# Monte Carlo Analysis of the Intensification Factor of Design Response Spectra for Hoisted Loads
## 5 Statistical Analysis of the Intensification Factor  
The objective of this study was to estimate the intensification factor of design response spectra (RS) for hoisted loads. This requires evaluating the peak acceleration of the hoisted load as a function of the resonant frequency of an equivalent linear system. The time response of the hoisted load is obtained by numerically integrating Equation (6) using a synthetic time history as input. The peak acceleration is then computed as follows:  
\[a_{peak}=\max\left(\left|\left(-k(s_{sol}(t)-l_{0})-cs_{sol}(t)\right)/m+g \right|\right) \tag{12}\]  
where \(s_{sol}(t)\) is the numerical solution of Equation (6). This process is repeated several times with a Monte Carlo approach where the input parameters are randomized or parameterized. The parameters we consider are as follows:  
1. Natural frequency of the hoisted load. The natural frequency of the load attached to the crane ranges in the bandwidth [0-6] Hz, which is justified by the range of stiffness and mass that can occur. 2. Seismic time history. The seismic time history is randomized according to Section 2 via the time envelope shape and the phase of the harmonic components. 3. Initial slope of the RS (or magnitude of the plateau). The magnitude of the acceleration in the plateau bandwidth, and, therefore, the slope of the linear part of the RS, is parameterized from a value of 1.8\(\times\)\(g\) to 3.6\(\times\)\(g\) with two equally spaced intermediate values (2.4\(\times\)\(g\) and 3.0\(\times\)\(g\)). These values are justified for a severe earthquake and account for an intensification of the load due to the structure where the crane is mounted. 4. Internal damping of the crane. The effect of the internal damping of the crane is studied parametrically for the most severe RS curves. 5. Shape of the RS. The effect of changing the RS shape, i.e., with a plateau in a smaller bandwidth, is studied independently and described at the end of this paper. The Monte Carlo analysis consists of 3000 simulated time histories with a crane internal damping of 2% and the RS with a plateau in the range [6-30] Hz. The 3000 simulations are equally distributed for the four RS plateau values. For each time history, we obtain the response of the hoisted load and derive its peak acceleration. The time history generation and numerical integration of motion equations were implemented in MATLAB R2020 [19]. Each case was simulated using both the non-linear system defined by Equation (6) and a corresponding linear oscillator with identical parameters for comparison. The entire process required several days of computation on a standard analysis machine. The peak accelerations obtained for the linear oscillator are presented in Figure 6, while the results for the non-linear crane are shown in Figures 7 and 8. The non-linear system exhibits a significantly larger dispersion in peak accelerations compared to the linear case. The computed intensification factors for the non-linear crane, relative to an equivalent linear system, are illustrated in Figure 9. Since the linear case (Figure 6) shows limited variability, its average value is used as the denominator. The estimated \(+1\sigma\) intensification factor is approximately 1.65, with the RS plateau magnitude having only a secondary effect. For accelerations below 1\(\times\)\(g\), the response remains linear. However, at higher acceleration levels, the non-linear effects become significant. The peak acceleration of the hoisted load initially decreases due to a reduction in energy transfer from the attachment point: when the hoisted load enters free fall, no external energy is introduced. In the second phase, starting approximately at 1.5\(\times\)\(g\), the intensification factor rises. We believe this is determined by a reduction in effective damping due to the slack rope, in line with what is discussed in Section 3. At around 2\(\times\)\(g\), the intensification factor starts decreasing again.