In our series we have shown in Part 1 and Part 2 how the spring stiffness „K” is determined for and edge and for an intermediate stiffener. In this part we will show how to proceed further.

The goal is to determine how efficiently can this stiffener support the connected compressed plate. To consider local buckling of the compressed plate the effective widths will be calculated. These widths can be either calculated for a plate supported at both ends or for a plate supported at one end only, using Table 4.1 and Table 4.2, respectively.

If a stiffener fulfills the minimum constructive requirements, we will first assume that it is rigid enough the act as a support. Based on this we work out the effective lengths for the connected compressed plates, assuming also that they are fully loaded up to the yield strength. Once we know the effective widths of the plate parts connected to the stiffener, we also know the load this stiffener is supposed to be able to carry without buckling, to justify the first assumption. Therefore, next we calculate the flexural buckling resistance of the stiffener. If we find out that it is lower than what was implicitly assumed previously, we go back to the first step, lower the stress to be compatible with the obtained buckling resistance of the stiffener (distortional buckling). This lower stress level may of course result different effective widths. An iterative procedure is started until the satisfactory result is found. After completion the final stiffener buckling resistance value will be incorporated into an equivalent effective thickness applicable for the stiffener and the connected effective plate parts.

In the above described iterative process the „K” value of the stiffener will be needed to determine the flexural buckling resistance of the stiffener. This resistance will be calculated using the help of a simple beam model, where the stabilization provided the remaining parts of the section is used as a continuous bedding of „K”. The cross section of the beam corresponds to the stiffener and connected plate parts and will be exposed to the actual stress level of the iteration, assumed as a simplification to be constant along the length. The length of the beam in this model is unknown at the moment, it should be equal to the half wavelength of the buckling shape which is expected to be around 1.5-3 times the height of the studied cold-formed section.

Distortional buckling will appear only if the length of the stiffener is much longer than this typical length range. As a further simplification Eurocode assumes infinite length as the possible worst case, because for this condition an analytic solution exists in the literature:

In this formula „K” is the spring value shown in Part 1 and Part 2 of this series of blog and it represents the spring stiffness value what the remaining section can provide to the stiffener. I_{s}and A_{s}are section properties of the corresponding stiffener together with connected effective widths of supported plates, namely area and inertia around an axis perpendicular to the expected direction of buckling of the stiffener. E is the Young modulus of the steel material.

The formula gives the critical stress level where elastic buckling would appear. Once this has been found, reduced slenderness is calculated, and the distortional buckling resistance can be obtained by using a special buckling curve given by (5.12a-c) formulas of EN 1993-1-3.

The validity of formula (5.15) can be easily demonstrated with a simple ConSteel model.

Let’s assume the Z section from our 1st blog exposed to compression force and let’s assume that we have already calculated effective width values. Yield strength of the material is 235 MPa and coating thickness is 0.04 mm. We found that the effective width of the lip and flange are c_{eff}=13 mm and b_{eff}=29 mm, respectively.

The section properties can be calculated with ConSteel

In the 1st part of this blog a „K” value of K_{1}=0.080 N/mm^{2}was found for this edge stiffener. Using the values calculated by ConSteel’s Section module the critical stress will be

σ_{cr}= 2*sqrt(K*E*Is)/As = 2*sqrt(0.080 * 210000 * 555)/49 = 124.63 MPa

In order to create a ConSteel model to demonstrate this calculation let’s assume an arbitrary model length of 3000 mm which seems to be long enough to form distortional buckling (flexural buckling in this case). A vertical continuous bedding of 0.08 N/mm/mm has been applied and a compression force of N_{Ed}=49*1=49 N corresponding to a unit compressive stress of 1 MPa.

For this model ConSteel finds the first critical multiplier as 111.67

This corresponds to a critical stress of σ_{cr}= 124.8*1 = 124.8 MPa which is almost equal to the value calculated with formula (5.15) of EN 1993-1-3.