NCCI Calculation of Alpha-cr

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This NCCI sets out

This NCCI sets out the basis for the calculation of the basis for the calculation of alpha-cr, the parameter that measuresalpha-cr, the parameter that measures the stability of the fra

the stability of the frame.me.

1. Methods for determining

cr

EN 1993-1-1 §5.2.1 concerns the checking of buildings for sway mode failures and defines the parameter α _cr_{as follows:}

Ed cr F F cr

=

α in which

F Ed is the design load on the structure

F cr is the elastic critical buckling load for the global instability mode.

For multi-storey buildings, the value of α _cr_{is calculated for each storey in turn and the}

criterion of expression (5.1) must be satisfied for each storey.

EN 1993-1-1 §5.2.1(4)B states: “α _cr_{may be calculated using the approximate formula (5.2)”,}

which is given as:

⎟

⎠

⎞

⎜

⎝

⎛

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

=

Ed H, Ed Ed cr δ α h V H where

H Ed is the (total) design value of the horizontal reaction at the bottom of the storey to the horizontal loads and fictitious horizontal loads

V Ed is the total design vertical load on the structure at the bottom of the storey

δ _H,Ed _{is the horizontal displacement at the top of the storey, relative to the bottom of the}

storey (due to the horizontal loads) h is the storey height

An illustration of the displacement of a multi-storey building frame under horizontal loads is given in Figure 1.1. b e r 1 0 , 2 0 0 9 r i g h t s r e s e r v e d . U s e o f t h i s d o c u m e n t i s s u b j e c t t o t h e t e r m s a n d c o n d i t i o n s o f t h e A c c e s s S t e e l L i c e n c e A g r e e m e n t

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Horizontal load applied as series of forces, calculated separately for each storey h δ H,Ed H₁ H₂ H3 H₄ H_Ed= H₁ HEd= H1+H2 H_Ed= H₁+ H₂+H₃ H_Ed= H₁+H₂+H₃+H₄ h δ H,Ed H₁ H₂ H3 H₄ H_Ed= H₁ HEd= H1+H2 H_Ed= H₁+ H₂+H₃ H_Ed= H₁+H₂+H₃+H₄

Figure 1.1 Displacement of a multi-storey frame due to horizontal loads (deflection parameters for second storey only illustrated)

As an alternative to formula (5.2), in certain cases other checks may be more convenient or more appropriate. The following three alternatives may be considered:

Al tern ati ve (1)

Use formula (5.2) with H Ed determined by the fictitious horizontal loads from the initial sway imperfections in 5.3.2(7) alone and with δ _H,Ed_{as the displacements arising from these}

fictitious horizontal loads (i.e. exclude the effects of any other horizontal loads, such as wind loads).

Al tern ati ve (2)

Calculateα _cr_{by computer by finding the first sway-mode from an eigenvalue analysis. When}

using this type of analysis, it is important to study the form of each buckling mode to see if it is a frame mode or a local column mode. In frames where sway stability is ensured by

discrete bays of bracing (often referred to as “braced frames”), it is common to find that the

m e n t i s s u b j e c t t o t h e t e r m s a n d c o n d i t i o n s o f t h e A c c e s s S t e e l L i c e n c e A g r e e m e n t

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2. Simplification of load distribution

In calculating F cr for normal multi-storey building frames, it is adequate to model the frame with the loading applied only at the nodes, thereby ignoring the bending moments caused by load distribution. However, for long span portal frames in which the bending moments in the members give rise to significant axial compression in the rafters, the distribution of the load must be modelled when calculating α _cr_{. According to Note 2B of EN 1993-1-1 §5.2.1(4)B,}

the axial compression in a beam or a rafter may be assumed to be significant if:

Ed y 3 , 0 N Af

≥

λ in which

N Ed is the design value of the compression force

λ _{is the in-plane non-dimensional slenderness calculated for the beam or rafter}

considered as hinged at its ends with the length equal to the system length measured along the beam or rafter.

3. Scope of application

The formula (5.2) in EN 1993-1-1 §5.2.1(4)B and alternatives (1) and (3) above apply to normal beam and column buildings and normal portals, because the global instability mode is a sway mode. For certain other forms of frame, such as arches, domes or pyramids, the

lowest mode of buckling is not a sway mode, so formula (5.2) will not give a safe value of

α _cr_. b e r 1 0 , 2 0 0 9 r i g h t s r e s e r v e d . U s e o f t h i s d o c u m e n t i s s u b j e c t t o t h e t e r m s a n d c o n d i t i o n s o f t h e A c c e s s S t e e l L i c e n c e A g r e e m e n t

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Quality Record

RESOURCE TITLE NCCI: Calculation of alpha-cr

Reference(s)

ORIGINAL DOCUMENT

Name Compan y Date

Created b y Charles King The Steel Construction

Institute

Technical cont ent checked by Martin Heywood The Steel Construction

Institute

Editorial content checked by D C Iles SCI 6/5/05

Technical content endorsed by t he following STEEL Partners:

1. UK G W Owens SCI 25/4/05

2. France A Bureau CTICM 25/4/05

3. Sweden A Olsson SBI 25/4/05

4. Germany C Mű_ller _RWTH _25/4/05

5. Spain J Chica Labein 25/4/05

Resource approved by Technical Coordinator

G W Owens SCI 22/4/06

TRANSLATED DOCUIMENT

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