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Oct 30, 2013 (3 years and 7 months ago)

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A.5.2.6.1 Global Buckling of Complete Launch Vehicle


1


Author:
Jessica Schoenbauer


A.5.2.6 Other Component Analysis


A.5.2.6.1

Global Buckling of Complete Launch Vehicle


Thin
-
walled structures are highly susceptible to buckling.
The design of our
launch
vehicle

is a thin
-
walled body and therefore, it is necessary to perform a buckling a
nalysis
to determine a critical buckling load.
In previous sections,
local
buckling has been
discussed. In this section
,

we consider
the overall global

buckling
of the launch vehicle
.
We determine the
critical buckling load of the
launch vehicle

by

usin
g an eigenanalysis.


The stability analyses, such as buckling, occur in two stages. These two stages are the
pre
-
buckle analysis and the buckling analysis.
2

It is important to
figure out

the geometric
stiffness matrix so that in
-
plane stresses can be dete
rmined.
2

We know it is necessary to
find the

in
-
plane stress because the presence of in
-
plane stress causes the onset of
buckling. However,
the in
-
plane stresses are
usually not known in advance.


T
herefore
,

it
is important to allow the degrees of freedom

to be such that the i
n
-
plane stresses can be
evaluated
. Then
, we solve

the ei
genvalue problem using the following equation
,




(A.5.2.6.1.1)


where K
E

is the elastic stiffness, K
G

is the geometric stiffness,
λ

is the
eigenvalue
, and
Φ
is the nodal displacements.
2

We complete the buckling analysis by solving for the values
of
λ that make the system u
nstable
.


In performing the buckling analysis of our
launch vehicle
,

we simplify the vehicle to
represent a column buckling problem
.
When the load on the column is applied through
the center of gravity of its cross s
ection,

the load is an axial load. In short columns
loaded axially, it is likely that it will fail due to the compression by the axial load before
it will fail due to bu
ckling. In long columns loaded axially, the failure will occur as
buckling.

The
critical load for buckling can be stated as the following,


A.5.2.6.1 Global Buckling of Complete Launch Vehicle


2


Author:
Jessica Schoenbauer




(A.5.2.6.1.2)


where P
c

is the critical buckling load, E is the modulus of elasticity
, I is the second
moment of inertia, and L is the length of the launch vehicle.
3

Now,

we analyze the ab
ove
equation using the

definition of the second moment of inertia of a hollow cylinder

in
Eq.

(
A.5.2.6.1.3
)
.




(A.5.2.6.1.3)


where D
O

is the outer diameter and D
I

is the inner diameter.

We perf
orm this analysis to
allow the e
ffect of the length and radius to be seen explicitly.
We see

that in order to
increase the critical buckling load, either the total length has to be decr
eased or the radius
has to be increased.

Now that we understand the necessity of a global buckling analysis
and understand what parameters affect t
he critical buckling load, we construct the
algorithm

employe
d in the global buckling analysis.


We model t
h
e three stage
launch vehicle

using fifteen elements and sixteen nodes. Each
stage contains five ele
ments of equal length. The

downfall to this approach is that the
element length is not consistent between various sized
launch vehicle
s or stages.

The
num
ber of elements and nodes are fixed quantities.

However, the number of elements
used in the analysis is enough to provide reliable values

for the first couple modes. In
Figure A.5.2.6.1.1, we sketch a rough
setup of the global buckling finite element mod
el
for the launch vehicle
.

A.5.2.6.1 Global Buckling of Complete Launch Vehicle


3


Author:
Jessica Schoenbauer



Fig. A.5.2.6.1.1.

Diagram of finite element model for the three stage
launch vehicle

applie
d for global
buckling analysis.

(Jessica Schoenbauer)


The figure shows that there are fifteen element and sixteen nodes.
We designate
elements one through five to the first

stage
. The second stage has elements six
through

ten alloc
ated to it, and t
he third stage has elements eleven through fifteen designated to it.

The figure illustrates that we assign a different material to
each stag
e to account for the
possi
bility of a different material for

every stage plus the different properties such as the
cross
-
section area and the moment of inertia. The figure also demonstrates the boundary
conditions and loading conditions applied to the
lau
nch vehicle

for the global buckling
analysis.
We fully fix t
he launch vehicle

at node one without any other degrees of
A.5.2.6.1 Global Buckling of Complete Launch Vehicle


4


Author:
Jessica Schoenbauer


freedom released on any of the other nodes in the model.
We then apply a

compressive
axial load
to the
model at node sixteen.
A simila
r model is

created for the two stage
launch vehicle

except

the
model
employ
s a total of fourteen elements,
which gives
seven
elements
per stage.
Similar to
the three stage
launch vehicle
, the two stage
launch
vehicle

also has a different material definiti
on for every stage.


We did not construct t
he finite el
ement model
in any
finite element modeling program.

Instead, we wrote a program

called

global_buck.m
,

using Matlab to generate

a
structure
data file that
we employ

in

StaDyn
4
, the executable
exercise
d
by QED
4
, to complete the
globa
l buckling analysis. T
he Matlab code
is written
so that a
tex
t file could be created

which

takes in several parameters.
The code i
s written so that the output from
main_once.m

can

be
applie
d as the input parameters to the gl
obal buckling code. These
input parameters include the number of stages in the launch vehicle, the length of each
stage, the diameter of each stage, the wall thickness of each stage, the material of each
stage, and the compressive axial load. The compres
sive axial load is the gross lift off
weight of the launch vehicle

multiplied by the maximum expected gravity loading, which
is 6 G’s
.


After these parameters are input into the code,

the code
employ
s the length of each stage
to

compute the node location
s. The code

then defines the material properties for each
stage and computes the area of the cross
-
sec
tion and

moment of inertia for each stage.
After
we calculate these values, the values

are sent to another code

called

editfiles2.m
,

which

applie
s them
to write the structure data file. Once
editfiles2.m

writes the structure
data file, it returns to the
global_buck.m
,

which
execute
s a command line along with a
command file to run the structure data file for buckling using the StaDyn program. After
StaDy
n

completes its analysis,
global_buck.m

reads in the output file written by StaDyn
with the results of the analysis. The output of interest is the first lambda value. After
the
code
read
s
in this value,
it

calculates
a maximum G loading that the structur
e can
withstand before buckling.

In Figure A.5.2.6.1.2, the process discussed above is written
out in a flow chart format so that the code algorithm can be followed more easily.

A.5.2.6.1 Global Buckling of Complete Launch Vehicle


5


Author:
Jessica Schoenbauer



Fig. A.5.2.6.1.
2
.

Flow chart of algorithm employe
d

for global buckling anal
ysis
.

(Jessica Schoenbauer)

A.5.2.6.1 Global Buckling of Complete Launch Vehicle


6


Author:
Jessica Schoenbauer



We ru
n
global_buck.m

for the final design of the 200 g payload, 1 kg payload, and 5 kg
payload launch vehicles.

T
he results from our analysis

are summarized

in Table
A.5.2.6.1.1.


Table A.5.2.6.1.1

Maximum G’s the Launch Vehic
le can
Withstand.



Launch Vehicle

Max G durability

Units

200 g payload

141.8

G’s

1 kg payload

246.7

G’s

5 kg payload

53.1

G’s






Dr. James Doyle, a professor of Aeronautics and Astronautics Engineering at Purdue
University
, suggests

using a kno
ckdown factor of 0.60 to account for

reduction
s

in
the
strength due to
manufacturing and imperfections of the material.
1

The values listed above
apply

a knockdown factor of 0.50 to account for the
topics brought to our

attention and to
allow some error for

apply
ing a simplified column buckling approach. In addition to the
knockdown factor, the factor of safety equal to 1.25 was also
employe
d in reporting the
results listed above. Although we do not
know if the knockdown factors

applied are
enough to allow

for error using the simplified appro
ach, the maximum withstanda
ble
gravity loading predicted by the analysis are much higher than the 6 G’s that we expect
the launch vehicle to experience. Therefore, we conclude that global buckling should not
present an
y problems for our launch vehicle.


When we first looked at the results from our analysis, we were interested to see the
difference in each launch vehicle that would produce the differences in the maximum
gravity loading durability. In the tables below, t
he length and diameter of each stage are
displayed to help grasp the results of the analysis.





A.5.2.6.1 Global Buckling of Complete Launch Vehicle


7


Author:
Jessica Schoenbauer


Table A.5.2.6.1.2

200 g P
ayload
Launch V
ehicle
Length and
Diameter of Each S
tage.



Stage

Length

Diameter

Units

First

4.94

0.83

m

Second

1.77

0.23

m

Thi
rd

0.99

0.06

m






Table A.5.2.6.1.
3

1 k
g P
ayload
Launch V
ehicle
Length and
Diameter of Each S
tage.



Stage

Length

Diameter

Units

First

4.16

0.66

m

Second

1.48

0.19

m

Third

1.06

0.0
7

m






Table A.5.2.6.1.
4

5 k
g P
ayload
Launch V
ehicle
Length
and
Diameter of Each S
tage.



Stage

Length

Diameter

Units

First

7.07

1.24

m

Second

2.19

0
.30

m

Third

1.00

0.06

m






Table A.5.2.6.1.5

Liftoff Mass for All Three Launch Vehicles.



Launch Vehicle

Liftoff Mass

Units

200 g payload

2890.33

kg

1

kg payload

1819.86

kg

5 kg payload

6372.71

kg






We look at these results
and recall equation A.5.2.6.1.3 to see what produced the
differences in the maximum gravity loading durability. We compare the 200 g payload
launch vehicle and the 1 kg paylo
ad launch vehicle and see that the 200g payload launch
vehicle is larger in its geometry and is more massive than the 1 kg payload launch vehicle
by 1.6 times. The fact that the vehicle geometry
increased in a smaller proportion than
the mass in the compa
rison of the two launch vehicles, results in the higher gravity
loading capability of the 1 kg payload launch vehicle over the 200 g payload launch
vehicle.

A.5.2.6.1 Global Buckling of Complete Launch Vehicle


8


Author:
Jessica Schoenbauer


Now we compare the 1 kg payload launch vehicle and the 5 kg payload launch vehicle.
As in the case

before, the ratio between th
e geometry of the two launch vehicles is less
than half
the ratio between the
masses of them. This, again, results in the higher gravity
loading capability of the 1 kg payload launch vehicle over the 5 kg payload launch
vehicl
e.


References:


1
James F. Doyle. Professor

of Aeronautics and Astronautics Engineering. Purdue
University.

2
Doyle, James F.,
Guided Explorations on the Mechanics of Solids & Structures:
strategies for learning and understanding
. Purdue University, West Lafayette, IN.
August 2007.

3
Doyle, James F.,
Structural Dynamics and Stability: a modern course of analysis and
applications
. Purdue University, West Lafayette, IN. August 2007.

4
ikayex Software Tools.
QED: Static, Dynamic, St
ability, and Nonlinear Analysis of
Solids and Structures
. Lafayette, IN. August 2007.