Seismic codes based equivalent nonlinear and stochastic soil structure interaction analysis

Mohamed Elhebib Guellil; Zamila Harichane; Erkan Çelebi

Open Access

Seismic codes based equivalent nonlinear and stochastic soil structure interaction analysis

Mohamed Elhebib Guellil

Zamila Harichane

and

Erkan Çelebi

| Oct 23, 2020

Studia Geotechnica et Mechanica

Volume 43 (2021): Issue 1 (April 2021)

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Cite

Published Online: Oct 23, 2020

Page range: 1 - 14

Received: Mar 31, 2020

Accepted: Aug 29, 2020

DOI: https://doi.org/10.2478/sgem-2020-0007

Keywords
Shear modulus, damping, uncertainties, linear iterative method, seismic code

© 2021 Mohamed Elhebib Guellil et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

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	$\begin{array}{l} \frac{S_{a}}{g} = 1, 25 . A . ⌊ 1 + \frac{T}{T_{1}} (2.5 η \frac{Q}{R} - 1) ⌋ & (T_{1} = T_{B}) \end{array}$ \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 }	$\begin{array}{l} \begin{array}{l} \frac{S_{a}}{g} = 1, 25 . A . 2, 5 . η . (\frac{Q}{R}) & (T_{1} \leq T \leq T_{2}) \end{array} \end{array}$ \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}	$\begin{array}{l} \frac{S_{a}}{g} = 1, 25 . A . 2, 5 . η . (\frac{Q}{R}) {(\frac{T_{2}}{T})}^{2 / 3} & (T_{2} \leq T \leq T_{3}) \end{array}$ \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 }	$V = \frac{A . D . Q}{R} . W {\begin{array}{l} A : area acceleration coefficient, \\ D : average dynamic amplification factor \\ Q : quality factor \\ R : overall behavior coefficient of the structure \\ W : The effective seismic weight of the structure \end{array}$ V = {{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			$\begin{array}{l} \frac{S_{a}}{g} = 1, 25 . A . 2, 5 . η . (\frac{Q}{R}) {(\frac{T_{2}}{3})}^{2 / 3} {(\frac{T}{3})}^{5 / 3} & (T_{2} \leq T \leq T_{3}) \end{array}$ \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	$S_{e} = a_{g} . S . ⌊ 1 + \frac{T}{T_{B}} (η 2.5 - 1) ⌋$ {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	$T_{C} \leq T \leq T_{D} \to S_{e} = 2.5 . a_{g} . S . η . [\frac{T_{C}}{T}]$ {T_C} \le T \le {T_D} \to {S_e} = 2.5.{a_g}.S.\eta .\left[ {{{{T_C}} \over T}} \right]	$F_{b} = S_{d} (T) \frac{W}{g} . λ {\begin{array}{l} λ = 0.85 if T_{1} \leq 2 T_{C} & or \\ λ = 1.00 otherwise \end{array}$ {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.
	$S_{d} = a_{g} . S . [\frac{2}{3} + \frac{T}{T_{B}} (\frac{2.5}{q} - \frac{2}{3})]$ {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	$T_{C} \leq T \leq T_{D} \to S_{d} {\begin{matrix} = \frac{2.5}{q} . a_{g} . S . [\frac{T_{C}}{T}] \\ \geq β . a_{g} \end{matrix}$ {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.
			$T_{D} \leq T \leq 4 s \to S_{e} = 2.5 . a_{g} . S . η . ⌊ \frac{T_{C} T_{D}}{T^{2}} ⌋$ {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
			$T \geq T_{D} \to S_{d} = \frac{2.5}{q} . a_{g} . S . ⌊ \frac{T_{C} T_{D}}{T^{2}} ⌋ \geq β . a_{g}$ T \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	$\begin{matrix} \frac{S_{a}}{g} = 1 + 15 T & (0.00 \leq T \leq 0.10) \end{matrix}$ \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)	$\begin{matrix} S_{a} = \frac{S_{DS}}{T} & (T > T_{s}) \end{matrix}$ \matrix{ {{S_a} = {{{S_{DS}}} \over T}} & {\left( {T > {T_s}} \right)} \cr }	$V = C_{s} . W {\begin{array}{l} C_{s} : The seismic response coefficient \\ W : The effective seismic weight of the structure \end{array}$ V = {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	$\begin{matrix} \frac{S_{a}}{g} = 1 + 15 T & (0.00 \leq T \leq 0.10) \end{matrix}$ \matrix{ {{{{S_a}} \over g} = 1 + 15T} & {\left( {0.00 \le T \le 0.10} \right)} \cr }	$\begin{matrix} \frac{S_{a}}{g} = 2.50 & (0.10 \leq T \leq 0.40) \end{matrix}$ \matrix{ {{{{S_a}} \over g} = 2.50} & {\left( {0.10 \le T \le 0.40} \right)} \cr }	$\begin{matrix} \frac{S_{a}}{g} = 1.00 / T & (0.40 \leq T \leq 4.00) \end{matrix}$ \matrix{ {{{{S_a}} \over g} = 1.00/T} & {\left( {0.40 \le T \le 4.00} \right)} \cr }	$V = \frac{Z . I . S_{a}}{2 . R . g} . W {\begin{array}{l} Z : zone factor \\ I : importance factor \\ S_{a} : average response acceleration coefficient \\ R : response reduction factor \\ g:acceleration due to gravity \\ W : Seismic weight of the structure \end{array}$ V = {{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%

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Seismic codes based equivalent nonlinear and stochastic soil structure interaction analysis

Article Category: Original Study

Published Online: Oct 23, 2020

Page range: 1 - 14

Received: Mar 31, 2020

Accepted: Aug 29, 2020

DOI: https://doi.org/10.2478/sgem-2020-0007

Keywords
Shear modulus, damping, uncertainties, linear iterative method, seismic code

© 2021 Mohamed Elhebib Guellil et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Figure 1

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Figure 3

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Figure 6

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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).

Seismic codes based equivalent nonlinear and stochastic soil structure interaction analysis

Article Category: Original Study

Published Online: Oct 23, 2020

Page range: 1 - 14

Received: Mar 31, 2020

Accepted: Aug 29, 2020

DOI: https://doi.org/10.2478/sgem-2020-0007

KeywordsShear modulus, damping, uncertainties, linear iterative method, seismic code

© 2021 Mohamed Elhebib Guellil et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

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).

Keywords
Shear modulus, damping, uncertainties, linear iterative method, seismic code