Tree statics

What´s about tree statics today ?
Today we are measuring the static of trees with traction forces, simulating the wind force. The inclination behaviour of the tree, measured at the same time, gives best results about it´s static situation - without damaging the tree. The way we do it you can see on the page "Die AfB-Methode". The AfB-method is spreaded by the Arbeitsstelle für Baumstatik to other tree-specialists in europe. Ask for further information.On the following pages we´ll give general information about Tree Statics in your language.

 

A contribution to the propper assessment of the strength and stability of trees - Summary
Prior to making any definite statement about the strength and stability of trees, exact details of occurring loads must be known, usually wind loads, the maximum stresses caused by these loads in the various parts of a tree and the anchorage potential of the roots. Previous methods of determining the condition of a tree, such as taking core samples and computer technology, in so far as they are relevant to TREE STATICS and a new non destructive type of survey method are discussed. The question of using supports to increase the safety factor against rupture or destruction is considered at the end of the paper.

Planungs-und Sachverständigenbüro
Günter Sinn
Sudetenstrasse 9
6368 Bad Vilbel 4
Federal Republic of Germany

Institut of Experimental Mechanics (Modellstatik)
Lothar Wessolly
University of Stuttgart
Pfaffenwaldring 7
7000 Stuttgart
Federal Republic of Germany.


A CONTRIBUTION TO THE PROPER ASSESSMENT OF THE STRENGTH AND STABILITY OF TREES

Recent growth in environmental awareness has significantly increased our interest in preserving the trees which shape our cities and countryside. Besides the importance of taking care of trees themselves, conservation has now significantly come to the fore when considering the question of traffic safety. Large town trees, optimal from the aesthetic point of view, were mostly planted at a time when there were very few, if any, automobiles. At this time the covering up of road and pedestrian surfaces with asphalt and concrete and the resulting restriction in growth room for street trees was not yet widespread. Due to later interference with tree roots resulting from, for example the laying of pipes and the oversalting and compression of the earth, as well as through additional mechanical damage to the upper parts of the tree and other stress factors, the vitality of our stock of older trees in particular has been significantly affected. These factors, combined with the effect of harmful organisms, primarily certain fungi types which attack the wood, often cause such trees to reach the limits of their strength and stability prematurely.

Assessing the stability of trees is the task performed by TREE STATICS. The importance of this task , whether its concern is the health of the trees themselves or the safety need of the general public, is not reflected in the methods of tree stabilization which have been applied up until now, methods founded on the skill and experience of the tree surgeon rather 1 than on a scientific basis. Without precisely determining both the stress factors mentioned above and also the natural forces at work on the tree such as wind loads, and without knowing the treels reaction to such forces, any assessment of strength and stability is, as a rule, based on assumptions which at best are empirically substantiated, and in certain cases cannot even be proven. The necessary expertise of dendrologically orientated authorities in TREE STATICS must also take account of recognized engineering fields, such as statics, dynamics, the testing of materials and surveying. When establishing strength and stability, the tree must be viewed in its entirety, using the principle of cause and effect. Dynamic factors and time dependent factors should also be included in the assessment, as should the individual characteristics of the tree. The correlation is shown in Fig. 1.

The Determination of Causes and Methods of Application

The determination of causes means the establishing of wind and snow loads and the factors which affect and control them. It is in built up areas in particular that the interaction between tree and building gains special importance. Thus it was recently discovered, through wind tunnel experiments, that the protection of trees can be quite substantial for the stability of the roofs of classical black forest houses. Then again, wind currents around buildings can place quite a considerable load upon the trees themselves. But there are also other regional and topographical peculiarities to be taken into account in order to 2 make a precise assessment of the forces which are at work on a tree. The general equation for the force Fw on an object resulting from wind is:

Fw = cw * A * p/2 * u²

Here cw stands for the coefficient of wind resistance, a term one may also recognize from the field of automotive engineering. When interacting with the projected tree surface area, A, it determines the resistance which the tree offers the wind. It is clear from observations that this cannot be a constant value; the tree behaves "cleverly" by giving way gradually as the wind load increases. when subjected to a wind current, it uses its flexibility to position itself more favourably, i.e. to minimize its total surface area exposed to the wind. Since the surface area can only be determined in the unloaded condition, both effects are included in the coefficient of resistance, the initial value of surface area A being entered. Survey results from MAYHEAD /2/ have clearly confirmed the coefficients of resistance, derived by SINN /l/ from equilibrium observations, of (cw = 0.1 ... 0.4, with a standard value of 0.3 for wind speeds of 12 Beaufort; this corresponds to 34 m/s (Fig. 2).

The tree´s flexibility, however, can also be of disadvantage if the wind induces it to sway at its natural frequency. To illustrate this point, imagine a child on a swing, throwing all his energy into the system just at the right moment. The process can only be taken so far until the swing rotates a full 360 degrees; this corresponds to uprooting or rupture in the tree's case. The effect can be considerable, particularly with the tall slim kind of tree which has a low natural frequency WESSOLLY /3/, AMTMANN /4/. This is also the reason why trees at the centre of a wooded area may be blown over even though they are not exposed to the full force of the wind.

Diagrams in which the additional loading due to natural swaying is taken into account have been developed for civil engineering purposes from surveys of the natural wind SCHLAICH /5/. With tall slim trees the increase can amount to more than 40%.
The other factor controlling the load is the wind pressure, q, which can be calculated be as follows:

q = p/2 * u²

Where
p = air clerisity
u = wind velocity

The two factors p and u are not constant either. Air density, p, is calculated at normal humidity from the local air pressure, the gas constant and the absolute temperature. Thus air density depends upon geographical situation, height above sea level, season and the weather conditions at that moment in time. Roughly speaking, an increase in air pressure of 100 mb causes a corresponding increase in air density of 11%. A fall in temperature from 300C to -250C causes air pressure and thus wind pressure to rise by 21%. The drop in air pressure from sea level to 1000 m amounts to 10%.

Of the two factors, the wind velocity, u, is of much greater importance in estimating the wind pressure, q, since its value is squared in the equation. This means that if the velocity is doubled, the resulting wind pressure is four times as great (RUSCHEWEY /6/, /7/, KAMEI u. MARUTA /8/, STATOPOULOS U. STORMS /9/ and WIESE /10/).

The wind velocity at the site of the tree is dependent upon:

- geographical situation
Wind load is not equal all over the earth. There are wind charts available for estimating the expected maximum wind force for a given period of time /11/, /12/. Weather stations have comprehensive documents on prevailing wind directions.

- topographical situation

The second factor controlling wind velocity is whether the tree in question is situated in lowland, close to the sea, on the brow of a hill or on the leeward side of a mountain group.

- the season and meteorological influences
The combined effect of seasonal weather conditions such as autumn storms and the fact that the tree may or may not have lost its leaves at the time can also be of significant importance.

- the wind path
The composition of the wind path has quite a considerable effect on wind velocity. It controls the course of the boundary layer. That is, the wind at ground level is not as strong as it is at greater heights. The wind velocity can be calculated approximately using a simple boundary layer equation, in which it is sufficient to take the structure of the path into account just by applying the factor alpha (Fig. 4).

- gusts of wind
Natural wind is not continuous. The air stream pulsates and rotates and thus is capable of exciting the tree at its natural frequency and injecting energy into the treels swaying system up to the point where it ruptures. This explains a proportion of the fallen trees within dense wooded areas.

- the close proximity of other large structures Every large structure in the wind stream influences wind velocity, direction and degree of turbulence. A first basic estimate of this fact is given in the boundary layer equation. This, however, only takes account of the wind path which leads directly to the tree and not of the area immediately surrounding it. The wind velocity increases significantly in alleyways and the immediate proximity of buildings due to its path being narrowed. There is an extensive amount of data available /6/, /7/, /8/, /9/ and /10/, which can be applied to determine the wind load on trees in built up areas, passages, etc. /3/.

Equations, diagrams and tables covering all these influences are available. They can be applied to obtain an exact analysis of wind conditions, should this be required. Thus it is possible to determine the loading on a tree situated at one particular location. Thanks to the advent of the computer these calculations are considerably less complicated than this description would lead one to believe.

THE STRUCTURAL SYSTEM OF A TREE AND THE TRANSFERENCE OF ITS FORCES INTO THE GROUND

Describing the structural system of a tree basically means answering the question: how does the tree assume loads and how does it transfer them into the ground via its internal stresses without collapsing? Answering this question enables us to determine its strength and stability.

Transference of Forces Underground - Stability
As a rule healthy trees standing freely are so well anchored in the ground by their root systems that they can easily withstand high wind velocities (e.g. 12 Beaufort) SINN /14/.

If, however, the root system has been interfered with and damaged to a considerable extent, then, particularly in strong winds, this can result in an unstable state of equilibrium between the crown, which acts almost like a sail, and the root system with its reduced static effectiveness, resulting in the uprooting and tipping over of the entire tree. Stability is treated in the following equation:

nk = (N * a) / (w * l) =
moment of resislance / tipping moment

- N is the weight of the tree, including its statically effective root system;
- a is the lever arm, the horizontal distance from the centroidal axis to the pivot point;
- w is the total wind load applied at the centroid of the crown taking account of the factors mentioned above;
- 1 is the lever arm due to the wind load, the vertical distance from the centroid of the crown to the pivot point SINN /15/, /16/.

In this way the self weight of the tree is calculated according to its measurements and the specific weight of the wood when wet SINN /15/. Subsequent investigations, such as those on the copper beech in the Villa Berg Park, have proven the workability of this method WESSOLLY /17/. The weight of the roots is calculated according to the root area necessary for the stability of the tree; the extent of this root area and its effectiveness is determined by a special examination (consisting of digging up a section of the root system and using a high pressure spray to reveal the roots themselves) The following illustrations show various possible tree root forms and sizes for a particular tree with a total surface area of 60 m² exposed to the wind, taking its self weight moment into account. Each of these root forms is stable enough to withstand force 12 gales. The form of the root system determines the standing room required by a tree to guarantee its stability SINN /18/.

The following parameters are included in the calculation of wind load, as already carried out above:

- coefficient of wind resistance, cw, taking particular note of the shape, form and structure of the crown.

- wind pressure, q, taking particular note of the actual location of the tree and wind properties.

- total surface area, A, exposed to the wind.

A simplified photo-optical process, combined with a method of measuring the tree´s height, has been developed by SINN /15/ in order to calculate precisely the treels inhomogenous surface area exposed to the wind.

Knowing the load values which are used to determine strength and stability is an indispensable requirement of any calculation in TREE STATICS.

When calculating stability, the determination of the wind load moments, taking into account the tree´s self weight moment, enables us to conclude the extent of the statically effective root system.

Should the root system have been damaged, the stability quotient produced in the stability equation describes the stability of the tree in question. In many cases, the tree can be restabilized by reducing the surface area exposed to the wind, i.e. trimming the crown SINN /25/. Occasionally, further stabilizing aids are necessary.

The methodical process of calculating a tree´s stability is demonstrated in the following example.

Transference of the Forces in a Tree of Safety against Rupture

The forces occurring in the crown are transferred to the root system via the parts of the tree above ground. In the course of this the internal stresses caused by the external forces must not exceed the rupture. stress of any part of the tree; if they do, then the overstressed part fails.

When making an assessment of the factor of safety against rupture, the basis and methods of calculation, the dimensions and material properties of the fresh wood of the tree type being examined must all be known. The study of dynamic behavior, theory of bending, computer modeling with truss and thick shell elements are all used for this purpose. Knowledge of the modulus of elasticity, shear modulus, compressive strength, tensile strength, shear strength and torsional strength are required too WESSOLLY /19/. Also important for the calculation is an exact knowledge of the loaded cross-section; in determining this, however, inaccuracies may have to be reckoned with, since there may be hidden weak points in the tree.

It must also be taken into account, that due to the low tensile strength of the wood parallel to the fibre, the area likely to fail can, particularly on the tensile side, cover quite a considerable section of the tree. The rupture occurs at the cross-section, which is weakest at that particular moment in time.

Survey Methods to determine a Tree´s Strength

Besides the TREE STATICS field of work, there are at present two other methods being used to determine the condition of a tree, neither of which is particularly meaningful or practical as far as statics is concerned, since what they are measuring is not actually material strength.

1. The Coring Method (Increment Borer) and Endoscopy
This process enables us to identify a weak point, however it does not enable us to determine precisely the loaded cross-section. It is impossible to take enough samples to allow us to determine the exact cross-section without causing lasting damage to the tree. The result of coring is a crisscrossing of so called "highways" (DUJESIEFKEN) leading into the tree and providing ideal access for harmful organisms. Besides, the samples obtained do not assist in determining the strength of the tree, since they are taken perpendicularly to the direction of the fibres, whereas the tree is loaded lengthwise to them. At most, the material density can provide a rough estimate of the treels strength WESSOLLY /19/. Nor does the method of endoscopy provide any information as to the loaded cross-section or material properties; it can merely serve to provide support for the findings from other test methods.

2. The Computer Tomograph (HABERMEHL et al /20/)
With the computer tomograph the density distribution of the wood can be clearly revealed and it is possible to locate rotten or weak points within the tree. However the strength values, which are used to determine the tree's factor of safety against rupture can not be obtained. The exact relation between the determined density and material strength is not clear (e.g. one problem is that the interior of the tree may be wet). Even if the strengths were known, it would still be necessary to calculate the stresses within the determined cross-section in order to arrive at the final safety factor against rupture. The extremely high price of the computer tomograph and its operational costs must also be taken into consideration. An ultrasonic system could possibly do the same job.

A New, Non-destructive Method of Directly Determining a Tree´s Strength
Since previous TREE STATICS methods have proven to be unsatisfactory, we are suggesting here a survey method by which the safety facture against rupture can be directly ascertained, without having to open up the tree or damage it in any way. Here, the trunk cross-section is arbitrary. The device works on a question-answer basis. A moderate force applied to the top of the trunk takes on the function of the question. The tree replies by yielding slightly. This yielding is then translated back to the questioner by a highly accurate sensor. By linking question, answer and translation, we are able to determine its condition, as regards strength, related to a clearly defined external load (see Eqn. 1) . This statement is possible because the maximum stress due to bending always occurs in the extreme fibre of the trunk cross-section.

Since the physical relations are very simple, it is easy to test the tree safely without having to approach its rupture point. Thus it can be ensured that the tests cause no danger f or the tree or the survey team.

This test may also be used to identify hidden weak spots caused by improper growth after grafting (HERBIG /2l/). The grafted area is tested first, and then areas directly above or below it are tested with the same load. The results are compared, and if a weak spot is present, this is indicated by a sharp jump in the reading from the sensor. This method will be fully discussed in a later publication, once detailed experiments have been carried out on various types of trees.

Each test only accounts for the condition of the tree at one particular moment in time, whereas the growth and deterioration processes of a tree are dynamic. The tests, therefore, should be repeated at intervals. Also of importance is an examination of fungal infections within the tree to determine the extent of their harmful effect.

TO WHAT EXTENT CAN STATIC SUPPORTS BE APPLIED TO MINIMIZE RUPTURES?

In the following Cases Static Supports are ineffective: Tensile experiments were conducted on the copper beech in the Villa Berg Park in Stuttgart in October 1987 and their subsequent evaluation was carried out using more than 50 different two and three dimensional computer models. All possible combinations of supports and soil conditions were considered, as was a comparison with torsional loading. The results are as follows: Supports within the trunk opening are statically ineffective (WESSOLLY /23/, /24/) if the opening does not extend up to the fork (Fig. 11). In any event, one should determine in advance whether the maximum forces expected are even strong enough to cause a rupture. Nor has anything been mentioned so far about the biological side effects of boring into the living wood, i.e. increased vulnerability to fungal attack. There is also the danger that the support could be rendered ineffective, should the bolt be punched through the wood as a result of extreme loading (WESSOLLY /17/). Since certain types of wood are significantly stronger than others, the maximum compression tolerated can vary considerably.

A Discussion of Cases where Supports can be successfully applied:

The above examination covered the cases in which it is not advisable, from a statics point of view, to use supports. Nevertheless, this does not mean that every application of supports has to be pointless.

Case 1
The tree trunk is split vertically and now consists of two halves.
Using the full cross-section, the moment of resistance, that is, the measure of geometrical bearing capacity, is equivalent to:

Wv = 0.785 * r³

With a tree which is split down the middle, the moment of resistance perpendicular to the split is only:

Wg = 0.382 * r³

This means that with the same wind load the safety factor of the solid trunk against rupture is twice as great:

Wv / Wg = 2.05

Thus any measures undertaken must aim to hold the two trunk halves together in such a way that they behave statically as though they were one trunk. Since it is, above all, shear, which is transferred via the broken connection, it is the shear bond which has to be reconstructed. It is even possible that the tree may grow back together again, if the living tissue is pressed back together and held in this position by a stabilizing aid, e.g. the way a plaster cast is applied in medicine. Only biologists can explain the exact details of this process. At any rate, the important point here is that the advantages of using a support outweigh the disadvantage of biologically harmful side effects.

Case 2
A hollow tree trunk has a large opening which extends up to the point where it forks into two branches. Here, what has to be prevented is the possibility of one or both of the branches tearing the main trunk apart. The application of stabilizing aids as measures against tearing apart may be appropriate. Nevertheless, prior to taking such measures, the external and internal forces should be established and compared with one another by performing load tests on the tree and determining the strength of the hollow tree trunk or by applying a special survey technique (to be examined in a later publication) Thus it is possible to determine whether the static system requires any supports. If so, the appropriate support to use is one which can transfer shear force. Horizontal and diagonal supports are connected together and to the trunk in such a way that the resulting shear bond can prevent the top of the trunk from tearing apart. This bond does not have to be continued into the lower part of the trunk, since there, this case overlaps with those shown in Fig. 11, for which it is ineffective and inadvisable to apply trunk supports as a means of increasing the safety factor against rupture caused by bending. Biological and aesthetical factors must also be taken into account when applying these measures.

Case 3
In the case of a tree trunk which splits into two branches to form a fork, practitioners are of the opinion, that secondary lateral growth could cause the collapse of one of the two branches. Assuming that, by limiting the motion of the two branches, both will grow in such a manner as to improve their ability to sustain tensile forces, and also that the formation of pockets of rotten wood within the trunk can be prevented, then, weighing all the pros and cons, a support in the upper part of the trunk may possibly be justified. Due to the unfavourable lever arm lengths, the forces occurring can be extremely high. Thus it should be taken into consideration that the bearing capacity of the wood perpendicular to the fibres is very low and it is possible that it may not be able to sustain the forces generated. The important thing here is knowing the maximum tolerable limit of compression, which, in extreme cases, can be seven times lower in one tree sort than in another (Table I).

In addition, it must be noted that rotting in the region where the support is bolted to the tree may considerably increase the possibility of the support being punched through the rotten wood. Surveys carried out on the supports of the copper beech proved that they had indeed punched through the wood prior to collapse.

As a rule, since wind loads may be considered to be applied at a point two thirds the height of the tree, then any supports should also be located here in order to intercept the loads at the point where they occur. A conventional cable anchorage could take on this support function, however, advantages and disadvantages regarding its effectiveness must be examined in detail, because the attaching of a cable to a tree restricts the natural swaying of its branches (the formation of reactionary wood may be prevented) and can cause impact loads on the system. It is possible to construct a simple damper for the system along the lines of those used in ship-building.

Fig. 15 shows the possible positioning of stabilizing aids. Two possible forms of loading can be differentiated:

In the case of load 1, wind and/or snow loads cause lateral out-of-phase motion in the branches (i.e. to and fro, towards and away from one another). The swaying motion which threatens to tear the two branches apart can be prevented by a cable anchorage, nevertheless, it must be taken into consideration that the restriction of motion due to the stiffness of a taut steel cable can be very abrupt. Thus the possibility exists that high forces can be introduced into the wood. A support at the fork (F) alone would not be capable of increasing the safety factor against tearing apart when subjected to a load such as this: the forces at this point are maximal, due to the lever arm relations.

In the case of load 2, wind and/or snow loads cause forward and backward out-of-phase motion in the branches (i.e. to and fro, forward and backward past one another). In this case, a bolt just above the fork (F) can be effective, since the cable hardly contributes anything towards load resistance here. The support is situated at the lowest possible point where it can contribute towards stability and transfer shear force. If it were any lower, the fork would be torn apart before the support could sustain any of the load. Another possibility is a support one level higher, where the section of the support between the two branches is so minimal as to allow the transfer of shear force, or a second parallel support at the same level. An extra support lower down in the trunk would not improve the safety factor against rupture.

RESULTS

Basically, supports are unnecessary in cases where the damage to the trunk and openings in the trunk do not reach up to the point where the wind load is introduced via the branches (Fig. 13).

In cases where the loads sustained by the branches are likely to cause the trunk to be torn apart, stabilizing aids can be effective. Since combined shear stresses often occur in such cases, it is advisable to use a shear bond. Such a bond is realized, in the case of a wide open trunk, by diagonal supports. This bond should be applied in such a way as to cause the least possible injury to the tree. This means the diagonal supports should be connected to the horizontal supports and not to the wood. Should parts of the trunk have to be fastened together, horizontal supports are sufficient.

It is possible to determine the forces occurring in a tree and its ability to transfer these forces into the ground, by proceeding carefully and responsibly prior to taking any measures.

CONCLUSION

Prior to making any def inite statement about the strength and stability of trees, one needs to know the exact details about the occurring loads, usually wind loads, the maximum stresses caused by these loads in the various parts of the tree and the anchorage situation of the roots. Previous methods of determining the condition of a tree, such as coring and computer tomography, in so far as they are relevant to the field of TREE STATICS, and also a new non-destructive type of survey method are discussed. The question of using supports to increase the factor of safety against rupture is considered at the end of this paper.

ACKNOWLEDGMENT

The studies reported in this paper were supported by the D.F.G. (Deutsche Forschungsgemeinschaft) 'specializing in the area of lightweight construction in nature and architecture. The writers are grateful to K. Chapman and S.K. Lo for translating this paper.

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Published in:
Arboricultural Journal
1989
Vol 13 pp. 45-65

AB Academic Publishers
1989
Great Britain

Günter Sinn
Lothar Wessolly