V. GB500017-2017 Tube section with Combined Axial and Bending

Verify the strength, stability, and slenderness of a tube section subject to combined axial and bending per GB50017-2017.

Reference

MOHURD. 2017. GB 50017-2017 Standard for design of steel structures . Beijing, China: Ministry of Housing and Urban-Rural Development

Problem

The section is a TUB 300x10.0 with a length of 4 m. The structure is a portal frame. Member #1 assigned with a tube section (TUB300x10.0) is designed per GB 50017-2017. The section has an unbraced length of 4 m in either direction. The governing load case #1 has the following ultimate loads on the member:

F_x = 30.55 kN
F_y = 25.06 kN
N = 66.69 kN
M_x = 83.44 kN·m (at member end)
M_y = 122.2 kN·m (at member end)

The material is Q235 type steel.

Design strength in tension, compression, and flexure: f_y = 215 MPa
Design strength in shear: f_v = 125 MPa

Calculations

Section Properties

Section depth, H = 300 mm
Section width, D = 300 mm
Wall thickness, t = 10 mm
Cross-sectional area, A = 11,260 mm²
Moment of inertia about x, I_x = I_y = 155,200,00 mm⁴
Radius of gyration about x, r_x = r_y = 118.2 mm
Section modulus about x, W_x = W_y = 1,035,000 mm³

Slenderness Ratio

The effective length is:

l_ox = μ_x×l_x = 1.402 × 4,000 mm = 5,188 mm

l_oy = μ_y×l_y = 2.0383 × 4,000 mm = 8,150 mm

The slenderness ratio is:

λ_{x} = \frac{l_{0 x}}{i_{x}} = \frac{μ_{x} l}{i_{x}} = \frac{1.402 \times 4,000}{117.4} = 47.77 < 150

λ_{y} = \frac{l_{0 y}}{i_{y}} = \frac{μ_{y} l}{i_{y}} = \frac{2.038 \times 4,000}{117.4} = 69.44 < 150

According to table 7.4.6 of Standard for design of steel structures, the allowable slenderness ratio of compression members is λ_clim = 150.

Ratio: λ_max / λ_clim = 69.44 / 150 = 0.46

According to table 7.4.7 of Standard for design of steel structures, the allowable slenderness ratio of tension members is λ_tlim = 300.

Ratio: λ_max / λ_tlim = 69.44 / 300 = 0.23

Steel Grade Correction Factor

The steel type is Q235. According to note 1 in table 3.5.1 of Standard for design of steel structures and table 1 of Standard for design of steel structures. Commentary 2.2, the steel grade correction coefficient is ε_k = 1.

Check Flange Width to Thickness Ratio of Compression Member

\frac{B - 2 t}{t} = \frac{300 - 2 \times 10}{10} = 28

According to table 3.5.1, note 2 of Standard for design of steel structures, and the width thickness ratio of cross-section plate is grade S3, so the limit value of height thickness ratio is:

{40 ε}_{k} = 40

Ratio:

\frac{28}{40} = 0.70

Note: Since the section is a square tube, the web heigh to flange thickness check is the same.

Plastic Development Coefficient

According to clause 8.1.1 of Standard for design of steel structures, and the width thickness ratio of cross-section plate is grade S3,

γ_x = 1.05

γ_y = 1.05

Check Member Strength

According to clause 8.1.1 of Standard for design of steel structures,

σ = \frac{N}{A_{n}} + \frac{M_{x}}{γ_{x} W_{nx}} + \frac{M_{y}}{γ_{y} W_{ny}} = \frac{N}{A} + \frac{M_{x}}{γ_{x} W_{x}} + \frac{M_{y}}{γ_{y} W_{y}}

= \frac{66.69 × 10^{3}}{11,260} + \frac{83.44 × 10^{6}}{1.05 × 1,034,600} + \frac{122.2 × 10^{6}}{1.05 × 1,034,600} = 195.2 {N/mm}^{2}

Ratio: $\frac{σ}{f} = \frac{195.2}{215} = 0.91$

Check In-Plane Stability of Member

According to table 7.2.1-1 of Standard for design of steel structures, the section is "b" for this section.

According to the formula (D.0.5) in Appendix D of Standard for design of steel structures, for the x-x direction:

λ_{n} = \frac{λ_{x}}{π} \sqrt{\frac{f_{y}}{E}} = \frac{47.77}{π} \sqrt{\frac{235}{206,000}} = 0.514

Class b, so, the coefficients are α1 = 0.650, α2 = 0.965, and α3 = 0.300.

Since $λ_{n} = 0.514 > 0.215$ , then the stability factor in x:

ϕ_{x} = \frac{1}{2 λ_{n}^{2}} [(α_{2} + α_{3} λ_{n} + λ_{n}^{2}) - \sqrt{{(α_{2} + α_{3} λ_{n} + λ_{n}^{2})}^{2} - 4 λ_{n}^{2}}]

= \frac{1}{2 \times {0.514}^{2}} [(0.965 + 0.300 \times 0.514 + {0.514}^{2}) - \sqrt{{(0.965 + 0.300 \times 0.514 + {0.514}^{2})}^{2} - 4 {\times 0.514}^{2}}] = 0.866

According to the formula 8.2.1-8 of Standard for design of steel structures,

N_{c r} = \frac{π^{2} E I}{{(μ l)}^{2}} = \frac{π^{2} E I_{x}}{{(μ_{x} l)}^{2}}

= \frac{π^{2} \times 206,000 \times 155,190,000}{{(1.402 \times 4,000)}^{}} = 10,033 kN

According to the formula 8.2.1-10 of Standard for design of steel structures,

β_{m x} = 1 - 0.36 N / N_{c r} = 1 - 0.36 \times 66.69 / (10,033) = 0.998

According to the formula 8.2.1-12 and 8.2.1-5 note of Standard for design of steel structures,

β_{t y} = 0.65 + 0.35 \times \frac{M_{2}}{M_{1}} = 0.65 + 0.35 \times \frac{6,000}{122,200,000} = 0.65

According to the formula 8.2.1-12 of Standard for design of steel structures,

N_{E x}^{'} = \frac{π^{2} E A}{1.1 λ_{x}^{2}} = \frac{π^{2} \times 206,000 \times 11,260}{1.1 \times {(47.77)}^{2}} = 9,121 kN

According to the formula 8.2.5, ϕ_by = 1.0.

According to the formula 8.2.1, η = 0.7.

According to the formula 8.2.5-1,

\frac{N}{ϕ_{x} A f} + \frac{β_{m x} M_{x}}{γ_{x} W_{x} (1 - 0.8 \frac{N}{N_{E x}^{'}}) f} + η \frac{β_{t y} M_{y}}{ϕ_{b y} W_{y} f}

\frac{66.69 \times 10^{3}}{0.866 \times 11,260 215} + \frac{0.998 \times 83.44 \times 10^{6}}{1.05 \times 1,034,600 (1 - 0.8 \times \frac{66.69}{9,121}) 215} + 0.7 \frac{0.65 \times 122.2 \times 10^{6}}{1.0 \times 1,034,600 \times 215} = 0.64

Check Out-of-Plane Stability of Member

According to table 7.2.1-1 of Standard for design of steel structures, the section is "b" for this section.

According to the formula (D.0.5) in Appendix D of Standard for design of steel structures, for the y-y direction:

λ_{n} = \frac{λ_{y}}{π} \sqrt{\frac{f_{y}}{E}} = \frac{69.44}{π} \sqrt{\frac{235}{206,000}} = 0.747

Class b, so, the coefficients are α1 = 0.650, α2 = 0.965, and α3 = 0.300.

Since $λ_{n} = 0.747 > 0.215$ , then the stability factor in x:

ϕ_{x} = \frac{1}{2 λ_{n}^{2}} [(α_{2} + α_{3} λ_{n} + λ_{n}^{2}) - \sqrt{{(α_{2} + α_{3} λ_{n} + λ_{n}^{2})}^{2} - 4 λ_{n}^{2}}]

= \frac{1}{2 \times {0.747}^{2}} [(0.965 + 0.300 \times 0.747 + {0.747}^{2}) - \sqrt{{(0.965 + 0.300 \times 0.747 + {0.747}^{2})}^{2} - 4 {\times 0.747}^{2}}] = 0.754

According to the formula 8.2.1-8 of Standard for design of steel structures,

N_{c r} = \frac{π^{2} E I}{{(μ l)}^{2}} = \frac{π^{2} E I_{y}}{{(μ_{y} l)}^{2}}

= \frac{π^{2} \times 206,000 \times 155,190,000}{{(2.038 \times 4,000)}^{}} = 4,748 kN

According to the formula 8.2.1-10 of Standard for design of steel structures,

β_{m y} = 1 - 0.36 N / N_{c r} = 1 - 0.36 \times 66.69 / (4,748) = 0.995

According to the formula 8.2.1-12 and 8.2.1-5 note of Standard for design of steel structures,

β_{t x} = 1.0

According to the formula 8.2.1-12 of Standard for design of steel structures,

N_{E y}^{'} = \frac{π^{2} E A}{1.1 λ_{y}^{2}} = \frac{π^{2} \times 206,000 \times 11,260}{1.1 \times {(69.44)}^{2}} = 4,316 kN

According to the formula 8.2.5, ϕ_bx = 1.0.

According to the formula 8.2.1, η = 0.7.

According to the formula 8.2.5-2,

\frac{N}{ϕ_{x} A f} + η \frac{β_{t x} M_{x}}{ϕ_{bx} W_{x} f} + \frac{β_{m y} M_{y}}{γ_{y} W_{y} (1 - 0.8 \frac{N}{N_{E y}^{'}}) f}

\frac{66.69 \times 10^{3}}{0.754 \times 11,260 215} + 0.7 \frac{1.0 \times 83.44 \times 10^{6}}{1.0 \times 1,034,600 \times 215} + \frac{0.995 \times 122.2 \times 10^{6}}{1.05 \times 1,034,600 (1 - 0.8 \times \frac{66.69}{4,316}) 215} = 0.83

Shear Strength

Take the neutral axis as the calculation point of shear stress, calculate the area moment:

S = 417,605 mm³

According to clause 6.1.3 of Standard for design of steel structures, shear stress

τ_{x} = \frac{V S}{I t_{w}} = \frac{V S_{x}}{I_{y} (2 t)} = \frac{30.55 \times 10^{3} \times 631,000}{155,190,000 \times 2 \times 10} = 6.21 {N/mm}^{2}

τ_{y} = \frac{V S}{I t_{w}} = \frac{V S_{x}}{I_{x} (2 t)} = \frac{25.06 \times 10^{3} \times 631,000}{155,190,000 \times 2 \times 10} = 5.09 {N/mm}^{2}

τ_max = 7.69 < f_v = 125 N/mm²

Therefore, the ratio is:

\frac{τ_{max}}{f_{v}} = \frac{6.21}{125} = 0.05

Comparison

Table 1. Comparison of results
Result Type	Reference	STAAD.Pro	Difference
Compression Slenderness	0.46	0.46	none
Tension Slenderness	0.23	0.23	none
Flange width to thickness ratio	0.70	0.70	none
Web height to thickness ratio	0.70	0.70	none
Column Strength	0.91	0.91	none
In-plane stability	0.64	0.64	none
Out-plane stability	0.83	0.83	none
Shear Strength	0.05	0.05	none

STAAD.Pro Input File

The file C:\Users\Public\Public Documents\STAAD.Pro CONNECT Edition\Samples\ Verification Models\09 Steel Design\China\GB500017-2017 Tube section with Combined Axial and Bending.std is typically installed with the program.

STAAD SPACE
START JOB INFORMATION
ENGINEER DATE 21-Aug-18
END JOB INFORMATION
INPUT WIDTH 79
UNIT METER KN
JOINT COORDINATES
1 0 0 0; 2 0 4 0; 3 6 4 0; 4 6 0 0;
MEMBER INCIDENCES
1 1 2; 2 2 3; 3 3 4;
DEFINE MATERIAL START
ISOTROPIC STEEL
E 2.05e+008
POISSON 0.3
DENSITY 76.8195
ALPHA 1.2e-005
DAMP 0.03
TYPE STEEL
STRENGTH FY 253200 FU 407800 RY 1.5 RT 1.2
END DEFINE MATERIAL
MEMBER PROPERTY CHINESE
1 TABLE ST TUB30030010.0
3 TABLE ST PIP299X10.0
2 TABLE ST HN300X150
CONSTANTS
MATERIAL STEEL ALL
SUPPORTS
1 4 FIXED
LOAD 1 LOADTYPE None  TITLE LOAD CASE 1
MEMBER LOAD
2 UNI GY -10
JOINT LOAD
2 3 FX 30 FY -50 FZ 30
PERFORM ANALYSIS
FINISH

Chinese steel design parameters (.gsp file):

[[version=2207]
*{ The below data is for code check general information, please do not modify it.
[CodeCheck]
SeismicGrade=None
BeamBendingStrength=1
BeamShearStrength=1
BeamEquivalentStress=1
BeamOverallStability=1
BeamSlendernessWeb=1
BeamSlendernessFlange=1
TrussStrength=1
TrussStability=1
TrussShearStrength=1
ColumnStrength=1
ColumnStabilityMzMy=1
ColumnStabilityMyMz=1
PressedTrussSlenderness=1
TensionTrussSlenderness=1
ColumnSlendernessFlange=1
ColumnSlendernessWeb=1
BeamDeflection=1
SelectAll=0
GroupOptimize=0
FastOptimize=0
Iteration=0
SecondaryMembers=
SectCollectionOrder=0
[CheckOptionAngle]
PrimaryAxis=60.000000
SecondaryAxis=60.000000
ExtendLine=10.000000
*{ The above data is for code check general information, please do not modify it.

[GROUP=1]
Name(Parameter Name)=TUBE
Type(Member Type)=3
Principle(Principle Rules)=0
SteelNo()=Q235
SectionSlendernessRatioGrade(Section Slenderness Ratio Grade)=3
Fatigue(Fatigue Calculation)=0
Optimization(Perform optimized design)=0
MaxFailure(Failure Ratio)=1
MinTooSafe(Safety Ratio)=0.3
CheckLoadCase(Force Loads Case No.)=1 
CheckDispLoadCase(Displacement Loads Case No.)=1 
BeamBendingStrength()=1
BeamShearStrength()=1
BeamEquivalentStress()=1
BeamOverallStability()=1
BeamSlendernessFlange(b/t on beam)=1
BeamSlendernessWeb(h0/tw on beam)=1
TrussStrength(Axial Force Strength)=1
SecondaryMoment(Secondary Moment of Truss)=0
TrussStability(Solid-web Axial Compression Stability)=1
TrussShearStrength(Axial Shear Strength)=1
PressedTrussSlenderness(Pressed Member Slenderness)=1
TensionTrussSlenderness(Tension Member Slenderness)=1
ColumnStrength(Column Member Strength)=1
ColumnStabilityMzMy(Column Stability In-plane)=1
ColumnStabilityMyMz(Column Stability Out-plane)=1
ColumnSlendernessFlange(b/t on column)=1
ColumnSlendernessWeb(h0/tw on column)=1
CheckItemAPPENDIX_B11(Beam Deflection)=1
UseAntiSeismic(Use Seismic Adjusting Factor)=0
GamaReStr(Seismic Adjusting Factor of Load-bearing Capacity for Strength)=0
GamaReSta(Seismic Adjusting Factor of Load-bearing Capacity for Stability)=0
SLevel(Grade of Seismic Resistance)=0
lmdc(Slenderness Limit of Compression Member)=0
lmdt(Slenderness Limit of Tension Member)=0
Lmd831(Slenderness of Seismic Column)=0
Lmd841(Slenderness of Seismic Brace)=0
Lmd9213(Slenderness of Seismic Single-story Plant)=0
LmdH28(Slenderness of Seismic Multi-story Plant)=0
rz(Plastic Development Factor in Major Axis)=0
ry(Plastic Development Factor in Minor Axis)=0
gamaSharp(Plastic Development Factor of sharp side)=0
betamz(the equivalent moment factor in Major Axis plane)=0
betamy(the equivalent moment factor in Minor Axis plane)=0
betatz(the equivalent moment factor out Major Axis plane)=1
betaty(the equivalent moment factor out Minor Axis plane)=0
HasHorLoadZ(Has Horizontal Load in Z-Axis)=0
HasHorLoadY(Has Horizontal Load in Y-Axis)=0
DFF(Deflection Limit of Beam)=150
DJ1(Start Node Number in Major Axis)=0
DJ2(End Node Number in Major Axis)=0
Horizontal(Check for Deflection in Minor Axis)=0
Cantilever(Cantilever Member)=0
fabz(Overall Stability Factor in Major Axis of Bending Member)=0
faby(Overall Stability Factor in Minor Axis of Bending Member)=0
StressFeature(Select the Stress Feature to calulate stability factor of beam)=1
faz(Overall Stability Factor in Major Axis of Axial Compression Member=0
fay(Overall Stability Factor in Minor Axis of Axial Compression Member)=0
lz(Unbraced Length in Major Axis)=0
ly(Unbraced Length in Minor Axis)=0
miuz(Effective Length Factor for Column in Major Axis)=0
miuy(Effective Length Factor for Column in Minor Axis)=0
Lateral(Member in Frame Without Sidesway or not)=0
APZ(Gyration Radius Calculation as Z-Axis Parallel Leg)=0
rFlange(Limit Ratio of Width to Thickness for Flange)=0
rWeb(Limit Ratio of High to Thickness for Web)=0
BucklingStrength(Axis forced member bulking strength)=0
ZSectType(Section Type in Z-Axis)=0
YSectType(Section Type in Y-Axis)=0
HSectWebInTrussPlane(Web of H in Truss Plane)=0
rAn(Net Factor of Section Area)=1
rWnz(Net Factor of Resistance Moment in Z-Axis)=1
rWny(Net Factor of Resistance Moment in Y-Axis)=1
CapReduce(Seismic Reduction Factor of Load-bearing Capacity for Brace)=1
AngleReduce(Angle Strength Reduce)=0
LAglConSta(Connect Type of unequal single angle)=0
LAngleStrength(Reduction Factor of Angle Strength)=0
LAngleStability(Reduction Factor of Angle Stability)=0
rTrussSectReduce(Effective Factor of Axial Force Section)=1
Members(Member Number)=1

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