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A design procedure is presented to estimate the load carrying capacity of beam-column channel sections. A reduced cross-section is used to compensate for the reduction in the post-buckling stiffness. The non-linear stress distribution acting on the entire channel width is replaced by simplified linear distributions. Using this simplified concept, the maximum stress in the post-buckling state, is assumed to be carried entirely by both edges while the central region of the channel remains unstressed. Thus a fraction of the channel section is considered in resisting the applied loading. This approximation enables the structural engineer to deal with a simplified stress distribution to compute the ultimate strength instead of the non-linear one.

Channel sections are extensively used in steel construction. The design of these members depends upon their industrial application. Channel sections are commonly used in light steel construction, such as false ceilings, catwalks, lighting partitions, curtain walls, cable trays, … etc. In this case excessive deformation is not favourable because the members support structures sensitive to deflection. Therefore, buckling would be the governing limit state. Channel sections are also used in the design of structures subjected to medium loadings, such as purlins, bracings, girts, …etc. This group is suitable for mild post-buckling deflection. Therefore serviceability limit state governs the design. The channel member in this case is braced in the lateral direction at regular intervals in order to reduce the effective length. Channel sections are also used in heavily loaded steel construction. Examples are column studs, primary beams framing, etc. A cost effective procedure is to design these members to accommodate full post-buckling capacity. Therefore, with proper understanding to the structural behaviour, the channel section can be grouped by the fabricators to provide cost-effective products in the marketplace.

The determination of the ultimate strength is a challenging in the design of channel sections. Several investigations were conducted by many researchers to provide understanding to the channel behaviour. Dinis and Camotim [

Objective of this paper is to provide simplified design procedure to estimate the load carrying capacity of thin walled beam-column channel sections. The non-linear stress distribution is approximated by a simplified linear distribution. A reduced cross-section is used to compensate for the reduction in the stiffness in the post-buckling range. This simplified procedure enables the structural engineer to compute the ultimate strength of the channel section and avoid the cost and effort that any numerical non-linear analysis may require.

Consider typical beam-column channel section of length (L), width (b) and thickness (t) as shown in _{cr}) is less than the yield stress (σ_{Y}). In the post-buckling state, the longitudinal membrane stresses (σ_{x}) shift from the center of the web and concentrate at the longitudinal edges. If the applied load is uniform compression, the maximum edge stresses are equal (σ_{max1} = σ_{max2}). With increasing load beyond buckling, the longitudinal edges carry, by equal amounts, most of the load compared to the center of the web. If the applied load is not uniform, the stresses at the edges are not equal (i.e. σ_{max1} ≠ σ_{max2}). Their magnitudes depend on the distribution of the applied load. Therefore, the stress, in the post-buckling range, carried by each edge differs and one anticipates that one edge will yield before the other.

The non-linear stress distribution acting on the entire width of the web can be replaced by equivalent uniform blocks distributed over two reduced effective widths (b_{e}_{1} and b_{e}_{2}). By using this simplified concept, the maximum stress in the post-buckling state, is assumed to be carried entirely by the longitudinal strips while the central region of the plate remains unstressed. Thus only a fraction of the channel width is used in the design.

Therefore, the non-uniform stress distribution shown schematically in _{e}_{1}) and (b_{e}_{2} ), such that

where η and ξ are non-dimensional parameters = y/b and x/L, respectively. (σ_{max1}) and (σ_{max2}) are the stresses at η = –0.5 and 0.5 respectively, and (b_{e}_{1}) and (b_{e}_{2}) are their corresponding effective widths. The right block can be assumed to carry the load from the center line of the channel, η = 0, to the edge η = 0.5, i.e.

Similarly, the left block is assumed to carry the load from η = 0 to the other edge, η = –1/2, i.e.

Note that (b_{e}_{1}) and (b_{e}_{2}) should satisfy the following inequality;

In some loading scenarios, the resulting non-linear stress distribution has the profile shown in

left edge of the web, η = –0.5. It is located in the vicinity of the channel center line. A reasonable approximation of (b_{e}) in this case, is to assume that the stress is completely carried by the right edge, η = 0.5, as shown in

The determination of the stress distribution (σ_{x}) can then be obtained by solving the following coupled compatibility and equilibrium differential

where

where β is the plate aspect ratio = L/b, W is the total out of plane deflection, E is Young’s modulus, D is the plate flexural rigidity given by Et^{3}/12(1 − v^{2}) and F is the stress function defined as;

For beam-column channel section subjected to combined compression and major axis bending (clockwise), the applied loading varies linearly in the transverse direction and can be expressed as;

where (Ψ) is the stress gradient coefficient that relates the applied stresses at the channel edges. In this case (s_{1}) is the compressive stress at the heavily loaded edge (η = 0.5). The applied stress at the opposite channel edge (η = –0.5) equals (Ψs_{1}). The post-buckling stress distribution is obtained by solving the coupled Equations (6) and (7), and is expressed as;

By substituting the values [σ_{max1}, σ_{max2}], into Equations (2) and (3) the width of each block [b_{e}_{1}, b_{e}_{2}] will have the following expressions;

_{e}_{1}) and (b_{e}_{2}). Note that the legends used in both diagrams are identical. For example, the solid curve represents stress ratio = 2.5. The dashed and long dashed curves represent stress ratios = 2 and 1.5, respectively. It can be observed that by increasing the applied load in the post-buckling state, the stress shifts from the central part of the channel and concentrates at the edges. The increase of the stresses at the heavily loaded edge is very rapid. The location of the minimum stress for this loading scenario shifts from the center line toward the left edge, η = –0.5 and is located at