V. EC3 German NA - Built up Section with UDL

Calculate the bending capacity of a beam using a built-up I section subject to a uniformly distributed load per the German NA to EC3.

Details

The section built up from 300 mm × 20.5 mm flange plates and a 279 mm × 11.5 mm web. Grade S275 steel. The member is a 5 m, fixed-fixed span subject to a 10 kN/m uniform load.

Validation

Section Properties

Moment of inertia about major axis,
$I_{z} = 2 [\frac{30 \times {(2.05)}^{3}}{12} + 30 (2.05) {(\frac{27.9 + 2.05}{2})}^{2}] + \frac{1.15 {(27.9)}^{3}}{12} = 29,707 {cm}^{4}$
Moment of inertia about minor axis,
$I_{z} = 2 [\frac{2.05 \times {(30)}^{3}}{12}] + \frac{27.9 {(1.15)}^{3}}{12} = 9,229 {cm}^{4}$
Elastic section modulus about major axis, $Z_{z} = \frac{29,707}{\frac{32}{2}} = 1,857 {cm}^{3}$
Elastic section modulus about minor axis, $Z_{y} = \frac{9,229}{\frac{30}{2}} = 615.2 {cm}^{3}$
Plastic section modulus about major axis,
$W_{ply} = \frac{30 {(32)}^{2}}{4} - \frac{(30 - 1.15) {(27.9)}^{2}}{4} = 2,066 {cm}^{3}$
Torsional constant,
$J = 2 [\frac{30 {(2.05)}^{3}}{3}] + \frac{27.9 {(1.15)}^{3}}{3} = 186.4 {cm}^{4}$
Warping constant,
$I_{w} = \frac{{(32 - 2.05)}^{2} {(30)}^{3} 2.05}{24} = 2,068,700 {cm}^{6}$

Moment capacity:

M_{c k d} = \frac{W_{p l y} f_{y}}{γ_{M 0}} = \frac{2,066 \times {(10)}^{3} \times 275}{1.0 \times {(10)}^{6}} = 568.1 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, $λ_{L T} = \sqrt{\frac{w_{y} f_{y}}{M_{c r}}} = \sqrt{\frac{2,066 \times {(10)}^{3} \times 275}{966.7 \times {(10)}^{6}}} = 0.767$

H/b = 320/300 = 1.067 < 2. So, use buckling curve "c". From Table 6.5 of Eurocode 3, α_LT = 0.49.

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 + 0.49×(0.767 - 0.2) +1 × 0.767²] = 0.810

So, $χ_{L T} = {(Ф_{L T} + \sqrt{{Ф_{L T}}^{2} - β {λ_{L T}}^{2}})}^{- 1} = 0.785$

Calculate k_c: $k_{c} = \sqrt{\frac{1}{C_{1}}} = \sqrt{\frac{1}{0.712}} = 1.185 > 1.0$ ; use the value from Table 6.6 of EC3: k_c = 0.9.

Modification factor: $f = 1 - 0.5 (1 - k_{c}) [1 - 2 {(λ_{L T} - 0.8)}^{2}] = 1 - 0.5 (1 - 0.9) [1 - 2 {(0.767 - 0.8)}^{2}] = 0.950$

$χ_{L T, m o d} = \frac{χ_{L T}}{f} = 0.883 < \frac{1}{{λ_{L T}}^{2}} = 1.702$ , OK

M_{B} = \frac{χ_{L T} w_{y} f_{y}}{γ_{M 1}} = \frac{0.826 \times 2,066 {(10)}^{3} \times 275}{1.1} = 426.5 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)	568.1	568.1	none
Critical moment, M_cr (kN·m)	966.7	967.3	negligible
Bending capacity, M_B (kN·m)	426.5	426.5	none

STAAD.Pro Input

The file C:\Users\Public\Public Documents\STAAD.Pro CONNECT Edition\Samples\ Verification Models\09 Steel Design\Europe\EC3 German NA - Built up 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 SBLT 1 indicates a built-up section
The parameter KC 0 instructs the program to calculate the value of k_c
The parameter PY 275000 indicates an f_y of 275 MPa

STAAD SPACE
INPUT WIDTH 79
UNIT METER KN
JOINT COORDINATES
1 0 0 0; 2 5 0 0;
MEMBER INCIDENCES
1 1 2;
START USER TABLE
TABLE 1
UNIT METER KN
WIDE FLANGE
BUILTUP1
0.01613 0.32 0.0115 0.3 0.0205 0.0003082 9.239e-005 2.251e-006 0.00368 0.0082
END
DEFINE MATERIAL START
ISOTROPIC STEEL
E 2.05e+008
POISSON 0.3
DENSITY 76.8195
ALPHA 1.2e-005
DAMP 0.03
END DEFINE MATERIAL
MEMBER PROPERTY AMERICAN
1 UPTABLE 1 BUILTUP1
CONSTANTS
MATERIAL STEEL ALL
SUPPORTS
1 2 FIXED
LOAD 1 LOADTYPE None  TITLE LOAD CASE 1
MEMBER LOAD
1 UNI GY -10
PERFORM ANALYSIS
PRINT MEMBER PROPERTIES ALL
PARAMETER 1
CODE EN 1993-1-1:2005
NA 10
C2 0.652 ALL
CMM 2 ALL
CMN 0.5 ALL
SBLT 1 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.    4
 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   BUILTUP1    (UPT)
                           PASS     EC-6.3.2 LTB       0.049         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        :    30820.002        9239.001
      Plastic modulus          :     2065.718         931.725
      Elastic modulus          :     1926.250         615.933
      Shear Area               :       82.000          36.800
      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         :       4180.9         4180.9
      Moment Capacity          :        568.1          256.2
      Reduced Moment Capacity  :        568.1          256.2
      Shear Capacity           :       1301.9          584.3
   BUCKLING CALCULATIONS (units - kN,m)
      Lateral Torsional Buckling Moment       MB =  426.5
      co-efficients C1 & K : C1 =0.712 K =0.5, Effective Length= 5.000
      Lateral Torsional Buckling Curve : Curve c
      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.    5
   CRITICAL LOADS FOR EACH CLAUSE CHECK (units- kN,m):
    CLAUSE        RATIO  LOAD     FX       VY      VZ      MZ      MY   
   EC-6.2.5       0.037     1     0.0     25.0     0.0    20.8     0.0
   EC-6.2.6-(Y)   0.043     1     0.0     25.0     0.0    20.8     0.0
   EC-6.3.2 LTB   0.049     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 **************