Elasticity of Wood

As a natural raw material, wood is characterised as both anisotropic, as well as volatile in its properties in the function of space; therefore, it is referred to as a non-homogeneous material. Below are the physical theories of elasticity for an anisotropic body and understood as an equation binding the component of the stress tensor with the components of a strain tensor. The most general form of such equations in the linear theory of elasticity is the equation (Nowacki 1970): where

Oij stress tensor,

Aijkl tensor of elasticity and Eki strain tensor.

Botanical name

Trade name

Symptoms of illness

Common use

Inadvisable

use

Pericopsis elata v. Meeuwen

Afrormosia, kokrodua, asamela

Skin irritation

To use inside and outside of buildings, furniture and veneers

Afzelia africana Sm.

Afzelia, doussi, lengue, ара

Skin irritation, irritation of mucous membranes

To use inside and outside of buildings

Kitchens

Gossweilerodendron balsamifentm Harms

Agba, tola, tola branca

Irritation of mucous membranes

To use inside and outside of buildings, substitutes oak

Antiaris africana Engl.

Antiaris, bonkonko, oro, kirundu, andoum, upas

Irritation of mucous membranes of the nose, throat and skin

Furniture and veneers

Turraeanthus africanus Pellegr.

Avodire

Irritation of mucous membranes, nosebleeds

Luxurious finishing of building interiors, children’s furniture and veneers

Distemonanthus

benthamianus

Baillon

Ayan, movingui

Mild allergies

Doors, windows, office furniture

Kitchens

laundry

rooms

Castanospermum austraie A. Cunn.

Black bean, moreton bay chestnut

Contains isoflavones

Finishing of building interiors,

furniture

and veneers

Fagus sylvatica L.

Beech

Possibility of eczema caused by particulates and sawdust

Furniture, food containers, tools, parts of musical instruments

Betula papyracea Ait

Birch

Possibility of recurring skin irritation during sanding of wood

Furniture and veneers

Acacia melanoxylon R. Br.

Blackwood

Nosebleeds asthma, skin irritation

High-quality furniture, finishing of building interiors, musical instruments

Guibourtia tessmannii J. Leonard

Bubinga, kevazingo

Skin irritation

High-quality furniture, finishing of building interiors, floors

Table 4.2 Major species of wood showing toxic, irritant or sensitising properties (Hausen 1981)

4.5 Materials Used in Furniture Design 205

Botanical name

Trade name

Symptoms of illness

Common use

Inadvisable

use

Machaerium scleroxylon Tul.

Caviuna vermelha, pao ferro, moradillo, jacaranda pardo, santos palisander

Possible cases of skin irritation

High-quality furniture, finishing of building interiors

Cedrelci odorata L.

Cedar, cedro

Skin irritation

Executive furniture, interior design

Castanea sativa Mill.

Chestnut, Spanish chestnut

Possible cases of skin irritation

Furniture, kitchen furniture, veneers

Brya ebenus DC.

Cocus, Jamaican ebony.

Skin irritation

Furniture, veneers

Cordia miUenii Baker

Cordia, canalete, freijo

Skin irritation

Furniture, finishing of building interiors

Hymenaea courbaril L.

Courbaril, locust

Skin irritation

Furniture

Pseudotsuga menziesii (Mirb.) Franco

Douglas fir, Oregon pine, douglasie

Skin irritation and dermatitis, eczema

Furniture, veneers

Entandrophmgma angolense DC.

Gedu nohor, tiama, edinam, kalungi

Skin irritation

Furniture

Guarea thompsonii Sprague & Hutch.

Guarea, bosse, obobo.

Irritation of mucous membranes of the nose, throat and skin

Furniture, finishing of building interiors

Liquidambar styraciflua F.

Gum, American sweetgum, red gum, bilsted, amberbaum

Skin irritation

Furniture, interior design

TenninaHa ivorensis A. Chev.

Idigbo, framire, emeri, black afara

Skin irritation

Furniture, finishing of building interiors

Kitchen

furniture

(continued)

206 4 Introduction to Engineering Design of Furniture

Botanical name

Trade name

Symptoms of illness

Common use

Inadvisable

use

Chlorophorci

excelsa Bentham and

Hooker

Iroko, kambala, mwule, odum

Skin irritation

Substitutes teak

Larix decidua Miller

Larch

Skin irritation

Furniture

Shorea

Lauan, red

Skin irritation

Furniture, interior design

Terminalia superba Engler and Diels

Limba, afara, korina

Skin irritation, nosebleeds

Chairs, interior design

Diospyros celebica Bakh.

Makassar Ebony, Coromandel

Skin irritation and dermatitis, eczema

Luxurious furniture, elements of musical instruments

Khaya grandifoliola DC.

African mahogany khaya, krala

Skin irritation

Furniture, executive offices, surfaces of worktops of office furniture

Swietenia macrophyllci King

American mahogany, tabasco, caoba

Skin irritation

Furniture, executive offices, interior design

Tieghemella heckelii Pierre ex Chev.

Makore, baku

Irritation of mucous membranes of the nose and upper respiratory tract

Furniture, veneers, high-quality interior design, doors

Man son ia altissima A. Chev.

Mansonia, bete

Skin irritation, cough, nosebleeds, headaches

Telecommunications engineering, furnishings of building interiors

Prosopis juliflora DC.

Mesquite

Skin irritation

To use inside and outside of buildings, furniture and interior design

Pterocarpus angolensis DC.

Muninga, kejaat

May cause allergies

Furniture, veneers, high-quality interior design

Triplochiton scleroxylon K. Schum.

Obeche, samba, wawa, abachi

Asthma, skin irritation and rash

Veneers, interior design

4.5 Materials Used in Furniture Design 207

Botanical name

Trade name

Symptoms of illness

Common use

Inadvisable

use

Nerium oleander L.

Oleander, laurier rose

Poisonings have toxic properties

Haberdashery

Kitchen, contact with food

Olea europaea L.

Olive wood

Severe skin irritation and paralysis

Interior design, jewelery, for turning

Direct

contact with body

Aspidospenna peroba Fr. All

Peroba rosa

Skin irritation. Particulates and sawdust cause irritation of mucous membranes of the nose, irritation of larynx and eyes, weakness, drowsiness, sweating, fainting

Outdoor furniture, hand tools

Pinus radiata D. Don

Radiata pine, Monterey pine

Skin irritation caused by the presence of resin and turpentine

Furniture

Pinus silvestris F.

Pine

Skin irritation

Furniture, stairs and many more

Gonystylus ban can us Baillon

Ramin, malawis

Skin irritation

Furniture and interior design

Dalbergia nigra All

Brazilian rosewood, Jacaranda, Rio palisandre

Eczema of hands and face, skin irritation

Furniture, interior design, veneers, musical instruments

Dalbergia latifolia Roxb.

East Indian rosewood, Indian palisandre

Eczema, skin irritation

High-quality furniture, interior design, elements of musical instruments

Entandrophragma cylindricum Sprague

Sapelli, sapele, sapeli

Skin irritation

Furniture, interior design, veneers

(continued)

208 4 Introduction to Engineering Design of Furniture

By entering the engineering markings of components of the stress and strain

tensor – @11 @x, @22 @y, @33 @z, T12 Txy, T23 Tyz, T31 Tzx, £11 £x, £22 £y,

£33 = «z, Y12 = Yxy, Y23 = Yyz and 731 = Yzx, and entering the markings: An = Ann, …, A16 = A1131, …, A21 = A2211, …, A26 = A2231, …, we obtain the following form of generalised Hooke’s law (Litewka 1997):

@x

‘A11

A12

A13

A14

A15

A16

_£x

ry

A21

A22

A23

A24

A25

A26

ey

@z

A31

A32

A33

A34

A35

A36

ez

sxy

A41

A42

A43

A44

A45

A46

Txy

syz

A51

A52

A53

A54

A55

A56

cyz

szx _

_A61

A62

A63

A64

A65

A66_

Jzx_

If the symmetry of stress and strain tensors is considered, the number of com­ponents is 36. But when we take into account the differentiating alternation of free energy function in relation to the tensor’s components:

dstjdski dSkidstj ’

where

V elastic energy,

then the number of the tensor A components will decrease to 21. In specific anisotropic cases, like for example an orthotropic body, tensor A has 9 components, and for a transversally isotropic body—5 components. By expressing the strains of an anisotropic body in the general form, the following equation can be written as follows:

eij aijkl ‘ rkl;

where

a compliance tensor and

aijki components of the compliance tensor determined by the measurement of the strains of planes of a three dimensional body, taking into account normal and shear strains (Fig. 4.9).

At the same time, the following dependencies apply: for the direction of X-axis

_ ey ez ex ex

mxy — — ; V* — — ; Pzr. z — “ ; fizy, x — “ ;

ex ex czx Izy

for the direction of Y-axis

Єх

Vyx = -; ЄУ

my z

_ £x ; ;

ey

lzy;y

;

У zy

ихУ;У

;

Уху

^х^у

_Єу

Уzx’

(4.21)

for the direction of Z-axis

Єх

mzx = ;

ez

mzy ‘

_ey;

;

ez

uzx, z =

ez ; Czx’

l zy ; z

ez ; У zy

иху, z

ez

Уху

(4.22)

in the XY plane

Txy ;

1Ууху = ~T~; ey

lxz

Ух; )

Єх

uz, xy

Уху ; ;

ez

игх, ху

Уху

Угх’

> игу, ху

Уху

yzy

(4.23)

in the YZ plane

_ czy ;

Uy, zy = “ ; ey

l x; zy

_ Угу ;

;

Єх

l z; zy

_ yzy ;

;

ez

иху, гу

_ yzy ;

УЛу

> u^,zy

_ Угу

Угх’

(4.24)

in the XZ plane

u = U ;

y, zx

ey

l x; zx

_Zzx; ;

Єх

l z; zx

_ух ; ;

Єх

иху, zx

= Угх, Уху

’ uzy, zx

= ь

Угу

(4.25)

Thus, it is easy to write, e. g. an equation of normal strains in the direction of the X-axis:

ex — E vyx£y vzxez + ^zyyX ■ czy ” 1ZX, X ■ yzx ” MryjX ■ Уху-

By substituting the now well-known dependencies e = a/E and у = т/G, we obtain, respectively:

e =-X – v^- v ^ + и ^ + и —+ и – Xh С42П

ex E vyx E vzx E Hzy, x G ‘ игх, х g г их^,х g ’ ( )

Ex Ey Ez Gyz Gxz Gxy

or

ex E (rx vxyry vxzrz Г 1x, yzsyz Г 1x, xzsxz Г Ax^xy) –

Ex

For shear strains, e. g. ^ the equation of the sum of partial strains will have the form:

xy

yxy G ‘ 1y? x^ ■ ey ‘ Mx, xy ■ ex “г ■ ez "T” 1^,xy ■ yxz ""I” 1zy, xy ■ Tyz’

Gxy

hence finally

yxy — G i1xy, xrx + Mxyjry г Mxy^z г Mxyz^xz г 1xy, zyTyz г Txy) .

Gx

In the above equations, Ex, Ey and Ez are the linear elasticity modules at stretching, Gxy, Gxz and Gyz are shear elasticity modules in planes that are parallel to the lines of direction coordinates x, y, z, vxy, vyx, and vzx, vxz, vyz, vzy are Poisson’s

ratios characterising elongation in the direction of the first axis and shortening in the direction of the second axis of the plane. Coefficients, Mxzyz… ,Mxz>xy, called Chentsov coefficients (Ashkenazi 1958; Lekhnickij 1977), characterise shear strains in planes that are parallel to the coordinate system, caused by tangential stresses acting in the second planes parallel to the coordinate system. Coefficients, wyz>x… ,Mxy>z, according to Rabinowicz (1946) called the coefficients of mutual impact of first degree, they express elongation in the direction of the axis of the coordinate system, caused by tangential stresses acting in planes parallel to the coordinate system. Coefficients, Mxyz… ,Mz>xy, express shear strains in planes parallel to the global coordinate system, caused by normal stresses acting in the direction of the axis of the system. They can be called coefficients of mutual impact of the second degree (Ashkenazi 1958).

The equations above correspond only to the given Cartesian system of coordi­nates. Changing this system will automatically change the values of the coefficients,

although their number remains constant. However, if at any point of the anisotropic elastic body, three mutually perpendicular planes of its internal structure can be led, such material can be called orthotropic. Wood, as an orthotropic body, in a spatial state of stresses is subject to normal strains (Fig. 4.10) and changes in shape (Fig. 4.11).

By summing up the value of partial strains in particular anatomical directions, we obtain expressions for total strains in the form:

where

aL, aR, aT vector of normal stresses, respectively, in the direction:

longitudinal, radial and tangential;

El, Er, Et linear elasticity modules of wood, respectively, in the

direction: longitudinal, radial and tangential; and vLR, vLT, vRT, vTR, Poisson’s ratios, respectively, in anatomical directions: longi – vRL, vTL tudinal-radial, longitudinal-tangential, radial-tangential, tan­

gential-radial, radial-longitudinal and tangential-longitudinal.

And expressions for shear strains are given as follows:

The generalised Hooke’s law in matrix convention has the form:

ri — Aijej; (4-34)

while for the discussed case, we will obtain:

_

ClEl

CtlEl

CrE,.

0

0

0

ffL

Ct (vrt+v, t vrl)

Ct (vrt+vltvrl)

Ct

Г7

vlt (1—vlrvrl )Et Ct

(1—vlrvrl )Et Ct

(vrt+v, t vrl)Et Ct

0

0

0

ffR

CrlEr

(1—vlrvrl)(1—vlt vtl)Er

(1—vlt vtl)Er

0

0

0

SLT

Ct (vrt+vlt vrl)

Ct (vrt+vltvrl)

Ct

0

0

0

glt

0

0

SLR

0

0

0

0

Glr

0

STR J

0

0

0

0

0

G7R

e,

eT

X £R ;

Jlt

Ilk

Ttr

(4.35)

where:

Ct (1 – vLTvTL)(1 – VlrVrl) – (vTR + VlrVlt)(Vrt + TltTrlX

CL CT(vRT + vLTvRL) + vLTvTL(1 – vLTRvRL)(vRT + vLTvRL) + vRL(vL7(1 – vLTvRL)

(1 – vLTvTL) – ^LT^Th CR vTL(vRT + vltVrL) + vRL(1 – vLTvTLX

CTL vTL(1 – vlrVrl)(Vrt + VltVrL) + vRL((1 – vLRvRL)(1 – vLTvTL) – CtX

CRL FltC1 – vLTvRL)(1 – vLTvTL) – vLTCT

Table 4.3 shows a list of elastic properties of selected wood species, commonly used in the furniture industry.