Effective length and allowable slenderness ratio

4 Calculation of flexural members

5.3 Effective length and allowable slenderness ratio

5.3.1 The effective length l0 of chord members and single system web members (connected

to chord members through gusset plate) of trusses shall be taken as those given in Table 5.3.1.

Table 5.3.1 Effective length l0 of truss chords and single system web members

Web members Item NO. Direction of bending Chord members

End posts Other web members 1

2 3

In truss plane Out of truss plane

In skew plane l l1 — l l l 0.8l l 0.9l

Note: 1 l—geometric length of the member (distance between panel points)；l1——distance

between laterally supported points of chord members.

2 The skew plane refers to the plane oblique to the truss plane. It is applicable to web members with single angle and double angle cruciform section whose principal planes are non-coincident with truss plane.

3 For web members without gusset plate, the effective length in any plane shall be their geomeric length (excepting tubular structures).

Fig. 5.3.1 Chord member of truss with unequal axial compression between laterally supported points

If the distance between laterally supported points of compression chord of a truss is equal to twice the panel length (Fig.5.3.1) and the two panels have unequal compressive forces, the effective length for out-of-plane buckling shall be taken as (but not less than 0.5l1):

⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ + = 1 2 1 0 0.75 0.25 N N l l (5.3.1) Where N1——The larger compressive force, taken as positive;

N2——The smaller compressive force or tensile force, taken as positive for

compression and negative for tension.

The main compression diagonals in subdivided web member system and the verticals in K-web member system shall also have their out-of-plane effective length determined by formula (5.3.1) (this length shall be l1 for main tension diagonals), while their in-plane

effective length shall be taken as distance between adjacent joints.

taken as the distance from a panel point to the crossing point for in-plane buckling, whereas for out-of-plane buckling, in the case of diagonals of equal length, it shall be taken as prescribed in the following:

1 Diagonal in compression.

1) In case the intersecting diagonal, is also in compression and that both diagonals, identical in section, are not interrupted at the crossing point:

⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + = N N 1 2 1 ₀ 0 l l

2) In case the intersecting diagonal is also in compression, but interrupted and spliced by a gusset plate at the crossing point:

N

·

12

1

0 2 0

π

+

= l

l

3) In case the intersecting diagonal is in tension and that both diagonals, identical in section, are not interrupted at the crossing point:

N

l

0 2 0

·

12 1+π

=

4) In case the intersecing diagonal is in tension, but interrupted and spliced by a gusset plate at the crossing point:

N N · 4 3 1 · 0 0 = l − l ≥0.5

In case this tension diagonal is continuous but the compression diagonal is interrupted and spliced by a gusset plate, and that N0≥N or the tension diagonal has its out-of-plane

rigidity EIy≥ ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ −1 4 3 0 2 2 0 N N π l N

, take l0=0.5l for the compression diagonal.

In the above formulae, l denotes distance between panel points of the truss (the crossing point is not counted in); N and N0 denote respectively the force in the diagonal under

consideration, and that in the intersecting diagonal, both in absolute value. When both diagonals are in compression, and have identical section, take N0≤N.

2 Diagonal in tension, l0 shall be taken equal to l.

The effective length of single angle cross diagonals in skew plane shall be taken as the distance between a panel point and the crossing point.

5.3.3 The in-plane effective length of uniform section columns of single-or multiple-story

frames shall be taken as the product of the column length (story height) by an effective length factor μ. Frames are classified as pure frame without bracing and braced frame. The latter is subdivided into strongly braced and weakly braced ones according to the stiffness against lateral translation.

1 Pure frames.

1) In case internal forces are computed by first-order elastic analysis, the column effective length factor μ shall be determined in accordance with Table D-2 in Appendix D as a

frame with sidesway.

2) In case internal forces are computed by second-order elastic analysis, and that notional horizontal forces Hni determined by Eq (3.2.8-1) are applied at the top of each story,

take μ= 1.0.

2 Braced frames.

1) In case the sway stiffness Sb (horizontal force that causes a leaning angle of unity)

of the bracing system (bracing truss, shear wall, elevator shaft) satisfies the requirement of the Formula (5.3.3-1), the frame is recognized as strongly braced, and the effective length factor μ of its columns shall be detemined in accordance with Table D-1 in Appendix D as a frame without sidesway.

(12

bi 0i

)

3 .

ΣN

S

≥

−

(5.3.3-1) Where

ΣN

_bi

、ΣN

_oi——Sum of the load carrying capacity of all the columns in the ith story calculated by making use of effective length factor of frames without sidesway and with sidesway respectively.

2) In case the sway stiffness Sb of the bracing system does not satisfy the requirement

of the formula (5.3.3-1), the frame is recognized as weakly braced and the relevant stability factor φ of axial compression member shall be determined according to Eqs (5.3.3-2).

(

_{) (}

₎

i 0 bi b 0 1 0

2 .

1

3 N

S

Σ

−

Σ

−

+

=ϕ

ϕ

(5.3.3-2) Where φ1, φ0 ——The stability factor of axially loaded columns corresponding to column

effective length factor of frames without and with sidesway respectively from Tables in Appendix D.

5.3.4 The in-plane effective length of a stepped column rigidly restrained at the base of a

single-story mill building shall be determined as follows: 1 Single-stepped column：

1) Effective length factor μ2 for the lower portion of the column: For column pin-connected to the girder, μ2 shall be the figures taken from Table D-3 (single stepped column with free upper end) in Appendix D, multiplied by an appropriate reduction factor given in Table 5.3.4 For columns rigidly connected to the girder, μ2 shall be the figures taken from Table D-4 (single-stepped column with translation-free and rotation-fixed upper end) in Appendix D, multiplied by an appropriate reduction factor given in Table 5.3.4.

Table 5.3.4 Reduction factor for the effective length of stepped columns in single story mill building

Type of the mill building

Number of bays

Number of columns in a longitudinal row within a section between expansion joints of the building

Roofing

Longitudinal horizontal

bracing on the two sides of the roof system

Reduction factor

Not provided Other than large-size

precast R. C. roofing slab Provided ＞6

Large-size precast R. C.

roofing slab —

Not provided

0.8 Other than large-size

precast R. C. roofing slab Provided Multiple —

Large-size precast R.C.

roofing slab —

0.7

Note: The reduction factor 0.9 may be used for outdoor structures with girder (e. g. drop ball mill, etc.)

2) Effective length factor μ1 for the upper portion of thee column shall be calculated by the following formula:

1 2

_η

μ

=

(5.3.4-1) Where η1——A parameter calculated by the formula shown in Table D-3 or D-4 in

Appendix D.

2 Double-stepped column:

1) Effective length factor μ3 of the lower portion of the column: For columns

pin-connected to the girder, μ3 shall be the figures taken from Table D-5 (double-stepped

column with free upper end), in Appendix D, multiplied by an appropriate reduction factor given in Table 5.3.4 For columns rigidly connected to the girder, μ3 shall be the figures

taken from Table D-6 (double-stepped column with translation-free and rotation-fixed upper end), in Appendix D, multiplied by an appropriate reduction factor given in Table 5.3.4. 2) Effective length factors μ1 and μ2 for the upper and middle portions shall be

calculated by the following formulae respectively:

1 3 1

_μ

μ

=

(5.3.4-2) 2 3 2

_μ

μ

=

(5.3.4-3) Where η1, η2——Parameters calculated by formulae shown in Tables D-5 and D-6 in

Appendix D.

Note: For the effective length of tapered columns, refer to the current national standard “Technical Code of Cold-formed Thin-wall Steel Structures” GB 50018.

5.3.5 In calculating the momemt of inertia of laced or battened columns of a frame and that

of lattice girders (trusses), the influence of the variation of section depth of columns or girders and that of the deformation of lacing bars or batten plates (or web members) shall be taken into account.

5.3.6 Special consideration to be taken in determining the effective length factor of frame

columns under the following situations:

1 The column slenderness ratio factor of pure frames and weakly braced frames adjoined with leaning columns (columns hinged at both ends) shall be multiplied by an

amplification factor η:

(

)

(

f f

)

l l

/H

N

Σ

/H

N

Σ

+

= 1

η

(5.3.6)

Where

Σ(N

/H

)

——Summation of the ratio of the factored axial compression to column height of all the frame columns;

Σ(N

/H

)

——Summation of the ratio of factored axial compression to column height of all the leaning columns.

The effective length of leaning columns is equal to their geometric length.

2 When other columns in the same story of the calculated column, or columns continuous with it in the upper and lower story, have potential in load-carrying capacity, the effective length factor of the column may be reduced by taking into account the restraining effect of these adjacent columns; whereas the effective length factor of the latter should be enlarged.

3 For columns joining beams by semi-rigid connections, the effective length shall be determined by taking the behavior of the connections into consideration.

5.3.7 The effective length of frame columns in the longitudinal direction of the building (out

of the frame plane) shall be taken as the distance between supporting points restraining the out-of-plane displacement.

5.3.8 The slenderness ratio of compression members should not exceed the allowable values

given in Table 5.3.8.

Table 5.3.8 Allowable values of slenderness ratio for compression members

Item

No. Nomenclature of members

Allowable values 1

Columns, members of trusses and monitors

Lacing of columns, column bracings beneath crane girders or crane trusses

150

Bracings (except column bracings beneath crane girders or crane trusses) Members used to reduce the slenderness ratio of compression members

200 Note: 1 A slenderness ratio of 200 may be allowed for compression web members in trusses (including space trusses) when they are stressed to or under 50％ of their capacities.

2 For single angle compression members, the least radius of gyration shall be used for calculation of slenderness ratio, but the radius of gyration about the axis parallel to the leg may be used for calculation of the out-of-plane slenderness ratio in the case of cross diagonals interconnected at the crossing point.

3 For trusses with span length equal to or larger than 60m, the allowable slenderness ratio should be taken as 100 for compression chords and end posts, and 150 (when subjected to static or indirect dynamic load), or 120 (when subjected to direct dynamic load) for other web members in compression.

4 In case member section is governed by allowable slenderness ratio, the effect of twisting may be neglected in calculating the slendeness ratio.

5.3.9 The slenderness ratio of tension members should not exceed the allowable values

given in Table 5.3.9.

Table 5.3.9 Allowable values of slenderness ratio for tension members

Structures subject to static or indirect dynamic loading Item No. Nomenclature of members

Common buildings

Mill buildings with heavy duty crane

Structures subject to direct dynamic loading

1 Members of Trusses 350 250 250

2 Column bracings beneath crane

girders or crane trusses 300 200 —

Other tension members, bracings

and ties (except pretensioned round bars)

400 350 —

Note: 1 A slenderness ratio of 200 may be allowed for compression web members in trusses (including space trusses) when they are stressed to or under 50％ of their capacities.

4 In case member section is governed by allowable slenderness ratio, the effect of twisting may be neglected in calculating the slendeness ratio of the bracings (except Item No.2 in the Table) should not exceed 300.

5 When tension members change into compression ones under the combined action of dead and wind loads, their slenderness ratio should not exceed 250.

6 For trusses with span length equal to or larger than 60m, the slenderness ratio of tension chords and tension web members should not exceed 300 (when subjected to static or indirect dynamic loading) or 250 (when subjected to direct dynamic loading).

In document GB 50017-2003 Code for Design Steel Structure, Chinese National Standard (Page 49-54)