For dynamic responses of highway bridges to moving vehicles, most of studies focused on singlefactor analysis or multifactor analysis based on full factorial design. The defect of the former one is that it has no consideration of interaction effects, while that of the latter one is that it has large calculation. To avoid these defects, simplified theoretical derivations are presented at first; then some numerical simulations based on the proposed method of the orthogonal experimental design in batches have been done by our own program VBCVA. According to simplified theoretical derivations, three factors (
Research on the dynamic analysis of the vehiclebridge coupled vibration system is an important issue in civil engineering [
As moving vehicles on the bridge vary in both time and space, the problem becomes more complex. And it has been noted more than 100 years before. A lot of researchers tried to obtain the effects of moving load on various elements, components, and structures based on theoretical derivations. It had been well reviewed by Frýba [
The specification about impact factor or dynamic load allowance in most of design codes was obtained based on the dynamic loading test. The first thorough investigation of highway bridge dynamic loading was conducted from 1922 to 1928 by an ASCE committee (10 bridges) [
However, not all researchers can study the problem by dynamic fielding tests due to the budgets limitations. In recent years, with the development of computers, numerical simulation studies are widely used. The dynamic behavior of simplespan [
According to the review of existing literature, the defects and shortages are listed as follows.
The vehiclebridge coupled model is oversimplified, especially in theoretical derivation. And it is largely different from the actual conditions.
The selection of influence factors is subjective, and it lacks sufficient theoretical basis. As for one problem, the understanding from different engineers may be largely different, even if opposite.
The interaction effects are seldom studied in the dynamic analysis of the bridge to moving vehicles. Most of studies focused on main effects.
Abundant calculations are needed in the method of full factorial design, which has been used for multifactors problem before.
The impact factor in current code is the function of single parameter. It has been proved not rational and should be revised.
Therefore, the method of orthogonal experimental design in batches has been proposed in this paper. It can be used for studying both the main effects and the interaction effects. Meanwhile, due to the processing in batches, it greatly reduces the calculation cost. To make a good understanding of the vehiclebridge system and to obtain the basis of the selection of some important factors, two simplified models are discussed at first. As a basic calculation tool, which will be more consistent with the actual condition, our own program VBCVA is described in detail. At last, using the proposed method, the influences of twelve common factors and some of their interaction effects on dynamic responses of the vehiclebridge coupled system are discussed.
For safety of bridges, the impact factor is usually adopted to account for the dynamic responses induced by moving vehicles. However, there are many different names [
In addition, for comfort analysis of pedestrians and passengers or drivers, the vibration accelerations of the bridge and the vehicle are adopted, because a large number of researchers have obtained the conclusion that the comfort of people is mainly determined by the acceleration [
Theoretical derivation on simplified models does not quite agree with the actual situation, but it is one of the best ways to determine the key parameters and their influences on the dynamic responses. And, according to the derivation, the physical meanings may be more obvious, which will be good guidance for design and evaluation on dynamic performance of the highway bridge to moving vehicular loads. Consequently, in the beginning of this study, the dynamic responses of a simply supported girder bridge traversed by a single moving constant load and a moving sprung mass which are deduced and discussed, respectively.
Of the wide range of problems involving vibration of structures subjected to a moving load, the easiest one to tackle is that of dynamic responses in a simply supported girder bridge (or a simply supported beam), traversed by a constant force moving at uniform speed [
If the weight of the vehicle is far smaller than that of the bridge, the inertia force of the vehicle can be ignored. Then the vehiclebridge coupled system can be simplified as a simply supported girder bridge traversed by a single moving constant load (Figure
A single moving constant load.
Based on the theory of the structural dynamics, the governing equation of the bridge can be given by
The initial displacement and velocity are assumed as zero. As for simply supported girder bridge, the sine functions can be assumed as the mode shapes
According to (
If the speed of moving load is zero, but the force vibration in (
In accordance with theory of structural statics [
Comparison of static displacement.
Figure
Due to neglect of the mass of the vehicle in the section above, coupled vibration in the vehiclebridge system is not considered. To make a good comprehension of this effect, a simply supported girder bridge traversed by a moving sprung mass has been adopted in this section [
A moving sprung mass.
In Figure
Respectively, based on the theory of structural dynamics, the governing equations of the vehicle body and the bridge can be given by
Substituting ((
In general, the mass of the wheel is much less than that of the vehicle body, so it can be ignored. And when the damping of the vehicle is neglected, (
Obviously, the fundamental frequency and the maximum static displacement of the vehicle and the bridge can be given by
In accordance with the current codes, the dynamic load allowance (DLA,
Then (
It can be seen from (
Now, the influence of the natural frequency of the vehicle is discussed in detail. Equation (
Totally, based on simplified model and derivation above, the most significant influence parameters may be
Based on their own advantages of existing generalized commercial software ANSYS and MATLAB, the program VBCVA (VehicleBridge Coupled Vibration Analysis) has been developed by our own research group. It is used for vibration and dynamic analysis of the vehiclebridge coupled system. Upon the modal synthesis method, the bridge model is found by ANSYS, while the vehicle model and the roughness model are established by MATLAB.
According to the theory of structural dynamics [
As there are various types of highway bridges and they are much more complicated with the increasing technology, the accurate modeling of the bridge may be difficult to realize by our own written program. In addition, this will hinder the generalization and the development of that program. And the limitation can be obviously seen when the equations of various types of bridges are different. Therefore, the modal synthesis method has been adopted. Another advantage of this translation is the reduction of the degree number:
Substituting (
For convenience in application, the matrix of mode shape obtained from ANSYS is normalized. Then the following matrixes are used in the program VBCVA:
Actually, the moving vehicle can be looked at as a vibration system with multidegrees of freedom. In this study, D’Alembert’s principle is used for deducing dynamic equations of spatial vehicle models. And the position induced by the static weight of the vehicle is selected as the reference position [
The schematic plot of the vehicle model can be seen in Figure
Schematic plot of the vehicle model.
Elevation view
Front view
There are some assumptions on the vehicle model. The wheel and the bridge will contact with each other all the time. Only vertical effects between the vehicle and the bridge are considered, while longitudinal and transverse effects are ignored. The vehicle body and all wheels are assumed as rigid bodies with corresponding mass, while the spring and the damper are linear [
Then the vehicle equation can be given by
The interaction force between the bridge and the vehicle can be considered from two aspects. One is the force of the vehicle
Coupled equations of the vehiclebridge vibration system can be obtained by the combination of the bridge equation and the vehicle equation above. They are listed as follows:
When a vehicle goes across the bridge, the position of the contact point changes with time. Therefore, the coupled equations are timevarying system of differential equations. And it is difficult to obtain the closedform solutions. But they can be solved by some numerical methods, such as the central difference method, Newmark method, Wilson
It is assumed that the acceleration during the period [
Based on the assumption of linear acceleration, the acceleration during this period [
Of course, at the time of
Substituting (
According to solving (
Similarly, using the designations
In both design and evaluation, pavement roughness is the primary factor affecting the dynamic performances of the bridge traversed by the designated vehicles [
There are two methods to take the roughness into account, field measurement and numerical simulation. As for the former way, it is measured generally by one of the following two methods, that is, (1) by using a profilometer or (2) by calculating pavement roughness backwards from vibration data of the wellresearched dynamic properties of the vehicle [
Typical PSD function can be approximated by an exponential function:
The process of generating pavement roughness is listed as follows:
Considering common seen speeds of vehicles in highway bridges, the lower and upper limits of the spatial frequency are specified as 0.05 m^{−1} and 3.00 m^{−1} [
Roughness coefficient
Condition  Roughness coefficient 

Very good 

Good 

Average 

Poor 

Very poor 

As for different conditions, five typical pavement roughness examples are obtained using the program VBCVA. And the results are plotted in Figure
Five pavement roughness examples.
Very good (
Good (
Average (
Poor (
Very poor (
In addition, the relation between the maximum amplitude of the roughness sample and the roughness coefficient is studied. It can be seen from Figure
Relation analysis of roughness.
Due to various types of bridges, the modal synthesis method is adopted in the program VBCVA for general use. At first, the dynamic characteristics of the highway bridge, including natural frequencies and mode shapes, are obtained based on the finite element model (FEM) built by the commercial software ANSYS. Meanwhile, the data files for vehicles are prepared by another commercial software MATLAB. Then the coupled equations are calculated using the Wilson
Flowchart of the program VBCVA.
To supplement the application of this program, it is necessary to emphasize that the data files preparation includes bridge data (frequencies, mode shapes, damping ratios, and coordinates of nodes), pavement roughness model, and vehicle data (vehicle type, speed, number, and initial location). Also, controlling parameters mainly means
The program can be used for calculating the cases of multilanes and multivehicles (both in longitudinal direction and in transverse direction). Also, the number of axles or vehicles is not limited. Furthermore, the validity and the rationality have been verified by some numerical and experimental results from the existing papers [
Concerning dynamic responses of highway bridges to moving vehicular loads, three methods are commonly adopted, including theoretical derivation, dynamic loading test, and numerical simulations. As we know, the first method can only be appropriate for some simplified models, while the last two methods can be applied to the actual situation in general. However, due to the economy and many other causes, most of studies focused on dynamic responses induced by much fewer factors. In addition, influence of interactions between these factors on dynamic responses is barely studied. Therefore, based on orthogonal experimental design, influences of many factors and their interactions on the dynamic responses of simply supported girder bridges to moving vehicular loads are discussed at length in this section.
As for the multifactor experimental problem, the full factorial design was widely accepted in the early stage. There are two primary benefits of this method. Firstly, it reveals whether the effect of each factor depends on the levels of other factors in the experiment. And one factorial experiment can show “interaction effects” that a series of experiments each involving a single factor cannot. Secondly, it provides excellent precision for the regression model parameter estimates that summarize the combined effects of the factors [
However, when the variables are more, the size of the full factorial design is much extremely larger. For example, with three levels of every factor, increasing the number of factors from 2 to 4 increases the size of the full factorial design from 3^{2} to 3^{4}, 9 times larger. So fractional factorial designs have come out to largely reduce the work in the analysis of multifactor problem. In statistics, fractional factorial designs consist of a carefully chosen subset of the experimental runs of a full factorial design. That is chosen so as to exploit the sparsityofeffects principle to expose information about the most important features of the problem studied, while using a fraction of the effort of a full factorial design in terms of experimental runs.
Obviously, there are two principal contradictions in the fractional factorial design. One is the contradiction between the larger runs in the full factorial design and the expected smaller runs in actual operation. The other one is that between the smaller runs and the expected whole information, which is just the same as that obtained from the full factorial design [
The orthogonal array is the basis of the orthogonal experimental design. And it has been constructed based on these two mathematical courses, combinatorics and probability. Of course, it is not necessary for engineers to know how to construct it. We only need to transplant existing orthogonal arrays into engineering areas. The orthogonal array is usually denoted by
It has been proved that the “interaction effects” are always seen in the multifactor experiment. They are defined as the influence of combined factors on the test index. In fact, the “interaction effects” are reflections of mutual promotion or inhibition, and these effects are more or less existed in all physical phenomenon. In the statistical analysis of the results obtained from factorial experiments, the sparsityofeffects principle states that a system is usually dominated by main effects (single factor) and loworder interactions. Therefore, it is most likely that main effects and twofactor interactions are the most significant responses in an experiment [
A majority of studies have shown that the dynamic responses of vehiclebridge system are influenced by so many factors, including vehicle characteristics, pavement roughness, and bridge characteristics [
As we know, there are many types of highway bridges, including the girder bridge, the arch bridge, the cablestayed bridge, and the suspension bridge. Of course, the simply supported girder bridge is the most common type. Based on the standard drawings issued by the Ministry of Transport of the People’s Republic of China in 2008, a 30 mspan bridge with smallbox section has been selected as the fundamental sample of bridges. The damping ratio is assumed as 0.05. The crosssection is in Figure
Parameters of the bridge.







30  12  1.98  6.23  34500  2600 
The crosssection of the bridge (cm).
In highway bridges, the types of moving vehicles are various, which is significantly different from the type of vehicles in railway bridges. And it makes the problem of the vehiclebridge coupled system become much more complex. Usually, there are three types of vehicles, including the small car, the bus or coach, and the loading truck. Apparently, due to their light weight, the first two types of vehicles may not be the main contributions on the dynamic responses. So the 3axle loading truck has to be selected in this study. Its weight is 30 tons. It is seen in Figure
Parameters of the threeaxle loading truck.
Parameters  Value  Parameters  Value 

Mass of truck body  31800 kg  Upper stiffness (front axle)  1200 kN·m^{−1} 
Mass of front wheel  400 kg  Upper stiffness (middle/rear axle)  2400 kN·m^{−1} 
Mass of middle/rear wheel  600 kg  Upper damping (front axle)  5 kN·s·m^{−1} 
Pitching moment of inertia  40000 kg·m^{2}  Upper damping (middle/rear axle)  10 kN·s·m^{−1} 
Rolling moment of inertia  10000 kg·m^{2}  Lower stiffness (front axle)  2400 kN·m^{−1} 
Distance (front axle to center)  4.60 m  Lower stiffness (middle/rear axle)  4800 kN·m^{−1} 
Distance (middle axle to center)  0.36 m  Lower damping (front axle)  6 kN·s·m^{−1} 
Distance (middle to rear axle)  1.40 m  Lower damping (middle/rear axle)  12 kN·s·m^{−1} 
Wheel base  1.80 m 
Threeaxle loading truck (cm).
Pavement roughness is the excitation source of the vehiclebridge coupled vibration. For new bridges, the condition of the pavement is very good or good. So the maximum amplitude of the pavement roughness is assumed as 1cm (
Based on so many existing research results [
Factors description.
Factor  Description 



Pavement roughness 

Span length of the bridge 

Width of the bridge 

Mass of the bridge 

Stiffness of the bridge 

Damping of the bridge 

Mass of the vehicle 

Upper stiffness of the vehicle 

Upper damping of the vehicle 

Lower stiffness of the vehicle 

Lower damping of the vehicle 

Speed of the vehicle 
Considering computational efficiency, three levels are given for every factor. The value of parameters in fundamental models described above is thought as the second level. Then the first level and the third level are taken by 20% lower and 20% higher, respectively. For instance, three levels of the span length are 24 m (
Firstly, the interaction has been ignored in this section. As for twelve factors and three levels for every factors, the orthogonal array
Orthogonal array
Run  1  2  3  4  5  6  7  8  9  10  11  12  13 













Non  
1  1  1  1  1  1  1  1  1  1  1  1  1  1 
2  1  1  1  1  2  2  2  2  2  2  2  2  2 
3  1  1  1  1  3  3  3  3  3  3  3  3  3 
4  1  2  2  2  1  1  1  2  2  2  3  3  3 
5  1  2  2  2  2  2  2  3  3  3  1  1  1 
6  1  2  2  2  3  3  3  1  1  1  2  2  2 
7  1  3  3  3  1  1  1  3  3  3  2  2  2 
8  1  3  3  3  2  2  2  1  1  1  3  3  3 
9  1  3  3  3  3  3  3  2  2  2  1  1  1 
10  2  1  2  3  1  2  3  1  2  3  1  2  3 
11  2  1  2  3  2  3  1  2  3  1  2  3  1 
12  2  1  2  3  3  1  2  3  1  2  3  1  2 
13  2  2  3  1  1  2  3  2  3  1  3  1  2 
14  2  2  3  1  2  3  1  3  1  2  1  2  3 
15  2  2  3  1  3  1  2  1  2  3  2  3  1 
16  2  3  1  2  1  2  3  3  1  2  2  3  1 
17  2  3  1  2  2  3  1  1  2  3  3  1  2 
18  2  3  1  2  3  1  2  2  3  1  1  2  3 
19  3  1  3  2  1  3  2  1  3  2  1  3  2 
20  3  1  3  2  2  1  3  2  1  3  2  1  3 
21  3  1  3  2  3  2  1  3  2  1  3  2  1 
22  3  2  1  3  1  3  2  2  1  3  3  2  1 
23  3  2  1  3  2  1  3  3  2  1  1  3  2 
24  3  2  1  3  3  2  1  1  3  2  2  1  3 
25  3  3  2  1  1  3  2  3  2  1  2  1  3 
26  3  3  2  1  2  1  3  1  3  2  3  2  1 
27  3  3  2  1  3  2  1  2  1  3  1  3  2 
1: 1st level, 2: 2nd level, 3: 3rd level, and Non: no factor in this column.
All of these runs are analyzed by our own program VBCVA. The fundamental frequency (
Results of numerical simulations without interaction.
Run 

DLA 



1  5.59  1.183  0.3867  2.3001 
2  6.25  1.133  0.7283  3.2982 
3  6.84  1.168  0.7333  2.4512 
4  3.20  1.345  1.4651  3.1712 
5  3.58  1.268  0.7086  2.5815 
6  3.92  1.120  0.4376  1.6981 
7  2.03  1.312  0.8765  3.8239 
8  2.27  1.095  0.6762  1.5517 
9  2.49  1.365  0.5886  1.9558 
10  4.56  1.194  0.6982  2.5255 
11  5.10  1.335  0.5796  3.9891 
12  5.59  1.290  0.3568  2.5697 
13  3.58  1.249  0.6848  2.8135 
14  4.00  1.426  1.0353  4.9386 
15  4.38  1.252  1.0250  2.6898 
16  2.22  1.194  1.5957  2.9421 
17  2.49  1.217  1.0797  2.7304 
18  2.72  1.575  1.3255  2.7280 
19  5.00  1.244  0.8916  2.7037 
20  5.59  1.236  0.9152  4.0151 
21  6.12  1.269  0.7024  5.5762 
22  2.92  1.482  1.6800  3.9525 
23  3.27  1.381  1.2634  2.6892 
24  3.58  1.351  0.8484  3.6659 
25  2.49  1.361  1.0922  2.9921 
26  2.78  1.439  1.3170  2.4652 
27  3.05  2.047  2.2685  5.3437 
In most of design codes of highway bridges, the dynamic load allowance (DLA) is defined as the function of the fundamental frequency of the bridge. Taking the current code (JTG D602004) [
Relation between the DLA and the fundamental frequency.
Figure
In statistics, two methods are always used for data processing, range analysis and variance analysis. The first method has advantage of low cost, simple thought, and convenience to be popularized and applied. But, compared with variance analysis, the range analysis has two defects. Firstly, the error is hardly estimated. Secondly, the reliability cannot be determined. And it also cannot be used for regression analysis and design. Of course, the method of various analyses can avoid both defects. Then these two methods are described as follows, respectively.
The range of analysis is completed by two steps. At first, the ranges of every factor are calculated by [
Range analysis of numerical simulations without interaction.
Index 













DLA  0.202  0.173  0.106  0.088  0.101  0.033  0.127  0.186  0.062  0.068  0.154  0.060 

0.486  0.536  0.249  0.189  0.121  0.090  0.112  0.319  0.154  0.315  0.119  0.426 

1.175  0.322  0.368  0.285  0.162  0.427  1.331  0.993  0.232  0.420  0.204  0.598 
Trend plot between factors and the test index.
Trend plot between factors and the
Trend plot between factors and the acceleration of the bridge.
Trend plot between factors and the acceleration of the vehicle body.
Variance analysis is more rigorous. There are four steps to realize the variance analysis [
Step 1: as for every factors, calculate the sum of square of deviations (
Step 2: estimate the variance of the error (
Step 3: obtain the test static
Step 4: for simplicity, the variance analysis table is listed, including the process and the results.
All of them are calculated as follows:
It has to be noted that the error is resulted from all of the vacant columns in the orthogonal array. Also, the accuracy increases with the increasing dof of the error. Therefore, if the significance level of one factor is larger than 0.25, it can be included as the error. Of course, that may not be the same in different situations. All of the results are listed in Table
Various analyses of numerical simulations without interaction.
Dynamic load allowance (DLA) of the bridge
Factor 

dof  Initial  Modification  






 

0.1861  2  0.0931  49.35  0.05  0.0931  41.55  0.01 

0.1344  2  0.0672  35.64  0.05  0.0672  30.00  0.01 

0.0545  2  0.0273  14.45  0.10  0.0273  12.17  0.05 

0.0351  2  0.0175  9.30  0.10  0.0175  7.83  0.05 

0.0586  2  0.0293  15.54  0.10  0.0293  13.09  0.05 

0.0052  2  0.0026  1.38  >0.25  —  —  — 

0.0754  2  0.0377  20.00  0.05  0.0377  16.84  0.05 

0.1605  2  0.0802  42.55  0.05  0.0802  35.82  0.01 

0.0188  2  0.0094  5.00  0.25  0.0094  4.21  0.25 

0.0211  2  0.0106  5.60  0.25  0.0106  4.72  0.10 

0.1214  2  0.0607  32.18  0.05  0.0607  27.10  0.01 

0.0182  2  0.0091  4.84  0.25  0.0091  4.07  0.25 
Error  0.0038  2  0.0019  0.0022  


Sum  0.8932  26 
Vibration accelerations of the bridge (
Factor 

dof  Bridge 
Vehicle 









 

1.0772  6.2119  2  0.5386  349.08  0.01  3.1060  182.75  0.01 

1.3357  0.4694  2  0.6678  432.86  0.01  0.2347  13.81  0.10 

0.2923  0.6948  2  0.1461  94.71  0.05  0.3474  20.44  0.05 

0.1977  0.3681  2  0.0989  64.07  0.05  0.1841  10.83  0.10 

0.0858  0.1244  2  0.0429  27.80  0.05  0.0622  3.66  0.25 

0.0478  0.8906  2  0.0239  15.49  0.10  0.4453  26.20  0.05 

0.0613  9.4647  2  0.0306  19.86  0.05  4.7323  278.44  0.01 

0.4733  5.4874  2  0.2366  153.37  0.01  2.7437  161.43  0.01 

0.1068  0.2728  2  0.0534  34.62  0.05  0.1364  8.03  0.25 

0.4521  0.8116  2  0.2260  146.50  0.01  0.4058  23.88  0.05 

0.0637  0.2004  2  0.0318  20.63  0.05  0.1002  5.90  0.25 

0.8217  1.6547  2  0.4108  266.29  0.01  0.8273  48.68  0.05 
Error  0.0031  0.0340  2  0.0015  0.0170  


Sum  5.0183  26.6849  26 
It can be concluded that the influence of factors on different indices are listed as follows.
For DLA of the bridge,
For vibration acceleration of the bridge,
For vibration acceleration of the vehicle,
Obviously, for different indices, the most important influence factors are different. The sensitive factors are the pavement roughness, the span length, and the mass of the vehicle for the indices of the DLA, the vibration accelerations of the bridge and vehicle, respectively. It has to be noted that the DLA is the relative value of the dynamic response and the static response, while the vibration acceleration is the absolute value. It is the result of influence differences between the DLA and the vibration acceleration of the bridge, even though they are correlated in some extent.
As mentioned earlier, the interaction almost exists in all of physical phenomena. When the interaction is so small, it can be ignored in application. However, as for research, we do not know whether the interaction can be ignored or not at first. As a result, the orthogonal design is introduced to solve this problem.
Based on the results from the above section, the first five most important factors influencing the DLA are selected. They are the pavement roughness (
To avoid the mixture, the most important factor has to be arranged at first. It can be seen in Table
Orthogonal array
Run  1  2  3  4  5  6  7  8  9  10  11  12  13  14  15 







Non 



Non 



 
1 


1 

1  1  1 

1  1  1  1  1  1 

2 


1 

1  1  1 

2  2  2  2  2  2 

3 


1 

2  2  2 

1  1  1  2  2  2 

4 


1 

2  2  2 

2  2  2  1  1  1 

5 


2 

1  2  2 

1  2  2  1  1  2 

6 


2 

1  2  2 

2  1  1  2  2  1 

7 


2 

2  1  1 

1  2  2  2  2  1 

8 


2 

2  1  1 

2  1  1  1  1  2 

9 


2 

2  1  2 

2  1  2  1  2  1 

10 


2 

2  1  2 

1  2  1  2  1  2 

11 


2 

1  2  1 

2  1  2  2  1  2 

12 


2 

1  2  1 

1  2  1  1  2  1 

13 


1 

2  2  1 

2  2  1  1  2  2 

14 


1 

2  2  1 

1  1  2  2  1  1 

15 


1 

1  1  2 

2  2  1  2  1  1 

16 


1 

1  1  2 

1  1  2  1  2  2 

1: 1st level, 2: 2nd level (be equal to the 2nd and the 3rd level in above section, resp.), and Non: no factor in this column.
The methods are the same as that in the above section. The results of range analysis and variance analysis are listed in Tables
Range analysis of numerical simulations considering interaction.
Index 














DLA  0.101  0.063  0.009  0.285  0.027  0.012  0.009  0.021  0.008  0.002  0.006  0.019  0.111 

0.222  0.117  0.013  0.607  0.053  0.010  0.018  0.013  0.001  0.001  0.036  0.014  0.099 

0.640  0.350  0.039  0.545  0.063  0.171  0.048  0.050  0.029  0.009  0.112  0.068  0.636 
Various analyses of numerical simulations considering interaction.
Dynamic load allowance (DLA) of the bridge
Factor 

dof  Initial  Modification  






 

0.0411  1  0.0411  26.04  0.05  0.0411  68.61  0.01 

0.0160  1  0.0160  10.12  0.10  0.0160  26.66  0.01 

0.0003  1  0.0003  0.21  >0.25  —  —  — 

0.3257  1  0.3257  206.42  0.01  0.3257  543.94  0.01 

0.0030  1  0.0030  1.90  >0.25  0.0030  5.00  0.10 

0.0006  1  0.0006  0.36  >0.25  —  —  — 

0.0003  1  0.0003  0.21  >0.25  —  —  — 

0.0018  1  0.0018  1.11  >0.25  0.0018  2.92  0.25 

0.0003  1  0.0003  0.17  >0.25  —  —  — 

0.0000  1  0.0000  0.01  >0.25  —  —  — 

0.0001  1  0.0001  0.08  >0.25  —  —  — 

0.0014  1  0.0014  0.90  >0.25  0.0014  2.38  0.25 

0.0493  1  0.0493  31.26  0.05  0.0493  82.37  0.01 
Error  0.0032  2  0.0016  0.0006  


Sum  0.4431  15 
Vibration acceleration of the bridge (
Factor 

dof  Initial  Modification  






 

0.1977  1  0.1977  32528.26  0.01  0.1977  36136.85  0.01 

0.0545  1  0.0545  8967.12  0.01  0.0545  9961.91  0.01 

0.0007  1  0.0007  107.79  0.01  0.0007  119.75  0.01 

1.4716  1  1.4716  242142.36  0.01  1.4716  269004.93  0.01 

0.0111  1  0.0111  1828.29  0.01  0.0111  2031.12  0.01 

0.0004  1  0.0004  65.65  0.01  0.0004  72.93  0.01 

0.0013  1  0.0013  207.47  0.01  0.0013  230.48  0.01 

0.0007  1  0.0007  119.36  0.01  0.0007  132.61  0.01 

0.0000  1  0.0000  1.10  >0.25  —  —  — 

0.0000  1  0.0000  0.50  >0.25  —  —  — 

0.0052  1  0.0052  848.12  0.01  0.0052  942.20  0.01 

0.0008  1  0.0008  129.02  0.01  0.0008  143.34  0.01 

0.0389  1  0.0389  6404.66  0.01  0.0389  7115.17  0.01 
Error  0.0000  2  0.0000  0.0000  


Sum  1.7828  15 
Vibration acceleration of the vehicle (
Factor 

dof  Initial  Modification  






 

1.6400  1  1.6400  14.80  0.10  1.6400  51.80  0.01 

0.4914  1  0.4914  4.44  0.25  0.4914  15.52  0.01 

0.0062  1  0.0062  0.06  >0.25  —  —  — 

1.1866  1  1.1866  10.71  0.10  1.1866  37.48  0.01 

0.0159  1  0.0159  0.14  >0.25  —  —  — 

0.1175  1  0.1175  1.06  >0.25  0.1175  3.71  0.10 

0.0094  1  0.0094  0.08  >0.25  —  —  — 

0.0099  1  0.0099  0.09  >0.25  —  —  — 

0.0034  1  0.0034  0.03  >0.25  —  —  — 

0.0003  1  0.0003  0.00  >0.25  —  —  — 

0.0498  1  0.0498  0.45  >0.25  0.0498  1.57  0.25 

0.0182  1  0.0182  0.16  >0.25  —  —  — 

1.6199  1  1.6199  14.62  0.10  1.6199  51.17  0.01 
Error  0.2216  2  0.1108  0.0317  


Sum  5.3900  15 
It can be concluded that the influence of factors on different indices are listed as follows.
For DLA of the bridge,
For vibration acceleration of the bridge,
For vibration acceleration of the vehicle,
Obviously, apart from the lower damping of the vehicle (
Similarly, for different testing indices, the influence of each factor is not the same.
According to numerical simulations, the sensitivity of each factor for different indices is not the same. Therefore, we should pay more attention to various factors for different purpose in the design of bridges and vehicles. For example, if the bridge is mainly for pedestrians, the acceleration of the bridge is related with their comfort. Then the span length and the pavement roughness should be noted. If the riding comfort is the primary consideration, maybe we should focus on the pavement roughness and the mass of the moving vehicle. However, as for safety of the bridge, the important factors include the pavement roughness, span length of the bridge, mass, and upper stiffness of the vehicle. In short, it has to be noted that the pavement roughness is the essential factor.
Comparing the acceleration of the bridge with that of the vehicle, the former one is more related with the span length, while the latter one is more related with the mass of the vehicle. Actually, the span length of the bridge is significantly related with its fundamental frequency. And the mass of the vehicle will influence natural frequencies of the vehicle. Therefore, in nature, the determined factors are natural frequencies of the bridge and moving vehicles. Also, it is consistent with the conclusion obtained from the simplified theoretical derivation in Section
For vehicle design, stiffness is more important than the damping of the suspension system, especially the upper stiffness. According to the structural dynamics, the damping of the vehicle has less influence on its natural frequencies than the stiffness. Meanwhile, it has been proved that the frequency is the main factor.
Using the method of orthogonal experimental design, the interaction effects are so small that they can be neglected in engineering application.
Furthermore, the plot of relation between DLA and the fundamental frequency shows that the DLA is dependent on many factors, and more synthetical and accurate formulas should be proposed in the code. In addition, for comfort of pedestrians and passengers, the empirical formulas of accelerations of the bridge and the vehicle could be added based on the proposed method in this study.
To get a good understanding of the dynamic responses of the simply supported girder bridge to moving vehicular loads or vehicles, theoretical derivations are presented based on simplified models of the vehiclebridge coupled vibration system; then some numerical simulations based on the proposed method of the orthogonal experimental design in batches have been done by our own program VBCVA.
Firstly, according to simplified theoretical derivations, three factors are proved as the most important factors to determine dynamic responses. They are the mass ratio of the vehicle and the bridge (
Secondly, based on the modal synthesis method, the thought of our own program VBCVA has been given in detail. Combining both advantages of the commercial software ANSYS and MATLAB, the bridge model is found by ANSYS, while the vehicle model and the roughness model are established by MATLAB. It can be used for calculating the cases of multilanes and multivehicles (both in longitudinal direction and transverse direction). Also, the number of axles or vehicles is not limited. Therefore, this program has to be thought of as convenient and powerful enough for the analysis of vehiclebridge coupled vibration problems.
Thirdly, on the basis of the orthogonal experimental design, both the main effects and the interaction effects can be studied. For different indices of dynamic responses, the influences of each factor are not the same. However, the pavement roughness is assumed as the most important factor. In addition, dynamic responses of the bridge and the vehicle are much more dependent on their own fundamental frequency, respectively. Furthermore, the interaction effects have proved to be so small that they can be neglected in engineering application.
In the end, it has to be emphasized that the proposed method of the orthogonal experimental design in batches greatly reduces calculation cost. And it is efficient and rational enough to study multifactor problems.
The proposed method is used for not only the analysis of influence factors, but also the analysis of regression. And it can be applied in all types of bridges, other than just the simply supported bridge. Furthermore, it provides a good way to obtain more rational empirical formulas of the DLA and other dynamic responses, which may be adopted in the codes of design and evaluation.
The authors declare that there is no conflict of interests regarding the publication of this paper.
The research reported herein was sponsored by the China Scholarship Council (the 2013 China StateSponsored Postgraduate Study Abroad Program) and the National Natural Science Foundation of China (no. 50678051, no. 51108132). The authors would like to express their deep gratitude to all the sponsors for the financial aid.