Instructions 4 – Safety evaluation

In order to evaluate stability the two key coefficients have to be determined first. They are:

  1. maximum breaking load and
  2. maximum tilting load.

hereinafter such terms are used without any further explanation.

But first a definition of both terms:

The breaking load describes the load on the top or the moment on the trunk that has to act on them so that the trunk breaks.
The tilting load describes the load on the top or the moment on the trunk that has to act on them so that the tree is uprooted and falls.

Next it is necessary to determine maximum wind load acting on the top and thus the maximum moment acting on the trunk base.
For this the procedure is as follows:
First you use the program ArWilo to determine top surface and its centre. It will be important in the subsequent calculation and is assumed to be the lever.

Calculation of wind load

Input values

Ak Top sail surface
LD Air density
LA Load centre
cw Cw value
v Wind velocity
ά Terrain coefficient
tu Gust coefficient
h(g) Height of end of soil border layer
w Frequency coefficient
v (=32.7m/s) Wind velocity for Bft. 12

1. First it is required to determine wind velocity:

For this the following equation is used:
v real = (tu*v*( (LA x / h(g))^ ά)) m/s

In words:
First the quotient from load centre and height of the soil border layer is established.
Such quotient is raised to the higher power of the terrain coefficient.
Then the result is multiplied with wind force and gust coefficient.

This result presents the actual wind velocity at the tree’s load centre.

2. The next step is establishing the moment acting on the tree or trunk base in relation to the established wind load.

For this the following equation is used:
Ftrunk base= Ak* LA*cw* LD *vreal^2 Nm

In words:
The wind velocity established previously is now squared (= multiplied with itself).
Such square now is multiplied with the top sail surface, the load centre (=lever), the tree’s cw value and air density.

This result now states the moment in Newton meter acting on the trunk base at a wind force of 12.

The obtained result now may be used as a reference value at any time. By comparing the tilting or breaking load the measure of safety is determined. If the tilting or breaking load is below the wind load (values below 1 in the result fields) there is a pressing need to act. Usually a safety of 1.5 is to be taken as a minimum value (+ 0.5 as safety coefficient for calculation of errors and other imponderables).

The tensile test

In the tensile test the load x is fed in the tree in an angle y. It then is observed by which angle z the tree tilts. By means of the generalized tilting curve it can be read off how many percent of the tilting load the initiated load is. The generalized tilting curve is a non-linear but exponential function. The y axis shows the straight line by which the tree is tilted, the percentage of the tilting load is displayed on the x axis. This means that e.g. 0.25° are 40 % of the tilting load.

Now the actual value has to be determined trigonometrically since we are not pulling in an angle of 90° to the tree. For this the load acting perpendicularly has to be established. This value then can be lead to 100 % by means of a simple percentage calculation. Now multiply such value with the measuring height where the load is fed, i.e. the lever, to establish the maximum moment at the trunk base causing breakage.

Then the calculated tilting load is correlated to the maximum wind load and thereby you obtain the tree’s safety in percent.