This book is intended as a textbook providing a deliberately simple introduction to finite element methods in a way that should be readily understandable to engineers, both students and practising professionals. Only the very simplest elements are considered, mainly two dimensional three-noded "constant strain triangles", with simple linear variation of the relevant variables. Chapters of the book deal with structural problems beams, classification of a broad range of engineering into harmonic and biharmonic types, finite element analysis of harmonic problems, and finite element analysis of biharmonic problems plane stress and plane strain. Full FORTRAN programs are listed and explained in detail, and a range of practical problems solved in the text. Despite being somewhat unfashionable for general programming purposes, the FORTRAN language remains very widely used in engineering. The programs listed, which were originally developed for use on mainframe computers, have been thoroughly updated for use on desktops and laptops. Unlike the first edition, the new edition has problems with solutions at the end of each chapter.Electronic copies of all the computer programs displayed in the book can be downloaded at: www.worldscientific.com/doi/suppl/10.1142/p847/suppl_file/p847_program.zip.