# Study on Stability of Elastic Compression Bending Bar in Viscoelastic Medium
## Appendix B  
Here is a detail introduction of Equation (9) based on basic theory from Mechanics of Structure [47]: Under the action of the critical force, the bar body maintains a state of limit equilibrium within the plane Cartesian coordinate system shown in Figure 5. Assuming that the deflection curve of the rod element, represented by \(w(x)\), undergoes slight bending in the critical state, the strain energy of the system in this state can be expressed by the following Equation (12):  
\[U= \frac{EI}{2}\int_{0}^{L}(w^{\prime\prime})^{2}dx+\frac{1}{2}\int_{0}^{L}b_{ 1}kw^{2}dx \tag{12}\]  
where: \(q(x)\) is passive load on elastic bar caused by constant load and soil self-weight  
\(b_{1}\) is calculated width of the elastic bar  
\(L\) is length of bar  
\(L_{p}\) is width of constant load  
\(p\) is magnitude of constant load  
\(d_{p}\) is distance from the bar to constant load When the critical force \(P_{cr}\) is applied to the end of the rod, it induces a slight displacement \(\Delta x\) in the vertical direction at the rod's end. The work performed by this displacement can be expressed as follows:  
\[U_{p}=-P_{cr}\Delta x \tag{10}\]  
where: \(P_{cr}\) is critical force  
\(\Delta x\) is a small displacement caused by constant load \(p\)  
Based on the energy criterion, when the rod element is in a critical state as \(U+U_{p}=0\), then:  
\[\frac{EI}{2}\int_{0}^{L}{(w^{\prime\prime})^{2}dx}+\frac{b_{1}}{2}\int_{0}^{L} kw^{2}dx-P_{cr}\Delta x=0 \tag{11}\]  
where: \(\Pi_{p}\) is total potential energy of both the bar and the surrounding soil  
\(E\) is elastic modulus of elastic bar  
\(I\) is moment of inertia of the cross section of elastic bar  
\(k\) is reaction force coefficient of elastic medium  
\(w\) is deflection of elastic bar as \(w(x)\)  
Then generalized function \(R(w)\) is Solved from Equation (11) as:  
\[R(w)=\frac{EI\int_{0}^{L}{(w^{\prime\prime})^{2}dx}+b_{1}\int_{0}^{L}{kw^{2}dx}} {2\Delta x} \tag{12}\]  
In Equation (12):  
\[\Delta x=\int_{0}^{L}{\sqrt{1+(w^{\prime})^{2}}dx}-L\approx\frac{1}{2}\int_{0 }^{L}{\big{(}w^{\prime}\big{)}^{2}dx} \tag{13}\]  
Considering boundary conditions in Figure 5 as shown in follow:  
\[x=0,\ w=0,\ w^{\prime}=0;\ x=L,\ w^{\prime\prime}=0 \tag{14}\]  
The deflection curve of the rod element in a state of slight bending can be approximated using a trial function in the form of a triangular series by Laplace transform as shown in Equation (13):  
\[w(x)=\sum_{n=1}^{i}w_{n}\bigg{(}1-\cos\frac{(2n-1)\pi x}{2L}\bigg{)} \tag{15}\]  
Take Equations (14) and (15) into Equation (13), the functional \(R(w)\) could be expressed as:  
\[R(w)=\frac{A(w_{1},w_{2},\ldots,w_{i})}{B(w_{1},w_{2},\ldots,w_{i})} \tag{16}\]  
In Equation (16):  
\[ \begin{array}{ll}A(w_{1},w_{2},\ldots,w_{i})=&EI\int_{0}^{L}{\bigg{(}\sum _{n=1}^{i}w_{n}\Big{(}\frac{(2n-1)\pi}{2L}\Big{)}^{2}\cos\frac{(2n-1)\pi x}{2 L}\bigg{)}^{2}dx}\\ &+b_{1}\int_{0}^{L}{\bigg{(}\sum_{n=1}^{i}w_{n}\Big{(}1-\cos\frac{(2n-1)\pi x}{2 L}\Big{)}\bigg{)}^{2}dx}\end{array}  \tag{17}\]  
\[ \begin{array}{ll}B(w_{1},w_{2},\ldots,w_{i})=\int_{0}^{L}{\bigg{(}\sum_{n=1 }^{i}w_{n}\bigg{(}\frac{(2n-1)\pi}{2L}\bigg{)}\sin\frac{(2n-1)\pi x}{2L}\bigg{)} ^{2}dx}\end{array}  \tag{18}\]  
when the minimum \(R(w_{1})\) is taken, \(P_{cr}\) will be equal to \(R(w)\), which means:  
\[\frac{\partial R}{\partial w_{r}}=\frac{1}{B}\frac{\partial A}{\partial w_{r}} -\frac{A}{B^{2}}\frac{\partial B}{\partial w_{r}}=0,r=1,2,3,\ldots,i \tag{19}\]In Equation (A12):  
\[ \begin{array}{ll}\frac{\partial A}{\partial w_{r}}=&2\sum\limits_{n=1}^{i}w_{n} \int_{0}^{L}\left(EI\frac{(2n-1)^{2}(2r-1)^{2}\pi^{4}}{16L^{4}}\cos\frac{(2n-1) \pi x}{2L}\cos\frac{(2r-1)\pi x}{2L}\right.\\ &\left.+b_{1}k\Big{(}1-\cos\frac{(2n-1)\pi x}{2L}\Big{)}\Big{(}1-\cos\frac{(2r- 1)\pi x}{2L}\Big{)}\Big{)}dx\\ \end{array}