V. IS 801-Beam with axial and major axis bending
Verification example for a cold-formed beam subject to axial compression and bending moment according to IS:801-1975.
Details
Verifies the calculations for an IS 60CU40x4 (Channel without lips) beam that is 2 m long and subject to axial compression and major axis bending moment. This example checks for compression, shear, bending and compression and bending interaction as per IS 801.
- E = 2,074,000 kgf/cm2 = 203,400 MPa
- Fyi = 350.0 MPa = 3,569 kgf/cm2
- Fu = 450 MPa
- G = 795,000 kgf/cm2 = 77,963 MPa
- P = 0.5 kN
- Mz = 0.748 kN·m
- Vy = 1.874 kN
Verification
Section Dimension Checks
Check for flat width ratio:
w = b - (r + t) = r - (0.4 + 0.6) = 3 mm
w/t = 3 / 0.4 = 7.5 < 60
Hence, OK (ref. cl 5.2.3(a))
hw/t = (d - 2×tf - 2× root radius) / t = (6 - 2×0.4 - 2×0.6)/0.4 = 10 < 500
Hence, OK (ref. cl 5.2.3(b))
Check for web height to thickness ratio:
h/t = (6 - 2×0.4) / 0.4 = 13 < 150
Hence, OK (ref. cl 5.2.4(a))
Check for limiting slenderness
Hence, OK (ref. cl 6.3.3)
Calculation of Allowable Compressive Stress
For calculation of axially loaded member, " Q " is an important factor. The definition and method of calculation for value of Q is provided in Clause no. 6.6.1.1 (a). Channel section without lips is a combination of stiffened & unstiffened elements.
As per clause no 6.1.1.1 of IS801, the increase of steel strength happens due to cold work of forming.
Total corner area, Acorner = 2(45.2 mm2) = 91.2 mm2 = 0.912 cm2
Total area of flanges, Aflange = 2×b×t = 2(0.4)(0.4) = 3.2 cm2
C = Acorner / Aflange = 0.931 / 3.2 = 0.285
Effective depth, he = h = 4 cm
Therefore, effective area, Ae = A = 4.91 cm2
Tensile yield point of corner,
Tensile yield point of flat portions, Fyt = Fy = 350.0 MPa
Average yield point of cold-forming for tension/compression members, Fya(compression) = (C × Fyc) + (1 - C)×Fyc = 381.8 MPa ( = 3,893 kgf/cm2)
Average yield point of cold-forming for flexural members, Fya(bending) = (C × Fyc) + (1 - C)×Fyc = 398.8 MPa
w = 3 cm
w / t = 7.5
As per cl. 6.2 of IS 801, compressive stress:
Fc = 0.6×Fya = 0.6 × 381.8 = 229.1 MPa
h = d - 2(r + t) = 4 cm
Q = Qs × Qa = 1
Allowable compression stress, Fa1 , for members braced against twisting (ref. cl 6.6.1.1)
Slenderness ratio KL/r = 158.3
Maximum allowable compressive stress (Fa2) for flexural-torsional buckling (ref. cl 6.6.1.2 of IS 801)
Fqy = Fy × Q = 381.8 MPa
Fa2 = 0.522 × σTF0 = 91.09 MPa
The allowable compressive stress, Fa is the minimum of Fa1 and Fa2 :
Fa = 41.06 MPa
Calculation of Allowable Bending Stress
As per clause number 6.1, maximum allowable stress on extreme fiber is:
F = 0.6 × Fya(compression) = 0.6 × 381.8 MPa = 229.1 MPa
As the section is channel without lips, the flanges are unstiffened. So, as per clause 6.2 we need to check allowable compressive stress on the unstiffened element.
Also, the yield strength of steel, Fy > 2,230 kgf/cm2 ( = 227.5 MPa).
Fc = 0.6 × Fya(compression) = 229.1 MPa
For the major axis bending, the allowable compressive stress, Fbc , is the minimum of F and Fc
Fbc = 229.1 MPa
Similarly, for major axis bending, the allowable tensile stress, Fbt=0.6 × Fya(compression) = 229.1 MPa
Calculate the allowable bending stress for laterally unbraced beams:
Allowable bending stress for laterally unbraced beams has been calculated as per clause 6.3 (a).
Unsupported length, L = 2 m (the UNL parameter can be used for this).
Sxc = Zxx = 8.93 cm3
Cb = 1.0 for a member under compression and bending.
Allowable bending stress in the web:
Per cl. 6.4.2, Fbw is the minimum of Fbw1 and 0.6 × Fya(bending) = 239.2 MPa
Fbw = 239.2 MPa
Calculation of Allowable Shear Stress
Per cl. 6.4:
Clear distance between flanges = h = d - 2t = 52 mm
(cl. 6.4.1(a) ) |
(cl. 6.4.1(b) ) |
Allowable shear stress, Fv is the minimum of Fv1 or 0.4 × Fy = 140.1 MPa
Fv = 140.1 MPa
Allowable combined bending and shear stress:
As , Fvc = Fvc1 = 577.1 MPa
Actual Stresses
Compression
fa = P/A = 0.5 kN / 4.91 mm2 = 1.018 MPa
Bending
fb = M / Zxx = 0.748 kN·m / 8.93 cm3 = 83.76 MPa
Bending in Web
- Actual bending stress in the web is calculated by interpolation of bending stress diagram:
Shear
Stress Ratio
Compression
fa / Fa = 1.018 / 41.06 = 0.024
Bending
- for bending compression: fb / Fbc = 83.76 / 229.1 = 0.366
- for bending tension: fb / Fbt = 83.76 / 229.1 = 0.366
- for unbraced bending: fb / Fb = 83.76 / 85.18 = 0.983
- for web bending: fbw / Fbw = 72.59 / 239.2 = 0.303
Shear
fv / Fv = 9.01 / 140.1 = 0.064
Combined bending and shear (ref. cl 6.4.3 of IS 801):
Interaction ratio for axial and bending
As Q = 1.0, Fa0 can be calculated using cl. 6.6.1.1(b) with L = 0:
(6.7.2(a) - 2nd eq) |
Results
Result Type | Reference | STAAD.Pro | Difference | Comments |
---|---|---|---|---|
Compression stress ratio | 0.024 | 0.024 | none | |
Bending Z (compressive) stress ratio | 0.366 | 0.365 | negligible | |
Bending Z (tensile) stress ratio | 0.366 | 0.365 | negligible | |
Bending unbraced | 0.983 | 0.983 | none | |
Bending at web/flange junction stress ratio | 0.303 | 0.303 | none | |
Shear Y stress ratio | 0.064 | 0.064 | none | |
Compression + Bending interaction | 0.371 | 0.371 | none | |
Bending + Shear interaction | 0.016 | 0.016 | none |
STAAD.Pro Input File
The file C:\Users\Public\Public Documents\STAAD.Pro 2023\Samples \Verification Models\09 Steel Design\India\IS 801-Beam with axial and major axis bending.STD is typically installed with the program.
STAAD SPACE
START JOB INFORMATION
ENGINEER DATE 27-Mar-19
END JOB INFORMATION
INPUT WIDTH 79
UNIT METER KN
JOINT COORDINATES
1 0 0 0; 2 2 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
TYPE STEEL
STRENGTH RY 1.5 RT 1.2
END DEFINE MATERIAL
MEMBER PROPERTY COLDFORMED INDIAN
1 TABLE ST 60CU40X4
CONSTANTS
MATERIAL STEEL ALL
SUPPORTS
1 FIXED
2 PINNED
LOAD 1 LOADTYPE None TITLE LOAD CASE 1
MEMBER LOAD
1 UNI GY -1.5
LOAD 2 LOADTYPE None TITLE LOAD CASE 3
MEMBER LOAD
1 CON GX -1
LOAD COMB 4 COMBINATION LOAD CASE 4
1 1.0 2 1.0
PERFORM ANALYSIS
PRINT MEMBER PROPERTIES ALL
LOAD LIST 4
PARAMETER 1
CODE IS801
CWY 1 ALL
FU 450000 ALL
FYLD 350000 ALL
RATIO 1 ALL
TRACK 2 ALL
CHECK CODE ALL
STEEL TAKE OFF ALL
FINISH
STAAD.Pro Output
STAAD.Pro CODE CHECKING - ( IS:801 ) v3.0 *********************** ALL UNITS ARE IN - METE KN (U.N.O.) |-----------------------------------------------------------------------------| | MEMBER: 1 SECTION: 60CU40X4 LEN: 2.000 LOC: 0.000 | | STATUS: PASS RATIO: 0.983 REF: 6.3 LTB LC: 4 | |-----------------------------------------------------------------------------| | DESIGN FORCES: | | Fx:(C) 0.500 Fy: 1.874 Fz: 0.000 | | Mx: 0.000 My: 0.000 Mz: 0.748 | |-----------------------------------------------------------------------------| | SECTION PROPERTIES: (Unit: CM) | | Ag: 4.91000 Az: 3.20000 Ay: 2.08000 | | Cz: 1.38000 Cy: 3.00000 Z0: 2.62000 | | Iz: 26.80000 Iy: 7.84000 J: 0.25500 | | Sz: 8.93000 Sy: 2.99000 | | Rz: 2.33629 Ry: 1.26362 Cw: 45.60001 | |-----------------------------------------------------------------------------| | MATERIAL INFO: (Unit: MPa) | | Fy: 350.025 Fu: 450.032 E: 203404.356 G: 77968.401 | | Fya(compression): 381.800 Fya(bending): 398.781 | |-----------------------------------------------------------------------------| | DESIGN PROPERTIES: | | Member Length: 2.000 Lz: 2.000 Ly: 2.000 Lb: 2.000 | | DESIGN PARAMETERS: | | Kz: 1.000 Ky: 1.000 NSF: 1.000 Cb: 0.000 | |-----------------------------------------------------------------------------| | CRITICAL SLENDERNESS: | | Actual: 158.275 Allowable: 200.000 Ratio: 0.791 | |-----------------------------------------------------------------------------| | CHECKS: | Stresses | | | | Loc. | Demand | L/C | Actual | Allow |Ratio | Ref CL | | |(MET) |(KN-MET)| | (MPa) | (MPa) | | | |--------------|------|--------|------|----------|----------|------|----------| | Tension | 1.167| -0.50| 4| 1.018 | 229.080 | 0.004| 6.1 | | Compression | 0.000| 0.50| 4| 1.018 | 41.835 | 0.024| 6.6.1.1 | | BendZComp | 0.000| 0.75| 4| 83.688 | 229.080 | 0.365| 6.3 | | BendZTens | 0.000| 0.75| 4| 83.688 | 229.080 | 0.365| 6.3 | | BendUnbraced | 0.000| 0.75| 4| 83.688 | 85.160 | 0.983| 6.3 LTB | | BendYComp | - | - | - | - | 239.268 | - | 6.3 | | BendYTens | - | - | - | - | 229.080 | - | 6.3 | | Bend Web | 0.000| 0.75| 4| 72.529 | 239.268 | 0.303| 6.4.2 | | Shear Z | - | - | - | - | 140.010 | - | 6.4.1 | | Shear Y | 0.000| 1.87| 4| 9.009 | 140.010 | 0.064| 6.4.1 | | Axial+Bend | 0.000| - | 4| - | - | 0.371| 6.7.2(a)2| | Bend+Shear | 0.000| - | 4| - | - | 0.016| 6.4.3 | |-----------------------------------------------------------------------------| | Effective Section Properties:(cm) | | Ae: 4.910 SzTop: 8.933 SzBot: 8.933 SyLeft: 5.681 SyRight: 2.992 | | Intermediate Results: Cb = 1.000 | |-----------------------------------------------------------------------------| NOTE: Torsion has not been considered in the design. STAAD SPACE -- PAGE NO. 5