Nonlinear vibration of beams and plates resting on elastic foundations having nonlinear stiffness properties - Feulefack, Songong Edwige (Department of Civil Engineering, 2024)
Beam elements are used to model components in which one dimension (the length) is significantly greater than the other two dimensions. On the other hand, a plate is a structural element char- acterised by its thickness being very small compared with the other two dimensions. Beams and plates find civil and mechanical engineering applications and are mostly used to construct build- ings, bridges, floors, pavements, and runways. Investigating the vibration of beams and plates is an ongoing topic, and researchers are becoming more interested in it. Because the structure foundation is the most important part of the building process, engineers and designers dedicate more attention to that part of the structure. This contributes to the desire to build strong, safe, and economic structures for a sustainable environment. Beams and plates on elastic foundations (BPoEFs) also find many applications in civil engi- neering. They are frequently used to design structural members of buildings, railroads, airports, highways, and railway tracks. For example, when studying railway track behaviour, a beam on an elastic foundation (BoEF) theory is often employed, and the study can be extended to track dynamics, noise, and vibration. Winkler (1867) was the first to introduce the topic of BPoEFs and developed these to analyse railroad tracks. Since then, many other researchers have extended the concept, giving rise to other foundation types. The foundation can be linear or nonlinear, depending on the purpose for which the structure will be used. For example, to model the interaction between the beam or plate and an elastic foundation, the entire beam or plate and the foundation are modelled without modelling the foun- dation itself. Then, the modelling can become easy or difficult depending on the foundation type and property (linear or nonlinear). Therefore, linear analysis is appropriate for easier cases, while nonlinear analysis is left for other complex cases. Nevertheless, it is important to note that linear analysis is limited to simple structures and is only valid for design. As a result, nonlinear analysis is recommended because it can be used for design and realisation purposes. Beams and plates on linear elastic foundations (BPoLEFs) have been extensively studied, whereas beams and plates resting on nonlinear elastic foundations (BPoNEFs) have been ne- glected and constitute the interest of the current thesis. A beam simply supported at both ends and a rectangular plate also simply supported are considered. Linear and nonlinear foundations are considered to capture the beam and plate behaviour and response. The foundation is assumed to be an assembly of discrete linear or nonlinear springs or connectors. First, the beam on a linear foundation is studied as the basis and then extended to the beam on a nonlinear foundation. Three cases of nonlinearities are discussed: quadratic, cubic, and the combination of quadratic and cubic. The straightforward expansion method (SEM) is used to find solutions to the governing nonlinear differential equation. Finally, the natural frequencies and the corresponding mode shapes of the system beam-foundation are derived. We note that the overall thesis only focuses on free vibration analyses. A rectangular plate on an elastic foundation (PoEF) is studied to extend the research on beams. Here also, linear and nonlinear foundations are considered. The same analyses as those of the beam are carried out on the plate to cover all structure types. Both studies on BoEFs and PoEFs revealed that the natural frequencies of the system increase with the increase of the nonlinear stiffness parameter of the foundation. Interestingly, the same increase in the nonlinear stiffness of the foundation tends to decrease the beam and the plate temporal vibration amplitude. Comparing linear to nonlinear results reveals that neglecting the nonlinearity of the foundation produces differences which are more significant at lower modes. This confirms the fact that, if the model of a beam or a plate resting on a nonlinear elastic foundation is simplified to a linear one, the results obtained from the linear analysis might not accurately represent the real system. Finite Element Analysis (FEA) in Abaqus is used for the numerical analysis. Here also, the natural frequencies and corresponding mode shapes of the beam and the plate with foundation are derived. Extensive parametric studies conducted on the beam and plate vibration reveal that the foundation's nonlinear stiffness is indispensable because it lowers the beam and plate vibration amplitude. Having validated the results from Abaqus, they are used as a benchmark to validate the analytical results. Satisfactory agreement is found between the results of the two methods, with details of each method shown. The results from this thesis show that the nonlinear foundation can accurately be used to control the vibrations of the beam and plate model. The findings further reveal that adding nonlinear quadratic or cubic or mixed parity stiffness properties in the foundation when studying BPoEFs is beneficial and should be adapted for design and realisation purposes.