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 http://prinpontimuratura.diseg.unige.it Research project partially supported by: RFI - Italian Railway Authority MURST- Dept. University and Sci.&Tech. Res. CNR - GNDT UIC – International Union of Raylways

Italian Railway Network European Railway Network (UIC data) About 12.000 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 100-150 years 69.5 % Seismic actions

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

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

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)

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

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

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

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

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)

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

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

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

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

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

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

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

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

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

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

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