Keywords: masonry arch bridge, finite element method, collapse analysis, joint element, contact problem, Bott-Duffin inverse.

The present study deals with the collapse analysis of masonry arch bridges by means of Finite Element Method (FEM). Many experimental results on masonry arch bridges show importance of tensile resistance of joints as well as the profile and boundary condition of bridges. In order to analyze and calculate masonry structures, there are several models such as theorem by Castigliano, concrete-like constitutive model, joint element, Bott-Duffin inverse, etc. In this paper, Bott-Duffin inverse is briefly introduced and by means of these models the results obtained from collapse analysis of the masonry arch bridge over Tanaro river, Alessandria in Italy, are discussed.

In order to analyze masonry structures, mathematical models are developed to describe their behaviours. While developing the mathematical models, some assumptions are made for simplification. Definitely masonry material can resist high compressive stresses but only feeble tensions. The theory of equilibrium of elastic system is applied to the conditions of imperfectly elastic stresses for the resisting section by Castigliano in 1879. As an extension of theorem by Castigliano, no-tensile resistant perfect elastic-plastic model is applied on masonry arch bridges [1].

There are mainly two approaches for the analysis of masonry structures by means of FEM, one is macro-modelling and the other is micro-modelling [2]. Cracks are assumed to form in planes perpendicular to the direction of maximum principal tensile stress which reaches the specified tensile strength. The cracked masonry is anisotropic and smeared crack model is adopted [3,4]. As has been shown by the analysis of discontinuous rocks, the joint element is effectively modelled for analyzing structures composed of two different materials with very different strength [5]. In masonry structures, only compressive stress is assumed to exist and to a certain extent they become contact problem. An automatic analytical method based on Bott-Duffin inverse to simulate masonry arches as contact problem are presented [6].

Three analytical models of the masonry arch bridge, that is Model 1, arch subjected to centric or eccentric load with fixed ends, Model 2, arch subjected to centric load is supported on pillars, Model 3, central arch subjected to centric load is supported on pillars and outer two arches are supported on both pillar and fixed end, are analyzed.

Collapse loads of Model 3 are approximately two to three times as much as those of Model 2. In so far as the boundary condition is concerned, Model 3 may be slightly over-idealized, while Model 2 is on the safe side from a structural point of view. From comparison of centric load with eccentric one in Model 1, the latter is more severe than the former in this profile. In the case of centric load, crack occurs, fracture develops, and at last, the collapse occurs at the center of arch in compression. But the portions at the fixed ends are still sound in this profile. On the other hand, in the case of eccentric load, the collapse mechanism due to four hinges occurs in tension. Collapse loads obtained from concrete-like constitutive model are larger than those of the other models. The Finite Element analysis (FEA) using the discrete crack model is more effective than that using the smeared crack model.

Bott-Duffin inverse enables us to present an automatic analytical method for a system of simultaneous linear equations with the subsidiary condition of unknowns. Comparison of those results suggests that the collapse mechanism can well by simulated by the FEA in terms of Bott-Duffin inverse presented herein. The main advantage of the present method is that it allows the procedure without rebuilding the stiffness matrix K even if the contact state changes.

1

A. Brencich, U.De Francesco, L. Gambarotta, "Elastic No Tensile Resistant - Plastic Analysis of Masonry Arch Bridges as an Extension of Castigliano's Method", Proc. of the 9th Canadian Masonry Symposium, pages 14, 2000.

2

P.B. Lourenço, J.G. Rots, J. Blaauwendraad, "Two approaches for the analysis of masonry structures: Micro and macro-modeling", HERON, Vol.40, No.4, pp.313-340, 1995.

3

E. Hinton, D.R.J. Owen, "Finite Element Software for Plates and Shells", Pineridge Press, 1984.

4

T. Aoki, D. Carpentieri, A. De Stefano, C. Genovese, D. Sabia, "Analisi Sismica di Parete in Muratura: Metodi e Tecniche a Confronto", Proc. of X Convegno Nazionale ANIDIS on L'ingegneria Sismica in Italia, Potenza, pages 12, 2001. (in Italian)

5

T. Aoki, "Formulation of Elastic-Plastic Joint Elements and their Application to Practical Structures", Computational Modelling of Masonry, Brickwork and Blockwork Structures, ed. by J.W. Bull, Saxe-Coburg Publications, Edinburgh, Scotland, pp.27-52, 2001. doi:10.4203/csets.6.2

6

T. Aoki, T. Sato, "Application of Bott-Duffin Inverse to Static and Dynamic Analysis of Masonry Structures", Proc. of the 8th Int. Conference on Structural Studies, Repairs and Maintenance of Heritage Architecture 2003, pp.277-286, 2003.

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