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<section id="mathematical-background">
<h1>Mathematical Background<a class="headerlink" href="#mathematical-background" title="Link to this heading">#</a></h1>
<section id="structural-analysis">
<h2><strong>Structural Analysis</strong><a class="headerlink" href="#structural-analysis" title="Link to this heading">#</a></h2>
<p>We summarize here basic concepts of structural dynamics which motivated the implementation of Namazu.
For deeper insights and special cases, the interested reader is referred
to <a class="footnote-reference brackets" href="#footcite-chopra2007dynamics" id="id1" role="doc-noteref"><span class="fn-bracket">[</span>1<span class="fn-bracket">]</span></a><a class="footnote-reference brackets" href="#footcite-li2010" id="id2" role="doc-noteref"><span class="fn-bracket">[</span>2<span class="fn-bracket">]</span></a>. An ideal linear
system is submitted to a time-dependent ground motion <span class="math notranslate nohighlight">\(x_g(t)\)</span> as
depicted in the figure below. The movement is such
that the inertia effects cannot be neglected. All supports of the
structure have the same displacement.</p>
<figure class="align-default" id="fig-ideal-frame-structure">
<img alt="_images/frame_sketch_a_v2.png" src="_images/frame_sketch_a_v2.png" />
<figcaption>
<p><span class="caption-text">Idealized linear single-degree-of-freedom system of a frame structure under seismic ground motion <span class="math notranslate nohighlight">\(x_g(t)\)</span>, <a class="footnote-reference brackets" href="#footcite-chopra2007dynamics" id="id3" role="doc-noteref"><span class="fn-bracket">[</span>1<span class="fn-bracket">]</span></a>.</span><a class="headerlink" href="#fig-ideal-frame-structure" title="Link to this image">#</a></p>
</figcaption>
</figure>
<p>Due to the ground’s movement, several reaction forces are occurring on
the structure, Newton’s second law of motion states that a dynamic
system is in equilibrium at each time instant. The occurring reaction
forces are shown in the figure below. The displacement <span class="math notranslate nohighlight">\(x(t)\)</span>
is in relation to the lateral resisting force <span class="math notranslate nohighlight">\(F_L = kx(t)\)</span>, where
<span class="math notranslate nohighlight">\(k\)</span> is the stiffness of the structure. Assuming damping is
present, the damping force can be obtained by
<span class="math notranslate nohighlight">\(F_D = c \dot{x}(t)\)</span>, with <span class="math notranslate nohighlight">\(c\)</span> being the damping
coefficient.</p>
<figure class="align-default" id="fig-dyn-eq">
<img alt="_images/frame_sketch_b_v2.png" src="_images/frame_sketch_b_v2.png" />
<figcaption>
<p><span class="caption-text">Free body system with reaction forces, dynamic equilibrium for each
time instance <span class="math notranslate nohighlight">\(t\)</span>.</span><a class="headerlink" href="#fig-dyn-eq" title="Link to this image">#</a></p>
</figcaption>
</figure>
<p>The resulting total displacements of the structure are denoted by
<span class="math notranslate nohighlight">\(x_{\Sigma}(t)\)</span>. Since the ground motion is time-dependent, the
following relation for the total displacement is given</p>
<div class="math notranslate nohighlight">
\[x_{\Sigma}(t) = x_g(t) + x(t).\]</div>
<p>The inertia force <span class="math notranslate nohighlight">\(F_I\)</span> depends on the acceleration and mass of
the structure, i.e. <span class="math notranslate nohighlight">\(F_I = m \ddot{x}_{\Sigma}(t)\)</span>. The dynamic
equilibrium gives</p>
<div class="math notranslate nohighlight">
\[F_I(t) + F_D(t) + F_L(t) = 0.\]</div>
<div class="float docutils container">
<figure class="align-default" id="fig-frame-b">
<img alt="_images/frame_sketch_b.png" src="_images/frame_sketch_b.png" />
<figcaption>
<p><span class="caption-text">Lateral resisting force, connected to displacement <span class="math notranslate nohighlight">\(x\)</span>. I
mean the arrow direction is kinda unintuitive</span><a class="headerlink" href="#fig-frame-b" title="Link to this image">#</a></p>
</figcaption>
</figure>
<figure class="align-default" id="fig-frame-c">
<img alt="_images/frame_sketch_c.png" src="_images/frame_sketch_c.png" />
<figcaption>
<p><span class="caption-text">Damping force, connected to velocity <span class="math notranslate nohighlight">\(\dot{x}\)</span>.</span><a class="headerlink" href="#fig-frame-c" title="Link to this image">#</a></p>
</figcaption>
</figure>
<figure class="align-default" id="fig-frame-d">
<img alt="_images/frame_sketch_d.png" src="_images/frame_sketch_d.png" />
<figcaption>
<p><span class="caption-text">Inertia force related to mass components, connected to
acceleration <span class="math notranslate nohighlight">\(\ddot{x}_t\)</span>.</span><a class="headerlink" href="#fig-frame-d" title="Link to this image">#</a></p>
</figcaption>
</figure>
</div>
<p>The equation of motion in a linear elastic case is thus given by</p>
<div class="math notranslate nohighlight">
\[m\ddot{x}(t) + c\dot{x}(t) + k x(t) = -m \ddot{x}_g(t).\]</div>
<p>Hence the idea of a displacement-controlled experimental setup. The
tunable shaking table should be able to model the seismic ground motion
<span class="math notranslate nohighlight">\(x_g(t)\)</span> present, which is represented by the shaking table’s
specimen plate movement. These motions can of course also be connected
to other motions, such as the vibration of machinery, slides, or
harmonic movements.</p>
</section>
<section id="signal-generation">
<span id="siggen"></span><h2><strong>Signal Generation</strong><a class="headerlink" href="#signal-generation" title="Link to this heading">#</a></h2>
<section id="harmonic-signals">
<h3><strong>Harmonic Signals</strong><a class="headerlink" href="#harmonic-signals" title="Link to this heading">#</a></h3>
<p>First, harmonic signals have been considered. They consist of a sum of frequency components and are closely related to the Fourier series. Choosing a number <span class="math notranslate nohighlight">\(n_{\Omega}\)</span> of frequencies, the position <span class="math notranslate nohighlight">\(x_{H}(t)\)</span> is calculated as the superposition of the harmonic frequencies and their respective amplitude factor by</p>
<div class="math notranslate nohighlight" id="eq-harmonic">
<span id="equation-eq-harmonic"></span><span class="eqno">(1)<a class="headerlink" href="#eq-harmonic" title="Link to this equation">#</a></span>\[x_H(t) = \sum_{k=1}^{n_{\Omega}} \mathbb{A}_k \sin(2\pi\mathbf{\Omega}_kt),\]</div>
<p>where <span class="math notranslate nohighlight">\(\mathbf{\Omega}_i \in \mathbb{R}_{\geq0}\)</span> are the frequencies of interest and <span class="math notranslate nohighlight">\(\mathbb{A}_i \in \mathbb{R}_{\geq0}\)</span>
the corresponding amplitude values with <span class="math notranslate nohighlight">\(i \in [1, n_{\Omega}]\)</span>. The figures below show a fixed harmonic signal with a single frequency and a mixed harmonic signal with three superposed frequency components. These signals are deterministic.</p>
<div class="float docutils container" id="fig-harmonic-mixed-harmonic">
<figure class="align-default" id="fig-input-signal-2hz" style="width: 60%">
<img alt="_images/fixed_harmonic_pre_rampup_2Hz_10s-crop.png" src="_images/fixed_harmonic_pre_rampup_2Hz_10s-crop.png" />
<figcaption>
<p><span class="caption-text">Fixed harmonic signal with 2 Hz</span><a class="headerlink" href="#fig-input-signal-2hz" title="Link to this image">#</a></p>
</figcaption>
</figure>
<figure class="align-default" id="fig-input-signal-fibonacci" style="width: 60%">
<img alt="_images/fixed_harmonic_pre_rampup_FibonacciHz_10s-crop.png" src="_images/fixed_harmonic_pre_rampup_FibonacciHz_10s-crop.png" />
<figcaption>
<p><span class="caption-text">Fixed harmonic signal with three overlapping frequencies</span><a class="headerlink" href="#fig-input-signal-fibonacci" title="Link to this image">#</a></p>
</figcaption>
</figure>
</div>
</section>
<section id="stationary-stochastic-signals">
<h3><strong>Stationary Stochastic Signals</strong><a class="headerlink" href="#stationary-stochastic-signals" title="Link to this heading">#</a></h3>
<p>Another class of signals of interest in structural engineering consists
of uni-variate stationary Gaussian stochastic processes. They can be
described by the <strong>Spectral Representation Method</strong> (SRM) from a
functional frequency content, the so-called Power Spectral Density (PSD)
function. The SRM and its variations, such as the Stochastic Harmonic
Function (SHF) representation <a class="footnote-reference brackets" href="#footcite-chen2017" id="id4" role="doc-noteref"><span class="fn-bracket">[</span>3<span class="fn-bracket">]</span></a>, are commonly
used to simulate artificial ground motion <a class="footnote-reference brackets" href="#footcite-lyu2023" id="id5" role="doc-noteref"><span class="fn-bracket">[</span>4<span class="fn-bracket">]</span></a><a class="footnote-reference brackets" href="#footcite-liu2022" id="id6" role="doc-noteref"><span class="fn-bracket">[</span>5<span class="fn-bracket">]</span></a>.</p>
<p>Any PSD function can be used in the OpenVIBE framework according to the
user preferences. Among possible PSD, the Shinozuka and Deodatis
frequency content <a class="footnote-reference brackets" href="#footcite-shinozuka1991simulation" id="id7" role="doc-noteref"><span class="fn-bracket">[</span>6<span class="fn-bracket">]</span></a> referred
to as the Shinozuka benchmark and defined by the following PSD</p>
<div class="math notranslate nohighlight" id="eq-sx">
<span id="equation-eq-sx"></span><span class="eqno">(2)<a class="headerlink" href="#eq-sx" title="Link to this equation">#</a></span>\[ S_{S}(\omega) = \frac{1}{4} \sigma^2 b^3 \omega^2 e^{-b |\omega|},\quad -\infty < \omega < \infty,\]</div>
<p>is chosen. The constant <span class="math notranslate nohighlight">\(\sigma\)</span> corresponds to the standard
deviation of the stochastic signal and <span class="math notranslate nohighlight">\(b\)</span> is a parameter that
models the correlation distance of the stochastic signals
<a class="footnote-reference brackets" href="#footcite-shinozuka1988response" id="id8" role="doc-noteref"><span class="fn-bracket">[</span>7<span class="fn-bracket">]</span></a>. The values <span class="math notranslate nohighlight">\(\sigma = 1\)</span>
and <span class="math notranslate nohighlight">\(b = 1\)</span> are chosen. From the reference PSD, a realization of
the stochastic signal denoted <span class="math notranslate nohighlight">\(x_S(t)\)</span> reads</p>
<div class="math notranslate nohighlight" id="eq-srm">
<span id="equation-eq-srm"></span><span class="eqno">(3)<a class="headerlink" href="#eq-srm" title="Link to this equation">#</a></span>\[x_S(t) = \mathbb{A}_s \sum \limits_{n = 0}^{N-1} \sqrt{4 S_{x_{S}}(\omega_n) \Delta \omega} \cos(\omega_n t + \varphi_n).\]</div>
<p>The angular frequencies <span class="math notranslate nohighlight">\(\omega_n\)</span> in <span class="math notranslate nohighlight">\(\text{rad/s}\)</span> are defined as
<span class="math notranslate nohighlight">\(\omega_n = n \Delta \omega\)</span> with <span class="math notranslate nohighlight">\(n \in [0,N-1]\)</span> and
<span class="math notranslate nohighlight">\(\Delta \omega = \omega_u/N\)</span>. The angular frequency
<span class="math notranslate nohighlight">\(\omega_u = 4\pi\ \text{rad/s}\)</span> is the upper cut-off
angular frequency, directly linked with the upper cut-off frequency as
<span class="math notranslate nohighlight">\(f_u = \omega_u /2\pi\)</span> beyond which the PSD function vanishes. The
corresponding time discretization is <span class="math notranslate nohighlight">\(\Delta t = \pi/\omega_u\)</span> in
<span class="math notranslate nohighlight">\([\text{s}]\)</span> . The number of terms <span class="math notranslate nohighlight">\(N\)</span> is a choice
which governs the accuracy of the approximation. The phase angles
<span class="math notranslate nohighlight">\(\varphi_n\)</span> are independent realizations of a uniform distribution
between <span class="math notranslate nohighlight">\(0\)</span> and <span class="math notranslate nohighlight">\(2 \pi\)</span>
<a class="footnote-reference brackets" href="#footcite-shinozuka1991simulation" id="id9" role="doc-noteref"><span class="fn-bracket">[</span>6<span class="fn-bracket">]</span></a>.</p>
<p>The signal magnitude is scaled by the factor
<span class="math notranslate nohighlight">\(\mathbb{A}_s \in \mathbb{R}_{\geq0}\)</span> to achieve the displacement
amplitude desired by the user. For instance, to reach a maximum
displacement around <span class="math notranslate nohighlight">\(17\ \text{mm}\)</span>, the scaling factor
<span class="math notranslate nohighlight">\(\mathbb{A}_s = 3\)</span> has been chosen. One realization, i.e. a
stochastic signal generated with the Shinozuka benchmark is shown here:</p>
<figure class="align-default" id="fig-shinozuka-signal">
<img alt="_images/23-12_shnzk_signal_unfiltered-crop.png" src="_images/23-12_shnzk_signal_unfiltered-crop.png" />
<figcaption>
<p><span class="caption-text">One realization of a random signal with the Shinozuka PSD <a class="reference internal" href="#equation-eq-srm">(3)</a> for <span class="math notranslate nohighlight">\(N=128, \omega_u = 4\pi, \Delta t = \pi/\omega_u, T = 64 \text{s}, \mathbb{A}_s = 3\)</span></span><a class="headerlink" href="#fig-shinozuka-signal" title="Link to this image">#</a></p>
</figcaption>
</figure>
<p>The periodogram <a class="footnote-reference brackets" href="#footcite-li2010" id="id10" role="doc-noteref"><span class="fn-bracket">[</span>2<span class="fn-bracket">]</span></a><a class="footnote-reference brackets" href="#footcite-newland2012introduction" id="id11" role="doc-noteref"><span class="fn-bracket">[</span>8<span class="fn-bracket">]</span></a><a class="footnote-reference brackets" href="#footcite-behrendt2022periodogram" id="id12" role="doc-noteref"><span class="fn-bracket">[</span>9<span class="fn-bracket">]</span></a>
is used to estimate the signal PSD based on the discrete Fourier
transformation of the random signals <a class="reference internal" href="#equation-eq-srm">(3)</a>, as
detailed in <a class="reference internal" href="#sec-periodogram"><span class="std std-ref">the Periodogram section</span></a>. The direct
approximation of the signal PSD by an estimator like the periodogram is
usually not possible, as the target PSD is not known exactly for natural
processes.</p>
<section id="shinozuka-benchmark">
<h4>Shinozuka benchmark<a class="headerlink" href="#shinozuka-benchmark" title="Link to this heading">#</a></h4>
<p>We introduced the Shinozuka benchmark as a way of testing the frequency accuracy of the table and the framework. Usually, a single realization from a stochastic signal does not produce a nice looking PSD, but for certain input PSD functions this can be the case (e.g. <a class="reference internal" href="#equation-eq-srm">(3)</a>). While looking for ways to ensure that the framework does what it is supposed to do, the idea came up to use this property. Generating a signal from this PSD, then measuring the movement of the table and estimating the frequency contents with the Periodogram, we can easily assess divergences in the table’s behavior.
The <a class="reference internal" href="#fig-shinozuka-psd"><span class="std std-ref">figure below</span></a> shows the approximation of the Shinozuka PSD by the periodogram compared to the target PSD originally used to simulate the signal <a class="reference internal" href="#equation-eq-sx">(2)</a>. All frequency components in the signal are preserved and thus the approximation is very accurate. The accuracy of the table can thus be evaluated by simulating one (or more for variance estimation) realizations with the Shinozuka PSD, measuring the displacement and then comparing the measured PSD to the original. For an accurate motion both PSDs should align. Therefore, the shaking table movement can be validated for the prescribed range of frequencies and accurate artificial ground motions of the shaking table will be expected for other stochastic signals generated by the SRM with similar frequency ranges to the Shinozuka benchmark.</p>
<figure class="align-default" id="fig-shinozuka-psd">
<img alt="_images/shinozuka_PSD_revamped_rad-crop.png" src="_images/shinozuka_PSD_revamped_rad-crop.png" />
<figcaption>
<p><span class="caption-text">Shinozuka benchmark: scaled periodogram of one arbitrary realization <a class="reference internal" href="#equation-eq-srm">(3)</a> and target PSD <a class="reference internal" href="#equation-eq-sx">(2)</a></span><a class="headerlink" href="#fig-shinozuka-psd" title="Link to this image">#</a></p>
</figcaption>
</figure>
</section>
</section>
</section>
<section id="post-processing-input-signals-for-the-shaking-table">
<span id="postprocessing"></span><h2>Post-processing input signals for the shaking table<a class="headerlink" href="#post-processing-input-signals-for-the-shaking-table" title="Link to this heading">#</a></h2>
<p>Some generated signals might require large initial accelerations or
velocities due to the underlying numerical procedures. To circumvent
these issues, we introduced post-processing procedures to ensure the
safe usage of the shaking table.</p>
<p>First, for the harmonic signals <span class="math notranslate nohighlight">\(x_H(t)\)</span>, artificial linear
acceleration and deceleration phases based on a window-function are
inserted. The acceleration phase lasts from <span class="math notranslate nohighlight">\(t=0\)</span> to <span class="math notranslate nohighlight">\(t=t_u\)</span>
and the deceleration phase’s duration lasts from <span class="math notranslate nohighlight">\(t=t_d\)</span> to
<span class="math notranslate nohighlight">\(t = T\)</span> such that the durations of acceleration and deceleration
are <span class="math notranslate nohighlight">\(t_u\)</span> and <span class="math notranslate nohighlight">\(T_d = T - t_d\)</span>, respectively. Then, the
harmonic signal <span class="math notranslate nohighlight">\(x_H(t)\)</span> is adjusted from these two windows as</p>
<div class="math notranslate nohighlight">
\[\begin{split}\hat{x}_H(t) =
\begin{cases}
\frac{t}{t_u} x_H, & t<t_u,\\
(1-\frac{t-T_d}{T-T_d}) x_H, &t>T_d,\\
x_H, &else.
\end{cases}\end{split}\]</div>
<p>Considering <span class="math notranslate nohighlight">\(t_u = t_d = 1\ \text{s}\)</span>, the harmonic signals from <a class="reference internal" href="#fig-input-signal-2hz"><span class="std std-ref">above</span></a> are
corrected to have adequate acceleration and deceleration:</p>
<div class="float docutils container" id="fig-post-processing-signals">
<figure class="align-default" id="fig-input-post-signal-2hz" style="width: 60%">
<img alt="_images/fixed_harmonic_post_rampup_2Hz_10s-crop.png" src="_images/fixed_harmonic_post_rampup_2Hz_10s-crop.png" />
</figure>
<figure class="align-default" id="fig-input-post-signal-fibonacci" style="width: 60%">
<img alt="_images/fixed_harmonic_post_rampup_FibonacciHz_10s-crop.png" src="_images/fixed_harmonic_post_rampup_FibonacciHz_10s-crop.png" />
</figure>
</div>
<p>Second, the position for the first time instance must always be zero,
whereas it may not be a priori the case for the stochastic signal. This
is achieved by zero padding for all signals, adding a zero position for
the time instance <span class="math notranslate nohighlight">\(t=0\)</span>. Only one time instant is added and not an
acceleration ramp because the frequency content of the signal has to be
preserved. It results in <span class="math notranslate nohighlight">\(\hat{x}_S(t) = [0, x_S(t)]\)</span> and the time
discretization is adjusted to <span class="math notranslate nohighlight">\(t \in [0,T+\Delta t]\)</span>:</p>
<figure class="align-default" id="fig-shinozuka-signal-zero-padding">
<img alt="_images/23-12_shnzk_signal_comparison_filter_unfilter-crop.png" src="_images/23-12_shnzk_signal_comparison_filter_unfilter-crop.png" />
<figcaption>
<p><span class="caption-text">Comparison of an unmodified SRM signal <span class="math notranslate nohighlight">\(x_S(t)\)</span> and the zero-padded signal <span class="math notranslate nohighlight">\(\hat{x}_S(t)\)</span>.</span><a class="headerlink" href="#fig-shinozuka-signal-zero-padding" title="Link to this image">#</a></p>
</figcaption>
</figure>
<p>Other post-processing procedures, such as different window-functions for
the ramp up and down are of course possible due to the open source
nature of OpvenVIBE, but these are currently not implemented.</p>
</section>
<section id="periodogram">
<span id="sec-periodogram"></span><h2>Periodogram<a class="headerlink" href="#periodogram" title="Link to this heading">#</a></h2>
<p>The periodogram <a class="footnote-reference brackets" href="#footcite-newland2012introduction" id="id13" role="doc-noteref"><span class="fn-bracket">[</span>8<span class="fn-bracket">]</span></a> is an
estimator to determine PSD functions from stochastic signals. It is
defined as the squared absolute value of the discrete Fourier transform
of a time signal <span class="math notranslate nohighlight">\(X(t)\)</span> as follows</p>
<div class="math notranslate nohighlight" id="eq-periodogram">
<span id="equation-eq-periodogram"></span><span class="eqno">(4)<a class="headerlink" href="#eq-periodogram" title="Link to this equation">#</a></span>\[\hat{S}_X(\omega_k) = \lim \limits_{T \rightarrow \infty} \frac{\Delta t^2}{T} \left|\sum \limits_{t = 0}^{T-1} X(t) e^{-2 \pi i k t / T}\right|^2,\]</div>
<p>where <span class="math notranslate nohighlight">\(T\)</span> is the total number of discretized signal points and
<span class="math notranslate nohighlight">\(\Delta t\)</span> the time increment. The time instant in the record
<span class="math notranslate nohighlight">\(t\)</span> is just an index and <span class="math notranslate nohighlight">\(k\)</span> is the angular frequency
discretization with <span class="math notranslate nohighlight">\(\omega_k = \frac{2 \pi k}{T}\)</span>.</p>
</section>
<section id="references">
<h2>References<a class="headerlink" href="#references" title="Link to this heading">#</a></h2>
<div class="docutils container" id="id14">
<aside class="footnote-list brackets">
<aside class="footnote brackets" id="footcite-chopra2007dynamics" role="doc-footnote">
<span class="label"><span class="fn-bracket">[</span>1<span class="fn-bracket">]</span></span>
<span class="backrefs">(<a role="doc-backlink" href="#id1">1</a>,<a role="doc-backlink" href="#id3">2</a>)</span>
<p>A.K. Chopra. <em>Dynamics of Structures</em>. Pearson, Great Britain, 2007.</p>
</aside>
<aside class="footnote brackets" id="footcite-li2010" role="doc-footnote">
<span class="label"><span class="fn-bracket">[</span>2<span class="fn-bracket">]</span></span>
<span class="backrefs">(<a role="doc-backlink" href="#id2">1</a>,<a role="doc-backlink" href="#id10">2</a>)</span>
<p>Jie Li and Jianbing Chen. <em>Stochastic Dynamics of Structures</em>. John Wiley & Sons, Ltd (Asia), United States, 1 edition, 2010. URL: <a class="reference external" href="https://www.wiley.com/en-us/Stochastic+Dynamics+of+Structures-p-9780470824252">https://www.wiley.com/en-us/Stochastic+Dynamics+of+Structures-p-9780470824252</a>.</p>
</aside>
<aside class="footnote brackets" id="footcite-chen2017" role="doc-footnote">
<span class="label"><span class="fn-bracket">[</span><a role="doc-backlink" href="#id4">3</a><span class="fn-bracket">]</span></span>
<p>Jianbing Chen, Fan Kong, and Yongbo Peng. A stochastic harmonic function representation for non-stationary stochastic processes. <em>Mechanical Systems and Signal Processing</em>, 96:31–44, 2017. URL: <a class="reference external" href="https://www.sciencedirect.com/science/article/pii/S0888327017301784">https://www.sciencedirect.com/science/article/pii/S0888327017301784</a>, <a class="reference external" href="https://doi.org/https://doi.org/10.1016/j.ymssp.2017.03.048">doi:https://doi.org/10.1016/j.ymssp.2017.03.048</a>.</p>
</aside>
<aside class="footnote brackets" id="footcite-lyu2023" role="doc-footnote">
<span class="label"><span class="fn-bracket">[</span><a role="doc-backlink" href="#id5">4</a><span class="fn-bracket">]</span></span>
<p>Meng-Ze Lyu, Jian-Bing Chen, and Jia-Xu Shen. Refined probabilistic response and seismic reliability evaluation of high-rise reinforced concrete structures via physically driven dimension-reduced probability density evolution equation. <em>Acta Mechanica</em>, pages 1619–6937, 2023. URL: <a class="reference external" href="https://www.sciencedirect.com/science/article/pii/S0267726122002858">https://www.sciencedirect.com/science/article/pii/S0267726122002858</a>, <a class="reference external" href="https://doi.org/https://doi.org/10.1007/s00707-023-03666-4">doi:https://doi.org/10.1007/s00707-023-03666-4</a>.</p>
</aside>
<aside class="footnote brackets" id="footcite-liu2022" role="doc-footnote">
<span class="label"><span class="fn-bracket">[</span><a role="doc-backlink" href="#id6">5</a><span class="fn-bracket">]</span></span>
<p>Shuoyu Liu, Liuliu Peng, Jun Liu, Shuang Zhao, and Zhongming Jiang. Spectral representation-based efficient simulation method for fully non-stationary spatially varying ground motions. <em>Soil Dynamics and Earthquake Engineering</em>, 161:107436, 2022. URL: <a class="reference external" href="https://www.sciencedirect.com/science/article/pii/S0267726122002858">https://www.sciencedirect.com/science/article/pii/S0267726122002858</a>, <a class="reference external" href="https://doi.org/https://doi.org/10.1016/j.soildyn.2022.107436">doi:https://doi.org/10.1016/j.soildyn.2022.107436</a>.</p>
</aside>
<aside class="footnote brackets" id="footcite-shinozuka1991simulation" role="doc-footnote">
<span class="label"><span class="fn-bracket">[</span>6<span class="fn-bracket">]</span></span>
<span class="backrefs">(<a role="doc-backlink" href="#id7">1</a>,<a role="doc-backlink" href="#id9">2</a>)</span>
<p>Masanobu Shinozuka and George Deodatis. Simulation of stochastic processes by spectral representation. <em>Applied Mechanics Reviews</em>, 44(4):191–204, 1991. <a class="reference external" href="https://doi.org/10.1115/1.3119501">doi:10.1115/1.3119501</a>.</p>
</aside>
<aside class="footnote brackets" id="footcite-shinozuka1988response" role="doc-footnote">
<span class="label"><span class="fn-bracket">[</span><a role="doc-backlink" href="#id8">7</a><span class="fn-bracket">]</span></span>
<p>M Shinozuka and G Deodatis. Response variability of stochastic finite element systems. <em>Journal of Engineering Mechanics</em>, 114(3):499–519, 1988.</p>
</aside>
<aside class="footnote brackets" id="footcite-newland2012introduction" role="doc-footnote">
<span class="label"><span class="fn-bracket">[</span>8<span class="fn-bracket">]</span></span>
<span class="backrefs">(<a role="doc-backlink" href="#id11">1</a>,<a role="doc-backlink" href="#id13">2</a>)</span>
<p>David Edward Newland. <em>An introduction to random vibrations, spectral & wavelet analysis</em>. Dover Publication, United States, 2012.</p>
</aside>
<aside class="footnote brackets" id="footcite-behrendt2022periodogram" role="doc-footnote">
<span class="label"><span class="fn-bracket">[</span><a role="doc-backlink" href="#id12">9</a><span class="fn-bracket">]</span></span>
<p>Marco Behrendt, Masaru Kitahara, Takeshi Kitahara, Liam Comerford, and Michael Beer. Data-driven reliability assessment of dynamic structures based on power spectrum classification. <em>Engineering Structures</em>, 268:114648, 2022. URL: <a class="reference external" href="https://www.sciencedirect.com/science/article/pii/S0141029622007453">https://www.sciencedirect.com/science/article/pii/S0141029622007453</a>, <a class="reference external" href="https://doi.org/https://doi.org/10.1016/j.engstruct.2022.114648">doi:https://doi.org/10.1016/j.engstruct.2022.114648</a>.</p>
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