# Monte Carlo Analysis of the Intensification Factor of Design Response Spectra for Hoisted Loads
## 2 Generation of Acceleration Time Histories  
Conducting a large number of simulations using a Monte Carlo approach requires an equally large set of generic acceleration time histories that must comply with a prescribed design response spectrum \(RS_{goal}\). The methodology used to generate the time histories follows the approach described in [PERSON] (1976) [8]. The Eurocode 8 (EC8) design code leaves significant freedom on how accelerograms should be selected [9]. As a trade-off, we use [8], which is very simple while still ensuring compliance with the given design spectrum. Discussing modern tools for the generation of spectrum-compatible accelerograms goes beyond the scope of this work and can be investigated in several other references [10; 11; 12]. The process can be summarized as follows:  
1. Initialization of the frequency vector, where the frequency step is defined as follows: \[\frac{\Delta f}{f}=2\zeta\] (1)
2. Derivation of the time envelope, \(F(t)\), from a recorded earthquake time history. 3. Construction of a synthetic time history as follows: \[a_{\text{rs,0}}(t)=F(t)\sum_{i=1}^{N}(-1)^{i}A_{0,i}\sin(2\pi f_{i}t)\] (2)where the initial amplitudes \(A_{0,i}\) are set proportional to the response spectrum at the corresponding frequencies \(f_{i}\). 4. Calculation of the response spectrum of the generated synthetic time history. The amplitudes are then adjusted iteratively using the relation: \[A_{1,i}=A_{0,i}\frac{RS_{\text{goal}}(f_{i})}{RS(H_{0},f_{i})}\] (3) The response spectrum is computed following the algorithm described in [13]. 5. Iteration of steps 3 and 4 until the synthetic response spectrum closely matches the target spectrum \(RS_{\text{goal}}\). As a reference earthquake time history, we used the L'Aquila earthquake (Abruzzo region, Italy), which occurred on 6 April 2009 at 01:32:40 UT (Figure 1). The accelerations were recorded by the AQK station and are available in the ITACA database (ITalian ACelerometric Archive, [14]). Using this time history ensures a realistic temporal evolution representative of an actual seismic event. However, since the frequency content is independently adjusted to match the design spectrum and the envelope function is randomized, the choice of the reference earthquake does not introduce any loss of generality. A few modifications are applied to the procedure above. First, the rescaling of Equation (3) in the first iterations is applied as follows:  
\[A_{1,i}=A_{0,i}\Bigg{(}\frac{RS_{goal}(f_{i})}{RS(H_{0},f_{i})}\Bigg{)}^{2} \tag{4}\]  
which empirically shows beneficial effects on the convergence. Meanwhile, the time envelope is randomized by a uniform \([-10,+10]\%\) factor at each of its points to limit its systematic effect. Finally, for each time history, each one of the harmonic functions of Equation (2) is assigned a random phase, with a \([-\pi,\pi]\) uniform distribution. This is a key aspect to be able to generate a large number of different time histories inducing different transient responses in the system under analysis. Figure 1: Time history and envelope for the L’Aquila earthquake. The envelope includes the 10% randomization and applies to both negative and positive values. The typical target RS is shown in Figure 2, where  
\[RS(f)\propto \begin{cases}f&0<f<f_{1}\\ \text{constant}&f_{1}<f<f_{2}\\ 1/f&f>f_{2}\end{cases}  \tag{5}\]  
The shape of this idealized RS is given in the Eurocode 8 [4] and is based on what is used for the Extremely Large Telescope [2; 3]. We omit here a first frequency range in which the RS is proportional to \(1/f^{2}\) because it concerns only very low frequencies (<1 Hz) that are not of interest for a slack rope in a crane, which can happen only with accelerations above \(1\times g\) typical of higher frequencies. We chose as \(f_{1}\) and \(f_{2}\) 6 Hz and 30 Hz, respectively. In Section 5, it is shown that these boundaries have a negligible effect on the intensification factor. The magnitude of the constant in the plateau is one of the parametrized values.