2C09 Design for seismic and climate changes Lecture 07: Seismic response of SDOF systems Aurel Stratan, Politehnica University of Timisoara 06/04/2017 European Erasmus Mundus Master Course Sustainable Constructions under Natural Hazards and Catastrophic Events 520121-1-2011-1-CZ-ERA MUNDUS-EMMC
Lecture outline 7.1 Time-history response of linear SDOF systems. 7.2 Elastic response spectra. 7.3 Time-history response of inelastic SDOF systems. 2
Seismic action Ground acceleration: accelerogram Properties of a SDOF system (m, c, k) + mu cu ku mu g ug () t Relative displacement, velocity and acceleration of a SDOF system 3
Seismic action North-south component of the El Centro, California record during Imperial Valley earthquake from 18.05.1940 4
Determination of seismic response Equation of motion: mu cu ku mu g /m: u u u u 2 2 n n g Numerical methods central difference method Newmark method... u u t, T, n Response depends on: natural circular frequency n (or natural period T n ) critical damping ratio ground motion u g 5
Seismic response 6
Elastic response spectra Response spectrum: representation of peak values of seismic response (displacement, velocity, acceleration) of a SDOF system versus natural period of vibration, for a given critical damping ratio u0 Tn, max u t, Tn, u0 Tn, max u t, Tn, t t u0 Tn, max u t, Tn, t t t 7
Elastic displacement response spectrum: Du 0 8
Pseudo-velocity and pseudo-acceleration Spectral pseudo-velocity: units of velocity different from peak velocity Strain energy E S 0 Spectral pseudo-acceleration: f ku m u ma 2 S0 0 n 0 units of acceleration different from peak acceleration 2 V nd D T n ku kd kv mv 2 2 2 2 2 2 2 2 0 n 2 2 2 A nu0 nd D Tn u(t) max = Sde(T) k m F =ks de (T) =msae(t) 2 u(t) max = Sde(T) k m a g (t) 9
Pseudo-velocity and pseudo-acceleration D 2 V nd D T n 2 2 A n D D Tn 2 10
Combined D-V-A spectrum Displacement, pseudo-velocity and pseudo-acceleration spectra: same information different physical meaning A Tn 2 V n D or A V D 2 T n T 2 A V lgt lg A lg2 lgv n n A line inclined at +45º for lga - lg2 = const. spectral pseudo-acceleration: an axis inclined to -45º n Similarly, spectral displacement: an axis inclined to +45º 11
Combined D-V-A spectrum 12
Characteristics of elastic response spectra 13
Characteristics of elastic response spectra 14
Characteristics of elastic response spectra For T n <T a pseudo-acceleration A is close to spectral displacement D is small u g 0 For T n >T f spectral displacement D is close to spectral pseudo-acceleration A is small Between T a and T c A > u g 0 u g 0 Between T b and T c A can be considered constant Between T d and T f D > u g 0 Between T d and T e D can be considered constant Between T c and T d V > u g 0 Between T c and T d V can be considered constant 15
Characteristics of elastic response spectra T n >T d response region sensible to displacements T n <T c response region sensible to accelerations T c <T n <T d response region sensible to velocity Larger damping: smaller values of displacements, pseudo-velocity and pseudoacceleration more "smooth" spectra Effect of damping: insignificant for T n 0 and T n, important for T b <T n <T d 16
Elastic design spectra Spectra of past ground motions: jagged shape variation of response for different earthquakes areas where previous data is not available 17
idealized "smooth" spectra Elastic design spectra based on statistical interpretation (median; median plus standard deviation) of several records characteristic for a given site 18
Elastic design spectra 19
Elastic design spectra PSA PSV SD T B T C T D T T B T C T D T T B T C T D T 20
Inelastic response of SDOF systems Most structures designed for seismic forces lower than the ones assuring an elastic response during the design earthquake design of structures in the elastic range for rare seismic events considered uneconomical in the past, structures designed for a fraction of the forces necessary for an elastic response, survived major earthquakes f S b c d a u u u u um f S f S f S f S a b c d 21
Inelastic response of SDOF systems Elasto-plastic system: stiffness k yield force f y yield displacement u y Elasto-plastic idealization: equal area under the actual and idealised curves up to the maximum displacement u m Cyclic response of the elasto-plastic system 22
Corresponding elastic system Corresponding elastic system: same stiffness same mass same damping the same period of vibration (at small def.) Inelastic response: yield force reduction factor R y f0 u0 Ry f u y y ductility factor um u y 23
Equation of motion: Equation of motion mu cu f u, u mu S g /m u u u f u u u 2 2 n n y S, g,, f u u f u u f S S y Seismic response of an inelastic SDOF system depends on: natural circular frequency of vibration n critical damping ratio yield displacement u y force-displacement shape fs u, u 24
Effects of inelastic force-displacement relationship 4 SDOF systems (El Centro): T n = 0.5 sec = 5% R y = 1, 2, 4, 8 Elastic system: vibr. about the initial position of equilibrium u p =0 Inelastic syst.: vibr. about a new position of equilibrium u p 0 25
Elastic inelastic Design of a structure responding in the elastic range: f 0 f Rd Design of a structure responding in the inelastic range: u m u Rd Rd ductility demand ductility capacity 26
u m /u 0 ratio El Centro ground motion = 5% R y = 1, 2, 4, 8 T n >T f u m independent of R y u m u 0 T n >T c u m depends on R y u m u 0 T n <T c u m depends on R y u m > u 0 27
R y - relationship: idealisation T n in the displacement- and velocity-sensitive region: "equal displacement" rule u m /u 0 =1 R y = T n in the acceleration-sensitive region: "equal energy" rule u m /u 0 >1 Ry 2 1 T n <T a : f S f 0 small deformations, elastic response R y =1 f S f 0 f S f 0 1 Tn Ta R 2 1 T T T Tn Tc y b n c' f y f y f y uy u uy=umu =um u u uy u uy um u um u u 28
R y - relationship: idealisation 1 Tn Ta R 2 1 T T T Tn Tc y b n c ' 29
References / additional reading Anil Chopra, "Dynamics of Structures: Theory and Applications to Earthquake Engineering", Prentice-Hall, Upper Saddle River, New Jersey, 2001. Clough, R.W. and Penzien, J. (2003). "Dynamics of structures", Third edition, Computers & Structures, Inc., Berkeley, USA 30
aurel.stratan@upt.ro http://steel.fsv.cvut.cz/suscos