Why do we need probability distributions with fat tails to describe the surface strain evolution in reinforced concrete flexural members?

Abstract

This paper, for the first time, justifies the use of alpha-stable distribution (a fat tailed distribution) for describing the statistical variations in surface strains in reinforced concrete flexural members. Justification is provided by viewing the cracking process in these members, that affects the strain evolution, as an emergent structure (in thermodynamical sense, Prigogine and Stengers in Order out of chaos: man’s new dialogue with nature, 1984; Carpinteri in J. Mech. Phys. Solids 37(5):567–582, 1989; van Mier et al. in Comput. Meth. Appl. Mech. Eng. 142(1–2):189–201, 1997). The proposal of the alpha-stable distribution is checked by comparing the estimated relevant parameters (namely, location and scale parameters which are representative of mean and dispersion) with the mean and standard deviation of experimentally measured surface strains both in extreme compression fibre and at the level of reinforcement (in the flexure zone), at different stages of loading. A satisfactory agreement is obtained between the respective experimental and estimated values. Also, comparison of the tails of estimated CDFs of normal and alpha-stable distributions with the cumulative distribution functions of experimentally observed strains suggested that alpha-stable distribution is able to fit the tails of compression and tensile strains better than normal distribution. For the purpose of the present study, the experimental results of three beams are considered. Also, in the present investigation the strains at the level of reinforcement and at the extreme compression fibre are considered. The practical consequence of the use alpha stable distribution in performance based design and future direction of research are also included.

Appendices

Appendix A

The aim of this appendix is to provide a basic introduction to alpha-stable distribution. This is included since in structural engineering applications, especially in the field of reliability analysis of reinforced concrete, use of this distribution is not very common.

1.1 A.1 Alpha-stable distribution

The alpha-stable distribution is described by its characteristic function (an explicit expression for probability density function generally does not exist) given by:

(A.1)

where X is the random variable, i is the imaginary unit, t is the argument of the characteristic function (t∈ℜ),E[exp(itX)] denotes the expected value of exp(itX),α is an index of stability or characteristic exponent (α∈(0,2]),β is the skewness parameter (β∈[−1,1]), c is a scale parameter (c>0),δ is a location parameter (δ∈ℜ), ln denotes the natural logarithm and sign(t) is a logical function which takes values −1, 0, 1 for t<0, t=0 and t>0, respectively. As α approaches 2, β loses its effect and the distribution approaches the normal distribution regardless of β [23]. A stable probability density function (PDF) is symmetrical when β=0.

1.2 A.2 Derivation of a alpha-stable distribution (which is also called Levy distribution)

The evolution of surface strains along the length of flexural member should be considered as a stochastic process. Based on the considerations presented in Sects. 2–6, a Levy’s process is proposed for describing this phenomenon. A brief derivation of Levy distribution is presented below.

Levy [24] reconsidered law of large numbers and formulated a new approach that can also be applied to distributions with infinite second moment. The importance of approach of Levy’s distributions and processes soon became clearer in many fields. Mandelbrot in 1982 [25] has presented numerous applications of the Levy’s distributions and coined the notion of Levy flights.

Let p _X (x) be normalized probability density function of random variable ‘X’. That is,

$$ \int_{ - \infty}^{\infty} p_{x}( x )dx = 1 $$

(A.2)

with characteristic function

$$ \varPhi_{X}( t ) = \int_{ - \infty}^{\infty} e^{itx}p_{x}( x )dx $$

(A.3)

Consider two random variables X ₁ and X ₂ and their linear combination as follows

$$ CX_{3} = C_{1}X_{1} + C_{2}X_{2}\quad C,\ C_{1},\ C_{2} > 0 $$

(A.4)

The linear combination law is stable if all X _i s (i=1,2,3) are distributed according to the same, $p_{x_{i}}( x_{i} )$; (i=1,2,3).

While Gaussian distribution satisfies the stable law, another class of distributions was given by Levy [24].

Let us write the density function of X ₃ in Eq. (A.4).

(A.5)

We know that the characteristic function of a random variable made up of summation of several random variables is simply multiplication of characteristic functions of individual random variables. Hence, we get

(A.6)

Taking logarithms on both the sides, we get,

(A.7)

Equations (A.6) and (A.7) are functional ones with a solution

$$ \ln\varPhi_{\alpha}( Ct ) = ( Ct )^{\alpha} = C^{\alpha}e^{ - \frac{i\pi\alpha}{2} ( 1 - \operatorname{signt} )}| t |^{\alpha} $$

(A.8)

Under the condition

$$ \biggl( \frac{C_{1}}{C} \biggr)^{\alpha} + \biggl( \frac{C_{2}}{C} \biggr)^{\alpha} = 1 $$

(A.9)

where α is an arbitrary parameter.

Any distribution p _X (x) with characteristic function

$$ \varPhi_{\alpha}( t ) = e^{ - C\vert t \vert ^{\alpha}} $$

(A.10)

is known as Levy distribution with Levy index α. An important condition imposed by Levy is 0<α≤2, which guarantees of positiveness of probability density function.

$$ p_{\alpha}( x ) = \int_{ - \infty}^{\infty} dt \cdot e^{itx} \cdot\varPhi_{\alpha}( t ) $$

(A.11)

The case of α=1 is known as Cauchy distribution

$$ p_{1}( x ) = \frac{C}{\pi} \frac{1}{( x^{2} + C^{2} )} $$

(A.12)

An important case is the asymptotic of large |x|.

(A.13)

It can be shown that the moments of p _α (x) of order ‘m’ diverge for m≥α.

1.3 A.3 Estimation of parameters of alpha-stable distribution

Different methods have been proposed in literature for the estimations of the parameters α,β, c and δ of the alpha-stable distribution. Fama and Roll [26] suggested a quantile-based method for estimation of characteristic exponent and scale parameter of symmetric alpha-stable distributions with δ=0. However, this method is applicable only for distributions with α>1. This method has been modified by McCulloch [27] to include even non-symmetric distributions with α in the range [0.6,2.0]. Koutrouvelis [28] proposed a characteristic function-based method involving an iterative regression procedure for estimation of the parameters of the alpha-stable distribution. Kogon and Williams [29] improved this method by eliminating the iterative procedure and simplifying the regression. Ma and Nikias [30] and Tsihrintzis and Nikias [31] proposed the use of fractional lower order moments (FLOMs) for estimating the parameters of symmetric alpha-stable distributions. Bates and McLaughlin [32] studied the performances of the methods proposed by McCulloch [27], Kogon and Williams [29], Ma and Nikias [30] and Tsihrintzis and Nikias [31] using two real data sets. They found that there are marked differences between the results obtained using the different methods. In the present study, the parameters α,β, c and δ of the alpha-stable distribution are estimated using an optimization procedure by minimizing the sum of squares of the difference between the observed cumulative distribution function (empirical distribution function) and the cumulative distribution function (CDF) of the alpha-stable distribution.

In the present study, the parameters α,β,c and δ of the alpha-stable distribution are estimated using an optimization procedure by minimizing the sum of squares of the difference between the observed cumulative distribution function (empirical distribution function) and the cumulative distribution function (CDF) of the alpha-stable distribution. The procedure used is as follows:

• 1. Given the N ordered observed data points x ₁,x ₂,x ₃,…,x _N , define the empirical distribution function as

  $$ E_{N}( i ) = \frac{n(i)}{N + 1} $$

  (A.14)

  where number of data points less than or equal to x _i , and x _i are ordered from smallest to largest value.

• 2. Define the objective function as

  $$ Z = \sum_{i = 1}^{N} \bigl( E_{N}( i ) - \tilde{F}_{\alpha S}( x_{i};\alpha,\beta,\gamma,\delta) \bigr)^{2} $$

  (A.15)

  where $\tilde{F}_{\alpha S}( x_{i};\alpha,\beta,\gamma,\delta)$ is the CDF of the alpha-stable distribution. ∼ denotes the fact that the form of distribution function is generally not available and has to be approximated numerically.

• 3. Determine α,β,c and δ by minimizing Z subject to the constraints 0<α≤2,−1≤β≤1 and c>0. In the present study, the minimization is carried out using the constrained nonlinear optimization function available in the software MATLAB. The CDF of the alpha-stable distribution, $\tilde{F}_{\alpha S}( x_{i};\alpha,\beta,\gamma,\delta)$, is computed by numerical integration [33].

Appendix B

The aim of this appendix is to bring out the importance of formulations presented in Dominguez et al. [5] and present the deterministic formulations for the strain field in concrete and then propose a stochastic counterpart of the deterministic model. It is shown in this appendix, for the first time, that WRW model naturally handles the fluctuations in both the components of strain field and that it brings out the need for the use of alpha stable distribution for handling fluctuations in strains of the singular part.

Sluys and de Borst [34] have researched upon the applicability of various types of constitutive relations for representing the tensile stress-strain behavior of concrete. They opined that gradient dependent model is more appropriate for finite element applications of reinforced concrete as this model incorporates the concept of characteristic length scale. Dyskin et al. [6] reviewed various models of size effect for concrete in tension and proposed a model which takes care of linear part of macroscopic stress distribution as a reasonably accurate model for predicting the size effect in concrete. Based on the results of investigations carried out by Dyskin and Gemanovicz [35], the authors [6] proposed a different mechanism of size effect based on the fact that random distribution of defects present in concrete can produce spatially (randomly) varying additional self equilibrating stress field. They further argued that the additional stress field may have an average stress zero but the actual stresses can vary from positive to negative values. Thus, this brings about large fluctuations in the body of material that the failure of the material may be governed by local stress exceeding the strength though macroscopic stress due to external loading is less than the material strength. The fact that the internal local stress (or strain) field can show large variations was further considered, recently, by Dominguez et al. [5] and developed a strong discontinuity model. This model seems very promising since it can handle both small and large fluctuations in strain field simultaneously and hence a brief account of the model is provided in the next section.

2.1 B.1 Strong discontinuity model by Dominguez et al. [5]

The authors formulate two level damage model which work in coupled manner in a given element. The two levels identified, based on dissipation mechanism, are: (i) continuum damage model and, (ii) discontinuity damage model. The first level considers, through continuous damage mechanics formulation, energy dissipation by formation of micro-cracks in the bulk material while the second level considers the formation of macro-cracks in localized zones resulting in surface dissipation of energy. The second level represents a strong discontinuity and both levels are made to work in coupled manner by introduction of a displacement discontinuity. In the following, the formulation related to displacement discontinuity is presented.

2.1.1 B.1.1 Kinematics of displacement discontinuity model

The authors [5] introduce a surface of displacement discontinuity on which are concentrated all localized dissipative mechanisms due to formation and development of localization zones. This is achieved by considering the domain Ω split into two sub-domains Ω ⁺ and Ω ⁻ by surface of discontinuity, denoted as Γ _S .

The surface of discontinuity Γ _S is characterized ‘at each point’ by the normal and tangential vectors (m and n). The discontinuous displacement field can then be written as,

(B.1)

where $H_{\varGamma_{S}}( X )$ is a Heaviside step function and is equal to one if X∈Ω ⁺ and zero if X∈Ω ⁻. φ(X) is ‘at least’ C ₀ function with its ‘boundary values’ defined according to

(B.2)

From Eq. (B.1) it is clear that $\bar{u}( X,t )$ has the same boundary values as the total displacement field u(X,t). Now, the strain field can be written as

(B.3)

In Eq. (B.3) the terms in square bracket represent the regular part of the strain field while those in the curly brackets represent the singular part. Representation of the strain field in the form of two parts allows us to write the damage evolution equation as

$$ \varepsilon= D:\sigma $$

(B.4)

The compliance D in Eq. (B.4) consists again of two parts, namely, the regular part and the other singular part. That is,

$$ D = \bar{D} + \bar{\bar{D}}\delta_{\varGamma_{S}}( X ) $$

(B.5)

It is noted that the first term in Eq. (B.5) corresponds to continuous damage/dissipative mechanism in the bulk continuum containing micro-cracks while the second term accounts for the discrete damage/dissipative mechanism describing the localized zones. The above equation allows stress to be finite. Dominguez et al. [5] formulated damage evolution equations, corresponding to the two parts, using thermodynamical considerations. While the above model is one of the best models available for modeling deterministic strain field in concrete/reinforced concrete members, to the best of authors knowledge incorporation of this information in making a choice of suitable probability distribution to describe the variations in strain field is still missing.

From the above discussion it is clear that any attempt to develop stochastic strain field model should take cognizance of the fact that the type of probability distribution function (pdf) to be selected for the two parts (in Eq. (B.3)) should be different. One of the expected features of the pdfs to be used is that the fluctuations in strains expected in regular part are going to conform to regular probabilistic laws such as Central Limit Theorem. Whereas, the fluctuations expected in singular part (of Eq. (B.3)) can be very wild (viz. see Dyskin et al. [6]) and hence a pdf with fat tails satisfying the generalized central limit theorem should be used.

2.2 B.2 Development of stochastic counterpart of Eq. (B.3)

The main aim of this paper is to develop a stochastic model for variations in the surface strain of reinforced concrete flexural members and compare the trend of predicted cumulative distribution function with the trend estimated based on experimental measurements.

Based on the formulations presented by Dominguez et al. [5] and the arguments presented in Sect. B.1, a Weierstrass random walk model is suggested in this paper to model the surface strain variations (in compression and tension). The reason for this selection would be clear at the end of the derivation.

2.2.1 B.2.1 Surface strain variations as Weierstrass random walk

In the following derivation, random variable jump length, a _J , at any time step J represents the value of strain at any loading stage J.

Random walk setting [36]:

• A one-dimensional lattice model

• Without loss of generality, lattice points are arranged periodically with spacing of ‘one’

• P{Jump of length a _J }=p _J

The aim is to estimate P{Jump of length l} given the above conditions.

Assuming the walk to be symmetrical, we have

$$ p ( l ) = \frac{1}{2}\sum_{J = 1}^{\infty} p_{j} \bigl[ \delta( l - a_{J} ) + \delta( l + a_{J} ) \bigr] $$

(B.6)

In a normal 1D random walk the walk would take a uniform step size and the probabilities of taking a right or left step is same at any point and remains constant throughout the walk. In the particular case we are going to consider now, let us have a Bernoulli scaling law. That is,

$$ \everymath{\displaystyle} \begin{array}{lll} p_{J} = cp^{J};\qquad a_{J} = a^{J} \cr\noalign{\vspace{3pt}} \sum p_{J} = 1 = c\sum_{0}^{\infty} p^{J} \cr\noalign{\vspace{3pt}} \hphantom{\sum p_{J}} = c(1 - p)^{ - 1} \cr\noalign{\vspace{3pt}} \hphantom{\sum p_{J}} \therefore c = (1 - p) \end{array} $$

(B.7)

Then,

$$ \everymath{\displaystyle} \begin{array}{@{}lll} P ( l ) = \frac{c}{2}\sum_{J = 0}^{\infty} p^{J} \bigl[ \delta\bigl( l - a^{J} \bigr) + \delta\bigl( l + a^{J} \bigr) \bigr] \cr\noalign{\vspace{3pt}} \hphantom{P ( l )} = \frac{(1 - p)}{2}\sum_{J = 0}^{\infty} p^{J} \bigl[ \delta\bigl( l - a^{J} \bigr) + \delta\bigl( l + a^{J} \bigr) \bigr] \cr\noalign{\vspace{3pt}} \int_{ - \infty }^{\infty} p(l)dl = 1 \end{array} $$

(B.8)

Thus the pdf of step length ‘l’ is made up of scaled probabilities and scaled step lengths.

$$ \everymath{\displaystyle} \begin{array}{@{}lll} \langle L \rangle= \int_{ - \infty}^{\infty} lp(l)dl \cr\noalign{\vspace{3pt}} \hphantom{\langle L \rangle} = \int_{ - \infty}^{\infty} l. \frac{1 - p}{2}\sum_{J = 0}^{\infty} p^{J} \bigl[ \delta\bigl( l - a^{J} \bigr) + \delta\bigl( l + a^{J} \bigr) \bigr] dl \cr\noalign{\vspace{3pt}} \hphantom{\langle L \rangle} = \frac{1 - p}{2}\sum _{J = 0}^{\infty} p^{J}.a^{J} \cr\noalign{\vspace{3pt}} \hphantom{\langle L \rangle} = \frac{1 - p}{2}\sum_{J = 0}^{\infty} ( pa )^{J} \cr\noalign{\vspace{3pt}} \langle L \rangle\mbox{ diverges if } pa \geq1 \cr\noalign{\vspace{3pt}} \bigl \langle L^{2} \bigr\rangle= \int_{ - \infty}^{\infty} l^{2}.\frac{1 - p}{2} \sum_{J = 0}^{\infty} p^{J} \bigl[ \delta\bigl( l - a^{J} \bigr) + \delta\bigl( l + a^{J} \bigr) \bigr] \cr\noalign{\vspace{3pt}} \hphantom{\langle L \rangle} = \frac{1 - p}{2}\sum _{J = 0}^{\infty} \bigl( p.a^{2} \bigr)^{J} \cr\noalign{\vspace{3pt}} \bigl\langle L^{2} \bigr\rangle\mbox{ diverges if } a^{2}p \geq1 \end{array} $$

(B.9)

The characteristic function of ‘L’ is

(B.10)

The characteristic function of walk is given by,

$$ \everymath{\displaystyle} \begin{array}[b]{@{}lll} P(\varOmega) &=& (1 - p)\sum_{J = 0}^{\infty} p^{J}\cos\bigl(\varOmega a^{J}\bigr) \cr\noalign{\vspace{3pt}} &=& (1 - p)\bigl\{ \cos(\varOmega a^{0}) + p.\cos(\varOmega a) \cr\noalign{\vspace{3pt}} &&{} + p^{2}\cos\bigl(\varOmega a^{2}\bigr) + \cdots\bigr\} \cr\noalign{\vspace{3pt}} &=& (1 - p)\cos(\varOmega) + (l - p)\bigl\{ p\cos(\varOmega a) \cr\noalign{\vspace{3pt}} &&{} + p^{2}\cos\bigl(\varOmega a^{2}\bigr) + \cdots\bigr\} \cr\noalign{\vspace{3pt}} &=& (1 - p)\cos(\varOmega) + p(l - p).\bigl\{ \cos(\varOmega a) \cr\noalign{\vspace{3pt}} &&{} + p\cos\bigl(\varOmega a^{2}\bigr) + \cdots\bigr\} \end{array} $$

(B.11)

$$\everymath{\displaystyle} \begin{array}{lll} &=& (1 - p)\cos(\varOmega) \cr\noalign{\vspace{3pt}} &&{} + p \Biggl[ (1 - p)\sum_{J = 0}^{\infty} p^{J}\cos \bigl(\varOmega a^{J + 1}\bigr) \Biggr] \cr\noalign{\vspace{3pt}} &=& (1 - p)\cos(\varOmega) + p.P(\varOmega a) \cr\noalign{\vspace{3pt}} P(\varOmega) &=& (1 - p)\cos(\varOmega) + p.P(\varOmega a) \end{array} $$

The solution of this equation can be written in the following form

$$ P(\varOmega) = \underbrace{P_{R}(\varOmega )}_{\mathrm{Regular\ part}} + \underbrace{P_{S}(\varOmega)}_{\mathrm{Singular\ part}} $$

(B.12)

The form of this solution is similar to the deterministic strain field model (i.e. Eq. (B.3)) proposed by Dominguez et al. [5].

The Singular part of the solution satisfies the condition

(B.13)

The renormalization equation has a singular behavior at Ω=0.

A qualitative analysis of the characteristic function P(Ω) requires that we treat P _R (Ω) and P _S (Ω) separately. The singular part represents the renormalization equation.

We will assume that P _S (Ω) has the following behavior at Ω=0

$$ P_{S}(\varOmega)=|\varOmega|^{\mu}Q(\varOmega) $$

(B.14)

where μ is an exponent and Q(Ω) is a non singular function.

It may be noted that the regular term of the solution contains ‘cos(Ω)’ term which caters to all integer powers of solution. Therefore, singular part should be so selected that the power/exponent μ is less that 2 and greater than ‘0’. [Note: we are looking at real solution.] Thus, 0<μ<2.

Thus, the probability distribution that satisfies the singular part of Eq. (B.12) is an alpha stable distribution; a distribution proposed in this paper to describe the stochastic surface strain variations in RC flexural members.