Cross-Section Analysis Examples

Contents

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 106 mm4.

RHS mesh.
Mesh discretisation for the closed 200 x 100 x 6 RHS.
RHS stress.
Shear stress due to torsion for the closed 200 x 100 x 6 RHS.
RHS vectors.
Shear stress vectors due to torsion for the closed 200 x 100 x 6 RHS at the top right of the section.

Split RHS results

The torsion constant was calculated by the python program to be J = 39.648 x 103 mm4, approximately 360 times less stiff than the closed section.

RHS_Split mesh.
Mesh discretisation for the split 200 x 100 x 6 RHS.
RHS_Split stress.
Shear stress due to torsion for the split 200 x 100 x 6 RHS, approximately 30 times higher than the closed section.
RHS_Split vectors.
Shear stress vectors due to torsion for the split 200 x 100 x 6 RHS at the top right of the section. The shear stress now has to flow within the thickness of the wall, rather than around the entire section.
RHS_Split vectors.
Shear stress vectors due to torsion for the split 200 x 100 x 6 RHS adjacent to the split.

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)
PFC centroids
Mesh discretisation for the 250 PFC with principal axes and centroids.

This produces the following output for the warping dependent properties:

These results, which use a mesh size of 4 mm2, 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 1 centroids
Mesh discretisation for the built-up section with principal axes and centroids.
Built-up 1 bending stress
Bending stress.
Built-up 1 torsion
Shear stress due to torsion.
Built-up 1 shear
Shear stress due to transverse shear force.
Built-up 1 vm
von Mises stress.

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:

Built-up 2 centroids
Mesh discretisation for the built-up section with principal axes and centroids.
Built-up 2 bending stress
Bending stress.
Built-up 2 torsion
Shear stress due to torsion.
Built-up 2 shear
Shear stress due to transverse shear force.
Built-up 2 vm
von Mises stress.

References

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

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