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TOTAL : ____________________

University of Illinois at Urbana-Champaign College of Engineering

CEE 570 – Finite Element Methods (in Solid and Structural Mechanics)

Spring Semester 2013

Quiz #1 February 25, 2013

Name: ______SOLUTION__________________ ID#: _____________________

PS.: ALL the pages must be stapled at all times.

PS2.: This is a closed book, closed notes, open minds exam.

Show ALL work on the exam sheets

SCORE:

Problem 1:_______20_______ / 20 points

Problem 2:_______20_______ / 20 points

Problem 3:_______10_______ / 10 points

Problem 4:_______10_______ / 10 points

Problem 5:_______40_______ / 40 points

100 / 100

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Solution)

(a) Gauss

a. Divergence Theorem

dV ds

u u n

b. Numerical Integration (Gauss Quadrature)

1

11

( ),

: Gauss point

: Gauss weigh

n

i i

i

i

i

I d w

w

c. Least Square Method

d. Other contributions: fundamental theorem of algebra, complex numbers, astronomy,

etc.

Problem 1 (20 points total)

Part I (8 points):

(a) The German scientist

shown at the left is Gauss

(prior to the introduction

of the Euro, his photo

used to be on the 10 mark

bill). State the relevance

of his work in the context

of the finite element

method (FEM). Please use

the language of

mathematics when stating

his contributions.

Acknowledgement: This question is similar to the one provided in the practice Quiz. Notice that

Part II of both quizzes are about the DT (Divergence Theorem)!

n

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Problem 1, Part II (12 points):

Use the famous divergence theorem (DT) to prove that (Green’s first identity):

dAvudVvuvu n)(2

where is an arbitrary three dimensional, smooth, bounded domain with boundary and outward

unit normal vector field n, u and v are scalar fields, dV is a differential volume component, dA is a

differential area component, and “ ” denotes the inner product.

Solution)

T

T

, where , , , , , , , , x y z x y zdV dA n n nx y z

φ φ n φ n

T

2 2 2

2 2 2

2

, ,

v v vu v u u u

x y z

u v

v v vu u u

x x x y z z

v v v u v u v u vu

x y z x x y y z z

u v u v

φ

φ

2 ,u v u v dV u v dA

n

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23

12 6

6 4

B

AB

B

ubEI

b bb

k

2

3

12 6 0

6 4 0

0 0 0

B

AB B

C

b uEI

b bb

k

Problem 2, Part I (10 points):

For the plane frame shown below, assume that members are slender and have same EI, and that

axial deformations are negligible in comparison with bending deformations. Let loads and

deformations be confined to the plane of the frame. Write the 3x3 structure stiffness matrix that

operates on the “active” DOFs (degrees of freedom) { } and the corresponding load

vector considering the load P applied at node C, as indicated in the figure below.

Solution)

- Element

P

L

EI

L

EI

L

EI

L

EIL

EI

L

EI

L

EI

L

EIL

EI

L

EI

L

EI

L

EIL

EI

L

EI

L

EI

L

EI

Ke

4626

612612

2646

612612

22

2323

22

2323

Acknowledgment: This problem was

extracted from HW#2, problem 2.5-5.

2 2

3 2 2

4 2

2 4

B

BC

C

a aEI

a a a

k

23

12 6

6 4

B

CD

C

ubEI

b bb

k

2 2

3

2 2

0 0 0

0 4 2

0 2 4

B

zBC B

C

uEI

a ab

a a

k

3

2

12 0 6

0 0 0

6 0 4

B

zCD B

C

b uEI

ab b

k

3 2 2

2

2

24 / 6 / 6 /

6 / 4 / 4 / 2 /

6 / 2 / 4 / 4 /

B

AB BC CD B

C

b b b u

EI b b a a

b a b a

k k k k 0

0

P

P

- Local [ ]

- Global [ ] (all DOFs)

- Load vector

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Problem 2, Part II (10 points):

The spring in the figure below is NONLINEAR and exerts a force proportional to the square of

its stretch. Write the expression for its potential energy and from it determine the equilibrium

value of the displacement .

Solution)

3

0

3

2

3

3

0

/

D

P

Peq

eq

kDU Fdu

PD

kDU PD

dkD P

dD

D P K

u

Pk

F

u

F=ku2

Acknowledgement: This problem was suggested as an OPTIONAL HOMEWORK during the

class of February 6, 2013 – the suggestion can also be found on the file CEE570-ppt10.pdf,

which is available on our class website.

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Problem 3 (10 points):

For the 2D beam element shown below, write the nodal displacement vector {d} that describes a

rigid body rotation of 1800

about node 1. For this vector, verify the product [k]{d}. Is the

resulting vector null or not? Why?

Solution)

The nodal displacement vector {d} T

1 1 1 2 2 2

T

{ } { , , , , , }

{0, 0, 180 , - 2 , 0, 180 )

u v u v

L

d

T

2 2

12 6 12 6{ } 2 , , , 2 , , [null vector]

EI EI EI EIEA EA

L L L L

K d

The resulting vector is not a null vector because rotation of is not small.

1 2

1

2

3 4

5

6 3 2 3 2

2 2

3 2 3 2

2 2

0 0 0 0

12 6 12 60 0

6 4 6 20 0

0 0 0 0

12 6 12 60 0

6 2 6 40 0

EA EA

L L

EI EI EI EI

L L L L

EI EI EI EI

L L L L

EA EA

L L

EI EI EI EI

L L L L

EI EI EI EI

L L L L

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For example, assume rigid body rotation of about node 1.

Displacement vector is

T

1 1 1 2 2 2

T

{ } { , , , , , }

{0, 0, , cos , sin , )

u v u v

L L L

d

Since the characteristic of the rigid body rotation is { } { }K d 0

2 3 2

2

2 3

2

0 0 0

6 12 60

4 6 20

( cos ) sin

0 0

6 120

2 60

EA

L

EI EI EI

L L L

EI EI E

L LL L L

EA

L

EI EI

L L

EI EI

L L

2

0

0

0

00

60

40

I

L

EI

L

EI

L

This work when small (note : sin , cos 1)

𝜃

𝐿 1

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Problem 4 (10 points):

Imagine that a curve φ = φ(x) is to be fitted to three data values: φ1 (actual field quantity) and

φ,x1 (slope of the field quantity) at x=0; and φ2 at x=L. Determine the FEM shape functions.

Hint: you may want to verify the properties of the shape functions to make sure that they are

correct.

Solution)

[ ]{ } [ ] {

}

[ ]{ } [ ] {

}

Evaluate given conditions (let )

{

} [ ] {

} [

] {

}

{

} [ ] { } [

] {

}

Using equation 3.2-3; [ ][ ] { } [ ]{ }, where

[ ] [

(

)

]

[ ] [

]

( ) ( ) ( ) ( )

Acknowledgement: This problem was assigned as part of Homework 2 (Problem 3.2-5). It is

also similar to the interpolation problem assigned in Homework #1 (Problem 1.3-3(c)).

Moreover, Problem 1.3-3(a) was discussed in detail in the Hints for HW#1.

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Problem 5 (40 points):

Consider the quarter of a plate with a hole as in Figure 1. The plate will be modeled using plane

stress, with a thickness of 0.1, a Young’s modulus E of 125 and Poisson ratio of 0.25. A

pressure load of is applied at the right end. The domain is discretized using

elements as in Figure 2. Fill the 14 missing spaces in the ABAQUS input file, and

answer questions (a) through (g). *HEADING

Quarter plate with hole problem - QUIZ 1

*NODE

1, 2.0000, 0.0000

2, 3.0000, 0.0000

3, 4.0000, 0.0000

4, 4.0000, 1.0000

5, 4.0000, 2.0000

6, 4.0000, 3.0000

7, 4.0000, 4.0000

8, 3.0000, 4.0000

9, 2.0000, 4.0000

10, 1.0000, 4.0000

11, 0.0000, 4.0000

12, 0.0000, 3.0000

13, 0.0000, 2.0000

14, 0.7654, 1.8478

15, 1.4142, 1.4142

16, 1.8478, 0.7654

17, 2.7071, 1.7071

18, 1.7071, 2.7071

*ELEMENT, TYPE=CPS8, ELSET=THINPLATE

1, 1, 3, 5, 15, 2, 4, 17, 16

2, 15, 5, 7, 9, 17, 6, 8, 18

3, 13, 15, 9, 11, 14, 18, 10, 12

*SOLID SECTION, ELSET=THINPLATE, MATERIAL=MAT01

0.1

*MATERIAL, NAME=MAT01

*ELASTIC

125, 0.25

*STEP, PERTURBATION

*STATIC

*BOUNDARY

1, 2

2, 2

3, 2

11, 1

12, 1

13, 1

*DLOAD

Figure 1

Figure 2

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1, P2, -3

2, P2, -3

*NODE PRINT

U, RF

*EL PRINT, POSITION=AVERAGED AT NODES

S, E

*END STEP

(a) What is the total size of the full stiffness matrix for the static analysis of the problem?

( )

Size of the full stiffness matrix is then

(b) What is the maximum semi-bandwidth of the stiffness matrix in (a)?

( ) ( ) ( )

The problem will be remodeled using elements. In doing so, the domain is partitioned

into three surfaces and meshed in PATRAN as in Figure 3.

Figure 3: Domain meshed using 3 surfaces with elements each

(c) What is the number of FE nodes after meshing? What is the number of FE nodes after

equivalencing?

After meshing: 27

After equivalencing: Counting the nodes in the "stitched together" so then model nodes = 21

(d) What is the total size of the full stiffness matrix for the static analysis of the problem now

modeled using elements?

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( )

Size of the full stiffness matrix is then

(e) Draw (neatly) the DUAL GRAPH (DG) and the COMMUNICATION GRAPH (CG)

associated to the finite element mesh of Figure 3. Hint: Notice that both DG and CG are

element graphs (not a nodal graph).

(f) Why is plane stress appropriate for this problem?

The plate is thin and out of plane stresses can be considered negligible

DUAL GRAPH COMMUNICATION GRAPH

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(g) The following plot represents for the quarter domain. Clearly mark the stress

concentration point in the plot and briefly explain whether this behavior makes sense or

not.

The hole causes a stress concentration that is maximum around the hole and where the

plate's connecting segment is at its thinnest.