# General

The following design considers EC2 (standard) and the National Annex (NA) for Denmark, Finland, Germany, Hungary, Norway, Sweden and United Kingdom.

With the steel module arbitrary structures in space can be designed with regard to a 1^{th} order or a 2^{nd} order analysis.

In the **Code Check** all checks prescribed in the codes depending on section type, section class and acting section forces are displayed

# Limitations

## Torsion

Only uniform torsion (St Venant torsion) is considered in the present version. For thin walled open sections the effect of warping torsion (Vlasov torsion) could be important and must then be considered separately.

## Crushing of the web

Crushing of an un-stiffened web due to a concentrated force is not checked in the present version.

# Global analysis

## General

The design can be performed using either:

- 1
^{th }order analysis, using the initial geometry of the structure or, - 2
^{nd}order analysis, taking into account the influence of the deformation of the structure.

### Choice between a 1^{th} order or a 2^{nd} order analysis

For structures not sensitive to buckling in a sway mode a 1^{th} order analysis is sufficient. The design for member stability should then be performed with non- sway buckling lengths.

In many cases it is easy to decide if a structure is sway or non-sway but in other cases it could be more difficult.

One way to estimate if the non-sway condition is fulfilled is described in EC3 part 1-1 with the following criterion:

α_{cr} = F_{cr }/ F_{Ed} > 10 => Non-sway

where:

**α _{cr}** is the critical parameter meaning the factor by which the design loading would have to be increased to cause elastic instability in a global mode,

**F**is the design loading on the structure,

_{Ed}**F**is the elastic critical buckling load for global instability mode based on initial elastic stiffness.

_{cr}An imperfection calculation in **FEM-Design** will display the critical parameters for the number of buckling shapes required by the user as shown below.

*Critical parameter αcr displayed for the three first buckling shapes with regard to load combination L1.*

As all critical parameters are > 10 a 1th order analysis and a design with non- sway buckling lengths would be sufficient in this case.

If the criterion above is not fulfilled 2nd order effects must be considered but a 1^{th} order analysis could still be used in most cases. This could be done either by amplifying the 1^{th} order moments or by using sway-mode buckling lengths. In FEM-Design the latter method should be used.

A full 2nd order analysis can be used for steel design in all cases.

## Structural stability

The calculations with regard to instability will be performed in different ways depending on the type of analysis.

### 1^{th} order theory

For a 1^{th} order design the following apply:

Both the flexural, lateral torsional and torsional buckling are calculated depending on the slenderness with respect to reduction factors specified in the appropriate code. The reduced slenderness for flexural and torsional buckling may be computed as:

λ = [A f_{y} / N_{cr}]^{1/2}

The critical normal force for flexural buckling N_{cr} will in this case be calculated using appropriate buckling lengths defined by the user in both directions for all members. The critical normal force Ncr for torsional buckling will be calculated according to support conditions defined by the user for all members. The reduced slenderness for lateral torsional buckling may be computed as:

λ_{LT} = [W f_{y} / M_{cr}]^{1/2}

The critical moment **M _{cr}** will be calculated as buckling of the compressed flange with regard to a buckling length defined by the user.

**Imperfections**

Initial bow imperfections may be neglected as these effects are included in the formulas for buckling resistance of the members.

For sway mode structures initial sway imperfections has to be considered. This could be done by changing the geometry before calculation or by defining systems of equivalent horizontal forces as described below

### 2^{nd} order theory

A 2^{nd} order calculation produces the critical normal force **N _{cr}** for flexural buckling and a corresponding stability check is not required. The critical normal force

**N**for torsional buckling or the critical moment

_{cr}**M**are not calculated since a basic finite element only contains the second order effects of the axial force and the effect of warping is neglected. The effect of the lateral torsional and torsional buckling will then have to be calculated as for 1

_{cr}^{th}order above.

**Imperfections**

For all structures initial local bow imperfections should be considered.

As the flexural buckling design is based on the 2^{nd} order effects of the bending moments it is vital that there is a moment distribution in all members. By considering local bow imperfections this is ensured also for hinged members without lateral load.

For sway mode structures also initial sway imperfections has to be considered. This could be done in the conventional way by changing the geometry before calculation or by defining systems of equivalent horizontal forces as described in the section below.

In **FEM-Design** both initial bow imperfections and initial sway imperfections are considered automatically by using an alternative method presented in EC3 1-1 as described below in the section below. The user has to connect each load combination to one of the calculated buckling shapes and the program will then calculate imperfections with regard to this shape. It is up to the user to decide how many buckling shapes that has to be considered and to which load combinations these shapes should be connected to receive an adequate result. No design based on a 2^{nd} order analysis for compressed members can be performed without considering imperfection for one of the available buckling shapes. Some examples describing this process are presented in the manual Useful examples.

## Imperfections for global analysis of frames

The following imperfections should be taken into account:

- Global imperfections for the structure as a whole.
- Local imperfections for individual members.

The assumed shape of global imperfections and local imperfections may be derived from the elastic buckling mode of a structure in the plane of buckling considered.

Both in and out of plane buckling including torsional buckling with symmetric and asymmetric buckling shapes should be taken into account in the most unfavorable direction and form.

For frames sensitive to buckling in a sway mode the effect of imperfections should be allowed for in frame analysis by means of an equivalent imperfection in the form of an initial sway imperfection and individual bow imperfections of members.

### Conventional method

Global initial sway imperfections

This effect could be considered in two ways.

- By changing the frame geometry before analysis with the slope as shown below. The slope φ will be calculated according to the relevant code.
- By defining a system of equivalent horizontal forces as shown below:

Where, in multi-storey beam and column building frames, equivalent forces are used they should be applied at each floor and roof level.

These initial sway imperfections should apply in all relevant horizontal directions, but need only be considered in one direction at a time.

The possible torsional effects on a structure caused by anti-symmetric sways at the two opposite faces, should also be considered

**Relative initial local bow imperfections of members for flexural buckling**

Equivalent horizontal forces introduced for each member as shown below could consider this effect.

The value**e**considering initial bow as well as residual stresses is calculated according to the relevant code._{o}

### Alternative method

As an alternative to the methods described above for calculating imperfections the shape of the

elastic critical buckling mode **η _{cr}** of the structure may be applied as a unique global and local imperfection according to EC3 1-1. This method is used in

**FEM-Design**when a 2

^{nd}order analysis together with imperfection for one of the available buckling shapes is chosen.

The amplitude of this imperfection may be determined from:

where:

**η _{init}** imperfection,

**η _{cr}** buckling shape.

where:

is the relative slenderness of the structure

**α** is the imperfection factor for the relevant buckling curve,

**χ** is reduction factor for the relevant buckling curve depending on the cross-section and the relevant code,

**M _{Rk}** is the characteristic moment resistance of the critical cross section,

**N _{Rk}** is the characteristic resistance to axial force of the critical cross-section,

**EI η _{cr,max}** is the bending moment due to η

_{cr}at the critical cross section,

where:

**α _{max}** is calculated from 1 st order theory and to be calculated at each cross-section.

**α _{ult,k}** is the maximum force amplifier for the axial force configuration N

_{Ed}in members to reach the characteristic resistance N

_{Rk}of the most axially stressed cross section without taking buckling into account,

**α _{cr}** is the minimum force amplifier for the axial force configuration

**N _{Ed}**

_{ }in members to reach the elastic critical buckling.

To be able to perform a design based on a 2 ^{nd} order analysis for compressed members imperfection for one of the available buckling shapes must be chosen by the user. It is important to decide which shapes that have to be considered for the current structure. Some examples are described in the manual **Useful examples**.

## Division of members

**Frame type structures**

A first calculation without division will show the force distribution in the structure. All compressed members should then be divided into an even member of elements e.g. four. It is recommended only to divide members in compression but as in reality members can be in compression for one loadcase and in tension for another this is often not possible.

If tensionend members are divided this could result in negative critical factors but they does not represent physical valid shapes and should be ignored. See also the manual **Useful examples, Example 2.**

**Truss type structures**

To perform a global stability check the grid members should not be divided but if local buckling is of interest they should. See the manual **Useful examples, Example 3.**

# EuroCode(EC3)

## Classification of cross-sections

The classification is made according to EN 1993-1-1 5.5-5.6.

A section is classified in section class 1, 2, 3 or 4 depending on the slenderness of the section, see EN 1993-1-1 5.5.2. Class 1 - The section can reach full plastic yielding, with sufficient rotation capacity for a plastic analysis.

Class 2 - The section can reach full plastic yielding, but the rotation capacity is limited.

Class 3 - The section can reach the yield limit without buckling. Class 4 - Local buckling will occur before the yield limit is reached.

In case of general cross-section the *line-topology* have to be determined to decide if the part is *internal* or *outstand*.

It can be decided based on the connections, see picture below. Parts between two nodes are *internal*, all other *outstand*.

If different parts of a section belongs to different classes the highest class will be chosen for the section.

## Axial force capacity

The capacity is calculated according to EN 1993-1-1 6.2.3-6.2.4.

### Tension force

The capacity is calculated as:

N_{t,Rd} = A f_{y} / γ_{M0}

where:

**A** gross area,**f _{y}** design strength,

**γ**partial factor.

_{M0}### Compression force

**Section Class 1, 2 and 3**

The capacity is calculated as:

N_{b,Rd} = χ A f_{y} / γ_{M1}

where:**χ **is the flexural buckling factor with regard to buckling around y-y axis and z-z axis respectively, **γ _{M1}** partial factor.

**Buckling factor**

The buckling factor is calculated as:

χ = 1 / ( + (^{2} - λ^{2})^{0,5}) ≤ 0,1

= 0,5 [1 + α(λ - 0,2) + λ^{2}]

λ is the slenderness calculated as:

λ = (A f_{y} / N_{cr})^{0,5}

**N _{cr}** is the elastic critical load for the relevant buckling mode.

The imperfection factor α is related to five groups according to the table below:

Group a_{o}, α = 0,13; group a, α = 0,21; group b, α = 0,34; group c, α = 0,49; group d, α = 0,76.

**Flexural buckling**

The slenderness parameter λ is calculated as:

λ = (A fy / Ncr)^{0,5}

**N _{cr}** is the critical load considering flexural buckling around the relevant axis.

N_{cry} = E I_{y} (π / L_{cry})^{2}; N_{crz} = E I_{z} (π / L_{crz})^{2}

Lcr is the buckling length.

**Torsional buckling and flexural-torsional buckling**

Torsional and flexural-torsional buckling is calculated according to EN 1993-1-1 6.3.1.4.

The slenderness parameter λ_{T} is calculated as:

λ_{T}= (A fy / Ncr )^{0,5}

where N_{cr} = N_{cr,TF} and N_{cr} < N_{cr,T}

**N _{cr,TF}** is the elastic torsional-flexural buckling force,

**N**is the elastic torsional buckling force,

_{cr,T}The critical load with regard to torsional buckling is calculated as:

N_{cr,T} = 1 / i_{p} 2 (G I_{T} + E I_{w} ( π / L_{cr} )^{2}

**G** is the shear modulus,**i _{p}** is the polar radius of gyration which in this case is ip = ((Iy + Iz) / A )0,5

**L**is the relevant buckling length,

_{cr}**N**is the critical load considering flexural-torsional buckling and is the lowest root to the third grade equation:

_{cr,TF}(N_{cry} – N_{cr,TF}) (N_{crz} – N_{cr,TF}) (N_{cr,T} – N_{cr,TF}) – (N_{cry} – N_{cr,TF}) N_{cr,TF}^{2} e_{z}^{2} / i_{p}^{2}

(N_{crz} – N_{cr,TF}) N_{cr,FT}^{2} e_{y}^{2} / i_{p}^{2} = 0

**N _{cry}** is the critical load with regard to flexural buckling around the y-axis as described above,

**N**is the critical load with regard to flexural buckling around the z-axis as described above,

_{crz}**N**is the critical load with regard to torsional buckling as described above,

_{cr,T}**e**is the distance between the centre of gravity and the shear centre in the y-direction,

_{y}**e**is the distance between the centre of gravity and the shear centre in the z-direction,

_{z}The polar radius of gyration is i

_{p}= (( I

_{y}+ I

_{z}) / A + e

_{y}

^{2}+ e

_{z}

^{2})

^{0.5}

## Bending moment capacity

The capacity is calculated according to EN 1993-1-1 6.2.5 and 6.3.2.4.

The capacity is calculated as:

**Section class 1 and 2**

M_{c,Rd} = W_{pl fy} / γ_{M0}

M_{b,Rd} = k_{fl χ} W_{pl} f_{y} / γ_{M1} (capacity with regard to lateral-torsional buckling)

where:

**W _{p}**

_{l}is plastic section modulus,

**f**is design strength,

_{y}**γ**

_{M0}, γ_{M}_{1}is partial factors,

**k**

_{fl}_{ }modification factor accounting for the conservatism of the equivalent compression flange method,

**χ**is reduction factor of the equivalent compression flange.

**Lateral-torsional buckling**

Lateral torsional buckling is calculated with the simplified assessment method according to EN 1993-1-1 6.3.2.4, flexural buckling of the com- pressed flange.

The reduction factor χ is calculated as shown above for flexural buckling of the compressed flange.

**Section class 3**

M_{c,Rd} = W_{el} f_{y} / γ_{M0}

M_{b,Rd} = k_{fl} χ W_{el} f_{y} / γ_{M1}**W _{el}** elastic section modulus.

## Shear capacity

Calculated according EN 1993-1-1 6.2.6, EN 1993-1-5 5.1-5.3.

The capacity is calculated as: V_{pl,Rd} = A_{v} f_{y} / 3^{0,5} / γ_{M0}

## Shear and Torsion

**Solid sections**

A_{v} = cross sectional area

**Rolled I and H sections, load parallel to web**

A_{v} = A - 2 b t_{f} + (t_{w} + 2r) tf ≥ η h_{w} t_{w}

**Rolled channel sections, load parallel to web**

A_{v} = A - 2 b t_{f} + (t_{w} + r) t_{f}

**Rolled T- section, load parallel to web**

A_{v} = 0,9 (A - b t_{f})

**Welded I, H and box sections, load parallel to web**

A_{v} = η Σ (hw t_{w})

**Welded I, H, channel and box sections, load parallel to flanges**

A_{v} = A Σ (hw t_{w})

**Rolled rectangular hollow sections of uniform thickness:**

load parallel to depth Av = A h / (b + h) load parallel to width Av = A b / (b + h)

**Circular hollow sections and tubes of uniform thickness**

A_{v} = 2 A / π

η is taken as 1,0.

In cases not included above the elastic capacity is calculated as:

τ_{Ed} / (f_{y} /(3^{0,5} γ_{M0})) ≤ 1,0

**Web buckling**

Calculated according to EN 1993-1-5 5.2

Web buckling is considered if: h_{w} / t_{w} >72 ε / η

ε = (235 / fy )^{0,5}

η = 1,2 for steel grades up to S460 then η = 1,0

The capacity is calculated as:

V_{b,Rd} = V_{bw,Rd} + V_{bf,Rd} ≤ η f_{yw} h_{w} t_{w} / (3^{0,5} γ_{M1}) Contribution from web:

V_{bw,Rd} = χ_{w} f_{yw} h_{w} t_{w} / (3^{0,5} γ_{M1})

The shear buckling factor χ_{w} is calculated as:

The program will understand that the member has a non-rigid end post if no stifeners are defined and as a rigid end post if a stiffener is defined at the very end of the member. For slender webs the capacity can be increased by defining additional stiffeners as shown below.

No transverse stiffeners defined:

Slenderness parameter λ_{w} = h_{w} /(86,4 t_{w} ε) else:

Slenderness parameter λ_{w} = h_{w} /(37,4 t_{w} ε κ_{τ}^{0.5})

κ_{τ} = 5,34 + 4,0 (h_{w} /a)^{2}; a / h_{w} ≥ 1

κ_{τ} = 4,0 + 5,34 (h_{w} /a)^{2}; a / h_{w} < 1

**a** = largest distance between transverse stiffeners in the member.

a = max (a1, a2, a3…)

**Contribution from flange**

When the flange resistance is not completely utilized in resisting the bending moment (MEd < Mf,Rd) the contribution from the flanges are calculated as:

V_{bf,Rd} = b_{f} t_{f} f_{yf} / (c γ_{M1}) (1 - (M_{Ed} / M_{f,Rd})^{2})

b_{f} ≤ 15 ε t_{f}

## Shear and Torsion

The capacity is calculated according to EN 1993-1-1 6.2.7.

The capacity is calculated as:

**I, H-sections or channel sections**

V_{pl,T,Rd} = [(1 - τ_{T,Ed} / (1,25 (f_{y} / 3^{0,5})/ γ_{M0})] V_{pl,Rd} where: V_{pl,Rd} is the shear capacity as above.

**Hollow sections**

V_{pl,T,Rd} = [(1 - τ_{T,Ed} / ((f_{y} /3^{0,5}) / γ_{Mo})] V_{pl,Rd}

## Warping torsion (Vlasov torsion)

Warping torsion is not considered in the present version.