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A Guide For TreeStability Analysis  1 
A Guide For
TreeStability Analysis
Peter Sterken
Second and expanded edition
A Guide For TreeStability Analysis  2 
© Peter Sterken 2005.
No part of this publication may be reproduced, in any form or by any means, without
the written permission of the author.
ISBN: 9090193774
Exemption from liability
The author expressively emphasises that each case is unique and has to be treated as
such, and that the guidance and recommendations published in the present paper
should be followed in accordance with the judgement, specialised knowledge and
expertise of the individual expert.
Therefore, the author is exempt from any liability for damage to persons, objects or
property resulting from the use of the herein published information.
The facts and results gained in the scope of the investigations that are mentioned in
this publication only relate to these cases and are not transferable to similar
circumstances.
A Guide For TreeStability Analysis  3 
Foreword
Ever since Prof. A. L. Shigo presented his CODIT model to the world,
knowledge on trees improved enormously. In stead of trying to heal a
tree, a case what seemed to be rather difficult, improvement of
growing conditions and preventing a tree to weaken and get infected
by pathogens of any kind became the new starting point for
arboriculture.
A lot of problems have been solved in the past, but still a lot of
questions are to be answered, and maybe have not been asked yet.
One of the main questions to be worked on is a question on the
stability and safety of trees, especially in an urban environment.
This book does not claim to give all the answers. It even will give no
answers that have not been given in other publications. The
advantage of this book is that it will bring together the answers that
are given on different aspects of treediagnosis.
A lot of research on different aspects of treestability is done, each
starting from one specific point of view. Trees have been researched,
on their dynamic, static and biological aspects. To get to results, it
was necessary to see a tree as just a static, dynamic or living mass.
All starting points to gather knowledge on trees were definitely good
ones, just because there was still a lot of knowledge to be gathered
and brought together.
So this book can in no way be seen as an attack to anybody who
ever did any kind of research on any aspect of trees, or on any of the
results they got.
But as a tree is not just a static, dynamic or biological mass, all the
knowledge gathered in these different scientific fields one day should
be brought together.
That is what this book tries to do: bring together all existing
knowledge on treestability, what should lead to a better diagnosis of
treeproblems. But as there is still a lot of knowledge on trees still
waiting to be discovered, this book can not claim to be complete, nor
to give all the answers. It is just what it claims to be: one small step
upwards the knowledge and understanding of trees. Treediagnosis
does not start with this book, neither does it stops here, it is just one
step upwards, hopefully in the right direction.
Wim Peeters
A Guide For TreeStability Analysis  4 
Contents
1. Introduction
2. Prognosis of the breakingsafety of hollow trees
3. A mathematical model for the prediction of the critical
wind speed for the failure of trees
3.1 Introduction
3.2 Structure of the V model
• Assessing the wind load in the crown
• Dynamics
• Breaking safety
• Uprooting safety
• Torsion safety
• Breaking safety of stems
• Critical wind velocity
• Necessary residual wallthickness
• Assessing the wind load in the crown cabling
3.3 Discussion
4. What is the necessary residual wall thickness for a tree?
4.1 Introduction
4.2 An example of treestatics
• Wind load analysis with the Vmodel
• Material properties
• Thickness of the residual wall: bending theory of the hollow
beam
4.3 Results: necessary trunk diameter and residual wall
thickness
4.4 Discussion
5. Visual assessment
5.1 Wooddecaying fungi
5.2 Assessing the breaking safety of hollow trunks
5.3 The body language of the tree
5.4 Assessment of vitality
5.5 Conclusion about visual assessment
A Guide For TreeStability Analysis  5 
6. Behaviours of the wooden body
6.1 Introduction
6.2 A brief description of some common mechanical
behaviours
• Summer Branch Drop
• Euler buckling
• Brazier buckling
• Shearing stresses
• Concentrations of stress
• Torsion and the complex combinations of forces
• Basal bell fracture
7. One of the secrets of current treestability assessment
8. Palms and stability
9. Protocol
Examples
• Eucalyptus camaldulensis. Parc Sant Jordi, Terrassa,
Cataluña.
• Eucalyptus globulus. Parc Sant Jordi, Terrassa, Cataluña.
• Statics vs. dynamics
10. Treesaving interventions
11. Conclusion
12. End notes
Important Literature
A Guide For TreeStability Analysis  6 
1. Introduction
This guide offers a deeper understanding of how a tree fails and how
to prevent it. A profound and intuitive knowledge of the inherent
perversity of materials and structures, mathematics and trees, is
one of the most precious qualities an arborist can have.
It is not necessary here to chew again on all types of symptoms of
structural defects that can be found of a tree. The current offer of
literature that deals with visual assessment is huge, and these works
are good references as far as common symptoms are concerned.
This small book briefly describes the principles upon which the
author’s protocol is based: mycology, biology, mathematics, wind
engineering, mechanical behaviours and visual assessment. But
above all pleads for the integration of current methods, criteria,
knowledge and common sense. The present publication can serve to
the reader as a continuous thread to which the following fine works
should be added:
The V.T.A. method as described in “The Body Language Of Trees”
by Mattheck & Breloer (1995), presents tree reactions and faults
leading to tree failures. The description of possible failures by
fracture is a delicious and obliged literature. The VTA method agrees
well with Gordon (1999) in how structures fail. Here, invasive
diagnostic procedures are reserved for trees assessed with serious
risks.
Currently, the SIA method (Wessolly & Erb, 1998) is a powerful tool
to assess the breaking strength of a hollow tree. This method,
available free of charge, calculates the diameter of a tree trunk and
the necessary residual wall that a given tree needs to withstand wind
gusts of 32,5 m/s. The results take into account tree height, crown
form, drag factor and strength of the wood, amongst other factors.
In the InclinoElasto method (Wessolly & Erb, 1998), the tree is
subjected to a wind simulation force by pulling and its behaviour is
recorded by devices recording stem angle and fibre length changes.
An inclinometer measures the inclination of the stem base in order to
assess the uprooting potential of a tree. An elastometer records the
longitudinal elongation or shortening of the most exterior stem fibres.
This method enables a better insight in the tree it’s stability. The
I.B.A. method (Reinartz & Schlag, 1997) describes the interaction of
mycology, vitality and stability. This visual method combines the
visual detection of wooddecaying fungi, sometimes long before the
first fruiting body appears, and the SIA method. The authors of the
A Guide For TreeStability Analysis  7 
IBA method state that not only many trees can deal easily with fungi,
but also that an instrumental diagnosis is very rarely necessary.
At the current time, Prof. Dr. Francis Schwarze is investigating the
interaction between the defence mechanisms of a tree and the
strategies that decay fungi employ to overcome the former.
The status quo, i.e. the current condition of decay in a tree, can be
assessed macroscopically. The above described methods can offer a
static recording of this current condition. Nevertheless, the dynamics
of the interaction, and hence the future safety of the tree, can only be
assessed microscopically. In this way, for the first time, it is possible
to incorporate a prognosis regarding the future development of
decay, and invasiveness of different decay fungi, in the sapwood of
trees.
Finally, the goal of the present publication is to blend components of
the above described methods. In this way, a reasonable and carefully
weighed guide is presented to assist treeprofessionals during the
treestability analysis, as a synthesis and an integration of these
previously published assessment methods.
Indepth studies, performed by the author in Spain on Mediterranean
trees and palms, were the “nursery” for the protocol. Another logical
step in the development of this guide was the study of specialised
literature on structures and their failure (e.g. Gordon, 1999).
The author of this current publication employs it successfully on
Mediterranean treespecies like Eucalyptus spp., Celtis spp.,
Brachychiton populneus, Tipuana spp., Cercis siliquastrum,
Cupressus sempervirens, Ficus spp. and Mediterranean pine trees.
For MiddleEuropean treespecies the protocol is equally accessible.
The message is that mostly a tree can be assessed very well with the
combination of a visual assessment and the estimation of its stability
by wind load analysis. Even many monumental trees can be
inspected very well on their stability with the help of the protocol
proposed by the author of this publication.
The instrumental diagnosis seems to be seldom necessary for the
assessment of the breaking safety of a tree. Should the tree in
question have an extraordinary monetary value and if there are
unsolved questions regarding its stability, a range of fine instrumental
methods can be chosen from.
A Guide For TreeStability Analysis  8 
2. Prognosis of the breakingsafety of hollow trees
Introduction
This publication presents a proposal for a new method for tree
stability analysis. This proposal is a synthesis and an integration of
previously published tree assessment methods. By integrating
knowledge from different research disciplines and methods, a higher
level of understanding, about how trees stand up or fall down, is
achieved.
Firstly, the present proposal is basically a visual assessment
combined with mathematical analysis of (wind) loads. By means of
the latter, the safety reserves of the hollow tree are calculated. Also
the critical wind velocity for bending fractures of the hollow trunk, the
bending frequency of the stem and uprooting, failure of the
anchorage roots, can be estimated.
In Sterken (2005) a proposal is formulated in order to assess whether
a hollow tree is hazardous or not. It is suggested that the residual
wall should have both enough safety reserves regarding wind loads
and a low risk of catastrophic mechanical behaviours.
Externally visible symptoms of internal decay, like adaptive growth
and growthdepressions, are to be recognised. This proposal can be
a more powerful tool if combined with the mapping of decay by
means of acoustic tomograms. For the interpretation of the latter, see
Schwarze et al. (2004).
This part of the proposal enables a momentary insight in the tree’s
stability (uprooting and breakage).
Secondly, a longterm prognosis regarding the future development of
decay in the sapwood, and hence the tree its safety, can be more
accurately predicted when knowledge of interactions at the host
fungus are considered. According to Schwarze (2001), while trees
attempt to resist the spread of decay with several defence
mechanisms, in many cases fungi possess strategies by which they
can overcome the host response system of the tree. The latter
counteraction between host response and invasiveness of decay
fungi have been studied extensively at the microscopic level by
Schwarze et al.. A summary of the most important findings have
been gathered in the present publication. For a more profound
understanding on this subject, a review of these recent findings are
recommended.
A Guide For TreeStability Analysis  9 
In the following lines the basic components are presented in a
resumed mode:
Basis components I: momentary assessment of stability or
status quo
The promising combination of opposed methods
For the first time, the present proposal situates the results of the
mathematics, of which the employed formulae are offered in Sterken
(2005), in the complete process of analysing treestability. Under
mathematics is understood here, for example, calculating safety
factors and the necessary thickness of the residual wall by wind load
analysis. Real trees and real winds usually do not fit themselves to
the mathematics.
Instead of the proposed mathematical model, the more accessible
SIA method (Wessolly and Erb, 1998) can be employed by the
practitioner, which might give information about the breaking safety of
the individual hollow tree. According to the SIA method, in the case of
many old trees, the trunk, if undamaged, can be many times thicker
and safer than necessary to withstand a hurricane. The stress,
induced by the hurricane load, would remain far below the
compression strength of the marginal woodfibres. Theoretically, and
according to the bending theory of the hollow beam, these old trees
only require very small residual wall thickness to withstand gusts of
117km/h.
Nevertheless, this can only be meant as an orientation for the
assessment, since very low t/R ratios can lead to different types of
collapse/failure due to the flexibility of the hollow shell. In this context
the analysis of catastrophic mechanical behaviours, according to
“The Body Language of Trees” (Mattheck & Breloer, 1995) is
introduced. According to these authors, very common mechanical
behaviours cannot be predicted by means of the bending theory of a
hollow beam and its mathematics (main point of criticism in regards
to the pullingtest or elastomethod of Wessolly and Erb (1998)). The
most important failure types are shellbuckling, shearing fractures,
torsion fractures and splitting due to transverse stresses. In the
present proposal these behaviours have to be analysed, by which the
mathematics are tempered.
The combination of the SIA method and the VTA method is very
accessible and proves to be highly efficient in the field.
A Guide For TreeStability Analysis  10 
Visual assessment
Visual assessment is the most important component of the proposal
presented here for assessing the stability analysis of trees.
The ring of sapwood is what gives the tree its main structural
strength, regarding for example torsion and bending, to a tree.
Therefore, if symptoms are detected that point towards damages in
the sapwood, e.g. due to wood decay fungi, then it is possible that
the crosssection suffers an important loss of its load bearing
capacity.
If the sapwood is damaged, then the water conducting capacity can
be diminished. This can lead to a loss of vitality. This loss of vitality
usually produces symptoms in the upper crown structures and the
bark. Other symptoms, like growth depressions or dead areas in the
bark also may indicate serious structural damages in the ring of
sapwood.
Therefore, it is possible during tree hazard assessment to take as a
starting point the state of the crown and bark, before attempting to
apply more technical resources. This method can be a very good
orientation, in order to assess visually if there is an important loss of
structural strength, caused by fungi, in the wooden body.
If there are still unsolved questions in regards to internal decay,
acoustic tomography can be introduced, facilitating more accurate
assessments of the fracturesafety of trees. According to Schwarze
et al. (2004), a zone of decay can be determined accurately for its
size and moderately accurately for its position by means of acoustic
tomograms.
Basis components II: What residual wall thickness is required
by a tree?
The safety assessment of hollow trees has always fascinated
arborists, and the criteria to be employed have led to severe public
discussions in the professional scene in Europe.
Based on Mattheck & Breloer (1995), many treeconsultants state
that the required thickness of the residual wall should not be below a
t/R ratio of 0,3 to prevent shellbuckling, crosssectional flattening
and hose pipekinking.
On the other hand Wessolly & Erb (1998) published an opposite
theory by which often much lower thickness are calculated and
accepted. Their methods are based on the bending theory of the
hollow beam.
A Guide For TreeStability Analysis  11 
A complete study was made on a 17,1m high eucalyptus tree
(Eucalyptus camaldulensis) in Spain. In accordance to the above
mentioned theory, which is also employed by Wessolly. For this
purpose the necessary trunk diameter and residual wall thickness
where calculated for different treeheights.
Afterwards, the results were contrasted throughout with criteria
regarding torsion, shear, stress peaks, crosssectional flattening and
shellbuckling.
A literature review suggests that the different types of failure of the
residual wall depend mainly on parameters like stiffness and strength
of the wood in different anatomical directions (wood is anisotropic),
geometry of the crosssection and different loads.
Hence, fixed t/R assessment limits should be approached carefully.
But:
Neither should the residual wall always be as thin in a real tree as
predicted mathematically with the formula employed in accordance
with the bending theory of the hollow beam.
The analysis by Sterken (2005) suggest that the truth lies somewhat
in the middle instead of in both extremes.
Basis components III: wind loads and bending frequency
Within the scope of the present publication, the author worked out a
mathematical model for the analysis of stability and breakage of
trees.
The goal of the model is to enable treespecialists appreciate better
the interaction between wind, treestability, biology, wooddecaying
fungi and mechanical behaviours.
The goal of the present proposal is to offer criteria that both situate
the results of the model in the treediagnosis process and temper the
mathematical calculations on which this model and similar methods
are based.
The critical wind velocity “V” – can be estimated with this model. “V”
is the wind velocity that would cause the stress in the outer fibres to
exceed the maximum compression strength and would hence
produce failure of those fibres. This computerised model calculates
the critical wind velocity for several types of failures (uprooting,
bending fractures of the sound or hollow trunk and torsion fractures
of closed and concentric cavities). Safety factors obtained by analysis
A Guide For TreeStability Analysis  12 
of the wind loads are incorporated as well, while the theoretical
necessary residual wall thickness for each tree can be calculated.
With these analysis, a basic idea of the safety of a tree can be
obtained although they can only be a small part of the stability
assessment, since real trees and real winds do not always fit in
mathematical models. Both the possibilities, as well as the limits of
these assessment methods for trees are discussed in depth in
Sterken (2005).
Assessing the wind load in the crown
In accordance with the Eurocode 1 (AENOR, 1998), which
recommends equations for predicting wind loads of structures, the
wind load in the treecrown is analysed. For trees, the following
formula is a logical adaptation of the equations:
F = ½*Cd*p*A* u
(z)
²
Where:
F = the force that a gust exerts in the crown
Cd = the aerodynamic coefficient describes the flexibility that the
tree employs in order to diminish the force of the wind
p = density of the air, which depends on the pressure and
humidity of the air, temperature and height above sealevel.
A = the exposed area of the crown to the wind.
u
(z)
= wind speed “u” at a certain height “
z
” above ground level.
Dynamics
In this model, the bendingfrequency of the bare trunk is represented
by the following equation (AENOR, 1998):
n = (el * d)/(h²) * v(Ws / Wt)
Where:
n = the bendingfrequency of the trunk, expressed in Hz
el = the factor of frequency
d = the diameter of the trunk
h = the height of the palm or tree
Ws = the weight of the structural parts that contribute to the
stiffness of the trunk
Wt = the total weight of the trunk
The natural bendingfrequency of the trunk is compared to its safety
factor (regarding static wind loads), allowing a better orientation for
the tree its stability.
A Guide For TreeStability Analysis  13 
Basic components IV: prognosis of the future development of
decay
Defence mechanisms of the tree
In Schwarze (2001) an excellent description can be found regarding
the current state of knowledge on compartmentalisation.
• Barrier zone
The barrier zone, which is very efficient regarding the delimitation of
decayed wood and healthy wood, is only formed after the cambium
has been injured. The latter can occur both after an injury is inflicted
from the outside (e.g. collision and pruning wounds) and/or when the
fungus parasitizes the cambium , from the inside towards the outside
(e.g. Inonotus hispidus).
The barrier zone (wall 4) has been demonstrated to be very efficient,
for example, in Tilia platyphyllos regarding Kretzschmaria deusta,
whereas the reaction zones are much weaker. Several authors
highlight the importance of the barrier zone (Wall 4) regarding the
assessment of the fracturesafety of a hollow tree.
• Reaction zones
An intact reaction zone can provide the tree with a natural defence,
often over a long period, against the spread of decay and the
resulting increase of probable fracture. While possibly being primarily
a means of maintaining the hydraulic integrity of the functional xylem.
Reaction zones are being formed exclusively in the sapwood. In a
tree, the formation of reaction zones is closely associated with the
enhancement of the moisture content, living parenchyma cells and its
energy reserves, the vitality of the tree, temperature and the season.
The reaction zone can be efficiently penetrated by several fungus
species. For example, the hyphae of I. hispidus can colonise the
living tissue in Platanus x hispanica during the dormant period, in
which the tree cannot respond effectively.
These reaction zones, nevertheless, seem to be important in slowing
down degradation processes and in preventing embolism in the water
conducting system (Schwarze & Baum, 2002).
Typically, the reaction zone is three to four centimetres wide on the
average when viewed in a transverse section. In some studies, a
width up to eight centimetres has been reported (Schwarze et al.,
2004).
A Guide For TreeStability Analysis  14 
According to Schwarze & Baum (2000), in beech and London plane,
some fungal species are able to extend into the functional sapwood
some way beyond the reaction zone, without stimulating the
simultaneous migration of the zone. Instead, a new reaction zone
forms later at the new colonisation front. This alternating event points
towards possible interaction phases that might be seasonal (cyclic
changes in carbohydrate reserves and accompanying changes in the
moisture content, i.e. a supraoptimal water content at the lesion
margin). Here, the reaction zone of the tree is envisaged as a static
boundary.
This type of discontinuous host response may be recognised by the
presence of reaction zone relics within decayed wood (Schwarze &
Baum, 2000). Inonotus hispidus for example causes these relics,
growing into the cell walls towards the healthy sapwood of Platanus x
hispanica, whereby the obstructed cell lumina, which form the
reaction zone, are avoided.
On the other hand, in Tilia platyphyllos, the reaction zone has been
demonstrated to be a dynamic boundary, whereby the reaction zone
migrates progressively into the previously healthy sapwood, ahead of
the penetration of hyphae (Schwarze & Baum, 2002).
Interaction at the hostfungus interface
Examples of strategies for reaction zonepenetration and
corresponding invasiveness of wooddecaying fungi.
• Ganoderma spp.
Depending on species, vitality and other factors, some trees have a
relatively good ability to form compensation wood. In this case there
are usually clear biomechanical signs, e.g. adaptive growth
(Mattheck & Breloer, 1995) and growthdepressions (Reinartz
&Schlag, 1997), of the decay caused by Ganoderma spp. before any
increased risk of failure occurs. (Schwarze & Ferner, 2003).
Nevertheless, it is very important to distinguish the invasiveness of
Ganoderma adspersum from that of G. applanatum and G.
resinaceum. Reaction zones can be not efficient at all when
Ganoderma adspersum colonises species of Tilia, Fagus, Acer and
Platanus.
A Guide For TreeStability Analysis  15 
At the time of the assessment, the tree might have adequate residual
walls of sound wood. This means that the residual wall has enough
safety reserves regarding wind loads (SIA method, Wessolly & Erb,
1998) and a low risk of catastrophic mechanical behaviours (VTA
method, Mattheck & Breloer, 1995). This proposal has been
formulated in depth in the book “A Guide for Treestability Analysis”.
Nevertheless, the longterm prognosis might be negative, even within
trees of high vitality.
In Fagus sylvatica, for example, the reaction zones can be used by
this fungus as a nutritional substrate, by digesting preferably the
polyphenols and suberized tyloses. Consequently, its invasiveness
can be classified as high in a standing tree.
On the other hand, if G. applanatum or G. resinaceum are identified,
and the stability assessment indicates that the tree is not a significant
hazard, internal investigations may be unnecessary (Schwarze &
Ferner, 2003). The decay can be delimited and kept under control
over a longer time in trees of high vitality.
Finally, the prediction, regarding the progress of decay for a given
fungus species, also will depend on the particular treespecies, and
not necessarily on the trees vitality.
• Kretzschmaria deusta
Due to the degradation of the sparse polyphenols in the cell lumina
and consequent spreading of the hyphae of K. deusta freely via the
pits in the cell walls, the ability to compartmentalise this fungus by
means of reaction zones is very low in T. platyphyllos. Even when
the tree is very vital and the momentary assessment of the residual
wall or the anchor roots points towards a “safe” tree.
Consequently, the absence of the barrier zone (Wall 4l of the CODIT
model) in this treespecies infected by K. deusta can pose a major
problem. This situation can occur for example, due to the
colonisation of the stem base through damaged major roots. During
the stabilityanalysis, this factor should be recognised since its very
high importance regarding the prediction of the tree’s stability in
future years..
In Fagus sylvatica, on the other hand, K. deusta can be very
effectively compartmentalised when the healthy tree has a high
vitality and high energy reserves. In this case the reaction zones are
A Guide For TreeStability Analysis  16 
moderately efficient, due to the abundant occlusion of natural
openings (pits) in the cell wall, the latter being the only pathway for
this fungus to spread from one cell to another.
If the stability assessment indicates that the tree is safe, then the tree
might be able to maintain an equilibrium between decay and stability
for a longer period. A longterm prognosis as regards its stability can
be formulated.
On the other hand, when the energy reserves are low (strongly
diminished vitality and/or growing under unfavourable conditions), the
very sparse antifungal occlusions in the lumina of the cells are not
able to stop the spread of the decay. Kretzschmaria. deusta is
regarded as particularly dangerous in this situation, impairing the tree
its longterm stability and fracturesafety even if the its current state is
satisfactory .
A Guide For TreeStability Analysis  17 
Conclusion
After assessing whether the existing sapwood, which gives the tree
its main structural strength, is able to bear the impacting loads, a
prognosis is made regarding the future development of decay in this
ring of sapwood. This allows a more accurate assessments of the
fracturesafety of the tree or defective branches.
During the stability assessment, it is very important to recognise
whether a barrier zone has developed in a tree or not. An intact
barrier zone might allow for an intact ring of sapwood, which can
behave according to the bending theory of a hollow beam. For the
necessary thickness of the residual wall, see Sterken (2005).
When the absence of a barrier zone is suspected in a tree, the
residual wall might not behave in the simplified manner, as if it was a
hollow beam under pure bending stresses. This due to the
invasiveness of certain fungi as regards reaction zones in the
sapwood, that can lead to an irregular geometry and a not clearly
defined ring of sapwood. Both the study on relics of breached
reaction zones in decayed wood (Schwarze, 2001), as the reporting
of externally visible symptoms of a degraded residual wall (Reinartz
& Schlag, 1997) seem to support this hypothesis.
Knowledge about the interactions in regards to hostfungusvitality is
also essential for making correct decisions on the appropriate
amount of pruning and proper cabling.
Crown or branch reductions can be necessary to reduce failure
components like endweight, wind loads, dynamics, torsion, bending
and buckling. But due to the foliar loss, they can also lead to a
reduction of vitality and hence diminished defence mechanisms
(compartmentalisation and compensation).
Properly designed cabling installations can prevent excessive
pruning and foliar loss, while still allowing the tree to compensate
defective parts.
Acknowledgements: The author wishes to thank Prof. Dr Francis
Schwarze for comments on the manuscript that was the basis for this
“Prognosis of the breakingsafety of hollow trees”.
A Guide For TreeStability Analysis  18 
Figure 1. Prognosis of the development of decay and the fracture
safety of hollow trees
.
Part I of the proposal: momentary assessment of strength and stability
Visual assessment:
Vitality: Visually assessable symptoms of the state of vitality in crown and bark.
Corresponding capacity of compartmentalisation and repairgrowth.
Wood decay fungi: Ring of sapwood and symptoms in the bark and crown.
Geometry: the form of the crosssection and structural defects (confined vs.
irregular rots) determine the behaviour of the wooden body under loads. This
factor might influence in whether the fracturesafety is predictable or not by
means of mathematics.
Mathematics: wind loads and safety. SIA method (Wessolly & Erb, 1998)
+ Reasons of mechanical failure and its visual assessment: The Body
Language of Trees (Mattheck & Breloer, 1995)
+
Part II of the proposal: long term prognosis of stability
Strategies for reactionzone /barrier zone penetration and corresponding
invasiveness of wood decay fungi (Schwarze et al.). Knowledge on the
development of decay allows for more accurate assessments of the longterm
fracturesafety of the tree or defective branches.
Synthesis and an integration of previously published tree
assessment methods.
A reasonable and accessible method that permits a better insight
in the trees safety, which proves to be highly efficient in the field.
A Guide For TreeStability Analysis  19 
3. A mathematical model for the prediction of the critical
wind speed for the failure of trees
Introduction
Within the scope of the present publication, the author worked out a
mathematical model for the analysis of stability and breakage of
trees.
The goal of the model is only to support the analysis that are
developed in this book. In this way, the model will also allow tree
specialists appreciate better the interaction between wind, tree
stability, biology, wooddecaying fungi and mechanical behaviours.
The goal of the present publication is to offer criteria that both situate
the results of the model in the treediagnosis process and temper the
mathematical calculations on which this model and similar methods
are based.
The Critical Wind Speed “V” – can be predicted with the V model.
“V” is the wind velocity that would cause the stress in the outer fibres
exceed the maximum compression strength and would hence
produce failure of those fibres. This computerised model calculates
the critical wind velocity “V” for several types of failures (uprooting,
bending fractures of the sound or hollow trunk and torsion fractures
of closed and concentric cavities). Safety factors obtained by analysis
of the wind loads are incorporated as well, while the theoretical
necessary residual wall thickness for each tree can be calculated.
Structure of the V model
• Assessing the wind load in the crown
In accordance with the Eurocode 1 (AENOR, 1998), which
recommend equations for predicting wind loads in structures, the
wind load in the treecrown is analysed. For trees, the following
formula is a logical adaptation of those equations:
F = ½*Cd*p*A* u
(z)
²
Where:
F = the force that a gust exerts in the crown
Cd = the aerodynamic coefficient describes the flexibility that the
tree employs in order to diminish the force of the wind
p = density of the air, which depends on the pressure and
humidity of the air, temperature and height above sealevel.
A = the exposed area of the crown to the wind.
u
(z)
= wind speed “u” at a certain height “
z
” above ground level.
A Guide For TreeStability Analysis  20 
Similar formulae are found in publications of scientists in the field of
arboriculture (Mattheck & Breloer, 1995; Wessolly & Erb, 1998;
Niklas, 1999; Peltola et al., 1999; Gaffrey, 2000; Spatz et al., 2000;
Ezquerra et al., 2001)
The powerlaw model is used to predict the wind speed at a given
height above ground level and is presented by the following formula
(Berneiser & König, 1996; Wessolly & Erb, 1998):
u
(z)
= tu * u
(g)
* (h
(z)
/ h
(g)
)
a
Where:
u
(z)
= wind speed “u” at a certain height “
z
” above ground level.
u
(g)
= maximum wind speed expected, not influenced by the
roughness of the terrain.
h
(z)
= height above ground level at which a certain wind speed is
reached (height of the analysis)
h
(g)
= height above ground level at which the maximum wind
speed is reached.
a = surface friction coefficient
tu = turbulence factor
A turbulence factor is incorporated to count with the influence of
strong incoming gusts.
• Dynamics
In the V method, the bendingfrequency of the bare trunk is
represented by the following equation (AENOR, 1998):
n = (el * d)/(h²) * v(Ws / Wt)
Where:
n = the bendingfrequency of the trunk, expressed in Hz
el = the factor of frequency
d = the diameter of the trunk
h = the height of the palm or tree
Ws = the weight of the structural parts that contribute to the
stiffness of the trunk
Wt = the total weight of the trunk
In the V model, the natural bendingfrequency of the trunk is
compared to its safety factor (regarding static wind loads), allowing a
better orientation for the tree its stability.
A Guide For TreeStability Analysis  21 
• Breaking safety
The calculation of the breaking safety of the tree is based on the
bending theory of the (hollow) flexible beam. The bending stress on
the surface of a hollow beam can be calculated by the following
equation (Mattheck & Kubler, 1995):
smax = (4 * M) / (pi * R³)
Where:
smax = the maximum bending stresses in the marginal fibres
M = the bending moment and is the sum of the forces (F) in the
crown, multiplied by the distances (P). This moment is calculated with
the above described wind load analysis.
R = the radius of the trunk
The geometrical moment of inertia (I) is the parameter that defines
the resistance against bending of the crosssection. Hereby I can be
defined, assuming the crosssection is ellipseshaped, as:
I = (pi / 4) * a * b³
Where:
I = the geometrical moment of inertia
b = the radius at 1 meter height parallel to the loading direction
a = the radius perpendicular to the first
The safety factor of the trunk is then found from:
S = s/s
Where:
S = the safety factor of the crosssection
s = the strength of the green wood
s max= the maximum bending stress in the marginal fibres
By these equations, it is clear that it becomes possible to calculate
the critical bending moment that would cause the stem to break,
when the radii of the trunk and the strength of the wood are known.
This moment is then expressed as:
Mcrit = I *s
Where:
Mcrit = the bending moment that would cause the stress in the outer
fibres of the trunk exceed the compressionstrength
I = the geometrical moment of inertia
s = the strength of the green wood as published by Lavers (1983)
and other publications.
A Guide For TreeStability Analysis  22 
In accordance with the bending theory of the hollow beam, correction
factors are employed in the case of cavities. These correction factors
diminish the moment of inertia, and hence the critical bending
moment. Finally, the critical force of the wind in the crown (Fcrit) is
obtained by dividing the critical bending moment (Mcrit) by the lever
(P).
• Uprooting safety
The present publication treats mainly the breaking safety of trees.
The safety against uprooting is a subject reserved for a future study
and publication. Nevertheless, should the urban tree in question have
unsolved questions regarding its root stability, the following method
can be recommended:
Wessolly & Erb (1998) state that with 0,25º of inclination of the stem
base, 40% of the overturning moment is reached to tip the tree over.
And this regardless of the type of soil, species and possible
damages. This assumption enables the prediction of the critical wind
load for uprooting in a relative simple manner.
At the time of this writing, a Dutch company has developed a very
interesting pullingtest for the prediction of the rootstability
(www.boomkcb.nl). Their method recognises that the tree its root
plate has a rather complex behaviour, even under a static and
unilateral load. Here, the angle of the root plate and stem base is
recorded in several directions by means of inclinometers and the load
exerted by a cablewinch is fully controlled with a dynamometer. The
wind load is simulated in order to reach a maximum inclination of the
stem base of 0,25º. Once the critical load for uprooting is known, the
wind speed which would cause failure of the root system can be
predicted.
The proposals of the present publication, regarding the assessment
of the breaking safety, could complete very well a pulling test as
proposed by these Dutch experts.
Finally, Peltola et al. (2000) describe two different models to predict
the failure of the rootsystem by windload analysis for forest trees
(GALES and HWINDmodels).
A Guide For TreeStability Analysis  23 
• Torsion safety
The formulas employed in the V model can predict the safety against
torsion of a hollow tree, provided that the residual wall is perfectly
closed and concentric. Although this idealised situation might seldom
occur in urban forests, in some cases it can provide good extra
information (in old but vital oak trees for example where the
heartwood is feasted away by Laetiporus sulphureus or Daedalea
quercina).
For tree species like Eucalyptus paniculata for example, Lavers
(1983) publishes both for maximum shearing strength parallel to
grain and resistance to splitting (cleavage) approximately the same
proportion of ¼ of the maximum compression strength parallel to
grain. Other Eucalyptus species demonstrate more benign
proportions, but the former value (¼) is employed here to assume the
shear and torque strength of the wood.
The torsion moment is calculated assuming that an offcentred wind
load can hit the crown of the tree even if the crown is very symmetric.
This is in accordance with the European Standards for signpost
boards (AENOR, 1998). Hence, for symmetric crowns, the
mathematical model assumes a torsion lever equal to:
Pt = Dc/4
Where:
Pt = torsion lever in metres
Dc = crown diameter
Nevertheless, the load centre and the resulting torsion lever in
eccentric crown forms can be calculated by means of a special
software and introduced afterwards as one of the parameters of the V
model. The critical moment regarding torsion of the closed residual
wall, is calculated in the same manner as for bending fractures (see
the explained above).
• Breaking safety of stems
The structural strength of codominant or damaged stems (e.g.
cavities or the influence of woundwood rolls) can be assessed with
this mathematical model.
The safety factor of the crosssection and necessary residual wall
thickness for the stems of the tree at a given height, regarding pure
bending fractures or torsion, is calculated. Correction factors can be
introduced to estimate the influence of rot, cracks, included bark or
cavities (see the explained above).
A Guide For TreeStability Analysis  24 
Also for the assessment of branch fracture, the present publication
might permit to employ more efficiently advanced technical
resources. The combination of visual assessment, wind load
analysis, proper cabling and common sense can do miracles.
• Critical wind velocity
When the critical moments both for bending fractures, torsion
fractures and uprooting are known, it becomes possible to predict the
wind velocity that would cause these moments, provided the above
exposed model parameters are known.
The formula employed for calculating “V” is a logical consequence of
calculating backwards the equation that defines the wind load in the
crown.
The wind speed at the height of the centre of pressure in the crown,
at which the above described failures would occur, is regarded as the
critical wind speed for each of those failures.
• Necessary residual wallthickness
The residual wall thickness that a tree needs in order to withstand
any given static wind speed can be calculated if assumed that the
hollow trunk behaves like a perfectly formed hollow tube.
The equation employed here for calculating the minimum residual
wall thickness, if completely concentrically and closed, is in
accordance with the bending theory of the hollow beam and is the
same as employed for the SIA method (Wessolly & Erb, 1998):
t =0,5*dm*(1(1(100/SF))^(1/3))
Whereby:
t = minimum residual wall thickness
dm = measured net diameter
SF = safety factor of the crosssection
• Assessing the wind load in the crown cabling
The static wind load in the crown, cabled branches and the cables
can be assessed with the V model, which enables to design tree
friendly cabling installations to prevent several types of failure.
A Guide For TreeStability Analysis  25 
Discussion
By these analysis, a basic idea of the safety of a tree can be obtained
although they can only be a small part of the stability assessment,
since real trees and real winds do not always fit in mathematical
models. It should also be acknowledged that these mathematics only
offer a momentary insight in the tree’s stability. This means that the
processes that determine for how much time the prognostic will be
valid, e.g. fungal infections, compartmentalisation, compensation and
others, have to be incorporated in the assessment.
In the V model, the natural bendingfrequency of the trunk is
compared to its safety factor (regarding static wind loads), allowing a
better orientation for the tree its stability. By introducing a bending
frequency, a rough and simple overview of the dynamics effects of
the wind in trees are incorporated in the V model.
For example, the trunk of a tall Cupressus sempervirens can have a
high safety factor regarding a simplified static wind load but, at the
same time, a bendingfrequency of 5,8 Hz. This means than that the
effect of sway should be considered seriously since it might lower
considerably the previously predicted theoretical safety factor.
It should be noted that, due to turbulence, the real dynamic behaviour
of a tree can differ from that of computer predictions. It can both sway
parallel and perpendicularly to the direction of the wind and produce
a rotational movement. Even when the tree is perfectly symmetrically
shaped.
According to James (2003), this assumption of the natural frequency
implies a simplification, which calculates only the sway period of a
bare trunk. In this way, the influence of mass damping is often
ignored. The latter is the dramatic influence of swaying side branches
that can prevent large sway amplitudes of the bare trunk.
According to AENOR (1998), the complete wind load analysis, as
presented in this publication, is a simplified model which should only
be employed for structures that have a dynamic coefficient of less
than 1,2. If the structure would be more susceptible than this to the
dynamic effects of the wind, then the formulae presented in this
section are not valid. The dynamic coefficient can higher the
theoretical bending moment in the tree with a factor >1, due to the
interaction of the frequency of nearby wind gusts and the frequency
of the structure.
A Guide For TreeStability Analysis  26 
One of the questions that has been remarked repeatedly by many
practitioners/ students is that it is normal to obtain deviations from the
reality, since multiple parameters have to be introduced in
mathematical models/ computers (e.g. strength, aerodynamic drag,
wind speeds, …) which might differ slightly from the real situation in
which the tree is found. For example:
• the wind force increases with the square of the wind speed.
In this way, small differences in the input of the latter will lead
to significantly lower or higher safety factors (both breakage
and uprooting) than the real stability of the tree. Doubling the
wind speed means that the calculated safety factor falls
down to one fourth!
• Within species, material properties can differ in the same
crosssection of the same tree, between parts of the same
tree or from tree to tree, and might easily deviate from the
value published in strength tables (Lavers, 1983).
• The combination of both deviations from the real situation will
lead to an even greater deviation.
Therefore, at the moment of this writing, mathematical equations, and
even the most refined methods, can only pretend to be an orientation
for reallife treestability assessment. Safety factors of 150200%
should be introduced to make up for the natural impossibility to target
precisely the chaotic interactions between treestructure, tree
geometry, material properties and loads.
If “V” is determined for a tree and its possible ways of collapsing, e.g.
92km/h, then it can be compared to the meteorological predictions or
hurricanes that would occur with a high probability. Peltola et al.
(2000) describe two similar models to predict the failure of the trunk
and rootsystem by wind load analyses (GALES and HWINDmodels)
for forest trees.
A Guide For TreeStability Analysis  27 
4. What is the necessary residual wall thickness for a tree?
Introduction
The safety assessment of hollow trees has always fascinated
arborists, and the criteria to be employed have led in Europe to
severe public discussions in the professional scene.
Based on Mattheck & Breloer (1995), many treeconsultants state
that the necessary thickness of the residual wall should not be below
a t/R ratio of 0,3 to prevent shellbuckling, crosssectional flattening
and hose pipekinking.
Wessolly & Erb (1998) on the other hand publish the opposite theory
by which often much lower thickness is calculated and accepted.
Their methods are based on the bending theory of the hollow beam.
A complete study was made on a 17,1m high eucalyptus tree
(Eucalyptus camaldulensis) in Spain. In accordance with the above
mentioned theory, which is also employed by Wessolly, the
necessary trunk diameter and residual wall thickness where
calculated for different treeheights.
The results are contrasted throughout the present publication with
criteria regarding torsion, shear, stress peaks, crosssectional
flattening and shellbuckling.
The above and the study of scientific literature suggest that the truth
lies somewhat in the middle instead of in both extremes.
A Guide For TreeStability Analysis  28 
An example of treestatics
• Wind load analysis with the Vmodel
The goal is to calculate the force that a hurricane would exert on the
inspected tree. Strong gusts of wind are the main reason for the
structural collapsing of trees, so they should be incorporated in the
stabilityassessment as well.
The powerlaw model is used to predict the wind speed at a given
height and is presented by the following formula (Berneiser & König,
1996; Wessolly & Erb, 1998):
u
(z)
= tu * u
(g)
* (h
(z)
/ h
(g)
)
a
Where:
u
(z)
= wind speed “u” at a certain height “
z
” above ground level.
u
(g)
= maximum wind speed expected, not influenced by the
roughness of the terrain.
h
(z)
= height above ground level at which a certain wind speed is
reached (height of the analysis)
h
(g)
= height above ground level at which the maximum wind
speed is reached.
a = surface friction coefficient
tu = turbulence factor
Different terrain categories are included with their corresponding
roughness parameters. The surface friction coefficient (
a
) can range
from 0,10 to 0,40 for urban areas with tall buildings. Hence, the
influence regarding wind speed of nearby constructions, other trees
or even short grass on untilled ground can be incorporated.
These mathematical models are used in the wind turbine and wind
engineering industries and it is known that the wind speed can be
overestimated for heights lower than 250m.
Nevertheless, the aerodynamic dragfactor employed for each tree
might provide a good equilibrium in the calculations and final result.
A Guide For TreeStability Analysis  29 
The force that the wind exerts in the crown is calculated with this
formula (Mattheck & Breloer, 1995; Wessolly & Erb, 1998; Niklas,
1999; Peltola et al., 1999; Gaffrey, 2000; Spatz et al., 2000; Ezquerra
et al., 2001):
F = ½*Cd*p*A* u
(z)
²
Where:
F = the force that a gust exerts in the crown
Cd = the aerodynamic coefficient describes the flexibility that the
tree employs in order to diminish the force of the wind and is
adaptable for each tree in accordance to his real crown
density.
p = density of the air, which depends on the pressure and
humidity of the air, temperature and height above sealevel.
A = the exposed area of the crown to the wind.
u
(z)
= wind speed “u” at a certain height “
z
” above ground level.
For Eucalyptus camaldulensis a aerodynamic drag of 0,25 was
chosen. The windexposed area of the original tree was calculated
with a specialised software. An air density (p) of 1,292 kg/ m³ was
used, which corresponds with an air temperature of 0ºC and a
standard atmospheric pressure (Windpower, 2003). In the computer
model, the impacting forces and resulting moments are derived from
simulating gusts of approximately 34m/s.
• Material properties
As with any engineering project, the material properties of the
structure (here: the wooden shell) have to be known.
For this study the important data was strength (s). The standard
deviation has been discounted to stay on the safe side for the
stabilitycalculation.
The data given by Lavers (1983) for Eucalyptus diversicolor (s =
3,15kN/cm² (standard deviation discounted) and MOE = 1340kN/cm²)
are employed in the Vmodel, since there is no data available yet for
the Spanish grown Eucalyptus camaldulensis and Eucalyptus
globulus.
A Guide For TreeStability Analysis  30 
• Geometry
The diameter and form of the trunk and the bark thickness were kept
the same, just as wind speed parameters, material properties and
others. In the datainput of the mathematical model, only the height
and corresponding vertical crown area varied.
Medium diameter of the trunk: 66,5cm
Medium bark thickness: 1,8cm
• Thickness of the residual wall: bending theory of the hollow
beam
The residual wall thickness that a tree needs in order to withstand
any given static wind speed can be calculated if assumed that the
hollow trunk behaves like a perfectly formed hollow tube.
The equation employed here for calculating the minimum residual
wall thickness, if completely concentrically and closed, is in
accordance with the bending theory of the hollow beam and is the
same as employed for the SIA method (Wessolly & Erb, 1998):
t =0,5*dm*(1(1(100/SF))^(1/3))
Where: t = minimum residual wall thickness
dm = measured net diameter
SF = safety factor of the tree
Results: necessary trunk diameter and residual wall thickness
With the help of the following diagram, the arborist will be able to
appreciate better the analysis of the breaking safety of this
Eucalyptus camaldulensis.
The safety factor (S) of the tree and the necessary thickness of the
wooden shell is determined. This in order to be safe against gusts of
117 km/h if growing in open field and surrounded by low shrubs.
The safety factor of this eucalyptus can be calculated with the
following geometrical formula (Wessolly & Erb, 1998):
S = 100 * (dm/dreq)³
Where:
S = safety factor of the trunk at 1m height, if completely
undamaged and sound
dm = measured net diameter of the tree (diameter without bark)
dreq = required diameter of the trunk if completely sound and
according to the theory of elasticity.
A Guide For TreeStability Analysis  31 
The required diameter and residual wall thickness at 1 meter height,
for several treeheights, can be read from the table. These results are
a function of the hurricane moment, compressionstrength and cross
section modulus:
For example: if this eucalyptus tree has a net diameter of 62,9 cm
and a height of 17,1m then the minimum required diameter is
44,9cm. The safety factor will be approximately 275%. This means
that the longitudinal stress in the marginal fibres of the sound trunk,
during gusts of 34 m/s, would reach a value of 1,14 kN/cm². This is
slightly more than one third of the assumed compressionstrength.
In this example, a value of 275% means that the tree needs a
minimum residual wall thickness of 4,4cm according to the formula
t =0,5*dm*(1(1(100/SF))^(1/3)), if the shell would be completely
closed and concentrically. Finally, a safety margin of 200% should be
incorporated, also for the required thickness of the residual wall since
the values of Lavers (1983) represent plastic failure of the wood
probes.
If the safety factor is higher than 100% the trunk can be damaged up
to some extent without being unsafe. If value is lower than 100%, the
height
required
required
theoretically
in m
diameter t
required
in cm
in cm
t/R
10
24,8
0,65 0,02
11
27,5
0,9
12
30,3
1,22
13
33,1
1,61
14
35,9
2,09
15
38,8
2,68
16
41,7
3,41
17
44,6
4,29
17,1
44,9
4,39 0,14
18
47,5
5,38
19
50,4
6,73
20
53,3
8,48
A Guide For TreeStability Analysis  32 
tree cannot permit any structural defect. In this case, its crown
should be reduced to heighten the safety. The extend to which the
crown has to be pruned can also be estimated with the mathematical
model.
Discussion
The study of specialised literature suggest that the different types of
failure of the residual wall depend mainly on parameters like stiffness
and strength of the wood in different anatomical directions (wood is
orthotropic), geometry of the crosssection and different loads.
Hence, fixed t/R assessment limits should be approached carefully.
But:
Neither should the residual wall always be as thin in a real tree as
predicted mathematically with the formula employed in accordance
with the bending theory of the hollow beam.
It is clear that the theory of the bending theory of hollow flexible
beams could reach indefinitely low t/R ratios, since it is nothing more
than just a geometrical formula. This can be observed in the table for
the 10m high tree that, theoretically, would need a residual wall
thickness of 0,65cm (t/R 0,02). Which is clearly unacceptable.
Logically, the theoretical result obtained by this formula cannot be
applied without limits in real trees! For the 17,1m high eucalyptus
tree, the bending theory of the hollow beam states that the example
tree needs only an t/R of 0,14 (fully exposed in open field, in the city
it would be even less).
Now: If the shell really experiments only pure bending stresses, this
would be enough, but care should be taken to exclude other types of
(treacherous) collapses. Amongst them are shellbuckling, torsion,
crosssectional flattening, shear or the combination of these and
other mechanical behaviours!
The reader has to be aware that mathematical models, measuring
devices and methods are only to be used as an aid for determining
the stability of trees. Hence, an assessment based solely on the
results obtained by the latter is inadequate.
Given the relatively small importance that can be given to
mathematics in reallife treeassessment, they can only pretend to be
a good orientation.
A Guide For TreeStability Analysis  33 
In the publication of Wessolly & Erb (1998), a diagram of Spatz is
shown where it can be concluded that primary failure in the
compression side occurs with lower loads than crosssectional
flattening if the t/R is higher than 0,0625 approximately.
This means that, regarding the risk of crosssectional flattening of a
perfectly concentric and closed residual wall, the calculations for the
residual wall thickness employed in the present study could then be
trusted if the tree has a higher t/R than this.
Nevertheless care should be taken, since with open or irregular
cavities and crosssections, and/ or the complex combination of
(wind)loads, this ratio might not be applicable, for the tree would
collapse without obeying the bending theory of the hollow beam.
It has to be said that if we focus only on the failure of the thin residual
wall itself, that this might be treacherous and give a wrong view about
the safety of the tree. The thin residual walls might collapse due a
combination of old prunedbranchholes and shearstresses
(Mattheck & Breloer, 1995).
And, for example, thick and heavy branches might break out of the
residual wall and cause severe damage while the shell keeps up
standing!
So, what is the necessary residual wall thickness of a tree?
Most of all seems to depend on: common sense.
A Guide For TreeStability Analysis  34 
5. Visual assessment
According to Gordon (1999), not only the thickness but also the
composition of each component that bears the load, is dimensioned
in a considerable measure by the use that it will be given and by the
forces it will have to withstand during it’s existence. By this, the
proportions of the living structures are designed to optimise their
strength. According to the author of the present publication, this
statement might define beautifully one of the main components of the
stabilityquestion of trees.
The basis for the author’s protocol
The SIA method (Wessolly & Erb, 1998) calculates the diameter of a
tree trunk and the necessary residual wall that a given tree needs to
withstand wind gusts of 32,5 m/s. The IBA method (Reinartz &
Schlag, 1997) describes the interaction of mycology, vitality and
stability. This visual method combines the visual detection of wood
decaying fungi, sometimes long before the first fruiting body appears,
and the SIA method.
At the current time, Prof. Dr. Francis Schwarze is investigating the
interaction between the defence mechanisms of a tree and the
strategies that decay fungi employ to overcome the former.
The status quo, i.e. the current condition of decay in a tree, can be
assessed macroscopically. The above described methods can offer a
static recording of this current condition. Nevertheless, the dynamics
of the interaction, and hence the future safety of the tree, can only be
assessed microscopically. In this way, for the first time, it is possible
to incorporate a prognosis regarding the future development of
decay, and invasiveness of different decay fungi, in the sapwood of
trees.
Finally, in “The Body Language Of Trees” (VTA method, Mattheck &
Breloer, 1995) a description of possible failures by fracture can be
found, which is a delicious and obliged literature. The description is
taken by the author of the present publication as the point of
departure, until the contrary can be demonstrated unquestionably.
A Guide For TreeStability Analysis  35 
Wooddecaying fungi
In the vast majority of current literature it is recommended to fell the
tree when fruiting bodies of a certain size or certain quantity appear .
But to the author of these lines, this seems to be a wrong perspective
that will only lead to the wrong answer.
The presence of wooddecaying fungi like Ganoderma sp., Meripilus
giganteus, Kretzschmeria deusta, Inonotus hispidus or Laetiporus
sulphureus, does not mean ipso facto that there is a strengthloss in
the wooden body. Premature conclusions should be avoided, since
the presence of fruiting bodies is only an indication that points
towards a diminished structural strength. But it is not an evidence.
A living tree is not a passive lifeless piece of wood; either it oxides as
if it was a helpless old iron tube.
The fact is that many times, a vital tree can be capable of
establishing an equilibrium between the formation of compensation
wood and the destruction of the material caused by the fungus.
Under compensation wood is understood here: laying down thicker
annual rings where higher strains and stresses are detected by the
cambium. According to the IBA method, the intact sapwood and
cambium guarantee that the tree can grow in diameter and
compensate in this way the destruction caused by the wooddecaying
fungus. Externally visible symptoms in the bark usually appear when
the growth of the annual rings and bark is affected. Apart from these
symptoms, damages, related to the low vitality of the tree due to
extended infections, will generally be seen in the crown (Reinartz &
Schlag, 1997).
It is also possible that most wooddecaying fungi have serious
difficulties to penetrate the sapwood and infect the cambium when
the tree is vigorous, due to the highly efficient living parenchyma
cells. So vitality, growth in diameter and bark condition are critical
factors when assessing the stability of a fungusinfected tree.
The wooddecaying infection, and its effects, should be understood in
the global context of each tree.
A Guide For TreeStability Analysis  36 
Assessing the breaking safety of hollow trunks
The fact is that in many old and thick, well compartmentalising trees,
the thickness of the sapwood can be estimated and that this ring of
sapwood can be enough to bear the load of the wind (the necessary
wallthickness is calculable with the wind load analysis).
But how can we know in a visual way how much strength loss the
wooddecaying fungi have caused?
The fourth wall of the CODITmodel (Shigo, 1986) can be a very
good barrier making it very difficult for fungi to penetrate the
sapwood. And if the tree is vigorous the parenchyma cells are very
efficient to compartmentalise infections. Many wood decaying fungi,
like Laetiporus sulphureus for example, might not be able to infect
the sapwood in vital trees. This is the wood that gives the main
structural strength to a tree.
So if we know the interaction between fungus, host and vitality we
can estimate if that ring of wood is intact or structurally weakened.
If that area is intact, we can assess if the stem is safe against
breaking, counting with this ring and by means of the calculations
according to the author’s protocol. The thickness of the sapwood is
estimated for each species and compared with the calculated
thickness that would be necessary to withstand the force of a storm
in its crown. The latter can be done with the SIA method or, in this
publication, with the V method. The thick wound woodrolls around
an open cavity can highten considerably the strength of the shell, not
only because of the optimised geometry but also due to a stronger
and stiffer, higher quality, wood.
By sounding with a hammer it is possible, with experience and
practice, to determine the point where the wall is thinnest (for
practising, removals of hollow trees can be very good opportunities!).
Another method is to compare the sounds made in the defective area
and nearby healthy crosssections. In that way higher precision can
be obtained to find the most interesting spot.
When assessing pure bending fractures, one possibility is to assume
that the residual wall is equally thin all the way round the tree in order
to stay on the safe side. This proposal accords beautifully with
Shigo’s model of compartmentalisation which assumes that the
decay can, at worst, occupy all the wood that was formed before the
barrierzone was formed! For open cavities, the loadbearing capacity
can be estimated with the help of correction factors. Sounding with a
hammer is not always reliable, but in many old trees it is quite easy to
estimate if the shell is thicker than the necessary t/R calculated.
A Guide For TreeStability Analysis  37 
The body language of the tree
The body language of the tree outdoes every instrumental diagnosis.
The bark tells us about the tree’s vitality and stability. Growth cracks
in the phloem (inner bark) also point towards its phase of vitality.
Certain anomalies in the tree’s skin enable sometimes to determine
the fungispecies and the extent of the infection, long before the first
conk appears. This means that the tree’s stability can be evaluated
with still a nice time margin before it “unexpectedly” would fall over.
A vital tree can also compensate an internal weakness.
Symptoms like a swollen butt (“Elephant foot”) or protuberances on
the stem do not necessarily mean that the tree has a problem. They
don’t have to worry ipsofacto. Symptoms of adaptive growth like
bulges, swellings or bottlebuts do not mean persé that the tree has a
problem. True, the problem might have been countered already by
this adaptive growth!
Compensation or adaptive growth is all about geometry and material.
Just as in civil engineering, the tree can obtain a better structural
efficiency laying down annual rings as far as possible from the
“Neutral Fibre”. In this way material, cost, is saved when the inner
material – heartwood for example – is eliminated, since it plays a
smaller role regarding stability than the outer ring.
A tree can compensate a defect if it has a good vitality and if the
higher strains or stresses are to be detected by the cambium (Shigo,
1986; Mattheck & Breloer, 1995; Reinartz & Schlag, 1996; Wessolly
& Erb, 1998).
Nevertheless, a tree might not always be vigorous enough to lay
down this extra wood. Maybe even the wood quality – strength and
stiffness – may not be as good as would be necessary if the tree
lacks vitality.
Some types of structural collapsing cannot be foreseen by the tree
itself. And it is possible that strong temporary gusts do not lead to
compensation either, although they might be the reason for failure.
If the symptoms are correctly interpreted against the background of
vitality, reaction capacity and safety factor, then a correct
assessment of a damaged tree can lead to a more efficient use of
apparatus.
The tree’s own language has to find a hearing!
A Guide For TreeStability Analysis  38 
Assessment of vitality
It is suspected that a visible loss of primary growth of the twigs
(upper part of the crown) are directly related to changes in the root
system or wooden body (Mattheck & Breloer, 1995; Reinartz &
Schlag, 1997; Wessolly & Erb, 1998; Roloff, 2001).
Therefore the assessment of the growth pattern of these twigs, and
the resulting crown structures, is of vital importance when assessing
the stability of a tree.
Roloff (2001) describes how the vitality of a tree (MiddleEuropean
species) can be classified in 4 phases. This author states that the
growthpatterns of the last 414 years will draw the real biological
situation of our assessed tree.
Temporary losses of vitality due to a droughtyear or mild pests do
not seem to have a great effect on the tree’s overall biological
situation. But they do lead to lesser growth or smaller or less greener
leaves in that small period. Mistakenly, the tree’s longterm reaction
capacity would be assessed as diminished although it isn’t. A healthy
tree can still catch up with life’s troubles with a loss up to 30% of it’s
green area. So one or twoyear crown appearances might finally
mislead the stabilityassessment.
The assessment of the vitality by this method is generally
reproducible by a third party, so objectivity and certainty is now better
attainable in the vitality assessment.
This method has been transferred experimentally by the author of the
present pages in Mediterranean species like Eucalyptus spp, Celtis
spp., Tipuana spp., and others.
The evaluation of the crown structures, laid down by the growth
patterns of the uppermost twigs, enables to classify the tree’s vitality.
By this, its capacity of reaction against wooddecaying fungi can be
assessed.
Conclusion about visual assessment
An expert visual assessment generally permits to recognise
damages caused by wooddecaying fungi, sometimes before the first
fruiting body appears, and to estimate their influence regarding tree
stability. In this way, many accidents can be prevented in time.
An expert visual assessment is the most powerful available
instrument. And probably the most noble one.
A Guide For TreeStability Analysis  39 
6. Behaviours of the wooden body
What is the real reason that leads structures to their failure?
Introduction
In the case of many old trees, the trunk, if undamaged, can be many
times thicker and safer than necessary to withstand a hurricane. The
stress, induced by the hurricane load, would stay far below the
compression strength of the marginal woodfibres. Theoretically, and
according to the bending theory of the hollow beam, these old trees
would need a very small residual wall thickness to withstand gusts of
117km/h.
Nevertheless, this can only be meant as an orientation for the
assessment, since very low t/R ratios can lead to different types of
collapse due to the flexibility of the hollow shell.
The calculations for stem safety against breakage as employed in
this publication and current treestatics, are based on the theory of
the elastic cantilever beam. A number of authors who examined stem
breakage are adamant regarding the usability of stem failure
calculations only under ideal circumstances. The interpretation of the
trunk of a tree, as if it was a homogeneous cylinder, would allow the
use of relatively simple mathematics.
But the very common structural problems that deal with torsion,
shearing stresses, delamination or structural failure may not obey to
this theory, at least in a tree, and should absolutely be taken into
account.
But then, how can treefailure due to torsion, delamination, shear, or
the propagation of cracks and defects be predicted? In “The Body
Language of Trees” (Mattheck & Breloer, 1995) very good
orientations are offered about how a thinwalled or even a defectfree
tree can fail. Some of them are mentioned here and should be taken
into account to temper the mathematical calculations that are based
on the bending theory of the hollow beam.
According to Mattheck & Breloer (1995), the majority of the following
failurecomponents cannot be predicted by means of the pullingtest
(elastomethod, Wessolly & Erb, 1998):
Fractures caused by bending stresses:
• Crosssectional flattening
• Shellbuckling of closed crosssections
• Local crosssectional flattening (hosepipe kinking)
A Guide For TreeStability Analysis  40 
• Shellbuckling of the walls of open cavities
• Harptree fractures
Fractures caused by shearing stresses:
• Basal bell fracture: shear cracks and delamination and local
crosssectional flattening
• Fractures at wound spindles and knot holes
Fractures caused by torsion:
• Opposing the direction of helical growth
• Opposing internal helical cracks (ribs!)
• Rotary fractures caused by flailing branches (complex
combination of forces)
Splitting due to transverse stresses:
• The hazard beam (so many in Pinus pinea trees for example)
• Rootbuttress delamination
• Included bark
Some criteria found in that work can be counterbalanced with the
publication of Wessolly & Erb (1998).
A brief description of some common mechanical behaviours
• Summer Branch Drop
It is suspected that Summer Branch Drop – sudden fracture of
branches on still and hot afternoons  occurs partially due to
stretching and relaxing of the pretensioned fibres (Mattheck &
Kubler, 1995; Wessolly & Erb, 1998). This pretensioning of the fibres
can heighten the resistance against compressionloads even up to a
140% of the original compression strength (Archer, 1996).
The combination of warming up the fibres, which causes relaxation of
the latter, and evaporation, which could influence pretensioning,
might reduce the loadbearing capacity of those fibres considerably.
Also the compression loads can be quite considerable in heavy,
leaning branches and, at least in dry wood, creeprupture can occur if
the duration of load is long (Wood Handbook 1999).
The stress caused by the weight of the branches would exceed the
compressionstrength of the wood, leading to sudden limb drop.
A Guide For TreeStability Analysis  41 
This phenomenon is very common in Spain in treespecies like
Eucalyptys spp., Celtis spp., and Ulmus minor in the hot and long
days of august. To the best of the author’s knowledge, in many cases
the risk can be lowered considerably by visual assessment and
specially designed cabling configurations. Maybe also by an extra
addition of water in periods of drought and high temperatures.
• Euler buckling
Slender trees can fail by forming a large wave along the length of
their stem. In reality, the tree “deviates itself” from the load and this
type of failure seems to depend principally on its modulus of
elasticity, height of the tree and the moment of inertia (Gordon,
1999).
This phenomenon could be visualised by leaning on a thin walking
stick which bends aside because of our weight. When we take away
the pressure, the stick turns back to its original shape. But if we force
it to bend too much – the stick escapes sideways  it breaks and we
fall on the floor. Its slenderness and capacity to deform elastically
determine how much of our weight the stick can take before it
buckles.
Usually the veteran trees do not represent this risk, as a result of
having a thick trunk in comparison with their height. There are of
course exceptions to this rule, e.g. slender trees and palms.
If the radius of the pipe is big and the wall thin, it can happen that the
pipe is safe against Euler buckling or the large wave; but that it fails
due to local folding of its skin. One of the ways in which it fails locally
is called the “Brazier buckling. (Gordon, 1999).
The same might be true for trees.
• Brazier buckling
According to Gordon (1999), the system of escaping that the
structure uses, will depend on its form and proportions and the
material from which it is composed.
A wall made of bricks (masonry) usually does not fail because of
primary failure – the blocks do not get crushed by the weight of the
wall – but the wall folds away and the whole comes down.
The same thing can happen with very hollow trees that get folded
because of being too flexible. The “Brazier buckling” is similar to
crushing an empty drink can under pure compression. The empty can
fails due to the formation of small waves, folding itself. The very thin
“residual wall” of the can crumples. It seems that a possible failure of
A Guide For TreeStability Analysis  42 
this type depends on the modulus of elasticity, thickness of the
residual wall and radius of the crosssection (Gordon
1999).
Mattheck & Breloer (1995) seem to refer to this failure as “shell
buckling” and transfer it to a hollow tree, adding that this type of
failure occurs with extraordinary thin shells under bending.
The author of this study observed similar behaviours in the open
hollow trunk of a eucalyptus tree with pullingtests (Elastomethod,
Wessolly & Erb, 1998). The different deformations of the damaged
structure during a wind load simulation were recorded with very
sensitive measuring instruments (Young’s modulus sensors).
It was astonishing to observe how the shell opened sideways. These
perpendicular deformations were much higher, with the same pulling
force, than the longitudinal stretching of the marginal fibres. The pure
longitudinal compression in the trunk was transformed in stresses
perpendicular to the fibres due to the forceflow deviation caused by
the open cavity.
This behaviour seems to suggest that this eucalyptus would fail due
to shellbuckling of the wall of the open cavity and delamination.
Which means that its real structural strength could be lower than the
one predicted with the bending theory of the hollow beam.
• Shearing stresses
Shearing stresses measure the tendency that one part of a body has
to slide over an adjacent part. This phenomenon can be visualised by
bending a book. The pages slide over each other due to the bending
of the whole. The strain due to shear obeys Hooke’s Law with
moderate stresses.
In depth studies were performed by the author of this work on a
datepalm (Phoenix dactylifera) which presented a very hollow and
open trunk. The deformations of the marginal fibres in different
orientations were recorded in the area of the open cavity during a
windsimulation with a cable winch (Elastomethod, Wessolly & Erb,
1998). With Young’s Modulus sensors extraordinary deformations
were recorded due to the sliding over each other of the two halves of
the stem. And they were many times higher than the longitudinal
deformations on which the theory of the Neutral Fibre is based.
This behaviour suggests that failure would occur due to shear,
splitting of the hollow stem in half, without obeying the bending theory
of a hollow beam.
A Guide For TreeStability Analysis  43 
• Concentrations of stress
According to Gordon (1999), the traditional way of calculating the
safety of a structure ignores virtually stress peaks.
The force flow deflection around round holes can cause an increase
of stresses by a factor of three (Mattheck & Breloer, 1995; Gordon,
1999). The effect, nevertheless, is much greater in notsoround
defects and at the end of a crack.
In this way it could be that the resistance against breaking as
calculated according to the simple bending theory of a beam seems
satisfying, although the stress increases dangerously in some points.
The treestructure would start to fail from here on without worrying
too much about the mathematical predictions as according to the
simple bending theory, e.g. due to crack propagation and
deformation energies.
“The locally thickened growth rings lead to a decrease of stresses in
the edges of a wound.” (Mattheck & Kubler, 1995).
A successful compensation  adaptive growth  could be the solution
in this case, although it is not known if a tree can react against
temporary loads (sudden gusts, not a continuous breeze). Neither
does the tree notice some mechanical defects or stresses.
• Torsion and the complex combinations of forces
“Así mismo, no es sorprendente que [la naturaleza] evita la torsión
como la peste” (Gordon, 1999).
In the case of trunks hollowed by fungi like Laetiporus sulphureus or
Daedalea quercina, a concentrically closed residual wall can
sometimes be found. The safety against fracture by torsion can be
estimated in these cases, by means of calculating the torsion forces
exerted by the wind and mathematical formulae. A closed residual
wall would be a very efficient “torsion box ” to resist torsion and shear
stresses. But open shells, just as the convertible oldtimers from
before the Second World War (Gordon, 1999), are not as stable
against this phenomenon.
The prediction of fracture by torsion or other collapses in the real life,
is very difficult due to the infinite morphological variations found in
trees. Several authors describe how a tree, or parts of it, can fail due
to the complex combination of flexion, torsion, his own weight, natural
oscillation, open cavities and so on.
This is why the failure of a tree is not always predictable, even with
the absence of structural defects.
A Guide For TreeStability Analysis  44 
• Basal bell fracture
According to the SIA method (Wessolly & Erb, 1998) and the V
model, a certain tree might only need, theoretically, very thin residual
walls to withstand a hurricane. Or the measurements of the fibre
elongation with elasticitysensors during a wind simulation, might
predict that the hollow buttress would be safe (Elastomethod,
Wessolly & Erb, 1998). According to the theory that supports the last
method, the structure would fail when the pure longitudinal
deformation of the wood fibres exceeds the elastic limit.
But then the tree might fail without obeying these methodologies,
simply because the latter do not contemplate torsion, shear or
delamination, or, the very common reasons that can lead to the
failure of a tree.
In the case of butt rots the tree may fail due to a combination of shear
and delamination. Even with perfectly unilateral wind loads, high
stresses can occur where the thin residual wall meets the root
buttress.
This behaviour was observed during a pullingtest (Elastomethod,
Wessolly & Erb, 1998) in Spain. There, a crack was caused
unexpectedly in the hollow buttress of a plane tree under the
simulated wind load. Afterwards, it was concluded that the tree it’s
structure failed even before the estimated point of material failure
(Detter, 2002).
This means that the tree was less safer than calculated with this
method and that it would have failed with a lower wind load than
predicted.
When the mathematics calculate that only a low t/R is needed in a
certain tree – even being a very good orientation  care should be
taken to exclude the possibility of delamination of the residual wall
nearby the root flares.
The basal bell fracture seems to gain importance when the ratio
between residual wall and radius of the crosssection gets very small.
This, again, will be true in the idealised case of perfectly closed,
intact and concentrically cavities. Nevertheless, open cavities or
highly curved root buttress for example might require a higher t/R.
It is possible that common sense and a good vitality – and hence a
high quality repair growth  might be the key, even in the future, to
solve this question.
A Guide For TreeStability Analysis  45 
7. One of the secrets of current treestability assessment
Merely from observation during treepruning and tree removals, it
seems to be clear that the heavy topend weight of the crown of
several treespecies probably add significantly to the swinging
movements.
This is quite clear for example in tall Cupressus sempervirens or
Aesculus hippocastanum when bearing their fruits. The reader can
experiment the difference when swinging back and forth a branch of
a Platanus spp. with leaves and afterwards a branch of a Pinus pinea
with cones.
Probably also the MOE (Modulus of Elasticity) of the material from
which the branch is made will influence this phenomenon, apart from
its weight and centre of gravity, its form or the presence/ absence of
leaves.
Naturally, and as with other methods and models, these factors do
not fit easily in mathematics and should be taken into account during
the visual stability assessment.
It is true that in the Vmethod, the bendingfrequency (in Hz) of the
bare trunk can be calculated from the height and circumference of
the trunk and the frequency factor. But then the crown form (swinging
“Liontails” for example), unequally positioned crown surfaces, crown
weight and the centre of gravity, the form of the trunk (if it’s not as
straight as a pole), previous structural defects, root/soil interactions
and other variables upset the calculations (Coder, 2000).
Even when leaving aside offcentred gusts and other chaotic
behaviours of the wind.
Therefore, real movements of the tree under wind loads can vary
highly from the ideal backandforth situation.
On the other hand, dangerous sway motions in trees can also be
minimised due the way in which the branches counteract each other
by complex swaying (James, 2003). The latter might especially be
the case when the drag area is relatively evenly distributed over the
whole tree.
Relatively flexible structures – structurally too damaged treeparts for
example – might not obey the mathematical calculations. This is also
the case with too slender trees.
The swinging can considerably lower the safety of a tree and, to the
best of the author’s knowledge, it does not fit easily in mathematics.
A Guide For TreeStability Analysis  46 
In consequence, slender trees (some pinetrees for example) and
palms, sensitive to swinging, require the contemplation of this
component.
Here is an excerpt of “Structural Engineering Systems Design”
(Sparling, 1997) that explains part of the mystery that involves tree
stability assessment, although it lays out about buildings:
“It is important to recognise that wind pressures, and hence wind
loads, are not static in nature but fluctuate constantly. The dynamic
nature of wind loads can, in fact, excite resonant motion in slender or
flexible structures, generating dynamic responses that can be far
larger than those that would be produced by equivalent static loads.
For low, relatively rigid buildings, on the other hand, the dynamic
response is less significant and the design can be safely based on
equivalent static loads.”
So to the author of the present publication, one of the secrets of
current treestability diagnosis is: a wind load analysis can safely be
employed in relatively low and rigid trees.
A Guide For TreeStability Analysis  47 
8. Palms and stability
“En todo lo natural hay un código y cada código tiene su clave”.
In the wood of a twocotyledon tree, the fibres are well glued
together, reducing the risk of the fibres sliding over each other
(Mattheck and Breloer, 1995). The geometry and the material many
times behave in a coherent manner, whereby one of the main
components of its load bearing capacity is the pure longitudinal
compressionstrength in the outermost fibres. This enables the
assumption that its breaking safety can be calculated by means of
the simple bending theory of a hollow beam.
Mere observation of fresh cut pieces of the trunks of palms, suggests
that the fibres of the palmtrunk do not seem to be glued together as
good as the fibres of a (not monocotyledonous) tree.
Therefore, the mass of palmfibres does not seem to behave as
coherently as the wood of a tree. The fibres can be torn apart easily
with the hands. Therefore, it is possible that the trunk of a palm is
more sensitive to splitting (shear and perpendicular stresses) than
the sapwood of a tree. Pullingtest experiments on palms in Spain
lead by the author of this publication also seem to suggest this
structural behaviour.
The fibrous structure might behave more as the hairs of a broom –
especially with small residual walls – instead of as it was a massive
wooden beam.
The respectable experience of specialists in palmcare points
towards the same direction.
Therefore, it is possible that the breaking safety of a palm is not
predictable with the bending theory of a hollow beam and that safety
factors of the trunk cannot be given. Neither the necessary residual
wall thickness, as according to the last theory, could be calculated
then.
At the current time, the basics of a possible solution have been
developed by the author of the present pages for the stability analysis
of palms.
A Guide For TreeStability Analysis  48 
A proposal for the stability analysis of palms
Little investigation has been performed about the structural behaviour
of palms. Hence, the only fixed rules that can be given until now are
intuition, experience and common sense. Nevertheless, the author of
the present publication developed the here presented proposal for
the stabilityanalysis of palms.
In this protocol, visual palm assessment is combined with
mathematics for the analysis of the wind loads and bending
frequencies of the palm trunk. With the V model, both the bending
frequency of the palmtrunk (in Hz) and the wind load in the crown
are estimated according to the Eurocode 1 (AENOR, 1998). The
equations for the analysis of the wind load and bendingfrequency
are described in the structure of the V model. The results are
combined with a rubber mallet to detect structural defects.
Within the limits of the present model, the following guidelines could
be given:
• A high bendingfrequency in combination with a sound
structure gives a lowrisk level.
• A high bendingfrequency in combination with structural
defects should demand attention and a closer look to the
situation.
• A medium frequency cannot allow damages.
• Finally, interventions should absolutely be taken to guarantee
the trafficsafety if the palmtrunk has a low frequency.
The closer the natural frequency of the palm gets to the frequency of
gusts of wind (1 Hz for example), the lower its stability. For example,
a sound Phoenix canariensis that has a bendingfrequency of 13,4
Hz might present very little risk.
On the other hand, a more slender Phoenix dactylifera can signify a
higher risk due to the combination of a half as high bending
frequency and higher wind loading. If this palm has a rot, for
example, then a more detailed investigation should be undertaken.
A Guide For TreeStability Analysis  49 
The limits are multiples and obvious: much more investigation is
required to establish acceptable levels of bendingfrequency and to
find the correct frequency factor for palm trunks.
The aerodynamic drag factor has to be estimated and is not
published yet. Finally, independent scientific investigation is
necessary to evaluate the value of the present proposal.
Within this method, the inspected palm can be cabled to other palms,
trees or structures. In this case, the wind load in the crown should be
estimated. The result enables a more efficient design of cabling
configurations, cable strength, cable elongation and loadtransfers.
P
hoenix canariensis. Casa Alegre. Terrassa, Cataluña.
(Load analysis in accordance with Eurocode 1, Part 24 + bending frequency of
the stem.)
Height: 14,15 m
Bendingfrequency: 13,4 Hz
Estimated wind load (F)
in the crown during gusts:
3,66 kN (373,32 kg)
If the bending theory of a
hollow beam would be feasible:
Safety factor at 1m
height: 394,8 %
t required: 2,5 cm
t/R required: 0,09
A Guide For TreeStability Analysis  50 
9. Protocol
Biology is one of the most important components of the protocol
presented in this publication.
The ring of sapwood is what gives the main structural strength,
regarding for example torsion and bending, to a tree.
Therefore, if symptoms are detected that point towards damages in
the sapwood, e.g. due to wooddecaying fungi, then it is possible that
the crosssection suffers an important loss of its load bearing
capacity.
If the sapwood is damaged, then the transporting capacity can be
diminished. This can lead to a loss of vitality. This loss of vitality
usually produces symptoms in the upper crown structures and the
bark. Other symptoms, like growth depressions or dead areas in the
bark also can point towards serious structural damages in the ring of
sapwood.
Therefore, it is possible to take as a starting point the state of the
crown and bark, instead of focusing immediately on conks or hollows.
This method can be a very good orientation, in order to assess
visually if there is an important loss of structural strength, caused by
fungi, in the wooden body.
Afterwards, the calculation of breaking safety factors might give
information about the importance of this strength loss for the
individual tree (V method or SIA method).
Thereby it is absolutely necessary, by means of Visual Tree
Assessment (VTA method), to take in account the failure components
that cannot be foreseen by mathematics.
The growth pattern of the upper crown structures and growth fissures
in the bark are a visible measure for the biology of the individual tree
and are an orientation for:
• the transport and vessels and hence the biological and
structural state of the sapwood
• the efficiency of compartmentalisation of decay
• the efficiency of compensation growth
• a long term prognosis instead of a momentary recording
A Guide For TreeStability Analysis  51 
The protocol offers treespecialists a carefully designed process,
including the following components:
• The wind load in a tree is analysed in accordance with
international engineering standards and scientifically
accepted mathematical procedures. For breaking safety, the
necessary residual wall thickness and safety factor of the
tree its trunk and stems is calculated.
• The reasons that cause the failure of a tree are very
complex. This complex combination is broken down in its
theoretical components. First the components have to be
understood and their influence in the tree assessed.
Amongst them, the most important seem to be: torsion,
bending, shear, oscillation, crackpropagation and stress
peaks in the wooden body, fatiguing of the anchorage roots
(!), architecture and stiffness of the loadbearing root system,
crown clashing and real wind behaviours.
• The results of the wind load calculations are compared with
the analysis of the stabilitycomponents that are relevant for
the assessed tree. The usefulness of the theoretical
calculations has to be tested against the real tree
behaviours.
• The interactions between host, fungi, vitality and tree
geometry have to be understood. Regarding biology, both
compartmentalisation – the formation of boundaries that slow
down the progress of wooddecaying fungi – and
compensation – position, amount and quality of the repair
growth  are directly correlated to the tree it’s vitality.
• After all, the continuous thread throughout the protocol
should be common sense.
A Guide For TreeStability Analysis  52 
T
h
e following components interact continuously and have to be assessed, not as
solitary, but as links of a complete chain:
Vitality: capacity of compartmentalisation and repair

growth. Visually assessable symptoms in crown and
bark
Geometry: form of
the crosssection
and structural
defects (confined
vs. irregular rots)
Wood

decaying
fungi: Ring of
sapwood and
symptoms in the
bark
Mathematics: V method
(
Sterken, 2004
)
and SIA method (
Wessolly & Erb, 1998
)
.
+ Reasons of mechanical failure and its visual assessment (VTA, Mattheck
& Breloer, 1995)
Many times, when the vitality of a tree is good enough, the inner
geometry of the ring of sapwood can have a clearly
defined form and a coherent mechanical behaviour.
This might permit to assess its breaking safety with
the combination of a wind load analysis and the
bending theory of a hollow beam.
On the other hand, when the tree is not vital enough,
the processes of compartmentalisation and
compensation a
re not as efficient regarding extending
rots. This can result in a irregular and treacherous
geometry that might not behave as according to the
bending theory of a hollow beam. Here, visual
assessment will be even more important to employ
more efficiently advanced technical resources.
A Guide For TreeStability Analysis  53 
Examples
• Eucalyptus camaldulensis. Parc Sant Jordi, Terrassa,
Cataluña.
Vitality: excellent
Height: 17,1 m
Bendingfrequency: 10,1 Hz
Safety factor for bending fractures: 275,7%
Required residual wall thickness: 4,39 cm
There are no symptoms that point towards a diminished transporting
capacity. The sapwood is in a very good state and the large wound is
very well limited by vigorous woundwood.
Possibly, and due to good quality compartmentalisation and
compensation, the mechanical behaviour of the residual wall is very
coherent. The tree its breaking safety can reliably be assessed with
the theory of elasticity and visual assessment.
The excellent vitality (biology) can counter the confined cavity
(stability) during several years more.
• Eucalyptus globulus. Parc Sant Jordi, Terrassa, Cataluña.
Vitality: very low, although the emergencygrowth, stimulated by
topping, provides a misleading view of the tree its vitality.
Height: 14,7 m
Bendingfrequency: 16,3 Hz
Safety factor for bending fractures: 1224,03%
Required residual wall thickness: 1,09 cm
Theoretically, this tree would need only 1,09 cm of residual wall to
withstand hurricanes of 117 km/h. The bendingfrequency suggests
that the influence of dynamics is very low.
Hence, these mathematics predict that the tree is extremely safe.
Nevertheless, in this tree, these results cannot be taken blindly as the
“happy” result. Several symptoms suggest a diminished transporting
capacity. Hence, he sapwood might not be in a good state, what
would lead to an important loss of structural strength. The large
cavity is not clearly limited by vigorous woundwood.
Therefore, and due to the irregular and not clearly confined rots, the
hollow trunk might not behave as an ideal and coherent hollow beam.
A Guide For TreeStability Analysis  54 
Sideways shell buckling due to the large openings and thin shell
(see Matheck and Breloer, 1995) might occur instead of obeying the
bending theory of a hollow beam. Probably also with much lower
wind speeds than predicted. The pullingtest experiment that was
performed in this tree also suggested the latter.
Thus in this case, visual assessment, a profound knowledge of the
possibilities and limits of the mathematics, a profound knowledge of
mechanical reasons of failure and a good dose of common sense,
will be vital to achieve the most reliable diagnosis.
• Statics vs. dynamics
In the V model, the natural bendingfrequency of the trunk is
compared to its safety factor (regarding static wind loads), allowing a
better orientation for the tree its stability.
The following examples are presented here:
Cupressus sempervirens
Parc Sant Jordi
Static safety:
217,4 %
t required: 3,47 cm
Bending frequency:
5,8 Hz
= the static safety factor and
theoretically residual wall
thickness cannot blindly be
trusted due to the high
influence of natural
oscillations
Eucalyptus camaldulensis
Parc Sant Jordi
Static safety:
275,7 %
t required: 4,39 cm
Bending frequency:
10,1 Hz
= the static safety factor and
theoretically residual wall
thickness can be a very good
orientation
A Guide For TreeStability Analysis  55 
10. Treesaving interventions
“…while remembering that the object is to retain the tree for as long
as possible with the minimum risk.” (Mattheck & Breloer, 1995)
The possibilities for keeping a tree as long as possible with a
acceptable risk level are virtually limitless.
Amongst them, the beautiful art of cabling trees is one of the most
interesting ways of lowering the risks of tree damage. Not only the
theory – the contemplation of the laws of gravity, angles, mechanical
energies and transfers of loads  but especially the practice is very
gratifying.
The protocol of the author employs several methods for designing
crown cabling configurations.
First of all the goal of the cabling has to be determined since, for
example, the prevention of breakage can require a different cabling
than for fallprevention in case the branch breaks.
In the first case the static wind load in the crown or cabled main
stems and cables can be assessed.
Meanwhile, the tree it’s overall stability should also be evaluated.
It has no sense for example to cable a tree if the next storm would
cause failure of the rootsystem that would make the complete tree to
fall down!
The stability question has to be solved first, on all levels, before
cabling can be installed!
The strength of the cabling should not exceed that of the supporting
treepart. Therefore, the critical load that would cause failure of the
supporting stem, if this load would pull or push at the height of
installation, can be assessed.
The maximum strength (Smax)that the cable should have, in order not
to break the supporting branch when the fault stem would fail, can be
represented by the following equation, which is derived from the
formulae employed in the Vmodel:
A Guide For TreeStability Analysis  56 
Smax = ((w*s)/100)/h
Where:
Smax = the maximum strength of the cable in kN
w = the crosssection modulus of the basis of the supporting
stem in cubic centimetres
s = the compressionstrength of the living wood in kN/cm²
h = the height of installation in metres
Nevertheless, the above equation does not incorporates the influence
of dynamics, which should absolutely been taken into account.
Also the influence of installationangles of the cables should be
incorporated. The angle of the cable is a critical factor regarding the
transfer of forces. With angles also the falling distance of a failed
branch can be made as short as possible.
It is very important to understand that stopping a falling branch with a
cable, even a steel one, is very unlikely to be successful due to the
high energy and force that grow enormously due to several reasons.
The supporting part of the tree or the cable, are usually not able to
withstand the high shock when the fall is stopped.
The following equation can be offered as an evidence (free after
Wessolly & Erb, 1998):
F = (m * g * h) / e * 2
Where:
F = the force in the cable at the elongation limit of the cable
m = the mass of the branch which is a product of the volume
and specific density of the latter
g = acceleration due to gravity and is expressed as 9,81 m/s²
h = falling height of the branch and can be the length of the
cable
e = the strain or elongation limit of the material from which the
cable is made and is expressed in %
It is important to underline that cabling is only a transfer of forces to
other parts of the tree e.g. the securing branch.
Currently, the use of flexible synthetic materials can be
recommended since they can help to stimulate the formation of
A Guide For TreeStability Analysis  57 
repairgrowth in the defective area. It is possible that in this way and
with vital trees, a properly cabled branch can regain complete
structural stability after some 1520 years, after which its cabling may
become unnecessary.
Damages due to perforation can be prevented by synthetic cabling
systems that incorporate slings which embrace the branch or stem
instead of passing trough it.
Also the risk of the socalled “karateeffect” can be lowered
considerably by flexible and shockabsorbing materials. Under this
effect is understood the failure of the cabled branches at the height of
the installation, due to the sudden stop (caused by rigid materials like
steel) of the natural swinging of the tree parts.
It is also important to study the architecture of the tree to take
advantage of it multiple dimensions. It is better to connect the main
stems ringwise than to draw a starlike single cable configuration,
especially regarding swinging and torsion.
The height of the installation will depend on each situation and the
oneandonly fist formula can really not be given. Profound training
by experts is thus required, whereby the given information should
always be contrasted with other sources.
Cabling of old trees and palms to other structures or trees has
already proven to be successfully done. Thereby, the analysis of the
wind load in the crown is undeniably a must to assess the loads in
the cabling system and the supporting structure.
In order to stimulate the vitality of a tree, and hence its capacities of
compensation and compartmentalisation, mulching and deep
watering are still unbeatable, the latter especially in periods of
drought. Also, in regions where summer branch drop occurs, the
extra addition of water might lower the risk for this type of failure.
The reaction capacity of the tree can be heightened by administering
potassium, phosphorus and magnesium during the dormant period.
On the other hand, nitrogen stimulates the extension of wood
decaying fungi.
After the tree or palm has been assessed and treated (in this
sequence) it is recommended to inspect it several times a year
visually. Broken branches, symptoms of structural defects or fungal
fruiting bodies can appear afterwards and sometimes depending on
A Guide For TreeStability Analysis  58 
the season. Symptoms of root stability loss can appear in the soil and
near the stem base after events of rain or wind. The morpho
physiology of the tree should absolutely be considered when a tree is
pruned, e.g. a crown reduction for reducing its sailarea. And the
works of Alex Shigo should still be considered as the basis for all our
actions upon trees. The author of the present study recommends his
reader to consult specialised literature, to contrast information and to
build up his own baggage of tree and human life saving knowledge.
11. Conclusion
In this publication the author only restricts himself to describing the
basics of the puzzle. To him, his personal idea of treeassessment is
what the “Philosophical Stone” is to the alchemists, an idea by which
the stability of a tree can be deciphered.
No attempt was made to write what others have already written better
prior to this study.
The case is that the body language of the tree, its life expectancy and
monetary value, many times require only an expert visual diagnosis.
Many old trees can be assessed with the present proposals.
Should the tree in question have an extraordinary monetary value
and if there are unsolved questions regarding its stability, a range of
exquisite instrumental methods can be chosen from.
And as Mattheck & Breloer (1995) state: “… do not trust technology
alone – it can only measure what you have seen.”
A Guide For TreeStability Analysis  59 
12. End notes
Gordon (1999) describes how several historical disasters occurred to
boats, bridges and other structures and which led to the loss of many
human lives. They were mainly the result of the application of just the
theory of elasticity.
Hence, we cannot rely solely on mathematics.
Tree and human lifesaving assessment should incorporate biology,
mycology, the experience and intuition of professional colleagues
and everything that we are capable to integrate.
After all, we should listen to our common sense.
And something to think about:
Real trees do not fit mathematical expressions perfectly. There are
differences in the growing conditions and loads on different trees and
palms under different conditions over time (Horacek, 2003).
A profound and intuitive knowledge of the inherent perversity of
materials, loads and geometry’s  mathematics and trees  is one of
the most precious qualities an arborist can have (free after Gordon,
1999).
A Guide For TreeStability Analysis  60 
Acknowledgments
The author wishes to thank especially his father Jan Sterken, his
family and María Dolores Infante Mera, Loli, for their relentless
support in all times. To Wim Peeters for his immediate enthusiast
support.
The author wishes to thank Prof. Dr Francis Schwarze for comments
on the manuscript that was the basis for “Prognosis of the breaking
safety of hollow trees”.
Dr. Ing. L. Wessolly is recognised for providing instrumentation for
the experiments in Terrassa and Mataró. The author of these lines
would like to express his acknowledgement to Josep Manel
Fernández for his patience and excellent expert opinion during the
Terrassa experiments. Dr. Petr Horacek is recognised for opening