The Structural Behavior
of AxiaUy Loaded Wood
Studs Exposed to Fire on
One Side
Trätek
THE STRUCTURAL BEHAVIOUR OF A X I A L L Y LOADED WOOD STUDS EXPOSED TO FIRE ON ONE SIDE
TräteknikCentrum, Rapport I 8808057
Nyckelord buckling
five vesistanae
load heaving oapacity stvuatuval hehaviouv studs
walls wood
rapporter betecknas med I eller P och numreras tillsammans med alla utgåvor från Träteknik-Centrum i löpande följd.
Rapporter kan som regel beställas kostnadsfritt i ett exemplar av medlemsföretag. Ytterligare be-ställda exemplar faktureras.
Citat tillätes om källan anges.
Reports issued by the Swedish Institute for Wood Technology Research comprise complete accounts for research results, or summaries, surveys and
stu-dies. Published reports bear the designation I or P and are numbered in consecutive order together with all the other publications from the Institute.
Member companies may generally order one copy of any report free of charge. A charge will be made for any further copies ordered.
Extracts from the text may be reproduced provided the source is acknowledged.
berskivor, spånskivor och plywood. Ett avtal om forskning och utveckling mellan industrin och Styrelsen för Teknisk Utveckling (STU) utgör grunden för verksamheten som utförs med egna, samverkande och externa resurser. Träteknik-Centrum har forskningsenheter, förutom i Stock-holm, även i Jönköping och Skellefteå.
The Swedish Institute for Wood Technology Re-search serves the five branches of the industry: saw-mills, manufacturing (joinery, wooden houses, fur-niture and other woodworking plants), fibre board, particle board and plywood. A research and deve-lopment agreement between the industry and the Swedish National Board for Technical Development (STU) forms the basis for the Institutes activities. The Institute utilises its own resources as well as those of its collaborators and other outside bodies. Apart from Stockholm, research units are also located in Jönköping and Skellefteå.
FOREWORD 3
SUMMARY 4
1 . BACKGROUND AND AIMS
2. EXPERIMENTAL INVESTIGATIONS 2.1 General
2.2 Specimens and t e s t apparatus 2.3 Test procedure and r e s u l t s 2.4 E v a l u a t i o n 2.41 Bending s t i f f n e s s 2.42 Stresses a t m i d s e c t i o n a t f a i l u r e 6 6 7 10 24 24 25 ANALYTICAL MODELS 27 3.1 Member w i t h p i n j o i n t e d end supports i n compression 27
3.2 Member w i t h c y l i n d r i c a l convex end surfaces i n compression 29
3.21 C r i t i c a l l o a d 29 3.22 E c c e n t r i c compressive f o r c e 29
3.23 D e t e r m i n a t i o n o f t h e i d e a l r a d i u s o f t h e studs and
surfaces 30 3.24 The i n f l u e n c e o f t h f r\uå rros.'^. s e c t i o n 36
3.25 The i n f l u e n c e o f t h e l e n g t h o f t h e s t u d 37 3.26 The i n f l u e n c e o f support i n c l i n a t i o n on l o a d b e a r i n g c a p a c i t y 38 3.27 S u p e r p o s i t i o n o f a x i a l f o r c e and t r a n s v e r s e l o a d i n g 39 CONCLUSIONS REFERENCES 41 43 APPENDIX A1 APPENDIX A2 APPENDIX A3 APPENDIX A4 APPENDIX A5 D e t e r m i n a t i o n o f t h e c r i t i c a l load f o r a member
w i t h c y l i n d r i c a l convex end surfaces 44 Member w i t h c y l i n d r i c a l convex end surfaces a c t e d
upon by e c c e n t r i c compressive load 47 The i n f l u e n c e o f i n c l i n e d support p l a t e s on a
mem-ber w i t h c y l i n d r i c a l convex end surfaces acted upon
by a compressive f o r c e 51 E c c e n t r i c a x i a l f o r c e and t r a n s v e r s e l o a d on a
mem-ber w i t h c y l i n d r i c a l convex end surfaces 52 Member w i t h i n i t i a l c u r v a t u r e and c y l i n d r i c a l convex end
The i n v e s t i g a t i o n s d e s c r i b e d i n t h i s r e p o r t were i n i t i a t e d by t h e wish o f the r e p r e s e n t a t i v e s o f i n d u s t r y t o improve p r e s e n t methods f o r t h e design w i t h r e s p e c t t o f i r e o f w a l l s f o r s i n g l e f a m i l y houses. The o r i g i n a l p l a n had been t h a t t h e problem would be s t u d i e d o n l y e x p e r i m e n t a l l y . However, d u r i n g e v a l u a t i o n o f t h e experiments a number o f phenomena arose which n e c e s s i t a t e d e l u c i d a t i o n by means o f t h e o r e t i c a l s t u d i e s . I wish t o thank B i r g i t östman f o r her s u p p o r t i n c a r r y i n g o u t t h i s work.
The t e s t s were made by Claes K u l l b e r g a t t h e Department o f S t e e l Construc-t i o n , Royal I n s Construc-t i Construc-t u Construc-t e o f Technology, SConstruc-tockholm. Joakim Norén a s s i s Construc-t e d i n the t e s t s and e v a l u a t i o n . Bo Källsner p r o v i d e d v a l u a b l e and c r i t i c a l obser-v a t i o n s .
The f i g u r e s were drawn by P i D r a g o j e v i c , and t h e manuscript was t y p e d by Yvonne Larsson.
The work was f i n a n c e d by funds from t h e t i m b e r i n d u s t r y and t h e Swedish Board f o r T e c h n i c a l Development (STU) t o t h e Swedish I n s t i t u t e f o r Wood Technology Research.
The Swedish o r i g i n a l was t r a n s l a t e d by L J Gruber BSc(Eng) MICE M I S t r u c t E . I n t h i s e d i t i o n s u b s e c t i o n 3.24 was shortened and a new appendix A5 was supplemented.
I wish t o extend my s i n c e r e thanks t o a l l who have c o n t r i b u t e d t o t h i s work.
Stockholm, August 1988
I n Sweden, t h e l o a d b e a r i n g w a l l s o f s i n g l e f a m i l y houses must a t present be designed w i t h respect t o f i r e i n order t o p r o v i d e t h e necessary degree o f s a f e t y a g a i n s t t h e spread o f f i r e . One o f t h e p r e c o n d i t i o n s i n order t h a t
the f i r e r e s i s t a n c e o f an e x t e r n a l w a l l should be secured i s t h a t i t s l o a d -b e a r i n g c a p a c i t y should -be maintained.
With t h e aim o f s t u d y i n g t h e s t r u c t u r a l behaviour o f wood s t u d s , a t e s t s e r i e s was c a r r i e d o u t on a x i a l l y loaded wood studs. I n these t e s t s , t h e e f f e c t o f f i r e on t h e wood studs was s i m u l a t e d by removing l a y e r s from t h e stud by p l a n i n g i t on t h e s i d e which was supposed t o be exposed t o f i r e . D i f f e r e n t support c o n d i t i o n s , such as t h e use o f c e l l u l a r rubber s o l e p l a t e s e a l i n g s t r i p s and i n c l i n e d base o r i n c l i n e d r o o f t r u s s r a f t e r , were
s t u d i e d . S i x o f t h e specimens c o n s i s t e d o f s o l i d 45 x 120 mm timber s t u d s , and two were l i g h t w e i g h t studs c o m p r i s i n g t i m b e r f l a n g e s and webs o f wood
f i b r e board. I n a l l cases t h e studs were j o i n e d by s h o r t pieces o f timber corresponding t o t h e s o l e p l a t e and t o p p l a t e . During t h e t e s t s , t h e s p e c i -mens were placed between r i g i d , non r o t a t i n g support p l a t e s , t h e i n t e n t i o n being t o reproduce c o n d i t i o n s i n a s i n g l e f a m i l y house c o n s t r u c t i o n .
In the t h e o r e t i c a l p a r t o f t h e i n v e s t i g a t i o n , two a n a l y t i c a l models a r e s t u d i e d . I n one of t h e models i t i s assumed t h a t t h e ends o f t h e s t u d are p i n j o i n t e d . The s t r u c t u r a l behaviour cannot be d e s c r i b e d s a t i s f a c t o r i l y by t h i s model. I n t h e o t h e r a n a l y t i c a l model, t h e s t u d i s assumed t o be placed between r i g i d end p l a t e s as i n t h e t e s t s . The end s u r f a c e s o f t h e s t u d a r e i d e a l i s e d as c y l i n d r i c a l convex s u r f a c e s , e n a b l i n g a r o l l i n g motion t o take place as l o a d i n g and deformations proceed. T h e o r e t i c a l s o l u t i o n s a r e de-r i v e d f o de-r d i f f e de-r e n t l o a d i n g cases and boundade-ry c o n d i t i o n s . I n ode-rdede-r t h a t i t should be p o s s i b l e t o apply these, i d e a l r a d i i are determined w i t h t h e a i d of t h e t e s t r e s u l t s t o d e s c r i b e t h e g e o m e t r i c a l shape o f t h e end s u r f a c e s .
kn approximate expression f o r the i d e a l r a d i u s i s formulated i n order t h a t
i t may a l s o be used f o r o t h e r cross s e c t i o n s and l e n g t h s .
With t h e a i d o f p a r a m e t r i c s t u d i e s , t h e i n f l u e n c e due t o i n c l i n a t i o n o f t h e base and t h e i n t e r a c t i o n between a x i a l f o r c e and t r a n s v e r s e load i s
s t u d i e d . The r e s u l t s show t h a t t h e two load components can be superimposed by a p p l y i n g l i n e a r i n t e r a c t i o n .
F i r e t e s t s can thus be c a r r i e d o u t s e p a r a t e l y f o r a x i a l and t r a n s v e r s e l o a d i n g . By d e t e r m i n i n g t h e l o a d b e a r i n g c a p a c i t y f o r a c e r t a i n f i r e r e s i -stance p e r i o d , i t w i l l be p o s s i b l e t o design a timber s t u d under f i r e expo-sure c o n d i t i o n s f o r a r b i t r a r y loads and load combinations.
Design o f the s t r u c t u r a l elements i n a b u i l d i n g w i t h r e s p e c t t o f i r e i s t o -day an obvious p a r t of o v e r a l l design. I n order t h a t the r e q u i r e d degree of s a f e t y a g a i n s t the spread of f i r e may be provided, i t i s e s s e n t i a l t h a t , i n p a r t i c u l a r , the e x t e r n a l w a l l s of houses should have adequate f i r e r e s i -stance .
The f i r e r e s i s t a n c e of a w a l l i s i t s a b i l i t y t o f u n c t i o n as a b a r r i e r a g a i n s t f i r e . I n the case o f w a l l s which have o n l y a space s e p a r a t i n g func-t i o n , f i r e r e s i s func-t a n c e depends on func-the i n s u l a func-t i o n of func-the w a l l func-t o l i m i func-t func-the f l o w of heat through the w a l l , and a l s o on i t s i n t e g r i t y t o prevent passage o f flames or hot gases. I n the case of l o a d b e a r i n g w a l l s , f i r e r e s i s t a n c e
i s i n a d d i t i o n l i m i t e d by the a b i l i t y of the w a l l t o c a r r y the imposed loads.
For the l o a d b e a r i n g e x t e r n a l w a l l s of s i n g l e f a m i l y houses and s i m i l a r b u i l d i n g s , l o a d i n g c o n s i s t s o f an a x i a l and a t r a n s v e r s e load. These loads are due, f o r i n s t a n c e , t o dead l o a d , snow load on t h e r o o f , imposed loads on the a t t i c s t o r e y , and wind loads on the w a l l s .
As a r u l e , the load on a w a l l v a r i e s from case t o case. The snow loads on the r o o f and the wind loads depend on where i n the c o u n t r y the b u i l d i n g i s s i t u a t e d , and the imposed loads depend on the type of b u i l d i n g i n
q u e s t i o n . I t i s t h e r e f o r e d e s i r a b l e t h a t design of w a l l s f o r the l o a d i n g case f i r e should be c a r r i e d o u t i n such a way t h a t the a c t u a l loads and d i f f e r e n t combinations of these can be taken i n t o c o n s i d e r a t i o n . A design method must a l s o f a c i l i t a t e the a p p l i c a t i o n of d i f f e r e n t p a r t i a l c o e f f i -c i e n t s , whi-ch i s i m p o r t a n t i n -c o n j u n -c t i o n w i t h the e x p o r t o f p r e f a b r i -c a t e d houses.
As a r u l e , the f i r e r e s i s t a n c e of l i g h t w e i g h t w a l l c o n s t r u c t i o n s such as timber stud w a l l s i s today determined i n Sweden by f i r e t e s t s a t f u l l s c a l e . I n order t o s i m p l i f y t e s t i n g , t h e w a l l i s g e n e r a l l y acted upon o n l y by a x i a l loads which are however, i n order t o take account of t h e a c t i o n of t r a n s v e r s e f o r c e s , a p p l i e d w i t h a c e r t a i n e c c e n t r i c i t y . The supports of the w a l l are p i n j o i n t e d / I / .
I n a number o f cases, t h i s procedure r e s u l t s i n c o n s i d e r a b l e underestima-t i o n of underestima-the l o a d b e a r i n g c a p a c i underestima-t y of underestima-the c o n s underestima-t r u c underestima-t i o n , since underestima-the s underestima-t r u c underestima-t u r a l behaviour of an a c t u a l c o n s t r u c t i o n i s not reproduced i n the c o r r e c t way. The aim of the experimental p a r t of the i n v e s t i g a t i o n d e s c r i b e d i n t h i s r e -p o r t was t o study t h e s t r u c t u r a l behaviour o f the s t u d s . The f i r e and i t s e f f e c t s , c h a r r i n g and the thermal e f f e c t s on the s t r e n g t h and s t i f f n e s s o f t h e remaining cross s e c t i o n , were s i m u l a t e d q u a l i t a t i v e l y by p l a n i n g the studs on the s i d e exposed t o f i r e .
The aim o f the t h e o r e t i c a l p a r t of the i n v e s t i g a t i o n was t o p r o v i d e an ex-p l a n a t i o n f o r a l l the ex-phenomena observed i n the t e s t s , and t o have the c a p a b i l i t y t o d e a l a l s o w i t h o t h e r types o f loads and load combinations and thus t o minimise the number of t e s t s .
The thermal e f f e c t s on s t r e n g t h and s t i f f n e s s d i d not come w i t h i n t h e terms of r e f e r e n c e of t h i s i n v e s t i g a t i o n . I t i s i m p o r t a n t t h a t these should be s t u d i e d s e p a r a t e l y so t h a t t h e r e s u l t s can be used i n a t h e o r e t i c a l t r e a t -ment of the problem, and the number of f i r e t e s t s can be reduced.
2.1 General
I n many cases, the f r a m i n g i n the l o a d b e a r i n g e x t e r n a l w a l l s f o r s i n g l e f a -m i l y houses today c o n s i s t s of s o l i d v e r t i c a l studs and h o r i z o n t a l studs attached t o the o u t s i d e faces of these. The c l a d d i n g on the o u t s i d e a c t s as wind p r o t e c t i o n and, when t h e r e are no h o r i z o n t a l s t u d s , a l s o has the func-t i o n of p r e v e n func-t i n g b u c k l i n g o f func-the sfunc-tuds i n func-the plane of func-the w a l l . Afunc-t
pre-sent, common stud dimensions are 45 x 120 mm and 45 x 170 mm. I n most cases, these studs are w i t h i n the c l a s s i f i c a t i o n "Ö-virke" a c c o r d i n g t o Swedish B u i l d i n g Code SBN 1980 /2/. Walls comprising l i g h t w e i g h t studs, which may be composite I - s e c t i o n s c o n s i s t i n g of f l a n g e s o f small dimension
timber and a web of wood f i b r e board, are becoming i n c r e a s i n g l y common. I n these w a l l s the c l a d d i n g which provides p r o t e c t i o n a g a i n s t the wind i s attached d i r e c t l y t o the e x t e r n a l f l a n g e s of the l i g h t w e i g h t studs. The w a l l s a r e f i l l e d w i t h m i n e r a l wool i n order t o p r o v i d e s a t i s f a c t o r y t h e r m a l
i n s u l a t i o n .
I n the event of f i r e , the i n t e r n a l board l i n i n g p r o v i d e s the i n i t i a l
b a r r i e r . A f t e r some time t h i s burns c o m p l e t e l y or f a l l s down from the w a l l , see e.g. /3/ (Noren and Östman, 1985). The l o a d b e a r i n g v e r t i c a l studs are then d i r e c t l y exposed t o f i r e . Since the m i n e r a l wool p r o t e c t s the sides of the studs from the f i r e , combustion takes place mainly on the i n s i d e of the w a l l . See Figure 2.1. The aim of the t e s t s r e p o r t e d here was t o study the mechanical d e f o r m a t i o n and f a i l u r e behaviour of a x i a l l y loaded cross
sec-t i o n s exposed sec-t o f i r e on one s i d e . For sec-t h i s reason, sec-the f i r e sequence was s i m u l a t e d by s u c c e s s i v e l y removing a p a r t of t h e cross s e c t i o n by p l a n i n g , so t h a t the e f f e c t i v e cross s e c t i o n of the s t u d was g r a d u a l l y reduced. The r a t e o f combustion i s somewhat g r e a t e r a t the c o r n e r s , and t h e boundary o f the e f f e c t i v e cross s e c t i o n i s t h e r e f o r e a l i t t l e rounded on the s i d e towards the f i r e . However, t h i s was not taken i n t o account i n the t e s t s s i n c e the general behaviour o f the specimens i s not a f f e c t e d . The cross s e c t i o n o f the studs and f l a n g e s was thus r e c t a n g u l a r a t a l l stages o f t h e t e s t .
F i g u r e 2.1.
Stage of f i r e a f t e r l o s s of the i n t e r n a l l i n i n g .
Since support c o n d i t i o n s can vary c o n s i d e r a b l y i n p r a c t i c e , the e f f e c t of these on t h e behaviour o f the studs was a l s o s t u d i e d i n t h e t e s t s . I n order
to ensure t h a t b u i l d i n g s a r e a i r t i g h t , s e a l i n g s t r i p s of c e l l u l a r rubber are a t present placed between the sole p l a t e or t o p p l a t e and a d j o i n i n g p a r t s of the b u i l d i n g . I n c l i n a t i o n o f the f o u n d a t i o n or d e f l e c t i o n of the r o o f t r u s s may have t h e e f f e c t t h a t l o a d i s a p p l i e d a t a l a r g e e c c e n t r i -c i t y .
Two types o f specimen were i n v e s t i g a t e d . Each specimen c o n s i s t e d o f a stud o f 2,400 mm l e n g t h and pieces o f t i m b e r 250 mm long and 45 mm t h i c k which
represented t h e sole p l a t e and t o p p l a t e i n t h e a c t u a l c o n s t r u c t i o n . The s i x specimens o f Type 1 comprised s o l i d t i m b e r studs o f pine (Pinus
s y l v e s t r i s ) i n t h e Swedish grade Ö-virke ( t h e c h a r a c t e r i s t i c bending s t r e n g t h i s 15 N/mm^), o f t h e dimensions 45 x 120 mm. The t i m b e r f o r t h e sole p l a t e and t o p p l a t e had t h e same dimensions and was i n t h e same
grade. The n a i l e d j o i n t s c o n s i s t e d o f 2 No 100 x 3.4 n a i l s d r i v e n s t r a i g h t through t h e sole p l a t e and t o p p l a t e i n t o t h e end g r a i n o f t h e studs. Four of t h e specimens were f i t t e d w i t h c e l l u l a r rubber s e a l i n g tapes (AB Värnamo Gummifabrik); these were a t t a c h e d t o t h e sole p l a t e and t o p p l a t e by
s t a p l e s . See t h e summary i n Table 2.1 and F i g u r e 2.2. I n t h e specimens where t h e s e a l i n g tape was narrower than t h e timber, i t was placed c e n t r a l l y .
Specimen Specimen
®
77.
Figure 2.2. Support c o n d i t i o n s i n t h e t e s t s . The upper support p l a t e i s h o r i z o n t a l i n a l l t e s t s .
Specimen Stud Sole p l a t e / Sealing I n c l i n a t i o n o f Remarks t o p p l a t e s t r i p bottom support p l a t e {%) 1 45 X 120 45 X 120
-
0 2 45 X 120 45 X 120 70 X 10 0 3 45 X 120 45 X 120 120 X 10 0 4 45 X 120 45 X 120-
3.5 5 45 X 120 45 X 120 120 X 10 3.5 6 45 X 120 45 X 120 70 X 10 3.5 7 H 200 MB H 200 MS 120 X loa) 0 L i g h t w e i g h t stud 8 H 200 MB H 200 MS 120 X loa) 3.5 - " - b) a) Cut and f i t t e d as i n Figure 2.2 / ^ ^ ^ " • ^ • — ab) S p l i c e i n web 260 mm from bottom end o f s t u d .
The two specimens o f Type 2 c o n s i s t e d o f Masonite Byggsystem l i g h t w e i g h t studs made by Swanboard Masonite AB, o f 2,400 mm l e n g t h and stud depth 200 mm. The dimensions o f t h e f l a n g e s were 45 x 45 mm and t h e t h i c k n e s s o f t h e wood f i b r e board web 6.4 mm. A l l dimensions a r e nominal. The f l a n g e s were o f spruce (Abies a l b a ) and, according t o i n f o r m a t i o n s u p p l i e d by t h e manufacturer, were a t l e a s t o f grade T 18, w h i l e t h e grade o f t h e web was a t l e a s t K 13. No checks were made t o f i n d whether t h e specimens complied w i t h these data. The s o l e p l a t e and t o p p l a t e c o n s i s t e d o f 45 x 70 mm
timber and t h e web o f wood f i b r e board 8 mm t h i c k . The minimum grades i n t h i s case a l s o were t h e same as f o r t h e studs. The 120 x 10 mm s e a l i n g tape was c u t i n t h e middle and was a t t a c h e d t o t h e s o l e p l a t e and head p l a t e a t a d i s t a n c e o f 10 mm from t h e end ( F i g u r e 2.2).
The t e s t i n g machine f o r a x i a l l o a d i n g was a h y d r a u l i c press (Losenhausen) of 6,000 kN load c a p a c i t y . I t was f i t t e d w i t h r i g i d end p l a t e s which were prevented t o r o t a t e . For f o u r o f t h e specimens t h e lower support p l a t e was i n c l i n e d 3.5 % so t h a t t h e undeformed specimen was i n c o n t a c t w i t h t h e base on t h e " f i r e " - s i d e .
The lower support p l a t e was placed on t h r e e load c e l l s (Alexen Load I n d i -c a t o r , -c a p a -c i t y 50 kN) i n order t h a t t h e p o s i t i o n o f t h e a x i a l f o r -c e may be determined. See Figure 2.3. Using t h e symbols i n t h e f i g u r e , t h i s i s
determined as-.
d = -aA - bB 4- cC N
The dimensions a, b and c a r e s e t o u t i n Table 2.2.
B u c k l i n g i n t h e d i r e c t i o n o f t h e minor a x i s o f t h e stud was prevented by means o f l a t e r a l supports w i t h s l i d i n g bearings o f t e f l o n s t r i p s , f i t t e d a t the midpoint and q u a r t e r p o i n t s . For specimens Nos 7 and 8 ( l i g h t w e i g h t s t u d s ) , o n l y t h e o u t e r f l a n g e ( t h a t u n a f f e c t e d by t h e " f i r e " ) was braced i n such a way.
accuracy b e t t e r than - 0.025 mm. :<A CX Figure 2.3. P l a c i n g o f load c e l l s f o r deter-m i n a t i o n o f t h e p o s i t i o n o f t h e a x i a l f o r c e . Table 2.2. P l a c i n g o f load c e l l s . Specimen a b c mm mm mm 1 - 6 154 33,9 33,4 7, 8 240 0 0
On a l l specimens, a t r a n s v e r s e load was a l s o a p p l i e d i n t h e s t i f f d i r e c t i o n i n order t h a t f l e x u r a l r i g i d i t y may be determined, see F i g u r e 2.4. When t h e compression f l a n g e had been removed from specimens Nos. 7 and 8, t h e load was t r a n s m i t t e d t o t h e t e n s i o n f l a n g e by means o f blocks o f wood on each
s i d e o f t h e web. Load was a p p l i e d by a h y d r a u l i c j a c k and measured w i t h a load c e l l . The measured load values had t h e accuracy - 20 N.
(200) 100 P/2 P/2
6^
6 6 6
125 700 > 600 700 >^^^^ 750 -T 750 750 22S0 'I 7 7 ^Figure 2.4. System and l o a d i n g f o r d e t e r m i n a t i o n o f f l e x u r a l r i g i d i t y . 2 . 3 Test procedure and r e s u l t s
The specimens were c o n d i t i o n e d i n a c o n t r o l l e d c l i m a t e room a t 20 *C and 65 \ RH f o r about one week. T e s t i n g commenced even i f t h e r e was no c e r t a i n -t y -t h a -t -t h e e q u i l i b r i u m mois-ture r a -t i o had been a -t -t a i n e d . The reason f o r t h i s was t h a t i t was p r i m a r i l y t h e behaviour a t f a i l u r e which was t o be s t u d i e d . The u l t i m a t e load o f a x i a l l y loaded studs o f l a r g e slenderness r a t i o i s mainly governed by f l e x u r a l r i g i d i t y , see t h e formula f o r t h e Euler load. Since t h e i n f l u e n c e o f t h e moisture r a t i o on f l e x u r a l r i g i d i t y
i s s u b s t a n t i a l l y less than on t h e s t r e n g t h , d e v i a t i o n s from t h e e q u i l i b r i u m moisture r a t i o can be i g n o r e d . The moisture r a t i o and d r y d e n s i t y were
de-termined on samples taken from t h e timber near t h e p o s i t i o n o f f r a c t u r e , and a r e s e t o u t i n Table 2.3.
Table 2.3. Type o f timber, moisture r a t i o and d r y d e n s i t y of specimens.
Specimen Timber Moisture
% r a t i o u Dry d e n s i t y QQU kg/m-^ 1 Pine 14 2 449 2 14 4 393 3 14 6 424 4 • 14 2 422 5 13 8 384 6 13 9 437 7 Spruce 14 0 369 8 n 14 1 355
The specimens were f i r s t t e s t e d under t r a n s v e r s e l o a d and then under a x i a l l o a d . A f t e r t h i s a l a y e r o f t h e m a t e r i a l was removed on t h e s i d e o f t h e stud exposed t o f i r e w i t h a j i g saw and e l e c t r i c plane. For specimen 1 r e -d u c t i o n o f t h e cross s e c t i o n began 200 mm from t h e en-ds o f t h e s t u -d , f o r o t h e r specimens r e d u c t i o n began 100 mm from t h e ends o f t h e s t u d , see F i g u r e 2.4. The m a t e r i a l was planed i n steps o f 5 mm on Specimen No. 1 and i n steps o f 10 mm on t h e o t h e r specimens. D e v i a t i o n s from t h i s , a t t h e f i -n a l stage -near f a i l u r e , a r e s e t o u t i -n Table 2.4. Reductio-n o f t h e cross
s e c t i o n i n Specimens Nos. 7 and 8 ( l i g h t w e i g h t studs) was as shown i n F i g u r e 2.5.
Table 2.4. R e s u l t s f o r t h e u l t i m a t e stage.
Speci- Depth o f cross s e c t i o n Maximum U l t i m a t e Time Distance o f N men a t
f a i l u r e
p r i o r t o f a i l u r e ^ ^
l o a d load from edge
h h ^max Nu d mm mm kN kN mm 1 55 60 13.0 12.80 0 18.8 2 50 60 13.0 9.22 0 15.2 3 55 60 13.0 12.36 0 17.9 4 60 60 13.0 13.00 2 h, 4 min 10.3c) 5 55 55 13.0 13.00 8 min 19.OC) 6 54.5 60 13.0 11.07 0 17.7 7 85 110 18.0 12.92 0 15.4 8 135 160b) 18.0 17.81 0 23.7
a) This cross s e c t i o n maintained t h e maximum load f o r a t l e a s t 5 minutes b) Without f l a n g e .
c) A t t h e end o f t h e creep stage, see F i g u r e 2.13.
h = 2 0 0 h= 160 h = 160 h = 85
During t h e t e s t s under t r a n s v e r s e load a c c o r d i n g t o F i g u r e 2.4, t h e load and d e f l e c t i o n were recorded f o r i n t e g r a l values o f t h e m i d p o i n t d e f l e c t i o n u n t i l t h i s was 5 mm, whereupon t h e specimen was immediately unloaded. The
r a t e o f l o a d i n g was such t h a t a p p l i c a t i o n o f load proceeded f o r a t l e a s t 2 minutes. An example o f t h e measured r e l a t i o n s h i p between l o a d and mid-p o i n t d e f l e c t i o n i s shown i n Figure 2.6. h= 120 Specimen no,1 110 105 0 5 10 mm W
Figure 2.6. Examples o f t r a n s v e r s e load - m i d p o i n t d e f l e c t i o n curves f o r d i f f e r e n t depths o f t h e cross s e c t i o n .
The specimens were then placed i n t h e t e s t apparatus f o r a x i a l l o a d i n g . Load was a p p l i e d a t a c o n s t a n t r a t e u n t i l t h e maximum load o f 13 kN and 18 kN r e s p e c t i v e l y was a t t a i n e d a f t e r about 4 minutes. The readings from the load c e l l s and displacement t r a n s d u c e r s were recorded f o r each l o a d i n g stage o f 1 kN. When t h e maximum load had been reached, i t was maintained constant f o r 5 minutes and t h e readings were recorded a f t e r every minute. The specimen was then immediately unloaded.
The maximum load chosen was a l i t t l e less than t h a t p e r m i t t e d a c c o r d i n g t o Swedish B u i l d i n g Code SBN 1980 /2/. This was c a l c u l a t e d w i t h r e s p e c t t o b u c k l i n g i n t h e s t i f f d i r e c t i o n f o r specimens o f Type 1 and had t h e magni-tude 13.7 kN. For these specimens, t h e p e r m i s s i b l e load w i t h r e s p e c t t o compression p e r p e n d i c u l a r t o t h e g r a i n i s 10.8 kN, which was thus exceeded i n t h e t e s t s . For specimens o f Type 2, i . e . t h e l i g h t w e i g h t beams, t h e per-m i s s i b l e a x i a l load i s 18.8 kN a c c o r d i n g t o data s u p p l i e d by t h e per-
manufactu-r e manufactu-r . I n t h i s case t h e design c manufactu-r i t e manufactu-r i o n i s compmanufactu-ression p e manufactu-r p e n d i c u l a manufactu-r t o t h e g r a i n i n t h e s o l e p l a t e and t o p p l a t e r e s p e c t i v e l y .
I n most o f t h e specimens f a i l u r e occurred a l r e a d y b e f o r e t h e whole o f t h e a x i a l load had been a p p l i e d . I n two o f t h e specimens, however, where t h e maximum load was k e p t c o n s t a n t f o r 5 minutes, i t was e v i d e n t t h a t t h e spe-cimen would creep t o f a i l u r e even w i t h o u t f u r t h e r r e d u c t i o n o f t h e cross s e c t i o n . I n specimen No. 4 t h i s creep process took over 2 hours. The values of t h e u l t i m a t e loads Nyr depth o f cross s e c t i o n h a t f a i l u r e and a stage of t h e cross s e c t i o n b e f o r e f a i l u r e , and t h e time over which t h e maximum
load Nu was maintained, a r e s e t o u t i n Table 2.4.
A x i a l load - m i d p o i n t d e f l e c t i o n curves a r e g i v e n i n F i g u r e 2.7 a-g. The r e s u l t s f o r specimen No. 8 a r e n o t g i v e n . The reason f o r t h i s i s t h a t t h e presence o f a b u t t j o i n t i n t h e web near t h e bottom support had such an e f f e c t on t h e d e f l e c t e d shape t h a t t h e maximum d e f l e c t i o n d i d n o t occur a t the c e n t r e . See below. I n o r d e r t o i l l u s t r a t e t h e i n f l u e n c e o f t h e d i f f e -r e n t bounda-ry c o n d i t i o n s on t h e d e f o -r m a t i o n s i n specimens Nos. 1-6, t h e curves f o r t h e l e a s t cross s e c t i o n s o f t h e specimens a r e a l s o g i v e n sepa-r a t e l y i n Figusepa-re 2.8. On compasepa-ring t h e sepa-reasonably s t sepa-r a i g h t p o sepa-r t i o n s o f t h e curves as load began t o be a p p l i e d , i t i s seen t h a t t h e s m a l l e s t d i s p l a c e -ments occurred i n specimens Nos. 1 and 4 which were n o t f i t t e d w i t h s e a l i n g s t r i p s . Specimens Nos. 3 and 5, w i t h 120 mm wide s e a l i n g s t r i p s , come next. The l a r g e s t displacements occurred i n specimens Nos. 2 and 6 which were f i t t e d w i t h 70 mm wide s t r i p s . I n a l l t h r e e p a i r s o f specimens, t h e l a r g e s t displacement was recorded f o r t h e specimen which had an i n c l i n e d support p l a t e . F i g u r e 2.7 a-g. A x i a l f o r c e - m i d p o i n t d e f l e c t i o n curves f o r d i f f e r e n t depths o f t h e cross s e c t i o n . h=120 115 110 105 100 95 90 85 SO 75 70 65 60 55 13 1 10 10 mm max F i g u r e 2.7 a.
halSO l i c ICQ 90 BO O 5 10 mm max F i g u r e 2.7 b. h=120 I I O I O O 90 80 70 60 55 13 1 10 O 5 10 mm max F i g u r e 2.7 c.
h = 1 2 0 n o lOO 9 0 8 0 7 0 6 0 13 1 10 Specimen no. 4 o 5 10 mm max Figure 2.7 d. h = i 2 0 n o l o o 9 o s o 7 0 13 1 10 5 5 5 Specimen no.5 o 5 10 mm max F i g u r e 2.7 e.
h s i a o I I O I O O 9 0 8 0 7 0 6 0 5 5 13 1 10 5 Specimen no.6 o 5 lOmm max Figure 2.7 f . h= 200 190 180 170 160 160 135 110 1 8 i 15H 85 mm :£ 10^ Specimen no. 7 O 5 10 mm max F i g u r e 2.7 g.
— 10 1-55
7/// '--^x
/.// /
I / ' / • " • x / ^ - ^ " 5 H/ / / / /
/•/ // /
i! i/ i
Ull I
/.//
/
////
6-54,5^1
50 100 V [ mm ] 130 Figure 2.8. A x i a l f o r c e - m i d p o i n t d e f l e c t i o n curves f o r t h e s m a l l e s t cross s e c t i o n s o f specimens Nos 1-6.As load was increased, i t was noted t h a t t h e ends o f t h e studs r o t a t e d so t h a t c o n t a c t w i t h t h e base p l a t e was concentrated i n c r e a s i n g l y towards t h e o u t e r edge o f t h e s t u d , w h i l e on t h e o p p o s i t e edge a gap opened between stud and s o l e p l a t e and between s o l e p l a t e and base p l a t e . See Figure 2.9 and 2.10. Near f a i l u r e , t h i s gap extended r i g h t up t o t h e middle o f t h e cross s e c t i o n . I t i s q u i t e e v i d e n t t h a t t h e p o s i t i o n o f t h e a x i a l load was d i s p l a c e d d u r i n g t h e t e s t towards t h e o u t e r edge o f t h e s t u d .
The general s t a t e o f a f f a i r s r e g a r d i n g t h e p o s i t i o n o f t h e a x i a l f o r c e and the d e f l e c t i o n o f t h e stud i s s e t o u t i n Figure 2.11. I n Figure 2.12, t h e p o s i t i o n o f t h e a x i a l f o r c e and t h e g e o m e t r i c a l c e n t r e o f g r a v i t y i n t h e c e n t r e i s shown w i t h f u l l curves f o r t h e maximum a x i a l load f o r t h e d i f f e r e n t stages o f t h e cross s e c t i o n . The stages where t h e maximum load had n o t been reached a r e marked w i t h dashed curves. The d i s t a n c e between the two curves i n t h e diagrams r e p r e s e n t s t h e l e v e r arm o f t h e bending moment i n t h e c e n t r e .
F i g u r e 2.9.
Deformations a t t h e upper h o r i z o n t a l support i n spe-cimen No. 4-60 a t maximum l o a d .
Figure 2.10.
Deformations a t t h e bottom i n c l i n e d support i n s p e c i -men No. 4-60 a t maximum
•
Figure 2.11. P o s i t i o n s o f t h e a x i a l f o r c e xj^ and t h e geomet-r i c a l c e n t geomet-r e o f g geomet-r a v i t y xcQ f o r t h e m i d s e c t i o n i n the d e f l e c t e d p o s i t i o n .Figure 2.12 a-h. The p o s i t i o n o f t h e a x i a l f o r c e xj^ a t t h e bottom
support o f t h e stud and t h e p o s i t i o n o f t h e g e o m e t r i c a l c e n t r e o f g r a v i t y XQQ i n t h e c e n t r e . Except f o r t h e dashed p o r t i o n s where t h e maximum load had n o t been reached, t h e curves r e l a t e t o t h e maximum a x i a l f o r c e .
Specimen no. 1 Specimen no.2
Z N - Z j - Q l m m ]
^N'^CG """"
Specimen no.3
F i g u r e 2.12 c.
Specimen 00.4
Figure 2.12 e. S p e c i m e n no.5 100 ZN'Z(-Q (mm] 120 n 100 -50 ^ Specimen no, 6 50 WO 2 ^ ,2(- Q ["W"] 150 200 F i g u r e 2.12 f .
200 1 100 60 Specimen no. 7 200 100 Specimen no. 8 100 200 100 200 [mm] Figure 2.12 g. Figure 2.12 h.
The diaqraias show t h a t the p o s i t i o n of the a x i a l f o r c e f o r the whole cross s e c t i o n i s dependent on support c o n d i t i o n s . For a h o r i z o n t a l base p l a t e , the load was very near the c e n t r e i n the cases where rubber s t r i p s were used (specimens Nos. 2 and 3 ) , w h i l e i n specimen No. 1 where t h e s t i f f n e s s p r o p e r t i e s of the s o l e p l a t e e x e r t e d a l a r g e i n f l u e n c e i t had a more eccen-t r i c p o s i eccen-t i o n . I n eccen-t e s eccen-t s w i eccen-t h an i n c l i n e d base p l a eccen-t e , eccen-the e c c e n eccen-t r i c i eccen-t y of
the load was c o n s i d e r a b l e but f a i r l y independent of the o t h e r support con-d i t i o n s .
Displacement o f the p o s i t i o n of the a x i a l l o a d d u r i n g a p p l i c a t i o n o f the load i s shown i n Figure 2.13 f o r the l a s t cross s e c t i o n a l stage of the spe-cimens. Specimen No. 8 i s not i n c l u d e d since the cause o f f a i l u r e was reduc-t i o n i n s reduc-t r e n g reduc-t h of reduc-the web near reduc-the boreduc-treduc-tom supporreduc-t. See below.
For specimen No.4, i n which creep proceeded f o r 2 hours before f a i l u r e occurred, the p o s i t i o n of the a x i a l load and the g e o m e t r i c a l c e n t r o i d of the cross s e c t i o n are a l s o p l o t t e d as a f u n c t i o n o f the t i m e , measured from the time when the maximum load was a p p l i e d . See F i g u r e 2.14.
I n the l i g h t w e i g h t studs, specimens Nos. 7 and 8, obvious buckles were noted i n the web a f t e r one f l a n g e had been removed and the web thus had an un-supported edge. A f t e r f u r t h e r r e d u c t i o n of the cross s e c t i o n t h e r e were no more v i s i b l e buckles since the slenderness r a t i o of the web decreased. At the f i n a l stage of the t e s t , c o l l a p s e was i n i t i a t e d by compressive f a i l u r e i n the web.
100 90 H 80 70 H 60 „ 50 -E -° 40 30 H 20 10 -\ n 1 -55 o 2 -50 A 3 -55 O 4 -60 5 -55 + 6 54,5 X 7 -85 5 10 N [KNl 15 Figure 2.13. P o s i t i o n o f t h e a x i a l f o r c e as a f u n c t i o n o f load f o r t h e s m a l l e s t cross s e c t i o n s i n t h e t e s t s . E L 3 180 160-^ 140 120 100 ^ 80 ^ 0,5 1.0 1,5 Time ( h ] 2,0 2.5
Figure 2.14. Creep p r i o r t o f a i l u r e i n specimen No. 4. P o s i t i o n o f t h e a x i a l f o r c e and t h e g e o m e t r i c a l c e n t r e o f g r a v i t y o f t h e beam cross s e c t i o n as a f u n c t i o n o f time from t h e a p p l i c a t i o n o f t h e maximum load.
The web b u t t j o i n t i n specimen No 8 which was placed 260 mm from t h e bottom end o f t h e stud had a c o n s i d e r a b l e i n f l u e n c e on d e f o r m a t i o n a l behaviour, and t h i s was f u r t h e r accentuated because o f t h e i n c l i n a t i o n o f t h e base
p l a t e . When 40 mm o f one o f t h e f l a n g e s had been removed by p l a n i n g , t h e web was no longer capable o f t r a n s m i t t i n g t h e load t o t h e o t h e r whole
f l a n g e which was n o t i n c o n t a c t w i t h t h e i n c l i n e d base p l a t e i n i t i a l l y . De-f o r m a t i o n s however became so l a r g e t h a t t h e e n t i r e De-f l a n g e was soon De-f o r c e d a g a i n s t t h e base p l a t e , and t h e a x i a l load had moved t o t h e f l a n g e . A f t e r the l a s t pieces o f t h e f l a n g e had been removed, t h e a x i a l load was i n
approximately t h e same p o s i t i o n as f o r specimen No 7, see Figure 2.12 g and h. However, due t o l o c a l weakening a t t h e web b u t t j o i n t , t h e cross s e c t i o n
could n o t be reduced t o t h e same e x t e n t as i n specimen No 7.
2.4 E v a l u a t i o n 2.41 Bending s t i f f n e s s
I n t h e t e s t under t r a n s v e r s e l o a d i n g d e f l e c t i o n was recorded a t t h e mid-p o i n t o f t h e beam and a t a f u r t h e r f o u r mid-p o i n t s , see Figure 2.4. With t h e a i d o f these displacements, t h e bending s t i f f n e s s can be determined as t h e mean value over t h e gauge l e n g t h s S.^ = 600 mm and = 2000 mm. Over t h e gauge l e n g t h s t h e bending moment i s c o n s t a n t . We thus have
p V i
1 81w^
where w-) i s t h e r i s e o f t h e e l a s t i c l i n e over t h e gauge l e n g t h JK-j.
Over t h e gauge l e n g t h S.2 ^^e bending moment v a r i e s . With t h e a i d o f t h e energy e q u a t i o n a p p l i e d t o t h e moment d i s t r i b u t i o n s i n Figure 2.15, we have
10 -3 Iw,
where W2 i s t h e r i s e o f t h e e l a s t i c l i n e over t h e gauge l e n g t h $.2-P/2 P/2 y.
:L
A y 750 o . 750 . 750 „ 2000 4 ^ 500 mm F i g u r e 2.15. Moment diagrams f o r d e t e r m i n a t i o n o f t h e mean f l e x u r a l r i g i d i t y ( E I ) 2 over t h e gauge l e n g t hThe moduli o f e l a s t i c i t y were determined f o r both gauge l e n g t h s and f o r a l l stages o f t h e cross s e c t i o n i n specimens of Type 1. The values were found to have no a p p r e c i a b l e dependence on t h e depth h of the cross s e c t i o n . The v a r i a t i o n s which d i d occur were l a r g e l y due t o measuring e r r o r s caused by the l i m i t e d accuracy of readings by the l o a d c e l l s . Only the mean values of t h e moduli of e l a s t i c i t y and t h e standard d e v i a t i o n s are t h e r e f o r e s e t out f o r each specimen i n Table 2.5. The s c a t t e r i n t h e values o f E. i s
g r e a t e r than i n the case o f E^, s i n c e t h e i n c i d e n c e and p o s i t i o n or d e f e c t s had a g r e a t e r i n f l u e n c e over t h e s h o r t e r gauge l e n g t h Zy
I n t h e case of specimens Nos. 7 and 8 i t was found t h a t t h e load l e v e l had been t o o low and t h e l o a d values f a r t o o u n r e l i a b l e f o r values of t h e
f l e x u r a l r i g i d i t y t o be g i v e n f o r the specimens.
Table 2.5. R e s u l t s from d e t e r m i n a t i o n of t h e moduli o f e l a s t i c i t y f o r t h e gauge l e n g t h s = 600 mm and fi^ = 2000 mm, and a s s o c i a t e d standard d e v i a t i o n s . Specimen ^1 ^1 '2 N/mm2 N/mm2 N/mm2 N/mm2 1 9117 767 10751 339 2 14270 1173 9107 268 3 17033 1567 10425 361 4 13664 2415 9085 341 5 17777 1797 12313 413 6 16881 2015 11056 614 2.42 S t r e s s e s a t m i d s e c t i o n a t f a i l u r e
Once the p o s i t i o n o f t h e a x i a l f o r c e and m i d p o i n t d e f l e c t i o n s of the s t u d are known, i t i s p o s s i b l e t o c a l c u l a t e the maximum extreme f i b r e s t r e s s e s and a t t h e m i d s e c t i o n . This was done f o r specimens Nos, 1, 4, 5, 6 f o r loads v e r y near t h e u l t i m a t e l o a d N^, i . e . when t h e f l a t p o r t i o n of the curve had been reached, and f o r specimens Nos. 2 and 3 f o r t h e u l t i m a t e load Nu where the curve e x h i b i t e d a more or l e s s pronounced maximum. See F i g u r e 2.8. I n t h e c a l c u l a t i o n s , l i n e a r d i s t r i b u t i o n of s t r e s s over t h e cross s e c t i o n was assumed. The a x i a l f o r c e was assumed t o be a t t h e same p o s i t i o n a t b o t h s u p p o r t s . The e r r o r s which a r i s e i n t h i s way are l i k e l y t o be small since t h e measured d i s t a n c e s of t h e l o a d from t h e edge e x h i b i t f a i r l y s m a l l v a r i a t i o n s i n t h e v i c i n i t y o f t h e u l t i m a t e l o a d and are con-s i d e r a b l y l e con-s con-s than t h e m i d p o i n t dicon-splacementcon-s of the con-s t u d , con-see
F i g u r e 2.11. The r e s u l t s of c a l c u l a t i o n s are s e t o u t i n Table 2.6. The c a l c u l a t e d compressive s t r e s s e s 0^0 are a t a l e v e l t o be expected when the c h a r a c t e r i s t i c value o f compressive s t r e n g t h i s 15 N/mm2. I t i s
obvious t h a t f a i l u r e commenced on t h e compression s i d e s o f t h e studs. Owing t o t h e p l a s t i c deformations on the compression s i d e near f a i l u r e , t e n s i l e s t r e s s e s a l s o rose r a p i d l y , which e x p l a i n s why a crack a l s o occurred a t
f a i l u r e a t a knot on t h e t e n s i l e s i d e near t h e m i d s e c t i o n . Table 2.6. Extreme f i b r e s t r e s s e s a t m i d s e c t i o n near f a i l u r e
c a l c u l a t e d w i t h t h e a i d o f measured values o f t h e e c c e n t r i c i t y . Specimen N e kN mm 1 12,70^) 19,7-^) 2 9,22 38,0 3 12,36 34,1 4 12,97 19,4 5 12,93 35,3 6 10,96 51,3 omc, N/mm' omt N/mm^ 16,2 22,8 23,6 14,1-32,4^) 25,3-34,9^) 29,7 5,9 14,6 13,6 4,5-22,7^) 14,9-24,4^) 20,8 Value somewhat t o o low s i n c e t h e s t r o k e o f t h e displacement
transducer had been exceeded.
^) The values r e l a t e t o t h e beginning and end o f t h e creep stage a t a constant load l e v e l .
3. ANALYTICAL MODELS
3.1 Member w i t h p i n j o i n t e d end supports i n compression
The stud i s assumed t o be p i n j o i n t e d a t t h e ends. The Euler b u c k l i n g load can be c a l c u l a t e d w i t h t h e a i d o f t h e moduli o f e l a s t i c i t y determined i n the bending t e s t s . The value f o r t h e gauge l e n g t h = 2000 mm i s more r e l e v a n t f o r b u c k l i n g , and t h i s i s t h e r e f o r e used i n t n e c a l c u l a t i o n s . The Euler b u c k l i n g load i s thus
2
IT E^I
N^ =
— f
(3.1)*k
where fij^ = 2490 mm.
The r e s u l t s f o r t h e cross s e c t i o n s a t f a i l u r e f o r specimens Nos. 1-6 a r e s e t out i n Table 3.1 and a r e compared w i t h t h e u l t i m a t e loads N^ obtained i n the t e s t s . I n a l l cases, t h e c a l c u l a t e d c r i t i c a l load was exceeded i n t h e t e s t s . I t t h e r e f o r e f o l l o w s t h a t t h e assumption o f p i n j o i n t e d end supports i s very c o n s e r v a t i v e . I n a c t u a l f a c t , t h e c r i t i c a l a x i a l f o r c e should
c o n s i d e r a b l y exceed t h e u l t i m a t e load since t h e r e i s no pure b u c k l i n g case i n these circumstances.
Table 3.1. Comparison o f t h e values o f c r i t i c a l a x i a l f o r c e c a l c u l a t e d f o r studs w i t h p i n j o i n t e d ends and experimental u l t i m a t e loads. Specimen h mm N N u N N u ^E 1 55 10677 12800 1,20 2 50 6795 9220 1,35 3 55 10354 12360 1,20 4 60 11714 13000 1,11 5 55 12229 13000 1,06 6 54,5 10684 11070 1,04
Owing t o t h e i m p e r f e c t i o n s o f the s t u d e t c . the design compressive f o r c e N^^ i s lower than t h e Euler load Ng. The design loadbearing c a p a c i t y i s determined as
N , = k f bh cd c c
According t o t h e CIB Code /4/, kc can be c a l c u l a t e d as
f / T : r
k^ = o . 5 [ ( i + (1 + • p ) ^ ^ ) - V (1 + (1 + nA f ^ ) ^ ^ ) - 4kg]
2^ °E ^ ^0 where k„ = -z- = T — E f f c c X = slenderness r a t i o = Äj^/i i = r a d i u s o f g y r a t i o n = Euler b u c k l i n g s t r e s s f ^ = compressive s t r e n g t h p a r a l l e l t o t h e g r a i n f = bending s t r e n g t h p a r a l l e l t o t h e g r a i n m The i n i t i a l c u r v a t u r e i s expressed as e = qrX where r = r a d i u s o f t h e core.
The design l o a d b e a r i n g c a p a c i t y was c a l c u l a t e d on t h e basis o f t h e f o l l o w i n g assumptions:
a) The maximum e x p e r i m e n t a l compressive s t r e s s e s a c c o r d i n g t o Table 2.6 were used as t h e compressive s t r e n g t h p a r a l l e l t o t h e g r a i n . For
specimens Nos. 4 and 5 t h e value a t t h e beginning o f t h e creep stage was used.
b) The r a t i o f c / f m P"^ equal t o 1.
c) The experimental values £^ a c c o r d i n g t o Table 2.5 were used as t h modulus o f e l a s t i c i t y .
d) The i n i t i a l c u r v a t u r e i s t h e same as t h e r i s e a t t h e c e n t r e o f t h e stud = Ij^/IOOO. The i n i t i a l c u r v a t u r e o f t h e specimens was n o t measured, b u t s i n c e they were v e r y s t r a i g h t , t h e value chosen i s a reasonable assumption.
The r e s u l t s o f c a l c u l a t i o n s a r e s e t o u t i n Table 3.2.
Table 3.2. Comparison o f t h e design l o a d b e a r i n g c a p a c i t y ti^å
c a l c u l a t e d a c c o r d i n g t o /4/ and t h e e x p e r i m e n t a l u l t i m a t e loads Ny. Specimen N , N ca u,exp cd kN 1 9746 1,313 2 6501 1,418 3 9788 1,263 A 10590 1,228 5 11483 1,132 6 10206 1,085
3.2 Member w i t h c y l i n d r i c a l convex end s u r f a c e s i n compression 3.21 C r i t i c a l load
During t h e t e s t s i t was noted t h a t t h e ends o f t h e studs n o t o n l y r o t a t e d but a l s o performed a r o l l i n g motion l i k e a wheel. I n order t h a t t h i s mechanism may be d e s c r i b e d i n a simple manner, i t may be assumed t h a t t h e
end surfaces o f t h e s t u d or t h e surfaces o f t h e base a r e convex and
c y l i n d r i c a l , see F i g u r e 3.1. The way t h i s i d e a l i s a t i o n i s c a r r i e d o u t i s d e s c r i b e d i n Subsection 3.23.
1'
F i g u r 3.1. Convex c y l i n d r i c a l c o n t a c t surfaces a t t h e end support of t h e member.
As an a n l y s i s o f a s t u d i n compression, o f such a shape, i s c a r r i e d o u t i n Appendices A1-A3. The c r i t i c a l a x i a l f o r c e , i . e . t h e b i f u r c a t i o n l o a d or Euler load Ng, i s c a l c u l a t e d as
2
The e f f e c t i v e l e n g t h JKj^ = 0£ i s dependent on t h e r a d i u s r o f t h e c o n t a c t s u r f a c e and i s g i v e n i n F i g u r e A3. The l e n g t h o f t h e member i s JK, see Figure A1. I t i s e v i d e n t t h a t t h e r e a r e two l i m i t i n g cases. The f i r s t occurs when r = 0 and corresponds t o E u l e r Case 2, w i t h p i n j o i n t e d ends and an e f f e c t i v e l e n g t h = JZ. The o t h e r extreme case i m p l i e s t h a t r = - and corresponds t o Euler Case 4, w i t h f i x e d ends and t h e e f f e c t i v e l e n g t h %Y = Ä/2.
3.22 E c c e n t r i c c o m p r e s s i v e f o r c e
During stages o f t h e cross s e c t i o n which occur when t h e member chars due t o exposure t o f i r e on one s i d e , l o a d i n g i s e c c e n t r i c w i t h an e c c e n t r i c i t y a w i t h r e s p e c t t o t h e remaining cross s e c t i o n . See F i g u r e A4. The s e c t i o n p r o p e r t i e s o f t h e member can be c a l c u l a t e d when i t s d e f l e c t i o n i s known. A p p l y i n g second order t h e o r y , see Appendix A2, t h e d e f l e c t i o n a t t h e m i d p o i n t i s
fl r a t a n —
^<f^ = % a x = - - ^> ^^-2) 1 + r a t a n ^ 2
and t h e r o t a t i o n a t t h e end o f t h e member i s aa t a n ^
1 + r a t a n — where a = / ^
For an i n c l i n e d support, see F i g u r e A6, t h e r e i s a l o a d e c c e n t r i c i t y o f magnitude 8r where 6 i s t h e magnitude o f t h e i n c l i n a t i o n . The same formulae as above can thus be used, t h e e c c e n t r i c u t y being r e p l a c e d by 8 r .
3.23 D e t e r n i n a t i o n o f t h e i ^ ^
I n a c t u a l f a c t , t h e shape o f t h e support o f t h e s t u d i s d i f f e r e n t from t h e i d e a l i s e d assumption t h a t t h e end s u r f a c e i s c y l i n d r i c a l . W i t h t h e a i d o f the experimental r e l a t i o n s h i p s between a x i a l f o r c e and m i d p o i n t d e f l e c t i o n , see F i g u r e 2.7, and t h e t h e o r e t i c a l e x p r e s s i o n f o r v^ax Equation
( 3 . 2 ) , c a l c u l a t i o n s were made o f t h e r a d i i r ^ o f t h e c o n t a c t s u r f a c e as a f u n c t i o n o f Vjj^jj which s a t i s f i e s t h e c o n d i t i o n
c a l c u l a t e d _ e x p e r i m e n t a l ^max " ^max
i . e . t h a t g e o m e t r i c a l shape o f t h e end s u r f a c e s o f t h e s t u d was c a l c u l a t e d which would have produced t h e same d e f l e c t i o n o f t h e s t u d as t h a t o b t a i n e d i n t h e t e s t s .
I n t h e c a l c u l a t i o n s t h e i n i t i a l c u r v a t u r e o f t h e s t u d was i g n o r e d s i n c e t h e specimens were very s t r a i g h t . The e c c e n t r i c i t y was thus
a = " 2 — + 8 r i
Since specimens Nos.4-6 had an i n c l i n e d s u p p o r t a t t h e bottom o f t h e s t u d and a h o r i z o n t a l s u p p o r t a t t h e upper end, t h e approximate v a l u e 8 = 0.0175 was used, i . e . t h e mean value o f t h e two support i n c l i n a t i o n s .
The r e s u l t s o f t h i s a n a l y s i s a r e s e t o u t i n F i g u r e 3.2 a-£. I t i s a common c h a r a c t e r i s t i c o f specimens Nos. 1-3 t h a t t h e r a d i u s r ^ assumes v e r y l a r g e values f o r s m a l l d e f l e c t i o n s . For specimens Nos. 4-6 t h e r a d i u s does n o t assume v e r y l a r g e values u n t i l t h e m i d p o i n t d e f l e c t i o n i s between 11 and
16 mm. A t t h i s p o i n t t h e r o t a t i o n v ' ( 0 ) o f t h e end o f t h e member i s
a p p r o x i m a t e l y 0.0175, i . e . t h e scune as t h e mean v a l u e o f t h e i n c l i n a t i o n s of t h e support p l a t e s . A f t e r t h i s t h e r a d i u s continues t o decrease, and a t t h e same time t h e m i d p o i n t d e f l e c t i o n Vj,ax i n c r e a s e s . This c h a r a c t e r i s t i c shape o f t h e curves i s p a r t i c u l a r l y pronounced i n specimens Nos. 1, 3, 4 and
5, i . e . t h e ones which were f i t t e d w i t h 120 fflin wide rubber s e a l i n g s t r i p s or had no s t r i p s a t a l l . I n specinens Nos. 2 and 6 which had t h e narrower,
o n l y 70 mm wide rubber p r o f i l e s , t h e curves i n some cases had a maximua which was however c l e a r l y d i f f e r e n t from t h e h i g h v a l u e s o b t a i n f o r t h e o t h e r specimens.
A t t h i s stage t h e s t u d was " b a l a n c i n g " on t h e narrow rubber p r o f i l e . For very l a r g e d e f l e c t i o n s , t h e curves f o r a l l specimens e x h i b i t a p p r o x i m a t e l y the same v a l u e s o f t h e i d e a l r a d i u s . I n t h i s p o s i t i o n , even i n t h e case o f specimens Nos. 2 and 6, t h e end o f t h e s t u d had r o t a t e d so much t h a t good c o n t a c t had been o b t a i n e d between t h e end o f t h e s o l e p l a t e and t h e end p l a t e . T h i s e v a l u a t i o n o f t h e t e s t r e s u l t s was n o t c a r r i e d o u t f o r
specimens Nos. 7 and 8 s i n c e t h e i r bending s t i f f n e s s was n o t determined w i t h s u f f i c i e n t accuracy. See Subsection 2.41.
I n some cases, s m a l l i d e a l r a d i i were c a l c u l a t e d f o r s m a l l m i d p o i n t d e f l e c t i o n s Vj^g^j^. The probable reason f o r t h i s i s t h a t t h e s m a l l a x i a l
f o r c e was n o t enough t o compress t h e rubber s e a l s u f f i c i e n t l y . See t h e r e s u l t s f o r specimens Nos. 1 and 2 i n F i g u r e 3.2a and 3.2c.
The e x p e r i m e n t a l r e l a t i o n s h i p s between t h e i d e a l r a d i u s and t h e m i d p o i n t d e f l e c t i o n , c a l c u l a t e d w i t h t h e a i d o f t h e a n a l y t i c a l model, e x h i b i t t h e same c h a r a c t e r i s t i c shape. P a r t i c u l a r l y f o r l a r g e v a l u e s o f Vj,ax» ^^e d i f f e r e n c e s between t h e d i f f e r e n t curves r e p r e s e n t i n g t h e l a s t c r o s s s e c t i o n a l stage a r e f a i r l y s m a l l . F i g u r e 3.2 a - f . I d e a l r a d i u s - m i d p o i n t d e f l e c t i o n curves f o r t h e s t u d s . 4000 3000^ 20001 lOOOi 500 Equation (3.4) F i g u r e 3.2. a 10 20 30 40 50 "max [mm]
4000^ 3500 3000^ 2500^ 2000-^ 1500^ 1000-^ F i g u r e 3.2. b 2 - 5 0 2 - 6 0 2 - 7 0 2 - 8 0 v 2 - 9 0 X 2-100 500 10 20 30 40 50 3500 41 3000 2500 ^ 2000-^ 1500
H
1000 500 Equation (3.4) 3 - 6 0 3 - 70 o 3 - 8 0 v 3 - 9 0 3-100 F i g u r e 3.2. c5000 4000 i 3000^ 2000H 1000^ Equation (3.4) O 10 20 30 40 50 60 70 80 D 4-60 ^ 4-70 : 4-80 5000 4000 3000 -2000 1000 Equation (3.4) o 5-55 5-60 'i 5-70 O 10 20 30 40 50 60 70 Figure 3.2. d F i g u r e 3.2. e 5000 4000-3000 2000 1000 Equation (3.4) o 6-54,5 6-60 ^ 6-70 o 6-80 O ~ 1 0 20 30 40 50 60 70 F i g u r e 3.2. f Vfnax''^'"'
I n order t h a t these curves may be used i n c a l c u l a t i n g t h e l o a d b e a r i n g
c a p a c i t y , an approximate expression was determined f o r t h e i d e a l r a d i u s by curve f i t t i n g . P a r t i c u l a r a t t e n t i o n was p a i d t o values i n t h e v i c i n i t y o f
the u l t i m a t e stage. The proposed expression i s
where v ^ ^ j ^ and r ^ a r e i n mm.
I n t h e f o r m u l a , t h e i n c l i n a t i o n o f t h e support i s taken i n t o c o n s i d e r a t i o n by u t i l i s i n g t h e r e l a t i o n s h i p between t h e r o t a t i o n a t t h e end support and the m i d p o i n t d e f l e c t i o n f o r an assumed s i n u s o i d a l e l a s t i c l i n e . The d e f l e c t i o n i s then d e s c r i b e d as (see Figure A1)
V = V s i n ^ X max 2
and t h e r o t a t i o n a t t h e end support i s V (0) = ^ V 2 max Thus V = V * ( 0 ) -max n W i t h S. = 2490 mm, we have ^m.vmax = "^^3 v ' ( 0 ) (3.5) Expression (3.4) f o r t h e i d e a l r a d i u s i s p l o t t e d w i t h dashed l i n e s i n F i g u r e 3.2 a - f .
With t h e a i d o f t h e approximate expression (3.4) f o r t h e i d e a l r a d i u s , t h e t h e o r e t i c a l r e l a t i o n s h i p between t h e a x i a l f o r c e and t h e m i d p o i n t
d e f l e c t i o n , and t h e t h e o r e t i c a l u l t i m a t e l o a d , was then determined f o r specimens Nos. 1-6. The t h e o r e t i c a l u l t i m a t e load i s d e f i n e d as t h e load a t which t h e maximum compressive s t r e s s a t t h e c e n t r e o f t h e member i s equal t o t h e e x p e r i m e n t a l values Ojg^^ s e t o u t i n Table 2.6. I n t h e c a l c u l a t i o n s , the moduli o f e l a s t i c i t y a c c o r d i n g t o Table 2.5 were used.
Since t h e i d e a l r a d i u s i s dependent on d e f l e c t i o n ( i n a c t u a l f a c t i t i s dependent on end r o t a t i o n ) , t h e c o r r e c t i d e a l r a d i u s i s c a l c u l a t e d by i t e r a t i o n . A f t e r choosing an i n i t i a l value f o r r ^ , v^^^ i s c a l c u l a t e d a c c o r d i n g t o ( 3 . 2 ) , whereupon a new value o f r j ^ i s determined a c c o r d i n g t o ( 3 . 4 ) , and t h e procedure i s repeated u n t i l t h e r e i s good agreement between t h e new and o l d v a l u e o f r ^ .
The r e s u l t s o f these c a l c u l a t i o n s a r e s e t o u t i n F i g u r e 3.3 a - f and i n Table 3.3 where t h e values o f t h e c r i t i c a l l o a d Ng and t h e t h e o r e t i c a l u l t i m a t e load t h e o r t a b u l a t e d . The l a t t e r i s a l s o marked on t h e a p p r o p r i a t e t h e o r e t i c a l l o a d - d e f l e c t i o n curve. The c r i t i c a l l o a d N£ i s
c a l c u l a t e d f o r t h e same i d e a l r a d i u s which occurs when t h e l o a d i s equal t o the u l t i m a t e l o a d .
The agreement between t h e experimental and c a l c u l a t e d l o a d - d e f l e c t i o n
curves i s good f o r specimens Nos. 1-3 and 5, and l e s s good f o r specimens Nos. 4 and 6. I t was o n l y i n t h e case o f specimen No. 6, f o r which t h e narrow, o n l y 70 mm wide, rubber p r o f i l e had been used, t h a t values which were o b v i o u s l y on t h e unsafe s i d e were o b t a i n e d .
Figure 3.3 a - f . C a l c u l a t e d and experimental l o a d - d e f l e c t i o n curves. The u l t i m a t e load a c c o r d i n g t o Table 3.3 i s marked on the t h e o r e t i c a l curve. 15 1 10 5 Specimen 1-55 15n 0 10 20 30 40 50 Specimen 2-50 0 10 20 30 40 50 60 vmax ^^^^ Figure 3.3. a Specimen 3-55 ° 0 10 20 30 40 50 60 vmax Inim] o - o F i g u r e 3.3. b Specimen 4-60 T 1 r 0 10 20 30 40 50 60 70 80 Vmax [mm] Figure 3.3. c Figure 3.3. d
-1 1 r
0 ^0 20 30 UO
50 60 70 80 'max [mm] 0 10 20 30 40 50 60 70 80 vmox l")m] F i g u r e 3.3. e F i g u r e 3.3. fTable 3.3. Conparison o f c a l c u l a t e d and experiiaental u l t i m a t e loads a c c o r d i n g t o t h e a n a l y t i c a l model "member w i t h c y l i n d -r i c a l end s u -r f a c e s " .
Specimen
N N
u,exp N u,theor N u,theor
u,theor mm kN kN kN 1 55 32416 10974 12800 1,157 0,339 2 50 17528 8177 9220 1,128 0,467 3 55 28575 12428 12360 0,995 0,435 4 60 39900 9915 13000 1,308 0,325 5 55 37441 12295 13000 1,052 0,345 6 54,5 29950 11975 11070 0,915 0,366
3.24 The i n f l u e n c e o f t h e stud cross s e c t i o n
Owing t o r o t a t i o n o f t h e stud a t t h e support, t h e p o i n t o f c o n t a c t w i t h t h e support, i . e . t h e p o s i t i o n o f t h e a x i a l f o r c e , i s i n c r e a s i n g l y d i s p l a c e d towards one o f the edges. At t h e o p p o s i t e edge an i n c r e a s i n g l y wide gap i s formed between t h e end s u r f a c e o f t h e stud and t h e base, and a t u l t i m a t e stage t h i s gap may extend as f a r as t h e middle o f t h e cross s e c t i o n , see F i g u r e 2.9 and 2.10. T h i s means t h a t t h e i . i t e r n a l f o r c e s o f t h e stud a t t h i s stage a r e independent o f t h e o r i g i n a l depth o f cross s e c t i o n , and t h a t
Equation (3.4) can be used a l s o when t h e o r i g i n a l depth i s d i f f e r e n t from 120 mm.
3.25 The i n f l u e n c e o f t h e l e n g t h o f _ t h ^
Expression (3.4) f o r t h e i d e a l r a d i u s was determined f o r t h e stud l e n g t h S. = 2490 mm. I n order t h a t t h i s formula may be used f o r o t h e r stud l e n g t h s , i t must be m o d i f i e d .
For a member w i t h rounded ends, t h e instantaneous r a d i u s which holds f o r a c e r t a i n load i s dependent o n l y on t h e end r o t a t i o n v ' ( 0 ) . When two members
(1) and (2) o f l e n g t h s fi-) and S.2 r e s p e c t i v e l y have t h e same end r o t a t i o n v ' ( 0 ) , see F i g u r e 3.4, we have t h e f o l l o w i n g r e l a t i o n s h i p when t h e d e f l e c t i o n curves v ( x ) a r e a f f i n e :
Vmax1 ^ h Vmax2 ^2
When t h e i d e a l r a d i u s i s expressed as a f u n c t i o n o f v,nax' i n Equation ( 3 . 4 ) , v^ax i s m u l t i p l i e d by t h e term fi/2490 where £ i s t h e a c t u a l s t u d l e n g t h i n mm. I n c l i n a t i o n o f t h e support i s taken i n t o c o n s i d e r a t i o n by m u l t i p l y i n g i t by fi/n. We thus have: r. = 1 (V 7300 max 2490 - ^ 8 IT 0,6 (3.8) I f t h e i d e a l r a d i u s i s expressed as a f u n c t i o n o f t h e end r o t a t i o n v ' ( 0 ) , a c o r r e c t i o n w i t h r e s p e c t t o d i f f e r e n t stud l e n g t h s i s n o t necessary. Figure 3.4. Maximum d e f l e c t i o n s o f members o f d i f f e r e n t lenghths f o r a f f i n e d e f l e c t i o n curves.
3.26 The i n f l u e n c e o f support !^ c^P§9i^5J In o r d e r t o e l u c i d a t e t h e i n f l u e n c e of support i n c l i n a t i o n on t h e l o a d
-b e a r i n g c a p a c i t y , t h e u l t i m a t e load was c a l c u l a t e d f o r a s t u d o f t h e dimen-s i o n dimen-s and m a t e r i a l valuedimen-s corredimen-sponding t o thodimen-se o f dimen-specimen No.6-54.5. The u l t i m a t e l o a d was c a l c u l a t e d f o r d i f f e r e n t support i n c l i n a t i o n s 8 and i s p l o t t e d i n F i g u r e 3.5 i n r e l a t i o n t o t h e u l t i m a t e l o a d a t 8 = 0. The maximum compressive s t r e s s o^q = 2 9 , 7 N/mm2 was used as t h e f a i l u r e c r i t e r i o n .
o
F i g u r e 3.5. The i n f l u e n c e o f t h e i n c l i n a t i o n o f t h e support p l a t e s on t h e u l t i m a t e load o f a w a l l s t u d w i t h t h e cross s e c t i o n a l and m a t e r i a l data a c c o r d i n g t o specimen No. 6-54.5.
The support may have an i n c l i n a t i o n f o r a number o f reasons. I t i s probable, however, t h a t l a c k o f dimensional accuracy i n t h e t i m b e r can be excluded as one o f these. Most studs a r e a t present c u t w i t h a saw on a bench. I t i s o n l y i n e x c e p t i o n a l cases t h a t t h e t i m b e r i s c u t by hand and t h e r e i s a r i s k of u n i n t e n t i o n a l skew c u t t i n g . Since s o l e p l a t e s a r e a l s o planed, they have v e r y good accuracy. On t h e o t h e r hand, i r r e g u l a r i t i e s i n t h e c o n c r e t e s l a b or a d e v i a t i o n i n i t s s u r f a c e from t h e h o r i z o n t a l may g i v e r i s e t o s u p p o r t i n c l i n a t i o n . The aim i n modern b u i l d i n g i s t o a v o i d such d e f e c t s . Concrete f l o o r s a r e g i v e n i n t e n t i o n a l slopes o n l y i n t h e v i c i n i t y o f f l o o r g u l l e y s . With studs spaced a t 600 mm, i t i s p o s s i b l e f o r two a d j a c e n t studs t o be
s i t u a t e d i n such an area. However, such an i n c l i n a t i o n has no u n f a v o u r a b l e e f f e c t on t h e l o a d b e a r i n g c a p a c i t y o f t h e e x t e r n a l w a l l i n t h e event o f f i r e , s i n c e t h e base i s i n c l i n e d towards t h e b u i l d i n g on t h e s i d e epxosed t o f i r e . I n t h e case o f core w a l l s , however, t h e f l o o r g u l l e y may be placed on the s i d e which i s n o t exposed t o f i r e , and i n such a case i t may have an u n f a v o u r a b l e e f f e c t .
A t y p i c a l v a l u e f o r t h e support i n c l i n a t i o n s which a r e p o s s i b l e i n p r a c t i c e due t o t h e d e f l e c t i o n s o f r o o f t r u s s e s can be g i v e n w i t h r e g a r d t o t h e f a c t t h a t these a r e o f t e n l i m i t e d t o L/300 where L i s t h e span o f t h e t r u s s . Such a d e f l e c t i o n i m p l i e s t h a t t h e r o t a t i o n o f t h e r a f t e r o f t h e t r u s s i s
8 « 0.013 a t i t s end s u p p o r t . I f t h e bottom s u p p o r t i s h o r i z o n t a l , t h e n t h e s t u d can be a p p r o x i m a t e l y replaced by one whose both supports have t h e i n c l i n a t i o n 6 = 0.065. According t o F i g u r e 3.5, t h e r a t i o Ny/Ny^O * 0.96, i . e . t h e i n f l u e n c e o f t h e support i n c l i n a t i o n i s v e r y s m a l l .
