Reconstruction policies: explicitating the link of decisions thresholds to safety level and costs for RC buildings

Abstract

In (FEMA 308 in Repair of earthquake damaged concrete and masonry wall buildings, Prepared by ATC for the Federal Emergency Management Agency, Washington DC, 1998) a general framework facilitating decisions on appropriate course of action (accept damage, restore, or upgrade) for damaged buildings after an earthquake was presented. Such Performance-Based Policy Framework relies on performance index of the building in its intact and damaged state and on the relative performance loss as significant indicators for repair and/or upgrade decisions; however, no specific guidance for the establishment of PL and IP thresholds governing damage acceptability was given. This paper proposes an improvement of the PBPF introduced in FEMA 308 by providing it with a set of performance thresholds that can be established based on a clear quantitative approach and presents a set of tools for its practical implementation with reference to Reinforced Concrete frame buildings. In addition to the Repair and Repair and Retrofit options, also the Demolish one is introduced, because it corresponds to a not infrequent decision in case of severely damaged buildings in European-Mediterranean regions. Criteria to establish decision thresholds IP and PL are proposed that allow to clearly connect them to reconstruction costs and probability of failure of the building in the intact and damaged state. Finally, a case study demonstrates the application of the hypothesized policy framework for a Municipality in southern Italy and investigates on the effects of varying decision thresholds on the total reconstruction costs and mean safety level for the considered building stock.

Appendices

Appendix 1

a _gC  = peak ground acceleration corresponding to collapse

a _gD  = peak ground acceleration corresponding to seismic demand

α = factor accounting for the willingness of having a new building with respect to a repaired/retrofitted one

β _TOT  = dispersion measure of the total uncertainty (uncertainty in ground motion and modeling)

$C_{b} = \frac{{V_{b} }}{{\varGamma \cdot m^{ * } \cdot g}}$ base shear coefficient

CN = normalized cost of a new construction (including demolition cost)

C _r  = repair cost normalized with respect to new construction cost (including demolition cost)

C _r,min , C _r,max  = minimum and maximum allowed repair costs normalized with respect to new construction cost (including demolition cost)

C _u  = upgrading costs normalized with respect to new construction cost (including demolition cost)

d* = displacement for the equivalent SDOF system

d _C , d _D  = global displacement capacity and demand for MDOF system

D = Demolish action

EA = elevation area determined as product of building surface area times the storey number

f = spectral amplification factor (see e.g. Equation (2))

f _c  = cylindrical concrete compressive strength

$\varGamma = \frac{{\sum {m_{i} \cdot \varPhi_{i} } }}{{\sum {m_{i} \cdot \varPhi_{i}^{2} } }}$ MDOF–SDOF transformation factor (= 1 in case of soft storey mechanism)

k ₀ , k = coefficient for linear regression of the hazard curve in the log–log plane

IM = intensity measure

IP = initial performance index

IP ₁  = minimum acceptable value of initial performance

IP ₂  = suitably established threshold for initial performance

$m^{ * } = \sum {m_{i} \cdot \varPhi_{i} }$ mass of the equivalent SDOF system

$\mu = \frac{{d_{{}}^{*} }}{{d_{y}^{*} }}$ global ductility demand

$\mu_{cap} = \frac{{d_{u}^{ * } }}{{d_{y}^{ * } }}$ global ductility capacity

µ _cap,b  = global ductility capacity corresponding to the attainment first brittle failure

P’ = performance index for the damaged building

P _C , P’ _C  = building’s probability of collapse for the intact and damaged structure

P _C,m  = mean collapse probability of collapse for the building stock

PBPF = performance based policy framework

PBPF ₀  = Policy with PL ₁  = 0.6; PL ₂  = 0.8 and C _rmax  = 0.7; C _rmin  = 0.41

PBPF ₁  = Policy with PL ₁  = 0.6; PL ₂  = 0.8 and C _rmax  = 0.8; C _rmin  = 0.51

PBPF ₂  = Policy with PL ₁  = 0.5; PL ₂  = 0.7 and C _rmax  = 0.7; C _rmin  = 0.41

PBPF ₃  = Policy with PL ₁  = 0.5; PL ₂  = 0.7 and C _rmax  = 0.8; C _rmin  = 0.51

R = Repair action

RR = Repair and Retrofit action

PL = performance loss index

PL ₁ , PL ₂  = performance loss corresponding to C _r,min and to C _r.max

REC _ag,0  = residual capacity in terms of peak ground acceleration a _g for intact structure

REC _Sa,0  = REsidual Capacity in terms of spectral acceleration for the intact structure

REC _ag,μ  = REsidual Capacity in terms of peak ground acceleration for damaged structure at ductility _μ

s _aC  = median spectral acceleration corresponding to the attainment of collapse capacity

$T_{eq} = 2\pi \sqrt {\frac{{\varGamma \cdot m^{ * } \cdot d_{y}^{ * } }}{{V_{b} }}}$ period of the equivalent SDOF system

ω = 2π/T_eq angular frequency of the equivalent SDOF system

η = spectral factor accounting for damping ratios different from 5 %

F ₀  = factor quantifying maximum spectral amplification with reference to rigid sub-soil

T _c  = spectral period corresponding to the transition from constant acceleration to constant velocity region

Appendix 2

This appendix resumes the main steps to evaluate CC for the intact building in the hypothesis of soft storey making reference to Fig. 11. It is an excerpt from (Polese et al. 2015a).

Fig. 11

The main steps to evaluate CC for the intact building (after Polese et al. 2015a)

Given the plastic mechanism, the corresponding base shear V _b may evaluated by equilibrium relations. Figure 11 (left panel) depicts the system of external and internal forces that should satisfy equilibrium for an hypothesized mechanism type (1st storey sway local mechanism) and uniform distribution of horizontal forces on the Multi Degree Of Freedom (MDOF) system.

Accordingly, the base shear V _b corresponding to equilibrium of internal and external forces may be computed with first equation on central panel in Fig. 11, where M ¹ _c (= M ¹ _c,y ) represents the generic yielding moment at the base or top section of the 1st floor columns (it is hypothesized that M _y,base  = M _y,top for the columns), and H₁ is the 1st storey height to foundation level. Once the base shear is calculated, the corresponding base shear coefficient C _b is easily determined dividing by equivalent mass m* of the SDOF system (that equals total building mass in case of uniform distribution of horizontal forces), MDOF-SDOF transformation factor Γ (= 1 in this case) and gravity acceleration, as it is shown in the right panel in Fig. 11. Adopting a similar approach to the one proposed in (Borzi et al. 2008), the yield displacement d _y at the roof level of the MDOF system is calculated assuming a linear deformed shape within the elastic range (see Fig. 11 left panel and second equation in central panel), while the ultimate displacement at the same level is given by the sum of d _y plus the plastic contribution d _pl =θ _pl ·H ₁ that is developed according to the hypothesized plastic mechanism (see Fig. 11 left panel and third equation in central panel). In the mentioned equations H _n represents the global building height to foundation level, θ _y corresponds to the maximum yield rotation of the base columns, while θ _pl is assumed as the minimum value of θ _u −θ _y among the hinges involved in the plastic mechanism, where θ _u is the rotation corresponding to Collapse Prevention CP limit state for the generic considered hinge.