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The Comparison of Different Approaches to Model Vehicle-Bridge Interaction
Available online at www.sciencedirect.com
ScienceDirect Procedia Engineering 190 (2017) 504 – 509
Structural and Physical Aspects of Construction Engineering
The Comparison of Different Approaches to Model Vehicle-Bridge Interaction Ľuboš Daniela, Ján Kortiša,* a
Department of Structural Mechanics and Applied Mathematics, Faculty of Civil Engineering, University of Zilina, Univerzitna 8215/1, 010 26, Slovakia
Abstract The results obtained from the numerical simulations of the vehicle-bridge interaction are appreciate sources of information that help to better understand the mechanism how the bridge responses to the moving vehicle. Several commercial computer programs that contain suitable contact algorithm are used for this purpose. Even though, there are often restrictions that define the limits for the simulation especially in the case when the irregularities on the road are treated. The development of the whole program which can solve this problem is difficult so there is good to use some open source applications that requires only some modifications to be used for this purpose. EasyDyn framework is a C++ library which can be used to create a multibody model of a vehicle. Then the development of the algorithm is only reduced to the finite element model of the bridge and the contact algorithm. In the paper there is presented this approach and the results are compared with the results obtained from the commercial software ADINA. © Published by Elsevier Ltd. This ©2017 2017The TheAuthors. Authors. Published by Elsevier Ltd. is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the issue editors. Peer-review under responsibility of the organizing committee of SPACE 2016 Keywords: ADINA; EasyDyn; vehicle-bridge interaction; contact algorithm; finite element method
1. Introduction The application of the commercial software in the investigation of the vehicle-bridge interaction is common as it is very fast way to create the model [3]. The graphics user interfaces which are used to build the model are userfriendly and the usage of them is comparable with the common CAD programs. This effective way has also some restrictions that can lead to the decision to develop a special own algorithm. The main reason is that they do not offer
* Corresponding author. Tel.: ++421 41 513 5616 E-mail address: [email protected]
1877-7058 © 2017 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license
(http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the organizing committee of SPACE 2016
doi:10.1016/j.proeng.2017.05.370
Ľuboš Daniel and Ján Kortiš / Procedia Engineering 190 (2017) 504 – 509
any possibility to modify their functionality for solution of special problems such as the vehicle-bridge interaction [1], [5], [6], [7]. The EasyDyn framework as a C++ library seems to be a good alternative to create a model for the solution of the vehicle-bridge interaction as there is not required to develop the whole algorithm. On the other hand it offers a free space to develop some special parts of the code which is an advantage in comparison with the commercial products where it is often impossible. It has been developed by the Department de Theoretical Mechanics, Dynamics and Vibrations of the University of Mons [2]. It offers functions that are used to create and solve the equations of motion of a multibody systems, which is very useful to define the model of the vehicle. It has been developed in the environment of a common programming language so there has been written also a part of a code which defines the model of the bridge. The interaction between the model of the bridge and the model of the vehicle is defined with the special contact algorithm. 2. The numerical model of the vehicle and the bridge The model of the vehicle is defined only in a plane so its spatial characteristics are neglected. The model contains three body parts that are used to define the main frame of the vehicle and the rear and the front axle. They are joined with linear springs that have stiffness and viscous damping characteristics derived from the previous experimental measurements of the vehicle Tatra 815.
Fig. 1. The model of the vehicle Tatra 815.
Geometrical characteristics of the vehicle are Lr = 1.075 m, Lf = 3.135 m and Lb= 1.32 m. Weight of the main mass of the model is m1 = 21.375 kg and weights of the front and rear axles are m 2 = 455 kg and m3 = 1070 kg. Inertial moment of the main mass and of the rear axle are I 1 = 1070 m4, I3= 466 kg. The damping and stiffness parameters of each part of the model are showed in the Tab. 1. Table 1. The stiffness and damping characteristics of the vehicle Tatra 815. Description of the vehicle part
label
value [-]
Stiffness of front springs
k1
143 716 [N/m]
Stiffness of rear springs
k2
761 256 [N/m]
Stiffness of front wheels
k3
1 250 000 [N/m]
Stiffness of rear wheels
k4, k5
2 500 000 [N/m]
Damping of front springs
b1
19 228 [kg/s]
Damping of rear springs
b2
260 197 [kg/s]
Damping of front wheels
b3
1 373 [kg/s]
Damping of rear wheels
b4, b5
2 746 [kg/s]
505
Ľuboš Daniel and Ján Kortiš / Procedia Engineering 190 (2017) 504 – 509
506
The same model of the vehicle was created in the program ADINA and with the employment of the EasyDyn framework. Modal parameters of both models were compared to check if they have equivalent dynamic behaviour. The comparison of natural frequencies (Tab. 2) shows that both models have the same dynamic characteristics. Table 2. Natural frequencies of the model of the vehicle. Natural frequency
ADINA
EasyDyn
f1
1.133 Hz
1.133 Hz
f2
1.451 Hz
1.450 Hz
f3
8.897 Hz
8.897 Hz
f4
10.900 Hz
10.901 Hz
f5
11.710 Hz
11.711 Hz
The mode shapes were solved to check if the parts of the model are defined properly. The springs defines the front axle are softer than the springs used to define the rear axle. This disproportionality also influences the mode shapes as it can be seen in the Fig. 2.
Fig. 2. The mode shapes and natural frequencies of the model of the vehicle.
The Euler-Bernoulli elements are used to define the planar model of the bridge. The material parameters are constantly defined along the bridge by the Young’s modulus (E=28GPa). The shape of the bridge cross-section is characterized by the geometrical moment of inertia in vertical plane (I=2.7m4) and area of the cross-section (A=9m2). The unit mass of the bridge is assumed to be 19000 kg.m -1.
Ľuboš Daniel and Ján Kortiš / Procedia Engineering 190 (2017) 504 – 509
507
Fig. 3. The scheme of the cross-section.
The damping characteristic is considered as Rayleigh damping defined by two parameters α = 1.597 s-1 and β= 0.0026 s. The values of the damping characteristics were obtained from a previous experimental analysis of the bridge. The associated damping matrix is therefore given by
>C@
D >M @ E >K @
(1)
The complete FEM of the bridge was created in the program ADINA as well as it was implemented in the algorithm that use EasyDyn framework through the corresponding mass, damping and stiffness matrices. The modal parameters (Fig. 4) of both models are compared.
Fig. 4. The mode shapes and natural frequencies of the model of the bridge.
3. Contact algorithm The contact algorithm that define the interaction between the model of the vehicle and the model of the bridge is one of the most crucial part of the whole model. The program ADINA use constraint-function method for implicit analysis. In this algorithm, constraint function is used to enforce the no-penetration and frictional contact conditions. The inequality constraints are replaced by the following normal constraint function 2
w( g , O )
g O §g O· ¨ ¸ HN , 2 © 2 ¹
(2)
where εN is a small user-defined parameter and default value of it is set to 1.0 x 10 -12. It is possible to set it 0.0. In this case ADINA automatically determines. [2] In the case of the vehicle-bridge interaction the default value is used. The system EasyDyn contains the general contact algorithm which is a part of its library. This algorithm is used to define the contact between vehicle (multi body EasyDyn model) and the bridge (finite element model). The main idea of the contact algorithm is based on the writing of a control flow statement in the environment of C++ programming language which transfers the contact forces from the vehicle to the bridge and vice versa (Fig. 5).
Ľuboš Daniel and Ján Kortiš / Procedia Engineering 190 (2017) 504 – 509
508
Fig. 5. The mode shapes and natural frequencies of the model of the bridge.
At the beginning of the statement the contforces in the contact points of the vehicle are generated. They represent gravity and initial forces of the vehicles. In the next step they are applied as a loading to the beams, which resents model of the bridge, in the positions where the contact points are generated. They are named applyforces. The response of the bridge is its deformation and there are calculated vertical deflections z beam in the contact points of the bridge. If there are also irregularities in that place, their height zireg is summed with the vertical deflection of the bridge. This is used in the next loop to define the position of the contact point of the wheel pos wheel. This is important to know to calculate new contforces in the contact points of the vehicle. These statements are repeated in each time steps.
4. Results of the solution The results of the solution of the vehicle-bridge interaction in the form of deflection vectors have been compared. The deflection of the bridge in the middle of the span is the most interesting so it is showed in the Fig. 6. There are the results obtained from the program ADINA and the algorithm that use EasyDyn framework where the vehicle is moved at the speed 3.5 m/s. The irregularities on the surface of the road are neglected. -4
Deflection in the midle of span of the bridge
x 10
deflection [m]
0
-5
-10
Speed of the vehicle v = 3.5 m/s Damping of the beam alfa = 1.597 beta = 0.0026
ADINA EasyDyn
-15 12
14
16
18
20 time [sec]
22
24
26
28
Fig. 6. The deflection of the bridge in the middle of the span with the smooth surface of the road.
The irregularities are defined in the following solution. The irregularities are derived from the 3D scan of the bridge. They also contain the obstacle positioned in the middle of the span which is commonly used for the dynamic experimental measurement of bridges. The dynamic response of the bridge increase as it can be seen in the Fig. 7 especially at the moment when the wheels bump to the obstacle.
Ľuboš Daniel and Ján Kortiš / Procedia Engineering 190 (2017) 504 – 509 -3
Deflection in the midle of the span of the beam
x 10 0
deflection [m]
-0.5
-1
-1.5
-2
Speed of the vehicle v = 3.5 m/s Damping of the beam alfa = 1.597 beta = 0.0026
ADINA EasyDyn
-2.5 13
14
15
16
17
18 time [sec]
19
20
21
22
23
Fig. 7. The deflection of the bridge in the middle of the span with the irregularities defined on the surface of the road.
The deflection in the middle of the bridge obtained from the solution in the program ADINA is slightly different in comparison with the solution obtained from the algorithm which uses EasyDyn framework. However, the amplitudes for both solution have the same values. 5. Conclusions There are two different approaches that can be used to solve the vehicle-bridge interaction. The first one is to use commercial program and the second one is to develop special algorithm that can be used for this purpose. In this case it is better to use some libraries which already contain functions that can be used. It is faster than developing the whole algorithm and it can be modified which is an advantage in comparison with the commercial program. The comparison of results obtained from the program ADINA and algorithm that use EasyDyn framework show that there are only small differences between them. However, there is difficult to find out the reason why there are small differences as it is not possible to know exactly the algorithm that is implemented in the program ADINA.
Acknowledgements This contribution is the result of the research supported by GA MŠVVaŠ SR VEGA, grant No. G1/0005/16. References [1] Frýba, L.,: Vibration of Solids and Structures Under Moving Loads. ACADEMIA, Praha, Nordhoff International Publishing, Groningen, 1972 [2] ADINA R&D, Inc. ; Theory and Modeling Guide, Volume I: ADINA; Report ARD 13-8; December 2013 [3] Daniel, Ľ., Kortiš, J.,: Moving load effect on bridges. CETRA 2014 : Road and rail infrastructure III : 3nd international conference on road and rail infrastructure - 28-30 April 2014, Split, Croatia. - ISSN 1848-9842. - Zagreb: Department of Transportation, Faculty of Civil Engineering University of Zagreb, 2014. - S. 535-539. [4] Kouroussis, G. and Verlinden, O. Prediction of railway induced ground vibration through multibody and finite element modelling, Mechanical Sciences, 4 (1), 167–183, (2013). [5] Kwasniewski, L., Li, H., Wekezer, J. and Malachowski, J. Finite element analysis of vehicle-bridge interaction, Finite Elements in Analysis and Design, 42 (11), 950–959, (2006). [6] Zhong, H., Yang, M. and Gao, Z. Dynamic responses of prestressed bridge and vehicle through bridge-vehicle interaction analysis, Engineering Structures, 87, 116–125, (2015). [7] Lipeng An, Dejian Li, Peng Yu, Peng Yuan.,: Numerical analysis of dynamic response of vehicle–bridge coupled system on long-span continuous girder bridge , Journal of Wind Engineering and Industrial Aerodynamics, Volume 155, August 2016, Pages 126-140
509
ScienceDirect Procedia Engineering 190 (2017) 504 – 509
Structural and Physical Aspects of Construction Engineering
The Comparison of Different Approaches to Model Vehicle-Bridge Interaction Ľuboš Daniela, Ján Kortiša,* a
Department of Structural Mechanics and Applied Mathematics, Faculty of Civil Engineering, University of Zilina, Univerzitna 8215/1, 010 26, Slovakia
Abstract The results obtained from the numerical simulations of the vehicle-bridge interaction are appreciate sources of information that help to better understand the mechanism how the bridge responses to the moving vehicle. Several commercial computer programs that contain suitable contact algorithm are used for this purpose. Even though, there are often restrictions that define the limits for the simulation especially in the case when the irregularities on the road are treated. The development of the whole program which can solve this problem is difficult so there is good to use some open source applications that requires only some modifications to be used for this purpose. EasyDyn framework is a C++ library which can be used to create a multibody model of a vehicle. Then the development of the algorithm is only reduced to the finite element model of the bridge and the contact algorithm. In the paper there is presented this approach and the results are compared with the results obtained from the commercial software ADINA. © Published by Elsevier Ltd. This ©2017 2017The TheAuthors. Authors. Published by Elsevier Ltd. is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the issue editors. Peer-review under responsibility of the organizing committee of SPACE 2016 Keywords: ADINA; EasyDyn; vehicle-bridge interaction; contact algorithm; finite element method
1. Introduction The application of the commercial software in the investigation of the vehicle-bridge interaction is common as it is very fast way to create the model [3]. The graphics user interfaces which are used to build the model are userfriendly and the usage of them is comparable with the common CAD programs. This effective way has also some restrictions that can lead to the decision to develop a special own algorithm. The main reason is that they do not offer
* Corresponding author. Tel.: ++421 41 513 5616 E-mail address: [email protected]
1877-7058 © 2017 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license
(http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the organizing committee of SPACE 2016
doi:10.1016/j.proeng.2017.05.370
Ľuboš Daniel and Ján Kortiš / Procedia Engineering 190 (2017) 504 – 509
any possibility to modify their functionality for solution of special problems such as the vehicle-bridge interaction [1], [5], [6], [7]. The EasyDyn framework as a C++ library seems to be a good alternative to create a model for the solution of the vehicle-bridge interaction as there is not required to develop the whole algorithm. On the other hand it offers a free space to develop some special parts of the code which is an advantage in comparison with the commercial products where it is often impossible. It has been developed by the Department de Theoretical Mechanics, Dynamics and Vibrations of the University of Mons [2]. It offers functions that are used to create and solve the equations of motion of a multibody systems, which is very useful to define the model of the vehicle. It has been developed in the environment of a common programming language so there has been written also a part of a code which defines the model of the bridge. The interaction between the model of the bridge and the model of the vehicle is defined with the special contact algorithm. 2. The numerical model of the vehicle and the bridge The model of the vehicle is defined only in a plane so its spatial characteristics are neglected. The model contains three body parts that are used to define the main frame of the vehicle and the rear and the front axle. They are joined with linear springs that have stiffness and viscous damping characteristics derived from the previous experimental measurements of the vehicle Tatra 815.
Fig. 1. The model of the vehicle Tatra 815.
Geometrical characteristics of the vehicle are Lr = 1.075 m, Lf = 3.135 m and Lb= 1.32 m. Weight of the main mass of the model is m1 = 21.375 kg and weights of the front and rear axles are m 2 = 455 kg and m3 = 1070 kg. Inertial moment of the main mass and of the rear axle are I 1 = 1070 m4, I3= 466 kg. The damping and stiffness parameters of each part of the model are showed in the Tab. 1. Table 1. The stiffness and damping characteristics of the vehicle Tatra 815. Description of the vehicle part
label
value [-]
Stiffness of front springs
k1
143 716 [N/m]
Stiffness of rear springs
k2
761 256 [N/m]
Stiffness of front wheels
k3
1 250 000 [N/m]
Stiffness of rear wheels
k4, k5
2 500 000 [N/m]
Damping of front springs
b1
19 228 [kg/s]
Damping of rear springs
b2
260 197 [kg/s]
Damping of front wheels
b3
1 373 [kg/s]
Damping of rear wheels
b4, b5
2 746 [kg/s]
505
Ľuboš Daniel and Ján Kortiš / Procedia Engineering 190 (2017) 504 – 509
506
The same model of the vehicle was created in the program ADINA and with the employment of the EasyDyn framework. Modal parameters of both models were compared to check if they have equivalent dynamic behaviour. The comparison of natural frequencies (Tab. 2) shows that both models have the same dynamic characteristics. Table 2. Natural frequencies of the model of the vehicle. Natural frequency
ADINA
EasyDyn
f1
1.133 Hz
1.133 Hz
f2
1.451 Hz
1.450 Hz
f3
8.897 Hz
8.897 Hz
f4
10.900 Hz
10.901 Hz
f5
11.710 Hz
11.711 Hz
The mode shapes were solved to check if the parts of the model are defined properly. The springs defines the front axle are softer than the springs used to define the rear axle. This disproportionality also influences the mode shapes as it can be seen in the Fig. 2.
Fig. 2. The mode shapes and natural frequencies of the model of the vehicle.
The Euler-Bernoulli elements are used to define the planar model of the bridge. The material parameters are constantly defined along the bridge by the Young’s modulus (E=28GPa). The shape of the bridge cross-section is characterized by the geometrical moment of inertia in vertical plane (I=2.7m4) and area of the cross-section (A=9m2). The unit mass of the bridge is assumed to be 19000 kg.m -1.
Ľuboš Daniel and Ján Kortiš / Procedia Engineering 190 (2017) 504 – 509
507
Fig. 3. The scheme of the cross-section.
The damping characteristic is considered as Rayleigh damping defined by two parameters α = 1.597 s-1 and β= 0.0026 s. The values of the damping characteristics were obtained from a previous experimental analysis of the bridge. The associated damping matrix is therefore given by
>C@
D >M @ E >K @
(1)
The complete FEM of the bridge was created in the program ADINA as well as it was implemented in the algorithm that use EasyDyn framework through the corresponding mass, damping and stiffness matrices. The modal parameters (Fig. 4) of both models are compared.
Fig. 4. The mode shapes and natural frequencies of the model of the bridge.
3. Contact algorithm The contact algorithm that define the interaction between the model of the vehicle and the model of the bridge is one of the most crucial part of the whole model. The program ADINA use constraint-function method for implicit analysis. In this algorithm, constraint function is used to enforce the no-penetration and frictional contact conditions. The inequality constraints are replaced by the following normal constraint function 2
w( g , O )
g O §g O· ¨ ¸ HN , 2 © 2 ¹
(2)
where εN is a small user-defined parameter and default value of it is set to 1.0 x 10 -12. It is possible to set it 0.0. In this case ADINA automatically determines. [2] In the case of the vehicle-bridge interaction the default value is used. The system EasyDyn contains the general contact algorithm which is a part of its library. This algorithm is used to define the contact between vehicle (multi body EasyDyn model) and the bridge (finite element model). The main idea of the contact algorithm is based on the writing of a control flow statement in the environment of C++ programming language which transfers the contact forces from the vehicle to the bridge and vice versa (Fig. 5).
Ľuboš Daniel and Ján Kortiš / Procedia Engineering 190 (2017) 504 – 509
508
Fig. 5. The mode shapes and natural frequencies of the model of the bridge.
At the beginning of the statement the contforces in the contact points of the vehicle are generated. They represent gravity and initial forces of the vehicles. In the next step they are applied as a loading to the beams, which resents model of the bridge, in the positions where the contact points are generated. They are named applyforces. The response of the bridge is its deformation and there are calculated vertical deflections z beam in the contact points of the bridge. If there are also irregularities in that place, their height zireg is summed with the vertical deflection of the bridge. This is used in the next loop to define the position of the contact point of the wheel pos wheel. This is important to know to calculate new contforces in the contact points of the vehicle. These statements are repeated in each time steps.
4. Results of the solution The results of the solution of the vehicle-bridge interaction in the form of deflection vectors have been compared. The deflection of the bridge in the middle of the span is the most interesting so it is showed in the Fig. 6. There are the results obtained from the program ADINA and the algorithm that use EasyDyn framework where the vehicle is moved at the speed 3.5 m/s. The irregularities on the surface of the road are neglected. -4
Deflection in the midle of span of the bridge
x 10
deflection [m]
0
-5
-10
Speed of the vehicle v = 3.5 m/s Damping of the beam alfa = 1.597 beta = 0.0026
ADINA EasyDyn
-15 12
14
16
18
20 time [sec]
22
24
26
28
Fig. 6. The deflection of the bridge in the middle of the span with the smooth surface of the road.
The irregularities are defined in the following solution. The irregularities are derived from the 3D scan of the bridge. They also contain the obstacle positioned in the middle of the span which is commonly used for the dynamic experimental measurement of bridges. The dynamic response of the bridge increase as it can be seen in the Fig. 7 especially at the moment when the wheels bump to the obstacle.
Ľuboš Daniel and Ján Kortiš / Procedia Engineering 190 (2017) 504 – 509 -3
Deflection in the midle of the span of the beam
x 10 0
deflection [m]
-0.5
-1
-1.5
-2
Speed of the vehicle v = 3.5 m/s Damping of the beam alfa = 1.597 beta = 0.0026
ADINA EasyDyn
-2.5 13
14
15
16
17
18 time [sec]
19
20
21
22
23
Fig. 7. The deflection of the bridge in the middle of the span with the irregularities defined on the surface of the road.
The deflection in the middle of the bridge obtained from the solution in the program ADINA is slightly different in comparison with the solution obtained from the algorithm which uses EasyDyn framework. However, the amplitudes for both solution have the same values. 5. Conclusions There are two different approaches that can be used to solve the vehicle-bridge interaction. The first one is to use commercial program and the second one is to develop special algorithm that can be used for this purpose. In this case it is better to use some libraries which already contain functions that can be used. It is faster than developing the whole algorithm and it can be modified which is an advantage in comparison with the commercial program. The comparison of results obtained from the program ADINA and algorithm that use EasyDyn framework show that there are only small differences between them. However, there is difficult to find out the reason why there are small differences as it is not possible to know exactly the algorithm that is implemented in the program ADINA.
Acknowledgements This contribution is the result of the research supported by GA MŠVVaŠ SR VEGA, grant No. G1/0005/16. References [1] Frýba, L.,: Vibration of Solids and Structures Under Moving Loads. ACADEMIA, Praha, Nordhoff International Publishing, Groningen, 1972 [2] ADINA R&D, Inc. ; Theory and Modeling Guide, Volume I: ADINA; Report ARD 13-8; December 2013 [3] Daniel, Ľ., Kortiš, J.,: Moving load effect on bridges. CETRA 2014 : Road and rail infrastructure III : 3nd international conference on road and rail infrastructure - 28-30 April 2014, Split, Croatia. - ISSN 1848-9842. - Zagreb: Department of Transportation, Faculty of Civil Engineering University of Zagreb, 2014. - S. 535-539. [4] Kouroussis, G. and Verlinden, O. Prediction of railway induced ground vibration through multibody and finite element modelling, Mechanical Sciences, 4 (1), 167–183, (2013). [5] Kwasniewski, L., Li, H., Wekezer, J. and Malachowski, J. Finite element analysis of vehicle-bridge interaction, Finite Elements in Analysis and Design, 42 (11), 950–959, (2006). [6] Zhong, H., Yang, M. and Gao, Z. Dynamic responses of prestressed bridge and vehicle through bridge-vehicle interaction analysis, Engineering Structures, 87, 116–125, (2015). [7] Lipeng An, Dejian Li, Peng Yu, Peng Yuan.,: Numerical analysis of dynamic response of vehicle–bridge coupled system on long-span continuous girder bridge , Journal of Wind Engineering and Industrial Aerodynamics, Volume 155, August 2016, Pages 126-140
509