D5.C.5.2 Members Subject to Bending Moments

The cross section capacity of a member subject to bending is checked as per Cl .6.2.5 of the code. The condition to be satisfied is:

\frac{M_{E d}}{M_{c, R d}} \leq 1.0

Where M_c,Rd is the is the design resistance given by:

$M_{c, R d} = M_{p l, R d} = \frac{W_{p l} f_{y}}{γ_{M 0}}$ for class 1 and 2 cross-sections
$M_{c, R d} = M_{e l, R d} = \frac{W_{e l, \min} f_{y}}{γ_{M 0}}$ for class 3 cross-sections
$M_{c, R d} = \frac{W_{e f f, \min} f_{y}}{γ_{M 0}}$ for class 4 cross-sections

Cross sectional bending capacity checks will be done for both major and minor axis bending moments.

Members subject to major axis bending will also be checked for lateral-torsional buckling resistance as per Section 6.3.2 of the code. The design buckling resistance moment M_b,Rd will be calculated as:

M_{b, R d} = χ_{L T} W_{y} \frac{f_{y}}{γ_{M 1}}

where

χ_LT

the reduction factor for lateral-torsional buckling. This reduction factor is evaluated per Cl. 6.3.2.2 or Cl 6.3.2.3 of the EN 1993 code depending on the section type. For I sections, the program will by default use Cl. 6.3.2.3 to evaluate χ_LT and for all other sections the program will resort to Cl 6.3.2.2. However, if a particular National Annex has been specified, the program will check if the National Annex expands on Cl.6.3.2.3 (Table 6.5) to include sections other than I sections. If so, the program will use Cl. 6.3.2.3 for the cross-section(s) included in Cl. 6.2.2.3 (or Table 6.5). For all other cases the program will use Cl. 6.3.2.2.

Note: You have the option to choose the clause to be used to calculate χ_LT through the MTH design parameter. Setting MTH to 0 (default value) will cause the program to choose Cl.6.3.2.3 for I Sections and Cl 6.2.3.2 for all other section types. As mentioned above, if the National Annex expands on Cl. 6.3.2.3 to include sections other than I Sections, the program will use Cl. 6.3.2.3 by default.

When using Cl. 6.3.2.3 to calculate χ_LT, the program will consider the correction factor kc (Table 6.6 of EN 1993-1-1:2006) based on the value of the KC parameter in the design input. By default the value of KC will be taken as 1.0. If you want the program to calculate kc, you must explicitly set the value of the KC parameter to zero.

Note: If the National Annex specifies a different method to calculate kc (e.g. the British, Singapore & Polish NAs), the program will use that method by default even if the KC parameter has not been explicitly set to zero. If the NA method does not deal with a specific condition while working out kc, the program will then fall back to table 6.6 of the code, thus ensuring that kc is considered for the particular NA.

The non-dimensional slenderness λ _LT (used to evaluate χ_LT) for both the above cases is evaluated as:

{\overset{⎯}{λ}}_{L T} = \sqrt{\frac{W_{y} f_{y}}{M_{c r}}}

where

M_cr

the elastic critical moment for lateral-torsional buckling. EN 1993-1-1 does not however specify a method to evaluate M_cr. Hence, the program will make use of the method specified in Annex F of DD ENV 1993-1-1 to evaluate M_cr by default.

Note: The method specified in Annex F will be used only when the raw EN 1993-1-1:2005 code is used without any National Annex. If a National Annex has been specified, the calculation of M_cr (and

{\overset{⎯}{λ}}_{L T}

) will be done based on the specific National Annex. (Refer to D5.D. European Codes - National Annexes to Eurocode 3 [EN 1993-1-1:2005] for specific details). If the National Annex does not specify a particular method or specify a reference document, the program will use the NCCI document SN-003a-EN-EU for doubly symmetric sections and SN030a-EN-EU for mono-symmetric sections that are symmetric about their weak axis. For all other sections types the program will use Annex F of DD ENV 1993-1-1 to calculate M_cr. In cases where Annex F does not provide an adequate method to evaluate M_cr, such as for Channel sections, the program will resort to the method as per Cl.4.3.6 of BS 5950-1:2000 to calculate the lateral-torsional buckling resistance moment (M_b,Rd) for the member.

Tapered Members

EN 1993-1-1 provides multiple methods for checking against lateral-torsional buckling in members with tapered I-shaped sections. The method given in Annex BB 3.2 of EN 1993-1-1 is used by STAAD.Pro. This method checks the unbraced length between lateral and torsional restraints against a calculated maximum length to ensure lateral-torsional stability. The tapered member is sub-divided into thirteen (13) analytical sections and bending design checks, including these for lateral-torsional buckling, are performed at each sub-section. While this approach is conservative for elastic analysis, it is necessary for plastic analysis.

The stable length between lateral restraints, L_m, is calculated as follows. This value must be greater than or equal to the design parameter LY.

L_{m} = 0.85 \frac{38 r_{zz}}{\sqrt{\frac{1}{57.4} (\frac{N_{Ed}}{A}) + \frac{1}{756 {C_{1}}^{2}} (\frac{{W_{pl,y}}^{2}}{A I_{T}}) {(\frac{f_{y}}{235})}^{2}}}

(Eqn. BB.5)

where

r_zz	=	the radius of gyration about the major axis ( notation i_z in EN 1993-1-1)
N_Ed	=	design value of compression force in the member
A	=	cross-sectional area of the member
W_pl,y	=	plastic section modulus of the member
I_T	=	torsional constant
f_y	=	yield strength
C₁	=	a factor depending on loading and end conditions; taken $= \frac{1}{{k_{c}}^{2}}$ , where k_c is taken from the `KC` parameter.

The stable length between torsional restraints, L_s, is calculated as follows. This value must be greater than or equal to the design parameter EFT.

L_{s} = 0.85 \frac{\sqrt{C_{n}} L_{k}}{c}

(Eqn. BB.12)

where

C_n

modification factor for non-linear moment gradient

= \frac{12 \times R_{max}}{[R_{1} + 3 R_{2} + 4 R_{3} + 3 R_{4} + R_{5} + 2 (R_{S} - R_{E})]} \leq 1.0

(Eqn. BB.14)

moment ratio calculated at ends, quarter points, and mid-point of member segment between torsional restraints, calculated as:

= \frac{M_{y,Ed} + a N_{Ed}}{f_{y} W_{pl,y}}

(Eqn. BB.15)

and R₁ is taken from the largest web depth.

M_y,Ed

design bending moment about the Y axis

the distanced between the centroid of the member and the centroid of the restraining members (e.g., purlins).

Note: This value is controlled in STAAD.Pro using the HGT parameter. To simplify the user input, the HGT parameter is specified in relation to the top of the member.

L_k

stable length between adjacent torsional restraints

= \frac{(5.4 + \frac{600 f_{y}}{E}) (\frac{h}{t_{f}}) r_{zz}}{\sqrt{5.4 (\frac{f_{y}}{E}) {(\frac{h}{t_{f}})}^{2} - 1}}

(Eqn. BB6)

and $\frac{h}{t_{f}}$ is taken from the shallowest web depth.

modulus of elasticity

taper factor:

= 1 + \frac{3}{(\frac{h}{t_{f}} - 9)} {(\frac{h_{\max}}{h_{\min}} - 1)}^{\frac{2}{3}}

(Eqn. BB.16)

depth of segment

t_f

thickness of the flange

h_max, h_min

the maximum and minimum depth of the cross-section within the length, L_y (LY), respectively.

Note: There are no provisions for lateral-torsional buckling in tapered hollow sections (i.e., tapered square or circular sections). As this is typically not a governing limit state, STAAD.Pro does not perform any such check.