Reliability-based code calibration of partial safety factors
P. Friis·Hansen∗ & J. Dalsgård Sørensen†
1
Introduction
Code calibration refers to that particular activity that is exercised when some superior method is
applied to assign values to the variables of a code format such that a specific design code is formulated.
For a code format of the partial safety factor type, the variables are characteristic values, partial
coefficients, and load reduction factors.
A code can be calibrated on different levels of superior methods. In the present study focus is on
a best fit to a superior formal reliability evaluation method. Other approaches of code calibration
are calibration by judgement and calibration to existing design practice. The pros and cons of these
alternative approaches will not be discussed in the present study.
Today, the highest level of code calibration consists of a best fit optimization aiming at approximating
results of a superior reliability analysis model. In recent years, several codes have been based on more
or less extended principles of such best fit optimization.
Having defined the goal of the code, i.e. the class of structures to which the code is meant to apply,
the next step is to define the code objective. In today’s state-of-the-art code calibration two different approaches are generally considered: One aims at maximizing the expected utility, the other at
achieving a constant target reliability level. Applying the utility-based approach requires identification of benefits, construction costs, repair and maintenance costs, and failure costs. If the benefits are
disregarded in the utility-based approach this then becomes a risk-based approach.
In the present study we will first review the utility-based approach and then the target reliability
approach. Thereafter we will based on the target reliability approach, compare and discuss the results
of two different procedures for calibrating a set of partial safety factors to a specific design equation.
The two different approaches that are used in the calibration to the target reliability level is partly
the design value format and partly a free format calibration. These methods will also be described
in the present paper. The two methods are compared in an illustrative example where partial safety
factors are calibrated in a code format similar to the Eurocode format.
2
Formulation of the code optimization
In principle the formulation of a code optimization involves six steps. These steps will shortly be
described in the following. For a fuller treatment reference is left to (Madsen, Krenk & Lind 1986)
and (Ditlevsen & Madsen 1996).
∗
†
Section for Maritime Engineering, MEK, DTU, DK-2800 Kgs. Lyngby. [email protected]
Department of Structural Engineering, Aalborg University, DK-8000 Aalborg. [email protected]
1
2.1
Structure class
The first step is to define the class of structures on which the code is intended to operate. This
definition may contain limitations of the code. An accurate definition of the domain of the code is a
necessity.
The simplest code format may consist of a specific set of characteristic values and a simple table
of corresponding partial safety factors. The more advanced the goal of the code becomes the more
advanced or complicated must the partial safety factors be given in order to properly meet the goal.
The more complicated partial safety factors can for instance be given as functions of different cases or
be defined implicitly in terms of mathematical expressions. It is obvious that the more complicated
the code format becomes the closer the goal of the code may generally be satisfied, that is, the closer
the code is to be optimal1 .
In order to perform the subsequent code format optimization, a set of representative example structures
belonging to the considered class must also be selected.
2.2
Probabilistic code
The second step is to formulate a probabilistic code format for the considered class of structures.
The preparation of a design code implies evaluation of characteristic values, design values (partial
safety factors (γ-values)), combination factors (ψ-values), formulation of load configurations, etc., by
calibration against the results of a set of representative examples of probabilistic reliability analyzes.
This means that a set of probability distribution models for the material parameters and the actions
must be established with due respect to the requirements of model realism. Moreover, the uncertainties
as regards the representative accuracy of the applied mathematical models (geometry, mechanics,
actions, strength, etc.) must be quantified in terms of probability distributions. When these models are
formulated, the probabilities of different relevant adverse events occurring within a pre-specified time
can be calculated. The relevant adverse events concern malfunctioning with respect to serviceability or
they concern regular collapse situations. The adverse events are represented in the reliability analysis
as different categories of limit states, and special reliability requirements are assigned to each category.
The usual limit state categories are serviceability limit state (SLS), ultimate limit state (ULS), and
accidental limit state (ALS).
2.3
Goal of the code
The third step is to define the goal of the code. For a partial safety factor format the goal can be
(1) maximization of the expected utility, or (2) to obtain a specified generalized reliability index. The
goal is thus defined in terms of a reliability method on a higher level. In the following section we will
describe the definition of the goal in more detail.
1
Consider a utility based code that directly takes into account the cost of failure and two elements that belong to
the same class and are exposed to independent but identical loading pattern. If the failure costs related to failure of
the two elements differ significantly for some reasons, then it should be required that the risk, i.e. P F CF , should be of
equivalent magnitude for the two elements; that is, the larger the failure cost the safer the element should be required to
be. A simple code format that do not take into account the failure costs cannot reflect the required difference of these
two designs.
2
2.4
Importance weighing
The fourth step is to obtain information of relative occurrence of different structural designs in practice
and related parameter values within the structural class defined in the first step. The two main
objectives of any code development and calibration of partial safety factors are the determination of
a code which is simple to use, and yet achieves a uniform safety level. Unfortunately simplicity and
safety are mostly two conflicting objectives. It is therefore necessary to define the most important
structural data for which the goal should be satisfied as well as possible.
2.5
Measure of fit
The fifth step is to define a measure for the degree of fit between a code and the goal of the code.
Subjected to analysis according to the probabilistic code (second step), the example structures from
the class of structures on which the code is intended to operate have reliability indices β that differ
from the target reliability index βT .
A possible choice of penalty function M is the simple square deviation
M (β, βT ) = (β − βT )2
(1)
Another commonly used penalty function is
M (β, βT ) = c(β − βT ) + exp[−c(β − βT )] − 1
(2)
in which c is a constant that can be chosen to 4.35, (Madsen et al. 1986). This value of c provides
a reasonable fitting of exp[−cβ] to Φ(−β) for β in the interval from about 3 to about 6. It is seen
that the penalty function equation (2) is skewed such that underdesign gives a larger penalty than
overdesign. In general it turns out that the final result is not very sensitive to the choice of the penalty
function, (Ditlevsen & Madsen 1996).
2.6
Code format optimization
With wi being the importance weight of design case i, the weighted average
X
∆=
wi M (βi , βT i )
(3)
all structures
of the penalty function is a measure of the degree of fit between the code and its goal. The sixth step
aims at minimizing this penalty measure ∆.
2.7
Defining the code calibration
As already mentioned, state-of-the-art code calibration today generally consider two different approaches: One aims at maximizing the expected utility, the other at achieving a constant target
reliability level. The two ways of performing the reliability-based, cost optimal calibration of partial
safety factors are:
3
• Direct calibration of partial safety factors from a cost-benefit formulation.
• A two step procedure, where first the target reliability level is determined from a cost-benefit
formulation and next the optimal partial safety factors are obtained by calibrating to the target
reliability level.
These procedures will be described in the following.
2.7.1
Direct calibration of partial safety factors
A general formulation based on decision theoretical concepts is obtained when the total expected
cost-benefits for a given class of structures are maximized with the partial safety factors as decision
variables, see e.g. (Sørensen & Faber 2000):
max W (z(γ)) =
γ
s.t.
L
X
j=1
wj [Bj − CIj (z(γ)) − CRj (z(γ)) − CF j PF j (z(γ))]
(4)
γ l ≤ γ ≤ γ u,
where z(γ) defines the design variables as a function of γ, γ = (γ1 , . . . , γm ) are the m partial safety
factors to be calibrated. γ l and γ u are lower and upper bounds on the partial safety factors. L is the
set of representative example structures belonging to the considered class. B j is the expected benefits
(in general for the society, but in some cases the benefits can be related to the owner of the structures
considered), CIj is the initial (or construction) costs, CRj is the repair/maintenance costs during the
design life time and CF j is the cost of failure. CF j is assumed to be independent of the partial safety
factors. PF j is the probability of failure for failure mode j if the structure is designed using given
partial safety factors.
The limit state functions related to the failure modes considered are written:
gj (X, pj , z) = 0 , j = 1, . . . , L
(5)
where X is the stochastic variables, pj a vector with deterministic parameters, and z the design
variables. The application area for the code is described by the set I of L different vectors p j . The set
I may e.g. contain different geometrical forms of the structure, different parameters for the stochastic
variables and different statistical models for the stochastic variables.
The deterministic design equation related to the limit state equation in equation (5) is written
Gj (xc , pj , z, γ) = 0 , j = 1, . . . , L
(6)
where xc are characteristic values.
The optimal design-to-limit of the example structures according to the model code is obtained by
minimizing the initial construction costs CIj (z(γ)), where z are the design variables (optimization
variables).
min CIj (z)
z
s.t. Gj (xc , pj , z, γ) ≥ 0
(7)
zl ≤ z ≤ zu
4
The objective function in equation (7) is the construction costs, and the constraints are related to
the design equations. It is noted that the (γ)-values are fixed within the optimization in equation
(7). The objective function in equation (7) may easily be extended to also include the repair costs
CRj (z(γ)) and the benefits.
Using the limit state equation in equation (5) the probability of failure of the structure and the
expected repair/maintenance costs CRj to be used in equation (4) are determined at the optimum
design point z ∗ . In cases where more than one failure mode is used to design a structure included in
the code calibration, the relevant design equations all have to be satisfied for the optimal design z ∗ .
It is recognized that when applying the optimization principle of CIj (z(γ)) to identify the designto-limit as given by equation (7), the optimization in equation (4) has an objective function with
a value determined by solving a second optimization. If the failure probability PF j is estimated by
first-order reliability methods, then this will also be determined by solving an optimization problem.
The optimization, equation (4), is therefore typically solved as a nested optimization problem. It
might be questioned why the inner optimization has to be carried to end when the outer optimization
is far from its optimum. This concern led to a proposal of combining the two optimization into one
simply by adding the Kuhn-Tucker conditions for the inner optimization as additional constraints to
equation (4) (Madsen & Friis·Hansen 1991) and (Friis·Hansen 1994). These ideas have been explored
further and extended in other studies, see (Rackwitz 2002) for details.
2.7.2
Calibration of target reliability level and partial safety factors
The optimal reliability level βT may be determined by considering the class of structures and then
maximizing the total expected benefits2 (with z as optimization variable):
max
z
W (z) =
L
X
j=1
£
¤
wj Bj − CIj (z) − CRj (z) − CF j PFT (z)
(8)
Next, a code format is chosen and the partial safety factors γ are calibrated such that the penalty
measure equation (3) is minimized. The choice of penalty function can either be equation (1) or
equation (2).
The optimal partial safety factors are obtained by numerical solution of the optimization problem
equation (3). The reliability index βj (γ) for combination j given the partial safety factors γ is obtained
as follows. First, for given partial safety factors γ the design-to-limit of the example structures are
obtained by minimizing the construction costs CIj as given in equation (7). For this particular optimal
design z the reliability index is next estimated by FORM/SORM or simulation on the basis of the
limit state equation (5).
A flow-chart of the general code calibration procedure is shown in figure 1. The lower part of the
figure illustrates the optimization scheme of the partial safety factors.
2
It is noted that if the benefits Bj are independent of the design parameters z then the optimum value of z is
independent of Bj . Since we are interested in the optimal value of z only, and not that of W (z), the benefits may be
removed from equation (8)
5
Figure 1: Flow-chart for code calibration procedure
6
Figure 2: Illustration of the design-value format
2.7.3
Design-value format
The most time consuming aspect in the code calibration procedure in figure 1 is the design-to-limit
and the resulting repeated reliability evaluation in the loop. Therefore, major computer time savings
could be achieved if the design-to-limit could be moved outside the optimization of the partial safety
factors. When the reliability analysis is based on the so-called single point FORM 3 the so-called
design-value format represents a fast alternative to the optimization outlined in figure 1.
In the following we will, following (Ditlevsen & Madsen 1996), give a condensed review of the designvalue format. Reference is left to (Ditlevsen & Madsen 1996) for a detailed treatment.
For an independent input variable X the relation between the design value xd in the physical formulation space and the corresponding coordinate βα to the globally most central limit-state point in the
standardized Gaussian space is determined by:
FX (xd ) = Φ(βα)
(9)
The design value xd then is
xd = FX−1 [Φ(βα)]
(10)
For the possible design cases within the domain of a code the directional vector α for a given target
reliability level βT will vary from case to case. The situation is illustrated in two dimensions in figure 2
with two different limit states 1 and 2. The radius β of the circle defines the common reliability level.
If this β is the target reliability level βT the value of the design variable in the limit-state problem is
determined such that the limit-state curve becomes tangential to the circle. For this particular design
there are infinitely many partial safety factor pairs applied to given characteristic values for X 1 and
X2 . All that is needed is arbitrarily to define a single design point Q on the limit-state curve. If D 1 is
3
Single-point FORM is the First Order Reliability Method that applies a linearization of the limit-state surface at
the most central limit-state point in the Gaussian space representation of the reliability analysis model.
7
Figure 3: Illustration of the limit-state surfaces and the reliability indicies for the ith design case.
chosen as design point, where D1 is the most central limit-state point, the partial safety factors that
produce the same design become
γ11 =
FX−11 [Φ(βT α11 )]
x1k
;
γ12 =
x2k
−1
FX2 [Φ(βT α12 )]
(11)
where x1k is an upper fractile characteristic value of X1 (load type variable) and x2k is a lower fractile
characteristic value of X2 (resistance type variable).
For the case with the two limit states in figure 2 it is immediately seen that if the intersection point
D between the two limit state curves is taken as the design point, then the partial safety factors are
common for the two limit states. The vector δ acts as a replacement vector for α 1 and α2 without
introducing any error of approximation. Clearly, with more than two limit states in the class of design
problems containing solely the two random variables X1 and X2 there is no replacement vector δ that
exactly reproduces all the designs. By a given superior requirement there will only be one design point
βδ for the class and all the limit-state curves are therefore adjusted to contain this point. Thus the
reliability indices will vary over the class. The strategy is then to determine the replacement vector
δ such that the expected loss (penalty) for the entire design class becomes minimal. It is noted that
in the optimization problem it is not required that δ should be a unit vector. By design-to-limit in
the ith design case the limit-state surface will, instead of the equation gi (u) = 0, get an equation
g̃i (u) = 0 which is satisfied for the point u = βT δ. In general this point is not necessarily the same
as the most central point on the surface g̃i (u) = 0. The situation is illustrated in figure 3. If the
limit-state surfaces defined by gi (u) = 0 and g̃i (u) = 0 are almost plane and parallel within a domain
that contains the points βT αi , βT δ i , βi α̃i , then it follows directly from the figure that the geometric
reliability index βi is approximately
βi ≈ βT α0i δ
(12)
The penalty function, equation (2), can hereafter with sufficient accuracy be modified to
µ
¶
βT (α0i δ − 1)
βT (α0i δ − 1)
+ exp
−1
M (βi , βT ) =
d
d
8
(13)
Figure 4: Flow-chart for code calibration according to the design value format
9
and the optimization of the weighed average ∆ can be made as shown in figure 4. Compared to the
optimization in figure 1 a major computer-time saving is gained because the box “Design-to-limit of
example structures according to new code” is moved outside the optimization procedure. Thus this
calculation is only made once.
It should be noted that the obtained optimal replacement vector δ for α can be used with good
accuracy for other specified values of βT in the vicinity of the value used in the optimization. By a
moderate change of the target reliability level it is therefore not necessary to make a new optimization
as is the case if the procedure of figure 1 is used.
2.7.4
Restriction on the replacement vector
It was observed in (Friis·Hansen & Ditlevsen 1998) that if no restrictions are made on the set of vectors
from which the δ-vector is chosen, it may happen that the minimization of ∆ by use of (12) leads to a
δ-vector far outside the cluster of α-vectors. To avoid this it is reasonable to require that the δ-vector
is inside the convex hull spanned by the total set of the α-vectors of the considered cluster. However,
the computational difficulties of satisfying this requirement makes it more convenient to be content
with the requirement that the δ-vector is inside a minimized elliptical hypercone that contains all the
α-vectors of the cluster.
Following (Friis·Hansen & Ditlevsen 1998) this n-dimensional cone is obtained by finding the matrix
A and the numbers λ1 , . . . , λn−1 that maximizes
min{λ1 , . . . , λn−1 }
(14)
under the constraints
∀i ∈ {1, . . . , N } : α0i Mαi ≤ 0
(15)
M = A ΛA
(16)
Λ = dλ1 − 1 . . . λn−1 − 1 − 1c (diagonal matrix)
(17)
AA=I
(18)
∀i ∈ {1, . . . , n − 1} : λi > 0
(19)
0
0
where I is the unit matrix and N is the number of α-vectors.
The constraint on the δ-vector is then
δ 0 Mδ ≤ 0
3
(20)
Example
In this example we will explore the differences in the calibrated partial safety factors obtained by
use of of the design-value format (section 2.7.3) and by use of the free format (section 2.7.2). An
important difference between the two approaches is that the design-value format seeks to find a single
replacement vector δ for all example structures encompassed by the code and then base the partial
safety factors on this vector. The traditional approach (what we here will call the free format) do not
10
Table 1: Stochastic model
Variable
Name Distribution Mean value
Permanent load
G
Normal
1
Variable load (environmental)
Q1
Gumbel
1
Variable load (imposed)
Q2
Gumbel
1
Strength Material 1
R1
Lognormal
1
Strength Material 1
R2
Lognormal
1
Model uncertainty
XR
Normal
1
COV
0.10
0.40
0.20
0.05
0.15
0.05
Fractile
50%
98%
98%
5%
5%
50%
focus on a joint vector but solely seeks to minimize the penalty function. The two approaches will in
this section be labelled DV and FREE, respectively.
To represent the modelling of the Eurocode the following representative limit state function is considered:
g(X, z) = zRXR − ((1 − η)G + ηQ)
R
XR
z
where
G
Q
η
(21)
strength
model uncertainty
design variable
permanent load
variable load
factor between 0 and 1, representing the relative fraction of variable load.
Two different materials with strengths R1 and R2 and two different variable loads Q1 and Q2 are
considered. The stochastic model is shown in table 1. The table also shows the fractile values used to
determine the characteristic value.
The design variable z = max(z1 , z2 ) is determined from the following two design equations corresponding to the equations (6.10a) and (6.10b) in Eurocodes, Basis of Structural Design:
LC 1: z1 Rc /γR − [(1 − η)γG Gc + ηγQ Qc ] = 0
LC 2: z2 Rc /γR − [(1 − η)ξG γG Gc + ηξQ γQ Qc ] = 0
where index c indicates characteristic value and
γG
γQ
ξQ
ξG
γR
partial safety factor for permanent load in LC 1 (variable load dominating)
partial safety factor for variable load in LC 1 (variable load dominating)
factor for variable load in LC 2 (permanent load dominating), ξQ ≤ 1
factor for permanent load in LC 2 (permanent load dominating), ξG ≥ 1
(=γR1 or γR2 ) partial safety factor for strength
It is chosen that γG = 1.
The following set of representative example structures are defined as follows:
• 2 materials
• 2 variable loads
11
(22)
(23)
Table 2: Calibrated optimal partial safety factors. DV indicates that the calibration is based on the
design-value format, FREE that it is based on the free format calibration.
βT = 4.2
βT = 4.6
βT = 5.0
Calibration DV Free DV Free DV Free
γG
1.00 1.00 1.00 1.00 1.00 1.00
γQ1
1.70 1.40 1.91 1.51 2.13 1.68
γQ2
1.46 1.19 1.59 1.24 1.74 1.36
γR1
1.14 1.29 1.16 1.36 1.18 1.39
γR2
1.17 1.34 1.21 1.44 1.24 1.50
ξG > 1
1.27 1.11 1.28 1.14 1.31 1.13
ξQ < 1
0.75 0.82 0.78 0.81 0.78 0.80
• 8 different η values: 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8 and 0.9
Thus, the considered structure class consists of L = 2 · 2 · 8 = 32 different failure modes. It is assumed
that all failure modes have identical weights, that is wj = 1/L, j = 1, . . . , L.
3.1
Result of calibration
In the calibration equation (2) has been used as the penalty measure in the calibration according to
the design-value format, whereas equation (1) was used in the free format calibration. The calibration
were performed to three different target reliability levels: βT = 4.2, βT = 4.6, and βT = 5.0. Table 2
shows the obtained optimal partial safety factors for the two procedures.
It is seen from Table 2 that the two different formats result in partial safety factors that are quite
different. Compared to the free format the design-value format finds larger γ-values for the loading
variables (Q1 , Q2 ) and smaller values for the resistance variables (R1 , R2 ) for all the considered
target reliability levels. Similar, the design-value format predicts larger ξ G -values compared to the
free format, and smaller ξQ -values.
In a comparison Table 3 shows the partial safety factors obtained by the design-value format where
no restriction on the partial safety factor is introduced on the permanent load. The partial safety
factors are moreover calibrated independently for the case of the variable and the permanent load
being dominating. Comparing the partial safety factors for the variable load being dominating with
the situation of the permanent load being dominating it is seen that the partial safety factors on both
the permanent load and the resistance increases, whereas the the partial safety factor on the variable
load is decreased.
To investigate the effect of the relatively large variation in the calibrated partial safety factors in Table
3, figure 5 shows the variation in the optimized reliability level of all example structures for the three
target reliability levels. The results are presented in four clusters, where each cluster is for a full
variation of the η-parameter. The first cluster is for the combination R 1 , Q1 , the second cluster for
R2 , Q1 , the third cluster for R1 , Q2 , and finally the fourth cluster for the combination R2 , Q2 .
It is seen that in the present example the design-value format results in slightly smaller variation in
the obtained reliability indices compared to the free format. This may, of course, may be attributed
12
Table 3: Partial safety factors obtained from the design value format β = 4.2. The partial safety
factors are given both for the case when the variable load is dominating and when the permanent load
is dominating.
Variable dom.
Permanent dom.
Combination
γG
γQ
γR
γG
γQ
γR
R1 , Q1
1.05 1.79 1.09 1.18 1.22 1.14
R2 , Q1
1.05 1.79 1.11 1.18 1.22 1.25
R1 , Q2
1.05 1.53 1.09 1.18 1.15 1.14
R2 , Q2
1.05 1.53 1.11 1.18 1.15 1.25
Calibration: βt=4.2
4.7
4.6
4.5
4.4
β
4.3
4.2
4.1
4
3.9
3.8
3.7
5
10
15
20
Case number
25
30
Calibration: β =5.0
Calibration: βt=4.6
5.4
t
5.6
5.2
5.4
5
β
β
5.2
4.8
5
4.6
4.8
4.4
4.6
4.2
5
10
15
20
Case number
25
30
5
10
15
20
Case number
25
30
Figure 5: Variation of optimal reliability index β. The squares represent the result from the free calibration and the diamonds the results from the design value format when the calibration is performed
according to the specified code format. The circles represents the result from design value format,
when all variables are allowed to have independent partial safety factors.
13
Comparing α−vector: βt=4.2
1
Comparing optimal design variables: βt=4.2
5
0.8
4.5
0.6
4
0.4
3.5
z
z
0.2
0
3
−0.2
−0.4
2.5
−0.6
2
−0.8
−1
5
10
15
20
Case number
25
1.5
30
5
10
15
20
Case number
25
30
Figure 6: Left: Comparison of elements of the unit normal vector α∗ to the failure surface at the
optimal design points. The legend is Strength: square; Model uncertainty: diamond; Permanent load:
circle; Variable load: star. Right: Comparison of the optimal value of the design parameter z. The
legend is Design value format: diamond; Free format: square.
Table 4: Optimal replacement vector δ for β = 4.2.
Variable dom. Permanent dom.
Variable load Q
0.88
0.62
Permanent load G
0.12
0.43
Model uncertainty XR
-0.34
-0.38
Strength R
-0.44
-0.62
to convergence requirements. The effect of applying the skewed penalty measure, equation (2), that
penalizes under design more than over design, is clearly reflected in the obtained reliability indices for
the design-value format, see figure 5.
For the considered example structures figure 6 left shows the comparison of the variation of the unit
normal vector α∗ to the failure surface at the optimal design points. The comparison is for β T = 4.2.
It is seen that both calibration routines results in an almost identical unit normal vector α ∗ . Figure
6 right shows the resulting value of the design variable z. It is seen that both the design-value format
and the free format result in almost identical variation of the design variable z. Note that the effect of
the design-value format penalizes under design more than over design is clearly reflected in the results.
It is recalled that the design variable z is obtained by design-to-limit according to the model code.
Therefore, it may be concluded that although the design-value format and the free format result in a
different optimal set of partial safety factors, the resulting design is not affected by the difference.
Finally, Table 4 presents the optimal replacement vector δ obtained from the design-value format. In
comparing the replacement vector δ to the variation of the unit normal vector α ∗ in figure 6 left, it
is seen that the δ-vector reflects the component of the variable load Q and the permanent load G
well. In an average sense the component of the model uncertainty XR appears slightly overestimated
and the strength R component slightly underestimated. However, it must be emphasized that the
14
replacement vector δ shall not be interpreted in an average sense but as the vector that best replaces
the design point of all limit states.
It is interesting to compare the proposed values in the Eurocode for the importance factors α V =
αP = 0.7 for variable and permanent loading and αS = −0.8 for strength with the values that is found
by application of the design-value format in the present study. From
√ Table 4 it is seen that when the
2
variable load is dominating then the pair αV = 0.88 and αS = − 0.442 + 0.34
√ = −0.56 is obtained,
and when the permanent load is dominating the pair αV = 0.62 and αS = − 0.622 + 0.382 = −0.73
is obtained. Thus a reasonable agreement is found.
4
Conclusion
The present study presented a resume of state-of-the art procedures for calibration of partial safety
factors, namely the utility-based approach and the target reliability approach. For the target reliability
approach two different procedures for calibrating the set of partial safety factors to a specific design
equation were presented. The two approaches presented were the design-value format and a free format
calibration. These two methods were finally compared in an illustrative example where partial safety
factors were calibrated in a code format similar to the Eurocode format.
The main conclusions from the illustrative example are:
• The design-value format and the free format result in a set of partial safety factors that are quite
different, see Table 2.
• In comparing the reliability indices for optimal design according to the code format the designvalue format results in slightly smaller variation compared to the free format, see figure 5.
• The comparison of the variation of the unit normal vector to the failure surface at the optimal
design point α∗ , showed that both calibration routines result in an almost identical vector α ∗ ,
figure 6 left.
• In comparing the resulting design variable z it was found that both procedures resulted in almost
identical values, figure 6 right.
• It is concluded, that the design-value format results in a set of partial safety factors that better
reflects the “joint design point” of all considered example structures and therefore may be more
appropriate when non-linear analysis is applied.
• Finally, it may be concluded that although the design-value format and the free format result
in a different optimal set of partial safety factors, the resulting design is not affected by the
difference.
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