V. IS 801-Column with axial and major axis bending

Verification example for a cold-formed column subject to axial compression and bending moment according to IS:801-1975.

Details

Verifies the calculations for an IS 60CS40x4 (channel with lips) column that is 2 m long and subject to axial compression and major axis bending moment. This example checks for compression, shear, bending, and compression and bending interaction as per IS 801.

Material properties:

E = 2,074,000 kgf/cm² = 203,400 MPa
F_y = 3,569 kgf/cm² = 353.0 MPa
F_u = 4,588 kgf/cm² = 450 MPa
G = 795,000 kgf/cm² = 77,963 MPa

Design forces:

P = 10 kN
M_z = 4.0 kN·m
V_y = 2 kN

Verification

Section Dimension Checks

Flanges (ref. cl 5.2.3):

w = 4 - 2× 0.4 - 2× 0.6 = 2

w/t = 2 / 0.4 = 5 < 60, Hence OK

Web (ref. cl 5.2.4):

W/t = (6 - 2×0.4 - 2× 0.6) / 0.4 = 10 < 500 and

h/t = (6 - 2× 0.4) / 0.4 = 13 < 150, Hence OK

Check for limiting slenderness (ref. cl 6.3.3):

K×L_y/r_y = 1.0× 200 mm / 1.453 mm = 137.6 < 200

Hence, OK

Allowable Compression Strength

Maximum allowable compressive stress for flexural buckling Y (ref. cl 6.6.1.1 of IS 801)

Design stress, F = 0.6× f_y = 2,160 kgf/cm² = 211.8 MPa

Effective width of flange, B_eff

W = 4 - 2×0.4 - 2× 0.6 = 2 cm

\frac{w}{t} = 5 < \frac{1,435}{\sqrt{F}} = 30.87

B_eff = w = 2 cm

Effective width of web, B_eff

W/t < 30.87

B_eff = w = 10 cm

A_eff = 5.8 cm²

Factor, Q = A_eff / A = 1.0=

C_{e} = \sqrt{\frac{2 π^{2} E}{F_{y}}} = 106.6

Slenderness limit $= \frac{C_{e}}{\sqrt{Q}} = 106.6 < 137.6$

F_{a 1} = \frac{10,680,000}{{(k L / r)}^{2}} = 564.1 k g f / {c m}^{2}

Maximum allowable compressive stress for flexural torsional buckling (ref. cl 6.6.1.2 of IS 801)

Polar radius of gyration, $r_{0} = \sqrt{{r_{x}}^{2} + {r_{y}}^{2} + {x_{0}}^{2}} = 5.181 c m$

β = 1 - {(x_{0} / r_{0})}^{2} = 0.259

σ_t = 2,036 kgf/cm²

σ_ex = 2,480 kgf/cm²

σ_{T F 0} = \frac{1}{2 β} [(σ_{e x} + σ_{t}) - \sqrt{{(σ_{e x} + σ_{t})}^{2} - 4 β \times σ_{e x} \times σ_{t}}] = 1,201 kgf {cm}^{2} = 117.8 MPa

1,201 kgf/cm² < 0.5×F = 1,800 kgf/cm²

F_a2 = 0.522×σ_TF0 = 626.9 kgf/cm²

The allowable compressive stress, F_a is the minimum of F_a1 and F_a1 .

F_a = 546.1 kgf/cm² = 55.32 MPa

Allowable Bending Strength

C = 1.0 for member under combined axial and bending.

F_{b} = \frac{2}{3} F_{y} - \frac{{F_{y}}^{2}}{54 π^{2} E ∙ C_{b}} (\frac{L^{2} S_{x c}}{d \times I_{y c}}) = 1,412 k g f / {c m}^{2} = 138.5 M P a

(ref. cl 6.3)

Maximum allowable compressive stress:

F_b1 = 0.6× F_y = 2,160 kgf/cm² = 211.8 MPa

Allowable Shear Strength

\frac{h}{t} = 13 < \frac{4,590}{\sqrt{F_{y}}} = 76.5

F_{v 1} = \frac{1,275 \sqrt{F_{y}}}{(h / t)} = 5,884 k g f / {c m}^{2}

F_{v 2} = 0.4 F_{y} = 1,440 k g f / {c m}^{2}

The allowable shear stress, F_v is the minimum of F_v1 and F_v2 :

F_v = 1,440 kgf/cm² = 141.3 MPa

Axial and Bending Interaction

f_a = P / A = 10 kN / 5.8 cm² = 1,019.7 kgf / 5.8 cm² = 175.8 kgf/cm²

f_b1 = M / Z = 4.0 kN·m / 9.4 cm³ = 40,789 kgf·cm / 9.4 cm³ = 4,339 kgf/cm²

C_mx = 0.6

F'_e = 1,294 kgf/cm²

\frac{f_{a}}{F_{a 1}} + \frac{f_{b 1} C_{m}}{F_{b 1} [1 - \frac{f_{a}}{{F'}_{e}}]} = \frac{175.8}{564.1} + \frac{4,339 (0.6)}{2,160 [1 - \frac{175.8}{1,294}]} = 0.312 + 1.395 = 1.705

Hence, not OK

F_a0 = 1,879 kgf/cm³

\frac{f_{a}}{F_{a 0}} + \frac{f_{b 1}}{F_{b 1}} = \frac{175.8}{1,879} + \frac{4339}{2,160} = 0.094 + 2.009 = 2.103

Results

Table 1. Comparison of results
Result Type	Reference	STAAD.Pro	Difference
Allowable compressive stress (MPa)	55.32	55.371	negligible
Allowable bending stress (MPa)	211.8	211.854	negligible
Allowable unbraced bending stress (MPa)	138.5	138.583	negligible
Allowable shear stress along Y (MPa)	141.3	141.236	negligible
Axial and Bending interaction	2.103	2.102	negligible

STAAD.Pro Input File

The file C:\Users\Public\Public Documents\STAAD.Pro CONNECT Edition\Samples\ Verification Models\09 Steel Design\India\IS 801-Column with axial and major axis bending.STD is typically installed with the program.

STAAD SPACE
START JOB INFORMATION
ENGINEER DATE 18-Mar-19
END JOB INFORMATION
INPUT WIDTH 79
UNIT METER KN
JOINT COORDINATES
1 0 0 0; 2 0 2 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 COLDFORMED INDIAN
1 TABLE ST 60CS40X4
CONSTANTS
MATERIAL STEEL ALL
SUPPORTS
1 FIXED
LOAD 1 LOADTYPE None  TITLE LOAD CASE 1
JOINT LOAD
2 FX 2 FY -10
PERFORM ANALYSIS PRINT STATICS CHECK
PARAMETER 1
CODE IS801
TRACK 2 ALL
CHECK CODE ALL
PRINT MEMBER PROPERTIES ALL
FINISH

STAAD.Pro Output

                        STAAD.Pro CODE CHECKING - ( IS:801 )   v3.0
                        ***********************
   ALL UNITS ARE IN - METE  KN   (U.N.O.)
|-----------------------------------------------------------------------------|
|  MEMBER:     1  SECTION: 60CS40X4              LEN:    2.000   LOC:   0.000   |
| STATUS: FAIL    RATIO:   3.071               REF: 6.3 LTB     LC:      1    |
|-----------------------------------------------------------------------------|
| DESIGN FORCES:                                                              |
|  Fx:(C)       10.000       Fy:         2.000          Fz:       0.000       |
|  Mx:           0.000       My:         0.000          Mz:       4.000       |
|-----------------------------------------------------------------------------|
| SECTION PROPERTIES:                                            (Unit:   CM) |
|  Ag:         5.82000       Az:     3.20000            Ay:     3.36000       |
|  Cz:         1.75000       Cy:     3.00000            Z0:     4.46000       |
|  Iz:        28.20000       Iy:    12.30000             J:     0.29600       |
|  Sz:         9.40000       Sy:     5.46000                                  |
|  Rz:         2.20122       Ry:     1.45375            Cw:   162.00003       |
|-----------------------------------------------------------------------------|
| MATERIAL INFO:                                                  (Unit: MPa) |
|  Fy:   353.090        Fu:   449.993      E: 203404.356       G:  77968.401  |
|  Fya(compression):   353.090             Fya(bending):   353.090            |
|-----------------------------------------------------------------------------|
| DESIGN PROPERTIES:                                                          |
|  Member Length:     2.000      Lz:     2.000 Ly:     2.000 Lb:     2.000    |
| DESIGN PARAMETERS:                                                          |
|  Kz:   1.000     Ky:   1.000   NSF:  1.000   Cb:   0.000                    |
|-----------------------------------------------------------------------------|
| CRITICAL SLENDERNESS:                                                       |
|  Actual:     137.575      Allowable:     200.000      Ratio:   0.688        |
|-----------------------------------------------------------------------------|
| CHECKS:                             |      Stresses       |                 |
|              | Loc. | Demand |  L/C |  Actual  |  Allow   |Ratio | Ref CL   |
|              |(MET) |(KN-MET)|      | (MPa)    | (MPa)    |      |          |
|--------------|------|--------|------|----------|----------|------|----------|
| Tension      |  -   |   -    |  -   |    -     |  211.854 |  -   | 6.1      | 
| Compression  | 0.000|   10.00|     1|   17.183 |   55.371 | 0.310| 6.6.1.1  | 
| BendZComp    | 0.000|    4.00|     1|  425.562 |  211.854 | 2.009| 6.3      | 
| BendZTens    | 0.000|    4.00|     1|  425.562 |  211.854 | 2.009| 6.3      | 
| BendUnbraced | 0.000|    4.00|     1|  425.562 |  138.583 | 3.071| 6.3 LTB  | 
| BendYComp    |  -   |   -    |  -   |    -     |  211.854 |  -   | 6.3      | 
| BendYTens    |  -   |   -    |  -   |    -     |  211.854 |  -   | 6.3      | 
| Bend Web     | 0.000|    4.00|     1|  368.820 |  211.854 | 1.741| 6.4.2    | 
| Shear Z      |  -   |   -    |  -   |    -     |  141.236 |  -   | 6.4.1    | 
| Shear Y      | 0.000|    2.00|     1|    5.953 |  141.236 | 0.042| 6.4.1    | 
| Axial+Bend   | 0.000|   -    |     1|    -     |     -    | 2.102| 6.7.2(a)2| 
| Bend+Shear   | 0.000|   -    |     1|    -     |     -    | 0.020| 6.4.3    | 
|-----------------------------------------------------------------------------|
| Effective Section Properties:(cm)                                           |
|  Ae:   5.820 SzTop:   9.400 SzBot:   9.400 SyLeft:   7.029 SyRight:   5.467 |
| Intermediate Results:  Cb =  1.000                                          |
|-----------------------------------------------------------------------------|
 NOTE: Torsion has not been considered in the design.
      STAAD SPACE                                              -- PAGE NO.    5