16 Vibration Testing and Analysis of A Monumental Stair 163 Fig. 16.1 Partial view of the staircase during the vibration tests 16.2 Description of the Staircase The staircase, which has a steel structure, is located at the Broad Art Museum in Lansing, Michigan. It connects the first and second levels of the building. Details of the initial static design of the structure and subsequent modifications to meet the dynamic requirements have been discussed by Setareh [19]. Since this paper presents the results of the dynamic analysis and testing of the structure after its installation and before the addition of the non-structural components such as cladding, plywood, cover, etc., only the structural elements of the stair will be described here. The staircase consists of stringers forming a truss with HSS12X3X5/16 for the top and bottom chords and HSS3X3X5/16 verticals at 1.22 m (4 ft) on center. The total depth (end to end) of the chords in the landing area is 1.10 m (43.5 in). Treads and risers are made of 6.4 mm (0.25 in) thick steel plates. Steel plates, 6.4 mm (0.25 in) thick, placed between the top and bottom chords of the stringers in the first two web panels from each end of the staircase. The plates are welded to the chords and verticals of the truss. In addition, HSS5X3X5/16 are used as diagonal members in the remaining truss panels. The bottom chords of the stair stringers are supported at their ends by the concrete beams through rigid connections. Figure 16.1 is a photo showing a partial view of the stair during the vibration tests, which occurred about two weeks after its installation. 16.3 Description of the Dynamic Tests A modal test using an electrodynamic shaker (APS 113) and a number of accelerometers (PCB393C and PCB393B04) were conducted to estimate the dynamic properties of the structure. In addition, a number of walking tests were performed to evaluate the vibration acceptability of the staircase. The shaker was placed on a force plate, which was located at the corner of the stair landing (see Fig. 16.2). Uniaxial accelerometers were oriented along three perpendicular directions. For vertical direction measurements, 393C accelerometers were placed on their bases, however, for lateral directions they were clamped to angle pieces as shown in Fig. 16.2. The accelerometers were placed along the exterior and interior stringers and at the middle of the steps (all along the truss verticals). Using a roving accelerometer approach, they were relocated from one position to another during the modal testing. A burst-chirp excitation of 6–19.5 Hz with 30 s on and 15 s off was selected. A measurement frequency resolution of 0.025 Hz (40 s duration) was used. In addition to the modal tests, a number of controlled vibration tests with the help of three human subjects were conducted. An individual and the entire group ascended and descended the stair at various step frequencies. Using a metronome, the movements of the subjects on the stair were synchronized. Two different step frequencies based on the apparent first mode resonance frequency were selected: 146 spm (steps per minute) and 195 spm, representing the fourth and third sub-harmonics of the first mode resonance frequency of 9.8 Hz, respectively. In addition, an average walking step frequency of 120 spm, representing the most common walking speed was used. The subjects were also asked to move the fastest possible in addition to random walks (three persons only), and ascending while skipping a step (one person only). For each test, the subject(s) ascended and descended the stair. At the end of each ascent/descent, the individual(s) stopped motionless while the measurement was completed.
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