Seismic design of bridges

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1 NAIONAL ECHNICAL UNIVERSIY OF AHENS LABORAORY FOR EARHQUAKE ENGINEERING Seismic design of bridges Lecture 4 Ioannis N. Psycharis

2 Seismic isolation of bridges I. N. Psycharis Seismic design of bridges 2

3 Se / Sηag Concept Application of special isolating system, aiming to reduce the response due to the horizontal seismic action. he isolating units are arranged over the isolation interface, usually located under the deck and over the top of the piers/abutments. Methods 3 Lengthening of the fundamental period of the structure reduces forces increases displacements Increasing the damping reduces displacements may reduce forces Combination of the two ects. 0 without isolation with isolation Period, I. N. Psycharis Seismic design of bridges 3

4 Basic requirements Isolators Each isolator unit must provide a single or a combination of the following functions: vertical load carrying capability combined with increased lateral flexibility and high vertical rigidity energy dissipation (hysteretic, viscous, frictional) re-centring capability horizontal restraint (sufficient elastic rigidity) under nonseismic service horizontal loads Increased reliability is required for the strength and integrity of the isolating system, due to the critical role of its displacement capability for the safety of the structure. System he seismic response of the superstructure and substructures shall remain essentially elastic. I. N. Psycharis Seismic design of bridges 4

5 Isolators with hysteretic behaviour d y = yield displacement d bd = design displacement of the isolator that corresponds to the design displacement d cd of the system F y = yield force at monotonic loading F max =force at maximum displacement d bd Κ e = elastic stiffness Κ p = post-elastic (tangent) stiffness K = ective (secant) stiffness at maximum displacement E D F F max F y d y K e K Dissipated energy per cycle at the design displacement d cd : E D = 4(F y d bd - F max d y ) K p d bd d I. N. Psycharis Seismic design of bridges 5

6 Lead Rubber Bearings (LRB) F max F y F Ly K p =K R total response Lead Elastomer K e =K R +K L Rubber Lead core K L K R d d y =d Ly d bd d y = d Ly = yield displacement of lead core F y = F Ly (1+K R /K L ) where F Ly = yield force of lead core Κ e = K R +K L = elastic stiffness Κ p = K R = post-elastic stiffness I. N. Psycharis Seismic design of bridges 6

7 Isolators with viscus behaviour he force is zero at the maximum displacement, therefore viscous isolators do not contribute to the ective stiffness of the isolating system F = Cv α For sinusoidal motion: d(t) = d bd sin(ωt) E D F max F(t) F α = 1 α < 1 d d(t) d bd F(t) = F max [cos(ωt)] α F max = C (d bd ω) α Dissipated energy per cycle: E D = λ(α)f max d bd λ(α) = 2 (2+α) Γ(1+0,5α)/Γ(2+α) Γ() is the gamma function I. N. Psycharis Seismic design of bridges 7

8 Isolators with friction behaviour A. Flat sliding surface N Sd F max = μ d N Sd where: μ d = dynamic friction coicient N Sd = normal force through the device E D F max F d d bd Such devices can result in substantial permanent offset displacements. herefore, they should be used in combination with devices providing adequate restoring force. Dissipated energy per cycle: E D = 4F max d bd I. N. Psycharis Seismic design of bridges 8

9 Isolators with friction behaviour F F max B. Spherical sliding surface of radius R b F 0 K p F 0 = μ d N Sd d bd d F max = μ d N Sd + Κ p d bd E D Κ p = Ν Sd /R b Dynamically, the device behaves as an inverted pendulum with period 2π R g b Friction Pendulum System (FPS) Dissipated energy per cycle: E D = 4F 0 d bd I. N. Psycharis Seismic design of bridges 9

10 Fundamental mode spectrum analysis he deck is assumed rigid he shear force transferred through the isolating interface shall be estimated considering the superstructure to behave as a single degree of freedom system using: the ective stiffness of the isolation system, K the ective damping of the isolation system, ζ the mass of the superstructure, m d = W d /g the spectral acceleration S e (, ζ ) that corresponds to the ective period and the ective damping ζ I. N. Psycharis Seismic design of bridges 10

11 Fundamental mode spectrum analysis Effective stiffness: K = ΣK,i where K,i is the composite stiffness of the isolator unit and the corresponding substructure (pier) i. In the calculation of the composite stiffness of each pier, the flexibility of the foundation must also be considered: Hi K K K K K,i where c,i b,,i t,i r,i Κ b, = the ective stiffness of the isolators of the pier Κ c,i = stiffness of the column of the pier Κ t,i = translational stiffness of the foundation in the horizontal direction Κ r,i = rotational stiffness of the foundation Η i = height of the pier measured from the level of the foundation I. N. Psycharis Seismic design of bridges 11

12 Fundamental mode spectrum analysis otal ective damping of the system: where ζ 1 2π K E D,i 2 dcd ΣE D,i is the sum of the dissipated energies of all isolators i in a full deformation cycle at the design displacement d cd. hen, the damping correction factor is: η 0,10 0,05 ζ Effective period of the system: m K d I. N. Psycharis Seismic design of bridges 12

13 Fundamental mode spectrum analysis Spectral acceleration and design displacement (for > C ) Τ S e d cd C < D a g S η 2.5 C C d C D a g S η 2.5 C D 2 D C d C where d C 0,625 π 2 a g S η 2 C I. N. Psycharis Seismic design of bridges 13

14 Multi-mode spectrum analysis he ective damping ζ is applied only to modes having periods higher than 0,8. For all other modes, the damping ratio corresponding to the structure without seismic isolation should be used. he ective damping ζ and the ective period are calculated as in the fundamental mode spectrum analysis. he design displacement, d d,m, and the shear force, V d,m, that are transferred through the isolation interface, calculated from the multi-mode spectrum analysis, are subject to lower bounds equal to 80% of the relevant ects d d,f and V d,f calculated in accordance with the fundamental mode spectrum analysis. In case that this condition is not met, the ects of the multimode spectrum analysis will be multiplied by 0,80 d d,f /d d,m and 0,80 V d,f /V d,m respectively. If the bridge cannot be approximated (even crudely) by a single degree of freedom model, the ects d d,f and V d,f can be obtained from the fundamental mode. I. N. Psycharis Seismic design of bridges 14

15 Isolating system Verifications In order to meet the required increased reliability, the isolating system shall be designed for increased displacements: d d,in = γ IS d d where γ IS = 1,50 is an amplification factor applicable only to the design displacements of the isolation system. All components of the isolating system shall be capable of functioning at the total maximum displacement: d max = d d,in + d G + ½d where d G is the displacement due to the permanent and quasipermanent actions and d is the displacement due to thermal movements. No lift-off of isolators carrying vertical force is allowed under the design seismic combination. I. N. Psycharis Seismic design of bridges 15

16 Verifications Substructures and superstructure Derive the internal seismic forces E E,A from an analysis with the seismic action for q = 1. Calculate the design seismic forces E E, due to seismic action alone, that correspond to limited ductile / essentially elastic behaviour, from the forces E E,A : E E = E E,A / q with q 1,5. All structure members should be verified for: Forces E E in bending with axial force Forces E E,A in shear he foundation will be verified for forces E E,A. I. N. Psycharis Seismic design of bridges 16

Seismic design of bridges

NATIONAL TECHNICAL UNIVERSITY OF ATHENS LABORATORY FOR EARTHQUAKE ENGINEERING Seismic design of bridges Lecture 3 Ioannis N. Psycharis Capacity design Purpose To design structures of ductile behaviour