## 4 HSI Formulation  
A fourth-order partial differential equation (PDE) describes the deflection \(u(x,t)\) at a point on the \(x\) axis of the beam at a given instant \(t\) (Equation (8)), where \(p(x,t)\) represents the load generated by a pedestrian traversing the structure, \(M_{br}\) represents the mass of the structure, and \(c\) represents the coefficient of the viscous damper. \[\frac{\partial^{2}}{\partial x^{2}}\bigg{[}EI(x)\frac{\partial^{2}u(x,t)}{ \partial x^{2}}\bigg{]}+m(x)\frac{\partial^{2}u(x,t)}{\partial x^{2}}+c(x) \frac{\partial u(x,t)}{\partial t}=p(x,t) \tag{8}\]  
Equation (8) was solved using the separation-of-variables method, which incorporates modal decomposition, as shown in Equation (9). In this context, the vibration modes are described by sinusoidal functions, as expressed in Equation (7). Equation (10) describes the motion (\(Y_{h}\)) of the \(\dot{n}th\) vibration mode (\(\phi_{h}\)) in modal coordinates, where the natural frequency (\(\omega_{h}=2\pi f_{h}\)) and damping ratio (\(\zeta\)) are defined using the parameters listed in Table 4. The mass (\(M_{h}\)) and modal loads (\(P_{h}\)) were determined using Equations (11) and (12), respectively. \[u(x,t)=\sum_{h=1}^{\infty}\phi_{h}(x)Y_{h}(t) \tag{9}\]  
\[\tilde{Y}_{h}(t)+2\xi\omega_{h}\tilde{Y}_{h}(t)+{\omega_{h}}^{2}Y_{h}(t)=\frac {P_{h}(t)}{M_{h}} \tag{10}\]  
\[M_{h}=\int_{0}^{L_{br}}[\phi_{h}(x)]^{2}m(x)dx \tag{11}\]  
\[P_{h}(t)=\int_{0}^{L_{br}}\phi_{h}(x)p(x,t)dx \tag{12}\]  
Drawing from the equations that detail the kinematics and kinetics of human gait, as outlined in Section 2, the human body and the structure interact and hence are dynamically connected. This connection was established by incorporating the force generated by walking as a function of time and position in the human system, as shown in Equation (13). The transition of the left and right foot forces, \(D(t)\), is defined as a function that is normalized to a range of 0 to 1. In single-stance zones, where only one foot is in contact with the ground, function D(t) is equal to 1 during the Begin (BGN) to Heel Contact Right (HCR) and Toe Off Left (TOL) to Heel Contact Left (HCL) periods. However, in the double-stance zones, specifically in the shaded region between the HCR to TOL and HCL to TOR, \(D(t)\) changes linearly from 0 to 1 and from 1 to 0, respectively. \[p(x,y,t)=\bigg{[}(1-D(t))\sum_{i=1}^{13}m_{i}\Big{(}\vec{a}_{i}(t)\ -\vec{g}+\vec{A}_{g}(t)\ \Big{)}\bigg{]}\delta(x-x_{L}(t)\ )+\bigg{[}D(t)\ \sum_{i=1}^{13}m_{i}\Big{(}\vec{a}_{i}(t)\ -\vec{g}+\vec{A}_{g}(t)\ \Big{)} \bigg{]}\delta(x-x_{R}(t)\ ) \tag{13}\]  
\[x_{L}(t)\ =x_{L}^{\prime}(t)\ +CoP_{l}(t) \tag{14}\]  
In Equation (13), the left and right feet are represented by \(x_{L}^{\prime}(t)\) and \(x_{R}^{\prime}(t)\), respectively, and the pressure centers for the left and right feet are represented by \(CoP_{l}(t)\) and \(CoP_{r}(t)\), respectively, in Equation (14). The accelerations induced by the structure owing to the human gait are represented by Equation (15), where \(\vec{A}_{g}(t)\) is calculated based on the positions of the feet and the vertical modes of vibration. The applied force of human walking in modal coordinates was obtained by combining Equations (12) and (13), resulting the Equation (16). \[A_{g}(t)=\sum_{h=1}^{3}\left[(1-D(t))sin\left(\frac{\hat{n}\pi x_{L}(t)}{L_{br}} \right)+D(t)sin\left(\frac{\hat{n}\pi x_{R}(t)}{L_{br}}\right)\right]\ddot{Y}_{h }(t) \tag{15}\]  
\[P_{h}(t)=-\bigg{[}F_{L}(t)sin\left(\frac{\hat{n}\pi x_{L}(t)}{L_{br}}\right)+F _{R}(t)sin\left(\frac{\hat{n}\pi x_{R}(t)}{L_{br}}\right)\bigg{]}-\bigg{[}(1-D(t ))m_{p}sin\left(\frac{\hat{n}\pi x_{L}(t)}{L_{br}}\right)+D(t)\,m_{p}sin\left( \frac{\hat{n}\pi x_{R}(t)}{L_{br}}\right)\bigg{]}\ddot{A}_{g}(t) \tag{16}\]  
Table 4.0  
where,  
Table 4.1  
Table 4.2  
Table 4.3  
Here, \(F_{L}(t)\) and \(F_{R}(t)\) represent the forces generated by walking on the left and right feet, respectively, without considering the influence of acceleration on the structure.