## 2 PB-DA Design and Fabrication  
### Derivation of Similarity Criteria  
This section derives the similarity criteria for a simply supported beam proportional model under concentrated load based on dimensional analysis. The similarity criteria are established to describe the bending state of a simply supported beam bridge under concentrated load, as shown in Equation (1):  
\[\pi=\sigma^{a}P^{b}M^{c}l^{d}W^{e} \tag{1}\]  
where \(\sigma\) represents the normal stress of the bridge's cross-section, \(l\) is the span of the bridge, \(M\) is the bending moment of the bridge, \(P\) is the load applied on the bridge, and \(W\) is the section modulus of the bridge against bending. The fundamental dimensions corresponding to these physical quantities are shown in Table 2. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Year** & **Author (s)** & **Mineral** & **Admixture and** & **Main Conclusions** \\  & **Admixtures** & **Connecting Strength** & & **Conduction** \\ \hline  & & & & For flexural fatigue behavior, \\
2006 & [PERSON] et al. [26] & Fly ash & C40: 29\% & C80 HPC bridges are better than C40 HPC bridges, and HPC bridges outperform ordinary concrete beams. \\ \hline  & & & C50: 0–40\% & Properly incorporating fly ash \\
2007 & [PERSON] et al. [27] & Fly ash & C60: 25\% & does not adversely affect the fatigue behavior of bridges. \\ \hline  & & & C50: 0–40\% & An admixture of 20\% to 40\% \\
2007 & [PERSON] et al. [28] & Fly ash & C60: 25\% & fly ash in prestressed simply supported beam bridges for \\  & & & C70: 25\% & railways is feasible. \\ \hline  & & & C50: 0–40\% & Prestressed concrete beam bridges with a 25\% fly ash \\
2008 & [PERSON] et al. [29] & Fly ash & C60: 25\% & admixture exhibit good fatigue behavior and are suitable for \\  & & & C80: 25\% & use in railway bridges. \\ \hline  & & & & After fatigue loading, HPC bridges maintain good \\
2010 & [PERSON] et al. [30] & Fly ash & C60: 17\% & mechanical properties. \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Scaled-model experiments for HPC bridge. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Physical Quantities** & **Fundamental Scale** \\ \hline \(\sigma\) & \([F][L]^{-2}\) \\ \(P\) & \([F]\) \\ \(M\) & \([F][L]\) \\ \(l\) & \([L]\) \\ \(W\) & \([L]^{3}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Fundamental scales of physical quantities. Through substituting the fundamental dimensions from Table 2 into Equation (1), the dimensional equation can be obtained, as shown in Equation (2). \[[\pi]=\left[L^{0}F^{0}\right]=\left[L^{-2}F\right]^{a}[F]^{b}[LF]^{c}[L]^{d}\left[ L^{3}\right]^{e} \tag{2}\]  
Based on the principle of dimensional homogeneity, the following set of equations can be derived from Equation (3):  
\[\left\{ \begin{array}{l}-2a+c+d+3e=0\\ a+b+c=0\end{array} \right. \tag{3}\]  
Equation (3) represents two equations with five unknowns. Thus, the unknowns cannot be directly solved. Therefore, in assuming \(a\), \(b\), and \(d\) as known values, \(c\) and \(e\) are determined as follows:  
\[\left\{ \begin{array}{l}c=-a-b\\ e=a+\frac{1}{3}b-\frac{1}{3}d\end{array} \right. \tag{4}\]  
In dimensional analysis, \(a\), \(b\), and \(d\) can be assumed as three sets of numbers, allowing for the solution of the five unknowns using Equation (4). Although different assumed values lead to different results, all conform to the similarity criteria. For an easy solution, \(a\), \(b\), and \(d\) are assumed to be the following three sets of numbers:  
\[\left\{ \begin{array}{l}a=1,b=0,d=0\\ a=0,b=1,d=0\\ a=0,b=0,d=1\end{array} \right. \tag{5}\]  
Substituting each of the three sets of numbers in Equation (5) into Equation (4) and the results obtained into Equation (1), the following three similarity criteria can be obtained:  
\[\left\{ \begin{array}{l}\pi_{1}=\sigma M^{-1}W\\ \pi_{2}=PM^{-1}W^{\frac{1}{3}}\\ \pi_{3}=LW^{-\frac{1}{3}}\end{array} \right. \tag{6}\]  
Based on the similarity criteria of Equation (6), the similarity indices can be obtained as  
\[\left\{