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ЗмістL = 2 m is loaded by a force of F = 2
Fig. 2 Internal forces on the original
R is the resistance of the structure function (yield strength, allowable deflection, stress) and S
RELIABILITY ANALYSIS OF A STRUCTURE
Slovak University of Technology
SK-813 68 Bratislava, Slovak Republic, Radlinskeho 11
Nowadays when design a structure it is necessary to take into account the possibility of saving material when the structure must be reliable to bear all anticipated loads, so designers use more and more new optimal and reliability analysis, mainly procedures of technical standards and methods of assessing reliability. Standards and assessments of reliability always changed related to the current development of science and technology. EUROCODE standard for steel structures for the first time allowed the assessment of reliability in construction design. We are currently witnessing the transition from the original, deterministic methods to new partially or fully probabilistic assessment of reliability design methods.
The First Order Method converts the optimization problem to an unconstrained one by adding penalty functions to the objective function. This one uses gradients of the dependent variables with respect to the design variables. For the each iteration variables are performed in order to determine a search direction, and a line search strategy is adopted to minimize the unconstrained problem. Thus, each iteration step is composed of a number of sub-iterations that include search direction and gradient computations. That is why one optimization iteration step for this method performs several analysis loops. First order iterations continue until either convergence is achieved or termination occurs.
esign variables are usually geometric parameters. They are restricted to positive values. Obviously, more ones demand more iteration steps and, therefore, more computer time. It is necessary specify a reasonable range of values for the design variables (MIN and MAX). State variables are usually response quantities that constrain the design. The objective function is the quantity that you are trying to minimize or maximize. An unconstrained version of the problem is formulated as follows:
where: Q is dimensionless, unconstrained objective function, Px, Pg, Ph, and Pw are penalties applied to the constrained design and state variables, f0 is reference objective function value that is selected from the current group of design.
The cantilever beam of a finite length ^ m is loaded by a force of F = 2 kN and a uniformly distributed load q = 5 kN/m. Cross-section dimensions of cantilever are b = 0,15 m and h = 0,20 m, with the modulus of elasticity E = 2,7.107 kPa. The aim is to find optimal dimensions b, h of the beam, so that the limit stress in bending σx,max = 20 000 kPa will not be exceeded. The load scheme is given in Fig.l.
cantilever (Ansys software)
The maximum deflection is 5.10-3m. Following parameters were declared as design variables: cross-section width b <0,01 m; 0,8 m>, cross-section depth b <0,02 m; 1,6 m>. The maximum stress σmax = 20 MPa is a state variable. The objective function is the volume. The most feasible design was set as follows: b = 0,141 m, h = 0,174 m, σmax = 19,379 MPa.
After design optimization of cross-section dimensions of cantilever final diagrams of bending moments M, shear forces V are presented in Fig. 2 and deflections w and rotations cp are presented in Fig. 3. The maximum deflection for this cantilever is 5,48.10-3 m.
Topological optimization is a special form of shape optimization. The goal of topological optimization is to find the best use of material for a body such that an objective criterion (i.e., global stiffness, etc.) takes out a maximum or minimum value subject to given constraints (i.e., volume reduction).
In topological optimization, the material distribution function over a body serves as an optimization parameter. The user needs to define the structural problem, the objective function to be minimized or maximized and the state variables. The theory of topological optimization seeks to minimize or maximize the objective function (f) subject to the constraints (gj) defined. The design variables (ηi) are internal, pseudo-densities that are assigned to each finite element (i) in the topological problem. The pseudo-density for each element varies from 0 to 1; where ηi, where ηi ≈ 0 represents material to be removed and ηi ≈ 1 represents material that should be kept.
Lets go doing topological optimization on the same beam as on Fig.l with original cross-section, with the Poisson's ratio v = 0,3. ANSYS software supports in topological optimization 2-D planar, 3-D solid, and shell elements. To use topology technique, the model contains only solid elements PLANE82 for plane stress. To topological optimization (domain inside) can be subjected only finite elements of type 1. The other area modeled by elements of type 2, is not optimized. The goal is to minimize the energy of structural compliances subject to a 50 and 60 percent reduction in volume of type 1 material.
The final topologic design represents the above topology, where the blue domain is created by elements, which may be removed from the beam, or made of ballast material.
he phenomenon may have different Ntotal mutually exclusive, possible outcomes. If the number N of these results has resulted in the execution of certain other characteristics of the phenomenon of f(Ntotal -f) excludes the results, then the probability of a phenomenon f noted as is given by:
With the increasing number of simulations decreases the error of determination Pf. To describe the random variables one can use histograms. Solution by Monte Carlo (MC) is a multiple repetitions of simulations and subsequent statistical processing of data for pre-estimated phenomenon - eg. behavior of the structure with the possibility of the collapse. Reliability design is the ability to maintain the required structural properties for a fixed period of the life. To determine the safety using reliability methods some performance criteria for the functional relationship between n-input variables, called basic random variables Xi, are first defined. This relationship is called the reliability function Fs, security, usability, reliability or function reserve function disorders
here the function g(X1, X2,..., Xn) represents a computational model (ie the idealization of reality). Reliability function can be defined by:
where ^ is the resistance of the structure function (yield strength, allowable deflection, stress) and S is a function of impact load (maximum tension in the structure, maximum deflection). The function is reliable if Fs≥0, failure occurs if Fs<0 and Fs= 0 divides the space in the area of reliable and unreliable. Probabilistic methods can be divided into both analytical and simulation methods. Among the simulation methods we mention SBRA - Simulation-Based Reliability Assessment.
Our aim was to determine the collapse of the beam loading by scheme in Fig.1. The assessment was transferred by comparing the maximum bending stress to the yield strength (allowable stress). Six mutually independent random variables, hvar = "area-of” bvar = "-m area." Lvar = nl-01 ", qvar =" dead ", Fvar =" dead ", Slvar = "area.m were used as stochastic inputs. The Anthill program using N = 3.106 steps in Monte Carlo simulation method was chosen for the reliability analysis. A comparison of the maximum bending stress σmax to a yield strength Rp=19667.slvar is presented in Fig. 5. From the picture it is clear that the bending stress exceeds the yield point with the probability of 7,92.10-3, which represents 0.792% probability of damage the structure.
There are several methods of optimization, which may be used for optimal design of structures. Using any of them the designer may find an optimal topology, shape, or dimensions, what leads to the save of material and energy. Nowadays it is also possible to solve the reliability of structures using the probability methods.0>
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