T h e v i s c o e l a s t i c i t y o f W o o d at v a r y i n g M o i s t u r e C o n t e n t

7/27/2019 T h e v i s c o e l a s t i c i t y o f W o o d at v a r y i n g M o i s t u r e C o n t e n t

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Wood Science and Technology Vol. 9 (1975) p. 189-2059 by Springer-Verlag 1975

T h e V i s c o e l a s t i c i t y o f W o o d a t V a r y i n g M o i s t u r e C o n t e n tBy Alpo Ranta-Maunus*Structural Mechanics Laboratory, Division of Building Technology and Community Development,Technical Research Centre of Finland, Espoo, Finland

Abstract. Wood is regarded as a viscoelastic material. Creep deformations that arise from vari-ations in the moisture content are described by a theory of hydroviscoelasticity developed bythe author. Two different types of behaviour have been appazent: one, arising from a continu-ously increasing strain with periodic variation in the moisture content, and another with nocumulative effect. The theory has been applied to previously published experimental resultsconcerned with beech, pine, hoop pine, klinki pine, along with birch and spruce plywood.Birch and spruce plywood have been used for experiments concerned with periodically-cyclingbending moment and moisture content. The results obtained have been compared with thetheory presented. Glue-laminated beams have been subjected to long-term out-door loadingextending for five years. A brief discussion is given of the results obtained.

I n t r o d u c t i o nWhen climatic conditions remain constant, wood behaves like a viscoelastic material.The stress level determines the que stio n of linearity. As a role, the line arity is as-sumed under working stresses. The constitutive equati on may then be writte n in theform

teij (t) = of Jijkl (t -- r) d Old (r ) , (1)where Jijkl is creep comp lia nce tenso r,

oij stress tenso r,% strain tensor, andt, r time-variables.

As has been mentioned in a number of papers, changes in moisture content andtempera ture induce an additional creep deformatio n. The creep of pine in thelongit udina l direction has been st udied by Bethe [1969], Eriksson and Nor en [ 1965],and Raczkowski [ 1969]. A com mo n observation is that the first change in moistur econtent results in an increasing step in creep strain for both drying and wettingchanges. Subse quentl y, after some variations of moist ure cont ent , a drying change

* The author is indebted to Professor Siimes and Mr. Saarelainen for the planning and carryingout of glue-laminated beam tests and for being kind enough to make the results available to theauthor for publication. The author acknowledges the valuable assistance of Mr. Kortesmaa inconnection with recovery experiments, and computation of the results.


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1 9 0 A. Ranta-M aunus Viscoelasticity of woo d at varying m. c.

i n d u c e s a n in c r e a s i n g s t e p i n c r e e p s t r a i n , a n d a w e t t i n g c h a n g e d i m i n i s h e s t h e c r e e ps t ra i n . T h e t o t a l e f f e c t o f c o n s e c u t i v e v a r i a t i o n s i n c r e a s e s t h e c r e e p s t ra i n . A r a u c a r i ah a s b e e n s t u d i e d b y A r m s t r o n g a n d C h r i s te n s e n [ 1 9 6 1 ]. T h e q u a l i t a ti v e b e h a v i o u r inb e n d i n g i s s i m i l ar t o t h a t d e s c r i b e d a b o v e f o r p i n e . R e c o v e r y is e s s e n t i a ll y d e p e n d e n tu p o n t h e m o i s t u r e c o n d i t i o n s a f t e r th e u n l o a d i n g . A d d i t i o n a l c o n s i d e r a t i o n i s g i ve nt o t h e o b s e r v a t i o n s o f s o m e a u t h o r s , t o t h e e f f e c t t h a t t h e t h e o l o g ic a l d e f o r m a t i o n si n d u c e d b y c h a n g e s in m o i s t u r e a n d t e m p e r a t u r e o c c u r s i m u l t a n e o u s l y w i t h t h e sec l i m a t i c c h a n g e s , a n d t h a t t h e r a t e o f c l i m a t i c c h a n g e s is n o t s i g n i fi c a n t. A c c o r d -i n g ly , m e n t i o n i s m a d e o f t h e b as i c c o n c e p t i o n s n e c e s s a r y t o e s t a b li s h t h e t h e o r y i nt h e f o l l o w i n g c h a p t e r .

T h e o r y o f h y d r o v i s c o e l a s t i e i t yS e v e r a l a u t h o r s h a v e o b s e r v e d t h a t t h e v a r i a t i o n s i n t e m p e r a t u r e , T , a n d m o i s t u r ec o n t e n t , u , e x e r t s o m e i n fl u e n c e u p o n t h e c re e p d e f o r m a t i o n . I n t h e t h e o r y o ft h e r m o v i s c o e l a s t i c i t y , a c c o u n t is t a k e n o f th e v a r i a ti o n i n t e m p e r a t u r e . I t i s n o wa s s u m e d t h a t t h e e f f e c t o f m o i s t u r e c h a n g e s c a n b e d e s c r ib e d b y a s im i l a r m a t h e -m a t i c a l f o r m u l a t i o n [ R a n t a - M a u n u s t 9 7 3 ] . C o n s i d e ri n g a n o n - a g e in g m a t e r i a l , w h i c hi s i n i ti a l ly u n l o a d e d u n t i l t h e t i m e t = 0 , t h e c o n s t i t u t i v e e q u a t i o n i s e x p a n d e d i n aF r 6 c h e t s e r i e s

te li ( t ) = i J i ~ ( t - r ) d O k l ( 7 - )t j .o .l o ( t - - r ) d u ( r )+ f t j0t+ f jO O l ( t - r ) d T 0 - )

0t t+f f0 0t t+f f( 1 0t t+f f0 0t t+f fO0t t+f fO0t t+f f0 0t t+f fO 0

200 d O ld ( r l ) d O mn (72 )Jijk/rm~ (t - 7-1, t -- 72)J ~i j . - - r l , t -- r 2 ) d U ( r l ) d u ( r 2 )

002J ij ( t - r l , t - r 2 ) d T ( r l ) d T ( r 2 )11oJ i jk l ( t - r l , t - r 2 ) d o l a ( r a ) d u ( r 2 )

j~ - 7"1, t - 72) d u ( r 1 ) d T (7 -2 )101J t fk l ( t - - r 1 , t - - 7 "2 ) d o l d ( 7" 1 ) d T ( r z )

t j12o [rf i ik l ~ - 7"1 , t - 7"2 , t - r 3) d o k l (7-1) d u (7-2) d u ( r 3 )0

(2 )


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A. Ranta-Maunus Viscoelasticity of wood at varying m.c. 191t t t O ~ - r l , t - r2 , t - % ) d OM (~ '1) d T( ' r2 ) d T( ' r3 )f f f ~o o o

+ . . .T he ke rne l s j i oo r ep resen t t he non l in ea r v i scoe las ti c behav iour und er equ i l ib r iumc o n d i t i o n s o f t e m p e r a t u r e a n d m o i s t u r e c o n t e n t . T h e c a s e o f l i n e a r v i s c o e l a s ti c i t y ( 1 )i s i n v o l ve d b y t h e f ir s t t e r m . T h e e x p a n s i o n o f a n u n l o a d e d b o d y d u e t o t e m p e r -a t u r e a n d m o i s t u r e c o n t e n t i s e x p r e s s e d b y k e r n e l s j o i o j o 0 i and j0 i j . T he poss ib i l -i t y t h a t c h a n g e s i n t h e m o i s t u r e a n d t e m p e r a t u r e o f a l o a d e d b o d y i n d u c e a n a d d i -t i o na I p a r t o f c r e ep d e f o r m a t i o n m a y b e i n c o r p o r a t e d i n t h e i n te g r al s w i t h k e r n e l sj i j0 , j i o j a n d ji i k . T h e t e r m s w i t h j ij k d i s a p p e a r u n d e r b o t h i s o t h e r m a l a n d c o n s t a n tm o i s t u r e c o n d i t i o n s .N ow , w hen no in t e res t is a t t ach ed to the m oi s tu re expan s ion , t he i so the rm al cons t i -tu t ive r e l a t ion i s w r i t t en in the fo rm

te i j ( t ) f 100= Jiiva (t - r ) d Ok, (r )0t t l l l O ( t - - r l , t - r 2 ) d a kl ( r l ) d u ( r 2 )f f qmo ot t t j 1 2 o / ' f+ f f f tjk ! v~ - - r l , t - - r 2 , t - -r3) d a k t ( r l ) d u ( r 2 ) d u ( r 3 ) .

0 0 o

(3)

T h e Eq. (3) implies a l i nea r s t r e s s -dependence . T o de r ive a conv en ie n t " m oi s t u rem e m o r y " , w e w r i t e , i n a c c o r d a n c e w i t h R a n t a - M a u n u s [ t 9 7 3 ]

lOOJ o k l ( t - - r ) =j l l O / f 3ijkl t.v, t - "/'2) =

~} 1110 (t - - r l , t -- r2 ) =T 1 ~ kl

J i jk t ( t - - r ) ,O ,

~1110 (t -- r l, t -- 72)ijkl{ K i j k l ( t - 7 ) fo r T l=T2=T

r l # r 212 0Ji jk l ( t - - T1 , t - r2, t - r3 ) = Li jk l ( t - r2) H (r 1 - r2) H (r 3 - r2 ) ,

1 f o r t > 0w here H ( t ) i s H eav i s ide s tep fu nc t io n = 0 t ~< 0 'B y the use o f t hese a s sum pt ions and no ta t io ns , t h e r e l a t ion (3 ) is r edu ced to

teli (t) = f J i jk l ( t - - r ) d Okl (T)0t+ f ( m i jk l ( t - r) Okl (r ) + t i j k l ( t - r ) [a u ( t ) - 17kl (T)] Xo

x [u ( t ) - u ( r ) ] } d u ( r ) . (4)


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1 9 2 A . R a n t a - M a u n u s V i s c o e la s t ic i ty o f w o o d a t v a r y i n g m . c .

I t i s a p p a r e n t t h a t K i i k l r e l a t e s t o t h e c r e e p t h a t a r i s e s f r o m a c o n s t a n t s t r e s s d i s t r i -b u t i o n o l d , a n d t h a t L i jk l i n d i c a t e s a r e c o v e r y p h e n o m e n o n .W h e n a n e x p e r i m e n t a l s t u d y is m a d e f o r d e t e r m i n a t i o n o f th e k e r n e l f u n c t i o n s ,c h o i c e m a y b e m a d e o f t h e o n l y s t r e s s - c o m p o n e n t , s a y ak l, w h i c h i s a c t in g , a n dd e n o t e d b y o . M e a s u r e d s t r a i n, s a y eij , i s d e n o t e d b y e . A n a l o g i c a l l y , s y m b o l s J ,K , L m a y b e e m p l o y e d .A c c o r d i n g l y ,

te ( t ) = f J ( t - T) d o (7)0t+ f { K ( t - r ) o ( ' O + L ( t - z ) [ o ( t ) - o ( r ) ] [ u ( t ) - u ( r ) ] } d u ( ~ ) . (5 )0

T h e d e r i v a t i o n o f k e r n e l f u n c t i o n s is d i v i d ed i n t o t w o p a r ts , a c c o r d i n g t o w h i c h t h eb e h a v i o u r i s t e r m e d b i r c h - l i k e o r s p r u c e - l i k e .Birch-like behaviourA l l o f t h e p a p e r s m e n t i o n e d i n t h e I n t r o d u c t i o n s t a t e t h a t a si ng le c h a n g e i n m o i s t u r ec o n t e n t b r i n g s a b o u t a c h a n g e i n s t r a in t h a t d o e s n o t d e p e n d s i g n i f i c a n t ly u p o n t h ei n s t a n t a t w h i c h t h e m o i s t u r e v a r i a t i o n a c ts . I n t h e s e te s t s, a a n d e a re t h e a x i als tr e ss a n d s t r a in in t h e s a m e d i r e c ti o n . M o r e o v e r , o re m a i n s u n c h a n g e d . T h u s i n ( 5 )i t i s w r i t t e n t h a t K ( t - T ) = K i s a c o n s t a n t , w h ic h a s s u m e s d i f f e r e n t v a lu e s , d e p e n d -in g u p o n w h e th e r d u ( ~- ) i s p o s i t i v e o r n e g a t i v e . L ( t ~-) s h o w s t h e a m o u n t o f Kw h i c h r e c o v e r s u n t i l t i m e t . O n e n e c e s s a r y c o n d i t i o n f o r t h e r e c o v e r y is t h a t t h e r eex is t s a m o m e n t ~ such th a t ~- ~< ~ ~< t a nd I o ( ~ ) l < I o ( r ) l , ( [ a l m e a n s t h e a b s o l u t ev a lu e o f a) . I t h a s b e e n o b s e r v e d b y A r m s t r o n g a n d C h r is t en s e n [ 1 9 6 1 ] a n d R a n t a -M a u n u s [ 1 9 7 3 ] t h a t t h e r e c o v e r y o f a n u n l o a d e d s p e c im e n o c c u r s i n p a r t i cu l a r w i t hi n c re a s e i n t h e m o i s t u r e c o n t e n t . T h u s , i t is a l so d e m a n d e d f o r ~ t h a t u ( ~) > u ( r ) .U n d e r m a n y u s u a l c o n d i t i o n s , t h e s e t w o r e q u i r e m e n t s a r e v a li d w h e n ~ is s o c h o s e nt h a tr ~< ~ ~< t a n d [ I o ( ~ ) l - I o ( ~ - ) l l u ( ~ ) ( 6 )r e a c h e s it s m i n i m u m v a l ue . W h e n e x a m i n a t i o n is m a d e o f t h e e f f e c t o f a s in g lec h a n g e o f m o i s t u r e c o n t e n t , d u ( r ) , a t t i m e t , t h e v a l u e t is t h u s r e p l a c e d b y ~ , d e -f i n e d i n ( 6 ) . I f th e i n c r e a s e i n m o i s t u r e c o n t e n t i s s u f f i c i e n t ly la r g e , t h e e n t i r ee f f e c t o f t h e m o i s t u r e c h a n g e w i l l d i s a p p ea r . T h i s cr i ti c a l m o i s t u r e i n t e r v a l is d e -n o t e d i n re g a r d t o t h e u n l o a d e d s t a te ( o ( ~) = 0 ) b y Uc- T h u s a n i m p u l s e f u n c t i o nf o r a s in gl e c h a n g e o f m o i s t u r e c o n t e n t is f o u n d a s f o l lo w s

u ( ~ ) - u ( r )

w here u c (~ ) i s the la rge r o f tw o va lue s u c o r u (~ ) - u (T ).


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A. Ranta-Maunus Viscoelast ic i ty of wo od at varying m.c . 193

W h e n m a n y s e q u e n t c h a n g e s o f m o i s t u r e c o n t e n t o c c u r d u r in g a p e r i o d o f u n c h a n g e dl o a d i n g , it i s f o u n d r e a s o n a b l e t o r e p l a c e th e c o m p a r i s o n m o i s t u r e c o n t e n t u ( r ) b yt h e v a l u e u (r? ). T h e t i m e r~ i s n o w s o c h o s e n t h a tr / / > r a n d I o ( r / ) l < I o ( r ) l ( 7 )t h e fi r s t t i m e . I n o t h e r w o r d s , t h e d i m i n u t i o n i n t h e l o a d b e g i n s a t m o m e n t r/ . I nt h e c a s e o f a s m a l l v a r i a t i o n in m o i s t u r e c o n t e n t d u r i n g t h e l o a d i n g t i m e , t h e e m p l o y -m e n t o f r/ d o e s n o t i n d u c e a n y p r a c t i c a l c h a n g e i n t h e n u m e r i c a l re s u l ts . W h e n t h ee f f e c t s o f s e v e ra l c h a n g e s i n m o i s t u r e c o n t e n t a r e s u p e r i m p o s e d , w e w r i t e , b y t h ea p p l i c a t i o n o f (5 ) t o a b i r c h -l i k e w o o d

te ( t ) : f J ( t - r ) d o ( r )0 ' [ ]f K o ( r ) + ( o ( ~ ) - o ( r ) ) u ( ~ ) - u ( r / ) d u ( r ) ( 8)

o

F i g . 1 i l l u s t r a t e s th e q u a n t i t i e s u s e d i n (8 ) , i n t h e c a s e o f t h e s t re s s h i s t o r yI a o fo r 0 < t ~< t*( t )I 0 o t h e r w i s e

T h e d i f f e r e n c e s d u ( r ) a r e t h e d i f f e r e n c e s i n th e e x t r e m e v a l u e s o f t h e m o i s t u r e c o n -t e n t : u i - u i _ 1. T h u s , a t t h e m o m e n t o f u n l o a d i n g , t * , t h e l a s t i n t e g r a l ta k e s t h e

U - - m -l ] I - - _ - -

0 t * ~ = q 2 = q 3 ~z tz t~:~3Fig. 1. Illus tratio n of quantit ies used in the theory of birch-like behaviour. To obtai n a strataof e ( t i ), there are neede d the maxim um value o f the m oisture co ntent u (~ i) , and the value a tthe unloading ins tant u ( rt i) , together w i th a l l the ext rem e values dur ing loading u 0 . . . u 4

4v a l u e o f th e s u m o o 2~ K ( u i - u i _ ~ ) . A f t e r t h e u n l o a d i n g , r e c o v e r y o c c u r s in r e g a r di=lt o t h e e f f e c t o f m o i s t u r e v a r i a t i o n w h e n t h e m o i s t u r e c o n t e n t a s s u m e s a v a l u e e x c e e d -i n g t h a t a t t h e u n l o a d i n g i n s t a n t t * = r~. T h r e e v a r i o u s m o m e n t s , t i, a re c o n s i d e r e dw i t h c o r r e s p o n d i n g v a lu e s o f ~ i. C o n s e q u e n t l y , t h e d o t t e d c u r v e r e p r e s e n t s t h e


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194 A . R a n t a - M a u n u s V i s c o e l a s t i c i t y o f w o o d a t v a r y i n g m . c .

m o n o t o n i c b e h a v i o u r o f u ( ~) w h e n ~ is c o n s i d e r e d a s a f u n c t i o n o f t . I n t h is c a se( F ig . 1 ) , E q . ( 8 ) i s r e d u c e d t o

[ ( ]( t ) = a o J ( t ) - J ( t - t * ) + 1 - u ( ~ ) - u ( t * ) ~ N K ( u i _ u i - 1 ) ,U c ( ~ ) / i = 1w h e n t > t * .Spruce-like behaviourC r e e p t e s ts o f s p r u c e v e n e e r h a v e s h o w n t h a t t h e d i f f e r e n c e b e t w e e n t h e i n it ia lm o i s t u r e c o n t e n t , U o , a n d t h e m a x i m u m m o i s t u r e c o n t e n t d u r i n g th e l o a d in g ti m e ,U m a x, is o f t h e g r e a t e s t s i g n i f i c a n c e ( F ig . 3 ) [ Ra n t a - M a u n u s , 1 9 7 3 ] . I n r e g a r d t ot h is , t h e t i m e O i s s o d e t e r m i n e d t h a t [ o ( O ) l u ( 0 ) a t t a i n s i t s m a x i m u m v a l ue . O nt h e a s s u m p t i o n t h a t t h e b e h a v i o u r o f t h e m e m o r y is si m il ar t o t h a t o f t h e b i r ch - li k em a t e r i a l , ( 5 ) i s w r i t t e n i n t h e f o r m

te ( t ) = f J ( t - r ) d o ( r )00 u ( ~ ) - u ( 7 ) ]+ f Ko [o f f )+ (o (~ ) - o ( r ) ) U c ( ~ j du(7) (9)0

H e r e , K o is c o n s t a n t , a n d ~ a n d ~ a r e d e f in e d a s i n E q . ( 8 ) .E q . ( 9 ) i s a p p l i e d t o t h e l o a d in g c a s e r e p r e s e n t e d i n F ig . 1 . E q . ( 9 ) n o w t a k e s t h ef o r me ( t ) = o o (J ( t ) - J ( t - t* )) + K o (Umax - u o) a o ( 1 U ( ~ u (t*)u c ( ~ ) ! 'w h e n t > t * . M o r e o v e r , U m ax h e r e t a k e s t h e v a lu e o f u 3 .Previous worksT h e p r i n c i p a l d i r e c t i o n s o f s o l id w o o d a r e s o c h o s e n t h a t s u b s c r i p t s1 i n d i c a t e s th e l o n g i t u d i n a l d i r e c t i o n ,2 t h e t a n g e n t i a l d i r e c t io n , a n d3 t h e r a d i a l d i r e c t i o n .W h e n p l y w o o d i s u n d e r d i s c u ss i o n , a c c o r d i n g l y , w e d e f i n e i n t h e p l a n e o f a v e n e e rt h e d i r e c t i o n1 a s p a r a l l e l t o t h e g r a in ,2 a s p e r p e n d i c u l a r t o t h e g r a in ,a n d t h e d i r e c t i o n3 a s t h e n o r m a l d i r e c t i o n t o t h e v e n e e r .I n t h e d e s ig n o f w o o d e n c o n s t r u c t i o n s , t h e b e h a v i o u r o f w o o d i n t h e l o n g it u d i n ald i r e c t i o n is o f t h e m o s t s i g n i f ic a n c e . C o n s e q u e n t l y , t h e n a t u r e o f J ~k 1 f k e r n e l s is o fp r a c t i c a l i n t e re s t . T h e d e t e r m i n a t i o n o f k e r n e l s r ij k a n d l i jk h o w e v e r , h a s a~ 2 2 2 2 ~ 3 3 3 3 '


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A. Ranta-Maunus Viscoelasticity of wood at varying m.c. 195s o l e ly p h y s i c a l m e a n i n g . T h e c o u p l i n g c o m p l i a n c e s r i jkUmm , a s w e l l a s t h e " s h e a r in gk e r n e l s " l ij k m a y a l s o b e o f p r a c t i c a l i m p o r t a n c e i n t h e d i s cu s s i o n o f c e r t a i n p l a t eOlmlm,a n d d i s k c o n s t r u c t i o n s .T h e f o l l o w i n g c o n t a i n s a s u m m a r y o f th e t e s t r es u l ts o b t a i n e d w i t h e i g h t v a r i o u sq u a l it i es o f w o o d . F o r b i r c h - li k e m a t e r i a ls , t h e n o t a t i o n s u s e d a r e as f o l l o w s

[ a - f o r d u < 0K _ ~ a + > 0 ( 1 0 )

J ( O ) [ a ++ f o r t h e fi r s t c h a n g e d u > 0 .T h e s y m b o l s a - , a a n d a 247 r e t h u s t h e m a te r i a l c o n s t a n t s c h o s e n .A n a lo g i c a l l y , f o r a s p r u c e - l i k e m a te r i a l t h e r e i s d e n o t e d

K o - b . ( 1 1 )J ( O )A b r i e f c o m p i l a t i o n o f e x p e r i m e n t a l r e s u lt s is l i st e d i n T a b l e 1 .Araucar&T h e l o n g it u d i n al b e h a v i o u r o f h o o p p i n e i n b e n d in g h a s b e e n s t u d ie d b y A r m s t r o n ga n d K i n g s t o n [ 1 9 6 2 ] , a n d t h a t o f k li n k i p i n e b y A r m s t r o n g a n d C h r i s t e n s e n [ 1 9 6 1 ] .T h e r e s u lt s a re gi v en i n T a b l e 1 . V e r y l a rg e d e f o r m a t i o n s w e r e i n d u c e d b y e x t r e m ec h a n g e s i n m o i s t u r e . I t s o o n b e c a m e a p p a r e n t t h a t t h e w e t t i n g o f a t e s t s p e c i m e n( a f t e r u n l o a d i n g ) i n d u c e d a n a l m o s t c o m p l e t e r e c o v e r y . T h e n u m e r i c a l v a l u es a r ec a l c u l a t e d b y th e a s s u m p t i o n t h a t m o i s t u r e c h a n g e s e x e r t e d a n e f f e c t u p o n t h e i n t er -v a l u = 0 . . . 0 . 4 , w i t h t h e v a lu e u c < 0 . 4 in d i c a t i n g o n l y t h a t c o m p l e t e r e c o v e r yo c c u r r e d w h e n u = 0 d u r i n g th e u n l o a d i n g , a n d a f t e r t h e w e t t i n g a t t a i n e d a v a l u e o fu = 0 .4 .BeechT h e l o n g i tu d i n a l b e h a v i o u r o f sm a ll b e e c h s p e c i m e n s ha s b e e n s t u d i e d b y H e a r m o na n d P a t o n [ 1 9 6 4 ] . O n v a r ia t io n i n t h e m o i s t u r e c o n t e n t f r o m 0 t o 0 . 3 ( r . h . = 9 0 % ) ,a q u i c k r u p t u r e o c c u r r e d w h e n t h e s tr e ss l e ve l e x c e e d e d 0 . 2 . F o r v a l u e 0 . 1 2 5 o f th es t re s s l ev e l, i t c a n b e e s t i m a t e d t h a t a + - a - = 1 .4 , o n t h e a s s u m p t i o n t h a t t h e w h o l em o i s t u r e i n t e r v a l e x e r t s a n e f f e c t .T a n g e n t i a l b e h a v i o u r i n t e n s i o n h a s b e e n s t u d i e d b y S c h n i e w i n d [ 1 9 6 6 ] . D r y i n gb e tw e e n u = 0 .2 6 a n d u = 0 .1 0 i n d u c e s a v e r y la r g e s t ra in w i th a - = 5 5 a t a s t r e s sl ev e l o f 0 . 3 0 a n d a t e m p e r a t u r e o f 2 0 ~ C . - T h e e f f e c t o f a s in gl e c h a n g e o f t e m p e r -a tu r e T w a s a l s o i n v e s t i g a t e d . A p p a r e n t l y i t i s p o s s ib l e t o w r i t e k e r n e l 1 1~ i n c lu d -e2222,i n g t h e f a c t o r a T, i n a w a y si m i la r t o t h a t f o r k e rn e l t i t 0 b y v i r tu e o f d e f i n i t i o n s~ l l l l '( 4 ) a n d ( 1 0 ) . O n t h e a p p l i c a t i o n o f a s tr e ss l e ve l 0 . 3 u n d e r m o i s t c o n d i t i o n s( u > 0 . 3 ) , a v a l u e o f a ~ = 0 . 0 7 1 / K w a s d er i v e d w i t h a c h a n g e i n t e m p e r a t u r e f r o m2 0 ~ C to 6 0 ~ C .R e c o v e r y a t v a r i o u s t e m p e r a t u r e s h a s b e e n s t u d i e d b y L a w n i c z a k a n d R a c z k o w s k i[ 1 9 6 1 ] .


8/17

Te1Acmpaoosmee

meac

phd

yow

ngcm3Dmeooszoesmeanmm

O~

Ah

Tsme

L

n

Cmae

R

Hmo

b

lounbn

T=?

Po1

2

s=01

u=+3

a+a=14

Sewn

b

ta

aeo

T=2

u=01o02

a=5(nec

[1

5x2x2

s=03

u~3T=2+6

a=00K

RaMa

b

v

P=06

[1

7x5x9

bns=02

T=2

u=00o03

a=-74a=43a+16uc=00

2x7x2

rons

s=01

T=2~Cu=00o03

b=5uc=00

Amso

h

pn

lounbn

T=2

Kno

1x1

s=03

u=00++s

a=2a+=1a+=2

1 Amso

knpn

lounbn

T=9

C

e

1x1x

s~02

u=0+>s

a=2a+=1a+=25uc~04

[1 Bh1

pnp=04

lounbn

T=2

7

s=1

u0+02

a+a=

R

w

pnp=05

lounbn

T=9

[1

113

Smx=02

u=s+01u=01+s

a=2a+=5

Pkn

pnsw

ta

acmp

o

T=?

[1

2

s=01o04

u=0o03

a+=1

RaMa

suv

p=04

[1

1

bns=02

T=2

u=00o03

b=1ue=00

.> < g O o 0

3x7x

rons

s=02

T=2~u=00o03

b=7ue=01


9/17

A. Ranta-Maunus Viscoelasticity of woo d at varying m. c. 197

Bi rch veneerThe behaviour of b irch veneer has been s tudied by the author [1973] . Tests weremade in the grain d irect ion by bending 5-ply s tr ips of b i rch p lywoo d. The temper-ature was 20 ~ C, and the mo istur e con ten t varied with in a range of 0 .05 . . . 0 .30.There was observed lower boundary value , u = 0 . 08 . . . 0 .10 , so that var iat ions ofu be low this l imi t had no ef fect . After unloading, the recovery was found to occurwith increasing moi stur e conte nt. Num erical results are given in detai l in Table 1.This theory and the values observed are i l lustrated in Fig. 2.

Z0b133ULLW0

LU

--3IJAt' V

3

2

f

0 - - , i

I l l l l III t: Lo observationtheory ~ ~ -~4__ _ . creep at constanthu mi di ty ..... 4 s

o

. . . . I

~,, 4. ~ 'e .......

I -ID


10/17

19 8 A. Ranta-iVlaunus Viscoelasticity of wo od at varying m. c.HinokiT h e e f f e c t o f t e m p e r a t u r e c h a n g e s h a s b e e n i n v e s t i g at e d i n t h e lo n g i t u d i n a l d i r e c t i o n( b e n d i n g ) b y K i t a h a r a a n d Y u k a w a [ 1 9 6 4 ] , a n d in ra d ia l c o m p r e s s i o n b y A r i m a[ 1 9 7 2 ] . I n b o t h c a s es , a n i n c r ea s i n g c h a n g e i n t e m p e r a t u r e w a s o b s e r v e d t o i n fl u e n c ea n i m m e d i a t e , i n c r e as i n g c r e e p d e f o r m a t i o n . T h e d e c r e a s e i n t e m p e r a t u r e b r i n gsa b o u t a d e c r e a s in g s te p i n c r e e p s t ra i n . T h e t e s t s p e c i m e n s w e r e p l a c e d i n w a t e r .T h e c h a n g e s w i t h i n t h e r a n g e o f 2 0 . . . 7 5 ~ C i n d ic a t e a n a p p a r e n t ly n o n l i n e a rr e l a t io n s h i p b e t w e e n c r e e p s t r a i n a n d t e m p e r a t u r e c h a n g e d T . I t m a y t h u s b e a p -1"102 r103p r o p r i a t e t o a p p ly t h e o r y ( 2 ) t o k e r n e l s " ij kl a n d o ijk l i n d i s c u s s io n o f t h e v a r i a t i o n si n t e m p e r a t u r e .T a k e m u r a [ 1 9 6 6 ] h a s p r e s e n t e d a th e o r y f o r c o n s i d e r a t io n o f t h e e f f e c t o f a si ng lec h a n g e o f m o i s t u r e c o n t e n t .

P in eB e t h e [ 1 9 6 9 ] h a s s t u d i e d t h e e f f e c t o f m o i s t u r e v a r i a t io n s b e t w e e n 0 . 0 9 a n d 0 . 2 1u p o n t h e d e f l e c t i o n o f l o n g i t u d i n a l p i n e b e a m s , w h e n t h e s t r es s l ev e l i ss = 0 . 1 6 . . . 0 . 4 0 , a n d s t a t e d t h a t t h e c h a n g e i n d e f l e c t i o n a f t e r s ix m o i s t u r e p e r -i o d s is r e l a t e d t o 0 3 . T h e v a lu e f o r a + - a - d e n o t e d i n T a b l e 1 i s d e r i v e d b y t h ea s s u m p t i o n t h a t c r e e p b e h a v i o u r c a n b e r e g a r d e d a s l i n e a r a s f a r as th e l o w e s t s t r es sl e v e l i n v e s t i g a t e d , s = 0 .1 6 . T h i s a r ti c l e f u r t h e r i l l u s t r a t e s t h e r e l a t i o n s h ip b e tw e e nr e l a t i v e c r e e p a n d w o o d d e n s i t y .R a c z k o w s k i [ 1 9 6 9 ] h a s s t u d ie d t h e e f f e c t u p o n t h e d e f l e ct i o n o f p i n e b e a m s o fs ing l e w e t t i n g o r d r y in g c h a n g e s . T h e v a lu e s a - = - 2 a n d a 247 5 i n T a b l e 1 a r ed e ri v ed b y t h e a s s u m p t i o n o f t h e e f f e c ti v e ch a n g e o f m o i s t u r e c o n t e n t i n th e i n te r va lb e i n g u = 0 . 1 . . . 0 . 4 .T h e l o n g it u d i n a l t e n s i o n h a s b e e n s tu d i e d b y E r i k s so n a n d N o r 6 n [ 1 9 6 5 ] . P e r k i t n y[ 1 9 6 5 ] h a s g i v en c o n s i d e r a t i o n t o r a d ia l a n d t a n g e n ti a l c o m p r e s s i o n . A s h a d b e e ne x p e c t e d , a s in g le w e t t i n g c h a n g e o f m o i s t u r e e x e r c i s e d a n i n f l u e n c e d e n o t e d b ya + = 1 7 ; h o w e v e r , b y r e a s o n o f a d r y i n g m o i s t u r e c h a n g e , t h e d e f o r m a t i o n w a sf o u n d p r o p o r t i o n a l t o t h e t e r m 0 3 .

Spruce veneerT h e a u t h o r [ 1 9 7 3 ] h a s s t u d i e d t h e b e h a v i o u r o f s p r u c e v e n e e r. T e s t s w e r e m a d e i nt h e gr a in d i r e c ti o n b y t h e b e n d i n g o f 5 - p ly s tr ip s o f p l y w o o d . T h e t e m p e r a t u r e w a s2 0 ~ C , a n d t h e m o i s t u r e c o n t e n t v a r ie d w i t h i n t h e r a ng e 0 . 0 6 . . . 0 . 3 0 . I t w a s o b -s e rv e d a l so h e r e t h a t m o i s t u r e v a ri a ti o n s b e l o w u = 0 . 0 8 . . . 0 . 1 0 d i d n o t e x e r t a n ye f f e c t . F o r n u m e r i c a l v a lu es , t h e r e w e r e o b t a in e d b = 1 5 a n d u r = 0 .0 5 . F ig . 3i l lu s t ra t e s a c o m p a r i s o n b e t w e e n t h e o r y a n d t h e r e s u lt s o f te s ts .R o l l i n g s h e a r s tr a i n w a s s t u d i e d b y t h e b e n d i n g o f s h o r t 1 5 - p ly s tr i ps ( s p a n =lijk2 0 0 m m ) . R e s u l t s o f b = 7 0 a n d u c = 0 . 1 2 i n re g a r d t o ~2 32 3 w e r e o b t a i n e d f o rs = 0 .26 .


11/17

A. Ranta-Maunus Viscoelasticity of woo d at varying m.c . 199

Z0

Ld

LI IQW2~

t.u

Q0_.J

3, I I I 1%~o Iooo~

0I0

~ 0 . 3Z0

w 0.2o~CO

i 0.2 3 O'ult

I I^ iA a t ~ ,

I0 100 200 dayst imeFig. 3. Creep and recovery o f spruce veneer in the grain direction

R e c o v e r y t e st s w i t h a n o n - z e r o s t re s sThe kernel funct io ns in the the ory of hydrov iscoelas t ic i ty are based up on tes ts inwhich va r i a t i ons occur i n t he moi s tu re con t en t , bu t t he l oad does no t change morethan tw ice, as is i l lustrated in Fig. 1. How ever, the load of pract ica l wo od en con-s t ruc ti ons may va ry i n a num ber o f ways . In many cases , t he t ime-dependence o floading i s express ible by a cycl ic funct ion, in which a given dead load acts for thewhole of the t ime. A fur ther series of exper im ents was made wi th a view to check-ing the val idi ty of resul t s (8) and (9) in the case of a per iodical ly changing load.Test material and conditionsBoth b i r ch and spruce p lyw ood were t es t ed under cond i t i ons o f pure bend ing. The5-p ly b i r ch p lywood s t r i ps sub j ec t ed t o t e s t , and 50mm x 850mm in s i ze , wereloaded b y a cons t an t bend ing mo me nt which was 7 % of t he shor t- t ime u l t ima tes t rength whe n the full load was act ing. Dur ing a per iod o f 81 days after the f i r s t


12/17

2 0 0 A . R a n t a - M a u n u s V i s c o e l a s ti c i ty o f w o o d a t v a r y in g m . c .

Ldr~

dLUn-"

3 Ic a s e 1

c a s e 2

0o

~ 0

o0

0

o

ou

' r r I

0 . 5 9 c a s e 10 . 2 5 c a s e 2 ' j

i i l Ji 9 Li I I i

zz

wc~

0 . 2 5

0 , 2 0

0 . 1 5

0 , 1 0 - - 4 , 81

v A fk / "

8 8

k J " k ]9 5 1 0 2 1 0 9 1 15 1 23

t

1 3 0 1 3 7 . 1 44 1 51 1 5 8 1 6 5 d a y st ~ m eF i g . 4 . R e c o v e r y e x p e r i m e n t s i n t h e p u r e b e n d i n g o f b ir c h p l y w o o d . E a c h p o i n t r e p r es e n tst h e m e a n o f f iv e o b s er v a t io n s . B o t h l o a d a n d c l im a t e w e r e u n c h a n g e d d u r i n g 81 f ir s t d a y s w i t hs = 0 . 0 7 , u = 0 . 2 2 a n d T = 2 0 ~

l o a d in g , t h e m o i s t u r e c o n t e n t o f b i r ch p l y w o o d w a s f o u n d t o b e c o n s t a n t , 0 . 2 2w i t h f u l l l o a d . S u b s e q u e n t l y , b o t h l o a d a n d m o i s t u r e w e r e v a r ie d , w i t h a c y c l e o f1 4 d a y s a s i n d i c a t e d i n F i g . 4 . T w o t e s t se r ie s , e a c h c o m p r i s i n g 5 s p e c i m e n s , w e r et e s t e d . I n o n e s e ri es , t h e m i n i m u m l o a d w a s 5 0 % , a n d in t h e o th e r 2 5 % o f t h e f u lll o a d . S i m i l a r t e s ts w e r e m a d e f o r sp r u c e p l y w o o d , w i t h a f u l l l o a d o f s = 0 . 2 3 .T h e t e m p e r a t u r e w a s c o n s t a n t a t 2 0 ~ C .


13/17

A. Ranta-Maunus Viscoelasticity of wood at varying m.c . 201

Results with birch plywoodThe three moisture cycles between t = 116 and 158 days are closely similar. There-fore, the cumulative effect o f moisture variation during these three cycles is examined.In the following, two parallel values observed are given in regard to two loadingcases: the recovery period with 50% and 25 % of the full load. Firstly, a discussionis given of the creep during each full-load period. The extent of elastic deformationplus creep under constant conditions in a week is assumed to be 0.59 and 0.89 ofthe elastic curvature induced by the full load, respectively, in the two cases of load-hag. We thus observe that the amount of creep strain arising from moisture variationon the average amounted to 0.28 and 0.33 during a full-load week (Fig. 4). Thesecorrespond to the values of a- and a which are about ~- of those given in Table 1.The difference can be explained by the low stress level s = 0.07.The recovery during part ly4oaded periods is discussed adapting the following as-sumptions:1. The recovery occurs in a linear relation to the decrease in load.2. The moisture variation does not induce any creep when s < 0.05, i.e. there is acreep limit for stress.3. During partly-loaded periods, the increase in moisture content is sufficiently largeto induce full recovery of moisture creep strain, if a perfect unloading were carriedout (u c ~< 0.05).While the moisture variation induces a creep amounting to 0.28 and 0.33 during athll-load week, the cumulative effect of three cycles is 0.42 and 0.25, in the actualcases of loading, as derived by application o f the assumptions mentioned. Corres-ponding values observed, the differences of the observations at the points 158 and116 days are 0.34 and 0.20, which principally arise from the moisture variation butinclude also the effect of linear viscoelasticity during a varying load. The lattereffect is small, but it is not simply evaluated trustworthily, because of the widerange of moisture variation and the fact that during 81 first days the averagemoisture content had a higher value than during the varying-load-period.In conclusion, it may be stated that the theory of hydroviscoetasticity is satisfacto-rily compatible with the results of the recovery tests with birch plywood, when it iscompleted by the adoption of the creep limit with respect to kernel function 1110-1111"The observat ion that u c may be less than 0.05 is extremely positive when the cumu-lative effect of snow-load is discussed.Results with spruce plywoodSpruce plywood behaved as had been expected: no perceptible creep deformationarose on variation in the moisture, as the maximum moisture content was fixed atthe start o f each full4oad-period. The observations made, the means of five testspecimens, are indicated in Fig. 5. These experiments were carried out in the sameroom as that for the birch plywood tests.In the test series with a minimum load, being 50% of the full load, the points ofmeasurement deviate to some extent from the theoretical curve. The principal rea-


14/17

202 A. Ranta-Maunus Viscoelasticity of wood at varying m. c.

2 .0

1 .0

02 . 0

1.0

c yc le : 1 0 0 % / 2 5 % o f fu l [ -t o a d

0

' I

Io

: 1 : 1e

cy c le : 100 ~ 'o/50% of fu l t - to ado

< 2 _ _ o81 88 95 102 109

0--0 o o o ]

f f f f0 o o

J ] I l116 123 130 137 144 151 158 dayst i m eFig. 5. Recovery experiments in pure bending of spruce plywood. Each point represents themean of five observations. Both load and climate were unchanged during 81 first days withs = 0.23, u = 0.195 and T = 20~ C. The curves indicated show the expected behaviour atu = 0.195 on the basis of the creep curve observed during 81 first daysson for this is that one of the five specimens behaved in an exceptional way duringthe changing loading. If the results of this test specimen were omitte d, the com-patibility between the theory and the experiments would improve. However, all thetest results have been taken into account in Fig. 5, because no acceptable reason foromitting was found.G l u e - l a m i n a t e d b e a m s u n d e r n a t u r a l c o n d i t i o n sMaterial and conditionsDuring the period 1962 to 68, tests were made with glue-lamina ted beams; pinebeams had dimensions of 708 0ram x t7 6m m x 95 mm for span, height and widthrespectively. A beam was glued from 8 boards. Highly classified heavy wood andunclassified light wood were utilized, so that each beam had 4 lamellae of bot hqualities. If the heavy lamellae were positioned as the two outermost lamellae ofthe tension and compression sides, the beam was called "heavy", and if the twooutermost lamellae were light, it was called "light".The beams of spruce, glued from 10 boards, had dimensions of 7600ram x 220ram xx 150 ram. The four heavy boards were posit ioned as the out ermo st or middlelamellae.The beams were subjected to a point-load at mid-beam, and the midpoint deflectionwas measured in each case. The ma xi mu m stresses, 8.2 N/ mm 2 for pine, a nd5.4 N/ mm 2 for spruce, were largely attri buta ble to the load, and to a lesser extentto the weight of the beam (15%).


15/17

A . Ranta-Maunus Viscoelasticity of wo od at varying m . c . 203A heavy pine beam and a l ight one were kept out of doors without any cover, whiletw o similar beam s were enc lose d in a plastic cover , relatively tight initially. Thespruce beams were tested in the enclosed form.

Resu l ts wi th p in eThe relative deflection was calculated in relation to the elastic deflection inducedby both the load and the weight of the beam itself. The values observed have beenindicate d in Fig. 6. N o appreciable difference s were discernible bet we en heav y andlight wo od . How ever, the cover was foun d to be valuable: the average increase inrelative deflection after the first year amounted to 0.25 per annum for the uncoveredbeams, and less than 0.1 for the covered ones.Thus the variation in moisture exercises a remarkable cumulative effect to deflectionin the case of uncovered pine beams. The major part of the creep deflection o fbeams enclosed in a plastic cover apparently consists of the conventional viscoelasticstrain (denoted by j ,o o in Eq. (2)) .This experiment did not permit any conclusion to be drawn as to a time-limit sub-sequent to which the moisture creep terminates, or changes.

o 2 .4

~ 2 . 2e3> 2 .0

~ 1 .8

3 . 23.0 %

2 h ~

2.6 9 ,.%,) ~\9

! . .t \ l ,,

t-~-o~'~ o h e a v y p i n e , u n c o v e r e d s = 0 . 21 . 6 k " 9 l i g h t p i n e , u n c o v e r e d s = 9 . 4

,..-+lio ,~ h e a vy p i n e , co ve r e d ~ s = 0 . 21 .4 r 2 + i g h t p i n e , c 0 v e r e d , s = O .z,1 . 2

1 .0 1963 1964 1965 1966 1967 y e ar 1968Fig. 6. Long-term loading of glue-laminated beams of pine, on the southern coast of Finland


16/17

20 4 A. Ranta-Maunus Viscoelast ic i ty of wo od at varying m. c .3 . 02 . 82 .~2 .4

r-9 2 . 20J

2 .0

1 . 81 .61.z ,1 .21 . 0

l i g h t s p r u c % c o v e r e dh e a v y s p r u c e , c o v e r e dtheore t ica l . ~t~ot~=1-~Q.66 ut ~247

_A/ / x

2 /

?,t h e o r e ~ i e o [ f o r u=0,20\

/

" ~ ~ he oro tie aL f or u*0,15

1963 1964 1965 1966 1967 t i m e 19 68 yea rFig. 7. Long-term loading of g lue- laminated beams of spruce, in Helsinki. Al l the beam s wereplastic-enclosed. The observ ation po ints indicate d are the me ans of two similar tests. Thetheoret ica l curves are the resul ts the au thor (1972) foun d for spruce plyw ood under c onsta ntcl imat ic co ndi t ions (T = 20 ~ C)Results with spruceT h e o b s e r v a t i o n s m a d e h a v e b e e n i l l u s t r a t e d i n F i g . 7 . I t is e v i d e n t t h a t , a l t h o u g ht h e a n n u a l m a x i m u m v a l u e o f d e f l e c t i o n i n c r e a s e s d u r i n g t h e f i r s t t h r e e y e a r s , t h ed i f fe r e n c e b e t w e e n t h e a n n u a l m i n i m u m v a lu e s o f d e f l e c ti o n , a n d t h e c r e e p cu r v ef o r a c o n s t a n t m o i s t u r e c o n t e n t o f 0 .1 5 , b e c a m e a n a l m o s t c o n s t a n t v a l u e a ft e rs p ri n g 1 96 4 . T h e a n n u a l m i n i m u m v a l u e s m a y a c c o r d in g l y b e d e s c r ib e d b y t h et h e o r y i f t h e e f f e c t i v e m e a n m o i s t u r e c o n t e n t i s 0 . 1 5 , a n d t h e i n c r e a s e i n m o i s t u r ec o n t e n t d u r i n g t h e f i r st y e a r a s s u m e s a v a l u e o f th e o r d e r o f 0 .0 1 t o 0 . 0 2 .A l t h o u g h t h e b e h a v i o u r o f t h e s p r u c e b e a m c a n n o t b e e x p l a i n e d in d e t a i l, o n e m a yc o n c l u d e t h a t n e i t h e r t h e v a r i a t i o n i n t e m p e r a t u r e , n o r t h e v a r i a t i o n i n m o i s t u r ec o n t e n t , e x e r c i s e s a c u m u l a t i v e e f f e c t u p o n t h e c r e e p s t ra i n o f s p r u c e . C o n s e q u e n t l y ,t h e l i n e a r t h e o r y o f v i s c o e l a s t i c i t y d e s c r i b e s th e l o n g - t e r m b e h a v i o u r o f s p r u c e i n it sp r i n c i p a l t r a it s . T h i s e x c l u d e s t h e c o n s i d e r a b l e v a r i a t i o n in a n n u a l d e f l e c t i o n , w h i c his a s s u m e d t o o r i g in a t e in t h e s h o r t - te r m e f f e c t s o f t e m p e r a t u r e a n d m o i s t u r e c o n t e n t .ConclusionsT h e t h e o r y o f h y d r o v i s c o e l a s t i c i t y e x p r e s s e s a f u n c t i o n a l r e l a t i o n s h i p b e t w e e n s t ra i n ,s t re s s a n d m o i s t u r e h i s t o r y . W o o d e n m a t e r i a l s a re d i v i d e d in t o t w o g r o u p s , b ir c h -a n d s p r u c e - li k e m a t e r i a ls , d e p e n d i n g o n w h e t h e r t h e v a r i a ti o n s in m o i s t u r e c o n t e n th a v e a c u m u l a t i v e e f f e c t o r n o t .


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A. Ranta-Maunus Viscoelasticity of wood at varying m.c. 205

In the planning of practical applications, the following questions arise:(i) What are the material constants in accordance with the theory of hydrovisco-elasticity?(ii) What are the limits of validity of the theory in time, stress and moisture content?(iii) What is the m oist ure-co ntent history in different places in the wood?(iv) What role does temperature play?In this paper, problems (i) and (ii) have been discussed as follows:- the material constan ts have been determine d for birch and spruce plywood in

laboratory tests (Table 1)- approxi mate values for the materia l cons tant s have also been calcula ted for certain

qualities of tree on the basis of data published earlier (Table 1)- in regard to the creep that arises from variation in the moist ure cont ent , it wasobserved that the creep limit in stress level for birch veneer is about s = 0.05,

and the lower creep limit in moisture content for spruce and birch veneer isu = 0.08 to 0.10.

In future, particular interest should be attached to laboratory experiments at temper-atures of - 20 ~ C to + 10~ C, along with measu res o f real-scale structures, with aduration of many years.

ReferencesArima, T. 1972. J. Japan Wood Res. Soc. 18 :3 49 -3 53Armstrong, L. D.; Christensen, G. 1961. Nature 19 1: 86 9-8 70Armstrong, L. D.; Kingston, R. S. T. 1962. Aust. J. appl. Sci. 13:257 -27 6Bethe, E. 1969. Holz Roh- u. Werkstoff 27: 29 1- 30 3Eriksson, L.; Noren, B. 1965. Holz Roh- u. Werkstoff 23:201-209Hearmon, R. F. S.; Paton, J. M. 1964. Forest Prod. J. 14 :3 57 -3 59Kitallara, K.; Yukawa, K. 1964. J. Japan Wood Res. Soc. 10 :1 69 -175Lawniczak, M.; Raczkowski, J. 1961. Nature 1 92 :5 83 -5 84Perkitny, T. 1965. Holz Roh- u. Werkstoff 23:173-182Raczkowski, J. t969. Holz Roh- u. Werkstoff 27:232-237Ranta-Maunus, A. 1972. Building Technology and Community Development, Technical ResearchCentre of Finland, Helsinki. Publication 3Ranta-Maunus, A. 1973. Publication 4.Schniewind, A. P. 1966. Holz Roh- u. Werkstoff 24:87-98Takemura, T. 1966. Coll. Agric. Kyoto Univ. 88. Forestry Ser. 1 :31-48

(Received August 20, 1974/April 14, 1975)Alpo Ranta-MaunusStructural Mechanics LaboratoryTechnical Research Centre of FinlandSF-02150 Otaniemi, FinlandPresent address of the author:Institute of Radiation ProtectionBOX 268SF-00101 Helsinki 10, Finland

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