You mean other than the assumption of the material behaviour?
The "order" that you speak of typically refers to the order (number of the highest order exponent in a polynomial equation) of the element shape function with respect to the primary variable.
these are types of computer generated analysis used to determine reaction forces of beams, columns, etc. my question is how are they related and how do they differ?
They are fundamentally different. Forget about the order of the element shape function.
The best way to approach your question is to look at how these analyses are used. Elastic analysis is typically used with an ASD (allowable stress design). Non-linear is typically used with LRFD (Load and Resistance Factor Design). Do you understand the difference between these two approaches?
Elastic analyses simply assume that the material of the frame or model being analyzed never exceeds it elastic limit and the modulus of elasticity stays the same throughout the entire analysis.
The non-linear elastic analysis does not assume a consistent E but checks throughout the analysis whether any portion of the model yields and then exhibits a different, lower modulus - adjusts the model to reflect this inelastic state, and then proceeds to continue the analysis through further steps in the iterative analysis.
Both are second order in that they take into account the models deflections and how those deflections affect the response of the model to the applied loads.
