dx w d EI M

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15 Νοε 2013 (πριν από 3 χρόνια και 11 μήνες)

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AE464
-
SPRING 2012

TAKE
-
HOME PART OF FINAL EXAMINATION

and HOMEWORK 5


TAKE
-
HOME PART

(40% of the final)


All parts contribute to
the
take home part

of the final examination. Part f will be also
counted for homework 5.


Consider the
rect
angular

cross
-
section (wxh)
cantilevered beam which is subjected to
triangular load distribution.

B
eam is modeled with a single beam element
as shown below.





















(a)

(
8
)

Determine the consistent element load vector

(nodal forces
and moments)



associated with the triangular loading

using the beam shape functions derived in class
.




(Consistent loads replace the distributed external load !)


(b)

(6)

Determine the free end vertical displacement

and rotation

of the beam by carr
ying
out FE solution. You can use the stiffness matrix you obtained in HW1

and consistent
load vector you determined in part a
.


(c)

(
8
)

Determine reaction moment at the wall (node 1) by the FE solution.
C
alculate the
reaction moment using equilibrium equation
.

Compare both results.

Now, post
-
process your displacement solution to determine the
bending moment at
x=0

utilizing the relation

2
2
dx
w
d
EI
M


(note that this relation gives
the internal
bending moment)
. U
se the interpolation w=[N]{d}. Is

it s
ame as the root bending
moment y
ou
calculate
d above?
Draw an infinitesemally small free body diagram at the
root of the beam and show the internal moment you calculated, consistent external
moment at the root calculated in part a and reaction moment calcul
ated. Show that
moment equilibrium is satisfied.

1 N/m

L=1 m

1

2

beam element

x

z,w

Width of the beam,w =1 cm=0.01 m

Height of the beam, h=2 cm=0.02 m

Beam material: E=70Gpa, v=0.3

Calculate the
maximum

stress
at the
from
I
Mc



. Note that from negative moment
(CW), curvature is also negative and for positive z, stress is positive, so sign
convention checks.

Which mome
nt do you have to use?



(4)

Now, determine the tip displacement and the maximum stress using mechanics of
materials approach. You can use the tables at the back of strength books. Compare the
FE solution with the strength solution for tip displacement, ro
tation and maximum
bending stress at the root. Make comments about the comparison.


Nastran Part:


(d)

(8)

Model the same beam in Patran

with single beam element just like you did above
.

To create the triangular load, first create a uniform load, carry out a s
olution, and in
the .bdf file change the PLOAD1 entry accordingly. An example of PLOAD1 card is
given below.
Once you modify the bdf file you can carry out a solution in two ways.

(i)

Double click Nastran.exe ico
n and point the bdf file
,

and on
ce the solution
is
finished
,

you can look at the .f06 file for solutions
. O
r in Patran
,

you can attach
the related .xdb file

but you don’t need that
.

(ii)

Import the modified bdf file into
Patran

and carry out the solution inside
Patran.
When you import the bdf file, change you
r load case from the load case
boundary condition_current load case menu.



Compare Nastran solution for tip displacement, tip rotation and maximum bending
stress at the root with the FE solution and mechanics of material solution you
determined above.





HOMEWORK 5

(Nastran application)


(e)

(6

points in final = 15 points in homework 5
)

Lastly, create
scalar fields for the width
and the height of the beam such that they vary according to:

w=0.01
-
0.005x and h=0.02
-
0.01x (both in meters)

As
sign them to the tapered beam section (select tapered beam section

in the pro
perties
menu).
Important:

Before the input properties make sure that you select the beam or
the curve for the application region and add it !!. After this step go to the input
pro
perties. In the beam section menu, you will see the scalar fields you have created.

You also need to select an e
valuation point. Select the glo
bal origin.

Show you beam
in 3D full span and show the taper.


Compare the tip displacement/rotation and root ben
ding stress of part (f) with those of
part (e). Comment on the root stresses. Are they equal to each other or different? If
they are equal
,

doe
s

it make sense? Comment.

How about the tip displacement and
rotation of this part compare with part e.? Comment.