# V. IS 801-Zee with lips having axial compression and bending

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

## Details

The section used is a IS 125ZS45x2.55 section. The beam is a 2 m span subject to axial compression and major axis bending moments. The span is a propped cantilever (one end fixed, the other pinned).

Material properties:
• E = 203,400 MPa = 2,074,000 kgf/cm2
• Fyi = 350 MPa = 3,569 kgf/cm2
• G = 77,968 MPa = 795,000 kgf/cm2
Design forces:
• P = 25 kN
• Mz = 2.71 kN·m
• Vy = 6.85 kN
Section properties:
• Depth of section, d = 125 mm
• Width of section, b = 45 mm
• Thickness, t = 2.55 mm
• Length of lips, c = 20 mm
• Fillet radius, r = 3.825 mm
• Area, A = 5.94 cm2
• Moment of inertia about the major axis, Izz = 135 cm3
• Moment of inertia about the minor axis, Iyy = 27.5 cm3
• Section modulus about the major axis, Zxx = 21.6 cm3
• Section modulus about the minor axis, Zyy = 6.3 cm3
• Torsion constant, J = 0.125 cm4
• Warping constant, Cw = 834 cm6
• Number of corners, Nc = 4
• Area of corner, Ac = 74 mm2

## Verification

Section Dimension Checks

Check flat width ratio per Cl. 5.2.3:

w = b - 2 × (r + t) = 4.5 - 2 × (0.3825 + 0.255) = 3.225 cm

$w t = 3.225 0.255 = 12.65 < 60$

Hence, OK.

Check web height to thickness ratio per Cl. 5.2.4:

h = D - 2(t + r) = 125 - 2×(2.55 + 3.825) = 112.3 mm

$h t = 11.23 0.255 = 44.02 < 150$

Hence, OK.

Check slenderness ratio limits per Cl. 6.3.3:

$K L r x = 1.0 ( 200 ) 4.77 = 41.95$

$K L r y = 1.0 ( 200 ) 2.15 = 92.95 < 200$

Hence, OK.

Compressive Stress

Actual stress in compression

Calculate the factor, Q, per Cl. 6.6.1.1(a):

$w t = 12.65 < 1,435 F c = 1,435 2,141 = 31.0$

Therefore, beff = w = 3.225 cm and Alost,f = (w - beff) × t = 0 cm2

Compression stress:

$Q s = F c 0.6 × F y = 1.0$
$h t = 44.02 > 1,435 F c = 31.0$

Calculate the effective depth of the section by re-arranging the flange ratio:

 (5.2.1.1)

and Alost,d = (h - heff) × t = (11.23 - 9.015) × 0.255 = 0.565 cm2

Therefore the effective area, Aeff = A - Alost,f - Alost,d = 5.94 - 0 - 0.565 = 5.38 cm2

$Q a = A eff A = 5.38 5.94 = 0.905$
$Q = Q s × Q a = 0.905$

Allowable compression stress:

$C e = 2 π 2 E F y = 2 π 2 × 203,404 350 = 107.1$
$C e Q = 106.6 1.0 = 107.1 < K L r = 1.0 ( 200 ) 2.15 = 92.95$

Allowable compression stress for members braced against twisting (Ref Cl. 6.6.1.1):

Maximum allowable compressive stress for flexural torsional buckling (Ref Cl.6.6.1.2). The section is symmetric, so the distance between the geometric and shear center, x0 = 0.

$β = 1 - ( x 0 / r 0 ) 2 = 1.0$
$σ TF0 = 1 2 β [ ( σ ex + σ t ) - ( σ ex + σ t ) 2 - 4 β σ ex σ t ]$

Therefore, the allowable compressive stress:

Stress ratio in compression: 42.1 / 108.9 = 0.387

Bending Stress

Actual bending stress:

Per Cl. 6.8, the maximum allowable stress on the extreme fiber:

Per Cl. 6.3(b) , the allowable bending stress for laterally unbraced beams:

$C b = 1.0$
$L 2 S sc d × I yc = ( 2,000 ) 2 ( 21.6 × 10 3 ) 125 ( 15.14 × 10 4 ) = 4,565$
$0.18 π 2 E × C b F y = 0.18 π 2 ( 203,400 ) 1.0 350 = 1,032$
$0.9 π 2 E × C b F y = 0.9 π 2 ( 203,400 ) 1.0 350 = 5,162$

Since $0.18 π 2 E × C b F y < L 2 S sc d × I yc < 0.9 π 2 E × C b F y$,

Stress ratio in bending: 125.5 / 130.1 = 0.963

Shear Stress

Shear area:

Clear distance between flanges, h = d - 2t = 12.5 - 2× 0.255 = 11.99 cm

$h t = 11.99 0.255 = 47.02 < 4,590 F y = 99.20$

Actual shear stress:

Stress ratio in shear: 17.36 / 140 = 0.124

Bending in Web Stress

The actual bending stress in the web is calculated by interpolating from the bending stress diagram:

The allowable bending stress in the web:
 (6.4.2)
Stress ratio in web bending: 120.4 / 211.9 = 0.568

Combined Bending and Shear Stress in Web

Per Cl. 6.4.3, use and

Stress ratio in combined web bending and shear: $( f bw F bw ) 2 + ( f v F vc ) 2 = ( 120.3 1,622 ) 2 + ( 17.35 159.6 ) 2 = 0.132$

## Results

Table 1. Comparison of results
Allowable compression stress (MPa) 108.9 109.052 negligible
Actual compression stress (MPa) 42.1 42.090 negligible
Compression stress ratio 0.387 0.386 negligible
Allowable bending stress (MPa) 130.1 130.131 negligible
Actual bending stress (MPa) 125.5 125.339 negligible
Bending stress ratio 0.963 0.963 none
Allowable shear stress (MPa) 140 140.01 negligible
Actual shear stress (MPa) 17.36 17.353 negligible
Shear stress ratio 0.124 0.124 none
Web bending stress ratio 0.568 0.572 negligible
Combined Bending and Shear Stress Ratio 0.132 0.132 none

## STAAD.Pro Input File

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

The following design parameters are used:
• The effect of cold work of forming strengthening is not considered by specifying the parameter CWY 0
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 125ZS45X2.55
CONSTANTS
MATERIAL STEEL ALL
SUPPORTS
1 FIXED
2 PINNED
1 UNI GY -5.5
1 CON GX -25 0
LOAD COMB 4 COMBINATION LOAD CASE 4
1 1.0 2 1.0
PERFORM ANALYSIS
PRINT MEMBER PROPERTIES ALL
PARAMETER 1
CODE IS801
FU 450000 ALL
FYLD 350000 ALL
RATIO 1 ALL
TRACK 2 ALL
CWY 0 ALL
CHECK CODE ALL
FINISH


                        STAAD.Pro CODE CHECKING - ( IS:801 )   v3.0
***********************
ALL UNITS ARE IN - METE  KN   (U.N.O.)
|-----------------------------------------------------------------------------|
|  MEMBER:     1  SECTION: 125ZS45X2.55          LEN:    2.000   LOC:   0.000   |
| STATUS: PASS    RATIO:   0.963               REF: 6.3 LTB     LC:      4    |
|-----------------------------------------------------------------------------|
| DESIGN FORCES:                                                              |
|  Fx:(C)       25.000       Fy:         6.854          Fz:       0.000       |
|  Mx:           0.000       My:         0.000          Mz:       2.707       |
|-----------------------------------------------------------------------------|
| SECTION PROPERTIES:                                            (Unit:   CM) |
|  Ag:         5.94000       Az:     2.29500            Ay:     3.94740       |
|  Cz:         4.37250       Cy:     6.25000            Z0:     0.00000       |
|  Iz:       135.00002       Iy:    27.50000             J:     0.12600       |
|  Sz:        21.60000       Sy:     6.30000                                  |
|  Rz:         4.76731       Ry:     2.15166            Cw:   834.00017       |
|-----------------------------------------------------------------------------|
| MATERIAL INFO:                                                  (Unit: MPa) |
|  Fy:   350.025        Fu:   450.032      E: 203404.356       G:  77968.401  |
|  Fya(compression):   350.025             Fya(bending):   350.025            |
|-----------------------------------------------------------------------------|
| 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:      92.952      Allowable:     200.000      Ratio:   0.465        |
|-----------------------------------------------------------------------------|
| CHECKS:                             |      Stresses       |                 |
|              | Loc. | Demand |  L/C |  Actual  |  Allow   |Ratio | Ref CL   |
|              |(MET) |(KN-MET)|      | (MPa)    | (MPa)    |      |          |
|--------------|------|--------|------|----------|----------|------|----------|
| Tension      | 2.000|   -0.00|     4|    0.000 |  210.015 | 0.000| 6.1      |
| Compression  | 0.000|   25.00|     4|   42.090 |  109.052 | 0.386| 6.6.1.1  |
| BendZComp    | 0.000|    2.71|     4|  125.339 |  210.015 | 0.597| 6.3      |
| BendZTens    | 0.000|    2.71|     4|  125.339 |  210.015 | 0.597| 6.3      |
| BendUnbraced | 0.000|    2.71|     4|  125.339 |  130.131 | 0.963| 6.3 LTB  |
| BendYComp    |  -   |   -    |  -   |    -     |  210.015 |  -   | 6.3      |
| BendYTens    |  -   |   -    |  -   |    -     |  210.015 |  -   | 6.3      |
| Bend Web     | 0.000|    2.71|     4|  120.225 |  210.015 | 0.572| 6.4.2    |
| Shear Z      |  -   |   -    |  -   |    -     |  140.010 |  -   | 6.4.1    |
| Shear Y      | 0.000|    6.85|     4|   17.363 |  140.010 | 0.124| 6.4.1    |
| Axial+Bend   | 0.000|   -    |     4|    -     |     -    | 0.851| 6.7.2(a)2|
| Bend+Shear   | 0.000|   -    |     4|    -     |     -    | 0.132| 6.4.3    |
|-----------------------------------------------------------------------------|
| Effective Section Properties:(cm)                                           |
|  Ae:   5.377 SzTop:  21.600 SzBot:  21.600 SyLeft:   6.275 SyRight:   6.275 |
| Intermediate Results:  Cb =  1.000                                          |
|-----------------------------------------------------------------------------|
NOTE: Torsion and deflections have not been considered in the design.