1. ARCH 04 – Fourth International Conference on Arch Bridges 17-19 November 2004
Upper bound limit analysis of multispan masonry bridges including arch-fill interaction
A. Cavicchi, L. Gambarotta
Dept. Structural and Geotechnical Engineering
University of Genoa, Italy
Research project partially supported by:
RFI - Italian Railway Authority
MURST- Dept. University and Sci.&Tech. Res.
CNR - GNDT
UIC – International Union of Raylways

2. Italian Railway Network European Railway Network (UIC data)
About masonry bridges (s > 2m)
15m<S<30m 2% 3%
S>30m
5m<S<15m
15%
(s: bridge span)
Lines to be updated to D4/type loads not later than:
2003
2004
2005
S<5m
80%
European Railway Network (UIC data)
Italy
France
18060
43.5 %
Main Issues
Portugal
874
49.3 %
Load increment
Germany
8653
27.5 %
Spain
3144
49.3 %
Material degradation …
Hydraulic actions
Age years
69.5 %
Seismic actions

3. Arch-Fill Interaction
Tests on full scale masonry bridges: Prestwood Bridge
(Page, 1987)

4. Arch-Fill Interaction
Tests on full scale masonry bridges: Prestwood Bridge
Lifting of arch and fill
Load position
Prestwood Bridge
Hinghes

5. Tests on model scale bridges (Royles & Hendry, 1991)
Total load applied F ( kN )
Complete bridge
Vault and fill
Vault
Crisfield (1985)
Choo et al. (1991)
v ( mm )
Owen et al. (1998)
Bicanic et al. (2003)

6. Two dimensional model of the bridge: arch-pier-fill interaction
Frictional-Cohesive
material
Vaults
Beams
Fill
NTR perfectly plastic in compression
Backing
Pier
Two dimensional domain under plane strain conditions
Arches and piers: no tensile resistant, compressive ductile beams
Fill: Mohr-Coulomb constitutive model with tension cut-off

7. Limit analysis – Upper Bound Theorem
Finite element discretization of the bridge
Rigid links
Triangular elements
Interface elements
Beam elements

8. Interface element (Sloan & Kleeman, 1995)
Normal and tangential velocity jump across the discontinuity
Stress components in the interface element s
Coulomb
Tension cut-off
Tension cut-off

9. Beam element: kinematics
Two noded (i,j) element with six degrees of freedom
Deformations localized in the end sections (1,2)

10. Discretization of the limit domain of the section
No tensile resistant and ductile in compression material
Element strain rate vector
Total relative axial velocity
Relative rotational velocities at the end sections
Discretized limit domain
(a)
(b)
(c)

11. Fill limit domain discretization
(Mohr-Coulomb)
Discretized limit domain
(Tension cut-off)
Cohesion
Tension cut-off
Angle of internal friction
number of planes

12. Upper bound collapse load
Dead load
Live load
Compatibility
Associated flow rule
Admissible domain
Constraints
Positivity of the power of live load
Dissipated power
Power of dead loads

13. Not resistant fill model: collapse mechanism
PRESTWOOD BRIDGE
Not resistant fill model: collapse mechanism
Hinge at impost
Hinge at haunch

14. Resistant fill model: collapse mechanism
PRESTWOOD BRIDGE
Resistant fill model: collapse mechanism
Hinge at haunch
(Page, 1993)

15. PRESTWOOD BRIDGE
Sensitivity of the collapse load Pu to the material parameters
Resistant fill
Not resistant fill
Not resistant fill
Experimental
Resistant fill
Compressive strength
Cohesion

16. Load\deflection curve and ductility demand
PRESTWOOD BRIDGE
Load\deflection curve and ductility demand
Upper bound
Incremental analysis
Masonry ductility:
Vertical displacement v

17. MULTI-SPAN BRIDGE Fill model properties: Arch model properties:
Fill density
Masonry density
Discrete domain planes
Discrete domain planes

18. MULTI-SPAN BRIDGE Collapse mechanism 30o Resistant fill
Not resistant fill
Collapse mechanism
Resistant fill
30o

19. MULTI-SPAN BRIDGE
Sensitivity of the collapse load Pu to the fill parameters
Resistant fill
Not resistant fill
Not resistant fill
Resistant fill
(rigid haunching)
(rigid haunching)
Cohesion
Angle of internal friction

20. Load\deflection curve and ductility demand
MULTI-SPAN BRIDGE
Load\deflection curve and ductility demand
Upper bound
Incremental analysis
Masonry ductility:
Vertical displacement v

21. Open Issues (some) Coming developments
Arch-fill interaction:
Modelling of velocity field discontinuities
Active and passive fill pressure description
Plain strain assumption consequences
…………………………….
Coming developments
Lower Bound approach
Optimization of the strengthening interventions by fill injections