 # V. IS 800 2007 - LSD Rod Compression and Bending

Verify the program-calculated design results for a 3 m long cantilever beam, subjected to axial compression and biaxial bending. The beam section used is a 50 mm diameter steel rod and is designed as per IS 800:2007.

## Details

The 3 m cantilever section is subject to a 10 kN axial compression load at the free end as well as 1 kN/m uniform loads along the length in both the "major" and "minor" axes. The cantilever is assumed to be laterally supported.

The steel yield stress is fy = 250 MPa, the specified minimum tensile strength is Fu = 420 MPa, and the modulus of elasticity is E = 205(10)3 MPa

## Validation

Section Properties

Gross area:

Moment of inertia:

Elastic section modulus:

Plastic section modulus:

Partial Safety Factors

Partial safety factor for failure in tension by yielding (per Table 5 of IS:800 2007), γm0 = 1.1.

Partial safety factor for failure at ultimate stress (per Table 5 of IS:800 2007), γm1 = 1.25.

Forces

Axial force,

Shear forces,

Moments,

Section Classification

Although Table 2 of IS800:2007 does not specify the classification for solid rod sections, the width to thickness ratio for a solid rod is 1.0. Hence, it is a Class 1 section (Plastic).

Check Slenderness Ratio

Ky = Kz = 1

The allowable slenderness ratio is 180.

Section fails in slenderness.

Check Axial Tension

The design strength in tension (or compression) due to yielding (per Section 6),

Assume that Anet = Ag and ɑ = 0.8, the design strength in rupture of critical section,

Check Axial Compression

Per Section 7 of IS: 800-2007

Euler buckling stress,

Non-dimensional effective slenderness ratio,

The imperfection factor given in Table 7; for buckling class (as per Table 10 for solid sections), ɑ = 0.49.

$ϕ = 0.5 [ 1 + ɑ ( λ - 0.2 ) + λ 2 ] = 4.654$
$χ = 1 ϕ + ϕ 2 - λ 2 = 0.118$

The design compressive stress,

 (Cl. 7.1.2.1)

The design compressive strength,

Design ratio in compression: $F x P d = 10 52.6 = 0.19$ (for both local z and y axes)

Check for Shear

Per Cl 8.4 and 8.4.1 of IS: 800-2007

Per Cl. 8.4.1.1, the shear area is taken as the gross area of the solid round section; Av = Ag = 19.6 cm2.

The shear force used for design is taken as the resultant SRSS combination of the "major" and "minor" axes:

The plastic under pure shear is taken as the nominal shear strength,

Design ratio in shear: $V u V d = 4.654 257.2 = 0.016$

Check for Bending

Per Section 8 of IS: 800-2007

The bending moment used for design is taken as the resultant SRSS combination of the "major" and "minor" axes:

This is used as the "major" axis bending moment and the "minor" axis bending moment is taken as zero.

When V < 0.6Vd on a cantilever beam, then the design bending strength is taken as:

$M d = β b × Z p × F y γ m0 < 1.5 × Z e × F y γ m0$
where
 βb = 1

Design ratio in bending: $M ′ M d = 6.364 4.186 = 1.52$

Check for Combined Axial Force

Check the reduced flexural strength under combined axial force and the respective uniaxial moment acting alone (Cl. 9.3.1.2).

The ratio for the compression yielding is $n = N N d = 0.022$

Use Md = 4.186 kN·m.

The combined bending ratio is taken as:

$( M y ′ M nd , y ) ɑ 1 + ( M z ′ M nd , z ) ɑ 2$
where
 ɑ1 = 1 ɑ2 = 1
$( 0 4.186 ) 1.0 + ( 6.364 4.186 ) 1.0 = 1.52$

Check for Combined Axial Force and Bending Moment

Per Cl 9.3.2.2 of IS: 800-2007

$K LT = 1.0$
$P P dy + K y ( C my × M y M dy ) + K LT M z M dz = 0.19 + 1.152 ( 0.9 × 4.5 4.186 ) + 1.0 4.5 4.186 =2.380$
$P P dy + K y ( C my M y 2 + M z 2 M dy ) + 0 = 0.19 + 1.152 ( 0.9 4.5 2 + 4.5 2 4.186 ) + 0 =1.766$
$P P dz + 0.6 K y ( C my × M y M dy ) + K z ( C mz × M z M dz ) = 0.19 + 0.6 × 1.152 ( 0.9 × 4.5 4.186 ) + 1.152 ( 0.9 × 4.5 4.186 ) =1.973$

## Results

Table 1. Comparison of results
Slenderness ratio 239.7 239.71 negligible
Yielding tension capacity (kN) 445.5 445.455 negligible
Rupture tension capacity (kN) 526.8 526.848 negligible
Axial compression capacity (kN) 52.6 52.598 negligible
Critical ratio for axial compression 0.19 0.190 none
Shear capacity (kN) 257.2 257.183 negligible
Shear ratio 0.016 0.016 none
Bending capacity (kN·m) 4.186 4.186 none
Bending ratio 1.52 1.520 none
Interaction ratio for Cl. 9.3.1.1 1.52 1.520 none
Interaction ratio for Cl. 9.3.2.2 1.766 1.766 none

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

The following design parameters are used:
• The lateral support condition is specified by LAT 1
• The cantilever member is specified by CAN 1
STAAD SPACE
START JOB INFORMATION
ENGINEER DATE 12-Jan-2021
ENGINEER NAME TK
END JOB INFORMATION
*****************************************************************************
*This problem has been created to validate the program calculated design
*results of rod section subjected to axial Compression and biaxial bending
*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 DUTCH
1 TABLE ST RD50
CONSTANTS
MATERIAL STEEL ALL
SUPPORTS
1 FIXED
2 FX -10
1 UNI GY -1
1 UNI GZ -1
PERFORM ANALYSIS
PRINT ANALYSIS RESULTS
PARAMETER 1
CODE IS800 LSD
LAT 1 ALL
CAN 1 ALL
TRACK 2 ALL
CHECK CODE ALL
FINISH


                       STAAD.PRO CODE CHECKING - IS-800-2007-LSD (V2.2)
****************************************************
|----------------------------------------------------------------------------------|
|*  Member Number:      1                                                          |
|  Member Section:  ST   RD50                  (DUTCH SECTIONS)                    |
|  Status: FAIL  Ratio:   1.766  Critical Load Case:     1  Location:       0.00   |
|  Critical Condition: Sec. 9.3.2.2                                                |
|  Critical Design Forces:  (Unit: KN    METE)                                     |
|      FX:      10.000E+00 C     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:      19.600E+00          IZZ:      30.700E+00          RZZ:       1.252E+00|
|  AYY:      19.600E+00          IYY:      30.700E+00          RYY:       1.252E+00|
|  AZZ:      19.600E+00          IXX:      61.400E+00           CW:       0.000E+00|
|  ZEZ:      12.280E+00          ZPZ:      20.833E+00                              |
|  ZEY:      12.280E+00          ZPY:      20.833E+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: 239.71 Allowable Ratio: 180.00  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   RD50                  (DUTCH SECTIONS)                    |
|----------------------------------------------------------------------------------|
|  Tension:    (Unit:  KN    METE)                                                 |
|  Parameters:     FYLD:      250.000E+03       FU:      420.000E+03               |
|                   NSF:            1.000    ALPHA:            0.800               |
|  Yielding   :  Design Force:        0.000E+00     LC:     0                      |
|                Capacity:          445.455E+00     As per:  Sec. 6.2              |
|  Rupture    :  Design Force:        0.000E+00     LC:     0                      |
|                Capacity:          526.848E+00     As per:  Sec. 6.3              |
|----------------------------------------------------------------------------------|
|  Compression:    (Unit:  KN    METE)                                             |
|  Buckling Class:   Major: c    Minor: c    As per:Cl. 7.1.2.2                    |
|  Major Axis:   Design Force:       10.000E+00    LC:     1  Loc:     0.000       |
|                Capacity:           52.598E+00    As per:  Sec. 7.1.2             |
|  Minor Axis:   Design Force:       10.000E+00    LC:     1  Loc:     0.000       |
|                Capacity:           52.598E+00    As per:  Sec. 7.1.2             |
|----------------------------------------------------------------------------------|
|  Shear:    (Unit:  KN  )                                                         |
|  Major Axis:   Design Force:        0.000E+00    LC:     0  Loc:     0.000       |
|                Capacity:          257.183E+00    As per:  Sec. 8.4               |
|  Minor Axis:   Design Force:        4.243E+00    LC:     1  Loc:     0.000       |
|                Capacity:          257.183E+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:       -6.364E+00    LC:     1  Loc:     0.000       |
|                Capacity:            4.186E+00    As per:  Sec. 8.2.1.2           |
|  Minor Axis:   Design Force:        0.000E+00    LC:     0  Loc:     0.000       |
|                Capacity:            4.186E+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:     1.520    As per:  Sec. 9.3.1.1                  |
|                       LC:            1    Loc:       0.000                       |
|  Overall Member Strength (Bending+Compression):                                  |
|  Equation 2:          Ratio:     1.766    As per:  Sec. 9.3.2.2   Ky :   1.1521  |
|                       LC:            1    Loc:       0.000        Kz :   1.1521  |
|----------------------------------------------------------------------------------|
|  Checks               Ratio   Load Case No.      Location from Start( METE)      |
|                                                                                  |
|  Tension                0.000       0                       0.000E+00            |
|  Compression            0.190       1                       0.000E+00            |
|  Shear Major            0.000       0                       0.000E+00            |
|  Shear Minor            0.016       1                       0.000E+00            |
|  Bend Major             1.520       1                       0.000E+00            |
|  Bend Minor             0.000       0                       0.000E+00            |
|  Sec. 9.3.1.1           1.520       1                       0.000E+00            |
|  Sec. 9.3.2.2(ii)       1.766       1                       0.000E+00            |
|----------------------------------------------------------------------------------|