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 Total Downloads : 479
 Authors : Sudhir Sastry Y B, , Y Krishna, Pattabhi R. Budarapu , Anirudh Koduganti
 Paper ID : IJERTV3IS031854
 Volume & Issue : Volume 03, Issue 03 (March 2014)
 Published (First Online): 22042014
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
Flexural Buckling Analysis of Thin Walled T Cross Section Beams with Variable Geometry
Sudhir Sastry Y B Y Krishna
Dept. of Aeronautical Engg scientist G, Head Institute of Aeronautical Engineering Structural Test Facility
Hyderabad, India DRDL, Hyderabad
India
Pattabhi R. Budarapu Anirudh Koduganti
Institute of Structural Mechanics Dept. of Aeronautical Engg
Bauhaus University of Weimar Institute of Aeronautical Engg 99423 Weimar, Germany Hyderabad, India
Abstract: Thin walled structure is a structure whose thickness is small compared to its other dimensions but which is capable of resisting bending in addition to membrane forces. Which is basic part of an aircraft structure, the structural components of an aircraft consist mainly of thin plates stiffened by arrangements of ribs and stringers. Thin plates (or thin sections or thin walled structures) under relatively small compressive loads are prone to buckle and so must be stiffened to prevent this. The determination of buckling loads for thin plates in isolation is relatively straightforward but when stiffened by ribs and stringers, the problem becomes complex and frequently relies on an empirical solution. The buckling of the thin plates is a phenomenon which could lead to destabilizing and failure of the aircraft; in this paper it is considered T cross section with variable geometry and length. The critical buckling stresses have been studied for several combinations of the geometry parameters of the beam with the help of ANSYS and drown the result plots
Keywords: Thin walled beams, buckling analysis, Finite element analysis
I INTRODUCTION
A great deal of attention has been focused on plates subjected to shear loading over the past decades. One main fact in design of such elements, which fall in the category of thin walled structures, is their buckling behavior. Plate girders and recently shear walls are being widely used by structural engineers, as well as ship and aircraft designers. The role of stiffeners is proved to be vital in design of such structures to minimize their weight and cost.
Xiaoting et al [1] presented an analytical model for predicting the lateral torsional buckling of thin walled channel section beams restrained by metal sheeting when subjected to an uplift load. And calculated the critical load from critical energy theory and showed that the critical buckling moment in the pure bending case is less than half of the critical moment, it is more effective to use the anti sag bars in the simply supported beams than in the fixed beams, the closer the loading point to the centre the lower the critical load. M.Ma et al [2] developed energy method for analyzing the lateral buckling behavior of the monosymmetric I beams
subjected to distributed vertical load, with full allowance for distortion of web. the method assumes that the flanges buckle as rigid the rectangular section beams, but the web distorts as an elastic plate during buckling. it is shown that the disparity between the distortional and classical critical load increases as h/l increases and that for short beams the classical method seriously over estimates the critical load. B. W. Schafer [3] worked on coldformed thinwalled open crosssection steel columns and provided local, distortional, and flexural torsional buckling. Experimental and numerical studies indicated that post buckling strength in the distortional mode is less than in the local mode. In pinended lipped channel and zed columns, local and Euler interaction is well established. A direct strength method is proposed for column design. The method uses separate column curves for local buckling and distortional buckling with the slenderness and maximum capacity in each mode controlled by consideration of Euler equation. Attard Mario et al [4] investigated lateral torsional buckling behavior of opensection thinwalled beams based on a geometrically nonlinear formulation, which considers the effects of shear deformations, also made Comparisons between the results based on fully nonlinear analysis and linearized buckling analysis in order to illustrate the effects of prebuckling deformations as well as the shear deformations on the buckling load predictions. Ing. Antonin pistek,[5] analytical method for limit load capacity Calculation Of thin walled aircraft structures focused on description and Comparison of different methods for limit load Capacity calculation of thin walled aircraft Structures considering all possible forms of Buckling and failures on nonlinear behavior of The structure under gradually increased Loading. Carine Louise Nilsen, et al [6] found that the behavior of thinwalled steel sections, including local buckling, distortional buckling, global buckling and shear buckling have been well understood and appropriate design methods existed. Foudil Mohria et al [7] derived analytical solutions Based on a nonlinear stability model, for simply supported beamcolumn elements with bisymmetric I sections under combined bending and axial forces. Jaehong Lee et al [8] explained lateral buckling of thinwalled composite beams with monosymmetric sections. A general geometrically nonlinear model for thin walled laminated
composites with arbitrary open crosssection and general laminate stacking sequences is given by using systematic variational formulation based on the classical lamination theory. The load capacity of coldformed thinwalled beams is usually restricted by their stability and postbuckling behaviour. Strength was considered by Cheng and Schafer [9], Trahair [10] Experimental investigations, stress and displacement distribution of coldformed beams were shown by Paczos et al [11] Other examples of papers directly connected with the subject of this work are Biegus et al [12] Magnucki et al [13] Magnucki et al [14] Paczos, Jeyaragan et al [15] Lawrence W. Rehfield et al [16] design methodology for buckling of thinwalled laminated composite beams shoed buckling by pure bending and bending torsion coupled modes can occur and that transition among modes are governed by structural parameters. Tomasz kubiak et al [17] presented analysis of local buckling of thinwalled beam columns, taking account Global pre critical bending within the first order approximation. Marco et al [18] published a paper on buckling of thinwalled structures is presented using
However, we investigate first the nature of buckling and the difference between theory and practice. It is common experience that if an increasing axial compressive load is applied to a slender column there is a value of the load at which the column will suddenly bow or buckle in some un predetermined direction. This load is patently the buckling load of the column or something very close to the buckling load. Clearly this displacement implies a degree of asymmetry in the plane of the buckle caused by geometrical and/or material imperfections of the column and its load. However, in our theoretical stipulation of a perfect column in which the load is applied precisely along the perfectly straight centroidal axis, there is perfect symmetry so that, theoretically, there can be no sudden bowing or buckling. We therefore require a precise definition of buckling load which may be used in our analysis of the perfect column. Assume that it is in the displaced state of neutral equilibrium associated with buckling so that the compressive load P has attained the critical value PCR. Simple bending theory
the 1D finite element based refined beam theory formulation Deepak et al [19 cold formed c and z sections because of their easy connections but they suffer fromcertain buckling
d 2v EI dz 2
M or
d 2v EI dz 2
PCRv
modes. here the structural behavior of c channel lipped beams due to lateral buckling and load carrying capacity is evaluated. Load Vs deflection curves are plotted in
so that the differential equation of bending of the column is
d 2v P
comparison with the experimental results attained through fea CR v 0
modeling of the software as part of results. Brad ford [20] lateraldistortional buckling of steel I section members presented how the usual types of buckling of steel members assumed in design are lateraltorsional and local buckling modes. In lateraltorsional buckling, the crosssections of the member translate and twist as rigid bodies. On the other hand, local buckling is characterized by localized distortions of the crosssection over a short wavelength in the absence of lateral translation.

BUCKLING
dz 2 EI
The wellknown solution v Acos z Bsin z
2 EI
Critical load PCR l 2
Other values of PCR corresponding to n=2, 3,…,
A Thin plate or a thin walled structure is a structure whose thickness is small compared with its other dimensions but which is capable of resisting bending in addition to
PCR
42 EI
l 2
92 EI
, l 2
,….
membrane forces. Such a plate forms a basic part of an aircraft structure, being, for example, the area of stressed skin bounded by adjacent stringers and frames in a fuselage. The
The total potential energy of the column in the neutral equilibrium of its buckled state is therefore
structural components of an aircraft consist mainly of thin
EI l d 2v 2 P l dv 2
plates stiffened by arrangements of ribs and stringers. Thin
U V dz CR
dz
0
0
plates (or thin sections or thin walled structures) under relatively small compressive loads are prone to buckle and so must be stiffened to prevent this. The determination of
2 dz 2
2 dz
buckling loads for thin plates in isolation is relatively straight forward but when stiffened by ribs and stringers, the problem becomes complex and frequently relies on an empirical solution. The buckling of the thin plates is a phenomenon which could lead to destabilizing and failure of the aircraft, hence we study the buckling phenomenon on thin plates or thin walled structures with the help of the finite element
And is capable, within the limits for which it is valid and if
suitable values for the constant coefficients are chosen, of representing any continuous curve. We are therefore in a position to find PCR exactly.
4 EI 4 2 P 2 2
2
U V n An CR n An
analysis software ANSYS.
The first significant contribution to the theory of the buckling of columns was made as early as 1744 by Euler. [22] His classical approach is still valid, and likely to remain so, for slender columns possessing a variety of end restraints. Our initial discussion is therefore a presentation of the Euler
4l 3
In general form
n22 EI
n1
4l n1
theory for the small elastic deflection of perfect columns.
PCR l 2
PCR
42EI 17l 2
2.471 EI
l 2
T section thin walled beam flanges are behave similar to plate buckling, the following equations are valid for thin plate simply supported along all four edges.
mx ny
w= Amn sin
m1 n1
sin
a b
Also, the total potential energy of the plate is Y
X
U+V=
1 a b
2 w
2 w 2
2 w 2 w
2 w 2 w 2
T section case 1
2 D x2 y2 21 v x2 y2 xy Nx x dxdy
0 0
ES
The total potential energy of the plate has a stationary value in the neutral equilibrium of its buckled state i.e. Nx=Nx,CR
k2 D
ES
NUM DE
Y
X
Z
N x,CR b2
where the plate buckling coefcient k is given by the minimum value of
k mb
a 2
a mb
Where a is length of the plate, b is width of the plate, m and n are the number of halfwaves in the x and y directions,
The critical stress of the plate is given by the equation
T section case 2
2 Y
X
k 2 E t
Z
CR 121 v 2 b
The local failure stress in longitudinally stiffened panels was determined by Gerard using a slightly modified form
f gt t
1 m
E 2
g sk st
cy
A cy
Where g is number of cuts + Flanges

MODELING
Modeled the six types of thin walled T cross sections by using ANSYS, all the cases are with constant area of cross sections (63.36 mm2) by changing the length, thickness parameters
T section case 3
ES NUM
DE TS
Y
X
Z
ES NUM
T section case 4
DE
Y
X
Z
MENT
44E+09
2033
Y
X
Z
(a)
Y
X
Z
ES NUM
T section case 5
DE
(b)
Y
X
Z
EMENT
166E+09
0292
Y
X
Z
T section case 6
Fig.1 the different shapes of T sections considered for analysis in this paper
(c)
+09
5
Y Z
0
(AVG) 05E11
01844
46629
Y
X
Z
MX
MN
(d)
.501844 .716167 .930489 1.14481 1.35913
.609005 .823328 1.03765 1.25197
MENT
D
69E+09
2792
Y
X
Z
21
(AVG) 205E11
458392
.47957 Y
X
Z
MX
(e)
.458392
MN
.685321
.912251
1.13918
1.36611
.571856 .798786 1.02572 1.25265
MENT
172E+09
02562
Y
X
Z
(AVG) 205E11
434777
.30821
OCT 2
21
MX
(f)
.434777
YMN
X
Z .628873
.531825
.725922
.82297
.920018
1.01707
1.11411
1.21116
Fig.2 (a) Meshed model (b)(c)(d)(e)(f) are Some of the mode shapes of modal analysis
Fig.3 some of the stress figures

RESULT
Combined graph for all cases of T section, Combined graph for all cases of T section, length
lengtp50 mm: : 300 mm
Combined graph for all cases of T section, Combined graph for all T sections,
length 150 mm: length 300 mm:
Combined graph for all cases of T section Combined graph for all cases of T section Length450mm: Length 600 mm:
Combined graph for all T sections Combined graph for all T sections Length450mm: Length 600 mm:
Combined graph for all cases of T section Critical moment Vs length to Length750 mm: height ratio:
Combined graph for all T sections: Critical moment Vs length to height ratio:

CONCLUSIONS

Thinwalled T sections with different crosssections have been considered for buckling analysis in this paper.

It is observed that in buckling analysis, at mode 1, the value of buckling value is higher for normal T section whereas at mode 4, the T section with lipped flanges is giving higher buckling value than all other cases.

Thin walled T sections consider the following lengths 150 mm, 300mm, 450mm, 600mm, and 750mm. It is observed that the buckling stress decreased gradually with increase lengths of the thin walled beams.

REFERENCES

Xiaoting Chu, Roger Kettle, Longyuan Li. "Lateraltorsion buckling analysis of partiallaterally restrained thinwalled channelsection beams. Journal of Constructional Steel Research 60 (2004) 1159 117.

M.Ma&O. Hughes. Lateral Distortional Buckling of Monosymmetric IBeams under Distributed Vertical Load. ThinWalled Structures Vol. 28, No. 2, pp.123145, 1996.

B. W. Schafer. Local, Distortional, and Euler Buckling of Thin Walled Columns Journal of Structural Engineering, Vol. 128, No. 3, March 1, 2002

R. Emre Erkmen, Mario M. Attard. Lateraltorsional buckling analysis of thinwalled beams including shear and prebuckling deformation effects. International Journal of Mechanical Sciences – INT J MECH SCI 01/2011; 53(10):918925.

Prof. Ing. Antonin pistek. Analytical Method for Limit Load Capacity Calculation of Thin Walled Aircraft Structures 28th International Congress of the Aeronautical Sciences

Carine Louise Nilsen, Md Azree Othuman Mydin and Mahyuddin Ramli 2012" Performance of lightweight thinwalled steel sections: theoretical and mathematical considerations." Advances in Applied Science Research, 2012, 3 (5):28472859.

Foudil Mohria, Cherif Bouzerirab, Michel PotierFerryc of France in the year 2008" Lateral buckling of thinwalled beamcolumn elements under combined axial and bending loads." ThinWalled Structures 46 (2008) 290302.

Jaehong Lee. Lateral buckling analysis of thinwalled laminated composite beams with monosymmetric sections Engineering Structures 01/2006.

Cheng Yu, Benjamin W. Schafer. Simulation of coldformed steel beams in local and distortional buckling with applications to the direct strength method. Journal of Constructional Steel Research 63 (2007) 581590.

N. S. Trahair. Buckling Analysis Design of Steel Frames. Journal of Constructional Steel Research, 65(7), 14591463.

K. Magnucki, P. Paczos. Theoretical shape optimization of cold formed thinwalled channel beams with drop flanges in pure bending. Journal of Constructional Steel Research 65 (2009) 17311737.

A. Biegus, D. Czepiak. Experimental investigations on combined resistance of corrugated sheets with strengthened crosssections under bending and concentrated load. ThinWalled Structures (Impact Factor: 1.23). 01/2008; 46(3):303309.

K. Magnucki, M. Rodak, J. Lewinski. Optimization of mono and antisymmetrical Isections of coldformed thinwalled beams Thin Walled Structures 44 (2006) 832836.

P. Paczos, P. Wasilewicz. Experimental investigations of buckling of lipped, coldformed thinwalled beams with Isection. ThinWalled Structures 47 (2009) 13541362.

S. Jeyaragan and M. Mahendran. Experimental Investigation of the New Builtup Litesteel Beams. Fifth International Conference on ThinWalled Structures Brisbane, Australia, 2008.

Lawrence W. Rehfield and Ulrich Mueller. Design Methodology for Buckling of ThinWalled Laminated Composite Beams. ICCM – 12 Europe 1999.

Tomasz Kubiak. Interactive Buckling in Thinwalled Beamcolumns with Widthwise Varying Orthotropy. Journal of Theoretical and Applied Mechanics 44, 1, pp. 7590, Warsaw 2006.

Syed Muhammad Ibrahim, Erasmo Carrera, Marco Petrolo, Enrico Zappino 2012. "Buckling of thinwalled beams by a refined theory." J Zhejiang UnivSci A (Appl Phys & Eng) 2012 13(10):747759.

M.S.Deepak, R.Kandasamy, Dr R.Thenmozhi. Investigation on Lateral Torsional Buckling Performance of Coldformed Steel C Channel Sections. International Journal of Emerging Trends in Engineering and Development Issue 2, Vol. 4 (May2012).

M.A. Bradford. LateralDistortional buckling of steel ISection members Journal of Constructional Steel Research 23, 97116.

Sudhir Sastry YB, Y Krishna, Pattabhi R. Budarapu Parametric studies on buckling of thin walled channel beams Manuscript submitted to Journal of Computational Materials Science, Elsevier Editorial System

T. H. G. Megson, AircraftStructuresforEngineeringStudents FourthEdition