SIMPLIFIED METHODS FOR ASSESSMENT OF MASONRY SHEAR-WALLS

Pere ROCA Professor Universitat Politècnica de Catalunya Barcelona Spain

ABSTRACT The paper presents a discussion on the possibility of using simple load-path or strut and tie models, based on equilibrium, to estimate the ultimate capacity of masonry shear-walls. Tentative rules and specific models are presented for elementary solid walls subjected to different load conditions. The performance of the suggested models is analysed by comparing their predictions with available experimental results. Tests carried-out on dry-joint and mortar-joint, true-scale or scaled laboratory walls are considered for this purpose. Some remarks are presented on the applicability of the simplified models. 1. INTRODUCTION Modelling the ultimate condition of reinforced concrete components by means of strut and tie models, where the struts describe the compression stress fields and ties represent the reinforcing bars, constitutes a well acknowledged approach accepted by modern concrete codes and commonly used for practical design and assessment. A comprehensive description of the technique and possibilities, in the case of reinforced concrete, can be found in [1]. The possibility of using a similar approach for the study of the ultimate response of plain masonry walls has not yet deserved much attention. Certainly, significant difficulties or limitations can be envisaged due to the fact that the principles which permit the use of the strut

102 SÍSMICA 2004 - 6º Congresso Nacional de Sismologia e Engenharia Sísmica and tie models in reinforced concrete may not be entirely applicable to a material of brittle nature such as un-reinforced clay brick or stone masonry. A specific difficulty comes from the lack, in this case, of identifiable, discrete tension-carrying components showing plastic behaviour, such as the reinforcing bars. However, the use of simple equilibrium methods is not completely unexplored. The working condition of a shear wall at maximum loading has been, in some occasions, described with the help of simple schemes involving a set or a continuum of diagonal struts. Ganz and Thürlimann [2] envisaged parallel or fan compression field models to estimate the ultimate load resisted by confined shear walls. De Tomasi et al. [3] have recently proposed an approach which considers the need for ties, in combination with the compression struts, to account for the deviation of the compression fields in masonry façades. A more specific proposal, oriented to the analysis of elementary shear walls subjected to a combination of vertical and horizontal forces, is described in this paper (sections 2 and 3). The ability of the proposed approach to estimate the ultimate capacity of masonry shear-walls has been analysed by comparing their predictions with the experimental results obtained for different walls tested in the laboratory, as described in section 4. These series include full scale dry-joint walls and one-forth-scaled cohesive walls subjected to different loading conditions. It must be remarked that the approach presented here constitutes only a first proposal still requiring further assessment. A more comprehensive calibration, based on both additional experimental evidence and numerical simulation by means of up-to-date computer methods, is currently being undertaken. Even if further calibration is carried out, the simple nature of the models prevent them from describing significant phenomena observed in plain stone or brick masonry, such as anisotropy, dilantancy, contact effects, or the acknowledged influence of the size and geometry of the blocks. Because of these possible limitations, the models proposed are only to provide an auxiliary tool intended for first-approach calculations. They can not be used for problems where the deformation of the wall is significant for the determination of the response; more specifically, the models can not be used to determine the strength contribution of wall panels confined in concrete or steel frames and subjected to imposed lateral displacements. 2. FEATURES OF ELEMENTARY EQUILIBRIUM MODELS 2.1 Proposed features Given the very limited tensile strength of the material, the ultimate capacity of shear walls can only be explained by the generation of diagonal fields of compression stresses in equilibrium with the external loads. It must be recognised, however, that the geometry of the wall and the particular loading conditions may not cause pure uniform diagonal fields but more disturbed ones experiencing significant deviations within the wall (Fig. 1,a); this occurs, in particular, in the case of a elementary solid wall subjected to vertical and horizontal loading. The deviation of the main compression stress fields produce inward or outward resulting thrusts which, in

Pere ROCA 103 turn, are balanced with complementary tension or compression internal forces. Where no confinement exists, the deviation of a compression stress field is only possible if a horizontal tensile force can be developed within the fabric. Observed damage in walls tested to failure under a combination of vertical and horizontal load seems compatible with such understanding of the internal distribution of forces (see section 4).

Figure 1: Features of models: (a) deviation of compression stress fields by horizontal tensile forces; (b) parallel distributed struts; (c) reverse “bottle neck” struts combined with ties; (d)

limitation of the maximum angle of the strut with respect to the vertical. The above considerations have been taken into account to envisage possible rules for the construction of the models and to propose specific simple models. In some cases, it is possible to conceive models which consist only of struts. In other cases, ties are necessary to explain the equilibrium or to improve the consistency of the model with the experimental evidence. Based on these ideas, the following rules are tentatively proposed:

≤α

tanα=tanϕ+c/σ

(a) (b)

(c) (d)

104 SÍSMICA 2004 - 6º Congresso Nacional de Sismologia e Engenharia Sísmica (1) The models must be as simple as possible to provide practical and efficient approaches.

The number of elements (struts, ties) is to be limited to the minimum amount required to obtain an acceptable description of the ultimate mechanism.

(2) The struts used in a model describe compression fields covering a certain volume. Because

of that, the effect of a distributed force on a wall is to be modelled as a distributed strut (Fig. 1, b) or, alternatively, as a set of discrete struts. A minimum amount of two struts should be used, in any case, to represent the effect of a distributed load on a wall.

(3) Concentrated or partial loads –as well as reactions- will cause compression fields to

experience a reverse “bottle-neck” effect which must be described by means of a mechanism combining a minimum of two opening struts with a balancing tie (Fig. 1,c).

(4) The maximum slope of a strut with respect to the vertical is limited by the frictional

response of the joints. If the Mohr-Coulomb criteria is adopted to describe the maximum shear force that can be transferred by the horizontal joints, the slope of the struts with respect to the vertical is limited to tanφ, where φ is the friction angle of the unit-mortar interface, in the case of a dry-joint wall. In a cohesive wall, the slope of struts is limited to tanα,

(1) where c is the cohesion and σn is the average vertical compression (Fig. 1,c). (5) Ties can only be admitted in the horizontal direction given the very low tensile strength of

the unit-mortar interfaces. The tensile force experienced by a tie is resisted thanks to the combined contribution of friction in joints and tensile strength of the units. The maximum tensile force T carried by a tie is estimated through the following two conditions:

(2) (3)

where Vi can be taken as the minimum of the vertical forces carried by the two struts linked to the tie, Ab is the sectional area of unit courses which are contributing to resist the tensional force and σbt is the average increment of tensile stress which can be resisted by the units.

(6) Two different types of nodes (or connections between linear elements) can be identified.

The first one is consists of the connection between two struts and a tie, in which a tensile internal force (T) is anchored by a deviating compression stress field. In that case, the following condition, also related to the transference of shear forces through the joints, must be satisfied:

tan tann

cα φ

σ= +

αtaniVT ≤

btbAT σ≤

Pere ROCA 105 (4)

where V1 is the vertical force carried by the strut. (7) The second type of node corresponds to the region where one or more compression forces

converge with a reaction. This type of node is represented as a finite region whose minimum dimensions are determined by the compression strength of the fabric.

(8) Last, but not least, the mechanisms should be consistent with the evidence obtained from

experiments and micro-modelling, in terms of distribution of stresses, cracking and other observable or measurable aspects. However, the way mechanisms are to be related to the observed cracking and distribution of stresses is not obvious.

Section 3 introduces a set of possible models stemming from the above rules. In these models, both the vertical and the horizontal load are assumed to be applied at the upper edge of the wall while the self-weight is considered negligible. 2.2 Discussion The afore mentioned features should not be regarded as a definite set of rules for the construction of mechanisms, but as a set of initial criteria still requiring refinement based on additional research. Significant difficulties may be foreseen in any attempt to apply them in a general manner. First, rule (4) does not yield complete objectivity since the slope of the struts may depend on the refinement of the models. Rule (4) is intended for struts representing the main compression fields. Note that rules (4), (5) -equation(2)- and (6) are in fact equivalent and simultaneously satisfied in the case of a strut with slope equal to tanα connected to a vertical strut and a horizontal tie. Equation (3) requires the determination of the area of the transverse section Ab, that is to say, of the number of unit courses contributing to resist the tensile force corresponding to the tie. It is suggested, as a first approach, to define Ab as the largest available area centred in the tie, of which, due to the vertical joints, only one half of the unit courses contribute simultaneously to carry the tensile force. Note that equation (2) is intended to ensure the friction needed to tie the courses transversally. The units experience complex stress states caused by the shear and vertical compression forces applied to them, to which their role as components of a tie contributes to additional tensile stress. If the Rankine criterion is adopted to characterize the strength response of the material of the unit (meaning that the first principal stress is limited to the tensile strength of the material fbt) the maximum acceptable stress σbt can be estimated from: (5)

αtan1VT ≤

( )22 4)(21

τσσσσ ++−= nbtnbtbtf

106 SÍSMICA 2004 - 6º Congresso Nacional de Sismologia e Engenharia Sísmica where σn and τ, are the average compression and shear acting on their faces. σn, and τ can be roughly estimated from the forces supported by the struts which are anchoring the tie. The difficulty in applying rule (5) lies in the determination of the compressive strength to be considered in the analysis of the compression node. The effective compression strength may be strongly influenced by the angle of the resulting compression forces with respect to the direction of the courses, due to anisotropy, and by the biaxial stress states produced by the converging struts. Such effects may be particularly significant in walls with small h/b ratio, or walls made with perforated or hollow bricks. 2.3 Theoretical background The proposal finds its theoretical justification in the lower bound theorem of plasticity. This implies that, if an equilibrated mechanism if found, the estimated ultimate load is a lower bound of the real load causing failure. The possibility of applying the bound theorems of plasticity stems from the fact that two physical mechanisms determining the ultimate condition in the models, namely, friction in joints and yielding of the fabric in compression, can be considered plastic. However, this not applicable in the case of walls for which the ultimate load is determined by cracking in tension. Nor is it applicable to walls for which crushing of the fabric in compression appears prior to the full development of the mechanism. In other words, the mechanism should be chosen in a way that (a) the deformation limit of the fabric in compression is not exceeded at any point and (b) the tensile forces due to the ties can be resisted by the fabric. An acceptable ultimate mechanism must be determined by the condition of maximum friction in joints (i.e., maximum slope of struts) or maximum compression in nodes. Rule (3), related to cracking of units, consists, in fact, of a requirement for the applicability of the mechanism. If rule (3) is not satisfied, cracking of units may prevent the full formation of the envisaged mechanism, meaning that a different mechanism –such as a residual one, see section 3.4- is to be considered. However, partial cracking may be compatible with this requirement. In any case, severe diagonal cracking is always expectable after the attainment of the peak loading due to secondary effects caused by the sliding of the units or by intense compression stresses close to loads and reactions. According to the lower-bound theorem, if two acceptable models are found, the more realistic of the two is the one providing the upper ultimate load. When different mechanisms can be found, all based on the aforementioned rules, the one providing the upper estimation is likely to be the more realistic one. 3. SUGGESTED MODELS FOR ELEMENTARY SOLID SHEAR WALLS 3.1 Walls subjected to partial or concentrate loading Several models, all representing reverse bottle-necked compression fields, and corresponding to different degrees of refinement, are envisaged for the case of walls subjected to partial or concentrated vertical load combined with horizontal load (Fig. 3).

Pere ROCA 107 The model indicated in Fig. 3,d is preferred because of its simplicity; compared with the, also simple, model of Fig. 3,c, it yields an upper estimation of the ultimate load. Mechanism of Fig. (3,b) leads to the following calculation of the ultimate horizontal load H:

(6) (7)

(8) where V is the total applied vertical load, b,h and t are the width, height and thickness of the wall and fc is the compression strength of the masonry; d is the exceeding dimension defined in Fig 2,c; m is de width of the minimum contact surface required to transfer the vertical load in plastic regime; tanα is defined in equation (1). Equation (6) refers the case of a failure initiated by the sliding of the units across the horizontal joints, while equation (7) corresponds to a failure caused by the yielding of the material in compression; in the first case, a stepped diagonal crack associated to the sliding of units along the joints is to be expected; in the second case, cracking of units due the tie action may be expected to initiate in the centre of the wall (where tension stresses are maximum) and then to develop toward the corners of the wall, to encounter additional cracks caused by the failure of the material in compression.

Figure 2: Models proposed for walls subjected to concentrated or partial vertical loads.

αtan2 hmdbVH −−

=

αtan32

1 VH =

{ }21 ,min HHH =

ctfVm =

a ≥ m=V/(tfc)

H

V

a

H

V

a

≤α

e

H

V

a

≤α

g g

d

b

g=(b-d)/4 g g

(a) (b) (c)

108 SÍSMICA 2004 - 6º Congresso Nacional de Sismologia e Engenharia Sísmica As mentioned in section 1, the models can not be used to determine the strength contribution of wall panels confined in concrete or steel frames subjected imposed lateral displacements, meaning that no possibility exists, based only in equilibrium, to determine the mobilised couple of V,H forces. However, similar models to those above described may be used to determine the relationship between the two forces. Models of Fig. 3, producing the same equations than those previously described, are proposed to determine this relationship in walls confined in the vertical direction. Model (b) shows satisfactory correspondence with the cracking scheme observed in real experiments (Fig. 11). 3.2 Walls subjected to vertical uniform loading The case of walls resisting vertical uniformly distributed loads requires the consideration of smeared struts arranged according to a parallel or fan distribution. Different envisaged models are described in Fig. 3. In this figure, thick solid lines are used to represent the struts, horizontal thin lines are used to represent the ties, and additional thin, discontinuous lines are included to indicate the stress fields associated to the struts. Note that in all the models the horizontal load can not be uniformly distributed along the upper edge of the wall. Model (a) of Fig. 3 -the “fan” model- recognizes the fact that the slope of the load paths developed along the wall must vary gradually in order to become compatible with the geometry of the wall. The width of the bottom distributed node is estimated as m= V/(t fc), where V is the total vertical load applied. Model (a) does not include a limitation on the maximum slope of the struts. Model (b) consists of a modified fan model which complies with the limitation of the maximum angle of the struts. In the case of narrow walls, or very compressed walls, with

(9) it becomes equivalent to the fan model. Models (c) and (d), Fig. 3, are simple variations of model (b), although more consistent with the type of failures observed in the experiments; they are acceptable if the fabric is resisting the tensile force carried by the tie at maximum loading. Note that models (c) and (d) predict a concentration of compression stresses close to the lateral edge (region A in Fig. 3, see also Figs. 7 and 9). Alternative and more continuous models, not causing such concentrations of compression stresses have been also considered; however, they produce lower estimations of the ultimate load. Using models (b), (c) or (d), the ultimate horizontal load is estimated as

if (10) h

mbVH2−

=

)tan(αhbm −≥ctf

Vm =

)tan(αhbm −≥

Pere ROCA 109

if (11) where (12)

Figure 3: Models proposed for walls subjected to uniform vertical load.

−

−=vb

hVH1

1tan2

1tan αα )tan(αhbm −<

ctbfV

bmv ==

h/2

fc fc

α

(a) (b)

(c) (d)

fc

α

fc

a a b-2a

αα

α

A

110 SÍSMICA 2004 - 6º Congresso Nacional de Sismologia e Engenharia Sísmica 3.3 Walls with openings Similar mechanisms can be considered for more complex walls or façades with openings or other geometric alterations. Fig. 4 shows an example of a mechanism describing the internal forces experienced by a wall with a central opening; in this example, almost no vertical compression exists close to the top and bottom edges of the opening, meaning that such a mechanism is only expectable in very cohesive walls; if the ties can not be mobilised, a more weak, marginal mechanisms similar to those described in section 3.4 is to be considered.

Figure 4: Primary mechanism for a Wall with opening. 3.4 Residual mechanisms As mentioned in section 2.3, cracking of units may prevent the full development of mechanisms involving horizontal ties. Whenever cracking of units is predicted (for instance, using equation (5) or another similar criteria), an alternative, secondary or residual mechanism, is to be considered. This is illustrated by the example of Fig. 5, consisting of a wall subjected to concentrated vertical load; if the main tie can no be mobilised –meaning that a diagonal crack “C” (Fig.5,a), developed through the units, is expected- alternative combinations of mechanisms such as those shown in Fig. 5,b-c must be considered. In case of direct loading, the combination of mechanisms A and B may be foreseen to describe the possible response of the cracked wall; however, mechanism B is not likely to be mobilised due to the deterioration of the region close to the wall, and only mechanism A may contribute to resist part of the load. In the case of a wall confined in the vertical direction, subjected to an imposed horizontal displacement, a secondary resisting scheme, consisting of the combination of mechanisms A and B’ (Fig. 5,c) may be envisaged. Mechanisms A and B’ maintain in this case the central symmetry of the problem.

≤α

≤α

Pere ROCA 111

Figure 5: Primary (a) and residual mechanisms of vertically free (b) and confined (c) walls.

4. COMPARISON WITH EXPERIMENTAL RESULTS 4.1 Summary of considered experimental series The predictions of the above models are compared with experimental results obtained for three different series of walls tested in the Laboratory of Technology of Structures of the Technical University of Catalonia. The geometry of the walls, load conditions and properties significant for the presented models are summarized in table 1.

Table 1: Summary of features of experimental wall series Series (n)

Joint Unit

b cm

h cm

t cm

fc N/mm2

tan φ c N/mm2

Vertical loading

1(7) 2(15) 3(20)

dry mortar mortar

Sandstone Clay brick Clay brick

100 30 30

100 25 25

20 3,5 3,5

30 15 12

0.66 0.8 0.8

0 0.22 0.20

uniform uniform partial

(n) = number of walls included in the series The dry-joint walls of series 1 were built with sandstone blocks with dimensions 20×20×10 cm (Fig. 6). The values considered for tanφ and fc (see in table 1) where determined in the laboratory with complementary experiments on units of the same type of sandstone and having the same dimensions than those used to build the walls; in particular, the value taken for fc was obtained by testing a specimen consisting of 4 staked blocs. Additional experiments showed that, in the case of the dry-joint walls tested, the value of fc, thus obtained, was significantly dependent upon the size of the blocks (Roca et al. [4]).

C

A

BB’

A

(a) (b) (c)

112 SÍSMICA 2004 - 6º Congresso Nacional de Sismologia e Engenharia Sísmica Walls of series 2 and 3 were made of on-purpose manufactured small solid clay bricks with dimensions 72,5×35×12.5 mm and 2.5 mm thick micro-mortar bed joints with measured uniaxial compression strength of 13.2 N/mm2 for series 2 and 8.6 N/mm2 for series 3. In all cases, the average compressive strength of the masonry fc, the cohesion c and the angle of friction φ where measured by testing specimens made of bricks with the same dimensions than those composing of the walls. The angle of friction was measured using the simple method described in [5]; it is intended to improve the measurement of the same parameter by using Van der Pluijm’s [6] more accurate procedure. 4.2 Comparison with experiments on dry joint walls with uniform vertical loading The experiments carried out on dry-joint sandstone unit walls by Oliveira [7] (series 1 of table 1, see also Roca et al. [4]) have been considered to carry out a first comparison with the predictions obtained by the proposed simple models introduced in section 3. These experiments have been already considered by Orduña and Lourenço [8] to validate a proposed cap model for limit analysis of masonry constructions.

Figure 6: Geometry of walls and loading system (series 1).

The main steps of the testing procedure consisted of the application of a vertical load by means of a hydraulic actuator kept under force control, resulting in a constant vertical load; and the gradual application of the horizontal load by small displacement increments by means of a second hydraulic actuator. The load was applied against the reinforced concrete beam built over the stone walls. The vertical actuator was provided with a triaxial hinge at each end. The damaged condition of two of the walls after the attainment of maximum horizontal load is presented in Fig. 7.

20

10

100

100

[cm]Reaction Slab

20

1010

10

Reinforced Concrete Beam

VerticalLoad

HorizontalLoad

Pere ROCA 113

Figure 7: Walls from series 1 after the experiment.

Fig. 8 shows the comparison between the experimental results and numerical predictions obtained from equations (10-11). Al the walls tested verified the condition corresponding to equation (11), meaning that in all cases the failure was due to the sliding of the joints.

Figure 8: Series 1. Comparison between experimental and predicted ultimate loading. 4.3 Comparison with experimental cohesive walls subjected to uniform vertical loading The experiments considered here where tested using the same procedure above described for the dry-joint walls. In the case of series 2, consisting of smaller, one-forth scaled walls, the tests were carried out in a press machine combined with a horizontal actuator. The damaged condition of two walls subjected to different vertical loads, after the attainment of maximum horizontal loading, is shown in Fig. 9.

0

20

40

60

80

100

120

0 50 100 150 200 250 300

Vertical load (kN)

Hor

izon

tal l

oad

(kN

)

experimental

simple model

cap model [8]

114 SÍSMICA 2004 - 6º Congresso Nacional de Sismologia e Engenharia Sísmica

Figure 9: Walls from series 2 after the experiment. Fig. 10 shows the comparison between the experimental load combinations leading to failure and the corresponding predictions of the simple models. Equations (10-11) corresponding to any of the models (b-c-d) of Fig. 4 are used.

Figure 10: Series 2. Comparison between experimental and predicted ultimate loading. 4.4 Comparison with experimental cohesive walls subjected to partial vertical loading In the case of series 3 the test was similar to those described in the above sections except for constraining the rotation of the top stiff beam. Consequently, applying horizontal load caused the vertical load to acquire a certain eccentricity which, in the ultimate condition, can be assumed to be the maximum possible one.

0

5

10

15

20

25

0 20 40 60 80 100 120 140

Vertical load (kN)

Hor

izon

tal L

oad

(kN

)

ExperimentalSimple modelSerie1

Pere ROCA 115 The damaged condition of two walls subjected to a different vertical load, after the attainment of maximum horizontal loading is presented in Fig. 12. Due to the expectable partialization of the contact length between the wall and the top and bottom elements (plate, top beam) which confine it the walls in this series are treated as walls subjected to concentrate or partial vertical load, according to section 3.1, with maximum compatible eccentricity. The ultimate horizontal force H is estimated with equations (6-8). Fig. 13 shows the comparison between the experimental vertical-horizontal load combinations leading to failure and the corresponding predictions of the simple models. The obtained predictions agree satisfactory both in the region determined by the sliding of joints (equation 7) and that governed by the yielding of the fabric in compression (equation 8).

Figure 11: Walls from series 3 after the experiment.

Figure 12: Series 3. Comparison between experimental and predicted ultimate loading.

0

5

10

1520

25

30

35

40

0 20 40 60 80 100 120 140

Vertical load (kN)

Hor

izon

tal l

oad

(kN

)

Simple modelExperimental

116 SÍSMICA 2004 - 6º Congresso Nacional de Sismologia e Engenharia Sísmica 5. FINAL REMARKS The possibility of using simple equilibrium models for the assessment of the in-plane ultimate response of shear masonry walls, such as those resulting from load-path or strut and tie combinations, has not yet deserved significantly attention. Certainly, the principles which encourage their application to other structural materials –such as reinforced concrete- are not so neatly present in the case of plain masonry shear walls. Aiming to contribute to the possible use of simple equilibrium models for the assessment of shear walls, a tentative set of rules for the construction of ultimate mechanisms has been proposed here. Additionally, some particular models have been suggested as a first solution for elementary cases of solid walls subjected to combined vertical and horizontal external loads. The predictions obtained using these models agreed acceptably with some available experimental measurements. However, such proposals are not the only possible, and the opportunity to envisage more realistic and accurate solutions remains fully open. The study carried-out suggests that the use simple models for first-approach calculations may be feasible. Such models may show a certain ability to predict ultimate loading conditions, in spite of neglecting important phenomena related to the real observed in-plane mechanical response of plain masonry walls. A further development of the concepts discussed should be based on significant additional experimental evidence combined with systematic numerical simulation. In particular, detailed micro-modelling is regarded as a very promising means to progress in the development of a more consistent technique of analysis based on simple equilibrium models. 6. ACKNOWLEDGEMENTS The experimental studies presented here were developed within the research project ARQ2002-04659, funded by DGE of the Spanish Ministry of Science and Technology, whose assistance is gratefully acknowledged. 7. REFERENCES [1] Schlaich, J., Schäfer, K., Jennewein M. - “Toward a consistent design of structural

concrete”. PCI Journal, Vol 32, Nº 3, 1987, p.72-150. [2] Ganz, H. R., Thürlimann, B. “Strength of brick walls under normal force and shear”. Proc.

8th Int. Symposium on load bearing brickwork, London, 1983, p 27-29. [3] De Tommasi, G., Monaco, P., Vitone, C. - “A first approach to the load-path method on

masonry structure behaviour”. Structural studies, repair and maintenance of heritage architecture VIII, WITpress, Southampton, 2003, p. 287-296.

Pere ROCA 117 [4] Roca, P., Oliveira, D., Lourenço, P., Carol, I. - “Mechanical response of dry joint

masonry”. Studies in Ancient Structures, Yildiz Teknik Universitesi, Istanbul, 2001, p. 291-300.

[5] Ghazali, M. Z., Riddington, J. R. – “Simple test method for masonry shear strength”. Proc. Instn. Civ. Engrs, Part2, Nº 85, 1988, p. 567-574.

[6] Pluijm, R., Van Der – “Shear behaviour of bed joints”. Proc. 6th North American Masonry Conf, Drexel University, Philadelphia, Pennsylvania, USA, 1993, p. 236-136.

[7] Oliveira, D. V. – Mechanical characterization of stone and brick masonry. Rep. No. 00-DEC/E-4, Univ. do Minho, Guimaraes, Portugal, 2000.

[8] Orduña, A., Lourenço, P. – “Cap model for limit analysis and strengthening of masonry structures”. Journal of Structural Engineering, V. 129 Nº 10, 2003, p.1367-1375.

118 SÍSMICA 2004 - 6º Congresso Nacional de Sismologia e Engenharia Sísmica