D12.A.5 Stability Check According to NPD

D12.A.5.1 Buckling of pipe members

Tubular beam-columns subjected to compression and lateral loading or end moments shall be designed in accordance with NPD 3.2.2

σ_{c} γ_{m k} + B σ_{b}^{*} + \sqrt{{(B_{z} σ_{b z})}^{2} + {(B_{y} σ_{b y})}^{2}} \leq \frac{f_{y}}{γ_{m}}

Where:

σ_c = N/A = axial compressive stress
ν_mk = structural coefficient
B = bending amplification factor = 1/ (1 - μ), B is taken as the larger of B_z and B_y
B_z = bending amplification factor about the Z-axis
B_y = bending amplification factor about the Y-axis
$μ = σ_{c} / f_{E}$
$f_{E} = \frac{π^{2} E}{l_{k}^{2}} i^{2}$
$i = \sqrt{I / A}$
$σ_{b}^{*} = σ_{c} (\frac{f_{y}}{f_{k}} - 1) (1 - \frac{f_{k}}{γ_{m} f_{E}})$
l_k = kl
k = effective length factor
f_k = characteristic buckling capacity according to NS fig. 5.4.1a, curve A.

D12.A.5.2 Interaction with local buckling, NPD 3.2.3

If the below conditions are not satisfied, the yield strength will be replaced with characteristic buckling stress given in NPD 3.4.

members subjected to axial compression and external pressure

$\frac{d}{t} \leq 0.5 \sqrt{\frac{E}{f_{y}}}$
members subjected to axial compression only

$\frac{d}{t} \leq 0.1 \frac{E}{f_{y}}$

D12.A.5.3 Calculation of buckling resistance of cylinders

The characteristic buckling resistance is defined in accordance with NPD 3.4.4

f_{k} = \frac{f_{y}}{\sqrt{1 + {\bar{λ}}^{4}}}

Where:

{\bar{λ}}^{2} = \frac{f_{y}}{σ_{j}} (\frac{σ_{a o}}{f_{e a}} + \frac{σ_{b 0}}{f_{e b}} + \frac{σ_{p 0}}{f_{e p}} + \frac{τ}{f_{e τ}})

σ_{j} = \sqrt{{(σ_{a} + σ_{b})}^{2} - (σ_{a} + σ_{b}) σ_{p} + σ_{p}^{2} + 3 τ^{2}}

σ_a ≥ 0 when σ_a0 = 0
σ_a < 0 when σ_a0 = -σ_a
σ_b ≥ 0 when σ_b0 = 0
σ_b < 0 when σ_b0 = -σ_b
σ_p ≥ 0 when σ_p0 = 0
σ_p < 0 when σ_p0 = σ_p
σ_a = design axial stress in the shell due to axial forces (tension positive)
σ_b = design bending stress in the shell due to global bending moment (tension
positive)
σ_p = σ_Θ = design circumferential stress in the shell due to external pressure (tension positive)
τ_S = design shear stress in the shell due to torsional moments and shear force.
f_ea, f_eb, f_ep and f_eι are the elastic buckling resistances of curved panels or circular cylindrical shells subjected to axial compression forces, global bending moments, lateral pressure, and torsional moments and/or shear forces respectively.

D12.A.5.4 Elastic buckling resistance for un-stiffened, closed cylinders

The elastic buckling resistance for un-stiffened closed cylinders according to NPD 3.4.6 is:

f_{e} = k \frac{π^{2} E}{12 (1 - ν^{2})} {(\frac{t}{l})}^{2}

where k is a buckling coefficient dependent on loading condition, aspect ratio, curvature, boundary conditions, and geometrical imperfections. The buckling coefficient is:

k = ψ \sqrt{1 + {(\frac{p ξ}{ψ})}^{2}}

The values of ψ, ζ, and p are given in Table 4.1 for the most important loading cases.

Table 1. Table 4.1 Buckling coefficients for un-stiffened cylindrical shells
	ψ	ζ	p
Axial or Bending stress	1	0.702 Z	$0.5 {(1 + \frac{r}{150 t})}^{- 0.5}$
Torsion and shear force	5.34	0.856 Z0.75	0.6
Lateral pressure	4	1.04 Z0.5	"
Hyrdostatic pressure	2	1.04 Z 0.5	"

The curvature parameter is defined by

Z = \frac{1^{2}}{r t} \sqrt{1 - ν^{2}}

For long shells the elastic buckling resistance against shear stresses is independent of shell length. For cases with:

\frac{1}{r} > 3.85 \sqrt{\frac{r}{t}}

the elastic buckling resistance may be taken as:

f_{e p} = 0.25 E {(\frac{t}{r})}^{2}

D12.A.5.5 Stability requirements

The stability requirement for curved panels and un-stiffened cylindrical shells subjected to axial compression or tension, bending, circumferential compression or tension, torsion or shear is given by NPD 3.4.7:

σ_j < f_kd

where the design buckling resistance is

f_{k d} = \frac{f_{k}}{γ_{m} γ_{m k}}

D12.A.5.6 Column buckling, NPD 3.4.9

For long cylindrical shells it is possible that interaction between shell buckling and overall column buckling may occur because second-order effects of axial compression alter the stress distribution as compared to that calculated from linear theory. It is necessary to take this effect into account in the shell buckling analysis when the reduced slenderness of the cylinder as a column exceeds 0,2 according to NPD 3.4.4.1.

σ_bshall be increased by an additional compressive stress which may be taken as:

Δ σ = B σ_{a} (\frac{f_{y}}{f_{k}} - 1) (1 - \frac{f_{k}}{f_{e}}) + (B - 1) σ_{b}

Where:

\bar{B} = \frac{1}{1 - μ}

\bar{λ} = \sqrt{f_{y} / f_{e}}

f_{e} = \frac{π^{2} E}{λ^{2}}

λ = slenderness of the cylinder as a column.

B, σ_a, σ_b, and μ are calculated in accordance with NPD 3.2.2.