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
I_z = I_y = 503 cm⁴
Z_z = Z_y = 97.6 cm³
C_w = 0 cm⁶
$r_{z} = r_{y} = \sqrt{\frac{I}{A}} = \sqrt{\frac{503}{22.9}} = 4.687 cm$

Partial safety factors:

Γ_M0 = 1.0

Γ_M1 = 1.0

Γ_M2 = 1.25

Validation

Section Classification

ε = \sqrt{\frac{235}{f_{y}}} = \sqrt{\frac{235}{275}} = 0.924

As per Table 5.2:

ɑ = 1

C = D - 2×t = 110 mm

C / t = \frac{110}{5} = 22 < \frac{396 \times ε}{13 ɑ - 1} = \frac{396 \times 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.

N_{pl,Rd} = A_{g} \times f_{y} / γ_{M0} = 2,290 \times 275 / 1.0 \times {(10)}^{-3} = 630.0 kN

N_{u,Rd} = 0.9 A_{net} \times f_{u} / γ_{M2} = 0.9 \times 2,290 \times 450 / 1.25 \times {(10)}^{-3} = 742.0 kN

The tensile capacity is the minimum of N_pl,Rd and N_u,Rd, thus: N_t,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.

N_{c,Rd} = A_{g} \times f_{y} / γ_{M0} = 2,290 \times 275 / 1.0 \times {(10)}^{-3} = 630.0 kN

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

N_{b,Rd} = χ \times A \times f_{y} / γ_{M1}

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

ƛ = \sqrt{A \times f_{y} / N_{cr}} = L_{cr} / i \times λ_{1}

where

L_cr	=	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 = \frac{1}{Ф + \sqrt{Ф^{2} - ƛ^{2}}} = 0.512

N_{b,Rd} = 0.512 \times 2,290 \times 275 / 1.0 = 322.4 kN

The compression capacity is the minimum of N_c,Rd and N_b,Rd, thus: N_c,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): M_Ed = 20.83 kN·m.

Check for bending capacity per Cl. 6.2.5:

For a Class 1 section:

M_{c,Rd} = W_{ply} \times f_{y} / γ_{M0} = 97.6 \times 275 / 1.0 \times {(10)}^{-3} = 26.84 kN·m

Ratio per Eq. 6.12: M_Ed / M_c,Rd = 20.83 / 26.84 = 0.776

Shear Capacity

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

A_{v} = \frac{A \times D}{D + B} = \frac{2,290 \times 120}{120 + 120} = 1,145 {mm}^{2}

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

V_{c,Rd} = V_{pl,Rd} = A_{v} \times (\frac{f_{y}}{\sqrt{3}}) / γ_{M0} = 1,145 \times (\frac{275}{\sqrt{3}}) / 1.0 = 181.8 kN

Ratio per Eq. 6.17: V_Ed / V_c,Rd = 25.0 / 181.8 = 0.138

Lateral Torsional Buckling

M_{c r} = C_{1} \frac{π^{2} E I}{{(k L)}^{2}} [\sqrt{{(\frac{k}{k_{w}})}^{2} \frac{I_{w}}{I_{z}} + \frac{{(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
$\frac{π^{2} E I_{y}}{{(k L)}^{2}}$	=	162,800
${(\frac{k}{k_{w}})}^{2} \frac{I_{w}}{I_{y}}$	=	0
$\frac{{(k L)}^{2} G I_{T}}{π^{2} E I_{y}}$	=	375,300
C₂Z_g	=	1.0×60 = 60

Therefore, $M_{cr} = 1.0 \times 162,800 [\sqrt{0 + 375,300 + 60^{2}} - 60] = 904.6 kN·m$

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

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

M_{b,Rd} = χ_{LT} W_{y} \times f_{y} / γ_{M1} = 1.0 \times 97.6 \times 275 / 1.0 \times {(10)}^{-3} = 26.84 kN·m

Ratio per Eq. 6.12: M_Ed / M_b,Rd = 20.83 / 26.84 = 0.776

Results

Table 1. Comparison of results
Result Type	Reference	STAAD.Pro	Difference
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
M_cr (kN·m)	904.6	905.7	negligible
M_B (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

STAAD.Pro Input

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 χ_LT: MTH 1
The values of C1 1.0 and C2 1.0 are specified.
The values of F_y and F_u 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
LOAD 1 LOADTYPE Dead  TITLE LOAD CASE 1
MEMBER LOAD
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 Output

                         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)
 MEMBER     TABLE       RESULT/   CRITICAL COND/     RATIO/     LOADING/
                          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                
      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 =   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     
      Squash Load              :    629.75
      Axial force/Squash load  :     0.000
      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 **************