# Eurocode 3. Example 6.6: Cross-section resistance under combined bending and compression

In this article we compare the resistance under combined bending and compression according to SDC Verifier and Designer’s Guide to Eurocode3: Design of steel buildings EN 1993-1-1.

All results are generated with SDC Verifier 4.5 and calculated with FEMAP v11.3.1

A member is to be designed to carry a combined major axis bending moment and an axial force. In this example, a cross-sectional check is performed to determine the maximum bending moment that can be carried by a UKB 457 x 191 x 98 in grade S275 steel, in the presence of an axial force of 1400 kN.

Figure 6.15. Section properties for a 457 × 191 × 98 UKB $$h = 467.2 mm$$ $$b = 192.8 mm$$ $$t_{w} \ = 11.4 mm$$ $$t_{f} \ = 19.6 mm$$ $$r = 10.2 mm$$ $$A \ = 12 500 mm^{2}$$ $$W_{pLy} \ = 2 230 000 mm^{3}$$

### Model

#### Materials

TitleYoung Modulus [Pa]Shear Modulus [Pa]Poisson RatioShear [Pa]Mass Density [kg/m^3]Tensile Strength [Pa]Yield Stress [Pa]
1..S2752.10e+1100.300007850.0000355.00e+6265.00e+6

#### Property “1..UKB 457x191x98”

PropertyValueProperty Shape
Type / ElementsBeam / 6 Material1..S275
Mass [kg]137.4
Gravity Center [m][0.70; 0.00; 0.00]
Area, [m^2]0.0125
I1, [m^4]2.346e-05
I2, [m^4]4.533e-04
I12, [m^4]0
Torsion Constant, [m^4]1.160e-06
Y Shear Area, [m^2]0.0081
Z Shear Area, [m^2]0.0051
Nonstructural Mass, [kg]0
Perimeter, [m]1.68
Warping Constant, [m^6]0
Y Neutral Axis Offset A, [m]0
Z Neutral Axis Offset A, [m]0
h [m]0.467
wt [m]0.193
wb [m]0.193
tf2 [m]0.020
tw [m]0.011
tf1 [m]0.020

### Boundary Conditions

#### Load “1..Shear -Y = 1050kN”

Force on NodeForceNodes: 4(0; -1050000; 0) Force on NodeForceNodes: 4(1400000; 0; 0) #### Constaint “1..fixed”

DefinitionCountDOF
Constraint on NodeNodes: 1Tx Ty Tz
Constraint on NodeNodes: 7Ty Rx Ry  ### Classificaton

 Standard 2..Eurocode 3 Member Checks Check 7..Classificaton Individual Load 1..Axial 1400kN Direction All Selection All Entities
LoadAlpha WebPhiLambda WebLimit1 WebLimit2 WebLimit3 WebWeb ClassClass
##### Cross-section classification under pure compression (clause 5.5.2)

$$\varepsilon =\sqrt{235/f_{y}} \ =\sqrt{235/265} \ =0.94$$

Outstand flanges (Table 5.2, sheet 2):

$$c_{f} \ = (b – t_{w} – 2r)/2 \ = 80.5 mm$$

$$c_{f}/t_{f} \ = 80.5/19.6 = 4.11$$

Limit for Class 1 flange = 9ε = 8.48

8.48 > 4.11      ∴ flange is Class 1

Web — internal part in compression (Table 5.2, sheet 1):

$$c_{w} \ = h – 2t_{f} – 2r \ = 407.6 mm$$

$$c_{w}/t_{w} \ = 407.6/11.4 = 35.75$$

Limit for Class 2 web = 38ε = 35.78

35.78 > 35.75      ∴ web is Class 2

Under pure compression, the overall cross-section classification is therefore  Class 2 . Consequently, unlike Example 5.1, nothing is to be gained by using the more complex approach of considering the actual stress distribution.

#### 11..Bending and Axial

 Standard 2..Eurocode 3 Member Checks Check 11..Bending and Axial Individual Load 1..Axial 1400kN Direction All Selection 1..Eurocode3 Shapes
ExtremenReduced Resistance YReduced Resistance ZUf Reduced YUf Reduced ZAlphaBetaUf Combined
Minimum0.42100016.5422113.00.000.002.002.110.00
Maximum0.42100016.5422113.00.000.002.002.110.00
Absolute0.42100016.5422113.00.000.002.002.110.00

Conclusion

In order to satisfy the cross-sectional checks of clause 6.2.9, the maximum bending moment that can be carried by a 457 x 191 x 98 UKB in grade S275 steel, in the presence of an axial force of 1400 kN is  425.3 kN m .

For class 2 elements in calculation take into account fillets. In shapes fillets are skipped and it gives the difference of 0.007% in results.