CE/ME 327: Finite Element Methods in Mechanics Midterm Exam

October 28, 2017

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CE/ME 327: Finite Element Methods in Mechanics
Midterm Exam: 10/26/2017
Problem 1 (85%)
Given is an one dimensional elastic bar of length ܮ with area ܣଵ from െܮ/2 ൑ ݔ ൏ 0
and area ܣଶ from 0 ൑ ݔ ൏ ܮ/2. The Young’s Modulus of the bar is ܧሺݔሻ. The left end of
the bar is connected to a spring with stiffness ݇, while the right end of the bar is fixed.
The bar is subjected to a temperature field ܶሺݔሻ and a body force ܾሺݔሻ. The strong form
is given by:
݀
݀ݔ ሺߪܣሻ ൅ ܾሺݔሻ ൌ 0
with boundary conditions:
ܮ ൬െ ߪ
2
൰ ൌ ݐ̅ൌ ݇
ଵܣ
ܮ ൬െ ݑ
2
൰ ܽ݊݀ ݑ ൬
ܮ
2
൰ ൌ 0
The stress‐strain relationship is given by:
ߪൌܧሺݔሻ ൬
ݑ݀
݀ݔ െ ߙሺݔሻܶሺݔሻ൰
where ߙ is the coefficient of thermal expansion. Suppose ܣଵ, ܣଶ, ܧሺݔሻ, ߙሺݔሻ, and ܾሺݔሻ
are all known functions. And ݇ is a known constant.

(a)
(b)
Figure 1
The weak form is given by:
ݓ݀ න
ܧܣ ݔ݀
ݑ݀
ݔ݀ ݔ݀
௅/ଶ
ି௅/ଶ
ൌ න ݀ݓ
ݔ݀ܶߙܧܣ ݔ݀
௅/ଶ
ି௅/ଶ
ଶ ௅/ݔܾ݀ݓ න൅
ି௅/ଶ
െ ሺݐܣݓ̅
ሻ௫ୀି௅/ଶ
∀ݓሺݔሻ with ݓ ൬
ܮ
2
൰ ൌ 0
For parts (a) ‐ (c) only substitute into the integrand terms. Do not evaluate the
integrals, and consider a mesh of a single element (not shown).
a) Using the variables, ࡷ ,࡯݂ ,ஐ
ሬሬሬሬԦ, ݂୻
ሬሬሬԦ, ݀Ԧ, ݎ Ԧand ܶሬԦ write down the matrix equation you
would need to solve if both ܶሬԦ and ݀Ԧ were assumed to be unknown. (8 points)
b) Discretize the weak form using general expressions of the element shape function
ܰሬԦ௘, its derivative ܤሬԦ௘, and any other functions/constants. (10 points)
c) Using the results from part b), define all the variables in the matrix equation that
was formulated in part a). [Hint: there will be a self‐defined term that arises from
the equilibrium of the bar]. (7 points)
For parts (d)‐(g), consider the mesh of two linear elements (figure 1(b)), and assume
all the following values are given:
ܮ ൌ 4, ܣଵ ൌ 2, ܣଶ ൌ 1, ܾሺݔሻ ൌ 2, ܧሺݔሻ ൌ 3, ߙሺݔሻ ൌ 2
ܶሬԦ ൌ ሾ1 0 െ 1ሿ்
d) Construct shape functions and its derivatives for elements 1 and 2. (15 points)
e) Evaluate the element matrices, ࢋࡷ ,for each linear element and assemble the global
matrix, ࡷ) .15 points)
f) Similarly evaluate the element vectors ݂Ԧ
ஐ
௘ and ݂Ԧ
୻
௘ and for each linear element and
assemble the global vectors ݂Ԧ
ஐ and ݂Ԧ
୻. [Hint: ܶሬԦ is no longer unknown] (15 points)
g) Solve for the nodal values of vectors ݀Ԧ and ݎ Ԧas functions of ݇. [Hint: One way to
double check yourself, is to see how your solutions are affected when ݇→∞] (10
points)
h) (Extra Credit) Derivate the weak form from the strong form. (up to 10 bonus points)
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Problem 2. (15%)
Drawn below is a one‐dimensional cubic element. Construct the shape functions for
nodes 2 and 4 ሺ݈ ൌ 2ሻ.
Figure 2
ݔ ൌ െ݈ ݔൌ0 ݔ ൌ 2݈ ݔ ൌ 5݈
1 2 3 4