Seismic codes based equivalent nonlinear and stochastic soil structure interaction analysis

Mohamed Elhebib Guellil; Zamila Harichane; Erkan Çelebi

doi:10.2478/sgem-2020-0007

Figures & Tables

Modelling of SSI: (a) schematic view of SSI system, (b) equivalent SDOF system.

Model of a rigid circular footing on a soil layer underlain by elastic half-space.

Impedance functions obtained before and after the correction of shear modulus and damping coefficient: (a) and (b) for horizontal motion, (c) and (d) for rotational motion.

SSI response with respect to the free field before and after the correction of shear modulus and damping coefficients: (a) displacement of the structure, (b) displacement of the mass and (c) total displacement of the base.

Comparison between stochastic (random G) and nonlinear (corrected G and ξ) SSI responses: (a) structure displacement, (b) mass displacement, (c) base displacement.

Design response spectrum: (a) RPA99-2003, (b) EC8-2004, (c) IBC-2015 and (d) IS-1893-2002.

Base shear forces: (a) RPA99-2003, (b) EC8-2004, (c) IBC-2015 and (d) IS-1893-2002.

Ordinate of elastic and inelastic design spectra for RPA99-2003, EC8-2004, IBC-2015 and IS-1893-2002_

	T ≤ T_B	T_B ≤ T ≤ TC	T ≥ T_C	Seismic base shear
RPA99v2003	Sag=1,25.A.⌊1+TT1(2.5ηQR−1)⌋(T1=TB)\matrix{ {{{{S_a}} \over g} = 1,25.A.\left\lfloor {1 + {T \over {{T_1}}}\left( {2.5\eta {Q \over R} - 1} \right)} \right\rfloor } \hfill & {\left( {{T_1} = {T_B}} \right)} \hfill \cr }	Sag=1,25.A.2,5.η.(QR)(T1≤T≤T2)\eqalign{ & \matrix{ {{{{S_a}} \over g} = 1,25.A.2,5.\eta .\left( {{Q \over R}} \right)} \hfill & {\left( {{T_1} \le T \le {T_2}} \right)} \hfill \cr } \cr & \cr}	Sag=1,25.A.2,5.η.(QR)(T2T)2/3(T2≤T≤T3)\matrix{ {{{{S_a}} \over g} = 1,25.A.2,5.\eta .\left( {{Q \over R}} \right){{\left( {{{{T_2}} \over T}} \right)}^{2/3}}} \hfill & {\left( {{T_2} \le T \le {T_3}} \right)} \hfill \cr }	V=A.D.QR.W{A:area acceleration coefficient,D:average dynamic amplification factorQ:quality factorR:overall behavior coefficient of the structureW: The effective seismic weight of the structureV = {{A.D.Q} \over R}.W\left\{ {\matrix{ {A:{\rm{area}}\,{\rm{acceleration}}\,{\rm{coefficient}},} \hfill \cr {D:{\rm{average}}\,{\rm{dynamic}}\,{\rm{amplification}}\,{\rm{factor}}} \hfill \cr {{\rm{Q}}:{\rm{quality}}\,{\rm{factor}}} \hfill \cr {{\rm{R}}:{\rm{overall}}\,{\rm{behavior}}\,{\rm{coefficient}}\,{\rm{of}}\,{\rm{the}}\,{\rm{structure}}} \hfill \cr {{\rm{W}}:\,{\rm{The}}\,{\rm{effective}}\,{\rm{seismic}}\,{\rm{weight}}\,{\rm{of}}\,{\rm{the}}\,{\rm{structure}}} \hfill \cr } } \right.
RPA99v2003			Sag=1,25.A.2,5.η.(QR)(T23)2/3(T3)5/3(T2≤T≤T3)\matrix{ {{{{S_a}} \over g} = 1,25.A.2,5.\eta .\left( {{Q \over R}} \right){{\left( {{{{T_2}} \over 3}} \right)}^{2/3}}{{\left( {{T \over 3}} \right)}^{5/3}}} \hfill & {\left( {{T_2} \le T \le {T_3}} \right)} \hfill \cr }
EC8-2004	Se=ag.S.⌊1+TTB(η2.5−1)⌋{S_e} = {a_g}.S.\left\lfloor {1 + {T \over {{T_B}}}\left( {\eta 2.5 - 1} \right)} \right\rfloor	S_e=2.5.a_g.S.η	TC≤T≤TD→Se=2.5.ag.S.η.[TCT]{T_C} \le T \le {T_D} \to {S_e} = 2.5.{a_g}.S.\eta .\left[ {{{{T_C}} \over T}} \right]	Fb=Sd(T)Wg.λ{λ=0.85 if T1≤2TCorλ=1.00 otherwise{F_b} = {S_d}\left( T \right){W \over g}.\lambda \left\{ {\matrix{ {\lambda = 0.85\,\,\,if\,\,\,{T_1} \le 2{T_C}} \hfill & {or} \hfill \cr {\lambda = 1.00\,otherwise} \hfill & {} \hfill \cr } } \right.
	Sd=ag.S.[23+TTB(2.5q−23)]{S_d} = {a_g}.S.\left[ {{2 \over 3} + {T \over {{T_B}}}\left( {{{2.5} \over q} - {2 \over 3}} \right)} \right]	S_d = 2.5.a_g.S.η	TC≤T≤TD→Sd{=2.5q.ag.S.[TCT]≥β.ag{T_C} \le T \le {T_D} \to {S_d}\left\{ {\matrix{ { = {{2.5} \over q}.{a_g}.S.\left[ {{{{T_C}} \over T}} \right]} \cr { \ge \beta .{a_g}} \cr } } \right.
			TD≤T≤4s→Se=2.5.ag.S.η.⌊TCTDT2⌋{T_D} \le T \le 4s \to {S_e} = 2.5.{a_g}.S.\eta .\left\lfloor {{{{T_C}{T_D}} \over {{T^2}}}} \right\rfloor
			T≥TD→Sd=2.5q.ag.S.⌊TCTDT2⌋≥β.agT \ge {T_D} \to {S_d} = {{2.5} \over q}.{a_g}.S.\left\lfloor {{{{T_C}{T_D}} \over {{T^2}}}} \right\rfloor \ge \beta .{a_g}
IBC-2015	Sag=1+15T(0.00≤T≤0.10)\matrix{ {{{{S_a}} \over g} = 1 + 15T} & {\left( {0.00 \le T \le 0.10} \right)} \cr }	S_a = S_DS (T₀≤T≤T_S)	Sa=SDST(T>Ts)\matrix{ {{S_a} = {{{S_{DS}}} \over T}} & {\left( {T > {T_s}} \right)} \cr }	V=Cs.W{Cs:The seismic response coefficientW: The effective seismic weight of the structureV = {C_s}.W\left\{ {\matrix{ {{C_s}:{\rm{The}}\,{\rm{seismic}}\,{\rm{response}}\,{\rm{coefficient}}} \hfill \cr {{\rm{W:}}\,{\rm{The}}\,{\rm{effective}}\,{\rm{seismic}}\,{\rm{weight}}\,{\rm{of}}\,{\rm{the}}\,{\rm{structure}}} \hfill \cr } } \right.
IS-1893-2002	Sag=1+15T(0.00≤T≤0.10)\matrix{ {{{{S_a}} \over g} = 1 + 15T} & {\left( {0.00 \le T \le 0.10} \right)} \cr }	Sag=2.50(0.10≤T≤0.40)\matrix{ {{{{S_a}} \over g} = 2.50} & {\left( {0.10 \le T \le 0.40} \right)} \cr }	Sag=1.00/T(0.40≤T≤4.00)\matrix{ {{{{S_a}} \over g} = 1.00/T} & {\left( {0.40 \le T \le 4.00} \right)} \cr }	V=Z.I.Sa2.R.g.W{Z:zone factorI:importance factorSa:average response acceleration coefficientR:response reduction factorg:acceleration due to gravityW:Seismic weight of the structureV = {{Z.I.{S_a}} \over {2.R.g}}.W\left\{ {\matrix{ {{\rm{Z:zone}}\,{\rm{factor}}} \hfill \cr {{\rm{I:importance}}\,{\rm{factor}}} \hfill \cr {{{\rm{S}}_{\rm{a}}}{\rm{:average}}\,{\rm{response}}\,{\rm{acceleration}}\,{\rm{coefficient}}} \hfill \cr {{\rm{R:}}\,{\rm{response}}\,{\rm{reduction}}\,{\rm{factor}}} \hfill \cr {{\rm{g:acceleration}}\,{\rm{due}}\,{\rm{to}}\,{\rm{gravity}}} \hfill \cr {{\rm{W}}:{\rm{Seismic}}\,{\rm{weight}}\,{\rm{of}}\,{\rm{the}}\,{\rm{structure}}} \hfill \cr } } \right.

Error margin (%) in displacements and frequency SSI system due to random G and corrected (G and ξ)_

SSI response	Deterministic	Stochastic (20% COV of G)	Nonlinear	Error margin (%)

				20% COV of G vs. initial G	Corrected (G & ξ) vs initial G
\|u\|/\|ug\|	1.81	1.51	1.21	− 16.6%	− 33.1%
\|u₀+hθ₀+u\|/\|ug	7.35	6.07	7.37	− 17.4%	+ 0.3%
\|u₀+ug\|/\|ug\|	0.86	0.86	0.95	±00.0%	+ 10.4%
Frequency (ω/ω_s)	0.5	0.5	0.41	±00.0%	−18.0%

Seismic codes based equivalent nonlinear and stochastic soil structure interaction analysis

Figures & Tables

Figure 1

Figure 2

Figure 3

Figure 4

Figure 5

Figure 6

Figure 7

Figure 8

Ordinate of elastic and inelastic design spectra for RPA99-2003, EC8-2004, IBC-2015 and IS-1893-2002_

Error margin (%) in displacements and frequency SSI system due to random G and corrected (G and ξ)_

Paradigm

My account