Open access peer-reviewed chapter

Discretization of Random Fields Representing Material Properties and Distributed Loads in FORM Analysis

By Ireneusz Czmoch

Submitted: March 27th 2017Reviewed: October 5th 2017Published: December 20th 2017

DOI: 10.5772/intechopen.71500

Downloaded: 225

Abstract

The reliability analysis of more complicated structures usually deals with the finite element method (FEM) models. The random fields (material properties and loads) have to be represented by random variables assigned to random field elements. The adequate distribution functions and covariance matrices should be determined for a chosen set of random variables. This procedure is called discretization of a random field. The chapter presents the discretization of random field for material properties with the help of the spatial averaging method of one-dimensional homogeneous random field and midpoint method of discretization of random field. The second part of the chapter deals with the discretization of random fields representing distributed loads. In particular, the discretization of distributed load imposed on a Bernoulli beam is presented in detail. Numerical example demonstrates very good agreement of the reliability indices computed with the help of stochastic finite element method (SFEM) and first-order reliability method (FORM) analyses with the results obtained from analytical formulae.

Keywords

  • FORM
  • SFEM
  • discretization
  • random fields
  • reliability

1. Introduction

In general, the safety of a structure is analyzed in the space ΩX=XRnof basic random variables X. For a given failure mode or serviceability requirement, represented by the limit state surface gX=0, the space ΩXis divided into the safe subset, ΩS=XRngX>0, and the failure subset, ΩF=XRngX0. If all random variables are continuous with the multivariate joint probability density function fXx, the failure probability is given by the integral

Pf=ΩFfxxdxE1

The integral (Eq. (1)) can be evaluated exactly for a few cases with the most important one: the linear limit state surface and multidimensional normal distribution function of variables X.

Development of reliability methods resulted in variety of powerful algorithms to estimate the probability of failure for complicated mechanical and statistical models of structures. The first-order reliability method (FORM) is the most popular approach applied in practice.

FORM algorithm starts with the nonlinear transformation. In general, non-normal random vector Xis transformed into a standard normal (Gaussian) vector Ywith zero mean and unit covariance matrix CYY=I. The limit state surface gx=0is mapped into a limit state surface Gy=0. Next, the design point y, that is, the point on the limit state surface with the minimum distance to the origin of the Yspace, is determined by solving the nonlinear optimization problem with a nonlinear constraint Gy=0

β=minyTyforyonGy=0E2

The hyperplane tangential to the limit state surface at the point yis given by the formula

βaTy=0E3

where ais a unit outward normal vector to the hyperplane and βis the distance between the hyperplane and the origin (Figure 1). Since the random vector Y=YXhas standard normal distribution, the first-order approximation of the failure probability is easily derived as follows

PfPβaTY0=ΦβE4

The nonlinear constrained optimization problem (Eq. (2)) can be solved with many standard procedures as well as algorithms developed especially for this purpose, for example, algorithm for the case of independent, non-normal random variables [1], algorithm for problems with incomplete probability information [2].

All such solvers are iterative: for the assumed value of design point xk, the values of limit state function gxkand its gradient gxkare determined. Next, a new position of design point xk+1is derived and the process continues until the convergence criteria are fulfilled. If the safety of mechanical problem is described by the limit state function with analytical form, then the gradient can be evaluated easily and one of the algorithms solving the optimization problem (Eq. (2)) can be applied directly. However, if the stochastic variability of material properties and loads is to be taken into account, SFEM approach must be applied.

Figure 1.

Definition of the FORM reliability index β .

In general, the limit state function gX=gRXSXcan be represented in terms of two vectors: resistance variables Rand load effects S. The elements of resistance variables vector R(e.g., yield stress, allowable strain or allowable displacement), are prescribed to finite elements or nodes and can be treated as deterministic or random variables. In the latter case, the vector Rcorresponds to the part of the vector of basic random variables X. The vector of load effects S(e.g., stresses, displacements and deformations) contains functions of basic random variables Xsuch as material properties, geometrical quantities or loads. The relation S=SXis called the mechanical transformation. In most practical cases, the load effects Shave to be evaluated by using numerical algorithms, for example, FEM.

Two main problems are to be solved in order to apply FEM in FORM analysis:

  • discretization of random fields of material properties and random fields of loads

  • determination of the gradient of the limit state function gx, when the load effect is defined by means of the implicit mechanical transformation S=SX

The solution of the second problem is presented in many papers and books [3].

2. Probabilistic description of random fields

2.1. Basics definitions

The spatial probabilistic variability of physical quantities such as Young's modulus, thickness of a plate and intensity of a distributed load can be described by means of random fields, wz, where zis the vector of space coordinates. One-dimensional random fields can be defined for beams, bars and columns, two-dimensional random fields for plates or shells, and three-dimensional random fields for bodies.

For any specific location z, random field wzis a random variable with the cumulative distribution function

Fwwz=PwzwE5

which is called the first-order distribution of the random field wz.

The mthorder distribution, that is, the joint cumulative distribution function of the random vector w=wz1wzmT, is defined as follows:

Fwww=Pwz1w1wzmwmE6

The first- and second-order probability density functions of the random field wzare defined accordingly

fwwz=ddwFwwzfwwwz1wz2=2wz1wz2Fwwwz1wz2E7

Following the well-known definition [4] with the help of the first-order probability density function (Eq. (6)) and the second-order probability density function (Eq. (7)), the second-order representation of the random field is defined by using the following functions: the mean value function μwz, the variance function σw2z, the covariance function, Cwz1z2and the correlation function ρwz1z2.

A random field wzis called strict-sense homogeneous, if its statistics are invariant to the translation of the origin and in particular, the nthorder density function has the property

fwww1z1wnzn=fwww1z1+zwnzn+zE8

for any separation vector z.

A random field wzis called wide-sense homogeneous or second-order homogeneous if its mean value and variance are constant,

μwz=μwσw2z=σw2E9

and its covariance function as well as correlation function depends only on the separation vector z,

Cwz,0=Cwz1,z1+z=CwzforanyzE10

A random field that is homogeneous in time is referred to as stationary process.

2.2. Ensemble average versus spatial average of random field

In order to estimate the statistical parameters of a random field, the sample (realization) must be collected in separate experiments. If the sample size is sufficiently large, the estimators of statistical parameters can be computed at each point of the random field domain. For example, at the location z1, the estimator of mean value and the estimator of variance are equal to ensemble averages

μ̂wz1=1ki=1kwiz1E11
σ̂w2z1=1k1i=1kwiz1μ̂wz12E12

where kis the number of realizations (also called as the sample size) and wizis the ithmeasurement of a random field.

Ensemble averages usually depend on the location vector. However, if the limit of the ensemble averages are invariant with respect to location

μ̂wz1=limk1ki=1kwiz1=limk1ki=1kwiz2=μwE13

then the random field can be considered as homogeneous, in strict- or wide-sense, which depends on the order of probability function invariant to location vector.

On the other hand, the spatial averages over the domain can be computed for every realization (measurement) of random field. For example, the average taken along with any single realization of a one-dimensional random field is equal to

μwiL=1L0LwizdzE14

and it usually depends on the character of field and the length of averaging interval L.

A homogeneous random field is called ergodic, if all statistical information can be obtained from one realization of the random field. This means that ensemble averages are invariant with respect to the location vector and the spatial averages are equal to the ensemble averages. Thus, in case of a homogeneous one-dimensional random field, the ensemble and spatial averages are equal in the limit

limk1ki=1kwiz=limL1L0LwizdzE15

In general, it is usually difficult to prove that a random field is homogeneous, and it is even more difficult to prove that a random field is ergodic. Great number of samples over a sufficiently large domain should be collected. These conditions are rarely fulfilled. Thus, the homogeneity and ergodicity is usually assumed. Most of the concepts and methods developed in the reliability analysis are based on these assumptions.

2.3. One-dimensional homogeneous random field

A one-dimensional homogeneous random field is often used in the reliability analysis of linear elements such as beams, bars and frames. All the above-mentioned definitions are valid in this case.

The variance reduction function γwL, which has been presented in detail in [5, 7], describes the correlation of the moving average wLzof one-dimensional homogeneous random field wz

wLz=1LzL/2z+L/2wldlE16

where Ldenotes the length of the averaging segment.

The mean value function and the variance function of the random field wLzare easy to determine

EwLz=Ewz=μwE17
VarwLz=σL2=γwLσw2E18

where γwLis the variance reduction function, and μwand σw2are mean value and variance value of the one-dimensional homogeneous random field wz.The variance reduction function γwLdemonstrates how fast the point variance σw2is reduced under local averaging. This dimensionless function has the following properties:

γwL=γwL0γw0=1E19

and is related to the correlation function ρwzof the one-dimensional homogeneous random field wzby the integral

γwL=1L20L0Lρwl1l2dl1dl2=2L0L1lLρwldlE20

Another useful scalar measure of the correlation is the scale of fluctuation θwdefined by the limit value of the variance reduction function

θw=limLLγwLE21

It can be proved that the scale of fluctuation θwis related to the correlation function ρwz

θw=20ρwldlE22

The variance reduction function and the scale of fluctuation are especially useful in the discretization procedure of the homogeneous random field.

Table 1 presents four correlation models. It should be noticed that the rectangular and triangular models are not proper correlation functions for the homogeneous random field, since they do not fulfill the basic condition of weak-homogeneity. However, they are quite often assumed, mostly as visualization tools. Triangular model demonstrates the meaning of the scale of fluctuation in a simple way, that is, correlation between values of random field at points separated by greater distance than the scale of fluctuation is equal to zero. The rectangular model constitutes the upper limit for variance reduction functions. The simple form of the exponential correlation function makes analytical computation of many integrals possible. On the other hand, similarity between the squared exponential model and the triangular model allows the simple physical interpretation of the scale of fluctuation, that is, the correlation functions are equal to zero for a separation interval greater than the scale of fluctuation.

ModelCorrelation functionVariance function
Rectangularρz=1zθ20z>θ2γL=1Lθ2θL1θ4LL>θ2
Triangularρz=1zθzθ0z>θγL=1L3θLθθL1θ3LL>θ
Exponentialρz=exp2zθγL=12θL22Lθ1+exp2Lθ
Squared exponentialρz=expπzθ2γL=2θLΦL2πθ05+A
A=θ2πL2expπLθ21

Table 1.

Description of four correlation models.

Note: θis the scale of fluctuation; Lis the separation (averaging) distance; and Φzis the Laplace function.

A special case of random field is the Gaussian random field, in which the random variables wz1,,wznfor any points z1,,znare jointly normal distributed. This random field is completely determined by two functions such as the mean value function and the covariance function. The n-th order probability density function has the joint normal density.

Figure 2 presents correlation functions and Figure 3 presents variance reduction functions for the correlation models described in Table 1.

Figure 2.

Four correlation functions for correlation models defined in Table 1.

Figure 3.

Four variance reduction functions for correlation models defined in Table 1.

3. Discretization of random fields representing material properties

A vast amount of papers deal with the problem how to develop the accurate and numerically efficient discretization methods for random fields of material properties.

The variability of random field is usually more accurately represented, if the number of random field elements or the number of series components is increased. However, greater number of random variables leads to longer computation time for realistic problems. Therefore, it has been an important issue to find out the optimal size of random field elements with respect to the scale of fluctuation, that is the scalar correlation measure. The accuracy of different methods is discussed by Zeldin and Spanos [9].

The reliability analysis of more complicated structures usually deals with FEM models. The random fields (material properties and loads) have to be represented by random variables assigned to random field elements. The adequate distribution functions and covariance matrices should be determined for a chosen set of random variables. This procedure is called discretization of a random field.

Two groups of methods for discretization of material random fields can be distinguished:

1. Random field elements

The value of a material property for any finite element is represented by a single random variable, constant within a random field element. Mean value, standard deviation and covariance as well as distribution function can be assigned to those random variables according to different procedures:

  • the spatial averaging method [10]

  • the midpoint method [8, 9, 11]

  • the interpolation method [13]

2. Random series

The random field is described in terms of series of deterministic functions and random coefficients. Two examples of this approach are as follows:

  • series composed of deterministic shape functions and random variables [16]

  • the Karhunen-Loeve orthogonal expansion [15, 17, 20]

Two discretization methods, namely the spatial averaging method and the midpoint method, are presented in detail.

3.1. The spatial averaging method of one-dimensional homogeneous random field

The spatial averaging method has been developed by Vanmarcke [5]. We consider a one-dimensional homogeneous random field wzthat represents the spatial random variability of a material property, for example, modulus of elasticity along beam. In general, the domain of the random field can be divided into finite elements of lengths Li. The material property within the ithelement is represented by a random variable which is assumed to be equal to the spatial average over the ithfinite element

Wi=1Li0LiwzdzE23

The mean value of a random variable Wiis equal to the mean value of the random field wz

EWi=1Li0LiEwzdz=μwE24

and the variance of random variable Wiis expressed in terms of the variance function γwLof the random field wz

VarWi=γwLiσw2E25

The formula for the covariance between two random variables Wiand Wjrelated to the i-th and j-th random elements is more complicated

CovWiWj=σw22LiLjk=031kLk2γLkE26

where the distances Lkare defined in Figure 4.

Figure 4.

Definition of intervals used in the calculation of covariance between the spatial averages related to two random field elements.

Eqs. (25 and 26) can be generalized for a random field defined in two- or three-dimensional spaces [5]. Eq. (26) depends on the variance reduction function γwL, which expresses a relation between the variance of the spatial average and the size of the averaging interval L.

Since full information about the variability of the random field is seldom available, Vanmarcke [56] suggested using in the practical analysis the approximation of the variance reduction function by its asymptotic form

γwL=1LθwθwLL>θwE27

where θwis the scale of fluctuation.

FORM analysis demands the knowledge about distribution functions of basic random variables. The spatial averaging method results in the normal random variables for the Gaussian random field wz, since the integration is a linear operation. However, for non-Gaussian random field, it is difficult to derive the distribution function of a random variable Widefined by Eq. (23).

Der Kiureghian [18] has suggested a heuristic model for the distribution of random variable Wi, which is based on the concept of the weighting the random field by the shape function, which results in the weighted variance reduction function. Figure 5 shows that the weighted variance function has much smaller values than the original variance reduction function. If the averaging interval Li=Llongis assumed many times longer than the scale of fluctuation θw, the distribution of random variable Witends to have the normal distribution according to the central limit theorem. Then, the variance reduction function can be approximated as follows:

γwLlongθwLlong1nforLlongθwE28

where the parameter n1should be determined by calculations or judgment.

According to Figure 3, which shows the variance functions of four models, as well as taking into account Figure 5, n10could be assumed, which means that for the element length which is 10 times longer than the scale of fluctuation, the normal distribution can be assigned to the random variable Wi. For shorter elements, non-normal distribution should be considered. On the other hand, for very short element length, the distribution of random variable Wiis close to the first-order distribution of random field. Taking into account both limits the approximate density distribution function of Wihas been proposed by Der Kiureghian [18]

fWiwi=αnσwφwiμwσw/n+1αfwzwiE29

where φuis the standard normal probability density function and fwzwis the first-order probability density function of random field wz.

Figure 5.

The original and weighted variance reduction functions for the random field with exponential correlation function.

The weight parameter 0α1can be determined for the current length Liof random field element by requiring that the variance of random variable Wiwith the approximate density function ((Eq. (29)) is equal to the variance of the spatial average of random field over the length Li, that is, γwLiσw2. It can be shown that for the element of length Li

α=nn11γwLiE30

3.2. Midpoint method of discretization of random field

The midpoint method [3] corresponds to the interpolation method [13] with constant interpolation function and is suitable for discretization of non-Gaussian random fields. In general, a nonhomogeneous random field wzis discretized with the help of a set of random variables defined as follows:

Wi=wzciE31

where zcidetermines the position of the centroid of the i-th element.

The mean value of random variable Wiand the covariance between random variables WiandWjare given below, correspondingly

EWi=EwzciCovWiWj=CwzcizcjE32

where Cwz1z2is the covariance function of nonhomogeneous random field.

In the midpoint discretization method, the probability distribution of the random variable Wiis equivalent to the first-order distribution of the random field wz

FWiwi=PWiwi=Pwzciwi=FwwizciE33

Thus, in case of a homogeneous random field, the probability distribution function does not depend on the location of the centroid of the i-th element, FWiwi=Fwwi.

3.3. Selection of the optimal size for random field mesh representing variability of a material property

Both discretization procedures described in the previous section are based on assumption that property (e.g., modulus of elasticity) is constant within a finite element, see [19].

In deterministic FEM, the variability of material properties is modeled by means of sufficient number of finite elements. The structural finite element size is chosen with respect to the gradient of the stress field.

In the same way, the variability of the random field is usually more accurately represented if many random field elements are used in the analysis. However, finer random field mesh increases the number of random variables, which leads to longer computation times for the reliability analysis. Therefore, it has been an important issue to find out the optimal size of the random field elements.

The scale of fluctuation of the random field has been shown to be a very important measure of the correlation, since it governs the optimal size of a random field mesh. Der Kiureghian and Ke [12] have shown that a sufficiently accurate value of the reliability index is obtained if the random field element size is between one-half and one-quarter of the scale of fluctuation for the midpoint method with the exponential correlation function. Hisada and Nakagiri [11] have presented similar results.

If random field elements shorter than one-quarter of the scale of fluctuation are chosen, then a singular correlation matrix can be obtained, indicating linear dependency of the random variables. Then, the nonhomogeneous linear transformation to the set of uncorrelated, normalized basic random variables must be proceeded by an extra transformation, which decreases the dimension of the random variable space. However, this extra transformation is not unique. Improper choice of the transformation can lead to numerical difficulties in the iteration procedure for determining the reliability index. Therefore, too small random field elements should be avoided. Liu and Liu [19] have derived a simple rule of thumb regarding the selection of an appropriate random field mesh: a coarse mesh should be assumed in an area where the gradient of the limit state function with respect to the random variable representing random field is small and a finer random field mesh in an area with large gradient.

However, in many case a random element size equal to the scale of fluctuation may be considered as adequate with respect to the reliability analysis accuracy.

4. Discretization of random fields representing distributed loads

Discretization of random field loads has not been a subject of many studies. The finite element modeling introduces the well-known procedure for representing distributed forces psby a set of equivalent nodal forces. The random field of distributed loads has to be discretized according to the structural finite element mesh. Thus, if the distributed loads are random, all equivalent nodal forces become random variables. This approach seems obvious and it can be applied directly to study the response variability of stochastic engineering problems [14]. However, if discretized random field load is a part of FORM calculations, then both the discretization procedure for random field loads as well as FORM/FEM analysis should be modified in order to get accurate results.

The general approach for the discretization procedure of the distributed body forces psis presented below [20]. A similar algorithm can be applied for other types of distributed loads (surface forces and initial stresses). For the purpose of FORM analysis, it is convenient to assume that the nodal forces for i-th element are random variables

Qpi=VNiTpsdVE34

where Niis the displacement interpolation matrix for the i-th element in the local coordinate system. Thus, the mean value vector of the load vector Qpiis equal to

EQpi=VNiTEpsdVE35

The covariance matrix of the vectors of equivalent forces QpiandQpjcorresponding to the i-th and j-th finite elements has the form

CovQpiQpjT=ViVjNiTCppNjdVidVjE36

where the matrix Cppcontains the cross-covariance functions between different components of the vector random field.

In general, a load effect can be an internal force, stress or strain at any point of structure which does not coincide with a nodal point (or Gaussian integration point). Thus, the load effect Sisat a point sof i-th element can be represented as a sum of a general solution, Sui(s) and a particular solution, Spi(s). The general solution Sui(s) is the load effect as a function of geometry, material properties and equivalent nodal forces applied at all nodal points. Whereas the particular solution Sfi(s) is the load effect at the point sof the i-th element due to the distributed body forces fisand reactions Qpiat the i-th element.

In the FORM analysis, the vector of basic random variables Xalso contains nodal equivalent forces Qpi(where the parameter "i"runs over all finite elements). Thus, in the search for the most likely failure point, the current values of equivalent nodal forces Qpihave to be determined at each iteration step of FORM algorithm. Those equivalent nodal forces, valid at a specific step of FORM iteration, correspond to unknown functions of distributed body forces ps. In order to determine the particular solution Spi(s), the function of the distributed body forces piswithin the ithfinite element must be known. One way to solve this problem is to assume that the distributed body forces can be approximated with the help of shape functions.

pis=NiTbE37

The matrix bcan be determined from the condition that the equivalent nodal force Qpiat the i-th finite element due to body forces pisshould be equal to the calculated equivalent nodal forces in the FORM algorithm. The vector bk, which is the k-th column of the matrix band corresponds to the k-th component of the vector pis, is determined by solving the system of linear equations

VNkiTNkidVbk=QpiE38

where Nkiis the k-th row of the matrix Ni.

In this way, the function of the distributed body forces pisas well as the particular solution Spi(s) is determined as functions of nodal equivalent forces Qpi.

For the distributed loads represented by the Gaussian random field, the components of vector Qpi, which are determined by means of a linear transformation (Eq. (34)), have the multidimensional normal distribution. For the non-Gaussian random field, the probability distribution function FQpirof the vector of nodal equivalent forces Qpicannot be determined easily. The first possible choice is to assume the normal distribution on the basis of the central limit theorem. Another approximate solution has been developed [20] on the basis of the approach presented by Der Kiureghian [18].

5. Discretization of transverse distributed load for a Bernoulli-Euler beam

In case of Bernoulli-Euler beam, four shape functions are applicable

N1s=13sL2+2sL3N2s=s1sL2N3s=sL232sLN4s=s2LsL1E39

The nodal forces equivalent to the transverse distributed load qsare defined by the integrals

Qj=0LNjsqsdsj=1,,4E40

The distributed load qsis assumed to be a homogeneous random field, with constant mean μqvalue, constant variance σq2and the covariance function for the correlation function ρq

Covqsqt=σq2ρqtsE41

Thus, the mean value of the nodal force Qiis just equal to

EQi=μq0LNisdsE42

and the covariance between nodal force Qimat the mthfinite element and the nodal force Qinat the nthfinite element is defined by the double integral

CovQimQjn=σq2s1ms2ms1ns2nNisNjtρqtsdsdtE43

where s1m,s2mand s1n,s2nare the coordinates of the two ends of the finite elements, defined in a common coordinate system.

If a beam is divided into N finite elements, then the random distributed load qsis modeled by 4Nrandom variables.

A typical limit state function for a beam can be defined at the mthfinite element

gms=RbmsMmsE44

where Rbmsis the bending resistance at cross-section s(usually assumed as a basic random variable) and Mmsis the bending moment due to external loads at cross-section s, which is a random function depending on the other basic random variables, for example, random nodal equivalent forces Q11,Q21,Q31,Q41,Q1N,Q2N,Q3N,Q4N.

The bending moment Mmscan be represented as a sum

Mms=Mums+MqmsE45

The general solution Mumsdepends on the nodal displacements, which are the functions of all nodal forces Q11,Q21,Q31,Q41,Q1N,Q2N,Q3N,Q4Nimposed to FEM model. The particular solution Mqmsis the bending moment within the mthelement due to the distributed load qsand reactions Q1m,Q2m,Q3m,Q4mat the mthelement,

Mqms=Q2mQ1mss1msqtstdtE46

At the kthiteration step of FORM algorithm, the function of the distributed load is unknown for the corresponding nodal forces. Therefore, the function of distributed load imposed on the mthfinite element is assumed as a linear combination of the shape functions

qsqms=i=14biNisE47

and the unknown coefficients bi, which have to be determined by requiring that the equivalent nodal forces (Eq. (40)) for the function qmsdefined by relation (Eq. (47)) should be equal to the current equivalent nodal forces. The coefficients bias well as the distributed load qmcan be obtained as functions of the current equivalent nodal forces Q1m,Q2m,Q3m,Q4mat the mthfinite element, by solving the system of linear equations

i=14s1ms2mNisNjsdsb=Qjmj=1,,4E48

Finally, the bending moment Mqmsis determined as a function of equivalent nodal forces

Mqms=m1sQ1m+m2sQ2m+m3sQ3m+m4sQ4mE49

where

m1s=s18sL+20sL220sL3+7sL4m2s=160sL2+200sL3225sL4+84sL5m3s=s2sL10sL2+15sL37sL4m4s=30sL2+140sL3195sL4+84sL5E50

In order to determine the reliability index βFORM, the gradient of the limit state function (Eq. (44)) has to be calculated. The partial derivatives of function (Eq. (44)) with respect to the nodal forces are given below

gmQin=j=14MumujmujmQinifmni=1,,4E51
gmQim=j=14MumujmujmQimmisi=1,,4E52

where ujmis the nodal displacements in the local coordinate system of the mthelement and the derivatives ujmQinare computed with the help of the SFEM algorithm (Liu Der Kiureghian, 1991).

5.1. Example of discretization of random distributed load for a simply supported beam

The deterministic simply supported beam of length L is subjected to the transverse homogeneous random load qswith mean value μqand variance σq2. Assuming the exponential correlation function with the scale of fluctuation θq,

ρτ=exp2τθqE53

the mean value and the variance of the bending moment function can be derived analytically:

EMs=μqsLs2VarMs=σq2Aθq48+BLθq34+CL3θq3E54

where

A=1sLsLexp2Lθqexp2sθq+sLsLexp2Lsθq1+1
B=1sLsLC=1sL2sL2

We consider the linear limit state function gs=RbMswhere Rbis the deterministic bending moment capacity, constant along the beam.

If the random field qsis Gaussian, then the FORM reliability index is equivalent to the Cornell reliability index

βFORMs=βCs=RbEMsVarMsE55

On the other hand, the FORM reliability indices have been computed for the finite element model of a simply supported beam. The distributed load random field qshas been discretized according to the procedure described earlier.

The calculations have been carried out for the following data:

μq=1000N/m, σq=200N/m, Rb=5000Nm, L=6mand 5 cross-sections: s=1.02;1.5;2.04;2.52;3m. Three finite element sizes have been considered: Le=0.5;1;3m. The scale of fluctuation has been assumed as: θq=0.5;1;4;100m

The results of the FORM analysis presented in Table 2 are in very good agreement with the reliability indices computed according to analytical formulae (Eq. (53–55)). Moreover, the reliability indices computed with the help of the described discretization procedure are insensitive to the scale of fluctuation and the finite element size.

Finite element lengthPosition of cross-section (m)
(m)1.021.502.042.523.00
Scale of fluctuation = 0.5
Analytical14.73507.30593.60402.12441.6821
0.514.73477.30553.60382.12431.6820
1.014.73457.30553.60382.12431.6820
3.014.75117.31663.60472.12471.6819
Scale of fluctuation = 1.0
Analytical10.72425.30372.61121.53781.2172
0.510.72405.30362.61121.53771.2172
1.010.72395.30362.61121.53771.2172
3.010.73145.30922.61161.53781.2172
Scale of fluctuation = 4.0
Analytical6.57743.25161.59970.94150.7451
0.56.57733.25171.59970.94150.7451
1.06.57733.25161.59970.94150.7451
3.06.57813.25231.59970.94150.7451
Scale of fluctuation = 100.0
Analytical4.91532.44231.20620.71140.5633
0.54.91532.44231.20620.71140.5633
1.04.91532.44231.20620.71140.5633
3.04.91532.44231.20620.71140.5633

Table 2.

Reliability indices computed according to analytical formula and FORM algorithm.

6. Conclusions

The reliability analysis of more complicated structures usually deals with the FEM models. The random fields (material properties and loads) have to be represented by random variables assigned to random field elements. The adequate distribution functions and covariance matrices should be determined for a chosen set of random variables. This procedure is called discretization of a random field.

The chapter presents the discretization of random field for material properties with the help of the spatial averaging method of one-dimensional homogeneous random field, and midpoint method of discretization of random field.

The second part of the chapter deals with the discretization of random fields representing distributed loads. The discretization of distributed load imposed on a Bernoulli beam is presented in detail. An example shows that the presented procedure for discretizing random fields representing distributed loads is very efficient, that is, the reliability indices computed with the help of SFEM and FORM analysis are in very good agreement with the results of analytical calculations.

© 2017 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution 3.0 License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Ireneusz Czmoch (December 20th 2017). Discretization of Random Fields Representing Material Properties and Distributed Loads in FORM Analysis, Dependability Engineering, Fausto Pedro García Márquez and Mayorkinos Papaelias, IntechOpen, DOI: 10.5772/intechopen.71500. Available from:

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