D12.B.1.5 Ultimate Limit State
Axial Tension
Clause 6.3.2 states that tubular members subject to axial tension shall satisfy the following condition:
N_{Sd} ≤ N_{t,Rd} = A⋅f_{y}/γ_{m} 
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Axial Compression
Clause 6.3.3 states that tubular members subject to axial compression shall satisfy the following condition:
N_{Sd} ≤ N_{c,Rd} = A⋅f_{c}/γ_{m} 
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The design axial compressive strength for a member that is not subject to any hydrostatic pressure will be taken as the smaller of in plane or out of plane buckling strengths determined by the equations given below:
f_{c} = [1.0  028⋅λ ^{2}]f_{y} when λ ≤ 1.34 
f_{c} = 0.9/λ ^{2}⋅f_{y} when λ > 1.34 
λ = √(f_{cl}/f_{E}) = k⋅l/(π⋅i)√(f_{cl}/E) 
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The characteristic local buckling strength is determined from:
f_{cl} = f_{y} when f_{y}/f_{cle} ≤ 0.170 (Plastic yielding) 
f_{cl} = [1.047  0.274⋅f_{y}/f_{cle}]⋅f_{y} when 0.170 < f_{y}/f_{cle} ≤ 1.911 (Elastic/Plastic) 
f_{cl} = f_{cle} when f_{y}/f_{cle} > 1.911 (Elastic buckling) 
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For a member that is subject to pure compression, if f_{y}/f_{cle} > 0.170, the section will be classed as a CLASS 4 (slender section). In such cases, the value of the material factor (γ_{m}) used in the above checks is increased according to equation 6.22 (Cl. 6.3.7) of the code.
Bending
Clause 6.3.4 states that tubular members subject to pure bending alone shall satisfy:
M_{Sd} ≤ M_{Rd} = f_{m}⋅W/γ_{m} 
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The bending strength f_{m} is calculated as:
f_{m} = Z/W⋅f_{y} when f_{y}D/(E⋅t) ≤ 0.0517 
f_{m} = [1.13  2.58⋅f_{y}D/(E⋅t)]⋅Z/W⋅f_{y} when 0.0517 < f_{y}D/(E⋅t) ≤ 0.1034 
f_{m} = [0.94  0.76⋅f_{y}D/(E⋅t)]⋅Z/W⋅f_{y} when 0.1034 < f_{y}D/(E⋅t) ≤ 120⋅f_{y}/E 
Shear
Clause 6.3.5 states that tubular members subject to shear shall satisfy:
V_{Sd} ≤ V_{Rd} = A⋅f_{y}/(2√3⋅γ_{m)}) 
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When torsional shear stresses are present, the following condition shall also be satisfied:
M_{T,Sd} ≤ M_{T,Rd} = 2⋅I_{p}f_{y}/(D√3⋅γ_{m)}) 
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Hydrostatic Pressure
Clause 6.3.6 states that tubular members subject to an external pressure shall primarily be checked for hoop buckling. The condition to be satisfied is:
σ_{p,Sd} ≤ f_{h,Rd} = f_{h}/γ_{m)} 
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The characteristic hoop buckling strength f_{h}, will be calculated as follows:
f_{h} = f_{y} when f_{he} > 2.44⋅f_{y} 
f_{h} = 0.7⋅f_{y}(f_{he}/f_{y})_{0.4} when 2.44⋅f_{y} ≥ f_{he} > 0.55⋅f_{y} 
f_{h} = f_{he} when f_{he} ≤ 0.55⋅f_{y} 
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The elastic hoop buckling strength, f_{he} , is evaluated as follows:
f_{he} = 2C_{h}E⋅t/D
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Combined Axial Tension and Bending (without Hydrostatic Pressure)
Clause 6.3.8.1 states that tubular members subject to axial tension and bending shall be designed to satisfy the following condition:
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Combined Axial Compression and Bending (without Hydrostatic Pressure)
Clause 6.3.8.2 states that tubular members subject to axial tension and bending shall be designed to satisfy the following conditions:
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=  ${N}_{Ey}=\frac{{\pi}^{2}EA}{{\left(\frac{k\ell}{i}\right)}_{y}^{2}}$
${N}_{Ez}=\frac{{\pi}^{2}EA}{{\left(\frac{k\ell}{i}\right)}_{z}^{2}}$
 
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=  ${N}_{cl,Rd}=\frac{{f}_{cl}A}{{\gamma}_{M}}$
 
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The reduction factors used in this clause depend on the "structural element type" and will be as given in Table 62 of N004. This requires the member to be classified under any one of the section types given in the table.
Combined Bending and Shear (without Hydrostatic Pressure)
Clauses 6.3.8.3 & 6.3.8.4 state that tubular members subject to beam shear force (excluding shear due to torsion) and bending moments shall satisfy:
M_{Sd}/M_{Rd} ≤ √(1.4  V_{Sd}/V_{Rd}) when V_{Sd}/V_{Rd}≥ 0.4 
M_{Sd}/M_{Rd} ≤ 1.0 when V_{Sd}/V_{Rd}< 0.4 
If the member is subject to shear forces due to torsion along with bending moments, the condition to be satisfied is:
M_{Sd}/M_{Red,Rd} ≤ √(1.4  V_{Sd}/V_{Rd}) when V_{Sd}/V_{Rd}≥ 0.4 
M_{Sd}/M_{Red,Rd} ≤ 1.0 when V_{Sd}/V_{Rd}< 0.4 
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Combined Loads with Hydrostatic Pressure
Clause 6.3.9 of NS004 describes two methods to check for members subject to combined forces in the presence of hydrostatic pressure: depending on whether the hydrostatic forces were included as nodal forces in the analysis or not. If the hydrostatic forces have not been included in the analysis as nodal forces, Method A given in the code is used. If, however, the hydrostatic forces have been included in the analysis, then Method B in the code is used. Prior to proceeding with the checks described in the sections below, the section is verified for hoop stress limit per clause 6.3.6 (see Hydrostatic Pressure above).
The choice of method for checking members subject to combined forces and hydrostatic pressure used by STAAD.Pro will depend on the HYD parameter specified as a design parameter. If the HYD parameter has been specified, then the program will assume that the hydrostatic forces have not been included in the analysis and will perform the necessary checks as per Method A in code. If, on the other hand, the HYD parameter has not been specified, the program will use the section forces and use Method B in the code.
Combined Axial Tension, Bending, and Hydrostatic Pressure
Checks per Clause 6.3.9.1:

When HYD is specified:
The following condition is to be satisfied:

For the net axial tension condition (σ_{a,Sd} ≥ σ_{q,Sd})
$\frac{{\sigma}_{a,Sd}{\sigma}_{q,Sd}}{{f}_{th,Rd}}+\frac{\sqrt{{\sigma}_{my,Sd}^{2}+{\sigma}_{mz,Sd}^{2}}}{{f}_{mh,Rd}}\le 1.0$where σ_{a,Sd}
=  the design axial stress, excluding any axial compression from hydrostatic pressure.
 σ_{q,Sd}
=  the design axial compressive stress due to hydrostatic pressure. (i.e., the axial load arising from the hydrostatic pressure being applied as nodal loads).
 σ_{my,Sd}
=  the out of plane bending stress
 σ_{mz,Sd}
=  the in plane bending stress
 f_{th,RD}
=  f_{y}/γ_{m}[√(1 + 0.09⋅B^{2}  B^{2η})  0.3B]
 f_{mh,RD}
=  f_{m}/γ_{m}[√(1 + 0.09⋅B^{2}  B^{2η})  0.3B]
 B
=  σ_{psd}/ f_{h,Rd}
 η
=  5  4⋅f_{h}/f_{y}

For the net axial compression condition (σ_{a,Sd} < σ_{q,Sd})
$\frac{{\sigma}_{a,Sd}{\sigma}_{q,Sd}}{{f}_{cl,Rd}}+\frac{\sqrt{{\sigma}_{my,Sd}^{2}+{\sigma}_{mz,Sd}^{2}}}{{f}_{mh,Rd}}\le 1.0$where f_{cl,Rd}
=  f_{cl}/γ_{m}
 f_{cl}
=  the characteristic local buckling strength (as determined by Clause 6.3.3)
Additionally, when:
σ_{c,Sd} > 0.5⋅f_{he}/γ_{m}
and
f_{cle} > 0.5⋅f_{he}
the following condition shall be satisfied in addition to the above check(s):
$\frac{{\sigma}_{c,Sd}0.5\frac{{f}_{he}}{{\gamma}_{M}}}{\frac{{f}_{cle}}{{\gamma}_{M}}0.5\frac{{f}_{he}}{{\gamma}_{M}}}+{\left(\frac{{\sigma}_{p,Sd}}{\frac{{f}_{he}}{{\gamma}_{M}}}\right)}^{2}\le 1.0$where σ_{c,Sd}
=  the maximum compressive stress at that section.


When HYD has not been specified:
$\frac{{\sigma}_{ac,Sd}}{{f}_{th,Rd}}+\frac{\sqrt{{\sigma}_{my,Sd}^{2}+{\sigma}_{mz,Sd}^{2}}}{{f}_{mh,Rd}}\le 1.0$where σ_{ac,Sd}
=  the axial stress in the member
Combined Axial Compression, Bending, and Hydrostatic Pressure
Checks per Clause 6.3.9.2:

Method used when HYD has been specified:
The following condition is to be satisfied:
$\frac{{\sigma}_{a,Sd}}{{f}_{ch,Rd}}+\frac{1}{{f}_{mh,Rd}}\sqrt{{\left(\frac{{C}_{my}{\sigma}_{my,Sd}}{1\frac{{\sigma}_{a,Sd}}{{f}_{Ey}}}\right)}^{2}+{\left(\frac{{C}_{mz}{\sigma}_{mz,Sd}}{1\frac{{\sigma}_{a,Sd}}{{f}_{Ez}}}\right)}^{2}}\le 1.0$and
$\frac{{\sigma}_{a,Sd}+{\sigma}_{q,Sd}}{{f}_{cl,Rd}}+\frac{\sqrt{{\sigma}_{my,Sd}^{2}+{\sigma}_{mz,Sd}^{2}}}{{f}_{mh,Rd}}\le 1.0$Where:
where σ_{a,Sd}
=  the design axial stress that excludes the stress from hydrostatic pressure
 fch,Rd
= $\frac{1}{2}\frac{{f}_{cl}}{{\gamma}_{M}}[\xi \frac{2{\sigma}_{q,Sd}}{{f}_{cl}}+\sqrt{{\xi}^{2}+1.12{\overline{\lambda}}^{2}\frac{{\sigma}_{q,Sd}}{{f}_{cl}}}]\text{\hspace{0.17em}}\text{when}\overline{\lambda}<1.34\sqrt{{(1\frac{2{\sigma}_{q,Sd}}{{f}_{fl}})}^{1}}$
$\frac{0.9{f}_{cl}}{{\overline{\lambda}}^{2}{\gamma}_{M}}\text{when}\overline{\lambda}\ge 1.34\sqrt{{(1\frac{2{\sigma}_{q,Sd}}{{f}_{fl}})}^{1}}$
 ξ
=  10.28λ ^{2}
Additionally, when:
σ_{c,Sd} > 0.5⋅f_{he}/γ_{m}
and
f_{cle} > 0.5⋅f_{he}
the following condition shall be satisfied in addition to the above check(s):
$\frac{{\sigma}_{c,Sd}0.5\frac{{f}_{he}}{{\gamma}_{M}}}{\frac{{f}_{cle}}{{\gamma}_{M}}0.5\frac{{f}_{he}}{{\gamma}_{M}}}+{\left(\frac{{\sigma}_{p,Sd}}{\frac{{f}_{he}}{{\gamma}_{M}}}\right)}^{2}\le 1.0$ 
Method used when HYD has not been specified:
The following condition is to be satisfied:

For the net axial tension condition (σ_{ac,Sd} ≥ σ_{q,Sd})
$\frac{{\sigma}_{ac,Sd}{\sigma}_{q,Sd}}{{f}_{ch,Rd}}+\frac{1}{{f}_{mh,Rd}}\sqrt{{\left(\frac{{C}_{my}{\sigma}_{my,Sd}}{1\frac{{\sigma}_{a,Sd}}{{f}_{Ey}}}\right)}^{2}+{\left(\frac{{C}_{mz}{\sigma}_{mz,Sd}}{1\frac{{\sigma}_{a,Sd}}{{f}_{Ez}}}\right)}^{2}}\le 1.0$and
$\frac{{\sigma}_{ac,Sd}}{{f}_{cl,Rd}}+\frac{\sqrt{{\sigma}_{my,Sd}^{2}+{\sigma}_{mz,Sd}^{2}}}{{f}_{mh,Rd}}\le 1.0$(Refer to the previous section for an explanation of these terms).

For the net axial compression condition (σ_{ac,Sd} < σ_{q,Sd})
$\frac{{\sigma}_{ac,Sd}}{{f}_{cl,Rd}}+\frac{\sqrt{{\sigma}_{my,Sd}^{2}+{\sigma}_{mz,Sd}^{2}}}{{f}_{mh,Rd}}\le 1.0$(Refer to the previous section for an explanation of these terms).
Additionally, when:
σ_{c,Sd} > 0.5⋅f_{he}/γ_{m}
and
f_{cle}/γ_{m} > 0.5⋅f_{he}/γ_{m}
the following condition shall be satisfied in addition to the above check(s):
$\frac{{\sigma}_{c,Sd}0.5\frac{{f}_{he}}{{\gamma}_{M}}}{\frac{{f}_{cle}}{{\gamma}_{M}}0.5\frac{{f}_{he}}{{\gamma}_{M}}}+{\left(\frac{{\sigma}_{p,Sd}}{\frac{{f}_{he}}{{\gamma}_{M}}}\right)}^{2}\le 1.0$where σ_{c,Sd}
=  the maximum compressive stress at that section.
