Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
National Examinations — May 2013 — 07-Str-B3 Applications of the Finite Element Method. Three hours, closed book; one of the two approved calculators (any Casio or Sharp model) and one 8.5 in by 11 in aid sheet written on both sides are permitted. The paper prints four problems, all of equal value, and instructs the candidate to answer only three (3) problems out of the four (4) proposed, the first three appearing in the answer book being the ones marked. Candidates are urged to submit a clear statement of any assumption made where a question admits more than one reading. All four problems are worked below, because this set is intended as a study resource rather than as a single exam sitting.
Reference texts: Logan, D.L., A First Course in the Finite Element Method (6th ed., Cengage) — bar and beam elements, the constant-strain triangle, and the isoparametric quadrilateral, in the same notation this paper uses; Cook, R.D., Malkus, D.S., Plesha, M.E. & Witt, R.J., Concepts and Applications of Finite Element Analysis (4th ed., Wiley) — quadrature, spurious zero-energy (hourglass) modes and element quality; Chandrupatla, T.R. & Belegundu, A.D., Introduction to Finite Elements in Engineering (4th ed., Pearson) — the CST gradient matrix in the beta/alpha form printed on page 4 of this paper; Bathe, K.-J., Finite Element Procedures (2nd ed., Prentice Hall) — isoparametric formulation and numerical integration; Zienkiewicz, O.C. & Taylor, R.L., The Finite Element Method: Its Basis and Fundamentals (7th ed., Butterworth-Heinemann) — general theory; Przemieniecki, J.S., Theory of Matrix Structural Analysis (Dover) — the plane-frame element stiffness matrix printed on page 3; Hibbeler, R.C., Mechanics of Materials (10th ed., Pearson) — statically indeterminate axially loaded members with an initial clearance; Ghali, A., Neville, A.M. & Brown, T.G., Structural Analysis: A Unified Classical and Matrix Approach (7th ed., CRC) — symmetry conditions in closed frames.
Check — the two lengths in Figure 1. Problem 1 dimensions the assembly once, as 10 in, and separately states that the copper rod is 0.005 in longer than the aluminium sleeve. The dimension line runs from the rigid support to the face of the rigid bearing plate, i.e. to the end of the rod, so the rod is taken as 10.000 in long and the sleeve as 9.995 in. Reading it the other way (sleeve 10.000 in, rod 10.005 in) changes every stress below by less than 0.1 per cent, which is far inside the precision of the data; the choice is therefore not load-bearing. Both members are treated as prismatic two-node bar elements sharing one axial degree of freedom at the loaded end.
