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The capacity that computer can solve more complex design problem is gradually increased. Progressive collapse simulation of masonry arch bridge needs a breakthrough in the current development limitations and then becomes more accurate and integrated. This paper proposes a theoretic framework and finite element implementation on progressive collapse simulation of masonry arch bridge. It is intended to develop a new large deformation element in OpenSees, which can be used for analyzing the collapse process of masonry arch bridge. A mathematical method for large deformation element is put forward by large deformation element. The feature model for bridge structure allows families of bridge components to be specified using constraints on geometry and topology. Geometric constraints are established in bridge components by feature dependence graph in the feature model for bridge. A bridge collapse simulation software system was developed according to such combined technologies. Results from our implementation show that the method can help to simulate the progressive collapse process of masonry arch bridge.

The masonry arch bridges, a conventional bridge type, are widely used throughout the world because of their low cost and beautiful appearance. However, there were a lot of arch bridges collapsed due to deficiencies in design, detailing, construction, maintenance, use of materials, and inadequate consideration of external events. The first four deficiencies represent integral roles in the building of a bridge [

A masonry arch bridge Dixituojiang Bridge under construction over the Dixituojiang River in Hunan, China, suddenly collapsed on the 13st of August 2007 and there were many victims in this disaster [

American Society of Civil Engineer (ASCE) defines progressive collapse as “the spread of an initial local failure from element to element, eventually resulting in the collapse of an entire structure or a disproportionately large part of it” [

Lau and Wibowo [

In China, there are many aging masonry arch bridges as well and they need prompt inspection, reinforcement, and maintenance. Many studies focused on the progressive collapse and clarified that local deficiencies may trigger collapse [

However, there are few researches on how to simulate the progressive collapse process of the masonry arch bridges. There are also few researches about how to use large deformation elements to simulate geometric nonlinearity properties during bridge collapse. In this research, it is intended to develop a new large deformation element, which can be used for analyzing the collapse process of masonry arch bridge. A mathematical method for large deformation element was put forward by large deformation element. Meanwhile, computer-graphics technology, feature modeling, and finite element method were combined to develop Bridge Collapse Software System (BCSS) for the progressive collapse process simulation.

Object in Cartesian Coordinates will continuously change its configuration under an external force, as shown in Figure

Geometrical nonlinear configuration.

At time

At time

Regard the change of object configuration as a mathematic change from

To obtain the relation between strain and displacement, the displacement field

We then have

In the large deformation problem, it is by cutting off a microelement from the deformed object that the equilibrium equation and its equivalent virtual work principle to be established. Accordingly we should cut off an element from the deformed object to define the stress tensor, that is, Euler stress tensor

Geometrical nonlinear strains.

If the Green strain tensor is defined by coordinates before deformation, then its corresponding stress tensor before deformation is required to be defined. Stress on the plane

In order to maintain the consistency of mathematics, there are usually two kinds of provisions for expressing the relationship between

Geometrical nonlinear stresses.

As for small deformation and linear elasticity problem under isothermal and adiabatic conditions, the relationship between stress and strain can be described by the following three kinds of equivalent methods:

If the materials are plastic, then

The coordinates and displacement of each point on the object at time 0:

The coordinates and displacement of each point on the object at time

The coordinates and displacement of each point on the object at time

The virtual displacement principle corresponding to the equilibrium condition of the configuration at time

For total Lagrange Formulation, we have

Disperse the solution domain by using isoparametric element; then the coordinates

Considering the computational cost, bbarBrick Element in OpenSees was applied to simulate the progressive collapse of bridge based on reduced integration. bbarBrick does not work because it cannot be applied under large deflection or large rotation, such as “snap through.” The process of bridge progressive collapse is the geometrically nonlinear problems with large rotation. On Figures

Collapse simulations by small strain elements in BCSS.

A snap through element in abutment

A snap through element in deck

By using the mathematical methods mentioned in Section

Collapse simulations by large strain elements in BCSS.

Initial failure

No snap through element during collapsing

The feature modeling provides a rapid method for parameterization of bridge models, through which a parametric model for bridges can be set up; and influencing factors analysis of bridge collapse can be carried out easily. The Object Oriented Program (OOP) was used to represent bridge elements.

An arch bridge can be regarded as a collection of various types of components, and every component can be stretched by one or more bridge sections. A section can be defined by changing the characteristic parameters which are familiar to engineers, such as the roof thickness of box-type section, web thickness, and floor thickness and flange length. Engineers can input few relevant parameters to complete fast the modeling of typical section, as shown in Figure

A single-box dual-room section using feature parameter methods.

There are many component section types in actual project. Each section has its unique mapping relationship from feature parameters to three-dimensional coordinate. Moreover, each section class has public functions or the methods, such as area calculation, reasonable judgment, section translation, section rotation, and eccentric conversion. And so a base class (parent class) supporting public functions or methods are established. Each section class (child class) of different types may call the methods and functions of the base class conveniently.

A section is encircled by several directed edges, according to which its binary space-partitioning (BSP) tree can be constructed. The commonly used method to construct a BSP tree for a section is to create a series of nodes, making each node represent a partition line that contains an edge of the section. Other polygon edges are separated by the partition line in the position of the nodes. The section geometric graphics definition is carried out by the Boolean operation on the section. By using difference operation, one of the Boolean operations, a section with hole can be defined conveniently, which provides a convenient and visual representation approach for the box section, a wildly used section in bridge engineering.

Figure

Definition of hollow sections with Boolean operation.

Let

The BSP tree of this hollow section.

As is shown in Figure

Examples of box section by “1 box and 3 rooms” and box section by “2 box and 1 room” in graphic platform are shown in Figure

The display examples in BCSS.

“1-box and 3-room” girder section

“2-box and 1-room” girder section

According to the modeling process, the cross wall will be built after the formation of the main arch ring. Since the cross wall is attached to the main arch, the bottom surface of each cross wall and the top surface of the main arch ring must be coplanar. And it is based on these surfaces that every 3D model of bridges including the cross wall can be formed. Make projections perpendicular to the horizontal plane for these surfaces to a specified height to the projection plane and then establish the cross wall model through connecting the top surfaces and the projection planes. To make projection of an objective plane onto assigned plane is essential to get the projection of the vertices of the objective plane onto the assigned plane. That is to say, when you get all of the projection of the vertices, you get the projection of the objective plane.

In order to obtain the projection of

The projection of point

In the picture,

Let the length of

So we have

After getting the projection of the objective plane onto assigned plane, the projection plane is formed, and then its 3D Solid Model can be established on the basis of the two planes. As shown in Figure

The 3D model established by projection in BCSS.

The 3D models by projection in BCSS.

Box girder model

Arch model

Arch bridges are a collection of various types of components, and the process of establishing the arch bridge model is also the organization of these components. There are three basic organizations that have been utilized in the design of organization for bridge components: master-slave; separate executive for each component; and symmetric or anonymous treatment of all components. For most bridge components, the first organization is availably operated in the master-slave mode. This type is certainly the easiest to implement and may often be produced by making relatively simple extensions to an organization that includes full multiprogramming capabilities.

Therefore we need to establish the feature dependence graph between master and slave which includes following two aspects.

Constraints on geometry of all components.

The organization of node coupling.

For so many arch bridge components, slave components are attached to one or more master components by position or geometry parameters. The master-slave component diagram can be established as in Figure

The master-slave component diagram.

In Figure

Bridge collapse simulation software system (BCSS) is developed by C# and OpenGL. C# is a massively popular programming language used by many thousands of software developers all over the world. The OpenGL graphics system is a software interface to graphics hardware. (The GL stands for Graphics Library.) It allows programmers to create interactive programs that produce color images of moving three-dimensional objects. With OpenGL, programmers can control computer-graphics technology to produce realistic pictures or ones that depart from reality in imaginative ways.

BSCC includes feature modeling module and bridge collapse visualization module, as shown in Figure

Steps on collapse simulation using BCSS.

At 4:00 p.m. on June 13, 2007, the Dixituojiang Bridge under construction collapsed suddenly in Fenghuang County of Hunan Province, China, and 59 people were killed. The full length of the Dixituojiang Bridge is 328.5 m. And it has four holes, each of which is 65 m in span length. The lane width of the bridge is 12 m. The main arch is equal in the cross section, and the type of curve is catenary. The rise-span ratio of main arch is 1/5, the arch-axis coefficient is 2.514, and the thickness is 1.35 m. There are seven Associate Arches on the top of each main arch. The span of Associate Arch is 5 m, and the rise-span ratio is 1/5. The thickness of the Associate Arch is 0.4 m. It is also the equal section component, and the type of curve is arc curve. The thickness of the cross wall is 1.1 m. The elevation of Dixituojiang Bridge is shown in Figure

Elevation of Dixituojiang Bridge.

The main arch is composed of No. 20 pebble and No. 60 rubble. The Associate Arch is composed by No. 12.5 mortar and No. 30 dressed stone. The No. 12.5 mortar and No. 30 stone are applied to bricklaying of the cross wall.

According to the characteristics of masonry arch bridge components, this feature modeling is divided into three modules: the module of main arch, the module of cross wall and spandrel arch, and the module of abutment and foundation. Take Dixituojiang Bridge as an example, the design parameters of each module are as shown in Table

The material parameters of Dixituojiang Bridge.

Components | Bulk modulus |
Shear modulus |
Density^{3}) |
---|---|---|---|

Main arch | 11078 | 5539 | 2400 |

Cross wall | 6083 | 3042 | 2400 |

Spandrel arch | 6083 | 3042 | 2400 |

Bridge pier | 9380 | 4690 | 2400 |

Floor system | 6083 | 3042 | 2400 |

With the material parameters above, a 3D FEM model of Dixituojiang Bridge is established by feature modeling method, as shown in Figure

The 3D solid simulation model of Dixituojiang Bridge in BSCC.

By the cause analysis of the accident: the bridge piers adopting spread foundation stand out in the slightly weathered rock. When providing vertical stress, the foundation can only partially constrain the displacement in the horizontal direction of bridge pier. When the horizontal thrust exceeds the allowable value, the pier may have lateral displacement or even overturn, which is different from fixed constraint. Therefore, in order to ensure the accuracy of the analysis, the zero-length contact element was set between nodes in foundation and in the bottom of the pier.

Due to the geometric nonlinear element analysis, and setting zero-length contact element to simulate the boundary of the pier and the foundation, and adopting the birth and death element technology in the process of analysis, all of which lead to the analysis having extremely strong nonlinearity.

The meshes of the original model are transversely coarsened, the final model is divided into 30,356 8-node hexahedral elements, and set 24 zero-length contact elements on the right side of the contact surface between the pier and ground, totally has 51086 nodes. The model sets up 950 calculation steps, and each step is 0.008 s. Before the first arch collapsed, 0.15 cm displacement is applied to the left abutment for each iteration step.

Considering the low quality of No. 1 main arch in construction, the ultimate tensile strain and ultimate compressive strain of No. 1 main arch should be reduced properly, where the former decreases 30% and the latter decreases about 20%. Meanwhile, set the rule that elements fail when

Collapse simulations for Dixituojiang Bridge in BCSS.

Initial failure is occurred on

The first main arch collapses on

The second main arch collapses on

The third main arch collapses on

The forth main arch collapses on

The whole bridge collapses on

The simulation of collapse process suggests that the continuous arch effect caused by the failure of No. 1 main arch leads to the collapse of No. 2 and No. 3 main arches in succession, which is consistent with the actual collapse process. From Figures

About

This paper proposes a theoretic framework and finite element implementation on progressive collapse simulation of masonry arch bridge. A new large deformation element in OpenSees was developed, which can be used for analyzing the collapse process of masonry arch bridge. A mathematical method for large deformation element is put forward based on large deformation element. The feature model which allows families of bridge components to be specified using constraints on geometry and topology facilitate the modeling of complicated bridge. Geometric constraints are established in bridge components by feature dependence graph in the feature model for bridge. Results from our implementation show that the method can help to simulate the progressive collapse process of masonry arch bridge.

The authors declared that they have no conflict of interests to this work.

This research is financially supported by the Zhejiang Provincial Natural Science Foundation of China (Grant no. LY13E080014). This support is gratefully acknowledged. Any opinions, findings, and conclusions or recommendations expressed in this material are those of authors.