# Cross-Section Analysis Examples

## Contents

- Split 200 x 100 x 6 RHS
- 250 PFC
- Built-up 200UB25 + 150 x 100 x 9 RHS
- Built-up 50 x 5 SHS + 100 x 50 x 5 RHS + 200 x 6 Flat

## Split 200 x 100 x 6 RHS

A box section provides torsional stiffness by providing a closed path for shear stresses to flow at a considerable distance from a rotational centre. Preventing this enclosed path dramatically reduces the torsional rigidity of the section. This is illustrated through the analysis and of a 200 x 100 x 6 RHS, and a comparison to that of the same section with a 1 mm wide cut in one of the sides. A torsion of 1 kN.m is applied to both sections.

The analysis can be carried out with the following python commands:

```
import main
import sectionGenerator
# Closed RHS
(points, facets, holes) = sectionGenerator.RHS(100, 200, 6, 15, 16)
mesh1 = main.crossSectionAnalysis(points, facets, holes, meshSize=2.5, nu=0.3)
# Open RHS
(points, facets, holes) = sectionGenerator.RHS_Split(100, 200, 1, 6, 15, 16)
mesh2 = main.crossSectionAnalysis(points, facets, holes, meshSize=2.5, nu=0.3)
# Perform stress analysis
main.stressAnalysis(mesh1, Nzz=0, Mxx=0, Myy=0, M11=0, M22=0, Mzz=1e6, Vx=0, Vy=0)
main.stressAnalysis(mesh2, Nzz=0, Mxx=0, Myy=0, M11=0, M22=0, Mzz=1e6, Vx=0, Vy=0)
```

### Closed RHS results

The torsion constant was calculated by the python program to be J = 14.236 x 10^{6} mm^{4}.

### Split RHS results

The torsion constant was calculated by the python program to be J = 39.648 x 10^{3} mm^{4}, approximately 360 times less stiff than the closed section.

## 250 PFC

The analysis of a 250 PFC (250 mm deep parallel flange channel) can be carried out with the following python commands:

```
import main
import sectionGenerator
(points, facets, holes) = sectionGenerator.PFC(d=250, b=90, tf=15, tw=8, r=12, n_r=16)
mesh = main.crossSectionAnalysis(points, facets, holes, meshSize=4, nu=0.3)
```

This produces the following output for the warping dependent properties:

These results, which use a mesh size of 4 mm^{2}, can be compared to the tabulated values in the OneSteel catalogue, and simple hand calculations. The OneSteel catalogue gives the following properties:

A mesh refinement study using the python cross-section program shows that the OneSteel value for the torsion constant is slightly (4%) overestimated, whereas the warping constant and shear centre show closer convergence. The numerical results obtained from the python program can be compared to simple hand calculations:

- Torsion Constant [1]

- Warping Constant [1]

- Shear Centre [2] (from centre of web to shear centre):

The above hand calculations align well with the results from the python program and the OneSteel catalogue.

## Built-up 200UB25 + 150 x 100 x 9 RHS

A steel section is fabricated by placing a 150 x 100 x 9 RHS on its side on top of a 200UB25. The section is subjected to a major axis bending moment of 50 kN.m, a torsion moment of 10 kN.m and a y-direction shear force of -25 kN.

The analysis can be carried out by using the section builder function with the following python commands:

```
import main
import sectionGenerator
(UBpoints, UBfacets, UBholes) = sectionGenerator.ISection(203, 133, 7.8, 5.8, 8.9, 8)
(RHSpoints, RHSfacets, RHSholes) = sectionGenerator.RHS(100, 150, 9, 22.5, 8)
UB = { 'points': UBpoints,
'facets': UBfacets,
'holes' : UBholes,
'x' : -0.5 * 133,
'y' : 0 }
RHS = { 'points': RHSpoints,
'facets': RHSfacets,
'holes' : RHSholes,
'x' : -75,
'y' : 203 }
(points, facets, holes) = sectionGenerator.combineShapes([UB, RHS])
mesh = main.crossSectionAnalysis(points, facets, holes, meshSize=5, nu=0.3)
# Perform stress analysis
main.stressAnalysis(mesh, Nzz=0, Mxx=50e6, Myy=0, M11=0, M22=0, Mzz=10e6, Vx=0, Vy=-25e3)
```

The mesh, centroids and stress contours are shown below:

## Built-up 50 x 5 SHS + 100 x 50 x 5 RHS + 200 x 6 Flat

A zed shaped steel section is fabriacted by welding a 50 x 5 SHS and a 100 x 50 x 5 RHS to a 200 x 6 flat section. The section is subjected to a major axis bending moment of 10 kN.m, a torsion moment of 5 kN.m and a y-direction shear force of -15 kN.

The analysis can be carried out by using the section builder function with the following python commands:

```
import main
import sectionGenerator
(Flatpoints, Flatfacets, Flatholes) = sectionGenerator.Flat(200, 6)
(SHSpoints, SHSfacets, SHSholes) = sectionGenerator.RHS(50, 50, 5, 12.5, 8)
(RHSpoints, RHSfacets, RHSholes) = sectionGenerator.RHS(50, 100, 5, 12.5, 8)
Flat = {'points': Flatpoints,
'facets': Flatfacets,
'holes' : Flatholes,
'x' : 50,
'y' : -100}
SHS = {'points' : SHSpoints,
'facets': SHSfacets,
'holes' : SHSholes,
'x' : 56,
'y' : -100}
RHS = {'points' : RHSpoints,
'facets': RHSfacets,
'holes' : RHSholes,
'x' : -50,
'y' : 50}
section = [Flat, RHS, SHS]
(points, facets, holes) = sectionGenerator.combineShapes(section)
mesh5 = main.crossSectionAnalysis(points, facets, holes, meshSize=1.5, nu=0.3)
# Perform stress analysis
main.stressAnalysis(mesh5, Nzz=0, Mxx=10e6, Myy=0, M11=0, M22=0, Mzz=5e6, Vx=0, Vy=-15e3)
```

The mesh, centroids and stress contours are shown below:

## References

- AS 4100-1998: Steel Structures
- W.D. Pilkey, Analysis and Design of Elastic Beams: Computational Methods, John Wiley & Sons, Inc., New York, 2002.

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