} = {c_1}KG - {c_1}{h_2}\,\cot \,{\alpha _{p2}},{C_{1\left( {MK} \right)}} = {c_1}MK = {c_1}{h_1}} \hfill \cr {{C_{1\left( {GD} \right)}} = {c_1}GD = {c_1}{h_1}\,\cos \,ec{\alpha _{p1}} - } \hfill \cr { - {c_1}\left\{ {{1 \over 2}\left( {{h_1}\,\cot \,{\alpha _{p1}} + {h_2}\,\cot \,{\alpha _{p2}} - {{{D_f}} \over {\tan \,i}} + {{{B_0}} \over 2}} \right)\sec \,{\alpha _{p1}}} \right\}} \hfill \cr }

Using limit equilibrium equations (19) ∑V=0⇒−C1(MK)−C1(GD)sin αp1++ Rp1 cos(ϕ1+αp1)−Pp1sin δ1++ (q2+γ1h1)KG(1−kv)−−(Q1+Wc)(1−kv)=0 \matrix{ {\sum {V = 0} } \hfill \cr { \Rightarrow - {C_{1\left( {MK} \right)}} - {C_{1\left( {GD} \right)}}\sin \,{\alpha _{p1}} + } \hfill \cr { + {R_{p1}}\,\cos \left( {{\phi _1} + {\alpha _{p1}}} \right) - {P_{p1}}\sin \,{\delta _1} + } \hfill \cr { + \left( {{q_2} + {\gamma _1}{h_1}} \right)KG\left( {1 - {k_v}} \right) - } \hfill \cr { - \left( {{Q_1} + {W_c}} \right)\left( {1 - {k_v}} \right) = 0} \hfill \cr } (20) ∑H=0⇒−C1(GD) cos αp1+C1(GK)−−Rp1 sin(ϕ1+αp1)+Pp1 cos δ1++ (q2+γ1h1)KGkh tan ϕ2++ (Q1+Wc)kh=0 \matrix{ {\sum {H = 0} } \hfill \cr { \Rightarrow - {C_{1\left( {GD} \right)}}\,\cos \,{\alpha _{p1}} + {C_{1\left( {GK} \right)}} - } \hfill \cr { - {R_{p1}}\,\sin \left( {{\phi _1} + {\alpha _{p1}}} \right) + {P_{p1}}\,\cos \,{\delta _1} + } \hfill \cr { + \left( {{q_2} + {\gamma _1}{h_1}} \right)KG{k_h}\,\tan \,{\phi _2} + } \hfill \cr { + \left( {{Q_1} + {W_c}} \right){k_h} = 0} \hfill \cr }

Solving Equations 19 and 20 and modifying both equations, passive resistance can be expressed as: (21) Pp1=[h1 cot αp1+h2 cot αp22h1−−14(h1 cot αp1+h2 cot αp2−Dftan i+B02)2tan αp1]γ1{(1−kv)sin(ϕ1+αp1)−kh cos(ϕ1+αp1)cos(ϕ1+αp1+δ1)}−γ1h1h2 cot αp2{(1−kv)sin(ϕ1+αp1)+kh tan ϕ2 cos(ϕ+αp1)cos(ϕ1+αp1+δ1)}−γ1tan(Dftan i−B02)(2Df−B0 tan i)2Df+tan i(2h1−B0)h2 cotαp2{(1−kv)sin(ϕ1+αp1)+kh tan ϕ2 cos(ϕ1+αp1)cos(ϕ1+αp1+δ1)}+ γ1 tan i(Dftan i−B02)2{(1−kv)sin(ϕ1+αp1)−kh cos(ϕ1+αp1)cos(ϕ1+αp1+δ1)}+c1h1{sin(ϕ1+αp1)cos(ϕ1+αp1+δ1)}+ c1 sin αp1[h12cos ecαp1−h22cot αp2sec αp1+12Dftan isec αp1−B04sec αp1]{sin(ϕ1+αp1)cos(ϕ1+αp1+δ1)}+ c1 cos αp1[h12cos ecαp1−h22cot αp2 secαp1+12Dftan isec αp1−B04sec αp1]cos(ϕ1+αp1)cos(ϕ1+αp1+δ1)−c1h2 cot αp2cos(ϕ1+αp1)cos(ϕ1+αp1+δ1) \matrix{ {{P_{p1}} = \left[ {{{{h_1}\,\cot \,{\alpha _{p1}} + {h_2}\,\cot \,{\alpha _{p2}}} \over 2}} \right.{h_1} - } \hfill \cr {\left. { - {1 \over 4}{{\left( {{h_1}\,\cot \,{\alpha _{p1}} + {h_2}\,\cot \,{\alpha _{p2}} - {{{D_f}} \over {\tan \,i}} + {{{B_0}} \over 2}} \right)}^2}\tan \,{\alpha _{p1}}} \right]{\gamma _1}} \hfill \cr {\left\{ {{{\left( {1 - {k_v}} \right)\sin \left( {{\phi _1} + {\alpha _{p1}}} \right) - {k_h}\,\cos \left( {{\phi _1} + {\alpha _{p1}}} \right)} \over {\cos \left( {{\phi _1} + {\alpha _{p1}} + {\delta _1}} \right)}}} \right\}} \hfill \cr { - {\gamma _1}{h_1}{h_2}\,\cot \,{\alpha _{p2}}\left\{ {{{\left( {1 - {k_v}} \right)\sin \left( {{\phi _1} + {\alpha _{p1}}} \right) + {k_h}\,\tan \,{\phi _2}\,\cos \left( {\phi + {\alpha _{p1}}} \right)} \over {\cos \left( {{\phi _1} + {\alpha _{p1}} + {\delta _1}} \right)}}} \right\}} \hfill \cr { - {\gamma _1}{{\tan \left( {{{{D_f}} \over {\tan \,i}} - {{{B_0}} \over 2}} \right)\left( {2{D_f} - {B_0}\,\tan \,i} \right)} \over {2{D_f} + \tan \,i\left( {2{h_1} - {B_0}} \right)}}{h_2}\,\cot {\alpha _{p2}}} \hfill \cr {\left\{ {{{\left( {1 - {k_v}} \right)\sin \left( {{\phi _1} + {\alpha _{p1}}} \right) + {k_h}\,\tan \,{\phi _2}\,\cos \left( {{\phi _1} + {\alpha _{p1}}} \right)} \over {\cos \left( {{\phi _1} + {\alpha _{p1}} + {\delta _1}} \right)}}} \right\}} \hfill \cr { + {\gamma _1}\,\tan \,i{{\left( {{{{D_f}} \over {\tan \,i}} - {{{B_0}} \over 2}} \right)}^2}\left\{ {{{\left( {1 - {k_v}} \right)\sin \left( {{\phi _1} + {\alpha _{p1}}} \right) - {k_h}\,\cos \left( {{\phi _1} + {\alpha _{p1}}} \right)} \over {\cos \left( {{\phi _1} + {\alpha _{p1}} + {\delta _1}} \right)}}} \right\} + } \hfill \cr {{c_1}{h_1}\left\{ {{{\sin \left( {{\phi _1} + {\alpha _{p1}}} \right)} \over {\cos \left( {{\phi _1} + {\alpha _{p1}} + {\delta _1}} \right)}}} \right\}} \hfill \cr { + {c_1}\,\sin \,{\alpha _{p1}}\left[ {{{{h_1}} \over 2}\cos \,ec{\alpha _{p1}} - {{{h_2}} \over 2}\cot \,{\alpha _{p2}}\sec \,{\alpha _{p1}} + {1 \over 2}{{{D_f}} \over {\tan \,i}}\sec \,{\alpha _{p1}} - {{{B_0}} \over 4}\sec \,{\alpha _{p1}}} \right]} \hfill \cr {\left\{ {{{\sin \left( {{\phi _1} + {\alpha _{p1}}} \right)} \over {\cos \left( {{\phi _1} + {\alpha _{p1}} + {\delta _1}} \right)}}} \right\}} \hfill \cr { + {c_1}\,\cos \,{\alpha _{p1}}\left[ {{{{h_1}} \over 2}\cos \,ec{\alpha _{p1}} - {{{h_2}} \over 2}\cot \,{\alpha _{p2}}\,\sec {\alpha _{p1}} + {1 \over 2}{{{D_f}} \over {\tan \,i}}\sec \,{\alpha _{p1}} - {{{B_0}} \over 4}\sec \,{\alpha _{p1}}} \right]} \hfill \cr {{{\cos \left( {{\phi _1} + {\alpha _{p1}}} \right)} \over {\cos \left( {{\phi _1} + {\alpha _{p1}} + {\delta _1}} \right)}} - {c_1}{h_2}\,\cot \,{\alpha _{p2}}{{\cos \left( {{\phi _1} + {\alpha _{p1}}} \right)} \over {\cos \left( {{\phi _1} + {\alpha _{p1}} + {\delta _1}} \right)}}} \hfill \cr }

Total weight of the passive wedge KGE, as shownin Fig. 6, is calculated as (22) WD=12h22 cot αp2γ2 {W_D} = {1 \over 2}{h_2}^2\,\cot \,{\alpha _{p2}}{\gamma _2}

Figure 6

Figure 7

According to 2 : 1 load distribution method, intensity of surcharge load at depth h₁ can be written as (23) q2=q1MR(MR+h1)=γ12tan i(Dtan i−B02)2(Dftan i−B02)+h1 \matrix{ {{q_2}} \hfill & { = {{{q_1}MR} \over {\left( {MR + {{h_1}}} \right)}}} \hfill \cr {} \hfill & { = {{{{{\gamma _1}} \over 2}\tan \,i{{\left( {{D \over {\tan \,i}} - {{{B_0}} \over 2}} \right)}^2}} \over {\left( {{{{D_f}} \over {\tan \,i}} - {{{B_0}} \over 2}} \right) + {h_1}}}} \hfill \cr }

Base shear between two passive wedges can be coded as (24) (q2+γ1h1)kh tan φ2+c1 and (q2+γ1h1)kh tan φ1+c1 \left( {{q_2} + {\gamma _1}{h_1}} \right){k_h}\,\,\tan \,{\varphi _2} + {c_1}\,\,{\rm{and}}\,\,\left( {{q_2} + {\gamma _1}{h_1}} \right){k_h}\,\,\tan \,{\varphi _1} + {c_1}

Cohesive forces in slip lines KE, KG and GE are given as

Case 1: Considering the effective area of active zones (33) γ¯=(2−h1B0cot αA1)2h1γ1+(1−h1B0cot αA1)2h2γ2(2−h1B0cot αA1)2h1+(1−h1B0cot αA1)2h2 \bar \gamma = {{{{\left( {2 - {{{h_1}} \over {{B_0}}}\cot \,{\alpha _{A1}}} \right)} \over 2}{h_1}{\gamma _1} + {{\left( {1 - {{{h_1}} \over {{B_0}}}\cot \,{\alpha _{A1}}} \right)} \over 2}{h_2}{\gamma _2}} \over {{{\left( {2 - {{{h_1}} \over {{B_0}}}\cot \,{\alpha _{A1}}} \right)} \over 2}{h_1} + {{\left( {1 - {{{h_1}} \over {{B_0}}}\cot \,{\alpha _{A1}}} \right)} \over 2}{h_2}}}

The bearing capacity factor (N_γ′) is a function of several parameters including cohesion, surcharge and unit weight. It can be expressed as: (35) Nγ″=(a1e1+b1e1+2c¯γ¯B0d1e1) {N_{\gamma ''}} = \left( {{{{a_1}} \over {{e_1}}} + {{{b_1}} \over {{e_1}}} + {{2\bar c} \over {\bar \gamma {B_0}}}{{{d_1}} \over {{e_1}}}} \right)

The details of the equations a₁, b₁, e₁ and d₁ are given in Appendix II.

Where c̅ is averaged cohesion in each layer in the slip line is shown by (36) c¯=c1h1+c2h2h1+h2 \bar c = {{{c_1}{h_1} + {c_2}{h_2}} \over {{h_1} + {h_2}}} a₁, b₁, e₁ and d₁ are dimensionless equations.

In seismic condition, (N_γ″) can be expressed as (N_γE) and in static condition, (N_γ″) can be expressed as (N_γS), whereas (N_γ″) is the unity factor for the simultaneous resistance of unit weight, surcharge and cohesion.

Merifield et al. (1999) investigated the bearing capacity of a strip footing resting on a two-layer clay deposit with a horizontal ground surface and proposed a modified bearing capacity factor N_c^*, which can be expressed as: (37) NC*=quc1 {N_C}^* = {{{q_u}} \over {{c_1}}} where q_u= ultimate bearing capacity and c₁ = undrained shear strength of the top layer.

Similarly, a dimensionless undrained bearing capacity factor N_cs is defined in the present study, which is a function of the parameters c_u/γB₀, D_f/B₀, c₁/c₂, i, k_h and it can be described by the following equation: (38) Ncs=qult/c1=f(cu/γB0,Df/B0,c1/c2,i,kh) {N_{cs}} = {q_{ult}}/{c_1} = f\left( {{c_u}/\gamma {B_0},{D_f}/{B_0},{c_1}/{c_2},i,{k_h}} \right)

Ranges of various parameters are given as follows: ϕ1ϕ2=0.6,0.8,1γ1γ2=0.6,0.8,1h1B0=0.1,0.25,0.5i=100,150,200,250kv=0, kh/2, kh \matrix{ {{{{\phi _1}} \over {{\phi _2}}} = 0.6,0.8,1{{{\gamma _1}} \over {{\gamma _2}}} = 0.6,0.8,1{{{h_1}} \over {{B_0}}} = 0.1,0.25,0.5} \cr {i = {{10}^0},{{15}^0},{{20}^0},{{25}^0}{k_v} = 0,\,{k_h}/2,\,{k_h}} \cr } δ1δ2 {{{\delta _1}} \over {{\delta _2}}} is the ratio of wall friction angles between the top and bottom layers.

Since the heuristic algorithms give us low ramification and high execution and these methods are relatively new, they can be applied in the geotechnical problem. Out of these methods, a brief discussion on PSO is given here as it is used in the analysis.

Kennedy and Eberhart (1995) developed PSO as a simulation of birds swarm. A swarm is a group of individuals with defined rules for individual behaviours and communication. The ability of each individual to deal with the previous experiences of the swarm is called swarm intelligence. This capability guides the swarm towards its optimum goal. PSO is a population-based search technique where a population of particles start their journey in a space concerning the current best position (Hossain and EI-Shafie 2014; Hajihassani et al. 2017). Reynolds (1987) described three simple rules for the behaviours of individuals inside a swarm, which were used as one of the basic concepts of PSO by Kennedy and Eberhart (1995). Although these simple rules model the behaviour of individuals, their combination produces a complicated behaviour for the swarm.

The process of decision-making related to individuals is another basic concept of PSO. Each individual of the swarm makes decision based on the following two factors:

Own experiences of the individual that is its bestresults so far

Figure 10 illustrates the standard flowchart of PSO. At the starting step of the original PSO, a certain number of individuals, called particles, are distributed in the search space by using a random pattern (Kennedy and Eberhart 1995; Cheng et al. 2007; Aote et al. 2013). Each particle is representative of a feasible solution. Figure 11 shows the schematic structure of a particle in PSO involving three divided parts as its current position, best position and velocity. The current position, best position and velocity of particles record the current coordinates, best coordinates and velocity vectors of a particle in D-dimensional space, respectively, where D starts from 1 (Kalatehjari 2013). Consequently, for a particle in D-dimensional space, a 3D-dimensional particle is desirable. PSO aims to meet the termination criteria which are defined as the criteria for terminating the iterative search process. To select an appropriate termination criterion, it should be noted that the termination condition does not cause a premature converge and it should protect against oversampling of the fitness (Engelbrecht 2007). The following termination criteria are frequently used in PSO:

Termination when the maximum number ofiterations is exceeded

Figure 10

Figure 11

These criteria are applied to ensure that PSO can converge on a feasible solution. Although PSO has some limitation which is explained by Gbenga et al. (2016) and Aote et al. (2013), PSO tries to make the objective function as a minimum or maximum dependingon the problem to be solved. To lead the swarm towards this aim, the fitness value of each particle is determined by evaluating its current position by the objective function. After evaluation offitness of all particles, Equation 39 (velocity equation) is used to calculate the velocity of particles based on their best position and the position of the best particle in the swarm. Using Equation 40, particle positions can be updated according to their current positions and velocities. This iterative process continues until reaching the termination criteria. Equations 39 and 40 are as follows (Kennedy and Eberhart 1995): (39) vn(i)=vn(i−1)+u(0,ϑ1)(bpn(i)−Xn(i))++ u(0,ϑ2)(bgn(i)−Xn(i)) \matrix{ {{v_{n\left( i \right)}} = {v_{n\left( {i - 1} \right)}} + u\left( {0,{\vartheta _1}} \right)\left( {{b_{{p_{n\left( i \right)}}}} - {X_{n\left( i \right)}}} \right) + } \hfill \cr { + u\left( {0,{\vartheta _2}} \right)\left( {{b_{{g_{n\left( i \right)}}}} - {X_{n\left( i \right)}}} \right)} \hfill \cr } (40) Xn(i+1)=Xn(i)+vn(i) {X_{n\left( {i + 1} \right)}} = {X_{n\left( i \right)}} + {v_{n\left( i \right)}} where v is the velocity of an nth particle in the past iteration and v_(n−1) is the velocity of the n^th particle in the current iteration. The vectors of random numbers of an n^th particle are presented by u(0, ϑ₁) and u(0, ϑ₂), b_pn(i) is the best position of the nth particle so far, b_gn(i) is the position of the best particle of the swarm so far, and X_n(i−1) and X_n(i) are the positions of the nth particle in the current and next iterations, respectively. Input parameters are taken for optimisation as follows: Input h1/B0,ϕ2,ϕ1/ϕ2,δ2,δ1/δ2,2c¯/γ¯B0,kh,kv,ξ,αA1=200−800 αA2=300−800αp1=300−800 αp2=300−600 \matrix{ {{\rm{Input}}\,{h_1}/{B_0},{\phi _2},{\phi _1}/{\phi _2},{\delta _2},{\delta _1}/{\delta _2},2\bar c/\bar \gamma {B_0},} \hfill \cr {{k_h},{k_v},\xi ,{\alpha _{A1}} = {{20}^0} - {{80}^0}\,{\alpha _{A2}} = {{30}^0} - {{80}^0}} \hfill \cr {{\alpha _{p1}} = {{30}^0} - {{80}^0}\,{\alpha _{p2}} = {{30}^0} - {{60}^0}} \hfill \cr }