A Review on Simulation Analysis in Lateral Torsional
Buckling of Channel Section by using Ansys Software
Akshay Kumar Rathore1, Nitesh Kushwah2
1Student, 2Assistant Professor, 1,2Millennium Institute of Technology, Bhopal, Madhya Pradesh, India
ABSTRACT
Design rules for eccentrically loaded beams with open channel cross- sections are not available in Indian code IS 800-2000 general construction in steel-code of practice (third revision). In this study Indian standard medium weight parallel flange ISMCP 175, ISMCP 200, ISMCP 3000 channel beams are used; different span length to section height ratio of the beams is taken. The type of loading and the uniformly distributed load application are limited through the web of the channels. General solutions like Elastic critical moment, Slenderness, Reduction factors ? and the Ultimate loads are determined by using formula given in ANNEX E (CL.8.2.2.1,IS 800:2007) for mono symmetric beams and compared with NEW DESIGN RULE (snijder) and with Finite Element (FE) simulations on the basis of a parameteric study using ANSYS software 14.0. It is was noticed that mono symmetric formula in code is giving elastic critical moment results upto 0.3% difference with ANSYS result for slender beams but showing larger difference for stocky beams. As the size of beam is increasing with constant cross section it is resulting in reduction in design capacity. The design curve for channel beam proposed by snijder seems to be a good choice , taking torsional effect into account, but it dosent claim to be correct for beams with a ratio L/h<15. The results obtained from the IS code formula is matching with ANSYS results for beams having length to depth ratio between range 20 to 40.
KEYWORDS: Channel Beam, Finite Element Modelling, Symmetric Beam, Lateral
Torsional Buckling, Elastic Critical Moment, Slenderness Factor, Reduction Facto
Rolled Channel steel beams are regularly used as Purlins to support roofs in truss members, Staging to support bridge decks, etc., As steel beams tend to be slender, lateral displacement and twisting of a member occurs when load is applied on it results to buckling, this phenomena is known as lateral torsional buckling. This failure is usually visible when a load is carried out to an unconstrained rolled steel channel beam in which two flanges performing differently, where upper flange is in compression and the bottom flange is in tension. In this flange under compression first tries to move laterally and then twist causes buckling in compression flange of simply supported beam. The twist happens in a case for channel sections, when the shear centre does not coincide with the vertical axis of the center of gravity of channel beam. The carried out load will unavoidably cause a torsional moment in the beam, which makes it tough to find elastic critical moment Mcr. Indian standard code IS 800-2000 General Construction in steel-code of practice (third revision) doesn’t provides any formula to calculate theoretical elastic critical moment for channel beams. The formula is given for symmetrical sections which is symmetrical about both the axis and for mono symmetric sections which is symmetrical about only minor axis.
But C channel is a mono-symmetrical section which is symmetric about minor axis. ANSYS Workbench as a modern approach to finite element method is design software is used for advance engineering simulation purpose. The process consists of three stages Preprocessing, Solution and post processing. The disadvantage of this approach is that it is able to be very time eating and consequently now not constantly costeffective. 1.2. SOFTWARE INTRODUCTION:
GENERAL OVERVIEW OF FEM-PROGRAM ANSYS
ANSYS, Inc. is a company in the USA that develops computer-aided engineering software (CAE) which gives the user the ability to analyse and simulate different situations concerning electronics, fluid dynamics and structural analysis. The software is centered on an instance called Workbench where the simulation is set up in a treelike manner, where different components are dropped to the canvas and interconnected to other components. In Engineering Data, which is present in each of the analysis components, the material is specified and can also be viewed in Workbench. The below figure shows that static structural is followed by Eigen value buckling for load factor analysis.
IJTSRD30036
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Figure1.1 shows that static structural is followed by Eigen value buckling for load factor analysis.
ELASTIC CRITICAL MOMENT IN ANSYS 14.0
ANSYS can evaluate the critical load in two ways; by using a linear buckling analysis (Eigen buckling) or by a non-linear buckling analysis. While doing a FEM analysis for a structure, generally an Eigenvalue buckling analysis is performed.
EIGENVALUE BUCKLING ANALYSIS
It predicts the theoretical buckling strength for an ideal linear elastic structure. This analysis is corresponds to the textbook approach to elastic buckling analysis: for instance, an eigenvalue buckling analysis of a column will match the classical Euler solution.
PROCEDURE FOR SIMULATION IN ANSYS In ANSYS, modeling and analysis include three steps as follows: 1. Preprocessing 2. Solution 3. Post processing � PREPROCESSING
It is the first step to analyze the physical problem. In this model first the engineering properties were given as shown in figure 1.2 then we go for making geometry
Figure1.2 Engineering library in ANSYS
Figure1.3 sketching of channel beam model
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� SOLUTION: In this boundary conditions, meshing ,and loading are been applied in static structural multiple system to the model.
Figure1.4 shows model with boundary conditions and loading
� POSTPROCESSING:
The load factor is obtained in Eigen value buckling by linking it with static structural.
Figure1.5 Eigen value buckling analysis
LITERATURE REVIEW 2.1. RELATED WORK
Teoman Pekozi [2018] verified the general solutions for elastic lateral buckling moment of singly symmetric section. The impact of the location of the load on the beam section as well as the impact of moment gradients on the lateral buckling moment. Design provisions are outlined for the case of moment gradients. General design procedure for calculating elastic lateral buckling moment of singly symmetric beams has been produced. F. Mohri et.al [2016] has studied the overall stability of unrestrained thin-walled open sections members. A model is evolved for beam lateral buckling balance evaluation. In keeping with the European steel beams. Stability is depends on bending distribution, on load height effect and on degree of the mono symmetry of the section referred as Wagner’s parameter. Coefficients C1, C2 and C3 are respectively affected to those parameters and are given for a few selected load instances. From the present model, coefficients were recomputed for simply supported beams. Ritz and Galerkin’s methods are applied. Following Galerkin’s method, the coefficients C1 and C2 are the same as those adopted in Euro code 3, but some of Wagner’s proposed C3 coefficients are very different from those usually used. The different solutions were calculated and then compared with finite element results. ABAQUS software was used for numerical simulations where three-dimensional beams with warping and shell elements are modelled for lateral buckling phenomena. The results are close to shell element results but solutions overestimate the lateral buckling resistance of mono-symmetric I beams. Coefficients C1, C2 and C3 are computed for some selected load cases and compared to the standard coefficients adopted in Euro code 3. It is found that few coefficients are the same as which adopted in Euro code 3, but the C3 are very different for some load cases. Numerical simulations were done to calculate critical buckling moment and critical loads. Three-dimensional beams including warping are use in the stability of bisymmetric I sections, but thin-walled shell elements are desired for the stability evaluation of mono-symmetric I sections. The comparison examples studied have proven that the modified solutions are in agreement with shell element outcomes and that most of the regular solutions overestimate the real lateral buckling moments of mono-symmetric I sections and consequently the resistance of the beams in lateral buckling behavior.
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Samanta et.al [2015] deals with distortional buckling of simply supported monosymmetric I- beams and they considered three types of load in this first a point load, second a uniformly distributed load and third a uniform sagging moment. This total investigation has done using ABACUS software. Load has been applied on top flange and bottom flange position for the first two load cases. Analysis has been done by calculating moment modification factors that accounts for distortion of web and then are compared with those based on SSRC guide lines, which are based on lateral torsional buckling analysis only. They concluded that in short beams buckling may be governed by distortion of web. And in short beams, provisions given in Structural Stability Research Council (SSRC) Guide-1998 overestimate the critical load. Snijder et.al [2013] propose new design rule for calculating new slenderness ratio and reduction factor. Authors has summarized five design rules, and also checked their validity by finite element analyses. Then the ultimate loads yielded by these five design rules were compared with geometrical and material nonlinear analyses of imperfect (GMNIA) beams with channel cross section. The study was performed by varying the dimensions of the cross section, and the beam length to section height ratio in the beam, the load was applied through the web of the channel. New design rule has been proposed, which can be safely applied for all vertical load application lies between shear center and center of the web. Hamid et.al [2010] presented the results of lateral torsional buckling obtained by finite element analyses and theoretically of I- girders with corrugated webs and lateral bracing, under uniform bending. They developed a three dimensional finite element model using ANSYS 10.0 of the I –girder with corrugated webs for the lateral torsional buckling analysis and the effects of lateral bracing stiffness on the critical moment of simply supported I- girders with corrugated webs under pure bending. It was found that for plastic and inelastic I-girder with corrugated webs, the effect of bracing initially is increased to some extent as the lateral unbraced length increases and then decreased until the beam behaves as an elastic beam. In other words, the effect of bracing depends not only on the stiffness of the restraint but also on the modified slenderness of the I-girder. Also, the results show that winter’s simplified method to determine full brace requirements cannot be applied to I- girders with corrugated webs. Therefore, a general equation is proposed to determine the value of optimum stiffness in terms of the I-girder's slenderness. Martin Ahnlén et.al [2008] study the limitations and assumptions of the various design software’s by doing a simulation analysis on a single spanned I beam. They conducted simulation by using computer programs like ADINA, COLBEAM, LTBEAM AND SAP 2000.In simulations the parametric study was carried out on single spanned IPE500 steel beam by calculating the C-factors using the Finite Element Analysis from different software programs and comparing it with the C- factors given for load cases in EURO CODE. They concluded that the observation of the study was C1 not only depend on the moment distribution, but also on the lateral restraints and the length of the beam. The main observation of the study was that C1 not only depends on the moment distribution, but also on the lateral restraints and the length of the beam. The general problem was to find accurate C- factor based on the 3 factor formula. It also observed that the point load application had more influence on elastic critical moment and this influence is of the greater magnitude when the beam is fixed about major axis. R. KANDASAMY et.al [2008] study the influence of warping and torsional restraints on flexural capacity and also the influence of buckling length for different boundary conditions to calculate critical flexural torsional buckling moment of lipped channel beams under restrained boundary conditions First flexural test for two point load have been applied to 3m span and uniform bending moment is obtained. The section size selected were 100x50x10mm, 100x50x15mm and 100x 50x20mm with 1.6mm and 2.0mm thickness for investigation. The test results were compared in the BS5950: part 5 and IS code 801-1975. It has been concluded that warping restraint results more increase in flexural capacity (i.e., 46.67%) to cold formed steel beams than torsional restraint under restraint boundary conditions. These results would be used to develop design rules for CFS lipped channel beams more accurately under restrained boundary conditions subjected to flexural bending. Hermann et.al [2007] calculate elastic critical moment for eccentrically loaded beams and compare them with different computer program software’s. First the elastic critical moments have been calculated by taking different beam lengths, different types of load and different eccentricities of the load, including centric loading was calculated using the five computer software’s they are ADINA, COLBEAM, LTBEAM, SAP2000 and STAAD pro and are compared with result got by an analytical expression called 3–factor formula taken from EURO code. They concluded that the methods used in the study were not successful in obtaining the elastic critical moment for the eccentrically loaded channel beams. The values obtained through different software are not accurately matching with each other. L. Dahmani et.al [2005] compare the ultimate loads based on the new rules for lateral torsional buckling of eccentrically loaded channel beams to the ultimate loads obtained from the Finite Element simulations. In this parametric study different lengths of beams with point load on different position of web is introduced and ultimate loads are calculated using the computer programming software i.e, ANSYS 14.0 and then validated with the Modified ��� Method. It has been concluded that new design rule method leaded to the underestimation of results even less than 0.5% of the ultimate lateral torsional buckling load of unrestrained beams obtained from Finite Element simulations. Jan Barnata et.al [2004] have performed experimental analysis of lateral torsional buckling on beams with single symmetric and slightly asymmetric cross section. First they have made 4 slightly welded differently I sections. Geometrical difference of cross section was made by slightly moving the flange at different positions test specimen were marked as H1A, H1B, H1C, H1D and H1E for mono symmetric cross-section and by the same way H2A to H4E for all asymmetric cross-sections. It was
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concluded that the real behavior does not change rapidly with the level of asymmetry of cross section rises. The important difference can be observed between H1 beam type, which was mono symmetric and all other type. Also concludes the presumption that final stress is a combination of stresses inflicted by bending and the torsion caused by the cross section torsional displacement.
Amin Mohebkhah et.al [2004] investigate the nonlinear Lateral Torsional Buckling analysis of unstiffened slender web plate girders (SWPGs) subjected to central loading. A 3d model is created which is an finite element model using the ABAQUS for the inelastic nonlinear flexural torsional analysis of SWPGs . The unbraced length and load height effects are evaluated for their moment gradient factor. Here they have concluded that the Cb facto provided by AISC in specification for structural steel buildings is generally conservative for the elastic and inelastic SWPGs under the considered point load cases. Also observed that applied load height doesn’t affect the relationships between moment resistance and unbraced length.
Carl-marcus [2003]calculate load carrying capacity of the of single span steel channel beams using computer programming software and compare it with different design approach like geometrically material nonlinear analysis (gmnia) and new design rule given by snijder. In this study first c-channel beam of different geometric sections, different lengths and load configuration were considered. While modelling in computer program i.e. Ansys using finite element method fork supported boundary conditions were given. The gmnia approach was utilised that takes geometric imperfections and material non linearity into account. The stress and deformation patterns were studied. It is concluded that plastic strength is not approached for stocky beams with eccentric load carrying capacity. Since the bottom flange moves laterally in the reverse direction to the upper flange. Beams failure mode is related to plastic capacity reached by cross section from major axis bending and restricted warping effects combined. The reduction factor χ increases for eccentric loading for smaller cross section size. The new design rule by snijder gives a curve which takes torsional effect also into account, except for the beams with l/h < 15. Extra care is taken for beams by limiting the reduction factor χ to 0.5.yielding started at top of the web for stocky beams, but for slender beams outer edge of the beam yield first. Karan Singh Saini [2000] Focus on elastic lateral torsional buckling moment for different beam sections, considering loading and a support condition are obtained and these are validated using FE modelling techniques. The effects of various parameters effecting lateral torsional buckling was studied and analyzed thoroughly. The results obtained in the thesis are validating the current ISCODE method for calculation of Mcr for doubly symmetric I section. The elastic critical moment obtained for Channel sections are not in complete agreement from Fem modelling technique. Hence it requires further modification for channel section. The software tool used for fem modelling technique is ANSYS version14.5. PROBLRM IDENTIFICATION & OBJECTIVES The purpose of the thesis is to analyze the authentication of the theoretical elastic critical moment, acquired from IS code via evaluating it with Finite element Modelling technique. And also to get further knowledge regarding behavior of lateral buckling of steel channel beams concerning the effects of slenderness, factor of load application and cross section size on deformations, stress patterns and load carrying capability.
3.1. PROBLEMS IDENTIFICATION
� Previous analysis is done on the lateral torsional buckling was manual based on Indian standard in which imagination of effect of lateral torsional on channel is difficult.
� No software analysis was done on channel section in designing of bridges.
3.2. OBJECTIVE OF DISSERTATION
The principle objective of this work is to analysis lateral torsional buckling. In order to achieve this further objective are despite here as follow: � To Study the factors which will affect the Lateral Torsional-Buckling of channel beam. � To find the Eigen buckling load factors using simulation software i.e., ANSYS WORKBENCH 14.0. � To compare elastic critical moment using formula which is given by 800:2007 clause 2.2.1 with Finite technique for
validation. � To present comparison by calculating slenderness and reduction factors by General method given in IS 800-2007 code and
with New Design Rule (Snijder) � To show bending capacity calculated using General method given in IS 800- 2007 code and with New Design Rule (Snijder)
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.
A parametric observation was conducted in which channel beams with specific dimensions, lengths and load conditions were modelled and analyzed in computer software i.e. ANSYS workbench 14.0.
Three cross sections were chosen ISMCP175, ISMCP200 and ISMCP300 of five different lengths i.e., 1600mm, 2200mm, 3000mm, 4000mm, and 5000mm. A uniformly distributed load of 100 kn/m is applied on each beam at the top, the middle and the bottom of the web respectively.
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Theoretical elastic critical moment was calculated from formula for monosymmetric section given in code IS: 800:2007, ANNEX E and Clause 8.2.2 and then validated using ANSYS workbench 14.0 by creating models, giving support conditions and loading. 4.1. COORDINATE SYSTEM ORIENTATION I have used here Cartesian coordinate system for orientation of 3 dimensional C channel as shown in figure 3.1 beams in which z axis indicates the direction along length of the beam. In the beam x–axis pointing up, y- axis pointing left side and z axis is pointing perpendicular to both axis. Here y axis indicates the major axis about which beam bends when uniformly distributed load or point load is acted upon perpendicular to beam and x-axis is the minor axis about which beam will laterally move due to load that results in buckling.
Figure4.1: X, Y, Z Coordinate orientation in C channel beam
4.2. CROSS SECTION FOR FEM MODEL The cross section details of the C channel beam have been acquired from “IS 808:1989 Dimensions for Hot Rolled Steel Beam” is in figure 3.2.
Figure 4.2 Notation of cross section from IS 808-1989 code Here � D: Depth of beam � t : Thickness of web � B: Width of flange on topand bottom R 1 : Radius of root 1 � R 2 : Radius of root2 � C y : Distance of center of gravity � e y : Distance of extreme fiber from y-y axis
4.3. CENTRE OF GRAGITY (Cg)
The center point of whole mass of the body where the twisting moment is zero is known as center of gravity. Can also be defined as the point where homogeneous mass of the body is directed at a point. The center of gravity in C channel section is located at within the section but not in the web. For channel section the distance from center of gravity of section to web center is denoted as Yg. As shown in figure 4.3
Figure4.3 Centre of gravity of C channel
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4.4. SHEAR CENTRE (SC) Shear center is the point only through which when the load passes results bending i.e, there no twisting of plane occurs only pure in- plane bending occurs. � In doubly symmetric sections the shear Centre will coincide with the Centre of gravity of member. � In single symmetric sections the shear Centre doesn’t coincide with the Centre of gravity of member.
C-channel section is a singly symmetric section whose shear Centre and center of gravity doesn’t coincide but available in same axis.
Figure4.4 Shear center Sc in channel section
Outside geometry shear center is located in channel beam. The horizontal distance between web center and shear center is calculated through equation.
�� = �2(�−2�)2��
………………………….3.1 4
4.5. DEGREES OF FREEDOM
Table 3.1 Refers to translation of beam joint in x,y, z direction and rotation about major and minor axis due to load application.
Translation in x, y, z direction
Rotation about x- axis
Rotation about y- axis
Rotation about z- axis
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4.6. TORSION The load acting on the beam which is restricted at ends by fork supported condition results a twisting nature due to increase in load but the cross section remains plain and undistorted, this phenomenon is called Torsion.
Figure4.6 Torsion of channel section in ansys
For channel beams the torsion equation is given in code IS 800-2007 ANNEX E clause (8.2.2.1)
b��2
��= ∑ …………………….3.2 3
Torsional formula for channel section �� = 2∗�∗�3+2∗ℎ����∗�3
3 ………………3.3
4.7. WARPING
In symmetrical and un-symmetrical sections elements will move out of plane when subjected to torsion. The distortion of flanges in flanges takes place due to warping as shown in figure 3.6. In this twisting of flanges will cause member to bend. In figure it shows the warping of a c channel section, in this flanges are not in plane.
Figure4.7 warping of channel section
The moment of inertia due to warping �� for c channel section is taken as
�� = ���3ℎ2 ( 3��� + 2ℎ�� )
12 6��� + ℎ�� ... … … … … … … 3.4 Where � ℎ = (� − ��) =height of web in excel � �� = thickness of web � �� = thickness of flange
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4.8. POINT LOAD APPLICATION Point load application is the distance between points where the load is applied in the beam to the shear center. The load can’t be place on the shear center point without any connection to other member. So the load applied on the top of web is making an eccentricity with center of gravity and shear center. The load on top of web results more torsion than the load on mid web and bottom of web.
Figure4.8 Point load on different position of web
4.9. BOUNDARY CONDITIONS
Buckling phenomenon for beam can be analyzed for simply supported condition by providing fork supports on each end as shown in figure. In a fork support translation in y and z direction not allowed and in x direction translation is restrained at one end and on another side is released. Rotation is allowed about y and z axis but prohibited about x axis.
Figure4.9 Fork support
4.10. EFFECTIVE LENGTH ������������ Effective length depends upon the different support conditions. For simply supported beam if it is not provided with any lateral restraints along the length then the effective length will be equal to its actual length. If lateral restraints and warping restraints are provided to its web and flanges then its effective length will be less than actual length. So effective length directly depends upon lateral rigidity and torsional restraints torestraints.
Table 3.2 is referred from IS 800-2007, table no 15 provides the effective length for simply supported beam with
fully restraint and not allowed to warp on compression flange and tension flange, given below table shows effective length
Torsional
restraint
Warping
Restraint
Destabilizing
load
Normal
load
Fully restrained Only Compression flange
restrained fully 0.90L 0.75L
Fully restrained Both the flanges restrained fully
0.85L 0.70L
Fully restrained Only Compression flange
restrained partially 1.0L 0.85L
Fully restrained Both flanges are
not restrained against warping 1.2L 1.00L
Partially restrained by bottom flange support connection
Warping not restrained in both flanges 1.2L+2�2 1L+2D
Partially restrained by flange
Bearing support at bottom Warping not restrained
in both flanges 1.4L+2D 1.2L+2D
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Note � Torsional restraint prevents rotation about the longitudinal axis. � Warping restraint prevents rotation of the flanges in its plane. � D is the overall depth of the beam. 4.11. CROSS SECTION CLASSIFICATION
The strength of the structural steel components requires the designer to consider first the cross section behavior weather the material is in elastic range or inelastic range. In code IS 800, four behavior classes are given for cross section depending upon material yield strength, the width to thickness ratios of the individual components within the cross section and the loading arrangement. The four classes as shown in figure3.9 of section are defined as follows
Plastic or class 1: cross sections which can develop plastic hinges and have the rotational capacity required for the failure of the structure by the formation of a plastic mechanism.
Compact or class 2: cross sections which have inadequate plastic hinge rotation capacity because of local buckling but can develop their plastic moment resistance.
Semi compact or class 3: cross section in which elastically calculated stress in extreme compression fiber of the steel member can reach yield strength but plastic moment is prevented.
Slender or class 4: cross section in which local buckling will occur even before the formation of yield stress in one or more parts of the cross section.
Figure 4.11 Section classification based on Moment Rotation Characteristics
THE THEORETICAL BACKGROUND TO STRUCTURAL INSTABILITY
INSTABILITY OF BEAMS
The most important design consideration in much structural system is buckling, this is due to instability of structural elements (HERMANN ÞÓR HAUKSSON, JÓN BJÖRN VILHJÁLMSSON 2014) which can lead to catastrophic consequences
Table 4.11: Structural elements that can be divided into four categories
1.Having three dimension approximatelyequal – compact solid component
Ex: cube
2.One dimension is small compared to other two dimension
Ex:plate
3.One dimension is larger compare to other two equal components
Rectangular beam
All three dimensions are of different sizes
I beam
� In this thesis study of instability of category 4 structural component is done.
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4.12. PRINCIPAL STATES OF STABILITY To understand the stability concept in a structural system let us consider a ball placed at different position on different curves given in fig 4.1. A. A ball at position A will be considered stable because if we apply any force on the ball it will regain its original position. B. A ball at position B will be considered unstable because if we apply any force on the ball it will not regain its original
position. C. A ball at position C will be considered neutral because if we apply any force on the ball it will go to a new position but not
regain its original position
Figure4.12 Three principal states of stability
4.13. EULER’S THEORY OF ELASTIC COLUMN BUCKLING
The first study on the column strength and stability were given by Erone of Alexandria (75 B.c) and then it was continued by Leonardo Da Vinci(1452- 1519),the theory of column buckling was first originated by Euler during 1744-1759 moshenko 1953: Euler 1759). Euler considered an ideal column with the following attributes. 4.13.1. Material is perfectly elastic, homogeneous and isotropic 4.13.2. The column is initially straight and the load acts along the centroidal axis (no eccentricity of loads) 4.13.3. Columns has no imperfections 4.13.4. Column ends are hinged
Figure4.13 both ends are hinged
Such columns are identified as Euler column and is shown in figure 4.13 Experiment has been conducted by placing a concentrated load p on the upper end of the member, which remains straight until buckling occurs, when it is slightly bent as shown in fig At any location x, the bending moment M on the member bent slightly about the principal axis is
� = � ∗ � … … … … … … … … .4.1 And since
�2 = − �
��2 � … … … … … … … . . … … .4.2
Where E is young’s modulus and I is the moment of inertia the differential equation for the member becomes
�2 + �
��2
( � ) = 0 … … … … … … 4.3 After letting %2=�/ �, the solution of this second order differential equation may be expressed as:
= &1'�(� + &2�)'%� … … … … … … ... 4.4
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Where &1and &2are unknown coefficients. Applying the boundary conditions (a) y=0 at x=0 and (b) y=0 at x=0 and (b) y=0 at x=L, one may obtain for condition (a),
&2 = 0 ………………….4.5
And for condition (b), 0 = &1'�(%�*2 …………………..4.6
The above equation is satisfied if KL = N+. Thus we get,
� = +2 � 4+2 � (2+2 �
,2 ,2 , … … . . , ,2 … … … … … … … … … . . . 4.7
The fundamental buckling mode, with a single curvature deflection (y=A1 (Sin∏ x/L)) will occur when N=1. Thus Euler critical load for column with both ends hinged is
��- = +2 � … … … … … 4.8 ,2 Or in terms of average critical stress, using
� = &0-2 … … … … … … … 4.9
��- = ��- = +2 � … … … … … .4.10
& *2
Where λ is the slenderness ratio defined by
, * = … … … … … … … . .4.11 - In below figure 4.3 shows a graph plotted between buckling force and column slenderness to get elastic critical buckling curve for columns proposed by Euler’s theory. The Py indicates the yielding force of entire crossecton.
Figure4.13 Eulers Curve
From this eulers curve it indicates that how column behave, how its stability gets effected due to increase or decrease in load P. � If P < Py only yielding of column takes axially no deflection. � If P> P y, column undergoes large deflection results to buckling. 4.14. RESIDUAL STRESS
Plays an important part in distortion of beam components. It is the stress which remains in the body before load is placed on it. This residual stresses inbuilt due to cooling and heating of beam while processing in manufacturing plants. Residual stress is not uniform in a I- section cross section and the stress formed due to loading is uniform over flanges and linear over web. The combination of the residual stress and stress due to load results in actual stress which is more over outer edge of flange that makes more possibility to fail first at flanges. The figure 4.4 represents the residualstress.
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Figure4.14 Combined effect of residual stresses to the left and normal stresses caused by bending moment from applied load in the middle
4.15. IDEAL AND REAL BEAMS BEHAVIOUR IN CASE
OF BUCKLING
The assumptions made theoretically are not realised generally in practice. This topic is about the behaviour of real beams which do not meet all the assumptions in the buckling theory. � Theoretically we consider beam as initially perfectly
straight undeformed as geometrically but actually in real beams it can be a deformed one, slightly deflected and also twisted about its plane.
� While loading, theoretically we consider loading applied in the plane of major axis only but actually in real cases the load may be not vertically applied and can be eccentrically applied.
� Theoretically we consider beams stress free, perfectly elastic and uniform cross section but we avoid residual stresses present, how much section is yielded partially and variation of yield stress along and across section variations in cross sectional dimensions.
So in order to understand the behaviour of real beams, it is necessary to consider the combined effects of instability and plasticity and also the role of factors such as residual stress and geometric imperfections. 4.16. INSTABILITY PHENOMENON IN BEAMS 4.6.1. FLEXURAL BUCKLING
Flexure is due to bending of beam deforming vertically downwards, here there is no twisting takes place. So the response to this behavior is in-plane bending. Flexural Buckling is represented in figure 4.5.
Figure4.5 Flexural buckling of I beam
4.6.2. TORSIONAL BUCKLING
The compression force on the beam which leads to twisting may generate torsion in beam as shown in figure 4.6. The support condition resisting torsion produces torsional buckling over an effective length.
Figure 4.6 Torsional buckling
4.6.3. DISTORTIONAL BUCKLING This type of buckling occurs when the upper flange which is carrying compressive forces when is restrained against lateral moment this may causes distortion on the bottom flange which is not restrained, as shown in figure 4.7
Figure4.7 Distortional buckling
CONCLUSION: In the current thesis 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 three different Indian standard medium weight channel section (ISMCP),cross section details taken from Hot rolled steel section given in IS:808-1989.Various mono symmetric channels have been modelled using ANSYS software tools and the beam is subjected to uniformly load for laterally unrestrained condition. The conclusions from this master’s thesis project are presented below: � It is observed that mono symmetric formula in code is
giving elastic critical moment results upto 0.3% difference with ANSYS result for slender beams but showing larger difference for stocky beams.
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� As the length of beam is increasing with constant cross section it is resulting in reduction in design capacity.
� The stocky beams seem to approach full plastic cross-section capacity for a load that the slender beams seem to approach elastic buckling.
� The stocky beams have much higher post yielding capacity than slender beams.
� The design curve for channel beam proposed by snijder seems to be a good choice , taking torsional effect into account, but it doesn’t claim to be correct for beams with a ratio L/h<15.
The results obtained from ISCODE stipulation are on the safer side for slender beams for design purpose. FUTURE SCOPE OF WORK:
� This research work is done on the use of Ansys software, In future researches other softwares can be used for analysis.
� Future researches can be carried out on different types of sections other than channel section.
In future study other type of bridges can be considered. REFERENCES:
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