Paper format. National Exams, December 2018 — 16-Civ-B9 The Finite
Element Method. Three hours; four pages; three problems of equal weight, all to be
attempted. Closed book, one aid sheet written on both sides, approved Casio or Sharp
calculator. Candidates are asked to state any interpretive assumptions with the answer
paper, which matters here because Problem 2 is a contact problem whose answer depends on
one such check.
Reference texts for this subject.
- Logan, D. L., A First Course in the Finite Element Method, 6th ed., Cengage
— beam elements and the Hermite shape functions (Ch. 4), the bilinear rectangle and
plane stress (Ch. 6 and Ch. 10), work-equivalent nodal loads (§4.5).
- Cook, R. D., Malkus, D. S., Plesha, M. E. and Witt, R. J., Concepts and
Applications of Finite Element Analysis, 4th ed., Wiley — completeness,
rigid-body modes and the row-sum property of a stiffness matrix (Ch. 3), consistent load
vectors (Ch. 4).
- Bathe, K.-J., Finite Element Procedures, 2nd ed. — the principle of
virtual work as the origin of the finite element equations (Ch. 4), and the treatment of
prescribed displacements and contact conditions.
- Hibbeler, R. C., Structural Analysis, 10th ed., Pearson — fixed-end
moments, shear and bending moment diagrams and the sign conventions used to plot
them.
Check — misprint on page 3 of the paper. The
examination prints the first Hermite shape function as
$N_1(s) = \dfrac{2s^3}{L^3} - \dfrac{2s^2}{L^2} + 1$. That expression gives
$N_1(L) = 1$ instead of $0$ and makes $N_1 + N_3 = 2$ at the far node, so it cannot be a
beam shape function. The middle coefficient must be $3$:
$N_1(s) = \dfrac{2s^3}{L^3} - \dfrac{3s^2}{L^2} + 1$, which is consistent with the
$N_3(s)$ printed immediately below it and with the $[K]$ matrix given on the same page.
The corrected form is used throughout. Nothing in the numerical answer depends on it,
because the element stiffness matrix is supplied directly.
