# V. IS 801-Beam with axial and major axis bending

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

## Details

Verifies the calculations for an IS 60CU40x4 (Channel without lips) beam 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
• Fyi = 350.0 MPa = 3,569 kgf/cm2
• Fu = 450 MPa
• G = 795,000 kgf/cm2 = 77,963 MPa
Design forces:
• P = 0.5 kN
• Mz = 0.748 kN·m
• Vy = 1.874 kN

## Verification

Section Dimension Checks

Check for flat width ratio:

w = b - (r + t) = r - (0.4 + 0.6) = 3 mm

w/t = 3 / 0.4 = 7.5 < 60

Hence, OK (ref. cl 5.2.3(a))

hw/t = (d - 2×tf - 2× root radius) / t = (6 - 2×0.4 - 2×0.6)/0.4 = 10 < 500

Hence, OK (ref. cl 5.2.3(b))

Check for web height to thickness ratio:

h/t = (6 - 2×0.4) / 0.4 = 13 < 150

Hence, OK (ref. cl 5.2.4(a))

Check for limiting slenderness

Hence, OK (ref. cl 6.3.3)

Calculation of Allowable Compressive Stress

For calculation of axially loaded member, " Q " is an important factor. The definition and method of calculation for value of Q is provided in Clause no. 6.6.1.1 (a). Channel section without lips is a combination of stiffened & unstiffened elements.

As per clause no 6.1.1.1 of IS801, the increase of steel strength happens due to cold work of forming.

Total corner area, Acorner = 2(45.2 mm2) = 91.2 mm2 = 0.912 cm2

Total area of flanges, Aflange = 2×b×t = 2(0.4)(0.4) = 3.2 cm2

C = Acorner / Aflange = 0.931 / 3.2 = 0.285

Effective depth, he = h = 4 cm

Therefore, effective area, Ae = A = 4.91 cm2

$B e = 3.69 F u F y - 0.819 F u F y 2 - 1.79 = 1.6004$
$m = 0.192 F u F y - 0.068 = 0.1789$

Tensile yield point of corner,

Tensile yield point of flat portions, Fyt = Fy = 350.0 MPa

Average yield point of cold-forming for tension/compression members, Fya(compression) = (C × Fyc) + (1 - C)×Fyc = 381.8 MPa ( = 3,893 kgf/cm2)

Average yield point of cold-forming for flexural members, Fya(bending) = (C × Fyc) + (1 - C)×Fyc = 398.8 MPa

w = 3 cm

w / t = 7.5

As per cl. 6.2 of IS 801, compressive stress:

Fc = 0.6×Fya = 0.6 × 381.8 = 229.1 MPa

$Q s = F c 0.6 F y a = 229.1 229.1 = 1$

h = d - 2(r + t) = 4 cm

$h t = 13 < 1435 F c = 29.69$

Q = Qs × Qa = 1

Allowable compression stress, Fa1 , for members braced against twisting (ref. cl 6.6.1.1)

$C e = 2 π E F y a = 102.5$
$C e Q = 102.5$

Slenderness ratio KL/r = 158.3

Maximum allowable compressive stress (Fa2) for flexural-torsional buckling (ref. cl 6.6.1.2 of IS 801)

$β = 1 - x 0 r x 2 = 0.5071$

Fqy = Fy × Q = 381.8 MPa

Fa2 = 0.522 × σTF0 = 91.09 MPa

The allowable compressive stress, Fa is the minimum of Fa1 and Fa2 :

Fa = 41.06 MPa

Calculation of Allowable Bending Stress

As per clause number 6.1, maximum allowable stress on extreme fiber is:

F = 0.6 × Fya(compression) = 0.6 × 381.8 MPa = 229.1 MPa

As the section is channel without lips, the flanges are unstiffened. So, as per clause 6.2 we need to check allowable compressive stress on the unstiffened element.

Also, the yield strength of steel, Fy > 2,230 kgf/cm2 ( = 227.5 MPa).

Fc = 0.6 × Fya(compression) = 229.1 MPa

For the major axis bending, the allowable compressive stress, Fbc , is the minimum of F and Fc

Fbc = 229.1 MPa

Similarly, for major axis bending, the allowable tensile stress, Fbt=0.6 × Fya(compression) = 229.1 MPa

Calculate the allowable bending stress for laterally unbraced beams:

Allowable bending stress for laterally unbraced beams has been calculated as per clause 6.3 (a).

Unsupported length, L = 2 m (the UNL parameter can be used for this).

Sxc = Zxx = 8.93 cm3

Cb = 1.0 for a member under compression and bending.

Allowable bending stress in the web:

Per cl. 6.4.2, Fbw is the minimum of Fbw1 and 0.6 × Fya(bending) = 239.2 MPa

Fbw = 239.2 MPa

Calculation of Allowable Shear Stress

Per cl. 6.4:

Clear distance between flanges = h = d - 2t = 52 mm

 (cl. 6.4.1(a) )
 (cl. 6.4.1(b) )
$h t = 13 < 4,590 F y = 76.8$

Allowable shear stress, Fv is the minimum of Fv1 or 0.4 × Fy = 140.1 MPa

Fv = 140.1 MPa

Allowable combined bending and shear stress:

As , Fvc = Fvc1 = 577.1 MPa

Actual Stresses

Compression

fa = P/A = 0.5 kN / 4.91 mm2 = 1.018 MPa

Bending

fb = M / Zxx = 0.748 kN·m / 8.93 cm3 = 83.76 MPa

Bending in Web

• Actual bending stress in the web is calculated by interpolation of bending stress diagram:

Shear

Stress Ratio

Compression

fa / Fa = 1.018 / 41.06 = 0.024

Bending

• for bending compression: fb / Fbc = 83.76 / 229.1 = 0.366
• for bending tension: fb / Fbt = 83.76 / 229.1 = 0.366
• for unbraced bending: fb / Fb = 83.76 / 85.18 = 0.983
• for web bending: fbw / Fbw = 72.59 / 239.2 = 0.303

Shear

fv / Fv = 9.01 / 140.1 = 0.064

Combined bending and shear (ref. cl 6.4.3 of IS 801):

$f bw F bw1 2 + f vy F vy1 2 = 75.59 21,215 2 + 9.01 577.1 2 = 0.016$

Interaction ratio for axial and bending

As Q = 1.0, Fa0 can be calculated using cl. 6.6.1.1(b) with L = 0:

 (6.7.2(a) - 2nd eq)

## Results

Table 1. Comparison of results
Compression stress ratio 0.024 0.024 none
Bending Z (compressive) stress ratio 0.366 0.365 negligible
Bending Z (tensile) stress ratio 0.366 0.365 negligible
Bending unbraced 0.983 0.983 none
Bending at web/flange junction stress ratio 0.303 0.303 none
Shear Y stress ratio 0.064 0.064 none
Compression + Bending interaction 0.371 0.371 none
Bending + Shear interaction 0.016 0.016 none

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

STAAD SPACE
START JOB INFORMATION
ENGINEER DATE 27-Mar-19
END JOB INFORMATION
INPUT WIDTH 79
UNIT METER KN
JOINT COORDINATES
1 0 0 0; 2 2 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 COLDFORMED INDIAN
1 TABLE ST 60CU40X4
CONSTANTS
MATERIAL STEEL ALL
SUPPORTS
1 FIXED
2 PINNED
1 UNI GY -1.5
1 CON GX -1
1 1.0 2 1.0
PERFORM ANALYSIS
PRINT MEMBER PROPERTIES ALL
PARAMETER 1
CODE IS801
CWY 1 ALL
FU 450000 ALL
FYLD 350000 ALL
RATIO 1 ALL
TRACK 2 ALL
CHECK CODE ALL
STEEL TAKE OFF ALL
FINISH


                        STAAD.Pro CODE CHECKING - ( IS:801 )   v3.0
***********************
ALL UNITS ARE IN - METE  KN   (U.N.O.)
|-----------------------------------------------------------------------------|
|  MEMBER:     1  SECTION: 60CU40X4              LEN:    2.000   LOC:   0.000   |
| STATUS: PASS    RATIO:   0.983               REF: 6.3 LTB     LC:      4    |
|-----------------------------------------------------------------------------|
| DESIGN FORCES:                                                              |
|  Fx:(C)        0.500       Fy:         1.874          Fz:       0.000       |
|  Mx:           0.000       My:         0.000          Mz:       0.748       |
|-----------------------------------------------------------------------------|
| SECTION PROPERTIES:                                            (Unit:   CM) |
|  Ag:         4.91000       Az:     3.20000            Ay:     2.08000       |
|  Cz:         1.38000       Cy:     3.00000            Z0:     2.62000       |
|  Iz:        26.80000       Iy:     7.84000             J:     0.25500       |
|  Sz:         8.93000       Sy:     2.99000                                  |
|  Rz:         2.33629       Ry:     1.26362            Cw:    45.60001       |
|-----------------------------------------------------------------------------|
| MATERIAL INFO:                                                  (Unit: MPa) |
|  Fy:   350.025        Fu:   450.032      E: 203404.356       G:  77968.401  |
|  Fya(compression):   381.800             Fya(bending):   398.781            |
|-----------------------------------------------------------------------------|
| 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:     158.275      Allowable:     200.000      Ratio:   0.791        |
|-----------------------------------------------------------------------------|
| CHECKS:                             |      Stresses       |                 |
|              | Loc. | Demand |  L/C |  Actual  |  Allow   |Ratio | Ref CL   |
|              |(MET) |(KN-MET)|      | (MPa)    | (MPa)    |      |          |
|--------------|------|--------|------|----------|----------|------|----------|
| Tension      | 1.167|   -0.50|     4|    1.018 |  229.080 | 0.004| 6.1      |
| Compression  | 0.000|    0.50|     4|    1.018 |   41.835 | 0.024| 6.6.1.1  |
| BendZComp    | 0.000|    0.75|     4|   83.688 |  229.080 | 0.365| 6.3      |
| BendZTens    | 0.000|    0.75|     4|   83.688 |  229.080 | 0.365| 6.3      |
| BendUnbraced | 0.000|    0.75|     4|   83.688 |   85.160 | 0.983| 6.3 LTB  |
| BendYComp    |  -   |   -    |  -   |    -     |  239.268 |  -   | 6.3      |
| BendYTens    |  -   |   -    |  -   |    -     |  229.080 |  -   | 6.3      |
| Bend Web     | 0.000|    0.75|     4|   72.529 |  239.268 | 0.303| 6.4.2    |
| Shear Z      |  -   |   -    |  -   |    -     |  140.010 |  -   | 6.4.1    |
| Shear Y      | 0.000|    1.87|     4|    9.009 |  140.010 | 0.064| 6.4.1    |
| Axial+Bend   | 0.000|   -    |     4|    -     |     -    | 0.371| 6.7.2(a)2|
| Bend+Shear   | 0.000|   -    |     4|    -     |     -    | 0.016| 6.4.3    |
|-----------------------------------------------------------------------------|
| Effective Section Properties:(cm)                                           |
|  Ae:   4.910 SzTop:   8.933 SzBot:   8.933 SyLeft:   5.681 SyRight:   2.992 |
| Intermediate Results:  Cb =  1.000                                          |
|-----------------------------------------------------------------------------|
NOTE: Torsion and deflections have not been considered in the design.
STAAD SPACE                                              -- PAGE NO.    5