 # V. EC3 - Tube Section with UDL

Verify the adequacy of a tube section used as a beam subject to a uniform load per EN 1993-1-1:2005 (no national annex used).

## Details

The member is a 5 m long beam with fixed ends. The member is subject to a 10 kN/m uniform load. The steel is grade S275. The section is a European TUB1201205.

Section Properties

• A = 22.9 cm2
• Depth, D = 120 mm
• Width, B = 120 mm
• t = 5 mm
• Iz = Iy = 503 cm4
• Zz = Zy = 97.6 cm3
• Cw = 0 cm6

Partial safety factors:

ΓM0 = 1.0

ΓM1 = 1.0

ΓM2 = 1.25

## Validation

Section Classification

$ε = 235 f y = 235 275 = 0.924$

As per Table 5.2:

ɑ = 1

C = D - 2×t = 110 mm

$C / t = 110 5 = 22 < 396 × ε 13 ɑ - 1 = 396 × 0.924 12 = 30.36$

Hence, this is a Class 1 section.

Slenderness Ratio

The slenderness ratio = kL / r = 1 × 5,000 / 46.87 = 106.7

Axial Tension

Determine axial tension capacity per Cl. 6.2.3.

The tensile capacity is the minimum of Npl,Rd and Nu,Rd, thus: Nt,Rd = 630.0 kN.

No axial tension in section, so by observation no need to check ratio.

Axial Compression

Determine axial compression capacity per Cl. 6.2.4 for a Class 1 section.

Next, check the flexural buckling resistance per Cl. 6.3.1.3:

$N b,Rd = χ × A × f y / γ M1$

From Table 6.1: the imperfection factor, ɑ = 0.21.

$ƛ = A × f y / N cr = L cr / i × λ 1$
where
 Lcr = 5,000 mm i = 46.87 mm λ1 = 93.9×ε = 93.9 × 0.924 = 86.8

ƛ = 1.229

$Ф = 0.5 [ 1 + ɑ ( ƛ - 0.2 ) + ƛ 2 ] = 0.5 [ 1 + 0.21 ( 1.229 - 0.2 ) + ( 1.229 ) 2 ] = 1.363$
$χ = 1 Ф + Ф 2 - ƛ 2 = 0.512$

The compression capacity is the minimum of Nc,Rd and Nb,Rd, thus: Nc,Rd = 322.4 kN.

No axial compression in section, so by observation no need to check ratio.

Bending Capacity

Maximum bending moment in the section (at fixed supports): MEd = 20.83 kN·m.

Check for bending capacity per Cl. 6.2.5:

For a Class 1 section:

Ratio per Eq. 6.12: MEd / Mc,Rd = 20.83 / 26.84 = 0.776

Shear Capacity

Maximum shear in the section (at fixed supports): VEd = 25 kN

Check for shear capacity per Cl. 6.2.6 for plastic design (Class 1):

Ratio per Eq. 6.17: VEd / Vc,Rd = 25.0 / 181.8 = 0.138

Lateral Torsional Buckling

$M c r = C 1 π 2 E I ( k L ) 2 [ ( k k w ) 2 I w I z + ( k L ) 2 G I t π 2 E I z + ( C 2 Z g ) 2 − C 2 Z g ]$
where
 C1 = 1.0 C2 = 1.0 $π 2 E I y k L 2$ = 162,800 $k k w 2 I w I y$ = 0 $k L 2 G I T π 2 E I y$ = 375,300 C2Zg = 1.0×60 = 60

Therefore,

From Table 6.4, buckling curve "d" is used. From Table 6.3, ɑLT = 0.76 for buckling curve d.

As per Cl. 6.3.2.1, tube sections are not susceptible to lateral-torsional buckling, so χLT = 1.0.

Ratio per Eq. 6.12: MEd / Mb,Rd = 20.83 / 26.84 = 0.776

## Results

Table 1. Comparison of results
Shear Area (cm2) 11.45 11.45 none
Section Class Class 1 Class 1 none
Slenderness Ratio 106.7 106.7 none
Tension Capacity (kN) 630.0 629.8 negligible
Compression Capacity (kN) 322.4 322.4 none
Moment Capacity (kN·m) 26.84 26.8 negligible
Shear Capacity (kN) 181.8 181.8 none
Mcr (kN·m) 904.6 905.7 negligible
MB (kN·m) 26.84 26.8 negligible
Ratio per Cl. 6.2.5 0.776 0.776 none
Ratio per Cl. 6.2.6 (Y) 0.138 0.138 none
Ratio per Cl. 6.3.2 LTB 0.776 0.776 none

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

The following design parameters are used:
• Fixed end supports with a uniform load: CMM 2
• Fixed end supports: CMN 0.5
• Use Cl. 6.3.2.2 to determine χLT: MTH 1
• The values of C1 1.0 and C2 1.0 are specified.
• The values of Fy and Fu are specified directly using PY 275000 and FU 295000.
STAAD PLANE
START JOB INFORMATION
ENGINEER DATE 04-May-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 TUB1201205
CONSTANTS
MATERIAL STEEL ALL
SUPPORTS
1 2 FIXED
1 UNI GY -10
PERFORM ANALYSIS
PARAMETER 1
CODE EN 1993-1-1:2005
C1 1 ALL
C2 1 ALL
CMM 2 ALL
CMN 0.5 ALL
MTH 1 ALL
PY 275000 ALL
FU 450000 ALL
KC 0 ALL
TRACK 2 ALL
CHECK CODE ALL
FINISH


                         STAAD.PRO CODE CHECKING - EN 1993-1-1:2005
********************************************
NATIONAL ANNEX - NOT USED
PROGRAM CODE REVISION V1.14 BS_EC3_2005/1
STAAD PLANE                                              -- PAGE NO.    3
ALL UNITS ARE - KN   METE (UNLESS OTHERWISE Noted)
FX            MY             MZ       LOCATION
=======================================================================
1 ST   TUB1201205  (EUROPEAN SECTIONS)
PASS     EC-6.2.5           0.776         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 =   22.90          Net Area =   22.90
z-axis          y-axis
Moment of inertia        :      503.000         503.000
Plastic modulus          :       97.600          97.600
Elastic modulus          :       83.833          83.833
Shear Area               :       11.450          11.450
Radius of gyration       :        4.687           4.687
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 :  1.25
z-axis          y-axis
Slenderness ratio (KL/r) :        106.7          106.7
Compression Capacity     :        322.4          322.4
Tension Capacity         :        629.8          629.8
Moment Capacity          :         26.8           26.8
Reduced Moment Capacity  :         26.8           26.8
Shear Capacity           :        181.8          181.8
BUCKLING CALCULATIONS (units - kN,m)
Lateral Torsional Buckling Moment       MB =   26.8
co-efficients C1 & K : C1 =1.000 K =0.5, Effective Length= 5.000
Lateral Torsional Buckling Curve : CURVE d
Elastic Critical Moment for LTB,               Mcr   =   905.7
Compression buckling curves:     z-z:  Curve a   y-y:  Curve a
STAAD PLANE                                              -- PAGE NO.    4
CRITICAL LOADS FOR EACH CLAUSE CHECK (units- kN,m):
CLAUSE        RATIO  LOAD     FX       VY      VZ      MZ      MY
EC-6.2.5       0.776     1     0.0     25.0     0.0    20.8     0.0
EC-6.2.6-(Y)   0.138     1     0.0     25.0     0.0    20.8     0.0
EC-6.3.2 LTB   0.776     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 **************