V. IS 800 2007 LSD - Tube - Tension and Bending

Verify the capacity of a tube section subject to axial tension and combined bending per the IS 800:2007 code.

Reference

Bureau of Indian Standards. General Construction in Steel - Code of Practice. IS 800:2007. New Delhi. 2007.
Bureau of Indian Standards. Hollow steel sections for structural use - Specification. IS 4923:1997. New Delhi. 1997. p.5

Details

A 3m cantilever member is subject to a 10 kN tensile force applied at the free end. The member is also subject to 1 kN/m uniformly distributed loads along the length in both the local y and z directions. The section used is a 125.0x125.0x6.0 square hollow section. The material is steel with a yield strength of 250 MPa and a rupture strength of 420 MPa. Consider the member as laterally supported along the length.

Validation

Section Properties

From Ref. 2:

Depth or width, D = 125.5 mm
Thickness, t = 6.0 mm
Area of section, A = 27.63 cm²
Moment of inertia, I = 640.89 cm⁴
Radius of gyration, r = 4.82 cm
Elastic modulus, Z_e = 102.54 cm³
Plastic modulus, Z_p = 121.87 cm³

Design Forces

The design tensile force (uniform along length):

T = 10 kN

The design shear force (occurs at support):

T = w×l = 1 (3) = 3 kN

The design bending moment (occurs at support):

M_{y} = M_{z} \frac{w l^{2}}{2} = \frac{1 {(3)}^{2}}{2} = 4.5 kN·m

Section Classification

ε = \sqrt{\frac{250}{f_{y}}} = 1.0

Per Figure 2 of IS 800:2007:

d = D - 2×t = 125.5 - 2×6 = 113.5 mm

Per Table 2 of Ref. 1:

\frac{d}{t} = \frac{113.5}{6.0} = 18.9 < 84 ε = 84

Therefore, the section is classified as "plastic".

From Table 10 of Ref. 1, the buckling class of a hot-rolled hollow section is "a".

Slenderness Ratio

Assume values of k_y = 1 and k_z = 1. Further assume similar bracing in both lateral directions (i.e., L_y = L_z = L = 3.0 m.

The slenderness ratio of the section is:

\frac{k L}{r} = \frac{1 (300)}{4.82} = 62.2

Tension Capacity

T_{dg} = A_{g} f_{y} / ɣ_{m0} = 27.63 (250) / 1.1 = 628 kN

(Cl. 6.2)

Assume a welded connection as the net section has not been specified: ɑ = 0.8.

T_{dn} = A_{n} f_{u} / ɣ_{m1} = 0.8 (27.63) (420) / 1.25 = 743 kN

(Cl. 6.3)

Critical stress ratio: T / T_dg = 10 / 628 = 0.016

Compression Capacity

(Checked for completeness even though the member is loaded in tension)

Non-dimensional slenderness ratio for section:

λ = \sqrt{\frac{f_{y} {(k L / r)}^{2}}{π^{2} E}} = \sqrt{\frac{250 {(62.2)}^{2}}{π^{2} 205,000}} = 0.692

For buckling class "a", the imperfection factor, ɑ = 0.21 (Table 7 of Ref. 1).

ϕ = 0.5 [1 + ɑ (λ - 0.2) + λ^{2}] = 0.5 [1 + 0.21 (0.692 - 0.2) + {(0.692)}^{2}] = 0.791

The stress reduction factor is:

χ = \frac{1}{ϕ + \sqrt{ϕ^{2} - λ^{2}}} = \frac{1}{0.791 + \sqrt{{(0.791)}^{2} - {(0.692)}^{2}}} = 0.852

f_{cd} = χ f_{y} / ɣ_{m0} = 0.852 (250) / 1.1 = 193.5 MPa

(Cl. 7.1.2.1)

Axial capacity: P_d = A×f_cd = 27.63 × 193.5 = 535 kN

Shear Capacity

The shear area, A_v, for a rectangular hollow section is $\frac{A h}{b + h} = \frac{27.63 (125.5)}{125.5 + 125.5} = 13.81 {cm}^{2}$ .

Since $\frac{d}{t} = 18.9 < 67 ε = 67$ , shear buckling check per Cl. 8.4.2.1 does not apply.

V_{p} = \frac{A_{v} f_{y}}{\sqrt{3}} = \frac{13.82 (250)}{\sqrt{3}} = 199 kN

V_{d} = V_{p} / ɣ_{m0} = 199 / 1.1 = 181 kN

Critical stress ratio: V / V_d = 3 / 181 = 0.017

Bending Capacity

For a "plastic" section, β_b = 1.

The section is subject to low shear (V = 3 kN < 0.6V_d = 0.6(181) = 109 kN).

M_{d} = β_{b} Z_{p} f_{y} / ɣ_{m0} = 1.0 (121.87) (250) / 1.1 = 27.7 kN·m

(Cl. 8.2.1.2)

Check against excessive deformation under service loads for a cantilever member:

M_{d} \leq 1.5 Z_{e} f_{y} / ɣ_{m0} = 1.5 (102.54) (250) / 1.1 = 35.0 kN·m

Does not govern.

Critical stress ratio: M / M_d = 4.5 / 27.7 = 0.162

Combined Tension and Bending Capacity

{(\frac{M_{y}}{M_{ndy}})}^{ɑ_{1}} + {(\frac{M_{z}}{M_{ndz}})}^{ɑ_{2}} \leq 1.0

(Cl. 9.3.1.1)

where

n	=	\|T\| / T_dg = 10 / 628 = 0.016
a_f, a_w	=	$(A - 2 b t_{f}) / A = [27.63 - 2 (12.55) (0.6) / 27.63] = 0.455$
M_ndy, M_ndz	=	${}_{min.}{\begin{cases} M_{d} = 27.7 kN ← \\ M_{d} (1 - n) / (1 - 0.5 a_{f}) = 27.7 (1 - 0.016) / [1 - 0.5 (0.455)] = 27.7 (1.274) = 34.5 kN \end{cases}$

where

ɑ₁, ɑ₂

1.66 / (1 - 1.13 n^{2}) = 1.66 < 6

(Table 17 in Ref. 1)

{(\frac{4.5}{27.7})}^{1.66} + {(\frac{4.5}{27.7})}^{1.66} = 0.098

Check the reduced effective moment under tension:

M_{eff} = M - ψ T Z_{e} / A

(Cl. 9.3.2.1)

where

1.0 (assume that moment and tension do not vary independently)

M_{eff} = 4.5 - 1.0 (10) (102.54) {(10)}^{-3} / 27.63 = 4.13 k·Nm

Critical stress ratio: M_eff / M_d = 4.13 / 27.7 = 0.149

Results

Result Type	Reference	STAAD.Pro	Difference
Slenderness ratio	62.2	62.25	negligible
Tension capacity, yielding of gross section (kN)	628	627.273	negligible
Tension capacity, rupture of net section (kN)	743	741.888	negligible
Critical stress ratio for tension capacity	0.016	0.016	none
Axial compression capacity (kN)	535	534.107	negligible
Shear capacity (kN)	181	181.078	negligible
Critical stress ratio for shear	0.017	0.017	none
Bending capacity (kN·m)	27.7	27.727	negligible
Critical stress ratio for bending	0.162	0.162	none
Ratio as per clause 9.3.1.1	0.098	0.098	none
Ratio as per clause 9.3.2.1	0.149	0.149	none

STAAD.Pro Input

The file C:\Users\Public\Public Documents\STAAD.Pro CONNECT Edition\Samples\ Verification Models\09 Steel Design\India\IS 800 2007 LSD - Tube - Tension and Bending.STD is typically installed with the program.

The following design parameters are used:

The value of CAN 1 is used to indicate a cantilever member.

The value of LAT 1 is used to indicate that member is laterally supported.

STAAD SPACE
START JOB INFORMATION
ENGINEER DATE 6-Sep-22
ENGINEER NAME TK
END JOB INFORMATION
*****************************************************************************
*This problem has been created to validate the program calculated design
*results of laterally supported tube section subjected to axial tension and
*biaxial bending, designed using IS 800 2007 (LSD)
*****************************************************************************
INPUT WIDTH 79
SET SHEAR
UNIT METER KN
JOINT COORDINATES
1 0 0 0; 2 3 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 INDIAN
1 TABLE ST TUB1251256
CONSTANTS
MATERIAL STEEL ALL
SUPPORTS
1 FIXED
LOAD 1 LOADTYPE Dead  TITLE LOAD CASE 1
JOINT LOAD
2 FX 10
MEMBER LOAD
1 UNI GY -1
1 UNI GZ -1
PERFORM ANALYSIS
PRINT ANALYSIS RESULTS
PARAMETER 1
CODE IS800 LSD
CAN 1 ALL
LAT 1 ALL
TRACK 2 ALL
CHECK CODE ALL
FINISH

STAAD.Pro Output

                       STAAD.PRO CODE CHECKING - IS-800-2007-LSD (V2.2)
                     ****************************************************
 |----------------------------------------------------------------------------------|
 |   Member Number:      1                                                            |
 |  Member Section:  ST   TUB1251256            (INDIAN SECTIONS)                   |
 |  Status: PASS  Ratio:   0.162  Critical Load Case:     1  Location:       0.00   |
 |  Critical Condition: Sec. 8.2.1.2                                                |
 |  Critical Design Forces:  (Unit: KN    METE)                                     |
 |      FX:     -10.000E+00 T     FY:       3.000E+00      FZ:       3.000E+00      |
 |      MX:       0.000E+00       MY:      -4.500E+00      MZ:       4.500E+00      |
 |----------------------------------------------------------------------------------|
 |  Section Properties:    (Unit:  CM  )                                            |
 |  AXX:      27.600E+00          IZZ:     641.000E+00          RZZ:       4.819E+00|
 |  AYY:      13.800E+00          IYY:     641.000E+00          RYY:       4.819E+00|
 |  AZZ:      13.800E+00          IXX:       1.038E+03           CW:       0.000E+00|
 |  ZEZ:     102.560E+00          ZPZ:     122.000E+00                              |
 |  ZEY:     102.560E+00          ZPY:     122.000E+00                              |
 |----------------------------------------------------------------------------------|
 |  Slenderness Check:    (Unit:  KN    METE)                                       |
 |  Actual Length:        3.000E+00                                                 |
 |  Parameters:     LZ:        3.000E+00  LY:        3.000E+00                      |
 |                  KZ:            1.000  KY:            1.000                      |
 |  Actual Ratio:  62.25 Allowable Ratio: 400.00  LOAD:     1                       |
 |----------------------------------------------------------------------------------|
 |  Section Class:      Plastic; Flange Class:      Plastic; Web Class:      Plastic|
 |----------------------------------------------------------------------------------|
      STAAD SPACE                                              -- PAGE NO.    7
                       STAAD.PRO CODE CHECKING - IS-800-2007-LSD (V2.2)
                     ****************************************************
 |----------------------------------------------------------------------------------|
 |  Member Number:      1                                                           |
 |  Member Section:  ST   TUB1251256            (INDIAN SECTIONS)                   |
 |----------------------------------------------------------------------------------|
 |  Tension:    (Unit:  KN    METE)                                                 |
 |  Parameters:     FYLD:      250.000E+03       FU:      420.000E+03               |
 |                   NSF:            1.000    ALPHA:            0.800               |
 |  Yielding   :  Design Force:       10.000E+00     LC:     1                      |
 |                Capacity:          627.273E+00     As per:  Sec. 6.2              |
 |  Rupture    :  Design Force:       10.000E+00     LC:     1                      |
 |                Capacity:          741.888E+00     As per:  Sec. 6.3              |
 |----------------------------------------------------------------------------------|
 |  Compression:    (Unit:  KN    METE)                                             |
 |  Buckling Class:   Major: a    Minor: a    As per:Cl. 7.1.2.2                    |
 |  Major Axis:   Design Force:        0.000E+00    LC:     0  Loc:     0.000       |
 |                Capacity:          534.107E+00    As per:  Sec. 7.1.2             |
 |  Minor Axis:   Design Force:        0.000E+00    LC:     0  Loc:     0.000       |
 |                Capacity:          534.107E+00    As per:  Sec. 7.1.2             |
 |----------------------------------------------------------------------------------|
 |  Shear:    (Unit:  KN  )                                                         |
 |  Major Axis:   Design Force:        3.000E+00    LC:     1  Loc:     0.000       |
 |                Capacity:          181.078E+00    As per:  Sec. 8.4               |
 |  Minor Axis:   Design Force:        3.000E+00    LC:     1  Loc:     0.000       |
 |                Capacity:          181.078E+00    As per:  Sec. 8.4               |
 |----------------------------------------------------------------------------------|
 |  Bending:    (Unit:  KN    METE)                                                 |
 |  Parameters:   Laterally Supported      KX:   1.00  LX:   3.000E+00  Cantilever  |
 |  Major Axis:   Design Force:        4.500E+00    LC:     1  Loc:     0.000       |
 |                Capacity:           27.727E+00    As per:  Sec. 8.2.1.2           |
 |  Minor Axis:   Design Force:       -4.500E+00    LC:     1  Loc:     0.000       |
 |                Capacity:           27.727E+00    As per:  Sec. 8.2.1.2           |
 |----------------------------------------------------------------------------------|
 |  Combined Interaction:                                                           |
 |  Parameters:  PSI: 1.00   CMX:    0.900   CMY:    0.900   CMZ:    0.900          |
 |  Section Strength:    Ratio:     0.098    As per:  Sec. 9.3.1.1                  |
 |                       LC:            1    Loc:       0.000                       |
 |  Overall Member Strength (Bending+Tension):                                      |
 |  Equation 1(Z):       Ratio:     0.149    As per:  Sec. 9.3.2.1                  |
 |                       LC:            1    Loc:       0.000                       |
 |  Equation 2(Y):       Ratio:     0.149    As per:  Sec. 9.3.2.1                  |
 |                       LC:            1    Loc:       0.000                       |
 |----------------------------------------------------------------------------------|
 |  Checks               Ratio   Load Case No.      Location from Start( METE)      |
 |                                                                                  |
 |  Tension                0.016       1                       0.000E+00            |
 |  Compression            0.000       0                       0.000E+00            |
 |  Shear Major            0.017       1                       0.000E+00            |
 |  Shear Minor            0.017       1                       0.000E+00            |
 |  Bend Major             0.162       1                       0.000E+00            |
 |  Bend Minor             0.162       1                       0.000E+00            |
 |  Sec. 9.3.1.1           0.098       1                       0.000E+00            |
 |  Sec. 9.3.2.1 (Z)       0.149       1                       0.000E+00            |
 |  Sec. 9.3.2.1 (Y)       0.149       1                       0.000E+00            |
 |----------------------------------------------------------------------------------|