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 f_y = 250 MPa, the specified minimum tensile strength is F_u = 420 MPa, and the modulus of elasticity is E = 205(10)³ MPa

Validation

Section Properties

Gross area:

A_{g} = \frac{π {(5 cm)}^{2}}{4} = 19.6 {cm}^{2}

Moment of inertia:

I_{y} = I_{z} = \frac{π {(5 cm)}^{4}}{64} = 30.7 {cm}^{4}

Elastic section modulus:

Z_{e} = \frac{I}{\frac{D}{2}} = 12.28 {cm}^{3}

Plastic section modulus:

Z_{p} = \frac{{(5 cm)}^{3}}{6} = 20.83 {cm}^{3}

Radius of gyration:

r_{yy} = r_{zz} = \sqrt{\frac{I}{A_{g}}} = 1.252 cm

Partial Safety Factors

Partial safety factor for failure in tension by yielding (per Table 5 of IS:800 2007), γ_m0 = 1.1.

Partial safety factor for failure at ultimate stress (per Table 5 of IS:800 2007), γ_m1 = 1.25.

Forces

Axial force, $F_{x} = 10 kN$

Shear forces, $F_{y} = F_{z} = 1 kN/m \times 3 m = 3 kN$

Moments, $M_{y} = M_{z} = \frac{(1 kN/m) \times {(3 m)}^{2}}{2} = 4.5 kN·m$

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

K_y = K_z = 1

The allowable slenderness ratio is 180.

\frac{k_{y} L}{r_{yy}} = \frac{k_{z} L}{r_{zz}} = \frac{(1.0) \times 300 cm}{1.252 cm} = 239.7 > 180

Section fails in slenderness.

Check Axial Tension

The design strength in tension (or compression) due to yielding (per Section 6), $T_{dg} = N_{d} = \frac{A_{g} \times F_{y}}{γ_{m0}} = \frac{19.6 \times {(10)}^{-4} m^{2} \times 250 \times {(10)}^{3} \frac{kN}{m^{2}}}{1.1} = 445.5 kN$

Assume that A_net = A_g and ɑ = 0.8, the design strength in rupture of critical section, $T_{dn} = ɑ \times A_{net} \frac{F_{u}}{γ_{m1}} = 0.8 \times 19.6 \times {(10)}^{-4} m^{2} \times \frac{420 \times {(10)}^{3} \frac{kN}{m^{2}}}{1.25} = 526.8 kN$

Check Axial Compression

Per Section 7 of IS: 800-2007

Euler buckling stress, $f_{cc} = \frac{π^{2} E}{{(\frac{K L}{r})}^{2}} = \frac{π^{2} \times 205 {(10)}^{3} MPa}{{(239.7)}^{2}} = 35.21 MPa$

Non-dimensional effective slenderness ratio, $λ = \sqrt{\frac{f_{y}}{f_{cc}}} = \sqrt{\frac{250 MPa}{35.21 MPa}} = 2.665$

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 = \frac{1}{ϕ + \sqrt{ϕ^{2} - λ^{2}}} = 0.118

The design compressive stress,

f_{cd} = χ \frac{f_{y}}{γ_{m0}} = 0.118 \frac{250}{1.1} = 26.84 MPa < \frac{f_{y}}{γ_{m0}} = \frac{250}{1.1} = 227.3 MPa

(Cl. 7.1.2.1)

The design compressive strength,

P_{d} = A_{g} f_{cd} = 19.6 \times {(10)}^{-4} m^{2} \times 26.84 \times {(10)}^{3} \frac{kN}{m^{2}} = 52.6 kN

Design ratio in compression: $\frac{F_{x}}{P_{d}} = \frac{10}{52.6} = 0.19$ (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; A_v = A_g = 19.6 cm².

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

V_{u} = \sqrt{{(F_{y})}^{2} + {(F_{z})}^{2}} = \sqrt{{(3 kN)}^{2} + {(3 kN)}^{2}} = 4.243 kN

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

V_{p} = A_{v} \frac{f_{yw}}{\sqrt{3}} = 19.6 \times {(10)}^{-4} m^{2} \times \frac{250 \times {(10)}^{3}}{\sqrt{3}} = 282.9 kN

V_{d} = \frac{V_{p}}{γ_{m0}} = \frac{282.9 kN}{1.1} = 257.2 kN

Design ratio in shear: $\frac{V_{u}}{V_{d}} = \frac{4.654}{257.2} = 0.016$

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:

M^{'} = \sqrt{{(M_{y})}^{2} + {(M_{z})}^{2}} = \sqrt{{(4.5 kN·m)}^{2} + {(4.5 kN·m)}^{2}} = 6.364 kN·m

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

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

V_{u} = 4.343 kN < 0.6 (257.2 kN) = 154.3 kN

M_{d} = \frac{β_{b} \times Z_{p} \times F_{y}}{γ_{m0}} < \frac{1.5 \times Z_{e} \times F_{y}}{γ_{m0}}

where

β_b

M_{d} = {}_{\min}{| \begin{matrix} \frac{1.0 (20.83 \times 10^{-6} m^{3}) (250 \times 10^{3} \frac{kN}{m^{2}})}{1.1} = 4.734 kN·m \\ \frac{1.5 (12.28 \times 10^{-6} m^{3}) (250 \times 10^{3} \frac{kN}{m^{2}})}{1.1} = 4.186 kN·m \end{matrix}}

Design ratio in bending:

\frac{M^{'}}{M_{d}} = \frac{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 $n = \frac{N}{N_{d}} = 0.022$

M_{nd, max} = 1.04 \times M_{d} (1 - n^{1.7}) = 1.04 \times 4.186 kN·m (1 - {0.022}^{1.7}) = 4.347 kN·m

Use M_d = 4.186 kN·m.

The combined bending ratio is taken as:

{(\frac{M_{y}^{'}}{M_{nd, y}})}^{ɑ_{1}} + {(\frac{M_{z}^{'}}{M_{nd, z}})}^{ɑ_{2}}

where

ɑ₁	=	1
ɑ₂	=	1

{(\frac{0}{4.186})}^{1.0} + {(\frac{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_{y} = K_{z} = {}_{\min}{| \begin{matrix} 1 + (λ - 0.2) n = 1 + (2.665 - 0.2) 0.19 = 1.468 \\ 1 + 0.8 n = 1 + 0.8 \times 0.19 = 1.152 \end{matrix}}

K_{LT} = 1.0

\frac{P}{P_{dy}} + K_{y} (\frac{C_{my} \times M_{y}}{M_{dy}}) + K_{LT} \frac{M_{z}}{M_{dz}} = 0.19 + 1.152 (\frac{0.9 \times 4.5}{4.186}) + 1.0 \frac{4.5}{4.186} =2.380

\frac{P}{P_{dy}} + K_{y} (\frac{C_{my} \sqrt{{M_{y}}^{2} + {M_{z}}^{2}}}{M_{dy}}) + 0 = 0.19 + 1.152 (\frac{0.9 \sqrt{{4.5}^{2} + {4.5}^{2}}}{4.186}) + 0 =1.766

\frac{P}{P_{dz}} + 0.6 K_{y} (\frac{C_{my} \times M_{y}}{M_{dy}}) + K_{z} (\frac{C_{mz} \times M_{z}}{M_{dz}}) = 0.19 + 0.6 \times 1.152 (\frac{0.9 \times 4.5}{4.186}) + 1.152 (\frac{0.9 \times 4.5}{4.186}) =1.973

Results

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

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 - 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
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
LAT 1 ALL
CAN 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   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            |
 |----------------------------------------------------------------------------------|