Characterisation of Large Fluctuations in Response Evolution of Reinforced Concrete Members

Abstract

Large fluctuations in surface strain at the level of steel are expected in reinforced concrete flexural members at a given stage of loading due to the emergent structure (emergence of new crack patterns). Thus, there is a need to use distributions with heavy tails to model these strains. The use of alpha-stable distribution for modelling the variations in strain in reinforced concrete flexural members is proposed for the first time in the present study. The applicability of alpha-stable distribution is studied using the results of experimental investigations, carried out at CSIR-SERC and obtained from literature, on seven reinforced concrete flexural members tested under four-point bending. It is found that alpha-stable distribution performs better than normal distribution for modelling the observed surface strains in reinforced concrete flexural members at a given stage of loading.

More details on motivation for the proposed probabilistic model are presented in the supplementary material provided for this chapter.

Appendix A

The central limit theorem states that the sum of independent, identical random variables with a finite variance converges to a normal distribution. Let ${{X}_1},{{X}_2}, \ldots, {{X}_n}$be independent identically distributed random variables. Define

$$ {{S}_n} = \begin{array}{lllllllllllll} {\text{Lim}} \\{n \to \infty } \\\end{array} \sum\limits_{{i = 1}}^n {{{X}_i}} $$

(A.1)

According to central limit theorem, ${{S}_n}$ follows a normal distribution with mean and variance given by

$$ \begin{array}{lllllllllllllll} \mu = \sum\limits_{{i = 1}}^n {{{\mu }_{{{{X}_i}}}}} \hfill \\{{\sigma }^2} = \sum\limits_{{i = 1}}^n {\sigma_{{{{X}_i}}}^2} \hfill \\ \end{array} $$

(A.2)

where ${{\mu }_{{{{X}_i}}}}$ and $\sigma_{{{{X}_i}}}^2$ are the mean and variance of ${{X}_i}$.

This suggests that as long as a macroscopic phenomenon is infinitely divisible into microscopic phenomena, which exhibits finite variance with exponential tails, the random variable associated with the macroscopic phenomenon can be represented using a normal distribution. However, wide variety of microscopic phenomena exhibit statistics that needs to be characterised with algebraic-tailed distributions. An example is the random variables associated with microcracks (size as well as geometry) in concrete which exhibit a power-law distribution [13] and hence may not have a finite mean and/or variance. In such cases, the random variable associated with the macroscopic phenomenon may not have a finite mean and/or variance (see Eq. A.2), and normal distribution will not be the limiting distribution of the sum $\sum\limits_{{i = 1}}^{\infty } {{{X}_i}}$.

1.1 An Important Asymptotic Property

A probability density $L(x)$ can be a limiting distribution of the sum $\sum\limits_{{i = 1}}^{\infty } {{{X}_i}}$ of independent and randomly distributed variables only if it is stable. A random variable X is stable or stable in the broad sense if for X ₁ and X ₂ independent copies of X and any positive constants a and b,

$$ a{{X}_1} + b{{X}_2}\mathop{ = }\limits^d cX + d $$

(A.3)

holds for some positive c and some $d \in \Re$ [30]. The symbol $\mathop{ = }\limits^d$ means equality in distribution, i.e. both expressions have the same probability law. The term stable is used because the shape is stable or unchanged under sums of the type given by Eq. (A.3).

The Gaussian – and Cauchy – distributions are potential limiting distributions, depending on the physical phenomenon that is being handled. However, there are many more distributions to which the summation series $\sum\limits_{{i = 1}}^{\infty } {{{X}_i}}$ is attracted to depending on the actual behaviour. The complete sets of stable distributions have been specified by Levy and Khinchine. A probability distribution is stable if its characteristic function is of the form as given in Eq. (12).

While Eq. (12) defines the general expression for all possible stable distributions, it does not specify the conditions which the probability density function (pdf) $p(l)$ has to satisfy so that the distribution of the normalised sum ${{\hat{S}}_n} = \sum\limits_{{i = 1}}^n {{{p}_i}(l)}$ converges to a particular ${{L}_{{\alpha, \beta }}}(x)$ in the limit $n \to \infty$. If this is the case, one can say ‘$p(l)$ belongs to the domain of attraction of ${{L}_{{\alpha, \beta }}}(x)$’. This problem has been solved completely, and the answer can be summarised by the following theorem:

Theorem:

The probability density $p(l)$ belongs to the domain of attraction of a stable density ${{L}_{{\alpha, \beta }}}(x)$ with characteristic exponent α ($\alpha \in \left( {0 < \alpha < 2} \right)$) if

$$ p(l)\sim \frac{{\alpha {{a}^{\alpha }}{{c}_{\pm }}}}{{{{{\left| l \right|}}^{{1 + \alpha }}}}}\quad {\text{for}}\quad l \to \pm \infty $$

(A.4)

where ${{c}_{ + }} \geq 0$, ${{c}_{ - }} \geq 0$ and a are constants. These constants are directly related to the scale parameter c and the skewness parameter β by

$$ c = \left\{ {\begin{array}{lllllllll} {\frac{{\pi \left( {{{c}_{ + }} + {{c}_{ - }}} \right)}}{{2\,\alpha \,\Gamma \left( \alpha \right)\,{ \sin }\left( {{{{\pi \,\alpha }} \left/ {2} \right.}} \right)}}} \hfill & {{\text{for}}\;\alpha \ne 1} \hfill \\ {\frac{\pi }{2}\left( {{{c}_{ + }} + {{c}_{ - }}} \right)} \hfill & {{\text{for}}\;\alpha = 1} \hfill \\ \end{array} } \right. $$

(A.5)

$$ \beta = \left\{ {\begin{array}{lllllllllllllll} {\frac{{{{c}_{ - }} - {{c}_{ + }}}}{{{{c}_{ + }} + {{c}_{ - }}}}} \hfill & {{\text{for}}\;\alpha \ne 1} \hfill \\ {\frac{{{{c}_{ + }} - {{c}_{ - }}}}{{{{c}_{ + }} + {{c}_{ - }}}}} \hfill & {{\text{for}}\;\alpha = 1} \hfill \\ \end{array} } \right. $$

(A.6)

Furthermore, if $p(l)$ belongs to the domain of attraction of a stable distribution, its absolute moments of order λ exist for $\lambda < \alpha$:

$$ \left\langle {{{{\left| l \right|}}^{\lambda }}}\right\rangle = \int\limits_{{ - \infty }}^{\infty }{{\text{d}}l{{{\left| l \right|}}^{\lambda }}\;p(l)} = \left\{{\begin{array}{llllllll} {< \infty \quad {\text{for}}\;0 \leq\lambda \leq \alpha \left( {\alpha \leq 2} \right)} \hfill \\{\infty \quad \quad {\text{for}}\;\lambda >\alpha \left( {\alpha< 2}\right)} \hfill\\\end{array} } \right. $$

(A.7)

The above discussion clearly brings out that the sum of independent random variables, as $n \to \infty$, may converge to an alpha-stable distribution, $\alpha = 2$ being a specific case, with $p(l)\sim {{{{1} \left/ {{\left| l \right|}} \right.}}^3}$, as a Gaussian distribution. For all other values of characteristic exponent, $\alpha$, $0 < \alpha < 2$, the sum would be attracted to ${{L}_{{\alpha, \beta }}}(x)$, and all these classes of stable distributions show the same asymptotic behaviour for large x. Thus, the central limit theorem can be generalised as follows:

The generalised central limit theorem states that the sum of a number of random variables with power-law tail distributions decreasing as ${{{{1} \left/ {{\left| x \right|}} \right.}}^{{\alpha + 1}}}$ where 0 < α < 2 (and therefore having infinite variance) will tend to a stable distribution as the number of variables grows.

The characteristic exponent α and the skewness (symmetry) parameter β have to be interpreted based on physical significance. As already mentioned, α defines the shape of the distribution and decides the order of moments available for a random variable. Longer power-law tails will lead to divergence of even lower-order moments. This should not be treated as a limitation, since, in some of the physical systems, the pdf of response quantities can have power-law tails. This may also be true of nonlinear response of engineering systems, especially at bifurcation points, where the system can exhibit longer tail behaviour. Though this can be brushed aside as a transient behaviour, for seeking performance of a system, this needs to be effectively handled. Hence, it is important to understand the pdfs and associated properties so that the systems can be modelled realistically.