Table 1 Dynamic characteristics of structures Structure f1 m1 ζ1 A1 6.0 Hz 6.82⋅103 kg 0.5 %cr A2 6.0 Hz 6.82⋅103 kg 1.0 %cr A3 6.0 Hz 6.82⋅103 kg 2.0 %cr B1 5.0 Hz 6.82⋅103 kg 1.0 %cr B2 6.0 Hz 6.82⋅103 kg 1.0 %cr B3 7.0 Hz 6.82⋅103 kg 1.0 %cr As it appears, the structures are split into two groups (A and B). In each group one of the modal characteristics are varied. For instance in group A, three different damping ratios are considered. In the studies, the structures will be modelled with and without a stationary crowd of people atop the structure. Table 2 gives the frequency and damping assumed for the crowd. Table 2 Dynamic characteristics of crowd f1 ζ1 6.0 Hz 0.35 The values are fairly close to those experimentally derived in [5] for a standing crowd of people. The modal mass of the crowd, m2, is varied in the studies, so it is not shown in Table 2. 3. LOAD MODEL AND RESPONSE PREDICTION Below the load assumed acting on the structure is defined. A simplistic model for jumping loads is assumed. The jumping frequency is denoted fl,, and load is modelled as follows: ( ) ( ) l n n n sin n f p t G π ϕ α + = ∑ = 2 5 1 (4) As it appears five harmonics are considered in the Fourier series expansion, where αn is the dynamic load factor and where ϕn is the phase lag. The factor G represents the weight of the jumper which is assumed to be 750 N, as only a single jumper is assumed. The dynamic load factors are determined from: ⎪ ⎩ ⎪ ⎨ ⎧ + = − 0.1 2.57 0.19 2 l n nf α 13Hz. 13Hz 3Hz 3Hz ≥ ≤ < < l l l if nf nf if if nf (5) This is a simplistic model for high jumping. For the jumping frequency and array of possibilities are considered namely 2.00 Hz, 2.01 Hz, 2.03 Hz up to 3.00 Hz. These jumping frequencies are quite realistic. For each jumping frequency assumption, the load is computed and so is the rms-value of structural accelerations. The maximum value of the rms-value is identified and is denoted Re where the subscript ‘e’ signals that this is the result obtained for the empty structure. Empty here refers to the situation where there are no stationary people on the structure. The only person on the structure is the jumper himself. However, the exercise of computing rms-values for different jumping frequency assumptions and identifying the maximum value is also carried out for situations where the structure is assumed occupied by a stationary crowd of people of mass m2. The maximum rms-value computed for a specific value of m2 is denoted Ro(m2), where the subscript ‘o’ signals that is a result obtained for the structure occupied by a stationary crowd of people. 447
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