Nghiên cứu thực nghiệm cơ chế phá hủy của kết cấu sàn phẳng bê tông cốt thép khi mất cột biên TT TIENG ANH

MINISTRY OF EDUCATION AND TRAINNING HANOI UNIVERSITY OF CIVIL ENGINEERING TRAN, QuocCuong EXPRIMENTAL STUDY ON THE COLLAPSE RESPONSE OF REINFORCED CONCRETE FLAT SLAB STRUCTURES SUBJECTED TO PERIMETER COLUMN LOSS Major: Construction engineering Code: 9580201 SUMMARY OF DOCTORAL THESIS Hanoi-2021 The dissertation is submitted at Hanoi University of Civil Engineering Supervisors 1: Associate Professor Ph.D PHAM XuanDat Supervisors 2: Associate Professor Ph.D NGUYEN TrungHieu Reviewer 1: Associate Professor Ph.D.TRAN Chung Reviewer 2: Associate Professor Ph.D TRAN The Truyen Reviewer 3: Ph.D NGUYEN Dai Minh The Phd thesis will be defended by School-level committee meeting at Hanoi University of Civil Engineering on day month year This thesis can be found at the National Library and the Library of Hanoi University of Civil Engineering GETTING STARTED Problem statement Flat slab structures are widely used in civil and industrial constructions, especially for supermarkets and sport centres Compared with the traditional building beam-slab structure, the advantage of the flat floor structures is that it is easy to construct, with greater clear floor heights However, flat slab structures are prone to progressive collapse due to their relatively heavy self-weight and susceptibility to punching shear failure Therefore, the current study with a tittle: “Exprimental research of collapse behavior on reinforced concrete flat slab structures subject to boundary column loss” was selected to investigate the structural behaviour of flat slab structures following a column removal so that the progressive collapse of such common building structures can be mitigated Research objectives • To construct innovative experimental models that can examine both static and dynamic behavior of flat slab structures subjected to a sudden column loss • To identify and evaluate the secondary load-carrying mechanisms in flat slab structures following a column removal • To identify and evaluate the dynamic increase factor (DIF) in flat slab structures following a column removal • To construct a simplified analytical model that can assess the collapse resistance of flat slab structures following a column removal Scope of work The collapse behaviour of reinforced concrete (RC) flat slab structures without drop panel The accidental scenarios selected for this research are the loss of either penultimate or perimeter column Academic basic of research (Justification of the experimental investigation) • Previous studies, both experimental and numerical, on the progressive collapse behavior of reinforced concrete building structures • Previous research, both experimental and numerical, on the large deformation response of reinforced concrete structures Research methodology The methodology used in this reasearch is the combination of experimental and theoretical studies scientific and practical significance of study This research has provided valuable insights in the collapse behaviour of flat slab structures following a column removal Also, it has provided a simplified analytical model that is able to assess the collapse resistance of flat slab structures in such accidental scenarios News research’s results • This research has successfully constructed innovative experimental models that are able to examine both static and dynamic responses of flat slab structures subjected to a sudden column loss • This research has successfully identified and evaluated the secondary load-carrying mechanisms in flat slab structures following a column removal • This research has successfully identified and evaluated the dynamic increase factor (DIF) in flat slab structures following a column removal • This research has successfully constructed a simplified analytical model that is able to assess the collapse resistance of flat slab structures following a column removal Outline getting starded : Chapter : OVERVIEW Chapter : EXPERIMENTAL STUDY Chapter : DISCUSSIONS ON THE TEST RESULTS Chapter : THE SIMPLIFIED APPROACH TO ASSESS THE COLLAPSE RESISTANCE OF FLAT SLAB STRUCTURES SUBJECTED TO COLUMN LOSS Conclusion : Conclusion and future works CHAPTER RESEARCH OVERVIEW ON RESPONSE OF REINFORCEMENT CONCRETE FLAT SLAB UNDER BEARING COLUMN LOSS This chapter will present the contents of building’s progress collapse due to the loss of a bearing column, the mechanisms of the structure in the large deformation and related studies updated in the world as well as in Vietnam 1.1 Progressive collapse of RC building 1.1.1 Progressive collapse concept Progressive collapse is a situation where local failure of a primary structural component leads to the collapse of adjoining members which, in turn, leads to additional collapse Hence, the total damage is disproportionate to the original cause [23] Figure 1.1 illustrates the Progressive collapse of the Murrah Building (1995) The community of structural engineers and professionals [21] both acknowledge that: (a) Location of the bomb explosion (b) After the explosion (photo Reuters) Figure 1.1: Murrah building before and after having a Progressive collapse [21] • The direct pressure of the car bomb can only destroy the G20 column and a small part of the floor area • The loss of bearing capacity of the transfer beam on the third floor due to the failure of the G20 column is the main cause of the collapse of half of the building 1.1.2 Mechanisms of progress collapse of RC structure in column loss case In column loss accident, the structure will be in a state of large deformation and it may lead to the risk of progress collapse due to the following effects: • The double span effect is shown in Figure 1.2 Figure 1.2: Increased internal force in two-span flat beams without support [45] • Dynamic effects is illustrated in Figure 1.3 The dynamic coefficient Ω (dynamic increast factor (DIF)) takes into account the amplification of the load (Mg gravity) due to the column loss situation Fd is the equivalent dynamic load acting on the system and causing displacement ∆m At equilibrium state, the dynamic coefficient is represented by equation (1.1) Fd Ω= (1.1) Mg Figure 1.3: Idealization of a typical floor [44] 1.2 Behavior of flexural reinforced concrete structures in high strain state and secondary load-bearing mechanisms 1.2.1 Behavior of flexural reinforced concrete structures in high strain state The secondary bearing mechanisms are illustrated in Fig 1.4 The formation and Figure 1.4: Compression arch action (CAA) and catenary action (CA) [37] development of compression arch action and catenary action in a state of great deformity is the main topic in research in the field of progressive collapse prevention 1.2.2 The secondary bearing mechanisms • Compression arch action (CAA) is illustrated with reinforced concrete beams restrained at both ends and subjected to uniformly distributed load q as shown in Figure 1.5(a) • Catenary action (CA), such as reinforced concrete beam subjected to uniformly distributed load W with restrained lateral displacement at both ends as shown in Figure 1.6 • Membrane action in reinforced concrete structure as shown in Figure 1.7(a) Figure 1.5: Compression arch action [43] Figure 1.6: Comparison of bearing capacity of catenary action and flexible action [43] The working mechanism of the slab when the deformation is large, the membrane effect can be classified into three different forms: • Compressive membrane action, Figure 1.9(a); • Tensile membrane action, Figure 1.9(b); • Tensile membrane action together with the development of a compressive ‘ring’ The conditions under which the membrane action occurs, including the tensile membrane action in the center, and the compression membrane action at the edges of the slab during large displacements, are summarized in Figure 1.8 Due to the reaction of the joint at the two ends of the floor, a developed compression arch can significantly improve the load-carrying capacity of the slab [25] and thereby increase the load resistance of the structural system 1.3 Research on reinforced concrete structures when losing bearing columns in the world and VietNam 1.3.1 Experimental research Sasani et all (2007) [48] studied with the load diagram shown in Figure 1.10(a) Tuan Pham (2017) [54] Research on 2D beam to clarify formation, development and load resistance of alternative load path, Figure 1.11(a) Dat Pham (2015) [45] tested 12 samples of reinforced concrete beam floor structure, Figure 1.11(b) Gouverneur (2014) [25], experimental research on three reinforced concrete slab samples, experimental model and loading diagram are illustrated as Figure 1.12 Figure 1.7: Formation of membrane action in RC flat slab structure [41] Figure 1.8: Types of membrane effects of flat slab structure [25] (a) Compressive membrance action (b) Tension membrance action Figure 1.9: Membrance action of restrained slab[25] Figure 1.10: Alternate load path in reinforced concrete beam [48] Russell Justin (2015) [46], University of Nottingham, UK Experimental study (static and dynamic) of flat floor structure without bearing columns QianKai and Li Binh (2016) [30] conducted an experiment on flat floor structures subjected to central column loss The test floor structure is loaded until it completely (a) Concrete beam structure [54] (b) RC floor beams Structure[44] Figure 1.11: Experiment of RC floor beams without a boundary column Figure 1.12: Experiment to evaluate the membrane action of flat floors [25] collapses to determine an alternative load path 1.3.2 Semi-empirical methods for calculation of building collapse resistance Izzudin and all proposed a computational model like Figure 1.13 [20] When Figure 1.13: Simple dynamic evaluation method [20] there is a maximum dynamic displacement Ud, the allowable impact load (P0) can be calculated by equalizing the area of the two figures shaded in Figure 1.13b, These areas represent the work of a static and dynamic external force Pham.X.D et al (2015)[44] have introduced a simple method to rapidly evaluate the load resistance of double span reinforced concrete beam-floor structures This equivalent one-degree-of-freedom system has a simplified stiffness and displacement relationship as elastic-plastic The load resistance function R of the structural system is built based on two specific parameters presented in Figure 1.14 Figure 1.14: Elastic-plastic relationship of RC structure at frame node[44] 1.4 Chapter summary and conclusion • Progressive collapse will be prevented if the collapse of the first floor structure continues after local failure does not occur • Miniature experimental model is a suitable method in laboratory conditions and is reliable within the scope of the doctoral thesis • Although there are quite a few empirical studies related to the reinforced concrete flat floor structure (the field of Progressive collapse) have been published recently, but these studies still have incomplete points • Developing a simple and reliable calculation method for flat-floor structures is necessary to assist engineers in practice during the structural planning phase CHAPTER EXPERIMENTAL RESEARCH ON MODEL OF RC FS SUBJECT TO BOUNDARY COLUMN LOSS The behavior of the reinforced concrete floor structure when losing a boundary column will be determined through experimental research on model objects in the laboratory Two case studies were including: Static test, carried out with two samples of reinforced concrete flat slab lost before a load-bearing column is slowly loaded until failure Dynamic test, conducted on a reinforced concrete floor slab with a sudden column loss while under the impact of the sevice load 12 (a) Column diagram (b) Column model Figure 2.3: Column system replaced by steel column 2.6 Measuring devices and instruments 2.6.1 Displacement measuring devices and instruments Diagram of displacement measuring device is arranged as in Figure 2.4 The LVDT1, LVDT3, LVDT5 measure vertical displacement, two LVDT2, LVDT4 measure horizontal displacement Đo chuyển vị ngang Figure 2.4: Layout of displacement measuring device (LVDT) 2.6.2 Equipment and tools for measuring steel strain Using a strain gauge (resistor plate) includes the following objectives: • Measure the axial strain of steel support columns; 13 • Measure the elongation strain of reinforcement in concrete to determine the working stages of the reinforcement under load; • Measure the strain in the temporary support steel column before and during the joint release of the dynamic test as shown in Figure 2.3 Mặt vị trí phiến điện trở trình bày Figure 2.5 Các phiến điện trở đánh số từ đến 15 The location plan of the resistor plates is presented as Figure 2.5 The resistor plates are numbered from to 15 Figure 2.6 shows how to Locaton of strain gauge lower layer Locaton of strain gauge upper layer (a) Specimens SP1, SP2 Locaton of strain gauge lower layer Locaton of strain gauge upper layer (b) specimen SP3 Figure 2.5: The layout of the resistor plates measure the internal force in each steel column (C-1 to C-5) The internal force in Figure 2.6: Layout diagram of steel column internal force measuring device the column is calculated from the deformation data in the experiment according to the formulas (2.1), (2.2), [42], [43], [44]: N1−1 = Es × As × (ϵ1 + ϵ2 + ϵ3 + ϵ4) /4 (2.1) M1−1 = EsIs × [ϵ4 − (ϵ1 + ϵ2 + ϵ3 + ϵ4) /4]/R (2.2) 14 Where: ϵ1, ϵ2, ϵ3, ϵ4 are the axial strain values of the steel column determined based on the readings of the resistor plates Es = 2∗ 106(N/mm2) is the young modulus of steel As, I, R are the cross-sectional area of the steel column, the moment of inertia and the outer radius of the steel column, respectively 2.7 Chapter conclusion • Miniature experimental model has been set up with 1/3 of the actual size • A suitable static and dynamic load method has been designed to allow static and dynamic tests to be performed • The measuring instruments and equipment have been arranged to allow the evaluation of the test sample behavior CHAPTER ANALYSIS AND ASSESSMENT OF EXPERIMENT RESULTS This chapter presents experimental results, including static and dynamic experiments Samples SP1, SP3 static testing Sample SP2 was dynamically tested due to sudden column loss 3.1 Analysis of static test results with samples SP1 and SP3 3.1.1 Crack distribution, pattern and failure mechanism of the test sample The crack distribution diagram of SP1, SP3 test samples is shown on Figure 3.1 The order of appearance of cracks (1), (3), (2), (4), (5), (6) Cracks on the underside Vết nứt 2,4,5,6 Vết nứt 1, 3, Vết nứt 2,4,5,6 (a) Sample SP1 Vết nứt 1, (b) Sample SP3 Figure 3.1: Diagram of crack development and failure mode of the test specimen of slab (1), (3), cracks on the top side of slab (2), (4), (5), (6) 15 3.1.2 Load-displacement relationship In Figure 3.2 is the largest load causing collapse of the test sample Figure 3.3 (a) Sample SP1 (b) Sample SP3 Figure 3.2: The load causes the test sample to collapse shows the detailed behavior of the test samples according to the load classes 20 20 18 16 P PH 14 Pmembrance Yiel state 10 Elastic-plastic state P GN SP-1 Elastic state fCP 10 20 30 40 50 60 (a) Specimen SP1 70 80 12 10 Elastic-plastic state P GN SP-3 Elastic state membrance action effect fCP 90 Tension membrance state Yiel state Displacement LVDT3 (mm ) Pmembrance PCP 14 PCP 12 PPH 16 Tension membrance state Load(kN/m2) Load (kN/m2) 18 membrance action effect 10 20 Displacement LVDT3 (mm ) 30 40 50 60 70 80 90 (b) Specimen SP3 Figure 3.3: Behavior of test samples according to load classes The behavior of the test sample is divided into four main stages Elastic phase, when the load-displacement line changes linearly (straight line) In the plastic elastic phase, the first small cracks appear, the load-displacement line still develops linearly (straight line) In the yielding phase, the nonlinear change charts show that the reinforcement is yielding continuously and not at the same time Segmentation (phase after yielding) has obvious involvement of the membrane action, corresponding to 20mm displacement at LVDT3 Sample SP1 crashes at destructive load PP H = 14kN/m2, sample SP3 at PP H = 18kN/m2 16 3.1.3 Membrane effect - behavior of flat slab under large deformation The membrane effect starts when the texture is in a state of great strain (double span effect, dynamic effect), Fig 3.4 In Figure 3.5, illustrating the formation of 3500 300 SP3 TSG11 SP1 TSG11 250 C1 column head displacement 150 100 displacement(mm) LVDT3 -50 20 40 60 -100 80 Yiel 2500 Change + 50 C1 column head displacement 3000 100 Biến dạng (10-6) Biến dạng (10-6) 200 2000 1500 SP3 BSG1 1000 SP1 BSG1 500 -150 displacement(mm) LVDT3 -200 (a) Upper layer steel 20 40 60 80 100 (b) Lower layer steel Figure 3.4: Reinforcement strain at the position of lost column of samples SP1, SP3 (a) Yield line diagram (b) Experimental moment balance diagram Figure 3.5: Schematic calculation by yield line method plastic lines in the diagram for calculating the load resistance of the test sample by the yieldine method Let the unit positive moment in the X and Y directions be mx, and my , respectively, and the negative unit moment m′x, and m′y The unit moment is determined by the following formula (Park and Paul, 1975) [38]: 0, 59 × As × fy (3.1) mx = As × fy × ds − fc′ Where: As is the area of tensile steel in the slab with unit width, ds is the calculated height of the slab During the yield phase, a vertical displacement at position LVDT3 (column loss position) will cause a displacement of ∆i at each rigid piece of the slab 17 and a rotation of θ at the ductile line The equation to balance the virtual work due to external force (actual load) with the virtual work caused by moment (internal force) isΣ as follows: Wult × ∆i = (mx × Ly + × my × Lx + mx′ × Ly + × m′y × Lx ) × θ (3.2) Create a virtual displacement y0 at the position where the column is lost, the work of the virtual external force due to the uniformly distributed load (Wult) is: Σ y0 Wult∆i = Wult × 2L (3.3) x When the virtual displacement y0 is small, the rotation3angle θ of the hard piece is y0 determined through y0 with θ = L So x y0 (3.4) (mxLy + 2myLx + m′xLy + 2m′y Lx )θ = W ult × 2Lx Therefore, the limiting load Wult calculated by the yield line method is determined by the equation (3.5) 6mx 6m′x mxLy + 2myLx m′xLy + 2m′yLx M′ Wult = + = + = W+ + 2 LyL2x/3 LyL x/3 LyL x/3 Lx Lx (3.5) When the test sample steel layout is averaged in two directions, then m′x = m′y = ′ mxy and mx = my = mxy Here M ′ = (2Lx + Ly )m′xy is the sum of negative 6mx moments along the edge of the test specimen, W + = is the contribution of L2x positive bending moment to the load resistance (tension membrance action) The magnitude of the critical load and moment according to the yield line formula is given in Table 3.1 Table 3.1: Calculation results of theoretical load capacity of fat slabs Lx Ly mxy m′xy Wult M’ W+ (m) (m) (kNm/m) (kNm/m) (kN/m2) (kNm) (kN/m2) 4,02 5,6 14,43 44,82 6,03 Figure 3.5 is a calculation diagram, showing that the total experimental negative moment MT′ N is determined by the equilibrium of: • The moment MS caused by the self-weight of the 80 mm thick slab (qS=2 kN/m); • Moment MT caused by ambient loads due to stacking of concrete blocks on the outer deck strip (700 mm wide) • The total bending moment MC in the supporting columns is determined by the equation (2.2), page 13 and illustrated in Figure 2.6 Table 3.2 presents a comparison of the actual total negative moment results with the theoretically calculated total negative torque The total theoretical negative moment is presented in Table 3.1 18 Table 3.2: Table of actual negative momen vs (and) theoretical kNm Ma′ ctual Ma′ ctual /M ′ Sample Mslab Mcube Mcolumn (0) (1) (2) (3) (4)=(1)+(2)+(3) (5) SP1 2,96 9,12 16,52 28,62 0,64 SP3 3,92 12,16 23,66 39,74 0,89 The tension membrance action WT MA can be determined by the equation (3.6) Ma′ ctual WTMA = Wactual — (3.6) LyL2x/3 ′ Where Ma ctual is the sum of experimental negative moments of samples SP1 and SP3 and Wactual is the experimental load causing floor slab collapse Table 3.3: Load and load resistance increase factor due to membrane effect Sample SP1 SP3 Ma′ ctual 28,62 39,74 Wactual(kN/m ) 14,0 18,0 WTMA(kN/m ) 6,85 8,07 W +(kN/m2) 6,03 6,03 WTMA/W 1,14 1,34 Table 3.3 shows a significant participation of the tension membrance action which significantly increases the load resistance Especially with the SP3 model, the load capacity increase factor is 1.34 times 3.2 Analyze and evaluate dynamic test results with SP2 sample SP2 sample test includes stages: • Stage 1: apply the static load to the value used (static test); • Stage 2: Dynamic test when sudden column loss occurs when the slab is under the effect of calculated load; • Stage 3: Static test, the static load continues to be applied until the sample collapses 3.2.1 Dynamic behavior of flat slab structure Figure 3.6(a) presents the deformation data of the steel support column at the time before, during and immediately after the situation where the column is lost Figure 3.6(b) shows a graph of the acceleration, characteristic of the dynamic effect 3.2.2 Deformation of reinforcement Summary of deformation variation of reinforcement is presented as Figure 3.7 Compared with the case of slab bearing static load, in case of dynamic effect Figure 3.7(a) shows a continuous change of reinforcement strain value in 400ms + 19 200 100 50 40 440 445 450 455 460 465 470 475 480 485 490 495 500 505 510 515 520 t (milliseconds) 30 -200 20 -300 10 Gia tốc (m/s2) Biến dạng (10-6) -100 -400 -500 -600 -800 -10 400 450 500 550 600 650 700 750 800 -20 BD14 BD15 -700 t (miliseconds) -30 ACC-1 ACC-2 -40 (a) Deformation of steel column (b) Acceleration oscillation graph Figure 3.6: Dynamic behavior of test sample SP2 2000 400 TSG-12 BSG-2 BSG-7 BSG-5 300 1600 1400 1200 1000 800 600 TSG-13 BSG-1 BSG-6 TSG-11 400 Biến dạng (10-6) Biến dạng (10-6 ) 1800 200 100 400 450 500 550 600 650 700 750 800 850 900 950 1000 -100 200 Thời gian (milliseconds) -200 400 450 500 550 600 650 700 750 800 850 900 950 1000 -200 Thời gian (milliseconds) -300 (a) Strain gauge 1,6,11 13 (b) Strain gauge 2,5,7 12 Figure 3.7: Deformation chart of reinforcement when losing columns 3.2.3 Determine the dynamic coefficient according to experimental data Dynamic effects are characterized by the Dynamic Increase Factor (DIF), which is calculated as follows: Vdyn DIF = (3.7) V stat Where: Vdyn is the maximum value of the survey object such as deformation, stress or displacement during the period under the influence of dynamic loads, Vstat is the stable value after switching off the oscillation and correspondingly under static load Vdyn Vstat DIF Table 3.4: Dynamic coefficients worksheet Rebar stress Longitudinal force in column C-5 Vertical displacement 54,5 kN 19,8 mm 1805 × 10−6 1608 × 10−6 42,57 kN 18,1 mm 1,12 1,28 1,09 3.2.4 Crack growth and failure mechanism of the SP2 sample Experimental results show that both SP1 and SP2 samples have the same failure form, the same crack form, see Figure 3.1(a) and Figure 3.8 20 (a) Sam SP1 (b) Sam SP2 Figure 3.8: The destructive form of the test sample When column C-1 has an inward displacement due to the tensile force from the sample center causing instability of the structural system, it is observed that slow failure of the rigid ring begins Before the complete collapse, at this stage the membrane effect has joined the bending mechanism, increasing the load capacity of the flat slab structure 3.3 Chapter conclusion • When the deformation is large, the tension membrance mechanism significantly increases the load resistance of the flat floor, the SP1=1.14 times (114%), SP3=1,34 times (134%) samples compared with the bending mechanism calculated according to the yieldine method; • When a bearing column is lost, the load is reallocated mainly to the columns near the loss of the column; • When the column is suddenly lost, the dynamic coefficient does not have a great influence on the load resistance and only happens in a short time The final failure pattern of the specimen in the dynamic test is similar to that in the static test; • A static test can be used to check the structural resistance to a sudden loss of a column; • The failure pattern of flat floors when the boundary column is lost tends to narrow the affected area around the position of the lost column (shown by the crack number (5)) CHAPTER SIMPLIFIED ANALYTICAL MODEL 4.1 Computational assumptions of the simple method • The progressive collapse resistance in the structural system can be assessed on a typical floor; • Determining the resistance according to two cases of loss of boundary columns; 21 • The behavior of the reinforced concrete flat slab is plastic elastic before collapsing; 4.2 Evaluation of dinamic effects The single-degree-of-freedom equivalent system (SDOF) is used to analyze the dynamic responses of a typical reinforced concrete slab structural system in case of sudden loss of penultimate column and of the middle boundary column that illustrated in Figure 4.1(a) Figure 4.1: Calculation model of dynamic effects when RC slab loses a column [44] ∆y, ∆p and ∆m are the displacements corresponding to the time of the beginning of yielding, the displacements during the yielding process and maximum displacement, Figure 4.1(b) Let the area SABCD=SAEF D, corresponding to the equation (4.1) 2µ∆ − (Meg)max = (4.1) Re 2µ∆ ∆m , Figure 4.1 Trong Re sức kháng tĩnh tải, µ∆ hệ số dẻo chuyển vị, µ∆ = ∆y Where: ∆mRe is the static load resistance, µ∆ is the displacement plasticity coefficient, µ = , Figure 4.1 (W )max is determined from the equation below (4.1): ∆ g ∆y 2µ∆ − W ult Wgmax = (4.2) 2µ∆ 4.3 Extreme load Wult of flat-floor structural system The ultimate yielding load Wult is predicted to be: 6mx 6m′ mx Ly + 2my Lx m′x Ly + 2m′y Lx M− + Wult = + 2x = + = W + LyL2x/3 LyL2x/3 LyL2x/3 Lx Lx (4.3) 22 4.4 Displacement plasticity coefficient µ∆ The vertical displacement plasticity coefficient µ∆ is calculated based on the rotational ductility coefficient µϕ of the [35] plastic joint The coefficient of rotational plasticity µϕ is determined by the formula (4.4) [35] ( 1/2) ρ′ d ′ 0, 8β1 Es ϵc fc′ ′ ′ 2 + (ρ − ρ )n − (ρ − ρ ) n + 2(ρ + µϕ = )n (4.4) d f 2y (ρ − ρ′) 4.4.1 Determine the plastic displacement ∆y according to the plastic curvature ϕy In Pham.X.D’s research, ∆y = ∆yb + ∆yj [44] The calculation diagram ∆y is illustrated as in Figure 4.2 (a) The finite element model calculates ∆yj (b) Deflection diagram ∆yb Figure 4.2: The diagram for calculating elastic displacement ∆y [44] Where ∆yb is the displacement at the position of the column loss at the beginning yield line of the slab ∆yj is the displacement at the position of the column loss caused by the curvature of the neighboring column ∆yb is calculated using the formula (4.5) q (2L)4 ∆yb = ∆max = (4.5) 384 (αEI) On the other hand, the curvature at the onset of yield at the most dangerous joint is written by the equation (4.6) q (2L)2 M′ = (4.6) ϕy = 12 (αEI) EI Where: M’ being the negative bending moment at the bearing where the joint has the smallest curvature coefficient From the two equations (4.5), (4.6) ∆yb is written as (4.7) L2 (4.7) ∆yb = ϕy With a flat slab structure, puncture failure is the most dangerous form of damage, corresponding to the structural form of stiff columns and weak beams Thus, the 23 Table 4.1: Experimental results ∆yj according to ∆yb [44] 250 × 300mm (L/d=20) 0,1 0,04 0,01 dimension beam Column 400 × 400mm 500 × 500mm 700 × 700mm 250 × 400mm (L/d=15) 0,13 0,06 0,02 250 × 500mm (L/d=12) 0,21 0,11 0,04 250 × 600mm (L/d=10) 0,31 0,16 0,06 Table 4.2: Determine the length of the flexible joint lp according to the member thickness d [35] Baker’s equation (k1 = 0, 9, k2 = 1, k3 = 0, 8) 1/3 lp = k1k2k3 (z/d) d z=10,55d (L/d=25) z=8,44d (L/d=20) z=6,33d (L/d=15) z=5,06d (L/d=12) z=4,22d (L/d=10) 1,58d 1,477d 1,33d 1,23d 1,16d 1,03d 1d 1d 0,922d 1d 0,922d 0,817d 1d 0,817d 0,753d 1d 0,753d 0,711d 1d 0,711d Mattock’s equation lp = 0, 5d + 0, 05Z ACI lp = 1, 0d value smallest coefficient in Table 4.1 can be used to calculate ∆yj in terms of ∆yb as (4.8) ∆yj = 0, 01∆yb (4.8) From the equations (4.7) and (4.8) the elastic displacement ∆y is determined by the equation (4.9) L2 (4.9) ∆y = ∆yb + ∆yj = 1, 01∆yb = 1, 01ϕy 4.4.2 Determine yield displacement ∆p according to yield curvature ϕp The calculation diagram ∆p is illustrated as Figure 4.3 Figure 4.3: Model for calculating yield displacement ∆p [44] The yield curvature ϕp only develops in the ductile length lp θp = ϕplp = (ϕm − ϕy) lp (4.10) The plastic displacement ∆p can be calculated using the equation (4.11) ∆p = θpL = (ϕm − ϕy) lpL (4.11) The smallest lp value is chosen in the simple method, corresponding to the smallest 24 plastic deformation, which is equal to 1d, according to the ACI criteria 4.4.3 Determine the plastic displacement coefficient µ∆ according to the rotational plasticity coefficient µϕ The plastic displacement coefficient µ∆ is determined by the equation (4.12) ∆p µ =1+ (4.12) ∆ ∆y Substitute the values of the equation (4.9) and the equation (4.11) into the equation (4.12) The displacement plasticity coefficient is written as (4.13) 8lp (4.13) µ∆ = + (µϕ — 1) 1, 01L Table 4.3: Displacement plasticity coefficient µ∆ according to rotational plasticity coefficient µϕ lp L µ∆ L/d=30 1d 30d 0,264µϕ+0,736 L/d=25 1d 25d 0,317µϕ+0,683 L/d=20 0,922d 20d 0,365µϕ+0,635 L/d=15 0,817d 15d 0,431µϕ+0,569 L/d=10 0,711 10d 0,563µϕ+0,437 20 18 Load W (kN/m2) 16 14 12 10 Wult-simple SP1 test SP2 test SP3 test 0 20 40 60 80 100 Displacement (LVDT3 mm) 120 140 160 Figure 4.4: Evaluating simple method compared to experiment Figure 4.4 is the result of calculating the static load resistance by a simple method and is determined by experiment The diagram shows a simple method in favor of safety 25 CONCLUSION • A suitable experimental model has been built to allow static and dynamic experiments The 1/3 scale model compared to the real structure, fabricated according to the instructions of the similar theory, has reflected the nonlinear static behavior as well as the nonlinear dynamics of the reinforced concrete slab no drop panel under the condition of real construction boundary • The thesis has proposed a semi-empirical calculation method through a static experiment that allows to quantify the contribution of the membrane effect to the load resistance of the flat slab • Based on the dynamic test results, the characteristic dynamic coefficients for longitudinal force in the supporting column: 1,28, reinforcement strain (stress in reinforcement): 1.12 and displacement at the position where the column is lost: 1.09 • A simplified analytical model has been developed that allows to quickly assess the progressive collapse resistance of the reinforced concrete flat slab structure without column caps when suddenly losing the boundary columns Wgmax = µ∆ = + 2µ∆ − W ult 2à 8lp 1, 01L (à 1) ã Provide a set of test data for the reinforced concrete flat floor under the effect of static and dynamic loads, especially in the state of large deformation PUBLISHED SCIENTIFIC ARTICLES [CT1] Tran Quoc Cuong, Nguyen Hoang Long, Pham Xuan Dat, Nguyen Trung Hieu (2017), "Dynamic behavior of reinforced concrete Flat slab Subjected to sudden loss of column", Vietnam Construction Magazine, ISSN 0866-0762, 167170 [CT2] Tran Quoc Cuong, Truong Ngoc Son, Pham Xuan Dat, Nguyen Trung Hieu (2017), "Experimental investigation on the behavior of reinforced concrete Flat slab subjected to static load and column removal", Vietnam Construction Magazine, ISSN 0866-0762, 143-147 [CT3] Tran Quoc Cuong, Nguyen Ngoc Linh, Ha Manh Hung Nguyen Trung Hieu, Pham Xuan Dat (2019), "Experiments on the collapse response of flat slab structures subjected to column loss", Magazine of Concrete Research, ISSN 0024-9831 Online E-ISSN 1751-763X Volume 71 Issue 5, March, 2019, pp 228-243 ... committee meeting at Hanoi University of Civil Engineering on day month year This thesis can be found at the National Library and the Library of Hanoi University of Civil Engineering 1 GETTING... pattern of the specimen in the dynamic test is similar to that in the static test; • A static test can be used to check the structural resistance to a sudden loss of a column; • The failure pattern... self-weight and susceptibility to punching shear failure Therefore, the current study with a tittle: “Exprimental research of collapse behavior on reinforced concrete flat slab structures subject