5th International Congress on Computational Mechanics and Simulation, 10-13 December 2014, India
NUMERICAL METHODS IN PROGRESSIVE COLLAPSE ASSESSMENT OF RC FRAMED STRUCTURES: FEM vs. AEM MIRCEA BOTEZ 1, LUCIAN BREDEAN 1 AND ADRIAN M. IOANI 1 1
Department of Structural Mechanics, Technical University of Cluj-Napoca, Cluj-Napoca, Romania. E-mail:
[email protected]
ABSTRACT
Progressive collapse is, according to GSA (2003), a situation where local failure of a primary structural component leads to the collapse of adjoining members which, in turn, leads to additional collapse. Since the experimental studies are costly and difficult to perform, numerical methods are a practical way to predict the structural response induced by the abnormal loads. Thus, this paper investigates the progressive collapse potential of two low-rise RC framed structures using two advanced numerical methods: Finite Element Method (FEM) and Applied Element Method (AEM). The confidence of these numerical methods is confirmed through numerical calibrations based on experimental study involving a nine panel reinforced concrete two-way slab subjected to gravity loads. Two structures of 3 and 6-story height are analyzed and the interior column damage scenario is considered. The structural response of the damaged models is assessed via the nonlinear dynamic analysis (NDA), in association with FEM, respectively AEM. The differences between the results in terms of vertical displacements and rotations, obtained via the two considered approaches, are compared and discussed. Keywords: Progressive collapse; nonlinear dynamic; dynamic; finite element; applied applied element.
Introduction Theoretical and experimental studies [Sasani and Sagiroglu (2008), (2010)] have shown that highrise RC framed buildings (20 stories) designed and detailed for high seismic zones have an inherent capacity to better resist progressive collapse. In order to evaluate the progressive collapse risk, the most important guidelines [GSA (2003), DoD (2009)] specify three different numerical procedures: linear static analysis (LSA), nonlinear static analysis (NSA), respectively nonlinear dynamic analysis (NDA). While the less complex procedure (LSA) indicates that the progressive collapse potential must be assessed considering c onsidering only the principal structural elements (beams and columns), the more accurate procedures (NSA, NDA) [McKay et al. (2010), al. (2010), Kokot et al. (2012)] al. (2012)] allow the option to also consider the secondary structural elements (slabs). Generally, the nonlinear procedures are applied to FEM numerical models based on linear elements [Tsai and Lin (2008),
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McKay et al. (2010)], associated with the concentrated plasticity concept (plastic hinges) [Kokot et al. (2012), Joshi et al. (2010)]. Apart from this method (FEM), recently a new numerical approach - Applied Element Method (AEM) - is also used in order to investigate the progressive collapse potential [Salem et al. (2011), Dinu and Dubina (2012)]. Starting from these considerations, in this paper two low-rise reinforced concrete framed structures (3 and 6 stories) placed in a low seismic zone are studied and their potential for progressive collapse is assessed. Both, primary (columns and beams) and secondary (slabs) structural elements are modelled. Two different advanced numerical approaches are used: Finite Element Method (FEM), respectively Applied Element Method (AEM). The results are compared and the main differences are underlined. The distribution and magnitude of inelastic demands, the influence of number of stories for the interior missing column scenario are also investigated.
Progressive collapse assessment method Most studies related to progressive collapse of RC structures are based on the methodology provided by the two main available guidelines: GSA (2003) and DoD (2009). In this paper, the progressive collapse potential of the considered structures is assessed based on the DoD (2009) provisions.
Fig. 1. Numerical models: a) 3-story structure, b) 6-story structure
Since the progressive collapse phenomenon is a nonlinear and dynamic event, in this paper the NDA is applied for both numerical approaches: FEM and AEM. The undamaged models are completely designed and detailed according to the provisions of the Romanian seismic design code P100-1/2006. Design is made for a low seismic zone (Cluj-Napoca) with a peak value of the ground acceleration is 0.08g, associated with the medium ductility class design options (DCM). Each structure consists of three 6.0m bays and five 6.0m spans, and has a story height of 3.15m (Figure 1). According to DoD (2009) Guidelines, the verdict regarding progressive collapse potential is obtained by applying several missing columns scenarios. In this study, one damage scenario is considered: an interior column placed at the ground floor of the structure is assumed to be missing. To determine the structural response of the damaged models, a nonlinear dynamic analysis (NDA) is performed: the ABAQUS software package is used for the finite element analysis and the Extreme Loading for Structures (ELS) is used for the Applied Element Method. The alternate path
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method (APM) is used in accordance with the DoD (2009) provisions. The basic procedure of the method consists in the sudden removal of one or more primary structural components (columns). The interior column placed on the ground floor is removed almost instantaneously, in less than 1/10 of the period associated with the structural response mode for the vertical motion of the bays above the removed column, as determined from the analytical model with the column removed. Thus, the considered removal time is 0.005 seconds. A 3 seconds total time is considered for the nonlinear dynamic analysis (NDA) along with a 0.005 seconds step size. Also, a 5% damping factor is taken into account [Tsai and Lin (2008)]. Both, geometric and material nonlinearity are considered. The compressive strength class of the concrete is C25/30 (f ck = 25N/mm2), and the steel for the longitudinal and transverse reinforcement is of S500 type (f yk = 500N/mm2). A strength increase factor of 1.25 and 1.50 is applied, according to DoD (2009) provisions, to the specified strengths of steel, respectively of concrete, to take into account the dynamic material properties. The material curves defined for C25/30 concrete and S500 type steel are illustrated in figure 2.
Fig. 2. Stress-strain curves for concrete and steel
The considered vertical load applied to the structure in this study, is computed in accordance with following equation (Eq. 1).
Load 1.2 D 0.5 L
(1)
where D represents the dead load (D=3.5 kN/m 2 + self-weight) and L represents the live load (L=2.0 kN/m2). According to the DoD (2009) provisions the progressive collapse potential is obtained by comparing the plastic rotations (ϴ p) of the primary (beam) and secondary (slabs) structural elements, under the specified loads (Eq. 1) with the corresponding allowable values: 0.063 rad for beams, respectively 0.05 rad for slabs. These limit values are compared to the plastic rotations obtained in a simplified manner via chord method specified by ASCE41 (2006), determined according to Eq. 2. / L (2) where Ɵ represents the plastic rotation, Δ is the corresponding vertical deflection and L is the clear span length.
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Numerical calibration The first phase of this study involves a numerical calibration of two software analysis tools, based on the FEM (Abaqus), respectively AEM (Extreme Loading for Structures - ELS). The Abaqus and ELS numerical calibrations are based on a well-known experimental study [Gamble et al. (1961)], involving a nine panel reinforced concrete two-way slab subjected to gravity loads. Force displacement curves were experimentally determined for multiple key points of the tested specimen: central point of the corner, marginal and central slab panels, respectively mid-point of two marginal and two interior beams. Both numerical approaches (FEM and AEM) indicate a good agreement between the experimentally obtained curves and numerical ones for each monitored key point described above. The curves obtained for the central point of the corner slab panel are illustrated in figure 3.
Fig. 3. Force - displacement curves: experimental vs. numerical model (FEM and AEM)
Progressive collapse analyses As it was mentioned before, numerical models representing two reinforced concrete framed structures of 3 and 6 stories affected by the failure of an interior column are analyzed in order to assess the progressive collapse potential. FEM approach (Abaqus)
The FEM based models use solid finite elements for concrete (C3D20R type with 20 integration points), respectively linear finite elements (T3D2 type with 2 integration points; one at each end) for reinforcement.
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When the 3-story structure is analyzed the maximum vertical displacement value is 4.23 cm (Figure 4). Based on this value, the obtained rotation is 0.0076 rad. The structure becomes dynamically stable for a vertical displacement of 4.01 cm as shown in figure 4. In this case, the results are obtained in 3060 minutes (51 hours). For the 6-story structure, the peek vertical displacement value is 2.60 cm (Figure 4) while the corresponding rotation is 0.0048 rad. The equilibrium state in this case is reached for a vertical displacement of 2.35 cm. The structural response is obtained for this structure in 17632 minutes (approximately 293 hours).
Fig. 4. Time - displacement curves: FEM models (3 and 6 stories structures) AEM approach (ELS)
By contrast to the previous approach, the AEM numerical models represents an assembly of 3D elements and one-dimensional elements, interconnected with two tangential and one normal spring. The spring’s behavior follow the stress-strain (σ-ε) defined proprieties. The nonlinear dynamic analysis (NDA) performed in 147 minutes (approximately 3 hours) for the 3-story indicates a maximum vertical displacement of 4.73 cm (Figure 5), respectively a rotation of 0.0084 rad. The structures reaches an equilibrium state for a vertical displacement of 4.10 cm as it is illustrated in figure 5. The damaged 6-story model reveals a maximum vertical displacement of 3.08 cm (Figure 5). The corresponding rotation value in this case is 0.0057 rad while the vertical displacement for which the structure becomes dynamically stable is 2.48 cm. A 276 minutes period (approximately 5 hours) is necessary to evaluate the structural response in this case.
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Fig. 5. Time - displacement curves: AEM models (3 and 6 stories structures)
Synthesis of results Using two distinct numerical approaches, Finite Element Method (FEM) and Applied Element Method (AEM), the progressive collapse potential for two low-rise reinforced concrete framed
Fig. 6. Time - displacement curves: FEM models vs. AEM models (3 and 6 stories structures)
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structures (3 and 6 stories) is assessed. The time-displacement curves obtained through the nonlinear dynamic analysis (NDA) for both studied structures are comparatively presented in figure 6. Also, the plastic strain distribution, revealed by the considered numerical approaches for the 3-story structure, is illustrated in figure 7.
Fig. 7. Plastic strain distribution for the 3-story structure: a) FEM model, b) AEM model
Conclusions The main goal of this paper is the comparison of two different numerical approaches (FEM and AEM) implemented in two structural analysis software packages (Abaqus and ELS) in evaluating the progressive collapse potential of reinforced concrete framed structures. The main conclusions of the study are: 1.
Both numerical approaches reveal consistent verdicts regarding the progressive collapse potential of the analyzed structures. Since the maximum plastic rotations obtained do not exceed the allowable DoD (2009) values, both structures are adequate to resist progressive collapse when the interior column placed at the ground floor is damaged.
2.
The obtained results for the 3-story structure indicate a difference of approximately 10% between the two numerical approaches. FEM model reveals a maximum vertical displacement of 4.23cm, which corresponds to a maximum rotation of 0.0076 rad while for the AEM model a maximum displacement of 4.73 cm, respectively a maximum rotation of 0.0084 rad are obtained.
3.
The nonlinear dynamic analysis (NDA) performed for the 6-story structure reveals a difference of approximately 15%. The FEM based model indicates a maximum vertical displacement of 2.60 cm, corresponding to a maximum rotation of 0.0048 rad. On the other hand, the AEM based model reveals a maximum vertical displacement of 3.08 cm which corresponds to a maximum rotation of 0.0057 rad.
4.
Since the vertical displacement level is not able to initiate the progressive collapse phenomenon, both structures reach an equilibrium state. The corresponding displacements
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differ with 2% for the 3-story structure (AEM model: 4.10 cm vs. FEM model: 4.01 cm), respectively with 5% for the 6-story structure (AEM model: 2.48 cm vs. FEM model: 2.35 cm). 5.
While both numerical approaches (FEM and AEM) reveal similar results regarding the progressive collapse verdict (maximum displacement values, maximum rotation values), in terms of efficiency (run-time costs) significant differences are obtained. The AEM model requires approximately 5% (147’ vs. 3060’) of the FEM model run-time in the case of 3-story structure, respectively 2% (276’ vs. 17632’) for the 6-story structure.
6.
As can be observed in figure 7 a similar plastic strain distribution is revealed for both numerical models: FEM and AEM. The plastic strain distribution is compatible with the plastic line theory applied in the slab limit analysis: the strain distribution along the slab panel diagonal indicate the action of positive bending moment while the stain distribution along the slab contour indicate the action of negative bending moment.
Acknowledgment This paper is supported by the Sectoral Operational Programme Human Resources Development POSDRU/159/1.5/S/137516 financed from the European Social Fund and by the Romanian Government.
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Tsai M.-H., Lin B.-H. (2008), “Investigation of Progressive Collapse Resistance and Inelastic Response for an Earthquake-Resistant RC Building Subjected to Column Failure”, Engineering Structures. Vol. 30, p. 3619-3628. Joshi D., Patel P.V., Tank S.J., “Linear and Nonlinear Analysis for Assessment of Progressive Collapse Potential of Multistoried Building”, ASCE Structures Congress, 2010. Salem H.M., El-Fouly A.K., Tagel-Din H.S., “Toward an economic design of reinforced concrete structures against progressive collapse”, Engineering Structures, Vol. 33, pp. 3341-3350, DOI: 10.1016/j.engstruct.2011.06.020, 2011. Dinu F., Dubină D., “Robustness based design of steel building frames under extreme loads”, STESSA 2012 Conference, ISBN 978-0-415-62105-2, 2012. P100-1/2006 (2006), Seismic design code – Part I: Design Rules for Buildings, MTCT, Bucharest. American Society of Civil Engineers. ASCE / SEI 41-06, Seismic Rehabilitation of Existing Building. Virginia, 2006. Gamble W.L., Sozen M.A., Siess C.P. An experimental study of a reinforced concrete two-way floor slab. Civil Engineering Studies – Structural Research Series No. 211. Illinois, 1961.
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