## Friday, 17 May 2019

### Secret formulas of EN 1993-1-3 – Part 3

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. Isand Asare 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 ceff=13 mm and beff=29 mm, respectively.

The section properties can be calculated with ConSteel

In the 1st part of this blog a „K” value of K1=0.080 N/mm2was found for this edge stiffener. Using the values calculated by ConSteel’s Section module the critical stress will be
σcr= 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 NEd=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 σcr= 124.8*1 = 124.8 MPa which is almost equal to the value calculated with formula (5.15) of EN 1993-1-3.

## Friday, 10 May 2019

### Secret formulas of EN 1993-1-3 – Part 2

In our series we continue with the second „secret” formula of EN 1993-1-3.

This formula (5.11) is used when the ability of an intermediate stiffener to stabilize a compressed web plate is studied.

When the intermediate stiffener is rigid enough, it divides efficiently the longer web plate into shorter plates. During buckling a rigid stiffener will not displace and the plate will buckle between the ends (stabilized by flanges) and this stiffener:

If the stiffener is not rigid enough, already at a much lower stress level the distortional buckling will appear which is characterized by a displacement of the stiffener and by the web plate buckling with a half-wave equal to the plate length:

In such a case Eurocode still accepts that web plate buckling will occur between ends and intermediate stiffener, but reduces the thickness of the intermediate stiffener zone, to compensate for the less than ideal rigidity. This compensation will depend on the buckling resistance of the stiffener when subject to compression.
The simplified model used by Eurocode in the procedure can be seen below. As a simplification it is assumed that the web plate is connected to the rest of the section with hinges. It means that it’s buckling shape is not influenced by the stiffness of connected parts, they do not encaster its ends, its ends can freely rotate. The ends of the plate are supported and there the plate can provide an elastic bedding to the stiffener, which will be considered as a „help” when its buckling resistance will be determined. This bedding is represented by the K spring stiffness:

The spring value which represents in this model the bedding provided by the plate is calculated as the ratio of the displacement obtained from the application of a unit load at the center of gravity of the stiffneer. In this case the unit load is applied perpendicularly to the web plate. Formula (5.11) gives the deflection (δ) from such unit load (u).

The ConSteel model shown below (dowonload the model here) reproduces the stiffness calculation for the web plate of a 1 m long Cee section where the web has been stiffened by a longitudinal intermediate web stiffener. As the formula (5.11) gives the deflection from a single concentrated load, the „V” shaped intermediate stiffener will not be modelled, only the flat plate with tthickness.
C channel, nominal thickness = 1.524 mm, 150 mm deep with an intermediate stiffener of 8 mm depth located right at the middle of the plate
in case of when axial compression force is applied on this section
δ= 1*75^2*75^2/(3*(75+75))*12*(1-0.3^2)/(210000*1.524^2) = 1.033 mm

The resulting horizontal displacement from the point load is 1.135 mm without the effect from tranverse strain. By multiplying this value by (1-ν2) gives 1.135 *(1-0.32) = 1.033 mm
This displacement will be converted into K spring stiffness value

resulting K = u/δ=1/1.033 = 0.968 N/mm2
In the next blog post we will show, how the buckling resistance of the stiffener is determined. For this the calculated spring values will be needed.

## Friday, 3 May 2019

### Secret formulas of EN 1993-1-3

EN 1993-1-3 contains 3 „secret” formulas. The first two are used to determine the effective cross section due to distortional buckling when edge or intermediate stiffeners are involved. The third is used to calculate the distortion of the whole cross section when analyzed with a connected sheeting.

The physical meaning of all three formulas can be easily shown with simple ConSteel models which helps designer to understand the underlaying mechanial model.

The first formula (5.10) is used when the ability of an edge stiffener to stabilize a compressed flange of Z or C section is studied. During distortional buckling the intersection point of flange with the lip (it is called as edge stiffener) is expected to move in a direction perpendicular to the flange. This formula gives the stiffness value provided by the Z or C section, when is assumed that during deformations the point of intersection of the web with the flange doesn’t move. This assumption corresponds to attaching supports to these nodes as seen on picture 5.6 of EN 1993-1-3.

When a compressed edge stiffener would buckle, it will be partially restrained by the section with these attached supports. Depending on the distribution of normal stresses on the section, one or two edge stiffeners might be under compression. If both stiffeners are under compression and tend to buckle, the restraining capacity of the section will have to be shared between them. This sharing requirement is reflected by the coefficent kf. The spring stiffness value will be used as a distributed spring support when the buckling resistance of the edge stiffener is calculated.

Stiffness values are typically calculated as the ratio of a displacements obtained from the application of a unit load. In this case the unit loads are applied parallelly to the expected displacement of the compressed edge stiffeners.

The ConSteel model shown below (download the model here) reproduces the stiffness calculation for a 1 m long Zee section (as a simplification the unit loads are placed at the intersection of the flange with the lip and not at the center of the gravity of the edge zone):
Z purlin, nominal thickness = 1.30 mm, 200 mm deep, 72 mm wide symmetric flanges, 15.5 mm deep lips
in case of My bending: kf = 0, b1 = b2 = 72 mm, hw = 198.7 mm, t = 1.26 mm
K1 = 210000*1.26^3/4*(1-0.3^2)*1/(72^2*198.7+72^3+0.5*72*72*198.7*1.0) = 0.08 N/mm2

The value of spring stiffness is: K = u / δ=1 kN/m / 12.9 mm = 0.0775 N/mm2 without the effect from tranverse strain. By multiplying this value by (1-ν2) gives 0.0775/(1-0.32) gives 0.08 N/mm2

in case of compression: kf = 1.0, b1 = 72 mm, hw = 198.7 mm, t = 1.26 mm
K1 = 210000*1.26^3/4*(1-0.3^2)*1/(72^2*198.7+72^3+0.5*72*72*198.7*1.0) = 0.06 N/mm2

The resulting vertical displacement from the point load is 17.7 mm.
The value of spring stiffness is: K = u / δ=1 kN/m / 17.7 mm = 0.056 N/mm2 without the effect from tranverse strain. By multiplying this value by (1-ν2) gives 0.056/(1-0.32) gives 0.06 N/mm2