V. EC3 French NA - Channel Section with Conc Load

Calculate the bending capacity of a channel section beam under subject to a pair of concentrated load per the French NA to EC3

Details

The section is a C15X50, grade S275 steel. The member is a 6 m span, simply supported beam is subject to two concentrated loads of 10 kN each at the 1/3 points (2 m from either support).

Validation

Moment capacity:

M_{c k d} = \frac{W_{p l y} f_{y}}{γ_{M 0}} = \frac{1,123 {(10)}^{3} 275}{1.0 {(10)}^{6}} = 308.7 kN·m

The critical moment is given by:

M_{c r} = C_{1} \frac{π^{2} E I}{{(k_{z} L)}^{2}} [\sqrt{{(\frac{k_{z}}{k_{w}})}^{2} \frac{I_{w}}{I_{z}} + \frac{{(k_{z} L)}^{2} G I_{T}}{π^{2} E I_{z}} + {(C_{2} Z_{g} - C_{3} Z_{j})}^{2}} - (C_{2} Z_{g} - C_{3} Z_{j})]

where

C₁	=	1.046
C₂	=	0.42
C₃	=	1.0
$\frac{π^{2} E I_{y}}{{(k L)}^{2}}$	=	257,320
${(\frac{k}{k_{w}})}^{2} \frac{I_{w}}{I_{y}}$	=	28,852
$\frac{{(k L)}^{2} G I_{T}}{π^{2} E I_{y}}$	=	330,055
C₂Z_g	=	0.42×0.5×381 = 80.01
β_f	=	$\frac{{b_{c}}^{3} + t_{c}}{{b_{c}}^{3} + t_{c} +- {b_{t}}^{3} + t_{t}} = 0.5$ (From SN030a-EN-EU)
Z_j	=	0.4×h_s (2×β_f - 1) = 0, since β_f ≥ 0.5
C₃Z_j	=	1.0 (0) = 0

Therefore, $M_{cr} = 1.046 \times 257,320 [\sqrt{28,852 + 330,055 + {(80.01)}^{2}} - (80.01)] = 141.2 kN·m$

Results

Table 1.
Result Type	Reference	STAAD.Pro	Difference	Comments
Moment capacity, M_ckd (kN·m)	308.7	308.7	none
Critical moment, M_cr (kN·m)	141.2	142.2	negligible

STAAD.Pro Input

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

The following design parameters are used:

The French NA is specified using NA 4
Simply-supported span: CMN 1 and C3 1
Concentrated loads at 1/3 points supports: CMM 5
The value of C2 0.42 is specified

STAAD PLANE
START JOB INFORMATION
ENGINEER DATE 15-June-20
END JOB INFORMATION
INPUT WIDTH 79
UNIT METER KN
JOINT COORDINATES
1 0 0 0; 2 6 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 AMERICAN
1 TABLE ST C15X50
CONSTANTS
MATERIAL STEEL ALL
SUPPORTS
1 2 PINNED
LOAD 1 LOADTYPE None  TITLE LOAD CASE 1
MEMBER LOAD
1 CON GY -10 2
1 CON GY -10 4
PERFORM ANALYSIS
PRINT MEMBER PROPERTIES ALL
PARAMETER 1
CODE EN 1993-1-1:2005
NA 4
C2 0.42 ALL
C3 1 ALL
CMM 5 ALL
CMN 1 ALL
FU 295000 ALL
PY 275000 ALL
TRACK 2 ALL
CHECK CODE ALL
FINISH

STAAD.Pro Output

                         STAAD.PRO CODE CHECKING - NF EN 1993-1-1:2005
                         ********************************************
                         NATIONAL ANNEX - NF EN 1993-1-1/NA
 PROGRAM CODE REVISION V1.14 BS_EC3_2005/1
      STAAD PLANE                                              -- PAGE NO.    4
 ALL UNITS ARE - KN   METE (UNLESS OTHERWISE Noted)
 MEMBER     TABLE       RESULT/   CRITICAL COND/     RATIO/     LOADING/
                          FX            MY             MZ       LOCATION
 =======================================================================
       1 ST   C15X50      (AISC SECTIONS)
                           PASS     EC-6.3.2 LTB       0.228         1
                        0.00            0.00         -20.00        2.00
 =======================================================================
   MATERIAL DATA                
      Grade of steel           =  USER          
      Modulus of elasticity    =  205 kN/mm2  
      Design Strength  (py)    =  275  N/mm2                         
   SECTION PROPERTIES (units - cm)
      Member Length =    600.00
      Gross Area =   94.84          Net Area =   94.84
                                      z-axis          y-axis
      Moment of inertia        :    16815.750         457.855
      Plastic modulus          :     1122.514         133.391
      Elastic modulus          :      882.716          61.711
      Shear Area               :       20.800          69.290
      Radius of gyration       :       13.316           2.197
      Effective Length         :      600.000         600.000
   DESIGN DATA (units - kN,m)   EUROCODE NO.3 /2005
      Section Class            :   CLASS 1     
      Squash Load              :   2608.06
      Axial force/Squash load  :     0.000
      GM0 :  1.00          GM1 :  1.00          GM2 :  1.25
                                      z-axis          y-axis
      Slenderness ratio (KL/r) :         45.1          273.1
      Compression Capacity     :       2170.6          227.2
      Tension Capacity         :       2014.4         2014.4
      Moment Capacity          :        308.7           36.7
      Reduced Moment Capacity  :        308.7           36.7
      Shear Capacity           :        330.2         1137.8
   BUCKLING CALCULATIONS (units - kN,m)
      Lateral Torsional Buckling Moment       MB =   87.6
      co-efficients C1 & K : C1 =1.046 K =1.0, Effective Length= 6.000
      Lateral Torsional Buckling Curve : Curve d
      Elastic Critical Moment for LTB,               Mcr   =   142.2
      Compression buckling curves:     z-z:  Curve c   y-y:  Curve c
      Critical Load For Torsional Buckling,          NcrT  =  4811.4
      Critical Load For Torsional-Flexural Buckling, NcrTF =  4544.2
      STAAD PLANE                                              -- PAGE NO.    5
   CRITICAL LOADS FOR EACH CLAUSE CHECK (units- kN,m):
    CLAUSE        RATIO  LOAD     FX       VY      VZ      MZ      MY   
   EC-6.2.5       0.065     1     0.0     10.0     0.0   -20.0     0.0
   EC-6.2.6-(Y)   0.009     1     0.0     10.0     0.0     0.0     0.0
   EC-6.3.2 LTB   0.228     1     0.0     10.0     0.0   -20.0     0.0
    Torsion has not been considered in the design.
                        _________________________
   ************** END OF TABULATED RESULT OF DESIGN **************