The main goal of learning finite element should be to solve engineering problems /continuum mechanics numerically by providing a sound methodological basis of the finite element method. In addition to numerical analysis of trusses, beams and plates, the spectrum of possible applications includes thermal problems, biomechanics, analyses of transport processes such as heat conduction and pollutant transport.
In the following a list of my favourite books to gain insights of finite element methods (FEM):
I remember my FEA course at Ruhr university of Bochum, one of the most challenging by Prof Meschke.
A picture is worth a thousand words: On the right side you can see me attending the fundamental course in vehicle dynamics by Thomas Gillespie itself. My journey on treating multibody codes started nearly in 2008 when i joined suspension department of MSIL. A notes on NPTEL course can be seen below. In these notes, i have tried my best to capture some of the physical concepts associated with the subject.
I especially recommend these books for beginners: i) A First Course in Finite Elements - Fish & Belytschko ii) A First Course in the Finite Element Method by Logan & iii) Fundamentals of Finite Element Analysis by Hutton. These books are primarily intended for undergraduate / graduate-level. Very easy to follow
Intermediate:
The last three books according to me come to intermediate level. These books cover fundamental theory and worked out examples in detail. The readers can apply the knowledge to their own engineering problems and understand more advanced applications.
Advanced:
In this category , I recommend The Finite Element Method: Linear Static and Dynamic Finite Element Analysis Book by Thomas J.R. Hughes. Readers who are strong in mathematics will enjoy reading it.