V. IS 800 2007 LSD - Channel - with LTB
Verify the design of a channel section with lateral-torsional buckling per the IS 800:2007 limit state method.
BIS. IS 800: General Construction in Steel - Code of Practice. 2007. Bureau of Indian Standards. New Delhi.
BIS. IS 808: Dimensions for Hot Rolled Steel Beam, Column, Channel, and Angle Sections. 1898. New Delhi. p.6
CISC. "Torsional Section Properties of steel Shapes". Canadian Institute of Steel Construction. 2002. Willowdale, Ont. p.7. http://dir.cisc-icca.ca/files/technical/techdocs/updates/torsionprop.pdf
Details
The member is a 3 m long cantilever subject to a 10 kN compressive force and uniform bending moments of 1 kN/m along both major and minor axes. The section used is a ISMC 200 and is unbraced against lateral-torsional buckling. The steel has a yield strength of 250 MPa, an ultimate tensile strength of 420 MPa, a modulus of elasticity of 205×(10)3 MPa, and a Poisson's ratio, ν = 0.3.
Section Properties
From IS:808- Cross-sectional area, Ax = 28.5 cm2
- Overall depth of the section, h = 200 mm
- Width of flange, bf = 75 mm
- Thickness of flange, tf = 11.4 mm
- Thickness of web, tw = 6.2 mm
- Distance from heel to centroid, cy = 2.20 cm
- Moment of inertia about major axis, Iz = 1,830 cm4
- Moment of inertia about minor axis, Iy = 139 cm4*
- Radius of gyration about major axis, rz = 8.02 cm
- Elastic section modulus about minor axis, Zey = 26.4 cm3
* This value have been taken from the built-in section library STAAD.Pro of Indian sections.
Additional section properties:
- Plastic section modulus about major axis, Zpz = 213 cm3
- Plastic section modulus about minor axis, Zpy = 51.1 cm3
- Torsional moment of inertia, Ix = 9.83 cm4
Validation
Constants
- γm0 = 1.1
- γm1 = 1.25
Section Classification
b = bf/2 = 37.5 mm
b / tf = 6.58 < 9.4ε = 9.4
dw/tw = 25.03 < 42ε = 42
Hence, from Table 2 of IS 800:2007, the flange is plastic but the web is non-slender (i.e., semi-compact).
Therefore, the overall classification of the section is "semi-compact".
Hence, from Table 10 of IS 800:2007, the buckling class about both axes is "c".
Check Slenderness Ratio
ky = kz = 1
The maximum allowable slenderness ratio is 180.
Slenderness ratio about the major axis, kzL / rz = 37.41 < 180
Radius of gyration about minor axis,
Slenderness ratio about the minor axis, kyL / ry = 135.8 < 180
So the slenderness is within the limit.
Axial Tension Capacity
Check for yielding of gross section:
Tdg = Ag × fy / γm0 = 647.7 kN
Check for rupture of critical section:
α = 0.8
Tdn = α × Anet × fu / γm1 = 766.1 kN
Block shear areas are not specified and therefore not checked.
No tension in the member, so critical ratio is 0.
Check Axial Compression
Axial compression capacity of major axis:
The non-dimensional slenderness ratio,
For buckling class "c", the imperfection factor, α = 0.49
So, as per Cl. 7.1.2.1 of IS 800:2007,
Therefore, axial compression capacity in the major axis: Pdz = Axfcdz = 575.6 kN
The critical ratio for axial compression in major axis = 10 / 575.6 = 0.017
Axial compression capacity of minor axis:
The non-dimensional slenderness ratio,
For buckling class "c", the imperfection factor, α = 0.49
So, as per Cl. 7.1.2.1 of IS 800:2007,
Therefore, axial compression capacity in the minor axis: Pdy = Axfcdy = 201.6 kN, which governs.
The critical ratio for axial compression in minor axis = 10 / 201.6 = 0.050
Check Shear Capacity
Major Axis
As per Cl. 8.4.1.1 of IS 800:2007, the shear area, Avz = 2×bftf = 1,710 mm2
So, as per Cl. 8.4.1,
Hence, shear capacity along major axis:
Critical shear ratio along major axis: 3 / 224.5 = 0.013
Minor Axis
As per Cl. 8.4.1.1 of IS 800:2007, the shear area, Avy = h×tw = 1,240 mm2
So, as per Cl. 8.4.1,
Hence, shear capacity along major axis:
As per Cl. 8.4.2, dw / tw = 25.03 < 67ε, so resistance to shear buckling need not be checked.
Shear force along both axes = 3 kN.
Critical shear ratio along minor axis: 3 / 162.7 = 0.018
Check Bending Capacity
Since the section has been classified as "semi-compact", βb = Zez / Zpz = 0.859.
Major Axis Bending
The elastic lateral-torsional buckling moment:
where= | ||
= | ||
= | ||
= | ||
= |
Since λLT > 0.4, per 8.2.2(c), Lateral Torsional Buckling must be considered.
So, the bending strength with respect to the major axis is Mdz = βbZpzfbd = 30.35 kN·m
Actual bending moment about the major axis is Mz = 4.5 kN·m.
The critical bending ratio in the major axis is 4.5 / 30.35 = 0.148
Minor Axis Bending
Since the section has been classified as "semi-compact", βb = Zey / Zpy = 0.517.Actual shear force in both axes, V = 3 kN < 0.6Vd = 89.75 kN. Hence, the section is subjected to low shear.
So, the governing bending strength with respect to the minor axis is Mdy = 5.938 kN·m
Actual bending moment about the minor axis is Mz = 4.5 kN·m.
The critical bending ratio in the minor axis is 4.5 / 5.938 = 0.758
Check Combined Interaction of Axial Compression and Bending
Section strength calculation as per Cl. 9.3.1.3:
Compression strength in yielding:Overall member strength in bending and axial compression
ny = Fx / Pdy = 0.050
nz = Fx / Pdz = 0.017
Results
Result Type | Reference | STAAD.Pro | Difference | Comments |
---|---|---|---|---|
Slenderness ratio | 135.8 | 135.84 | negligible | |
Tension capacity, gross yielding (kN) | 647.7 | 647.727 | none | |
Tension capacity, rupture of critical section (kN) | 766.1 | 766.08 | negligible | |
Axial compression capacity about major axis (kN) | 575.6 | 575.661 | negligible | |
Axial compression capacity about minor axis (kN) | 201.6 | 201.646 | negligible | |
Critical ratio for axial compression | 0.050 | 0.050 | none | |
Shear capacity in major axis (kN) | 224.4 | 224.379 | negligible | |
Shear capacity in minor axis (kN) | 162.7 | 162.708 | negligible | |
Ratio for shear in major axis | 0.013 | 0.013 | none | |
Ratio for shear in minor axis | 0.018 | 0.018 | none | |
Bending capacity about major axis (kN·m) | 30.35 | 30.382 | negligible | |
Bending capacity about minor axis (kN·m) | 5.938 | 5.938 | none | |
Ratio for bending about major axis | 0.148 | 0.148 | none | |
Ratio for bending about minor axis | 0.758 | 0.758 | none | |
Ratio per Cl. 9.3.1.3 | 0.921 | 0.921 | none | |
Ratio per Eq. 1 of Cl. 9.3.2.2 | 0.906 | 0.906 | none | |
Ratio per Eq.2 of Cl. 9.3.2.2 | 0.577 | 0.577 | none |
STAAD.Pro Input
The file C:\Users\Public\Public Documents\STAAD.Pro 2023\Samples \Verification Models\09 Steel Design\India\IS 800 2007 LSD - Channel - With LTB.STD is typically installed with the program.
The following design parameters are used:
- The default value of ALPHA is used.
- The value of KX 1.0 is directly specified.
- The default value of the maximum allowable slenderness ratio, MAIN, of 180 is used.
- The effective length for lateral-torsional buckling is specified as LX 3 (m).
- The beam type is a cantilever, specified by CAN 1
- The beam is laterally unsupported, specified by LAT 0
- The default values of CMX, CMY, and CMZ of 0.9 are used (CMZ 0.9 is directly specified).
STAAD SPACE
START JOB INFORMATION
ENGINEER DATE 23-Jun-19
END JOB INFORMATION
*****************************************************************************
*This problem has been created to validate the program calculated design
*results of laterally unsupported Channel section subjected to axial
*compression and biaxial bending, designed using IS 800 2007 (LSD)
*****************************************************************************
INPUT WIDTH 79
SET SHEAR
UNIT METER KN
JOINT COORDINATES
1 0 0 0; 2 3 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 INDIAN
1 TABLE ST ISMC200
CONSTANTS
MATERIAL STEEL ALL
SUPPORTS
1 FIXED
LOAD 1 LOADTYPE Dead TITLE LOAD CASE 1
JOINT LOAD
2 FX -10
MEMBER LOAD
1 UNI GY -1
1 UNI GZ -1
PERFORM ANALYSIS
PRINT MEMBER PROPERTIES
PARAMETER 1
CODE IS800 LSD
LX 3 ALL
KX 1 ALL
CAN 1 ALL
CMZ 0.9 ALL
LAT 0 ALL
TRACK 2 ALL
CHECK CODE ALL
FINISH
STAAD.Pro Output
STAAD.PRO CODE CHECKING - IS-800-2007-LSD (V2.2) **************************************************** |----------------------------------------------------------------------------------| | Member Number: 1 | | Member Section: ST ISMC200 (INDIAN SECTIONS) | | Status: PASS Ratio: 0.921 Critical Load Case: 1 Location: 0.00 | | Critical Condition: Sec. 9.3.1.3 | | Critical Design Forces: (Unit: KN METE) | | FX: 10.000E+00 C FY: 3.000E+00 FZ: 3.000E+00 | | MX: 0.000E+00 MY: -4.500E+00 MZ: 4.500E+00 | |----------------------------------------------------------------------------------| | Section Properties: (Unit: CM ) | | AXX: 28.500E+00 IZZ: 1.830E+03 RZZ: 8.013E+00| | AYY: 12.400E+00 IYY: 139.000E+00 RYY: 2.208E+00| | AZZ: 17.100E+00 IXX: 9.830E+00 CW: 9.899E+03| | ZEZ: 183.000E+00 ZPZ: 213.000E+00 | | ZEY: 63.761E+00 ZPY: 51.100E+00 | |----------------------------------------------------------------------------------| | Slenderness Check: (Unit: KN METE) | | Actual Length: 3.000E+00 | | Parameters: LZ: 3.000E+00 LY: 3.000E+00 | | KZ: 1.000 KY: 1.000 | | Actual Ratio: 135.84 Allowable Ratio: 180.00 LOAD: 1 | |----------------------------------------------------------------------------------| | Section Class: Semi-Compact; Flange Class: Plastic; Web Class: Semi-Compact| |----------------------------------------------------------------------------------| STAAD SPACE -- PAGE NO. 5 STAAD.PRO CODE CHECKING - IS-800-2007-LSD (V2.2) **************************************************** |----------------------------------------------------------------------------------| | Member Number: 1 | | Member Section: ST ISMC200 (INDIAN SECTIONS) | |----------------------------------------------------------------------------------| | Tension: (Unit: KN METE) | | Parameters: FYLD: 250.000E+03 FU: 420.000E+03 | | NSF: 1.000 ALPHA: 0.800 | | Yielding : Design Force: 0.000E+00 LC: 0 | | Capacity: 647.727E+00 As per: Sec. 6.2 | | Rupture : Design Force: 0.000E+00 LC: 0 | | Capacity: 766.080E+00 As per: Sec. 6.3 | |----------------------------------------------------------------------------------| | Compression: (Unit: KN METE) | | Buckling Class: Major: c Minor: c As per:Cl. 7.1.2.2 | | Major Axis: Design Force: 10.000E+00 LC: 1 Loc: 0.000 | | Capacity: 575.661E+00 As per: Sec. 7.1.2 | | Minor Axis: Design Force: 10.000E+00 LC: 1 Loc: 0.000 | | Capacity: 201.646E+00 As per: Sec. 7.1.2 | |----------------------------------------------------------------------------------| | Shear: (Unit: KN ) | | Major Axis: Design Force: 3.000E+00 LC: 1 Loc: 0.000 | | Capacity: 224.379E+00 As per: Sec. 8.4 | | Minor Axis: Design Force: 3.000E+00 LC: 1 Loc: 0.000 | | Capacity: 162.708E+00 As per: Sec. 8.4 | |----------------------------------------------------------------------------------| | Bending: (Unit: KN METE) | | Parameters: Laterally Unsupported KX: 1.00 LX: 3.000E+00 Cantilever | | Major Axis: Design Force: 4.500E+00 LC: 1 Loc: 0.000 | | Capacity: 30.382E+00 As per: Sec. 8.2.2 | | Minor Axis: Design Force: -4.500E+00 LC: 1 Loc: 0.000 | | Capacity: 5.938E+00 As per: Sec. 8.2.1.2 | |----------------------------------------------------------------------------------| | Combined Interaction: | | Parameters: PSI: 1.00 CMX: 0.900 CMY: 0.900 CMZ: 0.900 | | Section Strength: Ratio: 0.921 As per: Sec. 9.3.1.3 | | LC: 1 Loc: 0.000 | | Overall Member Strength (Bending+Compression): | | Equation 1: Ratio: 0.906 As per: Sec. 9.3.2.2 Ky : 1.0397 | | LC: 1 Loc: 0.000 KLT: 0.9931 | | Equation 2: Ratio: 0.577 As per: Sec. 9.3.2.2 Ky : 1.0397 | | LC: 1 Loc: 0.000 Kz : 1.0038 | |----------------------------------------------------------------------------------| | Checks Ratio Load Case No. Location from Start( METE) | | | | Tension 0.000 0 0.000E+00 | | Compression 0.050 1 0.000E+00 | | Shear Major 0.013 1 0.000E+00 | | Shear Minor 0.018 1 0.000E+00 | | Bend Major 0.148 1 0.000E+00 | | Bend Minor 0.758 1 0.000E+00 | | Sec. 9.3.1.3 0.921 1 0.000E+00 | | Sec. 9.3.2.2 (i) 0.906 1 0.000E+00 | | Sec. 9.3.2.2(ii) 0.577 1 0.000E+00 | |----------------------------------------------------------------------------------|