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CIB-W18/44-6-1
INTERNATIONAL COUNCIL FOR RESEARCH AND INNOVATION
IN BUILDING AND CONSTRUCTION
WORKING COMMISSION W18 - TIMBER STRUCTURES
IMPACT OF MATERIAL PROPERTIES ON THE FRACTURE MECHANICS DESIGN
APPROACH FOR NOTCHED BEAMS IN EUROCODE 5
R Jockwer
R Steiger
EMPA, Dübendorf
A Frangi
J Köhler
ETH Zürich, Institute of Structural Engineering IBK, Zürich
SWITZERLAND
MEETING FORTY FOUR
ALGHERO
ITALY
AUGUST 2011
1
Impact of material properties on the fracture
mechanics design approach
for notched beams in Eurocode 5
Robert Jockwer1,2)
René Steiger1)
Andrea Frangi2)
Jochen Köhler2)
1) EMPA, Swiss Federal Laboratories of Materials Science and Technology,
Wood Laboratory, Dübendorf, Switzerland
2) ETH Zürich, Institute of Structural Engineering IBK, Zürich, Switzerland
1 Introduction
At notch corners stress concentrations occur due to the sudden change in cross section.
Shear stresses and tensile stresses perpendicular to grain are leading to brittle failure of the
notch. Therefore special design considerations are needed to guarantee structural safety.
Generally notches should be avoided. If this is not possible they should be reinforced in an
adequate way to prevent failure. Within a small range of geometrical properties the design
of unreinforced notched beams is possible if the decrease in strength is taken into account.
This decrease in strength due to notches has been known for long time. Hence, several
design approaches were published, leading to, however, different results. The present paper
therefore aims at analysing the impact of material properties on the variation in strength of
notches.
1.1 Design approaches for notched beams
In 1935 Scholten [1] developed an empirical design approach with a reduction of the
strength proportional to the notch ratio α (Figure 1). A bilinear reduction of strength was
determined by Mistler [2] in 1979 from experimental tests and from a study on a stochastic
model. Leicester studied the theoretical stress distribution at notches [3] at the beginning of
the 1970’s. This led to the approach in the Australian Standard AS 1720 [4], which takes
into account also size effects.
At the end of the 1980’s Gustafsson developed a design approach for notched beams based
on fracture energy [5]. The application of the fracture mechanic concept in timber
engineering [6] was studied and later Gustafssons approach was implemented in
Eurocode 5 (EC5) [7].
1.2 Development of the EC5 design approach
Gustafsson described the equlibrium of energy at a notched beam during crack growth. By
rearranging the equation the average shear stresses were separated from a term including
material properties and geometric parameters (Equation (1)).
2
xxy
f
EG
d
G
hbV
11
6
1
6.0 22
(1)
Figure 1: Notched glulam beam
The beam’s height h, its notch ratio α and notch length ratio β, as specified in Figure 1, are
the geometric parameters in the model. Fracture energy Gf, modulus of elasticity (MOE) Ex
and shear modulus Gxy reflect the material properties.
Fracture energy is the only strength related material property in Equation (1). Gustafsson
proposed that it is sufficient to take into account only the part of opening of the crack
flanges (mode 1), since crack opening is the main failure mode ocurring at the notch.
Hence, the general fracture energy Gf depending on the stress state at the crack tip was
replaced by the fracture energy of mode 1 Gf,1 for crack opening only.
On the left hand side of Equation (1) the action effect and on the right hand side the
resistance during crack growth can be identified. The resistance of the notch on the right
hand side of Equation (1) increases for large notch ratios α. For a notch ratio of 1 the
resistance approaches infinity since no notch exists in that case. However, the resistance is
limited by the shear capacity of the beam. Hence, Equation (1) was extended by the shear
strength fv and the resulting reduction factor kv was limited to an upper value of 1
(Equation (2)). The material property values can be summarised in two material constants
A and B (Equation (3)). Riberholt, Enquist et al. [8] determined an additional term in a
large test series taking into account notch taper i.
vv kf
hbV
5.1
with
22
5.1
1
1.1
1
;1min
Bh
h
i
A
kv
(2)
and
x
xy
v
xf E
G
f
EG
A6.0
5.1 2
1,
,
x
xy
E
G
B10
(3)
In standards specifying material properties as e.g. EN 338 [9] and EN 1194 [10] all
material properties except fracture energy are given to apply Equation (2) for the relevant
strength classes of timber. For the implementation of Equation (2) in EC5 Larsen and
Gustafsson collected the values of fracture energy derived from experimental tests on
different wood species and timber grades [11]. For densities in a range of
300 kg/m3 ≤ ρ ≤ 450 kg/m3 Larsen et al. [12] suggested an approximation based on a linear
regression of the results [11]:
65.0
1,
f
G
.
(4)
Using this relationship and further test results on notched beams from literature the
material constants were found to be A = 5 for solid timber and A = 6.5 for glulam and
B = 0.8 for both solid timber and glulam [12]. In EC5 the material constant A is denoted kn.
α·h
bβ·h
h
V
i1
3
1.3 Test results from literature
The material constants found by Larsen et al. are used in different standards and
handbooks for the design of timber structures like EC5 [7], DIN 1052 [13] and SIA 265
[14]. However, all these standards assume different material property values, especially
different values of shear strength. Furthermore the characteristic (fifth percentile) values of
shear strength were subject of permanent modifications in the recent years. In Table 1
characteristic values of shear strength are summarized for common European strength
grades of solid timber and glulam. In Figure 2 the resulting estimated load bearing
capacities are compared with capacities determined in experimental tests by Franke [15]
and Rautenstrauch et al. [16].
Table 1: Charakteristic values of shear strength fv,k
in different standards.
Standard
fv,k [N/mm2]
C24
GL24h
EC5
EN 338 [9]
4
-
EN 1194 [10]
-
2.7
prEN 14080 [17]
-
3.5
DIN 1052 [13]
2
2.5
SIA 265 [14]
2.5
2.7
Figure 2: Ratio of mean notch capacity from
tests [15] (n= 40) and [16] (n= 32) to calculated
characteristic (fifth percentile) notch capacity
according to different standards.
The ratios between mean experimental results and estimated notch capacity on the
characteristic level in the order of 1 for EC5 are clearly too small and, hence, the notch
capacity is overestimated. The ratios determined according to the German and Swiss
standards DIN 1052 and SIA 265 are higher than according to EC5. However, it is not
clear if the targed safety is complied by the estimated capacities.
Hence, it is to be identified to what extend the material properties take an impact on the
notch capacity according to Equation (2) and if the material constants need to be updated
to guarantee the desired safety level the standards are based on.
2 Impact of material properties on the EC5 design approach
For the application of Equation (2) in the design of structures according to EC5 material
property values for shear strength fv, MOE E0 and shear modulus G are specified in
material standards (for solid timber in EN 338 [9], for glulam in EN 1194 [10] and in the
preliminary standard prEN 14080 [17]). Besides the given values material properties can
be determined from experimental tests as specified in EN 408 [18]. Information about
suitable distribution functions of the material properties can be found in part 3.5 of the
JCSS Probabilistic Model Code [19]. However, fracture energy values are not specified in
all these standards.
When evaluating the sensitivity of Equation (2) regarding the influence of any material
property, mean values and variations of these properties have to be estimated and
appropriate distribution functions have to be chosen.
0
0.5
1
1.5
2
2.5
4
2.1 Specification of material properties in standards and codes
Solid timber and glulam are assigned to strength classes as given in EN 338, EN 1194 and
prEN 14080 according to the basic material properties bending strength (MOR), bending
modulus of elasticity (MOE) and density. Other property values are not directly
determined but rather are derived from these three properties using prescribed
relationships.
MOE: JCSS assigns MOE to be lognormal distributed with a coefficient of variation
COV = 13% for both solid timber and glulam. This corresponds to the ratio between
fifth percentile and mean value of MOE in grain direction E0,g,0.05 / E0,g,mean = 0.81 given
for glulam in EN 1194 assuming lognormal distribution. The ratio of 0.67 for solid
timber in EN 384 leads to a higher COV = 23%.
Density: Mean values of density in the range of ρmean = 390-480 kg/m3 are given for
common strength classes between C20 and C35 in EN 338. JCSS assumes density to be
normal distributed with COV = 10% which is in line with the ratio of mean and fifth
percentile values given in EN 338. The mean and fifth percentile values of density with
ρmean = 370-500 kg/m3 for strength classes GL20h to GL32h according to prEN 14080
result in COV = 5-7%, which is lower than the variation stated by JCSS.
Shear strength: For glulam in EN 1194 and for solid timber in the 2004 version of
EN 384 [20] a relationship between the characteristic values of shear strength and
tensile strength of the lamella is stated. Glos and Denzler could not confirm this
relationship [21, 22]. In prEN 14080 and in the current version of EN 384 [23] constant
values therefore are given for shear strength of glulam and solid timber respectively.
Relationships between mean and characteristic values of shear strength are not given in
the cited standards. JCSS assumes a lognormal distribution with a coefficient of
variation COV = 25% for solid timber and COV = 15% for glulam [19].
Shear modulus: The shear modulus is correlated to MOE. In EN 384 and EN 1194 a
ratio between MOE and shear modulus of 16 is specified. In prEN 14080 a constant
value of shear modulus Gg,mean = 650 N/mm2 is assumed. This leads to ratios
E0,g,mean / Gg,mean between 12.3 and 21.5 since MOE is increasing for higher strength
classes. JCSS assumes the same distribution and variation for MOE and shear modulus.
With the lognormal distribution the ratio Gg,0.05 /Gg,mean = 5/6 given in prEN 14080 leads
to a COV = 11%.
Correlations between material properties: For the correlation of all these material
properties a medium correlation (0.6) is postulated in [19] except for the correlation
between MOE and shear strength. Here only low correlation (0.4) is assumed.
Table 2: PDF and COV of material properties.
Material
property
Distribution
function [19]
Solid Timber
Glulam
COV
[19]
COV
EN
COV
[19]
COV
EN
MOE
Lognormal
13%
23%
13%
13%1)
11%2)
Density
Normal
10%
10%
10%
5-7%2)
Shear strength
Lognormal
25%
-
15%
-
Shear modulus
Lognormal
13%
-
13%
11%2)
1) EN 1194:1999 2) prEN14080:2011
Table 3: Correlation of material
properties.
Property
E0
ρ
fv
G
MOE
-
0.6
0.4
0.6
Density
0.6
-
0.6
0.6
Shear strength
0.4
0.6
-
0.6
Shear modulus
0.6
0.6
0.6
-
5
2.2 Fracture energy
Different test methods exist to determine fracture mechanical properties of timber. Tests on
compact tension (CT) specimens, double cantilever beams (DCB) or a tension specimen
with a slit are used to measure fracture toughness or energy release rates [24-26]. A simple
method for determining fracture energy is making use of a single edge notched beam
(SENB) specified in a Draft Standard of CIB-W18 in Annex B of [11] and [27], also
known as Nordtest method. Fracture energy can be calculated from load displacement
curves without the need for detailed information about elastic properties.
2.2.1 Results from tests on SENB according to the Draft Standard CIB-W18
Results from tests on SENB according to the Draft Standard CIB-W18 [11] are
summarized in Table 4. Values were selected based on mean densities in the range of
ρmean = 369-506 kg/m3 reflecting the densities given in the material standards cited above.
A correlation between density and fracture energy can be found. However, this correlation
is low for the observed range of densities being of relevance for structural applications [11,
28]: The standard deviation of the linear regression of the data in Figure 3 (σε = 54 N/m) is
nearly identical to the standard deviation calculated for the whole data without regression
(σε = 55 N/m). Hence, no correlation of fracture energy and density is assumed in the
further parts of the present study.
Other authors found values different from those given in Table 4. Smith explains his values
(Gf,1,mean = 435 N/m with ρmean = 362 kg/m3 at a moisture content (MC) of 12%), being
rather high compared to other values, with the impact of the careful drying from green
wood [29]: Fast kiln drying can produce cracks and reduces the values of fracture energy.
This could explain the low values of fracture energy determined by Franke [15]
(Gf,1,mean = 171 N/m) and Daudeville [30] (Gf,1,mean = 205 N/m) for commercial timber.
However, no clear effect of MC can be identified. Smith [29] found highest fracture energy
values for MC = 18%, whereas Rug [31] found decreasing values for MCs higher than
12%. However, MCs lower than 12% provoke low values in fracture energy, since cracks
arise during storage at low r.H. [11] or during cylic climate at low level [29].
Other parameters with impact on fracture energy are knots and the growth ring orientation.
Knots severly increase fracture energy, due to their dowling effect [8]. Hence, most tests
were done at carefully selected clear wood. For fracture surfaces tangentially to the growth
rings higher values were determined than radialy to the growth rings [24, 30].
2.2.2 Mean value, distribution and variance of fracture energy
Parameters for distribution functions fitted to the fracture energy values of individual data
from Table 4 are summarized in Table 5 . A lognormal distribution with a mean value of
Gf,I,mean = 300 N/m and a COV = 20% is used for the further study, the fifth percentile
value then being Gf,I,0.05 = 216 N/m. A 3-parametric Weibull distribution in line with the
study [28] describes the data well. However, it is not used here since it inhibits fracture
energy values lower than the location parameter and hence is not adequate for the
prediction.
For a mean density of ρg,mean = 420 kg/m3 the regression of mean values of fracture energy
by Larsen and Gustafsson leads to a similar result of Gf,I,mean = 291 N/m as in Table 5
whereas for lower or higher densities the mean values deviate considerably. The proposed
simplification for characteristic values in Equation (3) for densities in the range of
ρg,c = 300 – 450 kg/m3 is not adequate to describe the fracture energy: For a density as
6
asked for C24 considerably higher fracture energies are resulting. A correlation of fracture
energy with other material parameters could not be found in literature. However, a
coefficient of correlation of 0.2 (very low correlation) can be assumed which is lower than
that of tensile strength perpendicular to the grain and all these other properties [19].
Table 4: Values of fracture energy Gf,I,mean from literature.
Reference:
n
[-]
ρmean (COV)
[kg/m3]
Gf,I,mean (COV)
[N/m]
Larsen and
Gustafsson
[11]
Annex 1
A. 4.3
A. 8
A. 10
A. 11
37
6
62
29
12
426 (10.3%)
369 ( 1.5%)
458 (8.15%)
506 (10.3%)
503 (10.8%)
323 (14.9%)
291 (12.1%)
286 (20.3%)
337 (15.3%)
263 (15.9%)
Riberholt
et al. [8]
Solid
Timber
Glulam
88
43
415 (7.9%)
436 (8.6%)
290 (17.5%)
311 (18% )
Gustafsson [5]
14
467 (11%)
294 (17.5%)
Aicher et al. [28]
83
457 (5.3%)
277 (27.2%)
Gustafsson et al. [32]
-
475 (8.8%)
272 (15.4%)
Table 5: Distribution parameters for fracture energy.
Individual data from Table 4
Aicher [28]
Normal
Logn.
2p-Weibull
3p-Weibull
Gf,I,mean [N/m]
300
300
299
283
Std [N/m]
55.2
57.3
59.4
74.9
COV
18.4%
19.1%
19.8%
26.5%
Gf,I,0.05 [N/m]
209
216
194
176
Figure 3: Fracture energy Gf,I in
dependency of density ρ from test
results [5, 8, 11] listed in Table 4.
Figure 4: Distribution of fracture energy
Gf,I from test results [5, 8, 11] listed in
Table 4 and PDFs from Table 5.
2.3 Impact of varying material properties on the EC5 design approach
The sensitivity of material properties in Equation (2) was analysed by means of the
structural reliability software COMREL [33] using the values and distributions listed in
Table 6. The impact of MOE and shear modulus depends on the ratios α and β as can be
seen in the denominator of Equation (1). The minimal notch length necessary for
preventing compression failure perpendicular to the grain is in the range of β = 0.1 - 0.5
(GL24h) and β = 0.2 - 0.7 (C24), respectively. Larger notch length should be prevented
since notch capacity is reduced considerably. Suitable notch ratios are not less than α = 0.5
for solid timber and higher for glulam. Depending on the structure smaller notch ratios lead
to uneconomical design due to the considerable decrease in beams capacity.
In the sensitivity analysis weighting factors αi are determined, giving information about the
relative impact of the respective parameter on the variation of notch capacity when
calculated according to Equation (2) and on the related reliability. Fracture energy is the
material property with the most impact on notch capacity as can be seen in Figure 5 and
Figure 6. The impact of variations in both MOE and shear modulus is almost constant for
different notch ratios α and notch length ratios β. However, depending on the values α and
0
100
200
300
400
500
300 350 400 450 500 550 600
Fracture energy Gf,I [N/m]
Density ρ [kg/m3 ]
Larsen
Riberholt
Gustafsson
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
0.11
100130160190 220 250 280 310 340 370 400 430 460 490 520 550
Probability density [-]
Fracture energy Gf,I [N/m]
Test results
Normal
Lognormal
2p-Weibull
7
β the impact is differently distributed in between MOE and shear modulus: the impact of
shear modulus increases for larger α and smaller β, respectively. In the practical range of
notch ratios 0.5 ≤ α ≤ 1.0 the influence of shear modulus and MOE is mostly depending on
notch length ratio β.
It has to be taken into account that shear strength originally was not part of Equation (1).
Equation (2) contains shear strength both in nummerator and denominator. That is why
shear strength has no influence on the estimated notch capacity.
The ratio of MOE and shear modulus has only minor effect on the notch capactiy as
estimated according to equation (2).
Table 6: Mean values and distributions of material properties in the sensitivity analysis.
Material property
Solid timber (C24)
Glulam (GL24h)
Mean
Distribution
COV
Mean
Distribution
COV
Fracture energy [N/m]
300
Lognormal
20%
300
Lognormal
20%
MOE [N/mm2]
11000
Lognormal
13%
11600
Lognormal
13%
Shear modulus [N/mm2]
690
Lognormal
13%
760
Lognormal
13%
Shear strength [N/mm2]
6.2
Lognormal
25%
3.5
Lognormal
15%
Figure 5: Impact of selected material properties
on Equation (2) for notch length ratio β = 0.25 in
glulam, range of suitable notch ratio α in grey.
Figure 6: Impact of selected material properties
on Equation (2) for notch ratio α = 0.75 in glulam,
range of suitable notch length ratio β in grey.
3 Revisiting material parameters in the EC5 design approach
Material constants A and B in Equation (3) can be calculated using the material property
values given in material standards EN 338 and EN 1194 or by analysing the results of
experimental tests on notched beams. For the use in structural design and for their
implementation in design codes these material parameters have to be set on a reliable and
safe base. Hence, a reliability analysis is required.
3.1 Evaluation of experimental data from tests on notched beams
To compare the results from experimental tests with different geometrical configurations it
is necessary to normalize the parameters. From the sensitivity analysis in chapter 2.3
fracture energy is found to be the key parameter with most impact on variation of notch
capacity. The overall impact of both MOE and shear modulus is almost constant for notch
ratios and notch length ratios in the common range in practise. For reasons of
simplification and for a better comparison a constant ratio of MOE to shear modulus of
Ex / Gxy = 16 in line with the ratios given in EN 384 and EN 1194 and Larsen et al. [12] is
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Sensitivity factor αi [-]
Notch ratio α[-]
α= 0.75
Ex
Gxy
Gf
Gf
Ex
Gxy
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.25 0.5 0.75 1 1.25 1.5 1.75 2 2.25
Sensitivity factor αi[-]
Notch length ratio β[-]
Gf
Ex
Gxy
β= 0.25
Ex
Gxy
Gf
β= 1.0
Ex
Gxy
Gf
β= 2.0
Ex
Gxy
Gf
8
assumed which leads to a parameter B = 0.8. Equation (2) can be solved for the remaining
material properties:
'
166.0 1
5.1
1.1
1
1
8.0
5.1 2
,
5.1
22
AAf
f
EG
Xf
h
i
h
hb
V
v
v
xIf
v
f
(5)
The model uncertainty of Equation (5) is covered by including an additional parameter X.
The parameter A’ neither depends on the geometrical parameters nor on the shear strength
and can therefore be determined independently from the values given in different
standards. It has the same unit [Nmm-3/2] as stress intensity factors (SIF) K have and hence
A’ can be seen as the fracture toughness of the notch and can therefore be called notch
strength. The corresponding critical SIF of mode 1 KI,c for the assumed crack opening is
related to the energy release rate Gc,I as follows:
IIccI EGK ,,
with
2
1
2
2
G
E
E
E
EEE x
y
x
yxI
(6)
This corresponds to the material property part of A’ neglecting the constant factors 1.5 and
0.6 and assuming that the energy release rate Gc,I is equal to the fracture energy Gf,I. Using
the ratios Ex / Gxy = 16 and Ex / Ey = 30 from EN 384 and EN 1194 and assuming a
Poissons ratio ν = 0.4 a good agreement between the material property part of Equation (5)
and KI,c = (Gf,I·Ex /14)1/2 is found despite the fact that orthotropic material behaviour was
not considered in Equation (1).
Table 7: Values of notch strength A’ [Nmm-3/2]
for glulam from experimental results
A’mean (COV)
Riberholt et al. [8]
25.5 (21.0%)
Rautenstrauch et al. [16]
20.2 (24.4%)
Gustafsson et al. [32]
26.2 (26.2%)
Möhler, Mistler [34]
30.5 (26%)
All
25.7 (27.6%)
Table 8: Distribution parameters to describe the
notch strength A’ [Nmm-3/2]
A’mean (COV)
A’0.05
Normal
25.7 (27.6%)
14.0
Lognormal
25.7 (28.2%)
15.7
2p-Weibull
25.6 (29.8%)
12.8
Figure 7: Distribution of notch strength A’ from
test results for glulam and PDFs as specified in
Table 8
The notch strength parameter A’ can be determined by analysing experimental data from
tests on notched beams. In Table 7 results from tests on glulam beams are summarized and
in Figure 7 the distribution of the notch strength parameter A’ is given. A lognormal
distribution with a mean value of A’mean = 25.7 Nmm-3/2 and COV = 28.2% fits the test data
well.
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
0.11
0.12
0.13
0.14
0 4 8 12 16 20 24 28 32 36 40 44 48
Probability density [-]
Notch strength A' [Nmm-3/2]
Test results
Normal
Lognormal
2p-Weibull
9
3.2 Evaluation of theoretical distributions of material properties
The notch strength parameter A’ depends only on fracture energy and shear modulus or
MOE and ratio of MOE to shear modulus, respectively. Hence, the estimated notch
strength can be determined by using the values and distributions as given in Table 6.
As can be seen in figure Figure 8 the mean value of the estimated strength is considerably
higher and its variation lower compared to the analysed test results if no model uncertainty
is assumed (X = 1). By chosing the model uncertainty to be lognormal distributed with
Xmean = 0.82 and COV = 24% the notch strength is well representing the experimental data.
The reason for considering this model uncertainty can be attributed to the assumption by
Gustafsson that mode 1 failure of the notch is dominating the notch failure [5]. The
fracture energy analysed in chapter 2.2 is pure mode 1 fracture energy. However, both
fracture modes 1 and 2 take impact on the notch strength as has been described in different
studies [15, 35]. In numerical studies and experimental tests Franke found that the sum of
the occurring fracture energies is constant. For Norway spruce a mean sum of fracture
energies from mode 1 and 2 fractures of
mNGG ff /210
2,1,
(7)
was determined, which corresponds to 65% of the fracture energy as assumed by Larsen et
al. in Equation (4) [12]. If the model uncertainty X is implemented in the fracture energy
by using equation (7), its coefficient of variation increases to COV ≈ 50%. Such high COV
of fracture energies were also observed by Franke [15] by means of Close Range
Photogrametry at notches.
Table 9: Distribution parameter to describe notch
strength A’ according to Equation (5) [Nmm-3/2].
A’mean
(COV)
A’0.05
Xmean
(COV)
Solid Timber
17.9
(28.7%)
12.4
0.66
(22.7%)
Glulam
22.8
(30.6%)
15.6
0.82
(24.4%)
Figure 8: Distribution of notch strength A’ from
test results for glulam and notch strength
according to Equation (5) with model
uncertainty X.
3.3 Reliability of the EC5 design approach
Adequate material constants for the EC5 design approach can be derived by means of a
reliability analysis. By taking into account the partial factors defined in EC5, the material
constants can be adapted to assure the reliability of the design approach.
The design equation can be expressed for a simplified case with characteristic values of
permanent (Gk) and variable (Qk) action effects and characteristic value of the resistance Rk
as follows [36]:
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
0.11
0.12
0.13
0.14
0 4 8 12 16 20 24 28 32 36 40 44 48
Probability density [-]
Notch strength A' [Nmm-3/2]
Test results
Eq. (5) X=0.82
Eq. (5) X=1
Lognormal
Test results
Eq. (5) X~LN(0.82,0.20)
Eq. (5) X=1
Lognormal
10
0 kQkG
m
kQG
Rz
(8)
Both action effects and resistance are factored by partial factors γi. The characteristic value
of the resistance Rk is reduced by the partial factor γm in order to account for model
uncertainties and dimensional variations [7]. In this study the modification factor to
account for duration of load and moisture content was set to kmod = 1 which corresponds to
short term loading and service class 1. The variable z takes into account the individual
geometric properties and configurations for a certain design and depends amongst others
on the partial factors used: z = f(γm, γG, γQ).
The ultimate limit state function g can be set up:
QGR zg
(9)
Partial factors γ are to be chosen that way that the limit state function (Equation (9)) does
not exceed a certain probability (here 10-5) according to Equation (10) [19] for a given
value of z and the distributed parameters resistance R and action effects G and Q.
5
10)0()0(
QGRzPgP ff
(10)
In EN 1990 [37] the partial factor is γG = 1.35 for permanent action effects and γQ = 1.5 for
variable action effects. Regarding the partial factor γm a general value of 1.3 is
recommended for solid timber and of 1.25 for glulam independently of the design situation
[7]. These values were determined for the verification of strength of beams subjected to
bending [38]. For other types of stresses like shear or tension, different γm values may be
obtained. If these values are larger than those recommended in EC5 the desired failure
probability is exceeded by applying the partial factors from EC5. If they are smaller failure
probability is below the target level but uneconomical design is the consequence.
For the combined design approach in Equation (2) both partial factors for shear and notch
capacities are to be considered. Since shear design shall not be affected by the partial factor
for notch design, γm = 1.3 is set for the shear strength fv. For the material constant A a
partial factor γNotch is determined in order to verify that the failure probability is below the
target probability for the notch capacity. However, the design value of the material
constant A should be implemented in the design approach to not confuse the user with
different partial factors. In the reliability analysis only the failure mode associated with the
notch capacity of kv is taken into account (the reliability of the system of failure modes
(shear and notch related capacity) of kv in Equation (2) is not studied). Therefore Equation
(2) is rearranged to receive a design equation according to the format of Equation (8):
0
1
8.0
1.1
15.1
22
5.1
05.0
,
kQkG
z
Notchm
kv QG
h
h
i
hb
A
f
(11)
The product of characteristic values of shear strength fv,k and material constant Ak can be
expressed by the fifth percentile value of notch strength A’0.05 according to Equation (5)
and Table 9. The corresponding limit state function according to Equation (9) is as follows:
11
QG
fEG
Xf v
xIf,
v
166.0
5.1 2
zg
(12)
In this limit state function all the properties shear strength fv, model uncertainty X, fracture
energy Gf,I and MOE Ex are distributed with parameters according to Table 6 and Table 9.
A ratio of Gmean / Qmean = 0.25 of self weight and live loads is assumed. Wind and snow
loads are neglected. The distribution parameter of G and Q are as given in Table 10
following JCSS recommendations [19].
The resulting notch strengths A’, partial factors γNotch and material constants A for solid
timber and glulam are summarized in Table 11. The relationship between partial factors
and characteristic and mean values are shown in Figure 9.
Table 10: Distribution characteristics for load types
according to JCSS [19] and partial factors.
Load type
Distr.
COV
char. level
γ
Self weight
Normal
10%
50%
1.35
Live Loads
Gamma
5%
98%
1.5
Table 11: Resistances, partial factor and
proposed material factors from reliability
analysis.
Solid
Timber
Glulam
A’mean [Nmm-3/2]
17.9
22.8
A’0.05 [Nmm-3/2]
12.4
15.6
A’d = fv,dAd [Nmm-3/2]
9.1
10.9
γm
1.3
1.25
γNotch = A‘0.05 / (γm A’d)
1.05
1.15
Ad = A’d / fv,d [mm1/2]
2.96 1)
3.17 2)
3.89 3)
1) EN 338:2009
2) EN 1194:1999
3) prEN 14080:2011
Figure 9: Illustration of mean-, characteristic-, fifth
percentile- and design values of action effect E and
restiance R.
The notch strength A’d is independent from the shear strength value and can be used for
different strength classes, similarly to the specified reaction force strength in the Canadian
standard CSA 083.1-94 [39]. It is particularly suitable to be applied in design codes, when
the material constants given for the design approaches should be independent of the shear
strength values in product standards as it is the case for EC5 and the corresponding
material standards EN 338, EN 1194 and prEN 14080, respectively. For the
implementation of material constant Ad as factor kn in EC5, the notch strength A’d is
divided by the corresponding shear strength values fv,d from valid material standards. If
different shear strength values are assigned to the strength classes the highest value is to be
used to determine the material factor to also provide the desired reliability for strength
classes with lower shear strength values. E.g. in the calculation of the material constant Ad
for glulam according to EN 1194 a characteristic value of shear strength fv,k = 4.3 N/mm2
assigned to GL32h is to be used.
The resulting values for the material constant Ad in Table 11 are up to two times smaller
than the existing values in the final version of EC5 (EN 1995-1-1:2004) [7]. Other
standards [39] and studies [40] declare similar values.
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
0.11
0.12
Probability density [-]
Logaritmic distribution of action effect Eand resistance R
Emean Rmean
γM
γNotch
γG,Q
R0.05
Ek
Ed=Rd
12
Conclusions
The impact of material properties on the fracture mechanical design approach for end
notched beams as given in EC5 was studied. Values and distributions of elastic material
properties included in the theoretical basis of the design approach are specified in
standards and codes whereas fracture energy can only be found in literature. In the
sensitivity analysis fracture energy is found to be the material property with the most
impact on notch capacity. A comparison of the theoretical distribution of the notch
capacity with data from experimental tests on notched beams shows a considerable model
uncertainty when taking into account only mode 1 fracture instead of both mode 1 and 2
fractures. Values of notch strength A’d were determined in a reliability analysis. They are
particularly suitable for being implemented in design codes due to their independency
towards shear strength. The revised design values of the material constants Ad, denoted kn
in EC5, were determined as well. Depending on the shear strength value used these
adapted values to be implementated in EC5 are up to two times smaller than the existing
values in the final 2004 version of EC5.
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