*Oscar Bustos Letelier*^{1}* and Walter Bussenius Cortada*^{2}

^{1}Forest Engineer, Lic., P.Ed., M.Sc. Forest Engineering Department, University of Talca, Chile.

^{2}Master in physics. Physics Department, University of Talca, Chile.

**Abstract**

This study presents the theory of load distribution on forest trucks and how to optimize the load to avoid damage to the soil.

Many forest industries have several problems due to poor load distribution on the trucks. The effect on the soil produces changes in its structure and increases maintenance costs.

A physical model is used to calculate how to distribute the load keeping the logs' weight in optimal position so that the total weight (truck weight plus load weight) is evenly distributed in order to obtain a homogeneous weight.

**Introduction**

The transport of the wood from the forests to the industries is made by means of trucks of different models and makes. The differences are usually given by the number of axles, weight of the truck, and engine position in relation to the front axle.

The load transported by each truck is controlled by the forest company based on measurements of load volume. Although this load measurement system presents the disadvantage that it is practically impossible to determine the real useful load, it gives the total weight carried on the trucks.

Since it is practically impossible to handle the relation weight/volume of the wood accurately, the way to load a truck depends on the operator's experience and skill and sometimes on a wrong economic judgement. As a result of this, many problems arise in the distribution of the load weight on the truck and the effect of this on the forest road structure.

The present research shows the way to make an accurate distribution of the load in order to increase the useful load and to decrease the harmful effects on the soil.

**Physical model development**

The physical model is based on a truck model of the following dimensions (Figure 1):

**Figure 1. Parameter and variables of length in a prototype truck**

| |

ED |
Distance from front bumper to front axle |

E |
Distance from front axle to rear axle 1 |

E |
Distance from front axle to rear axle 2 |

EE |
Distance from front axle to equivalent rear axle |

E1,2 |
Distance between rear axle 1 and rear axle 2 |

LC |
Cabin length |

LT |
Total length |

CM |
Distance from front bumper to mass centre of unloaded truck |

CM |
Distance from front bumper to mass centre with maximum load |

CC |
Distance from front bumper to load centre with maximum load |

| |

CM |
Distance from front bumper to mass centre with a determinate load. |

CC |
Distance from front bumper to load centre with a determinate load. |

| |

To |
Truck weight |

Co |
Maximum load of the truck |

Fod |
Maximum force on the front axle |

Fot |
Maximum force on the rear axle |

Fvd |
Force front axle with unloaded truck |

FVt |
Force rear axle with unloaded truck |

| |

Fd |
Force front axle |

Ft |
Force rear axle |

C |
Load truck |

**Equivalent axle**

When we have a truck with more than a rear axle, it is possible, by means of a mathematical abstraction, to replace them with just one equivalent axle that supports a force equal to the addition of the forces and located at a determined distance, so the pair of these forces are equal to the forces of the axles. This equality is practical to make calculations; however, we have to take into account that the maximum force of the equivalent axle is regulated by legislation.

As was stated above, if we define N_{i} as the number of axle wheels and E_{i}, the distance to that axle in relation to the front axle, the equivalent axle (EE) is:

EE = (S E_{i}N_{i})/ S N_{i})

In this way, now we will use the equivalent axle concept, without considering the number of rear axles that the truck has. Besides, we suppose that the trucks have only one front axle although, if the truck has a greater number of axles, we can do a similar reasoning as above.

**Determination of the mass centre of the log**

In order to determine the place where the central mass of the logs stay, we must think about a mathematical model that represents the real volume of the log and not its useful volume. We consider a cone model with a length L and a radius R1 in the low diameter and R2 in the high diameter of the log.

The volume is determined by the expression:

V = [R_{2}+ R_{1})^{2}- R_{1}R_{2}]p L/3

In this way, if we suppose an equal density in the log; its mass centre would be the distance

_{}

of the low diameter, so the coefficient s is more than 1.

**Distribution of the truck load**

One important thing in the distribution of the weight on the truck platform is an accurate distribution of the load. In relation to this, if we know the density (d) of each log (including the moisture content) we can determine the mass of each log by the above mathematical expressions.

So, when we load a truck with its whole load in logs with definite length and located in a definite place, the number of logs to load is

N = C_{o}/(dV)

Now, if we know that N is the total of logs to load and if we also know the place where they are put and the place where the maximum load must be located, then, it is possible to decide the number of logs that must be loaded on each side. in order that the Load Centre (CC) remains the nearest to the centre of load with the Maximum Load (CC_{o}).

**Figure 2. Log position in relation to the load centre**

If we suppose an equal geometry for every log, N is the total number of logs and in the logs placed with their low diameter toward the cabin (see Figure 2) then the condition is:

CC_{o}= CC

The mass centre of the log (load centre) is given by the following expression:

CC = (m(L_{o}+ s L/2) + (N-m)(L_{o}+ L - s L/2)]/N

If we replace CC in the former equality, we have:

m = N[(CC_{o}- L_{o}- L + s L/2)/(L(s - 1))]

**Determination of mass centre of a determined load**

To determine the place where must put the mass centre of a determined load, e.g. to determine the load centre, we can analyse the problem in three ways:

a) **Geometric model**

We suppose that the load centre for maximum load (CC_{o}) is situated exactly on the platform of the truck. In this case:

CC_{o}= (LT - LC)/2 + LC

Now, we can suppose that the truck has a small load, it must be at the rear extreme of the platform (in the rear bumper). So we can suppose that if we increase the load, the load centre will move from the extreme of the platform (LT) to the maximum load centre (CC_{o}).

The graphic below shows this idea.

Considering the straight-line equation, we have

_{}

where the load centre (CC) is

C = LT - (LT - CC_{o})C/C_{o}

Taking CC, the extreme values are

LT < CC < CC_{o}= (LT - LC)/2 + LC

b)** Mass centre model**

This model supposes the mass centre of the truck is moving in a straight line as a consequence of an increase of the mass. Starting at CM_{v} for unload truck we get to CM_{o} for truck with maximum load. So, when the load is increasing, the CM will go to a different position between these values.

The graphic below illustrates this idea.

In this case, the load centre position must vary since certain values are equal to CC_{o}. To find these values we can consider the straight-line equation

_{}

where

CM = CM_{v}+ (CM_{o}- CM_{v})C/C_{o}(b)

and considering the relation of torque

CM_{v}T +CC C = CM (T + C) (c)

then, replacing CM in equation (c), we have:

CC = CM_{v}+ (CM_{o}- CM_{v})(T + C)/C_{o}

which extreme values for CC are:

CM_{v}+ (CM_{o}- CM_{v})T/C_{o}< CC < CM_{o}+ (CM_{o}- CM_{v})T/C_{o}

c)** Axles forces model**

This third model consists in keeping a certain relation between the forces produced on both axles. According to this relation we can determine the point corresponding to the load centre (CC).

We can start from known values, such as, Fvd, Fvt, Fod and Fot. The first two values appear in the truck catalogue. The other two values Fod and Fot are values determined by the norms of the Ministry of Transport.

The following graph shows the relation between the values under consideration.

The relation between Fd and Ft would be the form of:

Fd = A Ft + B

where A and B are constants and are given by:

A = (Fod - Fvd)/(Fot - Fvt)B = (Fvd Fot - Fvt Fod)/(Fot - Fvt)

Considering now the addition of the forces equal to zero:

Fd + Ft - T - C=0

where T and C are the weight and load of the respective truck, replacing Fd in the last expression the following is obtained:

Ft = (T + C - B)/(1 + A)

and for Fd the following is found

Fd = [A(T + C) + B]/(1 + A)

Applying now the equation for torque with respect to the front bumper and equalling the addition of these to zero, the following is obtained:

Fd Ed + Ft(Ed + EE) - T CM_{v}- C CC = 0

With this, the position of the load centre of the truck is given by:

_{}

**Discussion**

The physical model developed over the distribution of the load on forest trucks was developed based on estimates regarding density factors and moisture content. The validation of the model will require detailed information on different harvesting areas.

**Conclusion**

The physical model developed represents the starting point of research about the distribution of the load on the trucks to obtain a maximum load with minimum effect on the trucks and on the forest roads.

**References**

**Baumgras, J.** 1976. Better load-weight distribution in needed for tandem-axle logging trucks. USDA Forest Service Research Paper NE-342.

**MacNally, J.** 1975. Trucks and trailers and their application to logging operations. Faculty of Forestry. University of New Brunswick. Fredericton. Canada.

**Skaar, R.** 1975 Truck transport systems and forest road planning. Department of forest operation. Agricultural University of Norway. N-1432. AS-NLH. Norway.

**Timson, F.** 1974. Weight and volume variation in truckloads of logs hauled in the Central Appalachians. USDA Forest Service Research Paper NE-300.