The Application of a Force Identification Method Based on Particle Swarm Optimization to Compression Steel Bars

Abstract

The non-destructive determination of internal forces in bars for existing building structures is a forward-looking challenge for the construction industry. The application of validated techniques would not only allow the detection of overloads or structural changes according to the load-bearing behavior, it would also allow the verification of the original structural analysis. Based on an experimental modal analysis, tried and tested methods to determine the tensile forces of cables in the operating condition have existed for many years. Due to the slenderness of the cables, even normal forces have a large influence on their natural frequencies: The higher the tensile forces, the higher the geometrical system stiffness, which increases natural frequencies. An equivalent effect occurs when considering load-bearing elements subjected to compression, whereby larger compressive forces lead to a reduction in the geometrical system stiffness and thereby to a reduction in natural frequencies. While various methods have been experimentally investigated for tension cables and tension rods, only a few elaborations are known with respect to structural compression rods. This research presents a system identification method based on the experimental determination of the modal parameters of a compression steel bar. Using an iterative optimization procedure, the design parameters of a basic model including the compressive force are iteratively adjusted until the best possible agreement between the experimental measurements and a theoretical analysis is found. For this purpose, a deviation function is formulated and an evolutional algorithm — in this case facilitating a Particle Swarm Optimization — is used to solve the optimization problem. The Particle Swarm Optimization is an innovative metaheuristic algorithm that allows searching for the global minimum even for complicated nonlinear functions. The method is experimentally validated on slender steel beams with varying compressive forces. Laboratory results from three different steel profiles combined with two different bearing conditions each are presented. Therefore, six specimens were constructed and tested with four different force levels each. The average deviation over all 24 tests between the optimized force and the actual force directly measured by a load cell was determined as 7.4%.

Appendix

In the following chapter, the results from test B3-C-T4 are illustrated in more detail. Figure 9.6 demonstrates the Frequency Response Function (FRF) from the acceleration sensors ACC-2 to ACC-13. A frequency range from 20 to 180 Hz is presented in the figure, while the full measurement contained a baseband to 200 Hz. The impact point was located at ACC-8 so that two modes could be determined sufficiently accurate. Figure 9.7 shows a more detailed view of the FRF between 25 Hz and 36 Hz demonstrating the first natural frequency at 30.26 Hz. Numerical results from the modal properties are provided in Table 9.5.

Fig. 9.6

Frequency Response Functions (FRF) from B3-C-T4 containing information from ACC-2 to ACC-13 (magnitude)

Fig. 9.7

Enlarged FRFs from B3-C-T4 containing information from ACC-2 to ACC-13 (magnitude)

Table 9.5 Experimental modal parameters from B3-C-T4

Fig. 9.8

Experimentally measured and theoretical optimized mode shape from test B3-C-T4

Fig. 9.9

Base beam with the optimized and the assumed parameters from the system identification method from test B3-C-T4 loaded with an actual compressive force of 70.00 kN

According to the modal parameters shown in Table 9.5, the system identification method was proceeded. A system length of 2.3975 m was chosen for the distance between ACC-2 and ACC-13. Figure 9.8 demonstrates the measured and optimized mode shape including the corresponding natural frequencies. The continuous line is the optimized mode shape; the markers illustrate the measured mode shape.

Figure 9.9 illustrates the optimized column based on the mode shape and frequency shown in Fig. 9.8. The presented beam contains the optimized and the assumed parameters from the system identification method. The optimized compressive force N had a deviation of 1.89 kN compared to the actual force. The percentage error was calculated to 2.7% for this specific test.