1. Seismic analysis of Bridges Part II

2. Analysis methods Simplified non-linear analysis: Simply supported Bridges
Most of the bridges are built using a simply supported deck configuration.
For the evaluation of the seismic behaviour of as-built condition of the retrofitted condition that maintain the static scheme (presence of discontinuity of the deck) a simplified methodology can be adopted. It is base on the following steps:
The pier can be considered a simple oscillator with a mass given by the tributary mass relative to the deck and the mass of the pier.
The non linearity can be considered as concentrated at the pier base and assumed of elasto-perfectly plastic type
The capacity of the pier represented by the shear displacement curve evaluated starting to the yielding and ultimate rotation on the plastic hinge at the pier base
The response is evaluated by Push-over analysis using the N2 method
The verification is made in term of ductile and fragile mechanisms

3. Analysis methods Simplified non-linear analysis: Simply supported Bridges
Transversal direction: determination of the SDOF characteristics
Mass of the simplified system
Effective height the pier

4. Analysis methods Simplified non-linear analysis: Simply supported Bridges
Transversal direction: determination of the capacity
Length of the plastic hinge
On the basis of the simplified model the capacity of the pier can be evaluated starting form the linearized Moment-Curvature relationship and evaluating the yielding and ultimate top displacements using the following formula

5. Analysis methods Simplified non-linear analysis: Simply supported Bridges
Transversal direction: Evaluation of the performance point by N2
Principle of equal displacement
Principle of equal Energy

6. Analysis methods Simplified non-linear analysis: Simply supported Bridges
Evaluation of strength reduction factors for earthquake-resistant design, Miranda Bertero ( 1994) – E. Spectra vol 10 n° 2
(behaviour factor)

7. Analysis methods Simplified non-linear analysis: Simply supported Bridges
As soon as we have found the global multilinear capacity curve of the bridge, built considering all the piers in parallel, the above procedure can be easily applied considering the global mass as the sum of the single masses

8. Analysis methods Simplified non-linear analysis: Continuous deck bridges
Mono-modal method
Bi-linearization
1st mode
Distribution of force
Participation
factor
V * u*

9. Analysis methods Simplified non-linear analysis: Continuous deck bridges
Mono-modal method
Evaluation of the maximum
displacement
V * u*
To be compared
With the capacity

10. Example

11. Simply supported bridge (ordinary bridge)
Geometry 2 3 4 5
Data 1
Class IV
Soil B
VS,30 > 500m/s
Concrete
fc = 25 MPa
Steel
fsy = 400 MPa

12. Simply supported bridge
Definition of the seismic input
Nominal life (ordinary bridge): VN= 50 years
Class: IV  Cu = 2
Reference life : VR = Cu  VN = 100 years
Soil B
Life safety level (SLV)
PR=10%  TR=949 y.
Collapse prevention (SLC)
PR=5%  TR=1950 y.
NTC08 provide discrete TR. Therefore
an interpolation of data is needed

13. Simply supported bridge
Definition of the seismic input
Life safety level (SLV)
PR=10%  TR=949 y.
Nominal life (ordinary bridge): VN= 50 years
Class: IV  Cu = 2
Reference life : VR = Cu  VN = 100 years
Soil B
Collapse prevention (SLC)
PR=5%  TR=1950 y.
NTC08 provide discrete TR. Therefore
an interpolation of data is needed

14. Simply supported bridge
Evaluation of weight and mass (pier 5)
Mass of the deck
Height of the deck respect to ground
Total weight of the pier cap
Height of the pier cap
Linear weight of the pier
Height of the pier
Total weight of the pier
Height of foundation
Axial force on the pier
Total mass
10.0 m

15. Simply supported bridge
Evaluation of Moment-rotation of the pier base
Concrete: Kent-Park model
Steel
N=5437 kN
Bi-linearization

16. Simply supported bridge
Evaluation of the capacity of the piers (Force-displacement law)
Pier 5

17. Simply supported bridge
Evaluation of the seismic Demand (transversal direction)
(T > Tc) equal
displacement
Fundamental period
Spectral acceleration (SLV)
Spectral Displacement
Base-shear
Base Moment

18. Simply supported bridge
Evaluation of the seismic Capacity (Longitudinal direction)
Bilinearization of the
Multilinear law

19. Simply supported bridge
Evaluation of the seismic Demand (Longitudinal direction)
(non-linear response)
For example for the pier 5

20. Simply supported bridge
Verification format (Ductile mechanisms)
Pier 5

21. Simply supported bridge
Verification format (Ductile mechanisms)

22. Simply supported bridge
Verification format (Fragile mechanisms)
Axial force contribution
Concrete contribution
Reinforcement contribution

23. Simply supported bridge
Verification format (Fragile mechanisms)

24. Simply supported bridge
Verification format (Fragile mechanisms)
Pier 5

25. Simply supported bridge
SLV
SLC
The level of safety is encountered both for SLV and SLC
In conclusion the bridge does not have an enough level of safety both with respect to ductile mechanism and to fragile one
A retrofitting action has to be adopted.