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
F_yi = 350 MPa = 3,569 kgf/cm²
G = 77,968 MPa = 795,000 kgf/cm²

Design forces:

P = 25 kN
M_z = 2.71 kN·m
V_y = 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, I_zz = 135 cm³
Moment of inertia about the minor axis, I_yy = 27.5 cm³
Section modulus about the major axis, Z_xx = 21.6 cm³
Section modulus about the minor axis, Z_yy = 6.3 cm³
Torsion constant, J = 0.125 cm⁴
Warping constant, C_w = 834 cm⁶
Number of corners, N_c = 4
Area of corner, A_c = 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

\frac{w}{t} = \frac{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

\frac{h}{t} = \frac{11.23}{0.255} = 44.02 < 150

Hence, OK.

Check slenderness ratio limits per Cl. 6.3.3:

Radius of gyration, about major axis $r_{x} = \sqrt{\frac{I_{zz}}{A}} = \sqrt{\frac{135}{5.94}} = 4.77 cm$

\frac{K L}{r_{x}} = \frac{1.0 (200)}{4.77} = 41.95

Radius of gyration, about minor axis $r_{y} = \sqrt{\frac{I_{yy}}{A}} = \sqrt{\frac{27.5}{5.94}} = 2.15 cm$

\frac{K L}{r_{y}} = \frac{1.0 (200)}{2.15} = 92.95 < 200

Hence, OK.

Compressive Stress

Actual stress in compression

f_{c} = \frac{P}{A} = \frac{25 \times 10^{3}}{5.94 \times 10^{2}} = 42.1 MPa = 429 kgf / {cm}^{2}

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

F_{c} = 0.6 \times F_{y} = 0.6 \times 350 = 210 MPa = 2,141 kgf / {cm}^{2}

\frac{w}{t} = 12.65 < \frac{1,435}{\sqrt{F_{c}}} = \frac{1,435}{\sqrt{2,141}} = 31.0

Therefore, b_eff = w = 3.225 cm and A_lost,f = (w - b_eff) × t = 0 cm²

Compression stress:

Q_{s} = \frac{F_{c}}{0.6 \times F_{y}} = 1.0

\frac{h}{t} = 44.02 > \frac{1,435}{\sqrt{F_{c}}} = 31.0

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

h_{eff} = t \times \frac{2,120}{\sqrt{f}} [1 - \frac{465}{(h / t) \sqrt{f}}] = 0.255 \times \frac{2,120}{\sqrt{2,141}} [1 - \frac{465}{(44.02) \sqrt{2,141}}] = 9.015 cm

(5.2.1.1)

and A_lost,d = (h - h_eff) × t = (11.23 - 9.015) × 0.255 = 0.565 cm²

Therefore the effective area, A_eff = A - A_lost,f - A_lost,d = 5.94 - 0 - 0.565 = 5.38 cm²

Q_{a} = \frac{A_{eff}}{A} = \frac{5.38}{5.94} = 0.905

Q = Q_{s} \times Q_{a} = 0.905

Allowable compression stress:

C_{e} = \sqrt{\frac{2 π^{2} E}{F_{y}}} = \sqrt{\frac{2 π^{2} \times 203,404}{350}} = 107.1

\frac{C_{e}}{\sqrt{Q}} = \frac{106.6}{\sqrt{1.0}} = 107.1 < \frac{K L}{r} = \frac{1.0 (200)}{2.15} = 92.95

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

F_{a1} = \frac{12}{23} Q F_{y} - \frac{3 {(Q F_{y})}^{2}}{23 π^{2} E} {(\frac{K L}{r})}^{2} = \frac{12}{23} \times 0.905 \times 3,569 - \frac{3 {(0.905 \times 3,569)}^{2}}{23 π^{2} (2,074,000)} {(92.95)}^{2} = 1,111 kgf / {cm}^{2} = 108.9 MPa

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.

r_{0} = \sqrt{{r_{x}}^{2} + {r_{x}}^{2} + {x_{0}}^{2}} = \sqrt{{(4.77)}^{2} + {(2.15)}^{2} + {(0)}^{2}} = 5.23 cm

σ_{x} = \frac{π^{2} E}{{(K L / r)}_{x}^{2}} = \frac{π^{2} (203,400)}{{(41.95)}^{2}} = 274.8 MPa

β = 1 - {(x_{0} / r_{0})}^{2} = 1.0

σ_{t} = \frac{1}{A \times {r_{0}}^{2}} \times \frac{G J + π^{2} E C_{w}}{{(K L)}^{2}} = \frac{1}{(594) \times {(52.3)}^{2}} \times \frac{77,968 (1,260) + π^{2} (203,400) (834 \times 10^{6})}{{(2,000)}^{2}} = 318.2 MPa

σ_{TF0} = \frac{1}{2 β} [(σ_{ex} + σ_{t}) - \sqrt{{(σ_{ex} + σ_{t})}^{2} - 4 β σ_{ex} σ_{t}}]

= \frac{1}{2} [(274.8 + 318.2) - \sqrt{{(274.8 + 318.2)}^{2} - 4 (274.8 \times 318.2)} = 274.8 MPa

> 0.5 \times Q \times F_{y} = 0.5 \times 0.905 \times 350 = 158.4 MPa

F_{a2} = 0.522 \times Q \times F_{y} - \frac{{(Q \times F_{y})}^{2}}{7.67 \times σ_{TF0}} = 165.3 - \frac{{(0.905 \times 350)}^{2}}{7.67 \times 274.8} = 118.3 MPa

Therefore, the allowable compressive stress:

F_{c} = min | \begin{matrix} F_{a1} = 108.9 MPa \\ F_{a2} = 118.3 MPa \end{matrix}

Stress ratio in compression: 42.1 / 108.9 = 0.387

Bending Stress

Actual bending stress:

f_{b} = \frac{M}{Z_{xx}} = \frac{2.71 \times 10^{3}}{21.60} = 125.5 MPa

Per Cl. 6.8, the maximum allowable stress on the extreme fiber: $F_{c} = 0.6 \times F_{y} = 210 MPa$

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

C_{b} = 1.0

\frac{L^{2} S_{sc}}{d \times I_{yc}} = \frac{{(2,000)}^{2} (21.6 \times 10^{3})}{125 (15.14 \times 10^{4})} = 4,565

\frac{0.18 π^{2} E \times C_{b}}{F_{y}} = \frac{0.18 π^{2} (203,400) 1.0}{350} = 1,032

\frac{0.9 π^{2} E \times C_{b}}{F_{y}} = \frac{0.9 π^{2} (203,400) 1.0}{350} = 5,162

Since $\frac{0.18 π^{2} E \times C_{b}}{F_{y}} < \frac{L^{2} S_{sc}}{d \times I_{yc}} < \frac{0.9 π^{2} E \times C_{b}}{F_{y}}$ ,

F_{b} = \frac{2}{3} π^{2} E \times C_{b} \frac{d \times I_{yc}}{L^{2} S_{xc}} = 130.1 MPa

Stress ratio in bending: 125.5 / 130.1 = 0.963

Shear Stress

Shear area:

A_{z} = 2 B t = 2.295 {cm}^{2}

A_{y} = t [(D - 2 t) + 2 (c - t)] = 3.947 {cm}^{2}

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

\frac{h}{t} = \frac{11.99}{0.255} = 47.02 < \frac{4,590}{\sqrt{F_{y}}} = 99.20

F_{v} = min | \begin{matrix} \frac{1,275 \sqrt{F_{y}}}{h / t} = \frac{1,275 \sqrt{3,600}}{47.02} = 1,627 kgf / {cm}^{2} = 159.6 MPa \\ 0.4 F_{y} = 0.4 x 350 = 140 MPa \end{matrix}

Actual shear stress:

f_{v} = \frac{V_{y}}{A_{z}} = \frac{6.85 (10)}{3.947} = 17.36 MPa

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:

f_{bw} = f_{b} (1 - \frac{t}{0.5 d}) = 125.5 (1 - \frac{0.255}{0.5 \times 12.5}) = 120.4 MPa

The allowable bending stress in the web:

F_{bw} = \frac{36,560,000}{{(\frac{h}{t})}^{2}} = 16,536 kgf / {cm}^{2} = 1622 MPa > 0.6 F_{y} = 211.9 MPa

(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 $F_{bw} = \frac{36,560,000}{{(\frac{h}{t})}^{2}} = 16,536 kgf / {cm}^{2} = 1622 MPa$ and $F_{v} = 159.6 MPa$

Stress ratio in combined web bending and shear:

\sqrt{{(\frac{f_{bw}}{F_{bw}})}^{2} + {(\frac{f_{v}}{F_{vc}})}^{2}} = \sqrt{{(\frac{120.3}{1,622})}^{2} + {(\frac{17.35}{159.6})}^{2}} = 0.132

Results

Table 1. Comparison of results
Result Type	Reference	STAAD.Pro	Difference
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
LOAD 1 LOADTYPE None  TITLE LOAD CASE 1
MEMBER LOAD
1 UNI GY -5.5
LOAD 2 LOADTYPE None  TITLE LOAD CASE 3
MEMBER LOAD
1 CON GX -25 0
LOAD COMB 4 COMBINATION LOAD CASE 4
1 1.0 2 1.0 
PERFORM ANALYSIS
PRINT MEMBER PROPERTIES ALL
LOAD LIST 4
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 Output

                        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 has not been considered in the design.