V. EC3 Polish NA - I Section with UDL

Calculate the bending capacity of a beam using an I section subject to a uniformly distributed load per the Polish 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	=	2.578
C₂	=	1.554
$\frac{π^{2} E I_{y}}{{(k L)}^{2}}$	=	7,477,200
${(\frac{k}{k_{w}})}^{2} \frac{I_{w}}{I_{y}}$	=	22,394
$\frac{{(k L)}^{2} G I_{t}}{π^{2} E I_{y}}$	=	23,737
C₂Z_g	=	1.554×160 = 248.6

Therefore, M_cr = 1,539 kN·m

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

So, $λ_{L T} = \sqrt{\frac{w_{y} f_{y}}{M_{c r}}} = \sqrt{\frac{2,149 \times {(10)}^{3} \times 275}{1,539 \times {(10)}^{6}}} = 0.620$

H/b = 320/300 = 1.067 < 2

So, from Table 6.5 of Eurocode 3, α_LT = 0.34 (buckling curve "b")

From Cl. 6.3.2.3 of Eurocode3:

Ф_{LT} = 0.5 [1 + ɑ_{LT} (λ_{LT} - λ_{LT,0}) + β \times {λ_{LT}}^{2}] = 0.5 [1 + 0.34 (0.620 - 0.4) + 0.75 \times {(0.620)}^{2}] = 0.681

So, $χ_{L T} = {(Ф_{L T} + \sqrt{{Ф_{L T}}^{2} - β \times {λ_{L T}}^{2}})}^{- 1} = \frac{1}{0.681 + \sqrt{{(0.681)}^{2} - 0.75 {(0.620)}^{2}}} = 0.908$

For the Polish NA, the value of k_c is evaluated as:

k_{c} = \sqrt{C_{mLT}}

where

ɑ_s	=	M_s / M_h = -10.42 / 20.83 = -0.5 for -1 ≤ ɑ_s ≤ 1; per Table B.3 of Annex B of EC3.
ψ	=	1 for 0 ≤ ψ ≤ 1
C_mLT	=	0.1 - 0.8×ɑ_s = 0.5 > 0.4, thus use C_mLT = 0.5

So, $k_{c} = \sqrt{0.5} = 0.707$

Modification factor: $f = 1 - 0.5 (1 - k_{c}) [1 - 2 {(λ_{L T} - 0.8)}^{2}] = 1 - 0.5 (1 - 0.707) [1 - 2 {(0.620 - 0.8)}^{2}] = 0.863$

$χ_{L T, m o d} = \frac{χ_{L T}}{f} = \frac{0.908}{0.863} = 1.05 > 1$ , thus use $χ_{L T, m o d} = 1$

M_{B} = \frac{χ_{LT} w_{y} f_{y}}{γ_{M 1}} = \frac{1.0 \times 2,149 \times {(10)}^{3} \times 275}{1.0 \times {(10)}^{6}} = 591 kN·m

Results

Table 1. Comparison of results for EC3 French 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)	1,539	1,541.5	negligible
Bending capacity, M_B (kN·m)	591.0	591.0	none

STAAD.Pro Input

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

The following design parameters are used:

The Polish NA is specified using NA 6
Uniformly distributed load w/ fixed-fixed supports: CMM 2
The value of C2 1.554 is specified
The yield strength is directly specified by PY 275000.

STAAD SPACE
START JOB INFORMATION
ENGINEER DATE 1-Feb-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 6
C2 1.554 ALL
CMM 2 ALL
PY 275000 ALL
TRACK 2 ALL
CHECK CODE ALL
FINISH

STAAD.Pro Output

 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   HD320X127   (EUROPEAN SECTIONS)
                           PASS     EC-6.2.5           0.035         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 :  1.10
                                      z-axis          y-axis
      Slenderness ratio (KL/r) :         36.2           66.1
      Compression Capacity     :       4078.2         3045.5
      Tension Capacity         :       4435.8         4435.8
      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 =  591.0
      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.    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   0.035     1     0.0     25.0     0.0    20.8     0.0
   ADDITIONAL CHECKS AS PER NATIONAL ANNEX [NA FOR PN-EN 1993-1-1:2006 ] (units- kN,m):
   EC CLAUSE           NA-CLAUSE        RATIO   LOAD      FX       VY       VZ     MZ       MY
    Torsion has not been considered in the design.
                        _________________________
   ************** END OF TABULATED RESULT OF DESIGN **************