LIMIT LOAD OF A MASONRY ARCH BRIDGE BASED ON FINITE ELEMENT FRICTIONAL CONTACT ANALYSIS

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1 5 th GRACM International Congress on Computational Mechanics Limassol, 29 June 1 July, 2005 LIMIT LOAD OF A MASONRY ARCH BRIDGE BASED ON FINITE ELEMENT FRICTIONAL CONTACT ANALYSIS G.A. Drosopoulos I, G.E. Stavroulakis II, C.V. Massalas III I Department of Material Science and Technology, University of Ioannina, Ioannina, Greece II Department of Mathematics, University of Ioannina, Ioannina, Greece and Depatment of Civil Engineering, Technical University of Braunschweig, Germany III Department of Material Science and Technology, University of Ioannina, Ioannina, Greece s: {me01122}{gestavr}{cmasalas}@cc.uoi.gr Keywords: masonry arch, limit load, unilateral contact Abstract. The limit load of a stone arch bridge can be identified by the lack of solvability of a finite element analysis including contact interfaces that simulate potential cracks. Opening or sliding at some of them indicates crack initiation. The ultimate load has been calculated by using a path - following (load incrementation) technique. The method is applied on the Strathmashie stone bridge and the results are comparable with the ultimate failure load prediction of the collapse mechanism method and with experimental data published in the literature. 1 INTRODUCTION The limit load of a masonry arch is mainly attributed to the interaction of unilateral (no-tension) effects and the structural form. In the present work the limit load of such a bridge is calculated by exploiting the solvability of the arising unilateral structural analysis problem. The finite element of the bridge includes a number of interfaces obeying unilateral contact with friction. The loading conditions of the bridge include the self weight and a concentrated load at the middle of the span or at the quarter-span. A parametric investigation demonstrates the influence of the location of the concentrated load, of stick or stick-slip conditions and of the number of interfaces on the limit load. The results are compared with the ultimate failure load of the traditional collapse mechanism method [10] and with the results of the experimental work reported in [9]. 2 UNILATERAL CONTACT FINITE ELEMENT ANALYSIS AND ESTIMATION OF THE LIMIT LOAD The structural analysis problem for an elastic structure with unilateral contact interfaces takes the form: Κu N r P P T + = O +λ Nu - g 0 r 0 T (Nu - g) r = 0 (1) (2) (3) (4) In the above equations, K is the stiffness matrix, u is the displacement vector, r represents the Lagrange multipliers and is equal to the normal pressure t n, P O is the self - weight and P is the live (applied) load of the structure. The problem described by equations (1) - (4) is a linear complementarity problem (LCP) [5]. For the theoretical study and the solution of the unilateral contact problem, the arising variational inequalities

2 and the available numerical techniques, the reader may consult [1], [2], [4], [5], [6], [8]. By considering a loading path with a scalar load parameter λ, the structural analysis problem is transformed to a parametric linear complementarity problem. The maximum value of λ for which problem (1)-(4) has a solution is related with the limit load. A path - following method is applied here, thus λ is a scalar loading factor which will be used in the sequel for the determination of the limit load; 0 λ λ failure 0, where λ failure represents the value of the loading factor in which the unilateral contact problem does not have a solution. For more information on the solvability conditions for unilateral contact problems the reader is refered to [3]. The behavior in the tangential direction is defined by a static version of the Coulomb friction model which takes into account stick - slip effects and, in analogy to the deformation theory of plasticity, is suitable for holonomic loading. This means that two contacting surfaces start sliding when the shear stress in the interface reaches a maximum critical equal to: n τ = µ t, (5) cr where t n is the contact pressure and µ is the friction coefficient. The variational inequality problem is transformed into a system of nonlinear equations by means of a suitable Lagrangian method. Finally, a set of nonlinear equations is solved by the Newton-Raphson incremental iterative procedure [4], [5], [6], [7]. 3 THE BRIDGE WITH FRICTIONAL CONTACT INTERFACES Quadrilateral, four - node, bilinear, plain strain elements are used for the finite element model of the bridge. The model has 3302 nodes and 3036 elements. Large displacement effects are neglected. Figure 1. Geometry of Strathmashie bridge

3 Figure 2. Mesh of the finite element model The load is applied in small steps. Termination of the analysis is caused from numerical singularity due to negative eigenvalues of the stiffness matrix at the onset of failure. In figure 3 the failure load is given for various number of unilateral contact interfaces equidistantly distributed along the bridge. The load is reduced for higher number of interfaces. If the force is applied in the middle span, the failure load is greater than if it is applied in the quarter span. In the model with 26 interfaces the failure load is approximately equal with that received from the experimental work [9], if a quarter span force is applied as it is shown in figure 4. If the friction coefficient is assumed to be equal to 0,4 instead of 0,3, the failure load is increased. Figure 3. Failure load - Number of interfaces Figure 4. Failure load

4 4 MECHANISM OF COLLAPSE A comparison between the results received from the collapse mechanism method [10] and the finite element method is presented. The program Archie-m has been used for the implementation of the collapse mechanism method. The failure load is found by solving the equilibrium equations of the arch; this is achieved by creating the thrust line of the arch [10]. When the thrust line in a cross section is adjacent to the ring of the arch, a hinge is opened in that point. According to the upper bound theorem from the theory of plasticity, the maximum load corresponding to some collapse mechanism is greater or equal to the maximum load corresponding to the real collapse mechanism. This theorem implies that when the thrust line is adjacent to the ring of the arch in four points then the arch is not safe. Friction is high enough between stones and sliding failure cannot occur. The masonry has an infinite compressive strength. A finite element model without sliding, equivalently with infinite friction coefficient is developed and used for the comparison. The failure load of this model with 26 interfaces and the quarter span load is four times greater than the experimental one. The deformation of the model at the onset of collapse is shown in figure 5. The load is applied at the quarter span of the bridge. Four hinges are opened. The comparison with the results of the collapse mechanism method given in figure 6, is satisfactory. Figure 5. Failure mode of the 26 interfaces contact model - Quarter span load Figure 6. Failure mode calculated by the collapse mechanism method - Quarter span load

5 5 CONCLUSIONS A model of unilateral contact with friction integrated into a finite element analysis can be used for the determination of the ultimate failure load of a masonry bridge. For a quite large number of potential interfaces, the computational ultimate load calculated by the model with friction with a quarter span load is equal with the experimental one. The model without frictional sliding, which is more compatible with the collapse mechanism method, led to higher values. The four hinge mechanism when failure occurs is confirmed both by the usage of the contact model and of the collapse mechanism method. One advantage of the proposed method is that it can be numerically implemented within every modern, general purpose, nonlinear finite element program (Abaqus has been used here), provided that no failure due to numerical instabilities arise before the activation of the failure mechanism. Otherwise a limit load problem must be formulated and solved, a task which would have required the use of specialized software. REFERENCES [1] Panagiotopoulos, P. D. (1985), Inequality problems in mechanics and applications. Convex and Nonconvex Energy Functions, Birkhauser, Boston, Basel, Stuttgart. [2] Stavroulaki, M. E., Stavroulakis, G. E. (2002), Unilateral contact applications using FEM software, Int. J. Appl. Comput. Sci., Vol.12, No.1, pp [3] Stavroulakis G. E., Panagiotopoulos P. D. and Al-Fahed A. M. (1991) On the rigid body displacements and rotations in unilateral contact problems and applications, Computers and Structures Vol. 40, No 3, pp [4] Bathe K. J. (1996), Finite Element Procedures, Prentice-Hall, New Jersey. [5] Mistakidis E. S. and Stavroulakis G. E. (1998), Nonconvex optimization in mechanics. Smooth and nonsmooth algorithms, heuristics and engineering applications, Kluwer Academic Publishers, Dordrecht. [6] Stavroulakis G. E and Antes H. (2000) Nonlinear equation approach for inequality elastostatics. A 2-D BEM implementation, Computers and Structures 75(6), pp [7] Zhowy H. W., He S. Y., Li X. S., Wriggers P. (2004) A new algorithm for numerical solution of 3-D elastoplastic contact problems with orthotropic friction law, Computational Mechanics, 34(1), pp [8] Hlavacek I., Haslinger J., Necas J., Lovisck J. (1988), Solution of variational inequalities in mechanics, Springer Verlay. [9] Page J. (1993), Masonry arch bridges - TRL state of the art review, HMSO, London. [10] Heyman J. (1980), The masonry arch, Ellis Horwood Series In Engineering Science.