D12.A.4 Stability Check According to NS 3472

The stability check is based on the assumption that both ends of the member are structural nodes. Buckling lengths and results for member with joints between the structural nodes have to be evaluated in each separate case.

Effects from local buckling or external hydrostatic pressure on pipes and tubes are not included.

The general stability criteria is: (ref. NS 12.3)

D12.A.4.1 Buckling

n_max + k^z × m_z + k_y × m_y ≤ 1

D12.A.4.2 Lateral Buckling

\frac{n}{χ_{y}} + k_{L T} \frac{m_{z}}{χ_{L T}} + k_{y} m_{y} \leq 1

Where:

i = z,y

n_max = n/χ_min

n = N_f/N_d

χ_min = min(χ_z,χ_y)

χ_i = N_kd,i/N_d

k_{i} = 1 - μ_{i} \frac{n}{χ_{i} γ_{m}} \leq 1.5

μ_i = λ _i(2·β_Mi - 4) ≤ 0.9

β_Mi ref. NS Tab. 12

k_{L T} = 1 - μ_{L T} \frac{n}{χ_{y} γ_{m}} \leq 1.0

μ_LT = 0.15(λ _y·β_M - 1) ≤ 0.9

λ _i = λ_i/λ₁

λ_i = L_ki/i_i

λ_{i} = π \sqrt{\frac{E}{f_{y}}}

χ_{i} = \frac{1}{ϕ + \sqrt{ϕ^{2} - {\bar{λ}}^{2}}}

φ = 0.5[1 + α(λ - 0.2) + λ ²]

α ref. NS Tab 10 & 11

χ_{L T} = \frac{1}{ϕ_{L T} + \sqrt{ϕ_{L T}^{2} - {\bar{λ}}_{L T}^{2}}}

ϕ_{L T} = 0.5 [1 + α ({\bar{λ}}_{L T} - 0.4) + {\bar{λ}}_{L T}^{2}]

α ref. NS sec. 12.3.4.1

{\bar{λ}}_{L T} = \sqrt{\frac{W_{z} f_{z}}{M_{c r}}}

M_cr = ψ·M_vio

ψ ref. NS2 A5.5.2 Sect. a - d

M_{v i o} = \frac{π}{L} \sqrt{E I_{z} G I_{T}} \sqrt{1 + \frac{π^{2}}{L^{2}} \frac{E C_{w}}{G I_{T}}}

D12.A.4.3 Determination of β_z and β_y

The equivalent moment factor β (for z and y) is calculated dependent on moment distributions as shown in the following table:

Table 1. β for different moment distributions
Moment diagram	β_M (β_LT)
	β_{M ψ} = 1.8 - 0.7ψ
	β_{M 0} = 1.3 β_{M 0} = 1.4
	$β_{M} = β_{M, ψ} + \frac{M_{0}}{Δ M} (β_{M, 0} - β_{M, ψ})$ M₀ = \|M_max\| due to transverse load only ΔM = \|M_max\| if the moment has the same sign ΔM = \|M_max\| + \|M_min\| if the moment changes sign

The user can override the calculated factor with the following parameters:

β_y=SSY

β_z=SSZ

D12.A.4.4 Lateral buckling

The Ideal lateral buckling moment is calculated according to NS2 A5.5.2

M_{v i} = ψ M_{v i o} = ψ L 95 \frac{E}{L} \sqrt{I_{y} I_{x}} \sqrt{1 + \frac{π^{2}}{L^{2}} \frac{2.6 C_{w}}{I_{x}}}

concern double symmetric cross sections where y is given in NS fig. A5.5.2, (input parameter CB), L = member length for lateral buckling (input parameter UNL), Cw and Ix , see section 5.

For single symmetric cross sections, the ideal lateral buckling moment is

M_{v i x} = ϕ \frac{π^{2} E I_{y}}{L^{2}} \sqrt{{(\frac{5 a}{π^{2}} + \frac{r_{x}}{3} - y_{s})}^{2} + C^{2}} - (\frac{5 a}{π^{2}} + \frac{r_{x}}{3} - y_{s})

Where:

C^{2} = \frac{C_{w} + 0.039 L^{2} I_{T}}{I_{y}}

α = distance from profile CoG to point where the load is acting, assumed to be on top flange.

The φ parameter (ref NS fig. A5.5.2.g) is controlled by the input parameter CB.

ψ-coefficients for a simple span beam

ψ-coefficients for a partially restrained beam

ψ-coefficients for a fully restrained beam

ψ-coefficients for the cantilevered beam with single loads and distributed loads. Dashed curves apply load on the surface.

D12.A.4.5 Stability check of pipe members

The stability criteria applied for members with pipe cross section is:

I R = \frac{N}{N_{k d}} + \sqrt{{[\frac{{\bar{M}}_{z}}{M_{d} (1 - \frac{N}{N_{E z d}})}]}^{2} + {[\frac{{\bar{M}}_{y}}{M_{d} (1 - \frac{N}{N_{E y d}})}]}^{2}} \leq 1.0

Where:

\frac{N}{N_{k d}} = \max (\frac{N}{N_{k z d}}, \frac{N}{N_{k y d}})

M _z and M _z are given in NS 5.4.2.

For the print output option TRACK 9.0 K_E ≡ 1.0 and M_vd ≡ M_d

D12.A.4.6 Angle profiles type RA (reverse angle)

The axial contribution to the total interaction ratio is checked according to the modified EECS-method, see NS A5.4.

The stability criterion is:

I R = \frac{N}{N_{k d}} + \frac{{\bar{M}}_{y}}{M_{y d} (1 - \frac{N}{N_{E y d}})} + \frac{{\bar{M}}_{z}}{M_{z d} (1 - \frac{N}{N_{E z d}})} \leq 1.0

Where:

\frac{N}{N_{k d}} = \max (\frac{N}{N_{k z d}}, \frac{N}{N_{k y d}})

N_kyd and N_kzd are found from NS 3472 fig. 5.4.la C-curve for y- and z-axis, respectively.

For λ ≤ √(2)

λ _eff = 0.60 + 0.57λ

For λ > √(2)

λ _eff = λ

Where:

\bar{λ} = \frac{λ_{k}}{π} \sqrt{\frac{f_{y}}{E}}

λ_k = l_k/i

i = \sqrt{I / A}

Possible lateral buckling effects and torsional buckling (NS A5.4.5) is not included in the code check. This has to be evaluated by the user separately.

D12.A.4.7 Stability check of members with tapered section

Stability of members with tapered cross section is calculated as described in section 3.1. The cross section properties used in the formulae are calculated based on the average profile height. (i.e., I_z, I_y values are taken from the middle of the member.)

D12.A.4.8 Lateral buckling for tension members

When compressive stress caused by large bending moment about strong axis is greater than tension stress from axial tension force, lateral buckling is considered as defined below.

σ_a = N/A (+ tension, - compression)

σ_bz = ± M_z/W_z

M_warp = | σ_a + σ_b | W_z for σ_a + σ_b < 0 (compression)

IR = M_warp/M_vd + M_y,max/M_yd ≤ 1.0

D12.A.4 Stability Check According to NS 3472

D12.A.4.1 Buckling

D12.A.4.2 Lateral Buckling

D12.A.4.3 Determination of βz and βy

D12.A.4.4 Lateral buckling

ψ-coefficients for a simple span beam

ψ-coefficients for a partially restrained beam

ψ-coefficients for a fully restrained beam

ψ-coefficients for the cantilevered beam with single loads and distributed loads. Dashed curves apply load on the surface.

D12.A.4.5 Stability check of pipe members

D12.A.4.6 Angle profiles type RA (reverse angle)

D12.A.4.7 Stability check of members with tapered section

D12.A.4.8 Lateral buckling for tension members

D12.A.4.3 Determination of β_z and β_y