Difference between Linear and Cubic Deformation in Bending?

[olski1]

I do not really understand this as you can probably tell, but What is the difference bewteen Linear and Cubic displacement deformation functions in beam bending? For example why would one be used over another to model or calculate beam bending? or how does either type effect what bending behaviour will occur?

All my notes just state we will assume linear or cubic for each example, so is it just the intrinsic properties of the material that determine the deformation function. For example is steel more likely to have a cubic relationship and wood a linear one?

 

[Navin A S]

The difference between linear and cubic displacement deformation functions in beam bending has to do with the shape of the function itself and the resulting bending behavior. In a linear displacement deformation function, the amount of displacement increases linearly with the applied force. This means that the displacement is proportional to the applied force, and the shape of the function will be a straight line. With a cubic displacement deformation function, the amount of displacement increases at a higher rate than linear displacement. This means that the displacement is proportional to the cube of the applied force, and the shape of the function will be a curved line. The choice of which deformation function to use in beam bending depends on the material properties of the beam. Steel, for example, is known to have a higher stiffness than wood, so it would typically require a cubic displacement deformation function. Wood, on the other hand, is usually softer and would require a linear displacement deformation function. The type of deformation function used will affect the bending behavior of the beam, with a cubic deformation function resulting in more pronounced bending behavior.