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Numerical dynamic tests of masonry-infilled RC frames
Engineering Structures 50 (2013) 43–55
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Numerical dynamic tests of masonry-infilled RC frames G. Baloevic´ ⇑, J. Radnic´, A. Harapin University of Split, Faculty of Civil Engineering, Architecture and Geodesy, 21000 Split, Croatia
a r t i c l e
i n f o
Article history: Available online 9 January 2013 Keywords: Numerical dynamic test Masonry-infilled reinforced concrete frame Earthquake
a b s t r a c t Several numerical dynamic tests of two-storey masonry-infilled reinforced concrete frames were performed by adopted numerical models. Bare frames, fully masonry-infilled frames and masonry-infilled frames with openings, with variants of strong or weak concrete frames and masonry, were studied. Uniform harmonic base excitations and base excitations by three real scaled earthquakes were applied. Among other, it is concluded that cross-sectional dimensions of frame, rigidity of masonry, openings in the masonry and type of dynamic base excitation have significant influence on behaviour of masonryinfilled reinforced concrete frames. Finally, some recommendations for practical application are given. Ó 2012 Elsevier Ltd. All rights reserved.
1. Introduction The use of masonry-infilled reinforced concrete frames is very often in practice. In these structures, infill masonry is usually made after frames and floor structures have been constructed, i.e. after activation of their gravity load. Constructing frame and infill masonry in different stages, especially due to their various stiffness, causes complex behaviour of these structures under horizontal static load and especially under dynamic load. Many experimental and numerical studies of masonry-infilled reinforced concrete frames under static and dynamic loads have been performed. Some of them can be found in [1–24]. Many parameters influence the behaviour of masonry-infilled reinforced concrete frames in seismic conditions. In this paper only some of them were numerically analysed, on certain types of frames and infill masonry. However, influence of the frame stiffness, masonry stiffness, openings in the wall and type of dynamic excitation has been researched on adopted types of real two-storey plane frames. In this paper, strong frame is considered to be a frame with relative large cross-sectional dimensions of beams and columns in relation to its length, with corresponding reinforcement calculated according to the existing standards. Analogously, weak frame is considered to be a frame with relatively small cross-sectional dimensions of beams and columns in relation to its length, with corresponding reinforcement calculated according to the existing standards. Therefore, both frames have reinforcement calculated for the same loads and according to the same standards, but for different cross-sectional dimensions of beams and columns.
Masonry of good quality in this paper is considered to be masonry with high compressive, tensile and shear strength, and high modulus of elasticity and shear modulus (further referred to as ‘‘strong masonry’’). The same analogy is applied for masonry of poor quality (further referred to as ‘‘weak masonry’’). In performed numerical tests, the ratio of the above-mentioned parameters for masonry was adopted as 5:1. Bare frames, fully masonry-infilled frames and masonry-infilled frames with openings were analysed, with variants of strong or weak masonry. Previously developed numerical models of authors for static and dynamic analysis of concrete and masonry structures [25–28] were used to analyse selected examples. These models can simulate many nonlinear effects of reinforced concrete, masonry and soil, as well as changes in system geometry. These models can also simulate different construction stages of structures. Numerical models for simulation of concrete structures [25,26] have been verified on many experimental tests, as well as on many practical structures. Numerical models for simulation of masonry structures [27,28] have been verified on several experimental tests. Appliance of previously mentioned numerical models for research of relative effects of observed parameters of concrete and masonry structures can be considered as sufficiently reliable. Harmonic base excitations and seismic base excitations were considered. Three real scaled earthquakes were applied. Only some numerical results are presented for every analysed case. Finally, the main conclusions of this research and recommendations for practical application are given.
2. Short presentation of the adopted numerical models
⇑ Corresponding author. Tel.: +385 21 303300; fax: +385 21 465117. E-mail address: [email protected] (G. Baloevic´). 0141-0296/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.engstruct.2012.11.034
Detailed presentation of the adopted numerical models can be found in [25–28], with short description below. They can simulate many nonlinear effects, such as:
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– Concrete yielding in compression, opening of cracks in concrete in tension, the mechanism of opening and closing of cracks in concrete under cyclic load, tensile and shear stiffness of cracked concrete. – Yielding of steel reinforcement. – The strain rate effect on mechanical properties of concrete and steel. – Yield of masonry in compression, opening of cracks in the masonry in tension, the mechanism of opening and closing of cracks under cyclic load, transfer of shear stresses, anisotropic properties of strength and stiffness of masonry in horizontal and vertical direction, tensile and shear stiffness of cracked masonry. – Soil nonlinearity, analogously to concrete and masonry. – Geometrical nonlinearity – large displacements. – Construction mode – the stages of masonry walls and infilled frames assembling.
Table 1 Newmark’s implicit algorithm of iterative problem solution. (1) (2)
nþ1 u1nþ1 ¼ u 1 _ ¼u u_ nþ1
Fig. 1. Adopted finite elements for structure.
nþ1
nþ1 Þ=ðbDt 2 Þ € 1nþ1 ¼ ðu1nþ1 u u (3)
Calculate effective residual forces (f⁄)i:
€ inþ1 Rðuinþ1 ; u_ inþ1 Þ ðf Þi ¼ f nþ1 Mu (4)
Calculate the effective stiffness matrix Ks (if required):
Ks ¼ (5)
Spatial discretization of the structure is approximated by the plane stress state and modelled by eight-node (‘‘serendipity’’) elements (Fig. 1a). Reinforcement is simulated using the 1D bar element, within the basic 2D element. It is assumed that there is no slipping between the reinforcing bars and the surrounding concrete. For contact modelling between the soil and foundations or between mortar and masonry units, contact elements can be used (Fig. 1b). Flat 2D six-node contact finite element of small thickness w (Fig. 1b1) can be used to simulate a continuous connection between the basic eight-node elements, or 1D (bar) two-node contact element (Fig. 1b2) for the simulation of the reinforcement which passes across the contact surface. 2D contact element can simulate sliding, separation and penetration in the contact surface, based on the adopted material model. 1D contact element can take both axial and shear forces, according to the adopted material models. For the solution of dynamic equation of structure motion, implicit Newmark’s algorithm, developed in iterative form by Hughes [29], is used [25] (Table 1). Graphical presentation of the adopted elasto-plastic concrete model is shown in Fig. 2. Here fc,c and fc,t are uniaxial compressive and tensile concrete strengths; ec,c and ec,t are uniaxial compressive and tensile limit concrete strains; Ec is elasticity concrete modulus.
For time step (n + 1), use iterative step i = 1 Calculate the vectors of the assumed displacement, velocity and acceleration at the beginning of time step using the known values from previous time step:
M Cs þc þ Ks bDt bDt 2
Calculate the displacement increment vector Dui :
Ks Dui ¼ ðf Þi (6)
Correct the assumed values of displacement, velocity and acceleration: i i uiþ1 nþ1 ¼ unþ1 þ Dunþ1 2 iþ1 € iþ1 u nþ1 ¼ ðunþ1 unþ1 Þ=ðbDt Þ
_i € iþ1 u_ iþ1 nþ1 ¼ unþ1 þ ðcDtÞunþ1 (7)
Control the convergence procedure:– criterion
if Dui satisfies the convergence
iþ1 kDui k=kunþ1 k 6 en
proceed to the next time step (replace ‘‘n’’ with ‘‘n + 1’’ and proceed to solution step (1)). The solution in time tnþ1 is:
unþ1 ¼ uiþ1 nþ1 u_ nþ1 ¼ u_ iþ1 nþ1 € iþ1 € nþ1 ¼ u u nþ1 – if the convergence criterion is not satisfied, the iteration procedure with correction of shear, velocity and acceleration continues (replace ‘‘i’’ with ‘‘i + 1’’, and proceed to solution step (3)).
The cracks are modelled as smeared, which disregards the actual displacement discontinuity and the topology of the idealised structure remains unchanged after concrete cracking. After opening of cracks, it is assumed that their position remains unchanged for the next loading and unloading. The concrete becomes anisotropic and the crack direction determines the main directions of concrete anisotropy. Partial or full closing of previously opened cracks is modelled, as well as reopening of previously closed cracks. The transfer of compressive stress across a fully closed crack is modelled as for concrete without crack. After crack reopening, tensile stiffness of cracked concrete is not considered any more. The effect of tensile stiffness of cracked concrete is simulated by gradual decreasing of tensile stress components perpendicular to the crack, in accordance with the stress–strain relationship for uniaxial stress state. The shear strength of concrete in tension is modelled by the linear decreasing of initial shear modulus of concrete G in accordance to the tensile concrete strain perpendicular to the crack plane. The adopted stress–strain relationship for reinforcement is shown in Fig. 3. Here fs,c and fs,t are uniaxial compressive and tensile steel strengths; es,c and es,t are uniaxial compressive and tensile limit steel strains; Es and E0s are elasticity steel modules.
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Fig. 2. Graphic presentation of the adopted concrete model.
Fig. 3. Stress–strain relationship for reinforcement.
Two models for masonry can be used: macro-model and micro-model (Fig. 4). In the macro-model of masonry (Fig. 4b), masonry is approximated by a representative material whose physical–mechanical properties adequately describe the real complex masonry properties. Such approach allows large finite elements (rough discretization), significantly reduces the number of unknown variables and rapidly accelerates the analysis. In the micro-model of masonry (Fig. 4c), the spatial discretization of
masonry can be performed at the level of masonry units and mortar (joints). In more accurate analysis, the connection between mortar and masonry units can be simulated by contact elements. It is possible to use various micro-models of masonry, with various precision and duration of analysis. In relation to the masonry macro-model, the masonry micro-model can provide more accurate description of the damage and failure of masonry, but with much more complex analysis. It is used mainly for smaller spatial problems, and for verification of experimental tests of the masonry structures. Graphical presentation of the adopted orthotropic constitutive masonry model in compression and tension is given in Fig. 5. The masonry parameters in the horizontal (h) and vertical (v) direch v are comprestions are: rhm and rvm are normal stresses, fm;c and fm;c h v are tensile strengths, Eh and Ev are sive strengths, fm;t and fm;t m m elasticity modules, ehm;c and evm;c are crushing compressive strains. As shown in Fig. 5, the effect of biaxial stresses on limit compressive strength of masonry is disregarded. It is possible to simulate the tensile stiffness of cracked masonry. Crack model for masonry is analogous to the crack model for concrete, where, according to the adopted assumption, the cracks in the masonry are horizontal and/or vertical. The transmitting of compression
Fig. 4. Macro- and micro-models of masonry.
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Fig. 5. Adopted orthotropic masonry model.
Fig. 6. Adopted shear failure of masonry.
bearing capacity and with equal behaviour factor (q = 2.5) [31]. Each frame was analysed for the case without infill, with full and perforated (with openings) strong masonry (SM, PSM), and with full and perforated weak masonry (WM, PWM). It should be noted that, with equal ultimate bearing capacity, the strong frame has substantially larger cross-sectional dimensions of beams and columns, and significantly lower reinforcement of these elements. To simplify analysis, isotropic properties of masonry were adopted. The adopted basic parameters of concrete, reinforcement and masonry are shown in Table 2. In addition to their self weight, the frames were loaded by floor slabs at the level of both floors. The total vertical load of the frame has a significant influence on its behaviour under horizontal static and especially dynamic load, which is not analysed in this paper. Possible lifting of the foundation from the rigid base and its possible sliding on the base was not prevented. The analysis was performed for three typical phases:
Fig. 7. Adopted shear stiffness of cracked masonry.
stresses over the closed crack is modelled as in homogeneous masonry. After reopening of the previously closed crack, the stiffness of the masonry is not taken into account. After the crushing in compression, it is assumed that the masonry has no stiffness. The collapse of the masonry due to the shear stress in the horizontal plane is modelled. The criterion of the shear failure of masonry is defined according to Fig. 6, where sxy is masonry shear stress from the numerical calculation, and shm is masonry shear strength. Shear stiffness of cracked masonry is simulated similar to shear stiffness of cracked concrete. Specifically, assuming that after cracking masonry remains a continuum, the initial shear modulus Gm of the masonry is reduced according to the value of the tensile strain perpendicular to the crack en;m , according to Fig. 7. Only the concrete model or masonry macro-model can be used for soil modelling, with corresponding material parameters. Which model is more reliable, depends on soil properties.
(i) First, the bare frame loaded by its own weight and uniform load on floor level q1 = 20 kN/m (self weight of the floor structure) was analysed. (ii) Taking into account the previously stated initial state, the infilled frame was then analysed including self weight of the infill and additional uniform load on floor level q2 = 20 kN/m (additional dead load and live load of the floor). (iii) Taking into account the initial states for all vertical loads (static analysis), finally a dynamic analysis of the bare frame (infill is taken only as load and additional mass) and frame with different infill was performed for different dynamic base excitations.
3. Numerical tests
The adopted spatial discretization of the considered frames is shown in Fig. 9. At the connection of the foundation and rigid base, thin contact elements were used to simulate the possible lifting and sliding of foundation, with material parameters according to Table 2. The observed dynamic excitations are shown in Fig. 10. Only the longitudinal base accelerations were considered, with maximum amplitude of 0.2g. The period of harmonic excitation corresponds to the first period of free oscillations of the observed structure. Table 3 shows first periods of the analysed cases using the model [27,28].
3.1. General
3.2. Numerical results
The basic data of the analysed two-storey frames are shown in Fig. 8. A strong frame (SF) and a weak frame (WF) were considered separately. The reinforcement of both frames without infill (bare frames) was determined according to [30,31], with equal estimated
Only next results of the numerical analysis are presented: (i) Horizontal displacement of the wall top for all considered cases.
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Fig. 8. Analysed masonry-infilled frames. Table 2 The adopted basic material parameters in the numerical analysis. Variable
Modulus of elasticity Poisson’s ratio Shear modulus Uniaxial compressive strength Uniaxial tensile strength Limit compressive strain Limit tensile strain
Unit
MPa – MPa MPa MPa – –
Material Concrete
Steel
Strong masonry
Weak masonry
Contact element
34,500 0.20 14,300 40.0 4.0 0.00350 0.00012
200,000 – – 560.0 560.0 0.02000 0.02000
5000 0.15 1000 5.0 0.15 0.01000 0.00003
1000 0.15 200 1.0 0.03 0.01000 0.00003
34,500 – 14,300 40.0 0.0 0.00350 0.00000
(ii) Reinforcement stress at the bottom of the column for all considered cases. (iii) The final state of cracks for some considered cases. The presentation of these results follows hereinafter.
Fig. 9. The adopted spatial discretization of the observed frames by finite elements.
3.2.1. Horizontal displacement of the top of the frame Horizontal displacement of the top of the bare frames (SF, WF) is shown in Fig. 11. It is mentioned once again that in this model the infill is not included in the strength capacity of overall structure, but only as load and additional mass. Horizontal displacement of the top of the frames with full strong masonry (SF-SM, WF-SM) is shown in Fig. 12, of the frames with full weak masonry (SF-WM, WF-WM) in Fig. 13, of the frames with perforated strong masonry (SF-PSM, WFPSM) in Fig. 14, and of the frames with perforated weak masonry (SF-PWM, WF-PWM) in Fig. 15. Maximal displacements of the top of the frame for all analysed cases are shown in Table 4. Based on the analysis results shown in Figs. 11–15 and Table 4, the following can be concluded:
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– Under the same dynamic excitations, bare reinforced concrete frames with larger cross-sectional dimensions of beams and columns have significantly smaller horizontal displacements compared to reinforced concrete frames with smaller cross-sectional dimensions of beams and columns, which both have reinforcement calculated for the same loads and according to the same standards. – Compared to bare frames, masonry-infilled frames have significantly smaller horizontal displacements under dynamic excitations. – Frames with weak masonry have significantly greater horizontal displacements compared to frames with strong masonry under same dynamic excitations. – For the fully infilled frames with the same quality of masonry, rigidity of frame does not influence significantly their horizontal displacements under same dynamic excitations. Openings in the masonry increase frame’s displacements under dynamic excitations. – For equal maximum base accelerations, horizontal displacements of bare frames and masonry-infilled frames are significantly influenced by type of dynamic excitation. In such cases, real earthquakes can be more unfavourable than resonant harmonic base excitations. 3.2.2. Reinforcement stress at the column bottom The stress in vertical reinforcement at the column bottom for the case of bare frames (SF, WF) is shown in Fig. 16, for the case of the frames with full strong masonry (SF-SM, WFSM) in Fig. 17, for the case of the frames with full weak masonry (SF-WM, WF-WM) in Fig. 18, for the case of the frames with perforated strong masonry (SF-PSM, WF-PSM) in Fig. 19, and for the case of the frames with perforated weak masonry (SF-PWM, WF-PWM) in Fig. 20. Maximal reinforcement stresses at the column bottom for all analysed cases are shown in Table 4. Based on the analysis results shown in Figs. 16–20 and Table 4, the following can be concluded:
Fig. 10. The considered dynamic base excitations.
Table 3 The first period of free oscillations of the observed frames. Mark
Case
T1 (s)
SF WF SF-SM WF-SM SF-PSM WF-PSM SF-WM WF-WM SF-PWM WF-PWM
Bare strong frame Bare weak frame Strong frame with full strong masonry Weak frame with full strong masonry Strong frame with perforated strong masonry Weak frame with perforated strong masonry Strong frame with full weak masonry Weak frame with full weak masonry Strong frame with perforated weak masonry Weak frame with perforated weak masonry
0.3392 0.5539 0.0793 0.0834 0.0886 0.0949 0.1449 0.1589 0.1595 0.1787
– Bare reinforced concrete frames with different cross-sectional dimensions of beams and columns and with corresponding reinforcement calculated for the same loads and according to the same standards, have approximate equal stresses of column reinforcement under same dynamic excitations. – Compared to bare frames, fully infilled frames have significantly lower reinforcement stresses in columns under dynamic excitations. – Under the same dynamic excitations and for fully infilled frames, maximum reinforcement stresses in columns are not greatly influenced by rigidity of frames and masonry. – Under the same dynamic excitations and for partially infilled frames, maximum reinforcement stresses in columns of frames are increasing. – Openings in the masonry of the frame increase reinforcement stresses in columns under dynamic excitations, and especially for cases with weak masonry. – For equal maximum base accelerations, reinforcement stresses of bare frames and masonry-infilled frames are significantly influenced by the type of dynamic excitation. In some cases, earthquakes can be more unfavourable than resonant harmonic base excitations. 3.2.3. Final state of the cracks Final state of the cracks (opened or closed) in the bare frame (SF, WF) under harmonic base excitation is shown in Fig. 21, in the frame with full strong masonry (SF-SM, WF-SM) under earthquake ‘‘Newhall’’ is shown in Fig. 22, in the frame with full weak masonry
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Fig. 11. Horizontal displacement of the top of the bare frames (SF, WF).
Fig. 12. Horizontal displacement of the top of the frames with full strong masonry (SF-SM, WF-SM).
(SF-WM, WF-WM) under earthquake ‘‘Santa Monica’’ is shown in Fig. 23 and in the frame with perforated strong masonry (SFPSM, WF-PSM) under earthquake ‘‘Kobe’’ is shown in Fig. 24.
Based on the analysis results, which are not completely shown in this paper, and based on the results shown in Figs. 21–24, the following can be concluded:
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Fig. 13. Horizontal displacement of the top of the frames with full weak masonry (SF-WM, WF-WM).
Fig. 14. Horizontal displacement of the top of the frames with perforated strong masonry (SF-PSM, WF-PSM).
– Compared to bare frames, masonry-infilled frames have significantly smaller crack zones in beams and columns under dynamic excitations.
– Size of crack zones in beams and columns of bare frames under dynamic excitations does not depend much on rigidity of the frame.
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Fig. 15. Horizontal displacement of the top of the frames with perforated weak masonry (SF-PWM, WF-PWM).
Table 4 Maximal displacements of the top of the frame and maximal reinforcement stresses at the bottom of the column. Case
SF WF SF-SM WF-SM SF-WM WF-WM SF-PSM WF-PSM SF-PWM WF-PWM a b
Horizontal displacementa (mm)
Reinforcement stressb (MPa)
Harmonic
Newhall
S. Monica
Kobe
Harmonic
Newhall
S. Monica
Kobe
22.2 67.5 2.1 2.5 4.7 5.2 2.3 2.6 5.3 5.6
40.3 58.7 0.6 0.8 6.2 10.5 0.8 1.2 20.2 21.8
18.6 29.2 0.9 1.2 5.0 6.1 2.4 3.0 6.2 18.3
32.2 88.4 0.7 0.6 6.7 7.1 1.0 1.7 13.8 29.3
117.1 219.1 67.7 61.0 52.3 55.6 63.6 58.7 52.4 58.6
155.0 164.5 32.5 32.3 68.2 77.8 31.9 38.4 110.0 101.1
112.1 88.0 33.9 46.4 59.4 65.5 54.3 66.2 76.2 91.8
220.1 236.3 32.3 32.7 59.6 65.5 32.0 46.6 99.8 101.2
Displacement in right direction ‘+’; displacement in left direction ‘’. Tensile stress ‘+’; compressive stress ‘’.
– Under the same dynamic excitation, weak masonry infill of the frame has wider crack zones than strong masonry infill. – Under the same dynamic excitations and for masonry-infilled frames, size of cracking zone in the masonry does not depend much on rigidity of the frame. 4. Conclusions Under the same dynamic base excitations, bare reinforced concrete frames with larger cross-sectional dimensions of beams and columns have significantly smaller horizontal displacements compared to reinforced concrete frames with smaller cross-sectional dimensions of beams and columns, which both have reinforcement calculated for the same loads and according to the same standards. In such cases, both types of frames have approximately equal reinforcement stresses and approximately equal width of crack zones in beams and columns.
In relation to bare frames, masonry-infilled frames have significantly smaller horizontal displacements, as well as lower reinforcement stresses in beams and columns, and smaller crack zones in the frames. In relation to the frames with strong masonry, frames with weak masonry have significantly greater horizontal displacements, as well as slightly higher reinforcement stresses in the frames and wider crack zones in the frames and masonry. Openings in the masonry-infilled frames increase their displacements, reinforcement stresses in the frames, as well as crack zones in the frames and masonry under dynamic excitations. For equal maximum base accelerations, horizontal displacements of bare frames and masonry-infilled frames, as well as reinforcement stresses and size of crack zones, are significantly influenced by the type of dynamic excitation. In such cases, earthquakes can be more unfavourable than resonant harmonic base excitations.
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Fig. 16. Reinforcement stress at the bottom of the columns of the bare frames (SF, WF).
Fig. 17. Reinforcement stress at the bottom of the frames’ columns with full strong masonry (SF-SM, WF-SM).
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Fig. 18. Reinforcement stress at the bottom of the frames’ columns with full weak masonry (SF-WM, WF-WM).
Fig. 19. Reinforcement stress at the bottom of the frames’ columns with perforated strong masonry (SF-PSM, WF-PSM).
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Fig. 20. Reinforcement stress at the bottom of the frames’ columns with perforated weak masonry (SF-PWM, WF-PWM).
Fig. 21. Final state of the cracks on the bare frames (SF, WF) – harmonic base acceleration.
Fig. 22. Final state of the cracks on the frames with full strong masonry (SF-SM, WF-SM) – earthquake ‘‘Newhall’’.
Fig. 23. Final state of the cracks on the frames with full weak masonry (SF-WM, WF-WM) – earthquake ‘‘Santa Monica’’.
Fig. 24. Final state of the cracks on the frames with perforated strong masonry (SFPSM, WF-PSM) – earthquake ‘‘Kobe’’.
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For the practical application of masonry-infilled reinforced concrete frame structures in seismic areas, it is recommended to use frames with large cross-sectional dimensions of beams and columns, as well as masonry with high mechanical properties. Acknowledgements This work was supported by the funds of the Ministry of Science, Education and Sport of Croatia. The authors appreciate their financial support. References [1] Al-Chaar G. Evaluating strength and stiffness of unreinforced masonry infill structures. Research report. US Army Corps of Engineering; 2002. [2] Al-Chaar G, Mehrabi AB, Manzouri T. Finite element interface modeling and experimental verification of masonry-infilled R/C frames. Masonry Soc J 2008;26(1):47–66. [3] Angel R, Abrams D, Shapiro D, Uzarski J, Webster M. Behavior of reinforced concrete frames with masonry infills. Research report. Department of Civil Engineering, University of Illinois at Urbana-Champaign; 1994. [4] Anil Ö, Altin S. An experimental study on reinforced concrete partially infilled frames. Eng Struct 2007;29:449–60. [5] Asteris PG, Antoniou ST, Sophianopoulos DS, Chrysostomou CZ. Mathematical macro-modeling of infilled frames: state of the art. ASCE J Struct Eng 2011;137(12):1508–17. [6] Blackard B, Willam K, Mettupalayam S. Experimental observations of masonry infilled RC frames with openings. ACI SP 265-9. American Concrete Institute; 2009. p. 199–222. [7] Colangelo F. Pseudo-dynamic seismic response of reinforced concrete frames infilled with non-structural brick masonry. Earthquake Eng Struct Dynam 2005;34:1219–41. [8] Crisafulli FJ, Carr AJ, Park R. Analytical modelling of infilled frames structures – a general review. Bull NZ Soc Earthquake Eng 2000;33:30–47. [9] Dolšek M, Fajfar P. Simplified non-linear seismic analysis of infilled reinforced concrete frames. Earthquake Eng Struct Dynam 2005;34:49–66. [10] Hashemi A, Mosalam KM. Shake-table experiment on reinforced concrete structure containing masonry infill wall. Earthquake Eng Struct D 2006;35(14):1827–52. [11] Kaushik HB, Rai DC, Jain SK. Code approaches to seismic design of masonryinfilled reinforced concrete frames: a state of the art review. Earthquake Spectra 2006;22:961–83. [12] Klingner RE, Rubiano NR, Bashandy TR, Sweeney SC. Evaluation and analytical verification of shaking table data from infilled frames. Masonry Soc J 1997;15(2):33–41.
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[13] Lotfi HR, Shing PB. An appraisal of smeared crack models for masonry shear wall analysis. Comput Struct 1991;41(3):413–25. [14] Madan A, Reinhorn AM, Mander JB, Valles RE. Modeling of masonry infill panels for structural analysis. ASCE J Struct Eng 1997;123(10):1295–302. [15] Mehrabi AB, Shing PB, Schuller M, Noland J. Experimental evaluation of masonry infilled RC frames. ASCE J Struct Eng 1996;122(3):228–37. [16] Mehrabi AB, Shing PB. Finite element modeling of masonry-infilled RC frames. ASCE J Struct Eng 1997;123(5):604–13. [17] Puglisi M, Uzcategui M, Florez-Lopez J. Modeling of masonry of infilled frames. Part I: the plastic concentrator. Eng Struct 2009;31(1):113–8. [18] Stavridis A. Analytical and experimental study of seismic performance of reinforced concrete frames infilled with masonry walls. PhD thesis. San Diego: Department of Structural Engineering, University of California; 2010. [19] Stavridis A, Shing PB. Finite element modeling of nonlinear behaviour of masonry-infilled RC frames. J Struct Eng 2010;136(3):285–96. [20] Stavridis A, Koutromanos I, Shing PB. Shake-table tests of a three-story reinforced concrete frame with masonry infill walls. Earthquake Eng Struct D 2012;41(6):1089–108. [21] Zarnic R, Tomazevic M. The behavior of masonry infilled reinforced concrete frames subjected to seismic loading. In: Proceedings of the 8th world conference on earthquake engineering, California; 1984. p. 863–70. [22] Zarnic R, Tomazevic M. Study of the behavior of masonry infilled reinforced concrete frames subjected to seismic loading. In: Proceedings of the 7th international brick masonry conference, Melbourne, Australia; 1985. [23] Zarnic R, Tomazevic M. Study of the behavior of masonry infilled reinforced concrete frames subjected to seismic loading. Part 2. ZRMK/IKPI – 8502. A report to the research community of Slovenia, Ljubljana; 1985. [24] Zarnic R, Gostic S, Crewe AJ, Taylor CA. Shaking table tests of 1:4 reduced scale models of masonry infilled reinforced concrete frame buildings. Earthquake Eng Struct D 2001:819–34. [25] Radnic´ J, Damjanic´ F. Numerical model for static and dynamic analysis of RC structures. Izgradnja 1989;10:5–14 [in Croatian]. [26] Radnic´ J. Modelling of the strain rate effects in dynamic analysis of R/C structures. Eng Modell 1990;3(1–2):13–20. [27] Radnic´ J, Harapin A, Matešan D, Trogrlic´ B, Smilovic´ M, Grgic´ N, et al. Numerical model for analysis of masonry structure. Gradevinar 2011;63(6):529–46 [in Croatian]. [28] Radnic´ J, Harapin A, Matešan D, Smilovic´ M, Grgic´ N. Numerical model for static and dynamic analysis of masonry structure. In: 5th International conference on advanced computational engineering and experimenting; 2011. [29] Hughes TJR, Pister KS, Taylor RL. Implicit–explicit finite elements in nonlinear transient analysis. Comput Methods Appl Mech Eng 1979;17(18):159–82. [30] Eurocode 2. Design of concrete structures – Part 1: general rules and rules for buildings ENV 1992-1-1. Brussels: European Committee for Standardization; 2002. [31] Eurocode 8. Design of structures for earthquake resistance – Part 1: general rules, seismic actions and rules for buildings ENV 1998-1-1/2/3. Brussels: European Committee for Standardization; 2004.
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Engineering Structures journal homepage: www.elsevier.com/locate/engstruct
Numerical dynamic tests of masonry-infilled RC frames G. Baloevic´ ⇑, J. Radnic´, A. Harapin University of Split, Faculty of Civil Engineering, Architecture and Geodesy, 21000 Split, Croatia
a r t i c l e
i n f o
Article history: Available online 9 January 2013 Keywords: Numerical dynamic test Masonry-infilled reinforced concrete frame Earthquake
a b s t r a c t Several numerical dynamic tests of two-storey masonry-infilled reinforced concrete frames were performed by adopted numerical models. Bare frames, fully masonry-infilled frames and masonry-infilled frames with openings, with variants of strong or weak concrete frames and masonry, were studied. Uniform harmonic base excitations and base excitations by three real scaled earthquakes were applied. Among other, it is concluded that cross-sectional dimensions of frame, rigidity of masonry, openings in the masonry and type of dynamic base excitation have significant influence on behaviour of masonryinfilled reinforced concrete frames. Finally, some recommendations for practical application are given. Ó 2012 Elsevier Ltd. All rights reserved.
1. Introduction The use of masonry-infilled reinforced concrete frames is very often in practice. In these structures, infill masonry is usually made after frames and floor structures have been constructed, i.e. after activation of their gravity load. Constructing frame and infill masonry in different stages, especially due to their various stiffness, causes complex behaviour of these structures under horizontal static load and especially under dynamic load. Many experimental and numerical studies of masonry-infilled reinforced concrete frames under static and dynamic loads have been performed. Some of them can be found in [1–24]. Many parameters influence the behaviour of masonry-infilled reinforced concrete frames in seismic conditions. In this paper only some of them were numerically analysed, on certain types of frames and infill masonry. However, influence of the frame stiffness, masonry stiffness, openings in the wall and type of dynamic excitation has been researched on adopted types of real two-storey plane frames. In this paper, strong frame is considered to be a frame with relative large cross-sectional dimensions of beams and columns in relation to its length, with corresponding reinforcement calculated according to the existing standards. Analogously, weak frame is considered to be a frame with relatively small cross-sectional dimensions of beams and columns in relation to its length, with corresponding reinforcement calculated according to the existing standards. Therefore, both frames have reinforcement calculated for the same loads and according to the same standards, but for different cross-sectional dimensions of beams and columns.
Masonry of good quality in this paper is considered to be masonry with high compressive, tensile and shear strength, and high modulus of elasticity and shear modulus (further referred to as ‘‘strong masonry’’). The same analogy is applied for masonry of poor quality (further referred to as ‘‘weak masonry’’). In performed numerical tests, the ratio of the above-mentioned parameters for masonry was adopted as 5:1. Bare frames, fully masonry-infilled frames and masonry-infilled frames with openings were analysed, with variants of strong or weak masonry. Previously developed numerical models of authors for static and dynamic analysis of concrete and masonry structures [25–28] were used to analyse selected examples. These models can simulate many nonlinear effects of reinforced concrete, masonry and soil, as well as changes in system geometry. These models can also simulate different construction stages of structures. Numerical models for simulation of concrete structures [25,26] have been verified on many experimental tests, as well as on many practical structures. Numerical models for simulation of masonry structures [27,28] have been verified on several experimental tests. Appliance of previously mentioned numerical models for research of relative effects of observed parameters of concrete and masonry structures can be considered as sufficiently reliable. Harmonic base excitations and seismic base excitations were considered. Three real scaled earthquakes were applied. Only some numerical results are presented for every analysed case. Finally, the main conclusions of this research and recommendations for practical application are given.
2. Short presentation of the adopted numerical models
⇑ Corresponding author. Tel.: +385 21 303300; fax: +385 21 465117. E-mail address: [email protected] (G. Baloevic´). 0141-0296/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.engstruct.2012.11.034
Detailed presentation of the adopted numerical models can be found in [25–28], with short description below. They can simulate many nonlinear effects, such as:
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– Concrete yielding in compression, opening of cracks in concrete in tension, the mechanism of opening and closing of cracks in concrete under cyclic load, tensile and shear stiffness of cracked concrete. – Yielding of steel reinforcement. – The strain rate effect on mechanical properties of concrete and steel. – Yield of masonry in compression, opening of cracks in the masonry in tension, the mechanism of opening and closing of cracks under cyclic load, transfer of shear stresses, anisotropic properties of strength and stiffness of masonry in horizontal and vertical direction, tensile and shear stiffness of cracked masonry. – Soil nonlinearity, analogously to concrete and masonry. – Geometrical nonlinearity – large displacements. – Construction mode – the stages of masonry walls and infilled frames assembling.
Table 1 Newmark’s implicit algorithm of iterative problem solution. (1) (2)
nþ1 u1nþ1 ¼ u 1 _ ¼u u_ nþ1
Fig. 1. Adopted finite elements for structure.
nþ1
nþ1 Þ=ðbDt 2 Þ € 1nþ1 ¼ ðu1nþ1 u u (3)
Calculate effective residual forces (f⁄)i:
€ inþ1 Rðuinþ1 ; u_ inþ1 Þ ðf Þi ¼ f nþ1 Mu (4)
Calculate the effective stiffness matrix Ks (if required):
Ks ¼ (5)
Spatial discretization of the structure is approximated by the plane stress state and modelled by eight-node (‘‘serendipity’’) elements (Fig. 1a). Reinforcement is simulated using the 1D bar element, within the basic 2D element. It is assumed that there is no slipping between the reinforcing bars and the surrounding concrete. For contact modelling between the soil and foundations or between mortar and masonry units, contact elements can be used (Fig. 1b). Flat 2D six-node contact finite element of small thickness w (Fig. 1b1) can be used to simulate a continuous connection between the basic eight-node elements, or 1D (bar) two-node contact element (Fig. 1b2) for the simulation of the reinforcement which passes across the contact surface. 2D contact element can simulate sliding, separation and penetration in the contact surface, based on the adopted material model. 1D contact element can take both axial and shear forces, according to the adopted material models. For the solution of dynamic equation of structure motion, implicit Newmark’s algorithm, developed in iterative form by Hughes [29], is used [25] (Table 1). Graphical presentation of the adopted elasto-plastic concrete model is shown in Fig. 2. Here fc,c and fc,t are uniaxial compressive and tensile concrete strengths; ec,c and ec,t are uniaxial compressive and tensile limit concrete strains; Ec is elasticity concrete modulus.
For time step (n + 1), use iterative step i = 1 Calculate the vectors of the assumed displacement, velocity and acceleration at the beginning of time step using the known values from previous time step:
M Cs þc þ Ks bDt bDt 2
Calculate the displacement increment vector Dui :
Ks Dui ¼ ðf Þi (6)
Correct the assumed values of displacement, velocity and acceleration: i i uiþ1 nþ1 ¼ unþ1 þ Dunþ1 2 iþ1 € iþ1 u nþ1 ¼ ðunþ1 unþ1 Þ=ðbDt Þ
_i € iþ1 u_ iþ1 nþ1 ¼ unþ1 þ ðcDtÞunþ1 (7)
Control the convergence procedure:– criterion
if Dui satisfies the convergence
iþ1 kDui k=kunþ1 k 6 en
proceed to the next time step (replace ‘‘n’’ with ‘‘n + 1’’ and proceed to solution step (1)). The solution in time tnþ1 is:
unþ1 ¼ uiþ1 nþ1 u_ nþ1 ¼ u_ iþ1 nþ1 € iþ1 € nþ1 ¼ u u nþ1 – if the convergence criterion is not satisfied, the iteration procedure with correction of shear, velocity and acceleration continues (replace ‘‘i’’ with ‘‘i + 1’’, and proceed to solution step (3)).
The cracks are modelled as smeared, which disregards the actual displacement discontinuity and the topology of the idealised structure remains unchanged after concrete cracking. After opening of cracks, it is assumed that their position remains unchanged for the next loading and unloading. The concrete becomes anisotropic and the crack direction determines the main directions of concrete anisotropy. Partial or full closing of previously opened cracks is modelled, as well as reopening of previously closed cracks. The transfer of compressive stress across a fully closed crack is modelled as for concrete without crack. After crack reopening, tensile stiffness of cracked concrete is not considered any more. The effect of tensile stiffness of cracked concrete is simulated by gradual decreasing of tensile stress components perpendicular to the crack, in accordance with the stress–strain relationship for uniaxial stress state. The shear strength of concrete in tension is modelled by the linear decreasing of initial shear modulus of concrete G in accordance to the tensile concrete strain perpendicular to the crack plane. The adopted stress–strain relationship for reinforcement is shown in Fig. 3. Here fs,c and fs,t are uniaxial compressive and tensile steel strengths; es,c and es,t are uniaxial compressive and tensile limit steel strains; Es and E0s are elasticity steel modules.
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Fig. 2. Graphic presentation of the adopted concrete model.
Fig. 3. Stress–strain relationship for reinforcement.
Two models for masonry can be used: macro-model and micro-model (Fig. 4). In the macro-model of masonry (Fig. 4b), masonry is approximated by a representative material whose physical–mechanical properties adequately describe the real complex masonry properties. Such approach allows large finite elements (rough discretization), significantly reduces the number of unknown variables and rapidly accelerates the analysis. In the micro-model of masonry (Fig. 4c), the spatial discretization of
masonry can be performed at the level of masonry units and mortar (joints). In more accurate analysis, the connection between mortar and masonry units can be simulated by contact elements. It is possible to use various micro-models of masonry, with various precision and duration of analysis. In relation to the masonry macro-model, the masonry micro-model can provide more accurate description of the damage and failure of masonry, but with much more complex analysis. It is used mainly for smaller spatial problems, and for verification of experimental tests of the masonry structures. Graphical presentation of the adopted orthotropic constitutive masonry model in compression and tension is given in Fig. 5. The masonry parameters in the horizontal (h) and vertical (v) direch v are comprestions are: rhm and rvm are normal stresses, fm;c and fm;c h v are tensile strengths, Eh and Ev are sive strengths, fm;t and fm;t m m elasticity modules, ehm;c and evm;c are crushing compressive strains. As shown in Fig. 5, the effect of biaxial stresses on limit compressive strength of masonry is disregarded. It is possible to simulate the tensile stiffness of cracked masonry. Crack model for masonry is analogous to the crack model for concrete, where, according to the adopted assumption, the cracks in the masonry are horizontal and/or vertical. The transmitting of compression
Fig. 4. Macro- and micro-models of masonry.
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Fig. 5. Adopted orthotropic masonry model.
Fig. 6. Adopted shear failure of masonry.
bearing capacity and with equal behaviour factor (q = 2.5) [31]. Each frame was analysed for the case without infill, with full and perforated (with openings) strong masonry (SM, PSM), and with full and perforated weak masonry (WM, PWM). It should be noted that, with equal ultimate bearing capacity, the strong frame has substantially larger cross-sectional dimensions of beams and columns, and significantly lower reinforcement of these elements. To simplify analysis, isotropic properties of masonry were adopted. The adopted basic parameters of concrete, reinforcement and masonry are shown in Table 2. In addition to their self weight, the frames were loaded by floor slabs at the level of both floors. The total vertical load of the frame has a significant influence on its behaviour under horizontal static and especially dynamic load, which is not analysed in this paper. Possible lifting of the foundation from the rigid base and its possible sliding on the base was not prevented. The analysis was performed for three typical phases:
Fig. 7. Adopted shear stiffness of cracked masonry.
stresses over the closed crack is modelled as in homogeneous masonry. After reopening of the previously closed crack, the stiffness of the masonry is not taken into account. After the crushing in compression, it is assumed that the masonry has no stiffness. The collapse of the masonry due to the shear stress in the horizontal plane is modelled. The criterion of the shear failure of masonry is defined according to Fig. 6, where sxy is masonry shear stress from the numerical calculation, and shm is masonry shear strength. Shear stiffness of cracked masonry is simulated similar to shear stiffness of cracked concrete. Specifically, assuming that after cracking masonry remains a continuum, the initial shear modulus Gm of the masonry is reduced according to the value of the tensile strain perpendicular to the crack en;m , according to Fig. 7. Only the concrete model or masonry macro-model can be used for soil modelling, with corresponding material parameters. Which model is more reliable, depends on soil properties.
(i) First, the bare frame loaded by its own weight and uniform load on floor level q1 = 20 kN/m (self weight of the floor structure) was analysed. (ii) Taking into account the previously stated initial state, the infilled frame was then analysed including self weight of the infill and additional uniform load on floor level q2 = 20 kN/m (additional dead load and live load of the floor). (iii) Taking into account the initial states for all vertical loads (static analysis), finally a dynamic analysis of the bare frame (infill is taken only as load and additional mass) and frame with different infill was performed for different dynamic base excitations.
3. Numerical tests
The adopted spatial discretization of the considered frames is shown in Fig. 9. At the connection of the foundation and rigid base, thin contact elements were used to simulate the possible lifting and sliding of foundation, with material parameters according to Table 2. The observed dynamic excitations are shown in Fig. 10. Only the longitudinal base accelerations were considered, with maximum amplitude of 0.2g. The period of harmonic excitation corresponds to the first period of free oscillations of the observed structure. Table 3 shows first periods of the analysed cases using the model [27,28].
3.1. General
3.2. Numerical results
The basic data of the analysed two-storey frames are shown in Fig. 8. A strong frame (SF) and a weak frame (WF) were considered separately. The reinforcement of both frames without infill (bare frames) was determined according to [30,31], with equal estimated
Only next results of the numerical analysis are presented: (i) Horizontal displacement of the wall top for all considered cases.
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Fig. 8. Analysed masonry-infilled frames. Table 2 The adopted basic material parameters in the numerical analysis. Variable
Modulus of elasticity Poisson’s ratio Shear modulus Uniaxial compressive strength Uniaxial tensile strength Limit compressive strain Limit tensile strain
Unit
MPa – MPa MPa MPa – –
Material Concrete
Steel
Strong masonry
Weak masonry
Contact element
34,500 0.20 14,300 40.0 4.0 0.00350 0.00012
200,000 – – 560.0 560.0 0.02000 0.02000
5000 0.15 1000 5.0 0.15 0.01000 0.00003
1000 0.15 200 1.0 0.03 0.01000 0.00003
34,500 – 14,300 40.0 0.0 0.00350 0.00000
(ii) Reinforcement stress at the bottom of the column for all considered cases. (iii) The final state of cracks for some considered cases. The presentation of these results follows hereinafter.
Fig. 9. The adopted spatial discretization of the observed frames by finite elements.
3.2.1. Horizontal displacement of the top of the frame Horizontal displacement of the top of the bare frames (SF, WF) is shown in Fig. 11. It is mentioned once again that in this model the infill is not included in the strength capacity of overall structure, but only as load and additional mass. Horizontal displacement of the top of the frames with full strong masonry (SF-SM, WF-SM) is shown in Fig. 12, of the frames with full weak masonry (SF-WM, WF-WM) in Fig. 13, of the frames with perforated strong masonry (SF-PSM, WFPSM) in Fig. 14, and of the frames with perforated weak masonry (SF-PWM, WF-PWM) in Fig. 15. Maximal displacements of the top of the frame for all analysed cases are shown in Table 4. Based on the analysis results shown in Figs. 11–15 and Table 4, the following can be concluded:
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– Under the same dynamic excitations, bare reinforced concrete frames with larger cross-sectional dimensions of beams and columns have significantly smaller horizontal displacements compared to reinforced concrete frames with smaller cross-sectional dimensions of beams and columns, which both have reinforcement calculated for the same loads and according to the same standards. – Compared to bare frames, masonry-infilled frames have significantly smaller horizontal displacements under dynamic excitations. – Frames with weak masonry have significantly greater horizontal displacements compared to frames with strong masonry under same dynamic excitations. – For the fully infilled frames with the same quality of masonry, rigidity of frame does not influence significantly their horizontal displacements under same dynamic excitations. Openings in the masonry increase frame’s displacements under dynamic excitations. – For equal maximum base accelerations, horizontal displacements of bare frames and masonry-infilled frames are significantly influenced by type of dynamic excitation. In such cases, real earthquakes can be more unfavourable than resonant harmonic base excitations. 3.2.2. Reinforcement stress at the column bottom The stress in vertical reinforcement at the column bottom for the case of bare frames (SF, WF) is shown in Fig. 16, for the case of the frames with full strong masonry (SF-SM, WFSM) in Fig. 17, for the case of the frames with full weak masonry (SF-WM, WF-WM) in Fig. 18, for the case of the frames with perforated strong masonry (SF-PSM, WF-PSM) in Fig. 19, and for the case of the frames with perforated weak masonry (SF-PWM, WF-PWM) in Fig. 20. Maximal reinforcement stresses at the column bottom for all analysed cases are shown in Table 4. Based on the analysis results shown in Figs. 16–20 and Table 4, the following can be concluded:
Fig. 10. The considered dynamic base excitations.
Table 3 The first period of free oscillations of the observed frames. Mark
Case
T1 (s)
SF WF SF-SM WF-SM SF-PSM WF-PSM SF-WM WF-WM SF-PWM WF-PWM
Bare strong frame Bare weak frame Strong frame with full strong masonry Weak frame with full strong masonry Strong frame with perforated strong masonry Weak frame with perforated strong masonry Strong frame with full weak masonry Weak frame with full weak masonry Strong frame with perforated weak masonry Weak frame with perforated weak masonry
0.3392 0.5539 0.0793 0.0834 0.0886 0.0949 0.1449 0.1589 0.1595 0.1787
– Bare reinforced concrete frames with different cross-sectional dimensions of beams and columns and with corresponding reinforcement calculated for the same loads and according to the same standards, have approximate equal stresses of column reinforcement under same dynamic excitations. – Compared to bare frames, fully infilled frames have significantly lower reinforcement stresses in columns under dynamic excitations. – Under the same dynamic excitations and for fully infilled frames, maximum reinforcement stresses in columns are not greatly influenced by rigidity of frames and masonry. – Under the same dynamic excitations and for partially infilled frames, maximum reinforcement stresses in columns of frames are increasing. – Openings in the masonry of the frame increase reinforcement stresses in columns under dynamic excitations, and especially for cases with weak masonry. – For equal maximum base accelerations, reinforcement stresses of bare frames and masonry-infilled frames are significantly influenced by the type of dynamic excitation. In some cases, earthquakes can be more unfavourable than resonant harmonic base excitations. 3.2.3. Final state of the cracks Final state of the cracks (opened or closed) in the bare frame (SF, WF) under harmonic base excitation is shown in Fig. 21, in the frame with full strong masonry (SF-SM, WF-SM) under earthquake ‘‘Newhall’’ is shown in Fig. 22, in the frame with full weak masonry
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Fig. 11. Horizontal displacement of the top of the bare frames (SF, WF).
Fig. 12. Horizontal displacement of the top of the frames with full strong masonry (SF-SM, WF-SM).
(SF-WM, WF-WM) under earthquake ‘‘Santa Monica’’ is shown in Fig. 23 and in the frame with perforated strong masonry (SFPSM, WF-PSM) under earthquake ‘‘Kobe’’ is shown in Fig. 24.
Based on the analysis results, which are not completely shown in this paper, and based on the results shown in Figs. 21–24, the following can be concluded:
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Fig. 13. Horizontal displacement of the top of the frames with full weak masonry (SF-WM, WF-WM).
Fig. 14. Horizontal displacement of the top of the frames with perforated strong masonry (SF-PSM, WF-PSM).
– Compared to bare frames, masonry-infilled frames have significantly smaller crack zones in beams and columns under dynamic excitations.
– Size of crack zones in beams and columns of bare frames under dynamic excitations does not depend much on rigidity of the frame.
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Fig. 15. Horizontal displacement of the top of the frames with perforated weak masonry (SF-PWM, WF-PWM).
Table 4 Maximal displacements of the top of the frame and maximal reinforcement stresses at the bottom of the column. Case
SF WF SF-SM WF-SM SF-WM WF-WM SF-PSM WF-PSM SF-PWM WF-PWM a b
Horizontal displacementa (mm)
Reinforcement stressb (MPa)
Harmonic
Newhall
S. Monica
Kobe
Harmonic
Newhall
S. Monica
Kobe
22.2 67.5 2.1 2.5 4.7 5.2 2.3 2.6 5.3 5.6
40.3 58.7 0.6 0.8 6.2 10.5 0.8 1.2 20.2 21.8
18.6 29.2 0.9 1.2 5.0 6.1 2.4 3.0 6.2 18.3
32.2 88.4 0.7 0.6 6.7 7.1 1.0 1.7 13.8 29.3
117.1 219.1 67.7 61.0 52.3 55.6 63.6 58.7 52.4 58.6
155.0 164.5 32.5 32.3 68.2 77.8 31.9 38.4 110.0 101.1
112.1 88.0 33.9 46.4 59.4 65.5 54.3 66.2 76.2 91.8
220.1 236.3 32.3 32.7 59.6 65.5 32.0 46.6 99.8 101.2
Displacement in right direction ‘+’; displacement in left direction ‘’. Tensile stress ‘+’; compressive stress ‘’.
– Under the same dynamic excitation, weak masonry infill of the frame has wider crack zones than strong masonry infill. – Under the same dynamic excitations and for masonry-infilled frames, size of cracking zone in the masonry does not depend much on rigidity of the frame. 4. Conclusions Under the same dynamic base excitations, bare reinforced concrete frames with larger cross-sectional dimensions of beams and columns have significantly smaller horizontal displacements compared to reinforced concrete frames with smaller cross-sectional dimensions of beams and columns, which both have reinforcement calculated for the same loads and according to the same standards. In such cases, both types of frames have approximately equal reinforcement stresses and approximately equal width of crack zones in beams and columns.
In relation to bare frames, masonry-infilled frames have significantly smaller horizontal displacements, as well as lower reinforcement stresses in beams and columns, and smaller crack zones in the frames. In relation to the frames with strong masonry, frames with weak masonry have significantly greater horizontal displacements, as well as slightly higher reinforcement stresses in the frames and wider crack zones in the frames and masonry. Openings in the masonry-infilled frames increase their displacements, reinforcement stresses in the frames, as well as crack zones in the frames and masonry under dynamic excitations. For equal maximum base accelerations, horizontal displacements of bare frames and masonry-infilled frames, as well as reinforcement stresses and size of crack zones, are significantly influenced by the type of dynamic excitation. In such cases, earthquakes can be more unfavourable than resonant harmonic base excitations.
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Fig. 16. Reinforcement stress at the bottom of the columns of the bare frames (SF, WF).
Fig. 17. Reinforcement stress at the bottom of the frames’ columns with full strong masonry (SF-SM, WF-SM).
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Fig. 18. Reinforcement stress at the bottom of the frames’ columns with full weak masonry (SF-WM, WF-WM).
Fig. 19. Reinforcement stress at the bottom of the frames’ columns with perforated strong masonry (SF-PSM, WF-PSM).
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Fig. 20. Reinforcement stress at the bottom of the frames’ columns with perforated weak masonry (SF-PWM, WF-PWM).
Fig. 21. Final state of the cracks on the bare frames (SF, WF) – harmonic base acceleration.
Fig. 22. Final state of the cracks on the frames with full strong masonry (SF-SM, WF-SM) – earthquake ‘‘Newhall’’.
Fig. 23. Final state of the cracks on the frames with full weak masonry (SF-WM, WF-WM) – earthquake ‘‘Santa Monica’’.
Fig. 24. Final state of the cracks on the frames with perforated strong masonry (SFPSM, WF-PSM) – earthquake ‘‘Kobe’’.
G. Baloevic´ et al. / Engineering Structures 50 (2013) 43–55
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