Tree Structure - Dynamics
Ken James
University of Melbourne, Australia
Tree Structure – Statics & Dynamics
Trees are optimised structures
• Statics – well covered in last
10 years
• Dynamics - difficult
Wind creates largest loads
Total loads on trees consist of
STATICS & DYNAMICS
Loads on Trees - Research
Statics
• Trees growth responds to loads
• “Axiom of Uniform stress” (Mattheck )
• Static approach good in still air
Dynamics
• Wind is dynamic,
creates large loads
• Tree dynamic response
is not known
Research Strategy
• Measure wind loads
• Measure tree response
Tree Structures and their loads
• Structure must be stronger than the loads applied.
• Failure occurs when
Applied stress at point > strength of material
Load > strength
Need data to assess structure - Research
• Data on strengths of trunks and limbs
• Data on loads on trunks and limbs in high winds
Tree structures
Auracaria
– with branches, STABLE,
- without branches, UNSTABLE
Loads on trees
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–
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–
Tension
Compression
Bending
Shear
Torsion
Growth
Static and Dynamic Loads
Loads applied as
• Static – weight of branch, snow, ice
• Dynamic – wind
-------------------Static and Dynamic loads ADD
Biggest loads occur during high winds
Difficult to measure actual loads during wind
storms, but need data on this!
Tree Structure
1. Data on strengths of trunks and limbs
Strength of a trunk/limb depends on
1. Size (area of cross section)
2. Shape (where material is positioned)
3. Material strength (k) - Young’s modulus
Strength depends on – 1. Size
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Trees – largest sections are the oldest and stiffest
Taper, gradually matches section to loads
Base trunks/branches stiffest
Ends smallest, most flexible
Bigger sections hold more load, but also
approach the limit of strength
Imperfections in wood reduce strength
so as trees get bigger they get nearer to failure.
Q. Must know what loads on section to assess how
close to failure!
Strength depends on – 1. Size
Strength depend on – 2. Shape
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Bending – compression and tension forces
on opposite sides of section
Bending – I beam shape
Torsion – twisting (may be significant in
small flexible sections)
- circular shape best
Load history of tree/branch seen in growth
rings and thickness variations
Loads on branches and trees
Bending - tree weaker in compression than tension
Strength depend on – 2. Shape
Howarth, 18th Century
Mattheck, 1994
Strength depend on – 2. Shape
Response to loads
Bending - tension, top & compression, bottom
Growth is not uniform from the centre
Strength depends on – 3. Material
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Strength of wood varies greatly
Tensile strength about twice compressive
strength
Measured by Young’s modulus
Young wood flexible
(7 year old Scot’s pine (Pinus sylvestris)1.7 GN m-2
•
Old wood stiffer
(27 year old Scot’s Pine 7.9 GN m-2
(Mencuccini, 1997)
Tree - base stiff, strong,
- tips flexible, not as strong
Strength depends on –
• Material elasticity
measured by (k) –
Young’s modulus
• Shows as slope of line
• k1 stiff
• k2 flexible
• Strength is different
• k2 flexible and strong
3. Material
Dynamic Loads on trees
1. Static loads – weight of limbs, foliage, snow, ice
2. Dynamic loads (wind) greatest (Mattheck 1994)
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•
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bending (tension and compression)
shear
torsion
Wind comes in gusts and pushes on tree canopy.
Gusts occur with period of 20 to 40 seconds
Complex sway motion of branches and tree
Pendulum ? How do trees sway?
Current dynamic tree models
Woods, C.J. 1995
Current dynamic tree models
Nield & Wood, 1998
Sanderson, et al.1999
Mass of canopy - rigid
Dynamic model
• Mass and spring
oscillator
• Cyclic period
• Damping
reduces motion
Dynamic model
Mass and spring oscillator
1. Mass (m)
2. Spring (k)
3. Damping (d)
Cyclic period defined
Tree sway motion
Complex sway motion of tree and limbs.
Dynamic model considers
1. Mass of trunk, branches and leaves
2. Spring – wood Young’s Modulus
3. Damping has three components
• aerodynamic drag – leaves in wind
• viscoelastic damping – stem/root/earth
• mass damping – limb sway interaction
A dynamic model of trees
• A mass (m) oscillates on a spring (k)
and motion is damped (d)
Model
Tree
Oscillation
Mass damping – effect of one branch
• A small mass (m) oscillates on a spring
and damper and “detunes” the structure
• The amplitude is greatly reduced
Model
Tree
Oscillation
Tuned mass damped Structure
Buildings
Poles
Bridges
Tuned mass damped Structure
First building using TMD, tuned mass damping
1987, Centrepoint Tower, Sydney – Soong, 1997
Mass damping – 2nd order branch
• further small branch (mass) oscillates on
larger branch and adds another mass damper
• Structure is “detuned” even more
• The amplitude is greatly reduced
Model
Tree
Oscillation
Mass damping – 5th order branch
• further small branch (mass) oscillates on
larger branch and adds another mass damper
• Structure is “detuned” even more
• The amplitude is greatly reduced
Model
Tree
Oscillation
A dynamic model of trees
Structure of trunk is damped by leaves, internal & branches
1. Branches – mass damping
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Large branches are first order mass dampers
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2nd, 3rd, 4th, 5th & 6th order branches
2. Damping (d) combination of leaves and viscoelastic
Mass (m) and stiffness (k) of each branch in model
Model
Tree
A dynamic
model for
urban trees
Spectrum data – Kerzenmacher & Gardiner, 1997
Spectrum data – Kerzenmacher & Gardiner, 1997
Spectrum data – Saunderson, et al. 1999.
Tree Structure
- Urban trees
Tree Structure – Wind effects
Measuring wind loads in trees and
branches
Wind map of Australia
AS 1170.2:2000
Wind Speeds
Tree
Windthrow
mph
Cullen, 2002
Hedden, R.L.
1995
m s-1
mph
Comment
m s-1
55-63 25-28 55-63 26-28 Wind scales and vel comparison
46
Spatz, 2000
20-30
Sanderson et
al. 1999
28
Coutts 1986
3-17
AS1170.2.
2000
Break
69
Winch tests, Sth Carolina,
hurricane 165 (max 249) km/h
Norway spruce, 56 y. 27 m high
20
Mathematical model, values seem
high (his comment)
Ref from Sanderson
48-60 m s-1. Code values for
return period of 100 years
Measuring wind loads - instrumentation
Wind Loads
on Branches
- Shigo
Branches in wind
• Deflection sideways and upwards
• Wind pushes branch
• Some sway but not back towards wind
direction
• Branch does not sway like a pendulum
Analysis of Tree Structures
1. Wind throw – whole tree
2. Limb/trunk failure – parts of the tree
Wind throw – whole tree analysis
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Overturning moment of wind resisted
by tree roots in soil
Wind throw – TREE PULL
TEST
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Pull tree to measure
resistance to
overturning
Determine wind
loads (difficult)
Verify strength of
tree in ground to
resist measured wind
loads
Overturning forces
Tree
Eucalypt -200mm dia. Erica
Eucalypt - 500 mm dia.
Burnley
Sitka spruce, 20 m high
kN.m
6
60
Comment
Winch test in forest, Aust. - failed
10-52
Bell et al., 1991
still stable though noticeable
movement
NZ trees, 7 sites x 13 trees, 939 years old, 28-35 m high
300
Max from winch tree pulls, Moore,
2000 PhD.
Calculated -Plane trees
18m high Parkville
600
Australian Wind Code (AS 1170.2)
- very high
Calculated from max wood
fibre stress
1219
Mattheck & Bethge, 2000
Tree Pulls
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200mm
Eucalypt
(Erica)
6 kN.m failed
Tree Pull - Burnley, 2002
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400 mm Eucalypt Burnley
- 60 kN.m still stable though noticeable movement
Overturning Force - calculated
University of Melbourne
Parkville 18 m plane trees
- calculated at 600 kN.m
(AS 1170.2) very high
Tree Pull Test – 4 directions
Pull Test Burnley
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Pull test – in 4 directions
Gives measure of resistance to overturning
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Need accurate wind load data (project to
measure overturning moments in wind storms)
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Provides data – for decisions
Modes of vibration
Dismantling trees
Examples
Examples
Examples
Examples
Conclusions
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Wind is dynamic, creates largest loads
Static and Dynamic loads ADD
Biggest loads occur during high winds
Complex sway motion of limbs modified by
damping
Damping has three components
• aerodynamic drag – leaves in wind
• viscoelastic damping – stem/root/earth
• mass damping – limb sway interaction
Mass damping minimises sway response
Further Work
Difficult to measure actual loads during wind storms,
but need data on this!
• Measure wind loads
• Measure tree response
Develop strength testing such as pull tests
Develop removal techniques to use natural
damping of tree to advantage.