# D5.D.3.4 Clause 6.2.10 – Bending shear and axial force

Requires the implementation of clauses 11.3.1.1 to 11.3.1.3 and 11.3.2.1 to 11.3.2.3 of NEN 6770 and clause 11.3 of NEN 6771

## Clause 11.3.1.2 (NEN 6770): Class 1 and class 2 circular hollow (CHS) profiles

Class 1 and class 2 sections with circular hollow profiles should satisfy the interaction equations given in table 12.

• Check #1 – If Vz;s;d ≤ 0.25 Vz;pl;d check equation 11.3.17
• Check #2 – If Vz;s;d > 0.25 Vz;pl;d check equation 11.3.18.

Refer to D5.D.3.3 Clause 6.2.8 – Bending and shear of this document for equations to derive Vz;s;d

• Vz;pl;d = Shear capacity of CHS sections
$V p l ; d = 2 A π f y ; d 3$

See equations 11.3-19 and 11.3-20 to evaluate Mv;y;u;d and N;v;u;d.

To check for these conditions about the y axis, substitute the index "z" in the above equations with ‘y’ (should be the same of CHS sections).

## Clause 11.3.2 ( NEN 6770)

Section 11.3.2 in general deals with Biaxial bending with axial force and shear. The general condition to be satisfied in this case is given by equation 11.3-31 of NEN 6770

$β 0 ( M y ; s ; d M N ; V ; y ; u ; d ) a 1 + β 1 ( M z ; s ; d M N ; V ; z ; u ; d ) a 2 ≤ 1$

## Clause 11.3.2.1 : Class 1 and class2 I-sections with biaxial bending + shear + axial force

The formula to evaluate M;N;v;y;u;d and M;N;v;z;u;d are to be taken from tables 14 and 15 of NEN 6770 respectively.

Checks for table 14:

1. Check #1 – If Vz;s;d ≤ 0.5 Vz;pl;d and Ns;d ≤ 0.5 x a1 x Npl;d use equation 11.3.32
2. Check #2 – If Vz;s;d ≤ 0.5 Vz;pl;d and Ns;d > 0.5 x a1 x Npl;d use equation 11.3.33
3. Check #3 – If Vz;s;d > 0.5 Vz;pl;d and Ns;d ≤ 0.5 x a2 x Nv;u;d use equation 11.3-34
4. Check #4 – If Vz;s;d > 0.5 Vz;pl;d and Ns;d > 0.5 x a2 x Nv;u;d use equation 11.3-35

Refer to D5.D.3.3 Clause 6.2.8 – Bending and shear for equations to evaluate Vz;s;d, My;pl;d, Npl;d, Mv;y;ud, N;v;u;d, a1 ,a2 and  Vz;pl;d.

Checks for table 15:

1. Check #1 – If Vy;s;d ≤ 0.25 Vy;pl;d and Ns;d ≤ 1.0 x a1 x Npl;d use equation 11.3.36
2. Check #2 – If Vy;s;d ≤ 0.25 Vy;pl;d and Ns;d > 1.0 x a1 x Npl;d use equation 11.3.37
3. Check #3 – If Vy;s;d > 0.25 Vy;pl;d and Ns;d ≤ 1.0 x a1 x Nv;u;d check equation 11.3-38
4. Check #4 – If Vy;s;d > 0.25 Vy;pl;d and Ns;d > 1.0 x a1 x Nv;u;d check equation 11.3-39

Refer to D5.D.3.3 Clause 6.2.8 – Bending and shear for equations to evaluate Vy;s;d, Mz;pl;d, Npl;d, Mv;z;ud, N;v;u;d, a1 ,a2 and  Vy;pl;d.

See table 16 for α1, α1, β0 and β1 use in tables 14 and 15.

## Clause 11.3.2.2 : Class 1 and Class 2 Circular hollow tubes

The formula to evaluate M;N;v;y;u;d and M;N;v;z;u;d (to be used in equation 11-3-31, see description of  clause 11.3.2 above) are to be taken from table 17 of NEN 6770.

1. Check #1 – If Vz;s;d ≤ 0.25 Vz;pl;d use equation 11.3.44
2. Check #2 – If Vz;s;d > 0.25 Vz;pl;d use equation 11.3.45.

Refer to D5.D.3.3 Clause 6.2.8 – Bending and shear for equations to evaluate Vz;pl;d, My;pl;d, and Npl;d use in equations 11.3.44 & 11.3.45.

For values to be used for α1, α2,  β1 and  β2 in this case refer to table 18 of NEN 6770.

## Clause 11.3.2.3 : Class 1 and class2 Rectangular and square hollow tubes

The formula to evaluate M;N;v;y;u;d and M;N;v;z;u;d (to be used in equation 11-3-31, see description of  clause 11.3.2 above) are to be taken from table 19 of NEN 6770.

1. Check #1 – If Vz;s;d ≤ 0.25 Vz;pl;d and Ns;d ≤ 0.5 x a3 x Npl;d use equation 11.3-48
2. Check #2 – If Vz;s;d ≤ 0.25 Vz;pl;d and Ns;d > 0.5 x a3 x Npl;d use equation 11.3.49
3. Check #3 – If Vz;s;d > 0.25 Vz;pl;d and Ns;d ≤ 0.5 x a4 x Nv;u;d use equation 11.3-50
4. Check #4 – If Vz;s;d > 0.25 Vz;pl;d and Ns;d > 0.5 x a4 x Nv;u;d check equation 11.3-51

Refer to D5.D.3.3 Clause 6.2.8 – Bending and shear for equations to evaluate Vz;pl;d, My;pl;d, Npl;d, Mv;y;ud, N;v;u;d, a3, a4 and  Vz;pl;d to be used in the above equations. For values to be used for α1, α2, β1 and β2 in this case refer to table 20 of NEN 6770.

To check for these conditions about the y axis, substitute the index "z" in the above equations with ‘y’.

## Clause 11.3 ( NEN 6771)

In general, this section deals with Biaxial bending with axial force and shear for class 3 and class 4 sections.

Check for class 3 sections: For class 3 sections use the method in section 11.3 NEN 6770. For class 3 sections the methods and equations discussed above can be used with the "plastic section modulus" being substituted with the ‘elastic modulus’.

Check for class 4 sections: Class 4 sections can be treated as class 3 sections if the effective section properties are used as given in clause 10.2.4.2.3 of NEN 6771. Working out the effective section properties for slender sections has already been done in STAAD.Pro.

For I- section profiles and tubular sections, the following cases are checked:

1. If M;y;s;d / MN;y;f;u;d ≤ 1 check equation 11.2-7 ( given below)

Vz;s;d/Vz;u;d ≤ 1

Where

• Vz;s;d is the shear for in the Z direction
• Vz;u;d is the shear capacity in the Z direction for ultimate limit state.

For an I section,

$V z ; u ; d = 2 3 A w ; e t f y ; d 3$

Where

• Aw,ef = effective web area as given in section 10.2.4.2.3.
• MN;y;f;u;d is the moment capacity about the Y axis for the effective section. = ( fy·W,eff)
2. If M;y;s;d / MN;y;f;u;d > 1 and M;y;s;d / M;y;f;u;d ≤1  check equation 11.2-13 (given below):

$M y ; s ; d M N ; y ; f ; u ; d + ( M N ; y ; u ; d − M N ; y ; f ; u ; d ) { 1 − ( 2 V z ; s ; d V z ; u ; d − 1 ) 2 } ≤ 1$