V. EC3 German NA - I Section with UDL

Calculate the bending capacity of a beam using an I section subject to a uniformly distributed load per the German 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} = \frac{W_{p l y} f_{y}}{γ_{M 0}} = \frac{2,149 \times {(10)}^{3} \times 275}{1.0 \times {(10)}^{6}} = 591.0 kN·m

The critical moment is given by:

M_{c r} = C_{1} \frac{π^{2} E I}{{(k L)}^{2}} {\sqrt{{(\frac{k}{k_{w}})}^{2} \frac{I_{w}}{I} + \frac{{(k L)}^{2} G I_{t}}{π^{2} E I} + {(C_{2} Z_{g})}^{2}} - C_{2} Z_{g}}

where

C₁	=	0.712
C₂	=	0.652
k	=	0.5
$\frac{π^{2} E I_{y}}{{(k L)}^{2}}$	=	29,909,000
${(\frac{k}{k_{w}})}^{2} \frac{I_{w}}{I_{y}}$	=	5,599
$\frac{{(k L)}^{2} G I_{t}}{π^{2} E I_{y}}$	=	5,934
C₂Z_g	=	0.652×160 = 104.3

Therefore, M_cr = 966.7 kN·m

From the German NA, λ_{LT, 0} = 0.4, β = 0.75

So, $λ_{L T} = \sqrt{\frac{w_{y} f_{y}}{M_{c r}}} = \sqrt{\frac{2,149 \times {(10)}^{3} \times 275}{966.7 \times {(10)}^{6}}} = 0.782$

H/b = 320/300 = 1.067 < 2. So, from Table 6.5 of Eurocode 3, α_LT = 0.34.

From Cl. 6.3.2.3 of Eurocode3:

Ф_LT = 0.5[1+α_LT (λ_LT- λ_{LT, 0}) + β× λ_LT²] = 0.5[1 + 0.34×(0.782 - 0.2) +1 × 0.782²] = 0.794

So, $χ_{L T} = {(Ф_{L T} + \sqrt{{Ф_{L T}}^{2} - β {λ_{L T}}^{2}})}^{- 1} = 0.827$

So, $k c = 1 / \sqrt{C_{1}} = 1 / \sqrt{0.712} = 1.182 > 1$ but this value cannot be used, so from Table 6.6 of the Eurocode 3, k_c = 0.9

Modification factor: $f = 1 - 0.5 (1 - k_{c}) [1 - 2 {(λ_{L T} - 0.8)}^{2}] = 1 - 0.5 (1 - 0.9) [1 - 2 {(0.782 - 0.8)}^{2}] = 0.950$

$χ_{L T, m o d} = \frac{χ_{L T}}{f} = 0.871 < \frac{1}{{λ_{L T}}^{2}} = 1.635$ , OK

M_{B} = \frac{χ_{L T} w_{y} f_{y}}{γ_{M 1}} = \frac{0.871 \times 2,149 {(10)}^{3} \times 275}{1.1} = 467.7 kN·m

Results

Table 1. Comparison of results for EC3 German NA - I Section with UDL
Result Type	Reference	STAAD.Pro	Difference
Moment capacity, M_ckd (kN·m)	591.0	591.0	none
Critical moment, M_cr (kN·m)	966.7	967.3	negligible
Bending capacity, M_B (kN·m)	467.7	467.8	negligible

STAAD.Pro Input

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

The following design parameters are used:

The German NA is specified using NA 10
Fully fixed span: CMN 0.5
Uniformly distributed load w/ fixed-fixed supports: CMM 2
The value of C2 0.652 is specified
The parameter KC 0 instructs the program to calculate the value of k_c
A second design parameter set is used with MTH 1 to use Cl 6.3.2.2 to determine Ф_LT
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 1-Aug-21
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 10
C2 0.652 ALL
CMM 2 ALL
CMN 0.5 ALL
FU 300000 ALL
KC 0 ALL
*MTH 1 ALL
PY 275000 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.    3
 ALL UNITS ARE - KN   METE (UNLESS OTHERWISE Noted)
 MEMBER     TABLE       RESULT/   CRITICAL COND/     RATIO/     LOADING/
                          FX            MY             MZ       LOCATION
 =======================================================================
 *** WARNING:CMN PARAM INVALID FOR NATIONAL ANNEX
 NATIONAL ANNEX ONLY DEALS WITH END RESTRAINT FACTORS OF K = KW = 1.
 HENCE WILL USE ANNEX F FROM DD ENV 1993-1-1:1992
       1 ST   HD320X127   (EUROPEAN SECTIONS)
                           PASS     EC-6.3.2 LTB       0.045         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 :  1.10          GM2 :  1.25
                                      z-axis          y-axis
      Slenderness ratio (KL/r) :         36.2           66.1
      Compression Capacity     :       3707.4         2768.6
      Tension Capacity         :       3484.1         3484.1
      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 =  467.8
      co-efficients C1 & K : C1 =0.712 K =0.5, Effective Length= 5.000
      Lateral Torsional Buckling Curve : Curve b
      Elastic Critical Moment for LTB,               Mcr   =   967.3
      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
   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   0.045     1     0.0     25.0     0.0    20.8     0.0
    Torsion has not been considered in the design.
                        _________________________
   ************** END OF TABULATED RESULT OF DESIGN **************