# 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:

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 $π 2 E I y k L 2$ = 7,477,200 $k k w 2 I w I y$ = 22,394 $k L 2 G I t π 2 E I y$ = 23,737 C2Zg = 1.554×160 = 248.6

Therefore, Mcr = 1,539 kN·m

From the Polish NA, λLT, 0 = 0.4, β = 0.75

So, $λ L T = w y f y M c r = 2,149 × ( 10 ) 3 × 275 1,539 × ( 10 ) 6 = 0.620$

H/b = 320/300 = 1.067 < 2

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

From Cl. 6.3.2.3 of Eurocode3:

$Ф LT = 0.5 [ 1 + ɑ LT ( λ LT - λ LT,0 ) + β × λ LT 2 ] = 0.5 [ 1 + 0.34 ( 0.620 - 0.4 ) + 0.75 × ( 0.620 ) 2 ] = 0.681$

So, $χ L T = Ф L T + Ф L T 2 - β × λ L T 2 - 1 = 1 0.681 + ( 0.681 ) 2 - 0.75 ( 0.620 ) 2 = 0.908$

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

$k c = C mLT$
where
 ɑs = Ms / Mh = -10.42 / 20.83 = -0.5 for -1 ≤ ɑs ≤ 1; per Table B.3 of Annex B of EC3. ψ = 1 for 0 ≤ ψ ≤ 1 CmLT = 0.1 - 0.8×ɑs = 0.5 > 0.4, thus use CmLT = 0.5

So, $k c = 0.5 = 0.707$

Modification factor: $f = 1 - 0.5 1 - k c 1 - 2 λ L T - 0.8 2 = 1 - 0.5 1 - 0.707 1 - 2 0.620 - 0.8 2 = 0.863$

$χ L T , m o d = χ L T f = 0.908 0.863 = 1.05 > 1$, thus use $χ L T , m o d = 1$

## Results

Table 1. Comparison of results for EC3 French NA - I Section with UDL
Moment capacity, Mckd (kN·m) 591.0 591.0 none
Critical moment, Mcr (kN·m) 1,539 1,541.5 negligible
Bending capacity, MB (kN·m) 591.0 591.0 none

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
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


 PROGRAM CODE REVISION V1.14 BS_EC3_2005/1
STAAD SPACE                                              -- PAGE NO.    3
ALL UNITS ARE - KN   METE (UNLESS OTHERWISE Noted)
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
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
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 **************