# What is roll stiffness

## ECE-R 111

<< To the ECE regulations**Safety of tank trucks from tipping over****Uniform provisions concerning the approval of tank vehicles of categories N and O with regard to rollover stability**

### Legal situation

According to the provisions of ADR, all tank vehicles that are registered for the first time after July 1, 2003 must meet the provisions of ECE-R 111. According to multilateral agreement No. 145, the proof of tilting stability according to ECE-R 111 was suspended in a number of European countries until June 30, 2004, but not in Germany. The proof of the stability against tipping is therefore part of the RSE. An "initial registration" is also required if an old tank is mounted on a new chassis.

According to ECE-R 111, proof must be provided that the fully loaded (!) Tanker does not tip over at a lateral acceleration of 4 m / s² or a tilting angle of 23 °. Whereby actually: arctan (4.0 / 9.81) = 22.2 °. With the required 23 ° you are on the safe side. The proof can be computationally or experimentally, on a tilting table (*tilt table*). In Germany, evidence is preferably provided by calculation, since tilting tables are rare and the tests are expensive. In the case of cross-border traffic in newer ADR countries (e.g. Poland), all (German) tankers in accordance with ECE-R 111 are allegedly placed on a tipping bridge by the local authorities in order to check that they are safe from tipping: one apparently does not trust the calculations and certifications carried out there. According to rumors, however, a (German) tanker with confirmation according to ECE-R 111 has never overturned there either. This also speaks for the practicability of the calculation of the tipping safety.

Incidentally, "fully loaded" is not always to be equated with the maximum permissible mass specified in the vehicle documents. If the vehicle manufacturer himself specifies a higher technically permissible total mass, this must be used as a basis for the calculation.

The calculation processes for mathematical proof are prescribed in detail in ECE-R 111. In principle, the calculation is based on a spring-loaded four-wheel model that tilts over the also spring-loaded wheels on one side. This model is also applied to the semi-trailer, although its tilting axes are known to run obliquely in space. In this case, the supporting torque of the fifth wheel is converted into the equivalent supporting torque of an axle mounted instead.

A lot of design data is required for the calculation, such as

- Track width
- Total mass
- unsprung mass
- Center of gravity of the sprung masses
- Center of gravity of the unsprung masses
- Height of the roll axis
- Tire spring stiffness
- Spring hardness of the construction springs

This data must

- in the case of chassis from the vehicle manufacturer
- for semi-trailers and trailers from the axle manufacturer (!)

made available, possibly supplemented by data from the body builder. The calculation must be checked by an officially recognized expert (aaS / aaSmT), which will, however, be limited to checking the bare calculation. The aaS can check the manufacturer's information on which the calculation is based, if necessary, with regard to the order of magnitude.

### The calculation method

The calculation method is the same for tank vehicles, tank trailers and tank trailers. In the semi-trailer, the fifth wheel is replaced by a fictitious axle, so to speak. The calculation pretends that the semitrailer, based on this fictitious axle, would tip over the wheels on one side.

The variable names of the ECE-R 111 are a bit strange, for example the axle load, i.e. the sum of the wheel contact**forces**, With *A.* designated. We therefore use a (hopefully) somewhat less familiar nomenclature in the following. The ECE-R 111 expresses all the formulas with the axle loads, so that we will also switch to this nomenclature in the last step. In the derivation itself, however, we fall back on the somewhat more familiar masses.

The sequence of the formulas in the following differs significantly from that of ECE-R 111, because here we are more closely oriented towards the sequence necessary for the derivation.

We do not use the sum formulas also mentioned in ECE-R 111 here. It is immediately apparent that

- the total weight is the sum of the axle loads
- the roll stiffness of the entire vehicle is the sum of the spring constants of the individual axles
- the mean track width is best calculated as the mean value weighted with the respective axle load.

### Roll stiffness

First the roll stiffness of the vehicle is calculated. Thereby the roll stiffness of the structure *c _{a}* and the roll stiffness caused by the vertical tire spring stiffness

*c*to an overall roll stiffness

_{r}*c*summarized. The springs are connected in series so that applies

1 ...... \ (\ frac1 c = \ frac1 {c_r} + \ frac1 {c_a} \)

and thus

2 ...... \ (c = \ frac {c_r c_a} {c_r + c_a} \)

About vertical tire spring stiffness *f* and gauge *t* the roll stiffness of the axles can be calculated

3 ...... \ (c_r = \ frac12 f t ^ 2 \)

The roll stiffnesses of the axle (unsprung mass) and body (sprung mass) relate to different roll centers: While the roll center of the unsprung masses is at road level, the roll center of the sprung masses is at the same level *H*. In order to get them according to Eq. (2) To be able to offset each other, you have to convert them to the same center of rotation. The ECE-R 111 provides for roll stiffness *c ' _{a}* of the superstructure (with the height of the center of gravity

*H*) to be converted to road level

4 ...... \ (c_a = \ left (\ frac H {H - h} \ right) ^ 2 c_a '\)

The quadratic increase in the torsional stiffness according to Eq. (4) follows from the fact that

- the roll angle is correspondingly smaller
- the lever arm corresponding to the inertia forces grows.

This is the (total) roll stiffness *c* now calculated. It is usually given in Nm / rad. If the axles are not identical, this calculation must be carried out for the individual axle types. The axes are then numbered consecutively (index *i*) and the individual roll stiffnesses *c _{i}* calculated.

For vehicles with double tires - not only when calculating the roll stiffness of the axle - initially an equivalent track width *t* from standard track width *T ' * and center-to-center distance of the double wheels *t* to calculate

5 ...... \ (T = \ sqrt {T '^ 2 + t ^ 2} \)

This track width roughly connects the inside of the outer wheels and, according to experimental studies, corresponds roughly to the tilt axis of a double-tire vehicle.

The equivalent torsional stiffness is calculated for tank trailers *c _{k}* of the kingpin (

*king pin*) from the tracking force

*F.*according to

_{k}6 ...... \ (c_k = 4 \ frac m {rad} F_k \)

This method of calculation is firmly specified in ECE-R 111. The contribution of the fifth wheel to the roll stiffness is only about 15% anyway, so that this rough calculation is apparently sufficient. The equivalent gauge *T _{k}* the fifth wheel is simply the mean value of the track widths of the other trailer axles. If these are identical, as is often the case, this value also results for the equivalent track width of the fifth wheel.

### Roll angle when lifting the axle (s)

If all axles are identical, all wheels on the inside of the curve lift off the roadway together. That through the roll angle *θ* The moment caused is then just as great as the weight *G* is only on the wheels on the outside of the curve, i.e. the wheels on the inside of the curve *G*/ 2 are relieved. The vehicle is therefore at the tipping limit

7 ...... \ (M = c \ theta = \ frac12 G T \)

The associated roll angle can therefore be derived from this condition *θ* calculate without the moment or the magnitude of the lateral acceleration causing it being known

8 ...... \ (\ theta = \ frac {G T} {2 c} \)

If the axles are not identical, the axles lift off at different angles, which must be calculated separately for each axle type. The axes are numbered with the index *i*, it results, depending on the axle load *F. _{i}* and torsion spring stiffness

*c*so

_{i}9 ...... \ (\ theta_i = \ frac {F_i T} {2 c_i} \)

The axis at which the smallest angle is calculated is the first to lift off the road.

### Lateral acceleration when tilting

In the rigid body model, the vehicle tilts when the resultant of lateral and gravitational acceleration crosses the tilting axis.

10 ..... \ (\ frac {m a_q} {m g} = \ frac H {\ frac {1} {2} T} \)

In the ECE-R 111, the transverse acceleration in g, i.e. the ratio *q = a _{q}/G* Voted. This applies

11 ..... \ (q = \ frac H {\ frac {1} {2} T} \)

The vehicle sways at the angle *θ*, this is the equilibrium condition

12 ..... \ (\ frac12 m g T = m a_q H + m_s g H_s \, sin \ theta \)

With *m _{s}* as sprung mass (

*jump mass*). Division by

*G*and sin

*θ*≈

*θ*results

13 ..... \ (\ frac12 m T = m q H + m_s H_s \ theta \)

The angle *θ* results from the roll stiffness and lateral acceleration

14 ..... \ (c \ theta = M = m_s a_q H_s + m g H \ theta \)

Since the torsional stiffness of the body and tires were initially offset against each other, the unsprung masses also rotate with the angle *θ* around the roll center, so that these masses must also be taken into account when the weight force contributes to the angle of rotation.

*Annotation:*

At this point, the ECE-R 111 apparently sets the approach

14a ..... \ (c \ theta = M = m_s a_q H_s + m g H_s \ theta \)

based on the total mass *m* with the (larger) center of gravity of the sprung masses *H _{s}* pairs. It is not clear why it does so. The result of this approach always gives a larger roll angle and reduces the stability against tipping, so you are on the safe side.

However, we will continue to count on our approach for now:

15 ..... \ ((c - m g H) \ theta = m_s a_q H_s \)

us finally

16 .....\ (\ theta = \ frac {m_s g H_s} {c - m g H} \)

Insertion into Eq. (13) gives

17 ..... \ (\ frac12 m T = m q H + m_s H_s \ frac {m_s g H_s} {c - m g H} q \)

Solving for q gives

18 ..... \ (q = \ frac12 \ frac {m T} {m H + g (m_s H_s) ^ 2 / (c - m g H)} \)

Expand with *G* results

19 ..... \ (q = \ frac12 \ frac {F T} {F H + (F_s H_s) ^ 2 / (c - F H)} \)

If the correction Eq. (14a) results in agreement with ECE-R 111

19a ..... \ (q = \ frac12 \ frac {F T} {F H + (F_s H_s) ^ 2 / (c - F H_s)} \)

### Pseudo tilt angle of the "weakest" axis

If the vehicle axles have different axle loads and spring stiffnesses, they lift off the road one after the other. As already explained above, the relatively stiffest axle lifts off first. According to ECE-R 111, a pseudo lateral acceleration is to be calculated with this axis. First is the ratio value *ξ* between the roll stiffness *c _{m}* this axis (index

*m*for maximum) and the roll stiffness

*c*of the entire vehicle

20 ..... \ (\ xi = \ frac {c_m} c \)

With this factor, the lateral acceleration becomes, analogous to Eq. (19a) *q _{m}* calculated when lifting the first axis

21 ..... \ (q_m = \ frac12 \, \ frac {F T} {\ xi F H + (\ xi F_s H_s) ^ 2 / (c_m - \ xi F H_s)} \)

So it is, with a slightly different nomenclature, in ECE-R 111. However, you could also exclude

22 ..... \ (q_m = \ frac1 {2 \ xi} \, \ frac {F T} {F H + (F_s H_s) ^ 2 / (c - F H_s)} = \ frac q \ xi \)

which would make the connections a little clearer.

### Relevant lateral acceleration for the verification

Now after the lateral acceleration *q _{m}* when lifting the first axis and the lateral acceleration

*q*are calculated when tilting the entire vehicle, should be interpolated between these two values

23 ..... \ (q '= q - (q - q_m) \ frac {F_m} G \)

Here too, you could save yourself all the intermediate steps and do the same

24 ..... \ (q '= \ left (1 - \ frac {\ xi - 1} \ xi \, \ frac {F_m} G \ right) q \)

calculate.

In any case it is *q ' * the transverse acceleration decisive for the computational proof and it must *q ' * > sin (23 °) apply.

### Calculation sheets for the proof of the stability against tipping

The spreadsheets for verifying the stability against tipping in accordance with ECE-R 111 are usually made available via the Internet, for example for

- MAN vehicles
- Mercedes-Benz Bodybuilder Portal Registration required for access! There you can find an Excel sheet
*ECE-R111_Tipping limit_de.xls*Download (with macros) and carry out a corresponding overturning limit calculation (of course only for an MB single vehicle!). The maximum permissible height of the center of gravity for the tipping condition according to ECE-R 111 results from the sum of the individual weights and the positions of the center of gravity for the (selected) vehicle, body and payload.

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