In-plane behavior of perforated brick masonry walls Alessia Vanin & Paolo Foraboschi Materials and Structures ISSN 1359-5997 Volume 45 Number 7 Mater Struct (2012) 45:1019-1034 DOI 10.1617/s11527-011-9814-x 1 23 Your article is protected by copyright and all rights are held exclusively by RILEM. This eoffprint is for personal use only and shall not be self-archived in electronic repositories. If you wish to self-archive your work, please use the accepted author’s version for posting to your own website or your institution’s repository. You may further deposit the accepted author’s version on a funder’s repository at a funder’s request, provided it is not made publicly available until 12 months after publication. 1 23 Author's personal copy Materials and Structures (2012) 45:1019–1034 DOI 10.1617/s11527-011-9814-x ORIGINAL ARTICLE In-plane behavior of perforated brick masonry walls Alessia Vanin • Paolo Foraboschi Received: 1 September 2010 / Accepted: 4 December 2011 / Published online: 3 January 2012 Ó RILEM 2011 Abstract This paper reports the results of experimental and theoretical research conducted on perforated brick masonry walls under in-plane loading. The walls’ structural behavior depends strongly on their specific features, e.g. geometry, mechanical properties of the masonry material, brick arrangement and loading conditions. The experimental program was designed to study the incidence of brick arrangement in the spandrels and piers, and of the acting vertical load on the failure mode and load-bearing capacity of the walls. Six specimens of brick masonry wall with a central opening were submitted to a constant vertical load and a monotonic horizontal force that was gradually increased until the kinematic mechanism condition was reached. The object of the theoretical research was to develop a simplified analytical model for describing the kinematic mechanism of the walls. The results of the experiments indicate that brick arrangement strongly influences the failure mode and load-bearing capacity of the walls. Proper a priori assessment of the failure mode of walls becomes fundamental to an accurate evaluation of their loadbearing capacity using the proposed model. A. Vanin (&)  P. Foraboschi Università IUAV di Venezia, Ex convento delle Terese, Dorsoduro 2206, 30123 Venezia, Italy e-mail: avanin@iuav.it P. Foraboschi e-mail: paofor@iuav.it Keywords Masonry walls  In-plane load  Brick arrangement  Failure mode  Kinematic mechanism 1 Introduction The assessment of the seismic safety of existing masonry buildings is an important issue nowadays. The historical centers of European cities are characterized by a very large number of masonry buildings, many of them an important cultural heritage, so it is our duty to conserve them for future generations. They must be preserved from the decay due to the natural aging of the materials and structures, but also from extraordinary destructive events, such as earthquakes. They may be preventively assessed in terms of their safety vis-à-vis any seismic action expected in the area where they stand. Such assessments are necessarily based on the reliable modeling of the buildings to avoid being too optimistic or pessimistic about their seismic safety. Overestimating their seismic safety may expose buildings to severe damage as a consequence of a seismic event, while underestimating may prompt recourse to unnecessary, highly invasive preventive solutions. As reported in the literature [1–4, 12], assessments of the seismic safety of masonry buildings must consider: – seismic actions orthogonal to the walls of the building (actions out of plane); Author's personal copy 1020 – Materials and Structures (2012) 45:1019–1034 seismic actions parallel to the walls of the building (actions in plane). 20 cm 84.5 cm It is common knowledge that actions out of plane are highly dangerous. A wall’s out-of-plane collapse can generally only be avoided by connecting it effectively to the orthogonal walls and floors [2, 6, 7]. Actions in plane, on the other hand, depend mainly on a wall’s strength and strain properties, which are known to derive from the wall’s geometry, materials, building technique and loading conditions [1, 3, 4, 17]. The in-plane behavior of masonry panels has been widely investigated from both the experimental and the theoretical points of view [8, 9, 11, 15, 18], but fewer experimental studies have been conducted on masonry walls with openings. Some results of experiments on perforated masonry walls are reported in Raijmakers and Vermeltfoort [14], and in Vermeltfoort and Raijmakers [19]. Probable failure modes have been defined, based on theoretical research and observation of masonry buildings damaged by seismic events [1–4, 7, 12, 17]. The present study is a contribution on the definition of the failure modes and the load-bearing capacity of masonry walls with an opening under inplane loading in relation to the brick arrangement in their spandrels and piers, and to specific loading conditions. This topic has been studied from both the experimental and the theoretical points of view. 2 Experiments The experiments were designed to evaluate the behavior of masonry walls with an opening submitted to a vertical load and a horizontal force acting in plane, in relation to: – – the arrangement of the bricks, or masonry texture, in the spandrels and piers; and the value of the acting vertical load. 2.1 Specimens Six masonry walls with a central opening were built. The overall dimensions of the walls are given in Fig. 1. 234 cm 279 cm 117 cm 32.5 cm 25 cm 93 cm 88 cm 93 cm 274 cm Fig. 1 Geometrical shape of the specimens Specimens 1, 2 and 3 Fig. 2 Masonry texture of specimens 1–3 The bricks were 24 by 11.5 by 5.5 cm in size. They were obtained from demolition of existing masonry buildings. The mortar was prepared mixing lime (3/12), Portland cement (1/12), sand (1/3) and water (1/3). The mortar joints were made flush the brick’s face and were 1 cm thick. The first three walls (specimens 1–3) were built using the same masonry texture for the spandrels and piers (Fig. 2). This arrangement (texture A) alternates one brick laid parallel to the wall’s plane (a stretcher) and one laid orthogonal to said plane (a header) in the same course (Fig. 4a). The headers are staggered by 18 cm from the first course to the second. The other three walls (specimens 4–6) were built using different arrangements of bricks for the spandrels and piers (Fig. 3). In particular, specimen 4 (Fig. 3a) was built using texture A for the spandrels, and for the piers a second arrangement (texture B), that Author's personal copy Materials and Structures (2012) 45:1019–1034 (a) Specimen4 1021 (b) Specimen5 (c) Specimen6 Fig. 3 Masonry texture of specimens 4–6 Header Stretcher (a) Texture A (b) Texture B Fig. 4 Masonry textures alternates three stretchers with one header in the same course (Fig. 4b). The headers are staggered by 18 cm from the first course to the second. Specimen 5 (Fig. 3b) was built using texture B for the spandrels and texture A for the piers. Specimen 6 (Fig. 3c) was built using a random arrangement of the bricks for both the spandrels and the piers. Two reinforced concrete beams were built at the top and bottom of the specimens (Figs. 2, 3). The beam at the bottom enables the specimens to be clamped to the floor of the laboratory, while the beam at the top allowed for the vertical load and the horizontal force to be applied to the specimens (see Sect. 2.2). Experimental tests were performed to determine the compressive strength, modulus of elasticity and indirect tensile strength of bricks and mortar. Experimental tests were also performed to determine the compressive strength and modulus of elasticity of masonry, while masonry tensile strength was defined on the base of tensile strength of bricks and mortar by means of the analytical formula proposed by Tassios [16]. Samples of mortar and masonry were tested 28 days after their construction. During the intervening period, they were submitted to the same environmental conditions as the walls. The mean values of the mechanical properties measured or analytically deduced are reported in Table 1. 2.2 Test setup The test setup is shown in Fig. 5. Two hydraulic jacks were placed along the vertical axis of each of the two piers of the wall and connected to each other and to the concrete beam at the bottom of the wall by means of iron chains. A steel plate was placed on the concrete beam at the top of the wall to prevent local crumbling due to the localized action of the iron chains. A load cell was connected to one of four hydraulic jacks used to apply the vertical load to the wall. Another steel plate was placed on the concrete beam at the top of the wall and two hydraulic jacks, connected to another load cell, were placed parallel to each other on this plate. These two jacks were used to apply the horizontal force to the wall; the particular shape of the steel plate enabled the transmission of the horizontal force exerted by the hydraulic jacks on the concrete beam to the top of the wall. Two transducers, installed on telescoping rods, were placed at two different heights between the laboratory wall and the specimens to record the horizontal displacements of the wall. Both the load cells were connected to a central hydraulic apparatus connected to an electronic system and a computer. Using this setup the vertical load was kept constant and the horizontal force was gradually increased according to the established loading steps. Author's personal copy 1022 Materials and Structures (2012) 45:1019–1034 Table 1 Material properties obtained experimentally or deduced Masonry Brick Mortar Compressive strength, fM (N/mm2) 1.21 Compressive strength, fb (N/mm2) 2.36 Compressive strength, fm (N/mm2) 0.77 Modulus of elasticity, EM (N/mm2) 1.360 Modulus of elasticity, Eb (N/mm2) 3.150 Modulus of elasticity, Em (N/mm2) 1.013 Tensile strength, fMt (N/mm2) 0.08 Tensile strength, fbt (N/mm2) 0.14 Tensile strength, fmt (N/mm2) 0.05 HYDRAULIC JACK STEEL PLATE STEEL PLATE LOAD CELL TRANSDUCER 1 LOAD CELL TRANSDUCER 2 CONTRAST WALL HYDRAULIC JACK HYDRAULIC JACK CONTRAST FLOOR Fig. 5 Test set-up The transducers were connected to the same electronic system to measure and record the horizontal displacements. 2.3 Test description The experimental test method is shown in Fig. 6. As mentioned previously, the specimens were submitted both to a vertical load and to a horizontal force. The vertical load was applied by means of the four hydraulic jacks placed along the vertical axes of the two piers. The values of the vertical load applied to each pier and to all six specimens are given in Table 2. The vertical load was kept constant throughout the tests. The monotonic horizontal force was applied by means of the two hydraulic jacks on the top of the wall and was gradually increased, starting from 0, until the wall reached the kinematic mechanism condition. At each step, the transducers recorded the horizontal displacements at the top of the wall and at the top of the two piers. 2.4 Experimental results During the tests, the development of the specimens’ cracking frame was observed (see Figs. 7, 8, 9, 10, 11, 12). Horizontal force–displacement curves were obtained for the specimens, based on the horizontal Author's personal copy Materials and Structures (2012) 45:1019–1034 1023 (see Sect. 2.1). The comparison of the experimental findings for these three specimens shows how the masonry texture affects the global behavior of the specimens. 2.4.1 Failure modes Fig. 6 Execution of the experiments Table 2 Values of the vertical load applied at the top of the specimens Specimens Vertical load on each pier (kN) Vertical load on the specimen (kN) Specimen 1 70 140 Specimen 2 40 80 Specimen 3 20 40 Specimen 4 20 40 Specimen 5 20 40 Specimen 6 20 40 forces and displacements recorded during the tests. These curves are shown in Figs. 13, 14, 15, 16, 17, 18. Specimens 1–3, built using the same geometry, materials and masonry texture (see Sect. 2.1), were submitted to different vertical loads, as shown in Table 1. Comparing the experimental results obtained for these three specimens illustrates the incidence of the vertical load on the global behavior of the specimens. Specimens 4–6 were submitted to the same vertical load, as shown in Table 1, and were built using the same geometry and materials, but different arrangements of the bricks in the spandrels and piers Specimens 1–3 exhibited the same development of the cracking frame (Figs. 7, 8, 9), meaning that the different vertical loads applied at the top of these specimens (see Table 1) did not give rise to different failure modes. The first crack appeared on the top spandrel, above the opening (Figs. 7a, 8a, 9a), and developed at an angle of 45 degrees with respect to the horizontal because of the arrangement of the bricks. As a consequence of the development of this crack, the two piers remained connected only by the concrete beam on the top of the wall. From the structural point of view, they therefore behaved like two poorly connected cantilevers. The second and third cracks appeared at the base of the right (Figs. 7b, 8b, 9b) and left piers (Figs. 7c, 8c, 9c), and both were indicative of the bending failure of the piers. The third crack also indicated that the wall had reached the kinematic mechanism condition. The uncoupled movements of the two piers involved in the kinematic mechanism prompted the loss of equilibrium of the flat arch, so other cracks developed in the flat arch, producing the detachment of several bricks, which slid towards the opening (Figs. 7d, 8d, 9d). Specimens 4–6 exhibited different developments of the cracking frame (Figs. 10, 11, 12): these different failure modes are very probably due to the different arrangements of the bricks in their spandrels and piers (see Sect. 2.1). In specimen 4, the first crack appeared at the upper left corner of the opening, where the top spandrel joins the left pier (Fig. 10a). This area marks the separation between parts of the wall (the top spandrel and piers) characterized by different masonry textures. The particular arrangement of the bricks probably made the top spandrel highly compact, so that it took the horizontal force from the concrete beam and transferred it to the bottom piers, thereby remaining intact. Author's personal copy 1024 (a) Cracking of the top spandrel Materials and Structures (2012) 45:1019–1034 (b) Cracking of the right pier (c) Cracking of the left pier (d) Cracking of the flat arch (c) Cracking of the left pier (d) Cracking of the flat arch (c) Cracking of the left pier (d) Cracking of the flat arch Fig. 7 Progression of cracking for specimen 1 (a) Cracking of the top spandrel (b) Cracking of the right pier Fig. 8 Progression of cracking for specimen 2 (a) Cracking of the top spandrel (b) Cracking of the right pier Fig. 9 Progression of cracking for specimen 3 (a) First cracking of the top spandrel (b) Cracking of the right pier (c) Second cracking of the top spandrel (d) Cracking of the flat arch Fig. 10 Progression of cracking for specimen 4 Because of the arrangement of the bricks, cracking developed in the top spandrel with a near-horizontal inclination, determining the rapid separation of the left pier from the rest of the wall. The second crack appeared at the base of the right pier (Fig. 10b), and indicated its bending failure. The third appeared in the right part of the top spandrel (Fig. 10c) and meant that the kinematic mechanism condition had been reached in the wall. Author's personal copy Materials and Structures (2012) 45:1019–1034 (a) Cracking of the top spandrel 1025 (b) Cracking of the right pier (c) Cracking of the left pier (d) Cracking of the flat arch Fig. 11 Progression of cracking for specimen 5 (a) First cracking of the top spandrel (b) Cracking of the right pier (c) Cracking of the left pier (d) Further cracking of the top spandrel and cracking of the flat arch Fig. 12 Progression of cracking for specimen 6 60 transducer 2 (b) (c) (a) (b) 50 40 (d) (a) (a) (d) (b) 30 20 (c) 10 10 20 30 40 50 60 horizontal force (kN) transducer 1 (a) transducer 2 (c) 50 40 (b) (b) (c) (d) (d) (b) 30 (a) 20 (a) (c) 10 0 0 40 10 20 30 40 (b) 30 20 (b) (a) 10 (a) 0 (c) (d)(c) (d) (b) (c) 10 20 50 60 horizontal displacement (mm) 70 (d) Fig. 14 Horizontal force–displacement curve for specimen 2 The movement of the right pier involved in the kinematic mechanism with respect to the left pier (which remained connected to the bottom of the wall) 30 40 50 60 70 horizontal displacement (mm) (d) Fig. 15 Horizontal force–displacement curve for specimen 3 SPECIMEN 4 70 SPECIMEN 2 (a) transducer 2 50 (d) Fig. 13 Horizontal force–displacement curve for specimen 1 60 transducer 1 60 70 horizontal displacement (mm) 70 SPECIMEN 3 70 0 0 0 horizontal force (kN) horizontal force (kN) (c) transducer 1 horizontal force (kN) SPECIMEN 1 70 transducer 1 60 (a) transducer 2 50 40 (b) 30 20 (c) (b) (a) (d) (a) (c) 10 0 0 10 20 30 40 50 horizontal displacement (mm) 60 70 (d) Fig. 16 Horizontal force–displacement curve for specimen 4 prompted the loss of equilibrium of the flat arch, making cracks develop in the flat arch, and causing the detachment of several bricks, which slid toward the opening (Fig. 10d). Author's personal copy 1026 Materials and Structures (2012) 45:1019–1034 SPECIMEN 5 horizontal force (kN) 70 transducer 1 60 (a) transducer 2 50 40 30 20 (a) (b) (c) (d) (d) (c) (a) 10 0 (c) (b)(b) 0 10 20 30 40 50 60 70 (d) horizontal displacement (mm) horizontal force (kN) Fig. 17 Horizontal force–displacement curve for specimen 5 SPECIMEN 6 70 transducer 1 60 (a) transducer 2 50 40 (c) (c) (b) 30 20 (b) (d) (d) (b) (a) (a) (c) 10 0 0 10 20 30 40 50 horizontal displacement (mm) 60 70 (d) Fig. 18 Horizontal force–displacement curve for specimen 6 In specimen 5, the first crack appeared on the top spandrel above the opening (Fig. 11a). The arrangement of the bricks probably gave rise to an inner weakness in the top spandrel, so that it took the horizontal force from the concrete beam and transferred it to the bottom piers, without remaining intact. Because of the masonry texture, cracking developed in the top spandrel at a near-horizontal angle and with a very marked ramification. As in specimens 1–3, the development of this crack meant that the two piers were only connected by the concrete beam placed on the top of the wall: from the structural point of view, they behaved like two poorly connected cantilevers. The second and third cracks appeared at the base of the right (Fig. 11b) and left piers (Fig. 11c); they both indicated the bending failure of the piers. The third crack was also evidence of the kinematic mechanism condition of the wall being reached. As in the previous specimens, the uncoupled movements of the two piers involved in the kinematic mechanism prompted the loss of equilibrium of the flat arch, making other cracks develop in the flat arch and causing the detachment of several bricks, which slid towards the opening (Fig. 11d). In specimen 6, the first crack appeared at the upper left-hand corner of the opening, where the top spandrel joins the left pier (Fig. 12a). The random arrangement of the bricks probably lent the top spandrel a better structural quality than that of the piers. The cracking consequently appeared in the same way as in specimen 4. Because of the masonry texture, the progression of this cracking was more complex, and the connection of the left pier to the rest of the wall was maintained as a result, unlike the situation in specimen 4. As in specimens 1, 2, 3 and 5, the cracking of the top spandrel in specimen 6 meant that the two piers remained connected only by means of the concrete beam on the top of the wall, i.e. they behaved like two weakly connected cantilevers. The second and third cracks appeared at the base of the right (Fig. 12b) and left piers (Fig. 12c), and both were indicative of the bending failure of the piers. The top spandrel also reached the failure condition, as shown by a crack on its right-hand side (Fig. 12d). This further crack in the top spandrel also meant that the wall had reached the kinematic mechanism condition. As in the previous specimens, the uncoupled movements of the two piers involved in the kinematic mechanism caused the loss of equilibrium of the flat arch, resulting in other cracks developing in the flat arch, and causing the detachment of several bricks, which slid towards the opening (Fig. 12d). 2.4.2 Horizontal force–displacement curves The horizontal force–displacement curves for specimens 1–3 (Figs. 13, 14, 15) show a first rising branch, that starts from the origin and continues until local crumbling becomes apparent. A second rising branch continues until the first cracks develop in the top spandrel above the opening (Figs. 13a, 14a, 15a); this cracking implies a first considerable reduction in the stiffness of the specimens. In specimens 2 and 3, the second rising branch is followed by a third, which continues until the right pier cracks (Figs. 14b, 15b), whereas in specimen 1 the second rising branch is followed first by a horizontal branch, then by a third rising branch, which continues until the right pier crack (Fig. 13b), Author's personal copy Materials and Structures (2012) 45:1019–1034 meaning a further reduction of the stiffness of the specimens. In both specimen 1 and specimens 2 and 3, a fourth rising branch continues until the peak load-bearing capacity is reached, marked by cracking in the left pier and the kinematic mechanism condition being reached as a result (Figs. 13c, 14c, 15c). A falling branch starts after from the load-bearing capacity peak has been reached and continues until the flat arch cracks (Figs. 13d, 14d, 15d). As in specimens 1–3, the horizontal force–displacement curves for specimens 4–6 (Figs. 16, 17, 18) show a first rising branch, which starts from the origin and continues until local crumbling occurs. A second rising branch continues until the first cracks develop, in the top spandrel above the opening in specimen 5 (Fig. 17a), and where the top spandrel joins the left pier in specimens 4 and 6 (Figs. 16a, 18a). The first cracking implies a very marked reduction in the stiffness of the specimens, so that the subsequent branches of the curves seem nearly horizontal. Both in specimen 5 and in specimens 4 and 6, the second rising branch is followed by a third, which continues until the right pier cracks (Figs. 16b, 17b, 18b). This third branch is followed by a fourth, that in specimen 4 continues until the right part of the top spandrel cracks (Fig. 16c), while in specimens 5 and 6 it continues until the left pier cracks (Figs. 17c, 18c). Both in specimen 4 and in specimens 5 and 6, the end of the fourth branch marks the load-bearing capacity peak and the reaching of the kinematic mechanism condition. Beyond the load-bearing capacity peak, a falling branch also develops and continues until the flat arch cracks (Figs. 16d, 17d, 18d). 2.5 Discussion of the experimental results An analysis of the experimental results obtained for all the specimens prompts several considerations. Concerning the failure mode of perforated masonry walls, our results show that: – if spandrels and piers are built using the same arrangement of bricks (e.g. specimens 1–3), the failure mode depends on the geometrical shape of the wall (the dimensions of the spandrels and piers, 1027 – the size of the opening) and on its loading conditions (the acting vertical load); if spandrels and piers are built using different arrangements of bricks (e.g. specimens 4–6), their masonry texture significantly influences their failure mode. In particular, the brick arrangement of the upper spandrel influences its mechanical behavior and the consequent level of coupling that it establishes between the two piers, so that: – – if the brick arrangement of the top spandrel is characterized by a prevalence of headers (see Sect. 2.1), the spandrel is very compact and consequently enables a very high coupling between the two piers; if the brick arrangement of the top spandrel is characterized by a prevalence of stretchers (see Sect. 2.1), the spandrel is less compact and consequently does not ensure a good coupling between the two piers. The type of coupling between the two piers influences the shape of the wall’s kinematic mechanism. In particular: – – a good coupling between the two piers encourages the activation of the kinematic mechanism shapes shown in Fig. 20a or b; a poor coupling between the two piers encourages the activation of the kinematic mechanism shapes shown in Fig. 20c or d. As explained in Sect. 3.2, the horizontal force that gives rise to the kinematic mechanism condition for the shapes represented in Fig. 20a or b is greater than the horizontal force needed to lead to the kinematic mechanism condition for the shapes represented in Fig. 20c or d. As for the mechanical behavior of the specimens deducible from the horizontal force–displacement curves, it is worth noting that: – – in all the specimens, the first crack occurs for a significantly lower horizontal force than the value corresponding to the load-bearing capacity peak; all the specimens exhibit a high deformation capacity, especially before reaching the loadbearing capacity peak; Author's personal copy 1028 – – – the length of the softening branch depends on the failure mode, i.e. on the progression of the cracking; the vertical load applied at the top of the specimens (see specimens 1–3) influences both their loadbearing capacity and their deformation capacity. The very similar progression of the cracking in specimens 1–3 coincides with different values of horizontal force and displacement: at each cracking step, the highest values are naturally recorded for specimen 1, which is submitted to a higher vertical load; the failure mode influences the load-bearing and deformation capacity of the specimens too (see specimens 4–6). In particular, the highest horizontal force and displacement values corresponding to the load-bearing capacity peak are recorded for specimen 6, which exhibited a very similar shape of kinematic mechanism to the one shown in Fig. 20b, and also a very wide spreading of the cracks. Specimen 6 also exhibited a significant softening branch. 3 Analytical modeling The theoretical study on the specimens was developed according to the kinematic approach of limit equilibrium analysis [6, 7, 10]. It is common knowledge that, for masonry material, limit analysis assumes an infinite compressive strength, an infinite sliding strength and a tensile strength of 0. The failure, or cracking, of masonry structures is attributed to tensile stresses, while their collapse is due to the kinematic mechanism condition being reached as a consequence of cracking [6, 7, 10]. The parts of a structure involved in the kinematic mechanism are modeled as rigid bodies. The equilibrium condition of these parts enables an assessment of the external forces prompting the activation of the kinematic mechanism. Nowadays, this approach is widely used in safety assessments on existing masonry structures, because it disregards the mechanical properties of the masonry material, which is often very uncertain and difficult to determine for such structures. Since the mechanical properties of the masonry material are not specifically considered, however, Materials and Structures (2012) 45:1019–1034 masonry structures of good and poor quality but with the same geometry and loading conditions are assumed to have the same load-bearing capacity, though this obviously does not reflect experimental evidence. Reference to the kinematic mechanism condition also requires a preliminary knowledge of the shape of a structure’s kinematic mechanism, which is very difficult to determine in many cases because it depends both on their geometrical features and on the building technique involved. All these limitations can be avoided by means of a numerical modeling approach. Numerical modeling is based on various strategies for modeling masonry material, depending on the accuracy and simplicity required of the model (detailed micro-modeling, simplified micro-modeling or macro-modeling). All these strategies enable the mechanical properties and the constitutive factors of masonry material to be specifically taken into account and do not require any prior definition of the structure’s failure mode. Accurate micro- or macro-modeling demands a thorough experimental characterization of the masonry material, however, because the mechanical properties of masonry are strongly influenced by a number of factors, such as the mechanical properties of the bricks or blocks and mortar, the arrangement of bed and head joints, joint width, quality of workmanship, environmental damage and age [11]. Existing masonry structures often have a very heterogeneous composition, making a thorough experimental characterization of the masonry material very expensive and difficult. Moreover, accurate micro- or macro-modeling entails a major computational effort, that becomes unsustainable in the case of large buildings. Lourenço [11] conducted important numerical experiments on perforated brick masonry walls, with comparisons and a discussion of the potential of the above-mentioned modeling strategies for masonry material. In the case of the perforated brick masonry walls considered here, the shape of the kinematic mechanism emerged from the experiments. Moreover, the quality of the masonry allowed for a rigid body schematization of the masonry panels. The modeling was consequently done according to the kinematic approach of limit equilibrium analysis. The numerical modeling of the experimental findings will be completed by the authors in the near future. Author's personal copy Materials and Structures (2012) 45:1019–1034 1029 3.1 Definition of the geometrical model The sides of ABCD are called l1, l2, l3 and l4, while the angles defining the inclination of l1, l2, l3 and l4 with respect to the horizontal are called a1, a2, a3 and a4 (Fig. 19a). The lengths of l1, l2, l3 and l4 and the amplitudes of a1, a2, a3 and a4 are independent of #1 ; so they can be predetermined starting from the coordinates of A–D. These coordinates obviously depend on the geometry of the wall and on the assumed shape of the kinematic mechanism. AB0 C0 D is the convex quadrilateral with vertexes A, B0 , C0 and D corresponding to the hinge points in the deformed condition of the wall (Fig. 19b, c), for which we can write: A geometrical model was developed to analyze the kinematic mechanism condition of masonry walls with openings. This model was drawn from the geometrical model developed by Focacci [5] to analyze the kinematic mechanism condition of the masonry arch. The wall was assumed to be loaded by horizontal forces and vertical forces acting in plane (Fig. 19a), and the generic cracked condition of the wall indicative of the kinematic mechanism condition was considered (Fig. 19a). The deformed shape of the wall in the kinematic mechanism phase is shown in Fig. 19b: the portions of masonry separated by cracks (called blocks) rotate in relation to each other around hinge points. The rotations of the blocks are correlated by geometrical relations that impose compliance with the necessary congruence in the displacements of the blocks, which can be expressed in analytical form. For this purpose, a Cartesian coordinate system was established, with its origin in the bottom left-hand corner of the wall (Fig. 19a, b). The angle #1 ; which quantifies the rotation of the block 1 (Fig. 19b), was assumed as the parameter that identifies the deformed shape of the wall. The angles #2 and #3 ; which quantify the rotations of the blocks 2 and 3 (Fig. 19b), may be determined as a function of #1 : ABCD is the convex quadrilateral with vertexes A, B, C and D corresponding to the hinge points in the undeformed condition of the wall (Fig. 19a). (a) (b) Y a1  #1 þ að#1 Þ þ bð#1 Þ þ a2  #2 ð#1 Þ ¼ p ð1Þ and then: #2 ð#1 Þ ¼ p þ a1  #1 þ að#1 Þ þ bð#1 Þ þ a2 ð2Þ where að#1 Þ and bð#1 Þ are obtained as follows. Referring to triangle AB0 D (Fig. 19c) we can write: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi l5 ð#1 Þ ¼ l21 þ l24  2  l1  l4  cosða4 þ a1  #1 Þ ð3Þ Referring to triangle B0 C0 D (Fig. 19c) we can write: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi l3 ¼ l22 þ l25 ð#1 Þ  2  l2  l5 ð#1 Þ  cos að#1 Þ ð4Þ so: að#1 Þ ¼ a cos l22 þ l25 ð#1 Þ  l23 2  l 2  l 5 ð# 1 Þ ð5Þ that, because of (3), becomes: (c) Y Y B' B l2 l2 C' C l1 l1 l3 l3 A A l4 l4 O D X O D' X O X Fig. 19 Geometrical model for analyzing the kinematic mechanism condition of the wall: a undeformed shape of the wall loaded in plane; b deformed shape of the wall with rotation angles of the blocks; c relationships between angles Author's personal copy 1030 Materials and Structures (2012) 45:1019–1034 #3 ð#1 Þ ¼ p  a3  eð#1 Þ þ a2  #2 ð#1 Þ að#1 Þ ¼ a cos l21 þ l22  l23 þ l24  2  l1  l4  cosða4 þ a1  #1 Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2  l2  l21 þ l24  2  l1  l4  cosða4 þ a1  #1 Þ ð6Þ Referring to triangle AB0 D (Fig. 19c) we can write: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi l4 ¼ l21 þ l25 ð#1 Þ  2  l1  l5 ð#1 Þ  cos bð#1 Þ ð7Þ so: bð#1 Þ ¼ a cos l21 þ l25 ð#1 Þ  l24 2  l1  l5 ð#1 Þ ð8Þ ð12Þ where eð#1 Þ is obtained as follows. Referring to triangle B0 C0 D (Fig. 19c) we can write: l5 ð#1 Þ ¼ so: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi l22 þ l23  2  l2  l3  cos eð#1 Þ eð#1 Þ ¼ a cos l22 þ l23  l25 ð#1 Þ 2  l2  l3 ð13Þ ð14Þ that, because of (3), becomes: that, because of (3), becomes: eð#1 Þ ¼ a cos bð#1 Þ ¼ a cos l21 þ l22 þ l23  l24 þ 2  l1  l4  cosða4 þ a1  #1 Þ 2  l2  l3 ð15Þ 2  l21  2  l1  l4  cosða4 þ a1  #1 Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð9Þ 2  l1  l21 þ l24  2  l1  l4  cosða4 þ a1  #1 Þ Substituting the values of #2 ð#1 Þ and eð#1 Þ given by (10) and (15) in (12), we obtain:  l21 þ l22 þ l23  l24 þ 2  l1  l4  cosða4 þ a1  #1 Þ #3 ð#1 Þ ¼ p  a3  a cos 2  l2  l3 " 2 l þ l22  l23 þ l24  2  l1  l4  cosða4 þ a1  #1 Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ a2  p þ a1  #1 þ a cos 1 2  l2  l21 þ l24  2  l1  l4  cosða4 þ a1  #1 Þ # 2  l21  2  l1  l4  cosða4 þ a1  #1 Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ a2 þa cos 2  l1  l21 þ l24  2  l1  l4  cosða4 þ a1  #1 Þ  Substituting the values of að#1 Þ and bð#1 Þ given by (6) and (9) in (2), we obtain: #2 ð#1 Þ ¼ p þ a1  #1 þ acos l21 þ l22  l23 þ l24  2  l1  l4  cosða4 þ a1  #1 Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2  l2  l21 þ l24  2  l1  l4  cosða4 þ a1  #1 Þ þ acos þ a2 2  l21  2  l1  l4 cosða4 þ a1  #1 Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2  l1  l21 þ l24  2  l1  l4  cosða4 þ a1  #1 Þ ð10Þ Referring to the quadrilateral AB0 C0 D (Fig. 19b, c) we can also write: a3 þ #3 ð#1 Þ þ eð#1 Þ  a2 þ #2 ð#1 Þ ¼ p so: ð11Þ ð16Þ The n horizontal and g vertical displacements of all the points of the masonry blocks in the wall may be expressed as a function of #1 ; #2 ð#1 Þ and #3 ð#1 Þ: In particular, the horizontal and vertical displacements of the generic point P1 ðxP1 ; yP1 Þ of the block 1 due to the rotation of #1 in relation to A(xA, yA) are: nP1 ð#1 Þ ¼ ðxP1  xA Þ þ ðxP1  xA Þ  cos #1 þ ðyP1  yA Þ  sen #1 gP1 ð#1 Þ ¼ ðyP1  yA Þ þ ðyP1  yA Þ  cos #1  ðxP1  xA Þ  sen #1 ð17Þ ð18Þ The horizontal and vertical displacements of the generic point P2 ðxP2 ; yP2 Þ of the block 2 Author's personal copy Materials and Structures (2012) 45:1019–1034 1031 due to the rotation of #2 ð#1 Þ in relation to B(xB, yB) are: nP2 ð#2 ð#1 ÞÞ ¼ nB ð#1 Þ  ðxP2  xB Þ þ ðxP2  xB Þ  cos #2 ð#1 Þ  ðyP2  yB Þ  sen #2 ð#1 Þ ð19Þ gP2 ð#2 ð#1 ÞÞ ¼ gB ð#1 Þ  ðyP2  yB Þ þ ðyP2  yB Þ  cos #2 ð#1 Þ þ ðxP2  xB Þ  sen #2 ð#1 Þ ð20Þ A virtual rotation is imposed on the first block of wall involved in the kinematic mechanism (Fig. 19b). The rotations of the other blocks are calculated using the geometrical model shown in Sect. 3.1. The displacements of the blocks’ barycenters and of the points where the acting horizontal and vertical forces come to bear are determined with reference to Eqs. 17–24, shown in Sect. 3.1. The condition that identifies the wall’s limit of equilibrium gives us: where: nB ð#1 Þ ¼ ðxB  xA Þ þ ðxB  xA Þ  cos #1 þ ðyB  yA Þ  sen #1 gB ð#1 Þ ¼ ðyB  yA Þ þ ðyB  yA Þ  cos #1  ðxB  xA Þ  sen #1  ð21Þ ð22Þ are the horizontal and vertical displacements of the hinge B(xB, yB) due to the rotation of the block 1 in relation to A(xA, yA). Finally, the horizontal and vertical displacements of the generic point P3 ðxP3 ; yP3 Þ of the block 3 due to the rotation of #3 ð#1 Þ in relation to D(xD, yD) are: nP3 ð#3 ð#1 ÞÞ ¼ ðxP3  xD Þ þ ðxP3  xD Þ  cos #3 ð#1 Þ þ ðyP3  yD Þ  sen#3 ð#1 Þ ð23Þ gP3 ð#3 ð#1 ÞÞ ¼ ðyP3  yD Þ þ ðyP3  yD Þ  cos #3 ð#1 Þ  ðxP3  xD Þ  sen #3 ð#1 Þ ð24Þ 3.2 Analytical assessment of the wall’s load-bearing capacity The horizontal force coinciding with the kinematic mechanism condition of the wall loaded in plane can be determined by applying the Virtual Work Principle. (a) Mech. shape 1 (b) Mech. shape 2 Fig. 20 Kinematic mechanism shapes of the wall loaded in plane n X WBi  gGBi  i¼1 m X k¼1 FVk  gPVk þ q X FHBj  nPHj ¼ 0 j¼1 ð25Þ where WBi is the weight of the i of n-blocks of wall involved in the kinematic mechanism, gGBi is the vertical displacement of the barycenter of the i-block, FVk and FHBj are respectively the k of m-vertical forces and the j of q-horizontal forces acting on the blocks involved in the kinematic mechanism, gPVk is the vertical displacement of the point where FVk is applied and nPHi is the horizontal displacement of the point where FHBj is applied. Figure 20 shows the possible shapes of the kinematic mechanism for the wall with an opening submitted to horizontal and vertical forces acting in plane as reported in the literature. For the specimens 1–6 used in the previous experiments, the theoretical values of the horizontal force FH that identifies the reaching of the kinematic mechanism condition for all the related shapes of kinematic mechanism were determined with reference to Eq. 25. These values are given in Table 3 (columns 2–5). For the same specimens, the theoretical values of the horizontal force FH coinciding with the kinematic mechanism condition for the shapes of kinematic mechanism obtained from the experiments (Fig. 21) were also determined with reference to Eq. 25. These values are given in Table 3 (column 6). (c) Mech. shape 3 (d) Mech. shape 4 Author's personal copy 1032 Materials and Structures (2012) 45:1019–1034 Table 3 Theoretical and experimental values of the horizontal force coinciding with the kinematic mechanism condition in the specimens Specimens Theoretical values of FH (kN) Experimental values of FH (kN) 1 mech. shape 2 mech. shape 3 mech. shape 4 mech. shape Experimental mech. shape 1 123.05 114.29 33.56 31.99 63.18 63 2 76.73 72.49 22.05 21.06 39.62 48 3 45.85 44.66 14.51 13.89 23.04 18 4 45.85 44.66 14.51 13.89 22.67 21.6 5 45.85 44.66 14.51 13.89 32.50 26 6 45.85 44.66 14.51 13.89 35.62 33.2 (a) Specimen1 (b) Specimen2 (c) Specimen3 (d) Specimen4 (e) Specimen5 (f) Specimen6 Fig. 21 Kinematic mechanism shapes of the specimens used in the experiments Table 3 also gives the values of the horizontal force coinciding with the load-bearing capacity peak of the specimens recorded during the experiments (column 7). Notice that all the theoretical values of the horizontal force FH were obtained with reference to the geometrical model presented in Sect. 3.1. This model is based on the assumption of an infinite value of masonry compressive strength. It could also take into account a finite value of masonry compressive strength by assuming a plastic distribution of compressive stresses (i.e. stress–block) at the toe of the wall, and so modifying the length of the contact surface between the wall and its base as suggested by Giuffrè [6]. For all the specimens and for all the considered shapes of kinematic mechanism the authors determined also the theoretical values of the horizontal force FHfcs that identifies the reaching of the kinematic mechanism condition in this second case (finite value of masonry compressive strength). These values exhibited only marginal differences from the theoretical values of the horizontal force FH previously determined (i.e. assuming an infinite value of masonry compressive strength). In particular, for specimens 3–6 values of FHfcs were equal to 98% of FH, while for Author's personal copy Materials and Structures (2012) 45:1019–1034 specimens 1 and 2 values of FHfcs were respectively equal to 94 and 96% of FH. 4 Comparison between experimental and theoretical results A first comparison was drawn between the theoretical values of FH identifying the reaching of the kinematic mechanism condition in the wall with reference to the shapes 1–4 of said mechanism and to the experimental values of FH (Table 3). For all the specimens, the experimentally found value of FH was higher than its theoretical value for the kinematic mechanism in shapes 3 and 4, i.e. the weakest shapes judging from the analytical assessment. This result shows that the weakest of all the possible mechanism shapes for a given specimen does not necessarily develop. As a consequence, a theoretical assessment of the load-bearing capacity of a wall that refers to the weakest of all possible mechanism shapes for safety’s sake is not always appropriate, because it might significantly underestimate the wall’s real loadbearing capacity. This also means that the shape of the kinematic mechanism that develops depends not only on the geometry of the wall and its loading conditions (and the acting vertical load in particular)—both considered in the analytical model described in Sects. 3.1 and 3.2—but also on its specific building features. As mentioned earlier (see Sect. 2.5), the experimental results obtained for specimens 4–6 show that the masonry texture strongly influences both the failure mode and the load-bearing and deformation capacity of the wall. More specifically, the bricks’ arrangement in the top spandrel influences the latter’s mechanical behavior and the coupling that it consequently establishes between the two piers. Our experiments were only conducted on a few specimens (and only one of each type of brick arrangement in the spandrels and piers), so further tests would be needed to validate our findings. The results that we obtained nonetheless give us some indication of the relationship existing between the brick arrangement in the spandrels and the coupling established between the two piers. As mentioned in Sect. 2.5, if the brick arrangement in the spandrels is characterized by a prevalence of headers, the spandrels are very compact and thus 1033 ensure a very strong coupling between the two piers. Vice versa, if the brick arrangement in the spandrels mainly contains stretchers, their cracking becomes highly likely and they cannot assure a good coupling between the two piers. The quality of the coupling between the piers influences the shape of the wall’s kinematic mechanism: in particular, a good coupling encourages the activation of the kinematic mechanism shapes 1 or 2, while a poor coupling supports the activation of the kinematic mechanism shapes 3 or 4. The former (shapes 1 or 2) demand a greater horizontal force to reach the kinematic mechanism condition than the latter (shapes 3 or 4). The Italian technical rules for the design and control of structures [13] assume that unreinforced masonry spandrels (as in the case of the spandrels of many historical masonry buildings) do not ensure a significant coupling between the supporting piers, so the load-bearing capacity of the walls should be estimated with reference to the kinematic mechanism shapes 3 or 4. This assumption is very restrictive and may mean that the real load-bearing capacity of masonry walls is underestimated. It is also worth recalling here that limit equilibrium analysis disregards the tensile strength of masonry material. In fact, this property is uncertain and generally very low, as reported in the literature. It would still influence the load-bearing and deformation capacities of masonry walls, however, so an accurate assessment of their mechanical behavior that takes their real strength and strain resources into account for conservation needs should consider it. For all these reasons, a definition of the most likely shape of the kinematic mechanism of masonry walls should be based on the specific features of masonry walls, such as the arrangement of the bricks in the spandrels and piers, and—wherever possible—the calculation of its load-bearing capacity should consider the contribution of the tensile strength of the masonry. A second comparison was drawn between the theoretical values of FH that identify the kinematic mechanism condition with reference to the experimental mechanism shape and the experimental values of FH (Table 3). The correspondence between the theoretical and experimental values was very good for specimens 1, 4 and 6, and fairly good for specimen 2. For specimens 3 and 5, the theoretical model slightly underestimates Author's personal copy 1034 the load-bearing capacity of the walls, so its prediction is not conservative. On this point, it is worth noting that the proposed model was developed in line with the limit analysis approach and assumes an infinite value for the sliding strength of masonry material, i.e. it does not take into account any sliding phenomena that may develop between the bricks and affect the assessment of the wall’s load-bearing capacity (as in the case of specimen 5, and possibly of specimen 3 as well). We conclude that the simplified model for assessing the load-bearing capacity of walls with an opening loaded in plane proposed in Sect. 3 is acceptable, but should be improved to take into account more realistic mechanical properties of masonry material. 5 Conclusions The outcomes of the present study confirm that assessing the in-plane behavior of masonry structures is a very complex problem. This complexity stems mainly from the nature of masonry material, consisting of a combination of small blocks. As shown by our experiments, different arrangements of the bricks in the spandrels and piers may give rise to different wall failure modes, and consequently different horizontal forces and displacements identifying their load-bearing capacity peak. An accurate knowledge of the real structure of the walls is therefore necessary for a preventive assessment of the seismic safety of a building. This is unfortunately not always possible in the case of existing masonry buildings due to objective problems implicit in the survey of the materials and building technique involved. Simplified analytical models such as ours, based on the kinematic approach of limit equilibrium analysis can certainly be useful in the seismic safety evaluation of existing masonry structures because they enable any consideration of the mechanical strength and strain parameters concerning the material (which are very uncertain in many cases) to be disregarded. Choosing the shape of the wall’s kinematic mechanism is very important, however: the right the choice enables us to take the real resources of masonry structures into account, thereby avoiding the risk of potentially harmful overestimations of their safety, which could lead to the adoption of preventive Materials and Structures (2012) 45:1019–1034 solutions that are highly invasive for the original structure of the building. References 1. Augenti N (2000) Il calcolo sismico degli edifici in muratura. UTET, Torino 2. Avorio A, Borri A, Cangi F (1999) Riparazione e consolidamento degli edifici in muratura. In: Gurrieri F (ed) Manuale per la riabilitazione e la ricostruzione post-sismica degli edifici. DEI Tipografia del Genio Civile, Roma, pp 227–287 3. Benedetti D, Tomaževič M (1984) Sulla verifica sismica di costruzioni in muratura. Ing Sismica 0:9–16 4. Braga F, Dolce M (1982) A method for the analysis of antiseismic masonry multi-storey buildings. In: Proceedings of sixth international brick masonry conference, Rome, pp 1088–1098 5. Focacci F (2003) Rinforzo di archi con cavi. In: Proceedings of 16th AIMETA Congress of Theoretical and Applied Mechanics, Ferrara, pp 1–10 6. Giuffrè A (1991) Letture sulla Meccanica delle Murature Storiche. Kappa, Roma 7. Giuffrè A (1993) Sicurezza e conservazione dei centri storici. Il caso Ortigia, Laterza 8. 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In: Proceedings of 6th Congresso Nacional de Sismologia e Engenharia Sı́smica, Guimaraes, pp 101–117 16. Tassios TP (1988) Meccanica delle murature. Liguori, Napoli 17. Tomaževič M, Turnšek V (1980) Lateral load distribution as a basis for the seismic resistance analysis of masonry buildings. In: Proceedings of the international research conference on earthquake engineering, Skopje, pp 455–488 18. Turnšek V, Cačovič F (1971) Some experimental results on the strength of brick masonry walls. In: Proceedings of second international brick masonry conference, Stoke-onTrent, pp 149–156 19. Vermeltfoort ATh, Raijmakers TMJ (1993) Deformation controlled tests in masonry shear walls. Part 2. In: Report TUE/BKO/93.08. Eindhoven University of Technology, Eindhoven