Hybrid Framework for Simulating Building Collapse and Ruin Scenarios Using Finite Element Method and Physics Engine

applied sciences

Article Hybrid Framework for Simulating Building Collapse and Ruin Scenarios Using Finite Element Method and Physics Engine

Zhe Zheng 1 , Yuan Tian 2 , Zhebiao Yang 1 and Xinzheng Lu 2,* 1 Beijing Engineering Research Center of Steel and Concrete Composite Structures, Tsinghua University, Beijing 100084, China; [email protected] (Z.Z.); [email protected] (Z.Y.) 2 Key Laboratory of Civil Engineering Safety and Durability of China Education Ministry, Department of Civil Engineering, Tsinghua University, Beijing 100084, China; [email protected] * Correspondence: [email protected]

Received: 26 May 2020; Accepted: 25 June 2020; Published: 26 June 2020

Abstract: Reliable and high-fidelity virtual ruin scenarios for collapsed buildings are essential for post-earthquake emergency search and rescue training. However, the existing research on the distribution of ruins caused by building collapse is insufficient for supporting post-earthquake rescue training. Therefore, this paper proposes a hybrid framework for simulating building collapse and ruin scenarios, using a finite element (FE) model and a physics engine. Based on this framework, the following methods are proposed: (1) geometric model conversion from the FE model to the physics engine; (2) determination of the initial moment of collapse; and (3) data mapping of the FE simulation results. In addition, a corresponding program, Finite Element Method to Rigid Body Dynamics (FEM2RBD), is developed for the hybrid framework. The proposed framework simulates the entire process of building collapse and the distribution of ruins. The accuracy of the framework is validated using a shaking table test of a three-story reinforced concrete frame. The collapse process and ruin scenario of a real-world library building is simulated as a case study. The results show that the proposed framework combines the advantages of the FE model during the small-deformation stage with the advantages of physics engines during the large-deformation stage. The proposed framework can be valuable in simulating building collapse and ruin scenarios for post-earthquake rescue training.

Keywords: collapse simulation; physics engine; ﬁnite element method; hybrid simulation; ruin scenario construction

1. Introduction Earthquakes can cause building collapse, and people may be buried by ruins [1,2]. Relevant statistics show that being buried is the most common cause of earthquake-related injuries [3]. Therefore, it is necessary to conduct training to improve rescue efficiency and mitigate casualties after earthquakes. To achieve this, the ability to accurately simulate ruin scenarios is required. The reliability of simulated ruins is also an essential factor for efficient training [4]. To date, several studies on the collapse simulation of buildings have been conducted using the finite element (FE) method. Such studies typically focus on the collapse resistance of structures, rather than the formation of ruins, such as the analyses of the progressive collapse resistance caused by accidental load [5–8], the simulations of seismic collapse processes, etc. [9–13]. Few studies that simulate the distribution of ruins induced by structural collapses have been conducted, due to the high computational workload, and the difficulty of simulating large deformations and nonlinear behaviors using the FE method [14,15].

Appl. Sci. 2020, 10, 4408; doi:10.3390/app10124408 www.mdpi.com/journal/applsci Appl. Sci. 2020, 10, 4408 2 of 19

Currently, there are two methods for performing studies on ruins induced by structural collapse: Shaking table test Shaking table tests have been used to study the ruin distribution in many • studies [16–20]. For example, Huang [20] analyzed the collapse process and distribution of ruins 1 using a 4 -scale shaking table test of a three-story reinforced concrete frame. In general, the structures used in such tests are highly simplified approximations of real-world structures. Owing to the high cost and limited experimental capacity of shaking table tests, small-scale models are often required, thereby limiting the reliability of the results. Numerical simulation methods The discrete element method (DEM) [21,22], applied element • method (AEM) [15,23] and physics engines are the most common numerical simulation methods for ruin simulations. A physics engine is a piece of computer software, based on Newton’s laws of dynamics, that provides an approximate simulation of complex physical behaviors, such as rigid body dynamics (including collision detection), soft body dynamics and fluid dynamics. [24]. Some physics engines have a relatively low computational demand [25]. In recent years, physics engines have been widely used to simulate structural collapse and ruin distributions [26–31]. For example, Xu et al. [28] simulated seismic damage in urban areas, based on a multi-story concentrated-mass shear model and PhysX. Bullet Constraints Builder (BCB) [29] is a building collapse simulation software, which is developed based on the open-source physics engine Bullet [32] and Blender Python script. Bullet [32] is an open-source, real-time physics engine developed by Erwin Couman in 2003, which can simulate collision detection, and soft and rigid body dynamics. The BCB can simulate the dynamic behavior of structures at the large-deformation stage, and the distribution of ruins. Furthermore, the simulation results are generally consistent with ruin scenarios in real-world earthquake events [30]. Although the physics engine-based building collapse simulation methods have advantages in simulating large structural deformations, the simulation results at the small-deformation stage are less reliable. For example, due to the difficulty of determining constraint damping in Bullet, and the oversimplification of the rebar model in BCB, the simulation results of the BCB and FE methods are quite different at the small-deformation stage. However, the process of the structural collapse is often sensitive to the initial state of the collapse. Therefore, ensuring the accuracy of collapse mode simulations using physics engine-based methods is difficult. To address the challenges mentioned above, a hybrid framework for simulating building collapse and ruin scenarios, based on an FE model and a physics engine, is proposed. For dynamic responses during the small-deformation stage, an FE model is used because of its accuracy in such simulations. For the large-deformation stage, a physics engine-based method is used. In this work, the MSC.Marc software [33] is used to perform the FE modeling and simulations, and the BCB is used to simulate the ruin scenario. The proposed framework is also applicable to other FE software and physics engines. In Section2, the proposed hybrid framework is introduced. Section3 will illustrate three key methods adopted in the proposed framework: (1) conversion of the geometric data from the FE method to the physics engine; (2) determination of the initial moment of collapse; and (3) data mapping of the FE simulation results. Section4 will validate the feasibility and accuracy of the proposed framework using a shaking table test of a three-story reinforced concrete frame. The collapse process and ruin scenario of a real-world library building are simulated as a case study to demonstrate the high-fidelity collapse process during an earthquake. The outcome of this research contributes to the construction of virtual building collapse and ruin scenarios, and supports post-earthquake emergency rescue training.

2. Hybrid Framework Based on FE Method and Physics Engine Figure1 shows the proposed hybrid framework, where the software programs are in bold italics, and the detailed functions of the program are in italics. The framework consists of four modules: Module 1: FE model establishment and conversion The FE model of the target structure is • established using the FE software MSC.Marc [33]. The geometric model is then converted into the Appl. Sci. 2020, 10, 4408 3 of 19

rigid body model in Blender, using the geometry mapping function that is integrated into the Blender Python script “Finite Element Method to Rigid Body Dynamics (FEM2RBD)”. In the rigid body model, the elements are rigid bodies, and no internal stresses and strains are considered. The deformations of the reinforced concrete structures are considered via the constraints/springs among diﬀerent rigid bodies [29]. Appl. Sci. 2020, 10, x FOR PEER REVIEW 4 of 20

1. Numerical Model Construction

FEM2RBD

Geometry Mapping

FE Model of a Rigid Body Model of Building a Building

BCB Processing BCB

2. Small FEM2RBD 3. Large Deformation Deformation Analysis Displacement Analysis and Velocity Mapping

Ti me Hi story Anal ysi s Analysis Model of a Building

Small Deformation Data

Physics Engine Computation Keyframe Data

FEM2RBD Displacement Mapping

4. Data Integration and Rendering

Rendering the Result using Blender

FigureFigure 1. 1.Hybrid Hybrid framework framework based based on on an an FE FE model model and and a a physics physics engine. engine.

Figure 1 shows the proposed hybrid framework, where the software programs are in bold italics, and the detailed functions of the program are in italics. The framework consists of four modules: • Module 1: FE model establishment and conversion The FE model of the target structure is established using the FE software MSC.Marc [Error! Reference source not found.]. The geometric model is then converted into the rigid body model in Blender, using the geometry mapping function that is integrated into the Blender Python script “Finite Element Method to Rigid Body Dynamics (FEM2RBD)”. In the rigid body model, Appl. Sci. 2020, 10, 4408 4 of 19

Module 2: Small deformation analysis and initial collapse moment determination A nonlinear • time history analysis is conducted using the FE model of the target building, to analyze the structural behavior during the small-deformation stage. Subsequently, the initial moment of collapse is determined, and the displacement and velocity data of each time step at, and before, the moment of initial collapse are extracted from the FE model. These data will be used in the large-deformation simulation based on the physics engine in Module 3 and the collapse process visualization in Module 4.

Module 3: Data mapping and large deformation analysis Based on the mapping method of the FE • results proposed in Section3, the displacement and velocity data from the moment of initial collapse are mapped into the BCB geometric model, using the displacement and velocity mapping functions of FEM2RBD. Subsequently, this model is modiﬁed using several BCB processing functions, in the following order: (1) establish the rigid body ground; (2) remove the overlapping portions of rigid bodies; (3) map the remaining ground motion; (4) deﬁne the constraint parameters; and (5) build the constraints among rigid bodies. The analysis model used for the physics engine simulation can then be established [29]. After establishing this model, simulations are performed using the Bullet physics engine to determine the structural dynamic behavior during the large-deformation stage.

Module 4: Data integration, rendering and visualization After the large-deformation stage • simulation, the render engines of Blender are used to render the data of the small and large-deformation stages. Subsequently, the rendered videos are integrated to visualize the entire structural collapse process.

3. Key Techniques To implement the proposed framework, the following three key issues need to be addressed.

Conversion of the geometric model from the FE method to the physics engine. The model in the • physics engine-based simulation component is a rigid body spring model, the elements of which are signiﬁcantly diﬀerent from the elements of the FE model. Therefore, it is necessary to propose a method for converting the geometric data from the FE model to the physics engine Determination of the initial moment of collapse. To achieve the proposed hybrid framework, the • switching moment (collapse keyframe) from the FE simulation to physics engine simulation needs to be determined based on the results of the FE simulation. Mapping the FE simulation data. After the FE simulation, the data computed for the • small-deformation stage need to be mapped into the physics engine.

3.1. Geometric Model and Material Mapping Method Fiber beam-column and multi-layered shell elements have been widely used to simulate beams, columns and shear walls during the nonlinear dynamic analyses of structures [10,34,35]. Compared with models that use solid elements, models that use fiber beam-column and multi-layered shell elements reduce the number of elements and the computational workload required, which makes the simulations of large-scale structures easier to conduct. In this study, soil–structure interaction (SSI) effects are neglected, assuming that that the soil underneath the foundations is infinitely rigid. It is worthwhile noting that previous studies have illustrated that SSI effects can significantly affect the lateral displacement demand and collapse capacity of building structures [36–38]. The classical Rayleigh damping with a damping ratio of 5% based on the initial stiffness of the building is used to model the inherent damping of the structure. Future studies should consider incorporating more accurate approaches to modeling inherent damping, as a few research works have identified drawbacks associated with the use of Rayleigh damping based on initial stiffness [39–42]. In the FE model, the geometric model of a fiber beam-column element is a line [a one-dimensional (1D) element], and the geometric model of a multi-layered shell element is a plane (a 2D element). However, the geometric Appl. Sci. 2020, 10, x FOR PEER REVIEW 6 of 20 structures [Error! Reference source not found.–Error! Reference source not found.]. The classical Rayleigh damping with a damping ratio of 5% based on the initial stiffness of the building is used to model the inherent damping of the structure. Future studies should consider incorporating more accurate approaches to modeling inherent damping, as a few research works have identified drawbacks associated with the use of Rayleigh damping based on initial stiffness [Error! Reference source not found.–Error! Reference source not found.]. In the FE model, the geometric model of a Appl.fiber Sci.beam-column2020, 10, 4408 element is a line [a one-dimensional (1D) element], and the geometric model5 of 19a multi-layered shell element is a plane (a 2D element). However, the geometric model in the physics engine is a solid (a 3D object). Because the geometric models of the physics engine and FE model are model in the physics engine is a solid (a 3D object). Because the geometric models of the physics engine different, a method is proposed to convert the FE models of the fiber beam-column and multi-layeredand FE model shell are di elementsfferent, a into method solid is objects proposed for the to convert physics the engine. FE models of the fiber beam-column and multi-layered shell elements into solid objects for the physics engine. 3.1.1. Solid Model Establishment Method in Blender 3.1.1. Solid Model Establishment Method in Blender The information available from FE models consists of geometric information (including the The information available from FE models consists of geometric information (including the coordinates of element nodes and sections) and material information about different elements. After coordinates of element nodes and sections) and material information about different elements. After data preparation, a physics engine model for large deformation analysis is established in Blender, as data preparation, a physics engine model for large deformation analysis is established in Blender, shown in Figure 2. as shown in Figure2.

Scale： Move & Rotate： Rigid Body： Change shape Change position Change Type column/beam

cube Geometry Model Rigid Body Model

floor/wall

Figure 2. SolidSolid model model establishment establishment method in Blender.

The element establishment method method in in Blender Blender is is di differentfferent from that in FE software. In In Blender, Blender, “primitive”“primitive” meshmesh shapes shapes should should be be established established first, firs thent, then all theall the elements elements should should be transformed be transformed from fromthe “primitive” the “primitive” mesh mesh shapes. shapes. However, However, as the as shape the shape and position and position of “primitive” of “primitive” mesh shapes mesh shapes do not docoincide not coincide with the with corresponding the corresponding FE elements, FE elem the shapeents, the and shape position and transformations position transformations are performed are performedto get the target to get elements the target in FEelements models. in Blender FE mo providesdels. Blender several provides “primitive” several mesh “primitive” shapes to beginmesh shapesmodeling, to includingbegin modeling, cubes, planes includin andg cylinders cubes, planes [43]. Because and cylinders a cube is [ aError! shape Reference that best approximates source not found.engineering]. Because components a cube is and a shape FE model that meshes, best approximates it is selected engineering as the basic meshcomponents shape inand Blender. FE model The meshes,geometric it modelis selected mapping as the is thenbasic performed mesh shape in twoin Blender. steps. The geometric model mapping is then performed in two steps. Step 1: Shape and position transformation (including translation and rotation). The shape and • Stepposition 1: Shape transformations and position aretransformation performed on(including the cube translation mesh provided and rotation). in Blender [43], to create geometricThe shape objectsand position with thetransformations same shapes are and perf positionsormed ason thethe elementscube mesh in provided the FE model. in Blender The rotation[Error! Reference method shown source in Figurenot found.3 is used], to tocreate model geometric inclined elementsobjects with in FE the models. same Specifically,shapes and thepositions rotation as anglethe elements is obtained in the by firstFE model. determining The rotation the endpoint method coordinates shown in of Figure the longest 3 is used edge to of anmodel FE element inclined (line elements segment inl 1FE) and models. the projection Specific ofally,l1 inthe the rotation XOY plane angle in theis obtained global coordinate by first α systemdetermining (line segmentthe endpointl2). The coordinates angle between of the lol2ngestand theedge X of axis, an andFE element the angle (lineβ between segmentl2 land1) andl1, arethe thenprojection calculated of l1 in using the theseXOY plane endpoint in the coordinates. global coordinate Subsequently, system the (line shape-transformed segment l2). The objectangle α isα rotatedbetween by l2 angleand thearound X axis,the and Z the axis angle in the β global between coordinate l2 and l1 system., are then The calculated object is thenusing rotated these byendpoint angle β coordinates.around the Y’Subsequently, axis in the rotated the shap coordinatee-transformed system. object The is rotated rotated coordinate by angle α system around is the localZ axis coordinate in the global system coordinate of the cube system. after The the firstobject rotation, is then where rotated the by X’O’ angle axis β isaround parallel the to Y’l2. Stepaxis in 2: Addthe rotated rigid bodies coordinate to the system. cubes. ConvertThe rotated the cubescoordinate into rigid system bodies is the after local the coordinate shape and • positionsystem of transformations the cube after the to obtainfirst rotati the rigidon, where body the model X’O’ for axis the is Bullet parallel physics to l2. engine simulation. • Step 2: Add rigid bodies to the cubes. Appl. Sci. 2020, 10, x FOR PEER REVIEW 7 of 20

Convert the cubes into rigid bodies after the shape and position transformations to obtain the

Appl. Sci.rigid2020 ,body10, 4408 model for the Bullet physics engine simulation. 6 of 19 Z

ߚ O Y l2 ߙ Z' Y'

X l2||O'X' O' X'

Figure 3. Rotation method for inclined elements. Figure 3. Rotation method for inclined elements. 3.1.2. Material Mapping Method in the BCB 3.1.2. Material Mapping Method in the BCB The BCB will group objects according to their names in Blender, and assign the same material propertiesThe BCB to the will objects group in objects each group according during to the their constraint names in building Blender, process. and assign When the establishing same material a geometricproperties model to the in objects Blender, in theeach object group names during need th toe constraint be chosen building so that objects process. of theWhen same establishing material in a thegeometric FE model model are mapped in Blender, into the the object same names material need group to be by chosen BCB. Theso that objects objects in Blenderof the same are namedmaterial accordingin the FE tomodel the rule are of mapped “element into type: the materialsame material type: element group by number.” BCB. The For objects example, in Blender an object are derived named fromaccording a beam to element the rule with of material“element number type: 1material and element type: numberelement 234 number.” is named For “Beam: example, 1: 234.” an Moreobject detailsderived are from available a beam in theelement BCB manualwith material [29]. number 1 and element number 234 is named “Beam: 1: 234.” More details are available in the BCB manual [Error! Reference source not found.]. 3.2. Determination of the Initial Moment of Collapse 3.2. Determination of the Initial Moment of Collapse According to the proposed hybrid framework, the FE simulation is conducted before the initial momentAccording of collapse to theTc ,proposed and the hybrid physics framework, engine simulation the FE simulation is conducted is conducted after Tc. before Therefore, the initial the determinationmoment of collapse of Tc is essential. Tc, and Severalthe physics collapse engine criteria simulation have been is proposed conducted [44 after–49]. ThereTc. Therefore, are various the structuraldetermination collapse of modes,Tc is essential. including Several lateral collapse collapse, criteria vertical have collapse, been and proposed so on. No[Error! matter Reference which collapsesource not mode found. is, the–Error! vertical Reference displacements source ofnot structural found.]. membersThere are willvarious exceed structural a specific collapse value modes, when theincluding structures lateral collapse collapse, [47,48 vertical]. Therefore, collapse, the definitionand so on. of No collapse matter dependswhich collapse on the momentmode is, whenthe vertical “the deformationdisplacements of the of structurestructural is members insufficient will to maintainexceed a a safespecific use space”value when [47]. Besides,the structures Zhao et collapse al. [46] studied[Error! fourReference collapse source criteria, not includingfound.,Error! the Reference following: source Criterion not 1) found. According]. Therefore, to Chinese the definition seismic codeof collapse [44], the depends earthquake-induced on the moment inter-story when drift “the ratio deformation of reinforced of concretethe structure frames is shouldinsufficient not be to greatermaintain than a 1 /safe50; Criterion use space” 2) Federal [Error! Emergency Reference Management source not Agency found. (FEMA)]. Besides, 356 [Zhao45] recommended et al. [Error! thatReference the inter-story source drift not ratiofound. limit] studied for the collapse-preventionfour collapse criteria, performance including levelthe following: of concrete Criterion structures 1) beAccording 4%; Criterion to Chinese 3) According seismic to code Villaverde [Error! [49 Reference], when the source tangent not slope found. of], the the incremental earthquake-induced dynamic analysisinter-story (IDA) drift curve ratio is lower of reinforced than 20% concrete of the slope fram ofes the should initial elasticnot be stage, greater the structurethan 1/50; will Criterion collapse; 2) CriterionFederal 4)Emergency “The vertical Management collapse displacement Agency (FEMA) of the main356 [Error! structural Reference members source exceed[ing] not found. 1 m”, ] isrecommended considered as that Criterion the inter-story 4 [46]. The drift work ratio of Zhaolimit for et al.the [46 collapse-prevention] shows that the conventional performance collapse level of criteriaconcrete (i.e., structures Criteria 1–3)be 4%; significantly Criterion 3) underestimate According to theVillaverde collapse [Error! resistance Reference of the frames.source not Compared found.], withwhen the the other tangent three slope collapse of the criteria, incremental Criterion dynamic 4 can better analysis reflect (IDA) the extremecurve is nonlinearlower than behavior 20% of ofthe theslope collapse of the limit initial state. elastic Therefore, stage, collapse the structure Criterion will 4, collapse; proposed Criterion by Zhao et4) al.“The [46], vertical is selected collapse as a ruledisplacement for determining of theT mainc in this structural work. members exceed[ing] 1 m”, is considered as Criterion 4 [Error! Reference source not found.]. The work of Zhao et al. [Error! Reference source not found.] shows that the conventional collapse criteria (i.e., Criteria 1–3) significantly underestimate the collapse Appl. Sci. 2020, 10, 4408 7 of 19

3.3. FE Simulation Data Mapping Method The published research on combining FE models and physics engines is limited [26,28]. These studies also lack applicable data mapping methods for converting FE-based numerical models that use ﬁber beam-column and multi-layered shell elements into physics engine models. Therefore, a method for mapping simulation data from the FE model into the Blender model is proposed.

3.3.1. Displacement Mapping Method The keyframe method of Blender [43] is used to achieve the displacement mapping. A keyframe is a location on a timeline that marks the beginning or end of a transition. For example, the displacement keyframe defines the beginning and endpoints of the displacement. Meanwhile, a kinematic keyframe defines the information about the kinematic type. There are two kinematic types in Blender: (1) the animation system, in which the motion of the object is controlled by the displacement keyframe, and (2) the physics engine system, in which the motion of the object is controlled by the simulation results obtained using the physics engine. Displacement mapping requires only the displacement keyframe. Before the initial moment of collapse Tc, the motions of objects in Blender are controlled by the displacement keyframes derived from the FE results. This means that only Blender is used to map the FE results; the Bullet physics engine is not involved. Specifically, the displacement mapping method is divided into three steps, as shown in Figure4. Appl. Sci. 2020, 10, x FOR PEER REVIEW 9 of 20

Begin

Step 1: Extract Extract FE Model Change FE Model Data Data Kinematic Type Step 3: Change Yes Kinematic Type

Frame No Frame Remains? Remains?

Yes No

Switch to Next End Step 2: Insert Keyframe Displacement Keyframe No Object Remains?

Ye s

Insert Displacement Keyframe

FigureFigure 4.4. FlowchartFlowchart ofof thethe displacementdisplacement mappingmapping method.method.

3.3.2.Step Velocity 1. Extract Mapping the FE Method model data. After the FE simulation, the displacement data of each vertex • atWhen each the time FE step simulation are extracted, is converted and then theto the deformed Bullet physics coordinates engine of each simulation vertex are at calculatedthe initial from the displacement data and the initial coordinates. The deformed centroid coordinates of each moment of collapse Tc, both displacement and velocity should be consistent between the two simulations.object at Consequently, each time step canvelocity be calculated mapping from is necessary. the deformed The velocity vertex coordinates mapping method of each object.requires both the displacement and the kinematic type keyframes [Error! Reference source not found.]. In Blender, the velocity of an object is defined as follows: At frame i, convert the kinematic type of an object to the animation system and insert the kinematic type as a keyframe. At frame j (where i < j), convert the kinematic type of the same object to the physics engine system, and insert the kinematic type as a keyframe again. Consequently, during frame i and frame j, the movement of the object will be controlled by the linear interpolation of the displacement at frame i and frame j, respectively. After frame j, the movement of the object will be controlled by the Bullet physics engine, and the

object will gain speed v j at frame j, as follows: − x j xi v = (1) j ( j − i) × dt

where i and j (i < j) denote the keyframe number; dt represents the time interval between two 1 adjacent keyframes, and in Blender, the default value is dt = s ; xi and xj represent the centroid 24

coordinates of an object at frame i and frame j, respectively. The speed v j is the average velocity represented by the secant line, rather than the actual instantaneous velocity represented by the tangent line, as shown in Figure 5. In Blender, the Bullet physics engine adopts the symplectic Euler algorithm to update the motion of a rigid body [Error! Reference source not found.], where the position of the object in the Appl. Sci. 2020, 10, 4408 8 of 19

Step 2. Insert the displacement keyframe. Initially, the keyframe that corresponds to the ﬁrst • time step in the FE model on the Blender timeline is selected. The deformed centroid coordinates obtained from the FE results for one object are then calculated. Subsequently, the deformed centroid coordinates are inserted as the displacement keyframe. When all the objects in the structure have been processed, the simulation moves to the next time step, and the aforementioned procedure is repeated until the initial moment of collapse Tc. Step 3: Change the kinematic type. After Step 2, the kinematic type of each object needs to be • converted to the animation system, otherwise the object motions will be controlled by the physics engine simulation, rather than by the FE result. Appl. Sci. 2020, 10, x FOR PEER REVIEW 10 of 20 3.3.2. Velocity Mapping Method next frame is determined by the velocity, position and force of the adjacent previous frame, as shownWhen in the Equations FE simulation (2) and is (3). converted to the Bullet physics engine simulation at the initial moment of collapse Tc, both displacement and velocity should be consistent between the two simulations. F + F Consequently, velocity mapping is necessary.= The+ velocityext c Δ mapping method requires both the vt+Δt vt t (2) displacement and the kinematic type keyframes [43]. In Blender,m the velocity of an object is deﬁned as follows: At frame i, convert the kinematic type of an object to the animation system and insert the x = x + v Δt kinematic type as a keyframe. At frame j (wheret+Δt i < j),t convertt+Δt the kinematic type of the same object to(3) thewhere physics Fext engine denotes system, external and forces insert defined the kinematic by the user; type F asc denotes a keyframe the constraint again. Consequently, forces determined during by framethe positioni and frame and jthe, the velocity movement of the of ob theject object in the will adjacent be controlled previous by frame. the linear interpolation of the displacement at frame i and frame j, respectively. After frame j, the movement of the object will be If the average velocity v j is used instead of the instantaneous velocity vj at the initial moment controlled by the Bullet physics engine, and the object will gain speed vj at frame j, as follows: of collapse Tc, an error due to approximating the velocity will be introduced. To improve the accuracy of the velocity mapping, this work proposesxj xi a velocity mapping method that uses a virtual vj = − (1) displacement vector xvirtual. The virtual displacement(j i) d ist obtained using Equation (4), based on the − × velocity and displacement at the collapse keyframe j. The proposed mapping method is shown in where i and j (i < j) denote the keyframe number; dt represents the time interval between two adjacent Figure 5. The blue curve is the displacement curve simulated by the FE method; ti and tj are the times 1 keyframes, and in Blender, the default value is dt = s; xi and xj represent the centroid coordinates of corresponding to the displacement keyframe insertion24 points; xi and xj are the displacements an object at frame i and frame j, respectively. The speed vj is the average velocity represented by the corresponding to the displacement keyframe insertion points; vj is the instantaneous velocity, and secant line, rather than the actual instantaneous velocity represented by the tangent line, as shown in xvirtual is the virtual displacement vector. Figure5.

Virtual vj displacement xvirtual ݒ

xi Original

Displacement displacement

ti tj Time

FigureFigure 5. Virtual5. Virtual displacement displacement vector vector and and velocity velocity mapping mapping method. method.

Consider the virtual displacement vector: = − × − × x virtual x j v j ( j k ) dt (4)

k = j − n(n ≥ 4) (5)

where xj and vj represent the displacement vector of the FE model and the velocity vector of the

collapse keyframe j, respectively; k is the keyframe number of xvirtual, and the value of k is defined by Equation (5). After testing, the velocity cannot be mapped successfully in Blender when n is less than four. The real instantaneous velocity and displacement of the FE model are mapped in the

Bullet physics engine by replacing xi with the virtual displacement vector xvirtual in Equation (1). In summary, the velocity mapping method can be divided into the following three steps, which are schematically diagrammed in Figure 6.

• Step 1: Extract the FE results at time step tj, including velocity vj and displacement xj. Appl. Sci. 2020, 10, 4408 9 of 19

In Blender, the Bullet physics engine adopts the symplectic Euler algorithm to update the motion of a rigid body [50], where the position of the object in the next frame is determined by the velocity, position and force of the adjacent previous frame, as shown in Equations (2) and (3).

Fext + Fc v + = v + ∆t (2) t ∆t t m

xt+∆t = xt + vt+∆t∆t (3) where Fext denotes external forces deﬁned by the user; Fc denotes the constraint forces determined by the position and the velocity of the object in the adjacent previous frame. If the average velocity vj is used instead of the instantaneous velocity vj at the initial moment of collapse Tc, an error due to approximating the velocity will be introduced. To improve the accuracy of the velocity mapping, this work proposes a velocity mapping method that uses a virtual displacement vector xvirtual. The virtual displacement is obtained using Equation (4), based on the velocity and displacement at the collapse keyframe j. The proposed mapping method is shown in Figure5. The blue curve is the displacement curve simulated by the FE method; ti and tj are the times corresponding to the displacement keyframe insertion points; xi and xj are the displacements corresponding to the displacement keyframe insertion points; vj is the instantaneous velocity, and xvirtual is the virtual displacement vector. Consider the virtual displacement vector:

x = x v (j k) dt (4) virtual j − j × − × k = j n(n 4) (5) − ≥ Appl. Sci. 2020, 10, x FOR PEER REVIEW 11 of 20 where xj and vj represent the displacement vector of the FE model and the velocity vector of the collapse keyframe j, respectively; k is the keyframe number of x , and the value of k is deﬁned • Step 2: Obtain xvirtual using Equations (4) and (5), and set the displacementvirtual of the frame k to xvirtual. by Equation (5). After testing, the velocity cannot be mapped successfully in Blender when n is less • Step 3: At frame k, convert the kinematic type to the animation system, and then insert the than four. The real instantaneous velocity and displacement of the FE model are mapped in the Bullet displacement and kinematic type keyframes for each object. Convert the frame from frame k to physics engine by replacing x with the virtual displacement vector x in Equation (1). frame j after the keyframei insertion at frame k. At frame j, convertvirtual the kinematic type to the In summary,physics engine the velocity system, mappingand then insert method the disp canlacement be divided and into kinematic the following type keyframes three steps,for each which are schematicallyobject. diagrammed in Figure6.

Keyframe j Keyframe k(k

Extract Data (vj , xj) Calculate xvirtual

Convert Kinematic Type to Convert Kinematic Type to Physics Engine System Animation System

Insert Displacement and Insert Displacement and Kinematic Type Keyframes Kinematic Type Keyframes

End

FigureFigure 6. 6.Flowchart Flowchart ofof thethe velocity mapping mapping method. method.

4. Validation and Case Study

4.1. Validation Using a 3D Shaking Table Test of a Three-Story Reinforced Concrete Frame

4.1.1. Basic Information To validate the reliability of the proposed hybrid framework, a 3D shaking table test was conducted with a three-story reinforced concrete frame [Error! Reference source not found.] as a case study. The collapse of the structure was simulated using three methods: (1) FE simulation, (2) BCB simulation, and (3) the proposed hybrid framework. The details of the structure are shown in Figure 7. The concrete information is shown in Table 1, and the rebar information is shown in Table 2. The structure was subjected to the amplitude-scaled El-Centro ground motion record during the shaking table test, and the load cases are shown in Table 3. The 5%-damped pseudo-acceleration spectra of Load Case 5 are shown in Figure 8. More details can be found in [Error! Reference source not found.]. The first mode is a translation with a fundamental period (elastic period) T1 = 0.316 s; the second mode is a translation with a fundamental period T2 = 0.316 s; and the third mode is the planar torsion, with a fundamental period T3 = 0.178 s. Appl. Sci. 2020, 10, 4408 10 of 19

Step 1: Extract the FE results at time step t , including velocity v and displacement x . • j j j Step 2: Obtain x using Equations (4) and (5), and set the displacement of the frame k to x . • virtual virtual Step 3: At frame k, convert the kinematic type to the animation system, and then insert the • displacement and kinematic type keyframes for each object. Convert the frame from frame k to frame j after the keyframe insertion at frame k. At frame j, convert the kinematic type to the physics engine system, and then insert the displacement and kinematic type keyframes for each object.

4. Validation and Case Study

4.1. Validation Using a 3D Shaking Table Test of a Three-Story Reinforced Concrete Frame

4.1.1. Basic Information To validate the reliability of the proposed hybrid framework, a 3D shaking table test was conducted with a three-story reinforced concrete frame [20] as a case study. The collapse of the structure was simulated using three methods: (1) FE simulation, (2) BCB simulation, and (3) the proposed hybrid framework. The details of the structure are shown in Figure7. The concrete information is shown in Table1, and the rebar information is shown in Table2. The structure was subjected to the amplitude-scaled El-Centro ground motion record during the shaking table test, and the load cases are shown in Table3. The 5%-damped pseudo-acceleration spectra of Load Case 5 are shown in Figure8. More details can be found in [20]. The ﬁrst mode is a translation with a fundamental period (elastic period) T1 = 0.316 s; the second mode is a translation with a fundamental period T2 = 0.316 s; and the third mode is the Appl.planar Sci. 2020 torsion,, 10, x FOR with PEER a fundamentalREVIEW period T3 = 0.178 s. 12 of 20

Figure 7. Dimensions and reinforcements of the model structure (units: mm). [20] (Reproduced with Figurepermission 7. Dimensions from Huang, and reinforcements Q., from Study onof spatialthe model collapse structure responses (units: of reinforced mm). concrete [Error! frameReference structures sourceunder not earthquake found.]; published(Reproduced by Tongjiwith permissi Univerity,on 2006.). from Huang, Q., from Study on spatial collapse responses of reinforced concrete frame structures under earthquake; published by Tongji Univerity, 2006.).

35 ) 2 30

a (m/s a 25 S X-direction 20 Y-direction 15 Z-direction 10

5 Spectral acceleration acceleration Spectral 0 01234

Periods (s) Figure 8. Pseudo-acceleration spectra of Load Case 5.

Table 1. Concrete information.

Floor No. Ec (GPa) fc (MPa) 1 25 22.5 2 27 23.9 3 23 15.6 Appl. Sci. 2020, 10, 4408 11 of 19 Appl. Sci. 2020, 10, x FOR PEER REVIEW 12 of 20

Table 1. Concrete information.

Floor No. Ec (GPa) f c (MPa) 1 25 22.5 2 27 23.9 3 23 15.6

Table 2. Rebar information.

Table Es (GPa) f s (MPa) 16# 200 486.4 10# 200 457.7

Table 3. Load cases.

Peak Ground Acceleration (PGA) (Unit: g) Load Case X-Direction Y-Direction Z-Direction 1 0.10 0.08 0.06 2 0.36 0.32 0.28 Figure 37. Dimensions and reinforcements 0.84 of the model structure 0.55 (units: mm). [Error! 0.54Reference source not found.] (Reproduced with permission from Huang, Q., from Study on spatial collapse 5 0.93 1.19 0.57 responses of reinforced concrete frame structures under earthquake; published by Tongji Univerity, 2006.).

35 ) 2 30

a (m/s a 25 S X-direction 20 Y-direction 15 Z-direction 10

5 Spectral acceleration acceleration Spectral 0 01234

Periods (s) FigureFigure 8. Pseudo-acceleration 8. Pseudo-acceleration spec spectratra of ofLoad Load Case Case 5. 5.

4.1.2. Comparison of Simulation ResultsTable 1. Concrete information.

Floor No. Ec (GPa) fc (MPa) Based on the information above, an FE1 model25 with ﬁber22.5 beam-column and multi-layered shell elements was established in MSC.Marc.2 Details27 of these23.9 element models have been reported in the published work [35]. The seismic responses3 of23 the structure15.6 under Load Cases 1–3 are used to validate the rationality of the FE model. Figure9 shows the comparison of the maximum horizontal displacement in the X-direction between the test results and the FE simulation. The results indicate that the FE simulation agrees well with the test results. In Load Case 5, a convergence problem occurred in the FE simulation (t = 4.16 s); therefore, the distribution of the ruins could not be obtained. The collapse process predicted by the FE simulation is shown in Figure 10. The colored contours in Figure 10 represent the longitudinal reinforcement yielding in the elements. The numerical simulation reveals that failures of the frame are initiated at both ends of the columns where the bending moment is large. Subsequent failures take place on the ground story, where the columns carry the largest lateral force. Large displacements occurred on the ground story, which leads to the collapse of the entire structure. In Figure 10c, the deformed shape of the structure at 4.16 s is shown. Appl. Sci. 2020, 10, x FOR PEER REVIEW 13 of 20

Table 2. Rebar information.

Table Es (GPa) fs (MPa) 16# 200 486.4 10# 200 457.7

Table 3. Load cases.

Peak Ground Acceleration (PGA) (unit: g) Load Case X-Direction Y-Direction Z-Direction 1 0.10 0.08 0.06 2 0.36 0.32 0.28 3 0.84 0.55 0.54 5 0.93 1.19 0.57

4.1.2. Comparison of Simulation Results Based on the information above, an FE model with fiber beam-column and multi-layered shell elements was established in MSC.Marc. Details of these element models have been reported in the published work [Error! Reference source not found.]. The seismic responses of the structure under Load Cases 1–3 are used to validate the rationality of the FE model. Figure 9 shows the comparison of the maximum horizontal displacement in the X-direction between the test results and the FE Appl. Sci. 2020, 10, 4408 12 of 19 simulation. The results indicate that the FE simulation agrees well with the test results.

20 Load Case 1

10 Load Case 2 Test Max Displacement (mm) Displacement Max Test Load Case 3 0 0 102030405060 FE Method Max Displacement (mm) Appl. Sci. 2020, 10, x FOR PEER REVIEW 14 of 20 FigureFigure 9. 9. MaximumMaximum horizontal horizontal displacement displacement in in the the X-direction: X-direction: FE FE method method versus versus test. test.

In Load Case 5, a convergence problem occurred in the FE simulation (t = 4.16 s); therefore, the distribution of the ruins could not be obtained. The collapse process predicted by the FE simulation is shown in Figure 10. The colored contours in Figure 10 represent the longitudinal reinforcement yielding in the elements. The numerical simulation reveals that failures of the frame are initiated at both ends of the columns where the bending moment is large. Subsequent failures take place on the ground story, where the columns carry the largest lateral force. Large displacements occurred on the ground story, which leads to the collapse of the entire structure. In Figure 10c, the deformed shape of the structure at 4.16 s is shown.

(a)

Plastic

Elastic

(b)

Plastic

Elastic

(c)

FigureFigure 10. Collapse 10. Collapse process process of the of reinforcedthe reinforced concrete concrete frame frame subjected subjected to theto the El-Centro El-Centro ground ground motion record:motion (a) t record:= 0.00 s;(a) ( bt =) t0.00= 3.88 s; (b s;) t (=c )3.88t = s;4.16 (c) t s. = 4.16 s.

From the FE results, the initial moment of collapse is 3.88 s, using the collapse criterion in Section 3.2. A ¼-scale model is adopted; thus, the vertical collapse displacement of the collapse criterion is 0.25 m. Subsequently, collapse simulation was performed using the proposed hybrid method. Figure 11b shows the distribution of ruins, which shows excellent agreement with the test results. When BCB is used for collapse simulation, the structure collapses vertically, which agrees poorly with the test results, due to the low accuracy of the BCB simulation of the small-deformation stage. As shown in Figure 11c, the final positions of the three slabs of the three-story reinforced concrete frame almost entirely overlap. Compared with the BCB results (Figure 11c), the distribution of the ruins obtained by the proposed hybrid method (Figure 11b) shows better agreement with the test results (Figure 11a). Appl. Sci. 2020, 10, 4408 13 of 19

From the FE results, the initial moment of collapse is 3.88 s, using the collapse criterion in 1 Section 3.2.A 4 -scale model is adopted; thus, the vertical collapse displacement of the collapse criterion is 0.25 m. Subsequently, collapse simulation was performed using the proposed hybrid method. Figure 11b shows the distribution of ruins, which shows excellent agreement with the test results. Appl. Sci. 2020, 10, x FOR PEER REVIEW 15 of 20

(a)

(b)

(c)

Figure 11.11. DistributionDistribution of of ruins: ruins: (a )(a test;) test; [20 []Error! (Reproduced Reference with source permission not found. from Huang,] (Reproduced Q., from Studywith permissionon spatial collapse from Huang, responses Q., of from reinforced Study concrete on spatial frame collapse structures responses under of reinforced earthquake concrete; published frame bystructures Tongji underUniverity, earthquake 2006.); published (b) the proposed by Tongji FE andUniverity, BCB hybrid 2006.) method; (b) the proposed (c) BCB. FE and BCB hybrid method; (c) BCB. When BCB is used for collapse simulation, the structure collapses vertically, which agrees poorly withThe the testfollowing results, observations due to the low are accuracy made from of the the BCB comparison simulation and of the analysis small-deformation of the simulation stage. Asmethods: shown in Figure 11c, the ﬁnal positions of the three slabs of the three-story reinforced concrete frame almost entirely overlap. • For FE models, a convergence problem occurs during the collapse simulation; therefore, the Compared with the BCB results (Figure 11c), the distribution of the ruins obtained by the proposed distribution of ruins cannot be easily obtained. hybrid method (Figure 11b) shows better agreement with the test results (Figure 11a). • The BCB simulation method is not sufficiently accurate for the small-deformation stage. Therefore, the collapse mode and the distribution of ruins differ substantially from the test results. • The proposed hybrid method achieves the most satisfactory simulation of the structural collapse mode and the distribution of ruins.

4.2. Collapse and Ruins Simulation of a Real-World Library Building Appl. Sci. 2020, 10, 4408 14 of 19

The following observations are made from the comparison and analysis of the simulation methods:

For FE models, a convergence problem occurs during the collapse simulation; therefore, the • distribution of ruins cannot be easily obtained. The BCB simulation method is not suﬃciently accurate for the small-deformation stage. Therefore, • the collapse mode and the distribution of ruins diﬀer substantially from the test results. The proposed hybrid method achieves the most satisfactory simulation of the structural collapse • mode and the distribution of ruins.

4.2. Collapse and Ruins Simulation of a Real-World Library Building To demonstrate the applicability of the proposed hybrid method to real-world complex structures, a case study of a multi-story reinforced concrete library building is performed. The building is a seven-story frame-shear wall structure, with a total height of 19.8 m and a building area of 15,938 m2. The total mass of the library building is 3.17 107 kg. All reinforcement consists of HRB400 reinforcing × bars, whose strength is 400 MPa. The material properties and dimensions of the main structural members are shown in Table4. Specifically, the library is located on a site class III in GB50011-2010 [ 44], with an approximate equivalent shear wave velocity of 200 m/s for 30-m soil (VS30). The characteristic period of the site is 0.45 s. The building has an 8-degree seismic design intensity, where the peak ground acceleration is 200 cm/s2 for a design basis earthquake (DBE) with a return period of 475 years. There is no R-factor, coefficient of displacement or overstrength factor in Chinese seismic design code. However, according to Lu et al. [51] and American Society of Civil Engineers (ASCE) 7–10 [52], the design information of the building is similar to the following seismic design parameters: R factor = 4.5; overstrength factor = 2.5 and deflection amplification factor = 4.

Table 4. The material properties and dimensions of the main structural members in the library building.

Element Member Location Strength of Concrete (MPa) Size of Elements (m) Walls 1st floor 40 8 2.25~2 2.25 × × 2nd floor 40 8 2.07~2 2.07 × × Others 35 8 2~2.5 2 × × Beams 1st floor 40 4~1.25 2nd floor 40 4~0.707 Others 35 4~0.707 Columns 1st floor 40 2.25 2nd floor 40 2.07 Others 35 2 Slabs All floors elastic 8 7.45~2 1.5 × × Note that the sizes of walls mean X-direction/Y-direction Z-direction. ×

Three element types are used in this model: (a) fiber beam elements for beams and columns, (b) multilayer shell elements for shear walls, and (c) membrane elements for floor slabs. Details of these element models have been reported in the published work [35]. Figure 12a illustrates the three-dimensional FE model. The first mode of the model is a translational mode, with a fundamental period (elastic period) T1 = 0.37 s. The widely used El-Centro ground motion record, the same as that in Section 4.1, is adopted. To simulate the collapse behavior, the PGAs of the input motions along the X- and Y-directions are scaled to 4000 cm/s2. The nonlinear time history analysis results show that the maximum vertical displacement of the structural component reaches 1 m at 1.76 s, and the corresponding maximum lateral displacement is 0.97 m. The corresponding deformed shape of the structure is shown in Figure 12b. The colored contours in Figure 12 represent the longitudinal reinforcement yielding in the elements. The amount of shear walls on the third story is much lower than on the first and second stories, which results in a sudden change of stiffness. Therefore, the third story is the weak story of the building. Appl. Sci. 2020, 10, 4408 15 of 19 Appl. Sci. 2020, 10, x FOR PEER REVIEW 17 of 20 Appl. Sci. 2020, 10, x FOR PEER REVIEW 17 of 20

(a) (a)

Soft Story Plastic Soft Story Plastic Elastic

(b) Elastic

Figure 12. FE model and simulation results of the(b) library building: (a) FE model of the library building; (b) simulation results (t = 1.76 s). FigureFigure 12. 12.FE FE model model and and simulation simulation results results of the of library the library building: building: (a) FE model (a) FE of model the library of the building; library (bbuilding;) simulationThe geometric (b) resultssimulation and (t = material 1.76results s). ( tinformation = 1.76 s). of the FE model, and its displacement and velocity results, are mapped into Blender using the mapping methods discussed in Section 3. After Thepre-processingThe geometric geometric with and and materialBCB, material a collapse information information simulation of the ofis FE performedthe model, FE model, and using its and displacementthe Bulletits displacement physics and engine. velocity and The results,velocity areresults, mappedcollapse are intoprocess mapped Blender of the into library using Blender thebuilding mapping using is then the methods determined, mapping discussed as methods shown in in Section Figurediscussed 313.. After in pre-processingSection 3. After withpre-processing BCB,The a collapse damage with simulationfirst BCB, occurs a collapse at is the performed soft simulation story using of the is thestructure, performed Bullet as physics shown using engine.in the Figure Bullet The 12b. physics collapse The structural engine. process The of deformation then increases substantially. Excessive deformation finally causes the collapse of the thecollapse library process building of is the then library determined, building as is shownthen determined, in Figure 13 as. shown in Figure 13. entire structure. The distribution of ruins is shown in Figure 14. The damage first occurs at the soft story of the structure, as shown in Figure 12b. The structural deformation then increases substantially. Excessive deformation finally causes the collapse of the entire structure. The distribution of ruins is shown in Figure 14.

(a) (b)

(e) (f)

Figure 13. Cont.

(e) (f) Appl. Sci. 2020, 10, 4408 16 of 19

Appl. Sci. 2020, 10, x FOR PEER REVIEW 18 of 20

(g) (h)

FigureFigure 13. Collapse 13. Collapse process process of theof the library library building: building:( a(a))t t == 2.01 s; s; ( (bb) )t t== 3.013.01 s; s;(c) ( ct )= t3.64= 3.64 s; (d s;) t ( d= )4.26t = 4.26 s; (e) s;t = (e5.51) t = 5.51 s; (f s;) t(f=) t6.97 = 6.97 s; s; (g ()gt) =t =14.26 14.26 s;s; (h) tt == 18.4318.43 s. s. (g) (h)

TheFigure damage 13. Collapse ﬁrst occurs process at the of the soft library story building of the structure,: (a) t = 2.01 as s; shown (b) t = 3.01 in Figure s; (c) t 12= 3.64b. The s; (d structural) t = 4.26 deformation then increases substantially. Excessive deformation ﬁnally causes the collapse of the entire s; (e) t = 5.51 s; (f) t = 6.97 s; (g) t = 14.26 s; (h) t = 18.43 s. structure. The distribution of ruins is shown in Figure 14.

Figure 14. Distribution of the library building ruins.

5. Conclusions In this work, a hybrid framework for simulating building collapse and ruin scenarios using an

FE method and physics engine is proposed, and the corresponding program FEM2RBD is developed. Shaking table tests of a three-story reinforced concrete frame and a real-world complex FigureFigure 14. 14.Distribution Distribution of theof the library library building building ruins. ruins. library building are performed to simulate the ruin scenarios. The following conclusions can be 5. Conclusionsdrawn: 5. Conclusions In• thisIn work,the proposed a hybrid hybrid framework framework, for simulatingthe FE method building simulates collapse structural and behaviors ruin scenarios during using the an In this work, a hybrid framework for simulating building collapse and ruin scenarios using an FE methodsmall-deformation and physics engine stage, is proposed,and the physics and theengine corresponding simulates structural program behaviors FEM2RBD during is developed. the FE method and physics engine is proposed, and the corresponding program FEM2RBD is Shaking tablelarge-deformation tests of a three-story stage. The reinforced proposed concrete framewor framek efficiently and a real-world combines the complex advantages library of buildingthe developed. Shaking table tests of a three-story reinforced concrete frame and a real-world complex are performedFE method to simulate and the thephysics ruin engine. scenarios. The following conclusions can be drawn: library building are performed to simulate the ruin scenarios. The following conclusions can be • Using a shaking table test of a three-story reinforced concrete frame, the proposed hybrid drawn:In the proposed hybrid framework, the FE method simulates structural behaviors during the • simulation method is demonstrated to be more accurate than an approach based on the physics small-deformation stage, and the physics engine simulates structural behaviors during the • Inengine the proposed alone. The hybrid case study framework, of a real-world the FE co mplexmeth odlibrary simulates building structural shows the behaviorshigh-fidelity during of the large-deformation stage. The proposed framework efficiently combines the advantages of the FE small-deformationthe collapse simulation. stage, and the physics engine simulates structural behaviors during the method and the physics engine. • large-deformationThe collision of structural stage. The components proposed and framewor the distributionk efficiently of ruins combines after thethe collapseadvantages are of the Using a shaking table test of a three-story reinforced concrete frame, the proposed hybrid • FEconsidered method and in the the proposed physics hybrid engine. method. The proposed framework has great significance for simulationsimulating method building is demonstrated collapse and ruin to bescenarios more accuratefor post-earthquake than an approach rescue training. based on the physics • engineUsing alone. a shaking The case table study test of of a real-worlda three-story complex reinforced library concrete building frame, shows the high-fidelityproposed hybrid of Note that the proposed framework is also applicable to other FE software and physics engines, thesimulation collapse simulation. method is demonstrated to be more accurate than an approach based on the physics besides the MSC.Marc software and the Bullet physics engine that were used in this work. Theengine collision alone. of structural The case components study of a real-world and the distribution complex library of ruins building after the shows collapse the are high-fidelity considered of • inAuthor thethe proposed collapseContributions: simulation. hybrid Conceptualization, method. The X.L. proposed and Z.Z.; framework data curation, has X.L. great and Z.Z.; significance formal analysis, for simulating Z.Z.; funding acquisition, X.L.; investigation, Y.T. and Z.Z.; methodology, Y.T. and Z.Z.; project administration, X.L.; • buildingThe collision collapse of and structural ruin scenarios components for post-earthquake and the distribution rescue training. of ruins after the collapse are Noteconsidered that the proposedin the proposed framework hybrid is alsomethod. applicable The proposed to other framework FE software has and great physics significance engines, for besidessimulating the MSC.Marc building software collapse and theand Bullet ruin scenarios physics engine for post-earthquake that were used rescue in this training. work. Note that the proposed framework is also applicable to other FE software and physics engines, besides the MSC.Marc software and the Bullet physics engine that were used in this work.

Author Contributions: Conceptualization, X.L. and Z.Z.; data curation, X.L. and Z.Z.; formal analysis, Z.Z.; funding acquisition, X.L.; investigation, Y.T. and Z.Z.; methodology, Y.T. and Z.Z.; project administration, X.L.; resources, X.L.; software, Z.Z.; supervision, X.L.; validation, Z.Z.; visualization, Z.Z.; writing—original draft, Z.Z.; writing—review and editing, X.L., Y.T., Z.Y. and Z.Z. All authors have read and agreed to the published version of the manuscript. Funding: This research was funded by BEIJING MUNICIPAL NATURAL SCIENCE FOUNDATION (grant number 8182025) Acknowledgments: The authors would like to thank Xianglin Gu and Qinghua Huang for providing the copyright permission. Conﬂicts of Interest: The authors declare no conﬂict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.

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