International Journal of Advanced Engineering Research and Studies E-ISSN

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1 Researh Paper FINIE ELEMEN ANALYSIS OF A CRACKED CANILEVER BEAM Mihir Kumar Sutar Address for Correspondene Researh Sholar, Department of Mehanial & Industrial Engineering Indian Institute of ehnology Roorkee, Roorkee, Uttrakhand, INDIA, ABSRAC his paper desribes the finite element analysis of a raked antilever and analyzes the relation beteen the modal natural frequenies ith rak depth, modal natural frequeny ith rak loation. Also the relation among the rak depth, rak loation and natural frequeny has been analyzed. Only single rak at different depth and at different loation are evaluated. And the analysis reveals a relationship beteen rak depth and modal natural frequeny. As e kno hen a struture suffers from damage its dynami property an hange and it as observed that rak aused a stiffness redution ith an inherent redution in modal natural frequenies. Consequently it leads to the hange in the dynami response of the beam. he analysis as performed using ALGOR softare. Modal natural frequeny as found to be dereasing ith inrease in rak depth. And the same as found to be inreasing ith inrease in rak loation from the fixed end. KEYWORDS: Free Vibration; Crak; antilever Beam; Modal Natural frequenies; ALGOR 1. INRODUCION Being very ommonly used in steel onstrution and mahinery industries, health monitoring and the analysis of damage in the form of rak in Beam strutures poses a vital mean. Sine long efforts are on their ay to find a feasible solution for rak detetion in beam strutures in this regard many approahes have so far being taken plae. When a struture suffers from damages, its dynami properties an hange. Crak damage leads to redution in stiffness also ith an inherent redution in natural frequeny and inrease in modal damping. he paper gives a feasible relationship beteen the modal natural frequeny and the rak depth at different loation. Sine free vibration analysis has frequently beome a topi of many studies therefore attention is foused it only. Crak loalization and sizing in a beam from the free and fored response measurements method is indiated by Karthikeyan et al. [1]. In the beam imoshenko beam theory is used for modeling transverse vibrations.fem is used for the free and fored vibration analysis of the raked beam and open transverse rak is seleted for the rak model. Being iterative in nature the iteration starts ith a guess for the rak depth ratio and iteratively estimates the rak loation and rak depth until the desired onvergene for both is reahed. he amount of literature related to damage detetion using shifts in resonant frequenies is quite large. Salau and Williams [] presents an exellent revie on the use of modal frequeny hanges for damage diagnostis. he observation that hanges in strutural properties ause hanges in vibration frequenies as the impetus for using modal methods for damage identifiation and health monitoring Kim and Zhao [3] proposed a novel rak detetion method using harmoni response. It as onluded in their paper that slope response has a sharp hange ith the rak loation and depth of the rak and therefore it an used as a rak detetion riterion. A fault diagnosis method based on geneti algorithms (GAs) and a model of damaged (raked) struture is proposed by aghi et al. []. In their approah the identifiation of the rak loation and depth in the antilever beam is formulated as an optimization problem, and binary and ontinuous geneti algorithms (BGA, CGA) are used to find the optimal loation and depth by minimizing the ost funtion. Ratliffe [5] performed the frequeny and urvaturebased experiments. Orhan [6] in his study analyzed the free and fored vibration of a antilever beam in order to identify the rak of a antilever beam. Single and to edged rak ere mainly evaluated in his study. he investigation reveals that free vibration analysis provides suitable information for the detetion of single and to raks; hereas fored vibration an detet only the single rak ondition. Hoever, dynami response of the fored vibration better desribes hanges in rak depth and loation than the free vibration Chang and Chen [7] presented tehnique for struture damage detetion based on spatial avelet analysis and the innovation of the proposed method is that both the positions and depths of multi-raks an be estimated from spatial avelet based method. First, the mode shapes of free vibration and natural frequenies of the multiple raked beams are obtained. It as observed from the analysis that the positions and depths of the raks an be predited ith aeptable preision even though there are many raks in the beam.. FINIE ELEMEN FORMULAION.1 Governing Equation of Free Vibration he free bending vibration of an Euler-Bernoulli beam of a onstant retangular ross-setion is given by the folloing differential equation as given in: d y EI mi y 0 dx = 1 Where m is the mass of the beam per unit length (kg/m), i is the natural frequeny of the ith mode (rad/se), E is the modulus of elastiity (N/m ) and I is the moment of inertia (m ). By defining λ = m / EI i equation (1) is rearranged as a fourth-order differential equation as follos: d y y 0 dx λ =... he general solution to equation () is

2 y= Aosλ x+ Bsinλ x+ Coshλ x+ Dsinhλ x i i i i 3 Where A, B, C, D are onstants and λi is a frequeny parameter. Sine the bending vibration is studied, edge rak is modeled as a rotational spring ith a lumped stiffness. he rak is assumed open. Based on this modeling, the beam is divided into to segments: the first and seond segments are left and right-hand side of the rak, respetively. Adopting Hermitian shape funtions, the stiffness matrix of the to-noded beam element ithout a rak is obtained using the standard integration based on the variation in flexural rigidity as e [ ] [ ] K = B(x) EI B(x) dx. Where[ ] { " " " " B(x) H } 1(x)H (x)h 3(x)H (x) =, 5 H (x), H (x), H 3(x), H (x) are the And 1 Hermitian shape funtions defined as Assuming the beam rigidity EI is onstant and is given by EI 0 ithin the element, and then the element stiffness is 1 6l 1 6l e EI0 6l l 6l l K = 3 l 1 6l 1 6l 6l l 6l l.10 Assuming the stiffness redution aused by as open rak falls ithin a single element, and then the stiffness matrix e K of the raked element an be ritten as e e K K = [ K ].11 Where [ K ] is the redution in the stiffness matrix due to the rak. Aording to Peng et al. [8], the K is matrix [ ] [ K ] k11 k1 k11 k1 k1 k k1 k = k11 k1 k11 k1 k1 k k1 k.1.18 l = 1.5d, and ξ is the distane beteen the left node and the rak. It is supposed that the rak does not affet the mass distribution of the beam. herefore, the onsistent mass matrix of the beam element an be formulated diretly as l e M = ρa[ H(x) ] [ H(x) ].19 0 And e have 156 l 5 13l e M = ρal l l 13l 3l l 156 l 13l 3l l l.0 Where [ H(x) ] = { H (x)h (x)h (x)h (x)} 1 3 In the dynami analysis, the system matrix is usually required to be inverted. From this aspet, a diagonalized mass matrix has a omputational advantage. In this study, a diagonalized mass matrix is adopted, hih is developed from the onsistent mass matrix using the approah e Al 0 l 0 0 = ρ M l.1 he natural frequeny then an be alulated from the relation [ ] [ ]{ } M + K q = 0. he natural frequeny of the ith mode for unraked and raked beams is finally obtained as follos Where

3 = EI i0 i = r ml EI i i i ml.3. Where i0 is the ith mode frequeny of the unraked beam and i is a onstant depending on the mode number and beam end onditions (for lamped-free beam, i is and.03 for the first and seond mode, respetively), i is the i th mode frequeny of the raked beam. r I is the ratio beteen the natural frequenies of the raked and unraked beam. l is the length of the beam.. Governing Equation of Fored Vibration he Euler-Bernoulli beam is disretized into finite beam element ithout rak and an be ritten as.. [ M] q(t) + [ K ] { q(t) } = { F(t) }.5 Where [ M ] is the element mass matrix, [ ] is the element stiffness matrix, { } F(t) is the element external fore vetor, { } q(t) is the element vetor of nodal degree of freedom and t is the time instant. he subsript represents ithout rak. And the subsript e represents element and the dot represents the derivative ith respet to the time. he rak is assumed to affet only the stiffness. So the equation of a raked beam element an be expressed as.. [ M] q (t) + [ K ] { q (t)} = { F(t) }.6 Where { q } (t) is the nodal degree of freedom of the raked element, the subsript represent the rak and [ K ] is the stiffness matrix of the raked element and is given as 1 [ K ] = [ ][ C] [ ].7 With [ C] = [ C ] [ ] 0 + C.8 Where is the flexibility matrix of the [ C ] 0 unraked beam element, [ C ] is the flexibility matrix of the rak, and [ C ] is the total flexibility matrix of the raked beam element. Equations of motion of the omplete system an be obtained by assembling the ontribution of all equations of motions motion for raked and unraked elements in the system. hen the system equation of motion beomes.. [ M ] q(t) + [ K]{ q(t) } = { F(t) }.9 K is Where [ M ] is the assembled mass matrix, [ ] the assembled stiffness matrix, { F(t) } is the K assembled external fore vetor, and { q(t) } is the assembled vetor of nodal degrees of freedom of the system. Let the fore vetor be defined as { F(t) } = { F} e jt.30 Where is the foring frequeny, { F } is the fore amplitude vetor (elements of hih are omplex quantities) and j= 1. hus, the response vetor an be assumed as { q(t) } = { q} e jt.31 Where { q } is the response amplitude vetor and their elements are omplex quantities. Substituting the equations (30) and (31) into equation (9), the system governing as follos: ( [ M ] + [ K ]){ q } = { F } For a given system properties (i.e. [ ].3 K ) M and[ ] the response an be simulated from equation (3) F. orresponding to a given fore { } 3. CONFIGURAIONS OF SIMULAED CRACK In this partiular approah the free vibration of a antilever beam having V-shaped edge rak are studied. he length of the beam is 800 mm and the ross-setion of the same is 50 x 6 mm. As per the material properties the modulus of elastiity (E) is 70 x 10 9 N/m and the mass density (ρ) is 710 kg/m 3. Different rak onfigurations of same depth and at different loations (from different distane from the fixed end) are prepared to find out ho the rak affets the dynami behavior of the beam. Crak depth as kept onstant at 0.0 mm and the rak loation from the fixed end as varied at instanes of 50 mm, 100 mm, 00 mm, 300 mm, 00 mm, 500 mm, 600 mm and 700 mm. And the effet of rak loation on the natural frequenies as investigated. Also the rak loation from the fixed end as kept onstant and rak depth as varied from 0.05 to mm at eah step in order to investigate the effet of rak depth on natural frequenies. Figure 1 shos on of the total nineteen rak depths at a partiular rak loations here rak depth d varies from 0.05 to mm. Fig. 1 Single rak on the antilever beam at 300 mm length from the fixed end. HE FINIE ELEMEN MODELING ALGOR softare as used for the free vibration analysis of the unraked and raked beams. For this purpose a model of the unraked and raked beams as prepared in the CAIAV5R15 and the same ere brought into the ALGOR environment and the mesh as generated. he beam as disretized into

4 850 elements ith 1176 nodes. he antilever boundary onditions ere modeled by onstraining all degrees of freedom of the nodes loated at the left hand side of the beam. Figure shos the finite element mesh model of the beam element. hree modes ere seleted to extrat and first 3 modes of natural frequenies ere alulated for unraked and raked beam. his proedure as thereafter repeated for different rak senarios. he rak zone portion ith arro mark is being highlighted in figure 3. Fig. 6 hird mode Natural Frequeny versus rak depth Fig. Finite Element Model of antilever Beam in ALGOR environment Fig. 3 Details of rak zone 6. RESULS he hange in natural frequenies ith rak depth for three different modes is plotted in figure, 5 and 6. And it as observed that in all the ases the modal natural frequenies derease ith inrease in rak depth. Fig. 7 Comparison of First Mode Natural Frequeny at 50 mm and 500 mm As shon in Figure 7 as the rak loation from the fixed end inreases the modal natural frequenies also inreases. A omparison has been made beteen the First Mode of natural frequenies at rak lengths of 50 and 500 mm from the fixed end respetively at subsequent rak depths. Fig. First Mode Natural Frequeny versus rak depth Fig. 5 Seond Mode Natural frequeny versus rak depth Fig. 8 hree dimensional um ontour plot for relative First Mode Natural Frequeny he relation among the rak depth, rak loation and natural frequeny has also being analyzed and is plotted as shon in the figure 8.Only First mode of vibration of vibration is onsidered for this omparison. And it as observed that as the rak loation from the fixed end inreases the modal natural frequeny also inreases. And hen the rak depth at a partiular rak loation inreases the modal natural frequeny dereases. REFERENCES 1. M. Karthikeyan, R. iari, S. alukdar (006). Crak loalization and sizing in a beam based on the free and fored response measurements. Mehanial Systems and Signal Proessing 1 (007). pp Salau, O. S. and Williams, C., 1993, Strutural Damage Detetion Using Experimental Modal Analysis

5 A Comparison Of Some Methods, in Pro. of 11th International Modal Analysis Conferene, pp Kim M-B, Zhao M. Study on rak detetion of beam using harmoni responses. Proeedings of the 00 international onferene on intelligent mehatronis and automation, August 00, Chengdu, China, p Mohammad-aghi Vakil-Baghmisheh, Mansour Peimani, Morteza Homayoun Sadeghi, Mir Mohammad Ettefagh (007). Crak detetion in beam-like strutures using geneti algorithms. Applied Soft Computing 8 (008). pp C.P. Ratliffe, Frequeny and urvature based experimental method for loating damage in strutures, J. Vibration Aousti. 1 (000) Sadettin Orhan (007). Analysis of free and fored vibration of a raked antilever beam. ND&E International 0 (007). pp Chih-Chieh Chang, Lien-Wen Chen(005). Detetion of the loation and size of raks in the multiple raked beam by spatial avelet based approah. Mehanial Systems and Signal Proessing. Vol.19. pp Peng, Z. K., Lang, Z. Q., Billings, S.A.(007), Crak Detetion using nonlinear output frequeny response funtions. Journal of Sound and Vibration, 301, pp