The reliability of multistory buildings with the effect of non-uniform settlements of foundation

The issue is the evaluation of reliability of construction considering the influence of the variation of the support settlement, which is changing during the lifetime of constructions due to the consolidation process of the ground. Recently, the specialists give special emphasis to the necessity to develop the methods for the estimation of reliability and durability of structures. The problem, the article considers, is the determination of the reliability of multistory buildings with non-uniform changing-in-time sediments caused by the consolidation process in soils. Failure of structures may occur before the draft reaches it`s stabilizing value, because of the violations of the conditions of normal use.


Introduction
The support settlement has a great effect on the distribution of stresses and deformations of the structure. The most dangerous effects on the behavior of the structure is produced by the difference of the settlement of separate parts of the structure's foundations, that difference causes some extra stresses and deformations of the structure's elements. It is known that the settlement (sagging) depends on the kind of the basement soil and the state of stress in the grounds the foundation consists of. Let us consider the example: statically indeterminate truss which is situated in the layer of the soil. Fig.1 The truss loaded uniformly Fig.2 The main system By the removal of one support, (B), the system is made statically determinate. Take the force of the interaction between the beam of the truss and the support (B) as the main unknown variable. The displacement of the point (B) in the main system in the direction of the unknown force x1(t), the force was caused only by the imposed load, is denoted with Δ1p, and the displacement of the same point (B) in the direction of the unit force x1(t)=1, denote with δ11.
The values Δ1p and δ11 do not depend on time but on the elasticity of the system. They can be determined by the well-known methods of frame-analysis.
The strain compatibility condition for the given statically indeterminate system can be written that way: -the function (yet unknown), which characterize the law of the variation of the supports` (B) settlement due to the unknown force x1(t).
The hydrodynamic stress in the soil skeleton, as we know, is determined by the integration of the equation: (2) with the following boundary limits: Solving the equation (2) with the Fourier's method, we have: here: 2 = 2 2 ℎ 2 с 2 с2 -the soil consolidation coefficient, the expression for it: с 2 = 0 2 1+ 1 here: k0 -the coefficient of filtration, and k0=const ξ1 -the porosity mid-index of the soil; α -the compacting factor under the compression of the soil; Δ -the volume weight of water The expression for the settlement (sagging) of the soil layer is After changing the order of integration in the last summand, we have: After partial integration of the right part: Substitute (5) in (7), so we have: Make a note that Finally, we get the following expression for s1(t): The expression (9) gives us the expression for the variation of the support (B) settlement for our statically indeterminate system as a function of the time t and the value of reaction x1(t).
The situation is that the limits of the fail-safe behavior area is determined by the condition: (Macc -the acceptable value of the moment) The condition could be transformed to Take only the first member of the raw in (17) and prelogarithmic both parts of (17), so we get: The probability features of ω depends on the random properties of all the variables the function z depends on. The limits of the fail-safe behavior area depends on the random value z=z0 (ω=ω0), which has mathematical expectation ω 0 ̅̅̅̅ and the dispersion σ 2 (ω0). Suppose that the failure happens then Mmax=Macc, the probabilistic properties of z0(ω0) are defined only by the probabilistic properties of 0 , and the probabilistic properties of s are defined by the variation of the consolidation coefficient.
The exact time of failure corresponds to the point where the stochastic process (21) and the probable limits of the no-failure behavior area cross. Suppose that then the failure happens s=ω0/T, the characters s and ω0 are independent and random and that they have a truncated Gaussian distribution in the intervale (0, ∞). This is so, for example, then the random value of z has a logarithmically normal distribution of the random value T, such as  (23) n = ω 0 , s The expression for the density of distribution for no-failure behavior is that: Then, in accordance with (4), it can be written down: