For a mathematical treatment of plant biomechanics see Tree Equations.
See also the different
cell types and the mechanics of leaves.

Introduction

Although much attention is frequently paid to the physiology and biochemistry of plants, their mechanical
properties are often overlooked. Adapting to life on land made it essential to develop strong stems, in order
to reach up to the light. Plants have a range of mechanical adaptations, whether green herbaceous
annuals, or woody trees. Trees are especially constructed to high mechanical specifications. Wood is very
strong, and yet relatively light and weight-for-weight is stronger than steel. This low weight construction is
important, as a tree must bear its own weight (static loading) in addition to wind loads (dynamic loadings)
and perhaps the weight of rain or snow. They must also be resistant to grazing and are often resistant to
lightning and fire damage. The tallest tree that ever lived, was probably about 130 m, certainly this is a safe
estimate for an upper limit (the tallest living trees are about 115 m) and may live for thousands of years -
trees are often built to last! How do they achieve this?

Tree Trunks

In woody stems lignin (a phenolic polymer) thickens the cell walls and increases compression resistance.
New tracheids are longitudinally contracted and so under compression (being squeezed along its axis). The
centre of the trunk is under compression, whilst the periphery of the trunk is under tension (being pulled or
stretched along its axis) and upon air-drying the shrinkage of wood is plane dependent and can approach
the breaking load (so that dry wood may split). When wood is under compression, slip planes appear at
about half of the breaking load. These slip planes relieve stresses and are possibly a normal part of growth
(?). The core of the trunk has a significant plastic component.

Hollow trunks and branches have a reduced weight per unit length for a given diameter. This increases
the second moment of area (
I) per unit of mass. (The second moment of area is a measure of the distance
material is spread-out from the central axis and the greater this is, the stiffer the column/stem or
beam/branch). The critical buckling length of a beam increases for beams with a lower weight per unit
length. This means that hollow beams can be longer. However, if the walls become too thin then the organ
becomes subject to brazier buckling or crimping (like a bent straw) in which upon bending the cross-section
becomes oval.

Tapering of tree trunks helps to minimise the maximum bending stress and uniformly distributes stress
along the trunk length. The taper of field saplings is close to the ideal theoretical taper. Taper also
increases the critical length before the trunk becomes unstable. Tall, slender columns fail by buckling, whilst
short, wide columns undergo crushing failure. The flexural stiffness or rigidity of a column or beam is given
by E
I. The elastic modulus, E, is determined by anatomy (tissue and cell properties) whilst the second
moment of area,
I, is determined by morphology (shape). Thicker trunks have a larger I.

Thigmomorphogenesis is the phenomenon of exhibiting growth responses to mechanical stimuli. Trees
growing in exposed places, where the trunk is buffeted by the wind, have shorter thicker trunks. Those
grown in sheltered forests have taller, thinner trunks, which enable them to reach the light for which they
compete with their neighbours. If trees are cut down around a neighbouring tree that grew in such sheltered
and crowded conditions, then the tree often fails because it is unable to resist the increased exposure to
stronger winds.

Resonance under dynamical wind loading is a major factor contributing to trunk failure. Tree trunks have
typical safety factors (ratio of maximum sustainable loading to the operational/normal loading) greater than
or equal to 4. Woody perennials have safety factors ~ 3-4.
Tree Branches

In cantilevered beams (beams supported at one end only, like tree branches) shear forces are very
important. Shear is the relative sliding of different 'layers' of material over one-another, like sliding cards
over each other in a deck. Maximum tension occurs in the upper surface whilst maximum compression
occurs in the lower surface. When branches fail they usually exhibit bending failure. Initially a tensile
fracture occurs, usually initiated in the outermost ring of wood on the upper surface. This is rapidly followed
by shear failure at the interface between the summer and spring xylem. Shearing between the wood and the
bark above and below the site of branch attachment to the trunk produces a large ellipsoidal wound
oriented along the length of the trunk. Willow branches usually shear far from their base, whilst oak
branches exhibit minimal shearing. In grape (
Vitis) mechanically failing young shoots usually snap off at
their base due to a shearing failure at the vascular/pith interface.

Taper of the beam, towards its tip, increases bending resistance and torsional (twisting) resistance at its
base. Hence, branches taper towards their tips. Branch collars are regional thickenings of wood at the
branch base with different wood grain. These collars serve to reduce stress concentrations and reduce
crack propagation because of their polylaminate construction.
Stems without Secondary Thickening

The stems of plants typically increase their resistance to bending under load by positioning the main
supporting tissues (collenchyma, vessels and fibres) toward the periphery. Many also have hollow stems,
which may also be square in cross-section, increasing the second moment of area for a given mass of
tissue. In grasses and horsetails, the hollow stems have transverse septa / diaphragms that increase
stiffness by 16-20% whilst increasing weight by only 2%. Annual stems typically have safety factors ~ 2.
In the Baltic onion there is a thin-walled inflation halfway up the stem. Later in the season when the stem is
dry and bearing the load of a flower-head now bearing developing plantlets, the stem undergoes Brazier
buckling at this inflation point and as the stem crimps and bends over it deposits the plantlets some distance
from the parent plant.
Epidermis

The epidermis is under longitudinal stress, whilst internal tissues are under compression. Excised epidermal
strips decrease in length and increase in girth. Thus, the epidermis experiences biaxial tension. Such an
arrangement of a hydrostatic core opposed by a tensed outer rind provides increased stiffness. Epidermal
cells often have wavy side-walls that increases the wall-wall contact area and binding strength.
Leaves

Leaves consist of an epidermis under tension, tightly bound to sclerenchyma, vascular fibres or spongy
mesophyll to form stress-skin panels. In the common iris the leaves are sandwich-laminate beams with fibre
composite faces and a low-density foam core. This gives a large second moment of area, low weight and
stiff construction with a mean stiffness of 0.465 Nmm-2. (Skis are constructed along equivalent principles).
Many leaves are folded/corrugated plates, enabling them to resist larger forces than simple plates. For
example, the corrugated leaves of the fan palm (
Corypha umbellata) are the world’s largest leaves.
Sclerenchyma cords may provide stiffness to leaves, whilst the mesophyll and epidermis serve as webbing.
In English rye grass (
Lolium perenne) sclerenchyma cords provide 90-95% of the longitudinal stiffness of
the leaves and account for only about 4% of the volume. (Vascular bundles also account for about 4% of
the volume). The optimum distance between neighbouring cords is 30x the cord width.

When banana (
Musca acuminata) leaves and traveller’s palm (Ravenala) leaves mechanically fail they
shear between parallel vascular bundles as wind pressure pulls the softer non-vascular tissues apart.
The swollen pulvinus at the petiole base functions in a similar fasion to tree branch collars and dissipate
stress and reduce crack propagation. V-shaped petioles resist bending, whilst there elliptical distal ends
resist torsion.

Leaves are particularly susceptible to resonance when under dynamic loading. This resonance can cause
damage if too severe, but also aids fruit and leaf fall. Resonant frequency increases as flexural stiffness (E
I)
increases. In
Populus and Birch the elliptical cross-sections of the petioles, in which the long axis is aligned
vertically, causes the leaves to tremble in slight breezes and increases gas exchange and leaf cooling.
Folding/lateral collapse of leaves, or of the leaflets of compound leaves, reduces drag and dynamic
loadings in high winds. The leaflets of compound leaves have differing natural frequencies that serve to
dampen the system and increase dissipation of kinetic energy, reducing the risk of resonant damage.
Old oak pollard
Plant Biomechanics
Hollow oak 2
Hollow oak
Bibliography - coming soon!

See also:

Alien Plants and Tree Equations

Biomechanics

Sequoia

Plant cell types and mechanical properties