Evaluation of the reliability of reinforced concrete beams on a stochastically inhomogeneous elastic foundation under the action of a non-stationary random load

The results of the evaluation of the reliability of reinforced concrete beams lying on an elastic foundation are presented. The load on the beam is considered as a non-stationary random function, the elastic properties of the foundation are described as a stationary random function. Beam stiffness is considered as a random variable depending on the cubic strength of concrete. To solve the beam bending equation on an elastic foundation with random properties and a loaded non-stationary random load, the small parameter method and the method of spectral representations are used. The obtained probability characteristics of the probability density distribution of bending moments allow us to find the probability of failure of a reinforced concrete beam on an elastic stochastically inhomogeneous foundation. By the reliability of the beam e, we mean the probability of nondestructive over a normal section of a reinforced concrete beam with random stiffness B(R) lying on an elastic stochastic heterogeneous foundation with a Winkler ratio C(x) and a random nonstationary load loaded q(x): e = 1 – ep . (1) Destruction of a reinforced concrete beam over a normal section occurs if M(x) > Mult(R, σт), where Mult(R, σт) is the random value of the bearing capacity of the beam, and M(x) is the normally distributed bending moments in characteristic sections beams depending on random parameters B(R), q(x), C(x). The probability of failure of a reinforced concrete beam in a normal section in this case will be equal to: ep = ∫∫∫ ∫ pσт(σт) pR(R) pq(q)pC (C) ∞ −∞ [ ∫ pM(M, R, q, C)dM] ∞ Mult(R,σт) dσтdRdqdC, (2) where pR(R), pσт(σт), pq(q), pC(C) probability density functions of cubic concrete strength R and the yield strength of reinforcement σт, random non-stationary load q(x) and random function of coefficient of the foundation (bed ratio) C(x); pM(M, R, q, C) probability density function of bending moments in the beam. * Corresponding author: kittaobao@mail.ru © The Authors, published by EDP Sciences. This is an open access article distributed under the terms of the Creative Commons Attribution License 4.0 (http://creativecommons.org/licenses/by/4.0/). E3S Web of Conferences 97, 04052 (2019) https://doi.org/10.1051/e3sconf/20199704052 FORM-2019

By the reliability of the beam e, we mean the probability of nondestructive over a normal section of a reinforced concrete beam with random stiffness ( ) lying on an elastic stochastic heterogeneous foundation with a Winkler ratio ( ) and a random nonstationary load loaded ( ): = 1 -.
(1) Destruction of a reinforced concrete beam over a normal section occurs if ( ) > ( , т ), where ( , т ) is the random value of the bearing capacity of the beam, and ( ) is the normally distributed bending moments in characteristic sections beams depending on random parameters ( ), ( ), ( ). The probability of failure of a reinforced concrete beam in a normal section in this case will be equal to: where ( ), т ( т ), ( ), ( ) -probability density functions of cubic concrete strength and the yield strength of reinforcement т , random non-stationary load ( ) and random function of coefficient of the foundation (bed ratio) ( ); ( , , , ) -probability density function of bending moments in the beam.
All random parameters are accepted distributed according to the normal Gaussian distribution.
In [7] the parameters of the distribution of deflections and bending moments in a reinforced concrete beam lying on an elastic foundation were constructed. In this case, the load on the beam was considered as a stationary random function, and the elastic properties of the foundation were described by the Winkler model, which was also considered as a stationary random function.
In the practice of designing foundation structures, the task of bending a reinforced concrete foundation beam under the action of a non-stationary load, which is a function of the coordinate, is quite often encountered.
The average values of the random Winkler coefficient of the foundation, the external load and deflections of the beam are denoted respectively by < >, < ( ) >, < ( ) > and the correlation functions of these parameters are denoted as ( − ′ ), ( , ′ ), ( , ′ ). Suppose also that the cube strength of concrete has received a random implementation, which corresponds to the flexural rigidity of the beam 0 .
The equation of the bend of a bar loaded with an arbitrary load and lying on an elastic foundation with Winkler coefficient has the well-known form: To solve equation (3), we will use the small parameter method in the form, which was done in [7].
We represent the function of the deflection of the beam ( ), as well as the function of repelling the foundation ( )and the load ( ) in the form: here = 0,1,2, … , → ∞(x); -a small parameter that, after all calculations are performed, is equal to 1. Substituting expressions (4), (5) and (6) into equation (3) and equating the terms with the same degree of small parameter, we turn to the system of differential equations: ……………….
We confine ourselves to solving the first two equations (7) and (8) from the above system of equations. In them, the functions 1 (x), 1 (x) и 1 ( ) are centered random functions. Obviously, solving the equation (7) of the above system of equations gives us the expectation of the beam deflections.
To solve equation (8) of the resulting system, we use the method of spectral representations. Given the well-known fact that the correlation function and the spectral density of a random process make up a pair of Fourier transforms, known as the Wiener-Khinchin theorem, we will have the following expression for the impedance of equation (8): The right-hand side of equation (8) can be written as: The generalized spectral density of a nonstationary random function ( ) is represented as follows: Ф ( , 1 ) = < ( ) > 2 ( ) + ( , 1 ).
(13) In expression (13), the first term is the spectral density of the base stiffness. The correlation function of the Winkler modulus can be taken in the form: where -variance of Winkler coefficient of foundation; и -coefficients, the method for determining which is given in [8]; − ′ -the distance between arbitrary correlated sections of the beam; , -coefficients that can be taken as equal to 0 or 1; coefficient taken as equal to 1 or 2, depending on the properties of the random base.
In [12], according to the results of soil testing, it was recommended to take the coefficients = 2, k = 1, m = 0, while the experimental and approximate curves shown in Fig. 1, as showed a good match. The second term in expression (13) is the spectral density of the load, which is determined by the generalized Wiener-Khinchin transform [9]: Applying the inverse generalized Wiener-Khinchin transform [10,11] with the expression for the impedance (10) of equation (8), we obtain the correlation function of the beam deflections: The correlation function of bending moments as the second derivative of the deflection function will be equal to: As expected, the variance of the bending moments turned out to be a function of the x coordinate, i.e. output random process is non-stationary.
Consider special cases of loading the beam. Let the load be specified in a deterministic form and is a function of the coordinate x, i.e. when = 0, in this case the correlation function of the beam deflections will be: and the variance of the deflections of the beam will be equal at = ′ : -distance between concentrated loads. The expected value of bending moments is obtained by differentiating twice (27) with respect x: Now we can construct the probability density function of bending moments, given that it was taken by us normal distribution and, thus, is completely determined by the expected value and variance: Substituting the expression for the probability distribution density of bending moments (29) into the expression for the probability of failure of a reinforced concrete beam over a normal cross section (2) and considering that all the above calculations were made for a concrete implementation of the concrete cube strength , which is a random variable with a Gaussian distribution ( ) with parameters: expectation < > and variance and on which the beam stiffness and the probability characteristics of bending moments depend, find the probability of jelly destruction reinforced concrete beams of normal cross section.