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V. EC3 Belgian NA - I Section with UDL

Calculate the bending capacity of a beam using an I section subject to a uniformly distributed load per the Belgian NA to EC3.

Details

The section is a HD320X127, Grade S275 steel. The member is a 5 m, fixed-fixed span subject to a 10 kN/m uniform load.

Validation

Moment capacity:

M c k d = W p l y f y γ M 0 = 2,149 × ( 10 ) 3 × 275 1.0 × ( 10 ) 6 = 591.0  kN·m

The critical moment is given by:

M c r = C 1 π 2 E I ( k L ) 2 { ( k k w ) 2 I w I + ( k L ) 2 G I t π 2 E I + ( C 2 Z g ) 2 C 2 Z g }
where
C1
=
2.578
C2
=
1.554
k
=
0.5
π 2 E I y k L 2
=
29,909,000
k k w 2 I w I y
=
5,599
k L 2 G I t π 2 E I y
=
5,934
C2Zg
=
1.554×160 = 248.6

Therefore, Mcr = 1,712 kN·m

From the Belgian NA, λLT, 0 = 0.2, β = 1.0

So, λ L T = w y f y M c r = 2,149 × ( 10 ) 3 × 275 1,712 × ( 10 ) 6 = 0.588

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+αLTLT- λLT, 0) + β× λLT2] = 0.5[1 + 0.34×(0.589 - 0.2) +1 × 0.5882] = 0.739

So, χ L T = Ф L T + Ф L T 2 - β λ L T 2 - 1 = 0.843

From Table 6.6 of the Eurocode 3, kc = 0.9

Modification factor: f = 1 - 0.5 1 - k c 1 - 2 λ L T - 0.8 2 = 1 - 0.5 1 - 0.9 1 - 2 0.588 - 0.8 2 = 0.955

χ L T , m o d = χ L T f = 0.883 < 1 λ L T 2 = 2.897 , OK

M B = χ L T w y f y γ M 1 = 0.883 × 2,149 ( 10 ) 3 × 275 1.0 = 521.8  kN·m

If Cl. 6.3.2.2 is used to determine χLT:

H/b = 320/300 = 1.067 < 2. So, from Table 6.3 of Eurocode 3, αLT = 0.21.

ФLT = 0.5[1+αLTLT- 0.2) + λLT2] = 0.5[1+0.21×(0.588 - 0.2) +1×0.5882] = 0.713

χ L T = Ф L T + Ф L T 2 - λ L T 2 - 1 = 0.895
M B = χ L T w y f y γ M 1 = 0.895 × 2,149 ( 10 ) 3 × 275 1.0 × ( 10 ) 6 = 528.9  kN·m

Results

Table 1. Comparison of results for EC3 Belgian NA - I Section with UDL
Result Type Reference STAAD.Pro Difference Comments
Moment capacity, Mckd (kN·m) 591.0 591.0 none  
Critical moment, Mcr (kN·m) 1,712 1,712.9 negligible  
Bending capacity, MB (kN·m) 521.8 522.1 negligible  
Bending capacity, MB (kN·m) 528.9 528.8 negligible  

STAAD.Pro Input

The file C:\Users\Public\Public Documents\STAAD.Pro CONNECT Edition\Samples\ Verification Models\09 Steel Design\Europe\EC3 National Annex\Belgian NA\EC3 Belgian NA - I Section with UDL.std is typically installed with the program.

The following design parameters are used:

  • The Belgian NA is specified using NA 8
  • Fully fixed span: CMN 0.5
  • Uniformly distributed load w/ fixed-fixed supports: CMM 2
  • The value of C2 1.554 is specified
  • The parameter KC instructs the program to calculate the value of kc
  • A second design parameter set is used with MTH 1 to use Cl 6.3.2.2 to determine ФLT
    Note: The previously set parameters are still used as they are not specified again with a different value, which is the default behavior in STAAD.Pro batch design.
STAAD SPACE
START JOB INFORMATION
ENGINEER DATE 14-Apr-20
END JOB INFORMATION
INPUT WIDTH 79
UNIT METER KN
JOINT COORDINATES
1 0 0 0; 2 5 0 0;
MEMBER INCIDENCES
1 1 2;
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
MEMBER PROPERTY EUROPEAN
1 TABLE ST HD320X127
CONSTANTS
MATERIAL STEEL ALL
SUPPORTS
1 2 FIXED
LOAD 1 LOADTYPE None  TITLE LOAD CASE 1
MEMBER LOAD
1 UNI GY -10
PERFORM ANALYSIS
PARAMETER 1
CODE EN 1993-1-1:2005
NA 8
C2 1.554 ALL
CMM 2 ALL
CMN 0.5 ALL
PY 275000 ALL
KC 0 ALL
TRACK 2 ALL
CHECK CODE ALL
PARAMETER 2
CODE EN 1993-1-1:2005
NA 8
MTH 1 ALL
CHECK CODE ALL
FINISH

STAAD.Pro Output

                         STAAD.PRO CODE CHECKING - NBN EN 1993-1-1:2005
                         ********************************************
                         NATIONAL ANNEX - NBN EN 1993-1-1 ANB : 2018
 PROGRAM CODE REVISION V1.13 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   HD320X127   (EUROPEAN SECTIONS)
                           PASS     EC-6.3.2 LTB       0.040         1                                                        
                        0.00            0.00          20.83        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               :       81.998          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  :     0.000
      GM0 :  1.00          GM1 :  1.00          GM2 :  1.25
                                      z-axis          y-axis
      Slenderness ratio (KL/r) :         36.2           66.1
      Compression Capacity     :       4078.2         3045.5
      Tension Capacity         :       4180.9         4180.9
      Moment Capacity          :        591.0          258.3
      Reduced Moment Capacity  :        591.0          258.3
      Shear Capacity           :       1301.9          821.3
   BUCKLING CALCULATIONS (units - kN,m)
      Lateral Torsional Buckling Moment       MB =  522.1
      co-efficients C1 & K : C1 =2.578 K =0.5, Effective Length= 5.000
      Lateral Torsional Buckling Curve : Curve b 
      Elastic Critical Moment for LTB,               Mcr   =  1712.9
      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.    4
 ALL UNITS ARE - KN   METE (UNLESS OTHERWISE Noted)
 MEMBER     TABLE       RESULT/   CRITICAL COND/     RATIO/     LOADING/
                          FX            MY             MZ       LOCATION
 =======================================================================
   CRITICAL LOADS FOR EACH CLAUSE CHECK (units- kN,m):
    CLAUSE        RATIO  LOAD     FX       VY      VZ      MZ      MY   
   EC-6.2.5       0.035     1     0.0     25.0     0.0    20.8     0.0
   EC-6.2.6-(Y)   0.030     1     0.0     25.0     0.0    20.8     0.0
   EC-6.3.2 LTB   0.040     1     0.0     25.0     0.0    20.8     0.0
    Torsion and deflections have not been considered in the design.
                        _________________________
   ************** END OF TABULATED RESULT OF DESIGN **************
    39. PARAMETER 2
    40. CODE EN 1993-1-1:2005
    41. NA 8
    42. MTH 1 ALL
    43. CHECK CODE ALL
 STEEL DESIGN    
                         STAAD.PRO CODE CHECKING - NBN EN 1993-1-1:2005
                         ********************************************
                         NATIONAL ANNEX - NBN EN 1993-1-1 ANB : 2018
 PROGRAM CODE REVISION V1.13 BS_EC3_2005/1
      STAAD SPACE                                              -- PAGE NO.    5
 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-6.3.2 LTB       0.039         1                                                        
                        0.00            0.00          20.83        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               :       81.998          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  :     0.000
      GM0 :  1.00          GM1 :  1.00          GM2 :  1.25
                                      z-axis          y-axis
      Slenderness ratio (KL/r) :         36.2           66.1
      Compression Capacity     :       4078.2         3045.5
      Tension Capacity         :       4180.9         4180.9
      Moment Capacity          :        591.0          258.3
      Reduced Moment Capacity  :        591.0          258.3
      Shear Capacity           :       1301.9          821.3
   BUCKLING CALCULATIONS (units - kN,m)
      Lateral Torsional Buckling Moment       MB =  528.8
      co-efficients C1 & K : C1 =2.578 K =0.5, Effective Length= 5.000
      Lateral Torsional Buckling Curve : Curve a 
      Elastic Critical Moment for LTB,               Mcr   =  1712.9
      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.    6
 ALL UNITS ARE - KN   METE (UNLESS OTHERWISE Noted)
 MEMBER     TABLE       RESULT/   CRITICAL COND/     RATIO/     LOADING/
                          FX            MY             MZ       LOCATION
 =======================================================================