UNE APPROCHE D ANÁLYSE LIMITE POUR DES STRUCTURES EN MAÇONNERIE. Tel: , Fax: ,

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1 UNE APPROCHE D ANÁLYSE LIMIE POUR DES SRUCURES EN MAÇONNERIE A LIMI ANALYSIS APPROACH FOR MASONRY SRUCURES A. ORDUÑA 1, P. B. LOURENÇO 1 1 Department of Civil Engineering, School of Engineering, University of Minho, Portugal. el: , Fax: , aord@eng.uminho.pt RESUME: Une approche d anályse limite pour des structures en maçonnerie construites avec blocs rígides est presentée et son utilisation sur l ordinateur est detaillée. Une loi d écrouissage associée est utilisée pour arriver à un probléme de programmation linéaire, qui est résolu avec la méthode Simplex. Arches à voussoir, soumises a des charges variables, sont modellées avec le programme développé, pour valider l approche utilisée. Les résultats sont comparés avec des essais expérimentaux obtenus sur la literature. Commentaires sur les avantages et désavantages de l approche sont inclus ci-oint avec suggestions d autres développements. ABSRAC: A limit analysis approach for 2D structures made from rigid blocs and its computer implementation are presented. In order to arrive to a linear programming problem, which is solved with the Simplex method, an associated flow rule is used. Masonry shear walls, under lateral live loads, and masonry voussoir arches, under vertical live loads, are modeled with the developed program, for the assessment of the adopted approach. he results of the arches analyses are compared with experimental tests extracted from the literature. Comments on the strength and weaness of this approach are included, along with suggestions on further possible developments. 1. INRODUCION he engineering assessment of ancient masonry structures, such as arch bridges, requires practical computational tools. Non-linear finite element approaches are a possible choice for masonry structures assessment. Both advanced continuous, anisotropic based models, and discrete (micro-) models for masonry structures have been developed in the last decades, e.g. Lourenço and Rots (1997) and Lourenço et al. (1998). Nevertheless, the drawbac of using nonlinear finite element analysis in practice might include: (a) requirement of adequate nowledge of sophisticated non-linear processes and advanced solution techniques by the practitioner, as well as (b) comprehensive mechanical characterization of the materials, and (c) large time requirements for modeling, for performing the analyses themselves, with a significant number of load combinations, and for reaching proper understanding of the results significance. Of course, for special cases, as complex, important or large arch bridges, non-linear analysis should not be disregarded as an analysis tool. Linear-elastic analysis can be assumed more practical, even if the time requirements of modeling are similar. Nevertheless, such an analysis fails to give an idea of the structural behavior beyond the beginning of cracing, and it is easy to arrive to strong under-estimations of the structural capacities. Limit analysis combines, on one side, sufficient insight into collapse mechanisms, ultimate stress distributions (at least on critical sections) and load capacities, and, on the other, simplicity to be cast in a practical computational tool. Another appealing feature of limit analysis is the reduced number of necessary material parameters (here, only the friction coefficient is needed), given the difficulties in obtaining masonry material parameters. Livesley (1978), was the first author to apply the limit analysis theory to rigid bloc arches, by means of the static theorem. More recently, Gilbert and Melbourne (1994) applied the inematics theorem to the same problem, aiming at analyzing masonry arch bridges. Begg and Fishwic (1995) have released the restriction of associated flow rule or normality condition for masonry arches, including sliding shear failure. his aspect was also addressed by Baggio and rovalusci (1998) in a general formulation for rigid bloc limit analysis. It is noted that extending the

2 formulation to non-associated flow results in a non-linear mathematical problem of a significantly larger size, in comparison with the lower size linear problem resulting from the classical theory. In the following, a classic limit analysis formulation for two-dimensional structures made from rigid blocs is presented. his formulation is implemented in a computer program that is suitable for engineering practice analysis of non-confined masonry structures. A discussion on the behavior of masonry shear walls is presented for the assessment of the program performance. he possibility of utilizing the program for the analysis of voussoir arches is demonstrated from a comparison with the results obtained experimentally by Pippard and Chitty (1951). 2. LIMI ANALYSIS FORMULAION he approach presented here follows the classical theory of limit analysis, meaning that small displacements, and associated flow rules are assumed to be valid. It is assumed that masonry structures are made from perfectly rigid blocs, the interface between two blocs cannot withstand tensile stresses, can withstand infinitely large compressive stresses, and the shear failure is of frictional nature, i.e. slipping on a oint is accompanied of separation. hese assumptions result in piecewise linear yield functions. he static as well as the inematics formulations are established, although only one of them needs to be solved, as they are dual formulations of the same linear programming problem (Grierson, 1977). he static variables, or generalized stresses, at an interface are selected to be the shear force, V, the normal force, N, and the moment at the center of the oint, M. Correspondingly, the inematic variables, or generalized strains, are the relative tangential, normal and angular velocities, δ n, δ s and δθ at the interface center, respectively. he degrees of freedom are the velocities in the x and y directions, and the angular velocity of the centroid of each bloc: δ ui, δ vi and δφ i for the bloc i. In the same way, the external loads are described by the forces in x and y directions, as well as the moment at the centroid of the bloc. he loads are split in a constant part (with a subscript c) and a variable part (with a subscript v): f cxi, fvxi for the forces in x direction, f cyi, f vyi for the forces in y direction, and m ci, mvi for the moments. hese variables are arranged in the vectors of generalized stresses Q, generalized strains δq, velocities δu, constant loads F c, and variable loads F v. Finally, the load factor α is defined, measuring the amount of the variable load vector applied, and whose limit value is looed for. In this way, the total load vector F is given by F = F c + αf v (1) he yield surface at a oint is composed by the combination of the hinge formation condition M + a N 0, and the sliding condition V + µ N 0 ; where a is the distance between the oint center and either of the edges; and µ is the friction coefficient, equal to the tangent of the friction angle. hese expressions define four planes delimiting the generalized stresses that the oint can stand. In matrix form, these conditions are expressed by eq. (2). Here, ϕ is the yield function; R is the vector of plastic resistances, defined as the distances between the origin and each one of the yield planes (in this case it is a zero vector); and N is a matrix which rows are vectors normal to each one of the yield planes. he extension of the eq. (2) for the whole structure gives eq. (3). Here, N is a macro-array formed by all the N arrays in diagonal form. ϕ N Q R 0 (2) N ϕ Q R 0 (3) he flow rule, according to the normality condition, is expressed, for oint by eq. (4), and for the whole structure by eq. (5). Where δλ and δλ are the vectors of the flow multipliers, for oint

3 and for the whole structure, respectively. Each flow multiplier corresponds to a yield plane, and must satisfy eqs. (6-7). Eq. (6) expresses that plastic flow must involve dissipation of energy, and eq. (7) specifies that plastic flow cannot occur unless the stresses have reached the yield surface. δq = Nδλ (4) δq = N δλ (5) δλ 0 (6) ϕ = 0 (7) Compatibility between oint generalized strains, and the velocities of the adacent blocs i and, is given in eq. (8), and its generalization to the whole structure in eq. (11), being the vector δu i, defined in eq. (9) and the matrix C, given in eq. (10). In this last equation α, β i, β, are the angles, with respect to the x axis, of the direction of oint, of the line defined by the centroid of bloc i and the center of oint, and of the line defined by the centroid of bloc to center of oint, respectively; and b i, b, are the distances between the center of oint to the centroid of the blocs i and, respectively. he matrix C is formed with the matrices C arranged accordingly to the blocs i and velocities positions in the δu vector. δq = C δu (8) i, i, i δu [ δu ] δu (9) cos( α ) sin( α ) bi sin( βi α ) cos( α ) sin( α ) b sin( β α ) C = sin( α ) cos( α ) bi cos( βi α ) sin( α ) cos( α ) b cos( β α ) (10) δq = C δu (11) Equilibrium is stated by applying the contragredience principle, see eq. (12). he principle of virtual wor can then be used to establish Q δq = F δu. Introducing here eq. (1), eq. (13) is obtained. Since, in the limit state, the velocities are arbitrarily large, it is necessary to introduce a restriction, as done, conveniently, by eq. (14). hen, starting from eq. (13) and using eqs. (3,5,7,14), eq. (15) is readily obtained. F = C Q (12) Q δq = F δu +αf δu (13) c v F v δu 1 (14) α = R δλ F δu (15) he static theorem is used, together with the eqs. (1,3,12), to establish the linear-programming problem stated in eq. (16). Whereas, the inematic theorem, together with the eqs. (5,6,11,14,15), results in the linear-programming problem stated in eq. (17). Only one of these problems is necessary to obtain the solution, as they are dual formulations. In the established form, the linear programming problem of eq. (16) is not directly solvable by the Simplex method. herefore, the problem stated in eq. (17) was chosen to obtain the sought solutions to the limit analysis problem. Maximize: [ ] c Q 0 1 (16) α C F Q = F v c Subect to: N 0 α R

4 δu Minimize:[ Fc R ] (17) δλ C N δu 0 Subect to: = ; δλ 0 Fv 0 δλ 1 he supports are taen into account by adding extra fixed blocs representing the foundations. his does not add new equations or variables, because only the movable blocs must appear in the formulation. 3. MASONRY SHEAR WALLS Masonry walls lie those presented in Baggio and rovalusci (1998) were analyzed, and a parametric study was carried out. he walls consist of regular dry assembled brics, subected to self weight and increasing lateral body loads provided by a tilting table, simulating a seismic action statically. he friction coefficient used is 0.75, the first row of the models starts with a incomplete piece, unless otherwise indicated. Numerical results are presented in able 1, where H and B are the wall height and length, respectively; h and b are the unit height and length; s is the overlap of units in consecutive courses, representing the bond (stac bond is given by s/b=1.0 and regular bond is given by s/b=0.5); the column labeled LF contains the load factors (horizontal to vertical ratio), and the column F the failure types (SL means slipping failure, and O means overturning failure). Several observations can be made from the results presented in able 1: (a) increasing the wall height to length (H/B) ratio, decreases the load factor and the failure type changes from slipping to overturning; (b) increasing the unit height to length (h/b) ratio, decreases the load factor and the failure type changes from slipping to overturning; (c) using smaller units (larger h/h ratios), decreases the load factor and the failure type changes from slipping to overturning; (d) the bond affects the load factor for monotonic loading, stac bond is not the worst, and regular bond is not the best, but, for cyclic loading, regular bond is the best; (e) for monotonic loading, if the first row starts with a full unit rather than a partial one, the load factor improves slightly. It must be stressed that homogenization techniques or continuum models are incapable of reproducing the present results, strongly dependent on the geometry and arrangement of units. In Fig. 1 three typical failure mechanism are presented. Fig. 1a presents a sliding failure type, where the separation that accompanies sliding is evident. It must be noted that the failure mode could be along almost any combination of horizontal oints, provided that all of them have a generalized stress state that lies on the yield surface, meaning that the sliding mechanism is not unique for such case. Figs. 1b and 1c present two different overturning failure mechanisms. able 1. Numerical results for masonry shear walls analyses. he shaded row represents the reference analysis and the bold items indicate the item subected to variation H/B=0.5 H/B=1.0 H/B=2.0 h/b h/h s/b LF F h/b h/h s/b LF F h/b h/h s/b LF F SL SL O SL SL O SL O O O O O O O O SL SL O O O O O O O SL O O O O O 0.33 * SL 0.33 * SL 0.33 * O * First row starts with a full piece

5 (a) (b) (c) Figure 1. Some failure mechanisms for masonry shear walls; (a) H/B=0.5, h/b=0.33, h/h=6, s/b=0.5, LF=0.75; (b) H/B=1.0, h/b=0.33, h/h=12, s/b=0.5, LF=0.622; (c) H/B=2.0, h/b=0.33, h/h=12, s/b=0.5, LF= MASONRY VOUSSOIR ARCHES UNDER VERICAL LOADS As an application, and at the same time a verification of the program developed with the formulation presented, the masonry voussoir arches tested by Pippard and Chitty (1951) were analyzed. he arches had 23 voussoirs, a span of 3048mm (10 ), a rise of 762mm (2-6 ), thicness of 254mm (10 ), and width of 152mm (6 ), see Fig. 2a. he arches were loaded with a dead weight simulating the bacfill of a bridge, and an increasingly point load at one of the voussoirs. he voussoirs were cast in concrete, two sets of voussoirs were made, the first one with limestone aggregates had a compressive strength of 12.0MPa (1740 lb/in 2 ); and the second one made with granite aggregates had a compressive strength of 46.2MPa (6700 lb/in 2 ). Six series of tests were carried out, series 1 through 3, with the limestone aggregate, and series 4 through 6, with the granite aggregate. Series 1 and 4 were ointed with a non-hydraulic lime mortar; in series 2 and 5 the voussoirs were in direct contact (the arch had 24 voussoirs); and series 3 and 6 were ointed with a rapid-hardening Portland cement mortar. Fig. 2b shows a comparison between the experimental and limit analysis results, while Fig. 3 shows the limit analysis results for the arch loaded on the 15 th voussoir. From Figs. 3a and 3b, it is possible to observe a clear correspondence between the oints were the thrust line touches the arch edge, and those where hinges are formed, as expected by the theory (Heyman, 1969). Limit load value series 1 series 2 series 3 series 4 series 5 series 6 limit analysis Load Cans of lead shot to represent filling load position (a) (b) Figure 2. Pippard and Chitty (1951) arch tests; (a) test configuration; (b) limit load against load position (as span rate) It can be seen, from Fig. 2b, that the analytical results limit load values always are lower than that of the mortar ointed arches (series 3 and 6). his can be explained by the tensile strength provided by the mortar, that allows the thrust line to lie outside the experimental arch, increasing its strength. On the other hand, the analytical limit load values are always higher than that of the dry ointed arches (series 1, 2, 4 and 5). his fact is explained because in the analysis, the thrust line is allowed to reach the edge of the oint were a hinge is formed, implying infinitely large stress values, see Fig. 3b. Furthermore, the thrust line lies outside the arch on the loaded voussoir in some cases, implying tensile stresses inside the voussoir, see Fig. 3c. he voussoir material indeed is capable of sustaining small tensile stresses, but the limit analysis approach does not restrain their amount. In the tests some spalling-slipping and crushing failures were observed

6 (Pippard and Chitty, 1951), particularly in those where the loaded voussoir was the 15 th or 16 th (two leftmost sets of points in Fig. 2b), that coincides with the greatest differences between analytical and experimental results, confirming this hypothesis. However, in real structures, where concentrated loads are very seldom (because of fill load distribution, for example), this drawbac of the analysis approach would be less meaningful, since the thrust line would be smoother. It is worth to mention that a limit analysis was made with the load on the mid span voussoir, the result was an infinitely large limit load. Obviously, the meaning of this result is that there exist a thrust line within the arch thicness for any value of the load, but what is not taen into account is that the voussoir material would fail by crushing for a sufficiently large load. (a) (b) (c) Figure 3. Pippard and Chitty (1951) arch with load on the 15 th voussoir, limit analysis results; (a) thrust line; (b) failure mechanism; (c) detail of the trust line on the loaded voussoir. 5. CONCLUSION he limit analysis formulation presented is general enough to handle rigid-bloc planar structures of any configuration. his type of approaches has a number of weanesses, not only that of frictional shear failure instead of sliding (without separation), but also the very simple failure criteria employed, which do not allow the prediction of failure modes as mortar or voussoir crushing, or voussoir spalling. A 3D extension of the program can be made by the addition of new variables and revisiting the failure modes. Including a non-associate sliding flow seems necessary and, as this improvement introduces non-linearity, more realistic yield functions can be introduced without increasing significantly the complexity of the mathematical programming problem. 6. ACKNOWLEGMENS he first author wishes to than the scholarship made available to pursue his PhD studies by the Conseo Nacional de Ciencia y ecnología of Mexico. he wor was also partially supported by proect SAPIENS funded by Fundação para a Ciência e ecnologia of Portugal. 7. REFERENCES Baggio, C. and rovalusci, P. (1998). Limit analysis for no-tension and frictional threedimensional discrete systems. Mechanics of Structures and Machines, 26(3), Begg, D.W. and Fishwic, R.J. (1995). Numerical Analysis of Rigid Bloc Structures Including Sliding. Proc. 3 rd Int. Symp. Comp. Meth. Struct. Mas., Portugal, eds. J. Middleton, G.N. Pande. Gilbert, M. and Melbourne, C. (1994). Rigid-bloc analysis of masonry structures. he Struct. Engineer, 72(21), Grierson D.E. (1977). Collapse load analysis. In: Engineering plasticity by mathematical programming. Proc. NAO Adv. Study Inst., Univ. Waterloo, Canada, eds. M.Z. Cohn, G. Maier. Heyman, J. (1969). he safety of masonry arches. Int. J. Mech. Sci., vol. 11, pp Livesley, R.K. (1978), Limit analysis of structures formed from rigid blocs. Int. J. Num. Meth. Engrg., 12, Lourenço, P.B., Rots, J.G. and Blaauwendraad, J. (1998). Continuum model for masonry: Parameter estimation and validation. J. Struct. Engrg., ASCE, 124(6), Lourenço, P.B. and Rots, J.G. (1997). A multi-surface interface model for the analysis of masonry structures. J. Engrg. Mech., ASCE, 123(7), Pippard A.J.S. and Chitty L. (1951), A study of the voussoir arch. National Building Studies, Research Paper No 11, London, England.