PAMM · Proc. Appl. Math. Mech. 11, 307 – 308 (2011) / DOI 10.1002/pamm.201110145

Seismic collapse capacity spectra for simple structures vulnerable todynamic instabilities

Christoph Adam1,∗ and Clemens Jäger1

1 University of Innsbruck, Unit of Applied Mechanics, Technikerstraße 13, 6020 Innsbruck, Austria

In this study collapse capacity spectra based on various definitions of the seismic intensity are set in contrast and evaluated.The presented collapse capacities for highly inelastic non-deteriorating single-degree-of-freedom (SDOF) systems, which arevulnerable to the destabilizing effect of gravity loads, are derived for a near-fault set of ground motions with distinct pulsecharacteristics.

c© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim

1 Fundamentals

Sidesway collapse of buildings during strong earthquakes can be led back either to the successive reduction of the load bearingcapacity due to cyclic component deterioration, to the amplification of deformations and stress resultants due to gravity loads(P-delta effect), or to the combined action of both sources. In the following the seismic collapse of highly inelastic single-degree-of-freedom (SDOF) structures vulnerable to the P-delta effect is studied.

As outlined e.g. in [1] a gravity load generates in an SDOF system a shearing of the hysteretic force-displacement relation-ship, as shown in Fig. 1 for a bilinear hysteretic loop with non-deteriorating characteristics. The slope of the curve is reduced,expressed by the stability coefficient θ. θ depends on the gravity load, the initial stiffness, and the geometry [1] . In the exam-ple of Fig. 1 the post-yield stiffness is negative because the stability coefficient θ is larger than the strength hardening ratio α.A negative slope of the post-tangential stiffness, expressed by the difference θ−α, is a precondition that the non-deterioratingstructure may collapse under severe earthquake excitation. In [1] it is shown that for a bilinear backbone curve the collapse ofan inelastic SDOF system vulnerable to P-delta is mainly governed by the negative slope of the post-tangential stiffness θ−α,the elastic structural period of vibration T , the viscous damping coefficient ζ , and the shape of the hysteretic loop.

The collapse capacity is defined as the maximum ground motion intensity at which the structure still maintains dynamicstability [2], and can be determined e.g. by the Incremental Dynamic Analysis (IDA) procedure. There is no unique definitionof intensity of an earthquake record. Examples of the intensity measure are the 5% damped spectral acceleration Sa atthe structure’s period T , Sa(T, ζ = 0.05), the peak ground acceleration (PGA), the peak ground velocity (PGV), and thepeak ground displacement (PGD), see e.g. [3]. The intensity measure at collapse is referred to as collapse capacity of theconsidered structure for this specific ground motion record (denoted by i). In non-dimensional form the relative collapsecapacity associated with a specific ground motion record may be defined according to the utilized intensity measure as [3]

CC|Sa(T ),i =Sa,i(T )

gγ

∣∣∣∣collapse

, CC|PGA,i =PGAi

gγ

∣∣∣∣c.

, CC|PGV,i =ωPGVigγ

∣∣∣∣c.

, CC|PGD,i =ω2PGDi

gγ

∣∣∣∣c.

(1)

γ represents the yield strength coefficient, γ = fy/(mg), where fy is the strength, m the lumped mass, g the acceleration ofgravity, and ω denotes the structural circular frequency, ω = 2π/T . T is the structural period. The inherent record-to-recordvariability leads to different collapse capacities for different ground motion records. Thus, the collapse capacity is determinedfor an entire set of n ground motion records, and subsequently evaluated statistically. The median CC of the individualcollapse capacities CCi may be considered as representative collapse capacity for the analyzed structure and set of groundmotion records. 16% and 84% percentiles characterize the distribution of the individual collapse capacities, [1] and [3]. Thecollapse capacities derived in this study are based on the ATC63 near-fault set of ground motions with pronounced pulsecharacteristics [4], abbreviated by ATC63-NFwP. The records of this set originate from severe seismic events of momentmagnitude between 6.5 and 7.6 and closest distance to fault rupture is smaller than 10 km. Thereby, only strike-slip andreverse sources are considered. For the computations of this paper only records with pronounced pulse characteristics areused. All 28 records of this set (two records from 14 earthquakes each) were recorded on NEHRP site classes C (soft rock)and D (stiff soil). It is noted that in a previous study [3] the quantities of Eq. (1) have been evaluated utilizing a set of far-fieldground motions. This far-field set denoted by ATC63-FF contains 44 records, and has been selected by the same criteria asthe near-fault set except that the closest distance to fault rupture is larger than 10 km [4].

∗ Corresponding author: email christoph.adam@uibk.ac.at, phone +43 512 507 6585, fax +43 512 507 2908

c© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim

308 Section 5: Nonlinear oscillations

2 Collapse capacity spectra

The representation of the collapse capacity of SDOF systems with assigned ζ and θ − α, and a particular hysteretic loop asa function of the initial period T results in collapse capacity spectra. In [1] the concept of collapse capacity spectra for theassessment of the collapse capacity of non-deteriorating SDOF systems vulnerable to the P-delta effect has been introduced.Subsequently, collapse capacity spectra according to the definitions of Eq. (1) are compared and evaluated. In Fig. 2a differentcharacteristics of median collapse capacity spectra are shown for a negative post-yield slope of θ − α = 0.10 and a dampingcoefficient of ζ = 0.05. It can be seen that rigid systems (T = 0) exhibit for acceleration dependent intensity measures (i.e.Sa(T ) and PGA) a collapse capacity of CC = 0, where as the application of PGV leads to a collapse capacity of infinity. Thiscan be attributed to the fact that the definition of the PGV based collapse capacity is multiplied by the structure’s fundamentalfrequency, which is infinity for T = 0, compare with Eq. (1). Collapse capacities, which rely on PGA as intensity measureshow a steep rise with increasing period. On the other hand, the graph of the median collapse capacity spectra based onSa(T, ζ = 0.05) exhibits a decreasing gradient with increasing period. These outcomes coincide with the findings presentedin [3], where the ATC63-FF set has been utilized.

Fig. 1 Normalized bilinear cyclic behavior of an SDOF structure without and with P-delta

0

2

4

6

8

10

0 1 2 3 4 5

PGD-scaling

PGV-scaling

ATC63-NFwP set

period T [s]

= 0.05bilinear hysteretic loop

CC

(median)

PGA-scaling

Sa(T)-scaling

= 0.10

a)

0

0.5

1

1.5

2

2.5

0 1 2 3 4 5

period T [s]

CC| A

TC63

-NFw

P /

CC| A

TC63

-FF PGD-scaling

PGV-scaling

PGA-scaling

Sa(T)-scaling

median

= 0.05bilinear hysteretic loop

= 0.10

b)

Fig. 2 Median collapse capacity spectra: different definitions of the intensity measure. a Median collapse capacity spectra based onnear-field ground motion set. b Ratios of median collapse capacity spectra based on near-fault and far-field ground motions.

Additionally, the different impact of near-fault and far-field ground motions on the collapse capacity is studied. Thus,Fig. 2b shows the collapse capacities of Fig.2a divided by the corresponding collapse capacities based on the ATC63 far-fieldset of ground motions [3]. In such representation a ratio larger than one means that the near-fault set is less hazardous than thefar-field set. No uniform trend of the ratios based on different definitions of the intensity measure can be observed with respectto their dependence on the period T . For PGA-scaling the collapse capacity for the far-field set is larger in the consideredperiod range. Contrary, PGD-scaling leads to ratios larger than one. The Sa and PGV based intensity measures are in differentperiod ranges smaller or larger than one with an opposed trend.

References[1] C. Adam and C. Jäger, Earthquake Engineering and Structural Dynamics (under review).[2] H. Krawinkler, F. Zareian, D.G. Lignos, and L.F. Ibarra, Proceedings Computational Methods in Structural Dyanmics and Earthquake

Engineering. Rhodes, Greece (M. Papadrakakis et al., 2009), paper no. CD449.[3] C. Adam and C. Jäger, in: Advanced Dynamics and Model-Based Control of Structures and Machines (Springer, Wien, 2011).[4] FEMA P695 - ATC63, Quantification of Building Seismic Performance Factors (Federal Emergency Management Agency, Washington

D.C., 2009).

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