CIB-W18/19-12-3

INTE RNATIONAL COUNCIL FOR BUI LDING RESEARC H STUDIES AND DOCUME NTATION

WORKING COMMI SS ION W18 - TIMBER STRUCTURES

INFLUENCE OF VOLUME AND STRESS DISTRIBUTION ON THE SHEAR STRENGTH AND TENSILE STRENGTH PERPENDICULAR TO GRAIN

by

F Coll i ng

University of Karlsruhe

Federal Republic of Germany

MEETING NINETEEN

FLORENCE ITALY

SEPTEMBER 1986

1. ABSTRACT

INFLUENCE OF VOLUME AND STRESS DISTRIBUTION ON

THE SHEAR STRENGTH AND

TENSILE STRENGTH PERPENDICULAR TO GRAIN

by

Fran~o i s Col1ing

Lehrstuhl für Ingenieurholzbau und Baukonstruktionen

Universität Karlsruhe

W. Germany

In 1939 WEIBULL [1} developed a theory, whieh allows to estimate the influence of the size of the stressed volume and the stress-distribution over t his volume on the strength of homogeneous, isotropie materials with brittle fracture behaviour. Although wood as material is neither homogeneous nor isotropic, it has been shown, that the application of ~Jeibull's theory is possible. \~eibull's theory has even entered several design codes (CIB-structural timber design code, Euro-Code 5, Canadian code .0.). It is used especially in the case of shear and tension perpendicular to the grain, not least because of the existing brittle fracture behaviour.

In this paper, the application of Weibull 's theory in the case of shear and tension perpendicular to the grain i5 discussed.

2. GENERAL

To estimate the influence of the stress-distribution and the size of the stressed volume on the strength. the two-parameter Weibull-distribution is used.

The cumulative frequency is defined by:

S = 1 - exp [- J (%,)k dV] (1) v

- 2 -

- 2 -

where 0" and kare the parameters of the Weibull-di stribution, and o = o(x ,y.z) is the stress distribution over the volume V.

The (constant) width W of a beam with rectangular cross section is assumed not to infl uence its strength , so tha t the integral in eq (1) may be written [2J

1 J( x) ( ) k (0 x ,y ) dy dx 0' y=o x=o

For a ~am with constant depth (D(x) = 0) eq. (2) may be written:

J a) k W (max 0) k J1 k ( ) (a I dV = • -or- . L • . f c dc. V c=o ~

AL

o . J t;=o ,"-"-----. ~---/ - v k

AD

or

where V = stressed volume

max 0 = maximum stress occuring over the volume V

(2 )

(3 )

(4)

f(E), f(t;) = dimensionless stress-distribution over the length and depth resp. related to max 0

= "fullness-parameters" to describe the fullness of the stress-distribution.

AL and A 0 are dependant on the stress-d; stribution and the exponent k of the 2-parameter Weibull-distribution. A value of A near 1 stands for a nearly constant stress distribution.

- 3 -

- 3 -

In .f.![:l equations for t he determination of the fullness -parameters A.

are given as a fu nc t i on of t he exponent k (see al so [3]) . The equation fo r the parabolic stress di stri bution (~ ) is only an approximation, because the integ ral can not be solved di rectly with a quadrati c stress di stribution . The exponent k only depends on the variation of the distribution and may approximately be determined by the following equation [2]:

(5 )

where v is the coeff icient of variation. In the case of shea r and tension perpendicular to t he grain , a value of k _ 5 may be assumed , corresponding to a coeffi cient of var iation of v _ 0,23 . For k = 5 the val ues of the ful1ness-parameters A are shown in fig . 2 and fig . 3 for any parabolic stress di stribution. Fig . 2 is used when the max imum stress occurs at an edge. whereas fig . 3 is used, when the maximum stress occurs al ong t he span.

If the fullness-parameter AL can not be determined directly by f ig. 1-3, due to the kind of loading or support, it is possible to divide t he gi ven stress di st r ibution in several fields (wi th the length 1i ), for which the fu l lness-paramete r s Ai can be determined according to fig . 1-3. The fullne ss­parameter AL of the beam may then be determined by:

where

1 . = 1

max G. 1

=

L ::

maX G =

AL . == ,1

1 . max G· = L:-' ,( 1

L max 0'

l ength of the ; - th f ield

maximum stress occur ing

length of the beam

maxi mum stress occu r i ng

5 ·AL .) ,1

al ong 1 . 1

along L

fullness-paramete r of the ;-th f ield

(6)

- 4 -

- 6 -

Eq. (8) may therefore be written as

(10)

This relationship is valid für beams with rectangular cross section and constant depth.

3.2 Shear stress in tapered beams

In the case of a tapered beam, the depth D is not a constant ( cf.eq.(2) and eq. (3 )).

With the substitution s = y/D(x) eq. (2) may be written

1 k ~ k ·L · J f (s ).h( s)dso D . J f (~) d~

S=O maxl' s=o . , v / ~

k k A A L,tap 0

or

where

o = depth of the beam where maXT occurs maXT

AL ,tap = fullness-parameter of the stress distribution over the length L, taking into account the variable depth

h(s) = dimensionless function of the depth related to DmaxT

( 11)

(12)

- 7 -

- 7 -

In the case of a uniformly stributed load l.e. a linear distribution of the shearforce, the values OfAL,tap are given in f;9. 6, assuming

a value of k = 5 for the exponent the Weibull-distribution.

According to eq.(6) of sect ion 2, the ful l ness-parameter A Lt , ap may be calculated knowing theAL,tap,i-values of each field of the beam (with the length 1, ) :

where

D maXT.

1

= depth of the beam in the i-th field • where the maximum shear stress max Ti occurs

( 13)

DmaXT = depth of the beam, where the maximum shear stress max T

occurs

Example

ITI I I I rr I ITDIJ I [UIII I I I [11 I I r D q

o (V)

Dt=~ ______________ ~~ __________ ~~~

s

s sheQr _ tor, 11

S D

)( 1 -- 8 -

- 8 -

f i e'id 51/S2 D1/D2 AL ~ tap,i (5,<52 ) see fig. 6

Q) 0 2,33 0 ,625

@ 0 0,875 0,710

® 0,2 2.0 0,658

with eq. (13):

AC,tap :: 0,5 .+ . (4.0,625)5 + 0,1 . 2 , ~67 (°1°75. 0 , 710)5

+ 0,4. 1,~33(~ . 0,658)5

== 0,0633

AL,tap == 0,576 and V = W . D . L

An approximat ion of this value for i\L t may be obtained with eq . (6): this , ap calculation is based on t he stress-distribution of the tapered beam , where the variab l e depth is considered in contrast with the distribution of the shear fo rce. This approximation is determi ned with the help of fig . 2 and eq. (6) :

field T 1h 2 TiT2 i\. L . , 1

CO 0 - 0,2 0 ,63

@ ° 0,033 0,71

® 0,1 - 0,15 0 ,66

- 9 -

9 -

AL 5 1 5 O,HO~075.0,71)5 + 0,4' (0175 . 0 ,66 )5 ::: O,5'(T . 0.63) +

'" 0 90615

AL :: 0 ,573 - 0,576 :::: A L ~ tap

Therefore the ful1ness-parameters for t apered beams may be calculated in good approximati on according to section 2 and section 3.1 resp. , based on the di stribution of the shear stress , whereas in the case of bearns with rectangular cross section and constant depth the ful1 ness-parameters may be ca1cul ated on the basis of the dis tribution of the shear forc e.

3.3 Design

If the characteristic shear strength T ;s deterrnined using the fol1owing o,k test specimen

1 F

D loo ~

~

lo Wo

v

'"

[]]]]I] IIIII~ 11111111 111[1 11 11111111]

Shear force

we have: AL o

1,0 (constant stress distribution)

AO = 0,82 (parabolic stress distribution) o

and

kL /0 according to eq. (9) o 0

= W . O · L. 000

- 10 -

- 10 -

Assuming a parabolic stress distribution over the depth for all beams, the characteristic shear strength of any beam may be determined by:

• T o,k

where Ä L ;:; fullness-parameter according to section 2 (fig. 1-3)

kL/ D := determined according to section 3.1 (eq. 9)

V == W • L . 0 maXT

W == Wi dth of the beam

L ;:; length of the beam

o :::: maXT depth of the beam where maXT occurs

~Tension perpendicular to the grain

4.1 Comparison of the theory with test results [61,

Kolb/Frech [6] tested three types of beams:

(14)

curved beams (type 1), cambered beams (type I I) and tapered cambered beams

(type 111 ). The beam configurations, test set-up and test results are

given in fig. 7.

The maximum tensi1e stress perpendicular to the grain is calculated according

to the fol1owing equations [7}:

6 M max 0' :: K • lif.1Y2 .L ap ( 15)

0 D K - A + B'(~) + C .(~)2

Rap Rap (16)

- 11 -

- 11 -

with

A = 0,2 . tan y

B 2 == 0 , 25 - 1 ,5·tan y + 2,6·tan y

C 2,1' tan y 2 ::: - 4· tan y

y = slope of the beam (upper border ) at the apex (for type I 'V ::: 0)

Dap ::; depth of the beam at the apex

Rap ::: R1 + Dap/2 ::; 6,0 m Dap

+ .".--L

The tensile strength perpendi cular to the grain is shown for an beams in fig. 8. This fig. clearly shows the decrease of the tensile strength perpen­

dicular to the grain within each type with increasing slope y and the different bearing capacity between the three types of beams:the tensi l e

strength perpendicular to the grain of type I is on average 0,75 times the strength of type II and 0,625 times the strength of type I I I. It sha 11 now

be shown how far these different bearing capacities can be explained by Weibull I s theory .

The distribution of the tens i le stress perpendicular to the grain was invest;­

gated with the method of finite elements. The following beams were considered:

beam 1.3

}} II.3 to investigate the differences between the beam types

I 11.3

II 1. 2 to investigate the differences within one beam type

I II . 1

In table 1 the K-values of eq. (16) are compared with the K-values, de­

termined by the method of finite elements.

- 12 -

.. 12 -

Table 1:

Beam Dap R K K ap mm mm eq,(16) FE

-I.3 1000 6500 0,0385 0,0394

I1.3 1450 6725 0,0942 0,0934 II 1.3 1450 6725 0,0942 0,0935

II 1. 2 1250 6625 0,0700 0,0705 II 1. 1 1110 6555 0,0535 0,0534

This comparison shows a good agreement (also for the tapered cambered beams,

if the slope of the upper borde r is used in eq. (1 6)). The ful1ness parameters

were determined by

AL 'AO tL: ( 0 i .1 ) 5. V.] 0 2 :;: max 0 J.. l'

Vtot ( 17)

where

V. := volume of a finite element with tensile stress 1

perp. to gra in

O; _.L ::: tensile stress perp . to grain of a finite element

with volume Vi

Vtot ::: L: Vi :;: stressed volume

max OJ..= maximum tensile stress perp. to grain in the volume Vtot

- 13 -

- 13 -

These values are given in table 2.

Table 2:

Beam Vtot 0; ..L 5 AL'An L:{-- ) .V. maxo.l. 1

m3 m3

1.3 0,922 O~05547 0,570 11.3 0,835 0,05715 0,585 I I 1. 3 0,420 0,05880 0,675 I 11.2 0,310 0,04800 0,689 I I I. 1 O~246 0.03320 0,670

In design calculations the volume Vc of the curved part of the beam(between the points of tangency)is used. This volume corresponds to the shaded area in fig. 7.

According to eq. (7) and eq . (8) resp., the fullness-parameters (AL,Au)c

corresponding to the volume Vc are calculated by

V (ALoAO)c = AL'An ' (V:ot )0,2

and given in table 3.

Table 3:

Beam Vc (AL'AO)C

m3

1----"

1.3 0,545 0,633 11.3 0,667 0,612

II 1.3 0,446 0,667

Ir 1. 2 0 ,28 7 0,700

II 1. 1 0.147 0 ,743

( 18)

- 14 -

h 11 ness··pa rameters whole volume

- 14 -

beam-type III may be explained by

of type III 15 located between the

load; ints (constant bending moment) whereas apart of the volumesVc of type I and II is located outside the lüading points (i.e, the stress in

this part 15 lower). Anotherfactorthat might explain the higher values of ,AD)C für type III is the influence of the loading points: in case of

beam type I and Ir a gt'eater part of the volume Vc is strained by a compressive

stress perpendicular to the gra in.

For the inves tiga t ion of the bearing capacity within beam type 111, the expected maxa·i ...L_ -values wete calculated accordingtoeq.(8) and compared with the maxa j..l..

test-values (see table 4).

Table 4:

I

0; .J...

o . ...L J

II!. 1 Ir!. 2 II 1.3

eq(8) test eq(8) test eq(8) test

-- -- .

II!. 1 1 ,0 1 ,0 0,928 0 ,914 0,892 0,864

IIL2 1,077 1,094 I 1 ,0 1 ,0 0,961 0,945

III.3 1 , 121 1 • 158 1 ,041 1,058 1 .0 1 .0

This comparison shows a good agreement between the theoretical values according

to Weibullis theory and the test values.

The decrease of the bearing capacity with increasing slope y can thus be

explained and numericallyevaluated with the help of Weibullis theory.

The expected ratios of the tensile strength perpendicular to the grain for

the different beam types are given in table 5 and are again compared with

the test resul

- 15 -

- 15 -

Table 5:

01'.1.

- -O·.!.

J

~ 1.3 11.3 I I 1.3

eq.(8 ) test eq.(8 ) test eq, (8) test

1.3 1 ,0 1,0 0,994 1 ,471 0 , 988 1,770

I1.3 1 ,006 0,680 1 ,0 1 ,0 0,994 1,204

III.3 1 ,012 0,565 1 ,0 06 0 ,831 1 ,0 1 ,0

According to Weibull's theory (eq . (8 )) an types would have the same tensile strength perp. to the grain: the value (,\,A.o)c·(V)O , 2 (cf. table 3) is

nearly a constant for all beam types, The oppositely oriented influences of

a greater volume and a lower fullness-parameter counteract , so that the ex­

pected strength is the same for all three beam types.

The tests however showed a clear tendency, that thetensile strength perp.

to the grain of the cambered beams is higher than the strength of the curved

beams with constant depth, and that the strength of the tapered cambered

beams is even high er than the strength of the other two beam types.

As too little is known about the tests described in [6J. no explanation could be found concerning the contradiction between the theoretical and the test-values ,

Therefore further (theoretical and experimen tal) investigation in this field

is required.

4.2 Design

The characteristic tens i le strength perpendicular to the grain 00.1. is

determi ned by a pure tension test (i .e. AL 'An = 1,0) wHh a test specimen of

volume Vo'

- 16 -

- 16 -

The characteristic strength of any beam can be calculated according to eq. (8):

• cr .l-o,k ( 19)

According to t he draft of Eurocode V (October 1985 ), the characteristic tensile strength perp. to grain of the strength class C3 (determined with a test

specimen of volume V = 0,02 m3 ) i s : o

In the case of beam 1.3 we could expect (according to eq. 19 and table 3) a

characteristic strength:

1 0 02 0, 2 2 Gk..l.. = 0,633 . (0:545) . 0,4 = 0,33 Nimm

Assuming a coefficient of variation of about 25%, the mean strength may be calculated approximately to

cr...l.. - 0,33· 1-1,6JS.O,25 ~ 0,56 N/mm2

\ (Gauss-distribution)

This value is approximately reached by beam I.3 (mean strength ~ 0,65 N/mm2 ).

With the assumption, that the decrease of strength with increasing slope y

within one beam type can be explained by Weibull's theory (see section 4.1),

eq. (19) may be used for beam type I (curved beam with constant depth).

Theoretically, eq. (19) is also valid for beam type II and Irr, but the higher bearing capacities of these beams (c.f. fig 8) might be taken into account

in the following way:

= 1,3 'cr .l.. o,k (20)

für beam type Ir (cambered beams)

- 17 -

- 17 -

and

0k L ~ ~ 1 '6~_L_.1_D __ • _(_~_O_) 0_,_2 _'_0_0_,_k _-L---II ( 21)

for beam type 111 (tapered cambered beams).

5. SU14rvtARY AND CONCLUSIONS

Weibull1s theory of brittle fro.cture is used to describe the influence of the stress distribution arid the size of the stressed volume on the strength of a beam.

The determination of the so called fullness-parameters A (which stand for the fullness of the stress-distribution) is shown. Also a mathematical relationsh ip between the expected ratio of the strength of two beams and their ful1ness-para­meters and their stressed volume has been deducted.

The application ofth i s theory arid a possible design method has been shown in the case of shear stress and tensile stress perpendicular to the grain.

Because of the differences between the theoretical and the experimental resul ts in the case of tensile stress perpendicular to the gra in, further investigations are required. The application of a modified weakest link theory (with weighted influences of the beam-length and depth) as well as the further dependency of

the strength on the wood-properties (density, growth rings, knots . . . ) will probably be investigated in a proposed research program in Karlsruhe.

1 vL 1939: Asta t ist ica 1 the strength of ma als. Ing. Veten . Akad. Handb. No. 151

[2] Co'll i 9 F. 1986: Einfluß des Volumens u[ld der Spannungs verteilung

die Festigkeit eines Rech , Herleitung einer allge-meinen Beziehung mit Hil der 2-parametrigen Weibull-Verteilung.

Holz als Roh- und Werkstoff 44. S. 121 - 125 .

[3J Colling, F. 1986: Einfluß des Volumens und der Spannugnsverteilung

auf die Festigkeit eines Rechteckträgers. Bestimmung der Völligkeits­

beiwerte. Anwendu ngsbeispiele. Holz als Roh- und Werkstoff 44,

S. 179 - 183

[4J Foschi~ R.O.; Barrett, J .D. 1976: Longitudinal shear strength of Douglas-fir.

Can. J. Civ. Eng., Vol. 3, p. 198-208

Fosch;, R.O.; Barrett 9 J,D, 1977: Longitudinal shear in wood beams:

ades ign methode

Can J. Civ. Eng •• Val. 4, p. 363-370

[6J Kalb, H; Frech, P. 1975: Biegeversuch zur Ermittlung der Querzug-

tigkeit von gekrümmten Trägern aus Brettschi chtholz ,

Prüfungsbericht Nr. 30667, FMPA Stuttgart. Germany

[7J Blllmer, H. 1972/79: Spannungsberechnungen an anisotropen Kreisbogen­

scheiben und Sattel trägern konstanter Dicke. VerÖffentlichung

des Leh I'S tuh 1 s filr Ingen i eu rho 1 zbau und Baukons trukti onen,

Universität Ka sruhe, Germany.

1 stress - distri bution j\ = [ J f k (€) de: ] lIk

€':O

I 1 .,

CD OmaÄ I I O'max 1J O

@ ~ama.)( (..J...... )l/k k+'

1 1-71 k+1 G) 17'Omox ~a )lIk (_. mal(

k+l 1- Tl

Ci) ~<1mQ'~ Ci'mol(

(_'_.~)l/k k ... , 1 -I> 171 1

® ~ma)l [ k1.l ( 1 • Oz" k ) r k

~

® ~~ [ ...l.. ( 1 ... 0)45 k -k+'

Ullk -Q027' k2 .. qOO13·k 3) ~!i~

f i g.I: fu l lness·-paramete r Xas a function of the exponent

k of the two-parameter We i bull-distribution

fig.2: fullness-pa r ameter A for a parabo l ic stress distribution, with t he maximum

stress occuring at the edge

0,95<1---+ 0,95

0,90

- 0,00

0,8 .... \ ,1S

3: f ullnes -parameter -~.-

for a parabolic

stess di tribu ion. i h h maximum

stress occuring o 9 the span

~I

a

Cl

--11

o

Uftihumly Dilt,i~ut" Loald

p I i i iI li j " e

LA l .. r

1 ----- -.--H i ! I •

4 6 8 10 12 14 16 18 0

LID fig .4: ß - values for uniformly distributed load

according to Foschi/Barrett [4]

_. -.~ accordin ,g to section 2 (i\:i\ = i\~01i\ ~'l't--,<"" "'1'1- =: 1> 742 ) , -- __ _ 1 742. LID L 0 v- D

• C71./=7

.1 .2 .3

al L

.!:J.JL~ß-va lu es for concentrated load

- - - accord i ng to Fosch i /Barrett [4]

_._- accord i ng to section 2

.4

A.L,tap

1,10

1,00

0,90

qso

0,70

0,60

0,50 0,5 ';0 ',5

Shear Force

Depth

51 UllllJlIlllJ 52

0, rrrrrroJJlJJ 02

2.0 2,5

fig . 6: fullness-parameter AL t for tapered beams , a p

51~ 52

),0 3;5 4j O °1 / °2