Elasticity of Wood | Furniture Design Ideas

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

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

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

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) .

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

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.

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