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Tree Structure - DynamicsKen James
University of Melbourne, Australia
Trees are optimised structures• Statics – well covered in last
10 years• Dynamics - difficultWind creates largest loadsTotal loads on trees consist ofSTATICS & DYNAMICS
Tree Structure – Statics & Dynamics
Loads on Trees - ResearchStatics• 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 materialLoad > 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
– Tension– Compression– Bending– Shear– Torsion– Growth
Loads applied as
• Static – weight of branch, snow, ice
• Dynamic – wind--------------------
Static and Dynamic loads ADD
Biggest loads occur during high windsDifficult to measure actual loads during wind
storms, but need data on this!
Static and Dynamic Loads
Tree Structure1. 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• 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• 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
Bending - tree weaker in compression than tension
Loads on branches and trees
Strength depend on – 2. Shape
Mattheck, 1994Howarth, 18th Century
Bending - tension, top & compression, bottom
Growth is not uniform from the centre
Strength depend on – 2. ShapeResponse to loads
Strength depends on – 3. Material • 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 – 3. Material
• Material elasticity measured by (k) –Young’s modulus
• Shows as slope of line• k1 stiff• k2 flexible• Strength is different• k2 flexible and strong
Dynamic Loads on trees
1. Static loads – weight of limbs, foliage, snow, ice2. Dynamic loads (wind) greatest (Mattheck 1994)
• bending (tension and compression)• shear• torsion
Wind comes in gusts and pushes on tree canopy.Gusts occur with period of 20 to 40 secondsComplex sway motion of branches and treePendulum ? How do trees sway?
Current dynamic tree models
Woods, C.J. 1995
Current dynamic tree models
Sanderson, et al.1999
Mass of canopy - rigid
Nield & Wood, 1998
Dynamic model
• Mass and spring oscillator
• Cyclic period• Damping
reduces motion
Dynamic model
Mass and spring oscillator1. Mass (m)2. Spring (k)3. Damping (d)
Cyclic period defined
Tree sway motionComplex sway motion of tree and limbs.Dynamic model considers1. Mass of trunk, branches and leaves2. Spring – wood Young’s Modulus3. 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 StructureBuildingsPolesBridges
Tuned mass damped StructureFirst building using TMD, tuned mass damping1987, 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 treesStructure of trunk is damped by leaves, internal & branches1. Branches – mass damping• Large branches are first order mass dampers• 2nd, 3rd, 4th, 5th & 6th order branches2. 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 AustraliaAS 1170.2:2000
Wind Speeds
48-60 m s-1. Code values for return period of 100 years
AS1170.2. 2000
55-63mph
55-63mph
3-17
28
20-30
46
25-28m s-1
Windthrow
Ref from SandersonCoutts 1986
Mathematical model, values seem high (his comment)
20Sanderson et al. 1999
Norway spruce, 56 y. 27 m highSpatz, 2000
CommentBreakTree
Winch tests, Sth Carolina, hurricane 165 (max 249) km/h
Wind scales and vel comparison69Hedden, R.L.
1995
26-28Cullen, 2002m s-1
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• Overturning moment of wind resisted
by tree roots in soil
Wind throw – TREE PULL TEST
• Pull tree to measure resistance to overturning
• Determine wind loads (difficult)
• Verify strength of tree in ground to resist measured wind loads
Overturning forces
Mattheck & Bethge, 20001219Calculated from max wood fibre stress
Australian Wind Code (AS 1170.2)- very high
600Calculated -Plane trees18m high Parkville
Max from winch tree pulls, Moore, 2000 PhD.
300NZ trees, 7 sites x 13 trees, 9-39 years old, 28-35 m high
CommentkN.mTree
Bell et al., 1991
still stable though noticeable movement
Winch test in forest, Aust. - failed
10-52Sitka spruce, 20 m high
60 Eucalypt - 500 mm dia. Burnley
6 Eucalypt -200mm dia. Erica
Tree Pulls• 200mm
Eucalypt (Erica)6 kN.m failed
Tree Pull - Burnley, 2002
• 400 mm Eucalypt Burnley- 60 kN.m still stable though noticeable movement
Overturning Force - calculated
University of MelbourneParkville 18 m plane trees- calculated at 600 kN.m(AS 1170.2) very high
Tree Pull Test – 4 directions
Pull Test Burnley• Pull test – in 4 directions
• Gives measure of resistance to overturning• Need accurate wind load data (project to
measure overturning moments in wind storms)
• Provides data – for decisions
Modes of vibration
Dismantling trees
Examples
Examples
Examples
Examples
Conclusions• 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 responseDevelop strength testing such as pull testsDevelop removal techniques to use natural
damping of tree to advantage.
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