Effects of masonry infills on the responses of reinforced concrete frame structures under seismic actions

hrough the modified Takeda model and its force-displacement curve is taken according to ASCE 41-13 (Figure 2.2). a) b) c) Figure 2.2. a) Plastic b) The modified Takeda c) Generalized M–θ deformation concentrated hysteresis rule relationship at plastic on the frame components hinges of RC frame components 2.3. ESTABLISH THE NONLINEAR BEHAVIOR MODEL OF THE MIs IN RC FRAMES 2.3.1. Setting up the force-displacement relationship of the model The behavior of the MIs in the frame is modeled as a curve shown in Figure 2.3. In the frame model, the MIs are shown in Figure 2.4. 7 2.3.2. Define the basic parameters of the model 2.3.2.1. The stiffness of MIs According to Nguyen Le Ninh (1980), the width (1 ) 0 m n m mw e w −= (2.1) with ( )0 m m h l dw h l kλ λ = + + (2.2); 4 42 2;4 4 m m m m m m h l c c m c b m E t l E t h E I h E I l λ λ= = (2.3) where n = H/Hu, H is the lateral force and Hu is the lateral force at the time when MI reaches the ultimate strength; m and k are coefficients depending on the type of masonry; other parameters indicate the geometric and mechanical properties of frames and MIs (Figure 1.8). From the width wm, determine the stiffness of the MI at the beginning of the crack (2.4) and when reaching to the ultimate strength (2.5): 0.4 20 cos m m m m my m e w t EK d θ= (2.4); 20 0.4cos mym m m mu m m Kw t EK d e θ∗ = = (2.5) 2.3.2.2. The strength of the MIs 1. The ultimate strength of masonry infill Vmu is determined from the condition ( )min ,mu ms mcV V V= (2.6), where: a) Vms is the sliding shear strength of MIs selected from approaches of following authors: Rosenblueth (1980); Smith and Coull (1991); Paulay and Priestley (1992); Decanini et al. (1993); Panagiotakos and Fardis (1994), Fardis (2009); Zarnic and Gostic (1997); FEMA 356 (2000), Al- Chaar (2002), ASCE 41-06, ASCE 41-13; Galanti et al. (1998), EN 1998- 1:2004; FEMA 306 (1998); EN 1996-1-1:2005; according to TCVN 5573:2011 (2.10). 11 0.72 bs m m ms f t lV n tgµ θ = − (2.10) Figure 2.3. The force-displacement relationship Figure 2.4. Position of plastic of the MI’s behavior model hinges in the model of infilled frames 8 b) Vmc is the diagonal compression strength of MIs selected from approaches of following authors: Smith and Coull (1991); Decanini et al. (1993); Galanti et al. (1998); FEMA 306; Al- Chaar (2002); Tucker (2007); ASCE 41-13. In order to select the appropriate strengths for MI’s model, comparative analyses are performed on the infilled RC frame consistent with the object and objectives of the research. The results are the curves representing relationships of Vms and Vmc associated with the common hm/lm ratios of MIs in Figures 2.9 and 2.10. Since then, choose the strength Vms according to TCVN 5573:2011 (2.10) and the strength Vmc according to ASCE 41-13: cos 3 m mc mc m hV f t θ= (2.11) 2. The yielding strength of the masonry infill Vmy is selected from approaches of following authors: Nguyen Le Ninh (1980), Dolsek and Fajfar (2008); Decanini et al. (1993); Panagiotakos and Fardis (1994); Saneinejad and Hobbs (1995), FEMA 306; Tucker (2007); Stavridis (2009). Similarly, from the results in Figure 2.12, choose Vmy = 0.6Vmu (2.12) as suggested by Nguyen Le Ninh and Dolsek and Fajfar (2008). 3. The residual strength of the masonry infill Vmr: 0 0.1mr myV V≤ ≤ (2.13) Figure 2.9. Variation of Vms determined by different approaches associated with hm/lm Figure 2.10. Variation of Vmc determined by different approaches associated with hm/lm Figure 2.12. Variation of Vmy determined by different approaches associated with hm/lm 9 2.3.2.3. Steps to establish the force-displacement curve of the model Step 1. Determine Kmy using (2.4). Step 2. Determine Vmu using (2.6). Step 3. Determine mu mu muV K ∗∆ = (2.14). Step 4. Determine Vmy using (2.12). Step 5. Determine my my myV K∆ = (2.15). Step 6. Determine Vmr using (2.13). Step 7. Determine ( ) /mr mu mr mu mrV V K∆ = ∆ + − (2.16) 2.3.2.4. Axial nonlinear response of equivalent diagonal strut Using the stress-deformation relationship of masonry proposed by Kaushik, Rai and Jain (2007) (Figure 2.13). 2.3.3. Calibrate the behavior model of the MI Calibration of the proposed model is performed based on the experimental data of infilled RC frame models designed according to EC8 and EC2 of Kakaletsis and Karayannis (2008) and Morandi, Hak and Magenes (2014- 2018). 2.3.3.1. Kakaletsis and Karayannis (2008) Based on the parameters of the experimental models, we establish the behavior models of the MIs based on the steps in section 2.3.2.3 (Figure 2.16). Using these models together with the behavior models of the RC materials and structural elements selected in section 2.2, performing a nonlinear pushover analysis of the experimental frame models. The capacity curves obtained from analyses are compared with the experimental force-displacement envelopment (Figure 2.18). The results show a good fit between them. Figure 2.18. Comparison between experimental results and analytical results using the proposed method a) Weak MI (S) b) Strong MI (IS) Figure 2.16. Force - displacement relationship of the proposed MIs’ models Figure 2.13. Idealized stress-strain relationship for masonry under uniaxial compression 10 2.3.3.2. Morandi et al. (2014-2018) Figure 2.20. Force - displacement relationship of the proposed MI’s model Similarly, set up a behavior model of the MI using the proposed method (Figure 2.20) and perform a nonlinear pushover analysis of the experimental frame models. The results show that the capacity curves obtained from the analyses are quite consistent with the experimental envelopment (Figure 2.21). 2.4. REMARKS ON CHAPTER 2 A simple model is established to simulate the behavior of the MIs in RC frames taking into account the decrease in strength and stiffness of frame and MIs. The verification results on the infilled RC frame models designed according to the current seismic conception exhibit good results. So, the calibration of the model is not necessary. CHAPTER 3 EFFECTS OF MASONRY INFILLS TO THE CONTROL OF THE FAILURE MECHANISM OF RC FRAME STRUCTURES UNDER SEISMIC ACTIONS 3.1. MODERN CONCEPTION AND DESIGN RULES FOR FRAMES IN THE CURRENT SEISMIC DESIGN STANDARDS 3.1.1. Modern conception in design of structures for earthquake resistance According to the current seismic conception, the design purpose of a building is to protect directly both human life and social properties. When a strong earthquake occurs, the buildings are allowed to work beyond the elastic limit, but they are not collapsed suddenly. 3.1.2. Basic design principles according to modern seismic conception From the aforementioned goals, the structure must be designed to experience plastic failure, and shear failure must happen after flexural failure when a strong earthquake occurs. Figure 2.21. Comparison between experimental results and analytical results using the proposed method 11 3.1.3. Design RC frames according to current seismic standards To carry out the above design principles, the capacity design method is used. By using this method, the forces used to design a frame must be as follows, for example, according to TCVN 9386:2012 (the “so-called” basic design principle of strong columns - weak beams): a) Beam: The bending moment M and the axial force N are taken from the results of structural analysis, while the shear force Q is determined from the bending resistance of the beam. b) Column: The bending moment M is redefined from the following condition: 1.3Rc RbM M≥∑ ∑ (3.1) in which: ΣMRc is the sum of the minimum design values of the moment resistances of the columns framing to the joint, taking into account the column axial force N in the seismic design situation; ΣMRb is the sum of the design values of the moment resistances of the beams framing to the joint. Shear force Q is redefined from the flexural strength of columns. Remarks: (i) The frame design process must follow a very strict process; (ii) Frame design rules do not take into account frame-MI interaction. 3.2. EFFECTS OF MIs TO THE BEAM RESPONSE Experimental studies on the infilled frames show that the interactive forces with the MIs make the beams behave more stiffly than that of bare frames. To clarify this phenomenon, consider a RC frame without MI (bare frame) as shown in Figure 3.2a. The external force H causes the bending moment at the ending section C of the beam: , 3 2 6 1bC H HhM ω ω = + (3.2) where: b c I h I l ω = (3.3) a) Bare frame; b) Infilled frame; c) Equivalent infilled frame Figure 3.2. Models for calculation of the frame The curvature of the beam at the end C has the following value: ,, 3 2 6 1 bC H bC H c b c b M Hh E I E I ωρ ω = = + (3.4) 12 When MI is available, the model to calculate an infilled frame is as shown in Figure 3.2b, where Rm is the compression force in the diagonal strut with the area of cross-section of wmtm. Replace the model in Figure 3.2b with the equivalent model in Figure 3.2c (Vm is the horizontal projection of the compression force Rm in the diagonal strut). With this model, we have the moment and curvature of the beam when taking into account the interactive force with MIs: ( ) , - - 3 2 6 1m m bC H V H V h M ω ω = + (3.5) , - , - , ( - ) 3 2 6 1 m m bC H V m bC H V bC H c b c b M H V h E I E I ω ρ ρ ω = = < + (3.6) Thus, the interaction with MIs makes the beam stiffer. Let Ibm ( >Ib) be the equivalent moment of inertia of the beam when considering interaction with MIs. Similarly (3.4), we will get the curvature of the beam in this case: * ,* , 3 2 6 1 bC H m bC H c bm c bm m M Hh E I E I ω ρ ω = = + (3.7), where: bmm c I h I l ω = (3.8) Considering (3.3) and (3.8), we obtain the coefficient bm mIb b Ik I ω ω = = (3.9) which indicates the increase in moment of inertia (flexural stiffness) of the beam when interacting with MIs. Balancing the curvatures (3.6) and (3.7), we establish the relationship: 6 1 6 1 m m H H V ω ω + = + − (3.11) From the relationship between horizontal force H and Vm established on the basis of the calculation diagrams in Figures 3.2b, 3.2c, and from (3.11), we set the ratio ωm/ω. With this result, determine the coefficient kIb (3.9) at the ultimate time (wm = wm0 when n = 1.0 see Chapter 2) when considering the interaction with MIs: 3 2 0w cos 3 21 72 bmu m m m Ibu b c c m I h t Ek I E I d θ ω ω + = = + (3.18) Equation (3.18) shows that, when considering the interaction with MIs, the moment of inertia of the beam Ibmu is increased by kIbu times: Ibmu = kIbuIb. This means that the cross-section height of the beam is increased to 3 bmu b Ibuh h k= (3.19) called the equivalent cross-section 13 height. The increase in the cross-section height of the beam leads to an increase in its bending resistance both positive RbM + and negative RbM − for the considered sense of the seismic action. In the general case, at any column-beam joint: Rbmu Rbmu Rbmu Rb Rb RbM M M M M M − + − += + > = +∑ ∑ (3.21) where ΣMRbmu and ΣMRb are the sums of the design values of the moments of resistance of the beams framing the joint when considering and not considering the interaction with MIs for the considered sense of the seismic action, respectively. Thus, when considering the interactive forces with MIs, the moments of resistance of the beams are increased by the following coefficient: - - 1 Rbmu Rbmu Rbmu Mb Rb Rb Rb M M Mk M M M + + + = = > + ∑ ∑ (3.22) 3.3. METHODS TO DESIGN THE RC FRAMES FOR EARTHQUAKE RESISTANCE WHEN CONSIDERING THE INTERACTION WITH THE MIs 3.3.1. Condition to control the failure mechanism of the RC frames From the above mentioned research results, in cases taking into account of the interaction with MIs, the condition to control the plastic failure mechanism (3.1) of a frame in TCVN 9386:2012 may not be accurate, because RbM∑ in the right-hand side is increased via kMb. This also means that the columns may be failed before the beam and the soft story failure mechanism may appear unintentionally. Therefore, to let the infilled frames be failed plastically as the design purpose, the design conditions (3.1) shall be rewritten as follows: 1.3Rcmu Mb RbM k M≥∑ ∑ (3.23) in which ΣMRcmu is the sum of the minimum design values of the moment resistances of the columns framing to the joint, taking into account the column axial force N in the seismic design situation at the ultimate limit state of MIs. With this condition, whether the MIs are available or not, the design principle of "strong columns - weak beams" will be guaranteed and the frames will be failed in plastic mechanisms under strong earthquakes. 3.3.2. The method to design RC frame structures under seismic actions when considering the interaction with the MIs Step 1. Design and detail of RC beams in accordance with current seismic design standards. 14 Step 2. Determine kIbu in (3.18) and the equivalent cross-section height of beam hbmu in (3.19). Then determine the moment resistances of the equivalent beams when considering the interaction with MIs RbmuM − and RbmuM + . Determine kMb in (3.22). Step 3. Determine the bending moment to design the columns RcmuM∑ in the proposed condition (3.23). Then design and detail the longitudinal reinforcement of columns according to the rules of the current seismic design standard. 3.4. CALCULATION EXAMPLES 3.4.1. The calculation data A 3-storey cast-in-place RC frame building with dimensions as shown in Figure 3.4. The exterior beams of 25x45 cm, the interior beams of 25x50 cm, the slab thickness of 15 cm. Materials: concrete B30, longitudinal reinforcement type CB400-V, stirrup reinforcement type CB240-T. The KB and KE frames are filled with solid brick masonry 20 cm thick, burnt clay bricks M100, cement mortar M75. Vertical load (permanent load g and imposed load q) at each floor (including roof): g + ψ2q = 9 kN/m2. The building is built in a region with the reference peak ground acceleration on type A ground (rock) agR = 0.1097g, ground type D, importance factor γI = 1.2; ductility class medium (DCM) according to TCVN 9386:2012. a) Plan view of the typical floor b) Elevation of the frame Figure 3.4. Models for the frame structure 3.4.2. Design the RC frame structures according to the regulations of TCVN 9386:2012 The reinforcement details of typical frame KE are shown in Figure 3.5. 15 Figure 3.5. Reinforcement details of frame KE 3.4.3. Determine responses of frame KE designed according to TCVN 9386: 2012 Using nonlinear pushover analysis is to determine responses of frame KE. Behavior models of materials and frame components are taken from EC2 and ASCE 41-13. The analysis results show that the frame is failed in agreement with the plastic mechanism as the design goal set out (Figure 3.6). The capacity curve is shown in Figure 3.7 (solid line). a) Step 6 b) Step 22 c) Step 48 d) Step 102 Figure 3.6. Behavior of frame KE designed according to TCVN 9386:2012 Figure 3.7. Capacity curves of frame KE in different cases 16 3.4.4. Determine responses of frame KE designed according to TCVN 9386:2012 when considering the interaction with MIs Calculation results for force-displacement relations of the MIs are shown in Figure 3.8. a) 1st-floor b) 2nd to 3rd floors Figure 3.8. Force-displacement relationship in the behavior model of MIs The pushover analysis in Figure 3.10 shows that the plastic deformation process starts from the MIs to beams and the bases of columns on the first floor. The capacity curve (dashed line) in Figure 3.7 shows that the frame stiffness drops suddenly and varies irregularly when the base shear force reaches its maximum value of V = 626.27 kN and ∆ = 0.023 m in step 10 because the MIs on the first and second floors are failed considerably. Until the target displacement ∆ = 4% H = 0.36 m (step 108) is achieved, the plastic deformations are almost focused on the bases of columns on the foundation surface and the top of columns on the first floor, the MIs on the first floor are no longer capable of bearing (Figure 3.10). The infilled RC frame is failed in agreement with the “soft storey” mechanism. Comparing the capacity curves of frame KE in Figure 3.7 (without considering (solid lines) and considering (dashed lines) the interaction with MIs) shows that the interaction with MIs has greatly increased the stiffness, horizontal bearing capacity, and energy dissipation capacity of the frame in the initial elastic phase. 3.4.5. Design and detail the RC frame structures considering the interaction with MIs using the proposed method The design of the frame structure shown in Figure 3.4 is implemented using the method proposed in section 3.3.2. Figure 3.10. Behavior of frame KE when considering the interaction with MIs a) Step 3 b) Step 10 c) Step 15 d) Step 108 17 Step 1: Calculate beam reinforcement of frame KE and calculate their flexural resistance MRb as design results in section 3.4.2 (Figure 3.5). Step 2: Determine kIbu = 2.508 and the equivalent cross-section height hbmu = 680 mm, thereby determine bending resistance of the beams and the coefficient kMb = 1.14. Step 3: Determine the required bending moment to design the columns yc RcmuM∑ from the proposed condition (3.23), from which design and arrange longitudinal reinforcement of the columns. Compared with the standard design results (Figure 3.5), the cross-section height of columns on the first floor (C1 and C4) must be increased by 50mm while the reinforcement of all columns remains the same. Performing pushover analysis is to determine responses of infilled frame KE with behavior models of materials, structural members, and MIs used in the calculation example in section 3.4.4. The analysis results show that the frame designed using the proposed method isn’t failed corresponding to the soft storey mechanism (Figure 3.11). The capacity curves in Figure 3.7 show that frame KE designed using the proposed method with the condition (3.23) (dashed double-dot line) has superior behavior compared with the case designed according to the condition (3.1) of TCVN 9386:2012. 3.5. REMARKS ON CHAPTER 3 1. The increasing coefficients of flexural stiffness kIbu and flexural resistance kMb of beams are quantified when considering the interaction with MIs. 2. On this basis, the condition that controls failure mechanism (3.23) is proposed to replace the condition (3.1) of TCVN 9386:2012 which is no longer accurate when considering the interaction with MIs. Then a method to design RC frame structures for earthquake resistance is proposed. 3. Specific calculation examples have demonstrated the reliability of the performed theoretical research: the model of MIs, the method to design RC frame structures for earthquake resistance when considering the interaction with MIs, etc. Figure 3.11. Behavior of frame KE designed using the proposed condition (3.23) a) Step 11 b) Step 17 c) Step 113 18 CHAPTER 4 CONTROL OF LOCAL FAILURES OF RC FRAMES UNDER SEISMIC ACTIONS CONSIDERING THE INTERACTION WITH THE MIs 4.1. CONTROL OF LOCAL FAILURES OF RC FRAMES IN CURRENT SEISMIC DESIGN STANDARDS 4.1.1. Control of shear failure in RC frames Shear failure is a brittle failure mode, so it must be prevented, and is not allowed to occur before flexural failure. For frame columns, according to TCVN 9386:2012, the design shear force is determined from the bending resistance of the column (called the capacity shear force) (Figure 4.1). ,1 ,2 1 2 , , Rb Rb Rd Rc Rc Rc Rc CD c cl c M M M M M M V l γ      +            = ∑ ∑ ∑ ∑ (4.3) where lcl,c is the clear length of the column; MRc,i is the design value of the column moment resistance at the end i (i = 1, 2) in the sense of the seismic bending moment under the considered sense of the seismic action; ( / )Rb Rc iM M∑ ∑ ≤ 1 where ∑MRc and ∑MRb are the sums of the design values of the moment resistances of the columns and the sum of the design values of the moment resistances of the beams framing into the joint, respectively; γRd is the factor accounting for overstrength. The values of MRc,i and ∑MRc should correspond to the column axial force in the seismic design situation for the considered sense of the seismic action. 4.1.2. Verification of column shear failure in seismic standards TCVN 9386:2012 and EN 1998-1:2004 require checking and detailing of columns in shear considering the interaction with MIs through the condition: , , ,Rd c Ed c lcV V≥ (4.4) Figure 4.1. Diagram of determining column shear force Figure 4.2. Acting shear on the columns due to MIs 19 in which VRd,c is the shear resistance at the ends of the columns designed according to the standard; VEd,c,lc is the increased design shear due to the horizontal strut force acting at the column ends (Figure 4.2): ( ), , , , , ,min ;Ed c lc Ed c ms Ed c MV V V= (4.5) where: (i) , ,Ed c ms m m mvV V A f= = (4.6) with Am = tmlm and fmv is the shear strength of the MIs; (ii) , , ,2Ed c M Rd Rd c cV M lγ= (4.7) Other countries' standards are the same. 4.1.3. Remarks on the rules for verification of shear failure 1. There is a high agreement among the standards: the interaction with the MIs is not considered when designing overall the frame but the interactive forces with the MIs are considered when verifying the columns in shear. 2. The instructions in TCVN 9386:2012 and EN 1998-1:2004 are rather ambiguous, leading to various interpretations when they are applied, e.g. the diagonal strut width wm, the length of contact regions lc, etc. 4.2. FRAME - MI INTERACTIVE FORCES AND LOCAL RESPONSE OF RC COLUMNS UNDER INTERACTIVE FORCES According to Nguyen Le Ninh, the contact lengths zh and zl between the MI and the frame change when the infilled frame is subjected to lateral load. At the ultimate time of MI (n = 1.0): 0 0 2h hz β π λ= and 0 0 2l lz β π λ= (4.14) with: ( )0 m mk h l d w h l k β λ λ = + + (4.13) Along the contact regions zh and zl, interactive stresses which are assumed to be linearly distributed appear, causing the force Rm in the equivalent diagonal strut (Figure 4.3). According to Tassios et al. (1988), it is possible to divide Rm into 3 parts as shown in Figure 4.4. At the ultimate state of the MI (n = 1.0), the interactive forces at the contact regions of column and beam with the MI are determined by the following expressions (Figure 4.5): Figure 4.4. The distribution of strut’s force on frame’s elements Figure 4.3. The interactive forces between frame and MI 20 0 00.8h mu hq V z= (4.17) 0 00.4l mu lq V tg zθ= (4.18) where Vmu is the horizontal projection of the force in the diagonal strut Rmu. From the force qh0, determine the column shear force due to the local interaction with the MI (Figure 4.5): 3 4 0 0 0 0 0 0 , 2 3 , ,2 4 10 h h h h h h c mA cl c cl c q z q z q zV l l = − + (4.19) 3 4 0 0 0 0 , 2 3 , ,4 10 h h h h c mB cl c cl c q z q zV l l   = − −     (4.20) 4.3. METHOD TO DESIGN THE RC COLUMNS IN SHEAR WHEN CONSIDERING THE FRAME-MI INTERACTIVE FORCES 4.3.1. Condition to control the column shear failure When considering the interactive forces with the MIs, the capacity design shear of columns, VCD,c,m is determined in (4.3) in which ∑MRcmu is determined by increasing kMb times using the proposed method in section 3.3, Chapter 3. So it will be greater than VCD,c determined according to TCVN 9386:2012. However, this increase in shear force is only caused by the stiffening effect of the beams, not counting the interactive forces between the MIs and the columns. Therefore, the design shear of columns will be determined from the following proposed condition: , , , , , ,max( ; )Ed c m CD c m c pt mV V V= (4.23) where , , , ,c pt m c pt c mV V V= + (4.24) is the column shear force determined from structural analysis considering the local interaction with the MIs; Vc,pt is the column shear force determined from the st