An extended probabilistic demand model with optimal intensity measures for seismic performance characterization of isolated bridges under coupled horizontal and vertical motions

Abstract

Intensity Measures (IMs) provide the relationship between Engineering Demand Parameters (EDPs) and seismic hazard characteristics for different structures and hence, their role in performance-based bridge design is significant. A few studies have investigated the optimal IMs for bridges under horizontal ground motions, however, detailed studies to explore the optimal IMs for isolated-bridges under the coupled vertical and horizontal records are very rare. This paper presents a procedure for the selection of optimal IMs for seismic-isolated bridges under the combined strong Horizontal Component (HC) and Vertical Component (VC) seismic excitations. Soil-Structure-Interaction (SSI) effects, the high level of uncertainties for soil properties, uncertainties for structural and geotechnical issues and advanced plasticity model for non-liquefied soil are included to explore the optimal vector-valued IMs. Four individual situations are considered to investigate the geometry effects on the selected optimal IMs. Optimal IMs criteria including efficiency, practicality, proficiency and sufficiency are investigated and developed to achieve a set of optimal vector-valued IMs for critical EDPs: drift ratio, pile-cap displacement and bearing displacement, which are affected by both HC and VC in isolated Soil-Pile-Bridge (SPB) systems. The results show that velocity-related IMs: peak ground velocity (PGV_H), Housner spectrum intensity (HI_H) and root mean square of velocity (V_RMSH) are optimal IMs as representative of HCs. In addition, structure-dependent spectral IMs: vertical spectral acceleration at T = 0.2 s (${\text{S}}_{\text{a}0.2}^{\text{V}}$) and square-root-of-the-sum-of-square of vertical spectral acceleration at the first and second vertical periods (${\text{S}}_{\text{a}}^{\text{V}}\left({\text{T}}_{\text{s}}\right)$) are the appropriate optimal IMs as representative of VCs. However, the displacement-related IM, peak ground displacement of HCs (PGD_H), may be treated as optimal IMs for SPB system supported by inclined pile foundations to investigate pile-cap displacement. In addition, the sufficiency term is investigated extensively with respect to seismological parameters.

Appendix

Number	Description	IM	Mathematical definition
1	Peak ground acceleration	PGA	Max $\left|\ddot{u}(t)\right|$
2	Peak ground velocity	PGV	Max $\left|\dot{u}(t)\right|$
3	Peak ground displacement	PGD	Max $\left|u(t)\right|$
4	Cumulative absolute velocity	CAV	${\int }_{0}^{{t}_{tot}}\left|\ddot{u}(t)\right|dt$
5	Cumulative absolute displacement	CAD	${\int }_{0}^{{t}_{tot}}\left|\dot{u}(t)\right| dt$
6	Arias intensity	${\mathrm{I}}_{\mathrm{A}}$	$\frac{\pi }{2g}{\int }_{0}^{{t}_{tot}}{[\ddot{u}(t)]}^{2}dt$
7	Root-mean square of acceleration	${\mathrm{A}}_{\mathrm{RMS}}$	$\sqrt{\frac{1}{{t}_{tot}}{\int }_{0}^{{t}_{tot}}{[\ddot{u}(t)]}^{2}dt}$
8	Root-mean square of velocity	${\mathrm{V}}_{\mathrm{RMS}}$	$\sqrt{\frac{1}{{t}_{tot}}{\int }_{0}^{{t}_{tot}}{[\dot{u}(t)]}^{2}dt}$
9	Root-mean square of displacement	${\mathrm{D}}_{\mathrm{RMS}}$	$\sqrt{\frac{1}{{t}_{tot}}{\int }_{0}^{{t}_{tot}}{[u(t)]}^{2}dt}$
10	Specific energy density	${\mathrm{S}}_{\mathrm{E}}$	${\int }_{0}^{{t}_{tot}}{[\dot{u}(t)]}^{2}dt$
11	Velocity spectrum density	VSI	${\int }_{0.1}^{2.5}{S}_{v}\left(T, \xi =5\%\right)dT$
12	Acceleration spectrum density	ASI	${\int }_{0.1}^{0.5}{S}_{a}\left(T, \xi =5\%\right)dT$
13	Housner intensity	${\mathrm{I}}_{\mathrm{H}}$	${\int }_{0.1}^{2.5}{PS}_{v}\left(T, \xi =5\%\right)dT$
14	Sustained maximum velocity	SMV	3rd largest peak in velocity
15	Horizontal spectral acceleration at the period of the first horizontal mode	${S}_{a}^{H}({T}_{H1})$	${S}_{a}^{H}({T}_{H1,} \xi =5\%)$
16	Vertical spectral acceleration at the period of the first vertical mode	${S}_{a}^{V}({T}_{V1})$	${S}_{a}^{V}({T}_{V1,} \xi =5\%)$
17	Horizontal spectral acceleration at the period of the second horizontal mode	${S}_{a}^{H}({T}_{H2})$	${S}_{a}^{H}({T}_{H2,} \xi =5\%)$
18	Vertical spectral acceleration at the period of the second vertical mode	${S}_{a}^{V}({T}_{V2})$	${S}_{a}^{V}({T}_{V2,} \xi =5\%)$
19	Square-root-of-the-sum-of-square (SRSS) of horizontal spectral acceleration at the first and second horizontal periods	${S}_{a}^{H}({T}_{S})$	$\sqrt{{{[S}_{a}^{H}({T}_{H1})]}^{2}+{{[S}_{a}^{H}({T}_{H2})]}^{2}}$
20	Square-root-of-the-sum-of-square (SRSS) of vertical spectral acceleration at the first and second vertical periods	${S}_{a}^{V}({T}_{S})$	$\sqrt{{{[S}_{a}^{V}({T}_{V1})]}^{2}+{{[S}_{a}^{V}({T}_{V2})]}^{2}}$
21	Vertical spectral acceleration at the predominant period	${S}_{a}^{V}({T}_{p})$	${S}_{a}^{V}({T}_{p,} \xi =5\%)$
22	Horizontal spectral acceleration at the predominant period	${S}_{a}^{H}({T}_{p})$	${S}_{a}^{H}({T}_{p1,} \xi =5\%)$
23	Combined-horizontal spectral acceleration	${S}_{a}^{H,1-5}$	$\sum_{\mathrm{i}=1}^{\text{N}}({S}_{a}^{H}{({\text{T}}_{\text{i}},\upxi ))}^{{{\alpha }}_{\text{i}}},$ ${{\alpha }}_{\text{i}}=\frac{{\text{m}}_{\hbox{i}}^{\text{eff}}}{{\sum }_{\text{j}=1}^{\hbox{N}}{\text{m}}_{\hbox{i}}^{\text{eff}}}$
24	Combined-vertical spectral acceleration	${S}_{a}^{V,1-5}$	$\sum_{\text{i}=1}^{\text{N}}({S}_{a}^{V}{({\text{T}}_{\text{i}},\upxi ))}^{{{\alpha }}_{\text{i}}},$ ${{\alpha }}_{\text{i}}=\frac{{\text{m}}_{\hbox{i}}^{\text{eff}}}{{\sum }_{\text{j}=1}^{\hbox{N}}{\text{m}}_{\hbox{i}}^{\text{eff}}}$
25	Horizontal spectral acceleration at the T = 2.0 s	${S}_{a2}^{H}$	${S}_{a}^{H}(T=2.0 s, \xi =5\%)$
26	Vertical spectral acceleration at the T = 2.0 s	${S}_{a0.2}^{V}$	${S}_{a}^{V}(T=0.2 s, \xi =5\%)$