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Pre-Analysis and Start-Up
In the Pre-Analysis step, we'll review the following:
- Mathematical model: We'll look at the governing equations + boundary conditions and the assumptions contained within the mathematical model.
- Hand-calculations of expected results: We'll use an analytical solution of the mathematical model to predict the expected stress field from ANSYS. We'll pay close attention to additional assumptions that have to be made in order to obtain an analytical solution.
- Numerical solution procedure in ANSYS: We'll briefly overview the solution strategy used by ANSYS and contrast it to the hand calculation approach.
Mathematical Model
We'll first list the assumptions in the mathematical model. Then, we'll review the governing equations and boundary conditions that form the mathematical model. Note that this type of a mathematical model where you have a set of differential equations together with a set of additional restraints at the boundaries is called a Boundary Value Problem (BVP). A lot of practical problems that are solved using ANSYS and other FEA software are BVP's. You should have encountered simple BVP's in your math courses, problems of the kind that involve solving a differential equation with a set of boundary conditions (I was never good at these math problems and it showed in my math grades to the displeasure of my parents .... fortunately that is now a distant memory!). You can think of the BVP considered in this tutorial as a souped-up version of simpler BVP's you have encountered in math courses (and either liked or hated!).
Assumptions
We'll assume that:
- Plane stress conditions apply since the bar is thin, thus we don't expect significant variation of stresses in the z direction:
Wiki Markup {latex}\[ \sigma_{z} = \tau_{xz} = \tau_{yz} = 0 \] {latex}
...
- Gravity
...
- effects
...
- can
...
- be
...
- neglected
...
- i.e.
...
- no
...
- body
...
- forces.
...
Wiki Markup {latex} \[ F_x = F_y =0 \] {latex}
...
Governing
...
Equations
...
Since
...
we
...
are
...
assuming
...
plane
...
stress
...
conditions,
...
we
...
can
...
use
...
the
...
2D
...
version
...
of
...
the
...
equilibrium
...
equations.
...
When
...
the
...
deformed
...
structure
...
reaches
...
equilibrium,
...
the
...
2D
...
stress
...
components
...
should
...
satisfy
...
the
...
2D
...
equilibrium
...
equations
...
with
...
zero
...
body
...
forces:
| Wiki Markup |
|---|
\\ {latex} \begin{eqnarray} {\partial \sigma_x \over \partial x} + {\partial \tau_{xy} \over \partial y} = 0 \nonumber \\ {\partial \tau_{xy} \over \partial x} + {\partial \sigma_y \over \partial y} = 0 \nonumber \end{eqnarray} {latex} h4. |
Boundary
...
Conditions
...
We
...
solve
...
these
...
equations
...
in
...
a
...
rectangular
...
domain
...
and
...
impose
...
the
...
appropriate
...
boundary
...
conditions.
...
At
...
every
...
point
...
on
...
the
...
boundary,
...
either
...
the
...
displacement
...
or
...
the
...
traction
...
must
...
be
...
prescribed.
The bottom and top edges are free. If a boundary location is not constrained and can move freely, it can expand and contract without incurring stress. Thus, traction on the free edges is zero and we get
| Wiki Markup |
|---|
_. [!Traction-displacement (1).png|width=500!|^Traction-displacement (1).png] The bottom and top edges are free. If a boundary location is not constrained and can move freely, it can expand and contract without incurring stress. Thus, traction on the free edges is zero and we get \\ {latex} \[ \sigma_y = \tau_{xy} = 0 \:\: at \: y = 0 \: and \: y = H \] {latex} |
The
...
left
...
end
...
is
...
fixed.
...
So
...
both
...
components
...
of
...
displacement
...
are
...
zero
...
at
...
this
...
end:
| Wiki Markup |
|---|
\\ {latex} \[ u = v = 0 \:\: at \: x = 0 \] {latex} |
The
...
boundary
...
condition
...
is
...
a
...
little
...
bit
...
more
...
complicated
...
at
...
the
...
right
...
end.
...
Here,
...
the
...
traction
...
is
...
specified
...
at
...
the
...
mid-point
...
where
...
the
...
point
...
load
...
is
...
applied.
...
The
...
applied
...
traction
...
at
...
all
...
other
...
points
...
on
...
the
...
right
...
boundary
...
is
...
zero.
...
For
...
brevity,
...
we
...
won't
...
write
...
out
...
the
...
corresponding
...
equations
...
at
...
the
...
right
...
boundary.
...
We'll
...
simplify
...
this
...
boundary
...
condition
...
in
...
our
...
hand
...
calculations
...
below
...
(to
...
make
...
the
...
problem
...
tractable)
...
but
...
the
...
ANSYS
...
solution
...
provided
...
uses
...
the
...
full
...
set
...
of
...
boundary
...
conditions.
...
Another
...
complication
...
is
...
that
...
since
...
we
...
have
...
a
...
point
...
load,
...
the
...
specified
...
traction
...
at
...
the
...
mid-point
...
of
...
the
...
right
...
end
...
is
...
infinite.
...
We'll
...
later
...
discuss
...
the
...
effect
...
of
...
this
...
in
...
the
...
ANSYS
...
solution.
...
Do
...
keep
...
in
...
mind
...
that
...
there
...
are
...
no
...
point
...
loads
...
in
...
practice,
...
it's
...
just
...
an
...
idealization
...
that
...
can
...
lead
...
to
...
weird
...
behavior
...
that
...
we
...
need
...
to
...
be
...
aware
...
of.
...
Hand
...
Calculations
...
Now
...
that
...
we
...
have
...
reviewed
...
the
...
mathematical
...
model
...
for
...
our
...
problem,
...
let's
...
hold
...
off
...
diving
...
into
...
ANSYS
...
just
...
yet
...
and
...
first
...
make
...
some
...
hand
...
calculations
...
of
...
expected
...
results.
...
We'll
...
use
...
these
...
hand
...
calculations
...
to
...
check
...
ANSYS
...
results
...
(like
...
an
...
expert
...
engineer
...
would
...
!).
...
In
...
order
...
to
...
make
...
the
...
problem
...
solvable
...
by
...
hand,
...
we
...
need
...
to
...
make
...
additional
...
assumptions.
...
The
...
ANSYS
...
solution
...
does
...
not
...
make
...
these
...
additional
...
assumptions.
...
Additional
...
Assumptions
...
in
...
Hand
...
Calculations
...
- We'll
...
- simplify
...
- the
...
- right
...
- boundary
...
- condition.
...
- Instead
...
- of
...
- a
...
- point
...
- load,
...
- we'll
...
- assume
...
- that
...
- the
...
- load
...
- is
...
- distributed
...
- over
...
- the
...
- entire
...
- right
...
- boundary.
...
- So
...
- the
...
- traction
...
- condition
...
- at
...
- the
...
- right
...
- boundary
...
- becomes
Wiki Markup {latex} \[ \sigma_x = P/(H \, t), \: \: \tau_{xy} = 0 \: \: \: at \: x = L \] {latex}
...
Here,
...
- t
...
- is
...
- the
...
- thickness.
...
The
...
- following
...
- schematic
...
- shows
...
- the
...
- process
...
- of
...
- simplifying
...
- the
...
- right
...
- boundary
...
- condition
...
- in
...
- the
...
- hand
...
- calculation.
- Away from the left and right ends, we expect a uni-axial state of stress with zero shear (OK, this is a bit of a leap of the imagination but it's plausible). So we'll assume that everywhere
Wiki Markup {latex} \[ \tau_{xy} = 0 \] {latex}
...
We
...
- don't
...
- expect
...
- this
...
- to
...
- hold
...
- near
...
- the
...
- left
...
- boundary
...
- or
...
- in
...
- the
...
- vicinity
...
- of
...
- the
...
- point
...
- load,
...
- so
...
- our
...
- hand
...
- calculations
...
- won't
...
- be
...
- valid
...
- there.
...
Analytical
...
Solution
...
With
...
these
...
additional
...
assumptions
...
in
...
hand,
...
we
...
can
...
easily
...
solve
...
the
...
BVP
...
and
...
we
...
get
...
the
...
following analytical solution:
| Wiki Markup |
|---|
analytical solution: \\ {latex} \[ \sigma_x = P/(H \, t), \: \: \: \sigma_y = 0 \] {latex} |
This
...
is
...
the
...
well
...
known
...
(P/A)
...
result
...
but
...
we
...
have
...
arrived
...
at
...
it
...
somewhat
...
carefully,
...
accounting
...
for
...
the
...
additional
...
assumptions
...
we
...
made
...
in
...
the
...
process.
...
We'll
...
need
...
to
...
keep
...
these
...
additional
...
assumptions
...
in
...
mind
...
when
...
comparing
...
the
...
hand
...
calculations
...
with
...
the
...
ANSYS
...
solution.
...
For
...
the
...
values
...
given
...
in
...
the
...
problem
...
statement,
...
we have
| Wiki Markup |
|---|
have \\ {latex} \begin{eqnarray} \sigma_x = 2000/(10*1) = 200 \ N/mm^2 = 200 \ MPa \nonumber \end{eqnarray} {latex} |
The
...
corresponding
...
strain
...
in
...
the
...
x-direction
...
can
...
be
...
calculated
...
from
...
Hooke'
...
law:
| Wiki Markup |
|---|
\\ {latex} \[ \epsilon_x = \frac{\sigma_x}{E} - \nu \, \frac{\sigma_y}{E} = 1 \times 10^{-6} \] {latex} |
The
...
strain
...
is
...
tiny
...
since
...
the
...
material
...
is
...
very
...
stiff
...
with
...
an
...
Young's
...
modulus
...
of
...
200
...
GPa.
...
The
...
displacement
...
at
...
the
...
right
...
end
...
can
...
be
...
estimated
...
by
...
integrating
...
the
...
constant
...
x-strain:
...
| Wiki Markup |
|---|
{latex}
\[
\epsilon_x = \frac{\partial u}{\partial x}
\]
{latex} |
| Wiki Markup |
|---|
\\ {latex} \[ u(x=L) = \int_0^L{\epsilon_x} \, dx = 0.05 \, mm \] {latex} |
The
...
above
...
hand
...
calculations
...
give
...
us
...
expected
...
values
...
of
...
stress,
...
strain
...
and
...
displacement
...
which
...
we'll
...
compare
...
with
...
the
...
ANSYS
...
results.
...
Numerical
...
Solution
...
Procedure
...
in
...
ANSYS
...
The
...
type
...
of
...
numerical
...
solution
...
procedure
...
used
...
by
...
ANSYS
...
is
...
called
...
finite-element
...
analysis
...
(FEA)
...
or
...
finite-element
...
method
...
(FEM).
...
In
...
FEA,
...
we
...
divide
...
or
...
"discretize"
...
the
...
domain
...
into
...
small
...
rectangles
...
or
...
"elements"
...
(hence
...
the
...
name
...
finite
...
element
...
analysis
...
).
...
ANSYS
...
obtains
...
the
...
numerical
...
solution
...
to
...
the
...
BVP
...
in
...
the
...
discrete
...
domain.
...
ANSYS
...
directly
...
solves
...
for
...
the
...
u
...
and
...
v
...
displacements
...
at
...
selected
...
points
...
called
...
"nodes".
...
Everything
...
else
...
such
...
as
...
the
...
stress
...
variation
...
is
...
derived
...
from
...
these
...
nodal
...
displacements
...
through
...
interpolation.
...
The
...
nodes
...
in
...
our
...
case
...
are
...
the
...
corners
...
of
...
the
...
elements
...
as
...
shown
...
below.
...
As
...
you
...
can
...
imagine,
...
the
...
numerical
...
solution
...
should
...
get
...
better
...
as
...
you
...
increase
...
the
...
number
...
of
...
elements.
...
The following figure summarizes the contrasts between the hand calculations and ANSYS's approach. One important point to keep in mind is that both start with the same mathematical model but use different assumptions and approximations to solve it. Also, in FEA, one always computes the displacement first and from that derives the stress. Contrast that to the hand calculations where we calculated the stress first and from that derived the displacement. The latter process works only for a few simple problems.
This brings us to the end of the Pre-Analysis section.
Start-Up: Load Solution into ANSYS
As mentioned before, we are providing the ANSYS solution so that you can focus on comparing the hand calculations with the ANSYS results (which is the goal of this exercise). Without further ado, let's download the ANSYS solution and load it into ANSYS.
1. Download "Tensile Bar Demo.zip" by clicking here
Unzip the file at a convenient location. You will see a folder called Tensile Bar Demo with the following contents:
- Tensile Bar Demo_files (this is a folder)
- Tensile Bar Demo.wbpj
Please make sure both these objects are in the unzipped folder, otherwise the solution will not load into ANSYS properly. (Note: The solution provided was created using ANSYS workbench 13.0 release, there may be compatibility issues when attempting to open with older versions).
2. Double click "Tensile Bar Demo.wbpj" - This should automatically open ANSYS Workbench (you have to twiddle your thumbs a bit before it opens up). You will then be presented with the ANSYS solution in the project page.
A tick mark against each step indicates that that step has been completed.
3. To look at the results, double click on Results - This should bring up a new window (again you have to twiddle your thumbs a bit before it opens up).
4. On the left-hand side there should be an Outline toolbar. Look for Solution (A6).
We'll investigate the items listed under Solution (A6) in the next step of this tutorial.