V. EC3 - Tube Section with Axial Load
Verify the adequacy of a cantilever tube section subject to axial force and bending per EN 1993-1-1:2005 (no national annex used).
Details
The member is a 5 m long cantilever. The member is subject to a 25 kN axial compressive load along with moments of 10 kN·m about the major axis and 5 kN·m about the minor axis at the free end. The steel is grade S275. The section is a European TUB120806.
Section Properties
- A = 23.4 cm2
- Depth, D = 120 mm
- Width, B = 80 mm
- t = 6.3 mm
- Iz= 447 cm4
- Iy = 234 cm4
- Zz = 91 cm3
- Zy = 68.2 cm3
- J = 486 cm4
- Cw = 0 cm6
Partial safety factors:
ΓM0 = 1.0
ΓM1 = 1.0
ΓM2 = 1.25
Validation
Section Classification
ɑ = 1
C = D - 2t = 120 - 2 × 6.3 = 107.4 mm
As per Table 5.2:
C / t = 107.4 / 6.3 = 17.05 < 50×ε2 = 50 (0.924)2 = 42.7
Hence, this is a Class 1 section.
Slenderness Ratio
The slenderness ratio = kL / r = 1 × 5,000 / 31.62 = 158.1
Axial Tension
Determine axial tension capacity per Cl. 6.2.3.
The tensile capacity is the minimum of Npl,Rd and Nu,Rd, thus: Nt,Rd = 497.0 kN.
No axial tension in section, so by observation no need to check ratio.
Axial Compression
Determine axial compression capacity per Cl. 6.2.4 for a Class 1 section.
Next, check the flexural buckling resistance per Cl. 6.3.1.3:
From Table 6.1: the imperfection factor, ɑ = 0.21.
Along the Z axis: where= | ||
= | ||
= |
ƛz = 1.318
Along the Y axis: where= |
ƛz = 1.822
The compression capacity is the minimum of Nc,Rd and Nb,Rd, thus: Nc,Rd = 170.2 kN.
Ratio per Eq. 6.9: MEd / Mc,Rd = 25.0 / 170.2 = 0.147
Bending Capacity
Along Z axis:
Maximum bending moment in the section: MEd,z = 10.0 kN·m.
Check for bending capacity per Cl. 6.2.5:
For a Class 1 section:
Ratio per Eq. 6.12: MEd / Mc,Rd = 10.0 / 25.03 = 0.400
Along Y axis:
Maximum bending moment in the section: MEd,y = 5.0 kN·m.
Check for bending capacity per Cl. 6.2.5:
For a Class 1 section:
Ratio per Eq. 6.12: MEd / Mc,Rd = 5.0 / 18.76 = 0.267
Shear Capacity
Along Z direction:
Check for shear capacity per Cl. 6.2.6 for plastic design (Class 1):
Along Y direction:
Check for shear capacity per Cl. 6.2.6 for plastic design (Class 1):
No shear in section, so by observation no need to check ratio.
Lateral Torsional Buckling
where= | ||
= | ||
= | ||
= | ||
= | ||
= |
Therefore,
As per Cl. 6.3.2.1, tube sections are not susceptible to lateral-torsional buckling, so χLT = 1.0.
Ratio per Eq. 6.12: MEd / Mb,Rd = 10.0 / 24.92 = 0.400
Check for Interaction
From Table B.3 in Annex B of EC3:
Ψ = 1.0
Cmy = Cmz = CmLT = 0.6 + 0.4×Ψ = 1.0
NRk = A × fy = 643.5 kN
From Table B.1 of Annex B, the interaction factors:
Kzz = 1.067
Kyy = 1.118
Kyz = 0.6×Kzz = 0.641
Kzy = 0.6×Kyy = 0.671
For Kyz, consider Table B.2 as well:
So, Kyz = 0.980
Check for Clause 6.3.3-661:
Check for Clause 6.3.3-662:
Results
Result Type | Reference | STAAD.Pro | Difference | Comments |
---|---|---|---|---|
Section Class | Class 1 | Class 1 | none | |
Slenderness Ratio | 158.1 | 158.1 | none | |
Tension Capacity (kN) | 497.0 | 497 | none | |
Compression Capacity (Major) (kN) | 296.3 | 296.3 | none | |
Compression Capacity (Minor) (kN) | 170.2 | 170.2 | none | |
Moment Capacity (Major) (kN·m) | 25.03 | 25 | negligible | |
Moment Capacity (Minor) (kN·m) | 18.76 | 18.8 | negligible | |
Shear Area (Major) (cm2) | 9.36 | 9.36 | none | |
Shear Area (Minor) (cm2) | 14.04 | 14.04 | none | |
Shear Capacity (Major) (kN) | 148.6 | 148.6 | none | |
Shear Capacity (Minor) (kN) | 222.9 | 222.9 | none | |
Mcr (kN·m) | 261.8 | 258.5 | negligible | |
MB (kN·m) | 25.03 | 25 | negligible | |
Ratio per Cl. 6.3.1.1 | 0.147 | 0.147 | none | |
Ratio per Cl. 6.2.9.1 | 0.400 | 0.4 | none | |
Ratio per Cl. 6.3.3-661 | 0.690 | 0.69 | none | |
Ratio per Cl. 6.3.3-662 | 0.837 | 0.837 | none |
STAAD.Pro Input
The file C:\Users\Public\Public Documents\STAAD.Pro 2023\Samples \Verification Models\09 Steel Design\Europe\EC3 - Tube Section with Axial Load.std is typically installed with the program.
- Fixed end supports: CMM 2
- Cantilever member: CMN 0.7
- Use Cl. 6.3.2.2 to determine χLT: MTH 1
- The values of C1 1.0 and C2 1.0 are specified.
- The values of Fy and Fu are specified directly using PY 275000 and FU 295000.
STAAD SPACE
START JOB INFORMATION
ENGINEER DATE 05-May-21
END JOB INFORMATION
INPUT WIDTH 79
UNIT METER KN
JOINT COORDINATES
1 0 0 0; 2 0 5 0;
MEMBER INCIDENCES
1 1 2;
MEMBER PROPERTY EUROPEAN
1 TABLE ST TUB120806
DEFINE MATERIAL START
ISOTROPIC STEEL
E 2.05e+08
POISSON 0.3
DENSITY 76.8195
ALPHA 1.2e-05
DAMP 0.03
END DEFINE MATERIAL
CONSTANTS
MATERIAL STEEL ALL
SUPPORTS
1 FIXED
LOAD 1 LOADTYPE Dead TITLE LOAD CASE 1
JOINT LOAD
2 FY -25 MX 5 MZ 10
PERFORM ANALYSIS
PARAMETER 1
CODE EN 1993-1-1:2005
CMM 2 ALL
C1 1 ALL
C2 1 ALL
FU 295000 ALL
PY 275000 ALL
MTH 1 ALL
CMN 0.7 ALL
TRACK 2 ALL
CHECK CODE ALL
PRINT MEMBER PROPERTIES
FINISH
STAAD.Pro Output
STAAD.PRO CODE CHECKING - EN 1993-1-1:2005 ******************************************** NATIONAL ANNEX - NOT USED PROGRAM CODE REVISION V1.14 BS_EC3_2005/1 STAAD SPACE -- PAGE NO. 3 ALL UNITS ARE - KN METE (UNLESS OTHERWISE Noted) MEMBER TABLE RESULT/ CRITICAL COND/ RATIO/ LOADING/ FX MY MZ LOCATION ======================================================================= 1 ST TUB120806 (EUROPEAN SECTIONS) PASS EC-6.3.3-662 0.837 1 25.00 C 5.00 -10.00 0.00 ======================================================================= MATERIAL DATA Grade of steel = USER Modulus of elasticity = 205 kN/mm2 Design Strength (py) = 275 N/mm2 SECTION PROPERTIES (units - cm) Member Length = 500.00 Gross Area = 23.40 Net Area = 23.40 z-axis y-axis Moment of inertia : 447.000 234.000 Plastic modulus : 91.000 68.200 Elastic modulus : 74.500 58.500 Shear Area : 9.360 14.040 Radius of gyration : 4.371 3.162 Effective Length : 500.000 500.000 DESIGN DATA (units - kN,m) EUROCODE NO.3 /2005 Section Class : CLASS 1 Squash Load : 643.50 Axial force/Squash load : 0.039 GM0 : 1.00 GM1 : 1.00 GM2 : 1.25 z-axis y-axis Slenderness ratio (KL/r) : 114.4 158.1 Compression Capacity : 296.3 170.2 Tension Capacity : 497.0 497.0 Moment Capacity : 25.0 18.8 Reduced Moment Capacity : 25.0 18.8 Shear Capacity : 148.6 222.9 BUCKLING CALCULATIONS (units - kN,m) Lateral Torsional Buckling Moment MB = 25.0 co-efficients C1 & K : C1 =1.000 K =1.0, Effective Length= 5.000 Lateral Torsional Buckling Curve : CURVE d Elastic Critical Moment for LTB, Mcr = 258.5 Compression buckling curves: z-z: Curve a y-y: Curve a STAAD SPACE -- PAGE NO. 4 CRITICAL LOADS FOR EACH CLAUSE CHECK (units- kN,m): CLAUSE RATIO LOAD FX VY VZ MZ MY EC-6.3.1.1 0.147 1 25.0 0.0 0.0 -10.0 5.0 EC-6.2.9.1 0.400 1 25.0 0.0 0.0 -10.0 5.0 EC-6.3.3-661 0.690 1 25.0 0.0 0.0 -10.0 5.0 EC-6.3.3-662 0.837 1 25.0 0.0 0.0 -10.0 5.0 EC-6.2.5 0.400 1 25.0 0.0 0.0 -10.0 5.0 Torsion has not been considered in the design. _________________________ ************** END OF TABULATED RESULT OF DESIGN **************