For glued-laminated timber (glulam) in bending the design equation for a beam ... In Figure 4 (left) the reliability indices for structural solid timber and glulam for ...
CIB-W18/45-102-1
INTERNATIONAL COUNCIL FOR RESEARCH AND INNOVATION IN BUILDING AND CONSTRUCTION WORKING COMMISSION W18 - TIMBER STRUCTURES
ASSESSMENT OF SELECTED EUROCODE BASED DESIGN EQUATIONS IN REGARD TO STRUCTURAL RELIABILITY
J Köhler Department of Structural Engineering, NTNU, Trondheim
NORWAY
R Steiger Empa, Structural Engineering Research Laboratory, Duebendorf
G Fink R Jockwer Swiss Federal Institute of Technology, Zurich
SWITZERLAND
MEETING FORTY FIVE VÄXJÖ SWEDEN AUGUST 2012
Assessment of selected Eurocode based design equations in regard to structural reliability Jochen Kohler Department of Structural Engineering, NTNU, Trondheim, Norway René Steiger Empa, Materials Science and Technology, Structural Engineering Research Laboratory Duebendorf, Switzerland Gerhard Fink Swiss Federal Institute of Technology Zurich, Switzerland Robert Jockwer Swiss Federal Institute of Technology Zurich, Switzerland
1 Introduction A large proportion of the societal wealth is invested in the continuous development and maintenance of the built infrastructure. It is therefore essential that decisions in this regard are made on a rational basis. A structural design code should be such a rationale that facilitates design solutions that balance expected adverse consequences (e.g. in case of failure or deterioration) with investments into more safety (e.g. larger cross-sections). Structural design codes are therefore calibrated on the basis of associated risks or, simplified, on the basis of associated failure probability. In this paper it is focused on the latter. Reliability based code calibration has been formulated by several researchers, see e.g. Ravindra and Galambos [1], Ellingwood et al. [2] and Rosenblueth and Esteva [3] and has already been implemented in several design codes, see e.g. OHBDC [4], NBCC [5], and more recently the Eurocodes [6]. However, the safety format of e.g. Eurocode 5 [7] still relies on a large extent on experience and engineering judgement. In the present paper, some selected design situations are assessed in regard to the reliability of the corresponding Eurocode 5 design solutions. A possible modification of the resistance related partial safety factor is discussed in the light of practical usability of the safety format.
2 Basic principles of reliability based code calibration Modern design codes are based on the so-called load and resistance factor design (LRFD) format. In the present paper selected situations are evaluated where timber structural elements are subjected to 1
two main loads (one constant and the other variable over time) as e.g. it is typical for roof structures (self-weight + ballast and snow load) or joists supporting ceilings/floors (self-weight + ballast and imposed load). For this case a LRFD equation could be written as: (1) is the characteristic value of the resistance, is the characteristic value of the permanent load (self-weight + ballast), is the characteristic value of the load variable over time (snow load, imposed load), are the corresponding partial safety factors for the resistance and for the load. is the so-called design variable, i.e. it is defined by the chosen dimensions of the structural component. The characteristic values for the resistance and the loads are conventionally defined as certain fractile values of the probability distributions of the random variables representing resistance and load, respectively. Within the Eurocodes, corresponds to the 5% fractile value of the random resistance [7], to a 50% fractile value (or median value) of the random load constant in time [6]. is the 98% fractile value of the random yearly maxima of the variable load [6]. Based on this conventional choice of , and the corresponding partial safety factors can be calibrated to provide design solutions ( ) with an acceptable failure probability. The failure probability is expressed as: { (
)
}
with
(2)
(
)
( ) is the Limit State Equation. , and are resistance and loads represented as random variables. ( ) is the design solution identified with Equation (1) as a function of the selected partial safety factors. is the model uncertainty. It is common to express structural reliability with the so called reliability index , which is defined as: ( where
)
(3)
is the standard normal operator.
In general, different design situations are relevant in terms of contributions of the permanent and variable load. With the following modification of Equations (1) and (2) this can be taken account of: ̂ (
̂
(
)̂
̂ ̂ ̂)
̂
̂
(4) (
)̂
(5)
Here, might take values between and including 0 and 1, representing different relative contributions of permanent and variable load. The hat “^” indicates that the variables , and are normalized to a mean value of 1, resulting in ̂ , ̂ and ̂ . With, e.g., [ ] 11 different design situations are numerically represented. The partial safety factors problem:
can be calibrated by solving the following optimisation
2
[∑(
) ]
(6)
different design situations =0, 0.1,
for the mentioned
.
represents the general requirement to the safety of the structure. In Table 1 target failure probabilities and corresponding target reliability indices are given for ultimate limit states based on the recommendations of JCSS [8]. Note that the values given correspond to a year reference period and the stochastic models recommended in JCSS [8]. Table 1: Target reliability indices (and associated target failure probabilities) related to a oneyear reference period and ultimate limit states (JCSS [9]). Relative cost of safety measure
Minor consequences of failure
Moderate consequences of failure
Large consequences of failure
High
=3.1 ( p f 10-3)
=3.3 ( p f 5 10-4)
=3.7 ( p f 10-4)
Normal
=3.7 ( p f 10-4)
=4.2 ( p f 10-5)
=4.4 ( p f 5 10-5)
Low
=4.2 ( p f 10-5)
=4.4 ( p f 10-5)
=4.7 ( p f 10-6)
The value for the most common design situation is indicated with grey shading in Table 1 ( ). Guidelines for the classifications in this table can be found in the probabilistic model code, JCSS [8].
3 Example The design equation for a beam subjected to bending can be calibrated according to the procedure described in chapter 2. The chosen formulations for the variables in Equations (4) and (5) are taken from the JCSS Probabilistic Model Code (PMC) [8, 9] and summarized in Table 2. Table 2: Chosen representation of the model uncertainty , the bending strength , the permanent load and the variable load (assumptions according to the JCSS Probabilistic Model Code (PMC) [8] and EN 1990 [6]). Mean value Standard deviation Distribution type Fractile Characteristic value
1 0.1 Lognormal
1 0.25 Lognormal 0.05 0.647
1 0.1 Normal 0.5 1
1 0.4 Gumbel 0.98 2.037
[ ] was chosen, i.e. the unrealistic design situations with less than A load range of 10% and more than 80% permanent load were excluded from the optimisation (Figure 1). The software CodeCal [10] was used to perform the calculations.
3
Rafters 1
0.9
0.9
0.8
0.8
0.7
0.7
Load ratio α [-]
Load ratio α [-]
Ceiling beams 1
0.6 0.5 0.4
0.6
0.5 0.4
0.3
0.3
0.2
0.2
0.1
0.1 0
0 A
B1/D1
B2/C1
C2
0
C3-C5/D2
100
500
1000
2000
Site altitude above sea level [m]
Category of imposed load (DIN EN 1991-1-1/NA)
Definition of markers: Span short large short or large
Roof trusses of halls 1
0.9 0.8
Weight of roof or ceiling heavy intermediate light
Load ratio α [-]
0.7 0.6
Snow load
1)
Imposed load
2)
0.5
Altitude [kN/m2] Category 0m 0.76 A 100 m 0.91 B1/D1 500 m 2.53 B2/C1 1000 m 3.73 C2 2000 m 31.76 C3-C5/D2 1) according to DIN EN 1991-1-3/NA 2) according to DIN EN 1991-1-1/NA
0.4 0.3 0.2 0.1 0
0
100
500
1000
2000
Site altitude above sea level [m]
[kN/m2] 1.5 2 3 4 5
Figure 1: Load ratio (permanent load / variable load) for ceiling beams, rafters and roof trusses of halls in function of their span, the type of construction of the roof / ceiling, the imposed load (on ceilings) or the snow load (on roofs). In Figure 2 the target reliability index of (red line) [6] is compared with the design solutions for structural solid timber obtained according to the recommended values in the current version of the Eurocodes [6, 7]; i.e. represented by the line with squares. It can be observed that the reliability indices of the design solutions according to the Eurocodes tend to be too low compared to the target reliability index, especially for small alphas. The line with the diamonds is obtained when all partial safety factors are subject to optimisation. The resulting set of partial safety factors is . However, it is the philosophy of the Eurocodes that the partial safety factors for the loads are material independent [6]; therefore it is reasonable to fix and and to perform the optimisation only subject to . The line with the circles in Figure 2 is representing the corresponding result ( ). An enhancement in reaching the target reliability level can be observed for both calibrated solutions.
4
Figure 2: Reliability index for different design situations alpha for solid timber in bending. The three black lines represent different sets of partial safety factors. The target reliability index equals 4.2 (red line).
4 Comparison of design solutions of different ultimate limit states for structural solid timber In the following the design solutions for different ultimate limit states are analysed for structural elements made from solid timber; i.e. uniaxial bending, tension parallel to the grain, tension perpendicular to the grain, compression parallel to the grain, compression perpendicular to the grain and shear. Modification factors ( , etc.) that are introduced to Eurocode 5 to account for e.g. moisture content, duration of load or size effects are not considered in this study, i.e. have been set to unity. In the Eurocodes the same design equation (Equation (1)) and the same set of partial safety factors ( ) is used for all considered design situations. However, structural timber shows different material behaviour for different load types (see e.g. Köhler [11]). In Table 3 the distribution functions and the coefficients of variation COV, to describe the resistance of structural solid timber for the different load types, are listed in accordance with the JCSS PMC [8]. This statistical representation of timber strength properties in the PMC had been subject of a collection of available experimental data and of extensive discussions along COST Action E24 “Reliability of Analysis of Timber Structures” [12]. Under the assumption that the resistance of structural solid timber for different loading modes can be represented as suggested by the JCSS PMC (Table 3) the reliability indices for different design situations alpha were calculated for the prescribed set of partial safety factors ( ). The results are shown in Figure 3 (left) and Table 4. A comparison of the target and the actual reliability indices shows a relatively good agreement for uniaxial bending (as already presented in the initial example above) and for shear (exactly the same results as for bending due to similar assumptions for the statistical representation of the material resistance in bending and shear). There, the mean value of the calculated reliability index is 5
which corresponds to a probability of failure . For the ultimate limit state design in tension parallel to the grain and in compression parallel to grain the differences between the target and the actual reliability indices are significantly larger; i.e. and , respectively. However, the calculated reliability indices for the ultimate limit state design in tension perpendicular to the grain and in compression perpendicular to grain are significantly different to those recommended by the JCSS PMC and by Eurocode 0. For compression perpendicular to the grain the present version of the Eurocode delivers design solutions that are too safe; i.e. the probability of failure is only one third of the value recommended by the JCSS. More problematic is the ultimate limit state design for tension perpendicular to the grain. Here, the Eurocode delivers design solutions that are not safe enough; i.e. the estimated probability of failure is order of magnitudes larger than the recommended. However, tension perpendicular to the grain is certainly a failure mode that is not sufficiently covered by the level of detail applied in the present analysis. The characteristic values prescribed in the EN 338 are of nominal character, i.e. a certain degree of conservatism is included due to the sensitivity of the failure mode to several aspects, e.g. moisture induced stresses, volume effects, deviation of test procedures to structural conditions, etc. It is also interesting to see that the chosen COV for tension perpendicular to the grain is similar to the COV for bending and shear. The large difference in regard to reliability indices is due to the different distribution type that is chosen to represent tension perpendicular to the grain. Table 3: Distribution type and coefficients of variation for different ultimate limit states for structural solid timber: assumptions according to the JCSS PMC [8]. Ultimate limit state Bending Tension parallel to the grain Tension perpendicular to the grain Compression parallel to the grain Compression perpendicular to the grain Shear
Distribution type Lognormal Lognormal 2-p Weibull Lognormal Normal Lognormal
COV 0.25 0.30 0.25 0.20 0.10 0.25
As described above, the philosophy of the Eurocodes [6] is that the partial safety factors for the loads are material independent. In the following the partial safety factor for the resistance is calibrated for the different limit states. For the calibration the partial safety factors for the loads are assumed constant with and . The calibrated partial safety factors for the resistance are listed in Table 4. In Figure 3 (right) the reliability indices calculated with the optimised for different design situations alpha are illustrated. Table 4: Calibrated partial safety factors for the resistance (for constant ). Ultimate limit state Bending Tension parallel to the grain Tension perp. to the grain Compression parallel to the grain Compression perp. to the grain Shear
6
1.33 1.40 3.05 1.24 1.20 1.33
and
5
5
4.8
4.8
4.6
4.6
4.4
4.4
4.2
4.2
4
4
3.8
3.8
3.6
3.6
3.4
3.4
3.2
3.2
3
3 0.1
0.2
0.3
0.4
Target
Bending
0.5
0.6
0.7
0.8
0.1
0.2
0.3
0.4
Tension par. to grain
Compression par. to grain
Tension perp. to grain
Compression perp. to grain
0.5
0.6
0.7
0.8
Shear
Figure 3: Reliability index for different design situations alpha. The different lines represent different ultimate limit states. Left: Calculated according to Eurocode. Right: Calculated with the optimised partial safety factor for the resistance (given in Table 4). Table 5: Reliability index (mean value, minimum and maximum) and the associated probability of failure for resistance for different ultimate limit states for structural solid timber calculated with the partial safety factors given in the Eurocodes [6, 7] Ultimate limit state
Mean value 4.17 4.01 3.22 4.32 4.47 4.17
Bending Tension parallel to the grain Tension perpendicular to the grain Compression parallel to the grain Compression perpendicular to the grain Shear
Min. value 4.06 3.88 2.93 4.12 4.14 4.06
Max. value 4.26 4.07 3.39 4.49 4.92 4.26
For the mean value of β 1.55 10 5 3.04 10 5 6.43 10 4 7.94 10 6 3.96 10 6 1.55 10 5
5 Comparison of design situations of structural solid timber and glued-laminated timber in bending For glued-laminated timber (glulam) in bending the design equation for a beam can similarly be calibrated according to the procedure described in chapter 2. The recommended partial safety factors are , and [6, 7]. Furthermore, it is assumed that the bending resistance can be represented by a lognormal distribution and a coefficient of variation COV=0.15 according to the JCSS PMC [8]. In Figure 4 (left) the reliability indices for structural solid timber and glulam for different design situations alpha are illustrated. It can be observed that the reliability indices of the design situations for glulam according to the Eurocodes tend to be too high, especially for large alphas.
7
4.7
4.7
4.6
4.6
4.5
4.5
4.4
4.4
4.3
4.3
4.2
4.2
4.1
4.1
4
4
3.9 0.1
0.2
0.3
0.4
0.5
0.6
0.7
Target
3.9 0.1
0.8
Structural timber
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Glulam
Figure 4 Reliability index for different design situations alpha. The different lines represent the ultimate limit state in bending of structural solid timber and glulam. Left: Calculated according to the Eurocodes. Right: Calculated with the optimised partial safety factor for the resistance . In the following the partial safety factor for the resistance of glulam subjected to bending is calibrated to . In Figure 4 (right) the reliability indices for different design situations alpha are illustrated. Glulam shows a significantly larger scatter of the reliability index for different design situations than solid timber. This results from the smaller variation of the bending strength. For the ultimate limite states tension parallel to the grain, compression parallel to the grain and shear the tendencies compared to uniaxial bending as observed for solid timber are qualitatively the same for glulam due to the same relations of coefficients of variation between the different ultimate limite states assumed in JCSS PMC [8]. However for tension perpendicular to the grain the estimated probability of failure is even larger as for structural solid timber. For compression perpendicular to the grain the probability of failure is lower than the target.
6 Conclusions From the assessment of the current design format of Eurocode 5 in regard to the reliability of the design solutions obtained by following the prescriptions in the code, it results that the safety level is rather different and very much dependent on:
the failure mode (i.e. bending, tension, shear etc.).
the importance of different load effects (constant and variable load, represented by the factor in the present study).
While the latter dependence is also observed for other building materials, the large dependence on the failure modes is a timber specific phenomenon. The load bearing capacity of structural solid timber and timber based materials as e.g. glued laminated timber is very much dependent on the loading mode. This results in rather different magnitudes and stochastic properties of the different load bearing capacities. This matter of fact is contrasted by the use of partial safety factors for the material related load bearing capacity, e.g. for solid timber. 8
The results as presented in this paper suggest a differentiated treatment of different failure modes in terms of different partial safety factors applied to different failure modes. This would enhance the consistency of the design format in regard to the structural reliability presented by the corresponding design solutions. However, such a development would add more complexity to the code and would not match the recent aim towards clarity and user friendliness of the Eurocodes [13]. A veritable alternative or a valuable extension respectively would be the presentation of “capacity values” in Eurocode 5, i.e. the characteristic values as presented in EN 338 (for solid timber) or EN 14080 (for glulam) multiplied with the failure mode specific partial safety factor. This has e.g. been done in Swiss code for the design of timber structures SIA 265 [14] where also duration of load effects are included into the simplified representation of capacity values.
7
References
1.
Ravindra, M. and T.V. Galambos, Load and resistance factor design for steel. ASCE, Journal of the Structural Division, 1978. 104(9): p. 1337-1353. Ellingwood, B., et al., Probability Based Load Criteria: Load Factors and Load Combinations. ASCE, Journal of the Structural Division, 1982. 108(5): p. 978-997. Rosenblueth, E. and L. Esteva, Reliability basis for some Mexican codes. ACI Publication, 1972. 31: p. 1-42. OHBDC, (Ontario Highway Bridge Design Code), Ontario Ministry of Transportation and Communication, Ontario. 1983. NBCC, (National Building Code of Canada), National Research Council of Canada. 1980. CEN, EN 1990: Basis of structural design. 2002, European Committee for Standardization: Brussels. CEN, EN 1995-1-1, Eurocode 5: Design of timber structures; part 1-1: general rules and rules for buildings. 2004, European Committee for Standardization: Brussels. JCSS Probabilistic Model Code. Probabilistic Model Code Part III - Resistance Models (3.05 Timber). 2006. Köhler, J., J.D. Sørensen, and M.H. Faber, Probabilistic modeling of timber structures. Structural Safety, 2007. 29(4): p. 255-267. JCSS, CodeCal - Reliability Based Code Calibration. 2004, http://www.jcss.ethz.ch/CodeCal/CodeCal.html. Köhler, J., Reliability of Timber Structures, in Department of Civil, Environmental and Geomatic Engineering. 2006, ETH Zurich: Zurich. p. 237. COST Action E 24. Reliability of timber structures. Several meetings and Publications,Internet Publication: http://www.km.fgg.uni-lj.si/coste24/coste24.htm. 2005. Dietsch, P. and S. Winter, Eurocode 5-Future Developments towards a More Comprehensive Code on Timber Structures. Structural Engineering International, 2012. 22(2): p. 223-231. Schweizerischer Ingenieur- und Architekten-Verein SIA, SIA 265: Timber Structures. 2003: SIA, Zürich, Switzerland.
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