D5.C.5.6 Design of Slender pipe sections to EN 1993-1-6

The design of Slender CHS sections is performed per EN 1993-1-6:2007 (hereafter, EC3-6). EC3-6 does not specify additional or modified safety factors. Therefore, the program uses the default safety factors from EN 1993-1-1.

Note: You can change these values through the GM0, GM1, & GM2 design parameters.

EC3-6 deals with four types of ultimate limits states: plastic limit state, cyclic capacity limit state, buckling limit state, and fatigue. The following are considered by STAAD.Pro:

LS1 – Plastic limit state: Deals with the condition when the capacity of the structure is exhausted by yielding of the material.
LS3 – Buckling Limit state: Deals with the condition in which the structure (or shell) develops large displacements normal to the shell surface, caused by loss of stability under compressive and/or shear membrane stresses.

The limit state verification is made based on the Stress design method described in EC3-6. The stress design approach takes into account three categories of stresses:

Primary stresses: Stresses that are generated for the member to be in equilibrium with the direct imposed loads.
Secondary stresses: Those that are generated for internal compatibility or for compatibility at supports due to imposed loads or displacements (e.g., temperature, settlement etc.)
Local stresses: Local stresses generated due to cyclic loading (or fatigue).

Only the primary stresses are considered the program. The primary stresses considered are those generated due to axial loads, bending, shear and /or a combination of these conditions.

Note: In the context of slender pipe section design for the Eurocode 3 module, the secondary and local stresses can be neglected since the loads and corresponding stresses dealt with in the design engine are largely direct and shear stresses.

The local axis coordinate system for a CHS is defined as:

circumferential: around the circumference of the circular cross section (θ)
meridional: along the length of the member (x)
normal: perpendicular to the tangential plane formed by the circumferential and meridional directions (n)

and the corresponding membrane stresses will follow the convention given below:

Nomenclature for membrane and transverse stresses in Slender CHS sections

Stress Design

Stress checks are made based on the Stress design method as per Section 8.5 of the code. This section deals with the buckling strength of the member (LS3). The principle is to evaluate the membrane stresses due to the applied loads and then compare that to the buckling strength, which is evaluated giving due consideration for local buckling effects.

The membrane stresses are evaluated as given in Annex A of the code. The pipe section is considered as an unstiffened cylindrical shell.

Meridional Stresses:
1. Axial load
  
  F_x = 2·π·r·P_x
  
  σ_x = -F_x/(2·π·r·t)
2. Axial stress from bending
  
  M = π·r²·P_x,max
  
  σ_x = ±M/(π²·r·t)
Shear Stress:
1. Transverse force, V
  
  V = π·r·P_θ,max
  
  τ_max = ±V/(π·r·t)
2. Shear from torsional moment, M
  
  M_t = 2π·r²·P_θ
  
  τ = M_t/(2π²·r²·t)

Where:

r is the radius of the middle surface of the shell wall.
t is the wall thickness of the cylinder

Calculation of Axial Buckling Stress

The buckling strength of A slender pipe section is evaluated using the method given in section 8.5.2 ofEC3-6. The design buckling stresses (buckling resistance) are calculated separately for axial, circumferential, and shear. The circumferential stresses are ignored in STAAD.Pro.

The naming convention and the coordinate axis used will be as given in the following diagram:

Naming convention and coordinate system used for the buckling stress of a slender CSH section

The axial buckling resistance is given by:

σ_x,Rd = σ_x,Rk/γ_M1

Note: Γ_M1 will have the same default value of 1.0 as in EN 1993-1-1.

σ_x,Rk is the characteristic buckling strength given by:

σ_x,Rk = Χ_x· f_yk

Where:

χ_x is the meridional buckling reduction factor. χ_x is evaluated per Section 8.5.2(4) of EC3-6 and is determined as a function of the relative shell slenderness given by:

λ_{x} = \sqrt{\frac{f_{y k}}{σ_{x, c r}}}

Where:

σ_x,cr is the elastic buckling critical stress.

Once the relative slenderness is evaluated, the reduction factor is calculated as follows:

χ = 1 when $\bar{λ} \leq {\bar{λ}}_{0}$
$χ = 1 - β {(\frac{\bar{λ} - {\bar{λ}}_{0}}{{\bar{λ}}_{P} - {\bar{λ}}_{0}})}^{η}$ when ${\bar{λ}}_{0} < \bar{λ} < {\bar{λ}}_{P}$
$χ = a / {\bar{λ}}^{2}$ when ${\bar{λ}}_{P} \leq \bar{λ}$

where

{\bar{λ}}_{P}

the plastic limit for slenderness given by:

\sqrt{\frac{α}{1 - β}}

The meridional buckling parameters the factors α and β are evaluated per section D.1.2.2 of EC3-6.

Note: A "Normal" fabrication quality will be assumed when evaluating the fabrication quality parameter as given in table D.2 of the code, unless the fabrication quality is set using the FAB design parameter. Refer to D5.C.6 Design Parameters

The elastic critical buckling stress, σ_x,cr and the factors α and β are evaluated per Annex D of EC3-6. The details are as given below:

The CHS section is classified based on the following criteria:

CHS Length Classification	Criteria
Short	ω ≤ 1.7
Medium	1.7 < ω ≤ 0.5· r/t
Long	ω > 0.5· r/t

Where:

ω = \frac{l}{\sqrt{r t}}

The elastic critical buckling critical stress is evaluated as:

σ_x,Rcr = 0.605·E·C_x·(t/r)

Where:

C_x is a factor dependent upon the CHS length classification as described in section D.1.2.1 of EC-3-6.
For a long cylinder, there are two separate methods that can be used to evaluate the C_x factor: Eqns D.9/10 and Eqn D.12. Initially the program evaluates C_x based on the maximum from equations D.9 and D.10. However, for long cylinders that satisfy the conditions in equation D.11, the program will also work out Cx based on equation D.12 and then choose the minimum obtained from D.12 and D.9/10.

Calculation of Shear Buckling Stress

The shear buckling resistance is given by:

τ_xθ,Rd = τ_xθ,Rk/γ_M1

Note: γ_M1 will have the same default value of 1.0 as in EN 1993-1-1.

τ_xθ,Rk is the characteristic buckling shear strength given by:

τ_xθ,Rk = Χ_θ· f_yk

Where:

χ_θ is the shear buckling reduction factor. χ_θ will be worked out as given in section 8.5.2(4) of En 1993-1-6 and is determined as a function of the relative shell slenderness given by:

λ_{θ} = \sqrt{\frac{f_{y k}}{τ_{x θ, c r}}}

Where:

τ_xθ,Rk is the elastic buckling critical stress.

The reduction factor, χ_θ, is then evaluated as described for the axial buckling stress, based on the same λ _p, α, and β parameters given in Annex D of EC3-6.

The CHS section is classified based on the following criteria:

CHS Length Classification	Criteria
Short	ω ≤ 10
Medium	10 < ω ≤ 8.7· r/t
Long	ω > 8.7· r/t

Where:

ω = \frac{l}{\sqrt{r t}}

The elastic critical buckling critical stress is evaluated as:

τ_{x θ, R c r} = 0.75 E C_{τ} \sqrt{\frac{1}{ω}} (\frac{t}{r})

Where:

C_τis a factor dependent upon whether the CHS length classification as described in section D.1.4.1 of EC-3-6.
A "Normal" fabrication quality will be assumed when working out the fabrication quality parameter as given in table D.6 of the code, unless the fabrication quality is set using the FAB design parameter.

Buckling Strength Verification

The buckling strength verification will be performed so as to satisfy the following conditions:

For axial stresses:

σ_x,Ed ≤ σ_x,Rd

For shear stresses:

τ_xθ,Ed ≤ τ_xθ,Rd

For a combined case of axial and shear stresses acting together, an interaction check will be done according to equation 8.19 of the code as below:

{(\frac{σ_{x, E d}}{σ_{x, R d}})}^{k_{x}} + {(\frac{τ_{x θ, E d}}{τ_{x θ, R d}})}^{k_{τ}} \leq 1

Where:

k_x and k_τ are the interaction factors as given in section D.1.6 of EN 1993-1-6:

k_x = 1.25 + 0.75 · χ_x

k_τ = 1.75 + 0.25 · χ_τ