In Tekla Structural Designer on the Analysis page of the Design Options dialog you have the choice of three analysis types. These are,

• First-order (Elastic) analysis,
• Amplified forces (k _amp ) method (uses first-order analysis),
• Second-order analysis.

For both steel and concrete, first-order analysis is only acceptable providing second order effects are small enough to be ignored - see: A practical approach to setting the analysis type below.

Both steel and concrete can use the amplified forces method to determine second-order effects although for steel this does have restrictions on use (basically regular frameworks with α _cr > 3 - see Validity of the amplified forces method below). Full second-order analysis is preferred for steelwork and since it is not precluded by EC2 it can be used for concrete.

The amplified forces method is described differently in EC3 compared to EC2, whilst the presentations are different, they are both based on the amplifier, k _amp given as,

k _amp = 1/(1 - 1/α _cr )

During the design process for both steel and concrete members, the member end forces from the analysis of the lateral loadcases are amplified by the 'appropriate' value of k _amp. Since the analysis is first-order this is carried out as part of summing the load effects from each loadcase (multiplied by their appropriate load factor given in the design combination). The 'appropriate' value is the worse of k _amp,Dir1 and k _amp,Dir2 based on the worst value of α _cr for all stacks in the building.

The k _amp results are summarised for each column in both directions. These can be viewed as follows:

1. Open a Review View, and select Tabular Data from the Review toolbar.
2. Select 'k _amp ' from the View Type drop list on the Review toolbar.
3. The k _amp results in both directions are tabulated for each column in the building.

EC3 Clause 5.2.2 (6)B lists limitations on the applicability of the Amp. Forces method. It is therefore your responsibility when selecting this method to ensure all of the following:

• all storeys have a similar distribution of vertical load
• all storeys have a similar distribution of horizontal load
• all storeys have a similar distribution of frame stiffness with respect to the applied storey shear forces

Also according to clause 5.2.1 (4)B limitation:

• roof slope shallow - not steeper than 1:2 (26 degs)
• axial compression in beams or rafters - N _cr / N _ed <= 11.1

Full second-order analysis is more widely applicable for steelwork structures and since it is not precluded by EC2 it can be used for concrete.

The accuracy of determination of the second-order effects for concrete structures is dependent upon a reasonable estimation of the concrete long term properties. This is a significant issue for both the amplified forces method and second-order analysis. It is therefore important that appropriate member type specific modification factors have been specified - see Use of modification factors

Unless α _cr is greater than 10 (in which case second-order effects can be ignored), it is essential that your final design utilizes one of the second-order analysis approaches. During the initial sizing process you may however choose to run a first-order analysis. Proceeding in this way you can obtain sections and an overall building performance with which you are satisfied, before switching to one of the P-∆ analysis methods.

The following approach to setting the analysis type is suggested:

1. On the Analysis page of the Design Options dialog, initially set the analysis type to First-order analysis.
2. Perform Design All (Gravity) using first-order analysis in order to size members for the gravity loads.
3. Once the members are adequately sized for the gravity combinations obtain a figure for the building's elastic critical load factor, α _cr (See: How do I assess the worst elastic critical load factor for the building? )
4. If the α _cr that has been determined is greater than 10 you can continue to perform Design All (Static) with the analysis type set to First-order analysis.
5. If α _cr is less than 10 you will need to proceed with one of the second-order approaches - and if it is very low (i.e. less than 2.0) some remodelling is required:
   • Either, refine the design until α _cr is greater than 2.0 to make the structure suitable for a final design using the full second-order approach, (which is the only method permitted if the structure contains non-linear members such as tension only braces),
   • Or, in order to use the amplified forces approach, refine the design further until α _cr is greater than 3.0.
6. When a suitable α _cr has been achieved change the analysis type to the full second-order, or the amplified forces method as appropriate.
7. With the analysis type still set to the full second-order, or the amplified forces method, perform Design All (Static).

Susceptibility to second order effects is a general characteristic and is not material specific, it has just been presented differently in EC3 and EC2:

• In EC3 a building can be considered as 'non-sway' when the elastic critical load factor α _cr ≥ 10, else the building is 'sway sensitive' and (global) second-order effects must be taken into account.
• In EC2 the definition is slightly different - it does not use the terms 'non-sway' and 'sway sensitive'. Rather it simply defines when second-order effects are small enough to be ignored. The principle is given in Clause 5.8.2 (6) which states that they can be ignored if they are less than 10% of the corresponding first order effects. Because of the way in which the amplification factor, k _amp is calculated the change point is at an α _cr of 11 not 10. (see: Derivation of the kamp formula for concrete structures ).

However, the intent is the same in both cases and so in Tekla Structural Designer α _cr ≥ 10 is taken as the change point. In any event, you are not restricted in your choice of analysis type irrespective of the value of α _cr (it is your call, although we will warn you about it)

To determine the sway sensitivity for the building as a whole, the worst stack (storey) in the worst column throughout the building in both directions has to be identified - this can be done as follows:

1. On completion of the analysis, open a Review View and select Tabular Data from the Review toolbar.
2. Select 'Sway' from the View Type drop list on the Review toolbar.
3. The elastic critical load factor in both directions (α _Dir1 & α _Dir2 ) is tabulated for each column in the building.
4. Make a note of the smallest elastic critical load factor from all of the columns in either direction - this is the α _cr value for your building.

Having determined an α _cr for your building, you then use it when deciding which is the most appropriate analysis type for design.

The elastic critical load factor, α _cr is calculated in the same manner for steel and concrete. The approach adopted is that for each loadcase containing gravity loads (Dead, Imposed, Roof Imposed, Snow) a set of Equivalent Horizontal Forces (EHF) are determined. These consist of 0.5% of the vertical load at each column node applied horizontally in two orthogonal directions separately (Direction 1 and Direction 2). From a first order analysis of the EHF loadcases the deflection at each storey node in every column is determined for both Direction 1 and Direction 2. The difference in deflection between the top and bottom of a given storey (storey drift) for all the loadcases in a particular combination along with the height of that storey provides a value of α _cr for that combination as follows,

α _cr = h/(200 * δ _EHF )

Where

h = the storey height

δ _EHF = the storey drift in the appropriate direction (1 or 2) for the particular column under the current combination of loads

EC2 provides two specific approaches to determine the change point below which second-order effects are small enough to be ignored:

The first specific approach is contained in Clause 5.8.3.3 which provides a pass/fail criterion to check whether the global second-order effects may be ignored. It is given as,

F _VEd = k ₁ * n _s /(n _s + 1.6) * S(E _cd * I _c )/L ²

where

F _VEd = the total vertical load (on 'braced' and 'bracing' members)

k ₁ = a factor that allows for cracking in the concrete of the LLRS and is a Nationally Determined Parameter (NDP)

* n _s = number of storeys

E _cd = the design value of the modulus of elasticity of the concrete

I _c = the second moment of area of the uncracked bracing members

L = the total height of the building

However, the above approach has a number of restrictions in its application and as a result it is not applied in Tekla Structural Designer.

The second specific approach is given in Annex H.

The method given in Annex H.1.2 is the background for the more limited method given in Clause 5.8.3.3 as described above, but it does not apply where there is significant shear deformation in the LLRS e.g. for shear walls with significant openings, hence again it is not considered in Tekla Structural Designer.

Instead, recourse is made to determining the level of second-order effect using Annex H.2. Using this approach, by rearranging Equation H.8 it is possible to provide a 'stability coefficient' 1/α _cr which can be applied as the change point between non-sway and sway sensitive structures.

F _HEd = F _H0Ed /(1-F _H1Ed/ F _H0Ed ) Equation H.8

Where:

F _H1Ed = fictitious horizontal force, giving the same bending moments as vertical load N _VEd acting on the deformed structure, with deformation caused by F _H0Ed (first order deformation), and calculated with nominal stiffness values according to 5.8.7.2

Considering how this definition of F _H1Ed might apply to an imaginary cantilever of height, h, we arrive at:

1. The moment due to F _H1Ed is the same as that due to the vertical load N _VEd , so:

F _H1Ed * h = N _VEd * δ

which can be rearranged to:

F _H1Ed = (N _VEd * δ)/h

2. Substituting for F _H1Ed in Equation H.8, we have:

F _HEd = F _H0Ed /(1 - (N _VEd * δ)/(F _H0Ed * h)

3. By defining k _amp = F _HEd /F _H0Ed the above can be rearranged to:

k _amp = 1/(1 - (N _VEd * δ)/(F _H0Ed * h)

4. Now, the EC3 Equation 5.2 for the elastic critical buckling load is:

α _cr = H _Ed /V _Ed * h/δ _HEd

which, when re-expressed in the terminology used in H.2 becomes:

α _cr = F _H0Ed /N _VEd * h/δ _HEd

and when further rearranged becomes:

1/α _cr = (N _VEd *δ _HEd )/(F _H0Ed * h)

5. Hence 1/α _cr can be substituted into the above equation for k _amp so that we arrive at the more well-known formula for amplification:

k _amp = 1/(1 - 1/α _cr )