# OS-E: 0915 Cantilever Beam using Equations

Demonstrates how to use the equation utility to define cross-sectional properties in a size (parameter) optimization.

## Model Files

Before you begin, copy the file(s) used in this example to
your working directory.

## Model Description

The structure is a cantilever beam of a length of 55 modeled with five
CBAR elements. The cross-section of the beam is a solid
circle. In each element, the diameter of the section is the design variable. Hence,
there are five design variables. The cross-sectional properties such as area, moment
of inertia, and torsional constant are calculated using the explicit formulas for a
circle. In terms of the DEQATN card, they appear
as:

```
$AREA
DEQATN,111,A(D)=PI(1)*D**2/4
$MOMENT OF INERTIA
DEQATN,122,I(D)=PI(1)*D**4/64
$TORSIONAL CONSTANT
DEQATN,133,J(D)=PI(1)*D**4/32
```

Using these equations, DVPREL2 statements are used to assign each
design variable to the respective PBAR property. The statements
that assign the diameter of the first bar element to the cross-sectional area of
that element look
like:

```
DESVAR,1,Diam1,10,1,20,0.5
DVPREL2,11,PBAR,1,4,,,111
+,DESVAR,1
```

The optimization problem to be solved is the minimization of the tip displacement with a volume constraint of 4000. Convergence was achieved after four iterations.

## Results

Diameter 1 | Diameter 2 | Diameter 3 | Diameter 4 | Diameter 5 | Tip-Displ | |
---|---|---|---|---|---|---|

Initial | 10.00 | 10.00 | 10.00 | 10.00 | 10.00 | 40.42 |

Final | 12.48 | 11.49 | 10.28 | 8.70 | 6.29 | 26.51 |