Probabilistic Gross Vehicle Weights and Associated Axle Loads for Military Vehicles in Bridge Evaluation and Code Calibration

Military vehicles frequently use civilian bridges. The loading effects of military vehicles, both wheeled and tracked, are specific and different than those of civilian vehicles in normal traffic. Calibration to determine appropriate load factors for military loading of civilian bridges has not been fully performed and the corresponding levels of safety have not been quantified. This is partially due to the lack of probabilistic information of the gross vehicle weights and corresponding axle loads of military vehicles while operating in real‐world conditions. This paper quantifies probabilistically the gross vehicle weight and axle loads for three military vehicles in use by NATO, each of which is representative of: military transport vehicles; armoured personnel carriers; and main battle tanks. A general means are proposed to quantify the probabilistic gross vehicle weight of military vehicles on the basis of maximum nominal payload as a proportion of the total nominal vehicle weight. Based on observed probabilistic gross vehicle weight of military vehicles, it is recommended to differentiate between military transport and military fighting vehicles as different categories of vehicles in bridge evaluation.


Introduction
Military vehicles frequently use civilian bridges in domestic, peacekeeping, stabilization and combat theatres of operation.The load effects of military vehicles, both wheeled and tracked, are unique and likely different than those of civilian vehicles in normal traffic.The probabilistic quantification of military vehicle bridge loading has not been fully performed.Calibration to determine appropriate load factors for military loading of bridges therefore cannot be undertaken and the corresponding reliability is unknown.This lack of calibration prevents the proper implementation of Limit States methods (e.g., Canadian Standards Association [1]) in military bridge design and evaluation.The objective of the research reported in this paper is to quantify probabilistically the gross vehicle weight and axle load of military vehicles and so partly address the shortfall of information needed for this calibration.
Three vehicles were investigated: the Armoured Heavy Support Vehicle System -Palletized Loading System (AHSVS-PLS); the Light Armour Vehicle III -Infantry Section Carrier (LAV III-ISC); and the Leopard 2A4M tank.They were selected because they represent three distinct loading categories: they have either transport or fighting functions and are either wheeled or tracked.This paper quantifies probabilistically the Gross Vehicle Weight (GVW), defined as the sum of the curb weight and payload.The curb weight is the weight of the fuelled vehicle and, if uparmoured (which relates to vehicles that have optional armour kits to achieve different levels of protection), additional armour including mine protection.The payload weight consists of cargo, crew, ammunition, communications equipment, consumables (i.e.extra fuel, water, food, etc.), secondary weapons, crew's personal equipment and mission-specific equipment.The nominal weight of the vehicle is the published combat weight, the sum of the nominal curb weight and nominal payload weight.

AHSVS-PLS (Wheeled-Transport)
Fig. 1 shows the AHSVS-PLS, a Palletized Loading System (PLS) variant of the militarized Mercedes-Benz Actros truck that fulfils various heavy logistics functions as a military transport vehicle.Fig. 2 shows the idealized axle loads in kg for the curb "weight" (above) and combat "weight" (below).The AHSVS-PLS facilitates loading / unloading of 6.1 m (20 ft) long intermodal shipping containers, its primary payload, without the need of an external lift by using its PLS.
The weights of intermodal shipping containers flown by the Canadian Armed Forces from Kandahar Afghanistan between 2006 and 2012 are assumed to be representative of intermodal shipping containers transported by the AHSVS-PLS.A query of the Department of National Defence (DND) National Material Distribution System (NMDS) for 6.1 m intermodal containers yielded 3 723 unique intermodal containers [2].The mean mass of these containers is 6 880 kg with a Coefficient of Variation (CoV), defined as the standard deviation divided by the mean, of 0.415.Using Weibull plotting positions sample data was fitted to Exponential (shifted), Normal, Log-Normal, Gumbel, Weibull, and Rayleigh (shifted) distributions.Linear regression of the transformed data was used to determine the best-fit slope and y-axis intercepts values, from which the parameters defining each distribution were computed.The fitted Log-Normal and Gumbel distributions were in closest agreement with the data.
The two corresponding root-mean-square errors are 0.0076 for the Log-Normal distribution and for the Gumbel distribution 0.0073.The fit of the Cumulative Distribution Function (CDF) to the data was tested using the Kolmogorov-Smirnov (K-S) test [3] at a significance level of 10 % (e.g.α = 0.10).Only the best-fit Log-Normal and Gumbel distributions passed this test.To simplify subsequent computations, the Probabilistic Gross Vehicle Weights and Associated Axle Loads for Military Vehicles in Bridge Evaluation and Code Calibration 131 best-fit Gumbel distribution with β = 2 247 kg and µ = 5 583 kg was selected to describe the "weight" of the intermodal shipping containers.

Fig. 1 AHSVS-PLS (image courtesy of Neil Peacock)
The bias coefficient and variability of the overall weight is quantified assuming curb "weight" to be deterministic, at 22 900 kg [4].It has been demonstrated by MacDonald [2], based on the flown weights of Canadian military vehicles, that this is a reasonable assumption for this vehicle type.Only the intermodal shipping container weights (i.e., the payload) therefore contribute to the overall vehicle weight variability.

Fig. 2 AHSVS-PLS idealized axle loads [kg] and spacing [m]
With these assumptions, the best-fit Gumbel distribution for the event "weight" of the AHSVS-PLS can be derived.Tab. 1 presents the central tendency and dispersion parameters, bias coefficients (defined as the mean GVW divided by the nominal combat weight) and CoV for the event vehicle, which represents the overall population of AHSVS-PLS vehicles.It is more common to quantify bridge traffic loadings for design or evaluation at Ultimate Limit States using the maximum load that would occur over a one-year period (e.g., Kennedy et al. [5]).Therefore, Tab. 1 also presents the parameters derived for annual traffic volumes of 100, 1 000, 10 000 and 100 000 vehicles per year.As the event data are assumed to follow a Gumbel distribution, the maximum annual "weights" also follow Gumbel distributions with the dispersion parameter, βA, given by: and the central tendency parameter, µA, given by: where: • β and µ are the dispersion and central tendency parameters of the event distribution; • n is the number of vehicles per year; and • ni is the number of vehicles for the reference population (in this case ni = 1 for the event distribution).

LAV III-ISC (Wheeled-Fighting)
Fig. 3 shows the uparmoured LAV III-ISC, a vehicle that primarily serves as an Armoured Personnel Carrier (APC) for one infantry section, but can also be armed to provide additional firepower.This vehicle is best categorized as a military fighting vehicle and so it serves a very different function than the AHSVS-PLS.Fig. 4 shows the idealized axle loads in kg for the curb "weight" (above) and combat "weight" (below).Nominally, the combat "weight" of a fully laden LAV III-ISC consists of a curb "weight", including uparmour, of 16 744 kg and a payload of 3 256 kg.It is assumed that the curb weight of the LAV III-ISC can be considered deterministic.Lacking field data, Tab. 2 presents the assumed parameters for the various operational payload components of the LAV III-ISC GVW.Unknown operational payloads are assumed to vary uniformly across the range of each parameter shown, which is intended to conservatively envelope, by neglecting vehicles with less than the nominal combat weight, the actual parameter range.Each component of the GVW was assumed independent.
Using the data summarized in Tab. 2, 10 000 vehicle weights were randomly generated, yielding the simulated event data shown in Tab. 3. The Log-Normal distribution best fits these simulated event data and was used to derive the CDF of the annual maximum weight using the mapping: where: Probabilistic Gross Vehicle Weights and Associated Axle Loads for Military Vehicles in Bridge Evaluation and Code Calibration 133 • FA(x) is the cumulative probability at weight x for the maximum observed value of n observations; • FE(x) is the event cumulative probability at x; • Several different annual traffic volumes were considered, yielding the statistical parameters summarized in Tab. 3.

Component of GVW Nominal Quantity
Mass [kg] Assumed "Weight" Variability for Idealization

Miscellaneous
Equipment / Stowage Crew and Personnel with Combat Gear 10 [6]  1 300 Mass of each soldier 136.5 kg [9]   Total (Combat Weight) 20 000 [6]   Note: Payload is normally distributed with parameters µ = 4 904 kg, σ = 643 kg When the annual traffic volume equals 100 or more vehicles per year, the weight of the maximum annual vehicle is best described by a Gumbel distribution.One might therefore expect that the dispersion factor β would remain constant.The dispersion factors shown in Tab. 3 change slightly for each value of n however, because, the Gumbel fit to the values determined using a Log-Normal event distribution is good but not perfect.

Leopard 2A4M Tank (Tracked-Fighting)
Fig. 5 shows the Leopard 2A4M tank.It is also a military fighting vehicle, primarily used to provide direct weapon fire support, and so designed primarily to ensure the mobility and survivability of the primary weapon system.When compared to the LAV III-ISC, a larger proportion of its GVW is the curb weight; mostly due to requirements for the primary weapon system and armoured protection.The curb weight of the Leopard 2A4M tank, 59 484 kg, can be assumed to be deterministic [2].It consists of the Leopard 2A4M tank chassis, main gun and turret, Add-on-Armour (AoA), slat armour system, and full fuel load.Tab. 4  assumed parameters for the various operational load components.The crew, consisting of four persons at 75 kg each, is also assumed deterministic.The nominal masses of the various operational weights are quantified from various DND sources and are sufficient to increase the nominal curb weight to the nominal combat weight.These operational weights are assumed to vary uniformly across the range of each parameter shown, which is again intended to conservatively envelope the actual parameter range.
The potential for an additional operational load of up to ten infantry riding on top of the tank is also considered.Each component of the GVW was assumed independent.Mass of each soldier 136.5 kg [9]  Total "Weight" 61 214 Using the data summarized in Tab. 4, 10 000 vehicle weights were randomly generated, yielding the event statistics shown in Tab. 5. Above the 35 th percentile, a Weibull distribution has an excellent fit to the simulated data, (passing the K-S test at a significance level of 10 %).The annual maximum statistical parameters for the Leopard 2A4M tank GVW are also shown in Tab. 5.

Relationship between Payload Weight Fraction and Vehicle Weight Variability
The assumption that curb weights are deterministic and only payload weights are stochastic causes the curb weight to influence the statistical parameters for the overall load.A particular payload may be associated with a vehicle depending upon its function.The statistical parameters for the GVW will therefore likely be related to the payload weight fraction, γ, defined as: where WP is the nominal payload and WV is the nominal vehicle combat weight.The nominal vehicle weight can be computed from the curb weight of the vehicle, WC, as: Since the curb weight is assumed deterministic, the mean vehicle weight, V W , is given by: where δP is the bias coefficient of the payload weight.Using Eq. ( 4) to eliminate WP and Eq. ( 5) to eliminate WV, Eq. ( 6) can be written as: The bias coefficient of the vehicle weight, δV, is simply the ratio of Eq. ( 7) to Eq. ( 5):

(
) Since all variability of the vehicle weight is due to the payload, the standard deviation of the vehicle weight, σV, equals the standard deviation of the payload weight, σP.After some manipulation, the standard deviation of the vehicle weight is:

Probabilistic Gross Vehicle Weights and Associated Axle Loads for Military
Vehicles in Bridge Evaluation and Code Calibration 137 where VP is the CoV of the payload.By dividing Eq. ( 9) by Eq. ( 7) the CoV of the vehicle weight, VV, is: The payload bias coefficient and CoV for the various levels of maximum annual volume of vehicles as calculated from Eq. ( 8) and Eq. ( 10) respectively is summarized in Tab. 6.The payload for the two fighting vehicles (LAV III-ISC and Leopard 2A4M) is characterized by a high bias and low CoV, while the AHSVS-PLS, a military transport, is characterized by a payload with a low bias and high CoV.

Axle Loads
To assess the reliability of short-span bridges, the statistical parameters for axle loads are required.In this section, suitable parameters are derived from the GVW parameters.

AHSVS-PLS (Wheeled-Transport) Axle Load
As shown in Fig. 6, the four axles of the AHSVS-PLS are in fact two tandem axles.Thus the axle loads can be estimated from the total load by idealizing the AHSVS-PLS as a simply supported span between the tandem axle centers.When the eccentricity of the payload resultant force extends beyond the rear tandem axle, a cantilever is assumed.Based on the nominal curb "weight" and axle loads for the AHSVS-PLS, the curb weight is represented as a point load (shown as black arrows labelled "C") located 1.52 m from the front support.The nominal maximum payloads are also represented as point loads (shown as white arrows labelled "P"), applied at 0.56 m in front of the rear support of the AHSVS-PLS.It is assumed that, if there is no eccentricity of the shipping container centers of gravity, the payload will act at this point for any given weight.Thus simple statics can estimate the loads on each axle, assuming that the tandem axle loads are shared equally.
To generate realistic axle loads, intermodal shipping containers were randomly generated by simulation given known eccentricities of the shipping container resultant force.Most of the available eccentricities are held by ETS Consulting, United Kingdom (http://ets-consulting.org/).The data made available by Brassington [11] for the current study indicated that for shipping containers less than 30 tonnes, the eccentricities are closely approximated by a Half-Normal distribution with standard deviation, σ, of 0.226 m.For shipping containers greater than 30 tonnes, a Half-Normal distribution with σ of 0.140 m is appropriate.Thus the variability of the eccentricity of the container resultant force is less for the heaviest shipping containers, perhaps because, for the heavily loaded containers it is more difficult in practice to load the container asymmetrically.It was therefore assumed that the longitudinal eccentricity of the container resultant force is normally distributed about the midpoint of the container.

Fig. 6 Idealization of AHSVS-PLS, spacing given in metres (a) Vehicle axle spacing; (b) Idealized representation
A total of 10 000 vehicles were generated and analysed to yield the event axle loads shown in Tab. 7. The loads on the first and second axles are primarily due to the curb weights and so the bias coefficient is close to 1.0 and the CoV is small.The loads on the third and fourth axles are primarily due to the shipping container weights and, although they average only 60 % of the nominal combat value, have a much greater CoV.

LAV III-ISC (Wheeled-Fighting) Axle Load
The bias coefficient and variability of the payload eccentricity for the LAV III-ISC is not available in the literature, so it was assumed that the payload only acts on the 3 rd and 4 th axle, with the load being shared equally.This results in changes to the payload only affecting the rear two axles.
Probabilistic Gross Vehicle Weights and Associated Axle Loads for Military Vehicles in Bridge Evaluation and Code Calibration 139 Tab. 8 shows the statistical parameters for the LAV III-ISC event axle loads, as generated by simulation.The bias and CoV are greater for the 3 rd and 4 th axles because the payload is assumed to be carried only by these two axles.Had the payload been assumed to act such that some of the payload would have been shared with the front axles the bias and CoV of the rear axles would have been lower.In considering shear and moment for short spans it is conservative to assume the payload acts only on the rear axles as this produces a greater bias and CoV in axle loads (which correspond to greater bias and CoV for the live load moment and shear acting on the span).In lieu of information relating to the center of gravity for vehicle payload and expected eccentricity of payload, this is a reasonable assumption when considering axle loads acting on short spans.

Leopard 2A4M Tank (Tracked-Fighting)
For tracked vehicles, the vehicle load is generally assumed to be uniformly distributed over the contact area of the tracks [12].In fact, there are peaks of pressure where roadwheels are located along the track [13].Given this, it is necessary to check the local load applied beneath the tracked vehicle roadwheel [12].Furthermore, the load in each roadwheel may not be equal, depending on how the vehicle is loaded.For long-span bridges the impact of these slight differences in roadwheel loads is negligible.For short-span bridges, particularly those with spans approaching the length of track, these differences may be significant.Case #1 in Fig. 7 shows the perfect case where loads are distributed equally between roadwheels, thus creating essentially a Uniformly Distributed Load (UDL) along contact surface of the tracks.In practice, some roadwheels may have heavier loads than others.If these heavier loads are near the middle of the vehicle, as represented by Case #2 in Fig. 7, a greater maximum moment than that caused by a UDL would be produced.The heavier loads might also be concentrated to one end of the vehicle as shown in Case #3 in Fig. 7, which would produce a greater maximum shear.In Cases #2 and #3 of Fig. 7, the load distribution would be caused by differences in the largest roadwheel load and smallest roadwheel load.For Case #2 (for moment) or Case #3 (for shear), if the largest magnitude of the distributed load is 35 % larger than the least magnitude, the increase in moment or shear with respect to Case #1 is less than 5 %.

Proposed Military Vehicle Categories
As previously shown, military transport vehicles have inherently different payloads than fighting vehicles.It is therefore reasonable to treat these vehicles differently when determining the probabilistic quantification of the GVW.As such, the following

Statistical Gross Vehicle Weight Parameters for Other Unsurveyed Vehicle
Populations Statistical GVW parameters have been derived for three specific military vehicles.These parameters may be applicable to similar military vehicles of interest.However, in lieu of better information, vehicle-specific statistical GVW parameters might be estimated using the payload weight fraction.This is beneficial because the statistical parameters for the vehicle loads on longer spans are similar to those for the vehicle weight [5].Tab. 10 presents typical payload statistical parameters by Military Vehicle Category for transports, armoured personnel carriers and tanks.It is derived using the bias and CoV from Tab. 6, and the average payload weight fractions from Tab. 9 for military transports, APCs and tanks.

Discussion
The method used to estimate variability of the AHSVS-PLS combat weight provides a good starting point for investigating other traffic populations.This could indicate if further resources are necessary to weigh military transport vehicles, and so obtain more reliable data to calibrate live load factors for military transport vehicles at Ultimate Limit States.
The methods used to quantify the LAV III-ISC and Leopard 2A4M tank weight variability are based on heuristic assumptions concerning different operational loads that affect the vehicle weight.Given the high level of control, itemized breakdown, and standardization of military fighting vehicle loads, it is possible to make these conservative assumptions with greater confidence than if the payload was uncontrolled.Even though vehicle weight data from the field are required to validate these assumptions, they still yield a useful method of comparing the expected weight variability in different categories of military vehicles.
It has generally been assumed that there is higher control in military vehicle loads than civilian vehicle loads [26], thus leading to lower weight variability.Based on the observed weights of intermodal containers from Afghanistan and qualitative descriptions of loading practices provided by military personnel while conducting research for this paper, this assumption may not always be valid, specifically during conflict operations.For example, the AHSVS-PLS military transport had similar statistical parameters for load as Canadian non-permit traffic, indicating no greater load control between the military and civilian traffic.In using conservative assumptions for the LAV III-ISC and Leopard 2A4M tank loadings, it was illustrated that the weights of these vehicles are less variable because the curb weight, assumed deterministic, is a significant portion of the GVW.

Summary and Conclusions
Using heuristic assumptions combined with available data, the statistical parameters for the Gross Vehicle Weight (GVW) and axle loads for three military vehicles have been quantified.This is a necessary prerequisite to employ Limit State Design (LSD) methods, including the assessment of existing bridges for military vehicles, at Ultimate Limit States.
The GVWs of military fighting and transport vehicles were found to have different probabilistic parameters.Given this, four vehicle classifications are recommended for military vehicles: Wheeled-Fighting (W-F); Wheeled-Transport (W-T); Tracked-Fighting (T-F); and Tracked-Transport (T-T).This reflects the difference in the payload weight fraction for fighting (0.02 to 0.25) and transport vehicles (0.35 to 0.60), and the associated differences in the parameters quantifying the GVW.
This paper illustrates that some military vehicles have large curb weights and light payloads, and so have total weights approaching the nominal vehicle combat weight.Reducing the payload weight fraction, i.e., the ratio of the payload to the nominal vehicle combat weight, reduces the overall vehicle weight variability.The following conclusions can be made: • The statistical parameters for the GVW of military vehicles differ depending on the general configuration and function of the specific vehicle.Specifically, military fighting vehicles have a lower CoV than military transport vehicles.
Probabilistic Gross Vehicle Weights and Associated Axle Loads for Military Vehicles in Bridge Evaluation and Code Calibration 143 • The lower weight variability of some military vehicles is less due to effective load control but rather is an inevitable outcome of the design and intended functionality of the vehicle itself.• The statistical parameters to quantify the GVW of military vehicles can be estimated using the payload weight fraction.
-Normal distribution dispersion parameter; and • x m ( is the Log-Normal distribution central tendency parameter.

Fig. 7
Fig. 7 Tracked vehicle load distribution cases: (a) Idealized load; (b) Worst case for moment; (c) Worst case for shear Tab. 9 identifies typical military vehicles based on their intended function, such as: Transport (Tpt.),Armoured Personnel Carrier (APC) or Tank.These three functions also pertain to the three vehicles investigated previously in this paper: AHSVS-PLS (Transport); LAV III-ISC (APC); and, Leopard 2A4M (Tank), respectively.The ranges of payload weight fractions, for these vehicles are 38 to 60 % for transport vehicles, 7 to 21 % for APCs, and 2 to 13 % for tanks.The payload weight fractions for transport vehicles are clearly distinct from those for the other two categories.It might therefore be appropriate to define Fighting vehicles (such as Tanks or APCs) as those with payload weight fractions less than 25 % and Transport vehicles as those with payload weight fractions greater than 35 %.Vehicles with payload weight fractions between 25 % and 35 %, such as an APC with a trailer attached, require additional investigation to be classified as Transport or Fighting.
presents the Probabilistic Gross Vehicle Weights and Associated Axle Loads for Military Vehicles in Bridge Evaluation and Code Calibration 135