Influence of vertical tie columns on bearing capacity of masonry walls

Numerical test results, defining influence of vertical tie columns on the performance of masonry walls subjected to static and dynamic load, are presented in the paper. The analysis focuses on two-storey walls without openings and with openings, with good and poor quality masonry, and with different boundary conditions at the contact between wall foundations and the subsoil. The influence of the profile of longitudinal bars of vertical tie beams on the performance of walls under horizontal static load, and harmonic and seismic acceleration of subsoil, is studied. The numerical model for static and dynamic analysis of in-plane masonry structures, previously developed by the authors, is used in the analysis.

GRAĐEVINAR 64 (2012) 4, 271-284 Marija Smilović, Jure Radnić, Alen Harapin equivalent material of representative mechanical properties.In the micro model of the masonry, the modelling at the level of masonry units and mortar (joints) is possible, as well as the simulation of connection of mortar and masonry units using contact elements.The isotropic model of masonry can be used, and the use can also be made of the anisotropic model of masonry with different strengths (compressive, tensile, shear), elasticity modulus, shear modulus, and limit strain of masonry in horizontal and vertical directions.The yield and failure of masonry in compression, tensile and shear stiffness of masonry with cracks, and shear failure of masonry, was modelled.The model of fixed orthogonal smeared cracks was used.The concrete behaviour is simulated with an isotropic material model.The yielding and crushing of concrete in compression, the opening and closing of cracks, as well as the tensile and shear stiffness of cracking concrete, were modelled.The fixed orthogonal crack model, with crack direction corresponding to the direction of principal tensile stresses, was also adopted.The effect of strain rate on mechanical properties of concrete and steel can be simulated through dynamic analyses.Nonlinear effects of reinforcement in compression and tension, with unloading effects, can be simulated.Material models for masonry or concrete can be applied for soil, based on appropriate material parameters.The geometric nonlinearity of the structure (large displacements) can be simulated.

General
The behaviour of masonry walls under load depends inter alia on their geometry (height, width, height and width ratio, the number and location of openings, connection with walls facing the opposite direction, the number and arrangement of ring beams, etc.).Simple geometry of walls was used to analyze the effect of vertical ring beams on the behaviour of masonry walls.Two-storey independent real walls without and with openings were analyzed.The walls are 6 m high, 3 m wide, and 0.24 m thick (Figure 1).All the walls have the same concrete foundation, which is supported by a rigid base.At the floor levels, all the walls were loaded with a constant continuous vertical load q.The weight of the walls, ring beams, and foundation, was directly included in the numerical model.A macro model of the masonry with isotropic material properties was used.The cases of the so-called good masonry (masonry of high strength, elasticity modulus and shear modulus) and the so-called poor masonry (masonry of low strength, elasticity modulus and shear modulus) were analyzed.
In the static analysis, the wall was loaded with a constant continuous vertical load, and with variable horizontal forces at floor levels.In the dynamic analysis, the horizontal harmonic base acceleration was first applied for all walls.The excitation

Introduction
Vertical and horizontal ring beams greatly influence the behaviour and strength of masonry walls under vertical load and especially under horizontal load.Their role is particularly important in cases when masonry structures are subjected to earthquake action.First of all, ring beams connect and stiffen the masonry.They contribute significantly to the strength capacity of masonry structures subjected to compression, bending and shear, both for loads in the wall plane, and loads perpendicular to the wall plane.Ring beams reduce the extent of deformation to masonry.In horizontal activities, ring beams allow formation of diagonal compression in masonry.Vertical ring beams dominantly transfer tensile stresses in masonry.They allow activation of concrete foundations when tension occurs at the wall -foundation interface.Horizontal ring beams redistribute vertical load on the masonry, especially in case of concentrated forces.Knowledge about the effect of ring beams in masonry structures is mostly qualitative.Although numerous experimental and numerical studies of masonry walls under static and dynamic load have been made (some can be found in [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20]), studies that quantitatively evaluate the effects of various parameters of vertical and horizontal ring beams on strength capacity and deformability of masonry walls were not available to the authors of this paper.Using the numerical model for static and dynamic analysis of masonry structures previously developed by the authors [21,22], this paper investigates the effect of several parameters of vertical ring beams on the behaviour and limit strength of masonry walls.Analyses were performed separately for static and dynamic (seismic) loads.Numerical tests were conducted for a simple wall geometry.Two-storey walls without openings and with openings were considered, with good and poor quality of masonry, and with different boundary conditions at the foundation -base interface.The effect of the profile of longitudinal bars of vertical ring beams on wall behaviour was analyzed.Finally, the most important conclusions of the research were presented.

Brief description of numerical model
A detailed description of the numerical model adopted for static and dynamic analysis of masonry structures can be found in [21,22], and will be only briefly described hereinafter.The model is intended for simulation of practical planar masonry structures loaded in their plane.The structure can be built of masonry and/or reinforced concrete, and geometry of the spatial structure can also contain subsoil.A simulation of all main nonlinear effects of the behaviour of the masonry, reinforced concrete, and soil, is possible.
A macro or micro model of masonry can be used.In the macro model of the masonry, the complex behaviour of the masonry (masonry units connected by mortar) is modelled by the GRAĐEVINAR 64 (2012) 4, 271-284 rigid surface were analyzed: (i) possible lifting and sliding of foundations, and (ii) prevented lifting and sliding of foundations.

Possible lifting and sliding of foundations
This case corresponds to numerous states of real masonry walls where possible lifting and sliding of foundations have not been prevented.Basic material parameters adopted in numerical analysis are presented in Table 1.Spatial discretisation of the walls is shown in Figure 2. A relatively coarse finite element mesh was adopted.At the foundation-soil interface, thin contact element connection of the foundations and the subsoil, thin contact elements were used for an adequate simulation of the lifting and sliding of foundations.

Static analysis
Firstly, the initial state of displacement, stresses and internal forces for the own weight and vertical load q was calculated.
Influence of vertical tie columns on bearing capacity of masonry walls period corresponded to the first period of free elastic oscillations of the corresponding wall.The resonant harmonic base acceleration was adopted in order to achieve higher levels of nonlinearity.It clearly illustrates the difference between the results obtained by the usual linear model and the adopted nonlinear material model [21,22].A dynamic analysis of all the walls was also performed for the Kobe earthquake.
The geometry of the analyzed masonry walls, with reinforcement of ring beams and foundations, is shown in Figure 1.After that, the walls were additionally loaded with horizontal forces H 1 = H 2 .The load was applied in successive increments until failure.
In addition to numerous other parameters, the strength and deformability of the walls loaded with horizontal load are highly dependent on their vertical load.To illustrate this effect, only the walls FW 2g and OW 2g (g = good masonry) with different vertical loads at the floor level (q): q 1 = 0 kN/m (wall self-weight only), q 2 = 40 kN/m (wall subjected to medium vertical load), and q 3 = 80 kN/m (wall subjected to stronger vertical load), were initially analyzed.The limit strength capacity of the walls depends on the loss of their stability (collapse) as a rigid body.Horizontal wall-top displacement is shown in Figure 3, while reinforcement stresses at points A and B are shown in Figure 4.As expected, the numerical results show that the walls can withstand small horizontal forces for small vertical loads, because the load quickly leads to the loss of the stability of the wall as a rigid body.In case of large vertical loads, walls are able to withstand greater horizontal forces.Tensile stresses in the longitudinal reinforcement of vertical ring beams are small.In the following examples, where the effect of vertical ring beams on the behaviour of the walls is analyzed, all walls are loaded with q = 40 kN/m at the floor level.
The effect of vertical ring beams on the horizontal wall-top displacement, for walls without openings, is shown in Figure 5, while the horizontal wall-top displacement, for walls with openings, is shown in Figure 6.It is evident that there is a big difference in the strength capacity and deformability of  Marija Smilović, Jure Radnić, Alen Harapin

Dynamical analysis
The walls analysed in Figure 1 were subjected to constant continuous vertical load q=40 kN/m.First the eigen problems were solved for every wall and its corresponding stiffness [21,22].The first and second periods of free oscillation of walls are shown in Table 2.It is evident that the opening in the wall softens the wall, and that the walls without ring beams (FW 1 , called OW 1 ) have significantly higher oscillation periods (lower stiffness) than the corresponding walls with vertical ring beams.The reinforcement of vertical ring beams does not greatly contribute to wall stiffness.Also, a large difference between the first (T 1 ) and second (T 2 ) periods of free oscillations of the walls is clearly visible, which is inherent to rigid structures.
In addition to wall analysis by nonlinear model [21,22], the analysis with linear-elastic material model was also performed so as to illustrate differences in numerical results.A 2 % viscous damping was adopted for all cases.

Harmonic base acceleration
The walls were subjected to harmonic base acceleration, as shown in Figure 10.Therefore, the excitation period corresponded to the first period of free oscillations of the elastic wall (T 1 ).The maximum base acceleration was 0,2 g, and the excitation period amounted to 7T 1 .The effect of vertical ring beams on the horizontal wall-top displacement for walls without openings is shown in Figure 11, while horizontal wall-top displacement for walls with openings is shown in Figure 12.It is obvious that the response of the walls with the nonlinear numerical model [21,22] differs completely from the response with a linear-elastic model.The well-known resonant motion of the wall is obtained for elastic behaviour and harmonic base excitation.In nonlinear models, the wall "falls out" from the resonant motion after the onset of nonlinearity (cracks) in the walls, and after lifting of foundations from the base.

Wall
There is also a big difference in maximum displacement of the walls.In nonlinear model, poor masonry walls without vertical

Seismic base acceleration
The dynamic analysis was carried out analogously to that presented in Section 3.2.2., using the horizontal acceleration component of the Kobe earthquake (Figure 16) and the nonlinear model only [21,22].Earthquake amplitudes were scaled, so that the maximum amplitude of acceleration is 0.2g (as in the harmonic base excitation).Some of the numerical results obtained are shown in

Prevented lifting and sliding of foundations
This case corresponds to some real masonry walls where the lifting and sliding of foundations are not possible.In this analysis, the walls from Figure 1 were re-examined, but with prevented horizontal and vertical displacement of the foundation bottom.The vertical load at the floor level was q = 40 kN/m.Only the nonlinear material model was considered according to [21,22].
during dynamic excitation.The state of cracks in the walls prior to failure is shown in Figures 14 and 15.Influence of vertical tie columns on bearing capacity of masonry walls

Static analysis
An analysis analogous to that presented in Item 3.2.1 was conducted.Some of the results obtained during the analysis are shown in Figures 20-24.By comparing the computational values from these Figures with the corresponding values from Figures 5-9, it can be concluded that the wall with prevented lifting and sliding of foundations has a significantly higher limit strength capacity, compared to a similar wall with enabled lifting and sliding of foundations.
The limit strength capacity depends on the strength capacity of the reinforced vertical ring beams (good masonry) or strength capacity of masonry (poor masonry).Here a greater difference can be noted in the limit strength capacity and deformability of the walls without vertical ring beams, when compared to the walls with vertical ring beams.The effect of profile of longitudinal bars of vertical ring beams on the limit strength capacity of the walls is also visible.A significant effect of the quality of masonry on the strength capacity and deformability of the walls can also be observed.

Dynamic analysis
A dynamic analysis analogous to that presented in Item 3.2.2 was conducted.It should be noted that the walls with prevented lifting and sliding of foundations also have periods of elastic system according to Table 2.    Influence of vertical tie columns on bearing capacity of masonry walls

Conclusion
The behaviour of each masonry wall under static and dynamic (seismic) load is specific and depends on many parameters.These are only global conclusions of the research, applicable to all masonry walls in real-life situations.In addition to other numerous parameters, the behaviour of masonry walls significantly depends on their total vertical load.Masonry walls with a higher total vertical load generally have a higher limit strength capacity when subjected to a horizontal static force.When these conditions are met, the walls with possible sliding and lifting of foundations generally have significantly lower limit strength capacity than similar walls with fixed foundations.In relation to the walls without openings, walls with openings can have significantly larger displacements and a significantly lower limit strength capacity, depending on the size and position of the openings.These differences increase with the decrease in the quality of masonry.The effect of ring beams in masonry walls with openings is greater than in the walls without openings.The walls without vertical ring beams have a significantly lower limit strength capacity when compared to similar walls with vertical ring beams.Walls with stronger reinforcement of vertical ring beams (bars of larger profile) have a higher limit strength capacity for horizontal static loads and during seismic activity -especially in cases when the wall strength capacity does not depend on the loss of their stability as a rigid body (sliding, overturning).Marija Smilović, Jure Radnić, Alen Harapin with reinforcement of pure reinforced concrete frames or reinforced concrete frames with masonry infill.The effect of real earthquakes may be less favourable than the resonant harmonic base acceleration of a similar max.acceleration.
The effect of vertical ring beams on the strength capacity and deformability of masonry walls is higher when the quality of the masonry is lower.Longitudinal and transverse reinforcement of vertical ring beams should be designed in practice in accordance

Figure 2 .
Figure 2. Finite element discretisation of analyzed masonry walls

Figure 1 .
Figure 1.Analyzed masonry walls The walls without openings (FW), and the walls with openings (OW), have the same horizontal ring beams (longitudinal bars 4Ø10).The walls FW 1 and OW 1 do not have vertical ring beams.Walls FW 2 and OW 2 have vertical ring beams with longitudinal reinforcing bars 4Ø10, walls FW 3 and OW 3 have vertical ring beams with longitudinal reinforcing bars 4Ø12, and walls FW 4 and OW 4 have vertical ring beams with longitudinal reinforcing bars 4Ø14.The stirrups of all vertical and horizontal ring beams are Ø6/25.The effect of transverse reinforcement of ring beams on the behaviour of confined masonry walls was not analyzed.All the walls have the same properties of concrete and reinforcement.Two cases of foundations supported by a

Figure 4 .Figure 3 .
Figure 4. Reinforcement stress of vertical ring beams at the bottom of the walls FW 2g and OW 2g , depending on vertical load q -freely supported foundations, static load

Figure 9 . 1 GRAĐEVINAR
Figure 9. Reinforcement stresses at the bottom of vertical ring beams -freely supported foundations static load

Figures 17 - 19 .
If the values given in these Figures are compared with the corresponding values from Figures 11-13, it can be concluded that the seismic base acceleration is here less favorable than the resonant harmonic base acceleration.

Figure 13 .Figure 14 .
Figure 13.Reinforcement stress at the bottom of vertical ring beams (point A) -freely supported foundations, harmonic base acceleration

Figure 24 .
Figure 24.Reinforcement stresses at the bottom of the vertical ring beams -fixed foundation, static load Harmonic base acceleration Some of the results obtained are shown in Figures 25-29.If computational values from these Figures are compared with the corresponding values from Figures 11-15, it can be concluded that these values are similar.For the observed harmonic base acceleration, the wall with fixed foundation behaves similarly to the wall with the lifting and sliding of foundations.The effect of vertical ring beams on reinforcement stress at the bottom of the

Figure 25 .
Figure 25.Horizontal wall-top displacement for walls top without openings -fixed foundation, harmonic base acceleration

Figure 32 .
Figure 32.Reinforcement stresses of vertical ring beams at the bottom of the walls -fixed foundation, seismic activity