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
F_yi = 350.0 MPa = 3,569 kgf/cm²
F_u = 450 MPa
G = 795,000 kgf/cm² = 77,963 MPa

Design forces:

P = 0.5 kN
M_z = 0.748 kN·m
V_y = 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))

h_w/t = (d - 2×t_f - 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

k \frac{L_{y}}{r_{y}} = 1.0 \frac{200 m m}{1.263 m m} = 158.3 < 200

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, A_corner = 2(45.2 mm²) = 91.2 mm² = 0.912 cm²

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

C = A_corner / A_flange = 0.931 / 3.2 = 0.285

Effective depth, h_e = h = 4 cm

Therefore, effective area, A_e = A = 4.91 cm²

B_{e} = 3.69 \frac{F_{u}}{F_{y}} - 0.819 {(\frac{F_{u}}{F_{y}})}^{2} - 1.79 = 1.6004

m = 0.192 \frac{F_{u}}{F_{y}} - 0.068 = 0.1789

Tensile yield point of corner, $F_{y c} = \frac{B_{e} \times F_{y}}{{(\frac{r}{t})}^{m}} = 521.0 M P a$

Tensile yield point of flat portions, F_yt = F_y = 350.0 MPa

Average yield point of cold-forming for tension/compression members, F_{ya(compression)} = (C × F_yc) + (1 - C)×F_yc = 381.8 MPa ( = 3,893 kgf/cm²)

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

w = 3 cm

w / t = 7.5

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

F_c = 0.6×F_ya = 0.6 × 381.8 = 229.1 MPa

Q_{s} = \frac{F_{c}}{0.6 F_{y a}} = \frac{229.1}{229.1} = 1

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

\frac{h}{t} = 13 < \frac{1435}{\sqrt{F c}} = 29.69

Q_{a} = \frac{A_{e}}{A} = 1

Q = Q_s × Q_a = 1

Allowable compression stress, F_a1 , for members braced against twisting (ref. cl 6.6.1.1)

C_{e} = \sqrt{\frac{2 π E}{F_{y a}}} = 102.5

\frac{C_{e}}{\sqrt{Q}} = 102.5

Slenderness ratio KL/r = 158.3

F_{a 1} = \frac{12}{23} Q \times F_{y a} - \frac{3 (Q \times F_{y a})}{23 π^{2} E} {(\frac{K L}{r})}^{2} = 41.06 M P a

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

r_{0} = \sqrt{{(r_{x}}^{2}) + {(r_{y}}^{2}) + {(x_{0}}^{2})} = 3.732 c m

σ_{x} = π^{2} \times \frac{E}{{(K_{x} \frac{L_{x}}{r_{x}})}^{2}} = 274.8 M P a

β = 1 - {(\frac{x_{0}}{r_{x}})}^{2} = 0.5071

σ_{t} = \frac{1}{A \times {r_{o}}^{2}} \times \frac{(G \times J) + {(π}^{2} \times E \times C_{w})}{{(K_{x} \times L_{x})}^{2}} = 324.2 M P a

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}}] = 174.5 MPa

F_qy = F_y × Q = 381.8 MPa

F_a2 = 0.522 × σ_TF0 = 91.09 MPa

The allowable compressive stress, F_a is the minimum of F_a1 and F_a2 :

F_a = 41.06 MPa

Calculation of Allowable Bending Stress

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

F = 0.6 × F_{ya(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.

\frac{w}{t} = 7.5 < \frac{530}{\sqrt{F_{y a}}} = \frac{530}{\sqrt{3,893 k g f / {c m}^{2}}} = 8.494

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

F_c = 0.6 × F_{ya(compression)} = 229.1 MPa

For the major axis bending, the allowable compressive stress, F_bc , is the minimum of F and F_c

F_bc = 229.1 MPa

Similarly, for major axis bending, the allowable tensile stress, F_bt=0.6 × F_{ya(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).

S_xc = Z_xx = 8.93 cm³

C_b = 1.0 for a member under compression and bending.

\frac{L^{2} \times S_{x c}}{d \times I_{y c}} = 14,140 > 1.8 π^{2} E \frac{C_{b}}{F_{y a (c o m p r e s s i o n)}} = 9,464

F_{b} = 0.6 π^{2} E \times C_{b} \frac{d \times I_{y c}}{L^{2} \times S_{x c}} = 85.18 M P a

Allowable bending stress in the web:

F_{b w 1} = \frac{36,560,000}{{(\frac{h}{t})}^{2}} = 216,330 \frac{k g f}{{c m}^{2}} = 21,215 M P a

Per cl. 6.4.2, F_bw is the minimum of F_bw1 and 0.6 × F_ya(bending) = 239.2 MPa

F_bw = 239.2 MPa

Calculation of Allowable Shear Stress

Per cl. 6.4:

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

F_{v 1} = \frac{1275 \times \sqrt{F_{y}}}{\frac{h}{t}} = 577.1042 M P a

(cl. 6.4.1(a) )

F_{v 2} = \frac{585000}{{(\frac{h}{t})}^{2}} = 3,395 M P a

(cl. 6.4.1(b) )

\frac{h}{t} = 13 < \frac{4,590}{\sqrt{F_{y}}} = 76.8

Allowable shear stress, F_v is the minimum of F_v1 or 0.4 × F_y = 140.1 MPa

F_v = 140.1 MPa

Allowable combined bending and shear stress:

As $\frac{h}{t} < \frac{4590}{\sqrt{F_{y}}}$ , F_vc = F_vc1 = 577.1 MPa

Actual Stresses

Compression

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

Bending

f_b = M / Z_xx = 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:

{f_{b w} = f}_{b} \times (1 - \frac{t}{0.5 \times d}) = 72.59 M P a

Shear

f_{v} = \frac{V_{y}}{N_{w} t (d - 2 t)} = \frac{1.874 {(10)}^{3}}{(1) 4 (60 - 2 \times 4)} = 9.01 M P a

Stress Ratio

Compression

f_a / F_a = 1.018 / 41.06 = 0.024

Bending

for bending compression: f_b / F_bc = 83.76 / 229.1 = 0.366
for bending tension: f_b / F_bt = 83.76 / 229.1 = 0.366
for unbraced bending: f_b / F_b = 83.76 / 85.18 = 0.983
for web bending: f_bw / F_bw = 72.59 / 239.2 = 0.303

Shear

f_v / F_v = 9.01 / 140.1 = 0.064

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

\sqrt{{(\frac{f_{bw}}{F_{bw1}})}^{2} + {(\frac{f_{vy}}{F_{vy1}})}^{2}} = \sqrt{{(\frac{75.59}{21,215})}^{2} + {(\frac{9.01}{577.1})}^{2}} = 0.016

Interaction ratio for axial and bending

As Q = 1.0, F_a0 can be calculated using cl. 6.6.1.1(b) with L = 0:

F_{a 0} = \frac{12}{23} Q \times F_{y a} - \frac{3 {(Q \times F_{y a})}^{2}}{23 π^{2} E} {(\frac{K * L}{r})}^{2} = 182.6 M P a

\frac{f_{a}}{F_{a 0}} + \frac{f_{b 1}}{F_{b 1}} = \frac{1.018}{182.6} + \frac{83.76}{229.1} = 0.371

(6.7.2(a) - 2nd eq)

Results

Table 1. Comparison of results
Result Type	Reference	STAAD.Pro	Difference
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

STAAD.Pro Input File

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
LOAD 1 LOADTYPE None  TITLE LOAD CASE 1
MEMBER LOAD
1 UNI GY -1.5
LOAD 2 LOADTYPE None  TITLE LOAD CASE 3
MEMBER LOAD
1 CON GX -1
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
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 Output

                        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 has not been considered in the design.
      STAAD SPACE                                              -- PAGE NO.    5