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.