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

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## Abstract

**:**

## 1. Introduction

- Shaking table testShaking table tests have been used to study the ruin distribution in many studies [16,17,18,19,20]. For example, Huang [20] analyzed the collapse process and distribution of ruins using a ¼-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 methodsThe 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 [26,27,28,29,30,31]. 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].

## 2. Hybrid Framework Based on FE Method and Physics Engine

**bold italics**, and the detailed functions of the program are in italics. The framework consists of four modules:

- Module 1: FE model establishment and conversionThe FE model of the target structure is established using the FE software MSC.Marc [33]. 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, 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].

- Module 2: Small deformation analysis and initial collapse moment determinationA 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 analysisBased on the mapping method of the FE results proposed in Section 3, 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 visualizationAfter 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

- 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

#### 3.1.1. Solid Model Establishment Method in Blender

- Step 1: Shape and position transformation (including translation and rotation).The shape and position transformations are performed on the cube mesh provided in Blender [43], to create geometric objects with the same shapes and positions as the elements in the FE model. The rotation method shown in Figure 3 is used to model inclined elements in FE models. Specifically, the rotation angle is obtained by first determining the endpoint coordinates of the longest edge of an FE element (line segment l
_{1}) and the projection of l_{1}in the XOY plane in the global coordinate system (line segment l_{2}). The angle α between l_{2}and the X axis, and the angle β between l_{2}and l_{1}, are then calculated using these endpoint coordinates. Subsequently, the shape-transformed object is rotated by angle α around the Z axis in the global coordinate system. The object is then rotated by angle β around the Y’ axis in the rotated coordinate system. The rotated coordinate system is the local coordinate system of the cube after the first rotation, where the X’O’ axis is parallel to l_{2}.

- Step 2: Add rigid bodies to the cubes.Convert the cubes into rigid bodies after the shape and position transformations to obtain the rigid body model for the Bullet physics engine simulation.

#### 3.1.2. Material Mapping Method in the BCB

#### 3.2. Determination of the Initial Moment of Collapse

_{c}, and the physics engine simulation is conducted after T

_{c}. Therefore, the determination of T

_{c}is essential. Several collapse criteria have been proposed [44,45,46,47,48,49]. There are various structural collapse modes, including lateral collapse, vertical collapse, and so on. No matter which collapse mode is, the vertical displacements of structural members will exceed a specific value when the structures collapse [47,48]. Therefore, the definition of collapse depends on the moment when “the deformation of the structure is insufficient to maintain a safe use space” [47]. Besides, Zhao et al. [46] studied four collapse criteria, including the following: Criterion 1) According to Chinese seismic code [44], the earthquake-induced inter-story drift ratio of reinforced concrete frames should not be greater than 1/50; Criterion 2) Federal Emergency Management Agency (FEMA) 356 [45] recommended that the inter-story drift ratio limit for the collapse-prevention performance level of concrete structures be 4%; Criterion 3) According to Villaverde [49], when the tangent slope of the incremental dynamic analysis (IDA) curve is lower than 20% of the slope of the initial elastic stage, the structure will collapse; Criterion 4) “The vertical collapse displacement of the main structural members exceed[ing] 1 m”, is considered as Criterion 4 [46]. The work of Zhao et al. [46] shows that the conventional collapse criteria (i.e., Criteria 1–3) significantly underestimate the collapse resistance of the frames. Compared with the other three collapse criteria, Criterion 4 can better reflect the extreme nonlinear behavior of the collapse limit state. Therefore, collapse Criterion 4, proposed by Zhao et al. [46], is selected as a rule for determining T

_{c}in this work.

#### 3.3. FE Simulation Data Mapping Method

#### 3.3.1. Displacement Mapping Method

_{c}, 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.

- Step 1. Extract the FE model data. After the FE simulation, the displacement data of each vertex at each time step are extracted, and then the deformed coordinates of each vertex are calculated from the displacement data and the initial coordinates. The deformed centroid coordinates of each object at each time step can be calculated from the deformed vertex coordinates of each object.
- 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 T
_{c}. - 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.

#### 3.3.2. Velocity Mapping Method

_{c}, both displacement and velocity should be consistent between the two simulations. Consequently, velocity mapping is necessary. The velocity mapping method requires both the displacement and the kinematic type keyframes [43]. 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 $\overline{{v}_{j}}$ at frame j, as follows:

_{i}and x

_{j}represent the centroid coordinates of an object at frame i and frame j, respectively. The speed $\overline{{v}_{j}}$ 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.

_{ext}denotes external forces defined by the user; F

_{c}denotes the constraint forces determined by the position and the velocity of the object in the adjacent previous frame.

_{j}at the initial moment of collapse T

_{c}, 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 x

_{virtual}. 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 Figure 5. The blue curve is the displacement curve simulated by the FE method; t

_{i}and t

_{j}are the times corresponding to the displacement keyframe insertion points; x

_{i}and x

_{j}are the displacements corresponding to the displacement keyframe insertion points; v

_{j}is the instantaneous velocity, and x

_{virtual}is the virtual displacement vector.

_{j}and v

_{j}represent the displacement vector of the FE model and the velocity vector of the collapse keyframe j, respectively; k is the keyframe number of ${x}_{virtual}$, 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 x

_{i}with the virtual displacement vector ${x}_{virtual}$ in Equation (1).

- Step 1: Extract the FE results at time step t
_{j}, including velocity v_{j}and displacement x_{j}. - Step 2: Obtain x
_{virtual}using Equations (4) and (5), and set the displacement of the frame k to x_{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

_{1}= 0.316 s; the second mode is a translation with a fundamental period T

_{2}= 0.316 s; and the third mode is the planar torsion, with a fundamental period T

_{3}= 0.178 s.

#### 4.1.2. Comparison of Simulation Results

- 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

^{2}. The total mass of the library building is 3.17 × 10

^{7}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 Table 4. 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/s

^{2}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.

_{1}= 0.37 s.

^{2}. 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.

## 5. Conclusions

- In the proposed hybrid framework, the FE method simulates structural behaviors during the small-deformation stage, and the physics engine simulates structural behaviors during the large-deformation stage. The proposed framework efficiently combines the advantages of the FE method and the physics engine.
- Using a shaking table test of a three-story reinforced concrete frame, the proposed hybrid simulation method is demonstrated to be more accurate than an approach based on the physics engine alone. The case study of a real-world complex library building shows the high-fidelity of the collapse simulation.
- The collision of structural components and the distribution of ruins after the collapse are considered in the proposed hybrid method. The proposed framework has great significance for simulating building collapse and ruin scenarios for post-earthquake rescue training.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 7.**Dimensions and reinforcements of the model structure (units: mm). [20] (Reproduced with permission from Huang, Q., from Study on spatial collapse responses of reinforced concrete frame structures under earthquake; published by Tongji Univerity, 2006.).

**Figure 10.**Collapse process of the reinforced concrete frame subjected to the El-Centro ground motion record: (

**a**) t = 0.00 s; (

**b**) t = 3.88 s; (

**c**) t = 4.16 s.

**Figure 11.**Distribution of ruins: (

**a**) test; [20] (Reproduced with permission from Huang, Q., from Study on spatial collapse responses of reinforced concrete frame structures under earthquake; published by Tongji Univerity, 2006.) (

**b**) the proposed FE and BCB hybrid method; (

**c**) BCB.

**Figure 12.**FE model and simulation results of the library building: (

**a**) FE model of the library building; (

**b**) simulation results (t = 1.76 s).

**Figure 13.**Collapse process of the library building: (

**a**) t = 2.01 s; (

**b**) t = 3.01 s; (

**c**) t = 3.64 s; (

**d**) t = 4.26 s; (

**e**) t = 5.51 s; (

**f**) t = 6.97 s; (

**g**) t = 14.26 s; (

**h**) t = 18.43 s.

Floor No. | E_{c} (GPa) | f_{c} (MPa) |
---|---|---|

1 | 25 | 22.5 |

2 | 27 | 23.9 |

3 | 23 | 15.6 |

Table | E_{s} (GPa) | f_{s} (MPa) |
---|---|---|

16# | 200 | 486.4 |

10# | 200 | 457.7 |

Load Case | Peak Ground Acceleration (PGA) (Unit: g) | ||
---|---|---|---|

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 |

**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 |

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## Share and Cite

**MDPI and ACS Style**

Zheng, Z.; Tian, Y.; Yang, Z.; Lu, X. Hybrid Framework for Simulating Building Collapse and Ruin Scenarios Using Finite Element Method and Physics Engine. *Appl. Sci.* **2020**, *10*, 4408.
https://doi.org/10.3390/app10124408

**AMA Style**

Zheng Z, Tian Y, Yang Z, Lu X. Hybrid Framework for Simulating Building Collapse and Ruin Scenarios Using Finite Element Method and Physics Engine. *Applied Sciences*. 2020; 10(12):4408.
https://doi.org/10.3390/app10124408

**Chicago/Turabian Style**

Zheng, Zhe, Yuan Tian, Zhebiao Yang, and Xinzheng Lu. 2020. "Hybrid Framework for Simulating Building Collapse and Ruin Scenarios Using Finite Element Method and Physics Engine" *Applied Sciences* 10, no. 12: 4408.
https://doi.org/10.3390/app10124408