Accuracy and Efficiency in Progressive Collapse Analysis: Real Structures vs. Successively Reduced Substructures *
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M. Botez , L. Bredean, I. Petran and A.M. Ioani Technical University of Cluj-Napoca, Romania
Abstract
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Due to the fact that the behavior of reinforced concrete structures is complex (nonlinear behavior) when they are subjected to gravitational or lateral loads, the computational cost when the entire model is considered is significant. Thus, the aim of the present study is to show a relation between a full modeled structure and several types of reduced substructures in terms of accuracy and cost efficiency (time, work volume) when the progressive collapse potential is assessed The considered structure has three stories and is placed in a low seismic area (ag = 0.08g). The design is made for a medium ductility class in accordance with the Romanian seismic code P100-1/2006. To evaluate the progressive collapse potential, only the corner column case specified in GSA(2003) Guidelines is considered. Three types of analysis are used. When the linear static analysis based on the GSA(2003) Guidelines is performed, the concept of DCR (Demand Capacity Ration) is applied. On the other hand, when the nonlinear static/dynamic analysis is used, the plastic hinge rotation will be compared with the allowable value specified in GSA(2003). As a conclusion, the reduction should be made preserving one bay/span adjacent with the affected bay in order to obtain accurate results. If the reduced model is obtained based on the previous statement, important run-time savings are obtained; for the analyzed structures these savings are 52% for NSA, respectively 46% for NDA. This study also emphasizes the importance of considering the slab in the structural model when progressive collapse potential is assessed.
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Keywords: progressive collapse, nonlinear analysis, corner column. ______________________________________ * Corresponding author: Technical University of Cluj-Napoca, Faculty of Civil Engineering. 15, C. Daicoviciu Street, 400020, Cluj-Napoca, Romania. E-mail:
[email protected].
Phone: +40 741 291 291
1.0 Introduction
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Progressive collapse has become an important topic for civil engineers after the catastrophic event that took place in 1968 at Ronan Point apartment building. The entire south-east corner of the building collapsed as a consequence of an explosion caused by a gas leak. Other significant progressive collapse events took place at Murrah Federal Building (Oklahoma, 1995) and at World Trade Center (New York, 2001). According to GSA (2003) Guidelines [1] 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. Progressive collapse potential of a building can be assessed using four different procedures. For preliminary studies the linear static (LS) or dynamic analysis (LD) is the most appropriate one due to its lower degree of complexity. When the acceptance criteria are not fulfilled or the structure is to complex (has more than ten stories or plan irregularities) a nonlinear static analysis (NS) should be carried out in order to obtain more accurate results. A nonlinear dynamic analysis (ND) can take into account in a more accurate manner the dynamic effect of sudden vertical elements failure. The results, obtained using the ND procedure, are considered to be the most accurate ones.
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As the complexity and dimensions of the model (number of bays/stories, types of FE, mesh size) increases, the necessary work time, computational power and professional expertise also increase. For this reason it would be helpful to study until what limit, the size and the complexity of a model can be reduced without significantly affecting the level of precision/accuracy of the results. In order to minimize the computational effort and the run time, US Department of Homeland Security, in a study [2] related to progressive collapse, simplified the original reinforced concrete structure of four spans and five bays, to a model of two and a half spans and bays.
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Starting from these remarks, in this paper a three story reinforced concrete framed structure is analyzed using three different procedures (LS, NS and ND). Previous studies [3, 4] have shown that low-rise reinforced concrete structures, and in particular those located in low seismic zones, are more vulnerable to progressive collapse compared to mid-rise structures. The original model (Figure 1a) with two spans and five bays is successively reduced with respect to its number of longitudinal bays, to a model with two bays (Figure 1b) and then, to a model with only one bay (Figure 1c). The main objective is to determine the maximum level of reduction that gives accurate results in the assessment of progressive collapse potential. In addition, the work time advantages offered by the simplified models will be presented and the final verdicts delivered by analyses with different degree of complexity (LS, NS and ND) are compared.
2.0 Structure and Sub-structures
Structural configuration of the original (unreduced) model consists of two spans and five bays of 6.0m each, and the model is designed according to the provisions of the Romanian seismic code P100/1-2006 [5], provisions that are similar to those specified by Eurocode 8 [6]. The structure was designed for a low seismic zone (ag = 0.08g) in order to minimize the positive influence of the seismic design and consequently, to capture the most unfavorable response of the building.
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Figure 1. Geometry and configuration of: a) original model; b) 2-bay reduced model; c) 1-bay reduced model.
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The story height is 2.75m except the first two floors were the story height is 3.6m. Dimensions of the cross section of beams and columns are detailed in Table 1. The substructures (reduced models) were obtained starting from the initial model (Figure 1a). The first reduced model (Figure 1b) consists of two spans and two bays while the second reduced model (Figure 1c) consists of two spans and one bay. Appropriate boundary conditions (blocked displacements in the longitudinal direction) were introduced in the joints, at all floor levels.
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Table 1: Cross-sectional dimensions of structural elements Column [mm]
Beam [mm]
Slab [mm]
3 -story
400 x 400
250 x 450
150
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Structure
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The beams and columns are modeled using one-dimensional (frame) finite elements. Each element is represented by a straight line connecting two joints. The frame element considers six degrees of freedom at both of its connected joints [7]. The slabs are modeled using two-dimensional (shell) finite elements available in the SAP2000 structural analysis software. This type of finite element has four joints and considers a total of twelve degrees of freedom. Kirchhoff formulation is used for shells, and transverse shear deformations are neglected [7]. The reinforcement of primary structural elements (beams, columns and slabs) was considered in the nonlinear analyses (NS and ND). The compressive strength class of the concrete is C25/30 (fck = 25N/mm2), and the steel for the longitudinal and transverse reinforcement is of S500 type (fyk = 500N/mm2) type. In the seismic design of the original model an average safety coefficient of 6 to 10% was considered when the amount of reinforcing steel was established.
3.0 Analysis Procedures In the assessment of the progressive collapse potential for the C3 damage column case (removal of the corner column) [1], three types of analyses are successively performed: Linear Static (LS), Nonlinear Static (NS) and Nonlinear Dynamic (ND). For both materials (concrete and reinforcing steel), a strength increase e factor of 1.25 is applied to the specified strengths, according to the provision of GSA Guidelines [1].
For the static analyses (LS, NS), the following vertical load is applied downward to the structure
Load 2( DL 0.25LL)
(1)
while for the dynamic analyses (ND), the applied load is
Load ( DL 0.25LL)
(2)
QUD QCE
(3)
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DCR
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where DL represents the dead load and LL represents the live load. When the LS procedure is used, the progressive collapse potential is assessed based on the DCR (Demand-Capacity-Ratios) concept. The magnitude and distribution of inelastic demands are indicated by the DCR values. For each structural component, DCR values are determined as follows
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where QUD is the acting force determined in member (moment, axial force, shear or combined forces), using the linear static analysis. QCE is the expected ultimate un-factored capacity of the member in terms of moment, axial force, shear or combined forces. Using the DCR criteria of the linear elastic approach, structural elements and connections that have DCR values that exceed the allowable value of 2.0 (for typical structural configuration) are considered to be severely damaged or collapsed. According to GSA (2003) Guidelines when the allowable DCR value (2.0) is exceeded, a five step procedure must be followed and successive iterations are required [1]. In the paper this procedure is not detailed, but the necessary iterations have been performed.
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Figure 2. Stress-strain curves for concrete and steel. The nonlinear procedures (NS and ND) take into account geometrical and material nonlinearity. While the LS procedure recommended by GSA Guidelines (2003) considers that an element is severely damaged or collapsed when DCR values are greater than 2.0, in nonlinear analyses the acceptance criteria specified by GSA (2003) for the plastic hinge rotation (Ɵ) are 0.035 rad / RC fames or 0.105 rad/RC beams [1]. According to the DoD (2009) [8] methodology, the acceptance criteria for RC structural elements, and also the level of the applied loads are different. In our study, for the nonlinear dynamic analysis a 5% damping factor is considered. Stress-strain curves used in this study are presented in Figure 2 and are based on EC-2 [9] specifications for concrete, respectively on the SAP2000 model for steel.
4.0 Progressive Collapse Analysis
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4.1. Linear Static Analysis
Figure 3. LSA - Flexural DCR values for: a) original model; b) 2-bay model; c) 1-bay model.
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The structural models considered in the LS procedure may have various levels of complexity. When the slab is not modeled as an independent primary component, there are two possibilities to introduce the beam in the frame response: as a beam with rectangular cross section or as a beam with T/L cross section. In this study the authors considered the following two cases: the less complex model (columns and rectangular beams) and the complete model that includes the columns, beams and slabs. GSA (2003) [1] and DoD (2009) [8] do not consider, in LS analyses, the slab as a primary component; for NS and ND analyses, DOD (2009) allows slabs to be modeled as primary components (user’s option). The results obtained for the model which does not include the slab in the analysis are displayed in red (values between brackets), while the results obtained when the slab is modeled are displayed in black (Figure 3). For all three models with rectangular beams (original model, two-bay reduced model and one-bay reduced model), the distribution of the DCR values that exceed the allowable value of 2.0 indicates the possibility of a 3-D mechanism occurrence (Figure 3). The progressive collapse risk in these cases is HIGH. When the slab effect is considered, the DCR values change significantly. If the whole (unreduced) model is analyzed, the allowable DCR value (2.0) is exceeded only in one beam section in the longitudinal direction (Figure 3a); the same phenomenon is observed at the two-bay reduced
model (Figure 3b), and the differences between DCR values obtained for those two models are practically negligible (max 1.5%). If the structure is reduced to one single bay, only DCR values in the longitudinal direction are influenced by this simplified approach. The final conclusion given by the LS analysis indicates that the progressive collapse potential of the original and reduced models is similar, and it is LOW. This conclusion is contrary to the verdict established when the contribution of the slabs is not considered.
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4.2. Nonlinear Static Analysis
Figure 4. NSA-Plastic hinges rotation (SAP2000) for: a) original model; b) 2-bay model; c) 1-bay model and vertical displacements for the original model (ABAQUS)
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The NS procedure was performed only for the structure and substructures where the slabs are introduced in the models. Values of the plastic hinges rotations furnished by the nonlinear static analysis are presented in Figure 4. For all the models, the maximum allowable rotation limit Ɵ=0.035rad = 35mrad, considered as a reference in this study, is not reached. For the original (unreduced) model, the maximum rotation value is only 3.2 mrad (Figure 4a), approximately ten times smaller than the limit. When the first level of reduction (from five to two bays) is applied to the original model, the maximum rotation value is Ɵ=3.3 mrad (Figure 4b). Compared with the original model the differences between the plastic hinges rotations are up to 3%. For the second level of reduction, the peak value of the plastic hinge rotation is Ɵ=3.6 mrad, and compared with the original model, the differences between the maximum rotations are up to 13%. The maximum vertical displacements for these three analyzed models are 2.802cm, 2.788cm respectively 2.999cm, and again, there are any noticeable differences between the analyzed models. For all the analyzed models (original and reduced models), the results obtained using NS
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analysis indicate that the risk for progressive collapse is LOW. If the elastic modeling of slabs with SAP2000 does not raise significant difficulties, the nonlinear modeling of slabs is not a common approach, it is much more complicate, involves time-consuming finite element analysis and there is still uncertainty about the results. In this paper, the nonlinear model of the beam-slab system was made using the offset option of the SAP2000 software (the mid-thickness of the slab is defined at certain distance from the beam cross-section axis). The results obtained with the offset option were compared to those obtained by the much more advanced finite element ABAQUS software (Figure 4), and also with results reported by Tsai [11] - for a similar damage case: Δmax=2.783cm (SAP2000-offset) vs. 4.07cm (ABAQUS) vs. 2.25cm [11]. If the slab is modeled without offset (the beam axis is defined to be at the mid-thickness of the slab) the vertical displacements obtained using SAP2000 software are higher (Δmax=5.91cm and Ɵmax=8.56 mrad), but the final verdict (LOW potential) remains unchanged. 4.3. Nonlinear Dynamic Analysis
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While it is preferable to remove the column instantaneously, the duration for removal must be 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 [8]. In the analysis, the authors have considered a removal time of 0.01 seconds, a total time of 3 second for the analysis starting from that moment and a step size of 0.01 seconds. The number of plastic hinges reduces significantly (Figure 5) compared with the nonlinear static procedure (Figure 4), and also the magnitude of plastic rotations.
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Figure 5. NDA - Plastic hinges rotation for original (unreduced) model. This fact may be explained by the reduced level of applied loads in NDA, where the amplification factor of 2.0 used in NSA is not considered anymore. The value for the maximum plastic hinge rotation is Ɵ=0.4 mrad when the entire model is analyzed. For the first level of reduction, the peak value of rotation is Ɵ=0.4mrad, equal to the previous value. When the structure is reduced to two spans, the maximum plastic hinge rotation value is Ɵ=0.5 mrad, that means 20% greater than the similar value obtained for the original model. In terms of maximum vertical displacement, the results obtained for these three models are 1.731cm, 1.608cm respectively 1.839cm (Figure 7). The final conclusion given by the NDA indicates that the progressive collapse potential of the original and reduced models is similar, and it is LOW. The nonlinear dynamic analysis conducted with SAP2000 software-offset option, are in very good agreement, for this level of loading, with
results furnished by ABAQUS software, and very close to results reported in literature by Tsai [11] : Δmax=1.73cm (SAP-offset) vs. 1.77cm (ABAQUS) vs. 1.83cm [11]
5.0 Synthesis of the Results
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Using three types of analyses (LS, NS and ND), results for the complete model and two reduced models have been obtained and compared. The contribution of the slabs, as independent primary components in the resisting mechanisms, was considered and discussed. The following graphics (Figures 6, 7) emphasize the results obtained using methods with various degrees of complexity.
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Figure 6. NSA: force-displacement curves for the original and reduced models
Figure 7. NDA: time-displacement curves for the original and reduced models. Force-displacement curves obtained for the NSA are presented in Figure 6. This graphic shows a good agreement between the unreduced model and the two-bay reduced model. A difference can
be observed between the original model and one-bay reduce model curves; the maximum vertical displacement varies with 0.5% for the two-bay reduced model and with 7% for the one-bay reduced model, compared to unreduced model. Based on the NDA, three time-displacement curves are obtained (Figure 7). As may be seen the curves have a similar shape and there are relative small differences when the reduced models are analyzed; in terms of maximum vertical displacements the differences are up to 6%.
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6.0 Conclusions
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In this paper the progressive collapse potential of a three story reinforced concrete framed structure is assessed using three different procedures: linear static (LS), nonlinear static (NS) and nonlinear dynamic (ND). For each one of these procedures, the entire model that has five bays is successively reduced to simplified models with two bays and one bay, respectively. Based on the results obtained the main conclusions of the study are:
For the LS analysis (Figure 3) the DCR values corresponding to the reduced structures vary from 1.50% to 10% with respect to the original model. The material and geometrical nonlinearity emphasizes the influence of the successive reduction of the model. Thus, the values of plastic hinge rotations obtained based on NSA (Figure 4) and NDA (Figure 5) differ with a ratio between 3% and 13%, respectively between 0% and 20% for the reduced models, compared with the original model. As a conclusion, it can be stated, that the progressive collapse risk is the same for all three types of analysis (LS, NS and ND) when the unreduced and successively reduced models are analyzed. Also, for accurate results the reduction should be made preserving one bay/span adjacent with the one containing the failed vertical structural element (2-bay reduced model).
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1. If the inelastic demands (DCR) are determined on a pure frame model (Figure 1-red values) these values are much greater than the DCR values obtained when the slab effect is taken into account (Figure 3-black values). The conclusion the LSA regarding the progressive collapse potential changes from HIGH to LOW if the slab is considered in the model. Thus, the slab has a significant influence and it must be taken into account in the assessment of the progressive collapse risk. A recent experimental works [10] shown increases of the ultimate load-carrying capacity up to 63%, if the slab was incorporated in the beam-column substructure.
The results show that for the LSA, the run-time of the FEM analysis is insignificant (1 minute) for all three models. The run-time for the NSA was 135 minutes for the entire model, 70 minutes for the first reduced model, respectively 31 minutes for the second reduced model. When the NDA was applied the run-time increased significantly at 1700 minutes for the initial model, 786 minutes for the first reduced model and 396 minutes for the second reduced model. All analyses were run on the same computer (i5 processor, 8 GB ram) and for an identic mesh size. Consequently, assessing the progressive collapse potential on reduced models gives important run time savings: 52% for NSA, respectively 46% for NDA for the two-bay reduced model, without losing the accuracy. For this level of reduction the results are accurate and lead to the same conclusion regarding progressive collapse potential.
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4. In the GSA static analyses (LS, NS) a dynamic increase factor (DIF) of 2.0 was considered [1].
From Figure 6 and 7, it can be seen that for the load applied in ND analysis 100% of (DL + 0.25LL), NDA shown a maximum vertical displacement of 1.731 cm for the original model; the same value for the vertical displacement was obtained in the NS analysis for a load
level corresponding to 77.5% of 2(DL+0.25LL). This is equivalent with an actual DIF of 1.55. This finding shows that the DIF value of 2.0 is over-estimated by 29% for the load level recommended by the GSA Guidelines (2(DL+0.25LL)). 5. For this level of applied loads 2(DL+0.25LL) for NSA, and (DL+0.25LL) for NDA, the
number of plastic hinges reduces from 12 (NSA) to 6 (NDA). These results are explained by the fact that the structure is not strongly pushed in the post-elastic domain and the loads are different in magnitude.
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6. As future research directions, the authors intend to study if the conclusions regarding the
7.0 Acknowledgments
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successive reductions of the models remain valid for different/higher level of applied loads. In addition, capacity curves for the models, corresponding to different missing column scenarios, should be obtained. Also, the accuracy of results shall be improved by considering a distributed plasticity model (plastic zones/ABAQUS) with respect with the concentrated plastic hinge model (SAP2000) used in this study. Similar analyses will be performed following the new DoD (2009) standard, and results will be compared to those obtained on the GSA (2003) basis.
8.0 References
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The authors gratefully acknowledge the funding by the Executive Agency for Higher Education, Research, Development and Innovation Funding (UEFISCDI), Romania, under grant PN II PCCA 55/2012 “Structural conception and Collapse control performance based Design of multistory structures under accidental actions” CODEC (2012-2015), made in the frame of the Partnerships Program Joint Applied Research Projects.
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1. General Service Administration, “Progressive Collapse Analysis and Design Guideline for New Federal Office Buildings and Major Modernization Projects”, Washington, USA, 2003. 2. U.S. Department of Homeland Security, “Preventing Structures from Collapsing”, USA, 2011. 3. Botez M., Bredean L., Ioani A.M., "Inelastic demands of RC Structures: Corner Column Case in the Progressive Collapse Analysis", GNP 2012, Žabljak, Montenegro, 2012. 4. Bredean, L., Botez, M., Ioani, A.M., “Progressive Collapse Risk and Robustness of Low-Rise Reinforced Concrete Buildings”. CST 2012, Dubrovnik, Croatia, 2012. 5. MTCT, P100-1/2006, “Seismic design code – Part I: Design Rules for Buildings”, Bucharest, Romania, 2006. 6. ASRO, SR EN 1998 -1/NA.,”Eurocode 8: Design of Structures for Earthquake Resistance Part 1: General Rules, Seismic Actions and Rules for Buildings”, Bucharest, Romania, 2008. 7. Computers and Structures, Inc., “CSI Analysis Reference Manual”, Berkeley, USA, 2011. 8. Department of Defense (DoD), “UFC 4-023-03 - Design of Buildings to Resist Progressive Collapse”, Washington, USA, 2009. 9. ASRO, SR EN 1992-1-1: 2004, “Eurocode 2: Design of Concrete Structures – Part 1-1: General Rules and Rules for Buildings”, Bucharest, 2004. 10. Qian K., Li B., “Slab Effects on Response of Reinforced Concrete Substructures after Loss of Corner Column”, ACI Structural Journal, Vol.109, 2012, pp.845-856. 11. Tsai M-H., Lin B-H., “Investigation of progressive collapse resistance and inelastic response for an earthquake-resistant RC building subjected to column failure”, Engineering Structures, Vol. 30, 2008, pp. 3619-3628.
