Probability analysis of an embedded water tank

The design of building structures must fulfil specific regulations, in our case, different building standards, among them Eurocodes. In addition to deterministic procedures in structural design, these standards also allow probabilistic procedures. The embedded tank loaded with soil and liquid is solved by the probability analysis using ANSYS, which contains several probabilistic methods. The reinforced concrete tank is solved by the RSM probabilistic method, which uses the well-known Monte-Carlo method in the background. Input parameters (material properties of soil and reinforced concrete, load pressure from water, geometric data change of both wall and tank bottom thickness) are entered into the calculation with certain aberrances allowed by standards in the construction and loading of structures. The results are also sets of probabilistic variables with a certain variance, as opposed to a deterministic calculation, where only one value results. These procedures, which use statistical methods, have been at the forefront in recent decades. At the end of the paper, some results of the analysis of embedded reinforced concrete water tank (deterministic and probabilistic procedure) in state of tank failure on the second limit state are presented.


Introduction
Each structure shall be designed and constructed in such a way as to serve the intended purpose and to withstand any loads and influences which may arise during its intended life. The design of each structure shall meet the requirements for load-bearing capacity, serviceability and durability laid down in the relevant national standards and Eurocodes. The design and assessment of the structure can be made by a deterministic and probabilistic approach. The required reliability is achieved by design according to EN 1990-EN 1999 and by appropriate design and management quality.
Current advances in computing technics and the widespread availability of information technology have led to the assumption of wider use of the probabilistic approach to assess the reliability of structures using various simulation methods.

Deterministic tank analysis
The storage tank built in Slovakia will be subjected to deterministic and probabilistic analysis. The cylindrical tank is fully embedded in the subsoil. It is made of reinforced concrete and is used for water storage.
The tank is made of concrete C 30-37. The outer diameter of the tank is 20 m, the total height of the tank is 3 m, the bottom thickness of the tank is 0.6 m, and the thickness of the jacket is 0.5 m. The subsoil consists of soil S3, sand with admixture of fine-grained soil, moderately facilitated. The

Reliability analyses
In the case of reliability analysis of structures, a probabilistic approach is used. Randomness in the loading, variability in materials and in geometrics of structures as well as inaccuracies in the model compared to reality can be included in the structural probability calculation. For this purpose, various methods for assessing the reliability have been developed.
At present it is possible to do probabilistic calculations even for complicated structures in programs based on stiffness variant of FEM.

Reliability Criteria
In the case of probabilistic approach, the measure of reliability should by identified with the survival probability where P f is the failure probability for the considered failure mode and within an appropriate period. The structure should be considered to be unsafe in the case when the calculated failure probability is larger than a present target value. The measure of reliability depends on the lifetime of the structure and it is defined by the reliability index β (Tab. 1).  [8].

Uncertainties of input parameters in probability analysis
The effect of the interaction of the structure with the subsoil can be introduced into probabilistic analysis through variability of the input data of the structure and the subsoil. The variability of the material properties of the soil is defined by the modulus of elasticity of the soil E_z from geological measurements and by the variable Evar_z. The stiffness of the reinforced concrete tank is determined through the characteristic value of the Young's modulus E_c and the variable Evar_c.
The load is determined by the characteristic value q_v and the variable qvar_v. These input parameters are described in more detail in Tab. 2. The diagram of some input parameters can be seen in Fig. 4.   Fig. 4. The diagrams of Evar_c and Qvar_c.

Reliability criteria for serviceability
Reliability assessment of the embedded water tank is realized in accordance of the Eurocode and national standard requirements (1-5) for serviceability an ultimate limit state. The serviceability of this structure is limited by maximum vertical displacement and rotation.
The function of structural failure in case of structural settlement is defined by the following relation: where w E is the maximum displacement in vertical direction and w R is the limit displacement.
The function of structural failure in case of structural rotation is: where fi E is the topic rotation and fi R is the limit rotation.
Reliability density function of vertical displacement and rotation are shown on the next figures.  The variability of the stiffness of soil and structure and also the variability of loads are important to the rotation of the structure.
In case of deterministic analysis the maximum deflection in the tank footing bottom is 19.73 mm, in the case of probabilistic analysis the maximum deflection is 24.45 mm. Further outputs from probabilistic analysis are given in Tab. 3.