V. EC3 - Simply supported laterally unrestrained beam
Verify the design of a pinned column per the EC3 2005 design code.
Details
The beam is a 5.7 m simply supported span. The member is a UKB 356 x171 x51shape with S275 grade steel. The member is uniformly loaded with 9.58 kN/m dead load and 6.25 kN/m live load.
Partial safety factors:
ΓM0 = 1.0
ΓM1 = 1.0
Validation
Factored Loads
w = 1.35 (9.58) + 1.5 (6.25) = 22.31 kN/m
Mz,Ed = 22.31 (5.7)2 / 8 = 90.6 kN·m
Vz,Ed = 22.31 (5.7)/2 = 63.58 kN
Yield Strength
Steel grade S275
The maximum thickness = 15.4 mm < 40 mm; therefore fy = 275 N/mm2 [EN 1993-1-1:2005 Table 3.1]
Section Classification
| ε = √(235/fy) = 0.92 | [EN 1993-1-1:2005 Table 5.2] |
c = (b-tw-2r)/2 = (171.5 – 7.4 – 2 × 10.2)/2 = 71.85 mm
c /tf = 71.85 / 11.5 = 6.25 < 9 ε
Class 1 element [EN 1993-1-1:2005 Table 5.2 Sheet 2]Internal compression part (web in bending):
c = h-2tf-2r = 355-2 × 11.5 – 2 × 10.2 = 311.6 mm
c / tw = 311.6 / 7.4 = 42.1 < 72 ε
Class 1 element [EN 1993-1-1:2005 Table 5.2 Sheet 1]
Therefore, section is a Class 2 section.
Moment Resistance
Design bending resistance:
| Mc, Rd = Wpl × fy/Γm0 = 896 cm3(103)×275 kN/mm2 / 1.0 ×10-6 = 246.4 kN·m > 90.6 kN·m | [EN 1993-1-1:2005 Cl. 6.2.4] |
Ratio 90.6 / 246.4 = 0.368
Lateral Torsional Buckling
Determine the elastic critical moment, Mcr:
| [EN 1993-1-1:2005 Annex F] |
| = | ||
| = | ||
| = | ||
| = |
Mcr = 121.74 kN·m
| [EN 1993-1-1:2005 Cl. 65.3.2.2] |
h/b = 355 / 171.5 = 2.07 > 2
Use buckling curve c. [EN 1993-1-1:2005 Cl. 6.3.2.3 & Table 6.5]
| αLT = 0.49 | [EN 1993-1-1:2005 Table 6.3] |
| ϕ = 0.5 [1 + α(λLT – λLT0) + βλ2] | [EN 1993-1-1:2005 Cl. 6.3.2.3] |
ϕ = 0.5 [1 + 0.49(1.423 – 0.4) +0.75×1.4232] = 1.510
| f = 1 - 0.5 (1 - kc)[1 - 2(λLT – 0.8)2] = 0.99 < 1 |
| = |
| Mb, Rd = χLTWpl,z × fy/Γm1 = 0.42×896 cm3(103)×275 kN/mm2 / 1.0 ×10-6 = 103.5 kN·m > 90.6 kN·m | [EN 1993-1-1:2005 Cl. 6.2.4] |
Ratio = 90.6 / 103.5 = 0.875
Check Shear Resistance
In the absence of torsion shear area
Av = A - 2btf + (tw + 2r) tf = 6490 - 2 × 171.5× 11.5 + (7.4+2 ×10.2) × 11. = 2,865 mm2
η hw tw = 1 × 332 × 7.4 = 2,457 mm2 > Av
Shear plastic resistance
Vpl,Rd = Av (fy/√3)/ ΓM0 = 2,865 × (275/√3) / 1.0 = 454.9 kN > 63.58 kN
Ratio = 63.58 / 454.9 = 0.140
| hw / tw = 332/7.4 = 44.86 < 66.24 (i.e. 72 ε) | [ EN 1993-1-1:2005 Cl. 6.2.6(6)] |
Hence shear buckling check not required.
| VEd = 63.58 < 0.5× 454.9 | [ EN 1993-1-1:2005 Cl. 6.2.8(2)] |
Hence combined shear and bending checks not required.
Check for Deflection Under Service Load
G + Q = 9.58 + 6.25 = 15.83 kN/m
STAAD Input
The file C:\Users\Public\Public Documents\STAAD.Pro CONNECT Edition\Samples\ Verification Models\09 Steel Design\Europe\EC3 - Simply supported laterally unrestrained beam.STD is typically installed with the program.
STAAD PLANE
START JOB INFORMATION
ENGINEER DATE 03-Jan-08
END JOB INFORMATION
INPUT WIDTH 79
UNIT METER KN
JOINT COORDINATES
1 0 0 0; 2 5.7 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 BRITISH
1 TABLE ST UB356X171X51
CONSTANTS
MATERIAL STEEL ALL
SUPPORTS
1 2 PINNED
LOAD 1 LOADTYPE Dead TITLE DL
MEMBER LOAD
1 UNI GY -9.58
LOAD 2 LOADTYPE Live TITLE LL
MEMBER LOAD
1 UNI GY -6.25
LOAD COMB 3 COMBINATION LOAD CASE 3
1 1.35 2 1.5
LOAD COMB 4 COMBINATION LOAD CASE 4
1 1.0 2 1.0
PERFORM ANALYSIS
PARAMETER 1
CODE EN 1993-1-1:2005
GM0 1 ALL
GM1 1 ALL
SGR 1 ALL
TRACK 2 ALL
CHECK CODE ALL
LOAD LIST 4
PARAMETER 2
CODE EN 1993-1-1:2005
SGR 1 ALL
DFF 300 ALL
TRACK 4 ALL
CHECK CODE ALL
FINISH
STAAD 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 UB356X171X51 (BRITISH SECTIONS) PASS EC-6.3.2 LTB 0.877 3 0.00 0.00 -90.60 2.85 ======================================================================= MATERIAL DATA Grade of steel = S 275 Modulus of elasticity = 205 kN/mm2 Design Strength (py) = 275 N/mm2 SECTION PROPERTIES (units - cm) Member Length = 570.00 Gross Area = 64.90 Net Area = 64.90 z-axis y-axis Moment of inertia : 14100.002 968.000 Plastic modulus : 896.000 174.000 Elastic modulus : 794.366 112.886 Shear Area : 35.500 28.652 Radius of gyration : 14.740 3.862 Effective Length : 570.000 570.000 DESIGN DATA (units - kN,m) EUROCODE NO.3 /2005 Section Class : CLASS 1 Squash Load : 1784.75 Axial force/Squash load : 0.000 GM0 : 1.00 GM1 : 1.00 GM2 : 1.25 z-axis y-axis Slenderness ratio (KL/r) : 38.7 147.6 Compression Capacity : 1678.4 496.1 Tension Capacity : 1784.8 1784.8 Moment Capacity : 246.4 47.9 Reduced Moment Capacity : 246.4 47.9 Shear Capacity : 563.6 454.9 BUCKLING CALCULATIONS (units - kN,m) Lateral Torsional Buckling Moment MB = 103.3 co-efficients C1 & K : C1 =1.132 K =1.0, Effective Length= 5.700 Lateral Torsional Buckling Curve : CURVE c Elastic Critical Moment for LTB, Mcr = 121.4 Compression buckling curves: z-z: Curve a y-y: Curve b Critical Load For Torsional Buckling, NcrT = 1574.5 Critical Load For Torsional-Flexural Buckling, NcrTF = 1574.5 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 0.368 3 0.0 0.0 0.0 -90.6 0.0 EC-6.2.6-(Y) 0.140 3 0.0 63.6 0.0 0.0 0.0 EC-6.3.2 LTB 0.877 3 0.0 0.0 0.0 -90.6 0.0 Torsion has not been considered in the design. _________________________ ************** END OF TABULATED RESULT OF DESIGN ************** 43. LOAD LIST 4 44. PARAMETER 2 45. CODE EN 1993-1-1:2005 46. SGR 1 ALL 47. DFF 300 ALL 48. TRACK 4 ALL 49. CHECK CODE ALL STEEL DESIGN 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. 5 DEFLECTION CHECKS (LENGTH UNITS - METE) MEMBER TABLE RESULT ACTUAL DEFL.LIMIT DEFL.LEN/ LOAD/ DEFL. RATIO DFF LOCATION ======================================================================= 1 ST UB356X171X51 (BRITISH SECTIONS) PASS 0.007 0.019 5.700 4 0.394 300.000 2.85 ************** END OF TABULATED RESULT OF DESIGN **************