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, A_x = 28.5 cm²
Overall depth of the section, h = 200 mm
Width of flange, b_f = 75 mm
Thickness of flange, t_f = 11.4 mm
Thickness of web, t_w = 6.2 mm
Distance from heel to centroid, c_y = 2.20 cm
Moment of inertia about major axis, I_z = 1,830 cm⁴
Moment of inertia about minor axis, I_y = 139 cm⁴*
Radius of gyration about major axis, r_z = 8.02 cm
Elastic section modulus about minor axis, Z_ey = 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, Z_pz = 213 cm³
Plastic section modulus about minor axis, Z_py = 51.1 cm³
Torsional moment of inertia, I_x = 9.83 cm⁴

Validation

Constants

γ_m0 = 1.1
γ_m1 = 1.25

Section Classification

ε = \sqrt{250 / f_{y}} = 1

b = b_f/2 = 37.5 mm

b / t_f = 6.58 < 9.4ε = 9.4

d_w/t_w = 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

k_y = k_z = 1

The maximum allowable slenderness ratio is 180.

Slenderness ratio about the major axis, k_zL / r_z = 37.41 < 180

Radius of gyration about minor axis, $r_{y} = \sqrt{\frac{I_{y}}{A_{x}}} = \sqrt{\frac{139}{28.5}} = 2.21 cm$

Slenderness ratio about the minor axis, k_yL / r_y = 135.8 < 180

So the slenderness is within the limit.

Axial Tension Capacity

Check for yielding of gross section:

T_dg = A_g × f_y / γ_m0 = 647.7 kN

Check for rupture of critical section:

α = 0.8

T_dn = α × A_net × f_u / γ_m1 = 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, $λ_{z} = \sqrt{\frac{f_{y} {(k_{z} L / r_{z})}^{2}}{π^{2} E}} = \sqrt{\frac{250 {(1.0 × 3,000 / 80.2)}^{2}}{π^{2} × 205,000}} = 0.416$

For buckling class "c", the imperfection factor, α = 0.49

ϕ_{z} = 0.5 [1 + α (λ_{z} - 0.2) + λ_{z}^{2}] = 0.639

χ_{z} = \frac{1}{ϕ_{z} + \sqrt{ϕ_{z}^{2} - λ_{z}^{2}}} = \frac{1}{0.639 + \sqrt{{0.639}^{2} - {0.416}^{2}}} = 0.889

So, as per Cl. 7.1.2.1 of IS 800:2007, $f_{c d z} = min (\frac{χ_{z} f_{y}}{γ_{m 0}}, \frac{f_{y}}{γ_{m 0}}) = 202.0 MPa$

Therefore, axial compression capacity in the major axis: P_dz = A_xf_cdz = 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, $λ_{y} = \sqrt{\frac{f_{y} {(k_{y} L / r_{y})}^{2}}{π^{2} E}} = \sqrt{\frac{205 {(1.0 \times 3,000 / 22.2)}^{2}}{π^{2} \times 205,000}} = 1.502$

For buckling class "c", the imperfection factor, α = 0.49

ϕ_{y} = 0.5 [1 + α (λ_{y} - 0.2) + λ_{y}^{2}] = 1.961

χ_{y} = \frac{1}{ϕ_{y} + \sqrt{ϕ_{y}^{2} - λ_{y}^{2}}} = 0.311

So, as per Cl. 7.1.2.1 of IS 800:2007, $f_{c d y} = min (\frac{χ_{y} f_{y}}{γ_{m 0}}, \frac{f_{y}}{γ_{m 0}}) = 70.75 MPa$

Therefore, axial compression capacity in the minor axis: P_dy = A_xf_cdy = 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, A_vz = 2×b_ft_f = 1,710 mm²

So, as per Cl. 8.4.1, $V_{p z} = \frac{A_{v z} f_{y w}}{\sqrt{3}} = 246.8 kN$

Hence, shear capacity along major axis: $V_{d z} = \frac{V_{p z}}{γ_{m 0}} = 224.4 kN$

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, A_vy = h×t_w = 1,240 mm²

So, as per Cl. 8.4.1, $V_{p y} = \frac{A_{v y} f_{y w}}{\sqrt{3}} = 179.0 kN$

Hence, shear capacity along major axis: $V_{d y} = \frac{V_{p y}}{γ_{m 0}} = 162.7 kN$

As per Cl. 8.4.2, d_w / t_w = 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", β_b = Z_ez / Z_pz = 0.859.

Major Axis Bending

The elastic lateral-torsional buckling moment:

M_{c r} = \sqrt{\frac{π^{2} E I_{y}}{{L_{L T}}^{2}} (G I_{t} + \frac{π^{2} E I_{w}}{{L_{L T}}^{2}})}

where

G	=	shear modulus, $= \frac{E}{2 (1 + ν)} = 78,850 MPa$
d'	=	d - t_f = 188.6 mm
b'	=	b - t_w/2 = 71.9 mm
ɑ	=	$\frac{1}{2 + \frac{d^{'} t_{w}}{3 b^{'} t_{f}}} = \frac{1}{2 + \frac{18.86 (0.62)}{3 (7.19) (1.14)}} = 0.404$
I_w	=	warping constant, as taken from the CISC document "Torsional Section Properties of Steel Shapes" (see reference), which is taken from SSRC 1998 and Galambos 1968. $I_{w} = {d^{'}}^{2} {b^{'}}^{3} t_{f} [\frac{1 - 3 (0.404)}{6} + \frac{ɑ^{2}}{2} (1 + \frac{d^{'} t_{w}}{6 b^{'} t_{f}})] = {(18.86)}^{2} {(7.19)}^{3} 1.14 [\frac{1 - 3 (0.404)}{6} + \frac{{(0.404)}^{2}}{2} (1 + \frac{d^{'} w}{6 b^{'} t_{f}})] = 9,899 {cm}^{6}$

M_{c r} = \sqrt{\frac{π^{2} (205,000) (141) 10^{4}}{{(3,000)}^{2}} [(78,850) (9.83) 10^{4} + \frac{π^{2} (205,000) (9,899) 10^{6}}{{(3,000)}^{2}}]} {(10)}^{-6} = 55.83 kN·m

λ_{L T} = min | \begin{array}{l} \sqrt{\frac{β_{b} Z_{p z} f_{y}}{M_{c r}}} = \sqrt{\frac{0.859 (213) (250)}{55.83 {(10)}^{3}}} = 0.905 \\ \sqrt{\frac{1.2 Z_{c z} f_{y}}{M_{c r}}} = \sqrt{\frac{1.2 (183) (250)}{55.83 {(10)}^{3}}} = 0.992 \end{array}

Since λ_LT > 0.4, per 8.2.2(c), Lateral Torsional Buckling must be considered.

ϕ_{LT} = 0.5 [1 + 0.21 (λ_{LT} - 0.2) + λ_{LT}^{2}] = 0.984

χ_{LT} = \frac{1}{ϕ_{LT} + \sqrt{ϕ_{LT}^{2} - λ_{LT}^{2}}} = 0.730

f_{b d} = \frac{χ_{L T} f_{y}}{γ_{m 0}} = 165.9 MPa

So, the bending strength with respect to the major axis is M_dz = β_bZ_pzf_bd = 30.35 kN·m

Actual bending moment about the major axis is M_z = 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", β_b = Z_ey / Z_py = 0.517.

Actual shear force in both axes, V = 3 kN < 0.6V_d = 89.75 kN. Hence, the section is subjected to low shear.

\frac{β_{b} Z_{p y} f_{y}}{γ_{m 0}} = 5.938 kN·m

\frac{1.2 Z_{e y} f_{y}}{γ_{m 0}} = 8.907 kN·m

So, the governing bending strength with respect to the minor axis is M_dy = 5.938 kN·m

Actual bending moment about the minor axis is M_z = 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:

N_{d} = \frac{A_{g} f_{y}}{γ_{m0}} = \frac{28.5 {(10)}^{-1} (250)}{1.1} = 647.7 kN

\frac{N}{N_{d}} + \frac{M_{y}}{M_{dy}} + \frac{M_{z}}{M_{dz}} = \frac{10}{647.7} + 0.758 + 0.148 = 0.921

Overall member strength in bending and axial compression

n_y = F_x / P_dy = 0.050

n_z = F_x / P_dz = 0.017

K_{y} = min | \begin{array}{l} 1 + (λ_{y} - 0.2) n_{y} = 1 + (1.51 - 0.2) \cdot 0.050 = 1.066 \\ 1 + 0.8 n_{y} = 1 + 0.8 \cdot 0.050 = 1.040 \end{array}

K_{z} = min | \begin{array}{l} 1 + (λ_{z} - 0.2) n_{z} = 1 + (0.416 - 0.2) \cdot 0.017 = 1.004 \\ 1 + 0.8 n_{z} = 1 + 0.8 \cdot 0.017 = 1.014 \end{array}

K_{L T} = max | \begin{array}{l} 1 - \frac{0.1 λ_{L T} n_{y}}{C_{m L T} - 0.25} = 1 - \frac{0.1 \times 0.905 \times 0.050}{0.9 - 0.25} = 0.993 \\ 1 - \frac{0.1 n_{y}}{C_{m L T} - 0.25} = 1 - \frac{0.1 \times 0.050}{0.9 - 0.25} = 0.992 \end{array}

\frac{P}{P_{d y}} + K_{y} \frac{C_{m y} M_{y}}{M_{d y}} + K_{L T} \frac{M_{z}}{M_{d z}} = 0.906

\frac{P}{P_{d z}} + 0.6 K_{y} \frac{C_{m y} M_{y}}{M_{d y}} + K_{z} \frac{C_{m z} M_{z}}{M_{d z}} = 0.577

Results

Table 1.
Result Type	Reference	STAAD.Pro	Difference
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 CONNECT Edition\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            |
 |----------------------------------------------------------------------------------|