Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
National Examinations — December 2013 — 07-Str-B3 Applications of Finite Elements. Three hours, closed book, with two 8.5 in by 11 in pages of handwritten notes and one approved non-communicating calculator. Four pages including the cover; three problems, all to be attempted, all of equal value. Every problem is answered in full below.
Reference texts: Logan, D.L., A First Course in the Finite Element Method (6th ed., Cengage) — the bar element, the Hermite beam element and its work-equivalent load vector, and the bilinear 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) — element strain fields, plane stress versus plane strain, and joining elements with dissimilar degrees of freedom; Chandrupatla, T.R. & Belegundu, A.D., Introduction to Finite Elements in Engineering (4th ed., Pearson) — the stepped-bar treatment of a tapered member; Bathe, K.-J., Finite Element Procedures (2nd ed., Prentice Hall) — convergence and constraint (multi-point) equations; 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 beam stiffness matrix printed on page 4; Hibbeler, R.C., Mechanics of Materials (10th ed., Pearson) — axially loaded members of varying section.
NOTE — sign convention used throughout Problem 1. The paper places the origin at the loaded tip and the built-in end at $x = L$, and states the boundary conditions as $u(L) = 0$ and $EA(x)\,du/dx|_{x=0} = P$. Figure 1 draws $P$ pulling the tip away from the wall, so with $x$ measured toward the wall the bar is in tension and every displacement is negative. Displacements are therefore reported as signed values, with the physical elongation quoted alongside; stresses are reported as positive tensile values. Nothing in the answers depends on this choice, only on its consistency.
