NCCI Calculation of Alpha-cr

NCCI: Calculation of alpha-cr

 NCCI: Calculation of alpha-cr  SN004a-EN-EU

NCCI NCCI:: Calcu Calcu latio n of o f alph a-cr  a-cr  This NCCI sets out the basis for the calculation of alpha-cr, the parameter that measures  the stability of the frame. frame.

Contents

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

1.

Methods for determining α cr  cr 

2

2.

Simplification of load distribution

4

3.

Scope of application

4

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NCCI: Calculation of alpha-cr

 NCCI: Calculation of alpha-cr  SN004a-EN-EU

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: α cr 

F cr 

=

F Ed 

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

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: α cr 

⎛  H   ⎞⎛  h  ⎞ ⎟ = ⎜⎜ Ed  ⎟⎟⎜⎜ ⎟ V  δ   ⎝  Ed   ⎠⎝  H,Ed  ⎠

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.

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NCCI: Calculation of alpha-cr

 NCCI: Calculation of alpha-cr  SN004a-EN-EU

Displacement due to horizontal Reaction at bottom loads of storey H1

Horizontal load applied as

H2

HEd = H1

H3

HEd = H1 + H2

series of forces, calculated separately for  each storey h HEd = H1 + H2+ H3

H4

HEd = H1

+ H2 + H3 +

H4

δ  H,Ed

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

 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) Calculate α 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 eigenvalues of the column buckling modes are lower than the eigenvalue of the first sway mode of the frame. Local column modes may also appear in unbraced frames at columns hinged at both ends or at columns that are much more slender than the average slenderness of  columns in the same storey.

 Al tern ati ve (3) F cr  may be found from design charts appropriate to the type of building considered.

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NCCI: Calculation of alpha-cr

 NCCI: Calculation of alpha-cr  SN004a-EN-EU

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: λ 

≥ 0,3

 Af y  N Ed 

in which  N Ed     t   n   e   m   e   e   r   g    A   e   c   n   e   c    i    L    l   e   e    t    S   s   s   e   c   c    A   e    h    t    f   o   s   n   o    i    t    i    d   n   o   c    d   n   a   s   m   r   e    t   e    h    t   o    t    t   c   e    j    b   u   s   s    i    t   n   e   m   u   c   o    d   s    i    h    t    f   o   e   s    U  .    d    9  e   v    0  r    0  e    2  s  ,   e    0  r   s    1   t   r   h   e   i    b  g   r    l   m    l   e  a    t     p   t   e   h    S   g  ,   i   y  r   a  y    d  p   o   s   r   c   u  s    h   i    l    T   a   n   i   r   o   t   e    d  a   e    t   m   a  s    i   e   r   h    C    T

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 .

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NCCI: Calculation of alpha-cr

 NCCI: Calculation of alpha-cr  SN004a-EN-EU

Quality Record RESOURCE TITLE

NCCI: Calculation of alpha-cr 

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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

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 Műller

RWTH

25/4/05

5. Spain

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Labein

25/4/05

G W Owens

SCI

22/4/06

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