V. IS 800 2007 LSD - Angle - Flexural Torsional Buckling

Verify the design of a single angle subject to axial compress per the IS 800:2007 limit state method.

References

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.12

Details

The member is a 2.5 m long cantilever subject to a 100 kN compressive force. The section used is a ISA 150x115x16. 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 of 0.3.

Section Properties

From IS: 808-1989, Table 6.1:

A_x = 40 cm²

c_z = 3.07 cm

c_y = 4.81 cm

I_x = 878 cm⁴

I_y = 447 cm⁴

r_u = 5.21 cm

r_y = 2.44 cm

Validation

Check for axial compression about the major principal axis

The non-dimensional slenderness ratio,

λ_{z} = \sqrt{\frac{f_{y} {(\frac{K_{z} L}{r_{z}})}^{2}}{π^{2} E}} = \sqrt{\frac{250 {(\frac{1.0 \times 250}{5.21})}^{2}}{π^{2} 205,000}} 0.533

Per Table 10, buckling class is "c" for an angle section. Per Table 7, for buckling class "c", ɑ = 0.49

ϕ_{z} = 0.5 [1 + ɑ (λ_{z} - 0.2) + {λ_{z}}^{2}] = 0.5 [1 + 0.49 (0.533 - 0.2) + {(0.533)}^{2}] = 0.724

χ_{z} = {(ϕ_{z} + \sqrt{{ϕ_{z}}^{2} - {λ_{z}}^{2}})}^{- 1} = \frac{1}{0.724 + \sqrt{{(0.724)}^{2} - {(0.533)}^{2}}} = 0.825

f_{cdz} = \frac{χ_{z} \times f_{y}}{γ_{m0}} = \frac{0.825 \times 250 {(10)}^{6}}{1.1} = 187.4 MPa

(Cl. 7.1.2.1)

The axial compression capacity with respect to the major principal axis is:

P_{dz} = A_{x} \times f_{cdz} = 40 {(10)}^{2} {cm}^{2} \times 187.4 MPa {(10)}^{-3} = 749.7 MPa

Check for axial compression about the minor principal axis

The non-dimensional slenderness ratio,

λ_{y} = \sqrt{\frac{f_{y} {(\frac{K_{z} L}{r_{z}})}^{2}}{π^{2} E}} = \sqrt{\frac{250 {(\frac{1.0 \times 250}{2.44})}^{2}}{π^{2} 205,000}} 1.139

ϕ_{y} = 0.5 [1 + ɑ (λ_{y} - 0.2) + {λ_{y}}^{2}] = 0.5 [1 + 0.49 (1.139 - 0.2) + {(1.139)}^{2}] = 1.379

χ_{y} = {(ϕ_{y} + \sqrt{{ϕ_{y}}^{2} - {λ_{y}}^{2}})}^{- 1} = \frac{1}{1.379 + \sqrt{{(1.379)}^{2} - {(1.139)}^{2}}} = 0.464

f_{cdy} = \frac{χ_{y} \times f_{y}}{γ_{m0}} = \frac{0.464 \times 250 {(10)}^{6}}{1.1} = 105.4 MPa

(Cl. 7.1.2.1)

The axial compression capacity with respect to the minor principal axis is:

P_{dy} = A_{x} \times f_{cdy} = 40 {(10)}^{2} {cm}^{2} \times 105.4 MPa {(10)}^{-3} = 421.8 MPa

Check for axial compression against flexural torsional buckling

Since the member being design is an angle loaded through one leg, the axial compression capacity with respect to flexural torsional buckling must be checked. The end connection has two or more bolts (or is welded) and is assumed to be unrestrained ("hinged"). Therefore, from Table 12: k₁ = 0.7, k₂ = 0.6, and k₃ = 5.

λ_{e} = \sqrt{k_{1} + k_{2} \times {λ_{vv}}^{2} + k_{3} \times {λ_{ϕ}}^{2}}

(Cl. 7.5.1.2)

ε = \sqrt{250 / f_{y}} = 1

λ_{ϕ} = \frac{(b + d) / 2 t}{ε \sqrt{\frac{π^{2} \times E}{250}}} = \frac{(150 + 115) / (2 \times 16)}{1.0 \sqrt{\frac{π^{2} \times 205,000}{250}}} = 0.092

λ_{vv} = \frac{L / r_{y}}{ε \sqrt{\frac{π^{2} \times E}{250}}} = \frac{250 / 2.44}{1.0 \sqrt{\frac{π^{2} \times 205,000}{250}}} = 1.139

So,

λ_{e} = \sqrt{0.7 + 0.6 \times {1.139}^{2} + 5 \times {0.092}^{2}} = 1.233

ϕ = 0.5 [1 + ɑ (λ_{e} - 0.2) + {λ_{e}}^{2}] = 0.5 [1 + 0.49 (1.233 - 0.2) + {(1.233)}^{2}] = 1.513

(Cl. 7.1.2.1)

χ = {(ϕ + \sqrt{ϕ^{2} - {λ_{e}}^{2}})}^{- 1} = \frac{1}{1.513 + \sqrt{{(1.513)}^{2} - {(1.233)}^{2}}} = 0.418

f_{cd} = \frac{χ \times f_{y}}{γ_{m0}} = \frac{0.418 \times 250 {(10)}^{6}}{1.1} = 95.06 MPa

The axial compression capacity under flexural torsional buckling is:

P_{dft} = A_{x} \times f_{cd} = 40 {(10)}^{2} {cm}^{2} \times 95.06 MPa {(10)}^{-3} = 380.2 MPa

The minimum capacity governs, thus flexural torsional buckling controls. The utilization ratio is $\frac{P}{P_{d}} = \frac{100}{380.2} = 0.263$

Results

Table 1.
Result Type	Reference	STAAD.Pro	Difference
Axial compression capacity,major axis (kN)	749.7	750.202	negligible
Axial compression , minor axis (kN)	421.8	421.773	negligible
Axial compression capacity, flexural torsional buckling (kN)	380.2	380.234	negligible
Ratio for compression	0.263	0.263	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 - Angle - Flexural Torsional Buckling.STD is typically installed with the program.

The following design parameters are used:

The beam type is a cantilever, specified by CAN 1
The member is connected by a single angle leg, so ANG 1
The connection is considered "hinged" using FXTY 0
Two or more bolts are assumed at the connection, thus NBL 1
The beam is laterally unsupported, specified by LAT 0

STAAD SPACE
START JOB INFORMATION
ENGINEER DATE 13-May-20
END JOB INFORMATION
INPUT WIDTH 79
SET SHEAR
*****************************************************************************
*This problem has been created to validate the program calculated compression
*capacity of a single angle section loaded eccentrically through one leg and
*designed using IS 800 2007 (LSD)
*****************************************************************************
UNIT METER KN
JOINT COORDINATES
1 0 0 0; 2 2.5 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 ISA150X115X16
CONSTANTS
MATERIAL STEEL ALL
SUPPORTS
1 FIXED
LOAD 1 LOADTYPE None  TITLE LOAD CASE 1
JOINT LOAD
2 FX -100
PERFORM ANALYSIS
PRINT MEMBER PROPERTIES ALL
PRINT ANALYSIS RESULTS
SECTION 0.25 0.5 0.75
PRINT MEMBER SECTION FORCES ALL
PARAMETER 1
CODE IS800 LSD
ANG 1 ALL
CAN 1 ALL
FXTY 0 ALL
NBL 1 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   ISA150X115X16         (INDIAN SECTIONS)                   |
 |  Status: PASS  Ratio:   0.569  Critical Load Case:     1  Location:       0.00   |
 |  Critical Condition: Slenderness (Compression)                                   |
 |  Critical Design Forces:  (Unit: KN    METE)                                     |
 |      FX:     100.000E+00 C     FY:       0.000E+00      FZ:       0.000E+00      |
 |      MX:       0.000E+00       MY:       0.000E+00      MZ:       0.000E+00      |
 |----------------------------------------------------------------------------------|
 |  Section Properties:    (Unit:  CM  )                                            |
 |  AXX:      40.000E+00          IZZ:     238.144E+00          RZZ:       2.440E+00|
 |  AYY:      24.000E+00          IYY:       1.094E+03          RYY:       5.229E+00|
 |  AZZ:      18.400E+00          IXX:      34.133E+00           CW:     465.161E+00|
 |  ZEZ:      47.777E+00          ZPZ:      82.555E+00                              |
 |  ZEY:     105.142E+00          ZPY:     181.532E+00                              |
 |----------------------------------------------------------------------------------|
 |  Slenderness Check:    (Unit:  KN    METE)                                       |
 |  Actual Length:        2.500E+00                                                 |
 |  Parameters:     LZ:        2.500E+00  LY:        2.500E+00                      |
 |                  KZ:            1.000  KY:            1.000                      |
 |  Actual Ratio: 102.46 Allowable Ratio: 180.00  LOAD:     1                       |
 |----------------------------------------------------------------------------------|
 |  Section Class: Semi-Compact; Flange Class: Semi-Compact; Web Class: Semi-Compact|
 |----------------------------------------------------------------------------------|
      STAAD SPACE                                              -- PAGE NO.    9
                       STAAD.PRO CODE CHECKING - IS-800-2007-LSD (V2.2)
                     ****************************************************
 |----------------------------------------------------------------------------------|
 |  Member Number:      1                                                           |
 |  Member Section:  ST   ISA150X115X16         (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:          909.091E+00     As per:  Sec. 6.2              |
 |  Rupture    :  Design Force:        0.000E+00     LC:     0                      |
 |                Capacity:            1.075E+03     As per:  Sec. 6.3              |
 |----------------------------------------------------------------------------------|
 |  Compression:    (Unit:  KN    METE)                                             |
 |  Buckling Class:   Major: c    Minor: c    As per:Cl. 7.1.2.2                    |
 |  Parameters:     ANG:  1       NBL:  1       FXTY:       0.000                   |
 |  Major Axis:   Design Force:      100.000E+00    LC:     1  Loc:     0.000       |
 |                Capacity:          750.202E+00    As per:  Sec. 7.1.2             |
 |  Minor Axis:   Design Force:      100.000E+00    LC:     1  Loc:     0.000       |
 |                Capacity:          421.773E+00    As per:  Sec. 7.1.2             |
 |  FlxTorBuck:   Design Force:      100.000E+00    LC:     1  Loc:     0.000       |
 |                Capacity:          380.234E+00    As per:  Sec. 7.1.2             |
 |----------------------------------------------------------------------------------|
 |  Shear:    (Unit:  KN  )                                                         |
 |  Major Axis:   Design Force:        0.000E+00    LC:     0  Loc:     0.000       |
 |                Capacity:          241.437E+00    As per:  Sec. 8.4               |
 |  Minor Axis:   Design Force:        0.000E+00    LC:     0  Loc:     0.000       |
 |                Capacity:          314.918E+00    As per:  Sec. 8.4               |
 |----------------------------------------------------------------------------------|
 |  Bending:    (Unit:  KN    METE)                                                 |
 |  Parameters:   Laterally Unsupported    KX:   1.00  LX:   2.500E+00  Cantilever  |
 |  Major Axis:   Design Force:        0.000E+00    LC:     0  Loc:     0.000       |
 |                Capacity:           22.597E+00    As per:  Sec. 8.2.2             |
 |  Minor Axis:   Design Force:        0.000E+00    LC:     0  Loc:     0.000       |
 |                Capacity:           10.858E+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:     0.110    As per:  Sec. 9.3.1.3                  |
 |                       LC:            1    Loc:       0.000                       |
 |----------------------------------------------------------------------------------|
 |  Checks               Ratio   Load Case No.      Location from Start( METE)      |
 |                                                                                  |
 |  Tension                0.000       0                       0.000E+00            |
 |  Compression            0.263       1                       0.000E+00            |
 |  Shear Major            0.000       0                       0.000E+00            |
 |  Shear Minor            0.000       0                       0.000E+00            |
 |  Bend Major             0.000       0                       0.000E+00            |
 |  Bend Minor             0.000       0                       0.000E+00            |
 |  Sec. 9.3.1.3           0.110       1                       0.000E+00            |
 |  Sec. 9.3.2.2 (i)       0.237       1                       0.000E+00            |
 |  Sec. 9.3.2.2(ii)       0.133       1                       0.000E+00            |
 |----------------------------------------------------------------------------------|