(PDF) Benchmark Numerical Model for Progressive Collapse Analysis of RC Beam-Column Sub-Assemblages

buildings Article Benchmark Numerical Model for Progressive Collapse Analysis of RC Beam-Column Sub-Assemblages Bilal El-Ariss 1, * , Said Elkholy 1,2 and Ahmed Shehada 1 1 Civil and Environmental Engineering Department, United Arab Emirates University, Al Ain 15551, United Arab Emirates;

(S.E.);

(A.S.) 2 Civil Engineering Department, Fayoum University, Faiyum 63514, Egypt * Correspondence:

Abstract: The pertinence of the fiber element approach to enable thorough numerical investigation on the potential for progressive collapse of reinforced concrete (RC) frame structures owing to interior column exclusion is examined using twenty-nine RC sub-assemblages with five different test setups and three different test scales. A qualitative examination of the results reveals a good agreement between the test results and the outcomes of the fiber-element-based numerical model using the finite element package SeismoStruct. Moreover, minor discrepancies between the test and numerical data demonstrate the capability of the fiber-element-based model to accurately simulate the behavior of RC elements with various boundary conditions and scales under the context of progressive collapse. Given the costly nature of experimental research, test errors, and the lengthy testing process, the proposed numerical model based on the fiber element approach can be considered a viable option for analyzing structures under progressive collapse due to the interior column exclusion scenario. Engineers and researchers can use the conclusions and comments highlighted in the study as a guide to create accurate models for the analysis of RC structures subjected to progressive collapse. Citation: El-Ariss, B.; Elkholy, S.; Keywords: progressive collapse; fiber-element-based model; numerical simulation; RC sub-assemblages; Shehada, A. Benchmark Numerical column removal Model for Progressive Collapse Analysis of RC Beam-Column Sub-Assemblages. Buildings 2022, 12, 122. https://doi.org/10.3390/ 1. Introduction buildings12020122 The severe damage caused by structural progressive collapse has increased research Academic Editor: Alessandra activities that focus on structural behavior prediction. The purpose of an experimental work Aprile carried out by Qiang et al. [1] was to undertake an experimental validation and analysis of Received: 27 November 2021 progressive collapse behaviors of RC frames with unique kinked rebar configurations. The Accepted: 23 January 2022 study verified that using kinked rebar configuration can improve the structure progressive Published: 25 January 2022 collapse resistance. Alogla et al. [2] described a mitigating system to improve the structure progressive collapse capacity by using extra steel bars in the middle layer of the beams. The Publisher’s Note: MDPI stays neutral proposed system significantly enlarged the ductility of the specimens. Kang and Tan [3] with regard to jurisdictional claims in experimentally examined the progressive collapse behavior of four precast concrete beam- published maps and institutional affil- column sub-assemblage specimens with 90◦ bend or lap-spliced bottom reinforcement iations. in the beam-column joint. The study showed that the improvement of compression arch action and catenary action depended primarily on the detailing. Yu and Tan [4] tested six RC beam-column sub-assemblage specimens to investigate alternate load paths to Copyright: © 2022 by the authors. alleviate progressive collapse. Test outcomes showed that compression arch action is Licensee MDPI, Basel, Switzerland. effective in short span beams with low reinforcement ratio, while the catenary action is This article is an open access article more active in large span beams with high reinforcement ratio. Specimens tested by Yu distributed under the terms and and Tan [5] had a special detailing design enhance the catenary action capability at big conditions of the Creative Commons deformations due to column removal. Additional reinforcement layer at the specimen mid Attribution (CC BY) license (https:// height section with partially unbonded bottom reinforcing bars in the joint region were creativecommons.org/licenses/by/ added. More tests were recommended to better investigate the effect of detailing on the 4.0/). progressive collapse behavior. Rashidian et al. [6] experimentally studied the impacts of Buildings 2022, 12, 122. https://doi.org/10.3390/buildings12020122 https://www.mdpi.com/journal/buildings Buildings 2022, 12, 122 2 of 22 transverse beams on the frame resistance to progressive collapse. The findings showed an overall improvement in the progressive collapse resistance, with a particular focus on the transverse beam flexural capacity. Furthermore, the transverse beam appeared to restrict the rotation of the middle joint, resulting in additional ductile behavior. Many more tests have been conducted to determine how resistant structures are to progressive collapse. Nevertheless, the existing tests have focused mostly on beams, plane frames, and beam-column substructures [7–9]. In comparison to typical RC framed structures, various structural systems and components, such as precast construction and pre-stressed concrete structures, receive less attention in the analysis of progressive collapse. The knowledge in this domain is restricted to a few studies, so more attention should be devoted to the progressive collapse analysis of such systems [10,11]. To explore the RC sub-assemblage structural behavior of, and the development of, alternate load path mechanisms during progressive collapse, Ahmadi et al. [12] tested scaled RC beam-column sub-assemblages in a quasi-static loading. As experimental testing of progressive collapse are time-demanding and costly, nu- merical and analytical models have been developed as viable options to resolve such issues and have gradually become the attention of research recently. Gross et al. [13] performed one of the earliest analytical studies on structural progressive collapse. This study pre- sented computer software capable of analyzing and tracing the performance of framed structures during progressive collapse. Casciati et al. [14] investigated seismic reliabil- ity in a progressive collapse of 2D reinforced concrete framed structures. Simulating progressive collapse, with simple models rather than sophisticated analysis, has been advo- cated in a number of recent research. For modeling bar breakage in RC frame structures, Sasani et al. [15] developed in depth models. Tsai et al. [16] used the static linear analysis approach specified by the US General Service Administration (GSA) [17] to conduct a pro- gressive collapse analysis. In their numerical examination, Rashidian et al. [6] proposed 2D and 3D models to evaluate the progressive collapse behavior of scaled RC sub-assemblages. The usage of the Dynamic Amplification Factor (DAF) was employed in inelastic dynamic progressive collapse analyses, as highlighted in DoD and GSA [17,18] guidelines, as well as in works by Grierson et al. and Mohajeri et al. [19,20]. With some simplifications, a non-linear static analysis was utilized to analyze pertinent dynamic effects in the investi- gations of Izzuddin et al., Powell, Vlassis, Marjanishvili et al., and Alashker et al. [21–25]. Ellingwood et al. [26] used the energy balance of interior and exterior forces to analyze progressive collapse. Luccioni et al. [27] employed comprehensive models to investigate structural breakdown of an existing RC structure due to a blast loading. Talaat et al. [28] generated a method using simplified models for simulating structural collapse of RC struc- tures under seismic loading. To test the severe load behavior of steel sub-assemblages in progressive collapse, pre-Northridge connections were considered and studied [29]. Kwasniewski [30] used the GSA guidelines and non-linear finite element simulations to investigate the progressive collapse response of an eight-story building subjected to ex- plosive loading. Hao et al. [31] and Shi et al. [32] also employed explosive loads in their studies and proposed a new methodology for analyzing the behavior of RC frames under the progressive collapse scenario. Elkholy and El-Ariss [33–36] as well as El-Ariss and Elkholy [37] presented a system and modeling algorithm for strengthening RC statically indeterminate beams to resist progressive collapse due to internal column failure. External unbounded unstressed straight wires attached to the beam at deviator and anchorage loca- tions were proposed as part of the technique. The proposed method improved the ultimate load-carrying capacity and energy-dissipation capabilities of the beam. Utilizing the fiber element technique, Shehada et al. [38] numerically examined the resistance of RC framed structures to progressive collapse due to column exclusion. The numerical results from the fiber element technique were compared to a database of 10 previously published tests of RC structures. In a study to simulate the inelastic dynamic response of RC buildings exposed to column failure, Brunesi and Nascimbene [39] detailed a numerical model using an open source fiber-based code, such as OpenSees and SeismoStruct, and verified it by comparison Buildings 2022, 12, 122 3 of 22 Buildings 2022, 12, x FOR PEER REVIEW 3 of 24 with a general purpose FE package, Ls-Dyna. Inelastic force-based fiber elements with five integration points accommodated for both geometric and material nonlinearities in buildings exposed to column failure, Brunesi and Nascimbene [39] detailed a numerical the fiber-based model. model A separate using an open purpose routine was source fiber-based derived code, such astoOpenSees simulateand theSeismoStruct, abrupt and column removal inverified OpenSees and SeismoStruct, and the tangent-stiffness-based Rayleigh it by comparison with a general purpose FE package, Ls-Dyna. Inelastic force- damping was assumed in the based fiber inelastic elements withdynamic analyses. five integration pointsSousa et al. [40] for accommodated usedboththe FE geometric and program SeismoStruct to numerically assess the nonlinear static and nonlinear dynamic material nonlinearities in the fiber-based model. A separate purpose routine was derived responses of an experimentally to simulate the tested abrupt three-story column removal RC building in OpenSees with andplan irregularity. SeismoStruct, andThethe tangent- conclusion was that the infill panels considerably increase the structure initial stiffness and Sousa stiffness-based Rayleigh damping was assumed in the inelastic dynamic analyses. et al. [40] used strength, and that modeling the the slabFEshould program beSeismoStruct given extra to numerically attention assessitthe because may nonlinear imposestatic and nonlinear dynamic responses of an experimentally tested three-story RC building with artificial beam strengthening. plan irregularity. The conclusion was that the infill panels considerably increase the struc- The above literature review of experimental and numerical studies on the progressive ture initial stiffness and strength, and that modeling the slab should be given extra atten- collapse analysis of structures tion because itwith failed interior may impose artificial column demonstrates that researchers beam strengthening. have typically carried outThe above literature review of experimentaldeveloped costly experimental testing and/or and numericaltimestudies consuming, on the progres- sophisticated finitesive element collapse numerical analysis ofmodels forwith structures theirfailed test interior structures. columnGiven the costlythat demonstrates na-research- ture of experimentalerstesting, possible have typically test errors, carried out costlyand the lengthytesting experimental testing process, and/or numerical developed time consum- ing, sophisticated models can be a viable alternative finite element numerical to experimental testing. models for their However, duetest to structures. the scarcity Given the of a comprehensive costly nature outline ofofthe experimental available testing, possible test finite element errors, and numerical the lengthy models, this testing study process, aims at utilizing a numerical large database modelsofcan be a viable alternative twenty-nine tests of RC to sub-assemblages experimental testing. toHowever, propose due a to the scarcity of a comprehensive outline of the available finite element numerical models, this simple fiber-element-based numerical model to be used by engineers and researchers as study aims at utilizing a large database of twenty-nine tests of RC sub-assemblages to a benchmark model for an accurate progressive collapse analysis of RC structures due to propose a simple fiber-element-based numerical model to be used by engineers and re- interior column removal. searchers as a benchmark model for an accurate progressive collapse analysis of RC struc- tures due to interior column removal. 2. Descriptions of Reference Specimens and Test Setups 2.1. Details of Sub-Assemblage Testof 2. Descriptions Specimens Reference Specimens and Test Setups Figure 1 shows2.1. a typical Details detailing of Test of Sub-Assemblage the Specimens RC sub-assemblage reference test spec- imens considered in this study. A total of twenty-nine Figure 1 shows a typical detailing of the RC sub-assemblage sub-assemblage reference test reference test specimens specimens with three different considered scales in this study.were A totalselected from sub-assemblage of twenty-nine twelve different experimental reference test specimens with threeTable investigations [1–12]. different scales were 1 shows the selected from twelve dimensions, different experimental reinforcement ratios, and investigations material [1–12]. Table 1 shows the dimensions, reinforcement ratios, and material properties of the scaled RC sub-assemblage specimens. More information can be found properties of theinscaled RC sub-assemblage the literature [1–12]. specimens. More information can be found in the literature [1–12]. Figure 1. Typical RCFigure 1. Typical sub-assemblage beam-column RC beam-column sub-assemblage detailing. detailing. 2.2. Sub-Assemblage Test Setup The test specimens considered in this study from various experimental investigations have five different test setups with various boundary conditions. This study labels the Buildings 2022, 12, 122 4 of 22 different setups as setup A, B, C, D, and E and describes their boundary conditions as illustrated in Table 2 and in Figure 2. Moreover, the specimen-generated numerical models with the different boundary conditions are displayed in Figure 2. Table 1. Specimen details (dimension symbols in the heading row are shown in Figure 1). Cross Section Dimension (mm × mm) Reference Specimen 0 ρtop at ρbot at fc for fy for Ls lb h1 h2 ID # Scale Side Mid- Section Section Beam Beam (mm) (mm) (mm) (mm) Beam Column Column A-A (%) A-A (%) (MPa) (MPa) Hb × B Hsc × Wsc Hmc × Wmc 1 [1] RCB 1/3 1900 664 185 185 170 × 85 N/A × 500 200 × 200 0.92 0.72 43.1 387 2 SS-1 1/2 2750 900 450 450 250 × 150 400 × 400 250 × 250 0.45 0.45 26.8 510 3 SS-2 1/2 2750 900 450 450 250 × 150 400 × 400 250 × 250 0.45 0.45 26.8 510 [2] 4 SS-3 1/2 2750 900 450 450 250 × 150 400 × 400 250 × 250 0.45 0.45 27.5 510 5 SS-4 1/2 2750 900 450 450 250 × 150 400 × 400 250 × 250 0.45 0.45 27.5 510 6 MJ-B-S 1/2 2750 900 450 450 300 × 150 N/A 250 × 250 0.55 0.37 27.9/35.8 462 [3] 7 MJ-L-S 1/2 2750 900 450 450 300 × 150 N/A 250 × 250 0.55 0.37 27.9/35.8 462 8 S4 1/2 2750 1000 450 450 250 × 150 450 × 400 250 × 250 1.24 0.82 38.2 494 9 S5 1/2 2750 1000 450 450 250 × 150 450 × 400 250 × 250 1.24 1.24 38.2 494 10 [4] S6 1/2 2750 1000 450 450 250 × 150 450 × 400 250 × 250 1.87 0.82 38.2 494/513 11 S7 1/2 2150 780 450 450 250 × 150 450 × 400 250 × 250 1.24 0.82 38.2 494 12 S8 1/2 1550 560 450 450 250 × 150 450 × 400 250 × 250 1.24 0.82 38.2 494 13 S1 1/2 2750 1000 450 450 250 × 150 400 × 400 250 × 250 0.9 0.49 31.2 511/527 [7] 14 S2 1/2 2750 925 450 450 250 × 150 400 × 400 250 × 250 0.73 0.49 31.2 511 15 NSC-8 1/2 2000 700 N/A N/A 250 × 150 400 × 400 250 × 250 0.96 0.64 31.7 577 16 NSC-11 1/2 2750 900 N/A N/A 250 × 150 400 × 400 250 × 250 0.96 0.64 31.1 577 17 NSC-13 1/2 3250 1100 N/A N/A 250 × 150 400 × 400 250 × 250 0.96 0.64 30.5 577 [9] 18 HSC-8 1/2 2000 750 N/A N/A 250 × 150 400 × 400 250 × 250 0.96 0.64 60.5 577 19 HSC-11 1/2 2750 900 N/A N/A 250 × 150 400 × 400 250 × 250 0.96 0.64 61.2 577 20 HSC-13 1/2 3250 1100 N/A N/A 250 × 150 400 × 400 250 × 250 0.96 0.64 59.3 577 21 F1 1/2 2750 1000 1100 1050 250 × 150 250 × 250 250 × 250 1.15 0.77 29.69 520/488 [5] 22 F2 1/2 2750 1000 1100 1050 250 × 150 250 × 250 250 × 250 1.15 0.77 26.69 520/488 23 IF-B 1/2 2750 1000 1325 1025 300 × 150 250 × 250 250 × 250 0.97 0.64 27.7/26.9 553.2/593.7 24 [8] IF-L 1/2 2750 1000 1325 1025 300 × 150 250 × 250 250 × 250 0.97 0.64 27.7/26.9 553.2/593.7 25 EF-B 1/2 2750 1000 1325 1025 300 × 150 250 × 250 250 × 250 0.97 0.64 27.7/26.9 553.2/593.7 26 [6] 2D 3/10 1400 500 520 820 140 × 200 200 × 200 200 × 200 0.57 0.38 26 530 27 [10] RC 1/2 2650 700 1200 2300 300 × 200 350 × 350 500 × 500 0.88 0.88 24.7 485 28 [12] Prototype 3/10 1500 500 720 820 140 × 200 200 × 200 200 × 200 0.39 0.39 26 530 29 [11] MS 1/3 1148 N/A 450 450 135 × 100 135 × 135 135 × 135 0.79 0.79 25 500 ρ = reinforcement ratio. Buildings 2022, 12, 122 5 of 22 Table 2. Types and descriptions of different test setups and boundary conditions. Specimen Test # Reference Scale Reference Findings Test Set-Up Description/Boundary Conditions ID Setup The kinked rebar arrangement proposed improves Setup • Restraints at Outer Beam Ends. 1 [1] RCB 1/3 the progressive collapse resistances of RC. A • Fixed Boundary Conditions at Beam Ends. 2 SS-1 1/2 3 SS-2 1/2 Concrete beam ductility and collapse load are greatly [2] 4 SS-3 1/2 improved by the proposed mitigation strategy. 5 SS-4 1/2 6 MJ-B-S 1/2 Joint details and beam reinforcement ratio have a big [3] role in improving CAA and catenary 7 MJ-L-S 1/2 action resistance. 8 S4 1/2 • Restraints at the Bottom of Lower Side Columns. 9 S5 1/2 Both compressive arch action and catenary action • Horizontal Restraints along the Side Columns. 10 [4] S6 1/2 might be enhanced with suitable axial constraints, • Axial/Horizontal Restraints of Beams with greatly boosting structural resistance beyond the Tension/Compression Load Cells at Beam 11 S7 1/2 beam flexural limit. Setup B Ends to Measure the Beam Axial/ 12 S8 1/2 Horizontal Reaction. • Rotational and Lateral Restraints of Beams at A component-based joint model made up of a and in the Vicinity of Failed Column. 13 S1 1/2 succession of springs was proposed and included [7] • Note: Ref. [9] Has only Axial Restraints into a macromodel-based finite element analysis that of Beams. 14 S2 1/2 used fiber elements to model beams. 15 NSC-8 1/2 16 NSC-11 1/2 The use of high-strength concrete was found to 17 NSC-13 1/2 improve the compressive arch action capacity, [9] particularly for frames with a low span-to-depth 18 HSC-8 1/2 ratio, while lowering the tensile catenary action 19 HSC-11 1/2 capacity at large deformations. 20 HSC-13 1/2 Distinct detailing includes adding an extra layer of reinforcement at the middle of beam sections, in part • Restraints at the Bottom of Lower Side Columns. 21 F1 1/2 debonding bottom reinforcing bars in the joint • Horizontal Restraints at Top of the Upper [5] Columns. location, and positioning partial hinges away from adjacent joint improve the catenary action capacity • Horizontal Restraints at the Outer Ends of Beams. 22 F2 1/2 of beams at large deformations. Setup • Horizontal Restraints at Top End of Side Columns. C • Pin Restraints at the Bottom End of Side Columns. In precast concrete frames, lap spliced reinforcing in • Axial/Horizontal Restraints of Beams with 23 IF-B 1/2 the joint facilitated the formation of higher catenary Tension/Compression Load Cells at Beam Ends activity than a 90◦ bend of beam bottom to Measure the Beam Axial/Horizontal Reaction. [8] 24 IF-L 1/2 reinforcement. Lap-splicing of beam bottom • Rotational and Lateral Restraints of Beams at reinforcement in the joint is recommended to and in the Vicinity of Failed Column. 25 EF-B 1/2 prevent progressive collapse. • Restraints at the Bottom of Lower Side Columns. Overall improvement in the progressive collapse • Horizontal Restraints at Top of the 26 [6] 2D 3/10 resistance as transverse beams appeared to restrict Upper Columns. the rotation of the middle joint. • Horizontal Restraints at Top End of Side Columns. More attention should be devoted to the progressive Setup • Fixed Boundary Conditions at Bottom End of 27 [10] RC 1/2 collapse analysis of precast construction and D the Side Columns. pre-stressed concrete structures. • Axial/Horizontal Restraints of Beams with Tension/Compression Load Cells at Beam The beam action mechanisms of the substructures Ends to Measure the Beam Axial/ are greatly increased when the beam height is Horizontal Reaction. 28 [12] Prototype 3/10 increased. it causes an early steel bar fracture and • Lateral Restraints of Beams in the Vicinity of reduces the catenary action of the sub-assemblage. Failed Column. More attention should be devoted to the progressive Setup • Restraints/Pin Support at the Top of Upper 29 [11] MS 1/3 collapse analysis of precast construction and E Side Columns. pre-stressed concrete structures. 27 [10] RC 1/2 gressive collapse analysis of precast construc- umns. tion and pre-stressed concrete structures. • Fixed Boundary Conditions at Bottom End of the Setup D Side Columns. The beam action mechanisms of the substruc- • Axial/Horizontal Restraints of Beams with Ten- tures are greatly increased when the beam sion/Compression Load Cells at Beam Ends to 28 [12] Prototype 3/10 height is increased. it causes an early steel bar Measure the Beam Axial/Horizontal Reaction. fracture and reduces the catenary action of the • Lateral Restraints of Beams in the Vicinity of sub-assemblage. Failed Column. Buildings 2022, 12, 122 More attention should be devoted to the pro- 6 of 22 • Restraints/Pin Support at the Top of Upper Side 29 [11] MS 1/3 gressive collapse analysis of precast construc- Setup E Columns. tion and pre-stressed concrete structures. Specimen test setup and boudary conditions SeismoStruct model and boundary conditions Buildings 2022, 12, x FOR PEER REVIEW 6 of 24 Test Setup A Test Setup B Test Setup C Test Setup D Test Setup E Figure 2. Test setups (as per Table 2) and their corresponding numerically generated models in Figure 2. Test setups (as per Table 2) and their corresponding numerically generated models in this study. this study. Buildings 2022, 12, 122 7 of 22 3. Sensitivity Analysis of Model Key Parameters Using SeismoStruct The sensitivity of key parameters impacting the performance of the fiber element- based numerical model using the finite element package SeismoStruct [41] is studied in this section. This sensitivity study is, however, classified in the following subsections based on the model different key parameters. The key model parameters that were studied and discussed are the element types, concrete material modeling, steel material modeling, num- ber of section fibers, plastic hinge length, number of segments, and boundary conditions of different test setups. Extensive numerical analyses and simulations, based on the behaviors of the twenty- nine sub-assemblage test specimens detailed in the previous section, were performed. To achieve the optimal numerical model that accurately simulates the behavior of each test specimen, the consequences of using different material (concrete and steel) modeling, vari- ous number of elements, numerous number of section fibers, several plastic hinge lengths, and different boundary conditions on the accuracy of the outcomes of each specimen model were examined and systematized for tuning and calibrating the numerical model to be utilized as a benchmark model in the progressive collapse analysis of RC structures. 3.1. Element Types There are two types of frame fiber element modeling available in SeismoStruct: in- ealastic force-based plastic-hinge frame elements (infrmFBPH) and inealastic displacement- based plastic-hinge frame elements (infrmDBPH). The two types are founded on distribut- ing the plasticity throughout the element length either by breaking the infrmFBPH element into a number of sections for numerical integration or by defining different nodes and connecting them together in the case of infrmDBPH element. In general, infrmDBPH element shows slower convergence than that of the infrmFBPH element. Therefore, in this study the RC sub-assemblage members were modeled using the infrmFBPH elements. 3.2. Concrete Material Modeling A variety of concrete material models is provided by the software library: Trilinear concrete model (con_tl); Mander nonlinear concrete model (con_ma); Chang-Mander non- linear concrete model (con_cm); Kappos and Konstantinidis nonlinear concrete model (con_hs); Engineered cementitious composites material (con_ecc); and Kent-Scott-Park con- crete model (con_ksp). More information on these models can be found in the SiesmoStruct user manual. An expanded task of numerically simulating the test behavior of each one of the twenty-nine sub-assemblage test specimens, using the reported test concrete material properties, revealed that the optimal concrete material model to simulate the test be- haviors was model con_ma. An example of this expanded simulation task is shown in Figure 3. The figure shows the experimental load-displacement curve of only one specimen, specimen SS-3 [2], and its corresponding numerical load-displacement curves, using the different concrete material models above. It is clear from the figure that most of the concrete material models produced fairly accurate simulations of the test performance, but the most accurate simulation was for the concrete material model con_ma. The same concrete model, con_ma, produced the best simulation of each test behavior of the twenty-eight reference specimens reported in the previous section. Hence, the model con_ma for concrete was adopted and used throughout this study. Moreover, for this adopted model a strain value between 0.2 and 0.6% at peak stress as well as an ultimate strain value of 0.7% were found to be the optimal values for modeling the above reported twenty-nine test specimens, whereas a value between 1.5 and 3.5 MPa proved to be suitable for the concrete tensile strength. 3.3. Steel Material Modeling Like the concrete material, various steel material models are provided by the software library to choose from: bilinear steel model (stl_bl); bilinear steel model with isotropic strain Buildings 2022, 12, 122 8 of 22 Buildings 2022, 12, x FOR PEER REVIEW hardening (stl_bl2); Menegotto-Pinto steel model (stl_mp); Dodd-Restrepo steel 8model of 24 (stl_dr); Monti-Nuti steel model (stl_mn); Giuffre-Menegotto-Pinto steel model (stl_gmp); and Ramberg-Osgood steel model (stl_ro). 80 60 Experimental Applied load (kN) con_ma con_tl 40 con_cm con_hs 20 con_ecc con_ksp 0 0 100 200 300 400 500 600 Vertical displacement (mm) (a) (b) Figure Figure3.3.Test TestSS-3 SS-3[2][2] and numerical and load-displacement numerical load-displacementcurves with curves concrete with material concrete modeling. material (a) modeling. Applied load versus vertical displacement (at failed column location); (b) Concrete material model (a) Applied load versus vertical displacement (at failed column location); (b) Concrete material model con_ma. con_ma. 3.3. Steel Material Similar to theModeling previous section, and with the concrete material model con_ma being adopted, Like all thethe above steel concrete material material, models various were steel tested in material an effort models aretoprovided determineby thethe optimal soft- steel material ware library to model choosethatfrom: best simulates the test bilinear steel behavior model of each (stl_bl); one of bilinear themodel steel twenty-nine test with iso- specimens tropic straininhardening the previous section.Menegotto-Pinto (stl_bl2); The reported teststeel steelmodel material properties (stl_mp); were applied Dodd-Restrepo in themodel steel models. An example (stl_dr); of thissteel Monti-Nuti effortmodel is presented (stl_mn);in Figure 4. The figure shows Giuffre-Menegotto-Pinto the steel experimental load versus displacement curve model (stl_gmp); and Ramberg-Osgood steel model (stl_ro). of one specimen, specimen SS-3 [2], and its corresponding numericalsection, Similar to the previous load-displacement and with thecurvesconcreteusing the different material steel material model con_ma being adopted, all the above steel material models were tested in an effort to determine thefairly models above. It can be concluded that all the steel material models produced op- comparable timal Buildings 2022, 12, x FOR PEER REVIEW simulations steel material modelof thebest that testsimulates performance, butbehavior the test the mostofaccurate each onesimulation 9 ofwas of the twenty- 24 for the nine teststeel material specimens inmodel stl_mp.section. the previous The same Thesteel model, reported teststl_mp, produced steel material the best properties simulation were applied ofin each theone of theAn models. other twenty-eight example test behaviors. of this effort is presented in Figure 4. The figure 80 shows the experimental load versus displacement curve of one specimen, specimen SS-3 [2], and its corresponding numerical load-displacement curves using the different steel material models above. It can be concluded that all the steel material models produced 60 fairly comparable simulations of theExperimental test performance, but the most accurate simulation Applied load (kN) was for the steel material model stl_mp. stl_mpThe same steel model, stl_mp, produced the best simulation of each one of the other twenty-eight stl_bl test behaviors. 40 Hereafter, the model stl_mp for steel was adopted this study. Furthermore, for this stl_bl2 adopted model, a strain hardening ratio of 0.5% and a fracture strain value between 6 and stl_gmp 20 11% were found to be appropriate for stl_ro modeling the twenty-nine test specimens. Table 3 shows a summary of thestl_dr input parameters related to the steel bar and concrete stl_mn The material parameters are extracted from the materials used in the numerical modeling. 0 reported test data and are shown in Table 1. The material parameters that are not provided 0 100 200 in the 300 reported 400 test500 600taken form the associated references shown in Table 3. data are Vertical displacement (mm) (a) (b) Figure 4. Test SS-3 [2] and numerical load-displacement curves with various steel material models. (a) Figure 4. Test SS-3 [2] and numerical load-displacement curves with various steel material mod- Applied load versus vertical displacement (at failed column location); (b) Steel material model stl_mp. els. (a) Applied load versus vertical displacement (at failed column location); (b) Steel material model stl_mp. Table 3. Summary of input material parameters in the numerical model. Menegotto-Pinto Steel Material Model, stl_mp [42] Mander et al. Nonlinear Concrete Model, con_ma [43] Modulus of elasticity, 𝐸𝑠 (GPa) Compressive strength, 𝑓𝑐′ (MPa) Yield strength, 𝑓𝑦 (MPa) Tensile strength, 𝑓𝑡 (MPa) Strain hardening parameter, 𝜇 (mm/mm) Modulus of elasticity, 𝐸𝑐 (GPa) stl_mp Applied load ( stl_bl 40 stl_bl2 stl_gmp 20 stl_ro Buildings 2022, 12, 122 stl_dr 9 of 22 stl_mn 0 0 100 200 300Hereafter, 400 the 500model 600 stl_mp for steel was adopted this study. Furthermore, for this adopted model, Vertical displacement (mm) a strain hardening ratio of 0.5% and a fracture strain value between 6 and 11%(a) were found to be appropriate for modeling the twenty-nine (b) test specimens. Table 3 shows a summary of the input parameters related to the steel bar and concrete Figure 4. Test SS-3 [2] and numerical load-displacement curves with various steel material models. (a) materials used in the numerical modeling. The material parameters are extracted from the Applied load versus vertical displacement (at failed column location); (b) Steel material model stl_mp. reported test data and are shown in Table 1. The material parameters that are not provided in the3.reported Table Summarytest data material of input are taken form thein parameters associated references the numerical model. shown in Table 3. Menegotto-Pinto Steel Material Summary Table 3.Model, of input stl_mp [42]materialMander parameters in Nonlinear et al. the numerical model. Model, con_ma [43] Concrete Modulus of elasticity, 𝐸𝑠 (GPa) Compressive strength, 𝑓𝑐′ (MPa) Menegotto-Pinto Steel Material Model, stl_mp [42] Mander et al. Nonlinear Concrete Model, con_ma [43] Yield strength, 𝑓𝑦 (MPa) Tensile strength, 𝑓𝑡 (MPa) 0 Modulus of elasticity, E (GPa) Compressive strength, f (MPa) Strain hardening parameter, s𝜇 (mm/mm) Modulus of elasticity, 𝐸c𝑐 (GPa) Yield strength, f y (MPa) Tensile strength, f t (MPa) Transition curve initial Strain hardening parameter, (mm/mm) 𝑅0 shape µparameter, Strain at peakofstress, Modulus 𝜀𝑐 E(mm/mm) elasticity, c (GPa) TransitionTransition curve shape curvecalibrating parameter, R0𝑎1 and 𝑎2 initial shapecoefficients, Specific Strain stress,𝛾ε c (kN/m weight, at peak (mm/mm)3) Transition curve shape calibrating coefficients, a and a Specific weight, (kN/m 3) Isotropic hardening calibrating coefficients, 𝑎13 and 𝑎 24 γ Isotropic hardening calibrating coefficients, a3 and a4 Fracture/buckling strain, 𝜀𝑢𝑙𝑡 (mm/mm) Fracture/buckling strain, ε ult (mm/mm) Specific weight, γ𝛾 (kN/m Specificweight, (kN/m3 )3) 3.4. Number of Section Fibers 3.4. Number of Section Fibers The RC sub-assemblage members were modeled in this study using the infrmFBPH The RC elements. Thesub-assemblage members member cross-sections arewere modeledby represented in athis studyof number using the infrmFBPH discrete fibers that elements. The member cross-sections are represented by a number represent concrete and steel independently. Figure 5 shows the discretization of discrete fibers of a repre- that represent concrete and steel independently. Figure 5 shows the discretization of a sentative RC cross section. Integration of the inelastic stress-strain relationship of the sep- representative RC cross section. Integration of the inelastic stress-strain relationship of the arate fibers yields the cross-section stress-strain state. A two Gauss point nonlinear cubic separate fibers yields the cross-section stress-strain state. A two Gauss point nonlinear formulation, which is used to numerically integrate the equations of equilibrium, results cubic formulation, which is used to numerically integrate the equations of equilibrium, in a spread of plasticity along the element. results in a spread of plasticity along the element. Figure 5. Discretization of the structural element in fiber type modeling. Figure 5. Discretization of the structural element in fiber type modeling. To examine the effect of the number of section fibers on the accuracy of the numerical results, the discretization of the sub-assemblage members was carried out by dividing the member section into a number of fibers that varied from 29 to 249 fibers, as shown in Figure 6. The adopted material models con_ma and stl_mp were employed in the numerical model. Despite the fact that all the numerical behaviors were close to the test specimen SS-3 behavior [2], Figure 6 demonstrates that when the number of section fibers was 149 the predicted numerical behavior was the most representative of test results. The same conclusion was arrived at when simulating the test behaviors of the other twenty- eight specimens. As a result, 149 section fibers were adopted in the model utilized in this study. member section into a number of fibers that varied from 29 to 249 fibers, as shown in Figure 6. The adopted material models con_ma and stl_mp were employed in the numer- ical model. Despite the fact that all the numerical behaviors were close to the test specimen SS-3 behavior [2], Figure 6 demonstrates that when the number of section fibers was 149 the predicted numerical behavior was the most representative of test results. The same Buildings 2022, 12, 122 conclusion was arrived at when simulating the test behaviors of the other twenty-eight 10 of 22 specimens. As a result, 149 section fibers were adopted in the model utilized in this study. 80 Experimental 149 Section Fibres 29 Section Fibres 60 29 Section Fibres Applied load (kN) 49 Section Fibres 77 Section Fibres 101 Section Fibres 40 249 Section Fibres 29 49 77 101 149 249 20 Number of section fibers 0 0 100 200 300 400 500 600 Vertical displacement in mm (at failed column location) Figure6.6.Test Figure TestSS-3 SS-3[2][2] and numerical and load-displacement numerical curves load-displacement with with curves different numbers different of section numbers fibers. of section fibers. 3.5. Plastic Hinge Length 3.5. Plastic Another Hinge key Length model parameter that has an impact of the numerical results is the plastic hingeAnother length. Tokeyexamine the impactthat model parameter of this hasparameter an impact on numerical results of the numerical outcomes, plastic is the plas- hinge lengths tic hinge (Lp )To length. of 20, 25, 35,the examine 40, impact 45, and of50% of parameter this the elementon length (L) were considered, the numerical outcomes, in addition plastic hingetolengths the software (Lp) of default length 20, 25, 35, of and 40, 45, 16.67%. 50% The material of the elementmodels, con_ma length (L) were and con- stl_mp, and number of section fibers, 149, adopted in this study were sidered, in addition to the software default length of 16.67%. The material models, con_ma employed. When compared and stl_mp, with and thenumber test result of specimen of section SS-3 fibers, [2],adopted 149, Figure 7inshows clearlywere this study that the plastic employed. hinge Whenlength comparedspecified with in thethe testmodel resultsignificantly of specimen affected SS-3 [2],the ductility Figure 7 showsandclearly the capability that the to predict plastic the length hinge catenary action of specified in the the test modelspecimen. The software significantly default affected the valueand ductility of 16.67% the ca- produced pability to predict the catenary action of the test specimen. The software default valuethe the least ductility without capturing the catenary action behavior, whereas of ductility and catenary 16.67% produced the action improved least ductility with the without increase the capturing in the plastic hinge catenary action length, behavior,as shown Buildings 2022, 12, x FOR PEER REVIEW whereas in the Figure 7. It can ductility andbecatenary concluded from action the figure improved thatthe with using a plastic increase hinge in the length plastic of of 11hinge 24 50% of the element length produced the most accurate simulation of test length, as shown in Figure 7. It can be concluded from the figure that using a plastic hinge specimen SS-3 [2]. A plastic length ofhinge 50% of length of 50% of the element the element length produced length was adopted the most accuratein this study.of test speci- simulation men SS-3 [2]. A plastic hinge 90 length of 50% of the element length was adopted in this Experimental PHL = 50% study. 80 PHL = 16.67% PHL = 20% 70 PHL = 25% PHL = 35% PHL = 40% PHL = 45% Applied load (kN) 60 50 40 30 LP LP L 20 10 0 𝐿𝑃 𝑃𝐻𝑅 = x 100 0 100 200 300 400 500 600 𝐿 Vertical displacement in mm (at failed column location) Figure 7. Test Figure 7. TestSS-3 SS-3[2] [2]and andnumerical numericalload-displacement curves load-displacement using curves various using plastic various hingehinge plastic lengths, PHL. lengths, PHL. 3.6. Number of Segments 3.6. Number The beamsof Segments of the sub-assemblage test specimens were modeled by dividing them into a number The beams of thefollowing of segments their physical sub-assemblage cross-section test specimens dimensions were modeled and reinforcement by dividing them ratios in their hogging (A-A) and sagging (B-B) regions, as displayed into a number of segments following their physical cross-section dimensions in scheme 1 ofand Figure 8. rein- Each segment is identified by one element with constant cross section and reinforcement forcement ratios in their hogging (A-A) and sagging (B-B) regions, as displayed in scheme ratio along the 1 of Figure length 8. Each of the element. segment In thebynumerical is identified model, one element withallconstant the above-adopted cross sectionmodel and reinforcement ratio along the length of the element. In the numerical model, all the above- adopted model key parameters were employed, and the generated numerical simulation was in good agreement with the specimen HSC-8 [9] test behavior, as exhibited in Figure 8. In order to verify whether or not the number of segments hah any impact on the numerical results, the same reinforcement ratio, A-A, was assumed in both hogging and reinforcement ratio along the length of the element. In the numerical model, all the above- adopted model key parameters were employed, and the generated numerical simulation was in good agreement with the specimen HSC-8 [9] test behavior, as exhibited in Figure 8. In order to verify whether or not the number of segments hah any impact on the numerical results, the same reinforcement ratio, A-A, was assumed in both hogging and Buildings 2022, 12, 122 11 of 22 sagging regions, and the beam was, therefore, modeled as one segment, as shown in scheme 2 in Figure 8. Scheme 2 established an underestimation of the arch action, overes- timation of the catenary action, and more ductile behavior, which were not in agreement key parameters with were the test results employed,HSC-8 of specimen and the [9],generated as shown innumerical Figure 8. simulation was in good agreement with the specimen HSC-8 [9] test behavior, as exhibited in Figure 8. 175 Experimental 150 A-A A-A Scheme 1 A-A A-A Vertical Load (kN) 125 Scheme 2 B-B B-B 100 Scheme 1 75 50 A-A A-A 25 Scheme 2 0 0 100 200 300 400 500 600 700 800 Vertical Displacement in mm (at failed column location) Figure 8. Test HSC-8 [9] and numerical load-displacement curves with different numbers of segments. Figure 8. Test HSC-8 [9] and numerical load-displacement curves with different numbers of seg- ments.In order to verify whether or not the number of segments hah any impact on the numer- ical results, the same reinforcement ratio, A-A, was assumed in both hogging and sagging regions, and the beam was, therefore, modeled as one segment, as shown in scheme 2 in Figure 8. Scheme 2 established an underestimation of the arch action, overestimation of the catenary action, and more ductile behavior, which were not in agreement with the test results of specimen HSC-8 [9], as shown in Figure 8. 3.7. Numerical Damage and Failure Detection It is advantageous to set numerous limit state criteria in order to detect and identify dif- ferent types of damages to the structure when subjected to progressive collapse scenario. It is equally critical to recognizing when different damages such as yielding of reinforcement, structural damage, and collapse are reached in the framework of structural performance. These limit states are realized by SeismoStruct and can be set by defining so-called Perfor- mance Criteria, which allow the program to automatically monitor the achievement of a certain limit state value (such as cracking in the element, concrete cover spalling, concrete core crushing, yielding of steel, steel fracture, member section curvature, or element chord rotation) throughout the analysis of the structure. The built-in Performance Criteria have limits, in accordance with different design codes, which when reached the corresponding damage is detected and identified in a log file. Therefore, to further validate the numerical model outcomes using all of the above proposed optimal key parameters, the model was put to test by defining different perfor- mance criteria and comparing the predicted, detected different damage types with those observed during the physical testing of specimen SS-3 [2]. Specimen SS-3 was selected for this comparison, since it provided the development sequence of the different physical dam- age types as the test progressed up to the complete collapse. This comparison is displayed in Figures 9 and 10. In these figures, the instant at which a specific physical test damage was reached is identified by a letter of the alphabet (i.e., A–F); while the instant at which a specific matching numerical damage was reached is labeled as a numeral (i.e., 1–7). Figure 9 clearly shows that the proposed model not only numerically predicted the same type of damage developed in the test specimen but also closely predicted the test instant/load at which the test damage was developed. The schematic different damage types observed in the test and their corresponding model results are identified in Figure 10. From the figure, it can be deduced that the model estimated the test vertical displacement at the removed column location and predicted the experimental formation of flexural cracks, bar yielding, chord rotation, concrete spalling, concrete crushing, and rupture of bars towards damage was reached is identified by a letter of the alphabet (i.e., A–F); while the instant at which a specific matching numerical damage was reached is labeled as a numeral (i.e., 1–7). Figure 9 clearly shows that the proposed model not only numerically predicted the same type of damage developed in the test specimen but also closely predicted the test instant/load at which the test damage was developed. The schematic different damage Buildings 2022, 12, 122 types observed in the test and their corresponding model results are identified in Figure 12 of 22 10. From the figure, it can be deduced that the model estimated the test vertical displace- ment at the removed column location and predicted the experimental formation of flex- ural cracks, bar yielding, chord rotation, concrete spalling, concrete crushing, and rupture the of beam bars endsthe towards fairly beamaccurately. ends fairly Only the specimen accurately. right endright Only the specimen schematic damages are shown end schematic in Figure damages are10. shown in Figure 10. 90 Experimental 80 Numerical F E 70 Experimental Important Tracked Points Numerical Performance Criteria Points 7 60 6 Applied load (kN) 50 B 5 40 A C D 30 23 1 4 20 10 0 0 100 200 300 400 500 600 Vertical Buildings 2022, 12, x FOR PEER REVIEW displacement in mm (at failed column location) 13 of 24 Figure Figure9. 9. Test SS-3 Test [2] and SS-3 numerical [2] and load-displacement numerical curves with load-displacement different curves performance with differentcriteria. performance criteria. VD = 42mm A VD = 48 mm 1 Damage type: bar yielding Damage type: bar yielding VD = 98mm 2 Damage type: B chord rotation capacity VD = 87 mm (code-based) VD = 112 mm 3 Damage type: chord rotation Damage type: chord rotation rapacity (code-based) and concrete crushing C 4 VD = 160 mm VD = 161mm Damage type: Damage type: concrete crushing concrete crushing E 5 VD = 406mm VD = 417 mm Damage type: Damage type: concrete spalling concrete spalling Figure 10. Cont. Buildings 2022, 12, 122 Buildings 2022, 12, x FOR PEER REVIEW 14 of1324of 22 6 VD = 518mm F Damage type: severe VD = 550 mm concrete crushing Damage type: concrete spalling Damage type: severe concrete crushing 7 VD = 567mm and bar rupture Damage type: bar rupture and failure of sub-assemblage Figure 10. Damage types and sequence in specimen SS-3 [2]: test, left, and numerical predictions, Figure 10. Damage types and sequence in specimen SS-3 [2]: test, left, and numerical predictions, right (letters of theright alphabet and (letters of the numerals alphabet andare the same numerals are theas those same in Figure as those in Figure9); 9); VD VD isisthe the vertical vertical displacement displacement at location at location of failed of failed column. column. Overall, it can be Overall, it canthat inferred be inferred that adopting adopting the optimal the optimal key key parameters recommended parameters recommended above in the model can produce numerical behaviors that would be in good agreement above in the model can produce numerical behaviors that would be in good agreement with their test counterparts. with their test counterparts. It should be further noted that the number of elements in each model was low and It should beranged further noted between that 7 and 11 the number elements, of elements depending in each model on the reinforcement detailingwas of thelow spec-and imen. This remarkably low number of elements in the SeismoStruct analysis, when com- ranged between 7 and 11 elements, depending on the reinforcement detailing of the speci- pared with other 2D and 3D numerical methods, is advantageous, as it not only signifi- men. This remarkably low number cantly reduces of elements the modeling in the SeismoStruct process, computational analysis, running time, and computer when com- storage pared with other 2D spaceandbut3D alsonumerical demonstratesmethods, is advantageous, excellent agreement as it not and between the numerical only significantly experimental test results. reduces the modeling process, computational running time, and computer storage space but also demonstrates3.8. excellent agreement between the numerical and experimental test results. Boundary Conditions of Different Test Setups To complement the numerical model verification in this study, the effects of the 3.8. Boundary Conditions of Different Test Setups boundary conditions on the numerical progressive collapse simulation of the twenty-nine To complementtest specimens the numericalwere examined. The various boundary model verification in thisconditions study, the of the considered effects of thespeci- bound- ary conditions onmens theare described in Table 2 and Figure 2 above. The boundary conditions depend on numerical progressive collapse simulation of the twenty-nine test the specimen support types, i.e., fixation, roller, or pin support and, accordingly, the cor- specimens were examined. responding degreeThe various of freedom boundary in the model conditions was released. of As theforconsidered the interface specimens between are described in Table 2 and different Figure materials, this2study above. Thethe assumes boundary model defaultconditions dependbetween full bond interface on thethespeci- concrete and steel bar. The boundary conditions of the test specimens men support types, i.e., fixation, roller, or pin support and, accordingly, the corresponding were simulated, and their effects on the numerical results were plotted in Figure 11, which compares the nu- degree of freedommerical in theand model was released. As for the interface between different materi- test load versus vertical displacement curves of all twenty-nine specimens. In als, this study assumes general,the model the figure default shows fullcapability the high bond interface between of the proposed modeltheto concrete and steel consider different bar. The boundary conditions boundary of the conditions andtest specimens to predict were the flexure simulated, action, compressionandarchtheir effects action, on the and ten- numerical resultssionwere catenary plottedaction in of the test11, Figure results. which Figure 11a clearly compares thereveals that simulating numerical and testtheload boundary conditions of test setup A produced less numerical ductility but was able to versus vertical displacement accurately predict curves of all the failure load.twenty-nine Figure 11b–t show specimens. that simulatingIn general, the boundarythecon- figure shows the high capability of the proposed model to consider different boundary ditions of test setup B produced numerical behaviors that are overall in good agreement conditions and to predict thewith their test flexure counterparts, action, except forarch compression Figure 11f,i,k,q.and action, Figure 11f demonstrates tension catenarythat the of action boundary conditions resulted in a more numerical ductile behavior, accompanied with the test results. Figure 11a clearly reveals that simulating the boundary conditions of test higher numerical failure load than those of the test, whereas Figure 11i,q show less nu- setup A produced less ductility merical numerical ductility associated butfailure with lesser was able to accurately load than predictItthe their test counterparts. failure is worth load. Figure 11b–tnoting show that that simulating the boundary the boundary conditions conditions of setup B appears to haveofantest setup impact B produced on the simula- numerical behaviorstion where that arethe overall numericalininitial good behavior agreementof the specimens with their were stiffer test than those of except counterparts, the for Figure 11f,i,k,q. Figure 11f demonstrates that the boundary conditions resulted in a more numerical ductile behavior, accompanied with higher numerical failure load than those of the test, whereas Figure 11i,q show less numerical ductility associated with lesser failure load than their test counterparts. It is worth noting that the boundary conditions of setup B appears to have an impact on the simulation where the numerical initial behavior of the specimens were stiffer than those of the experimental results. The boundary conditions of test setups C and E gave accurate numerical simulations of the test behaviors as shown in Figure 11u–y,ac. However, for test setup D, simulating the boundary conditions of one test demonstrated an accurate simulation, Figure 11z, while the other two tests instigated more numerical ductility and accurate numerical failure load when compared with the test results as shown in Figure 11aa,ab. setup D, simulating the boundary conditions of one test demonstrated an accurate simu- lation, Figure 11z, while the other two tests instigated more numerical ductility and accurate numerical failure load when compared with the test results as shown in Figure 11aa,ab. Overall, the boundary conditions in all the test setups were implemented in the Buildings 2022, 12, 122 model that showed a capability that could generate fairly accurate responses in 14 compari- of 22 son with the test data. 60 45 Applied load (kN) Applied load (kN) 40 30 20 15 Experimental Experimental Numerical Numerical 0 0 0 100 200 300 400 500 0 100 200 300 400 500 Vertical displacement (mm) Vertical displacement (mm) (a) specimen RCB [1]-Setup A–1/3 scale test (b) specimen SS-1 [2]-Setup B–½ scale test 80 80 Experimental Experimental Applied load (kN) Applied load (kN) 60 Numerical 60 Numerical 40 40 20 20 0 0 0 100 200 300 400 500 600 0 100 200 300 400 500 600 Vertical displacement (mm) Vertical displacement (mm) (c) specimen SS-2 [2]-Setup B–½ scale test (d) specimen SS-3 [2]-Setup B–½ scale test 80 60 Experimental Experimental Numerical Applied load (kN) 60 Numerical Applied load (kN) 40 40 20 20 0 0 0 100 200 300 400 500 600 0 100 200 300 400 500 600 700 Vertical displacement (mm) Vertical displacement (mm) (e) specimen SS-4 [2]-Setup B–½ scale test (f) specimen MJ-B-S [3]-Setup B–½ scale test 80 120 Experime… Experime… 60 90 Applied load (kN) Applied load (kN) 40 60 20 30 0 0 0 100 200 300 400 500 600 700 0 100 200 300 400 500 600 Vertical displacement (mm) Vertical displacement (mm) (g) specimen MJ-L-S [3]-Setup B–½ scale test (h) specimen S4 [4]-Setup B–½ scale test Figure 11. Cont. Buildings 2022, 12, x FOR PEER REVIEW 16 of 24 Buildings 2022, 12, 122 15 of 22 120 180 Experimental Experimental 150 Applied load (kN) 90 Numerical Numerical Applied load (kN) 120 60 90 60 30 30 0 0 0 100 200 300 400 500 600 700 0 100 200 300 400 500 600 700 Vertical displacement (mm) Vertical displacement (mm) (i) specimen S5 [4]-Setup B–½ scale test (j) specimen S6 [4]-Setup B–½ scale test 160 140 Experimental 120 Experimental 120 Numerical Numerical 100 Applied load (kN) Applied load (kN) 80 80 60 40 40 20 0 0 0 100 200 300 400 500 600 700 0 100 200 300 400 500 600 Vertical displacement (mm) Vertical displacement (mm) (k) specimen S7 [4]-Setup B–½ scale test (l) specimen S8 [4]-Setup B–½ scale test 80 Experimental Experimental 60 Applied load (kN) 60 Numerical Numerical Applied load (kN) 40 40 20 20 0 0 0 100 200 300 400 500 600 0 100 200 300 400 500 600 700 Vertical displacement (mm) Vertical displacement (mm) (m) specimen S1 [7]-Setup B–½ scale test (n) specimen S2 [7]-Setup B–½ scale test 80 100 Experimental 80 Numerical Applied load (kN) Applied load (kN) 60 60 40 40 20 Experimental 20 Numerical 0 0 0 100 200 300 400 500 600 0 100 200 300 400 500 600 700 800 Vertical displacement (mm) Vertical displacement (mm) (o) specimen NSC-8 [9]-Setup B–½ scale test (p) specimen NSC-11 [9]-Setup B–½ scale test Figure 11. Cont. Buildings 2022, 12, x FOR PEER REVIEW 17 of 24 Buildings 2022, 12, 122 16 of 22 80 100 Experimental 80 Applied load (kN) 60 Numerical Applied load (kN) 60 40 40 20 Experimental 20 Numerical 0 0 0 100 200 300 400 500 600 700 0 100 200 300 400 500 600 Vertical displacement (mm) Vertical displacement (mm) (q) specimen NSC-13 [9]-Setup B–½ scale test (r) specimen HSC-8 [9]-Setup B–½ scale test 100 100 Experimental Experimental 80 Applied load (kN) 80 Numerical Numerical Applied load (kN) 60 60 40 40 20 20 0 0 0 100 200 300 400 500 600 700 0 150 300 450 600 750 Vertical displacement (mm) Vertical displacement (mm) (s) specimen HSC-11 [9]-Setup B–½ scale test (t) specimen HSC-13 [9]-Setup B–½ scale test 60 120 Experimental 100 Numerical Applied load (kN) Applied load (kN) 40 80 60 20 40 Experimental Numerical 20 0 0 0 100 200 300 400 500 600 0 100 200 300 400 500 600 Vertical displacement (mm) Vertical displacement (mm) (u) specimen F1 [5]-Setup C–½ scale test (v) specimen F2 [5]-Setup C–½ scale test 70 150 60 Experimental Applied load (kN) 120 Applied load (kN) 50 Numerical 40 90 30 60 20 Experimental 10 Numerical 30 0 0 0 100 200 300 400 500 0 200 400 600 Vertical displacement (mm) Vertical displacement (mm) (w) specimen IF-B [8]-Setup C–½ scale test (x) specimen IF-L [8]-Setup C–½ scale test Figure 11. Cont. Buildings 2022, 12, x FOR PEER REVIEW 18 of 24 Buildings 2022, 12, 122 17 of 22 80 40 Applied load (kN) 60 30 Applied load (kN) 40 20 20 Experimental 10 Experimental Numerical Numerical 0 0 0 100 200 300 400 500 0 50 100 150 200 250 300 350 Vertical displacement (mm) Vertical displacement (mm) (y) specimen EF-B [8]-Setup C–½ scale test (z) specimen 2D [6]-Setup D–3/10 scale test 150 20 120 Applied load (kN) 15 Applied load (kN) 90 10 60 Experimental 5 Experimental 30 Numerical Numerical 0 0 0 150 300 450 600 750 900 0 100 200 300 400 Vertical displacement (mm) Vertical displacement (mm) (aa) specimen RC [10]-Setup D–½ scale test (ab) specimen MS [12]-Setup D–3/10 scale test 60 50 Experimental Applied load (kN) Numerical 40 30 20 10 0 0 50 100 150 200 250 300 350 Vertical displacement (mm) (ac) specimen Prototype [11]-Setup E–1/3 scale test Figure 11.11.Test Figure Testand andnumerical load-displacementcurves numerical load-displacement curves forfor thethe considered considered twenty-nine twenty-nine test speci- test speci- mens (all(all mens displacements displacementsare are measured measured atatfailed failed column column location). location). Overall, the boundary conditions in all the test setups were implemented in the model 4. Discussion that showed a capability that could generate fairly accurate responses in comparison with theTo testhave data.a better comparison of the predicted behavior of each test specimen, this section presents the comparison of different numerically predicted key points and their 4. Discussion test values. The key points selected for this comparison process are the corresponding To have a flexure initial stiffness, better comparison of the predicted load, displacements behavior at flexure of each load, peaktest specimen, load, this displacement at section presents the comparison of different numerically predicted key points peak load, failure load, displacement at failure load, and energy, as shown in Figure 12. and their corresponding test values. The key points selected for this comparison process are the The figure also shows the different behavior stages of flexural action, compression arch, initial stiffness, flexure load, displacements at flexure load, peak load, displacement at peak and tension catenary action. load, failure load, displacement at failure load, and energy, as shown in Figure 12. The A total figure also of twenty-nine shows data the different points stages behavior were extracted of flexural for eachcompression action, property, numerical arch, and and test, and plotted tension catenaryinaction. the following figures with trend lines of maximum difference of ±15%. A maximum difference ±15% can be acceptable in progressive collapse analysis and, there- fore, was considered in this study. Buildings 2022, 12, x FOR PEER REVIEW 19 of 24 Buildings 2022, 12, 122 18 of 22 Figure Figure 12. 12. Selected key points for comparison (test specimen SS-3 [2]). A total 13a Figure of twenty-nine data points were shows the comparison extracted between for each property, the numerical and test numerical and test, initial stiffness for and plotted in the following figures with trend lines of maximum all the specimens. It is clear from Figure 13a that the proposed numerical model was ca- difference of ± 15%. A maximum difference ± 15% can be acceptable in progressive collapse pable to efficiently estimate the initial stiffness of the specimens as all the data points, analysis and, therefore, was considered except three, areinwithin this study. the maximum difference buffer zone of ±15%. Two of these three data Figure points 13a shows that indicated the comparison betweenover-evaluated the model slightly the numerical and the test initial initial stiffness stiffness. for all Most of specimens. the data It is clear points that from Figure are within 13azone the buffer that the are proposed slightly above numerical model was the diagonal capable line indicat- to efficiently ing the initial estimate stiffness is thea little initialover-evaluated; stiffness of thenevertheless, specimens asitall the data is still within points, the 15% except dif- three, ference. are within the maximum difference buffer zone of ± 15%. Two of these three data points indicatedthe Similarly, that the modelofslightly evaluation over-evaluated the numerical the initial flexure loads withstiffness. reference Most of test to the the data val- points that are within the buffer zone are slightly above the ues is presented in Figure 13b, which demonstrates that the proposed model accurately diagonal line indicating the initial stiffness predicted is a little the flexure over-evaluated; load, nevertheless, since all the data points areitnot is still onlywithin the 15% well within difference. a ±15% maxi- Similarly, the evaluation of the numerical flexure mum difference but also they are very close to the diagonal line. loads with reference to the test values is presented In Figurein13c, Figure 13b, which demonstrates the numerically that the proposed predicted displacements model at flexure loadaccurately are compared pre- dicted the flexure load, since all the data points are not only well with the corresponding experimentally obtained values. It is evident from the figure that within a ± 15% maximum difference there but also they are very is a slight-to-moderate close to of dispersion thethediagonal line. some of which fell within the data points, In Figure 13c, the numerically predicted maximum difference of ±15% while other points fell outside displacements at flexure load aredifference. the maximum compared Most of the dispersed data points indicate that the model moderately under-evaluatedthat with the corresponding experimentally obtained values. It is evident from the figure the there is a slight-to-moderate dispersion of the data points, some of which fell within the displacements at flexure load for some specimens, whereas the data points within the maximum difference of ±15% while other points fell outside the maximum difference. maximum difference demonstrates that the model accurately evaluated the displacements Most of the dispersed data points indicate that the model moderately under-evaluated at flexure load for the other specimens. This dispersion can be contributed to the higher the displacements at flexure load for some specimens, whereas the data points within the numerical initial stiffness of the relevant specimens. maximum difference demonstrates that the model accurately evaluated the displacements By the same comparison, Figure 13d reveals that the proposed model accurately cap- at flexure load for the other specimens. This dispersion can be contributed to the higher tured the peak load of the specimens as all the data points, except five, are within the numerical initial stiffness of the relevant specimens. maximum difference of ±15%. The five data points are slightly dispersed outside the max- By the same comparison, Figure 13d reveals that the proposed model accurately imum difference, where the model seemed to mostly and marginally over-predicted the captured the peak load of the specimens as all the data points, except five, are within peak load. the maximum difference of ±15%. The five data points are slightly dispersed outside the Likewise, Figure 13e discloses that the proposed model adequately predicted the dis- maximum difference, where the model seemed to mostly and marginally over-predicted placement the peak load. at peak load of the test specimens as all the data points, apart from three, fell within the Likewise, maximum Figure difference 13e discloses of ±15%. These that the three data proposed model points, however, adequately are moder- predicted the ately displacement at peak load of the test specimens as all the data points, apart fromover- spread outside the maximum difference zone and isolated between the three,andfell under-predicted within the maximum displacements. difference of ±15%. These three data points, however, are moderately spreadSimilar to Figure outside 13d, Figure the maximum 13f unveils difference zonethatandthe numerical isolated failure between theloads over-of andtheunder- spec- imens were accurately predicted displacements. evaluated by the model since all the data points, except six, are and just a few a bit dispersed. This indicates that the numerical model can produce a reli- able estimation of the total energy generated as the structure undergoes a progressive col- lapse due to an interior column failure. In general, it can be concluded that the above-predicted values by the proposed model are reasonably matching with their test counterparts. The proposed model seems to be most accurate in predicting the capacity of the test specimens, i.e., ultimate load and failure load, and the total energy within ±15% maximum difference trend lines. Slight to Buildings 2022, 12, 122 19 of 22 moderate dispersion away from the trend lines is observed in the predicted values of ver- tical displacement at the removed column location in some specimens. 4 150 Ref. [1] Ref. [1] Ref. [2] −15% Ref. [2] −15% Ref. [3] Ref. [3] Ref. [4] Ref. [4] 3 Ref. [5] Ref. [5] Ref. [6] Ref. [6] +15% 100 +15% Ref. [7] Test (kN/mm) Ref. [7] Ref. [8] Test (kN) Ref. [8] Ref. [9] Ref. [9] 2 Ref. [10] Ref. [10] Ref. [11] Ref. [11] Ref. [12] Ref. [12] 50 1 0 Buildings 2022, 12, x FOR PEER REVIEW 0 21 of 24 0 1 2 3 4 0 50 100 150 Numerical (kN/mm) Numerical (kN) (a) initial stiffness (b) flexure load 150 200 −15% Ref. [1] Ref. [2] Ref. [3] Ref. [4] −15% Ref. [5] Ref. [6] 150 Ref. [7] Ref. [8] Ref. [9] Ref. [10] 100 +15% Ref. [12] +15% Test (kN) Test (mm) 100 50 Ref. [1] Ref. [2] Ref. [3] Ref. [4] 50 Ref. [5] Ref. [6] Ref. [7] Ref. [8] Ref. [9] Ref. [10] Ref. [11] Ref. [12] 0 0 0 50 100 150 0 50 100 150 200 Numerical (mm) Numerical (kN) (c) displacement at flexure load (d) peak load 800 200 −15% Ref. [1] Ref. [2] −15% Ref. [3] Ref. [4] Ref. [5] Ref. [6] 600 150 Ref. [7] Ref. [8] Ref. [9] Ref. [10] +15% +15% Ref. [11] Ref. [12] Test (mm) Test (kN) 400 100 Ref. [1] Ref. [2] Ref. [3] Ref. [4] 200 Ref. [5] Ref. [6] 50 Ref. [7] Ref. [8] Ref. [9] Ref. [10] Ref. [12] 0 0 0 200 400 600 800 0 50 100 150 200 Numerical (kN) Numerical (mm) (e) displacement at peak load (f) failure load 1000 90 Ref. [1] Ref. [2] Ref. [1] Ref. [2] Ref. [3] Ref. [4] Ref. [3] Ref. [4] −15% −15% 80 Ref. [5] Ref. [6] Ref. [5] Ref. [6] 800 Ref. [7] Ref. [8] 70 Ref. [7] Ref. [8] Ref. [9] Ref. [10] Ref. [9] Ref. [10] 60 Ref. [11] Ref. [12] Ref. [11] Ref. [12] Test (kN-m) 600 +15% +15% Test (mm) 50 40 400 30 20 200 10 0 0 0 200 400 600 800 1000 0 10 20 30 40 50 60 70 80 90 Numerical (mm) Numerical (kN-m) (g) displacement at failure load (h) total energy Figure 13. Numerical and test comparison. Figure 13. Numerical and test comparison. Buildings 2022, 12, 122 20 of 22 Similar to Figure 13d, Figure 13f unveils that the numerical failure loads of the speci- mens were accurately evaluated by the model since all the data points, except six, are within the maximum difference of a ±15% buffer zone. The six data points are slightly dispersed outside the buffer zone, where the model appeared to mostly and faintly over-predict the failure load. Figure 13g demonstrates that all the data points not only are within a ±15% maximum difference but also are almost on the diagonal line, which indicate that the numerical model accurately evaluated the specimen displacements at failure load. However, two data points are away from the 15% maximum difference, indicating that the model predicted much more ductile behavior for these two specimens than their test results. In contrast, four data points are slightly dispersed due to the model marginally under-evaluating the maximum displacement of the relevant specimens. For the predicted total energy at failure, Figure 13h exhibits that almost all the data points are within a ±15% maximum difference, with some of them on the diagonal line and just a few a bit dispersed. This indicates that the numerical model can produce a reliable estimation of the total energy generated as the structure undergoes a progressive collapse due to an interior column failure. In general, it can be concluded that the above-predicted values by the proposed model are reasonably matching with their test counterparts. The proposed model seems to be most accurate in predicting the capacity of the test specimens, i.e., ultimate load and failure load, and the total energy within ±15% maximum difference trend lines. Slight to moderate dispersion away from the trend lines is observed in the predicted values of vertical displacement at the removed column location in some specimens. 5. Conclusions This paper presents a numerical model based on a fiber element approach using SeismoStruct software to be utilized as a benchmark model for an accurate progressive collapse analysis of RC structures due to inner column removal. The model was based on an extensive validation process and wide parameter sensitivity analysis to accurately simulate test behaviors of twenty-nine large-scale RC sub-assemblage specimens. A qualitative examination of the load displacement relationships and different per- formance criteria such as yielding of reinforcing bars, spalling and crushing of concrete, and bar rupture for twenty-nine specimens with different dimensions, material properties, reinforcement detailing, test setups, and boundary conditions, plus test scales showed a good agreement between the numerical and experimental results. This confirms the efficacy of the proposed benchmark model, which can be applied in structure progressive collapse analysis within the framework outlined in this study. The proposed benchmark model required only seven to eleven elements to accurately simulate the test behaviors and correctly predict the different damage types and locations of the considered tests. With such a low element number, the time of analysis and computation was significantly reduced. Reducing the computation time would allow further parametric study, and using low number of elements would permit full structure 2D or 3D progressive collapse analysis with high accuracy. The framework of the proposed model recommended key parameters such as ele- ment types, material models, number of section fibers, plastic hinge length, number of segments, and boundary conditions that best optimized the numerical simulations of the test behaviors. The proposed model appears to be most accurate in predicting the capacity of the test specimens within ±15% maximum difference trend lines. Slight discrepancies outside the trend lines were observed in the predicted values of vertical displacement at the removed column location in some specimens. Given the costly nature of experimental research, test errors, and the lengthy testing procedure, the proposed model using SeismoStruct software can be considered as a bench- mark model to accurately analyze structures under progressive collapse due to interior Buildings 2022, 12, 122 21 of 22 column removal. Engineers and researchers can use the model’s recommended optimal key parameters that are highlighted in this study as a guide to create accurate numerical models for the progressive collapse analysis of RC structures with an internal failed column. Author Contributions: Conceptualization, B.E.-A. and S.E.; methodology, S.E.; software, A.S.; vali- dation, B.E.-A., S.E. and A.S.; formal analysis, B.E.-A. and S.E.; investigation, B.E.-A., S.E. and A.S.; resources, S.E. and A.S.; data curation, B.E.-A. and S.E.; writing—original draft preparation, B.E.-A.; writing—review and editing, B.E.-A.; visualization, B.E.-A. and S.E.; supervision, B.E.-A. and S.E.; project administration, B.E.-A. and S.E. All authors have read and agreed to the published version of the manuscript. Funding: This research received no external funding. Institutional Review Board Statement: Not Applicable. Informed Consent Statement: Not Applicable. Data Availability Statement: The data that support the findings of this study are available from the authors upon reasonable request. Acknowledgments: The authors would also like to acknowledge SEiSMOSOFT for the free academic license for the SeismoStruct software. 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