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MAE 456 FINITE ELEMENT ANALYSIS

FINAL EXAM Practice Questions

Name: ______________________ You are allowed two sheets of notes.

1. Given: - constant strain triangle element shown

- plane stress formulation

- nodal displacements shown

- E = 200 GPa and ν = 0.27

- sy = 250 MPa (ductile material)

Find: (a) planar strain at center

(b) planar stress at center

(c) principle stresses at center

(d) von Mises stress at center

(e) factor of safety, if the stress from (d)

represents the most extreme situation.

30 6

3

3

2

2

1

1

10

1

1

0

2

0

0

−×

−

=

=

v

u

v

u

v

u

d

u1

v1

v3

v2

u2

u3

(0, 0)

(20, 30)

(30, 10)

(coordinates in mm)

y

x

2

Question 1 cont’d

3

Question 1 cont’d

4

2. Given: - linear quadrilateral element shown

- plane strain formulation

- a = b = 10 mm

- nodal displacements shown

- E = 200 GPa and ν = 0.27

- sy = 250 MPa (ductile material)

Find: (a) displacement at the center

(b) planar strain at center

(c) planar stress at center

(d) principle stresses at center

(e) von Mises stress at center

(f) factor of safety, if the stress from (e)

represents the most extreme situation

Note that for a linear quadrilateral element:

Nd=

v

u

d

=

4321

4321

0000

0000

NNNN

NNNN

ab

ybxaN

ab

ybxaN

ab

ybxaN

ab

ybxaN

4

))((

4

))((

4

))((

4

))((

4

3

2

1

+−=

++=

−+=

−−=

d

NdNdBdε

+−−++−+−−−−−

−++−−−

+−+−−−

=

∂∂

∂∂

∂∂

∂∂

=∂==

)()()()()()()()(

)(0)(0)(0)(0

0)(0)(0)(0)(

4

1

0

0

ybxaybxaybxaybxa

xaxaxaxa

ybybybyb

ab

xy

y

x

30 m 10

0

0

1

1

0

2

0

0

6

4

4

3

3

2

2

1

1

−×

−

=

=

v

u

v

u

v

u

v

u

d

5

Question 2 cont’d

6

Question 2 cont’d

7

3. Sketch the shape function for the edge node of a quadratic quadrilateral.

4. Draw approximate lines of constant stress for σx, if nodal averaging is used and the stresses

of the constant strain triangles are as shown. Draw the lines for σx = 1, 2, 3, …8 kPA.

Assume that the averaged stress is interpolated linearly.

σx = 0

σx = 0 σx = 8 kPa

σx = 0

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5. Identify four problems with the mesh shown.

6. The figure on the right shows the displacement results after inputting the problem on the

left. Has all the information been entered correctly? If not, what is wrong?

P

9

7. State the type of finite element(s) that are best to use when performing the structural

analysis for each of the following situations.

a. A calculator housing under load from being sat on

b. The floor of a house loaded with furniture. The floor has wooden joists (beams)

and plywood flooring.

c. A coffee cup loaded with coffee, where we are interested in the stresses where the

handle joins the cup.

8. What problem is inherent in “stiff” systems? What happens when you try to solve the

global system of equations?

9. Name 4 causes of singular stiffness matrices.

10. What is the difference between “sub-structuring” and “sub-modeling”?

30

10

11. What is the criterion for the solution to converge when the mesh is refined?

12. Draw a picture to show the correct support conditions that should be used in a structural FE

analysis of the following situations:

a. An office chair with rollers, loaded with the weight of one person.

b. One shelf in a bookcase loaded with books.

c. A bookend holding books from falling over.

d. A flagpole cemented into the ground, analyzed using beam elements.

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13. What is the difference between a “mistake” and an “error,” as discussed in class? Name

three types of “errors.”

14. Identify a situation in which a non-linear structural FE analysis should be used.

15. What is the advantage of using isoparametric elements?

16. The figure below shows the von Mises stress contours that are presumed to exist in the part.

Draw a mesh that can be used to accurately analyze this part.

10 MPa

20 MPa

30 MPa

40 MPa

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17. What are the criteria for setting up a patch test? What is the criterion for an element to pass

a patch test?

18. Give the work equivalent nodal loads for the following distributed load.

6 m

1 kN/m

5 kN/m

1

2

3

4

5 y

x

=

=

5

5

5

4

4

4

3

3

3

2

2

2

M

F

F

M

F

F

M

F

F

M

F

F

M

F

F

y

y

y

y

y

x

x

x

x

1

1

1x

R