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International Journal of Civil Engineering and Technology (IJCIET)

Volume 9, Issue 10, October 2018, pp. 388–399, Article ID: IJCIET_09_10_040

Available online at http://www.iaeme.com/ijciet/issues.asp?JType=IJCIET&VType=9&IType=10

ISSN Print: 0976-6308 and ISSN Online: 0976-6316

©IAEME Publication Scopus Indexed

COMPATIBILITY CHECK OF ELASTIC

CRITICAL MOMENT EVALUATION OF

ROLLED CHANNEL SECTION BEAM

Juned Raheem

Civil Engineering Department, Maulana Azad National Institute of Technology,

Bhopal - 462003, Madhya Pradesh, India

Dr. S.K Dubey



Dr. Nitin Dindorkar



ABSTRACT

Design policies for eccentrically loaded beams with open channel cross- sections

aren't available in Indian code IS 800-2000 General Construction in Steel-Code of

Practice (THIRD REVISION). General solution for Elastic Critical Moment, Mcr has

determined with the aid of the use of expression given in ANNEX-E (CL.8.2.2.1,IS

800:2007) for mono symmetric beams. Results so acquired are tested with Finite

Element (FE) simulations on the idea of a parametric take a look at the use of ANSYS

software program14.0.

A variant of maximum 0.3% is observed between the analytical end result and

ANSYS result for slender beams however a massive distinction was observed for

stocky beams. There is a reduction in design strength with an increase in span of the

beam. The strength curve for channel beam proposed by means of IS 800: 2007 seems

to be an awesome choice, however it does no longer claim to be accurate for beams

with a ratio L/h<20.

Key words: Elastic Critical Moment, Slenderness, Reduction factors and Design

beam capacity.

Cite this Article: Juned Raheem, Dr. S.K Dubey and Dr. Nitin Dindorkar,

Compatibility Check of Elastic Critical Moment Evaluation of Rolled Channel Section

Beam, International Journal of Civil Engineering and Technology (IJCIET) 9(10),

2018, pp. 388–399.

http://www.iaeme.com/IJCIET/issues.asp?JType=IJCIET&VType=9&IType=10

Compatibility Check of Elastic Critical Moment Evaluation of Rolled Channel Section Beam


1. INTRODUCTION

Beside mono symmetric I sections, rolled channel steel section are often used as beam to

guide light loads in the shape of purlin to support roof over truss shape, staging to support

bridge decks, etc[20] The structural conduct of channel section is different from doubly

symmetric section or mono symmetric I section because its shear centre and centre of gravity

do no longer coincide.[12] In steel structure layout of laterally unsupported beam is quite

complex because of diverse reasons consisting of lateral buckling of complete beam between

the supports, nearby buckling of flanges and longitudinal buckling of web.[19] When a beam

which is extra rigid about its major axis than its minor axis subjected to bending about its

principal axis, lateral torsional buckling will occur about the minor axis of the beam. Lateral

torsion buckling will generally tend the compression flange to buckle in the course transverse

to the load earlier than the steel yields results in pulling beam sideways, whereas the tension

flange will hold the beam in its plane.[1]

The lateral torsional buckling happens in a case for channel sections, where the shear

centre does not coincide with the vertical axis of the centre of gravity of the section.[8] The

load is generally subjected eccentrically on the web which reasons a torsional moment in the

beam, which makes it tough to discover elastic critical moment Mcr.[4] Indian standard code

for General Construction In Steel-Code Of Practice (Third Revision) IS 800-2007 has

furnished guidelines to calculate elastic lateral torsional buckling moment or theoretical

elastic critical moment for doubly symmetric section and cross section mono symmetric about

its minor axis subjected to bend about its principal axis. Indian standard code IS 800-2007

doesn’t offers any components to calculate theoretical elastic critical moment for channel

beams.[17] Whereas channel section is a mono-symmetrical section that's symmetric about its

major axis. However new design rule was already proposed for design bending ability for

channel section subjected to eccentric loading, however with restriction of span to section

depth ratio.[8]

Stability of steel beam conjointly depends on bending distribution, point of application of

load and degree of the mono symmetry of the section referred as Wagner’s parameter.[5]

Coefficients C1, C2 and C3 are respectively influenced by these parameters that are given for

a few selected load circumstances.[5] Since, because the slenderness ratio decreases, extra

fibers of the beam become inelastic (increasing the degree of plasticity) and only the elastic

element of the cross section remains potent in supplying resistance to lateral buckling.[6] In

stocky beams buckling may be ruled by distorsion of web.[7] The effect of bracing relies not

only on the stiffness of the restraint but also on the modified slenderness of the section. The

load application had more influence on elastic critical moment and this impact is of the larger

magnitude when the beam is rigid about major axis.[9] Channel section has been analysed for

Mcr, utilising 3 factor formula of Eurocode-3 and verified with diverse softwares akin to

ADINA, COLBEAM, LTBEAM, SAP2000 and STAAD pro however failed to establish any

conclusion.[11] In present state of affairs, most of the commercial Structural engineering

program consider lateral torsional buckling during the analysis of potential of steel beams.

ANSYS application as a modern day approach established on FEM simulation and modeling

is used for evolved engineering simulation rationale. The method has three phases

preprocessing, solution and postprocessing.[22].

2. METHODOLOGY

Initially, a literature study on the theory behind various instability phenomena for steel beams

was made, including study of formula given in Indian Standard codes IS: 800:2007, ANNEX

E and Clause 8.2.2 treats lateral-torsional buckling and establishes the elastic critical moment,

Mcr.

Juned Raheem, Dr. S.K Dubey and Dr. Nitin Dindorkar


A parametric study has been carried out wherein channel beams with targeted dimensions,

lengths and load conditions has been modeled and analyzed using FEM simulation based

computer program ANSYS workbench 14.0. Four cross sections were chosen ISMCP 125,

ISMCP150, ISMCP175 and ISMCP200 of 5 exceptional lengths i.e., 1600mm, 2200mm,

3000mm, 4000mm, and 5000mm. A uniformly dispensed load of 1KN/m is applied on each

beam on the top flange, the center and the bottom of the web respectively. Theoretical elastic

critical moment was calculated using method given in code IS: 800:2007, ANNEX E and

Clause 8.2.2 for mono symmetric section and then validated making use of ANSYS

workbench 14.0.

2.1. Elastic Critical Moment of a Section Symmetrical About Minor Axis[17]

The guideline is only valid for major axis bending where the cross-section is uniform and

symmetrical about the minor axis. It will for that reason no longer be 100% correct for

channel beams.

{[(

)

( )

( )

( )]

( )}

Where:

C1 = Coefficient depending on the shape of the moment diagram

C2 = Coefficient depending on the point of load application relative to the shear centre

C3 = Coefficient depending on the symmetry of the cross-section around the weak axis

K, Kw = effective length factors of the unsupported length accounting for boundary conditions

at the end lateral supports.

K = 0.5 (For complete restraint against rotation about weak axis

= 1.0 (For free rotation about weak axis)

=0.7 (For one end fixed and other end free)

Kw = Factor for warping restraint. Unless special provisions to restrain warping of the section

at the end lateral supports are made, Kw should be taken as 1.0.

yg =distance between the point of application of the load and the shear centre of the cross

section in y- direction and is positive when the load is acting towards the shear centre from the

point of application.

∫ ( )

ys = coordinate of the shear centre with respect to centroid and is positive when the shear

centre is on the compression side of the centroid.

y, z = coordinates of the elemental area with respect to centroid of the section

h = overall height of the section

Iyy = Moment of inertia around y axis ( minor axis)

Iw = Warping constant

It = Torsion constant

G= Shear Modulus

E= Youngs Modulus



Loading and support conditions Bending moment diagram Value of

K

Constants

C1 C2 C3

1.0

0.5

1.365

1.070

0.553

0.432

1.780

3.050

1.0

0.5

1.132

0.972

0.459

0.304

0.525

0.980

2.2. Design Strength of Laterally Unsupported Beams [17]

Beam experiencing bending about its major axis and when its compression flange is not

restrained against lateral buckling, may just fail through lateral torsional buckling earlier than

reaching its bending strength. The effect of lateral torsional buckling on flexural strength

doesn’t need to be viewed when the non dimensional slenderness ratio for lateral torsional

buckling is less than 0.4. The design bending capacity of a laterally unsupported beam as

governed via lateral torsional buckling, according to the Indian code (IS 800:2007) can be

obtained from the expression mentioned below.

With

= 1.0 for plastic and compact sections

=

for semi –compact sections

Where and are the plastic section modulus and elastic section modulus with respect

to extreme compression fibre and is the design bending compressive stress.

The design bending compressive stress is given by

yield stress

Where is the reduction factor to account for lateral torsional buckling given by

* , - +

In which , ( ) -

The values of imperfection factor for lateral torsional buckling of beams is given by

= 0.76 for rolled channel section

= 0.49 for welded section

= partial safety factor for material=1.10

The non- dimentional slenderness ratio , is given by

√



Where

= elastic critical moment and

= plastic moment resistance

3. PARAMETRIC STUDY

The study was conducted by considering four (Indian standard medium weight parallel flange

channel) ISMCP beams with different cross section size of ISMCP 125, ISMCP 150, ISMCP

175 and ISMCP 200.Three beams having different length of 1600mm, 2200mm, 3000mm,

4000mm and 5000mm.

The physical properties considered in the study are

Unit mass of steel, ρ =7850 kg/m

Modulus of elasticity, E=2.0 x N/ (Mpa)

Poisson ratio, μ=0.5

Modulus of rigidity, G=0.769 x N/ (Mpa)

Co-efficient of thermal expansion, x / C

3.1. Sectional Properties of Hot Rolled Beams

a) ISMCP 125 b) ISMCP 150 c) ISMCP 175 d) ISMCP 200

Table 1 Showing sectional properties if ISMCP channel sections

ISMCP 125 ISMCP 150 ISMCP 175 ISMCP 200

Depth, D

mm 125 150 175 200

Breadth, B

mm 65 75 75 75

Web thickness, t

mm 5.3 5.7 6 6.2

Flange thickness, T

mm 8.1 9.0 10.2 11.4

Root 1

mm 9.5 10 10.5 11

Root 2

mm 5 5 6 6

I xx

321 794 1240 1840

I yy

69.8 120 138 156

Mass, M

kg/m 13.1 16.8 19.6 22.3

Sectional area, a

16.7 21.3 24.9 28.5



3.2. Load Case

Uniformly distributed load has been considered and applied at different levels of web at top,

middle and bottom. Load of 100 kN/m is applied for total of five different spans of beam.

Figure 1 Linearly distributed load application [15]

3.3. Boundary Condition

Beams are modelled with the boundary conditions known as fork support conditions. This

type of boundary conditions is given to allow warping in flanges and to resist torsion in the

webs at the supports.

Figure 2 Fork support condition [15]

3.4. Validation using ANSYS 14.0

Figure 3 Eigen value of ISMCP 175, 1600mm beam length [22]



ANSYS is Finite Element software used for simulation purpose. In this study Eigen values

are evaluated for an Indian standard medium weight parallel flange channel beam by creating

its model, applying boundary condition and a uniformly distributed load of 100 KN/m. the

figure shows the buckling load factors of ISMCP 175 for different lengths

1600,2200,3000,4000 and 5000mm obtained in ANSYS.

Table 2 Variation (%M) of Elastic Critical Moment obtained theoretically, Mcr (KN-m) and using FEM

Simulation software ANSYS, M’cr (KN-m) through Eigen buckling factors, (x) w.r.t different span for different

channel section beams subjected to UDL of 1KN/m applied on Top web of the beam.

Length(mm)


x Mcr M’cr %M x Mcr M’cr %M x Mcr M’cr %M x Mcr M’cr %M

1600 0.1030 32.92 32.96 -0.12 0.1756 58.86 56.19 4.75 0.223 78.52 71.36 9.12 0.279 98.11 89.28 9.89

2200 0.0400 23.72 24.20

-

1.98 0.0653 40.38 39.51 2.20 0.0838 53.04 50.7 4.41 0.102 66.04 61.71 7.02

3000 0.0161 17.67 18.15

-

2.64 0.02574 29.207 28.96 0.85 0.033 37.95 37.13 2.16 0.04 47.14 45 4.76

4000 0.0070 13.57 14.00 -3.07 0.01089 22.11 21.78 1.52 0.01422 28.55 28.44 0.39 0.0174 35.44 34.8 1.84

5000 0.0037 11.06 11.44

-

3.32 0.005975 17.95 18.67

-

3.86 0.00744 23.11 23.25

-

0.61 0.00913 28.69 28.53 0.56

Figure 4 Graph showing variation in Elastic critical moment calculated theoretically and using ANSYS w.r.t

different span and cross section of channel section beam subjected to UDL on Top web of the beam.



channel section beams subjected to UDL of 1KN/m applied on Mid web of the beam.

Length(mm)



1600 0.147 49.07 47.04 4.32 0.263 87.83 84.16 4.36 0.339 117.34 108.48 7.55 0.428 147.94 136.96 8.02

2200 0.053 32.4 32.07 1.03 0.0916 56.32 55.42 1.62 0.1186 74.37 71.75 3.52 0.147 93.42 88.94 5.04

3000

0.02 22.4 22.5 -

0.44 0.0336 38.11 37.8 0.82 0.0435 49.855 48.94 1.84 0.0541 62.44 60.86 2.60

4000

0.00828 16.26 16.56 -

1.81 0.01373 27.29 27.46

-

0.62 0.01766 35.46 35.32 0.40 0.02191 44.44 43.82 1.41

5000 0.004181 12.805 13.07 -

2.03 0.007 21.33 21.88 -

2.51 0.0089 27.63 27.81 -

0.65 0.0101 34.56 34.38 0.52

0.00

20.00

40.00

60.00

80.00

100.00

120.00

0 1000 2000 3000 4000 5000 6000

Elas

tic

Cri

tica

l Mo

me

nt

(KN

-m)

Length(mm)

ISMCP125 Mcr

ISMCP125 M’cr

ISMCP150 Mcr

ISMCP150 M’cr

ISMCP175 Mcr

ISMCP175 M’cr

ISMCP200 Mcr

ISMCP200 M’cr



Figure 5 Graph showing variation in Elastic critical moment calculated theoretically and using

ANSYS w.r.t different span and cross section of channel section beam subjected to UDL on Mid web

of the beam.



channel section beams subjected to UDL of 1KN/m applied on Bottom web of the beam.

Length(mm)



1600 0.206 67.91 65.92 3.02 0.382 131.04 122.24 7.20 0.49 175.35 156.8 10.58 0.621 223.1 198.72 10.93

2200 0.0697 42.22 42.17 0.12 0.126 78.56 76.23 3.06 0.164 104.26 99.22 4.83 0.207 132.16 125.24 5.24

3000

0.0247 27.62 27.79 -

0.61 0.04369 49.74 49.15 1.20 0.057 65.49 64.13 2.08 0.0716 82.7 80.55 2.60

4000

0.00978 19.16 19.56 -

2.04 0.01669 33.66 33.38 0.84 0.0219 44.05 43.8 0.57 0.0274 55.4 54.8 1.08

5000 0.004814 14.651 15.04 -

2.59 0.008344 23.35 26.08 -

10.47 0.0106 33.03 33.13 -0.30 0.01322 41.49 41.31 0.43

Figure-6 Graph showing variation in Elastic critical moment calculated theoretically and using

ANSYS w.r.t different span and cross section of channel section beam subjected to UDL on Bottom

web of the beam.

0

20

40

60

80

100

120

140

160

0 1000 2000 3000 4000 5000 6000

Elas

tic

Cri

tica

l Mo

me

nt(

KN

-m)

Length(mm)

ISMCP125 Mcr

ISMCP125 M’cr

ISMCP150 Mcr

ISMCP150 M’cr

ISMCP175 Mcr

ISMCP175 M’cr

ISMCP200 Mcr

ISMCP200 M’cr

0

50

100

150

200

250

0 1000 2000 3000 4000 5000 6000

Elas

tic

Cri

tica

l Mo

me

nt(

KN

-m)

Length(mm)

ISMCP125 Mcr

ISMCP125 M’cr

ISMCP150 Mcr

ISMCP150 M’cr

ISMCP175 Mcr

ISMCP175 M’cr

ISMCP200 Mcr

ISMCP200 M’cr



Table 5 Variation of Elastic Critical Moment (%M) obtained theoretically and using FEM Simulation

software ANSYS w.r.t different L/D ratio for different channel section beams subjected to UDL of

1KN/m applied on Top web of the beam.

L/D

RATIO

ISMCP125

%M L/D

ISMCP150

%M L/D

ISMCP175

%M L/D

ISMCP200

%M

12.80 -0.12 10.67 4.75 9.14 9.12 8.00 9.89

17.60 -1.98 14.67 2.20 12.57 4.41 11.00 7.02

24.00 -2.64 20.00 0.85 17.14 2.16 15.00 4.76

32.00 -3.07 26.67 1.52 22.86 0.39 20.00 1.84

40.00 -3.32 33.33 -3.86 28.57 -0.61 25.00 0.56


different L/D ratio of channel section beam subjected to UDL on Top web of the beam.

Table 6 Variation of Elastic Critical Moment (%M) obtained theoretically and using FEM Simulation software

ANSYS w.r.t different L/D ratio for different channel section beams subjected to UDL of 1KN/m applied on

Mid web of the beam.

L/D

ISMCP125

%M L/D

ISMCP150

%M L/D

ISMCP175

%M L/D

ISMCP200

%M

12.80 4.32 10.67 4.36 9.14 7.55 8.00 8.02

17.60 1.03 14.67 1.62 12.57 3.52 11.00 5.04

24.00 -0.44 20.00 0.82 17.14 1.84 15.00 2.60

32.00 -1.81 26.67 -0.62 22.86 0.40 20.00 1.41

40.00 -2.03 33.33 -2.51 28.57 -0.65 25.00 0.52


different L/D ratio of channel section beam subjected to UDL on Mid web of the beam.

-6.00

-4.00

-2.00

0.00

2.00

4.00

6.00

8.00

10.00

12.00

0.00 10.00 20.00 30.00 40.00 50.00%V

AR

IATI

ON

of

Mcr

L/D RATIO

ISMCP125 %M

ISMCP150 %M

ISMCP175 %M

ISMCP200%M

-4.00

-2.00

0.00

2.00

4.00

6.00

8.00

10.00

0.00 10.00 20.00 30.00 40.00 50.00% V

AR

IATI

ON

of

Mcr

L/D RATIO

ISMCP125 %M

ISMCP150 %M

ISMCP175 %M

ISMCP200%M



Table 7 Variation of Elastic Critical Moment (%M) obtained theoretically and using FEM Simulation

software ANSYS w.r.t different L/D ratio for different channel section beams subjected to UDL of

1KN/m applied on Bottom web of the beam.

L/D

ISMCP125

%M L/D

ISMCP150

%M L/D

ISMCP175

%M L/D

ISMCP200

%M

12.80 3.02 10.67 7.20 9.14 10.58 8.00 10.93

17.60 0.12 14.67 3.06 12.57 4.83 11.00 5.24

24.00 -0.61 20.00 1.20 17.14 2.08 15.00 2.60

32.00 -2.04 26.67 0.84 22.86 0.57 20.00 1.08

40.00 -2.59 33.33 -10.47 28.57 -0.30 25.00 0.43


different L/D ratio of channel section beam subjected to UDL on Bottom web of the beam.

Note:

Theoretical formula of Elastic critical moment is calculated using mono symmetric beam

formula

{[(

)

( )

( )

( )]

( )}

Elastic critical moment (ANSYS) =( )

( )

4. CONCLUSIONS

In the current study various factors which will affect the lateral torsional buckling have been

analyzed using codal formula given in IS: 800: 2007 ANNEX E in Clause 8.2.2.1 and

validated with ANSYS simulation program which works on Finite element method. After

analyzing the factors, the elastic critical moment, Mcr, have been evaluated for the four

different Indian standard medium weight channel section (ISMCP),cross section details taken

from Hot rolled steel section given in IS:808-1989.

The conclusions are presented below:

It is observed that mono symmetric formula in IS 800:2007 is giving elastic critical moment

conservative results vis-à-vis ANSYS result for slender beams but showing large variation for

stocky beams.

The results obtained from the IS code formula is compatible with ANSYS results for beams

having length to depth ratio between range 20 to 40.

-15.00

-10.00

-5.00

0.00

5.00

10.00

15.00

0.00 10.00 20.00 30.00 40.00 50.00

% V

AR

IATI

ON

of

Mcr

L/D RATIO

ISMCP125 %M

ISMCP150 %M

ISMCP175 %M

ISMCP200%M



Stocky beams having length to depth ratio less than 20 is showing a percentage variation more

than 5%.

The stocky beams have much higher post yielding capacity than slender beams.

The results obtained from ISCODE stipulation are on the safer side for slender beams for

design purpose.

REFERENCES

[1] Clark, J. W. and Hill, H. N. [1960], “Lateral Buckling of Beams”, J. of the Structural

Division, Vol. 86, No. ST7, July 1960, ASCE, pp. 175-196.

[2] Peköz, T, “Lateral Buckling Of Singly Symmetric Beams” Eleventh International

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[3] Trahair N.S. (1993): Flexural-Torsional Buckling of Structures, CRC Press, Boca Raton,

1993

[4] Salmon, C.G., and J.E. Jhonson[1996], Steel Structures, Design and Behavior, 4th edn,

Harper Collins, NY, 1996, P.1024.

[5] F. Mohri , A. Brouki, J.C. Roth, “Theoretical and numerical stability analyses of

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[6] Amin Mohebkhah, “The moment-gradient factor in lateral– torsional buckling on

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[7] Avik Samanta, Ashwini Kumar, “Distortional buckling in monosymmetric I-beams”

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[8] H.H. (Bert) Snijder , J.C.D. (Hans) Hoenderkamp , M.C.M (Monique) Bakker

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[9] Martin Ahnlén, Jonas Westlund ”Lateral Torsional Buckling of I-beams” Division of

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[10] R. Kandasamy , R. Thenmozhi , L.S.Jeyagopal “Flexural -Torsional Buckling Tests of

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(2014), pp. 1176-1187

[11] Hermann Þór Hauksson, Jón Björn Vilhjálmsson “Lateral-Torsional Buckling of Steel

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[12] L. Dahmani, S. Drizi, M. Djemai, A. Boudjemia, M. O. Mechiche ”Lateral Torsional

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[13] Jan Barnata, Miroslav Bajera, Martin Vilda, Jindřich Melchera, Marcela Karmazínováa,

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[14] Amin Mohebkhah , Mojtaba G.Azandariani “Lateral-torsional buckling resistance of

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[15] Carl-Marcus Ekström, David Wesley, ”Lateral-torsional Buckling of Steel Channel

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[16] Karan Singh Saini “Lateral Torsional Buckling Of Hot Rolled Steel Beams” Division of

Structural Engineering Maulana Azad National Institute of Technology, Bhopal, India

Master’s Thesis (2017)

[17] IS 800 : 2007 General construction in steel – code of practice (third edition)

[18] Dimensions for Hot rolled steel beam, column, channel and angle sections ( Third

Revision ) IS 808-1989

[19] Subramanian, ”Design Of steel Structures”, Textbook

[20] .Ramchandra, ”Design Of steel Structures”, Textbook

[21] Timoshenko S.P. and Gere J. (1961): Theory of Elastic Stability (2nd ed.), McGraw-Hill,

New York, 1961

[22] ANSYSSoftwarehttp://www.ansys.stuba.sk/html/guide_55/g-str/GSTR7.htm

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