For a member subjected to combined bending moment and axial force, the bending moments are amplified by the presence of axial force. This occurs for both isolated, statically determinate members and members in a statically indeterminate frame. A first-order elastic analysis alone does not consider second-order effects, however, moment amplification can be used to account for second-order effects. The moment amplification factor is calculated differently for braced and sway members as explained in the following sub-section.
4.2.1 Braced Members
In a braced member the transverse displacement of one end of the member relative to the other is effectively prevented. The moment amplification factor for a braced member is bb.
If a first-order elastic analysis is carried out then bb is used to amplify the bending moments between the ends of the member (Clause 4 . 4 . 2 . 2 of AS 4100 ). A first-order elastic analysis with moment amplification cannot be used if bb is greater than 1.4. If bb is greater than 1.4, it may be practical to alter the member sizes or connections so that bb )1 . 4 . Alternatively a second-order elastic analysis in accordance with Appendix E of AS 4100 may be used.
bb can be calculated from the flow chart in Figure 4 . 1 . The design bending moment (M*) is then given by:
M* = M*m (for braced members subject to axial tension or with zero axial force) M* = bbM*m (for braced members subject to compression)
where M*m is the maximum design bending moment calculated from a first-order analysis.
Part 4
METHODS OF STRUCTURAL ANALYSIS
Australian Tube Mills A.B.N. 21 123 666 679. PO Box 246 Sunnybank, Queensland 4109 Australia Telephone +61 7 3909 6600 Facsimile +61 7 3909 6660 E-mail info@austubemills.com Internet www.austubemills.com Figure 4.1: Flow Chart for the calculation of the moment amplification factor for a braced member, bb
Calculate Member Effective Length keL; Clauses 4 . 6 . 3 . 3 , 4 . 6 . 3 . 4
and Figure 4 . 6 . 3 . 3 (a) of AS 4100 Calculate Member
Effective Length keL;
Figure 4 . 6 . 3 . 2 of AS 4100 or Figure 6.1 of this publication
Compute Nomb from Clause 4 . 6 . 2 of AS 4100
Compute cm from Clause 4 . 4 . 2 . 2 of AS 4100
Members within Frames;
Clause 4 . 6 . 3 . 3 of AS 4100 Members with
Idealised End Restraints;
Clause 4 . 6 . 3 . 2 of AS 4100
bb= cm 1< N*
Nomb
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Part 4
METHODS OF STRUCTURAL ANALYSIS
Calculation of bb 4.2.1.1 Calculation of cm
The factor for unequal moments (cm) is used in the calculation of bb. If a braced member is subject only to end moments then the factor cm is calculated as follows:
cm = 0 . 6 – 0 . 4`m ) 1 . 0 (Clause 4 . 4 . 2 . 2 of AS 4100 ) where `m is the ratio of the smaller to the larger bending moment at the ends of the member, taken as positive when the member is bent in reverse curvature.
If the member is subjected to transverse loading, the same expression for cm shall be used provided `m is calculated using one of the following methods:
a) `m = - 1 . 0 (conservative) (Clause 4 . 4 . 2 . 2 (a) of AS 4100 ) b) `m is obtained by matching the moment distribution
options shown in Figure 4 . 4 . 2 . 2 of AS 4100 (Clause 4.4.2.2(b) of AS 4100) c) `m is based on the midspan deflection. (Clause 4 . 4 . 2 . 2 (c) of AS 4100 )
4.2.2 Sway Members
In a sway member the transverse displacement of one end of the member relative to the other is not effectively prevented. The moment amplification factor for a sway member is bs.
The bending moments calculated from a first-order elastic analysis are modified by the moment amplification factor (bm) which is the greater of bb (see Section 4 . 2 . 1 ) and bs (Clause 4 . 4 . 2 . 3 of AS 4100 ). If bm is greater than 1 . 4, a second-order elastic analysis must be used in accordance with Appendix E of AS 4100. A detailed explanation of the procedure for calculating bs may be found in Ref.[4.2].
bb and bs are calculated from the flow charts shown in Figures 4 . 1 and 4 . 2. The design bending moment is given by:
M* = bmM*m
Australian Tube Mills A.B.N. 21 123 666 679. PO Box 246 Sunnybank, Queensland 4109 Australia Telephone +61 7 3909 6600 Facsimile +61 7 3909 6660 E-mail info@austubemills.com Internet www.austubemills.com
4.2.3 Elastic Flexural Buckling Loads
Elastic flexural buckling loads (Nomx, Nomy) are required for the calculation of bb and bm. Values of Nom are determined from Clause 4 . 6 . 2 of AS 4100 using the expression:
Nom = /2EI keL
2
where keL = Le = effective length. ke is given in Figure 6 . 1 for members with idealised end restraints or Clause 4.6.3 of AS 4100 for other end restraint conditions. For braced or sway members in frames, ke depends on the ratio (a) of the compression member stiffness to the end restraint stiffness, calculated at each end of the member. Refs. [ 4 . 1 , 4 . 3 ] provide worked examples for the calculation of effective lengths, elastic flexural buckling loads and moment amplification factors for members in those instances.
For a specific effective length, reference can be made to the Dimensions and Properties Tables in Part 3 (i.e. Tables 3 . 1 - 1 to 3 . 1 - 6 as appropriate) to determine I (i.e. Ix or Iy ) and then simply evaluate the above equation for Nom. No tables relating Nom to effective length are provided in this publication.
Part 4
METHODS OF STRUCTURAL ANALYSIS
Compute hms from Clause 4 . 7 . 2 . 2 of AS 4100
bs 1 1< 1
hms
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Figure 4.2: Flow Chart for the calculation of the moment amplification factor for a sway member, bs
Members in Frames;
Clause 4 . 6 . 3 . 3 of AS 4100 Any Member:
Appendix F of AS 4100 Members with Idealised
End Restraints; Clause 4 . 6 . 3 . 2 of AS 4100
Calculate Member Effective Length keL; Figure 4 . 6 . 3 . 2 of AS 4100 or
Figure 6 . 1 of this Publication
Calculate hc from Rational Buckling Analysis
Compute Noms from Clause 4 . 6 . 2 of AS 4100
Non-Rectangular Frames; Clause
4 . 4 . 2 . 3 (b) of AS 4100
“P6” Analysis Clause 4 . 4 . 2 . 3 (a)(i)
of AS 4100
b 1
1< 1 hc
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b 1 s
1< 6s
hs YN*
YV*
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Calculation of bs
Rectangular Frames with Negligible Axial Forces in the
Beams; Clause 4 . 4 . 2 . 3 (a) of AS 4100
Calculate Member Effective Length keL; Clauses 4 . 6 . 3 . 3 , 4 . 6 . 3 . 4 and
Figure 4 . 6 . 3 . 3 (b) of AS 4100
Australian Tube Mills A.B.N. 21 123 666 679. PO Box 246 Sunnybank, Queensland 4109 Australia Telephone +61 7 3909 6600 Facsimile +61 7 3909 6660 E-mail info@austubemills.com Internet www.austubemills.com