Stress-strain state analysis and optimization of rod system under periodic pulse load

The paper considers the problem of analysis and optimization of rod systems subjected to combined static and periodic pulse load. As a result of the study the analysis method was developed based on traditional approach to solving homogeneous matrix equations of state and a special algorithm for developing a particular solution. The influence of pulse parameters variations on stress-strain state of a rod system was analyzed. Algorithms for rod systems optimization were developed basing on strength recalculation and statement and solution of optimization problem as a problem of nonlinear mathematical programming. Recommendations are developed for efficient organization of process for optimization of rod systems under static and periodic pulse load.


Introduction
To date, multiple investigations have been performed in the field of design and optimization of rod systems under static and dynamic load [1,2,8,[10][11][12][13][14][15][16][17]. In the majority of them, harmonic loads serve as dynamic ones. In a number of studies [2,3,5,6] the emphasis is put on determination of optimal systems properties in optimization problems with constraints of strength, stiffness and lowest frequency of natural vibrations.
Previous research [7,[9][10][11][12][13]15] were focused on decrease of labor intensity for optimization algorithms, first of all, due to more efficient organization (effective analysis and optimization methods, effective methods of structural state parameters approximation in iteration optimization processes, division of optimization process into levels and stages, etc.). In particular, the work [11] suggests combining the numerical procedure of dynamic analysis and the algorithm of finite difference approximation of structural state parameters to improve efficiency of optimization process for systems dynamic loading.
In recent years, a tendency to build processes of structural optimization without using approximation of structural state parameters appeared [16,17]. However, development of efficient methods for analysis and optimization of structures under static and periodic pulse load is a relevant task. The purpose of this research is development of convenient and effective methods for analysis and optimization of rod systems under combined static and periodic pulse load.

Brief description of dynamic analysis procedure
Equation of equilibrium of a discrete dynamic system (without the account of damping and with regard to the method of finite elements displacement) is written as follows: where m r , e rmatrices of mass and stiffness,   F R tvector of dynamic load. Dependence of load value on time of interpulse interval is approximated by invariable, linear and sinusoidal components (Fig. 1). In case when the interpulse interval exceeds the pulse duration, there are time intervals in which the system executes forced and free vibrations (Fig. 2).
In the first case (interval 1) the solution is as follows [1]: where -  Z tgeneral solution of the homogeneous equation.
  L tvector of approximating functions, Qnumerical matrix, constants of integration are found from the initial conditions.
In the second case (interval 2), the general solution is the solution of the homogeneous equation (3), whereas constants of integration are found from the conditions of the solutions interface in the intervals 1 and 2. Then, the recalculation procedure repeats, in which the initial conditions for defining constants of integration on interval 1 for the next period are the calculation results on interval 2 for the previous period.
A single-span pin-supported beam with its own mass ( m  0.13659 t/m) evenly distributed along the longitudinal axis and with three concentrated masses ( m  0.5 t) fixed on the longitudinal axis was used as an experimental structure subjected to periodic pulse load. Length of the beam is l  4 m, its stiffness is EI  838400 kN·m 2 . Pulse loading was applied to the beam in the form of concentrated force. The size of pulse stayed the same while its shape and duration were varied. Two variants of pulse load application are considered: force at quarter-span (Fig. 3a); and force at mid-span (Fig. 3b).

Fig. 3.
Design schemes for beam: a) pulse action at quarter-span; b) pulse action at mid-span.

Analysis of effect of pulse frequency
Influence of pulse repetition frequency rep n (with repetition period rep Т ) on the maximum value of bending moment in the beam cross-section was analyzed at the initial stage of research. Sinusoidal shape of pulse was accepted for analysis ( Fig. 2). First 40 pulses were studied at determined intervals. The analysis was performed with the help of DINAM software based on the method of finite elements displacement and developed at the Structural Mechanics Department of Novosibirsk State University of Architecture and Civil Engineering (Sibstrin). The software allows analyzing rod systems for arbitrary dynamic loads with calculation of natural vibration frequencies.
In the process of the beam vibration under pulse action the maximum values of bending moments were selected in the cross-sections taken every 0.2 m along the beam.
Analysis was performed at the constant duration and amplitude of force action ( imp Т  0.008 s, max F  10 kN). Frequency of pulse load varied from 3 Hz to 125 Hz, which corresponds to the pulse interval ( rep Т from 0.33 s up to 0.008 s). The value of the force pulse was 0.05055 kN·s.
The considered frequency range of periodic pulse load contained 2 natural frequencies of the beam: f 1 = 15.12 Hz, f 2 = 60.13 Hz. Figure 4 demonstrates a fragment of the dependence graph for bending moment amplitude on pulse frequency. The green line corresponds to the pulse applied to quarterspan, the pink oneto the pulse applied to mid-span. As it was assumed, the resonance effect occurred near the natural frequency regions, but the peak values were slightly displaced with regard to natural frequencies. Beside, resonance phenomena were observed at the pulse frequencies divisible by frequencies of natural vibrations. These peaks of amplitudes are obviously caused by harmonic resonance.

Analysis of the effect of pulse shape and duration
The effect of pulse shape was analyzed on the example of the beam under load at quarterspan. Two pulse shapes were considered (Fig. 5. The pulse size is constant. Figure 6 demonstrates the graph with the analysis results under action of wave pulse (gray line) and sinusoidal pulse (red line). The graph clearly shows that wave pulse is the most dangerous. The maximum value of force under wave pulse action appears to be 30% higher than that under sinusoidal pulse action.
Then, the effect of pulse duration was investigated on the example of a beam under load at quarter-span. Figure 7 presents dependences of the maximum bending moment amplitude for various relations /  As the result of the investigation wave pulse was found more dangerous than sinusoidal one. Drop of maximum bending moment amplitude is observed in the weakest cross-section when duration of pulse with set frequency was increased; however, in case of two pulses overlap the amplitude shows a sharp rise. Variation of pulse frequency also has a remarkable effect on the values of bending moments amplitudes.
These research outcomes were further used in statement of the frame structure optimization problem with the account of the most negative cases of periodic pulse loading.

Solution of optimization problem for a steel frame subjected to static and periodic pulse load
A five-floor double-span steel frame made of welded I-shaped elements was taken as an example for optimization problem statement for a structure subjected to combined static and periodic pulse load   q t (Fig.8). The floor height is 3 m, section span is 6 m. Wave pulse is considered.

Admitted design with identical cross-sections of frame elements
The function of frame material volume was accepted as an objective function. With identical cross-sections of frame elements the function has the following form: where col llength of columns; cb llength of cross-beams. The value of 1 X   was recalculated based on fulfillment of the following condition:

Optimum design for two groups of frame elements (columns, cross-beams)
The function of frame material volume was accepted as an objective function. 1.00 m  f X .

Optimum design for ten groups of frame elements (columns and cross-beams on each floor)
While performing analysis the elements were divided into 10 groups with respect to the floors (5 groups of columns, i=3,5,7,9; 5 groups of cross-beams, i=2, 4,6,8,10;number of groups). As before, the function of the frame material volume was accepted as objective function f(X).  0.808 m  f X .
The following conclusions can be made after the data analysis: 1. Parameters of column cross-sections decrease from first to fifth floor. Cross-sections of columns of fourth and fifth floors are slightly off the dependence, the cross-section of fifth floor column being slightly bigger than that of fourth floor column; 2. The parameters of cross-beam sections are quite stable and fluctuate between 6.61 mm and 6.44 mm. Due to this fact, cross-beams can be integrated in one group for all floors and unified.