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##### 2  Conventional rules and calculation methods
[German version]

Conventional calculation methods distinguish between direct lashing and tie-down lashing and apply both kinds of securing to the aims of securing items against sliding and tipping. Virtually no account is taken in the calculations of compaction, which is often encountered in road freight transport in the form of strapping or bundling.

Conventional calculation methods are briefly presented below, with emphasis on the general conditions and simplifying assumptions which apply. In order to clarify the most recent trends, the calculation conventions from three regulatory texts will be presented and, if necessary, compared:

Source [1] VDI 2700, Sheet 2, November 2002,

Source [2] DIN EN 12195-1, April 2004,

Source [3] DIN EN 12195-1, January 2009.

The systems of notation for the operands in the formulae vary between the stated regulatory texts. In order to facilitate comparability, the following standard system is used for the purposes of this presentation:

F = force in the securing device assumed in the calculation [kN]

Fx, Fy, Fz = force components in the system of coordinates of the loading area [kN]

L = length of the securing device [m]

X, Y, Z = geometric components of length L [m]

m = cargo mass [t]

fx, fy = coefficients of acceleration in the longitudinal and transverse directions

m = coefficient of friction

n = number of parallel securing devices

Figure 10: Spatial coordinates of a securing device

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###### 2.1  Direct lashing

Direct securing connects the cargo and the vehicle with securing devices which are capable of transferring forces directly by tensile, compressive or shear stress. According to conventional assessment, this type of securing is limited solely by the strength capacity of these securing devices and the participating fastening points on cargo unit and vehicle.

###### 2.1.1  Securing against sliding

The balances compare the load assumption related to the cargo mass with the friction plus the action of the securing devices. Friction is generally calculated using the coefficient of dynamic friction and the normal force = cargo weight. The action of the securing devices is made up of the horizontal force component plus the vertical force component multiplied by the coefficient of dynamic friction.

In Source [1] , the balance for the transverse direction reads:

The balances in the longitudinal direction look similar. The balances are solved to find n or F in order to determine the necessary amount of securing.

The stated approach is basically also used in the other sources. Sources [2] and [3] do not, however, specify force components with the assistance of the length components, but instead with corresponding angular functions of the lashing angles a and bx or by. The relations are::

, ,

Sources [2] and [3] additionally indicate a variant of the balance for securing against sliding with direct lashing and blocking, in which the blocking force BC is added to the securing forces without taking account of the stiffness of the blocking.

As to coefficients of friction, Source [3] makes use of "standard values" which are reduced by a factor of 0.85 in the sliding balance. These standard values are means from series of measurements of coefficients of static friction, which were multiplied by 0.925, and coefficients of dynamic friction, which were divided by 0.925, in each case for the same material pair. The balance in the transverse direction then reads:

###### 2.1.2  Securing against tipping

Securing against tipping is only tested if the inherent stableness of a cargo unit is insufficient. The test criteria for inherent stableness are thus an integral part of the calculation model.

According to Source [1], the test criteria for sufficient inherent stableness, where L, B, H = length, breadth, height of a (cubical) cargo unit with a center of gravity in the geometric center and fw = 0.2 (rolling factor) are:

Testing of tipping stableness in transverse direction B : H > (fy + fw),

Testing of tipping stableness in longitudinal direction L : H > fx

The balance in the transverse direction reads:

The balances in the longitudinal direction look similar, but without the rolling factor. The balances are solved to obtain n or F in order to determine the necessary amount of securing. The possibility of an asymmetric center of gravity is not addressed separately.

Source [2] does not provide an adequate treatment of securing against tipping with the assistance of direct lashing. The test criteria for tipping stableness are as in [1], but with lack of clarity with regard to the coefficient of transverse acceleration to be used in the test. No separate tipping balance is stated, however, but instead a system of inequalities, which are intended to demonstrate both sliding and tipping resistance in the event of securing with diagonal lashing combined with blocking.

The system of inequalities is, however, only appropriate for demonstrating securing against sliding, albeit while disregarding the different load generation of lashing and blocking (see 2.1.1). It is unusable for demonstrating securing against tipping and readily leads to erroneous results. In the original text, the formulae for the transverse direction with n = 2 lashings per side are:

Formel 17:

Formel 18:

Formel 19:

BC = blocking force [kN]

a = vertical lashing angle

b = horizontal lashing angle

mD = coefficient of dynamic friction

LC = lashing capacity (admissible lashing force) [kN]

cy = coefficient of transverse acceleration

cz = coefficient of vertical acceleration

m = cargo mass [t]

g = acceleration due to gravity [m/s2]

The variables d, b, w and h are illustrated in Figure 11. Figure 11 shows a securing situation as presented in VDI Guideline 2700, Sheet 2, Figure 14.

Figure 11: Application of testing of securing against tipping to DIN EN 12195-1

Formula 17 corresponds to the conventional approach to demonstrating sliding resistance. Formula 18 is intended to demonstrate securing against tipping by lashing. Blocking makes no contribution to tipping resistance. Formula 19 is superfluous in this respect.

An example calculation shows the unsuitability of formula 18 with the values m = 10 t, cy = 0,7, cz = 1, h = 3,0 m, d = 1,5 m, b = 0,25 m w = 0,5 m, a = 64°, by = 0°,mD = 0,4, n = 2

According to Source [1], the tipping balance reads:

The following replacements are made for comparability: H/2 = d, B/2 = b, H = h, B = w. Der Winkel a liefert die Größen Y, Z und L. Z = h = 3,0 m, L = h/sina = 3,34 m und Y = L × cosa = 1,46 m.

The formula according to source [2] provides a result in this example which is substantially too small. The difference becomes all the more serious, the greater is the coefficient of friction mD, which fundamentally has no place in a tipping balance.

Source [3] contains a reduced rolling factor, the calculation being intended to be carried out with a coefficient of acceleration cy = 0.6 for cargo units at risk of tipping and direct lashing. Testing of tipping stableness is, however, calculated with cy = 0.5 and cz = 1:

Testing of tipping stableness in transverse direction b : d > cy : cz,

The recently included tipping balance is equivalent to the one stated in source [1]. The partially unsuitable system of inequalities, which is already to be found in source [2], is however additionally still present.

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###### 2.2  Tie-down lashing

Tie-down lashing is conventionally treated for the most part such that only the vertical component of the pre-tensioning force is regarded either as enhancing friction or as increasing tipping stableness. Tie-down lashings generally do not have a horizontal lashing angle and are moreover virtually always applied in the transverse direction of the vehicle.

###### 2.2.1  Securing against sliding

Source [1] provides the sliding balance in the notation agreed above:

The balance may be solved to find n or F. A minimum pre-tensioning force is recommended for F, but it should not exceed 50% of LC. In the case of one-sided pre-tensioning, it is recommended initially to apply a higher force on the tensioning side so that, on equalization during the journey, the overall loss of pretension is not so high. No k-factor for friction losses during pre-tensioning is provided. The coefficient of dynamic friction is used for m.

Source [2] adopts this approach, but in the case of one-sided pretension uses the k-factor which replaces the factor 2 (two legs to be tied down per string of lashing).

In the case of one-sided pre-tensioning, k = 1.5, in the case of two-sided pre-tensioning k = 2. The coefficient of dynamic friction is likewise used. In this approach, the two different horizontal components of the lashing-loops are disregarded. The difference between these forces could be introduced into the balance. The two forces amount to:

Pre-tensioning side:                       Opposite side:

Source [3] again turns away from the k-factor, but does introduce a safety factor fs = 1.1, which increases the necessary pre-tensioning force by 10%. The balance reads:

This agreement corresponds to a k-factor of 1.82. The reason stated for the safety factor in [3] is, however, not pre-tension loss by friction but instead calculation uncertainty.

Source [3] moreover contains a sliding balance for the combination of tie-down lashing and blocking, again disregarding the load-bearing behavior of the two different securing means.

###### 2.2.2  Securing against tipping

Source [1] interprets the effect of the tie-down lashing as increasing the normal force onto the loading area, which increases the stabilizing moment with the half breadth as lever. Horizontal force components of the tie-down lashings here cancel each other out.

A similar formula is stated for the longitudinal direction, which however assumes longitudinally oriented lashing loops. Securing effects against tipping in the longitudinal direction by transverse tie-down lashings are not addressed.

Source [2] treats the forces on the two sides of the cargo unit separately in the tipping balance and assumes the less favorable case in which the external force acts towards the pre-tensioned side. The expanded balance in the agreed notation reads:

If this balance is solved to get n × F, the following is obtained:

This formula for determining the necessary amount of securing has the unfortunate characteristic that, on the right-hand side, the denominator of the fraction may readily assume a value of zero. This gives rise to a result tending towards infinity on the left-hand side. If the denominator is equal to zero, then a combination of the variables B, Z, H and Y is present in which each further added tie-down lashing cancels out the vertical component, which increases tipping stability, due to the difference between its horizontal components, i.e. it has no effect.

Anticipating section 3, it should be noted at this point that "permitting" a small offset, shift or tipping of the cargo unit under the external load reverses the forces. The balance then reads:

Once solved for n × F, the following is obtained:

The difference in the results is demonstrated with an example. The values are: H = Z = 2.75 m, B = 1.5 m, Y = 0.5 m, L = 2.8 m, F = 2.5 kN, m = 6 t

Figure 12: Tipping balance according to source [2] on the left; alternative on the right

According to source [2] on the left in Figure 12, 10 tie-down lashings are required for securing against tipping. If the calculation is performed with changed belt tensions as on the right in Figure 12, 3 tie-down lashings are enough. In this case too, the distribution of belt tensions corresponds to the decline in force due to friction at the top edges of the cargo unit. Elongation of the belts as a result of the slight shift of the cargo unit and the favorable increase in force has again not been taken into account in this comparison.

If the breadth B is reduced to 0.5 m, the number of tie-down lashings required according to the calculation in source [2] tends towards infinity, while taking a small movement of the cargo into account results in 7 tie-down lashings.

Source [3] no longer uses the k-factor and so avoids the unfortunate calculation for securing against tipping. The approach from source [1] is adopted with the following modifications:

coefficient of transverse acceleration fy = 0,5, if pretension FT = STF

coefficient of transverse acceleration fy = 0,6, if pretension FT = 0,5 × LC.

a safety factor fs = 1,1 leads to a required increase in pretension or the number n.

Source [3] additionally contains a calculation approach which tests the compacting action of tie-down lashings on a group of tall, narrow unit loads standing adjacent one another with regard to securing against tipping. This approach may be regarded as pointing the way towards the computational evaluation of compaction measures.

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