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

Verify the design of a pipe section subject to axial tension and biaxial bending per the IS 800:2007 limit state method.

Details

The member is a 3 m long cantilever subject to a 10 kN tensile force and uniform bending moments of 1 kN/m along both major and minor axes. The section used is a 90 mm nominal bore with a wall thickness of 3.6 mm (STAAD.Pro section designation PIP10106L) and is braced to prevent lateral-torsional buckling. The steel has a yield strength of 250 MPa, an ultimate tensile strength of 420 MPa, and a modulus of elasticity of 205×(10)³ MPa.

Validation

Section Properties

Outer diameter, OD = 101.6 mm

Wall thickness, t = 3.6 mm

Inner diameter, ID = 101.6 - 2 × 3.6 = 94.4 mm

Cross-section area, $A_{x} = \frac{π}{4} ({101.6}^{2} - {94.4}^{2}) = 11.08 {cm}^{2}$

Moment of inertia about both axes, $I = \frac{π}{64} ({101.6}^{4} - {94.4}^{4}) = 133.2 {cm}^{4}$

Radius of gyration about both axes, $r = \sqrt{I / A_{x}} = 3.47 cm$

Elastic section modulus about both axes, $Z_{e} = \frac{I}{O D / 2} = 26.23 {cm}^{3}$

Plastic section modulus about both axes, $Z_{p} = \frac{1}{6} ({101.6}^{3} - {94.4}^{3}) = 34.6 {cm}^{3}$

Constants

γ_m0 = 1.1
γ_m1 = 1.25

Section Classification

ε = \sqrt{250 / f_{y}} = 1

For circular, hollow tubes subjected to moment, OD/t = 28.22 < 42ε² = 42

Therefore, the classification of the section is "plastic".

Hence, from Table 10 of IS 800:2007, the buckling class about both axes is "a".

Check Slenderness Ratio

k_y = k_z = 1

The maximum allowable slenderness ratio is 180.

Slenderness ratio about both axes, k_zL / r_z = 86.67 < 180

So the slenderness is within the limit.

Axial Tension Capacity

Check for yielding of gross section per Cl. 6.2:

T_dg = A_g × f_y / γ_m0 = 11.08 × 250 / 1.1 = 251.8 kN

Check for rupture of critical section per Cl. 6.3:

α = 0.8

T_dn = α × A_net × f_u / γ_m1 = 0.8 × 11.08 × 420 / 1.25 = 297.8 kN

Block shear areas are not specified and therefore not checked.

The critical tension ratio 10 / 251.8 = 0.040

Check Axial Compression

The non-dimensional slenderness ratio, $λ_{z} = \sqrt{\frac{f_{y} {(k_{z} L / r_{z})}^{2}}{π^{2} E}} = 0.961$

For buckling class "a", the imperfection factor, α = 0.21

ϕ_{z} = 0.5 [1 + α (λ_{z} - 0.2) + λ_{z}^{2}] = 1.042

χ_{z} = \frac{1}{ϕ_{z} + \sqrt{ϕ_{z}^{2} - λ_{z}^{2}}} = 0.692

So, as per Cl. 7.1.2.1 of IS 800:2007, $f_{c d z} = min (\frac{χ_{z} f_{y}}{γ_{m 0}}, \frac{f_{y}}{γ_{m 0}}) = 157.3 MPa$

Therefore, axial compression capacity for both axes: P_dz = A_xf_cdz = 147.3 kN

The critical ratio for axial compression is 0 (i.e., no compression).

Check Shear Capacity

As per Cl. 8.4.1.1 of IS 800:2007, the shear area, A_vz = 2×A_x / π = 7.054 cm²

So, as per Cl. 8.4.1, $V_{p z} = \frac{A_{v z} f_{y w}}{\sqrt{3}} = 101.8 kN$

Hence, shear capacity along major axis: $V_{d z} = \frac{V_{p z}}{γ_{m 0}} = 92.56 kN$

Shear force along both axes = 3 kN.

Critical shear ratio along both axes: 3 / 92.56 = 0.032

Check Bending Capacity

Since the section has been classified as "plastic", β_b = 1.

Actual shear force in both axes, V = 3 kN < 0.6V_d = 55.53 kN. Hence, the section is subjected to low shear.

\frac{β_{b} Z_{p y} f_{y}}{γ_{m 0}} = 7.861 kN·m

\frac{1.5 Z_{e} f_{y}}{γ_{m 0}} = 8.925 kN·m

So, the governing bending strength with respect to the minor axis is M_dy = 7.861 kN·m

Actual bending moment about the minor axis is M_z = 4.5 kN·m.

The critical bending ratio in the minor axis is 4.5 / 7.861 = 0.572

Check Combined Interaction of Axial Compression and Bending

Section strength calculation as per Cl. 9.3.1:

n = N / N_d = 0.004

Hence, from Table 17:

α₁ = 2

α₂ = 2

M_ndy = M_dy = 9.341 kN·m

(Cl. 9.3.1.2(c) )

1.04·M_dz·(1 - n^1.7) = 8.175 kN·m > M_dz

M_ndz = M_dz = 7.861 kN·m

{(\frac{M_{y}}{M_{n d y}})}^{α_{1}} + {(\frac{M_{y z}}{M_{n d z}})}^{α_{2}} = {(\frac{4.5}{7.861})}^{2} + {(\frac{4.5}{7.861})}^{2} = 0.655

Overall member strength in bending and axial compression

M_{eff} = M_{z} - \frac{ψ F_{x} Z_{e z}}{A_{x}} = 4.5 - \frac{1.0 \times 10 \times 26.23}{11.08 \times 10^{2}} = 4.264 kN·m

M_eff / M_dz = 4.263 / 7.861 = 0.542

The loading and section is the same in both axes.

Results

Table 1.
Result Type	Reference	STAAD.Pro	Difference
Slenderness ratio	86.46	86.67	negligible
Tension capacity, gross yielding (kN)	251.8	252.273	negligible
Tension capacity, rupture of critical section (kN)	297.8	298.368	negligible
Axial compression capacity about both axes (kN)	174.3	174.338	negligible
Shear capacity in both axes (kN)	92.56	92.723	negligible
Ratio for shear	0.032	0.032	none
Bending capacity about both axes (kN·m)	7.861	7.864	negligible
Ratio for bending about both axes	0.572	0.572	none
Ratio per Cl. 9.3.1.1	0.655	0.655	none
Ratio for both axes per Cl. 9.3.2.1	0.542	0.542	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 - Pipe - Tension and Bending.STD is typically installed with the program.

The following design parameters are used:

The default value of ALPHA is used.
The default values of KY and KZ are used.
The default value of the maximum allowable slenderness ratio, MAIN, of 180 is used.
The default value of Ψ.
The beam type is a cantilever, specified by CAN 1
The beam is laterally supported, specified by LAT 1

STAAD SPACE
START JOB INFORMATION
ENGINEER DATE 29-Jul-20
END JOB INFORMATION
*****************************************************************************
*This problem has been created to validate the program calculated design
*results of laterally supported pipe 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 PIP1016L
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   PIP1016L              (INDIAN SECTIONS)                   |
 |  Status: PASS  Ratio:   0.655  Critical Load Case:     1  Location:       0.00   |
 |  Critical Condition: Sec. 9.3.1.1                                                |
 |  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:      11.100E+00          IZZ:     133.000E+00          RZZ:       3.462E+00|
 |  AYY:       7.066E+00          IYY:     133.000E+00          RYY:       3.462E+00|
 |  AZZ:       7.066E+00          IXX:     266.000E+00           CW:       0.000E+00|
 |  ZEZ:      26.181E+00          ZPZ:      34.600E+00                              |
 |  ZEY:      26.181E+00          ZPY:      34.600E+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:  86.67 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   PIP1016L              (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:          252.273E+00     As per:  Sec. 6.2              |
 |  Rupture    :  Design Force:       10.000E+00     LC:     1                      |
 |                Capacity:          298.368E+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:          174.338E+00    As per:  Sec. 7.1.2             |
 |  Minor Axis:   Design Force:        0.000E+00    LC:     0  Loc:     0.000       |
 |                Capacity:          174.338E+00    As per:  Sec. 7.1.2             |
 |----------------------------------------------------------------------------------|
 |  Shear:    (Unit:  KN  )                                                         |
 |  Major Axis:   Design Force:        3.000E+00    LC:     1  Loc:     0.000       |
 |                Capacity:           92.723E+00    As per:  Sec. 8.4               |
 |  Minor Axis:   Design Force:        3.000E+00    LC:     1  Loc:     0.000       |
 |                Capacity:           92.723E+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:            7.864E+00    As per:  Sec. 8.2.1.2           |
 |  Minor Axis:   Design Force:       -4.500E+00    LC:     1  Loc:     0.000       |
 |                Capacity:            7.864E+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:     0.655    As per:  Sec. 9.3.1.1                  |
 |                       LC:            1    Loc:       0.000                       |
 |  Overall Member Strength (Bending+Tension):                                      |
 |  Equation 1(Z):       Ratio:     0.542    As per:  Sec. 9.3.2.1                  |
 |                       LC:            1    Loc:       0.000                       |
 |  Equation 2(Y):       Ratio:     0.542    As per:  Sec. 9.3.2.1                  |
 |                       LC:            1    Loc:       0.000                       |
 |----------------------------------------------------------------------------------|
 |  Checks               Ratio   Load Case No.      Location from Start( METE)      |
 |                                                                                  |
 |  Tension                0.040       1                       0.000E+00            |
 |  Compression            0.000       0                       0.000E+00            |
 |  Shear Major            0.032       1                       0.000E+00            |
 |  Shear Minor            0.032       1                       0.000E+00            |
 |  Bend Major             0.572       1                       0.000E+00            |
 |  Bend Minor             0.572       1                       0.000E+00            |
 |  Sec. 9.3.1.1           0.655       1                       0.000E+00            |
 |  Sec. 9.3.2.1 (Z)       0.542       1                       0.000E+00            |
 |  Sec. 9.3.2.1 (Y)       0.542       1                       0.000E+00            |
 |----------------------------------------------------------------------------------|