V. EC3 - Pipe Section with Conc Load

Verify the adequacy of a circular hollow section used as a beam subject to concentrated loads per EN 1993-1-1:2005 (no national annex used).

Details

The member is a 6 m long beam with pinned ends. The beam supports 10kN concentrated loads at the third points. The steel is grade S275. The section is a European 114.3X8CHS.

Section Properties

A = 26.7 cm²
Outer diameter = 114.3 mm
t = 8 mm
I_z = I_y = 379 cm⁴
Z_z = Z_y = 90.6 cm³
C_w = 0 cm⁶
$r_{z} = r_{y} = \sqrt{\frac{I}{A}} = \sqrt{\frac{379}{26.7}} = 3.768 cm$

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:

d / t = 114.3 / 8 = 14.3 < 50×ε² = 50 (0.924)² = 42.7

Hence, this is a Class 1 section.

Slenderness Ratio

The slenderness ratio = kL / r = 1 × 6,000 / 37.68 = 159.3

Axial Tension

Determine axial tension capacity per Cl. 6.2.3.

N_{pl,Rd} = A_{g} \times f_{y} / γ_{M0} = 2,670 \times 275 / 1.0 \times {(10)}^{-3} = 734.3 kN

N_{u,Rd} = 0.9 A_{net} \times f_{u} / γ_{M2} = 0.9 \times 2,670 \times 295 / 1.25 \times {(10)}^{-3} = 567.1 kN

The tensile capacity is the minimum of N_pl,Rd and N_u,Rd, thus: N_t,Rd = 567.1 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} = 2,670 \times 275 / 1.0 \times {(10)}^{-3} = 734.3 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	=	6,000 mm
i	=	37.68 mm
λ₁	=	93.9×ε = 93.9 × 0.924 = 86.8

ƛ = 1.834

Ф = 0.5 [1 + ɑ (ƛ - 0.2) + ƛ^{2}] = 0.5 [1 + 0.21 (1.834 - 0.2) + {(1.834)}^{2}] = 2.354

χ = \frac{1}{Ф + \sqrt{Ф^{2} - ƛ^{2}}} = 0.2611

N_{b,Rd} = 0.2611 \times 2,670 \times 275 / 1.0 = 191.7 kN

The compression capacity is the minimum of N_c,Rd and N_b,Rd, thus: N_c,Rd = 191.7 kN.

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

Bending Capacity

Maximum bending moment in the section (center third of span): M_Ed = 20.0 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} = 90.6 \times 275 / 1.0 \times {(10)}^{-3} = 24.92 kN·m

Ratio per Eq. 6.12: M_Ed / M_c,Rd = 20.0 / 24.92 = 0.803

Shear Capacity

Maximum shear in the section: V_Ed = 10 kN

A_{v} = 2 A / π = 1,700 {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} = 1,700 \times (\frac{275}{\sqrt{3}}) / 1.0 = 269.9 kN

Ratio per Eq. 6.17: V_Ed / V_c,Rd = 10.0 / 26.9 = 0.037

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
$\frac{π^{2} E I_{y}}{{(k L)}^{2}}$	=	213,000
${(\frac{k}{k_{w}})}^{2} \frac{I_{w}}{I_{y}}$	=	0
$\frac{{(k L)}^{2} G I_{T}}{π^{2} E I_{y}}$	=	2,806,000
C₂Z_g	=	1.0×114.3 / 2 = 57.15

Therefore, $M_{cr} = 1.0 \times 213,000 [\sqrt{0 + 2,806,000 + {57.15}^{2}} - 57.15] = 344.8 kN·m$

For CHS sections, χ_LT = 1.0.

M_{b,Rd} = χ_{LT} W_{y} \times f_{y} / γ_{M1} = 1.0 \times 90.6 \times 275 / 1.0 \times {(10)}^{-3} = 24.92 kN·m

Ratio per Eq. 6.12: M_Ed / M_b,Rd = 20.0 / 24.92 = 0.803

Results

Table 1. Comparison of results
Result Type	Reference	STAAD.Pro	Difference
Section Class	Class 1	Class 1	none
Slenderness Ratio	159.3	159.3	none
Tension Capacity (kN)	567.1	567.1	none
Compression Capacity (kN)	191.7	191.7	none
Moment Capacity (kN·m)	24.92	24.9	negligible
Shear Area (cm²)	17.0	16.99	negligible
Shear Capacity (kN)	269.9	269.9	none
M_B (kN·m)	24.92	24.9	negligible
Ratio per Cl. 6.2.5	0.803	0.803	none
Ratio per Cl. 6.2.6 (Y)	0.037	0.037	none
Ratio per Cl. 6.3.2 LTB	0.803	0.803	none

STAAD.Pro Input

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

The following design parameters are used:

Symmetric point loads about mid-span: CMM 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 07-May-21
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 EUROPEAN
1 TABLE ST 114.3X8CHS
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
PARAMETER 1
CODE EN 1993-1-1:2005
C1 1 ALL
C2 1 ALL
CMM 5 ALL
MTH 1 ALL
FU 295000 ALL
PY 275000 ALL
TRACK 2 ALL
CHECK CODE ALL
PRINT MEMBER PROPERTIES 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   114.3X8CHS  (EUROPEAN SECTIONS)
                           PASS     EC-6.2.5           0.803         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 =   26.70          Net Area =   26.70
                                      z-axis          y-axis
      Moment of inertia        :      379.000         379.000
      Plastic modulus          :       90.600          90.600
      Elastic modulus          :       66.317          66.317
      Shear Area               :       16.998          16.998
      Radius of gyration       :        3.768           3.768
      Effective Length         :      600.000         600.000
   DESIGN DATA (units - kN,m)   EUROCODE NO.3 /2005
      Section Class            :   CLASS 1     
      Squash Load              :    734.25
      Axial force/Squash load  :     0.000
      GM0 :  1.00          GM1 :  1.00          GM2 :  1.25
                                      z-axis          y-axis
      Slenderness ratio (KL/r) :        159.3          159.3
      Compression Capacity     :        191.7          191.7
      Tension Capacity         :        567.1          567.1
      Moment Capacity          :         24.9           24.9
      Reduced Moment Capacity  :         24.9           24.9
      Shear Capacity           :        269.9          269.9
   BUCKLING CALCULATIONS (units - kN,m)
      Lateral Torsional Buckling Moment       MB =   24.9
      co-efficients C1 & K : C1 =1.000 K =1.0, Effective Length= 6.000
      Lateral Torsional Buckling Curve :        
      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       0.803     1     0.0     10.0     0.0    20.0     0.0
   EC-6.2.6-(Y)   0.037     1     0.0     10.0     0.0     0.0     0.0
   EC-6.3.2 LTB   0.803     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 **************