# A Study on the Mechanism of Impact between Curved Bridge Segments Using Nonsmooth Dynamics

Authors: Wenshan Li; Yong Huang; Guangming Xie
Journal: Shock and Vibration
 (2020)

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## Abstract

It has been observed in many previous earthquakes that impact often occurs between the main girders in curved bridges. An earthquake can result in deck-unseating leading to catastrophic destruction of the structure. In this paper, the nonsmooth multirigid body dynamics method and the set-valued formulation were used to model and analyze the mechanism of impact between the curved bridge segments. The analysis demonstrated that these impacts are the major cause of segment rotation. The main contribution of this paper is to use Newton’s impact law and Coulomb’s friction law to describe the interaction between the curved bridge segments in the form of a set-valued function and to express impacts with friction as a linear complementary problem. For frictionless and frictional contact, the paper considers the single-point and multipoint impacts using the linear complementary formula to detect the unique actual slip-stick conditions of these states. A variety of criteria for distinguishing each case are presented and the results provide the kinetic characteristics of each contact case. The analysis has shown that the impact between the segments of a curved bridge and the tendency of the segments to rotate (and thus detach) are related to the overall geometry, the coefficient of restitution, the coefficient of friction, and the preimpact conditions in the plane of motion. Finally, a theoretical relationship diagram of the impact, rotation slip, and stick condition of the curved bridge segments at the contact point is given. The presented results will be useful for the seismic design of curved bridges.


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## Body

## 1. Introduction

Impact occurs when two or more bodies contact for a brief time, which is mainly determined by the material characteristics, geometry, and preimpact conditions of the bodies. Pounding between the adjacent segments, in-plane deck rotation, and deck-abutment interaction make the curved bridges very susceptible during such activities, resulting in deck-unseating, shear failure, torsional failure, and interfacial concrete failure [1–4].Impacts between curved bridge segments during strong earthquakes have been reported for many structures, for example, the I-5/SR-14 interchange bridge during the 1971 San Fernando earthquake, the I-10/I-215 interchange bridge during the 1991 San Francisco earthquake in the United States [5], Southbound Separation Overpass during the United States Northridge earthquake [6], Kobe Line and Shibukawa Interchange Bridge No. 3 during the 1995 Kobe earthquake in Japan [7], and the Shiwei Bridge and Maoluoxi Bridge during the 1999 Taiwan Chi-Chi earthquake. The Baihua Bridge [8, 9], shown in Figure 1, sustained torsional damage of the pier and deck-unseating during the 2008 Wenchuan earthquake in China. The same 2008 Wenchuan earthquake damaged curved ramp bridges of the Huilan interchange [10].Figure 1
Unseating of Baihua Bridge caused the Wenchuan earthquake.The reviewed earthquake damage cases demonstrate that it is necessary to investigate the impact between the curved bridge superstructures during strong ground motion events, and a significant number of relevant studies have already been conducted. Using the Karnopp friction model, Amjadian and Agrawal [11] investigated the influence of pounding on rigid-body motion of horizontally curved bridges during strong earthquakes. Banerjee et al. [12] studied the dynamic response sensitivity of horizontally curved bridges under seismic impact and assessed the influence of columns, bent arrangement, and gap distances on the structural response. Julian et al. [13] studied the rotations and displacements of curved bridge superstructures caused by flexural-torsional interaction under seismic loading. A two-fifths scale curved bridge model was constructed at the University of Nevada, Reno Large Scale Structures Laboratory, and tested on a shake table to observe the effects of earthquake shocks and develop a contact element for design [14].There are many impact simulation methods [4, 15], such as the restitution coefficient method, the stereo-mechanical approach, the compliance method, and the linear complementarity problem (LCP) method. The restitution coefficient method [16] uses the principle of conservation of momentum to modify the velocity after impact according to the precontact state of the colliding bodies. The restitution coefficient method has the advantages of conceptually clarity and calculation simplicity. However, the method only uses the coefficient of restitution to model the changes in structural strength and speed before and after the impact and not during the whole impact process. Thus, the method can simulate the impact process of rigid bodies, but not deformable objects. Although the stereo-mechanical approach has the advantages of conceptual clarity and simple numerical implementation, it cannot consider the compression deformations of the contact areas. The stereo-mechanical approach can be applied when the stiffness of the impacting objects is very large and the impact time is very short. However, when the stiffness of the objects is very small, they will be more prone to large deformations and the impact between them will last longer. In the compliance method, contact elements are used to simulate the contact impact. The commonly used contact elements are Kelvin elements [17, 18] and Hertz elements [19, 20]. The advantage of the compliance method is the convenient calculation and clear physical meaning and that the approach can be used in combination with finite element software. However, in the process of a simulating impact, the model will be affected by the material and structural shape, and the choice of these parameters is left to the subjective judgment of the analyst. If the contact stiffness is large, convergence becomes relatively quick, but the calculated impact force will be unrealistic. On the other hand, if the contact stiffness is small, the impact time will be longer, and the calculated impact force is relatively small. Because of its uncertainty, the damping coefficient cannot accurately simulate the energy loss of adjacent structures during impact. In 1988, Moreau, known as the father of nonsmooth mechanics, published his seminal paper [21]. Panagiotopoulos [22, 23] improved this new theory by introducing a nonconvex variational inequality. After the Delassus problem [24] was posed in 1917, researchers realized that as the number of contact points increases, and it is impossible to make a real state in a large number of possible contact states. However, the LCP method can detect the hysteresis and contact states in the nonsmooth events unambiguously and effectively; thus, it is convenient to use the LCP method to study nonsmooth dynamics phenomena. Pfeifer and Glocker [25] successfully combined multibody system dynamics with nonsmooth mechanics theory, which laid the theoretical foundation for nonsmooth multibody system dynamics. Dimitrakopoulos [26, 27] proposed a method for analyzing abutment-to-deck and deck-to-deck impact in skewed bridges. This paper studies in detail the impact on curved bridges based on their method.In this formalism, the deck is assumed to be a rigid body. Therefore, any impacts that may occur between the decks are locally convex. The problem of impacts in the curved bridge in the LCP method is transformed by a complementary form of the normal Newton impact law [25], the tangential Coulomb friction law, and the set-valued function. By solving the LCP equation, the normal and transverse stick-slip contact cases of curved bridge segments are studied for different preimpact conditions. The overall geometry influences significantly the postimpact deck responses, which again depended on the preimpact conditions. This complex phenomenon is analytically studied for a single-impact case during deck-to-deck pounding between two curved bridge segments. Based on a previous investigation [12] on the impact between the abutment and the deck of a curved bridge, this paper studies the problem of impact between bridge segments of a curved bridge. The main contributions are as follows: first, the mechanism of deck rotation caused by the impact between bridge segments of a curved bridge is explained. Next, the specific conditions to be satisfied to avoid the rotation of the deck when friction is ignored are given. Finally, the classification boundary of slip and stick after two adjacent bridge segments make contact is established from a kinematic consideration.


## 2. Research Methods

### 2.1. Linear Complementarity Problem

The LCP method can encapsulate many of the physical phenomena occurring during bridge segment impacts in a curved bridge. In this paper, by solving the LCP equation, the deck of the bridge is assumed to be a fully rigid body moving only in the horizontal plane [25]. This study further assumes that the interaction between adjacent parts is a unilateral contact [25]. This interaction is described in the form of the set-valued function using Newton’s impact law in the normal direction and Coulomb’s friction law in the transverse direction. Only two impact parameters are required to describe the contact, namely, the coefficient of restitution, εN, and the coefficient of friction, μ. The coefficient of restitution, εN, is defined as the ratio of postimpact normal relative velocity g˙N+, to preimpact normal relative velocity, g˙N−. It can vary between 0 and 1; that is, εN∈0,1. The case εN=0 corresponds to the completely inelastic impact and εN=1 to the fully elastic impact. In the transverse direction, a zero coefficient of restitution is assumed, εT=0. Newton’s impact law can be represented as g˙N+=−εNg˙N− [25].The set-valued function is an extension of the ordinary function concept. The set-valued functionF: Rn⟶Rn is a multivalued mapping between the domain of definition and the range, which maps point x∈Rn in the domain of definition to the aggregate Fx∈Rn in the value domain [25].The normal impulse,ΛNi, and the transverse impulse, ΛTi, use two set-valued mappings, the unilateral primitive function, Upr (Figure 2, right) and the sgn function (Figure 2, left), respectively:(1)−ΛNi∈UprvNi−ΛTi∈μiΛNisgnvTi.Figure 2
Set-valued friction force law.The sgn function is an extension of the normal signum function. The difference is that the function maps the parameterx = 0 to the set [−1, +1]. The image of the sgn function at x = 0 is a vertical line segment. The sgn function can be decomposed into two unilateral primitive functions [25]. The following decomposition is also illustrated in Figure 2:(2)−ΛTRi∈UprvTRi,ΛTRi=μΛNi+ΛTi,−ΛTLi∈UprvTLi,ΛTLi=μΛNi−ΛTi,g˙Ti+=vTRi−vTLi.Expressing the above system of equations is a matrix form yielding(3)ΛTL=μ¯¯ΛN−ΛT=2μ¯¯ΛN−ΛTR,where μ¯¯=diagμi. The LCP method can parametrize the system using a linear system of equations, y=Ax+b, with a known matrix A and vector b. Vectors y and x are unknown. The additional complementarity is defined by y≥0, x≥0, and yTx=0. More detailed information about the LCP, the solvability of the numerical LCP, and the associated numerical algorithms are available in [28, 29].Using Newton’s impact law and Coulomb’s friction law, a coupled set of linear complementary equations can be expressed as follows [30]:(4)vNvTRΛTL=GNN−GNTμ¯¯GNT0GTN−GTTμ¯¯GTTE2μ¯¯−E0ΛNΛTRvTL+ε¯¯N+Eg˙T−E⋅g˙T−0,(5)vNvTRΛTL≥0,ΛNΛTRvTL≥0,vNvTRΛTLTΛNΛTRvTL=0,where ε¯¯N=diagεNi, μ¯¯=μμ, and E is an identity matrix.


### 2.2. Geometric Analysis

In the kinematic analysis of bridge impact, we can represent the possible contact point relative distance,gi, and the relative speed, g˙t, as generalized coordinates qi. Since a rigid body on a plane has three degrees of freedom, the generalized coordinate vector can be expressed as qT=x1y1θ1x2y2θ2. Figure 3 illustrates the position of the left deck (deck-1) and the right deck (deck-2).Figure 3
Deck geometry of curved bridge showing the position of the center of mass and different impact parameters without in-plane deck rotation for deck-1 and deck-2.The distance from the center of mass to the center of the bridge curvature is(6)dmi=2+k26sinαi/2αiR,i=1,2,where k=W/R and i = 1, 2 indicate deck-1 and deck-2, respectively. From this formula, the moment of inertia can be calculated as follows:(7)Ii=1+14k2LiR2,i=1,2,where Li=kαiR2. Figure 4 shows that the distance vector, rD, between deck-1 and deck-2 is determined by the parameters σ1 and σ2:(8)rD=δ+x2−x1+Rsinθ1−sinθ2+σ1⋅sinθ1−σ2⋅sinθ2+dm2⋅sinθ2−α22+dm2⋅sinα22−dm1⋅sinα12+cosθ1−α12y2−y1+Rcosθ2−cosθ1+σ2⋅cosθ2−σ1⋅cosθ1−dm2⋅cosθ2−α22+dm2⋅cosα22−dm1⋅cosα12−cosθ1−α12.Figure 4
Contact between deck-1 and deck-2. (a) Case 1. (b) Case 2.
(a)(b)In this formula,σi describes the location of the contact point at the contour of the deck. The range of σi is between −0.5 W and +0.5 W. The tangential and normal vectors of the contour of deck-1 and deck-2 can now be calculated as follows:(9)t1=−sinθ1cosθ1,t2=sinθ2−cosθ2,n1=−cosθ1−sinθ1,n2=cosθ2sinθ2,where t1, t2, n1, and n2 represent the tangential and normal direction vectors of the two contours, as shown in Figure 4.


## 2.1. Linear Complementarity Problem

The LCP method can encapsulate many of the physical phenomena occurring during bridge segment impacts in a curved bridge. In this paper, by solving the LCP equation, the deck of the bridge is assumed to be a fully rigid body moving only in the horizontal plane [25]. This study further assumes that the interaction between adjacent parts is a unilateral contact [25]. This interaction is described in the form of the set-valued function using Newton’s impact law in the normal direction and Coulomb’s friction law in the transverse direction. Only two impact parameters are required to describe the contact, namely, the coefficient of restitution, εN, and the coefficient of friction, μ. The coefficient of restitution, εN, is defined as the ratio of postimpact normal relative velocity g˙N+, to preimpact normal relative velocity, g˙N−. It can vary between 0 and 1; that is, εN∈0,1. The case εN=0 corresponds to the completely inelastic impact and εN=1 to the fully elastic impact. In the transverse direction, a zero coefficient of restitution is assumed, εT=0. Newton’s impact law can be represented as g˙N+=−εNg˙N− [25].The set-valued function is an extension of the ordinary function concept. The set-valued functionF: Rn⟶Rn is a multivalued mapping between the domain of definition and the range, which maps point x∈Rn in the domain of definition to the aggregate Fx∈Rn in the value domain [25].The normal impulse,ΛNi, and the transverse impulse, ΛTi, use two set-valued mappings, the unilateral primitive function, Upr (Figure 2, right) and the sgn function (Figure 2, left), respectively:(1)−ΛNi∈UprvNi−ΛTi∈μiΛNisgnvTi.Figure 2
Set-valued friction force law.The sgn function is an extension of the normal signum function. The difference is that the function maps the parameterx = 0 to the set [−1, +1]. The image of the sgn function at x = 0 is a vertical line segment. The sgn function can be decomposed into two unilateral primitive functions [25]. The following decomposition is also illustrated in Figure 2:(2)−ΛTRi∈UprvTRi,ΛTRi=μΛNi+ΛTi,−ΛTLi∈UprvTLi,ΛTLi=μΛNi−ΛTi,g˙Ti+=vTRi−vTLi.Expressing the above system of equations is a matrix form yielding(3)ΛTL=μ¯¯ΛN−ΛT=2μ¯¯ΛN−ΛTR,where μ¯¯=diagμi. The LCP method can parametrize the system using a linear system of equations, y=Ax+b, with a known matrix A and vector b. Vectors y and x are unknown. The additional complementarity is defined by y≥0, x≥0, and yTx=0. More detailed information about the LCP, the solvability of the numerical LCP, and the associated numerical algorithms are available in [28, 29].Using Newton’s impact law and Coulomb’s friction law, a coupled set of linear complementary equations can be expressed as follows [30]:(4)vNvTRΛTL=GNN−GNTμ¯¯GNT0GTN−GTTμ¯¯GTTE2μ¯¯−E0ΛNΛTRvTL+ε¯¯N+Eg˙T−E⋅g˙T−0,(5)vNvTRΛTL≥0,ΛNΛTRvTL≥0,vNvTRΛTLTΛNΛTRvTL=0,where ε¯¯N=diagεNi, μ¯¯=μμ, and E is an identity matrix.


## 2.2. Geometric Analysis

In the kinematic analysis of bridge impact, we can represent the possible contact point relative distance,gi, and the relative speed, g˙t, as generalized coordinates qi. Since a rigid body on a plane has three degrees of freedom, the generalized coordinate vector can be expressed as qT=x1y1θ1x2y2θ2. Figure 3 illustrates the position of the left deck (deck-1) and the right deck (deck-2).Figure 3
Deck geometry of curved bridge showing the position of the center of mass and different impact parameters without in-plane deck rotation for deck-1 and deck-2.The distance from the center of mass to the center of the bridge curvature is(6)dmi=2+k26sinαi/2αiR,i=1,2,where k=W/R and i = 1, 2 indicate deck-1 and deck-2, respectively. From this formula, the moment of inertia can be calculated as follows:(7)Ii=1+14k2LiR2,i=1,2,where Li=kαiR2. Figure 4 shows that the distance vector, rD, between deck-1 and deck-2 is determined by the parameters σ1 and σ2:(8)rD=δ+x2−x1+Rsinθ1−sinθ2+σ1⋅sinθ1−σ2⋅sinθ2+dm2⋅sinθ2−α22+dm2⋅sinα22−dm1⋅sinα12+cosθ1−α12y2−y1+Rcosθ2−cosθ1+σ2⋅cosθ2−σ1⋅cosθ1−dm2⋅cosθ2−α22+dm2⋅cosα22−dm1⋅cosα12−cosθ1−α12.Figure 4
Contact between deck-1 and deck-2. (a) Case 1. (b) Case 2.
(a)(b)In this formula,σi describes the location of the contact point at the contour of the deck. The range of σi is between −0.5 W and +0.5 W. The tangential and normal vectors of the contour of deck-1 and deck-2 can now be calculated as follows:(9)t1=−sinθ1cosθ1,t2=sinθ2−cosθ2,n1=−cosθ1−sinθ1,n2=cosθ2sinθ2,where t1, t2, n1, and n2 represent the tangential and normal direction vectors of the two contours, as shown in Figure 4.


## 3. Kinematic Analysis

This section discusses the kinematic problem between two adjacent bridge segments of a curved bridge. In the following analysis, the contact between deck-1 and deck-2 is divided into multiple-point contact (the impact takes place along a contour) and single-point contact (the impact takes place in a corner). We can then analyze the single-point contact in two specific cases.Case 1.
A point on deck-2 touches the contour of deck-1.
When this happens,rDTt1=0 is satisfied, as shown in Figure 4(a). In this case, we can calculate the normal relative displacement, gN, as follows:(10)gN=−rDTn1=δ+x2−x1⋅cosθ1+y2−y1⋅sinθ1+R+σ2⋅sinθ1−θ2+dm2⋅sinθ1+α22+sinθ2−θ1−α22−dm1⋅sinθ1+α12−sinα12.
Usingg˙N=WNTq, calculate the normal constrained Jacobian matrix, WNT, as follows:(11)WNT=−cosθ1−sinθ1r˜N1cosθ1sinθ1r˜N2,(12)r˜N1=x1−x2−δsinθ1+y2−y1cosθ1+R+σ2cosθ1−θ2+dm2,cosθ1+a22−cosθ2−θ1−a22−dm1⋅cosθ1+a12,r˜N2=−R+σ2cosθ1−θ2+dm2⋅cosθ2−θ1−α22.Case 2.
A point on deck-1 touches the contour of deck-2.
This happens whenrDTt2=0 is satisfied and is illustrated in Figure 4(b). One can calculate the normal relative displacement, gN, as follows:(13)gN=rDTn2=δ+x2−x1⋅cosθ2+y2−y1⋅sinθ2+R+σ1⋅sinθ1−θ2+dm2⋅sinθ2+α22−sinα22−dm1⋅sinθ2+α12−sinθ1−θ2−α12.
Usingg˙N=WNTq, calculate the normal constrained Jacobian matrix, WNT, as follows:(14)WNT=−cosθ2−sinθ2r˜N1cosθ2sinθ2r˜N2,(15)r˜N1=R+σ1cosθ1−θ2−dm1⋅cosθ1−θ2−α12,r˜N2=x1−x2−δsinθ2+y2−y1cosθ2−R+σ1cosθ1−θ2+dm2⋅cosθ2+α22−dm1⋅cosθ2+α12+cosθ1−θ2−α12.


### 3.1. Frictionless Impact Analysis

Since our model is composed of rigid bodies, we can convert the full-edge impact (Figure5) at the contact of the two sides into a two-point impact. When a two-point impact occurs, we know that n1Tt2=0 or n2Tt1=0. Further, set sinθ1−θ2=0, without loss of generality. For this model, it is assumed that θ1−=θ2−=0.Figure 5
Full-edge frictionless collision between deck-1 and deck-2.MatrixWN can be simplified using equations (11) and (14):(16)WNT=WN1TWN2T=−10r110r3−10r210r4,where r1, r2, r3, and r4 are distances shown in Figure 3, which can be calculated as follows:(17)r1=R+W2−dm1⋅cosα12,r2=R−W2−dm1⋅cosα12,r3=R+W2−dm2⋅cosα22,r4=R−W2−dm2⋅cosα22.We then calculateGNN and GNN−1 as follows:(18)GNN=WNTM−1WN=1m1+1m2+r12I1+r32I21m1+1m2+r1r2I1+r3r4I21m1+1m2+r1r2I1+r3r4I21m1+1m2+r22I1+r42I2,GNN−1=1GNN=1m1+1m2+r22I1+r42I2−1m1+1m2+r1r2I1+r3r4I2−1m1+1m2+r1r2I1+r3r4I21m1+1m2+r12I1+r32I2.According to Newton’s impact law [25], the normal impulse, ΛN, is defined as follows:(19)ΛN=ΛN1ΛN2=−GNN−1I2+ε¯¯Ng˙N−.In this equation,I2 is an identity matrix and the Newton impact coefficients of the two impact points are the same, εN1=εN2=εN. We then decompose the above matrix as follows:(20)ΛN1=−g˙N−GNN⋅1+εNr2r2−r1I1+r4r4−r3I2,ΛN2=−g˙N−GNN⋅1+εNr1r1−r2I1+r3r3−r4I2.Since1+εN>0,GNN>0, and the contact occurs if and only if g˙N−<0, one obtains the following:(21)ΛN1>0⟺r2r2−r1I1+r4r4−r3I2>0,(22)ΛN2>0⟺r1r1−r2I1+r3r3−r4I2>0.In order to prevent the main girders from rotating during full-edge impact, equations (21) and (22) must be satisfied at the same time, so that both ΛN1 and ΛN2 are positive. After examining all the cases of r1, r2, r3, and r4 in equation (17), it is concluded that only the following two cases can satisfy that both ΛN1 and ΛN2 are positive. In the following analysis, two variables, si and sj, which are only related to geometric parameters, are introduced for the convenience of analyzing the following two cases:Case 1.
−W/2<si<W/2, sj<−W/2(23)2si⋅αj⋅W/2+siW⋅αi⋅W/2+sj<−1,2si⋅αj⋅W/2−siW⋅αi⋅W/2−sj<1.Case 2.
−W/2<si<W/2,−W/2<sj<W/2(24)si⋅αj⋅W/2+sisj⋅αi⋅W/2+sj<−1,si⋅αj⋅W/2−sisj⋅αi⋅W/2−sj>−1.
For these cases,si=dmi⋅cosαi/2−R and sj=dmj⋅cosαj/2−R, where si and sj represent deck-1 and deck-2, respectively. According to equations (21) and (22), it can be concluded that the different contact conditions are determined by the geometric characteristics and inertial conditions of the bridge segments. This conclusion can be used to guide engineering practice.


## 3.1. Frictionless Impact Analysis

Since our model is composed of rigid bodies, we can convert the full-edge impact (Figure5) at the contact of the two sides into a two-point impact. When a two-point impact occurs, we know that n1Tt2=0 or n2Tt1=0. Further, set sinθ1−θ2=0, without loss of generality. For this model, it is assumed that θ1−=θ2−=0.Figure 5
Full-edge frictionless collision between deck-1 and deck-2.MatrixWN can be simplified using equations (11) and (14):(16)WNT=WN1TWN2T=−10r110r3−10r210r4,where r1, r2, r3, and r4 are distances shown in Figure 3, which can be calculated as follows:(17)r1=R+W2−dm1⋅cosα12,r2=R−W2−dm1⋅cosα12,r3=R+W2−dm2⋅cosα22,r4=R−W2−dm2⋅cosα22.We then calculateGNN and GNN−1 as follows:(18)GNN=WNTM−1WN=1m1+1m2+r12I1+r32I21m1+1m2+r1r2I1+r3r4I21m1+1m2+r1r2I1+r3r4I21m1+1m2+r22I1+r42I2,GNN−1=1GNN=1m1+1m2+r22I1+r42I2−1m1+1m2+r1r2I1+r3r4I2−1m1+1m2+r1r2I1+r3r4I21m1+1m2+r12I1+r32I2.According to Newton’s impact law [25], the normal impulse, ΛN, is defined as follows:(19)ΛN=ΛN1ΛN2=−GNN−1I2+ε¯¯Ng˙N−.In this equation,I2 is an identity matrix and the Newton impact coefficients of the two impact points are the same, εN1=εN2=εN. We then decompose the above matrix as follows:(20)ΛN1=−g˙N−GNN⋅1+εNr2r2−r1I1+r4r4−r3I2,ΛN2=−g˙N−GNN⋅1+εNr1r1−r2I1+r3r3−r4I2.Since1+εN>0,GNN>0, and the contact occurs if and only if g˙N−<0, one obtains the following:(21)ΛN1>0⟺r2r2−r1I1+r4r4−r3I2>0,(22)ΛN2>0⟺r1r1−r2I1+r3r3−r4I2>0.In order to prevent the main girders from rotating during full-edge impact, equations (21) and (22) must be satisfied at the same time, so that both ΛN1 and ΛN2 are positive. After examining all the cases of r1, r2, r3, and r4 in equation (17), it is concluded that only the following two cases can satisfy that both ΛN1 and ΛN2 are positive. In the following analysis, two variables, si and sj, which are only related to geometric parameters, are introduced for the convenience of analyzing the following two cases:Case 1.
−W/2<si<W/2, sj<−W/2(23)2si⋅αj⋅W/2+siW⋅αi⋅W/2+sj<−1,2si⋅αj⋅W/2−siW⋅αi⋅W/2−sj<1.Case 2.
−W/2<si<W/2,−W/2<sj<W/2(24)si⋅αj⋅W/2+sisj⋅αi⋅W/2+sj<−1,si⋅αj⋅W/2−sisj⋅αi⋅W/2−sj>−1.
For these cases,si=dmi⋅cosαi/2−R and sj=dmj⋅cosαj/2−R, where si and sj represent deck-1 and deck-2, respectively. According to equations (21) and (22), it can be concluded that the different contact conditions are determined by the geometric characteristics and inertial conditions of the bridge segments. This conclusion can be used to guide engineering practice.


## 4. Frictional Impact Analysis

When multiple points with frictional contact impacts are considered, the traditional approach is to consider the normal and tangential impulses of all possible contacts. For this analysis,ΛN=ΛNi and ΛT=ΛTi, but GNT, GTN, and GTT are all singular matrices. In this paper, we use the alternative description of multipoint impact proposed by Dimitrakopoulos [27] (Figure 6) to solve the overconstrained problem of multibody dynamic impact and therefore avoid the closed solutions of all singular points.Figure 6
Full-edge frictional impact between deck-1 and deck-2.Using the results of Dimitrakopoulos [27], we can calculate WTT, GNT, GTN, and GTT as follows:(25)WTT=01rT10−1rT201rT10−1rT2,ΛT1+ΛT2=μΛN1+μΛN2=μμΛN1ΛN2=μ¯¯ΛN=ΛT,GNT=WNTM−1WT=r1rT1I1+r3rT2I2r2rT1I1+r4rT2I2=GTNT,GTT=WTTM−1WT=1m1+1m2+rT12I1+rT22I2.Because of the existence of matricesGNT, GTN, and GTT [27], one can avoid the tedious calculations caused by irreversibility, and the discussion of the subsequent analysis of frictional impact can proceed. All the possible cases of slip and stick, when the two furthest points of each deck are in contact with the other deck contour, are illustrated in Figure 7.Figure 7
Classification of different frictional postimpact states of the curved bridge.


### 4.1. Multipoint Impact

#### 4.1.1. Deck-1 Slides down Relative to Deck-2

When deck-1 slides down relative to deck-2,ΛN1>0, ΛN2>0vN1=vN2=vTR1=vTR2=0, ΛTR1=2μ¯¯ΛN1, ΛTR2=2μ¯¯ΛN2, and ΛTL1=ΛTL2=0.It follows from equation (4) that the impulses in the normal direction are as follows:(26)ΛN=−GNN+GNTμ¯¯−1ε¯¯N+Eg˙N−,(27)vTL=−GTN+GTTμ¯¯ΛN−Eg˙T−.After integrating the Newton–Euler equations [26], one obtains(28)Mu+−u−=WNΛN+WTΛT.From equations (27) and (28), we have(29)u+=u−−M−1WN+μ¯¯WTGNN+GNTμ¯¯−1εN+Eg˙N−.From equation (4), we know that g˙T+=vTR−vTL=−vTL≤0. The preimpact tangential normal velocity rate is denoted as η and plays an important role in the analysis of the following cases. Using this inequality, we have(30)η=g˙T−g˙N−≥1+εNGNN+GNTμ¯¯⋅W2I1I2⋅μI1+I2m1+m2m1m2+rT1−rT2μrT1−μrT2+1Wr2r3−r1r4=ηmax.


#### 4.1.2. Deck-1 Slides up Relative to Deck-2

When deck-1 slides up relative to deck-2,ΛN1>0,ΛN2>0, vN1=vN2=vTL1=vTL2=0, ΛTL1=2μ¯¯ΛN1, ΛTL2=2μ¯¯ΛN2, and ΛTR1=ΛTR2=0.It follows from equation (4) that the impulses in the normal direction are as follows:(31)ΛN=−1+εNg˙N−GNN−GNTμ¯¯−111.From equations (28) and (31), we have(32)u+=u−−M−1WN−μ¯¯WTGNN−GNTμ¯¯−1εN+Eg˙N−.From equation (4), we know that g˙T+=vTR−vTL=−vTL≤0 and obtain the following:(33)η=g˙T−g˙N−≤1+εNGNN−GNTμ¯¯⋅W2I1I2⋅μI1+I2m1+m2m1m2+rT1−rT2μrT1−μrT2+1Wr1r4−r2r3=ηmin.


#### 4.1.3. Stick

When deck-1 sticks to deck-2,ΛN1>0,ΛN2>0, and ΛT1<μΛN1, ΛT2<μΛN2, vN1=vN2=0, and the following constraints can be obtained:(34)ΛTR1=μΛN1+ΛT1>0,ΛTR2=μΛN2+ΛT2>0,ΛTL1=μΛN1−ΛT1>0,ΛTL2=μΛN2−ΛT2>0.Then, we havevTL1=vTL2=vTR1=vTR2=0. It follows from equation (4) that the impulses in the normal and tangential direction are as follows:(35)GNN−GNTμ¯¯ΛN+GNTΛTR+ε¯¯N+Eg˙T−=0,GTN−GTTμ¯¯ΛN+GTTΛTR+E⋅g˙T−=0.From the above analysis, we list the conditions for the occurrence of the multipoint stick and multipoint slip using the preimpact tangential normal velocity rate,η:(1)
When deck-1 slides down relative to deck-2:η≥ηmax(2)
When deck-1 slides up relative to deck-2:η≤ηmin(3)
When deck-1 sticks to deck-2:ηmin≤η≤ηmaxwhere(36)ηmin=1+εNGNN−GNTμ¯¯⋅W2I1I2⋅μI1+I2m1+m2m1m2+rT1−rT2⋅μrT1−μrT2+1Wr1r4−r2r3,ηmax=1+εNGNN+GNTμ¯¯⋅W2I1I2⋅μI1+I2m1+m2m1m2+rT1−rT2⋅μrT1−μrT2+1Wr2r3−r1r4.Two conclusions can be drawn from the above analysis. First, from equations (21) and (22), it can be concluded that the different contact conditions are determined by the geometric characteristics and inertial parameters of the bridge segments. Secondly, according to equation (36), the preimpact tangential normal velocity rate, η, the normal impact recovery coefficient, εN, and the transverse friction coefficient, μ, determine the state of motion between the decks after contact.


### 4.2. Single-Point Frictional Impact Analysis

#### 4.2.1. Stick or Slip at the Outer Point of Bridge Segments

When deck-1 slides down relative to deck-2, the following can be obtained from equation (30):(37)η=g˙T−g˙N−≥μm1+m2/m1m2+r1rT1+μrT12/I1+r3rT2+μrT22/I2m1+m2/m1m2+r12−μr1rT1/I1+r32−μr3rT2/I2×1+εN=ηmax.When deck-1 slides up relative to deck-2, we can obtain from equation (33) the following:(38)η=g˙T−g˙N−≤μm1+m2/m1m2+r1rT1−μrT12/I1+r3rT2−μrT22/I2m1m2/m1m2+r12−μr1rT1/I1+r32−μr3rT2/I2×1+εN=ηmin.According to the above analyses, we list the conditions for the occurrence of the single-point stick and single-point slip at the outer point of the bridge segments, respectively, based on the preimpact tangential normal velocity rate,η:(1)
When deck-1 slides down relative to deck-2:η≥ηmax(2)
When deck-1 slides up relative to deck-2:η≤ηmin(3)
When deck-1 sticks to deck-2:ηmin≤η≤ηmaxwhere(39)ηmin=μm1+m2/m1m2+r1rT1−μrT12/I1+r3rT2−μrT22/I2m1+m2/m1m2+r12−μr1rT1/I1+r32−μr3rT2/I2×1+εN,(40)ηmax=μm1+m2/m1m2+r1rT1+μrT12/I1+r3rT2+μrT22/I2m1+m2/m1m2+r12−μr1rT1/I1+r32−μr3rT2/I2×1+εN.


#### 4.2.2. Stick or Slip at the Inner Point of Bridge Segments

When deck-1 slides down relative to deck-2, we can obtain from equation (30) the following:(41)η=g˙T−g˙N−≥μm1+m2/m1m2+r2rT1+μrT12/I1+r4rT2+μrT22/I2m1+m2/m1m2+r22−μr2rT1/I1+r42−μr4rT2/I2×1+εN=ηmax.When deck-1 slides up relative to deck-2, we can obtain from equation (33) the following:(42)η=g˙T−g˙N−≤−μm1+m2/m1m2+r2rT1−μrT12/I1+r4rT2−μrT22/I2m1+m2/m1m2+r22−μr2rT1/I1+r42−μr4rT2/I2×1+εN=ηmin.According to the above analyses, we list the conditions for the occurrence of the single-point stick and single-point slip at the inner point of the bridge segments, respectively, based on the preimpact tangential normal velocity rate,η:(1)
When deck-1 slides down relative to deck-2:η≥ηmax(2)
When deck-1 slides up relative to deck-2:η≤ηmin(3)
When deck-1 sticks to deck-2:ηmin≤η≤ηmaxwhere(43)ηmax=μm1+m2/m1m2+r2rT1+μrT12/I1+r4rT2+μrT22/I2m1+m2/m1m2+r22−μr2rT1/I1+r42−μr4rT2/I2×1+εN,(44)ηmin=−μm1+m2/m1m2+r2rT1−μrT12/I1+r4rT2−μrT22/I2m1+m2/m1m2+r22−μr2rT1/I1+r42−μr4rT2/I2×1+εN.From equations (39), (40), (43), and (44), it is concluded that the preimpact tangential normal velocity rate, η, the normal impact recovery coefficient, εN, and the transverse friction coefficient, μ, determine the state of motion between the decks after contact (sliding upward, sliding downward, or stick). It is necessary to ensure the two decks stick after impact (without relative sliding) to avoid the rotation of the decks due to the impact.In the following analyses and figures, “upward” refers to deck-1 sliding up relative to deck-2 and “downward” to deck-1 sliding down relative to deck-2. Potential contact points are indicated in Figure3. A planar rectangular coordinate system is established considering only the rotation of the main girder in the plane. The preimpact tangential normal velocity rate, η, is calculated by using the parameters modelling the process of impact. In the process of impact, the stick-slip between the main girders is affected not only by the geometric parameters of the main girders but also by the preimpact tangential normal velocity, η. The influencing geometric factors are the curvature of the two decks, the ratio of width to radius of the main girder, and the rotation angle of the two decks, αi. From these parameters, the transverse and normal constrained Jacobian matrices can be calculated from the formulas in this paper. Without loss of generality, we assume that the width is 15 m and the radius is 100 m. The normal recovery coefficient of the main girder is 0.5, and we assume that the geometry satisfies R−W/2−dmi⋅cosαi/2>0. In Figures 8–11, it is assumed that σ1=W/2 and σ2=W/2.Figure 8
Identification of stick and upward and downward slip states for different preimpact transverse and normal velocity rates for various preimpact rotations of center angle of girder at 60° and 120°.Figure 9
Identification of stick and upward and downward slip states for different preimpact transverse and normal velocity rates for various preimpact rotations of the center angle of girder at 60° and 90°.Figure 10
Identification of stick and upward and downward slip states for different preimpact transverse and normal velocity rates for various preimpact rotations of the center angle of girder at 90° and 120°.Figure 11
Identification of stick and upward and downward slip states for different preimpact transverse and normal velocity rates for various preimpact rotations of the center angle of girder at 60° and 60° (upper part) and 120° and 120° (lower part).The variables analyzed include the tangential friction coefficient and the center angle of the girders. We assume that the tangential friction coefficient is 0.5 and 1, respectively. Figures8–11 show the different center angles of the girders. When a point of one deck contacts the edge of the other deck, this case is called a single-point contact. When friction is considered, the main girder will slip and rotate when the friction exceeds the maximum static sliding friction. Figure 8 is a diagram drawn for the center angles of the girders of 60° and 120°, respectively. It can be seen from the diagram that, with the increase in friction coefficient, the area of adhesion increases because more force is needed to overcome it.In Figures8–11, the area between the upper and lower lines represents the stick condition, [−10°, 10°] represents the rotation angle range of the two decks, and [−10, 10] represents the range of preimpact tangential normal velocity ratio, η. The upper and lower parts of the adhesion area in these figures represent different sliding cases, including the inward and outward sliding cases of the inner and outer points of the two main girders. The area and situation of different motion states can clearly be observed in Figures 8–11.


## 4.1. Multipoint Impact

### 4.1.1. Deck-1 Slides down Relative to Deck-2

When deck-1 slides down relative to deck-2,ΛN1>0, ΛN2>0vN1=vN2=vTR1=vTR2=0, ΛTR1=2μ¯¯ΛN1, ΛTR2=2μ¯¯ΛN2, and ΛTL1=ΛTL2=0.It follows from equation (4) that the impulses in the normal direction are as follows:(26)ΛN=−GNN+GNTμ¯¯−1ε¯¯N+Eg˙N−,(27)vTL=−GTN+GTTμ¯¯ΛN−Eg˙T−.After integrating the Newton–Euler equations [26], one obtains(28)Mu+−u−=WNΛN+WTΛT.From equations (27) and (28), we have(29)u+=u−−M−1WN+μ¯¯WTGNN+GNTμ¯¯−1εN+Eg˙N−.From equation (4), we know that g˙T+=vTR−vTL=−vTL≤0. The preimpact tangential normal velocity rate is denoted as η and plays an important role in the analysis of the following cases. Using this inequality, we have(30)η=g˙T−g˙N−≥1+εNGNN+GNTμ¯¯⋅W2I1I2⋅μI1+I2m1+m2m1m2+rT1−rT2μrT1−μrT2+1Wr2r3−r1r4=ηmax.


### 4.1.2. Deck-1 Slides up Relative to Deck-2

When deck-1 slides up relative to deck-2,ΛN1>0,ΛN2>0, vN1=vN2=vTL1=vTL2=0, ΛTL1=2μ¯¯ΛN1, ΛTL2=2μ¯¯ΛN2, and ΛTR1=ΛTR2=0.It follows from equation (4) that the impulses in the normal direction are as follows:(31)ΛN=−1+εNg˙N−GNN−GNTμ¯¯−111.From equations (28) and (31), we have(32)u+=u−−M−1WN−μ¯¯WTGNN−GNTμ¯¯−1εN+Eg˙N−.From equation (4), we know that g˙T+=vTR−vTL=−vTL≤0 and obtain the following:(33)η=g˙T−g˙N−≤1+εNGNN−GNTμ¯¯⋅W2I1I2⋅μI1+I2m1+m2m1m2+rT1−rT2μrT1−μrT2+1Wr1r4−r2r3=ηmin.


### 4.1.3. Stick

When deck-1 sticks to deck-2,ΛN1>0,ΛN2>0, and ΛT1<μΛN1, ΛT2<μΛN2, vN1=vN2=0, and the following constraints can be obtained:(34)ΛTR1=μΛN1+ΛT1>0,ΛTR2=μΛN2+ΛT2>0,ΛTL1=μΛN1−ΛT1>0,ΛTL2=μΛN2−ΛT2>0.Then, we havevTL1=vTL2=vTR1=vTR2=0. It follows from equation (4) that the impulses in the normal and tangential direction are as follows:(35)GNN−GNTμ¯¯ΛN+GNTΛTR+ε¯¯N+Eg˙T−=0,GTN−GTTμ¯¯ΛN+GTTΛTR+E⋅g˙T−=0.From the above analysis, we list the conditions for the occurrence of the multipoint stick and multipoint slip using the preimpact tangential normal velocity rate,η:(1)
When deck-1 slides down relative to deck-2:η≥ηmax(2)
When deck-1 slides up relative to deck-2:η≤ηmin(3)
When deck-1 sticks to deck-2:ηmin≤η≤ηmaxwhere(36)ηmin=1+εNGNN−GNTμ¯¯⋅W2I1I2⋅μI1+I2m1+m2m1m2+rT1−rT2⋅μrT1−μrT2+1Wr1r4−r2r3,ηmax=1+εNGNN+GNTμ¯¯⋅W2I1I2⋅μI1+I2m1+m2m1m2+rT1−rT2⋅μrT1−μrT2+1Wr2r3−r1r4.Two conclusions can be drawn from the above analysis. First, from equations (21) and (22), it can be concluded that the different contact conditions are determined by the geometric characteristics and inertial parameters of the bridge segments. Secondly, according to equation (36), the preimpact tangential normal velocity rate, η, the normal impact recovery coefficient, εN, and the transverse friction coefficient, μ, determine the state of motion between the decks after contact.


## 4.1.1. Deck-1 Slides down Relative to Deck-2

When deck-1 slides down relative to deck-2,ΛN1>0, ΛN2>0vN1=vN2=vTR1=vTR2=0, ΛTR1=2μ¯¯ΛN1, ΛTR2=2μ¯¯ΛN2, and ΛTL1=ΛTL2=0.It follows from equation (4) that the impulses in the normal direction are as follows:(26)ΛN=−GNN+GNTμ¯¯−1ε¯¯N+Eg˙N−,(27)vTL=−GTN+GTTμ¯¯ΛN−Eg˙T−.After integrating the Newton–Euler equations [26], one obtains(28)Mu+−u−=WNΛN+WTΛT.From equations (27) and (28), we have(29)u+=u−−M−1WN+μ¯¯WTGNN+GNTμ¯¯−1εN+Eg˙N−.From equation (4), we know that g˙T+=vTR−vTL=−vTL≤0. The preimpact tangential normal velocity rate is denoted as η and plays an important role in the analysis of the following cases. Using this inequality, we have(30)η=g˙T−g˙N−≥1+εNGNN+GNTμ¯¯⋅W2I1I2⋅μI1+I2m1+m2m1m2+rT1−rT2μrT1−μrT2+1Wr2r3−r1r4=ηmax.


## 4.1.2. Deck-1 Slides up Relative to Deck-2

When deck-1 slides up relative to deck-2,ΛN1>0,ΛN2>0, vN1=vN2=vTL1=vTL2=0, ΛTL1=2μ¯¯ΛN1, ΛTL2=2μ¯¯ΛN2, and ΛTR1=ΛTR2=0.It follows from equation (4) that the impulses in the normal direction are as follows:(31)ΛN=−1+εNg˙N−GNN−GNTμ¯¯−111.From equations (28) and (31), we have(32)u+=u−−M−1WN−μ¯¯WTGNN−GNTμ¯¯−1εN+Eg˙N−.From equation (4), we know that g˙T+=vTR−vTL=−vTL≤0 and obtain the following:(33)η=g˙T−g˙N−≤1+εNGNN−GNTμ¯¯⋅W2I1I2⋅μI1+I2m1+m2m1m2+rT1−rT2μrT1−μrT2+1Wr1r4−r2r3=ηmin.


## 4.1.3. Stick

When deck-1 sticks to deck-2,ΛN1>0,ΛN2>0, and ΛT1<μΛN1, ΛT2<μΛN2, vN1=vN2=0, and the following constraints can be obtained:(34)ΛTR1=μΛN1+ΛT1>0,ΛTR2=μΛN2+ΛT2>0,ΛTL1=μΛN1−ΛT1>0,ΛTL2=μΛN2−ΛT2>0.Then, we havevTL1=vTL2=vTR1=vTR2=0. It follows from equation (4) that the impulses in the normal and tangential direction are as follows:(35)GNN−GNTμ¯¯ΛN+GNTΛTR+ε¯¯N+Eg˙T−=0,GTN−GTTμ¯¯ΛN+GTTΛTR+E⋅g˙T−=0.From the above analysis, we list the conditions for the occurrence of the multipoint stick and multipoint slip using the preimpact tangential normal velocity rate,η:(1)
When deck-1 slides down relative to deck-2:η≥ηmax(2)
When deck-1 slides up relative to deck-2:η≤ηmin(3)
When deck-1 sticks to deck-2:ηmin≤η≤ηmaxwhere(36)ηmin=1+εNGNN−GNTμ¯¯⋅W2I1I2⋅μI1+I2m1+m2m1m2+rT1−rT2⋅μrT1−μrT2+1Wr1r4−r2r3,ηmax=1+εNGNN+GNTμ¯¯⋅W2I1I2⋅μI1+I2m1+m2m1m2+rT1−rT2⋅μrT1−μrT2+1Wr2r3−r1r4.Two conclusions can be drawn from the above analysis. First, from equations (21) and (22), it can be concluded that the different contact conditions are determined by the geometric characteristics and inertial parameters of the bridge segments. Secondly, according to equation (36), the preimpact tangential normal velocity rate, η, the normal impact recovery coefficient, εN, and the transverse friction coefficient, μ, determine the state of motion between the decks after contact.


## 4.2. Single-Point Frictional Impact Analysis

### 4.2.1. Stick or Slip at the Outer Point of Bridge Segments

When deck-1 slides down relative to deck-2, the following can be obtained from equation (30):(37)η=g˙T−g˙N−≥μm1+m2/m1m2+r1rT1+μrT12/I1+r3rT2+μrT22/I2m1+m2/m1m2+r12−μr1rT1/I1+r32−μr3rT2/I2×1+εN=ηmax.When deck-1 slides up relative to deck-2, we can obtain from equation (33) the following:(38)η=g˙T−g˙N−≤μm1+m2/m1m2+r1rT1−μrT12/I1+r3rT2−μrT22/I2m1m2/m1m2+r12−μr1rT1/I1+r32−μr3rT2/I2×1+εN=ηmin.According to the above analyses, we list the conditions for the occurrence of the single-point stick and single-point slip at the outer point of the bridge segments, respectively, based on the preimpact tangential normal velocity rate,η:(1)
When deck-1 slides down relative to deck-2:η≥ηmax(2)
When deck-1 slides up relative to deck-2:η≤ηmin(3)
When deck-1 sticks to deck-2:ηmin≤η≤ηmaxwhere(39)ηmin=μm1+m2/m1m2+r1rT1−μrT12/I1+r3rT2−μrT22/I2m1+m2/m1m2+r12−μr1rT1/I1+r32−μr3rT2/I2×1+εN,(40)ηmax=μm1+m2/m1m2+r1rT1+μrT12/I1+r3rT2+μrT22/I2m1+m2/m1m2+r12−μr1rT1/I1+r32−μr3rT2/I2×1+εN.


### 4.2.2. Stick or Slip at the Inner Point of Bridge Segments

When deck-1 slides down relative to deck-2, we can obtain from equation (30) the following:(41)η=g˙T−g˙N−≥μm1+m2/m1m2+r2rT1+μrT12/I1+r4rT2+μrT22/I2m1+m2/m1m2+r22−μr2rT1/I1+r42−μr4rT2/I2×1+εN=ηmax.When deck-1 slides up relative to deck-2, we can obtain from equation (33) the following:(42)η=g˙T−g˙N−≤−μm1+m2/m1m2+r2rT1−μrT12/I1+r4rT2−μrT22/I2m1+m2/m1m2+r22−μr2rT1/I1+r42−μr4rT2/I2×1+εN=ηmin.According to the above analyses, we list the conditions for the occurrence of the single-point stick and single-point slip at the inner point of the bridge segments, respectively, based on the preimpact tangential normal velocity rate,η:(1)
When deck-1 slides down relative to deck-2:η≥ηmax(2)
When deck-1 slides up relative to deck-2:η≤ηmin(3)
When deck-1 sticks to deck-2:ηmin≤η≤ηmaxwhere(43)ηmax=μm1+m2/m1m2+r2rT1+μrT12/I1+r4rT2+μrT22/I2m1+m2/m1m2+r22−μr2rT1/I1+r42−μr4rT2/I2×1+εN,(44)ηmin=−μm1+m2/m1m2+r2rT1−μrT12/I1+r4rT2−μrT22/I2m1+m2/m1m2+r22−μr2rT1/I1+r42−μr4rT2/I2×1+εN.From equations (39), (40), (43), and (44), it is concluded that the preimpact tangential normal velocity rate, η, the normal impact recovery coefficient, εN, and the transverse friction coefficient, μ, determine the state of motion between the decks after contact (sliding upward, sliding downward, or stick). It is necessary to ensure the two decks stick after impact (without relative sliding) to avoid the rotation of the decks due to the impact.In the following analyses and figures, “upward” refers to deck-1 sliding up relative to deck-2 and “downward” to deck-1 sliding down relative to deck-2. Potential contact points are indicated in Figure3. A planar rectangular coordinate system is established considering only the rotation of the main girder in the plane. The preimpact tangential normal velocity rate, η, is calculated by using the parameters modelling the process of impact. In the process of impact, the stick-slip between the main girders is affected not only by the geometric parameters of the main girders but also by the preimpact tangential normal velocity, η. The influencing geometric factors are the curvature of the two decks, the ratio of width to radius of the main girder, and the rotation angle of the two decks, αi. From these parameters, the transverse and normal constrained Jacobian matrices can be calculated from the formulas in this paper. Without loss of generality, we assume that the width is 15 m and the radius is 100 m. The normal recovery coefficient of the main girder is 0.5, and we assume that the geometry satisfies R−W/2−dmi⋅cosαi/2>0. In Figures 8–11, it is assumed that σ1=W/2 and σ2=W/2.Figure 8
Identification of stick and upward and downward slip states for different preimpact transverse and normal velocity rates for various preimpact rotations of center angle of girder at 60° and 120°.Figure 9
Identification of stick and upward and downward slip states for different preimpact transverse and normal velocity rates for various preimpact rotations of the center angle of girder at 60° and 90°.Figure 10
Identification of stick and upward and downward slip states for different preimpact transverse and normal velocity rates for various preimpact rotations of the center angle of girder at 90° and 120°.Figure 11
Identification of stick and upward and downward slip states for different preimpact transverse and normal velocity rates for various preimpact rotations of the center angle of girder at 60° and 60° (upper part) and 120° and 120° (lower part).The variables analyzed include the tangential friction coefficient and the center angle of the girders. We assume that the tangential friction coefficient is 0.5 and 1, respectively. Figures8–11 show the different center angles of the girders. When a point of one deck contacts the edge of the other deck, this case is called a single-point contact. When friction is considered, the main girder will slip and rotate when the friction exceeds the maximum static sliding friction. Figure 8 is a diagram drawn for the center angles of the girders of 60° and 120°, respectively. It can be seen from the diagram that, with the increase in friction coefficient, the area of adhesion increases because more force is needed to overcome it.In Figures8–11, the area between the upper and lower lines represents the stick condition, [−10°, 10°] represents the rotation angle range of the two decks, and [−10, 10] represents the range of preimpact tangential normal velocity ratio, η. The upper and lower parts of the adhesion area in these figures represent different sliding cases, including the inward and outward sliding cases of the inner and outer points of the two main girders. The area and situation of different motion states can clearly be observed in Figures 8–11.


## 4.2.1. Stick or Slip at the Outer Point of Bridge Segments

When deck-1 slides down relative to deck-2, the following can be obtained from equation (30):(37)η=g˙T−g˙N−≥μm1+m2/m1m2+r1rT1+μrT12/I1+r3rT2+μrT22/I2m1+m2/m1m2+r12−μr1rT1/I1+r32−μr3rT2/I2×1+εN=ηmax.When deck-1 slides up relative to deck-2, we can obtain from equation (33) the following:(38)η=g˙T−g˙N−≤μm1+m2/m1m2+r1rT1−μrT12/I1+r3rT2−μrT22/I2m1m2/m1m2+r12−μr1rT1/I1+r32−μr3rT2/I2×1+εN=ηmin.According to the above analyses, we list the conditions for the occurrence of the single-point stick and single-point slip at the outer point of the bridge segments, respectively, based on the preimpact tangential normal velocity rate,η:(1)
When deck-1 slides down relative to deck-2:η≥ηmax(2)
When deck-1 slides up relative to deck-2:η≤ηmin(3)
When deck-1 sticks to deck-2:ηmin≤η≤ηmaxwhere(39)ηmin=μm1+m2/m1m2+r1rT1−μrT12/I1+r3rT2−μrT22/I2m1+m2/m1m2+r12−μr1rT1/I1+r32−μr3rT2/I2×1+εN,(40)ηmax=μm1+m2/m1m2+r1rT1+μrT12/I1+r3rT2+μrT22/I2m1+m2/m1m2+r12−μr1rT1/I1+r32−μr3rT2/I2×1+εN.


## 4.2.2. Stick or Slip at the Inner Point of Bridge Segments

When deck-1 slides down relative to deck-2, we can obtain from equation (30) the following:(41)η=g˙T−g˙N−≥μm1+m2/m1m2+r2rT1+μrT12/I1+r4rT2+μrT22/I2m1+m2/m1m2+r22−μr2rT1/I1+r42−μr4rT2/I2×1+εN=ηmax.When deck-1 slides up relative to deck-2, we can obtain from equation (33) the following:(42)η=g˙T−g˙N−≤−μm1+m2/m1m2+r2rT1−μrT12/I1+r4rT2−μrT22/I2m1+m2/m1m2+r22−μr2rT1/I1+r42−μr4rT2/I2×1+εN=ηmin.According to the above analyses, we list the conditions for the occurrence of the single-point stick and single-point slip at the inner point of the bridge segments, respectively, based on the preimpact tangential normal velocity rate,η:(1)
When deck-1 slides down relative to deck-2:η≥ηmax(2)
When deck-1 slides up relative to deck-2:η≤ηmin(3)
When deck-1 sticks to deck-2:ηmin≤η≤ηmaxwhere(43)ηmax=μm1+m2/m1m2+r2rT1+μrT12/I1+r4rT2+μrT22/I2m1+m2/m1m2+r22−μr2rT1/I1+r42−μr4rT2/I2×1+εN,(44)ηmin=−μm1+m2/m1m2+r2rT1−μrT12/I1+r4rT2−μrT22/I2m1+m2/m1m2+r22−μr2rT1/I1+r42−μr4rT2/I2×1+εN.From equations (39), (40), (43), and (44), it is concluded that the preimpact tangential normal velocity rate, η, the normal impact recovery coefficient, εN, and the transverse friction coefficient, μ, determine the state of motion between the decks after contact (sliding upward, sliding downward, or stick). It is necessary to ensure the two decks stick after impact (without relative sliding) to avoid the rotation of the decks due to the impact.In the following analyses and figures, “upward” refers to deck-1 sliding up relative to deck-2 and “downward” to deck-1 sliding down relative to deck-2. Potential contact points are indicated in Figure3. A planar rectangular coordinate system is established considering only the rotation of the main girder in the plane. The preimpact tangential normal velocity rate, η, is calculated by using the parameters modelling the process of impact. In the process of impact, the stick-slip between the main girders is affected not only by the geometric parameters of the main girders but also by the preimpact tangential normal velocity, η. The influencing geometric factors are the curvature of the two decks, the ratio of width to radius of the main girder, and the rotation angle of the two decks, αi. From these parameters, the transverse and normal constrained Jacobian matrices can be calculated from the formulas in this paper. Without loss of generality, we assume that the width is 15 m and the radius is 100 m. The normal recovery coefficient of the main girder is 0.5, and we assume that the geometry satisfies R−W/2−dmi⋅cosαi/2>0. In Figures 8–11, it is assumed that σ1=W/2 and σ2=W/2.Figure 8
Identification of stick and upward and downward slip states for different preimpact transverse and normal velocity rates for various preimpact rotations of center angle of girder at 60° and 120°.Figure 9
Identification of stick and upward and downward slip states for different preimpact transverse and normal velocity rates for various preimpact rotations of the center angle of girder at 60° and 90°.Figure 10
Identification of stick and upward and downward slip states for different preimpact transverse and normal velocity rates for various preimpact rotations of the center angle of girder at 90° and 120°.Figure 11
Identification of stick and upward and downward slip states for different preimpact transverse and normal velocity rates for various preimpact rotations of the center angle of girder at 60° and 60° (upper part) and 120° and 120° (lower part).The variables analyzed include the tangential friction coefficient and the center angle of the girders. We assume that the tangential friction coefficient is 0.5 and 1, respectively. Figures8–11 show the different center angles of the girders. When a point of one deck contacts the edge of the other deck, this case is called a single-point contact. When friction is considered, the main girder will slip and rotate when the friction exceeds the maximum static sliding friction. Figure 8 is a diagram drawn for the center angles of the girders of 60° and 120°, respectively. It can be seen from the diagram that, with the increase in friction coefficient, the area of adhesion increases because more force is needed to overcome it.In Figures8–11, the area between the upper and lower lines represents the stick condition, [−10°, 10°] represents the rotation angle range of the two decks, and [−10, 10] represents the range of preimpact tangential normal velocity ratio, η. The upper and lower parts of the adhesion area in these figures represent different sliding cases, including the inward and outward sliding cases of the inner and outer points of the two main girders. The area and situation of different motion states can clearly be observed in Figures 8–11.


## 5. Conclusions

In this paper, the nonsmooth rigid body dynamics method and set-valued formulation are used to model all possible postimpact states of a multisegment curved bridge by using the linear complimentary formula. The single-point and multipoint impacts are described, and the preconditions and corresponding closed-form solutions for all postimpact cases are calculated.It is concluded that the geometric characteristics and inertial conditions of the bridge segments define the location of the contact points. The preimpact tangential normal velocity rate,η, the normal impact recovery coefficient, εN, and the transverse friction coefficient, μ, determine the state of motion between the decks after impact. This paper also proposes equations for solving the contact kinematics between two adjacent curved planar rigid bodies in all possible situations. From the conducted analyses, it is found that the impact between the segments of a curved bridge is the major factor of segment rotation.



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*Source: 5217165-2020-12-17.xml*