# ELASTIC NO TENSILE RESISTANT â€“ PLASTIC ANALYSIS OF canad .ELASTIC NO TENSILE...

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ELASTIC NO TENSILE RESISTANT PLASTIC ANALYSIS OF MASONRY ARCH BRIDGES AS AN EXTENSION OF CASTIGLIANOS METHOD

A. Brencich1, U. De Francesco2, L. Gambarotta3

ABSTRACT On the basis of the classical Castiglianos theory, a step-wise procedure for the non-linear analysis of multi-span masonry arch bridges is developed and implemented by standard programming of a commercial F.E. code. The mechanical model for masonry is assumed perfectly elasto-plastic in compression and no tensile resistant (NTR). The iterative procedure is set following the standard scheme of an elastic prevision and subsequent non-linear correction of the nodal forces. Tensile stresses are not allowed to develop on the mortar joint by means of an adequate re-meshing of the arch, while the plastic response is taken into account via additional fictitious external forces. The procedure is applied to a flat arch with different models: a No Tensile Resistant (NTR) model that accounts for the original Castiglianos method, and models with different compressive strength. The structural response is estimated for different loading positions, pointing out the change of the collapse mechanism as a function of the geometric and mechanical parameters. The flexibility of the procedure allows the analysis of twin-span and three-span models, so that the effect of abutment compliance and material crushing in complete bridge models is taken into account and quantified.

Key words : arch collapse analysis; multi-span arch bridges; non-linear interface model; elasto-plastic mortar joints.

Common address: Department of Structural and Geotechnical Engineering University of Genoa - ITALY DISEG, via Montallegro 1 16145 Genova - ITALY

1brencich@diseg.unige.it 2defrancesco@diseg.unige.it 3gambarotta@diseg.unige.it

INTRODUCTION

A large number of ancient arch bridges is still in service in their original configuration, facing increasing vehicle loads and speed; for this reason reliable estimates of the structural response of these bridges are needed.

The classical methods of assessment (Heyman, 1982) refer to a single arch, assuming that the skewbacks are perfect built-in ends and considering simplified loading conditions. These procedures do not give any estimate of the structural response but either ensure the existence of an equilibrium configuration or give rough estimates of the limit load. The parameters relevant to the structural response, such as masonry strength, mechanical characteristics of the fill, adjacent spans, etc., are taken into account by means of corrective factors (Department of Transport, 1993.a,b, Hughes and Blackler, 1997) of uncertain heuristic origin.

To get a deeper insight in this kind of structures several different procedures have been developed (Hughes and Blackler, 1997) on the basis of a relevant series of experimental tests both on models and on real structures (Page, 1987, 1993).

The mechanism approach to arch collapse, originated from the first work of Pippard and Ashby (1939) and Pippard (1948), looks for the minimum load, once its position is defined, needed to introduce a number of hinges at the arch intradoes and extradoes large enough to transform the arch into a mechanism. The limit load is obtained through an application of the kinematic theorem (Heyman, 1982) that takes the position of the hinges as the unknowns of the problem. This approach finds its latest results in the work by Criesfield and Packham (1988), Harvey (1988), Blasi and Foraboschi (1994), Falconer (1994), Gilbert and Melbourne (1994), Hughes (1995) and Como (1998).

Such an approach is suitable for semicircular arches provided that the mechanism activates when the stress state is still rather limited, but this is not so for flat arches or for arches with very weak mortar joints. In this case the compressive stresses turn out to be significantly high when a collapse mechanism is still far from activating, so that the non linear elasto-plastic response of masonry, i.e. of the mortar joints, becomes a relevant parameter of the structural behaviour of the arch.

The elasto-plastic collapse can be dealt with in several ways. On the basis of simplifying assumptions for the plastic stress distribution in the mortar joints (Clemente et al., 1995) or of experimental tests (Taylor and Mallinder, 1993, Boothby, 1997) yield surfaces are obtained following the classical approach to plasticity; the limit load is obtained assuming that collapse is met when the axial thrust and the bending moment from the thrust line theory lie on the limit surface.

Another popular method of assessment makes use of F.E. procedures. The local collapse condition is usually derived from experimental tests and introduced into a F.E. model in which the arch is considered a mono-dimensional structure (Criesfield, 1985, Bridle and Hughes, 1990, Choo et al., 1991, Molins and Roca, 1998.a,b) a bi-dimensional (Loo and Yang, 1991, Falconer, 1994, Boothby et al., 1998, Owen et al., 1998, Ng et al. 1999, Loureno and Rots, 2000, among the latest results) or three-dimensional one (Rosson et al., 1998). While mono-dimensional models proved to be reliable and flexible enough to be used in assessment and design procedures for arch bridges, requiring limited

computational effort, bi- and three-dimensional models are able of giving detailed information on local phenomena at expenses of a relevant complexity of the model and of long computing times. For this reason analyses of multi-span arch bridges have been performed, up to present, mainly by means of monodimensional finite elements (Molins and Roca, 1998.b), while bi-dimensional analyses had to be limited to quite simple models (Falconer, 1994).

The effect of the adjacent spans on the loaded arch had also been studied experimentally on 1:5 multi-span bridge models (Royles and Hendry, 1991, Melbourne and Wagstaff, 1993, and Melbourne et al., 1995, Ponniah and Prentice, 1998) and of mechanism ones (Hughes, 1995). The results, neglecting the elasto-plastic response of the joints, show a reduction of the limit load due to the presence of the adjacent arches somewhere in-between 20% to 50% of the limit load for a single span arch bridge.

In this paper, following the approach by Bridle and Hughes (1990) and Choo et al. (1991), Castiglianos elastic method (1879) is implemented in a mono-dimensional F.E. procedure and is extended in order to take into account the plastic response of the mortar joints. The flexibility of the procedure allows the analysis of several single, twin and multi-span arch bridges pointing out the way the structural response of the bridge is affected by the different mechanical and geometrical factors, i.e. masonry strength and arch barrel geometry. The obtained results give further information on the arch collapse mechanisms, allowing some considerations on the multi-span arch bridge structural behaviour.

ELASTO-PLASTIC EXTENSION OF CASTIGLIANOS APPROACH

According to Castiglianos approach, the interface between adjacent voussoirs can be represented by means of an elastic unilateral contact surface. In this way no traction is allowed to develop on the interface and no limit is set to the compressive stresses developed in the joint, Fig. 1.

N

MMN

G

NMx

h

compression

no-tensile resitant area

'c

Fig. 1. No-tensile resistant model for the voussoirs interface, Castigliano (1879)

The constitutive equations for the no-tensile resistant model can be derived in terms of the effective section height x:

2bx

'Nc

E = ,

=

=

32322xh

Nxhbx

'M Ec

E , (1.a,b)

where the superscript E stands for the forces equilibrated by a No Tensile Resistant material fully elastic in compression.

In Castiglianos original work this model is assumed to represent dry assemblages of voussoirs or weak mortar joints. In the first case the model is somewhat questionable because it does not take into account the spalling collapse of the compressed edge (Taylor and Mallinder, 1993). On the other side mortar joints are subjected to a bi-axial compressive stress state due to the confinement effect of the voussoirs and, therefore, can exhibit quite high compressive stresses and significant plastic strains.

An improved mechanical model of the interface should assume a limit stress c in compression beyond which plastic strains are allowed. Even though the proposed procedure is general and not related to a specific elasto-plastic model for the mortar joint, in the following reference is made to a perfect elasto-plastic constitutive model.

Under the plane section hypothesis, i.e. a linear distribution of strains, Fig. 2, the constitutive equations for the joints can be derived:

( )yxbNc

EP +=2

( ) ( )

+++= 22

3

1

22yxyxyx

hbM

cEP (2.a,b)

G

'c

c

MN

compression

hx

y

Fig. 2. No-tensile resistant perfect compressive elasto-plastic model for the joint interface.

The plastic cut-off of Fig. 2 can be computed as the difference between the elastic response of the effective section x eq.s (1) - and the elasto-plastic response eq.s (2):

yx

ybNNN c

EPEEP

==

2

2 (3.a)

yx

ybhN

yx

y

yx

yhbMMM

cEP

cEPE

EP =

==

332

623

1

22 (3.b)

The constitutive eq.s (2) and (3) are suitable for setting up an iterative procedure for the analysis of masonry arch-type structures. Even though this interface model has been developed with reference to a perfectly elasto-plastic response of the mortar joint but is suitable to

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