Solution | Moving Load Analysis

1. What is Moving Load?

A moving load is a load that moves over a certain distance. Let’s narrow it down to a moving load acting on a structure. In case of buildings, the loads that affect the floor of the building, such as people and furniture, can be all called moving loads. So then, what are the moving loads on civil engineering structures such as bridges? In case of bridges, there are four types of moving loads: vehicle loads, train loads, pedestrian loads, and special loads.

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Fig. Moving load of structures
Fig. Moving load on a bridge

2. Moving Loads on a Bridge

When designing a bridge, each country’s design code divides moving loads into four categories: automobiles, special vehicles, trains, and people. Let’s find out the characteristics of each type of moving load.

A. Vehicle Loads

Vehicle loads applied to Highway Bridge designs are proposed to bridge designers in various ways according to the design conditions of each country. To begin with, there are various types of vehicles with different sizes and shapes. Therefore, the load on the actual vehicle is modeled as a Notional load and applied as a Design load. A model of the effect of one vehicle is referred to as the design truck load(or other vehicle load). Although the load is smaller than the design truck load, the modelling effect when these vehicles are carried is called a design lane load. A truck load is mainly expressed as a concentrated load acting on each wheel while a lane load is expressed as a uniformly distributed load.

Fig. Example of design truck load
Fig. Example of design lane load

B. Train Loads

Train load, like vehicle loads, are loads frequently used in bridge design. It is characterized by being very load compared to vehicle loads. Moreover, trains travel only on the installed railroad tracks, concentrating the train load above them. Train loads acting on railway bridges are larger than live loads acting on highway bridges, and the variation in the stress at each member of the bridge is greater. The variation of the stress greatly affects the fatigue strength of the reinforced concrete. Therefore, in railway bridges, checking the fatigue strength is important, and at the same time, the vertical deflection of the bridge is very strictly limited. In the same manner as vehicle loads, train loads are characterized by the design conditions of each country.

Fig. A train above a railway road, Train load in design codes

C. Pedestrian Loads

A Pedestrian load is represented by the weight of a person. It is also referred to as crowd load. Not only vertical loads, but also loads in the transverse direction are transmitted due to the characteristics of a person’s gait. The size of the load is small compared to a vehicle or train load, but caution is required when many people are up on a bridge. The reason is that if the vibration of a person’s gait and the natural frequency of a bridge match, a resonance phenomenon occurs, causing large vibrations to the bridge.

Fig. Pedestrian loads

D. Special Loads

Tracked vehicles such as tanks, heavy equipment vehicles, and vehicles for bridge inspection are all examples of vehicles that generate special loads. These vehicles have a greater effect on bridge design because the magnitude of the loads are greater than that of moving loads, and have a greater volume compared to other vehicles.

Fig. Special vehicle loads

3. Moving Load Analysis for Bridge Design

A moving load is considered as a live load or variable load when designing a bridge and it must always be considered for bridge design after dead loads. Moving loads have a moving characteristic, so it is necessary to find the maximum member force that occurs in the bridge by changing the position of the load in the direction of the bridge axis or in the direction perpendicular to the bridge axis.

Fig. Moving load analysis for bridges

A. Existing Method

The analysis result is obtained by creating several model files for the moving load analysis and changing the location of the moving load. The longer the bridge, the more model cases that take into account the location of moving loads, elongating the time spent on modeling. In addition, since the interval at which the moving load is moved is set by the designer, there is a risk that the result values will may vary as the spacing is dense. For the moving load, the load type suggested in the design standards is applied, and depending on the type of Element Analysis model, the load is selected and applied.

Fig. Models for moving load analysis

B. Influence Line Analysis Method

When a structure is under live loads or moving loads, the change in shear forces and bending moments in a particular section, depending on the loading position of the load, can be best represented by influence lines. An influence line is defined as a line (straight or curve) that represents the variations in reaction forces, shear forces, bending moments, or deflection at any particular point in the structure when as a single concentrated load moves along the structure. Once the influence line is drawn, one can easily find the location at which the moving load has the greatest impact on the structure. Furthermore, the magnitude of the reaction force, shear force, and moment involved at that point can be calculated from the ordinate of the influence line. For this reason, influence lines play an important role in the design of bridge structures in which loads move over the entire span of the structure.

Fig. Calculation of maximum minimum moment using the Influence lines

The figures below show a possibility that we could obtain more critical results in the case which has fewer loads. Therefore, influence lines analysis enables an efficient review of the maximum or minimum member force.

Fig. Comparison of results between full loading case and alternate loading case

The influence line analysis differs depending on the elements constituting the model. A beam element uses influence line analysis while a plate element uses influence surface analysis.

B-1. Influence Line Analysis

Bridge behaviors governed by main girders or 2-dimensional elevation analysis of bridge(Steel box girder bridge, etc.).

Influence line presented along the traffic lane elements(beam elements).

Fig. Influence line in a program

B-2. Influence Surface Analysis

Large variation of structural behaviors under moving loads in the transverse direction(Slab bridge, rigid frame bridge, etc.).

Influence surface presented on the traffic surface lane elements(Plate elements).

Fig. Influence Surface in a program

4. Moving Load Analysis Post-Processing

What should the designer check after performing the Moving Load Analysis? Just as with other Static analysis results, designers should check the analysis results for member forces, deflections, or stresses. If the Moving Load Analysis was performed as a Static Load Analysis for multiple models using either the Nodal or Beam load, the analysis results of each analytical model can be checked. If the designer used a program that supports the Influence Analysis, then the maximum and minimum results should be checked and also the corresponding results as well.

Corresponding Forces / Concurrent Forces

A moving load is one of the many variable loads. Therefore, the maximum/minimum analysis results for all elements may not be calculated from loads in the same position. This characteristic appears when checking the analysis results for variable loads. Programs that support the Moving Load Analysis sometimes show only the maximum/minimum analysis result values for ease of review. Corresponding forces separately provide the information users want. The following example in an example of Corresponding forces.

Fig. Enveloped results and corresponding results

5. Considerations for Moving Load Analysis

Country-specific design standards summarize and outline the points to be considered for Moving Load Analysis. The basic framework is similar, but some specific details follow the standards of each country. Let’s find out briefly below.

A. Definition of Carriageways

Bridges are often part of road projects, so one can refer to the design lane width and number of lanes planned by the road planning department. However, the design standard provides an equation for determining the total number of lanes using the total road width and the design lane width.

Fig. Number of notional lanes

B. Lane Factors

The Simultaneous Loading Factor is a coefficient introduced based on the fact that the probability of simultaneous loading of a design vehicle load on two or more lanes is relatively small compared to that of a single lane. In order to take this into account in Japan, the UK, and the Eurocode, the magnitude of the vehicle loads is different for the main loading lane and longitudinal loading lane. On the other hand, in Korea, the United States, and Canada, vehicle loads of the same magnitude, multiplied by the simultaneous loading factor, are applied to all loading lanes. This same principles also applies to the loading of Pedestrian Loads.

Since the probability that the vehicle load and pedestrian load are simultaneously loaded is relatively small compared to the case where only the vehicle load is loaded, it is reasonable to apply the simultaneous loading factor even if the pedestrian load and the vehicle load are included at the same time.

Fig. Multi lane loading factor example

C. Live Loads

For the moving load analysis, it is important to know which vehicle load is applied to the analysis. Fortunately, the design standard proposes the ideal vehicle load to the designer. Designers simply apply these loads to the bridge model and proceed with the structural analysis. The vehicle load model is designed according to the standards of each country. In this article, we will briefly explore the vehicle models proposed by the two most representative design standards: the Eurocode and the AASHTO LRFD design standards.

C-1. Eurocode

The Eurocode proposes vehicle loads to be divided into Vertical and Horizontal forces. There are four load models in Vertical forces. Load model 1 includes uniformly distributed loads and wheel axle loads. Load model 2 includes single wheel axle loads. Load model 3 includes SUVs and Special loads. Load model 4 includes crowd loads. Horizontal forces include braking & acceleration forces and centrifugal & transverse forces. In addition, pedestrian and bicycle loads are included as uniformly distributed loads. The above three types of loads are combined into a total of 6 Group of loads and are used as design loads.

Fig. Vertical loads of traffic loads in Eurocode

Fig. Transverse loads of traffic loads in Eurocode

Fig. Groups of traffic loads in Eurocode

C-2. AASHTO LRFD

AASHTO LRFD proposes vehicle loads to be divided into Design truck, Design tandem, and Design lane load. These loads are classified as Live loads (LL). These loads are able to adjust to suit design environments. For pedestrian loads, a uniformly distributed load is applied and carried along with vehicle loads. These pedestrian loads are also called PL. Unlike the Eurocode, there are no groups of vehicle loads, and the above loads must be properly combined to achieve maximum results.

Fig. Vertical loads of traffic loads in AASHTO LRFD

D. Dynamic Effect

According to the design code, Dynamic effect can also be called dynamic allowance or dynamic amplification. This is an increment value applied to the static wheel load to account for the wheel load impact caused by moving vehicles. According to national design standards, this value must be included in the vehicle load or must be incremented separately.

Fig. Dynamic allowance presented by the AASHTO LRFD

Fig. Additional dynamic amplification presented by the Eurocode

E. Centrifugal Forces

Vehicles driving on curved bridges generate centrifugal force, which is a transverse load. The equation for calculating the centrifugal force is specified for each design standard, and the main parameter is the radius of the curve.

Fig. Centrifugal force

F. Braking & Acceleration Forces

Braking force is the longitudinal force generated on the bridge deck, in the direction of the vehicle headway direction, when the vehicle comes to a sudden stop. Acceleration force is the longitudinal force generated on the bridge deck, in the direction of the vehicle headway direction, when the vehicle suddenly accelerates. Braking and acceleration forces mainly occur due to friction between wheels and a surface of a carriageway. And those forces have the same magnitude of the force but the direction is opposite each other. Those forces may or may not be considered depending on the design environment. The magnitude of the load is calculated according to the equation in the design standards of each country.

Fig. Braking and acceleration force

G. Fatigue

Fatigue is a phenomenon in which the strength of a material is weakened when repeatedly subjected to a load that has a magnitude less than the yield strength of the material. Fatigue fracture is when a structure suddenly fails after being subjected to cyclic loads. For example, when a wire is repeatedly bent and straightened, it will fail after reaching a certain number. Bridges are also structures that are subjected to fatigue loads. The most common fatigue loads present in bridges is caused by the moving vehicles. Bridges that do not consider the fatigue strength caused by repeated vibrations caused by the vehicle loads often experience fatigue failure. Therefore, to do the fatigue design of a bridge, vehicle loads must be considered in the design process. For fatigue analysis, the magnitude of the vehicle load used in the moving load analysis is not used, but a reduced value. For AASHTO LRFD, the magnitude of the load for the fatigue analysis as well as the dynamic allowance are implemented. In the case of the Eurocode, 5 load models are implemented for fatigue analysis.