3  Extended approaches

An extended approach should certainly take account of the movement of cargo units associated with direct securing, without which a deformation of the securing devices and thus the necessary generation of force is not possible. If it is possible in this way to define "acceptable" cargo movements, identical movements could also be permitted for arrangements with frictional securing (tie-down lashings). This could in turn lead to a re-assessment of this securing method, possibly with more favorable results.

3.1  Cargo movement

Cargo movements which result in necessary deformation (usually changes in length) of securing devices, primarily involve sliding or slight tilting and many and varied changes in the shape of the unit itself which may be superimposed on the first two types of movement mentioned. Sliding is generally irreversible, while slight tilting is reversed once the external tipping moment has disappeared.

Changes in the shape of the cargo unit may be elastic, but there is usually a considerable plastic component to the change. Since extreme loads occur as individual events in road transport, permanent deformation of the loading arrangement is more readily acceptable because checking and remedying the securing arrangement is immediately and straightforwardly possible.

This assessment may be explained by making reference to maritime transport, where extreme loads are associated with storms and rough seas. These conditions may last for an extended period, as a result of which an unforeseeable succession of extreme loads may occur and checking and remedying cargo securing arrangements during this period can often only be carried out at risk to life.

Figures 13 and 14 show the basic cargo movements, while some further cases could certainly be added to the deformations shown in Figure 14.

Figure 13: Types of movement of rigid cargo units

Figure 14: Types of movement of flexible loading arrangements

Which kinds and amounts of movement of secured cargo can be tolerated in road transport has not previously been specified or recommended in any regulatory texts, guidelines or similar documents. Some influencing variables which can play a part in such considerations will thus firstly be mentioned.

The frequency of a load causing cargo movement could influence the tolerable extent such, that rare events, such as full braking or extreme centrifugal forces, are allowed larger movements than would be accepted in normal travel, since it is reasonable after an extreme event to drive into a parking place and check the securing arrangements.

The nature of the movement likewise has an influence on its tolerable extent. A sliding offset movement shifts the center of gravity of the cargo and may moreover exceed the loading area limit. Racking (shear deformation) of compact cargo units, on the other hand, may remain within the elastic deformation range and is therefore less critical. Racking of bundled units, however, is quasi-plastic and ought therefore to be limited in a similar way to sliding offset. Tilting of a cargo unit should be limited to very small angles due to the low damping of the tilting process.

The direction of movement is influential. Movements in the longitudinal direction are less critical than movements in the transverse direction, because the latter may exceed the admissible breadth of the vehicle and may also have a major effect on the transverse center of gravity.

In order to gain an impression of the magnitude of cargo movements which have not previously been scientifically accepted, it is worthwhile analyzing direct securing methods which have conventionally been regarded as "good". This does not, however, mean that the identified movements may thus generally be recommended as being tolerable.

The proper and rational way of defining and recommending tolerable cargo movements should ultimately proceed by applying numerical criteria:

• admissible sliding/racking in the longitudinal direction and transverse direction with regard to cargo geometry and center of gravity,
• admissible tilting angle with regard to dynamic loading of the securing devices,
• general dynamic overstressing the capacity of securing devices.

If this is to be achieved, typical loading situations must be fully calculated. In order at this point to provide an initial impression of the order of magnitude of previously accepted cargo movement, one example of conventional direct lashing will be presented, using some formulae taken from section 3.2 below.

A cargo unit with the dimensions breadth = 2.1 m, height = 2.5 m is secured in the transverse direction onto the vehicle with diagonal synthetic fiber belts which are attached to the upper corners of the unit. The geometric components of the belts are X = 1.0 m, Y = 2.3 m, Z = 2.5 m, while the loaded length L = 3.54 m. The belts have an LC = 25 kN and an elongation of 3.75% once the LC is reached. The spring constant of the belt is thus:

The belts are pre-tensioned to 2.5 kN. The sliding balance (not shown here) shows that LC of the belts is just reached. The change in length of the belts must then be:

In order to achieve this elongation, the upper corner of the cargo unit must be deflected to the side by the amountD Y by undergoing a sliding or racking movement or by a combination of both movements:

The lateral movement of 0.18 m transverse relative to the vehicle under extreme loading would appear to be acceptable and, under certain circumstances, for example with elastic deformation, an even larger deformation could be accepted.

3.2 Deformation and force development of securing devices

Breaking the direction of action of securing devices down into Cartesian components has already been presented in section 2. In geometric terms, each securing device has the components X, Y and Z assigned to it, with the Pythagorean relation to its length L:

Cargo movement or deformation is expressed by specific, small changes DX, DY and DZ in these components. Change in length DL is then precisely calculated by:

If the individual changes are small relative to the total length L, the change in length DL may in most cases sufficiently accurately also be determined by an approximation equation as follows:

This approximation equation should, however, not be used if for example component Y is close to zero and a cargo shift involving a DY is to be investigated. This case in particular applies to steep tie-down lashings.

If force development by securing devices as a result of cargo movement or deformation is to be determined, it is advisable to assign a "personal" factor to each securing device which makes it possible to calculate the force change DF directly from the change in length DL. This factor is the spring constant D conventionally used in industrial mechanics. The relation

applies.

The spring constant includes the cross-section, modulus of elasticity and length of the securing device as influencing variables in accordance with Hooke's law:

DF = force change in the securing device [daN] or [kN]

A = cross-section of securing device [cm²]

E = modulus of elasticity [daN/cm²] or [kN/cm²]

L = loaded length of securing device [m]

DL = change in length of securing device [m]

The spring constant is not generally stated in data provided by manufacturers of securing equipment. There are several ways of determining the spring constant depending on the information which is available about the securing material.

Manufacturers of ropes, chains and belts often state that the elongation of the material is P % when the material is loaded with a specific force F (LC is usually stated here). The force F stated corresponds to the force change DF starting from zero load and assuming approximately linear load/elongation behavior, which is generally the case in the limited load range between pretension F₀ and admissible load LC. The spring constant is thus:

If the length of the lashings is initially unknown, a normalized spring constant D_N for the unit length of 1 m may be used for the lashing material, the following applying:

The spring constants of pressure transferring element, e.g. squared lumber, may be estimated from the three variables cross-section A, modulus of elasticity E and length L:

The spring constant of end walls and stanchions on a truck loading area may be estimated by regarding them as beams clamped on one end. The spring constant is then calculated on the basis of the bending equation for cantilevers:

E = modulus of elasticity [daN/cm²] or [kN/cm²]

I = geometrical moment of inertia in clamping point [cm⁴]

d = lever length of clamped beam [m]

In practice, however, this solution will provide excessively large values for D, since the wall or stanchion is not clamped absolutely rigidly, the load-bearing substructure instead likewise deforming. It is therefore advisable to establish a correction factor by making representative measurements or to determine the entire spring constant experimentally. The same also applies to sidewalls of truck loading areas which are differently designed and often receive some support over their length from the roof structure. A simple formula can no longer be stated in this case.

The following formula applies to devices arranged in parallel: D = D₁ + D₂ + ... + D_n

The following formula applies to devices arranged in series: 1/D = 1/D₁ + 1/D₂ + ... + 1/D_n

3.3  Horizontal components of tie-down lashings

Almost all conventional approaches to calculation disregard the horizontal components of an inclined tie-down lashing. If identical pre-tensioning on both sides is assumed, these components cancel each other out. A one-sided tensioning device provides a resultant which, while acting favorably on one side, acts unfavorably on the other. If, for safety's sake, the calculation is carried out with the unfavorable side, the peculiarity occurs which does not correspond to reality, as was described in section 2.2.2.

Tied-down cargo units do in fact move under the influence of external forces, so changing the geometry of the tie-down lashing with various consequences:

• The tie-down lashing is (slightly) lengthened with a (small) increase in overall forces.
• In the event of transverse movement of the cargo unit, distribution of forces on both sides of the tie-down lashing adapts to the ratio of forces determined by friction at the deflection points. A "favorable" resultant of the transverse components is inevitably obtained.
• In the event of longitudinal movement of the cargo unit (assuming a transverse tie-down lashing), a "favorable" horizontal component of the lashing force likewise arises on both sides, which increases continuously with the movement and only remains constant when the belt slips on top of the cargo unit.

The smallest possible ratio of forces between the two sides of a transverse tie-down lashing and thus the transverse component with the greatest possible securing effect may be sufficiently reliably calculated using the known Euler's relation.

F and F₀ = forces on both sides of the tie-down lashing [daN] or [kN]

e = Euler's constant (2,718281828)

m = coefficient of friction at the deflection point

g = angle of deflection (change of direction) of the tie-down lashing [rad]

The magnitude of the transverse component, which has a securing effect, of a tie-down lashing is, however, crucially determined by the vertical lashing angle a. At m = 0.25, the transverse component is at its greatest with a = 45° on both sides. The overall ideal vertical lashing angle a is, however, always at distinctly higher values, since the main action of a tie-down lashing depends on the friction-enhancing vertical component which, as is known, increases with the sine of the lashing angle a.

There is no transverse component with a purely vertical tie-down lashing. In such a case, only once an appreciable displacement is reached, are favorable transverse components obtained on both sides, the magnitude of which is, however, not limited by the friction between lashing device and the cargo. This is a borderline case of direct securing, i.e. it is based on a different effect compared to the transverse component from Euler's friction.

Figure 15: Transverse components of transverse tie-down lashings

The transverse component of the tie-down lashing on the left in Figure 15 amounts to F₀ × (1 – e^-m×2×a) × cosa. The transverse component of the tie-down lashing on the right in Figure 15 amounts to F₀ × (1 + e^-m×p) × cos g. In order to clarify the orders of magnitude, an example with plausible values is obtained with: F₀ = 2 kN, m = 0,25, a = 75°, g = 85°.

Transverse component on left in Figure 15 = 0.25 kN; transverse component on right in Figure 15 = 0.13 kN.

Neither value takes account of the possible elongation of the lashing belt by the change in geometric conditions. This entails more complex calculation, which is not shown here.

Figure 16: Securing effect in the transverse direction of a tie-down lashing with a = 80°

Using a tie-down lashing with a = 80° by way of example, Figure 16 compares the securing effect according to a conventional assessment and the enhanced securing effect taking account of the transverse component and the increase in force due to a change in the geometry as a result of a sliding or racking movement.

Up to a transverse movement of approx 1.9 cm, the effects differ depending on the direction of loading. In the event of loading towards the pre-tensioned side, smaller values are initially obtained. From 1.9 cm, the belt slips and the greatest possible transverse component, arising from Euler's friction between belt and cargo, takes effect. In the event of loading on the opposite side, the belt slips from the start, so providing the greatest possible transverse component. It is thus clear that while the usual one-sided tensioning device does overall impair securing, so entirely justifying the k-factor, it does not bring about any detrimental asymmetry in the securing effect.

The further increase is attributable to increasing elongation of the belt. Already with a transverse movement of 15 cm, the securing effect achieved in this example is a good twice that obtained in the conventional assessment.

3.4  Semi-dynamic approach to calculation

Taking account of cargo movement and elastic deformation of the securing devices makes approaches to calculation other than those conventionally used necessary, since the input variables for simple equilibrium methods are initially unknown. They are only obtained in the course of a relatively long calculation process. A distinction may be drawn between a semi-dynamic and a fully dynamic approach.

The term "semi-dynamic" is here intended to indicate that while cargo movements are indeed taken into consideration, they are only used to determine the different loads assumed by the securing devices up until equilibrium with the external force. Further dynamic effects arising from the cargo having started to move are ignored and are still to be covered by the safety margin between LC and breaking load.

3.4.1  Iterative method

In an iterative calculation method, the cargo is moved stepwise under the action of an external force in a predetermined manner (sliding, tipping, racking). For each step, on the basis of this movement, the deformation of the securing devices involved and the force it has developed is determined. The developed force is added to the initial pretension and is broken down as a securing force into Cartesian components.

These components are included in the securing balances against horizontal sliding/racking and against tipping. Horizontal components act directly against sliding/racking and vertical components act via the coefficient of friction, while against tipping the horizontal and the vertical components are introduced into the calculation with the associated levers to the applicable tipping axis. If the securing arrangement consists of tie-down lashings, a similar procedure must be applied which also takes account of Euler's friction at the edges of the cargo unit.

The calculation is terminated once equilibrium with the external force or with the external moment is reached. It may then be established which loads have been assumed by the individual securing devices and how large the cargo movement or deformation has become. The suitability or admissibility of the securing arrangement in question may be assessed on the basis of these two items of information. In addition, the results may point to possible improvements and increases in efficiency for a securing arrangement.

If reliable experience relating to additional dynamic loads is available, the iterative calculation procedure may also be terminated a little later than when static equilibrium is reached. The amount of this allowance could be selected to be approximately proportional to the cargo movement which has occurred up to equilibrium. This would enable a pragmatic step to be taken towards a dynamic approach to calculation.

Obviously, this calculation process can only be carried out with a programmed computer and thus cannot be considered for on the spot dimensioning or verification of a securing arrangement, but it may be considered for individual planning of critical transport operations or for designing standard securing arrangements for long-term use.

3.4.2  Selective methods

A selective approach to direct securing starts from the securing device in the arrangement in question which is definitely the first to reach its admissible load (LC). On the basis of the deformation of the selected securing device, this loading is converted into a cargo movement/deformation. The latter is used to determine the deformations and loads developed by all further securing devices and these values are input into a balance.

The balance indicates whether the securing arrangement is adequate or requires further improvement. It also makes it possible to identify which securing means may possibly be only inadequately contributing to securing.

For frictional securing, i.e. tie-down lashing, the selective approach should be modified such that the calculation starts from the maximum tolerable cargo movement and uses this as a basis for determining the changes in length, forces and geometric components of the lashings and inputs these values into the sliding and tipping balances.

These selective methods involve less complex calculation and may in most cases also be calculated manually. They are thus also suitable for staff training. A result, one has oneself calculated, carries more weight than a result passively accepted from a computer.

3.5  Fully dynamic approach to calculation

A fully dynamic approach to calculation should not only determine the different loads assumed by the securing devices until equilibrium with the external force, but should also identify the additional forces which are required to bring a cargo unit which has started to move/deform back to a standstill. Such an approach can virtually only be implemented with the assistance of a simulation which models the limited time period of the critical driving situation. The effects of buildup phases and of pitching and rolling oscillations outlined in section 1 can be taken into account in this way.

One essential item of information from typical, calculated cases is the magnitude of the stated additional forces and their dependency on the parameters involved, such as friction coefficients and the elasticity of cargo securing devices. Evaluating such items of information should allow for all-inclusive allowances to be defined and recommendations for laying out securing arrangements to be drafted.