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  VDI 2700, Sheet 2, November 2002,
Source  DIN EN 12195-1, April 2004,
Source  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]
cargo mass [t]
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
Top of page
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
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  , 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  and  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  and  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  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:
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 , 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
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  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 , 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
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:
blocking force [kN]
a = vertical
b = horizontal
mD = coefficient of dynamic friction
lashing capacity (admissible lashing force) [kN]
= coefficient of transverse acceleration
= coefficient of vertical acceleration
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.
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 , 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 
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  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:
tipping stableness in transverse direction b : d > cy : cz,
The recently included tipping balance is
equivalent to the one stated in source . The partially unsuitable system of
inequalities, which is already to be found in source , is however additionally
Top of page
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.
Securing against sliding
Source  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  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:
Source  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
This agreement corresponds to a k-factor
of 1.82. The reason stated for the safety factor in  is, however, not
pre-tension loss by friction but instead calculation uncertainty.
Source  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.
Securing against tipping
Source  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  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
12: Tipping balance according to source  on the left; alternative on the
According to source  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 
tends towards infinity, while taking a small movement of the cargo into account
results in 7 tie-down lashings.
Source  no longer
uses the k-factor and so avoids the unfortunate calculation for securing
against tipping. The approach from source  is adopted with the following
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  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.