STEP lecture B
To explain volume and stress distribution effects as a consequence of the weakest Technique Bois theory for brittle To describe the options of Eurocode 5 and
supporting standards for deriving characteristic values and evaluating design stresses.
Prerequisites
A7 Solid timber
-
Strength classesA8 Glued laminated timber - Production and strength classes Tension and compression
Bending Members B4 Shear and torsion Summary
The lecture begins with a presentation of the weakest link theory, for tension in brittle materials, and explains volume effects. This is expanded to other stress fields, with attention to bending, tension, shear and tension perpendicular to grain. Research results are The options of EC5 for bending and tension perpendicular to grain are explained. Some examples calcularions are given.
Theory
The weakest link theory has been developed by Pierce and (1939) who studied brittle materials, including concrete. This theory says
"when subjected to tension, a chain as strong as its weakest To explain this theory, consider a reference volume subjected to tension. The probability of failure this volume is defined by:
= = Probability
where F is the distribution of the strength, illustrated in Figure 1
of for a
Now consider a series assembly of reference volumes. This survives if of the survives,
an initiative EU Programme
is the of survival of the system and is the probability of survival of an individual i. Equation (2) and
the of events
in the probability failure of' the can
be evaluated:
,
Now, assume that the lower of been fitted by a power model,
= a ( a -a,)"
The probability of failure is then expressed by:
This is as 3 parameters Weibull It is also well as the 2 parameters Weibull when
a,,
= The and can bethe mean of and the coefficient of variation of a by the following equations:
where is the
can be to explain the size effect in tension. Consider a volume V, which has a given probability of failure at level 0, and a which a given probability of failure at IF the characteristic strengths of these two volumes are compared, the following is obtained:
This equation is the basic of size effect. In the case of stress than these equations are modifiecl to take into account the stress variations:
where is stress in volume a distribution function
model is then written:
( a ) =
where is defined by:
an
For example, a supported with rectangular cross-section and at the by a force gives following value for
method of calculating the stress distribution effect has been used by ( 1 986) and Colling (1986) to evaluate the volume and stress distribution effects on the shear tension perpendicular to for- curved, tapered and beams. In paper, the "distribution factor" is used, where:
is used to the design tension strength
for different load configurations:
refers to a reference stress.
In Colling the following notation is used:
and are called parameters".
Research results
A amount of has been published to explain size effect for size timber. results are conflicting (Barrett and Lam, 1992;
and be due to the following reasons:
- The effect is by a brittle failure theory, is applicable to tension parallel to grain (Barrett, 1974; Colling, and to shear and Barrett, 1976; Foschi, 1985; Colling, 1986). But in the case of compression, and in bending which is a of failure between tension and the use of theory is debatable.
The size effect is based an probability of failure of the "reference volumes". This assumption is nor verified for all the species, especially for pines in which knots are not randomly located.
- For graded defect sizes increase with sizc of the member. This that the material changes size, which can mask a pure size effect. In particular, when size effect is investigated in a mixture of grades, effect of will an influence on the size effect.
- When are for constant span to ratios in the
size effect is a combination of effect and a length effect (Barrett and 1990). These effects cannot be identified separately.
The following tables these They some discrepancies, which have been explained by and (1
an EU B
In Table 1, different factors for bending size effects are recorded:
- a length factor (for beams tested at constant depths) is calculated from:
a depth factor (for beams tested at constant spans) which is calculated from:
- a "size factor" (for beams tested at span to depth ratio, = which, according to the combination of equations (16) is calculated from:
Author
and Larsen, 1992
and 1990
Size
Additional results are reported for (Ehlbeck and but are based on a size which was much The size effects for are Iower than for solid timber, probably due to a lamination effect which increases strength.
In Table 3, load configuration factors for different bending cases reported according to Johnson (1953). These configuration factors are derived according to Equations (10) and normalized to reference four points bending case.
Tension results are slightly different from those for bending. This might be to a pure brittle failure mechanism (see Table 2).
Author
and Larsen, 992 1992
2 Size for
EU
Load
For results of the different studies are in general agreement:
For tension perpendicular to grain and for shear, a been derived by who also derived configuration factors for tension perpendicular grain
These I-esults subject to different opinions but show evidence of size effects for stresses, together with a stress distribution effect which be as significant as the size itself. For code purposes, the approach has been
especially in the case stress distribution effects.
Size
and
stress distribution related to338: 1991 first application of' size concerns the modification characteristic given in "Structural
-
Strength classes".strengths in bending and in tension given for a reference depth of for solid limber nnci for For depths less these reference strengths are multiplied by a size factor, which has a fixed upper limit. This means size is only applied in one direction, as shown in Figure 2.
solid
.
where It is the beam in
ECECS: Part
-
: 3.3.2 := min.
2 size iir (solid to
line),
Part 5.1.3 For tension perpendicular to grain and characteristic strengths are also given for a reference volume. But, for simplicity, a size factor is only proposed for tension perpendicular in The designer is required to verify the following equation:
where is a reference volume of
1 - 1 : 5.2.4 For curved and cambered beams, an additional requirement is included to account of the stress distribution effects. The designer must verify the following equation in the apex zone:
is a stress distribution factor has been fixed for special cases:
= for double tapered and curved beams
for beams.
For simplicity, aspects size and stress distribution effects like compression size and load configuration factors not been taken into account.
- EU
Calculation
1: Bending Strength of a solid timber beam of cross-section 40x100 strength class C24.
class provides = 24
= (150 =
<
(modified) 26
2: Design of a double tapered beam. Verification of perpendicular lo grain.
Span: L = 20 It = = = 150
194 : 1993 Strength Class: GL 36
GL 36 strength class provides =
To design strength, take = = I
This = 0,277
Part 1- 1: 5.2.4 The of the apex zone is equal to: = 0,2097
= 0,544
For a double tapered =
The maximum design stress perpendicular to the is equal to :
Concluding
- Size and stress effects are explained by theory.
- results show discrepancy, especially depth effect in bending.
- provides approach to size and stress distribution to aid the designer.
References
J.D.. A.R. Tor the bending structural
lumber. Proc, W18 Paper 0-3.
J.D., (1992). in visually structural Proc. of
CIB Swcdcn.
J.D. (1974). of on lo Fir. Wood
F. of' stress on
groin. Proc, CIB Florence,
an Comctt
Rending of beams, proposal. Proc.
the W 18 Meeting, Lisbon. Portugal, Paper
R.O. (1385). of Proc. of
Meeting.
R.O., and J.D. (1975). Longitudinal Douglas Fir. Canadian Journal Civil Engincering, 198-208.
Johnson, A.I. (1953). safety of State
Committee for Building Bulletin n. 22, p. 159.
Larsen, (1986). 5 and design of
Florence, Italy. 10-
B. (1902). Engineering Vancouver, B.C.,
Canada.
Pierce, F.T. (1926). Tension tests for cotton yarn. Journal of the
Tucker, A study of of with
of Franklin 304: 751-781.
Weibull, W. (1939). A statistical theory of strength of for
Engineering N.
phenomenon in solids. In: Royal Swedish for Engineering
Research, 153,