## Appendix A  
An approach, called the method of work equivalent load, is proposed to treat the locally distributed load shown in Figure 1. The method is similar to the one presented in [16] (pp. 98-99, 125-127) but the derivation is more rigorous. Multiplying both sides of Equation (6) by \(w^{c}(x)/2\) and then integrating the resultant equation from \(-L/2\) to \(L/2\) yield  
\[ \begin{split}&\frac{1}{2}\int_{-L/2}^{L/2}\left[\frac{f_{1}^{3}}{12}E_ {1}^{t}w_{xxxx}^{c}-\frac{f_{1}Q_{66}}{2}\left(\frac{\partial^{2}w}{\partial x ^{2}}+\frac{\partial^{2}u}{\partial x\partial z}\right)_{z=c}+\left(Q_{13} \frac{\partial u}{\partial x}+Q_{33}\frac{\partial w}{\partial z}\right)_{z=c }\right]w^{c}dx\\ &\quad=\frac{1}{2}\int_{-L/2}^{L/2}q(x)w^{c}dx=\frac{1}{2}\int_{ -L_{p}/2}^{L_{p}/2}q(x)w^{c}dx=W_{p}\end{split}  \tag{10}\]  
The major difference between present method to the one in [16] (pp. 98-99, 125-127) is that \(w^{c}(x)/2\) is multiplied to Equation (6). This makes the derivations of the work equivalent load more rigorous and clearer. Equation (10) is then numerically integrated. For solution accuracy considerations, Gauss quadrature is used. After doing so, one obtains  
\[ \begin{split}\sum\limits_{j=2}^{N-1}\frac{LH_{j}}{2}\left\{\frac{ f_{1}^{3}}{12}E_{1}^{t}w_{xxxx}^{c}-\frac{f_{1}Q_{66}}{2}\left(\frac{\partial^{2}w}{ \partial x^{2}}+\frac{\partial^{2}u}{\partial x\partial z}\right)_{z=c}+\left( Q_{13}\frac{\partial u}{\partial x}+Q_{33}\frac{\partial w}{\partial z}\right)_{z=c }\right\}_{x_{j}}w_{j}^{c}=\sum\limits_{j=2}^{N-1}P_{j}w_{j}^{c}\end{split}  \tag{11}\]  
where \(x_{j}\) is the Gauss quadrature point and \(P_{j}\) is the work equivalent load applied at \(x_{j}\). From Equation (11) one has  
\[ \begin{split}\left\{\frac{f_{1}^{3}}{12}E_{1}^{t}w_{xxxx}^{c}- \frac{f_{1}Q_{66}}{2}\left(\frac{\partial^{2}w}{\partial x^{2}}+\frac{ \partial^{2}u}{\partial x\partial z}\right)_{z=c}+\left(Q_{13}\frac{\partial u }{\partial x}+Q_{33}\frac{\partial w}{\partial z}\right)_{z=c}\right\}_{x_{j} }=\frac{2P_{j}}{H_{j}L}\\ (j=2,3,...,N-1)\end{split}  \tag{12}\]  
For any type of load \(q(x)\), whether it is continuous or discontinuous distributed or even concentrated loads, the work equivalent load \(P_{j}\) can be obtained by [16] (pp. 98-99, 125-127)  
\[ \begin{split} W_{p}&=\frac{1}{2}\sum\limits_{j=2}^{ N-1}P_{j}w_{j}^{c}=\frac{1}{2}\int_{-L_{p}/2}^{L_{p}/2}q(x)w^{c}(x)dx=\frac{1}{2} \sum\limits_{j=2}^{N-1}\frac{L_{p}w_{j}^{c}}{2}\int_{-1}^{1}q(x)l_{j}(x)d\xi,\\ &=\frac{1}{2}\sum\limits_{j=2}^{N-1}\frac{L_{p}}{2}\left(\sum \limits_{k=2}^{N-1}q(L_{p}\xi_{k}/2)l_{j}(L_{p}\xi_{k}/2)H_{k}\right)w_{j}^{c} \end{split}  \tag{13}\]  
where \(\xi_{k}\) and \(H_{k}\) (\(k=2,3,...,N-1\)) are the abscissas and corresponding weights in Gauss quadrature, and \(w_{j}^{c}\) is the displacement component in \(z\) direction at point \(x_{j}\). Therefore, \(P_{j}\) is given by  
\[P_{j}=\frac{L_{p}}{2}\sum\limits_{k=2}^{N-1}q(L_{p}\xi_{k}/2)l_{j}(L_{p}\xi_{k }/2)H_{k}(j=2,3,...,N-1) \tag{14}\]  
where \(l_{j}(L_{p}\xi_{k}/2)\) is calculated by  
\[l_{j}(L_{p}\xi_{k}/2)=\prod\limits_{\begin{subarray}{c}i=2\\ i\
eq j\since \(L_{P}=L\) and \(l_{j}(L_{P}\xi_{k}/2)=l_{j}(L\xi_{k}/2)=\delta_{kj}\), where \(\delta_{kj}\) is the Kronecker symbol. The correctness of the equivalent loads is thus proved.  
For the case considered in this paper, \(q(x)=P/L_{P}\) and thus the equivalent point load \(P_{j}\) becomes  
\[P_{j}=\frac{P}{2}\sum\limits_{k=2}^{N-1}l_{j}(L_{P}\xi_{k}/2)H_{k}(j=2,3,...,N-1) \tag{10}\]  
Therefore, Equation (11) can be rewritten as  
\[\left\{\frac{f_{1}^{3}}{12}E_{1}^{l}w_{xxxx}^{c}-\frac{f_{1}Q_{66}}{2}\left( \frac{\partial^{2}w}{\partial x^{2}}+\frac{\partial^{2}u}{\partial x\partial z }\right)_{z=c}+\left(Q_{13}\frac{\partial u}{\partial x}+Q_{33}\frac{\partial w }{\partial z}\right)_{z=c}\right\}_{x_{j}}=\frac{P\sum\limits_{k=2}^{N-1}l_{ j}(L_{P}\xi_{k}/2)H_{k}}{H_{j}L} \tag{11}\]  
The term on the right hand side of Equation (11) is the load \(q_{Ml}\) in Equation (20) with \(j\) replaced by \(l\).