1 2 Milan Holicky and Ton Vrouwenvelder
1Klokner Institute, Czech Technical University in Prague, Czech Republic 2Delft University of Technology, TNO BOUW, The Netherlands
Elementary methods of structural reliability are described considering a fundamental case of two random variables when the limit state function is formulated as a difference between the resulting structural resistance and load effect. The initial assumption of normal distribution of both resulting variables is generalised to any type of probability distribution. The described computational procedures are illustrated by a number of numerical examples, which are supplemented by MATHCAD and EXCEL sheets. An extension of the elementary methods of structural reliability is presented in Annex B.
Fundamental concepts and procedures of structural reliability are well described in a number of national standards, in the new European document EN 1990  and International Standard ISO 2394 . Additional information may be found in the background document developed by JCSS  and in recently published handbook to EN 1990 . Guidance on application of the probabilistic methods of structural reliability may be found in publications and working materials developed by JCSS  and in relevant literature listed in  and .
The theory of structural reliability considers all basic variables as random quantities having appropriate types of probability distribution. Different types of distributions should be considered for actions, material properties and geometrical data. In addition, model uncertainties of actions and resistance models should be taken into account. Prior theoretical models of basic variables and procedures for probabilistic analysis are indicated in JCSS documents .
2 FUNDAMENTAL CASES OF STRUCTURAL RELIBILITY 2.1 General
The fundamental task of the theory of structural reliability concerns a basic requirement for the relation between the action effect E and the structural resistance R written in the form of inequality
Condition (1) describes a desirable (satisfactory, safe) state of a considered structural component. It is assumed that structural failure occurs when the condition (1) is not satisfied. Thus, an assumed sharp (unambiguous) distinction between a desirable (safe) and undesirable (failure) state of the structure is given as
Equation (2) represents a fundamental form of the failure boundary called the limit state (performance) function (see also Chapter I of this Handbook 2). It should be noted that for some structural members and materials the assumption of sharp failure boundary might be rather artificial and can be accepted as an approximation only. Such a case is indicated in the following Example 1.
A steel rod indicated in Figure 1 has a tensile resistance R = n d fy / 4, where d denotes the diameter of the rod and fy the yield point. The rod is loaded by a weight E = Vp, where V denotes the volume and p the bulk weight density of the load. Thus the inequality (1) has the form
Vp< n d2fy / 4 The limit state function (2) can be then written as n d2fy / 4 - Vp= 0
In this example, the limit state is defined as the state when the stress in the rod reaches the yield point fy. This simplification is accepted in many common cases, but (depending on a type of structural steel) it may not correspond to the actual failure of the rod. In particular when structural steel with significant ductility and strain hardening is used, then a failure (rupture) will occur when the stress reaches the ultimate strength of the steel, which is a considerably greater than the yield point.
Attached MATHCAD sheets SteelRod.mcd, DesVRod.mcd numerical calculations.
Both the variables E and R are generally random variables and the validity of inequality (1) cannot be guaranteed absolutely, i.e. with the probability equal to 1 (the total certainty). Therefore, it is necessary to accept the fact that the limit state described by equation (2) may be exceeded and failure may occur with a certain small probability. The essential objective of the reliability theory is to assess the probability of failure Pf and to find the necessary conditions for its limited magnitude. For the simple condition in the form of inequality (1), the probability of failure may be formally written as
The random character of the action effect E and the resistance R, both expressed in terms of a suitable variable (performance indicator) X (i.e. stress, force, bending moment, deflection) is usually described by appropriate distribution function, i.e. by distribution functions 0E(x), 0R(x) and by corresponding probability density functions ^E(x), ^R(x), where x denotes a general point of the considered variable Xused to express both the variables E and R. Distributions of variables E and R further depend on appropriate parameters, e.g. on
may be used to make all moment parameters /E, ge, oE, Mr, gr and or. Let us further assume that E and R are mutually independent (which may be provided by appropriate transformation).
Figure 2 shows an example of the probability density functions of both the variables E and R and their mutual location. Types of distribution and their parameters shown in Figure 2 are just indicative information. In particular, the moment parameters (the means and standard deviations) may be considered as relative values given as a percentage of the resistance mean /ur (i.e. normalised by /R).
Note, that the probability density functions ^E(x) and ^R(x) shown in Figure 2 overlap each other and, therefore, it is clear that unfavourable realizations of variables E and R, denoted by small letters e and r, may occur in such a way that e > r, i.e. the load effect is greater than the resistance and failure will occur. Obviously in order to keep the failure probability Pf = P(E>R) within an acceptable limits, the parameters of variables E and R must satisfy certain conditions (concerning the mutual position and variances of both distributions) depending on the types of distribution.
40 60 80 100 120 140
Random variable X
Figure 2. Action effect E and resistance R as random variables.
The desired conditions will certainly include the trivial inequality /E < /ur (see Figure 2). Obviously, this "requirement for mutual position" of both distributions is not sufficient to ensure specified failure probability Pf. The correct conditions should certainly include also conditions for variances of both variables. This will be clarified by the following discussion of fundamental cases of structural reliability.
First, consider a special case when one of the variables E and R, say the action effect E, has a very low (negligible) variability comparing to the variability of resistance R. Then E may be considered as non-random (deterministic) variable, i.e. such a variable that attains a certain fixed value E0 (E = E0) in its every realization. This assumption may certainly be considered as an approximation of some practical cases. One of these cases is the loaded steel
Load effect E,
_ UE = 70, CE = 7
Mr = 100, GR = 10
rod from Example 1, where the weight of the suspended mass can be determined with sufficient accuracy (i.e. without any significant uncertainty). This special case is illustrated in Figure 3, where the action effect is indicated by a fixed value e0 = 80 (ue = 80, oe = 0) and the resistance by the lognormal distribution having the mean jur = 100, <jR = 10 (all numerical data being normalised to dimensionless quantities).
The probability of failure Pf for the special case of deterministic load effect of actions shown in Figure 3 may be assessed directly from the distribution function OR(x) similarly as in the case of a fractile. The value e0 may be simply considered as the fractile of the resistance R for which the probability Pf may be calculated using equation
The value of distribution function OR(E0) is usually assessed from tables for a standardized random variable U, for which the value u0 corresponding to E0 is computed. It follows from the general transformation formula
Probability density ^«(x)
Probability density ^«(x)
Figure 3. Deterministic effect of actions E and random resistance R.
The probability of failure is then given as
where OU(u0) is the value of distribution function of a standardized random variable of the appropriate distribution (e.g. normal or log-normal).
Note that the value -u0 is the distance of the fixed value E0 of action effect E from the mean juR of resistance R expressed in the units of standard deviation or. If the distribution of resistance R is normal, then the defined distance is called the reliability index J
and the probability of failure may be expressed by the relation
In general the reliability index J is defined as the negative value of a standardized normal variable corresponding to the probability of failure Pf. Thus, the following relationship is accepted as a definition (see Chapter I in this Handbok)
where - 0-:( p f) denotes the inverse standardised normal distribution function. At present the reliability index J defined by equation (9) is a commonly used measure of structural reliability in several national and international documents (see also previous Chapter I of this Handbook 2). Note, however, that the probability distribution of the resistance R may differ from the normal distribution.
Consider that resistance R has the mean juR = 100 (expressed in dimensionless units), standard deviation aR =10 (the coefficient of variation is VR = 0,10). For the deterministic action effect it holds that e0 = 80 (see Figure 3). If R has normal distribution, then the reliability index follows directly from equation (7)
and probability of failure follows from relation (8)
where OU(-2) is the value of the distribution function of the standardized normal distribution for u = -2. However, if the distribution of R is not normal but lognormal with the lower limit
at zero (skewness a>R = 3 VR + V R = 0,301 ), then it follows from equation (5)
The probability of failure Pf is then given as
where Oln,U-2) is the distribution function of the standardized random variable U with lognormal distribution having the lower bound at zero (the skewness a> = 0,301). The resulting probabilities do not much differ but their values are rather high.
If the fixed value of the action effect decreases to e0 = 70, then for normal distribution of resistance R the reliability index is JJ= 3 and probability of failure is
If the distribution of resistance R is log-normal with the lower limit at zero, then
The reliability index defined by equation (9) is then JJ= - OU1(0,00021) = 3,53, i.e. greater than the value 3, which holds if normal distribution of resistance R is assumed.
Obviously, when the load effect is only e0 = 70 the resulting failure probabilities are remarkably lower than in the case when e0 = 80. Furthermore, the numerical example also shows that the assumption concerning the type of distribution plays an important role and may be, in some cases, decisive.
2.3 Fundamental case of two random variables
Assume that both basic variables, the action effect E and the resistance R are random variables. Then it is generally more complicated to assess the probability of failure defined by equation (3). A simple solution can be obtained assuming a normal distribution for both E and R. Then also the difference
called the safety margin, has the normal distribution with parameters
where pRE is the coefficient of correlation of R and E. It is often assumed that R and E are mutually independent and pRE = 0. Equation (3) for the probability of failure Pf can now be modified to
and the whole problem is reduced to determining the distribution function OZ(z) for z = 0, which leads to the probabilities of the safety margin Z being negative. The distribution function 0Z(0) is usually determined by transformation of the variable Z to standardised random variable U. Using this equation, the value u0 corresponding to the value g = 0 is
The probability of failure is then given as
The probability density function ^Z(z) of the safety margin Z is shown in Figure 4, where the grey area under the curve ^Z(z) corresponds to the failure probability Pf.
Safety margin Z Figure 4. Distribution of the safety margin Z.
Assuming that Z has a normal distribution, the value -u0 is called the reliability index, which is commonly denoted by the symbol J. In case of a normal distribution of the safety margin Z, it follows from equations (11), (12) and (14) that the reliability index J is given by a simple relationship
If the quantities R and E are mutually independent, then the coefficient of correlation pRE vanishes (pRE=0). Thus, the reliability index J is the distance of the mean /Z of the safety margin Z to the origin, given in the units of the standard deviation az.
Consider again the Example 2, in which the resistance R and the load effect E are mutually independent random variables (pRE=0) having normal distribution. The resistance R has the mean /R = 100, variance aR = 10 (coefficient of variation is therefore only w = 0,10), and the effect of actions E has the mean /E = 80 and aE = 8 (all expressed in dimensionless units). It follows from equation (11) and (12) that
As both the basic variables R and E have normal distributions, the reliability index J follows directly from equation (16)
J= 20 / 12,81 = 1,56 and the probability of failure follows from relation (8)
If the variables E and R are not normal, then the distribution of the safety margin G is not normal either and then the above-described procedure has to be modified. In a general case, numerical integration or transformation of both variables into variables with normal distribution can be used. The transformation into a normal distribution is primarily used in software products.
An approximate simple procedure can be used for a first assessment of the failure probability Pf. The safety margin Z may be approximated by a three-parameter lognormal distribution. Assume that the distributions of E and R depend on the moment parameters /iE, aE, a>E, /R, aR and cor. The mean and variance of the safety margin Z may be assessed from the previous equations (11) and (12), which hold for variables with an arbitrary distribution. Assuming mutual independence of E and R, the skewness <oZ of the safety Z may be estimated using the approximate formula (see Annex A - Basic statistical concepts and techniques in this Handbook 2)
Then it is assumed that the safety margin Z can be described with sufficient accuracy by a log-normal distribution with determined moment parameters /Z, aZ and <oZ (equations (11), (12) and (17)). It shows that this approximation offers satisfactory results if the probability of failure is not too small.
Consider a tie rod having a resistance R under a suspended load of weight E. Let R be a log-normal variable with origin at zero having the parameters (expressed again in relative dimensionless units) juR = 100 and <jr = 10 (and therefore a>R = 0,301), E has Gumbel distribution with moment parameters juE = 50 and <E = 10 (for Gumbel distribution  has the positive skewness o>E = 1,14).
The moment parameters of the safety margin are assessed according to equations (11), (12) and (17)
Or-OE®E 103 x 0,301 -103 x 1,14 ®z =-——i)-=———2)2—=-0,30
For a standardized random variable it follows from equation (14) that
For a log-normal distribution having the skewness /Z = - 0,30 it holds that
which corresponds to the reliability index ( = 3,09. A more precise result obtained by application of the software VaP  is Pf = 0,00189.
However, when skewness is not taken into account in the assessment of failure probability and the normal distribution is assumed, it follows that
which differs significantly from the result when the log-normal distribution was assumed.
Attached MATHCAD sheets StRod.mcd, DesVRod.mcd may be used to make all numerical calculations.
3 EXACT SOLUTION FOR TWO RANDOM VARIABLES
In the case of two random variables E and R having any distribution, the exact determination of the failure probability Pf, defined by equation (3), may be obtained by probability integration. Figure 5 is used to explain this procedure. Let the event A denote the occurrence of the action effect E in the differential interval <x, x+dx>. Probability of the event A is given as
Let us denote B as the event that resistance R occurs within the interval <-ro, x>. Probability of the event B is  given as
The differential increment of failure probability dPf corresponding to the occurrence of the variable E in the interval <x, x+dx> is given by the probability of the simultaneous occurrence of the events A and B, i.e. by the probability of their intersection A n B. According to the principle of multiplication of probabilities , it holds that
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