Mass Minimizing of Truss and Beam Structures Subjected to Cumulative Fatigue Damage

The work presents the methods and solutions of the optimizing design of the truss and beam structures subjected to the cumulative fatigue damage. Two basic approaches used in the optimization of mechanical systems are presented in the paper the so-called direct search methods and the gradient methods. A short comparative and gradient approach was made. The aim was to minimize the weight of the truss or beam structure with restrictions affecting the prescribed fatigue life. The computational model assumed only quasi-static loading with different number of cycles.


Introduction
The aim of the structural optimization is to find the technical realizable suggestion. It is the best solution of all possible projects. The optimal result gives the extreme of the objective function with regards to defined constrain conditions. The objective function often includes prize, weight structure, deformation, maximal stress or prescribed some eigenvalues and so on [2,5,17,21,22]. The article presents some discrete optimization methods for mass minimization of the truss and beam structures subjected to described fatigue life. The loading forces are static, with prescribed the number of cycles and loading duration for more variation loading. The damage accumulation was solved by Corten-Dolans hypothesis [1,3,4,8,18].

Structural Mass Minimizing Subjected to Prescribed Fatigue Life
Let's consider the optimizing process of the mass minimizing of the truss and frame structures subjected to prescribed fatigue life of each element. Assuming the data processing of the objective function and constrain conditions, we know following optimizing methods [2,6,7,19]:  Direct search methods -simplex's method and its modifications, Hooke-Jeeves method, Monte-Carlo and others. The mentioned methods are working only with function's values of objective function and constrain conditions. They make no use of gradient information.  Gradient methodsthey use the information about gradient of the objective function and constrain conditions. The most known and the simplest gradient method of all is the steepest descent method.  Newton's methodsthey use the information about first and second differentiation of objective functions and constrains conditions (Fletcher-Powell, conjugate gradient method, etc.). Let's define the optimizing problem of the optimal mass design subjected to the prescribed fatigue life [2,14,16,23,24]. For a structure of multiple elements, the optimization problem of discrete variables can be stated mathematically as minimizing an objective function where n is number of the elements, m is number of the element groups (one design variable X i ), T iP is prescribed fatigue life in hours. Problem (1) consists of a linear objective function and non-linear constraints. Let's form a new objective function with penalty function where penalty function  is The objective function (2) is analyzed by the Gauss-Seidel and gradient method in discrete form. These methods were implemented into MATLAB [13].

Formulation of the Mathematical Model for S-N Curve
The Wöhler's curve is statistically evaluate experimental fatigue curve, that is obtained from amplitude of nominal stress  a with the number of oscillations N for failure of sample. The  a -N relation can be written as follows where f is the fatigue stress coefficient, 2N f is number of cycles to failure, b is fatigue strength exponent and  a is stress amplitude to failure [1,8,9,19].
In the case of higher stress amplitudes the fatigue test with controlled deformation is better to realize [3,11,15,20]. The total ε a or only plastic ε ap deformation vs. cycles to failure is usually presented as a well-known Manson-Coffin curve, which mathematic formulation is following where E is Young modulus, ε f is the fatigue ductility coefficient, c represents the fatigue ductility exponent. Using the number of cycles to failure 2N f and linear Palmgren-Miner law we can calculate cumulative damage as follows If we suggest that maximum value of stress amplitude in critical places will be lower than real endurance strength, the next step of computational analysis will be define a safety factor, which mathematic formulation is following where  a and  m are computed or experimental obtained stress amplitude and mean stress,  A and  M are limit stress amplitude and mean stress from Haigh diagram, or its modified version (Goodman, Soderberg, Geber). Therefore it can be written as follows The  C is conventional fatigue limit. Using equation (7) and (8) we assumed If we suggest that in the regime of the giga-cycle fatigue loading the value of fatigue limit decrease k g times, so safety factor can be written as From experimental measuring in the giga-cycle region of loading, it can be obtained polynomial approximation of S-N curve (Fig. 1). A mathematical formulation of multistage S-N curve for i th part will be where b i exponent can be written as follows Correction of the total damage with the respect to mean stress  m is given by and final fatigue life prediction can be calculate as follows where T r is realization time interval of simulation, D r is damage in realization time interval T r .

Description of Program Toolbox
This section presents the program package for the linear static analyze and chosen discrete optimizing methods for finite element models created by MATLAB software [10,12,13]. The program STATIC.M was used for static analyzes and programs GSM.M and DISKGRAD.M were used in the optimization process (Fig. 2).   Let's design cross sections A i of the simple frame structure excited by forces F 1 , F 2 (Tab. 1). The geometry of the structure is presented in Fig. 3. The used cross sections are presented in Table 2. The prescribed fatigue life will consider T p = 2·10 5 [hours].

Example No. 2
Let's minimize the mass of truss structure on Fig. 6 subjected to the prescribed minimum fatigue life. Consider the prescribed fatigue life T p =2·10 5 hours. The structural loading is presented in Table 4. The material properties are: Young's modulus E = 2.

Cross section X1
Cross section X2 Cross section X3    The results are presented in Table 6.

Conclusions
The work presents the methods and solutions of the optimizing design of the truss and beam structures subjected to the damage accumulation. Considering the solution of the numerical examples, it was compared the so-called direct search methods and the gradient methods.
The aim was to minimize the weight of the truss or beam structure with restrictions affecting the prescribed fatigue life. The computational model assumed only quasi-static loading with different number of cycles. It has been shown that the direct search methods are preferable for discrete optimization problems. The discrete Gauss-Seidel method (direct search method) is suitable for problems with complicated objective function respectively constrain conditions (e.g. fatigue life) and for the structures with the less number of design variables (e.g. a few sections).The discrete gradient method (gradient method) shows less advantageous due to irregular gradient calculation andthe problem of incorrect gradient follows on the penalty function definition.
The first method belongs to direct search methods and it's advantageous for application in discrete optimisation process. Therefore, we can advise it to use for mass structures minimisation with fatigue life constraints.