Conventional calculation methods
distinguish between direct lashing and tiedown 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 121951, April 2004,
Source [3] DIN EN 121951, 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]
f_{x},
f_{y} = 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 b_{x} or b_{y}. 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 f_{w} = 0.2 (rolling factor) are:
Testing of tipping stableness in transverse direction B : H > (f_{y} + f_{w}),
Testing of tipping stableness in longitudinal direction L : H > f_{x}
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
m_{D} = coefficient of dynamic friction
LC =
lashing capacity (admissible lashing force) [kN]
c_{y}
= coefficient of transverse acceleration
c_{z}
= coefficient of vertical acceleration
m =
cargo mass [t]
g = acceleration due to gravity [m/s^{2}]
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 121951
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, c_{y} = 0,7, c_{z} = 1, h = 3,0 m, d =
1,5 m, b = 0,25 m w = 0,5 m, a = 64°, b_{y} = 0°,m_{D} = 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 m_{D}, 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 c_{y} = 0.6 for cargo units at risk of tipping and direct
lashing. Testing of tipping stableness is, however, calculated with c_{y}
= 0.5 and c_{z} = 1:
Testing of
tipping stableness in transverse direction b : d > c_{y} : c_{z},
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
Tiedown lashing
Tiedown lashing is conventionally
treated for the most part such that only the vertical component of the
pretensioning force is regarded either as enhancing friction or as increasing
tipping stableness. Tiedown 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 pretensioning force is recommended for F, but it should not exceed
50% of LC. In the case of onesided pretensioning, 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 kfactor for
friction losses during pretensioning is provided. The coefficient of dynamic
friction is used for m.
Source [2] adopts this approach, but in
the case of onesided pretension uses the kfactor which replaces the factor 2
(two legs to be tied down per string of lashing).
In the case of onesided pretensioning,
k = 1.5, in the case of twosided pretensioning k = 2. The coefficient of
dynamic friction is likewise used. In this approach, the two different horizontal
components of the lashingloops are disregarded. The difference between these
forces could be introduced into the balance. The two forces amount to:
Pretensioning side:
Opposite side:
Source [3] again turns away from the kfactor, but does introduce a safety factor f_{s}
= 1.1, which increases the necessary pretensioning force by 10%. The balance
reads:
This agreement corresponds to a kfactor
of 1.82. The reason stated for the safety factor in [3] is, however, not
pretension loss by friction but instead calculation uncertainty.
Source [3] moreover contains a sliding
balance for the combination of tiedown lashing and blocking, again
disregarding the loadbearing behavior of the two different securing means.
2.2.2
Securing against tipping
Source [1] interprets
the effect of the tiedown 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 tiedown
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 tiedown 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 pretensioned 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
righthand side, the denominator of the fraction may readily assume a value of
zero. This gives rise to a result tending towards infinity on the lefthand
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 tiedown 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 tiedown lashings are required for securing against tipping. If
the calculation is performed with changed belt tensions as on the right in
Figure 12, 3 tiedown 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 tiedown 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 tiedown lashings.
Source [3] no longer
uses the kfactor 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 f_{y} = 0,5, if pretension
F_{T} = S_{TF}.
coefficient of transverse acceleration f_{y} = 0,6, if pretension
F_{T} = 0,5 × LC.
a safety factor f_{s} = 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 tiedown 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.
