## 3 The Theory of Composite Plate without Holes
### The Transformed Flexural Rigidity \(D_{t}\)  
In 1959, [PERSON] [15] proposed a solution for the transformed flexural rigidity \(D_{t}\) of the perfectly bonded layered plate where each layer is assumed to be isotropic, a plate element with the associated coordinate system is shown in Figure 3, and the calculation formulas of \(D_{t}\) are in Equations (3)-(6).  
\[D_{t}\ =\ C-\frac{B^{2}}{A} \tag{3}\]  
\[A\ =\ \sum_{k}\frac{E_{k}}{1-v_{k}^{2}}t_{k} \tag{4}\]  
\[B\ =\ \sum_{k}\frac{E_{k}}{1-v_{k}^{2}}\frac{t_{k}^{2}-t_{k-1}^{2}}{2} \tag{5}\]  
\[C\ =\ \sum_{k}\frac{E_{k}}{1-v_{k}^{2}}\frac{t_{k}^{3}-t_{k-1}^{3}}{3} \tag{6}\]  
To a pure steel plate, the flexural rigidity of \(D_{0}\) can be calculated by Equations (4)-(6), and the rigidity \(D_{0}\) can be expressed as Equation (7), which is the same with Equation (2), so the method proposed by [PERSON] can also be used to calculate the flexural rigidity of the pure steel plate.  
\[D_{0}=C_{0}-\frac{B_{0}^{2}}{A_{0}}=\frac{E_{s}t_{s}^{3}}{12(1-v_{s}^{2})} \tag{7}\]  
Figure 3: A composite plate with the associated system.  
Considering that the steel and CFRP are perfectly bonded up to the ultimate load [18] and the Young's modulus of the adhesive layers is about 1/10 of those of steel and FRP, the effect of adhesive layers is not taken into account for conservative design in this paper. Therefore, a plate strengthened with FRP can be seen as FRP-steel layered plates. Two reinforcement index \(\alpha\), \(\gamma\) are introduced as given by Equations (8) and (9).  
\[\alpha\ =\ \frac{E_{f}}{1-v_{f}^{2}}/\frac{E_{s}}{1-v_{s}^{2}} \tag{8}\]  
\[\gamma\ =\ \frac{t_{f}}{t_{s}} \tag{9}\]  
Substitution of Equations (8) and (9) into Equations (4)-(6) and then substitution of Equations (4)-(6) into (3) leads to Equation (10), where \(D_{t}\) is the flexural rigidity of plate strengthened with FRP. Therefore, \(D_{t}/D_{0}\) can be expressed as Equation (10), and shown in Figure 4. Equation (10) can be approximate simplified as Equation (11) for calculation convenient, and shown in Figure 5.  
\[D_{t}/D_{0}\ =\ 1+3\alpha\gamma\left(1+\frac{4}{3}\gamma^{2}+2\gamma-\frac{ \alpha\gamma(\gamma+1)^{2}}{1+\alpha\gamma}\right) \tag{10}\]  
\[D_{t}/D_{0}\ =\ \left\{\begin{aligned} & 1+3\alpha\gamma\,,\,\gamma\leq 0.2\\ &\min\left(1+3\alpha\gamma,\,\frac{(\alpha+2)(\gamma+1)^{3}}{3} \right),\,between\\ &\frac{(\alpha+2)(\gamma+1)^{3}}{3},\ 0.3\leq\gamma\leq 1\end{aligned}\right. \tag{11}\]  
From the equations, figures and discussion above, the following can be concluded:  
* The method proposed by [PERSON] to calculate the transformed flexural rigidity of the perfectly bonded layered plate is available to calculate the flexural rigidity of the pure steel plate.
* When \(\gamma\) is in a range of 0 to 0.2 and \(\alpha\) is in a range of 0.8 to 2.4, \(D_{t}/D_{0}\) almost had a linearly related with \(\alpha\) and \(\gamma\). For \(\gamma\) = 0.2, \(D_{t}/D_{0}\) will increase from 1.48 to 2.44 as \(\alpha\) enlarges from 0.8 to 2.4.
* \(D_{t}/D_{0}\) will have a non-linear increase as \(\gamma\) is enlarged from 0.2 to 0.3.
* When \(\gamma\) is in a range of 0.3 to 1 and \(\alpha\) is in a range of 0.8 to 2.4: for a fixed \(\alpha\), \(D_{t}/D_{0}\) will had a cubic curvilinear increase as \(\gamma\) is enlarged from 0.3 to 1.0, and for a fixed \(\gamma\), \(D_{t}/D_{0}\) will have a linear change as \(\alpha\) enlarges from 0.8 to 2.4. When \(\alpha\) = 0.8, \(D_{t}/D_{0}\) will increase from 2.05 to 7.47 as \(\gamma\) enlarges from 0.3 to 1, when \(\alpha\) = 2.4, \(D_{t}/D_{0}\) will increase from 3.22 to 11.73 as \(\gamma\) enlarges from 0.3 to 1.  
Figure 4: Variation of Stiffness enhancing ratio \(D_{t}/D_{0}\) calculated by Equation (10) as a function of \(\gamma\).