Shear resistance of masonry walls and Eurocode 6: shear versus tensile strength of masonry

Transcription

1 DOI /s ORIGINAL ARTICLE Shear resistance of masonry walls and Eurocode 6: shear versus tensile strength of masonry Miha Tomaževič Received: 7 April 2008 / Accepted: 17 September 2008 Ó RILEM 2008 Abstract In the case of masonry structures subjected to seismic loads, shear failure mechanism of walls, characterised by the formation of diagonal cracks, by far predominates the sliding shear failure mechanism. However, as assumed by Eurocode 6, the latter represents the critical mechanism for the assessment of the shear resistance of structural walls. The results of a series of laboratory tests are analysed to show that in the case of the diagonal tension shear failure the results of the Eurocode 6 based calculations are not in agreement with the actual resistance of masonry walls. The results of calculations, where the diagonal tension shear mechanism and tensile strength of masonry are considered as the critical parameters, are more realistic. Since the results of seismic resistance verification, based on the Eurocode 6 assumed sliding shear mechanism, are not in favour of structural safety, it is proposed that in addition to sliding shear, the diagonal tension shear mechanism be also considered. Besides, in order to avoid misleading distribution of seismic actions on the resisting shear walls, the deformability characteristics of masonry at shear should be determined on the basis of experiments and not by taking into account the Eurocode 6 recommended G/E ratio. M. Tomaževič (&) Slovenian National Building and Civil Engineering Institute, Dimičeva 12, 1000 Ljubljana, Slovenia miha.tomazevic@zag.si Keywords Masonry structures Seismic resistance Shear Sliding shear mechanism Diagonal tension shear mechanism Shear strength Tensile strength Shear resistance Eurocodes 1 Introduction Masonry is a typical composite construction material, which is suitable to carry the compressive loads; however its capacity to carry the tension and shear is relatively low. As a result of non-homogeneity and anisotropy of masonry, the relationships between the mechanical characteristics of masonry at shear and compression are significantly different than in the case of the homogeneous and isotropic materials. Since the walls and piers represent the basic structural elements of masonry structures, shear mechanisms prevail in the case where the masonry walls are subjected to in-plane lateral loads. Flexural mechanisms are rarely observed. Therefore, the parameters which define the behaviour of masonry walls at shear are of relevant importance for the seismic resistance verification of buildings in seismic-prone areas. Because of specific characteristics of each constituent material, it is not easy to predict the mechanical properties of a specific masonry construction type by knowing only the characteristics of its constituents. The values, which determine the strength characteristics of masonry, do not represent the actual stresses

2 in materials at failure but the average values, calculated on the basis of the gross sectional areas of individual structural elements. For example, stresses in material at compressive failure in the case of a solid brick are not the same as in the case of a hollow block, although the declared strength of both units is equal. Although the normalized values, determined in accordance with EN [1] are used, significant differences exist between the actual compressive stresses in masonry material and the design values, obtained on the basis of the gross sectional area of the units. Similarly, in order to simplify the numerical procedures, the sectional stresses and forces are used and the gross dimensions of masonry walls are taken into consideration in the case of the structural analysis, assuming that masonry is elastic, homogeneous and isotropic construction material. However, the equations of the elastic theory of structures and methods of calculation are modified in order to take into account the specific characteristics of masonry materials. Correlation of experimental results with Eurocode 6 [2] recommended values of parameters, which determine the strength and deformability characteristics of masonry at compression, indicates that the values of the compressive strength f and modulus of elasticity E of masonry can be predicted reasonably well on the basis of the known compressive strength of individual units and masonry mortar. However, the experiments indicate that the relationships are not straightforward in the case where the walls are subjected to lateral loads and different failure mechanisms are possible. In this contribution, the results of a recent study, carried out at Slovenian National Building and Civil Engineering Institute in Ljubljana, Slovenia, aimed at providing the values of national parameters regarding the shear resistance of unreinforced masonry walls to be recommended by Slovenian National Annex to Eurocode 6, will be presented and discussed. Besides, the behaviour depends on the level of precompression, i.e. the ratio between the working stresses in the wall due to gravity loads and compressive strength of masonry, as well as on the direction of action of horizontal loads (in-plane, out-of-plane). Consequently, various types of failure mechanism are possible. In this contribution, however, only the shear failure mechanism of unreinforced masonry walls subjected to in-plane action of lateral loads will be discussed. If the vertical compressive stresses in the wall are low and the quality of mortar is poor, seismic forces may cause sliding of a part of the wall along one of the bed-joints (Fig. 1a). Sliding shear failure of unreinforced walls usually takes place in the upper parts of masonry buildings below rigid roof structures, where the compressive stresses are low and the response accelerations are high. However, this phenomenon is seldom observed in the buildings bottom parts, where, typically, diagonally oriented cracks develop in the walls when subjected to seismic loads (Fig. 1b). Because of the orientation of cracks, the failure of the wall in such a case is also called diagonal tension shear failure. Depending on the quality of masonry units and mortar, diagonally oriented cracks may either follow the bed- and head-joints or pass through the units or partly follow the joints and partly pass through the units. Typical examples of diagonal shear cracks in the load-bearing walls caused by the earthquakes are shown in Figs. 2 and 3. Although the resistance to lateral loads is the key parameter, other parameters, such as deformability, ductility and energy dissipation capacity, strength and stiffness degradation at repeated lateral load reversals, are also important for the assessment of the seismic resistance of the structure. Therefore, decades ago the experimental tests for the evaluation of the seismic resistance of masonry walls have been 2 Behaviour of masonry walls subjected to in-plane acting seismic loads and testing The behaviour of masonry walls subjected to a combination of vertical and horizontal loads depends on the geometry of the walls (height/length ratio), mechanical characteristics of masonry and reinforcement, if any, as well as on the boundary conditions. Fig. 1 Shear failure mechanisms: a shear sliding on the bedjoint, b shear failure characterized by formation of diagonal cracks

3 Fig. 2 Typical shear failure of brick masonry piers of a three storey building after the earthquake Fig. 3 Shear cracks in stone-masonry walls of a historic building after the earthquake designed to simulate the cyclic character of lateral loading and actual boundary restraints (for example [3 7]). Such tests made possible the evaluation of all important parameters, influencing the seismic resistance of masonry structures. Horizontal and vertical actions, which act on individual walls in a masonry structure during the earthquake, change in an alternate, cyclic way. Since the wall is restrained by horizontal elements, such as parapets, lintels and floors, which hinder its rotation at large lateral displacements, additional compressive stresses develop in the wall at each cycle, which prevent the formation of horizontal tension cracks at the wall s end sections. When tested in the laboratory, however, the simulation of actual restraints would increase the costs of testing. Therefore, the walls are tested at a controlled, usually constant level of vertical load, as well as at controlled conditions of boundary supports either as symmetrically fixed or as vertical cantilevers. The specimens are constructed on a reinforced-concrete (r.c.) foundation block, whereas vertical and cyclic lateral load act on an r.c. bond beam, located on the top of the walls. If unreinforced masonry walls are tested, horizontal cracks develop at the most stressed bed-joints as a result of low axial tensile strength of masonry, so that rocking of the wall on the support takes place. In order to prevent the rotation, the vertical steel ties, which take the tension forces developed on the tensioned side of the wall, are used in the case of the so called racking test [8]. In the case of cyclic testing, however, this is not the practice. As a result, the phenomena, typical for flexural mechanism can be observed in the initial phase of testing (Fig. 4). Before the formation of diagonal shear cracks in the central part of the wall, the horizontal tensile cracks develop in the tensioned part of the bed-joints at the supports and the crushing of masonry units at the compressed corners takes place. Although the flexural effects prevail in the beginning of the test, and the compressive stresses at the compressed corners are near to the compressive strength of masonry units, this is not the flexural failure of the wall. The resistance increases until the diagonal cracks develop in the central part of the wall and the wall finally fails in shear. No such phenomena take place if the wall is tested in situ, where the specimen is separated from the surrounding masonry by two vertical cuts. Although in the particular case, shown in Fig. 5, the level of vertical stresses has been relatively low (estimated compressive stresses r o = 0.15 MPa represented about 7.5% of the masonry s compressive strength) neither the horizontal cracks nor the crushing of bricks have been observed at supports [9]. In their recommendations for the design of masonry structures, CIB recommended three methods of testing the masonry walls for assessing the values of parameters needed for the earthquake resistant design of masonry structures ( design by testing ; [10]): cyclic lateral resistance tests of symmetrically fixed or cantilever walls at constant vertical load, as well as diagonal compression test of the walls (Fig. 6). 3 Shear strength of masonry Shear strength is the mechanical property of masonry, which defines the resistance of masonry wall to

4 Fig. 4 Damage to masonry walls during laboratory testing. a Hollow clay units type B2: shear cracks are passing through the units. b Perforated clay units type B6: shear cracks pass partly through the joints and partly through the units. In both cases, tensile cracks and crushing of units at support have been observed before the shear failure Fig. 5 In-situ shear resistance test of a brick masonry wall: neither horizontal cracks nor crushing of bricks is observed at supports (adapted from [9]) lateral in-plane loads in the case that the wall fails in shear. As there are several modes of such failure, the definition of the shear strength is not straightforward. The parameter, which determines the shear resistance of a masonry wall, depends on the physical model describing the failure mechanism. In the case of the sliding shear mechanism, which is characterized by the formation of horizontal cracks, masonry units slide upon one of the bedjoints as soon as the shear stresses exceed the value, called the shear strength of masonry (friction analogy). In the case of the shear mechanism, however, characterised by the formation of diagonally oriented cracks, shear cracks are caused by the principal tensile stresses developed in the wall under the combination of vertical and lateral load. When the principal tensile stresses exceed the value called the tensile or diagonal tensile strength of masonry, diagonal cracks occur in the wall (tensile strength hypothesis). A clear distinction should be made between both mechanisms [11, 12], and the resistance of a masonry wall should be checked for both of them. Whereas the tests for the determination of initial shear strength of masonry are standardized, the procedure for obtaining the tensile strength is not. However, statistical correlation analysis, carried out on the basis of the results of tests of a number of masonry walls of the same type, tested by using testing methods, recommended by CIB, has shown that any method is suitable to determine the values of tensile strength [13]. It is recommended that the walls having the geometry aspect ratio h/l = 1.5 or smaller are tested, where h is the height and l is the length of the wall.

5 Fig. 6 Schematic presentation of different types of tests suitable for evaluation of parameters of seismic resistance of masonry walls. a cyclic test of a fixed-ended wall, b cyclic or racking test of a cantilever wall, c diagonal compression test (after [10]) 3.1 Tensile strength of masonry Turnšek and Čačovič [14] found that it is not possible to explain the formation of diagonally oriented cracks in the walls by using the friction theory. Assuming that the masonry wall behaves as an ideal elastic, homogeneous and isotropic panel all the way up to the failure, they called the principal tensile stress at the attained maximum resistance of the wall the tensile, or better the referential tensile strength of masonry, f t. On the basis of such, purely conventional definition, the equation for the calculation of the shear resistance of masonry walls has been proposed [14], modified by various other authors in the following years (e.g. [15, 16]). The equations based on the idea that the tensile strength governs the shear resistance of masonry walls have been implemented in several recommendations (e.g. [17]) and seismic codes in former Yugoslavia [18] and other countries. By taking into account the assumption that masonry wall is an elastic, homogeneous and isotropic panel, the basic equation can be derived on the basis of the elementary theory of elasticity. If the vertical, N, and horizontal (shear) load, H, are acting on the wall, the principal compressive and tensile stresses develop in the middle section of the wall: rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r o 2þðbsÞ 2 r P ¼ 2 r o 2 ; ð1þ oriented in the directions of both diagonals of the wall: / c ¼ / t ¼ 0:5 arc tg 2s : ð2þ r o The meaning of the symbols in Eqs. 1 and 2 is as follows: r o = N/A w the average compressive stress in the horizontal section of the walls due to constant vertical load N; s = H/A w the average shear stress in the horizontal section of the wall due to horizontal load H; A w the area of the horizontal cross-section of the wall; b the shear stress distribution factor, which depends on the geometry of the wall and the ratio between the vertical load N and maximum horizontal load H max. In case that the aspect ratio is equal to or greater than h/l = 1.5, the value of b = 1.5 can be assumed. The value decreases in the case of squat walls. Factor b is not the shear stress distribution factor j, used in the theory of the strength of materials. Assuming the elastic, homogeneous and isotropic behaviour of the wall panel all the way up to the attained maximum value of horizontal load, H max, the idealised principal tensile stress at that instant is conventionally called the tensile or referential tensile strength of masonry, f t : rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r o 2þðbsmax f t ¼ r t ¼ Þ 2 2 r o 2 ; ð3þ where f t the tensile strength of masonry; s max the average shear stress in the horizontal section of the wall at the attained maximum horizontal load H max (at maximum lateral resistance). A substantial number of test results of fixed-ended and cantilever walls have been evaluated using the Eq. 3 in the last decades. Typical values have been recommended for the design in seismic codes. The values of the tensile strength, recently evaluated on the basis of cyclic lateral resistance tests of wall specimens, made of different types of hollow clay blocks, which have been also used for the determination of the initial shear strength of masonry at zero compression, discussed in the following, are given in Table 1. Surprisingly, in this series of tests the masonry units strength did not significantly influence the tensile strength of masonry.

6 Table 1 Mean, f t, and characteristic values of tensile strength of hollow clay unit masonry, f tk, obtained by lateral resistance tests of walls (adapted from [19]) Units Normalized compressive strength of unit f b (MPa) Mean compressive strength of mortar f m (MPa) Tensile strength of masonry f t (MPa) f tk (MPa) B B B B B Shear strength according to Eurocode 6 According to Eurocode 6, the shear strength of masonry is defined as a sum of the initial shear strength (shear strength at zero compressive stress) and a contribution due to the design compressive stress perpendicular to shear at the level under consideration. Characteristic initial shear strength at zero compression, f vko, is determined by testing specimens made of three masonry units according to standard EN ([20], Figs. 7 and 8). As can be seen in Fig. 7, the standard does not define the geometry aspect ratio of the specimen. The scheme, shown in Fig. 7, is presented for the case of testing the specimens made of bricks, whereas the specimens made of hollow blocks with different geometrical proportions have been actually tested (Fig. 8). During the test, it should be ensured that pure shear stresses develop in the connecting planes between the units and mortar. Six specimens of each type are tested. As the characteristic, the lesser value of the minimal obtained or 80% of the mean value is considered. Characteristic shear strength of masonry, f vk, made of any mortar, at the condition that all, bed- and headjoints are fully filled with mortar, is determined by: f vk ¼ f vko þ 0:4r d : ð4þ Equation is modified in the case where the vertical joints are not filled with mortar: f vk ¼ 0:5f vko þ 0:4r d ; ð4aþ where r d is the design compressive stress in the wall s section. Since the value depends on the stress state in the particular wall under consideration, the shear strength, as defined by the Eurocode, cannot be considered as the mechanical characteristic of masonry. The shear strength represents the average shear stress in the horizontal section of a wall subjected to specific axial load at sliding shear failure. The coefficient defining the contribution of the shear strength due to compressive stresses in the wall, 0.4, is taken as a constant for all types of masonry, although the procedure for the determination of the internal Fig. 7 Schematic presentation of initial shear strength test according to EN Fig. 8 Initial shear strength test according to EN in the laboratory

7 Table 2 Characteristic initial shear strength of masonry f vko (EN :2005) Material f vko (MPa) General purpose mortar of the strength class given Thin layer mortar (bed joint C0.5 mm and B3 mm) Lightweight mortar Clay M10 M M2.5 M Calcium silicate M10 M M2.5 M Concrete M10 M Autoclaved aerated concrete M2.5 M Manufactured and dimensioned natural stone M1 M friction angle is specified by standard EN According to Eurocode 6, in no case the characteristic shear strength should be greater than either 0.065f b (6.5% of the units compressive strength) or the limit value f vlt, which should be determined by the National Annex. In the case that the experimental values of f vko are not available, recommended values of the initial shear strength can be taken into consideration. As can be seen in Table 2, the Eurocode 6 recommended values depend only on the units materials and mortar strength class, but not on the strength of the units. Recently, the characteristic initial shear strength has been determined by testing a series of masonry specimens prepared with six different hollow clay unit types and two mortar classes. Altogether 72 specimens have been tested. The shape of the units is shown in Figs. 9 and 10, whereas their dimensions and physical properties are given in Table 3. The actual test layout and typical specimens after the test can be seen in Figs. 8 and 11, respectively. Factory made, pre-batched mortar of strength classes M5 and M10 (brand name Omalt MzZ type M5 and M10, produced by Cinkarna Celje, Ltd.) has been used to prepare the specimens. The values of initial shear strength obtained by testing are given in Table 4. Shear failure along the mortar joints occurred in all cases. As can be seen, failure is the result of the exhausted bond between mortar and units where, as a rule, the mortar delaminated from the units (see Fig. 11). In no case the failure occurred through the units. In the particular case studied, EN tests indicated that the initial shear strength values do not depend on the strength of the mortar. Also, no direct correlation could be observed between the initial shear strength and geometry (volume of holes) or compressive strength of the unit. The values obtained by testing the specimens made with units B5 are significantly higher than those obtained by testing other types of units. Since the differences could not be explained by comparing neither the mechanical and geometrical characteristics of the units (see Table 3) nor the failure modes, the values have not been considered in the calculation of the average values of the initial shear strength of the tested series of specimens. Fig. 9 Hollow clay units B1, B2 and B3, used for construction of walls for cyclic seismic resistance tests and initial shear strength tests according to EN

8 Fig. 10 Hollow clay units B4, B5 and B6, used for construction of walls for cyclic seismic resistance tests and initial shear strength tests according to EN Table 3 Dimensions and compressive strength of hollow clay masonry units, used for the construction of walls for lateral resistance tests and initial shear strength tests of masonry (adapted from [19]) a Normalized mean values Units Length (mm) Width (mm) Height (mm) Volume of holes (%) Thickness of shells (mm) Thickness of webs (mm) B B B B B B Compressive strength a (MPa) Fig. 11 Typical view on failure planes after the completed initial shear strength tests of specimens made of units B3, B5 and B6 The tests did not confirm the recommendations of Eurocode 6 that the initial shear strength depends on the mortar s strength class (Table 2). As can be seen in Table 4, the experimental characteristic values are close to those recommended only for the case where the specimens have been prepared with the mortar of declared strength class M5 (actually 17.9 MPa). 3.3 Correlation between the shear and tensile strength If the shear strength and tensile strength were the parameters which determine the same property, i.e. the shear resistance of a masonry wall, there should be a correlation between them. At least there should be a correlation between the initial shear strength at zero vertical stress, f vko, and the tensile strength of masonry, f tk, since these parameters obviously represent the characteristics of masonry materials. If this were the case, then the average shear stress in the section at shear failure could have been the common denominator. Assuming that s max in Eq. 3 actually represents an equivalent of the shear strength f vk, determined by Eq. 4: s max ¼ f vk ; ð5þ which would be the case if the wall is under compression along the whole length of the wall s horizontal section, and by introducing this

9 Table 4 Characteristic, f vko, and mean values of initial shear strength of masonry, f vo, obtained by testing specimens according to EN (values in MPa) Units Compressive strength of units a Strength class of mortar 5 MPa b 10 MPa c f vko f vo f vko f vo B B B B B B Average d a Normalized mean value b Actual mean value of compressive strength is f m = 17.9 MPa c Actual mean value of compressive strength is f m = 23.2 MPa d The values obtained for units B5 are not considered assumption into Eq. 4, the equivalent tensile strength, ftk 0, can be expressed as: rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ftk 0 ¼ r d 2þðbfvk Þ 2 r d 2 2 : ð6þ Taking into consideration the Eurocode s 6 recommended value of f vko from Table 2 (f vko = 0.2 MPa) and a series of values of design compressive stresses r d, expressed in terms of the ratio between the design stress and characteristic compressive strength of masonry, equivalent characteristic tensile strength of masonry, ftk 0, can be calculated. However, as can be seen in Table 5, such values are unacceptably high and are much higher than the values, obtained by testing the considered types of masonry walls (see Table 1). Although the theoretical relationship between the quantities seems correct, there is actually no correlation between the initial shear strength and tensile strength of masonry. The quantities have different physical meanings and define two different failure mechanisms. Whereas the shear strength, f v (Eq. 4), is defined on the basis of the assumption that the shear failure of the wall takes place because of sliding of the units along the bed-joint, and is therefore depending on the design compressive stresses in each particular wall under consideration, the tensile strength, f t (Eq. 3), is considered as one of the mechanical characteristics of masonry, not depending on the stress state in the wall panel. Therefore, the Table 5 Correlation between the characteristic initial shear strength, f vko, and corresponding characteristic tensile strength of masonry, ftk 0, at different levels of design compressive stresses, r d, in the walls (values in MPa) r d a 0.1 f k a 0.2 f k a 0.3 f k a 0.4 f k a 0.5 f k f vko ftk 0 ftk 0 ftk 0 ftk 0 ftk a f k = 5.0 MPa transformation from the Eurocode s shear strength to tensile strength is even not possible. 4 Shear resistance of unreinforced masonry walls According to Eurocode 6, the design shear resistance of the wall is calculated by simply multiplying the characteristic shear strength of masonry by the area of the cross-section of the wall, which carries the shear. Characteristic shear strength is reduced by the partial safety factor for masonry, c M, so that the design shear resistance of an unreinforced masonry wall, R ds,w, is calculated by: R ds;w ¼ f vk tl c ; ð7þ c M where t the thickness of the wall, and l c the length of the compressed part of the wall, ignoring any part of the wall that is in tension, and calculated assuming a linear stress distribution of the compressive stresses, and taking into account any openings, chases or recesses. It can be shown that in the case where the eccentricity of axial load exceeds 1/6 of the wall s length, the length of the compressed part of the wall is expressed by: l c ¼ 3 l 2 e ; ð8þ where e = Hah/N is the eccentricity of the vertical load, ah is the arm of the horizontal load, which depends on restraints, i.e. boundary conditions at the bottom and the top of the wall (a = 1.0 in the case of a cantilever and a = 0.5 in the case of a fixed ended wall). Obviously, when using Eq. 7, the seismic shear should be already distributed onto the walls: to

10 calculate the length of the compressed part of the wall, the design vertical and design seismic loads should be known. Therefore, Eq. 7 is only useful in the case of traditional safety verification procedures, where for each structural element and for the structure as a whole, the design resistance capacity is compared with the design action effects. In the case of the non-linear push-over procedures, iterations would be required due to the changes in lateral load distribution in the non-linear range. By taking into consideration the same structural safety requirements and reducing the characteristic value of the tensile strength by partial safety factor for masonry, c M, the shear resistance of an unreinforced masonry wall in the case of the diagonal tension shear failure can be expressed by: rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi f tk 1 c R ds;w ¼ A M w r d þ 1: ð9þ c M b f tk A series of unreinforced masonry walls, built of different types of hollow clay units, have been recently tested under a combination of constant vertical and cyclic lateral load [19]. The same units as used for the initial shear strength tests (see Table 3), have been used for the construction of walls. Disposition of tests is shown in Fig. 12, whereas the dimensions of the walls and vertical load, V, acting on the walls during the lateral resistance tests and respective compressive stress, r o, in the horizontal section of the walls are given in Table 6. In the same table, the main experimental results, such as the maximum horizontal load, measured during the tests, H max,exp, and Fig. 12 Disposition of cyclic lateral resistance test of a cantilever wall respective average values of the shear stresses in the walls sections, s max, are summarized. All walls failed in shear, characterized by the formation of diagonal cracks, with the initial tension cracks and crushing of units occurring at the support (Fig. 4). Test results have been used to compare the shear resistance of the walls, calculated by assuming that either the sliding shear (Eq. 7) or diagonal tension shear (Eq. 9) mechanisms govern the failure mode. In the first case, the shear strength of masonry has been determined by Eq. 4. Instead of design, mean values of the shear strength, calculated on the basis of the mean values of the initial shear strength, given in Table 4 (mortar class M5), and actual compressive stresses in the walls during the tests have been considered in the calculations. In the second case, mean values of the tensile strength, given in Table 1, and actual compressive stresses in the walls have been considered in assessing the shear resistance of the walls. No reduction with partial safety factor for masonry, c M, has been considered. In other words, it has been assumed that c M = 1.0. Actual ratio between the vertical and lateral load at failure, observed during the tests, has been taken into account when determining the compressed part of the walls length. The walls have been tested as vertical cantilevers, so that, obviously, the bottom most section should have been considered. However, as the calculated compressed length at the foundation was unrealistically short (in two cases, the walls should have overturned during the test, although no such phenomenon has been observed), the section at the mid-height of the walls has been also considered. Compressive stresses in the compressed section, used to determine the shear strength, have been calculated by taking into account the compressed length of the wall. The results, obtained by considering the compressed length of the walls at both, support and midheight sections, are summarized in Table 7. It can be seen that in all cases the shear strength of the walls, relevant for the support section, f vk, exceeds the allowable limit value, i.e f b. Therefore, in the calculation of the shear resistance at support, the limit value of the shear strength has been taken into account. The calculated values of the shear resistance of the tested walls are compared with the experimentally obtained maximal values of horizontal load in Table 8. It should be noted that the diagonal tension

11 Table 6 Characteristics of tested walls and results of lateral resistance tests (adapted from [19]) Units Wall Dimensions of walls l/h/t (cm) A w (m 2 ) f k (MPa) V (kn) r o (MPa) r o /f k H max,exp (kn) s max (MPa) B1 B1/1 100/143/ B1/ B2 B2/1 102/151/ B2/ B2/ B3 B3/1 101/142/ B3/ B4 B4/1 99/142/ B4/ B6 B6/1 107/147/ B6/ shear failure, characterised by the formation of diagonal cracks, has been observed in the case of all tests. Therefore, good agreement between the experimental results and calculations, based on the diagonal tension shear failure mechanism, is obvious. It should be noticed, however, that, in the particular case studied, the calculated resistance is slightly overestimated in the case of the low precompression. However, no correlation between the experimental values and calculations can be observed in the case where the shear resistance of the walls has been calculated on the basis of the sliding shear mechanism and using methods, required by Eurocode 6. In the case where the requirements of Eurocode 6 have been strictly respected, i.e. where the support sections and the values of the shear strength limited by the units strength have been taken into account, any agreement can be considered as a mere coincidence. In the case where the mid-height section has been considered as critical, the calculations by times overestimate the experimentally obtained values. The meaning of the symbols in Table 8 is as follows: Table 7 Mean values of the tensile strength of masonry, f t, length of the compressed section, l c, and corresponding mean values of the shear strength of the tested walls, f v, evaluated by taking into account the compressed length of the wall at the support a and middle of the height b Wall f t (MPa) l c a (cm) f v a (MPa) l c b (cm) f v b (MPa) 0.065f b (MPa) B1/ B1/ B2/ B2/ B2/ B3/ B3/ B4/ B4/ B6/ B6/ a Bottom section, b Mid-height section Table 8 Comparison of experimentally obtained and calculated values of the shear resistance of the tested walls Wall H max,exp (kn) R s,w-ft (kn) R s,w-fv a (kn) R s,w-fv b (kn) B1/ c B1/ c B2/ c c B2/ B2/ c c B3/ c B3/ c B4/ c c B4/ c B6/ c B6/ c a Bottom section b Mid-height section c f v = 0.065f b (see Table 7)

12 H max,exp the experimentally obtained maximal value of lateral load, representing the shear resistance of the tested wall, R s,w-ft the shear resistance of the wall, calculated by taking into account the diagonal tension shear failure mechanism and mean values of the tensile strength, R s,w-fv the shear resistance of the wall, calculated by taking into account the sliding shear failure mechanism and mean values of the shear strength. 5 Shear modulus of masonry Mechanical characteristics of masonry at shear have predominant effect on the resistance and deformability of load-resisting elements of masonry structures. Eurocode 6 recommends that the shear modulus, G, of masonry be evaluated on the basis of the known modulus of elasticity, E, of masonry as follows: G ¼ 0:4E; ð10þ where the modulus of elasticity E is determined by either testing the walls according to EN [21] or using equations, based on the known compressive strength of units and mortar. However, the experiments indicate that, because of inelastic, nonhomogeneous and anisotropic characteristics of masonry, the actual relationships are quite different. The tests to determine the shear modulus G of masonry are not standardized. However, modulus G can be evaluated on the basis of lateral displacements, measured during the lateral resistance tests of wall specimens. In this, purely conventional procedure, the definition of the lateral stiffness of the wall, K, which is defined as the lateral load, H, causing unit displacement of the wall, is used: K ¼ H=d: ð11þ In the case of the wall, fixed at both ends and subjected to horizontal load, H, acting at the top, the displacement, d, at the top is due partly to bending and partly to shear: d ¼ Hh3 þ jhh ; ð12þ 12EI w GA w where I w = tl3 12 the moment of inertia of the wall s horizontal cross-section; j = 1.2 the shear coefficient for rectangular section. On the experimentally obtained resistance curve, the equivalent elastic stiffness of the wall (called also initial, or effective stiffness), K, is defined by the slope of a secant, connecting the origin with the point on the curve where the first cracks occur in the wall. If the modulus of elasticity of masonry E had been determined by compression tests according to EN , shear modulus G can be evaluated by simply introducing Eq. 11 into Eq. 12 and rearranging Eq 12: G ¼ K A w 1:2h a0 K E 2 ; h l ð13þ where a 0 is the coefficient of boundary restraints (a 0 = 0.83 for a fixed-ended and a = 3.33 for a cantilever wall). It has to be noted, that such definition of the shear modulus G is purely conventional. As the experiments indicate, the value slightly depends on the level of compressive stresses in the wall s section. Conventionally, shear modulus G is determined at the precompression level between 0.20 and 0.33 of the masonry s compressive strength. Experimentally obtained values of the shear modulus G and resulting ratio between the shear modulus G and modulus of elasticity E are given in Table 9. As can be seen, the actual values are within the range of 6 13% of the value of modulus of elasticity E. In no case the values close to 40% of E, as recommended by Eurocode 6, have been observed. It can be therefore concluded, that the use of Eurocode 6 recommended G/E ratio results into unrealistic distribution of seismic loads onto the shear walls. In order to avoid inadequate distribution, it is recommended that instead of Eurocode 6 proposed value G = 0.4E, either the values obtained Table 9 Correlation between the experimentally obtained and Eurocode 6 recommended values of the shear modulus of masonry G Unit Experimental Eurocode 6 E (MPa) G (MPa) G/E G = 0.4E a (MPa) B1 6, ,388 B2 7, ,757 B3 5, ,950 B4 6, ,680 B6 4, ,669 a E = 1,000 Kf b a f m b ; see Table 1 for f b and f m

13 by testing or the value G = 0.10E be considered in the calculations. 6 Verification of the seismic resistance of unreinforced masonry structures Various methods have been developed for the seismic resistance verification of masonry structures. In Slovenia, for example, a simplified non-linear, push-over type method for the seismic resistance verification of unreinforced masonry buildings named POR has been proposed after the earthquake of Friuli in 1976 [22, 23]. The original method has been improved and other methods of the same push-over type have been developed, like method SAM [12]. In all cases, the lateral resistance of individual shear walls is checked for different possible failure mechanisms, like the diagonal tension shear and flexural failure. The critical mechanism, yielding the lowest value of the lateral resistance of the wall, is taken into account in further analysis. Resistance curve of the critical storey is calculated on the basis of the idealised resistance curves of all resisting walls in the storey. The seismic resistance of the building is verified by comparing the calculated maximum resistance and ductility of the structure with the design seismic loads and ductility demand, required by the structural behaviour factor, taken into consideration for the determination of the design seismic loads. The results of such calculations have been verified by experiments and correlations with earthquake damage observations. According to the principles of Eurocodes, the following general relationship shall be satisfied for all structural elements and the structure as a whole: E d R d ; ð14þ where E d is the design action effect and R d is the design resistance capacity of a structural element under consideration. When considering a limit state of transformation of the structure into a mechanism, it should be verified that a mechanism does not occur unless the actions exceed their design values. In the case of the simplified non-linear methods, the requirement is verified for the structure as whole. In the case where the elastic structural models are used for the distribution of design action effects on individual elements, the resistance of the structure is verified by comparing the design resistance of each individual structural element with the corresponding design seismic action effect. In the following, the results of the seismic resistance verification of a typical threestorey confined masonry building, shown in Fig. 13, Fig. 13 Floor plan of masonry building, used for seismic resistance analysis

14 carried out by using this principle, will be discussed. In the analysis, a simple elastic structural model has been used for the distribution of the design seismic shear on individual shear walls. Storey mechanism of the seismic behaviour, i.e. the pier action of shear walls, fixed at both ends, has been assumed and the lateral stiffnesses of the walls have been calculated accordingly. The dimensions of structural walls, considered in the calculation (see Fig. 13), are given in Table 10. The values of the design compressive stresses in the wall s section, r d, have been taken from the actual analysis of the building under consideration. The values of the lateral stiffnesses of the walls, K, calculated by rearranging Eq. 13, are also given in Table 10: GA w K ¼ h 1:2h 1 þ a 0 G h 2 i: ð15þ E l Since the shear resistance, calculated on the basis of Eq. 7, depends on the compressed length of the wall s section, i.e. the lateral/vertical load ratio, the influence of G/E ratio on the distribution of the design base shear on the walls, and, hence, on the calculated shear resistance values, has been also analysed. Therefore, the lateral stiffness of the i-th wall, K i, has been calculated by considering either the experimentally obtained values of modules E and G (K i,test ), or the Eurocode 6 recommended G/E ratio (K i,ec6 ). It can be seen that, although quantitative values of individual stiffnesses differ significantly, the differences in distribution factors K i /RK i are not so great. Mechanical characteristics of masonry, taken into account in the calculations of the shear resistance and lateral stiffness of the walls, are given in Table 11. Walls type B1 have been considered. To determine the design values, partial material safety factor for masonry c M = 1.5 has been taken into account. In the case where the design shear resistance has been calculated on the basis of the sliding shear failure mechanism (R ds,w-fv ), the characteristic values of Table 10 Dimensions of walls, design compressive stresses and calculated values of lateral stiffnesses Wall no. l (m) t (m) h (m) r d (MPa) K i,test (kn/m) (K i /RK i ) test (%) K i,ec6 (kn/m) (K i /RK i ) EC6 (%) Note: K i,test, values of E and G obtained by testing: E = 6,826 MPa, G = 551 MPa; K i,ec6, values of E and G calculated according to Eurocode 6: E = 5,971 MPa, G = 0.4E = 2,388 MPa

15 Table 11 Mechanical characteristics of masonry, used in the calculations of seismic resistance (walls type B1, f b = 20.7 MPa, f m = 4.7 MPa) Quantity Test (MPa) Recommended by Eurocode 6 Equation Value Compressive strength f k 4.78 f k = Kf a b b f m 5.97 MPa Modulus of elasticity E 6,826 1,000 f k 5,971 MPa Shear strength f vk f vk = 0.20? 0.4 r d Calculated for each wall Tensile strength f tk 0.19 Shear modulus G 551 G = 0.4E 2,388 MPa mechanical properties of masonry have been calculated on the basis of the known strength characteristics of masonry units and mortar using equations given in Eurocode 6. For the distribution of design seismic loads, lateral stiffnesses K i,test and K i,ec6 have been taken into account. In the case where the design shear resistance of individual walls has been calculated on the basis of diagonal tension shear failure mechanism (R ds,w-ft ), experimentally obtained characteristic values of mechanical properties of masonry have been considered. For the distribution of design seismic loads, lateral stiffnesses of individual walls K i,test have been taken into account. The analysis has been carried out for the x-direction of the building. According to the requirements of Eurocode 6, the walls perpendicular to the direction of seismic action have not been considered. Design seismic loads have been determined in accordance with the requirements of Eurocode 8 [24], following the response spectrum approach, where the design spectral value is calculated by: Sa g 2:5 S d ðtþ ¼c I ; ð16þ q and the design base shear by: F Bd ¼ S d ðtþw; ð17þ where S d (T) the design spectrum value; in the specific case considered, S d (T) = g; c I the importance factor; c I = 1.0 for residential buildings; a g the design ground acceleration; in the specific case considered, a g = 0.15 g; S the soil type coefficient; in the specific case considered, S = 1.2 for soil type B; 2.5 the spectral amplification factor assumed to be constant in the range of typical natural periods of vibration, T, of masonry buildings; q the structural behavior factor; q = 2.0 for confined masonry structures; F Bd the design base shear, and W the weight of the building above the analysed section. Assuming that the weight of the building above the analysed section is W = MN (the value has been taken from actual seismic analysis of the building under consideration), the design seismic base shear attains the value of F Bd = 2.89 MN. The design seismic base shear has been distributed on the structural shear walls in proportion with their stiffnesses: F Bd;i ¼ K i P Ki F Bd : ð18þ In the case where the design shear resistance of the walls has been calculated on the basis of the sliding shear failure mechanism (R ds,w-fv ), the compressed part of the wall s length and the resulting shear strength values have been determined on the basis of the calculated relationship between the corresponding part of the design base shear F Bd,i and design vertical load V d,i = r d,i A w,i, acting on the i-th wall. In the case where the eccentricity of vertical load would theoretically cause the overturning of the wall (compressed part of the wall s length resulted negative), the wall has not been considered as lateral load resisting element. The design seismic shear was redistributed to remaining walls and the calculation repeated. The results of calculations are given in Table 12.It can be seen that, although the distribution factors K i /RK i did not differ significantly, the differences between the experimentally obtained and Eurocode 6 recommended G/E ratios influenced the lateral/vertical load ratio, and, consequently, the design shear resistance of the walls, calculated in accordance with Eurocode 6. Consequently, the verification of the shear resistance of individual walls according to rule (14) may lead to different conclusions, depending on the data used for the calculation of the lateral stiffness of the walls. Although not all walls in the story comply with the requirement (14), a conclusion can be made that the

16 Table 12 Design seismic shear acting on individual walls, F Bdi, and design shear resistance of structural walls, calculated on the basis of the sliding shear, R ds,wi-fv, and diagonal tension shear failure mechanism, R ds,wi-ft Wall no. Sliding shear mechanism Eurocode 6 Diagonal tension failure Distribution by K i-test Distribution by K i-ec6 Distribution by K i-test F Bdi (kn) R ds,wi-fv (kn) F Bdi (kn) R ds,wi-fv (kn) F Bdi (kn) R ds,wi-ft (kn) R (kn) seismic resistance of the building under consideration, assessed as proposed by Eurocode 6, is adequate. Namely, the sum of the design shear resistances of all walls in the storey, which can be used as an indicator of the seismic resistance of the building, is greater than the design base shear. This, however, is not the case if the design resistance of the walls is determined by taking into account the diagonal tension shear failure mechanism (R ds,w-ft ). In the latter case, the sum of the design resistances of all walls in the storey does not attain the required value of the design base shear. By comparing the values, given in Table 12, it can be seen that for all walls in the storey, except where the overturning is theoretically expected, the resistance of the walls to diagonal tension is smaller than the resistance to sliding shear. Generally speaking, the differences are not as great as those obtained by correlating the calculations with the results of tests of individual walls (Table 8). However, they are significant. In the particular case studied, the ratio between the sliding shear and diagonal tension shear based calculated lateral resistances of individual walls exceeds 1.5. Moreover, if calculated in accordance with Eurocode 6, the shear resistance of the same wall in different seismic situations does not remain the same. Namely, if the design seismic shear, acting on the wall, changes, the lateral/vertical load ratio, hence the compressed part of the wall s length, and, consequently, the design shear resistance also change. To assess the possible differences, the seismic resistance of the same building has been verified for varying seismic loads. The results of this analysis are presented in Table 13, where again the sum of resistances of all walls in the storey is considered as an indicator of the seismic resistance of the building under consideration. As can be seen, significantly different values are obtained for the same