# C-Channel Bending Stress and Torsional Stress Spreadsheet

## Where to Find a  C-Channel Bending Stress and Torsional Stress Spreadsheet

For a C-channel bending stress and torsional stress spreadsheetclick here to visit our spreadsheet store.  Obtain a convenient, easy to use C-channel bending stress and torsional stress spreadsheet at a reasonable price. Read on for information about calculating combined bending and torsional warping stresses in light-gage steel C-channel bending members.

## Background for Calculations with Combined Bending and Torsional Warping Stresses.

The usual calculation of stresses in bending members applies to loads that act along one of the principal axes passing through both the centroid and the shear center. For symmetrical sections, the centroid and the shear center coincide and this requirement is easily accomplished.  If this is the case, the beam will deflect in the plane of these applied loads without any rotational displacements. Many light-gage structural bending members are shaped into configurations that have cross sections where the shear center and the centroid do not coincide.  In these cases, if the loads do not pass through the shear center, the beam will rotate as well as displace along the direction of the applied loads.  C-shaped channel sections fall into this category.  A realistic analysis of these shapes must calculate both the usual bending stresses and torsional stresses.  The torsional warping stresses act in the same direction as the bending stresses.  The bending stresses vary from top to bottom of the section with the maximum stresses equally distributed across the top and bottom flanges .The torsional moments will cause sideways bending of the flanges with maximum warping stresses occurring at the ends of the flanges. This is shown in the figure at the left above.  The maximum combined stresses therefore may occur at either of the points A. B or C. a

## Calculations for a  C-Channel Bending Stress and Torsional Stress Spreadsheet

The current edition of the AISI Specification for the Design of Cold-Formed Steel Members includes a requirement that combined bending and warping stresses be calculated in order to determine the maximum combined stress.  From these combined stresses a reduction factor is calculated.  This factor reduces the moment capacity established by bending alone.  This reduction factor assures that the maximum combined stresses will not cause premature failure of the beam.

The calculation of these maximum combined stresses is time consuming.  The spreadsheet available at www.EngineeringExcelSpreadsheets.com does these calculations for C-shaped members subject to the common load case of a uniformly distributed load.  Three possibilities are considered:

1. Top flange of the member braced against torsion only at its end supports.
2. Top flange of the member braced at its ends and mid-span.
3. Top flange braced at its ends and third-points.

For a C-shape, the centroid of the cross-section is on the same side of the web as the flanges.  The shear center is on the opposite side of the web from the flanges and hence the centroid.

Vertical loads applied to the top flange of the channel will not pass through the shear center.  The stresses in this case may be calculated by analyzing the member for bending alone then combining these with the torsional warping stresses calculated independently.  The torsional warping stresses are in line with the member similar to the bending stresses. They can be combined algebraically.

## Equations for a  C-Channel Bending Stress and Torsional Stress Spreadsheet

Torsional analysis requires calculation of the angle of rotation and its second derivative along the length of the beam.  Derivation of this approach may be found in the references. Reference 1 provides formulas for the angle of rotation for 12 loading and boundary support conditions.  In this spreadsheet, we assume simple torsional support for the boundary conditions at the end supports of the beam. This means the cross-section can warp freely at the ends of the member and the warping normal stresses are zero. If the member is part of a continuous beam, the user may input the beam end moments from separate analysis. In Reference 1, Case 4 applies to a uniformly distributed applied torsional moment along the span.  Case 3 applies to a concentrated applied torsion moment applied anywhere along the length of the beam.  These are the basis of this spreadsheet.

The formulas for the angle of rotation, Ɵ, are as follows:

Case 3 Concentrated torsional moment applied at a point αL from the left support with pinned end boundary condition:

Ɵ = TL/GJ [(1.0 – α)(z/L) + ((sinh αL/a)/(tanh L/a) – (cosh αL)) (a/L)sinh z/a)]

Case 4 Uniformly distributed torsional moment along length of beam with pinned end boundary condition:

Ɵ = (t a2/GJ )[(L2/a2)(z/L – z2/L2) + cosh z/a – (tanh L/2a) (sinh z/a – 1.0)]

The terms in these formulas and their units are described in the spreadsheet.

The spreadsheet consists of four sheets.  Each is for one of the loadings shown above.  Only the first sheet requires input data.  The user must place this required data in the amber colored cells.  All other cells are locked to protect the integrity of the spreadsheet. The final results for all possible bracing conditions are summarized on the last sheet.  The user selects the proper solution for the actual bracing condition.  This c-channel bending stress and torsional stress spreadsheet is available at www.EngineeringExcelTemplates.com.

References:

1. Heins, C.P. and P.A. Seaburg, “Torsional Analysis of Rolled Steel Sections”, Bethlehem Steel Design File, 1963.

2. Galambos, T.V., “Structural Members and Frames”, Prentice Hall, 1968.

3. Yu, Wei-Wen, “Cold-Formed Steel Design”, John Wiley & Sons, 1991.

4. Seaburg, P.A., “C Channel Beam Design Spreadsheet“, an online blog article.