NATIONAL TECHNICAL UNIVERSITY OF ATHENS...

NATIONAL TECHNICAL UNIVERSITY OF ATHENS LABORATORY FOR EARTHQUAKE ENGINEERING

Seismic design of bridges

Lecture 2

Ioannis N. Psycharis

I. N. Psycharis “Seismic design of bridges” 2

Piers with elastic bearings

P

Δ

Py = Pc,y

Δc,y

Column

Bearing

Δb Δtot,y

Δb

Δb

Δc,u Δtot,u

total response

Ktot Kb Kc

bc

bctot

cbtot KK

KKKor

K

1

K

1

K

1

Δc,y

Δc,y



System ductility:

Let f be the ratio of the total (deck) displacement over the corresponding column displacement in the elastic region (up to the yield of the pier). At yield:

Then

where is the ductility of the column

by,c

bu,c

y,tot

u,totf,Δ

ΔΔ

ΔΔ

Δ

Δμ

1fΔ

Δ

Δ

Δ1

Δ

ΔΔ

Δ

Δf

y,c

b

y,c

b

y,c

by,c

y,c

y,tot

f

1fμμ c,Δ

f,Δ

y,c

u,cc,Δ

Δ

Δμ



The above relation can be written as:

The value of f depends on the relative stiffness of the bearings w.r.t. the stiffness of the pier: For “soft” bearings, f is large and the ductility of the pier can attain large values.

Example

● f=5 and μΔ,f =3.5 (design ductility). Then:

μΔ,c = 1+ 5∙(3.5-1) = 13.5

● f=5 and we want μΔ,c =3.5. Then, the design must be performed for:

μΔ,f = (3.5+5-1)/5 = 1.5

)1μ(f1μ f,Δc,Δ



P

Δ

Py = Pc,y

Δc,y

Column

Bearing

Δb Δtot,y

Δb

Δb

Δc,u Δtot,u

total response

Ktot Kb Kc

Δc,y Δc,p

Δtot,y Δtot,p = Δc,p

𝜇𝛥,𝑓 =𝛥𝑡𝑜𝑡,𝑢

𝛥𝑡𝑜𝑡,𝑦 𝜇𝛥,𝑐 =

𝛥𝑐,𝑢

𝛥𝑐,𝑦

Δc,u

Δtot,u



For this reason, EC8 – part 2 (bridges) considers that:

When the main part of the design seismic action is resisted by elastomeric bearings, the flexibility of the bearings imposes a practically elastic behaviour of the system.

The allowed values of q are:

q 1.5 according to EC8

q = 1.0 according to E39/99


Displacement ductility & Curvature ductility

Moments Curvatures Displacements

Lh

Cantilever pier


Ροπή

Καμπυλότητα

C

0 C C

M

M y

p

y

u

u


During elastic response (P < Py)

and where and M = P L

which leads to

Moment

K

PΔ

Δύναμη

Μετακίνηση

Δ

0 Δ Δ

P

P

u

p

y

u

y

Force

Curvature Displacement

IE

MC

3L

IE3K

3

LCΔ

2y

y



After yield

Cp = Cu – Cy and Δp = Δu – Δy

θp = Lh Cp = Lh (Cu-Cy)

Δp = Δp,1 + Δp,2

(hardening)

(plastic rotation)

Therefore

1

M

MΔΔ

y

uy1,p

)L5.0L()CC(L)L5.0L(θΔ hyuhhp2,p

)L5.0L()CC(L1M

MΔΔ hyuh

y

uyp



By definition

Finally

where λ = Lh / L

Example: λ = 0.1, μΔ = 3.0 μc = 7.0

y

p

y

uΔ

Δ

Δ1

Δ

Δμ

y

p

y

uC

C

C1

C

Cμ

)λ5.01()1μ(λ3M

Mμ C

y

uΔ

)λ5.01(λ3

1μ1μ Δ

C


Chord rotation ductility

Moments Curvatures Displacements

θu

L

L = length between the plastic hinge and the section of zero moment

θy = Δy/L = CyL/3

θp,u= Lh(Cu-Cy)(1-λ/2)

θu = θy + θp,u

μθ = θu/θy

For longitudinal reinforcement of characteristic yield stress fyk (in MPa) and bar diameter ds, the plastic hinge length Lh may be assumed as follows:

Lh = 0,10L + 0,015fykds


Model of analysis

Masses

● Permanent masses (with their characteristic value)

● Quasi-permanent values of the masses corresponding to the variable actions: ψ2,1 Qk,1, where Qk,1 is the characteristic value of traffic load and

♦ ψ2,1 = 0,2 for road bridges

♦ ψ2,1 = 0,3 for railway bridges

● When the piers are immersed in water, an added mass of entrained water acting in the horizontal directions per unit length of the immersed pier shall be considered (see Annex F of EC8-2)


Model of analysis

Stiffness of reinforced concrete ductile members

● The effective stiffness of ductile concrete components used in linear seismic analysis should be equal to the secant stiffness at the theoretical yield point.

● In the absence of a more accurate assessment, the approximate methods proposed in Appendix C of EC8-2 may be used.

● E39/99 proposes:

♦ At piers expected to yield, (ΕΙ)eff=300MRd,hd, where ΜRd,h is the moment of resistance and d is the height of the cross section at the place of the plastic hinge.

♦ At piers expected to respond elastically, the mean value between the above value and the one that corresponds to the uncracked cross section.


Model of analysis

Stiffness of reinforced concrete ductile members - Appendix C of EC8-2

● Method 1

Jeff = 0,08Jun + Jcr

♦ Jun = moment of inertia of uncracked section

♦ Jcr = My/(EcCy) = moment of inertia of cracked section

● Method 2

EcJeff = νMRd/Cy

♦ ν = 1,20 = correction coefficient reflecting the stiffening effect of the uncracked parts of the pier

♦ Cy = 2,1 εsy/d for rectangular sections Cy = 2,4 εsy/d for circular sections

where d is the effective depth of the section


Model of analysis

Stiffness of elastomeric bearings

where:

G = shear modulus

G varies with time, temperature etc. Two cases:

(a) Gmin = Gb; (b) Gmax = 1,5 Gb

where Gb = 1.1 Gg with Gg = 0.9 N/mm2 = 900 KPa.

Ab = effective area of bearing

Αb = (bx-dEd,x)( by-dEd,y) for rectangular bearings of dimensions bxby, where dEdx, dEdy are the seismic design displacements of the bearing in x- and y-direction, respectively.

T = total thickness of elastomer

T

AGK bb


Design displacements

Design seismic displacement

where

dEe = displacement from the analysis

η = damping correction factor

q for T T0 = 1,25 TC μd = for T < T0

Total design displacement

where

dG = displacement due to the permanent and quasi-permanent actions

dT = displacement due to thermal movements

EedE dμηd

4q51T

T)1q( 0

T2GEEd dψddd


Second order effects

Approximate methods may be used for estimating the influence of second order effects on the critical sections (plastic hinges):

where NEd is the axial force and dEd is the relative transverse displacement of the ends of the member

EdEd Nd2

q1MΔ


Design seismic combination

The design action effects Ed in the seismic design situation shall be derived from the following combination of actions:

Gk "+" Pk "+" AEd "+" ψ21Q1k "+" Q2

where “+” means “to be combined with” and

Gk = the permanent loads with their characteristic values

Pk = the characteristic value of prestressing after all losses

Aed = is the most unfavourable combination of the components of the earthquake action

Q1k = the characteristic value of the traffic load

Ψ21 = the combination factor

Q2 = the quasi permanent value of actions of long duration (e.g. earth pressure, buoyancy, currents etc.)


Design seismic combination

● Seismic action effects need not be combined with action effects due to imposed deformations (temperature variation, shrinkage, settlements of supports, ground residual movements due to seismic faulting).

● An exception to the rule stated above is the case of bridges in which the seismic action is resisted by elastomeric laminated bearings. In this case, elastic behaviour of the system shall be assumed and the action effects due to imposed deformations shall be accounted for.

● For wind and snow actions, the value ψ21 = 0 shall be assumed.

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