# Design and Experimental Analyses of Hybrid Piston Rods Used in Hydraulic Cylinders under Axial Load
## 2 Materials and Methods  
### Analysis Method  
The buckling resistance of the piston rod is the most important parameter in hydraulic cylinders. There are three types of failure when hydraulic rods are subjected to compressive loading, either crushing, buckling or both. Generally, the hydraulic rod with shorter length fails by crushing, while the longer rod fails by buckling, and the intermediate stages usually fail by both (see Table 1) [11]. The shortness and length of rod are classified based on slenderness ratio, which is the ratio between the effective length of rod and the smallest gyration radius. \[\textit{i.e.,}\quad\textit{Senderness radio}\ (S)=\frac{\mathbf{L_{e}}}{\mathbf{K}}= \frac{\textit{Effective length of the rod}}{\textit{Least radius of gyration}}\]  
Buckling failure in longer rods is mainly due to the stiffness of material and geometry and is independent of material strength. The failure usually occurs within the elastic range of the material. [PERSON]'s theory is usually used to predict the buckling load of rods in hydraulic cylinders. It is based on the assumption that a structure is initially straight and the compressive load is applied axially at the center of gravity of the structure [12]. The critical buckling load of a structure is determined by the following equation  
\[\mathbf{P_{cr}}=\frac{\pi^{2}\mathbf{E}\mathbf{I}}{(\mathbf{K}\mathbf{L_{e}})^{2}}\]  
where \(\mathbf{P_{cr}}\) is the critical buckling load, \(\mathbf{E}\) is the elastic modulus of the material, \(\mathbf{I}\) is the moment of inertia resisting the buckling direction, \(\mathbf{K}\) is the effective length factor that depends on type of support the structure is under during loading, and \(\mathbf{L_{e}}\) is the length of the structure. However, yielding considerations cannot be completely ignored since the rod will suffer deflection (due to the above imperfections) that will increase with load, leading to a bending moment that will cause failure before the critical Euler load is reached. It can be assumed that the actual failure is due to bending stress rather than buckling. For any value of slenderness ratio less than 120 (\(\mathbf{L}\mathbf{K}<120\)), the error is too large for the application of [PERSON]'s theory. Therefore, [PERSON] extended [PERSON]'s theory to include the crushing load generated in short columns [13; 14]. The stress that leads to buckling according to [PERSON]'s formula for a fixed-fixed condition is  
[PERSON]'s stress,  
\[\mathbf{\sigma_{e}}=\frac{\mathbf{P_{e}}}{\mathbf{A}}=\frac{4\pi^{2}\mathbf{E}\mathbf{I}}{\mathbf{A} \mathbf{L_{e}^{2}}}\]  
since \(\mathbf{I}=\mathbf{A}\mathbf{K}^{2}\), then  
\[\mathbf{\sigma_{e}}=4\left(\frac{\pi^{2}\mathbf{E}}{(\mathbf{L_{e}}/\mathbf{K})^{2}}\right)\]  
where  
\[\frac{\mathbf{L_{e}}}{\mathbf{K}}=\mathbf{S}\ (\textit{Senderness radio})\]  
then  
\[\mathbf{\sigma_{e}}=4\left(\frac{\pi^{2}\mathbf{E}}{\mathbf{S}^{2}}\right)\] \begin{table}
\begin{tabular}{c c c c} \hline \hline
**S.No** & **Slenderness Ratio** & **Type of Column** & **Mode of Failure** \\ \hline
1 & Between 0 to 40 & Short Column & Crushing \\
2 & Between 40 to 125 & Intermediate Column & Combined crushing \& Buckling \\
3 & Greater than 125 & Long Column & Buckling \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Senderness ratio. This value \(\sigma_{e}\) is compared with yield strength of material to check if yielding occurs before buckling in a structure. \[P_{c}=\sigma_{c}A\]  
where \(\sigma_{c}\) is the crushing stress; then,  
\[\frac{1}{P_{R}}=\frac{1}{P_{E}}+\frac{1}{P_{C}}\]  
where  
\[P_{R}=Rankine-Gordan\text{ buckling load}\]  
\[P_{C}=Euler\text{ buckling load}\]  
Then,  
\[\frac{1}{\sigma}=\frac{1}{\sigma_{e}}+\frac{1}{\sigma_{c}}\]  
In Figure 5 the slenderness ratio is plotted against the buckling stress (\(\sigma_{e}\)) and is shown in the curve ABC. The actual value at failure should be below and within line CBD considering all imperfections that occur during loading and in the structure [15].