# 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:

 NSd ≤ Nt,Rd = A⋅fy/γm

where
 NSd = Design axial force (tension positive) fy = Characteristic yield strength A = Cross section area γm = Default material factor = 1.15

## Axial Compression

Clause 6.3.3 states that tubular members subject to axial compression shall satisfy the following condition:

 NSd ≤ Nc,Rd = A⋅fc/γm

where
 NSd = Design axial force (compression positive) fc = Characteristic axial compressive strength γm = Refer to clause 6.3.7

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:

 fc = [1.0 - 028⋅λ 2]fy when λ ≤ 1.34

 fc = 0.9/λ 2⋅fy when λ > 1.34

 λ = √(fcl/fE) = k⋅l/(π⋅i)√(fcl/E)

where
 fcl = Characteristic local buckling strength λ = Column slenderness parameter fE = Smaller Euler buckling strength in y or z direction E = Young's modulus of elasticity = 2.1x105 MPa k = Effective length factor, refer to Clause 6.3.8.2 l = Longer unbraced length in y or z direction i = Radius of gyration

The characteristic local buckling strength is determined from:

 fcl = fy when fy/fcle ≤ 0.170 (Plastic yielding)

 fcl = [1.047 - 0.274⋅fy/fcle]⋅fy when 0.170 < fy/fcle ≤ 1.911 (Elastic/Plastic)

 fcl = fcle when fy/fcle > 1.911 (Elastic buckling)

where
 fcle = 2CeE⋅t/D (Characteristic elastic local buckling strength) Ce = 0.3 (Critical elastic buckling coefficient) D = Outside diameter t = wall thickness

For a member that is subject to pure compression, if fy/fcle > 0.170, the section will be classed as a CLASS 4 (slender section). In such cases, the value of the material factor (γm) 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:

 MSd ≤ MRd = fm⋅W/γm

where
 MSd = Design bending moment fm = Characteristic bending strength W = Elastic section modulus γm = Refer to clause 6.3.7

The bending strength fm is calculated as:

 fm = Z/W⋅fy when fyD/(E⋅t) ≤ 0.0517

 fm = [1.13 - 2.58⋅fyD/(E⋅t)]⋅Z/W⋅fy when 0.0517 < fyD/(E⋅t) ≤ 0.1034

 fm = [0.94 - 0.76⋅fyD/(E⋅t)]⋅Z/W⋅fy when 0.1034 < fyD/(E⋅t) ≤ 120⋅fy/E

## Shear

Clause 6.3.5 states that tubular members subject to shear shall satisfy:

 VSd ≤ VRd = A⋅fy/(2√3⋅γm))

where
 VSd = Design shear force fy = Yield strength A = Cross section area γm = Default material factor = 1.15

When torsional shear stresses are present, the following condition shall also be satisfied:

 MT,Sd ≤ MT,Rd = 2⋅Ipfy/(D√3⋅γm))

where
 MT,Sd = Design bending moment Ip = Polar moment of inertia

## 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:

 σp,Sd ≤ fh,Rd = fh/γm)

where
 σp,Sd = pSd⋅D/(2⋅t) pSd = Design hydrostatic pressure fh = Characteristic hoop buckling strength γm) = Refer to clause 6.3.7

The characteristic hoop buckling strength fh, will be calculated as follows:

 fh = fy when fhe > 2.44⋅fy

 fh = 0.7⋅fy(fhe/fy)0.4 when 2.44⋅fy ≥ fhe > 0.55⋅fy

 fh = fhe when fhe ≤ 0.55⋅fy

where
 =

The elastic hoop buckling strength, fhe , is evaluated as follows:

fhe = 2ChE⋅t/D

where
 Ch = 0.44⋅t/D when μ ≥1.6⋅D/t Ch = 0.44⋅t/D + 0.21⋅(D/t)3/μ4 when 0.825⋅D/t ≤ μ <1.6⋅D/t Ch = 0.737/(μ - 0.579) when 1.5 ≤ μ < 0.825⋅D/t Ch = 0.8 when μ <1.5 μ = Geometric Parameter = L/D√(2⋅D/t) L = Length of tubular member between stiffening rings, diaphragms, or end connections.

## 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:

$( N S d N t , R d ) 1.75 + M y , S d 2 + M z , S d 2 M R d ≤ 1.0$
where
 My,Sd = the design bending moment about the y axis (out-of plane axis) Mz,Sd = the design bending moment about the z axis (in plane axis) NSd = the design axial force MRd = the moment resistance (as determined by Clause 6.3.4) Nt,Rd = the tension capacity of the section (as determined by Clause 6.3.2)

## 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:

$N S d N c , R d + 1 M R d ( C m y M y , S d 1 − N S d N E y ) 2 + ( C m z M z , S d 1 − N S d N E z ) 2 ≤ 1.0$

and

$N S d N c l , R d + M y , S d 2 + M z , S d 2 M R d ≤ 1.0$
where
 NSd = the design axial compression Cmy and Cmz = the reduction factors corresponding to the Y and Z axes, respectively. You may specify a value for these using the CMY and CMZ design parameters, respectively (default0.85 for both). Ney and Nez = the Euler buckling loads about y & z axes and are given by: $N E y = π 2 E A ( k ℓ i ) y 2$ $N E z = π 2 E A ( k ℓ i ) z 2$ k = the effective length factor and is given in table 6-2 of the code. Ncl, Rd = the design axial local buckling resistance given by: $N c l , R d = f c l A γ M$ fcl = the characteristic local buckling strength (as determined by Clause 6.3.3)

The reduction factors used in this clause depend on the "structural element type" and will be as given in Table 6-2 of N-004. 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:

 MSd/MRd ≤ √(1.4 - VSd/VRd) when VSd/VRd≥ 0.4

 MSd/MRd ≤ 1.0 when VSd/VRd< 0.4

If the member is subject to shear forces due to torsion along with bending moments, the condition to be satisfied is:

 MSd/MRed,Rd ≤ √(1.4 - VSd/VRd) when VSd/VRd≥ 0.4

 MSd/MRed,Rd ≤ 1.0 when VSd/VRd< 0.4

where
 MRed,Rd = W⋅fm,Red/γm fm,Red = fm√[1 - 3(τT,Sd/fd)2] τT,Sd = MT,Sd/(2π⋅R2⋅t) fd = fy/γm R = Radius of the tubular member γm = Refer to clause 6.3.7

## Combined Loads with Hydrostatic Pressure

Clause 6.3.9 of NS-004 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:

1. When HYD is specified:

The following condition is to be satisfied:

1. For the net axial tension condition (σa,Sd ≥ σq,Sd)

$σ a , S d − σ q , S d f t h , R d + σ m y , S d 2 + σ m z , S d 2 f m h , R d ≤ 1.0$
where
 σa,Sd = the design axial stress, excluding any axial compression from hydrostatic pressure. σq,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). σmy,Sd = the out of plane bending stress σmz,Sd = the in plane bending stress fth,RD = fy/γm[√(1 + 0.09⋅B2 - B2η) - 0.3B] fmh,RD = fm/γm[√(1 + 0.09⋅B2 - B2η) - 0.3B] B = σpsd/ fh,Rd η = 5 - 4⋅fh/fy
2. For the net axial compression condition (σa,Sd < σq,Sd)

$| σ a , S d − σ q , S d | f c l , R d + σ m y , S d 2 + σ m z , S d 2 f m h , R d ≤ 1.0$
where
 fcl,Rd = fcl/γm fcl = the characteristic local buckling strength (as determined by Clause 6.3.3)

σc,Sd > 0.5⋅fhem

and

fcle > 0.5⋅fhe

the following condition shall be satisfied in addition to the above check(s):

$σ c , S d − 0.5 f h e γ M f c l e γ M − 0.5 f h e γ M + ( σ p , S d f h e γ M ) 2 ≤ 1.0$
where
 σc,Sd = the maximum compressive stress at that section.
2. When HYD has not been specified:

$σ a c , S d f t h , R d + σ m y , S d 2 + σ m z , S d 2 f m h , R d ≤ 1.0$
where
 σac,Sd = the axial stress in the member

## Combined Axial Compression, Bending, and Hydrostatic Pressure

Checks per Clause 6.3.9.2:

1. Method used when HYD has been specified:

The following condition is to be satisfied:

$σ a , S d f c h , R d + 1 f m h , R d ( C m y σ m y , S d 1 − σ a , S d f E y ) 2 + ( C m z σ m z , S d 1 − σ a , S d f E z ) 2 ≤ 1.0$

and

$σ a , S d + σ q , S d f c l , R d + σ m y , S d 2 + σ m z , S d 2 f m h , R d ≤ 1.0$

Where:

where
 σa,Sd = the design axial stress that excludes the stress from hydrostatic pressure fch,Rd = $1 2 f c l γ M [ ξ − 2 σ q , S d f c l + ξ 2 + 1.12 λ ¯ 2 σ q , S d f c l ] when λ ¯ < 1.34 ( 1 − 2 σ q , S d f f l ) − 1$ $0.9 f c l λ ¯ 2 γ M when λ ¯ ≥ 1.34 ( 1 − 2 σ q , S d f f l ) − 1$ ξ = 1-0.28λ 2

σc,Sd > 0.5⋅fhem

and

fcle > 0.5⋅fhe

the following condition shall be satisfied in addition to the above check(s):

$σ c , S d − 0.5 f h e γ M f c l e γ M − 0.5 f h e γ M + ( σ p , S d f h e γ M ) 2 ≤ 1.0$
2. Method used when HYD has not been specified:

The following condition is to be satisfied:

1. For the net axial tension condition (σac,Sd ≥ σq,Sd)

$σ a c , S d − σ q , S d f c h , R d + 1 f m h , R d ( C m y σ m y , S d 1 − σ a , S d f E y ) 2 + ( C m z σ m z , S d 1 − σ a , S d f E z ) 2 ≤ 1.0$

and

$σ a c , S d f c l , R d + σ m y , S d 2 + σ m z , S d 2 f m h , R d ≤ 1.0$

(Refer to the previous section for an explanation of these terms).

2. For the net axial compression condition (σac,Sd < σq,Sd)

$σ a c , S d f c l , R d + σ m y , S d 2 + σ m z , S d 2 f m h , R d ≤ 1.0$

(Refer to the previous section for an explanation of these terms).

$σ c , S d − 0.5 f h e γ M f c l e γ M − 0.5 f h e γ M + ( σ p , S d f h e γ M ) 2 ≤ 1.0$