Stochastic Methods in Structural Mechanics

MSc. Thesis, Structural Engineering and Mechanics

Student : Codor Khodr Supervisor : Dr. Stefanos Papanicolopulos (UoE) Moderator : Dr. Pankaj Pankaj (UoE)

I. Introduction

The main interest of Stochastic Structural Mechanics is to deal with structural problems that contain some randomness : random geometry, random material and/or random boundary conditions (including external forces). The first person to do so was Shinozuka & Jan (1972). Unlike the deterministic case (no randomness), the probabilistic structural problems have random responses. The objective of this thesis is to study the statistical properties (mean value, deviation, probability of exceeding thresholds, etc.) of the responses of several stochastic structural problems.

II. Stochastic Structural Problem

In Structural Mechanics, as in other Engineering domains, the random quantities X (e.g. random Young’s Modulus E) are modeled by Gaussian random processes N(µ;σ2), determined by their mean values µ and variances σ2 :

Randomness can be tackled by doing M observations θk of the random input parameter θ, which gives us m deterministic equations.

IV. Failure probabilities of stochastic structural elements

Three simple problems where investigated in order to introduce the concept of randomness in structural mechanic. The structure is assumed to fail when the failure criterion is met. Safety is reached when the probability of failure Pf = Nf/m does not exceed 0.05, where Nf is number of sample responses that meet the failure criterion among the m samples gnerated.

- Bending with stochastic Young’s Modulus E :

- Reasonance with stochastic load frequency w :

- Buckling with stochastic imperfection γ :

IV. Stochastic Earthquake response of a 3-storey shear building

Stochastic methods are often used in earthquake engineering. Here, we studied the response (which is a random process) of a 3-story shear building submitted to a stationary and non-stationary white noise ground acceleration a(t)=e(t)W(t) at its basis. We determined the evolution of the mean value and standard deviation of the relative (local) displacements zi of each floor. 3 simulations were done for different damping coefficients. We also used the more realistic and conservative Kanai-Tajimi approach (Kanai (1957))

- Non-Stationary earthquake model (modulated white noise + 3 simulations)

- Stationary earthquake model : (white noise + 3 simulations)

Fig1. Probability density of the normal distribution N(µ, σ2)

Fig2. Probability of failure as a function of mean Modulus, for different values of the mean force E(F).

M=10000 Monte Carlo samples

Fig3. Probability of failure as a function of the loading frequency for different values of the Young’s Modulus

M=10000 Monte Carlo samples

References : (1) Shinozuka and Jan, 1972, ‘Digital Simulation of Random Processes and its applications’. Journal of Sound and Vibration. 25(1), 111-128 (2) Kanai, 1957, ‘Some empirical formulas for the seismic characteristics of the ground’. Bull. Earthquake. Res. Institute Univ. Tokyo. 35, 309-325 (3) Ghanem and Spanos, 1991, ‘Stochastic Finite Elements : A Spectral Approach’. Springer-Verlag, New-York. (4) Bucher. 2009, 2009, ‘Computational Analysis of Randomness in Structural mechnaics’. Structures and Infrastructures series, Vol. 3.

Fig4. Probability of failure as a function of the loading mean value for different values of the Imperfection

M=10000 Monte Carlo samples

By using standard solving methods (Newton-Raphson, Newmark, etc.) we obtain M deterministic responses uk which form a space of samples from which we can deduce the mean value, the variance, etc. These samples are generated by the Monte Carlo Method. (Bucher (2009))

Fig2. Steps of a stochastic analysis