# Collision Study on New Aluminum Alloy W-Beam Guardrail
## 2 Structure Optimization of W-Beam Guardrail  
### Theoretical Analysis  
W-beam guardrails typically consist of three components, namely, beams, posts, and connectors, with the wave beam serving as the primary load-bearing structure. The performance of the wave beam during vehicle collisions is analogous to the bending deformation of a beam, and its bending stiffness directly affects the extent of deformation the guardrail undergoes upon impact. Therefore, in the derivation of the theoretical formula, the guardrail is assumed to be a simply supported beam model. The partial differential equation of the vertical motion of the beam is introduced to describe the nonlinear behavior of the wave beam during a collision, as shown in Equation (1). \[m\frac{\partial^{2}u}{\partial t^{2}}+EI\frac{\partial^{4}u}{\partial x^{4}}= \delta(t_{0}) \tag{1}\]  
where \(m\) is the mass per unit length of the wave beam, \(EI\) is the flexural rigidity of the wave beam, and \(\delta(t_{0})\) is the Dirac delta function, representing the impact force on the specimen during the collision. The wave beam is connected to the supports in a simply supported manner. Therefore, the boundary conditions can be expressed as follows. \[u(0)=u(l)=0,\ \frac{\partial^{2}u}{\partial x^{2}}\bigg{|}_{x=0}=\left.\frac{ \partial^{2}u}{\partial x^{2}}\right|_{x=l}=0 \tag{2}\]Under Boundary Conditions (2), the solution to Equation (1) can be assumed as follows:  
\[u(x,\ t)=\sum_{i=1}^{\infty}\varphi_{i}(t)\sin\frac{i\tau\text{rx}}{l} \tag{3}\]  
where \(u(x,t)\) represents the deflection at any point on the wave beam, and \(l\) is the span of the wave beam. Substituting Equation (3) into Equation (1), multiplying both sides of the equation by \(\sin\frac{i\eta\text{rx}}{l}(n=1,\ 2,\ 3)\) and integrating over the span \((0,\ l)\), and then applying the [PERSON] method with a three-mode truncation [17], yields the following system of equations:  
\[\left\{ \begin{array}{l}m\ddot{\varphi}_{1}+\frac{\pi^{4}EI}{l}\varphi_{1}= \frac{4}{\pi}\delta(t_{0})\\ m\ddot{\varphi}_{2}+\frac{16\pi^{4}EI}{l}\varphi_{2}=0\\ m\ddot{\varphi}_{3}+\frac{81\pi^{4}EI}{l}\varphi_{3}=\frac{4}{3\pi}\delta(t_{0} )\end{array} \right. \tag{4}\]  
From the above equations, it is evident that the modes are uncoupled. Therefore, Equation (3) can be simplified to:  
\[u(x,\ t)=\sum_{i=1}^{\infty}C_{i}\sin(\omega_{i}t)\sin\frac{i\tau\text{rx}}{l} \tag{5}\]  
where \(\omega_{i}=\sqrt{\frac{EI}{m}}(\frac{i\pi}{l})^{2}\). Based on the model, the following initial conditions are set  
\[\left\{ \begin{array}{l}u(t_{0}^{+})=0\\ \int_{0}^{l}m\frac{\partial u(t_{0}^{+})}{\partial t}=p\end{array} \right. \tag{6}\]  
Here, \(p\) represents the momentum of the colliding object. According to Equation (5), we obtain:  
\[mC_{1}\omega_{1}\frac{2l}{\pi}+mC_{3}\omega_{3}\frac{2l}{3\pi}=p \tag{7}\]  
From Equation (4), we know that \(C_{1}>>C_{3}\), so Equation (7) can consider only the first mode shape, as the collision impact point is located at the mid-span (\(x=\frac{l}{2}\)). The midpoint displacement is not related to even mode shapes and is approximately equal to the amplitude of the first mode shape. Therefore,  
\[C_{1}=\frac{pl}{2\pi\sqrt{mEI}} \tag{8}\]  
According to the theoretical formula, the deformation of the corrugated guardrail is inversely proportional to the square root of the moment of inertia of its cross-section. Although this formula is primarily applicable to the elastic stage, this relationship also holds in the elastoplastic stage. Therefore, using the moment of inertia of the cross-section as the basis for optimization offers a simple yet effective approach to enhancing the guardrail's protective capability. The elastoplastic stage calculations can be referenced from Ref. [18], which developed a comprehensive elastoplastic beam finite element analysis model. During collisions, the guardrail absorbs impact energy through plastic deformation, which is crucial for the guardrail's durability and impact resistance and will be discussed in detail in the article's discussion section.