Collect HW HW: 5.1,9 by hand and 2 meshes w/NENastran, # 15 (use 2 elements by hand, 4, &

8 elements w/CST.xls) Due April 14, 2005.

What if we have a 2-d problem with surface tractions, point loads, acting in the plane of a relatively thin structure?

With

We can take the preceding problem and Discretize it:

How can we model the problem better?

Constant Strain Triangle Element 2 dof per node (x,y), 3 nodes per element (6 x 6 stiffness matrix)

Shape Functions

Shape functions are just ratios of areas:

Representing Displacements:

or

Finding the Stiffness Matrix, and assembling the Force Vector:

the element stiffness matrix:

The point loads are given by:

The surface Tractions are given by:

Kreduce

2.435 106

×

0

1.258− 106

×

7.102− 105

×

0

7.162 106

×

6.684− 105

×

3.7− 106

×

1.258− 106

×

6.684− 105

×

1.259 106

×

7.123 105

×

7.102− 105

×

3.7− 106

×

7.123 105

×

2.119 106

×

=Kreduce

K2 2,K3 2,K4 2,K5 2,

K2 3,K3 3,K4 3,K5 3,

K2 4,K3 4,K4 4,K5 4,

K2 5,K3 5,K4 5,K5 5,

:=

Looking at the figure, we see that node 1, in element 1, and node 1 and 4, in element 2 are restrained. If we use the elimination method, the reduced stiffness matrix for element 1 is:

Note that the initial row and column start at 0 rather than 1, so K00 is the first stiffness value, rather than K11

K

1.178 106

×

6.664− 105

×

1.177− 106

×

6.684 105

×

1.174− 103

×

2.089− 103

×

6.664− 105

×

1.88 106

×

7.102 105

×

3.462− 106

×

4.387− 104

×

1.581 106

×

1.177− 106

×

7.102 105

×

2.435 106

×

0

1.258− 106

×

7.102− 105

×

6.684 105

×

3.462− 106

×

0

7.162 106

×

6.684− 105

×

3.7− 106

×

1.174− 103

×

4.387− 104

×

1.258− 106

×

6.684− 105

×

1.259 106

×

7.123 105

×

2.089− 103

×

1.581 106

×

7.102− 105

×

3.7− 106

×

7.123 105

×

2.119 106

×

=

KE t⋅

4 A1⋅ 1 v2

−

⋅

y23

0

y31

0

y12

0

0

x32

0

x13

0

x21

x32

y23

x13

y31

x21

y12

⋅

1v

0

v

1

0

001 v−

2

y23

0x32

0x32

y23

y31

0x13

0x13

y31

y12

0x21

0x21

y12

⋅

⋅:=

y21 y12−:=y12 1.75:=y32 y23−:=y23 1.75−:=y31 y13−:=y13 0:=

x31 x13−:=x21 x12−:=x12 3.1−:=x13 6−:=x32 x23−:=x23 2.9−:=A1 5.25:=v .32:=t .125:=E 30000000:=

Example: 2 Constant strain triangular elements, Steel plate

K2

1.215 106

×

6.874− 105

×

1.206− 106

×

7.102 105

×

2.532− 104

×

1.292− 104

×

6.874− 105

×

2.008 106

×

6.684 105

×

4.101− 105

×

2.828 104

×

1.604− 106

×

1.206− 106

×

6.684 105

×

2.419 106

×

0

1.179− 106

×

6.684− 105

×

7.102 105

×

4.101− 105

×

0

8.224 105

×

7.102− 105

×

4.009− 105

×

2.532− 104

×

2.828 104

×

1.179− 106

×

7.102− 105

×

1.188 106

×

6.72 105

×

1.292− 104

×

1.604− 106

×

6.684− 105

×

4.009− 105

×

6.72 105

×

1.999 106

×

=

Combining Elements 1 and 2, and reducing:Defining the Force Vector:

Kall Kreduce

K22 2,

K23 2,00

K22 3,

K23 3,00

0

0

00

0

0

00

+:= Kall

4.854 106

×

0

1.258− 106

×

7.102− 105

×

0

7.984 106

×

6.684− 105

×

3.7− 106

×

1.258− 106

×

6.684− 105

×

1.259 106

×

7.123 105

×

7.102− 105

×

3.7− 106

×

7.123 105

×

2.119 106

×

= Uall

u2

v2

u3

v3

:=

u2

Fall

00025−

:= Uall Kall1−Fall⋅:= Uall

8.319− 106−

×

4.502− 105−

×

2.533 105−

×

1.017− 104−

×

=

What if we assume that there is only a body force from the weight of each element, 0.5 lbs per cubic inch? What if we assume that a body force from the weight of each

element, 0.5 lbs per cubic inch, is added to the 25 lbs?

Fall

0000

0

0.5 A1 t⋅

3A2 t⋅

3+

⋅

0

0.5 A1 t⋅

3

⋅

+:= Uall Kall1−Fall⋅:= Uall

7.515 108−

×

4.066 107−

×

1.904− 107−

×

8.508 107−

×

=Fall

00025−

0

0.5 A1 t⋅

3A2 t⋅

3+

⋅

0

0.5 A1 t⋅

3

⋅

+:= Uall Kall1−Fall⋅:= Uall

8.244− 106−

×

4.462− 105−

×

2.514 105−

×

1.009− 104−

×

=

Looking at Element 2 now:

E 30000000:= t .125:= v .32:= A2 5.565:= x23 3.1:= x32 x23−:= x13 0:= x12 3.1−:= x21 x12−:= x31 x13−:=

y13 3.59:= y31 y13−:= y23 1.79:= y32 y23−:= y12 1.75:= y21 y12−:=

K2E t⋅

4 A2⋅ 1 v2

−

⋅

y23

0

y31

0

y12

0

0

x32

0

x13

0

x21

x32

y23

x13

y31

x21

y12

⋅

1v

0

v

1

0

001 v−

2

y23

0x32

0x32

y23

y31

0x13

0x13

y31

y12

0x21

0x21

y12

⋅

⋅:=