2.5 Design approach of different specifications
2.5.1 SANS 10162-1:2011
The design for laterally unsupported members is given in §13.6 in SANS 10162-1:2011. The code specifies an equation, equation 2.6, for the critical elastic moment similar to that of Timoshenko et al. (1961). The difference lies in how the code takes account for the loading conditions and support conditions at the beam ends.
Mcr= ω2π KL s EIyGJ + πE KL 2 IyCw (2.6)
The ω2 parameter in equation 2.6 is the equivalent moment factor, more generally known as Cb. This
parameter is calculated with the equation for an unbraced beam segment subjected to end moments, equation 2.2. The only difference in SANS 10162-1:2011 in comparison to the equation developed by Salvadori is that the upper limit is 2.5 instead of 2.3. The code states that ω2 is equal to equation 2.2 for
unbraced beam segments subjected to end moments, or ω2= 1.0 when the bending moment at any point
within the unbraced length is larger than the larger end moment or when there is no effective lateral support for the compression flange at one of the ends of the unsupported length (SABS, 2011).
As mentioned in Section 2.4.1, κ is the ratio of the smaller end moment to the larger end moment at opposite ends of the unbraced length and positive for double curvature and negative for single curva- ture (SABS, 2011). The code however gives no indication of how to account for triple curvature, which typically occurs when a uniformly distributed load is applied along with two equal end moments, as in figure 2.11.
(a) Load configuration (b) Bending moment distribution
Figure 2.11: Uniformly distributed load with equal end moments
Equation 2.6 also accounts for the level of the applied load and the end restraints of the beam by using the K parameter, which is known as the effective length factor. This parameter can be found in Table 1 in SANS 10162-1:2011.
Table 2.3: Table 1 in SANS 10162-1:2011
Restraint Against Lateral Bending at Supports
Effective Length Factor K
Loading Condition
Normal Destabilizing
Unrestrained 1.00 1.20
Partially Restrained 0.85 1.00
Practically Fixed 0.70 0.85
Table 2.3 offers a series of end restraints, which range from unrestrained to fully fixed, as discussed in Section 2.4.3. As the fixity of the end supports increases K reduces, in other words the buckling length of the beam decreases as indicated by Timoshenko et al. (1961). In SANS 10162-1:2011 Section 10.2.1, it is also stated that the K factor in table 2.3 must be increased by 20% where the beam ends are not restrained against torsion (SABS, 2011).
Table 2.3 also take into consideration the level of application of the load. The K factor for normal loading basically means that the load is applied at the shear centre of the section or below the shear centre, i.e.
the load doesn’t cause an additional moment about the centroid. The K factor for destabilising loading is for the case where the load is applied above the shear centre and the load causes an additional moment about the shear centre and increases the rotation of the beam.
When continuous lateral support is not provided to the compression flange of a member subjected to uni-axial strong axis bending, the factored moment resistance, Mr, may be taken as follows (SABS, 2011):
a) for doubly symmetric class 1 and 2 sections, except closed square and circular sections
• when Mcr> 0.67Mp Mr= 1.15φMp 1 − 0.28Mp Mcr
but not greater than φMr
• when Mcr≤ 0.67Mp
Mr= φMcr
b) for doubly symmetric class 3 and class 4, except closed square and circular sections, and for channels:
• when Mcr> 0.67My Mr= 1.15φMy 1 − 0.28My Mcr
but not greater than φMr
• when Mcr≤ 0.67My
Mr= φMcr
The 0.67 factor in the equations accounts for the effects of residual stresses that initiate inelastic buckling and is based on the research by Galambos (1963).
2.5.2
ANSI/AISC 360-05
Chapter F in ANSI/AISC 360-05 contains provisions for calculating the flexural strength of members subject to simple bending about one principle axis (AISC, 2005). In this study only members with a compact section will be considered, i.e. members that classify as class 1 sections. The ANSI/AISC 360-05 provides ten different procedures (F2-F12) for members in bending, where each of these procedures is applicable to a certain type of cross section, i.e. compact, slender, non-compact and square and rectangular circular hollow sections.
Under general provisions (F1) the procedures for determining the load and resistance factor design (LRFD) and the allowable flexural strength (ASD) are given. They are as follows:
• LRFD = φbMn, where φb= 0.90 • ASD = Mn
The equation for determining the LTB modification factor, or more commonly known, the equivalent moment factor Cb, is stated as in section F1:
Cb=
12.5Mmax
2.5Mmax+ 3MA+ 4MB+ 3MCRm≤ 3.0 (2.7)
Equation 2.7 is very similar to equation 2.3 that was developed by Kirby et al. (1979), with the only difference being the coefficients for the moments that are slightly higher. Rm in equation 2.7 is the cross section’s mono-symmetry parameter, which equals 1.0 for doubly-symmetric sections. In section F2 provisions are specified to determine the nominal flexural strength (Mn) of doubly-symmetric compact I- shaped members and channels bent about their major axis. This section provides three different equations for Mn, they are:
Mn = Mp= FySx (2.8) Mn= Cb Mp− (Mp− 0.7FySx) Lb− Lp Lr− Lp ≤ Mp (2.9) Mn= FcrSx≤ Mp (2.10) Fcr= Cbπ2E Lb rts 2 s 1 + 0.078 J Sxho Lb rts 2 (2.11) where
Fy is the specified minimum yield stress of the type of steel being used.
Sx is the plastic section modulus about the x-axis.
J is the St. Venant’s torsion constant. rts is the effective radius of gyration.
h0 is the distance between flange centroids.
Lp is the limiting laterally unbraced length for the limit state of yielding.
Lb is the length between points that are either braced against lateral displacement of compression flange or braced against twist of the cross section.
Lr is the limiting laterally unbraced length for the limit state of inelastic LTB (AISC, 2005). The limiting lengths Lp and Lr are determined as follows:
Lp= 1.76ry s
E Fy
Lr= 1.95rts E 0.7Fy r J c Sxho v u u t1 + s 1 + 6.76 0.7Fy E Sxho J c 2 (2.13) Eqn. 2-8 Eqn. 2-9 Eqn. 2-10 Mn 0.7FySx Lp Lr I N
Figure 2.12: Beam strength vs. Unbraced length (AISC, 2005)
As can be seen in figure 2.12 equations 2.8-10 are each applicable for a certain length of the beam. First, equation 2.8 applies when Lb ≤ Lp, where the limit state of LTB does not apply. This equation is generally used when stocky beams are under consideration and the beam behaves plastically. Secondly, equation 2.9 applies only when Lp< Lb ≤ Lr. In this region the strength is limited by inelastic buckling. Lastly, on the far right of the curve equation 2.10 is applicable, where Lb > Lr. In this region elastic buckling is limiting the strength of the beam.
2.5.3
EN 1993-1-1
In EN 1993-1-1 two methods are given for the determination of the lateral-buckling resistance of a beam namely the general case (GC) and the special case (SC). The GC can be used for all sections, while the SC is specifically used for rolled sections of standard dimensions. As with any design of a structural element the maximum applied moment (MEd) first needs to be determined by an elastic analysis (if the beam is statically indeterminate), or by statics (if the beam is statically determinate) (Trahair et al., 2008). In section 6.3.2.1 of EN 1993-1-1 it is stated that a laterally unrestrained member subjected to major axis bending should be verified against lateral-torsional buckling as follows for both cases:
MEd Mb,Rd ≤ 1, 0 (2.14) λLT = r Wyfy Mcr (2.15)
Mcr= C1 π2EI z (kL)2 s k kw 2I w Iz +(kL) 2GI t π2EI z + (C2zg)2− C2zg (2.16) where
k is an effective length factor for out-of-plane bending (usually 1.0). kw is an effective length factor for warping (usually 1.0).
zg is the distance between the point of load application and the shear centre, as shown in figure 2.13.
C1 and C2 are coefficients depending on the loading and end restraint conditions (NCCI, 2008).
S S
F
F
zg> 0 zg< 0
Figure 2.13: Point of application of transverse load
The elastic critical moment is then calculated by using equation 2.16 (Trahair et al., 2008). EN 1993- 1-1 does not provide an explicit expression for the Mcr, but equation 2.16 is found in NCCI (2008). This document is part of a set of non-contradicting complementary information provided by the Steel Construction Institute (SCI) in the UK. The modified slenderness ratio is then calculated afterwards by using equation 2.15 (CEN, 2005). For the rest of the procedure for both the GC and SC it is necessary to calculate αLT, β and λLT ,0. The steps and equations necessary to determine these parameters for each method will be discussed separately.
2.5.3.1 C1 and C2 factors
The C1 and C2 factors depend on various parameters that include section properties, support conditions
and the moment diagram (NCCI, 2008). There are three different categories for which a beam can be classified when determining these factors, namely:
• Member with end moments only;
• Member with transverse loading;
• Member with end moments and transverse loading.
For each of these categories separate values for C1 and C2 need to be determined according to the
bending moment diagram for the beam under consideration. In NCCI (2008) various bending moment distributions are presented with corresponding graphs for the determination of C1and C2 values.
2.5.3.2 General case
In equation 2.14 MEd is the design value of the moment and Mb,Rd is the design buckling resistance moment (CEN, 2005). To calculate the resistance moment of a beam, the reduction factor (χLT) and section modulus (Wy) are required as shown in equation 2.17.
Mb,Rd= χLTWy
fy
γM 1
(2.17)
Section 6.3.2.2 in EN 1993-1-1 describes the procedures for the reduction factor. For bending members of constant cross section, the value of χLT for the appropriate non-dimensional slenderness λLT, should be determined from equation 6.56 in EN 1993-1-1:
χLT = 1 ΦLT+ p Φ2 LT−λ 2 LT but χLT ≤ 1, 0 where ΦLT = 0.5[1 + αLT(λLT − 0, 2) + λ2LT]
The imperfection factor (αLT) is determined from the buckling curves provided in EN 1993-1-1. The buckling curves are a function of the reduction factor and non-dimensional slenderness, λLT. In the clause two tables, reproduced here as tables 2.4 and 2.5, are given to determine the imperfection factor.
Table 2.4: Recommended values for imperfection factor for lateral-torsional buckling curves
Buckling Curve a b c d
Imperfection factor αLT 0.21 0.34 0.49 0.76
Table 2.5: Recommended values for lateral buckling curves for cross sections using eq. (6.56)
Cross section Limits Buckling Curve
Rolled I-sections h/b ≤ 2 a h/b > 2 b Welded I-sections h/b ≤ 2 c h/b > 2 d Other cross sections - d
2.5.3.3 Special case
As mentioned before, the SC is similar to the GC. The difference between the two methods becomes apparent in the determination of the reduction factor, χLT. In section 6.3.2.3 equation 6.57 is used for the determination of the reduction factor for the SC. The equation is as follows:
χLT = 1 ΦLT+ p Φ2 LT−βλ 2 LT but {χLT ≤ 1, 0 and χLT ≤ 1 λ2LT} where ΦLT = 0.5[1 + αLT(λLT− λLT ,0) + βλ 2 LT]
The parameters λLT ,0and β and any limitation of validity concerning the beam depth or h/b ratio may be given in the National Annex. The following values are recommended for rolled sections or equivalent welded sections (CEN, 2005):
λLT ,0= 0.4 (maximum value)
β = 0.75 (minimum value)
Table 2.6 gives the recommendations for buckling curves to determine the imperfection factor αLT in a similar fashion as with the GC.
For both of these methods LTB effects may be ignored and only cross-sectional checks apply, if the following statements are true:
λLT ≤ λLT ,0 or MEd Mcr ≤ λ 2 LT ,0
Table 2.6: Selection of lateral-torsional buckling curve for cross sections using equation. (6.57)
Cross section Limits Buckling Curve
Rolled I-sections h/b ≤ 2 b h/b > 2 c Welded I-sections h/b ≤ 2 c h/b > 2 d