SUPPLEMENTARY INFORMATION 4

Buckling lengths of compression members

1. Basis

(1) The buckling length Lcr of a compression member is the length of an otherwise similar member with “pinned ends” (ends restrained against lateral movement but free to rotate in the plane of buckling) which has the same buckling resistance.

(2) In the absence of better information, the theoretical buckling length for elastic critical buckling may conservatively be adopted.

(3) An equivalent buckling length may be used to relate the buckling resistance of a member subject to non-uniform loading to that of an otherwise similar member subject to uniform loading.

(4) An equivalent buckling length may also be used to relate the buckling resistance of a non-uniform member to that of a uniform member under similar conditions of loading and restraint.

2. Columns in building frames

(1) The buckling length Lcr of a column in a non-sway mode may be obtained from Figure 1.

Figure 1: Buckling length ratio Lcr / L for a column in a non-sway mode

(2) The buckling length Lcr of a column in a sway mode may be obtained from Figure 2.

Figure2: Buckling length ratio Lcr / L for a column in a sway mode

(3) For the theoretical models shown in Figure 3 the distribution factors η1 and η2 are obtained from:

111 12

c

c

KK K K

η =+ +

(1)

221 22

c

c

KK K K

η =+ +

(2)

where Kc is the column stiffness coefficient I / L and Kij is the effective beam stiffness coefficient.

Figure 3: Distribution factors for columns

(4) These models may be adapted to the design of continuous column, by assuming that each length of column is loaded to the same value ratio (N / Ncr). In the general case where (N / Ncr) varies, this leads to a conservative value of Lcr / L for the most critical length of column.

(5) For each length of a continuous column the assumption made in (4) may be introduced by using the model shown in Figure 4 and obtaining the distribution factors η1 and η2 from:

11

1 11 1

c

c

K KK K K K

η2

+=

+ + + (3)

22

2 21 2

c

c

K KK K K K

η2

+=

+ + + (4)

where K1 and K2 are the stiffness coefficients for the adjacent lengths of column.

Figure 4: Distribution factors for continuous columns

(6) Where the beams are not subject to axial forces, their effective stiffness coefficients may be deter vided tha st esign moments.

Table 1: Effective stiffness coefficient for a beam

Conditions of rotational restraint at far end of beam Effective beam stiffness coefficient K (provided that beam remains elastic)

mined by reference to Table 1, pro t they remain ela ic under the d

Fixed at far end 1,0 IL

Pinned at far end 0,75 IL

Rotation as at near end 1,5 I(double curvature) L

Rotation equal and opp 0,5 IL

osite to that at near end (single curvature)

General case Rotation θ at near end and

1 0,5 b

a

ILθ

+⎜ ⎟⎝ ⎠

θ⎛ ⎞

a θb at far end

(7) For building frames with concrete floor slabs, provided that the frame is of regular layout and the loading is uniform, it is normally sufficiently accurate to assume that the effective stiffness coefficients of the beams are as shown in Table 2.

Table 2: Effective stiffness coefficient for a beam in a building frame with concrete floor slabs

Loading conditions for the beam Non-sway mode Sway mode

Beams directly supporting concrete floor slabs 1,0 IL

1,0 IL

Other beams with direct loads 0,75 IL

1,0 IL

Beams with end moments only 0,5 IL

1,5 IL

(8) moment in any of the beam exceeds Wel fy / γM0, the be ed to be pinned at the point or points concerned.

(9) Where a beam has nominally pinned joints, it should be assumed to be pinned at the point or points concerned.

(10) Where a beam has semi-rigid joints, its effective stiffness coefficient should be

(11) Where the beams are subject to axial forces, their effective stiffness coefficients y. Stability functions may be used. As a simple alternative, the

increased stiffness coefficient due to axial tension may be neglected and the effects of axial ompression may be allowed for by using the conservative approxim

Table 3: Approximate formulae for reduced beam stiffness coefficients due to axial compression

restraint at far end of beam Effective beam stiffness coefficient K (provided that beam remains elastic)

Where, for the same load case, the design s am should be assum

reduced accordingly.

should be adjusted accordingl

c ations given in Table 3.

Conditions of rotational

Fixed 1,0 1 0, 4I N⎛ ⎞−⎜ ⎟

EL N⎝ ⎠

Pinned 0,75 1 0, 4E

I NL N⎛ ⎞−⎜ ⎟

⎝ ⎠

Rotation as at near end (double curvature)

1,5 1 0, 4E

I NL N⎛ ⎞−⎜ ⎟

⎝ ⎠

Rotation equal and opposite to that at near end (single curvature)

0,5 1 0, 4E

I NL N⎛ ⎞−⎜ ⎟

⎝ ⎠

In this table 2

2EEIN

Lπ

=

(12) The following empirical expressions may be used as conservative approximations instead of reading values from Figure 1 and Figure 2:

a) non-sway mode (Figure 1)

( ) ( )21 2 1 20,5 0,14 0,055crL

Lη η η η= + + + + (5)

or alternatively: ( )( )

1 2 1 2

1 2 1 2

1 0,145 0, 2652 0,364 0, 247

crLL

η η η ηη η η η

⎡ ⎤+ + −= ⎢ ⎥− + −⎣ ⎦

(6)

b) sway mode (Figure 2)

( )( )

1 2 1 2

1 2 1 2

1 0,2 0,121 0,8 0,60

crLL

η η η ηη η η η

− + −=

− + + (7)