V. EC3 German NA - Column with Axial Load
Calculate the axial and bending capacities and interaction ratios of a column using the German NA to EC3.
Details
The 5 m tall column as a fixed base and is free to translate and rotate at the top. The column is subject to a 25 kN compressive load and moments of 5 kN·m about the X axis and 10 kN·m about the Z axis. The section is an HD320X127, grade S275 steel.
Validation
Bending Capacity
Moment capacity:
The critical moment is given by:
where= | ||
= | ||
= | ||
= | ||
= | ||
= |
Therefore, Mcr = 1,540.6 kN·m
From the German NA, λLT, 0 = 0.4, β = 0.75
So,
H/b = 320/300 = 1.067 < 2. So, from Table 6.5 of Eurocode 3, αLT = 0.34.
From Cl. 6.3.2.3 of Eurocode3:
ФLT = 0.5[1+αLT (λLT- λLT, 0) + β× λLT2] = 0.5[1 + 0.34×(0.619 - 0.4) +0.75 × 0.6192] = 0.681
So,
and thus modification factor, f = 1 - 0.5(1 - 0.623)×[1 -2(0.619 - 0.8)2] = 0.824
So,
Compression Capacity
Critical compressive values:
Nc,Rd = Agfy / γM0 = 16,130 × 275 / 1.0 (10)-3 = 4,436 kN for a Class 1 section.
Check the flexural buckling resistance per Cl. 6.3.1.1.
Nb,Rd = χ×A×fy / γM1 for a Class 1 section.
Buckling curves for I section are found in Table 6.2. Imperfection factor, ɑ, values are found in Table 6.1.
Along Z
h/b = 1.067 < 2
tf = 20.5 < 100
So, for the z-z axis, use curve "b". Hence, ɑ = 0.34 (EC3 Table 6.2).
where= | ||
= | ||
= |
Thus,
ф = 0.5[1+α (λz - 0.2) +λ2)] = 0.624
Compression capacity about Z:Nb,Rd = χz×A×fy / γM1 = 0.919×16,130×275 / 1.1 = 3,707.5 kN
Along Y
For the y-y axis, use curve "c". Hence, ɑ = 0.49 (EC3 Table 6.2).
where= | ||
= | ||
= |
Thus,
ф = 0.5[1+α (λz - 0.2) +λ2)] = 0.827
Compression capacity about Z:Nb,Rd = χz×A×fy / γM1 = 0.687×16,130×275 / 1.1 = 2,768.6 kN
Thus, the compression capacity, Nb,Rd = 2,768.6 kN
Compression ratio: 25 kN / 2,768.6 kN = 0.009
Critical Axial Loads for Flexure and Flexural Torsional Buckling
From NCCI document SN001a-EN-FU:
where= | iy = 75.68 mm and iz = 138.23 mm = 24,835 mm2 |
Interaction Check
From Annex B, Table B.3 of EN 1993:1-1:2005,
Wz = Wplz / Welz = 2,149 / 1,926 = 1.116
Wy = Wply / Wely = 939 / 615.9 = 1.525 > 1.5; Wy = 1.5
ψ = 1.0
So, Cmz = 0.6 + 0.4ψ = 1.0
CmLT = Cmz = 1.0
Cmy = 0.6 + 0.4ψ = 1.0
Interaction factors:
So, Kyz = maximum(Kyz1, Kyz2) = 0.999
Kzy = 0.6×Kyy = 0.6(1.008) = 0.605
Check for Clause 6.3.3-661:
Check for Clause 6.3.3-662:
Results
Result Type | Reference | STAAD.Pro | Difference | Comments |
---|---|---|---|---|
Moment capacity, Mckd (kN·m) | 591.0 | 591.0 | none | |
Critical moment, Mcr (kN·m) | 1,540.6 | 1,541.5 | negligible | |
Bending capacity, MB (kN·m) | 537.3 | 537.2 | negligible | |
Critical load for torsional buckling, Ncr,T (kN) | 13,889 | 13,898.0 | negligible | |
Critical load for torsional-flexural buckling, Ncr,TF | 13,889 | 13,898.0 | negligible | |
Compression interaction, Cl. 6.3.1.1 | 0.009 | 0.009 | none | |
Bending and compression interaction, Cl 6.3.3-661 | 0.038 | 0.038 | none | |
Bending and compression interaction, Cl. 6.3.3-662 | 0.049 | 0.049 | none |
STAAD.Pro Input
The file C:\Users\Public\Public Documents\STAAD.Pro 2023\Samples \Verification Models\09 Steel Design\Europe\EC3 German NA - Column with Axial Load.std is typically installed with the program.
The following design parameters are used:
- The German NA is specified using NA 10
- Fixed support: CMM 2.0
- Fixed base and free at other end: CMN 0.7
- Program calculated kc per Table 6.6 of EN 1993-1-1:2005: KC 0
STAAD SPACE
START JOB INFORMATION
ENGINEER DATE 05-Aug-2021
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 HD320X127
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 None TITLE LOAD CASE 1
JOINT LOAD
2 FY -25 MX 5 MZ 10
PERFORM ANALYSIS
PRINT MEMBER PROPERTIES ALL
PARAMETER 1
CODE EN 1993-1-1:2005
NA 10
CMM 2 ALL
FU 295000 ALL
PY 275000 ALL
KC 0 ALL
CMN 0.7 ALL
TRACK 2 ALL
CHECK CODE ALL
FINISH
STAAD.Pro Output
STAAD.PRO CODE CHECKING - DIN EN 1993-1-1:2010-12 ******************************************** NATIONAL ANNEX - DIN EN 1993-1-1/NA:2010-12 PROGRAM CODE REVISION V1.14 BS_EC3_2005/1 STAAD SPACE -- PAGE NO. 4 ALL UNITS ARE - KN METE (UNLESS OTHERWISE Noted) MEMBER TABLE RESULT/ CRITICAL COND/ RATIO/ LOADING/ FX MY MZ LOCATION ======================================================================= 1 ST HD320X127 (EUROPEAN SECTIONS) PASS EC-6.3.3-662 0.049 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 = 161.30 Net Area = 161.30 z-axis y-axis Moment of inertia : 30820.004 9239.001 Plastic modulus : 2149.000 939.100 Elastic modulus : 1926.250 615.933 Shear Area : 109.573 51.728 Radius of gyration : 13.823 7.568 Effective Length : 500.000 500.000 DESIGN DATA (units - kN,m) EUROCODE NO.3 /2005 Section Class : CLASS 1 Squash Load : 4435.75 Axial force/Squash load : 0.006 GM0 : 1.00 GM1 : 1.10 GM2 : 1.25 z-axis y-axis Slenderness ratio (KL/r) : 36.2 66.1 Compression Capacity : 3707.4 2768.6 Tension Capacity : 3426.0 3426.0 Moment Capacity : 591.0 258.3 Reduced Moment Capacity : 591.0 258.3 Shear Capacity : 1739.7 821.3 BUCKLING CALCULATIONS (units - kN,m) Lateral Torsional Buckling Moment MB = 537.2 co-efficients C1 & K : C1 =2.578 K =1.0, Effective Length= 5.000 Lateral Torsional Buckling Curve : Curve b Elastic Critical Moment for LTB, Mcr = 1541.5 Compression buckling curves: z-z: Curve b y-y: Curve c Critical Load For Torsional Buckling, NcrT = 13898.0 Critical Load For Torsional-Flexural Buckling, NcrTF = 13898.0 STAAD SPACE -- PAGE NO. 5 CRITICAL LOADS FOR EACH CLAUSE CHECK (units- kN,m): CLAUSE RATIO LOAD FX VY VZ MZ MY EC-6.3.1.1 0.009 1 25.0 0.0 0.0 -10.0 5.0 EC-6.2.9.1 0.020 1 25.0 0.0 0.0 -10.0 5.0 EC-6.3.3-661 0.038 1 25.0 0.0 0.0 -10.0 5.0 EC-6.3.3-662 0.049 1 25.0 0.0 0.0 -10.0 5.0 EC-6.2.5 0.019 1 25.0 0.0 0.0 -10.0 5.0 EC-6.3.2 LTB 0.019 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 **************