V. IS 800 2007 LSD - I Section - with LTB
Verify the design of an I 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.
CISC. "Torsional Section Properties of steel Shapes". Canadian Institute of Steel Construction. 2002. Willowdale, Ont.
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 ISMB200 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.
Validation
Constants
- γm0 = 1.1
- γm1 = 1.25
Section Classification
b = bf/2 = 50 mm
b / tf = 5 < 9.4ε = 9.4
Since the section is symmetric about both axes, the neutral axis is at mid-depth.
dw / tw = 27.72 < 84ε = 84
Hence, from Table 2 of IS 800:2007, both the flange and web are plastic.
Therefore, the overall classification of the section is "plastic".
h / bf = 2 > 1.2 and tf = 10 mm < 40 mm
Hence, from Table 10 of IS 800:2007, the buckling class about the major axis is "a" and the same about the minor axis is "b".
Check Slenderness Ratio
ky = kz = 1
The maximum allowable slenderness ratio is 180.
Slenderness ratio about the major axis, kzL / rz = 36.20 < 180
Slenderness ratio about the minor axis, kyL / ry = 142.2 < 180
So the slenderness is within the limit.
Axial Tension Capacity
Check for yielding of gross section:
Tdg = Ag × fy / γm0 = 700 kN
Check for rupture of critical section:
α = 0.8
Tdn = α × Anet × fu / γm1 = 827.9 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 "a", the imperfection factor, α = 0.21
So, as per Cl. 7.1.2.1 of IS 800:2007,
Therefore, axial compression capacity in the major axis: Pdz = Axfcdz = 666.5 kN
The critical ratio for axial compression in major axis = 10 / 666.5 = 0.015
Axial compression capacity of minor axis:
The non-dimensional slenderness ratio,
For buckling class "b", the imperfection factor, α = 0.34
So, as per Cl. 7.1.2.1 of IS 800:2007,
Therefore, axial compression capacity in the minor axis: Pdy = Axfcdy = 219.8 kN, which governs.
The critical ratio for axial compression in minor axis = 10 / 219.8 = 0.045
Check Shear Capacity
Major Axis
As per Cl. 8.4.1.1 of IS 800:2007, the shear area, Avz = 2×bftf = 2,000 mm2
So, as per Cl. 8.4.1,
Hence, shear capacity along major axis:
Minor Axis
As per Cl. 8.4.1.1 of IS 800:2007, the shear area, Avy = h×tw = 1,140 mm2
So, as per Cl. 8.4.1,
Hence, shear capacity along major axis:
As per Cl. 8.4.2, dw / tw = 27.72 < 67ε, so resistance to shear buckling need not be checked.
Shear force along both axes = 3 kN.
Critical shear ratio along major axis: 3 / 149.6 = 0.020
Critical shear ratio along minor axis: 3 / 262.4 = 0.011
Check Bending Capacity
Since the section has been classified as "plastic", βb = 1.
Major Axis Bending
The elastic lateral-torsional buckling moment:
where= | ||
= | ||
= | ||
= |
Mcr = 60.13 kN·m
So, the bending strength with respect to the major axis is Mdz = βbZpzfbd = 36.35 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 / 36.35 = 0.124
Minor Axis Bending
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 = 9.341 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 / 9.341 = 0.482
Check Combined Interaction of Axial Compression and Bending
Section strength calculation as per Cl. 9.3.1:
n = N / Nd = 0.014 < 0.2
Hence, from Table 17:
α1 = max(5n , 1) = 1
α2 = 2
Mndy = Mdy = 9.341 kN·m | (Cl. 9.3.1.2(c) ) |
1.11·Mdz·(1 - n) = 39.78 kN·m > Mdz
Mndz = Mdz = 36.35 kN·m
Overall member strength in bending and axial compression
ny = Fx / Pdy = 0.045
nz = Fx / Pdz = 0.015
Results
Result Type | Reference | STAAD.Pro | Difference | Comments |
---|---|---|---|---|
Slenderness ratio | 142.3 | 142.24 | negligible | |
Tension capacity, gross yielding (kN) | 700 | 700 | none | |
Tension capacity, rupture of critical section (kN) | 827.9 | 827.904 | negligible | |
Axial compression capacity about major axis (kN) | 666.5 | 666.502 | negligible | |
Axial compression capacity about minor axis (kN) | 219.8 | 219.804 | negligible | |
Critical ratio for axial compression | 0.045 | 0.045 | none | |
Shear capacity in major axis (kN) | 262.4 | 262.432 | negligible | |
Shear capacity in minor axis (kN) | 149.6 | 149.586 | negligible | |
Ratio for shear in major axis | 0.011 | 0.011 | none | |
Ratio for shear in minor axis | 0.020 | 0.020 | none | |
Bending capacity about major axis (kN·m) | 36.35 | 36.347 | negligible | |
Bending capacity about minor axis (kN·m) | 9.341 | 9.341 | none | |
Ratio for bending about major axis | 0.124 | 0.124 | none | |
Ratio for bending about minor axis | 0.482 | 0.482 | none | |
Ratio per Cl. 9.3.1.1 | 0.497 | 0.497 | none | |
Ratio per Eq. 1 of Cl. 9.3.2.2 | 0.618 | 0.618 | none | |
Ratio per Eq.2 of Cl. 9.3.2.2 | 0.397 | 0.397 | none |
STAAD.Pro Input
The file C:\Users\Public\Public Documents\STAAD.Pro CONNECT Edition\Samples\ Verification Models\09 Steel Design\India\IS 800 2007 LSD - I Section - with LTB.STD is typically installed with the program.
The following design parameters are used:
- The default value of ALPHA is used.
- The default values of KY and KZ are used (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 05-Jun-2020
END JOB INFORMATION
*****************************************************************************
*This problem has been created to validate the program calculated design
*results of laterally unsupported I 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 ISMB200
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 PROPERTY
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 ISMB200 (INDIAN SECTIONS) | | Status: PASS Ratio: 0.790 Critical Load Case: 1 Location: 0.00 | | Critical Condition: Slenderness (Compression) | | 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: 30.800E+00 IZZ: 2.115E+03 RZZ: 8.287E+00| | AYY: 11.400E+00 IYY: 137.000E+00 RYY: 2.109E+00| | AZZ: 20.000E+00 IXX: 10.600E+00 CW: 15.042E+03| | ZEZ: 211.500E+00 ZPZ: 240.000E+00 | | ZEY: 27.400E+00 ZPY: 46.000E+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: 142.24 Allowable Ratio: 180.00 LOAD: 1 | |----------------------------------------------------------------------------------| | Section Class: Plastic; Flange Class: Plastic; Web Class: Plastic| |----------------------------------------------------------------------------------| STAAD SPACE -- PAGE NO. 5 STAAD.PRO CODE CHECKING - IS-800-2007-LSD (V2.2) **************************************************** |----------------------------------------------------------------------------------| | Member Number: 1 | | Member Section: ST ISMB200 (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: 700.000E+00 As per: Sec. 6.2 | | Rupture : Design Force: 0.000E+00 LC: 0 | | Capacity: 827.904E+00 As per: Sec. 6.3 | |----------------------------------------------------------------------------------| | Compression: (Unit: KN METE) | | Buckling Class: Major: a Minor: b As per:Cl. 7.1.2.2 | | Major Axis: Design Force: 10.000E+00 LC: 1 Loc: 0.000 | | Capacity: 666.502E+00 As per: Sec. 7.1.2 | | Minor Axis: Design Force: 10.000E+00 LC: 1 Loc: 0.000 | | Capacity: 219.804E+00 As per: Sec. 7.1.2 | |----------------------------------------------------------------------------------| | Shear: (Unit: KN ) | | Major Axis: Design Force: 3.000E+00 LC: 1 Loc: 0.000 | | Capacity: 262.432E+00 As per: Sec. 8.4 | | Minor Axis: Design Force: 3.000E+00 LC: 1 Loc: 0.000 | | Capacity: 149.586E+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: 36.347E+00 As per: Sec. 8.2.2 | | Minor Axis: Design Force: -4.500E+00 LC: 1 Loc: 0.000 | | Capacity: 9.341E+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: 0.497 As per: Sec. 9.3.1.1 | | LC: 1 Loc: 0.000 | | Overall Member Strength (Bending+Compression): | | Equation 1: Ratio: 0.618 As per: Sec. 9.3.2.2 Ky : 1.0364 | | LC: 1 Loc: 0.000 KLT: 0.9930 | | Equation 2: Ratio: 0.396 As per: Sec. 9.3.2.2 Ky : 1.0364 | | LC: 1 Loc: 0.000 Kz : 1.0030 | |----------------------------------------------------------------------------------| | Checks Ratio Load Case No. Location from Start( METE) | | | | Tension 0.000 0 0.000E+00 | | Compression 0.045 1 0.000E+00 | | Shear Major 0.011 1 0.000E+00 | | Shear Minor 0.020 1 0.000E+00 | | Bend Major 0.124 1 0.000E+00 | | Bend Minor 0.482 1 0.000E+00 | | Sec. 9.3.1.1 0.497 1 0.000E+00 | | Sec. 9.3.2.2 (i) 0.618 1 0.000E+00 | | Sec. 9.3.2.2(ii) 0.396 1 0.000E+00 | |----------------------------------------------------------------------------------|