Pergamon Compwrs d Srructwes Vol. 49, No. 5, pp. 837-842, 1993

0 1994 Elwier Science Ltd Printed in Great Britain.

004s7949/93 s6.00 + 0.00

FINITE ELEMENT ANALYSIS OF HIGHWAY BRIDGES SUBJECTED TO MOVING LOADS

M. A. SAADEGIWAZIRI Department of Civil Engineering, New Jersey Institute of Technology, Newark, NJ 07102, U.S.A.

(Received I July 1992)

Abstract-A proper estimate of the dynamic effect of traffic load on the response of highway bridges is becoming increasingly important. Reduction in the ratio of dead load to total load which makes the effect of the live load more pronounced, and the adoption of ultimate-strength design method by many design codes are among those factors that make an accurate evaluation of the live load very important. This paper illustrates how a general purpose finite element package can be used to consider the dynamic effect of a moving load traversing a highway bridge. This objective is achieved by employing the load arrival time option in ADINA. It is shown that, if the problem is modelled properly, the results are as good as the exact solution. Note that the exact solution is practically possible only for a simply supported bridge considering only the effect of the fundamental mode. Thus, using finite element a practical research study can be performed to develop graphical design aides for more accurate evaluation of the impact factor. The versatility and power of tinite element technique will also make it possible to easily investigate the spatial nature of response to traffic load.

INTRODUCTION

Although the behavior of highway bridges has been the subject of numerous investigations in the past, the dynamic effect of a moving load crossing a bridge is still accounted for through a relationship adopted in 1927 [l]. Literature surveys of works on the dynamic effect of a traffic load point to two factors as the reasons for the lack of any change since late 1920s: (i) need for a simple relationship to account for a complex phenomenon, and (ii) adequacy of the adopted equation in light of the fact that few (if any) failures of highway bridges can be attributed directly to the dynamic effect of a moving load.

However, now the U.S.A. is faced with an essential and urgent need to rehabilitate and maintain the infrastructure to provide the required level of service and to cope with growing demand on the highway system. Resources available to address this issue are limited and to optimize the allocation of these fund- ing it is essential to implement new technologies. Finite element technology can play a significant role in management of highway bridges as it relates to maintenance, rehabilitation as well as replacement activities. The advancement in this area has resulted in the ability to rationally analyze and evaluate the load capacity of structural systems with great accu- racy. Saving from accurate load ratings of highway bridges can be hundreds of millions annually. For example, FHWA estimates that the additional user costs associated with commercial traffic detouring around load-restricted bridges can cost from $150,000 annually on the low side, to $200,000 a day on high capacity bridges. Proper estimate of the dynamic effect of a moving load on the response of

highway bridges is an essential component to accu- rate evaluation of load capacity of existing bridges. Furthermore, due to recent advancement in the devel- opment of high strength materials bridges are becom- ing more efficient, while the live load carried by these bridges is increasing. Consequently, the ratio of the dead load to total load is decreasing and the dynamic effect of the live load is becoming more important, especially as it relates to evaluation of the fatigue life of the structure.

Previous analytical studies on the dynamic response of highway bridges subjected to moving load have implemented special mathematical models (analytical programs) developed for this purpose that are not widely available to design engineers [2-4]. However, commercial finite element packages with many capabilities are becoming increasingly avail- able to even small firms. It is the purpose of this paper to demonstrate the use of the load arrival time option, available in programs such as ADINA [5], in modeling a moving load on a bridge. The accu- racy of the method along with possible pitfalls are discussed.

BACKGROUND

The current design of highway bridges uses the following equation to account for the dynamic effect of a moving load (L in feet)

I = 5O/(L + 125) < 0.3. (1)

This equation was adopted in 1927 by a joint com- mittee of American Railway Engineering Association and AASHTO.

837

838 M.A. SAADEGHVAZIRI

Obviously, this simple equation can not account for many parameters that influence the vehicle-bridge dynamic interaction. Among those parameters which influence the vehicle-bridge interaction the most important ones are: dynamic characteristics of the bridge (frequencies, damping, etc.) dynamic charac- teristics of the vehicle, initial condition of the vehicle and the bridge, roughness of the bridge deck and irregularities in the approach. Extensive analytical as well as experimental work conducted at the Uni- versity of Illinois [24] has addressed many of these parameters in details. As a result of this work, it was determined that for a smoothly rolling load crossing a span the speed parameter, a, controls the dynamic amplification. That is, the dynamic deflec- tion is increased over the static value by an amount that depends on a. The speed parameter is defined as follows:

ci = V/ZLf,

where V is the speed of the load crossing the bridge (ft/sec), L is the span length (ft), and f is the natural frequency of the bridge (Hz).

Of course other parameters such as spacing of axles relative to the span length were also identified to be important. The response of three-span con- tinuous bridges was also examined. Consequently, an alternative impact factor was suggested as follows:

I=O.i5+cc.

The constant term represents the effect of the initial condition and the speed parameter accounts for the dynamic interaction between the vehicle and the bridge. However, this proposed formula which has a much greater relation than eqn (1) to the observed behavior of highway bridges under traffic load was not adopted in any design code.

The need for change is essential for the fact that bridges are becoming more slender and trucks are

I,(?)

T I;

getting heavier. Furthermore, most design codes are adopting ultimate-strength criteria and a more realistic estimate of the dynamic effect of live load is essential to the application of this design methodology. Indeed, the Ontario Highway Bridge Design Code has taken the first step by relating the Dynamic Load Allowance factor to the natural frequency of the structure rather than the span length [6]. In a report by the ASCE committee on loads and forces on bridges [7] the need for research in areas of bridge loading is examined. As it relates to impact, it is stated that: there is as much need to bring uniform practical application in a coherent way to the large body of fundamental knowledge that has already been acquired on the subject of bridge impact, as there is to further expand the field of knowledge in this area. One of the research problems is identified as: Preparation of Design Aids Based on Rational Methods of Analysis for Dynamic Live Load Allowances due to Traffic Loading on Highway Bridges. The finite element technique can provide the means to convert the rational method- ologies for calculating dynamic load increments to usable charts and other graphical design aids.

FINITE ELEMENT MODEL

Load arrival time option

The arrival time flag in ADINA allows for acti- vation of the load at a time that may be different than the solution time [5]. That is, with a nonzero arrival time (ARTM) the loading time function will be shifted equal to the arrival time specified. Thus, the load will be zero for those solution time t ,< ARTM, and it is active for solution time t > ARTM. Note that for t = ARTM the loading is not active. There- fore, for a solution time starting from zero and an ARTM equal to twice of the solution time step (i.e. 2At) the corresponding time function and load values are shown in Fig. I.

' SOLUTION PER100 c

Fig. 1. The arrival time option: associated time function and resulting load values.

1 7At i 0.0 1

At = time step

No. of Steps = 8

Finite element analysis of highway bridges 839

# of elements = 24

# of nodes = 25

T IS varyng lo give different 0

I 3 I 0.06

0.96

Fig. 2. A typical mesh and corresponding ARTM corresponding to V = 1000 in/set

Thus, by defining time functions and concentrate loads with appropriate duration, 7p, and arrival times one can easily simulate the effect of smoothly moving truck with multiple axles.

Finite element mesh

Although one can use 3-D elements to model the bridge and investigate the spatial effect of traffic load, in this work the bridge is modelled using beam elements. The number of elements is controlled by the slowest velocity of the moving load to be simulated and the period of the bridge.

The length of each element, AL, is equal to L/N, where L is the span length and N is the number of elements. Then, the time to traverse one element (i.e. to go from node i to node i + l), TV, is equal to AL/V, where V is the velocity of the moving load. The ratio of 7c, to I, period of the highest mode with significant contribution to the response of the sys- tem, must be as small as possible in order to avoid unrealistic impact amplification. Furthermore, to avoid causing a random type of loading the value of 7p must be always equal to 7,. That is, the load must be always acting on the system, although it will be at different locations at any given time. Other truck speeds, faster than the speed used in evaluation of 7,) can be modelled by applying the load to every several other node depending on the desired speed to be represented. These points are discussed further in a following section under modeling considerations.

Therefore, the mesh to model a single axle vehicle (say 40 kips weight) traversing a bridge with span equal to 960 at a velocity of 1000 in/set (56.8 mph) will be as shown in Fig. 2. Faster vehicular speed such as 2000 in/set can be modelled by applying the load to every other node (i.e. nodes 1, 3, 5, . ., 25 with arrival times equal to 0, 0.04, 0.08, . . ., 0.48 set). For slower velocity of the truck either finer mesh can be used or the load duration can be elongated to accom- modate lower velocity. The later option can be employed as long as the ratio of the load duration, 7,,,

to system period, T, is low enough for accuracy as discussed in the following section.

RESULTS

In the following examples effects of surface rough- ness as well as the initial conditions of the bridge and the truck are not being considered. Furthermore, it is assumed that the weight of the moving load is negligible compared to the weight of the bridge, and the magnitude of the moving load is constant.

Simple-span bridge

The mid-span deflection and moment of a simply supported beam traversed by a constant force ignor- ing damping is determined using finite element and the results are compared to the exact solution. The exact solution for deflection, Y as given by Walker and Veletsos [2], is as follows:

2PL3 m Y= -1 714EL.=,

1 t

sin 2nm- t

sin 2an 2-

X T, Tb nnt

n2(n2-cz) -Cl

n3(n2--cx2) 1 sin - ,

L

where P is the magnitude of the moving load, EI is the bridge Youngs modulus and moment of inertia, L is the span length, c( is the speed parameter, Tb is the fundamental period of the bridge, and n is the mode number.

To simplify the exact solution only the fundamen- tal mode is considered. Furthermore, the quantity 2PL3/n4EI which represents the first term series approximation to the static mid-span deflection is replaced by the exact value of mid-span static deflection, y,, . Thus, the central deflection takes the following form

[

t sin 27ca -

Tb sin 2n L

T, Y, = .u2 = YSI l-u2 -a1_a2 I

840 M. A. SAADEGHVAZIRI

Table 1. Dynamic amplification factors for exact and FE solutions

Speed parameter, a 0.1 0.2 0.3 0.4 0.5 0.6

DAD exact solution 1.09 1.08 1.41 1.62 1.73 1.76 DAD finite element 1.09 1.07 1.40 1.60 1.70 1.74 DAM/DAD FE 0.822 0.822 0.822 0.822 0.822 0.822

DAD = Dynamic amplification factor for deflection; DAM = Dynamic amplification factor for moment.

which is maximum when

cos 27ru + = cos 2n f b b

For different values of CI the exact solution is compared to the finite element solution employing modal superposition method and considering only the fundamental mode (note that for typical high- way bridges tl = 0.06 to 0.2). The values of dynamic amplification factor (i.e. ~/JJ,,) for both methods are given in Table 1. The finite element solution compares very well to the exact solution. The ratio of moment amplification factor to that for deflection for the finite element solution is also given in this table which is

exactly the same as the exact value of rt r/l2 = 0.8224 [2] for all values of speed parameter.

An interesting point to consider is the contribution of higher modes to the value of the reactions. As seen from Figs 3 and 4, the mid-span deflection and moment are not affected by consideration of higher modes, however, the time history of the left reaction (Fig. 5) does indicate significant contribution by the higher modes. Note that considering single mode response there is a sudden drop in the reaction as the load passes the abutment and enters the span. This is due to low contribution of the first mode (sinusoidal shape) to reaction when the load is close to the supported end. Considering only the first mode the

Fig. 3. Center span deflection.

f 4@JO t .4 /Multi modt

Time, WC

Fig. 4. Center span moment.

Finite element analysis of highway bridges 841

Time, set

Fig. 5. Reaction at entering abutment.

L

a = 0.8: cu = VC?Lf, f, = first mode frequency

Fig. 6. A three-span bridge.

Table 2. Dynamic amplification factors for deflection (three-span bridge)

0.1 Speed parameter, a

0.2 0.3 0.4 0.5 0.6

Exact solution [3] 1.06 1.14 1.30 1.20 1.74 2.02 Finite element solution 1.07 1.15 1.27 1.18 1.74 2.02

maximum reaction is equal to 41.4 kips. With this value the ratio of shear amplification factor to that of deflection is equal to 0.646 which is exactly equal to the theoretical value of 7~ /48 given by Walker and Veletsos [2]. For a multi-mode solution the maximum reaction is 46.35 kips which is 12% higher than the value obtained by single mode solution. Note that a similar observation has been made by Walker and Veletsos 121.

The effect of higher modes is included simply by considering more than one mode in the modal super- position analysis, or one can simply consider the effect of all modes by performing direct time history analysis rather than modal supe~osition. This ihus- trates the power of the finite element method and the contribution that it can make to improvement of design guidelines.

Multiple-swan bridge

The three-span bridge shown in Fig. 6 was also analyzed for different values of CL The results for a = 0.8 are given in Table 2 along with the exact solution evaluated by Nieto-Ramirez and Veletsos [3] using a special-purpose program.

Similar to the simple-span bridge the finite element results are in very good agreement with the exact solution. Some points to be considered for a good balance between accuracy and practicality are discussed in the next section.

~OD~LLING CONSIDERATIONS

Through two examples it was demonstrated that general purpose finite element packages can be used to model the effect of moving loads traversing a bridge. However, there are a couple of points that need to be considered in developing the finite element model.

The first point to consider is the duration of the loading. This is determined by the velocity that is being represented, which in turn is related to the size of the elements used. For example, if a simply supported bridge of length 960 is to be traversed by a truck with lOOOin/sec velocity and the mesh con- sists of only four elements (i.e. five nodes), then the duration of loading must be 0.24 sec. However, for most cases this will not give accurate results due to high ratio of Q. In this example, for a = 0.2 the

842 M. A. SAADEGHVAZIRI

system period would be 0.384, which gives a 0.625 ratio for rp/T. The analysis of the bridge results in a deflection amplification factor equal to 1.4, which is significantly different than the exact value of 1.08 as shown in Table 1. If the mesh is refined to eight elements (i.e. 9 nodes) the load duration will decrease to 0.12 sec. With this mesh the deflection amplifica- tion factor will be 1.089, which is a very good estimte of the actual value. Further refinements of the mesh changed the results slightly. The number given in Table 1 is for a mesh with 24 elements. This is taken as the finite element solution because further refine- ment of the mesh (up to 192 elements) did not make any difference in the value of the dynamic amplifi- cation factor. So, the results indicate that a mesh must be used such that the ratio of to/T is less than 0.1, however, this ratio does not have to be very small.

Another point to consider is the fact that the load must be always acting on the system. That is one can not reduce the load duration rather than refinement of the mesh. For example, for the problem discussed, one may decide to use tP = 0.12 with a four elements mesh. Therefore, in order to represent the same velocity the arrival times are 0.0, 0.24, 0.48, ., 0.96 at the first, second, third, . ., and last node. This gives a DAF equal to 0.57, which is grossly in error. So, the load duration must be always equal to the time required to traverse one element.

CONCLUSIONS

It is shown that general purpose finite element packages can be used to model the dynamic effect

of moving loads traversing highway bridges. With the power and versatility of finite element, an analyst can easily investigate the dynamic effect of multi-axles trucks coming from both directions at different intervals. Furthermore, using 3-D elements, the spatial nature of response can be also investigated.

REFERENCES

I. AASHTO, Standard specifications for bridges. Ameri- can Association of State Highway and Transportation Officials, Washington, DC (1989).

2. W. H. Walker and A. S. Veletsos, Response of simple- span highway bridges to moving vehicle. SRS No. 272, Department of Civil Engineering, University of Illinois, Urbana, Illinois, Sept (1963).

3. Nieto-Ramirez and A. S. Veletsos, Response of three- span continuous highway bridges to moving vehicles. Engflg Experi. Station Bull. 489, University of Illinois, Urbana, IL (1966).

4. A. S. Veletsos and T. Huang, Analysis of dynamic response of highway bridges. J. Engng Mech. Diu., ASCE %, EM5 (1970).

5. ADINA Engineering, Automatic dynamic incremen- tal nonlinear analysis. Report ARD 87-1, ADINA Engineering, December (1984).

6. Ontario Highway Bridge Design Code, 2nd Edn. Ontario Ministry of Transportation and Communi- cation, Downsview, Ontario, Canada (1983).

7. Bridge loading: research needed. Committee on Loads and Forces on Bridges of the Committee on Bridges of the Structural Division. J. Strucf. Die., AXE 108, ST5 (1982).