Application of Fiber Model for Progressive Collapse Analysis of Reinforced Concrete Frames^{*} Xinzheng Lu, Yi Li, Lieping Ye and Yi Liang Department of Civil Engineering, Tsinghua University, Beijing 100084, P.R. China Proc. 12th Int. Conf. on Computing in Civil and Building Engineering, Oct. 2008, Beijing, CDROM Keywords£ºfiber model; progressive collapse; simulation; reinforced concrete frame; slabs 
1 Introduction The progressive collapse of a building is initiated by an event that causes local damages which the structural system cannot absorb or contain, and that subsequently propagates throughout the structural system, or a major portion of it, leading to a final damage state that is disproportionate to the local damage that initiated it^{[1]}. Although some advanced regulations^{[2]} to resist progressive collapse were presented, the progressive collapse design is still lack in China. One of major difficulties is that normal finite element (FE) analysis with solid elements lead to large computational cost in collapse simulation of whole RC structures. On the contrary, fiber beam model is a kind of high efficiency computing element which associates the nodal force/deformation of beams or columns with the nonlinearity of material. So it has bright application future in collapse researches and designs. In this study, a fiber beam model for RC structures (referred as THUFIBER) is developed, which is based on the generalpurpose finite element package of MSC.MARC that has significant capacity of solving nonlinear problems. Dynamic timehistory analysis for a RC planar frame experiment is carried out to proof the capacity of the proposed fiber beam model, and a farther research on the influence of slabs on the progressive collapse behavior of structures is also presented. 2 THUFIBER Program 2.1 Discretization in section and longitudinal direction In the THUFIBER program, each section of the RC member is divided into 36 concrete and 4 reinforcement fibers as shown in Figure 1. Users can define the position, area and constitutive model of each fiber. The program calculates the strain of each fiber by assuming plane remains plane and can insure that the stresses on the section are in equilibrium by iteration^{ [3}^{}^{4]}. Users can also define the quantity and position of sections (integration points) in a beam element, as shown in Figure 2. 
2.2 Constitutive relation of fibers In THUFIBER program, the constitutive relation used for concrete fiber is shown as Figure 3. Users input 6 parameters to define the uniaxial stressstrain relation: the initial Young¡¯s Modulus of the concrete, the peak compressive strength f_{c} and the corresponding strain ¦Å_{o}, the ultimate compressive strength ¦Ò_{u} and the corresponding strain ¦Å_{u}, and the ultimate tensile strength f_{t}. The compressive constitutive model of concrete has a envelop curve containing two parts. The ascending part of the envelop curve is expressed by Equation 1, and the descending part is a straight line. The program can simulate confined and unconfined concrete behavior as well. The constitutive relation of rebar fiber is based on the L¨¦geron model and Bauschinger effect is also considered^{[}^{5}^{]}, as shown in Figure 4. _{ } (1) 2.3 Test verification THUFIBER program with the above mentioned constitutive models for the concrete and the rebar fiber agrees well in the dynamic simulation for the collapse experiment of RC planar frame ^{[}^{6}^{]}, as shown in Figure 5. Figure 6 shows the relation between the axial force and displacement of the removed column. Obviously, the program has enough accuracy in collapse simulation. Besides, only 156 fiber beams with one integration point are used in dynamic timehistory analysis which accomplishes within 1 minute on a general personal computer. That indicates great efficiency of fiber model.

3 Computational Models for Progressive Collapse 3.1 Geometric model According to Chinese concrete code^{ [}^{7}^{]}, an 8stories RC frame is designed without special collapse design to check the collapse resistance of typical Chinese frames, as shown in Fig 7 and Fig 8. The ground storey is 4.2m high and all the other stories are 3.6m high. The sections of columns, longitudinal beams and transverse beams are 550¡Á550mm, 300mm¡Á500mm and 300mm¡Á550mm respectively. The columns have a fixed end on the ground. 3.2 Material parameters The selected parameters for the materials are as follows. For concrete material, Young¡¯s Modulus E_{0}=30GPa, peak compressive strength f_{c}=30MPa, peak compressive strain ¦Å_{0}=0.002, ultimate compressive strength f_{u}=20MPa and ultimate compressive strain ¦Å_{u}=0.004. For the reinforcement, Young¡¯s Modulus E_{s}=200GPa, yield strength=335MPa and the broken tensile strain=0.1.

3.3 Slabs modeling In order to study the influence of slabs on the progressive collapse behavior of RC frames, the numerical models of the frame with and without slabs are compared: Model A: Frame model with columns and beams only; Model B: Frame model including columns, beams and slabs. When progressive collapse happens, slabs are usually in large deformation stage, and therefore, the bending bearing capacity of slabs can be ignored comparing with their suspending capacity. So in this paper, the reinforcement in slabs is modeled with truss elements to consider the tensile effect of rebars, and the concrete in slabs is not taken into account. And moreover, when tensile strain of reinforcement reaches 10%, the reinforcement is considered to be broken, and then the procedure will delete it automatically. Thus, the contribution of slabs is conservatively considered by means of this method. 4 Analysis procedure 4.1 Load case According to the specification in DoD2005^{ [}^{2}^{]}, the following factored load combination is applied to the entire structures for nonlinear dynamic analyses: 1.2G_{k}£«0.5Q_{k}£«0.2W_{k} (2) where, G_{k} =standard value of dead load; Q_{k}_{ }=standard value of live load; W_{k} =standard value of wind load. 4.2 Analysis procedure Alternate Path (AP) method based on the dynamic nonlinear analysis is used in this paper to analyze the progressive collapse resistance capacity because of its high accuracy. In AP method, one vertical loadbearing element is removed instantaneously from Model A and Model B in every load case to check the bridge over capacity of the residual structures. The columns are near the middle of the short side, near the middle of the long side and at the corner of the building at each floor are removed. Besides, an internal column at ground floor is also removed, as shown in Figure 9.
According to the DoD2005^{ [}^{2}^{]}, the beams are considered to be failed (or collapsed) when their deflections reach 10% of their spans or the tensile strain of reinforcement in them exceeds 10%. If collapse area exceeds the value specified in DoD2005^{ [}^{2}^{]}, the structure will be considered that its progressive collapse resistance is insufficient and the collapsed part need be strengthened. Both the results of collapse resistance and strengthening reinforcement of Model A and Model B are compared to analysis the effect of slabs in collapse process. 5 Results 5.1 Collapse resistance capacity The results of simulations for collapse of Model A and Model B are listed in Table 1. It can be indicated that Chinese typical RC frames have insufficient collapse resistance. For the columns in different stories located at the same planar position, the columns in higher stories have more collapse possibility, as shown in Table 1, because the higher floor the removed column Table 1 the results of simulations for collapse of Model A and Model B

locates, the less alternate loading paths the residual frame has. In addition, slabs enhance the resistance of progressive collapse (the examples marked by ¡°**¡±). According to the simulation results as shown in Figure 10, it can be concluded that the tensile effect of slabs improves integrated loadbearing capacity which enhances the collapse resistance. Actually, the slabs still hold the broken beams though collapse happens, as shown in Figure 10(b).
Table 2 Comparison for consumption of longitudinal reinforcement in beams
5.2 Consumption of strengthening reinforcement Both Model A and Model B are redesigned until they do not collapse after any column is removed. Then the consumption of strengthening reinforcement in frames including original frame and strengthened frames of Model A and Model B are compared, as shown in Table 2. Columns are strong enough but the reinforcement in beams should be increased. And moreover, Model A, which considers the contribution of slabs, just cost 21.4% of the strengthened reinforcement in the Model B, which ignores the effect of slabs. That means that slabs greatly enhance the collapse resistance of frame. It is also indicated that the design method in DoD2005^{ [2]} which does not take the contribution of slabs into account is conservative but has some redundancy. 6 Conclusions Fiber beam model is a high efficient element to simulate the columns and beams in collapse. In this paper, THUFIBER model embedded in commercial software MSC.MARC is developed. Because of the accurate constitutive laws of fibers and suitable discretization on section, THUFIBER performs well in simulating the collapse process of RC frames. And the simulation also shows that the slabs will obviously enhance the progressive collapse resistance of the structure. References [1] Ellingwood B R. Mitigating risk from abnormal loads and progressive collapse. Journal of Performance of Constructed Facilities, 2006, 20(4): 315323. [2] DoD. UFC 402303. Design of Structures to Resist Progressive Collapse. 2005. [3] Spacone E, Filippou F, Taucer F. Fiber beamcolumn modeling for nonlinear analysis of R/C frames. Journal of Earthquake Engineering and Structural Dynamics, 1996, 25(7): 711~725. [4] D¡¯Ambrisi A, Filippou F. Modeling of cyclic shear behavior in RC members. Journal of Structural Engineering, ASCE, 1999, 125(10): 1143~1149. [5] L¨¦geron F, Paultre P, Mazar J. Damage mechanics modeling of nonlinear seismic behavior of concrete structures. Journal of Structural Engineering, ASCE, 2005. 131(6): 946~954. [6] Yi W J, He Q F, Xiao Y. Collapse performance of RC frame structure. Journal of Building Structures, 2007, 28(5): 104109. (in Chinese) [7] GB500102002. Code for Design of Concrete Structures, Beijing: China Building Industry Press, China. 2002 