Stress analysis of masonry structures

Structural Analysis of Historic Construction – D’Ayala & Fodde (eds) © 2008 Taylor & Francis Group, London, ISBN 978-0-415-46872-5 Stress analysis of masonry structures: Arches, walls and vaults A. Baratta, I. Corbi & O. Corbi Department of Structural Engineering, University of Naples “Federico II”, Naples, Italy ABSTRACT: In the paper, the authors present a synthetic overview of some results obtained by means of a number of theoretical and experimental studies developed on some classical masonry typologies such as arches, panels and vaults. The proper implementation of the analyzed structural problem, specialized to the specific case, which derives from the extension of classical structural approaches to structures made of masonry material, is shown to provide a reliable approach to the problem itself, also in the case of some reinforcement, as demonstrated by experimental data which are in perfect agreement with numerical results. 1 INTRODUCTION The basic assumption of no-tension masonry model coincides with the hypothesis that the tensile resistance is null. Under this hypothesis, no-tension stress fields are selected by the body through the activation of an additional strain field, the fractures (see Baratta, 1991, Baratta et al. 1981, Baratta & Toscano 1982, Bazant 1996, Heyman 1966). The behavior in compression can be modeled in a number of different ways (elastic linear, elastic non-linear, elastic-plastic; isotropic, anisotropic; etc.), without altering substantially neither the results nor the mathematical treatment of the problem; some convenience exists for practical applications in assuming a isotropic linearly elastic model, in order to keep limited the number of mechanical parameters to be identified for masonry, since increasing the number of data causes increasing uncertainty in the results. Because of these reasons, and being clearly understood that there is no difficulty in introducing more sophisticated models, it is convenient to set up the fundamental theory on the basis of the assumption that the behavior in compression is indefinitely linearly elastic. Analysis of NRT (Not Resisting Tension) bodies proves that the stress, strain and displacement fields obey extreme principles of the basic energy functionals. Therefore the behaviour of NRT solids under ordinary loading conditions can be investigated by means of some extensions of basic energy approaches to NRT bodies (Baratta 1984, Baratta & O. Corbi 2005a, 2007, Baratta & al. 1981, Baratta & Toscano 1982). In details, the solution of the NRT structural problems can be referred to the two main variational approaches: – the minimum principle of the Potential Energy functional; – the minimum principle of the Complementary Energy functional. In the first case the displacements and the fractures are assumed as independent variables; the solution displacement and fracture strain fields are found as the constrained minimum point of the Potential Energy functional, under the constraint that the fracture field is positively semi-definite at any point. The approach based on the minimization of the Complementary Energy functional assumes the stresses as independent variable. The complementary approach is widely adopted since the existence and uniqueness of the NRT solution are always assured in terms of stress, if some conditions on the compatibility of the loads are satisfied. The stress field can, then, be found as the constrained minimum of the Complementary Energy functional, under the condition that the stress field is in equilibrium with the applied loads and is compressive everywhere in the body. The solution of both problems can be numerically pursued by means of Operational Research methods (see i.e. Rao 1978) suitably operating a discretization of the analyzed NRT continuum (Baratta & I. Corbi 2004, Baratta & O. Corbi 2003b, 2005a). One should notice that discussion about existence of the solution actually can be led back to some Limit Analysis of the considered NRT continua (Baratta & O. Corbi 2005a). 321 2,3 m 0,124 m 1 0,200 m 2,23 m 2 3 4 architrave 1,322 m FRP 0,382 m 0,775 m Transducer 0,750 m (a) Strain-gauge (b) Figure 1. (a) Masonry panel geometry, (b) with the applications of a light reinforcement by means of FRP strips. In this regard, a special formulation of Limit Analysis for No-tension structures has been performed, allowing the set up of theorems analogous to the basic kinetic and static theorems of classical Limit Analysis; thus, one can establish efficient procedures to assess structural safety versus the collapse limit state (see e.g. Como & Grimaldi 1983) by specializing and applying fundamental theorems of Limit Analysis to NRT continua (Baratta 1991, Baratta & O. Corbi 2003a, 2005a, Bazant 1996, Como & Grimaldi 1983, Khludnev & Kovtinenko 2000). In details, the individuation of the collapse (live) load multiplier for NRT continua can be referred to the approaches relying on the two main limit analysis theorems: – the static theorem; – the kinetic theorem. This means that, after defining the classes of statically admissible and kinetically sufficient load patterns, Limit Analysis allows individuating the value of the live load multiplier limiting the loading capacity of the body, i.e. evaluating the collapse live load and/or the safety factor versus collapse. One should note that in a NRT structure its own weight (the dead load) is an essential factor of stability, while collapse can be produced by not-admissible additions of the variable component of the load pattern. Duality tools may also be successfully applied in order to check the relationships between the two theorems of Limit Analysis (Baratta & O. Corbi 2004). In the study of plane mono-dimensional structures featuring a low degree of redundancy, the force/stress approach appears the most convenient to be adopted if compared with the displacement/strain approach, whose number of governing variables is higher and, moreover, increasing with the order of discretization (Baratta & I. Corbi 2003, 2004, 2006). The following section reports some results showing how the proper implementation of the described theoretical approach for classical masonry structural typologies such as panels, arches and vaults, produces results that are in a very good agreement with experimental data, demonstrating the overall reliability of the mentioned approach, for whose details one should refer to cited references. As shown in the following, results can also be successfully extended to the case of reinforcements with fiber-reinforced polymers (FRP). 2 TESTS ON PROTOTYPES OF MASONRY PANELS 2.1 Experimental investigation This section reports some of the results of the wide experimental campaign developed at the Laboratory of Materials and Structural Testing of the University of Naples “Federico II” on masonry panels, which are symmetrical, with a central hole covered by a steel architrave, and having upper part characterized by a 322 concrete fascia lightly reinforced by steel (Baratta & I. Corbi 2003, 2004, 2006). The geometry of the panels is shown in Figure 1.a. A laboratory prototype of masonry panel is referred to, made of tuff bricks (type “yellow tuff of Naples”, Italy) jointed to each other by a pozzolana mortar in order to confer a light additional resistance to the masonry; the masonry itself is characterized by unit weight γ = 10300 Nm and Young modulus E = 5.5 GPa. As regards to the loading condition, a varying force is applied in the middle of the left side of the panel, in such a way to mitigate the proneness of the panel to sliding of bricks with respect to each other, and some loading/unloading cycles are developed up to the collapse condition. Once reached the crisis, the panel is reinforced by directly laminating on the masonry some FRP strips according to the provision scheme shown in Figure 1.b, at the same time with the impregnation of the fibers by means of a special bi-component epoxy resin, and a further experimental investigation is developed on the reinforced structure by re-executing some loading/unloading cycles. The adopted reinforcement, produced by FTS, is a BETONTEX system GV330 U-HT, made of 12 K carbon fiber, jointed by an ultra light net of thermo-welded glass. The mechanical characteristics of the employed carbon fibers are: tensile limit stress σfrp = 4.89 GPa, elastic modulus in traction Efrp = 244 GPa, limit elongation σfrp = 2%. The FRP strip is characterized by thickness of 0.177 mm and depth of 200 mm. The induced displacements at some selected points [the transducers 1, 2, 3 and 4 in Figure 1.a] of the panel both for the not reinforced and for the lightly reinforced panel are recorded by a monitoring equipment consisting of: 4 transducers, placed at different locations of the panel in order to record the absolute displacements, and 15 strain-gauges, arranged in 3 blocks of 5 strain-gauges, in such a way that each block is devoted to record the related strain situation. In details two transducers are located horizontally at two different heights on the panel right side (transducers 1 and 2), and two are placed in correspondence of the opening, one in horizontal position at the top of the left side of the hole (transducer 3) and the other one under the architrave, which is devoted exclusively to control the panel deflection (transducer 4). The displacements s(mm) versus the varying force F(N) monitored by the transducers during the experiment in the not-reinforced and in the reinforced case with some horizontally applied C-FRP strips are shown in Figures 2.a–c and 2.d–f respectively, as regards to the first loading cycle. By the diagrams in Figure 2, which report the displacements s(mm) vs the varying force F(N) read by the transducers 1–3, some considerations can been made. With reference to the panel’s reinforcement by means of the application of some C-FRP strips, the major effect of the C-FRP intervention is the reduction of the stress in the masonry. In general lower displacements at the locations monitored by the transducers can be recorded in the consolidated case with comparison to the unconsolidated case. To this regard, the pretty light type of reinforcement allows to read the influence of even a small provision on the panel response, which, on the counterpart, cannot be expected to be macroscopic. One should emphasize that the first objective of this application is, then, to show the sensitivity of the NRT model even to small changes in the structural response, very differently from the elastic model, which, on the contrary, for the specific case, is unable to detect any difference in the behaviour of the wall. A number of more effective reinforcements have also been tested by the authors obviously resulting in more appreciable results and a much higher performance (Baratta & I. Corbi 2006). In the specific case, one can notice that, with reference to the same load intensity [e.g. in correspondence of the load value 3000 N in Figures 2.a–c], lower displacements can be recorded in case of FRP insertions. Moreover, the increase of the overall stiffness of the panel results in a higher loading capacity with respect to the not-reinforced wall. In particular the trend of each curve, shows that it is closer to the x-axis (representing the load variable), thus indicating an increase in the stiffness which is also related to an higher collapse value of the load. 2.2 Experimental/theoretical comparison Actually the application of the general theory of NRT structures to the considered case of the masonry panel, also in the presence of FRP reinforcements, can produce numerical results which are in good agreement with the results obtained by the above reported experimental campaign (Baratta & I. Corbi 2006). The specialization of the general problem to the case of masonry walls requires the definition of a discrete model coupled to the real structural model, the set up of the energetic problem (in the case of masonry panels the potential energy approach is to be preferred) for the discrete problem, which, for masonry material, results in a Non Linear programming problem to be solved by means of Operational Research tools, and, finally, the search of the numerical solution of the set up OR problem by means of a suitably implemented calculus code (Baratta & I. Corbi 2004, 2006). Once followed the above described steps, the numerical results can be compared to the ones coming out from the experimental investigation, for the final validation of the theoretical set up. 323 Transducer 1 Transducer 3 Transducer 2 (b) (a) (c) Experimental data Experimental data 1 Theoretical results 1 Experimental data 1 Theoretical results 0,8 0,6 0,6 0,6 u1 (mm) u3 (mm) 0,8 u2 (mm) 0,8 0,4 0,4 0,4 0,2 0,2 0,2 0 0 0 0 100 200 300 F (Kg) 400 0 500 Theoretical results 100 Transducer 1 200 300 400 F (Kg) 500 0 600 100 300 400 500 600 F (Kg) Transducer 3 Transducer 2 (d) 200 (e) (f) 1 1 Experimental data Experimental data Theoretical results Experimental data 1 Theoretical results Theoretical results 0,8 0,4 0,2 0 0 100 200 300 400 500 0,8 0,6 0,6 u3 (mm) u2 (mm) u1 (mm) 0,6 0,8 0,4 0,4 0,2 0,2 0 0 0 100 200 F (Kg) 300 F (Kg) 400 500 0 100 200 300 F (Kg) 400 500 Figure 2. Comparison between the numerical (continuous line) and the experimental (dotted line) at the monitored positions 1, 2, 3 for the not reinforced panel (a, b, c) and for the reinforced panel (d, e, f). For the specific case one may compare the results relevant to the first loading cycle with those related to experiments. As shown in Figures 2, the theoretical data (continuous lines) are in good agreement wit the experimental ones (dotted lines) both as regards to the not reinforced case (Figs 2.a–c) and to the consolidated case (Figs 2.d–f). In the first unconsolidated case the masonry exhibits a behaviour which appears lightly stiffer than in the theoretical model: this effect is maybe due to the micro-fractures present at the first stage of the computational procedure, which are probably absent in the real behaviour of the masonry. The transducers 2 and 3 show an overall pretty good agreement between numerical and experimental data even if also other phenomena as sliding between bricks, micro-fractures, etc., should be taken into account, which cause the not perfect agreement of the diagrams relevant to the first transducer 1 (Figs 2.a–c). It is indeed because of these reasons that the numerical/experimental agreement is higher, almost perfect, in the reinforced case (Figs 2.d–f). In this case, the sliding between bricks are reduced and do not influence the overall characteristics of deformability and stiffness of the masonry panel. 3 TESTS ON PROTOTYPES OF MASONRY ARCHES 3.1 Experimental investigation This section reports some of the results of the wide experimental campaign developed at the Laboratory of Materials and Structural Testing of the University of 324 d 13 d 12 d 11 d 14 d 15 d 20 I1 d 10 d 21 d 22 d9 d 23 d 24 d 19 d 16 G1 d8 d 18 d 17 d7 d6 d5 d 25 d4 d 26 d 27 d 28 d 29 E1 T1 T2 d3 d2 F d 30 d 1 Dial Gauge G1 Inclinometer I1 Transducers T1,T 2 Extesemeter E1 Deformometric cells dk, k=1…30 Figure 3. The portal arch model with the monitoring equipment: sketch of the monitoring equipment. Naples “Federico II” on masonry arches, consolidated or not by means of FRP strips (Baratta & O. Corbi 2003b, 2005b). The geometry of the portal arch (Figs 3 and 4) is symmetrical and is characterized by span L = 1900 mm, rise f = 660 mm, arch thickness d = 240 mm, piles thickness b = 385 mm, piles height h = 1700 mm; the arch shape is a semi-ellipse. The arch depth is 400 mm, whilst the two abutments are 480 mm deep. The masonry is characterized by unit weight γ = 12300 N · m−3 and Young modulus E = 5.5 GPa. As mentioned, in the above, in the second stage of the experimental campaign one also considers some FRP continuous reinforcement applied on the arch length. In this case, the FRP reinforcements consist of continuous mono-directional FRP strips applied on the extrados of the arcade. The adopted reinforcement, produced by FTS, is a BETONTEX system GV330 U-HT, made of 12 K carbon fibre, jointed by an ultra light net of thermowelded glass. The mechanical characteristics of the employed fibres are: tensile limit stress σfrp = 4.89 GPa, elastic modulus in traction Efrp = 244 GPa, limit elongation εfrp = 2%. The FRP strip is characterized by thickness of 0.177 mm and depth of 100 mm. After roughly preparing the masonry support in order to render the application surface smoother, the FRP is directly laminated on the masonry, at the same time with the impregnation of the fibres by means of a special bi-component epoxy resin. As regards the execution the tests, the structure is subject to its constant own weight and to a lumped horizontal force F, applied on the top right side of the right abutment in the rightward direction in the increasing phase (Figs 3 and 4), which is transmitted Figure 4. The portal arch model with the monitoring equipment: picture from laboratory tests. by means of a loading equipment consisting of a load cell placed on the right side of the portal arch. This force is able to potentially produce collapse of the structure according to a mechanism that is typical of earthquake failures of arch-portals (Fig. 6), and it is intended to represent a pseudo-seismic action, able to yield a measure of the structure attitude to sustain earthquake shaking. The monitoring stuff (Figs 3 and 4) consists of: – 1 dial gauge G1 , placed on the left side of the left abutment, finalized to the monitoring of the absolute displacement of the pile; – 2 transducers T1 and T2 , vertically placed on the front side of the left abutment, finalized to the monitoring of the length variation of both edges of the pile; – 1 inclinometer I1 , placed on the top of the left abutment, finalized to the monitoring of the pile average rotation; – 1 extensometer E1 , placed between the two abutments, finalized to the monitoring of the relative piles’ displacement; – 30 deformometer cells, placed on the front of the arch, finalized to the monitoring of the arcade deformation. For the un-reinforced structure (Baratta and Corbi, 2003a,b) the critical condition is related to the activation of a collapse mechanism composed by four hinges distributed as follows: 325 – 1 at the top of the left pile on the intrados, – 1 at the keystone on the extrados, – 1 under the load cell on the intrados of the right pile (where shear occurs), – 1 at the bottom of the right pile on the extrados. ∆ u (mm) 10 Experimental Data 8 Trend Line The collapse is reached at F∼800 N with an increase in the loading capacity of the portal arch of approximately 10 times with respect to the unconsolidated case. The experimental force-displacement diagram is reported in Figure 7. 6 Numerical Data 4 3.2 2 F (N) 0 0 20 40 60 80 100 Figure 5. Unreinforced portal arch: pile displacement u versus load F-numerical/experimental comparison. – 1 at the keystone on the extrados, – 2 at the reins on the intrados, – 1 at the bottom of the right pile on the extrados. The collapse condition is reached at F∼80 N; the low failure value of the force shows that, due to the chosen elliptical shape of the arch, the funicular line compatible with the applied loads and admissible (i.e. interior to the arch profile) is already very close to the upper and lower bounds of the arch profile at the rest condition. The experimental force-displacement diagram is reported in Figure 5. After reaching the collapse condition, the portal arch is then unloaded in order to be prepared for the subsequent experimental tests on FRP reinforcements. After completing the unloading process, the portal arch is prepared for laboratory tests on FRP reinforcements, which are finalized to the evaluation of the benefits induced on the model response by the application of carbon fibre strips. The reinforcement consists of a continuous FRP strip bonded on the extrados of the arch. Since the collapse mechanism of the not reinforced simple portal arch is characterized, as described in the above, by the formation of two intrados hinges at the reins of the arch, corresponding to the fractures d4 –d5 and d12 –d13 at the extrados, the major effect of this intervention is supposed to be the prevention of these fractures, and, therefore, a wide increase in the model loading capacity. The funicular line is now free to exceed the lower contour of the portal arch cross section. In this case the critical condition is related to the activation of a collapse mechanism composed by four hinges, distributed as follows: Experimental/theoretical comparison Actually the application of the general theory of NRT structures to the considered case of the masonry portal arch, also in the presence of FRP reinforcements, can produce numerical results which are in good agreement with the results obtained by the above reported experimental campaign (Baratta & O. Corbi 2005a, 2005b, 2007). The specialization of the general problem to the case of masonry arches requires the definition of a discrete model coupled to the real structural model, the set up of the energetic problem (in the case of masonry arches the complementary energy approach is to be preferred) for the discrete problem, which, for masonry material, results in a Non Linear programming problem (which in the specific case can be reduced to a Linear Programming problem) to be solved by means of Operational Research tools, and, finally, the search of the numerical solution of the set up OR problem by means of a suitably implemented calculus code (Baratta & O. Corbi 2003a, 2003b, 2005a). Once followed the above described steps, the numerical results can be compared to the ones coming out from the experimental investigation, for the final validation of the theoretical set up. Numerical investigation on the portal arch model experimentally tested results in the possibility of appreciating the skill of the NRT model to capture the major features of the structure behaviour. Moreover also the correct modelling of the reinforcement and of its coupling with the main structure can be evaluated. Figure 5 reports the numerical/experimental comparison relevant to the right pile top displacement u (mm) versus the varying load F (N) for the considered un-reinforced arch. A very good agreement between the numerical and experimental data can be observed. The calculus code is demonstrated to be able to capture the behaviour of the portal arch following the whole loading path up to collapse; Figure 6 depicts the collapse mechanism of the structure as it appears directly from the calculus code, clearly due to the formation of four hinges: one at the keystone on the extrados, two at the reins on the intrados, one at the bottom of the right pile on the extrados. 326 Figure 6. Unreinforced portal arch: picture of the collapse mechanism captured from the calculus code. Figure 8. Portal arch with extrados reinforcement: picture of the collapse mechanism captured from the calculus code. u (mm) 10 Experimental Data 8 Trend Line 6 Numerical Data 4 Figure 9. Barrel vault with horizontal directrix. 2 4 PROTOTYPES OF MASONRY VAULTS F (N) 0 0 200 400 600 800 4.1 The problem of barrel vaults with indefinite length 1000 Figure 7. Portal arch with extrados reinforcement: pile displacement u vs. load F-numerical/experimental comparison. Moreover one reports in Figure 7 the numerical/experimental comparison relevant to the right pile top displacement u (mm) versus the varying load F (N) for the arch reinforced with an extrados FRP reinforcement. Again a very good agreement can be observed between the numerical and experimental data. The calculus code is demonstrated to be able to capture the behaviour of the portal arch ; Figure 8 depicts the collapse mechanism of the structure as it appears directly from the calculus code, clearly due to the formation of four hinges: one at the top of the left pile on the intrados, one at the keystone on the extrados, one under the load cell on the intrados of the right pile, one at the bottom of the right pile on the extrados. Both numerical and experimental data agree in assessing at approximately ten times the original value the increment of the loading capacity of the structure due to the extrados FRP reinforcement. As regards to barrel vaults (Baratta & O. Corbi 2007), first of all, one should consider that, since the vault geometrically derives by the translation along a directrix of a generating arch curve, in this case, the meridian lines coincide with the generatrix in their shapes; if one considers a rectilinear directrix, the vault parallels are horizontal and rectilinear as well (Fig. 9). The surface of the shell of the mid-surface of the vault may be defined by the equation z = f(x). Because of the vault geometry, one has where dsx and dsy denote the length of the sides of the generic vault element ABCD of area dA dx and 327 dy the length of the corresponding sides on the element A′ B′ C′ D′ projected in the xy-plane, and ϕ and θ denote the angles formed by the meridian sides AB and DC of the element with the x-axis and by the parallel sides AD and BC with the y-axis, respectively. As concerns equilibrium, hypothesizing that the vault is in a membrane state of stress, a correspondence can be set between forces acting on the element ABCD (stresses Nx , Ny , Nxy = Nyx and applied load for unit area, px , py , pz ) and projected forces acting on the associated element A′ B′ C′ D′ (Nx , Ny , Nxy = Nyx and px , py , pz ) in the xy-plane (Baratta & O. Corbi 2007). In absence of horizontal loads and if the vertical load is not dependent on “y”, as it happens when the vault is subject to only vertical loads due to the selfweight (i.e. pz = pz (x) ≥ 0), and assuming that the vault has an indefinite length in the direction y, equilibrium may be expressed in the form where z0 and z1 are arbitrary ordinates, conditioned by the fact that z(t) should be contained in the interior of the profile of the vault. After this result, it is possible to calculate the internal forces Nx ≤ 0, Ny = Nxy = 0 and Nx ≤ 0, Ny = Nxy = 0 It is also possible to realize that the equilibrium solution allows the structure to behave as a sequence of identical independent arches. From this result, one may refer to the results reported in the previous section for the portal arch model, reinforced or not with some FRP strips, whose analytical problem implementation has been shown to give theoretical results in perfect agreement with the produced experimental data, also exhibiting very effective results in the reinforced case. 5 which reduces the problem to the determination of stress function ψ(y). Assuming that the directrix curve of the vault is a circular arch (Fig. 10) of radius R, with constant thickness “s” and unit weight γ, and imposing suitable constraint conditions, one yields the final solution (Baratta & O. Corbi 2007) with Figure 10. Cross section of a barrel vault with circular arch generatrix. CONCLUSIONS The paper reports some results proving the successful application of a correct theoretical treatment, based on the NRT material assumption, of structural problems relevant to classical masonry typologies such as arches, walls and vaults. The set up of the general energetic approach for analyzing masonry structures under live loads, its specialization to the relevant discrete models, the implementation of ad hoc built up calculus codes are demonstrated to produce numerical results in very good agreement with data produced by experimental investigation. One should emphasize that, differently from many models which require a number of parameters allowing a certain adaptation of the shape of the numerical curve to the experimental one, the NRT model has the big advantage that the only mechanical parameter to be evaluated is the masonry elastic modulus. Since the tuning of the theoretical model is pretty simple, there would be no possibility to force it to produce theoretical results fitting with such a good agreement the experimental data, because the tuning operation itself cannot influence the shape of the numerical diagram but only the displacements scale. As a point of fact, the sensitivity of the modelling to material assumptions reduces to the inverse proportionality between the material elastic modulus and displacements, without any influence on the load capacity and on the evolution of displacements with the loads. 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