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Homework answers / question archive / MNE 490/591- FEM for Additive Manufacturing Fall 2020 Homework 5 (100 Points) Due: Nov

MNE 490/591- FEM for Additive Manufacturing Fall 2020 Homework 5 (100 Points) Due: Nov

Mechanical Engineering

MNE 490/591- FEM for Additive Manufacturing Fall 2020
Homework 5
(100 Points) Due: Nov. 17 at 3:30pm
Problem 1 (20 Points)
Consider a five-node element in one dimension. The element length is 4, with node | at x = 2, and the
remaining nodes are equally spaced along the x-axis.
a. Construct the shape functions for the element.
b. The temperatures at the nodes are given by 7; = 3°C. T2 = 1°C.7T;3 =0°C.Ty = —1°C. Ts = 2°C.
Find the temperature field at x = 3.5 using shape functions constructed 1n (a).

Problem 2 (30 Points)
Considera four-node cubic elementin one dimension. The element lengthis 3 withx, = —1;the remaining
nodes are equally spaced.
a. Construct the element shape functions.
b. Find the displacement field in the element when

ly l

a = |"? | = 103] 9].

U3 l

lly 4
c. Evaluate the B® matrix and find the strain for the above displacement field.
d. Plot the displacement u(x) and strain ¢(x).
e. Find the strain field when the nodal displacements are dé’ = [1 1 1 1]. Why is this result

expected?

Note: strain e(x) = = = — d° = B“d’, where N* are element shape functions.

Problem 3 (50 Points)
Consider an element shown in Figure 5.26 with a quadratic displacement field u(x) = ay + dox + 43x.
a. Express the displacement field in terms of the nodal displacements 1. to. 3. (Hint: Use the
Lagrangian interpolants and the local coordinate € to derive shape functions. )
b. Fora linear body force field b(€) = b,(1/2)(1 —&) +. b3(1/2)(1 + €) determine the external
body force matrix.
l Lae ae
c. Develop the B® matrix such that ¢ = — = Bed’, d“! = (uy). wh. up).
d. Determine the element stiffness * matrix K,= _ [ BOlTES ASB? dQ.
O,
Note: the next two questions are independent of the above questions and have a different body force
field using the global coordinate x:
og . d-u
e. Use one three-node quadratic displacement element to solve by finite elements Ea —b(x) = —cx,
u(—L/2) = u(L/2) = 0. “O
f. Compare the FEM results to the exact solution for w(x). At what locations they
give the same results? Discuss and justify your understandings. (Hint: Use the
integration by twice to get the exact solution.)
A
€=2x/L
| 2 3
XL¢
y= 0 °
+— L/2 —>}+— L/2 ——+|
Figure 5.26 A single quadratic element for Problem 3.

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