1  Investigation of load assumptions

Due to the considerable variation in vehicle characteristics, road conditions and behavior of individual drivers, current calculation practice makes use of all-inclusive, generalized acceleration values for road freight traffic. According to the current consensus in Germany and a number of other European countries, these are as follows:

Forward:	0,8 g
Rearward:	0,5 g
Sideways:	0,5 g

In several other European countries, the forward acceleration is assumed to be 1.0 g.

The force acting perpendicularly to the loading area that is important both for the friction on the loading area and for the stabilizing moment of a cargo unit is taken as 1.0 g, i.e. the full force of acceleration due to gravity.

In order to get a clearer picture of the situation in reality, typical borderline cases for external loads will be investigated. Accident events will be excluded.

1.1  Full braking

Full braking is the greatest load to which a forward securing arrangement is exposed. Recent developments in the field of truck tires, coupled with modern brake systems and asphalt roads, permit braking deceleration values that are perfectly capable of approaching 0.8 g (1). Other factors, such as the distribution of axle weights, also play a role in this context.

The connection between the loading area of a truck or semitrailer and the tyre footprints is not rigid but resilient, which means that the inertial force of the cargo does not follow directly from the braking deceleration, but instead initially brings about a forward tilting of the loading area. This "pitching angle" is not at a steady state throughout full braking, but has pitching oscillations superimposed on it. The amplitude of the pitching oscillations is very highly dependent on buildup time, i.e. the time taken for the braking force to increase to its full value.

During full braking, the following forces act forwards on the cargo in the coordinate system of the loading area (parallel to the loading area):

• inertial force component from the braking maneuver,
• downhill force (weight component) arising from the geodetic inclination of the loading area (pitching angle and gradient of road),
• inertial force arising from tangential acceleration from superimposed pitching oscillation.

The normal force acting from the cargo on the loading area is generally reduced by two causes, namely, as a result of the inclination of the loading area, by the

• upwardly directed vertical component of the inertial force,
• reduced normal component of the weight-force.

The upwardly directed vertical component of the inertial force, and the reduced normal force arising from the geodetic inclination of the loading area reduce both the friction relative to the loading area and the moment of stableness of a cargo unit.

Figure 1: Full braking on downward sloping road

Figure 2: Full braking on a level road from 90 km/h with 0.8 g braking deceleration
and 0.3 s buildup time; stopping distance = 42.9 m

Figure 2 shows the numerical solution of the equations of motion over a period of 6 seconds. The forces acting on the Cargo have been converted into units of g. The vehicle is stationary after approx. 3.3 seconds.

The truck is loaded in such a way that, at 0.8 g deceleration, a steady-state pitching angle of 4° is obtained. The maximum pitching angle after 0.9 seconds amounts to 5.5° as a result of the superimposed pitching oscillation. This oscillation is strongly damped and largely subsides by the time the vehicle is at a standstill, but is re-excited by the familiar jerk at the end of the braking maneuver.

The maximum longitudinal load on the cargo at 0.9 seconds amounts to 0.98 g, at which point the normal force has simultaneously declined to 0.92 g.

Numerous further simulated full braking maneuvers at other speeds, uphill and downhill road gradients and other vehicle types (e.g. semi trailer with a smaller pitching angle) reveal similar profiles. The following general conclusions may be drawn:

• Calculating on the basis of braking force transfer corresponding to 0.8 g, cargo securing must be designed for just about 1.0 g, because the downhill force from the pitching angle plus the tangential force from the superimposed pitching oscillation add about 0.2 g.
• Full braking from lower initial speeds results in only insignificantly more favorable results. Only at speeds of below 15 km/h may it happen that the vehicle is already stationary before the maximum longitudinal force has been reached.
• Semi trailers, which are assumed to have half the pitching angle, experience approx. 3% lower longitudinal forces and a 4% lower reduction in normal force. The outcome is no more favorable than this because the pitching oscillation period simultaneously becomes shorter and the amplitudes of the pitching oscillations are only insignificantly smaller than in a vehicle with a 4° steady-state pitching angle.
• The more rigidly is a loading area mounted, i.e. the less it responds to deceleration with a pitching angle and with pitching oscillations, the closer the longitudinal force acting on the cargo approximates to the pure inertial force from the braking deceleration.
• Gentler braking maneuvers with buildup times of longer than 2 seconds result in virtually no superimposed pitching oscillations. Calculating on the basis of 0.8 g maximum braking deceleration, the only further allowance which need be made is for the parallel component of the force of gravity from a steady-state pitching angle. The allowance is obtained from the sine of this angle.
• On full braking uphill from a speed of 50 km/h, the braking force is increased by the backward downhill force and, as a result, the braking distance is distinctly shorter than on a level road. The effective pitching angle is, however, reduced by the rearwardly directed inclination of the road, such that the difference in longitudinal force on the cargo is almost equalized compared to the situation on a level road. Under the selected conditions according to Figure 2, the cargo should be secured against acceleration of 0.99 g.
• On full braking downhill from a speed of 50 km/h, the longitudinal force on the cargo is somewhat smaller than in the event of full braking on a level street. The effective braking force is smaller and the braking distance greater. The downhill force is, however, increased by the inclination of the road. Under the selected conditions, the cargo should be secured against acceleration of 0.96 g.
• Calculation methods for dimensioning longitudinal cargo securing should take suitable account of the decrease in normal force (weight).

1.2  Cornering

Similar phenomena occur on tight cornering as occur on full braking. In the steady-state phase of cornering, the loading area is inclined laterally by a rolling angle. A rapid buildup in centrifugal force up to its maximum value gives rise to a rolling oscillation with amplitudes which are superimposed on the steady-state rolling angle. The transverse force parallel to the loading area acting on the cargo is therefore made up of:

• centrifugal force component from cornering,
• downhill force arising from the geodetic inclination of the loading area,
• inertial force arising from tangential acceleration from a rolling oscillation.

In this case too, the normal force acting from the cargo on the loading area is reduced by two causes, namely, as a result of the inclination of the loading area, by the

• upwardly directed vertical component of the centrifugal force of cornering,
• reduced normal component of the weight-force.

Figure 3: Cornering with unfavorable inclination b of the road

Unlike in the longitudinal direction, the centrifugal force is oriented horizontally in the geodetic reference system, i.e. not parallel to the inclination of the road. Thus the inclination of the road has a direct impact on the centrifugal force components through the downhill force.

Figure 4: Cornering on a level road with 0.42 g centrifugal acceleration and
0.54 g maximum transverse acceleration, maximum rolling amplitude = 5.8°.

Figure 4 shows the numerical solution of the equations of motion over a period of 6 seconds. The forces acting on the cargo have been converted into units of g.

Maximum centrifugal acceleration was deliberately selected at 0.42 g such that, once the rolling oscillations have subsided, a steady-state transverse acceleration of 0.50 g is established. As a consequence, the first rolling amplitude gives rise to a maximum transverse acceleration of 0.54 g. This value increases if the buildup time is shortened or damping of the rolling oscillations is reduced.

Further simulated cornering maneuvers with other loading area spring constants and with favorable and unfavorable corner inclination of the road reveal comparable profiles. The following general conclusions may be drawn::

• The generally accepted assumption of transverse acceleration of 0.5 g for dimensioning cargo securing against sideways sliding must not be interpreted in such a way that this value could be solely attributable to the centrifugal force. Instead, between 20 and 30% of this value must be reserved for the downhill force from the inclination of the loading area and the tangential forces from superimposed rolling oscillations.
• In steady-state cornering, the inclination of the loading area is still present even after the rolling oscillations have subsided and contributes just about 20% to transverse acceleration.
• The transverse force allowances from the downhill force and tangential forces have nothing to do with the "rolling factor", which is required in VDI Guideline 2700 Sheet 2. The rolling factor takes account of dynamic tipping moments, while the stated allowances are forces acting at the center of gravity.
• In favorably constructed curves (road inclined towards the center point of the curve), the parallel component of the force of gravity is partially offset by the inclination of the road. The opposite applies when the road is inclined unfavorably.
• As previously with full braking, stiffer loading area suspension gives rise to smaller rolling angles and the transverse forces thus approximate to pure centrifugal forces.
• Starting to corner more slowly with buildup times of distinctly more than two seconds allows the superimposed rolling oscillations to become insignificant if damping is adequate, because the initial amplitudes fall within the range of the still increasing centrifugal force.
• The normal force from a given cargo unit is reduced by an order of magnitude of around 5%. This has a negative impact both on friction and on stableness.

1.3  Lane changing

Rapid lane changing is also included among problematic driving situations. It may be concluded from publications (2) that conventional lane changing for a truck involves a lateral offset of 3.75 m and a lane change lasting 4 seconds may be regarded as very rapid.

Analysis of such a lane change is based on the driven geometric contour of the vehicle's center of gravity. This contour is often represented by an "inclined sine curve". There are, however, also other meaningful mapping functions. The second time derivative of the mapping function reveals the profile of the transverse acceleration of the vehicle, which may be equated with a good approximation to the centrifugal force. Representing centrifugal force more accurately with the assistance of the curve radius does not reveal any appreciable difference with the conventionally slender curve profiles.

Inclined sine curve:                             

Alternative lane changing profile:         

Figure 5: Lane changing according to the alternative calculation profile

The lane change shown in Figure 5 was calculated using the alternative, asymmetric profile. In order to maintain comparability with the inclined sine curve in the first half, the lane changing time of 4 seconds is determined by doubling the time required for half the transverse distance.

With a half-value period of 2 seconds, this is already a very rapid lane change. The transverse force acting on the cargo reaches just about 0.3 g after around one second and is already appreciably out of phase with the distinctly smaller centrifugal force. The rolling oscillations are pronounced. There is, however, no discernible resonant buildup.

A series of test runs with modified input variables allows the following general conclusions to be drawn:

• Even extremely short lane changing times do not give rise to transverse acceleration values of greater than 0.5 g. A lane changing time of for example 3 seconds (half-value period 1.5 seconds) generates 0.48 g. However, according to available sources, this time cannot be achieved with a truck.
• Reduced damping of rolling oscillations brings about a slight increase in transverse acceleration values, while greater rolling stiffness reduces the values.
• The reduction in normal force is negligibly small because the rolling angles reach only small values.
• The distinct phase shift is indicative of the onset of resonance between the rolling oscillations and steering movements. Resonance ought to become distinctly perceptible once half the rolling period is equal to the half-value period of the lane change. The result would be a larger transverse acceleration value in the second semi-oscillation of the rolling process.

1.4 Obstacle avoidance

The stated observation of the onset of resonance during quick lane changes led to an investigation of an "obstacle avoidance" maneuver similar to lane changing in which the transverse distance is distinctly smaller, meaning that the achievable lane changing time may also be shorter. Since no observations under practical conditions were available, experienced truck drivers were asked whether it would be possible to change lanes in a loaded truck with a sideways offset of one meter within 1.5 seconds. The answers were in the affirmative, but the drivers' "gut feeling" was that this was a borderline maneuver.

Figure 6: Obstacle avoidance action of 1.0 m with a half-value period of 0.75 s

The results are in line with expectations. Figure 6 shows the evasive maneuver of 1 meter sideways with a half-value period of 0.75 seconds. The forces are shown normalized to acceleration values in the unit g. The transverse force on the cargo is distinctly out of phase with the centrifugal force with an oscillation offset of approx. p/2. This means that at least the first maximum of the transverse force, corresponding to 0.32 g, is made up solely of the downhill force and tangential forces, because the centrifugal force is equal to zero at that point. The second maximum is greater in absolute terms than the first and reaches 0.41 g. At this maximum too, the tangential acceleration from the rolling oscillation predominates. The rise from 0.32 g to 0.41 g is attributable to resonant excitation.

It may thus be concluded that while extremely short avoidance maneuvers similar to lane changing (objects on the roadway) do not necessarily cause transverse acceleration values of greater than 0.5 g, they may involve significant rolling acceleration components which must be taken into account inter alia for assessing the rolling factor.

1.5  Rolling factor

The rolling factor was introduced by VDI Guideline 2700 and "takes account of dynamic tipping moments brought about by a non-steady-state lateral inclination or by angular acceleration from rolling oscillations of the vehicle about its longitudinal axis". This description is unambiguous and complete. It relates to dynamic tipping moments.

The quasistatic tipping moment on a cargo unit is calculated from the force F_x or F_y acting at its center of gravity multiplied by the distance d of this force vector from the effective tipping axis (Figure 7 left). The force F_x or F_y is also the force which must be taken into account when securing the cargo against sliding.

Dynamic tipping moments in the longitudinal or transverse directions arise from the rotational inertia of the cargo mass against the angular acceleration caused by pitching or rolling oscillations. These give rise to a "dynamic" turning moment on the cargo unit in question which is independent of the location of the tipping axis and center of gravity of the unit (Figure 7 right). The following applies to the magnitude of this additional moment:

Transverse direction : 
Longitudinal direction:

For homogeneous or hollow cubical cargo units, the moment of rotational inertia J about an axis through the center of gravity may be approximately determined by:

Figure 7: Static and dynamic tipping moment

The applicable VDI Guideline 2700, Sheet 2 requires an allowance of 0.2 g for securing cargo units at risk of tipping on the assumption of sideways acceleration of 0.5 g. The rolling factor is explicitly not used for testing and dimensioning the securing against sliding of these cargo units at risk of tipping.

In order to verify the reasonableness of the order of magnitude of this rolling factor, the rotational inertia of a cargo unit at risk of tipping (h ³ 2 × b) of maximum height (road transport: h = 3 m) is converted into an allowance w for transverse acceleration.

Figure 8: Conversion of a rotational tipping moment into a rolling factor

Tipping moment from rotational inertia:

Equivalent tipping moment:
 
 

Acceleration allowance, homogeneous:       

Acceleration allowance, hollow:               

The angular acceleration from rolling oscillations or pitching oscillations to be used may be estimated from the described simulation calculations. Since these simulations may all be considered borderline situations, the results are usable for the stated purpose.

Full braking, cornering and lane changing exhibit maximum angular acceleration values of 0.5 s^-2 or lower (3). Only the avoidance maneuver exhibits distinctly larger values of up to 1.3 s^-2. It must, however, be borne in mind that, due to phase shift, these elevated angular acceleration values without exception occur together with relatively small centrifugal forces, such that the effective transverse acceleration values still remain far below 0.5 g. This offsets the need for a larger rolling factor.

If generous angular accelerations of up to 1 s^-2 are used for the calculation, this justifies an acceleration allowance for securing against tipping of 0.064 to 0.115 g. In the vast majority of cases of full braking, cornering or lane changing maneuvers, however, half this value would be sufficient. A requirement still remains for real world measurements.

Figure 9: Angular acceleration values for pitching or rolling oscillations

For securing unstable cargo units against tipping, the relatively recent draft of DIN EN 12195-1 (01/2009) only requires 0.6 g to be assumed as transverse acceleration, i.e. an allowance of 0.1 g. An equivalent pitching factor does not apply in the longitudinal direction of the vehicle.

The conclusion which may be drawn from the above considerations is as follows: It is possible to support the assumption of 0.1 g as a reasonable acceleration allowance for securing cargo units at risk of tipping against tipping. The allowance should, however, also be used for securing such cargo units against tipping in the longitudinal direction.