Progressive Collapse Resistance Demand of Reinforced Concrete Frames Under Catenary Mechanism

Transcription

1 Progressive Collapse esistane Demand of einfored Conrete Frames Under Catenary Mehanism Author Li, Yi, Lu, Xinzheng, Guan, Hong, Ye, Lieping Published 014 Journal Title ACI Strutural Journal Copyright Statement 014 Amerian Conrete Institute. The attahed file is reprodued here in aordane with the opyright poliy of the publisher. Please refer to the journal's website for aess to the definitive, published version. Downloaded from Link to published version mdetails&i Griffith esearh Online

2 ACI STUCTUAL JOUNAL TECHNICAL PAPE Title No. 111-S104 Progressive Collapse esistane Demand of einfored Conrete Frames under Catenary Mehanism by Yi Li, Xinzheng Lu, Hong Guan, and Lieping Ye Progressive ollapses are resisted by the atenary mehanism in reinfored onrete (C) frame strutures undergoing large deformations. esearh to date has mainly foused on the nonlinear dynami progressive ollapse resistane demand of this type of strutures under the beam mehanism (that is, for small deformations), and that the atenary mehanism is laking. As a first attempt, this study establishes a dynami amplifiation fator for evaluating the resistane demands of C frames under the atenary mehanism. To ahieve this, an energy-based, theoretial framework is proposed for alulating the aforementioned demands. Based on this framework, the analytial solution for the ollapse resistane demands of regular C frames under the atenary mehanism is readily obtained. Numerial validation indiates that the proposed equations an aurately desribe the progressive ollapse demand of C frames undergoing large deformations. Keywords: atenary mehanism; energy onservation priniple; progressive ollapse; reinfored onrete frame; resistane demand. INTODUCTION Progressive ollapse of building strutures is defined as loal damage aused by an aidental event propagating throughout the entire strutural system. 1 In the past few deades, progressive ollapse events triggered by gas explosions, bombing attaks, fire, and vehiular ollisions have been ontinuously reported. Extensive researh has been onduted to investigate and improve progressive ollapse resistane of building strutures. Existing experimental -5 and numerial 6-8 investigations have demonstrated that progressive ollapses of reinfored onrete (C) frame strutures are resisted by end moments of the beams for small deformations (namely, the beam mehanism) and by axial tensile fores in the beams for large deformations (namely, the atenary mehanism). The latter is a one-way load-transfer mehanism where the external load is resisted by the beams in one diretion. When the frame strutures undergo large deformations, ast-in-plae C floor slabs exhibit a two-way membrane tension effet 9 in whih the load transfer in eah diretion is equivalent to a atenary mehanism. The atenary mehanism is the prototype model of the tie fore (TF) method adopted in design odes, 10-1 by whih the axial fore demand of beams (in one diretion) and floor slabs (in two diretions) is alulated. Progressive ollapse of an C frame struture an be effetively prevented through atenary ation, whih enhanes the strutural integrity after a part of the struture has beome a mehanism. Note in the existing TF method that only the nonlinear stati () demand under the atenary mehanism is onsidered. The dynami effet of the atual progressive ollapse proess is, however, negleted, and this may lead to an unsafe design. 13 ACI Strutural Journal/September-Otober 014 The nonlinear dynami () demand is a more realisti representation of the progressive ollapse resistane demand, whih an be alulated using dynami equilibrium equations or energy onservation equations. However, to solve dynami equilibrium equations, nonlinear dynami finite element analysis is ommonly performed, whih is time-onsuming. Hene, dynami analyses are generally used for buildings with high seurity risks. 10,11 To solve energy onservation equations based on the onventional methods, both energy dissipation and work done by the unbalaned gravity load must be alulated for every single strutural member, whih is also a tedious task for engineering design. In view of the aforementioned, more engineer-friendly methods, suh as linear stati or nonlinear stati () methods, have been widely adopted in the existing progressive ollapse design ode. 10 This is followed by the orretion of the aforementioned demand to onveniently approximate the demand using a DAF. Based on this onept, the progressive ollapse resistane demand under the beam mehanism has been studied in the past, using the dynami method 17 and the energy method However, little researh has been devoted to the demand evaluation under the atenary mehanism for C frame strutures. In addition to the aforementioned researh gap, another ritial problem assoiated with the existing energy-based approahes is the energy distribution in different strutural omponents In these approahes, the single-degree-offreedom (SDOF) model is widely adopted to represent the mehanial harateristis, inluding ollapse resistane and deformation, of the ollapse-resisting substrutures, as shown in Fig. 1. It should be noted that the resistane of the SDOF model refers to the ollapse-resisting apaity of the overall struture, namely the strutural resistane. Note that the strutural resistane is provided by the strutural elements in the substruture, but is not a simple summation of their individual elemental resistane due to ompliated load-redistribution harateristis in the strutural system. Aordingly, the DAF for the strutural resistane is different from that for the elemental resistane. The DAF 18-0 for the beam mehanism obtained based on the SDOF model is the strutural-level DAF, DAF s. However, in engineering designs the elemental-level DAF, DAF e, is required to determine the resistane demand of the strutural elements. ACI Strutural Journal, V. 111, No. 5, September-Otober 014. MS No. S , doi: / , was reeived Otober 30, 013, and reviewed under Institute publiation poliies. Copyright 014, Amerian Conrete Institute. All rights reserved, inluding the making of opies unless permission is obtained from the opyright proprietors. Pertinent disussion inluding author s losure, if any, will be published ten months from this journal s date if the disussion is reeived within four months of the paper s print publiation. 15

3 Fig. 1 C frame struture under atenary mehanism. Further, the relationships between the DAF s and the DAF e desribe the distribution of the total energy dissipation in resisting progressive ollapse throughout the strutural system. However, the values of DAF s and DAF e have not been identified in the existing studies and, more speifially, the elemental-level DAF e has not yet been studied before. This paper thus aims to present an energy-based, theoretial framework for alulating the strutural and elemental demand relationships between the and demands of C frame strutures under the atenary mehanism. Based on this framework, the DAF s and DAF e are derived for regular C frames undergoing large deformations. The two DAFs are validated through a series of numerial examples with varying parameters. ESEACH SIGNIFICANCE This study investigates for the first time the dynami amplifiation fator (DAF) for C frames under atenary mehanism. The method proposed is useful for the effetive assessment of the progressive ollapse resistane demand of C frames undergoing large deformations, whih is a simpler alternative to the diret nonlinear dynami approah. Unlike existing work, the proposed demand relationship (DAF) is lassified into a strutural DAF and an elemental DAF, whih an be evaluated using an energy-based, theoretial framework developed in this study. This framework will failitate further studies on the nonlinear dynami effet of other types of strutural systems. FUAMENTAL CONCEPTS The aforementioned resistane demands an be lassified by the analysis targets (strutural or elemental demand) and the analysis tehniques ( or method) as shown in Table 1, whih are disussed as follows. Strutural and elemental demands For C frame strutures under the atenary mehanism, the ollapse resistane demands inlude the strutural demand of the substrutures and the elemental demand F of the beam members. and demands The progressive ollapse resistane demands under the atenary mehanism an be alulated using the or Table 1 Types of demands under atenary mehanism Analysis targets Analysis tehniques Strutural-level Elemental-level Nonlinear Stati Dynami Nonlinear stati strutural demand (SD, ) Nonlinear dynami strutural demand (SD, ) Nonlinear stati elemental demand (ED, F ) Nonlinear dynami elemental demand (ED, F ) method. The orresponding outomes of these methods are, respetively, referred to as the and demands. Detailing requirement Under large deformations, the atenary mehanism of a C frame is substantiated by the detailing requirement to provide effetive axial fores through the beams. For example, reinforing steel bars designed to provide the progressive ollapse resistane demand are required to be ontinuous from one edge to the other and be effetively embedded or anhored. 10 Note that this study fouses on the progressive ollapse resistane demand under the atenary mehanism. Therefore, C frame strutures with adequate dutility and ontinuous reinforement arrangement are onsidered. THEOETICAL FAMEWOK FO DEMA ANALYSIS The demand relationships an be ategorized into three types (Fig. (a)). Type 1 is the relationship between the and the demands. In this study, these demand relationships at the strutural and elemental levels for the atenary mehanism are designated as DAF s and DAF e, respetively. Type is the relationship between the strutural and elemental demands, whih an be desribed by the struturalto-elemental demand ratio (SE). In this study, SE and SE are used to desribe the and strutural-toelemental demand ratios, respetively. Type 3 is the relationship between the DAF s and DAF e, whih is referred to as the SE DAF. Figure (b) presents the proposed theoretial framework of the energy-based progressive ollapse demand analysis whih inludes the following four major steps: 16 ACI Strutural Journal/September-Otober 014

4 Fig. Framework of progressive ollapse resistane demand analysis. 1. Take the progressive ollapse-resisting substrutures as the study objet (idealized as a SDOF system, refer to Fig. 1) and establish the expression of the DAF s based on the energy onservation priniple;. Analyze the mehanial mode of substrutures under the atenary mehanism and establish the SE DAF, based on whih DAF s an be onverted into DAF e; 3. Determine the ED using the analysis; and 4. Corret the ED to approximate the ED using the DAF e obtained in Step. The framework shown in Fig. (b) provides a feasible energy-based method of alulating the DAF e in whih the alulation of the energy dissipation of every member in the substruture is not required. ESISTANCE CUVE FO C FAMES UE CATENAY MECHANISM For the regular substruture shown in Fig. 1, when the progressive ollapse resistane demands of the beams are satisfied at every story, the demands of the substrutures are also satisfied. In this study, the two framed beams are isolated from the substruture to evaluate the demands of the beam elements under the atenary mehanism. This is presented in Fig. 3. A progressive ollapse in the framed beams an be resisted by the strutural resistane and the elemental resistane; that is, the axial tensile fores (F 1 and F ) in the beams. The framed beams may deform into two different shapes, depending on the type of load and the horizontal stiffness of the support onstraints. Under a uniformly distributed load q, the beams deform into a urvetype atenary, as shown in Fig. 3(a). This is the alulation model for the ode speified TF method, 10-1 as desribed in Appendix B of UFC. 1 On the other hand, when subjeted to a onentrated load P, the beams deform into a straighttype atenary, as shown in Fig. 3(b). The latter has been observed in the published laboratory test results. -5 In pratie, when a disproportionate ollapse ours in the intermediate floors of a multi-story building due to the loal failure of a vertial element, a large onentrated load from the upper story olumns is expeted on top of the missing element. This results in a straight-type atenary mehanism. For the top floor, on the other hand, the existene of the uniformly distributed load would lead to a urve type atenary mehanism. Curve-type atenary mehanism For the two beams in Fig. 3(a), the maximum vertial displaement Δ ours at the midspan (L 1 +L )/ of the ACI Strutural Journal/September-Otober 014 Fig. 3 Shemati diagram of substruture under atenary mehanism. two beams. For both small and large deformations,, F 1, and F at the loation of maximum deformation must satisfy the following equations, whih are obtained through moment equilibrium of the left or the right symmetrial freebody with respet to the support point 8D F1 ( L + L ) F 1 F (1a) (1b) For C beams, the axial fores are provided by the longitudinal reinforing bars embedded in the beams. Before yielding of the steel bars, the beam ends undergo primarily the plasti hinge rotations, with a small amount of axial deformation. At this stage, the axial fore F 1 an be alulated from the following equation, aording to the deformation mode of the beams F L + L αr E A L1 + L D EA 1 1 3( L + L ) where E 1 and A 1 are the modulus of elastiity and the ross-setional area of the longitudinal reinforing bars, respetively. As shown in Fig. 3(a), r is the radius of the atenary ar and α is the subtended angle of the half ar. Note that α an be expressed as arsin[(l 1 +L )/r] and the first two terms of the Taylor s polynomials of α are given in Eq. (). () 17

5 Fig. 4 esistane urve of C frame strutures under atenary mehanism. Substituting Eq. () into Eq. (1) gives the strutural resistane under the atenary mehanism L before the beams yield in tension. Fig. 5 Entire resistane urve of C frame strutures. L EA 1 1( L1 + L) 3 3 D (5a) LL 1 L 64EA 1 1 D 3 3( L + L ) 3 (3a) N ( L + L ) F 1y LL D (5b) Corresponding to the expression in Eq. (3a), the urve OAB in Fig. 4 represents the behavior of the atenary ation before the beams yield. After yielding of the longitudinal reinforing bars in the beams, assuming that Beam 1 has a relatively smaller tensile yield strength than Beam, the axial fore F 1 remains the same as the yield fore F 1y of Beam 1. Based on Eq. (1), the strutural resistane under the atenary mehanism N after yielding of the beams beomes the following N 8F 1 y ( L + L ) D (3b) The expression in Eq. (3b) refers to the straight line OC in Fig. 4, whih represents the atenary ation following tension yielding in the beams. Straight-type atenary mehanism Aording to the straight-type atenary mehanism shown in Fig. 3(b),, F 1, and F satisfy the following equations ( L1 + L) D F1 LL F 1 F when Δ 0. min (L 1, L ) (4a) (4b) Note that the disrepany in the axial fores in two beams is less than % if the beam length ratio beomes as large as 10, whih is rarely pratial. Similar derivations as presented previously are adopted herein. Under the straight-type atenary mehanism, the strutural resistanes L and N before and after yielding of the longitudinal reinforing bars are given by Eq. (5a) and (5b), respetively Equations (5a) and (5b) indiate that C frames under the straight-type atenary mehanism exhibit similar mehanial behavior, as illustrated in Fig. 4. Effet of beam mehanism It is well aepted that C-framed beams behave in the form of a beam mehanism before exhibiting the atenary mehanism. -8 Therefore, as Fig. 5 shows, the entire resistane urve of an C frame struture is defined by the polyline ODEFA under the beam mehanism and the straight line AC under the atenary mehanism. The polyline onnets suh key points as the referene point O (0, 0), the yield point D (Δ b y, b y), the peak point E (Δ b p, b p), the ultimate point F (Δ b u, b u), and the failure point G (Δ b f, 0) of the resistane urve of the C frame strutures under the beam mehanism. 13 In the figure, Δ f and f are the ultimate displaement and the ultimate strutural resistane, respetively, of an C frame under the atenary mehanism. In this paper, the strutural and elemental resistane demands of the C frames exhibiting the atenary mehanism are studied first. This is followed by further analysis of the effet of the beam mehanism on the demands under the atenary mehanism. STUCTUAL DEMA ELATIOHIP OF C FAMES UE CATENAY MECHANISM The analytial expression of the DAF s under the urve-type atenary mehanism, DAF s, is derived in this setion. The DAF s under the straight-type mehanism will be disussed at the end of the setion. The SD and SD are notionally illustrated in Fig. 6. The SD,, is represented by Point B, whih satisfies the stati equilibrium ondition under the unbalaned gravity load G and an be evaluated by the preliminary analysis. The SD,, is represented by Point E, whih satisfies an additional demand indued by the nonlinear and dynami effets during the 18 ACI Strutural Journal/September-Otober 014

6 ollapse proess, and it an be determined by orreting the analysis through the DAF s. In the existing odes of pratie, 10-1 the ultimate displaement Δ f of the C-framed beams under the atenary mehanism is speified as 0.L, where L is the span length of the beams. Therefore, the and displaements, Δ and Δ, are idential, as shown in Fig. 6. To simplify the theoretial disussion, it is assumed that the yield displaements Δ y1 and Δ y, orresponding to the yield resistanes y1 and y, respetively, are equal, as indiated in Fig. 6. Note that the energy dissipated before reahing Δ y1 and Δ y only onsumes a very small proportion of the total energy. For example, the framed beams disussed later in Fig. 7 only dissipate 3.3% of the total energy before reahing Δ y1 or Δ y. Therefore, under the aforementioned assumption, slight disrepanies are expeted in the solution. In aordane with the energy onservation ondition, in whih the work done by the unbalaned gravity load (the area OFBC) and is equal to the energy dissipation by the struture (the area ODEC), the energy equation of the C frames under the urve-type atenary mehanism is as follows Dy EA GD 64 1 D dd D D L + L ( ) ( 3 y y ) (6) 0 ( ) where A is the ross-setional area of the longitudinal reinforing bars in the framed beams for the demand. For y and Δ y to satisfy Eq. (3a), we have the following y 64EA 1 3 ( D y ) 3 (7) 3( L + L ) The yield fator β of the substrutures is defined as b y1 (8a) Considering the aforementioned disussion on Δ, Δ, Δ y1, and Δ y, Eq. (3b) and (8a) produe the following Fig. 6 esistane demands of C frame strutures under atenary mehanism. b y (8b) Fig. 7 Shemati diagram of validation method for progressive ollapse resistane demand under atenary mehanism. Parameters of demonstrated example: q 31.5 kn/m, L 8 m, dimensions 600 x 300 mm, uniformly distributed load, no seismi design. (Note: 1 kn.m kip.ft 1 ; 1 m 3.81 ft; 1 mm in.; 1 kn 0.5 kip.) ACI Strutural Journal/September-Otober

7 and Dy1 Dy b D b D (9) Substituting Eq. (7) to (9) into Eq. (6) yields the SD,, under the urve-type atenary mehanism. 4G ( b ) (10) Considering the DAF s as defined previously and Eq. (10), the strutural demand relationship DAF s of an C frame struture under the urve-type atenary mehanism an be expressed as follows 4 DAF s ( b ) (11) It should be noted that the initial energy an be dissipated by the beam mehanism. Thus, the resistane demand under the atenary mehanism is dereased. Assuming that the energy dissipation U b by the beam mehanism is as follows U b ygd (1) where ψ is the energy dissipation fator representing the ratio of the partial (by the beam mehanism) to the total energy dissipations, we then have the DAF s, onsidering the energy dissipation of the beam mehanism, as follows 41 ( y) DAF s ( b ) (13) Equation (13) desribes the relationship between SD and SD of an C frame struture under the urve-type atenary mehanism. The expressions for DAF s for the straight-type atenary mehanism an be derived following the same proedure as desribed previously, exept that the expressions assoiated with the straight-type atenary mehanism should be adopted in Eq. (6) and (7). This indiates that the relationships between the and demands for the two mehanisms are idential. ELATIOHIP BETWEEN STUCTUAL A ELEMENTAL DEMAS OF C FAME STUCTUES UE CATENAY MECHANISM Aording to Eq. (3b), the relationships between ED, (F 1y ), and SD,, and those between ED, (F 1y ), and SD,, satisfy Eq. (14a) and (14b), respetively: 8D L + L 8D L + L ( F ) (14a) 1y ( F ) (14b) 1y Thus, the relationship between DAF e and DAF s, SE DAF, satisfies Eq. (15) DAF ( F ) ( F ) 1y e 1y D D DAF s (15) For the straight-type atenary mehanism, an idential relationship between DAF e and DAF s an be established following the proess outlined previously. Note that the aforementioned theoretial derivations for the DAF e and DAF s are based on the regular C frame strutures shown in Fig. 1 and 3. The derived Eq. (15) indiates that the strutural DAF equals the elemental DAF for a regular C frame under the atenary mehanism. Eq. (15) an be transformed into the following equation SE SE ( F ) (F ) 1y 1y (16) The physial signifiane of Eq. (16) is that, for regular C frame strutures, the distributions of the total stati demand and the total energy dissipation demand of all the strutural elements in the ollapse-resisting substruture are idential. It should be noted, however, for the irregular frame strutures defined in GSA003, that the harateristis of the DAF e, DAF s, SE and SE do not neessarily follow the aforementioned observation and that they must be evaluated ase by ase using the proposed methodology. NUMEICAL VALIDATION Validation method The numerial validation is aomplished using the nonlinear finite element program THUFIBE. 13 Published literatures show that C frames exhibiting a flexural and axial behavior an be well simulated using THUFIBE. 13,3 The independent and analysis are onduted to obtain the and demands respetively. Figure 7(a) shows a theoretial model of one-story two-span ontinuous framed beams used to validate the proposed demand relationships. Details of the model are given in the next setion, and the validation proedure is summarized below in five stages. Stage 1: Calulate the atual ED and SD An pushdown analysis is onduted herein. The amount of longitudinal reinforing steel is iteratively adjusted until the pushdown load equals the unbalaned gravity load G when the joint displaement reahes the target displaement that is, 0.L The obtained frame satisfies the demand the stati equilibrium ondition under the unbalaned gravity load G. The orresponding axial fore in the beams is therefore taken as the atual ED, (F 1y ). Note that the atual SD equals to the gravity load G. Stage : Calulate the atual ED and DAF e For the framed beams established in Stage 1, a gravity load G is initially onsidered to be applied to the beams, and a stati analysis is onduted until the stati equilibrium state is reahed. This state orresponds to 1s in Fig. 7(b), whih is followed by a sudden removal of the middle support. A 130 ACI Strutural Journal/September-Otober 014

8 transient analysis is subsequently onduted to simulate the dynami response of the framed beams, as shown in Fig. 7(b). During the analysis, the amount of longitudinal reinforing steel is iteratively adjusted until the atenary mehanism takes plae and the orresponding displaement reahes the target displaement speified in the existing odes, 0.L The obtained frame satisfies the demand the energy equilibrium ondition under the unbalaned gravity load G. The orresponding maximum axial fore in the beams is reorded as the atual ED, (F 1y ). Thus, the atual value of DAF e an be expressed as follows (DAF ) e ACI Strutural Journal/September-Otober 014 at ( F ( F 1y 1y ) ) (17) Stage 3: Calulate the atual SD and DAF s An pushdown analysis is onduted again for the framed beams obtained in Stage, satisfying the demand, to alulate the atual SD, as shown in Fig. 7(). The target displaement, 0.L, and the load pattern in the pushdown analysis are idential to those in the transient analysis (Stage ). The pushdown load at the target displaement 0.L is taken as the atual SD,. Thus, the atual value of DAF s, (DAF s) at, is given by the following (DAF ) s at G (18) Stage 4: Calulate and validate the predited DAF s and ompare the strutural and elemental DAF s For the numerial model being analyzed, the atual strutural DAF, (DAF s) at obtained in Stage 3, is ompared with the predited DAF, (DAF s) pre. Note that (DAF s) pre is alulated using Eq. (13), in whih the energy dissipation fator ψ under the beam mehanism is determined by the ratio of the area below ODEFA to that below ODEFAC of the pushdown urve alulated in Stage, as illustrated in Fig. 7(). The yield fator β in Eq. (13) is given by Eq. (8), in whih the yield displaement Δ y of the atenary mehanism is determined by the transformation point from the beam mehanism to the atenary mehanism (Point A in Fig. 7()). The atual strutural and DAFs, (DAF s) at and (DAF e) at, are also ompared in the following setion to validate Eq. (15), desribing their theoretial relationship. Stage 5: Calulate the predited ED and validate the framework in Fig. (b) Following the proposed framework in Fig. (b), the predited ED, ED pre, is alulated by orreting the atual ED obtained in Stage 1 using the predited (DAF s) pre obtained in Stage 4. Then, the ED pre is ompared with the atual ED, ED at, obtained by the diret analysis in Stage. Model parameters To ensure the representativeness of the validation, different parameters, inluding the load type and magnitude, the span length, and the ross-setional dimension of the beams, are onsidered, as summarized in Table. This Table Parameters of validation examples Load type Conrete Longitudinal reinforing steel Longitudinal steel bars at top setions of beam ends Uniformly distributed load q Two mm bars C30 (f k 0.1 MPa) HB335 (f y 335 MPa) Conentrated load P (ql) Six mm bars Area load q, kn.m Beam span length L, m Crosssetional dimensions, mm x mm x x x x x 300 Notes: 1 MPa ksi; 1 mm in.; 1 kn.m kip.ft 1 ; 1 m 3.81 ft. results in 180 models in total. A uniformly distributed load and a onentrated load are applied to investigate the resistane demand under the urve- and straight-type atenary mehanisms, respetively. To investigate the effet of the beam mehanism, two different reinforing senarios are onsidered, as shown in Fig. 8. In the first senario, the struture is reinfored with two steel bars with diameters of mm (0.866 in.) at the top of the beam ends. This orresponds to a 0. to 0.33% reinforement ratio, whih is greater than the minimum ratio of 0.% speified in the Chinese Design Code of Conrete Strutures. 4 This minimum ratio is widely adopted in frames designed without onsideration for seismi effets. In the seond senario, the struture is reinfored with six steel bars with diameters of mm (0.866 in.) at the top of the beam ends, whih orresponds to a 1.3 to.1% reinforement ratio. This ratio is smaller than the ode-speified maximum ratio of.5% 5 and is ommonly used in frames designed with onsideration for seismi effets. The aforementioned two senarios over the upper and lower limits of the energy dissipation effets of the beam mehanism. For all the models, two additional bars with diameters of mm (0.866 in.) are also plaed at the top setions at the midspan of the beams. The bottom bars are positioned along the entire length of the beams. The ross-setional areas of the bottom bars are iteratively adjusted until the target displaement of 0.L is reahed for the atenary mehanism. Validation results The validation for the strutural demand relationship, DAF s, is illustrated in Fig. 9. It is evident that the predited values of (DAF s) pre are very lose to the atual values of (DAF s) at, with the majority of the errors within ±10% and the mean absolute perentage errors are.61% and.44% for the urve-type and straight-type atenary mehanisms, respetively. This indiates that the proposed Eq. (13) is able to aurately predit the relationship of the strutural 131

9 Fig. 8 Arrangement of steel bars in validation models. (Note: 1 mm in.) resistane demands of C frame strutures under the atenary mehanism. It is also observed that the DAF s under the atenary mehanism ranges from approximately 1.5 to.0. This value is larger than the DAF b s under the beam mehanism, whih is below 1.34 for regular C frame strutures with a dutility ratio greater than 1.0 in the analysis. 10 In the existing design odes, the elemental demand is diretly adopted for the TF design. The validation results reveal that the atual elemental demand is approximately 1.5 to.0 times the elemental demand. This indiates that the demand alulated by the existing TF method is insuffiient for ollapse prevention; therefore, the method should be further modified to ensure safe design for the atenary mehanism. It is also notied that only the DAF for the beam mehanism is regulated in the existing design odes, whih is muh lower than that for the atenary mehanism. If the urrently reommended DAF is used to orret the demand under the atenary mehanism, the solution would be underestimated. To improve the demand alulation method for the atenary mehanism, the proposed equations (Eq. (13) and (15)) an be used to orret the demand in the existing TF design. The effet of the beam mehanism on the strutural demand under the atenary mehanism is illustrated in Fig. 10. The square dots denote the relationship between the energy dissipation fator ψ and (DAF s) at for all 180 numerial models. Substituting the average value of β of the validation models into Eq. (13) gives the nominal average value of (DAF s) at as a funtion of ψ, whih is represented by the solid line in Fig. 10. It is apparent that when the energy dissipation by the beam mehanism inreases, the strutural demand under the atenary mehanism dereases. It is well-known that the beams in C frame strutures onstruted in earthquake zones are designed to have higher bending moment apaities than those onstruted in non-earthquake zones. As a result, C frame strutures with seismi design onsiderations have higher energy-dissipating apaities under the beam mehanism in a progressive ollapse proess. Consequently, the ollapse resistane demand of these strutures under the atenary mehanism dereases. In Fig. 11, the atual values of the elemental (DAF e) at are ompared to those of the strutural (DAF s) at for all 180 models. Again, the mean ratios of (DAF e) at and (DAF s) at are well within ±10% and the mean absolute perentage errors are.49% and.64% for the urve-type and straight-type atenary mehanisms, respetively. This further validates the proposed relationship (Eq. (15)) under the atenary mehanism. Fig. 9 Comparison between atual and predited values of DAF s. The ED alulated based on the framework proposed in Fig. (b) is validated by the results presented in Fig. 1. The majority of the mean ratios of ED at and ED pre are within ±10% and the mean absolute perentage errors are 3.39% and 1.79% for the urve-type and straight-type atenary mehanisms, respetively. That demonstrates that the ED of the numerial examples an be well assessed by the proposed framework. CONCLUSIO Under large deformations, C frame strutures resist progressive ollapse through the atenary mehanism. An energy-based framework is proposed for alulating the progressive ollapse resistane demands under the atenary mehanism at both the strutural and elemental levels. The following onlusions an be drawn. 1. Two types of atenary mehanisms, the urved type and the straight type, are similar mehanisms and are idential in their DAF values.. For regular C frame strutures, the elemental DAF equals the strutural DAF. These two values may not be idential for irregular frame strutures with varied strutural arrangement from story to story. This aspet merits further investigation. 3. For strutures with seismi design onsiderations, the beam mehanism apaity inreases, whih subsequently redues the atenary mehanism demand, due to 13 ACI Strutural Journal/September-Otober 014

10 Fig. 10 Effet of beam mehanism on strutural demand under atenary mehanism. the notieable ontribution of the energy dissipation of the beam mehanism. 4. Numerial validation using a total of 180 framed beam models demonstrates that the proposed relationships are aurate and that the DAF s under the atenary mehanism is larger than the DAF b s under the beam mehanism. The proposed DAF e an be used to orret the elemental demand to approximate the elemental demand required for engineering design. AUTHO BIOS Yi Li is an Assistant Professor in the Department of Civil Engineering, Beijing University of Tehnology, Beijing, China. His researh interests inlude the progressive ollapse of onrete strutures. Xinzheng Lu is a Professor in the Department of Civil Engineering, Tsinghua University, Beijing, China. His researh interests inlude disaster prevention of engineering strutures. Hong Guan is an Assoiate Professor in the Griffith Shool of Engineering, Griffith University Gold Coast Campus, Brisbane, Australia. Her researh interests inlude strutural engineering and omputational mehanis. Lieping Ye is a Professor in the Department of Civil Engineering, Tsinghua University. His researh interests inlude earthquake engineering and onrete strutures. ACKNOWLEDGMENTS The authors are grateful for the finanial support reeived from the National Basi esearh Program of China (No. 01CB719703), the National Siene Foundation of China (No , ), and the esearh Program of Beijing Muniipal Commission of Eduation (KM ). Fig. 11 Comparison between atual values of DAF e and DAF s. NOTATION A 1, E 1, ross-setional area and elasti modulus of longitudinal reinforing bars, respetively A ross-setional area of longitudinal reinforing bars for demand DAF dynami amplifiation fator DAF s, DAF e strutural DAF and elemental DAF, respetively DAF s, DAF e strutural and elemental DAFs under atenary mehanism, respetively (DAF s) at, (DAF s) pre atual and predited values of DAF s, respetively (DAF e) at, (DAF e) pre atual and predited values of DAF e, respetively F, F 1, F elemental demand under atenary mehanism F 1y yield fore of Beam 1 F, (F 1y ) nonlinear stati elemental demand F, (F 1y ) nonlinear dynami elemental demand G total unbalaned gravity load g balaned, uniformly distributed gravity load L, L 1, L span length of beam P unbalaned onentrated gravity load q unbalaned, uniformly distributed gravity load q p uniformly distributed pushdown load strutural demand b, strutural demand under beam mehanism and atenary mehanism, respetively L, N before and after beam yield is in tension, respetively y1, y when beam yield in tension, nonlinear stati and nonlinear dynami, respetively r radius of atenary ar SE, SE, SE DAF strutural-to-elemental ratio for demand, demand, and DAF, respetively α subtended angle of the half ar ACI Strutural Journal/September-Otober

11 Fig. 1 Comparison between atual and predited values of ED. β yield fator under atenary mehanism Δ joint displaement Δ, Δ nonlinear stati and dynami joint displaements under atenary mehanism, respetively Δ y, Δ y1, Δ y yield displaement of substruture under atenary mehanism Δ f failure displaement of substruture under atenary mehanism Δ max target displaement ψ energy dissipation fator of beam mehanism EFEENCES 1. Ellingwood, B.., Mitigating isk from Abnormal Loads and Progressive Collapse, Journal of Performane of Construted Failities, ASCE, V. 0, No. 4, Nov. 006, pp Yi, W. J.; He, Q. F.; Xiao, Y.; and Kunnath, S. K., Experimental Study on Progressive Collapse-esistant Behavior of einfored Conrete Frame Strutures, ACI Strutural Journal, V. 105, No. 4, July-Aug. 008, pp Su, Y.; Tian, Y.; and Song, X., Progressive Collapse esistane of Axially-estrained Frame Beams, ACI Strutural Journal, V. 106, No. 5, Sep.-Ot. 009, pp Sadek, F.; Main, J. A.; Lew, H. S.; and Bao, Y., Testing and Analysis of Steel and Conrete Beam-Column Assemblies under a Column emoval Senario, Journal of Strutural Engineering, ASCE, V. 137, No. 9, Sept. 011, pp Yu, J., and Tan, K., Experimental and Numerial Investigation on Progressive Collapse esistane of einfored Conrete Beam Column Sub-Assemblages, Engineering Strutures, V. 55, Ot. 013, pp Sasani, M.; Wernera, A.; and Kazemia, A., Bar Frature Modeling in Progressive Collapse Analysis of einfored Conrete Strutures, Engineering Strutures, V. 33, No., Feb. 011, pp Bao, Y.; Kunnath, S. K.; El-Tawil, S.; and Lew, H. S., Maromodel-Based Simulation of Progressive Collapse: C Frame Strutures, Journal of Strutural Engineering, ASCE, V. 134, No. 7, July 008, pp Valipour, H.., and Foster, S. J., Finite Element Modelling of einfored Conrete Framed Strutures inluding Catenary Ation, Computers & Strutures, V. 88, No. 9-10, 010, pp Mansur, M. A.; Ahmad, I.; and Paramasivam, P., Punhing Shear Strength of Simply Supported Ferroement Slabs, Journal of Materials in Civil Engineering, ASCE, V. 13, No. 6, Nov.-De. 001, pp UFC , Design of Buildings to esist Progressive Collapse, Department of Defense, Washington, DC, 010, 181 pp. 11. EN :004, Euroode : Design of Conrete Strutures. Part 1: General ules and ules for Buildings, European Committee for Standardization, Brussels, Belgium, 004, 6 pp. 1. BS 8110, Strutural Use of Conrete, Part 1: Code of Pratie for Design and Constrution, British Standards Institute, London, UK, 1997, 164 pp. 13. Li, Y.; Lu, X. Z.; Guan, H.; and Ye, L. P., An Improved Tie Fore Method for Progressive Collapse esistane Design of einfored Conrete Frame Strutures, Engineering Strutures, V. 33, No. 10, Ot. 011, pp Dusenberry, D. O., and Hamburger,. O., Pratial Means for Energy-Based Analyses of Disproportionate Collapse Potential, Journal of Performane of Construted Failities, ASCE, V. 0, No. 4, Nov. 006, pp Izzuddin, B. A.; Vlassis, A. G.; Elghazouli, A. Y.; and Netherot, D. A., Progressive Collapse of Multi-Storey Buildings due to Sudden Column Loss Part I: Simplified Assessment Framework, Engineering Strutures, V. 30, No. 5, May 008, pp Xu, G., and Ellingwood, B.., An Energy-Based Partial Pushdown Analysis Proedure for Assessment of Disproportionate Collapse Potential, Journal of Construtional Steel esearh, V. 67, No. 3, Mar. 011, pp Marhand, K. A.; MKay, A. E.; and Stevens, D. J., Development and Appliation of Linear and Non-Linear Stati Approahes in UFC , Proeedings of the 009 Strutures Congress, Austin, TX, 009, pp Pujol, S., and Smith-Pardo, J. P., A New Perspetive on the Effets of Abrupt Column emoval, Engineering Strutures, V. 31, No. 4, Apr. 009, pp Tsai, M. H., and Lin, B. H., Investigation of Progressive Collapse esistane and Inelasti esponse for an Earthquake-esistant C Building Subjeted to Column Failure, Engineering Strutures, V. 30, No. 1, De. 008, pp Tsai, M. H., An Analytial Methodology for the Dynami Amplifiation Fator in Progressive Collapse Evaluation of Building Strutures, Mehanis esearh Communiations, V. 37, No. 1, Jan. 010, pp UFC , Design of Strutures to esist Progressive Collapse, Department of Defense, Washington, DC, 005, 176 pp.. U. S. General Servies Administration, Progressive Collapse Analysis and Design Guidelines for New Federal Offie Buildings and Major Modernization Projets, Washington, DC, 003, 119 pp. 3. Lu, X.; Lu, X. Z.; Guan, H.; and Ye, L. P., Collapse Simulation of einfored Conrete High-ise Building Indued by Extreme Earthquakes, Earthquake Engineering & Strutural Dynamis, V. 4, No. 5, Apr. 013, pp GB , Code for Design of Conrete Strutures, The Ministry of Constrution of the People s epubli of China, Beijing, China, 00, 45 pp. 5. GB , Code for Seismi Design of Buildings, The Ministry of Constrution of the People s epubli of China, Beijing, China, 001, 483 pp. APPEIX POOF OF UNIVESALITY OF DEMA ELATIOHIPS The theoretial derivations presented in the aforementioned setions are based on the simplified model illustrated in Fig. 3, with pre-defined beam dimensions and material parameters. When the geometri dimensions and material parameters of Beam 1 and Beam vary independently, E, A, L, Δ, and F y in Eq. (1) to (5) would hange aordingly, whih would in turn lead to a variation of the ratios of the strutural to elemental demand (SE and SE ) regulated by Eq. (14). It should be noted, however, that the parameters L, Δ and F y are eliminated in the alulation proess as evident in Eq. (15). This indiates that the geometri dimensions and material parameters have no influene on the DAF s and DAF e. The aforementioned disussion learly demonstrates that the strutural and elemental demand relationships derived from the substruture in Fig. 3 are universal for all regular C frame strutures. 134 ACI Strutural Journal/September-Otober 014

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