V. IS 800 2007 LSD - I Section - without LTB

Verify the design of an I section subject to axial force and biaxial bending 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.

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 braced to prevent lateral-torsional buckling. The steel has a yield strength of 250 MPa, an ultimate tensile strength of 420 MPa, and a modulus of elasticity of 205×(10)³ MPa.

Validation

Constants

γ_m0 = 1.1
γ_m1 = 1.25

Section Classification

ε = \sqrt{250 / f_{y}} = 1

b = b_f/2 = 50 mm

b / t_f = 5 < 9.4ε = 9.4

Since the section is symmetric about both axes, the neutral axis is at mid-depth.

d_w / t_w = 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 / b_f = 2 > 1.2 and t_f = 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

k_y = k_z = 1

The maximum allowable slenderness ratio is 180.

Slenderness ratio about the major axis, k_zL / r_z = 36.20 < 180

Slenderness ratio about the minor axis, k_yL / r_y = 142.2 < 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 = 700 kN

Check for rupture of critical section:

α = 0.8

T_dn = α × A_net × f_u / γ_m1 = 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, $λ_{z} = \sqrt{\frac{f_{y} {(k_{z} L / r_{z})}^{2}}{π^{2} E}} = 0.402$

For buckling class "a", the imperfection factor, α = 0.21

ϕ_{z} = 0.5 [1 + α (λ_{z} - 0.2) + λ_{z}^{2}] = 0.602

χ_{z} = \frac{1}{ϕ_{z} + \sqrt{ϕ_{z}^{2} - λ_{z}^{2}}} = 0.952

So, as per Cl. 7.1.2.1 of IS 800:2007, $f_{c d z} = min (\frac{χ_{z} f_{y}}{γ_{m 0}}, \frac{f_{y}}{γ_{m 0}}) = 216.4 MPa$

Therefore, axial compression capacity in the major axis: P_dz = A_xf_cdz = 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, $λ_{y} = \sqrt{\frac{f_{y} {(k_{y} L / r_{y})}^{2}}{π^{2} E}} = 1.581$

For buckling class "b", the imperfection factor, α = 0.34

ϕ_{y} = 0.5 [1 + α (λ_{y} - 0.2) + λ_{y}^{2}] = 1.985

χ_{y} = \frac{1}{ϕ_{y} + \sqrt{ϕ_{y}^{2} - λ_{y}^{2}}} = 0.314

So, as per Cl. 7.1.2.1 of IS 800:2007, $f_{c d y} = min (\frac{χ_{y} f_{y}}{γ_{m 0}}, \frac{f_{y}}{γ_{m 0}}) = 71.37 MPa$

Therefore, axial compression capacity in the minor axis: P_dy = A_xf_cdy = 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, A_vz = 2×b_ft_f = 2,000 mm²

So, as per Cl. 8.4.1, $V_{p z} = \frac{A_{v z} f_{y w}}{\sqrt{3}} = 288.7 kN$

Hence, shear capacity along major axis: $V_{d z} = \frac{V_{p z}}{γ_{m 0}} = 262.4 kN$

Minor Axis

As per Cl. 8.4.1.1 of IS 800:2007, the shear area, A_vy = h×t_w = 1,140 mm²

So, as per Cl. 8.4.1, $V_{p y} = \frac{A_{v y} f_{y w}}{\sqrt{3}} = 164.5 kN$

Hence, shear capacity along major axis: $V_{d y} = \frac{V_{p y}}{γ_{m 0}} = 149.6 kN$

As per Cl. 8.4.2, d_w / t_w = 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", β_b = 1.

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

Major Axis Bending

\frac{β_{b} Z_{p y} f_{y}}{γ_{m 0}} = 54.55 kN·m

\frac{1.2 Z_{e y} f_{y}}{γ_{m 0}} = 72.10 kN·m

So, the governing bending strength with respect to the minor axis is M_dy = 54.55 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 / 54.55 = 0.083

Minor Axis Bending

\frac{β_{b} Z_{p y} f_{y}}{γ_{m 0}} = 10.46 kN·m

\frac{1.2 Z_{e y} f_{y}}{γ_{m 0}} = 9.341 kN·m

So, the governing bending strength with respect to the minor axis is M_dy = 9.341 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 / 9.341 = 0.482

Check Combined Interaction of Axial Compression and Bending

Section strength calculation as per Cl. 9.3.1:

n = N / N_d = 0.014 < 0.2

Hence, from Table 17:

α₁ = max(5n , 1) = 1

α₂ = 2

M_ndy = M_dy = 9.341 kN·m

(Cl. 9.3.1.2(c) )

1.11·M_dz·(1 - n) = 59.70 kN·m > M_dz

M_ndz = M_dz = 54.55 kN·m

{(\frac{M_{y}}{M_{n d y}})}^{α_{1}} + {(\frac{M_{z}}{M_{n d z}})}^{α_{2}} = {(\frac{4.5}{9.341})}^{1} + {(\frac{4.5}{54.55})}^{2} = 0.489

Overall member strength in bending and axial compression

n_y = F_x / P_dy = 0.045

n_z = F_x / P_dz = 0.015

K_{y} = min | \begin{array}{l} 1 + (λ_{y} - 0.2) n_{y} = 1 + (1.581 - 0.2) \cdot 0.045 = 1.063 \\ 1 + 0.8 n_{y} = 1 + 0.8 \cdot 0.045 = 1.036 \end{array}

K_{z} = min | \begin{array}{l} 1 + (λ_{z} - 0.2) n_{z} = 1 + (0.402 - 0.2) \cdot 0.015 = 1.003 \\ 1 + 0.8 n_{z} = 1 + 0.8 \cdot 0.015 = 1.012 \end{array}

\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}} = 0.359

Results

Table 1.
Result Type	Reference	STAAD.Pro	Difference
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)	54.55	54.545	negligible
Bending capacity about minor axis (kN·m)	9.341	9.341	none
Ratio for bending about major axis	0.082	0.082	none
Ratio for bending about minor axis	0.482	0.482	none
Ratio per Cl. 9.3.1.1	0.489	0.489	none
Ratio per Eq.2 of Cl. 9.3.2.2	0.359	0.359	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 - without 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 supported, specified by LAT 1
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 supported 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 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   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 Supported      KX:   1.00  LX:   3.000E+00  Cantilever  |
 |  Major Axis:   Design Force:        4.500E+00    LC:     1  Loc:     0.000       |
 |                Capacity:           54.545E+00    As per:  Sec. 8.2.1.2           |
 |  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:     0.489    As per:  Sec. 9.3.1.1                  |
 |                       LC:            1    Loc:       0.000                       |
 |  Overall Member Strength (Bending+Compression):                                  |
 |  Equation 2:          Ratio:     0.359    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             0.082       1                       0.000E+00            |
 |  Bend Minor             0.482       1                       0.000E+00            |
 |  Sec. 9.3.1.1           0.489       1                       0.000E+00            |
 |  Sec. 9.3.2.2(ii)       0.359       1                       0.000E+00            |
 |----------------------------------------------------------------------------------|