Jurgen König
Thin-walled Wood-based
Ranges in Composite Beams
Paper Presented at CIB- W18A,
Meeting 22, Berlin, September 1989
Trätek
s T I T U T E T FÖR TRÄTEKNISK F O R S K N I IJiirgen König
THIN-WALI.ED WOOD-BASED FLANGES I N COMPOSITE BEAMS
Paper presented a t CIB-W18A, Meeting 22, B e r l i n , September 1989 TrateknikCentrum, Rapport I 8911052 Nyckelord beams buckling oomposi-te aoti-on flange curling wood Stockholm, September 1989
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SAMMANFATTNING
Denna r a p p o r t s y f t a r t i l l e t t e f f e k t i v a r e och säkrare u t n y t t j a n d e av tunna träbaserade d e l a r i bärande k o n s t r u k t i o n e r .
Bärförmågan av tunnväggiga flänsar av träbaserat s k i v m a t e r i a l i sammansatta balktvärsnitt kan reduceras på grund av s k j u v d e f o r m a t i o n e r (shear l a g ) , b u c k l i n g och skålning ( f l a n g e c u r l i n g ) . Tryckta flänsar har dock en be-tydande bärförmåga även när flänsen b u c k l a t . E f f e k t i v a - b r e d d - k o n c e p t e t som är väletablerat inom tunnplåtsområdet kan användas även v i d dimensionering med avseende på b u c k l i n g av träbaserade s k i v o r . Detta v i s a s genom en utvär-d e r i n g av försök genomförautvär-d v i utvär-d NNI i D e l f t , H o l l a n utvär-d . Dimensioneringsform-l e r för skåDimensioneringsform-lningens i n v e r k a n härDimensioneringsform-leds och d i m e n s i o n e r i n g s k r i t e r i e r föresDimensioneringsform-lås.
CIB-W18A/22-10-3
INTERNATIONAL COUNCIL FOR BUILDING RESEARCH STUDIES AND DOCUMENTATION
WORKING COMMISSION W18A - TIMBER STRUCTURES
THIN-WALLED WOOD-BASED FLANGES IN COMPOSITE BEAMS
by J König
Swedish I n s t i t u t e f o r Wood Technology Research Sweden
MEETING TWENTY-TWO BERLIN
CONTENT Summary Page 1 1. I n t r o d u c t i o n 1 2. P o s t c r i t i c a l l o a d b e a r i n g c a p a c i t y a f t e r b u c k l i n g o f a f l a n g e i n compression 2 2.1 D e f i n i t i o n o f the e f f e c t i v e w i d t h 2 2.2 E v a l u a t i o n o f t e s t s c a r r i e d o u t a t D e l f t 3 2.3 Proposed design r e g u l a t i o n s 6 3. Flange c u r l i n g ^ References 13
SUMMARY
The l o a d b e a r i n g c a p a c i t y of t h i n - w a l l e d wood-based f l a n g e s i n composite beams can be reduced due t o shear l a g , b u c k l i n g and f l a n g e c u r l i n g . How-ever, f l a n g e s i n compression have c o n s i d e r a b l e l o a d b e a r i n g c a p a c i t y even a f t e r the f l a n g e has buckled. The e f f e c t i v e w i d t h concept which i s w e l l e s t a b l i s h e d i n the f i e l d of sheet metal c o n s t r u c t i o n can a l s o be used i n d e s i g n i n g wood-based board m a t e r i a l w i t h respect t o b u c k l i n g . This i s shown by an e v a l u a t i o n o f t e s t s c a r r i e d out a t D e l f t . Design formulae f o r t h e e f f e c t of f l a n g e c u r l i n g are d e r i v e d , and design c r i t e r i a are proposed.
1. INTRODUCTION
I n the f i e l d of timber c o n s t r u c t i o n , s t r u c t u r a l elements comprising t h i n -w a l l e d components of -wood-based board M a t e r i a l have been used f o r some time. These s t r u c t u r a l elements are c h a r a c t e r i s t i c f o r l i g h t w e i g h t
c o n s t r u c t i o n , and i t i s expected t h a t they w i l l be used more e x t e n s i v e l y i n the f u t u r e .
T y p i c a l examples are composite s e c t i o n s comprising webs of s o l i d wood and f l a n g e s of board m a t e r i a l . The webs may a l s o be made up of components of board m a t e r i a l and s o l i d wood, see f i g u r e 1. However, owing t o the small t h i c k n e s s o f the board m a t e r i a l , t h e r e are a number of p r o p e r t i e s which demand more accurate a n a l y s i s i f the m a t e r i a l i s t o be u t i l i s e d i n the optimum manner. The f l a n g e o f t e n has the d u t y of d i s t r i b u t i n g load i n the t r a n s v e r s e d i r e c t i o n . I n f l o o r s , c o n s i d e r a t i o n must a l s o be given t o the s t i f f n e s s requirement which determines the t h i c k n e s s of t h e f l a n g e . I n these s t r u c t u r e s i t i s g e n e r a l l y o n l y the shear l a g i n the board m a t e r i a l which l i m i t s the loadbearing c a p a c i t y of the f l a n g e when the f l a n g e s are very wide i n p r o p o r t i o n t o the span o f the beam. I n o t h e r s t r u c t u r e s , f o r
instance i n r o o f c a s s e t t e s , the load d i s t r i b u t i o n task may be of
subordinate importance and c e r t a i n deformations may be t o l e r a t e d . F i n a l l y ,
d i r e c t a p p l i c a t i o n of load on the f l a n g e can o f t e n be avoided, f o r i n s t a n c e by a p p l i c a t i o n of the load over the web of the composite s e c t i o n or on the bottom f l a n g e of t h e s e c t i o n . Flanges i n compression may a l s o be s u b j e c t t o b u c k l i n g . When the beam i s curved, or becomes curved owing t o the e x t e r n a l l o a d , t h e f l a n g e s are deformed inwards towards the n e u t r a l a x i s o f the cross s e c t i o n ( f l a n g e c u r l i n g ) , and the c o n t r i b u t i o n of the f l a n g e s t o the l o a d b e a r i n g c a p a c i t y and s t i f f n e s s of the box s e c t i o n i s t h e r e f o r e reduced. I t i s w e l l known t h a t t h i n - w a l l e d p l a t e s i n compression have c o n s i d e r a b l e p o s t c r i t i c a l loadbearing c a p a c i t y , i . e . the load can be f u r t h e r increased a f t e r the c r i t i c a l b u c k l i n g s t r e s s of the p l a t e has been reached. This f a v o u r a b l e behaviour has been u t i l i s e d f o r a long t i m e , f i r s t i n aeroplane c o n s t r u c t i o n and since the end of the 1940s a l s o i n b u i l d i n g c o n s t r u c t i o n , mainly i n t h i n - w a l l e d s t r u c t u r a l elements of s t e e l and aluminium s h e e t i n g .
I n the f i e l d of timber c o n s t r u c t i o n a l s o the p o s t c r i t i c a l loadbearing c a p a c i t y o f buckled p l a t e elements has been known. I n order t h a t t h i s phenomenon may be u t i l i s e d t o some e x t e n t , the c o e f f i c i e n t of s a f e t y i s s l i g h t l y reduced i n the s t r u c t u r a l r e g u l a t i o n s f o r timber i n most
c o u n t r i e s .
I n accordance w i t h the CIB S t r u c t u r a l Timber Code / 1 / , i t i s p e r m i s s i b l e t o u t i l i s e the p o s t c r i t i c a l loadbearing c a p a c i t y . The code l a y s down some values r e g a r d i n g the e f f e c t i v e w i d t h of the f l a n g e , but these values are u n d i f f e r e n t i a t e d since n e i t h e r the c r i t i c a l b u c k l i n g load of the f l a n g e nor i t s compressive s t r e n g t h i s taken i n t o c o n s i d e r a t i o n . As f a r as the author i s aware, the e f f e c t of these i s r e f l e c t e d o n l y i n the Swiss timber code SIA 164 (1981) /5/. I t i s however not known whether the r u l e s i n the code had been v e r i f i e d f o r wood-based board m a t e r i a l s when the code was
w r i t t e n . I n /6/, which gives t h e background t o the Swiss code, no reference i s made t o t h i s .
The research r e p o r t s which deal w i t h the b u c k l i n g of wood-based boards or t h i n - w a l l e d components i n s t r u c t u r a l elements mainly c o n f i n e them-selves t o the d e t e r m i n a t i o n o f the c r i t i c a l b u c k l i n g l o a d , see e.g. /2/, /3/ and /4/. While most s t r u c t u r a l r e g u l a t i o n s f o r timber s t r u c t u r e s take account of the f a c t t h a t the l o a d b e a r i n g c a p a c i t y of t h i n - w a l l e d f l a n g e s i s l i m i t e d due t o shear l a g i n the board m a t e r i a l , t h e r e i s no requirement, as f a r as the author i s aware, t h a t the e f f e c t of f l a n g e c u r l i n g on the l o a d b e a r i n g capa-c i t y of the capa-composite s e capa-c t i o n should be capa-checapa-cked.
I t i s shown below t h a t t h e e f f e c t i v e w i d t h concept i n u t i l i s i n g the post-c r i t i post-c a l r e g i o n post-can a l s o be a p p l i e d t o wood-based t h i n - w a l l e d iflanges. Design r u l e s are a l s o g i v e n which take the e f f e c t of f l a n g e c u r l i n g i n t o c o n s i d e r a t i o n . The e f f e c t of shear l a g i n wood-based board m a t e r i a l s i s a l r e a d y w e l l documented and r e q u i r e s no f u r t h e r e l u c i d a t i o n .
2. POSTCRITICAL LOADBEARING CAPACITY AFTER BUCKLING OF A FLANGE IN COMPRESSION
2.1 D e f i n i t i o n of the e f f e c t i v e w i d t h
-The e f f e c t i v e w i d t h concept i s w e l l e s t a b l i s h e d i n t h e f i e l d of sheet metal c o n s t r u c t i o n . The e f f e c t i v e w i d t h bgf i s d e f i n e d so t h a t the f o l l o w i n g r e l a t i o n s h i p holds (see f i g u r e 2 ) :
The formula which i s most w i d e l y used a t present f o r d e t e r m i n a t i o n o f t h e e f f e c t i v e w i d t h i s t h a t determined e m p i r i c a l l y by Winter /7/ and l a t e r somewhat m o d i f i e d i n t h e A I S I Code (1968) /8/:
^ (1 - 0.227
e V e
(2) where o cr i s the c r i t i c a l buckling stress
Ö i s t h e compressive edge s t r e s s i n t h e buckled s t a t e , see
f i g u r e 1.
Expression (1) i s a development o f t h e formula according t o von Karman: b
b cr (3)
-A
Figure 2. D e f i n i t i o n o f t h e e f f e c t i v e w i d t h be. The c o n d i t i o n b
^ a (y)dy - b must be s a t i s f i e d .
According t o t h i s formula, t h e l o a d b e a r i n g c a p a c i t y o f a p l a t e i n
compression i s n o t reduced u n t i l t h e c r i t i c a l b u c k l i n g s t r e s s i s reached, w h i l e Winters formula takes account o f i m p e r f e c t i o n s which reduce t h e
l o a d b e a r i n g c a p a c i t y a t compressive s t r e s s e s which a r e lower than t h e c r i t i c a l b u c k l i n g s t r e s s .
2.2 E v a l u a t i o n o f t e s t s c a r r i e d o u t a t D e l f t
Dekker e t a l (1978) /9/ c a r r i e d o u t a t e s t s e r i e s on 100 sheets o f plywood loaded i n compression. The o b j e c t o f t h i s t e s t s e r i e s was t o
i n v e s t i g a t e whether t h e l i n e a r b u c k l i n g t h e o r y could be a p p l i e d t o plywood a l s o . I n t h e f o l l o w i n g , o n l y t h e data and r e s u l t s which a r e i m p o r t a n t w i t h regard t o e v a l u a t i o n w i t h r e s p e c t t o t h e p o s t c r i t i c a l l o a d b e a r i n g c a p a c i t y are given.
The m a t e r i a l i n t h e t e s t s was Canadian Oregon Pipe Plywood, Select
Sheathing, E x t . 1". Two d i f f e r e n t w i d t h s , 600 and 400 mm, and two d i f f e r e n t nominal t h i c k n e s s e s , 8 and 13 mm, were used. The aspect r a t i o a/b v a r i e d between 0.5 and 4 5 . One h a l f o f t h e t e s t specimens were simply supported on f o u r s i d e s , w h i l e t h e unloaded edges o f t h e other h a l f were subjected t o some r e s t r a i n t . Since t h e r e are no data r e g a r d i n g t h e degree o f r e s t r a i n t , these specimens a r e n o t i n c l u d e d i n t h i s e v a l u a t i o n .
The t e s t r e p o r t sets o u t t h e measured t h i c k n e s s t o f t h e t e s t specimens and t h e i r s t i f f n e s s e s N^/ Ny and N^y which a r e d e f i n e d as f o l l o w s :
N X X 12(1-v V ) X y N E V y 12(1-v^Vy) ct^ 1 N =xy 6 2 + ( v X X N + y y V N )
The t h e o r e t i c a l c r i t i c a l b u c k l i n g s t r e s s can then be c a l c u l a t e d as . 2 0 = k / N N (4) «-2^ / X y ^ ^ w i t h t h e b u c k l i n g c o e f f i c i e n t 2 a ^ ^ = - V ' 2 ^ % (5) 4a 4m V
where m i s t h e number o f h a l f wavelengths i n t h e l o n g i t u d i n a l d i r e c t i o n i n the buckled s t a t e . I t i s assumed t h a t i n t h e t r a n s v e r s e d i r e c t i o n t h e r e i s only one h a l f wave. Wo f u r t h e r have
«v " h i / N ( 6 )
N xy
n = , — — - (7)
The c r i t i c a l b u c k l i n g load N^r was determined f o r each t e s t specimen i n accordance w i t h Equation (4) using t h e experimental s t i f f n e s s values. The experimental c r i t i c a l b u c k l i n g s t r e s s was a l s o determined by means o f t h e r e l a t i o n s h i p between t h e load and t h e p e r p e n d i c u l a r displacement o f t h e p l a t e . The c r i t i c a l load was d e f i n e d by t h e p o i n t o f i n t e r s e c t i o n o f the load a x i s and t h e e x t e n s i o n o f t h e p r a c t i c a l l y s t r a i g h t p o r t i o n o f t h e curve i n t h e p o s t c r i t i c a l r e g i o n .
The u l t i m a t e load N^ was a l s o recorded f o r each t e s t specimen, but t h e compressive s t r e n g t h s f^, Q and f ^ 90 o f t h e sheets of plywood were n o t recorded. I n s p i t e o f t h i s omission i n t h e i n v e s t i g a t i o n , i t i s p o s s i b l e t o study the p r o p e r t i e s of t h e specimens i n t h e p o s t c r i t i c a l r e g i o n as shown below.
I n t h e e v a l u a t i o n , o n l y the specimens whose m o d i f i e d aspect r a t i o s according t o Equation (6) are l a r g e r than 1 were used. The e x p e r i m e n t a l e f f e c t i v e w i d t h of the t e s t specimens i n the u l t i m a t e s t a t e was determined as
ef u b ~ f
c where a = T — u b t
I n c a l c u l a t i n g t h e slenderness r a t i o a = JOQ/OQJ-, t h e extreme f i b r e
s t r e s s OQ was p u t equal t o t h e compressive s t r e n g t h f ^ 0 0^ ^c 90-The compressive s t r e n g t h here i s an e f f e c t i v e v a l u e , i . e . plywood i s considered t o be homogeneous. I n view of the u n c e r t a i n t y i n d e t e r m i n i n g the e x p e r i m e n t a l c r i t i c a l b u c k l i n g s t r e s s a^,^ test» values o^j. c a l c c a l c u l a t e d using the e x p e r i m e n t a l s t i f f n e s s values Njj, Ny and N^y m accordance w i t h Equation (4 ) were used i n s t e a d . When o^r c a l c
°cr,test compared i n Table 1, i t i s seen t h a t t h e l a t t e r e x h i b i t a c o n s i d e r a l y l a r g e r s c a t t e r . I n order t o s i m p l i f y c a l c u l a t i o n s , t h e mean values of each t e s t s e r i e s , s e t o u t i n Table 1, were used. The slenderness r a t i o a thus c o n t a i n s a small e r r o r which i s however n e g l i g i b l e .
Since the compressive s t r e n g t h of t h e plywood m a t e r i a l had n o t been determined, both b^f and a were c a l c u l a t e d f o r s e v e r a l assumed values of f ^ i see Table 2. The r e s u l t s are s e t out i n f i g u r e 3. Curves i n
accordance w i t h Equations (2) and (3) are a l s o p l o t t e d i n t h i s f i g u r e . An e s t i m a t e of the compressive s t r e n g t h of the plywood m a t e r i a l can be made on the b a s i s of the c h a r a c t e r i s t i c compressive s t r e n g t h s of s i m i l a r
m a t e r i a l s which i t i s understood /9/ w i l l be i n c l u d e d i n the Canadian code f o r t i m b e r s t r u c t u r e s t o be p u b l i s h e d i n 1990. The c o e f f i c i e n t of v a r i a t i o n of the s t r e n g t h values i s a p p r o x i m a t e l y 0 13. This means t h a t the mean values are a p p r o x i m a t e l y 25% g r e a t e r than t h e c h a r a c t e r i s t i c v a l u e s , which a p p r o x i m a t e l y corresponds t o the 5 t h p e r c e n t i l e . Table 2 a l s o s e t s o u t the e f f e c t i v e w i d t h s and slenderness r a t i o s f o r t h e mean values of t h e
compressive s t r e n g t h determined i n t h i s way.
I t can be seen t h a t t h e e x p e r i m e n t a l r e s u l t s are i n good agreement w i t h the curves and t h a t the r e s u l t s are not p a r t i c u l a r l y s e n s i t i v e t o t h e choice of the c o r r e c t compressive s t r e n g t h . Most of t h e values l i e above t h e curve i n accordance w i t h Equation (3) by von Karman. I t i s o n l y f o r slenderness r a t i o s g r e a t e r than 3.6 t h a t t h e r e are values n o t i c e a b l y below t h e curve. One reason f o r t h i s may be t h a t t h e value o^r c a l c "^^^ somewhat too
l a r g e owing t o some e r r o r s i n d e t e r m i n i n g the s t i f f n e s s values of the
plywood sheets. I n t e s t s e r i e s 600/8 i n which l o a d was a p p l i e d p a r a l l e l and p e r p e n d i c u l a r t o the d i r e c t i o n of the g r a i n , c o n s i d e r a b l y s m a l l e r c r i t i c a l b u c k l i n g s t r e s s e s o^r t e s t were measured. The corresponding slenderness r a t i o a i s then much h i g h e r , so t h a t even t h e lowest values o f bgf/b w i l l be very near the curve ( 3 ) .
von Kdrmdn ( 3) Winter (2)
a = 'cr
Figure 3. E f f e c t i v e w i d t h s determined u s i n g t e s t r e s u l t s by Dekker e t a l (1978) /8/.
In t h e o t h e r t e s t s e r i e s o^j- c a l c ^ l i t t l e .smaller than o^-.^- t e s t - ^'"^ these cases the v a l u e of t h e slenderness r a t i o i s reduced t o some e x t e n t ,
i.e. t h e p l o t s i n t h e diagram a r e moved t o the l e f t by a small d i s t a n c e . The d i f f e r e n c e i s however small and has no s i g n i f i c a n c e .
2.3 Proposed design r e g u l a t i o n s
For l a r g e values of t h e slenderness r a t i o , t h e r e i s very l i t t l e d i f f e r e n c e between t h e formulae a c c o r d i n g t o Winter and von Karman. However, f o r
slenderness r a t i o s near 1 and below, t h e d i f f e r e n c e i s l a r g e . I n t h e t e s t s e r i e s under c o n s i d e r a t i o n there are no t e s t specimens w i t h slenderness r a t i o s as low as t h i s , and i t i s t h e r e f o r e n o t p o s s i b l e t o v e r i f y Winter's formula i n t h i s r e g i o n .
With r e f e r e n c e t o t h e procedure a t present a p p l i e d f o r sheet metal
s t r u c t u r e s , i t i s proposed t h a t i n c o n j u n c t i o n w i t h design i n t h e u l t i m a t e l i m i t s t a t e t h e e f f e c t i v e w i d t h o f t h i n - w a l l e d wood-based f l a n g e s supported on two webs should be c a l c u l a t e d as
b - b ef for a < 0.67
where a ^ J a /a
i e c r
TABLE 1. Mean v a l u e s , standard d e v i a t i o n s and c o e f f i c i e n t s o f v a r i a t i o n f o r t h e t e s t data. Only t e s t specimens w i t h > 1 a r e
i n c l u d e d . b / t nom D i r e c t i o n o f load No o f spe-cimens w i t h a > 1 u °cr,calc N/mm °cr,test N/mm^ N/mm^ t mm X 5.02 5.20 13.00 8.01 400/8 II 7 s 0.545 1.34 2. 91 0.113 6 0.109 0.258 0. 224 0.014 X 12.11 12.26 19. 03 11.97 400/13 II 7 s 0.876 1.82 1. 05 0.111 6 0,072 0.148 0.052 0,093 X 1.47 1,13 5, 65 7.55 600/8 II 3 s 0,176 0.234 0, 522 0 035 Ö 0.120 0.207 0, 092 0.005 X 5,14 5.21 11 , 88 12.2 600/13 II 4 s 0. 16 1.06 1 , 66 0 6 0.032 0.203 0, 140 0 X 4.15 4.44 9, 74 7.78 400/8 J. 5 s 0.488 0.937 0.466 0. 129 6 0.117 0.211 0, 048 0.017 X 10.82 12.76 16, 86 12.24 400/13 5 s 0.576 1.36 1, ,39 0.152 Ö 0.053 0.107 0 .083 0.012 X 1.51 1.073 4 72 7.85 600/8 j - 3 s 0.097 0,158 0 544 0.551 Ö 0.064 0.147 0 ,115 0 070 X 4.66 5:26 10 23 12.37 600/13 j . 3 s 0 099 0.108 1 .069 0.058 6 0.021 0.020 0 . 104 0.005
TABLE 2. Experimental e f f e c t i v e w i d t h bgf and slenderness r a t i o a f o r d i f f e r e n t values o f t h e compressive s t r e n g t h f ^ b -b / t D i r e c t i o n nom of load 400/8 400/13 600/8 600/13 400/8 400/13 600/8 600/13 f c N/mm2 ^ f a 20 0, ,650 1 ,996 25 0, 520 2.232 30 a) 0, 433 2.445 21.7 a) 0.600 2.079 20 0 ,952 1 .285 25 0 761 1.437 30 a) 0 634 1.574 17.0 a) 1, ,119 1. 185 20 0, 282 3,685 25 0, 226 4.120 30 a) 0, 188 4.513 21.7 a) 0, 260 3.842 20 0. ,594 1.974 25 0. 475 2,207 30 a) 0, 396 2.417 17.0 a) 0, 699 1.819 15 0 ,649 1 .901 20 0 487 2-195 25 a) 0, 390 2.454 6.7 a) 1, ,454 1.271 15 1 ,177 1 . 121 20 0, 843 1.360 25 0,675 1.520 7.9 a) 2. 130 0 855 10 0. 472 2.576 15 0 315 3. 155 20 0, 236 3.643 25 a) 0. 189 4.073 6.7 a) 0, 705 2.106 15 0, 682 1.794 20 0. 512 2.071 25 a) 0. 409 2.316 7.9 a) 1. 295 1-302 Estimated mean s t r e n g t h a c c o r d i n g t o /9/
When t h e r e i s a glue j o i n t between the f l a n g e and the webs, b i s put equal t o b f , i . e . the c l e a r d i s t a n c e between the webs. For n a i l e d or screwed connections, b = bf + t„ where t„ i s t h e t h i c k n e s s of the web.
I n a composite s e c t i o n a c c o r d i n g t o f i g u r e 1, the s t r e n g t h s of t h e f l a n g e and web m a t e r i a l s are n o r m a l l y d i f f e r e n t . When the compressive s t r e n g t h of the f l a n g e m a t e r i a l i s lower than t h a t of the web m a t e r i a l , i n t h e above formula i s put equal t o the compressive s t r e n g t h f ^ of the f l a n g e m a t e r i a l . When, on the o t h e r hand, u l t i m a t e s t r e s s i n the web i s reached before t h i s occurs i n the f l a n g e , i s put equal t o the compressive edge s t r e s s which i s o b t a i n e d i n the f l a n g e . I n v e s t i g a t i o n s on sheet metal s t r u c t u r e s have shown t h a t the formula a c c o r d i n g t o Winter y i e l d s good accuracy i n the u l t i m a t e l i m i t s t a t e w h i l e i t i s c o n s e r v a t i v e f o r lower
loads, see e.g. / I I / , /12/ and /13/. Pending the a v a i l a b i l i t y of t e s t r e s u l t s f o r wood-based boards, i t i s proposed however t h a t the above
formula should a l s o be used f o r s t r e s s e s which are lower than the compressive s t r e n g t h .
T h i n - w a l l e d f l a n g e s which are supported along one edge and have a f r e e edge a l s o have a p o s t c r i t i c a l l o a d b e a r i n g c a p a c i t y . For wood-based f l a n g e s no t e s t r e s u l t s are a v a i l a b l e a t present. I t i s proposed t h e r e f o r e t h a t the p o s t c r i t i c a l l o a d b e a r i n g c a p a c i t y should not be u t i l i s e d i n such cases. The compressive s t r e n g t h of plywood i s n o r m a l l y quoted as l o a d b e a r i n g c a p a c i t y per u n i t w i d t h (e.g. N/mm). The value of f ^ can be o b t a i n e d by d i v i d i n g t h i s value by the t h i c k n e s s of the plywood.
I t i s proposed t h a t i n d e t e r m i n i n g the c r i t i c a l b u c k l i n g s t r e s s o^j-' simply supported c o n d i t i o n s should be assumed unless a more accurate i n v e s t i g a t i o n i s c a r r i e d o u t . F u l l f i x i t y i s very d i f f i c u l t t o achieve i n sheet metal s t r u c t u r e s . However, r e s t r a i n t has a f a v o u r a b l e e f f e c t i n the p o s t c r i t i c a l r e g i o n a l s o . This was shown i n / I I / f o r hinged supports. When
t h i n - w a l l e d f l a n g e s are bonded t o s o l i d wood webs, r e s t r a i n t i s l i k e l y t o be very e f f e c t i v e . I t i s t h e r e f o r e proposed t h a t some r e s t r a i n t should be p e r m i t t e d i n d e t e r m i n i n g Ocj- i f the degree of r e s t r a i n t can be
a s c e r t a i n e d .
I n c o n j u n c t i o n w i t h design f o r t h e s e r v i c e a b i l i t y l i m i t s t a t e , i t i s
proposed t h a t f ^ i n the above expression should be r e p l a c e d by the a c t u a l compressive s t r e s s i n the f l a n g e . I m p e r f e c t i o n s which can g i v e r i s e t o v i s i b l e b u c k l i n g a l r e a d y i n the s u b c r i t i c a l r e g i o n can be allowed f o r i n t h i s way.
3. FLANGE CURLING
I n a curved beam, or beam which becomes curved owing t o the e x t e r n a l l o a d , components of f o r c e which are d i r e c t e d towards the n e u t r a l l a y e r a r i s e i n the a x i a l l y loaded f l a n g e s ( f i g u r e 4 ) . These f o r c e s cause both the f l a n g e i n compression and the f l a n g e i n t e n s i o n t o c u r l inwards towards the c e n t r e of the s e c t i o n ( f i g u r e 5 ) . When t h i s d e f o r m a t i o n of the f l a n g e becomes excessive, the loadbearing c a p a c i t y of the composite s e c t i o n as a whole i s a f f e c t e d . The Swedish Code f o r L i g h t Gauge Metal S t r u c t u r e s , StBK- N5 /14/ s p e c i f i e s t h a t d e f l e c t i o n s h a l l not exceed 5% of the depth of the cross s e c t i o n i n order t h a t f l a n g e c u r l i n g may be i g n o r e d i n c a l c u l a t i n g the loadbearing c a p a c i t y .
10
Figure 4. Forces i n t h e f l a n g e of a curved beam.
tw
Fi.gure 5. C u r l i n g o f t h i n w a l l e d f l a n g e s .
The inward f o r c e component may be determined as ( f i g u r e 4)
0 j t d i p ~ d ^
1 M
With dJ2 = rd(p and - ~ „ - , we have
IT ki« X« b b q = Ogt O j t M ~ r ~ " ETI b b M and w i t h ^ — z we f i n a l l y have ^b q = E^z (9)
11 where z i s t h e d i s t a n c e between t h e n e u t r a l l a y e r and t h e midplane o f t h e f l a n g e ( f i g u r e 5 ) . The moduli o f e l a s t i c i t y o f t h e web and f l a n g e i n a
composite s e c t i o n a r e u s u a l l y d i f f e r e n t . C o n v e n t i o n a l l y , t h i s i s taken i n t o c o n s i d e r a t i o n by e.g. p u t t i n g t h e modulus o f e l a s t i c i t y o f t h e web equal t o
and using a m o d i f i e d f l a n g e area i n c a l c u l a t i n g t h e second moment o f are a o f t h e composite s e c t i o n .
This load causes t h e f l a n g e t o d e f l e c t , and under simply supported c o n d i t i o n s t h e value o f t h i s a t t h e midpoint o f t h e f l a n g e i s 5 q b l u = 384 ( E I ) ^ where ( E I ) f i s t h e f l e x u r a l r i g i d i t y o f t h e f l a n g e i n t h e t r a n s v e r s e d i r e c t i o n o f t h e beam. S u b s t i t u t i o n o f (9) y i e l d s 5 °f u = 384 Ej^ ( E D j z (10
I n t h e same way, when t h e web i s f u l l y r e s t r a i n e d , we have
u = -T^n
o J t b
384 Ej^ ( E I ) ^ (11)
I n box s e c t i o n s w i t h s o l i d webs and glue j o i n t s between webs and f l a n g e s , f u l l f i x i t y can probably be assumed. When t h e webs and f l a n g e s a r e j o i n e d by mechanical f a s t e n e r s , t h e degree o f r e s t r a i n t i s very u n c e r t a i n , and
Equation (9) f o r t h e simply supported case should be used.
For open s e c t i o n s , i . e . where t h e s e c t i o n comprises o n l y a compression or t e n s i o n f l a n g e which i s bonded t o t h e web, t h e degree o f r e s t r a i n t depends on t h e t o r s i o n a l r i g i d i t y o f t h e web. I t should be p o s s i b l e t o make use o f some f i x i t y by t a k i n g t h e mean value o f Equations (10) and ( 1 1 ) .
I n t h e same way as i n StBK-N5, t h e f o l l o w i n g i s proposed as t h e l i m i t i n g value o f t h e maximum f l a n g e displacement where t h i s must be taken i n t o c o n s i d e r a t i o n :
u = 0.05 h max
where h = h^^ + t f o r a box s e c t i o n and
12
When t h e maximum displacement, o f t h e f l a n g e exceeds U m ^ ^ ' ^^'^^^ must be
allowed f o r i n c a l c u l a t i n g t h e second moment o f area o f the composite s e c t i o n . Approximately, t h e e f f e c t i v e d i s t a n c e between t h e f l a n g e and t h e n e u t r a l l a y e r o f t h e cross s e c t i o n i s
^ f i
= M "
See f i g u r e 6.
CNi|m
Figure 6. E f f e c t i v e cross s e c t i o n when f l a n g e c u r l i n g i s taken i n t o c o n s i d e r a t i o n .
When 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 f l a n g e i s reduced due t o b u c k l i n g or the e f f e c t o f shear deformations, t h e displacement u o f the f l a n g e f o r t h e mean s t r e s s i n t h e f l a n g e i s determined as
f c b
When t h e f l a n g e i s i n t h e u l t i m a t e l i m i t s t a t e , t h e compressive edge s t r e s s i n t h e f l a n g e , OQ, i s p u t equal t o t h e compressive s t r e n g t h f j ^ .
13 REFERENCES
/ 1 / CIB S t r u c t u r a l Timber Code. CIB Report, P u b l i c a t i o n 66, 1983.
121 B u c k l i n g o f F l a t Plywood P l a t e s i n Compression, Shear, o r Combined
Compression and Shear. USDA, F o r e s t Products L a b o r a t o r y , Madison, Report No. 1316 w i t h Supplements 1316A-1316J.
/3/ Foschi, R.O. 1969. B u c k l i n g o f t h e compressed s k i n o f a plywood s t r e s s e d - s k i n panel w i t h l o n g i t u d i n a l s t i f f e n e r s . Canadian F o r e s t r y S e r v i c e , Publ. No. 1265.
/4/ Halasz, R. von & C z i e s i e l s k i , E.: Berechnung und K o n s t r u k t i o n g e l e i m t e r Träger m i t Stegen aus F u r n i e r p l a t t e n . B e r i c h t e aus d e r Bauforschung, H e f t 47.
/5/ SIA 164, 1981, Swiss Timber Code, Schweizerischer I n g e n i e u r - und A r c h i t e k t e n - V e r e i n , Z u r i c h .
/6/ E i n f u h r u n g i n d i e Norm SIA 164 (1981), Holzbau, P u b l i k a t i o n Nr 21-1, B a u s t a t i k und Stahlbau, Eidg. Techn. Hochschule, Z i i r i c h .
ni Winter, G. 1947. S t r e n g t h o f Thin S t e e l Compression Flanges. Trans.
ASCE, V o l . 112, 1947.
/8/ A I S I . 1968. S p e c i f i c a t i o n f o r t h e Design o f Cold-formed S t e e l
S t r u c t u r a l Members. 1968 E d i t i o n . American I r o n and S t e e l I n s t i t u t e , Washington.
/9/ Dekker, J., K u i p e r s , J. S. Ploos van Amstel, H. (1978). B u c k l i n g s t r e n g t h o f plywood, S t e v i n - L a b o r a t o r i u m , D e l f t .
/10/ F o s c h i , R., p r i v a t e communication.
/ I I / König, J. 1978. T r a n s v e r s a l l y loaded t h i n w a l l e d C shaped panels w i t h i n t e r m e d i a t e s t i f f e n e r s . Swedish Council f o r B u i l d i n g Research,
Stockholm. Document D7:1978.
/12/ Graves-Smith, T.R. 1971. A v a r i a t i o n a l method f o r l a r g e d e f l e c t i o n e l a s t o - p l a s t i c t h e o r y i n i t s a p p l i c a t i o n t o a r b i t r a r y f l a t p l a t e s . Proc. I n t . Conf. on S t r u c t u r e s , S o l i d Mechanics and E n g i n e e r i n g Design, Southampton U n i v e r s i t y 1969, John Wiley, London (1971), pp 1249-1256.
/13/ Thomasson, P-0. 1978. T h i n w a l l e d C-shaped panels i n a x i a l compression. Swedish C o u n c i l f o r B u i l d i n g Research, Stockholm. Document D1:1978.
/14/ StBK-N5, Swedish Code f o r L i g h t Gauge Metal S t r u c t u r e s , Stockholm, 1980.
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