Pankaj Charan Jena 1, Dayal R. Parhi 2, G. Pohit 3. Pankaj Charan Jena et al. / International Journal of Engineering and Technology (IJET)

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1 Theoretical, Nmerical (FEM) and Experimental Analsis of composite cracked beams of different bondar conditions sing vibration mode shape crvatres Pankaj Charan Jena, Daal R. Parhi, G. Pohit 3 Department of Mechanical Engineering, Orissa Engineering College, Bhbaneswar, Odisha, India, 75050, Tel: , pankajcharanjena@ahoo.com. Department of Mechanical Engineering, N.I.T, Rorkela, Odisha, India, , Tel: , Fax: , daalparhi@ahoo.com. 3 Department of Mechanical Engineering, Jadavpr Universit, Jadavpr, Kolkata, India, Tel: (O), gpohit@vsnl.net, gpohit@gmail.com. Abstract: This manscript presents theoretical, experimental and nmerical (finite element methods) analsis of fibers composite beams with transverse crack sbjected to free vibration. Epox- glass fibers (nidirectional) composite has been taken as the material nder std. Vibration parameters have been evalated sing varios bondar conditions of beam. The beam sbjected to different bondar conditions has a principal effect on dnamic characteristics of composite beam. The crrent work presents the evalation of changes in natral freqencies and corresponding mode shapes crvatre for different bondar conditions b varing crack positions and crack depth. The nmerical reslts were fond in good agreement with experimental and analtical reslts. The std concldes that strctre with crack can be diagnosed b sing vibration signatres and it help to monitoring the health of beam tpe strctres. Kewords: crack, composite, beams, natral freqenc, mode shapes; finite element. Nomenclatre: A C ij E F i, H h I = cross-sectional area of the beam = flexibilit inflence coefficient of flexible matrix = ong s modls of elasticit of the beam material (i =, ) = experimentall determined fnction = thickness of the beam = depth of crack = moment of inertia i, j = variables J C = strain-energ release rate KIi,,(i =, ) = stress intensit factors for P i loads Kij = local flexibilit matrix elements = length of the beam = location (length) of the crack from one end Mi,(i=,4) = compliance constant Pi,(i=,) =P i = axial force (i=) and P j = bending load (i=) Ω = stiffness matrix (i=,) = additional displacement fnctions i U c W = strain energ de to crack = Breadth of the composite beam X, Y, and Z = co-ordinate of the composite beam = co-ordinate of the composite beam Y 0 = amplitde of the exciting vibration ISSN : Vol 6 No Apr-Ma

2 i (i=,) ωn( ω c = normal fnctions (transverse) i (x) = natral freqenc of ncracked composite beam = natral freqenc of cracked composite beam =Relative crack Depth (RCD) χ = /= relative crack position (RCP) ρ = mass-densit of the composite beam. Introdction Higher strain to weight ratio is a desirable characteristics in its applications to light weight strctres, aircraft, ships, atomobiles, Composite strctres have seen an increased applications in atomobiles, ship bilding, aerospace, mechanical, and civil strctre etc in the recent past. It has been established that the presence of crack, changes the natral freqenc of the strctre and its mode shape. In the present manscript efforts have been made to detect the crack localit and depth sing theoretical, experimental and nmerical analsis. Vision et al. [] have examined that the fibers composite orthotropic material properties exhibit to exemplif its sitabilit in high speed prposes of civil strctres and mechanical engineering components. Erdeli et al. [] have described mostl composites have different characteristics, sch as high strength to weight ratio, good bckling resistance, and high stiffness. Irwin [3] has explained that existence of cracks in beam kind strctres proves variation of local stiffness that considerabl affects on natral freqenc and mode shape of the beam. Gpta [4] has given an approach to std the stress vale arond the crack tip interface. Nikpr et al. [5] have evalated compliance matrix of composite cracked bodies. Sreekanth et al. [6] have stdied on composite strctre taking into accont varios forms of transverse cracks like, fibers fractre, matrix cracking and srface-breaking cracks with b spectral finite element methods. The established FEM-D model is more sitable for complex cracked beam to detect crack location and crack depth. Pande et al. [7] have stdied to identif the crack in beam sing Eigen parameters and conclded that mode shape crvatre is sitable to find ot damage position. Marek [8] has sed finite element approach on non propagating middle cracked beam to find ot its dnamic and static behavior. Amara et al. [9] have analzed mechanical properties of angle-pl composites laminates and stdied the effect of transverse cracks on the stiffness of laminated composite beam with different fibers orientations. Ramanamrth et al.[0] have presented the detection of damage in a composite strctre sing finite element approach. Nag et al. [] have modeled a laminate cracked composite sing node based D finite element approach based on spectral wave scattering. akshmi et al.[] have worked on detection of crack location b the first for torsional modes and bending modes and calclated corresponding changes in natral freqencies of a cracked beam. Mrat [3] has investigated the effects of cracks on the vibration characteristics of beam made of graphite fiber-reinforced polamide sing finite element methods. Sh et al. [4] have done analsis work on delaminated non overlapping composite beam sing free vibration techniqes b theoretical and experimental procedre to find the locations of delaminations. The have conclded that the mode shape and freqenc will change significantl in presence of delaminations. Singh et al. [5] have sed free vibration mode shape procedre on non niform composite to evalate its phsical parameter. The objective of this paper is to establish the mode shape crvatres behavior of a composite beam with a transverse open crack sbjected to free vibration. In the present std, a composite beam made p epox glass fibres is considered for analsis. The effort is made to std the changes in ω n i.e. natral freqenc and mode shapes in presence and absence of cracks in composite strctres, and the effort is also made for recognition of the cracks throgh non destrctive and inexpensive was.. Analtical Approaches In the present article, composite beam with varios end conditions has been considered for analsis with a single transverse crack. The mathematical analsis in the present std will lead to the derivation of stiffness matrix of the cracked beam. Referring to Irwin theor [3], the strain energ release rate at the cracked section can be expressed as, (K K ) E' J = C I + () I Where, the stress intensit factors are K I, K I of mode I (opening of the crack) nder load P and P respectivel. ν = (For plane strain condition) and E ' = (for plane stress condition) E E ' E ISSN : Vol 6 No Apr-Ma 04 50

3 The expression for stress intensit factors from earlier stdies Irwin et al. [3], are P h 6P h KI= πhf ( ( )), KI= πhf ( ( )) () WH H WH H Where parameters F and F can be presented as bellow, h H π h 3 ( ), ( tan( )) (h/ H) ( sin( π h/ H)) cos( π h/ H) F = H π h H h H π h ( sin( π h/ H)) ( ), ( tan( ) π cos( π h/ H) F = H h H et Uc be the strain energ de to the crack, then from Castiglione s theorem, the additional displacement along U c i = P the direction of force P i is (5) i The strain energ (Uc) can be related to strain energ release rate (J c ) as, h U c Uc = dh = Jc( h) dh h h c Where J C = 0 0 U h (6) From eqations (5) and (6), ths we have h i = Jc(h) dh P (7) i 0 Defining the flexibilit inflence co-efficient C ij per nit depth, i Cij = = Jc(h) dh P P P j i j h 0 Integrating over the breadth W, the final flexibilit matrix element can be obtained as, + W / h i Cij = = Jc (h) dh dz P P P (9) j i j W / 0 Using the vale of strain energ release rate from eqation () and ptting in eqation (8), h B Cij = ( KI+ KI ) dh E Pi Pj 0 (0) The local stiffness matrix can be obtained b taking the inversion of compliance matrix. i. e. K K = K K K C = C C C Figre shows the relative crack depth to that of variation of dimensionless compliances. 4 () (3) (4) (8) ISSN : Vol 6 No Apr-Ma 04 5

4 Figre. Relative crack depth (h /H) vs. dimensionless compliance (ln ( C )) x.. Analsis of Vibration Characteristics of the Cracked Composite Beam A composite beam of length width W and depth H, with a crack of depth h at a distance from the one end is considered as shown in figre. Taking (x, t) and (x, t) as the amplitdes of longitdinal vibration for the sections before and after the crack and (x, t), (x, t) are the amplitdes of bending vibration for the same sections as shown in fig. (). Figre : Geometr of composite beam: (a) Composite beam (b) cross-sectional view of the beam (c) crack depth and crack location. The normal fnction for the sstem can be defined as = () (x) A cos (K x) + A (x) A 3 cos (K x) + A 4 sin(k sin(k x) = (3) (x) A 5 cosh (K x) + A 6 sinh(k x) + A 7 cos(k x) + A x) = (4) (x) A 9 cosh (K x) + A0 sinh(k x) + A cos(k x) + A Where, 8 sin(k = (5) =, x x =, =, χ = sin(k x) x) ISSN : Vol 6 No Apr-Ma 04 5

5 K A i ω =, K C (0) = 0 / ω =, C C / E =, ρ C / EI =, µ = Aρ µ, (i=, ) Constants are to be determined, from bondar conditions. The bondar conditions of the cantilever beam in consideration are: ; (0) 0 ; () = 0 ; " () = 0 ; () = 0 ; (0) 0 = = ( χ) = ( χ) At the cracked section: ( χ) = ( χ) ; ( χ) = ( χ) ; ( χ) = ( χ) ; Also at the cracked section at distance from fixed end of cantilever beam, we have: d AE ( dx ) = K ( ( ) ( )) + K d ( dx ) d( dx AE Mltipling both sides of the above eqation b we get; K K MM ( χ) = M ( ( χ) ( χ)) + M( ( χ) ( χ)) (7) ) Similarl, d EI () = K dx ( ( ) ( )) + K d ( dx ) d( dx EI Mltipling both sides of the above eqation b we get, K K MM ( χ) = M( ( χ) ( χ)) + M( ( χ) ( χ)) (9) Where, AE M =, K AE M =, K EI M 3 =, K EI M 4 = K The normal fnctions, eqation () along with the bondar conditions as mentioned above, ield the characteristic eqation of the sstem as: Ω= 0 (0) Where, Ω is a Matrix. This determinant is a fnction of natral circlar freqenc (ω), the relative location of the crack (χ) and the local stiffness matrix (K) which in trn is a fnction of the relative crack depth (h /H). 3. Nmerical analsis of cracked composite beam sing ANSYS Finite element approach (FEA) is widel sed nmerical methods for vibration analsis of strctres. ANSYS is a software tool based on FEA is sed here for free vibration analsis of a composite beam nder consideration. A Composite beam of dimensions 650mm 60mm 6mm with eleven laers of epox glass fibers for determinations of natral freqenc and mode shapes nder varios end conditions with and withot crack were analzed sing ANSYS. The crack depth and crack locations were varied in a sstematic manner for analsis. Following procedres for a cracked composite beam were followed for the analsis: Step-: geometr modeling of composite beam. Step-: selection of element tpe (here solid 86 laered element for 3D modeling) for composite beam. 0 nodes Solid 86 laered element is sed for solid modeling. Step-3: material modeling of test composite beam (properties of orthotropic material). Step-4: section lap of model (here 0 0 fibers orientation). With a shell section or laered composite specified with laer thickness, orientation, nmber of integration points throgh the thickness of laer. Step-5: model meshing Step-6: bondar conditions and modal soltion Step-7: reslt and plotting ) (6) (8) ISSN : Vol 6 No Apr-Ma 04 53

6 Figre 4: Epox E-glass fibre composite cracked beam meshing Figre 5: Epox E-glass fibre composite cracked beam meshing (EXX) in KN/mm Table:. Mechanical properties of prepared Epox E-glass with 0 0 fiber orientation composite beam: (E KN/mm YY) in (E KN/mm ZZ) in (G KN/mm XY) in (G KN/mm YZ) in (GZX) in KN/mm ρ in Kg/m Experimental Procedre Composite beam of epox glass fibres with dimensions 650mm 60mm 6mm were prepared sing hand laot techniqes. Crack depths with respect to fixed different locations are taken separatel for experimental analsis. A total of eleven laers (nidirectional fibers) of niform thickness with matrix pt alternatel were prepared sing hand laot techniqes. The mechanical properties of the composite beam recorded sing Instron setp machine are given in Table. The transverse cracks were created at different locations of the beam individall b inserting sharp thin smooth plate at time of preparation of composite beams. The transverse cracks were located at different locations from one end of the beam to std the changes in dnamic characteristics. The reslts were compared with a beam withot crack. The composite beam with different bondar fixations i.e. fix-free, fix-fix, were considered. For each experiment the crack location were changed for different bondar fixations keeping the crack depth ratio constant. Pre fixed distance of ranging from 50mm to 500mm with 50mm steps along the length were considered for crack location for each experiment with a specified bondar fixation. The experiments were repeated for varios bondar fixations and investigation. The experimental setp is shown in figre (6) the beams were pt into the holder for test with different bondar fixations. External excitation was created in the beam with a impact hammer initiall at mid position of beam. ISSN : Vol 6 No Apr-Ma 04 54

7 Fig.6: Schematic diagram of experimental setp The excitation created b hammer is transmitted in the form of a signal to the FFT (model plse lite Device) throgh amplifier. The readings of the amplitde of transverse vibration and resonant freqencies at different crack locations along the length are recorded b sing vibration set p. The experimental set p with beam nder test is shown in the fig. (6). 5. Reslts and Discssions Analsis: Theoretical, Finite element (nmerical) and Experimental analsis of a cracked composite beam were carried ot. Relative natral freqenc was calclated varing the crack depth and crack locations. Finite element analsis (ANSYS) was sed to extract natral freqenc and mode shapes of varios end spports and reslts were compared with experimental and theoretical reslts. The relative crack depth to that of variation of dimensionless compliances is given in figre () and in figre () is present the Geometr of composite cracked beam. 0 nodes Solid 86 laered element is sed for solid modelling in nmerical analsis. In figre (4-5) are shown the mesh form of (epox E-glass fiber) composite beam. The schematic diagram of experimental setp of beam nder test is given in figre (6). In figres (7-9) are presented the first three mode shape of composite cracked beam. It is observed from st mode of vibration that relative freqenc decreases with increase in crack depth for a given crack position. The slop of the crve is more significant between 0.4 to 0.6 of crack depth ratio and indicating a steep fall in relative freqenc. Similar trend is observed for the corresponding relative freqenc of second mode of vibration. However the drop in relative freqenc is higher in 3 rd mode figre (0) as compared to st and nd mode of vibration in respect to crack depth. Similar trend is observed for variation in relative freqenc of C-C end spport composite beam figre (). Variation in relative freqenc are recorded b varing the crack positions for a given crack depth ratio. It is observed, st mode of vibration that the relative freqenc increases with the shifting of crack positions from one end spport with respect to st mode of vibration. However similar trend is not observed for nd and 3 rd mode of vibration for given crack depth ratio. For Clamp-Clamp beam figre (3), a drop in relative freqenc is significant as compared to Clamp-Free beam. The relative freqenc variation in nd mode of vibration is more prononced as compared to its corresponding 3 rd mode of vibration.in general, the natral freqenc decreases with increase in crack depth de to beam losses its stiffness. ISSN : Vol 6 No Apr-Ma 04 55

8 (a). Mode shape of epox glass fiber Composite cantilever(c-f) Beam sing ANSYS at Relative crack position ( χ = ) 0.08 and Relative crack depth(h /H)=0.58: Figre 7. First mode shape of C-F cracked (epox glass fiber) composite beam Figre 8. nd mode shape of C-F cracked (epox glass fiber) composite beam Figre 9. Third mode shape of C-F cracked (epox glass fiber) composite beam χ = (b). (h /H) vs. (ω c /ω n ) at fixed ( ) of C-F Composite beam: (ωc/ωn) h /H Figre 0. (h /H) vs. (ωc/ωn) at χ = 0.08 of C-F Composite beam in 3rd mode ISSN : Vol 6 No Apr-Ma 04 56

9 ( χ = ) (c). (h /H) vs. (ω c /ω n ) at = 0.08 of C-C Composite beam (d). ( χ = ) vs. (ω c /ω n ) of C-F at fixed (h /H) : (e). ( χ = ) vs. (ω c /ω n ) of C-C at fixed (h /H): (ωc/ωn) (ωc/ωn) h /H Figre. (h /H) vs. (ωc/ωn) at χ = 0.08 of C-C Composite beam in st mode (ωc/ωn) χ Figre. ( χ ) vs. (ω c /ω n ) of C-F at crack depth Ratio h /H= 0.48 in st Mode χ Figre 3. ( χ) vs. (ω c /ω n ) of C-C at crack depth Ratio ( h /H)= 0. in st Mode. ISSN : Vol 6 No Apr-Ma 04 57

10 6. Conclsions: The reslts of experimental, analtical and finite element analsis investigation are given and the observed that with increase in crack depth, the relative freqencies redce in order de to drop in stiffness of the composite beam. Frther it is observed that the relative crack position affects the relative freqenc of the composite beam. The relative freqenc increases with higher relative crack location. Frther attempts will be made to investigate the effect of freqenc for composite beams with variation in fibre angles and volme fractions. Reference: [] Vision, J. R. and Cho, T. W., Composite Materials and their ses in strctres, John Wile, First edition, 975. [] Erdeli, N. H. and Hashemi Seed, M., A Dnamic Stiffness Element for Free Vibration Analsis of Delaminated aered Beams, Modelling and Simlation in Engineering, Vol. 0, -8, 0. [3] Irwin, G. R., Analsis of stresses and strains near the end of a crack transverse in a plate, Jornal of Applied Mechanics, 4, , 956. [4] Gpta, G. D., A laered composite with a broken laminate, International Jornal of Solids Strctres. Vol. 9, 4 54, 973. [5] Nikpr, K., and Dimarogonas, A., ocal compliance of composite cracked bodies, Composite science and Technolog, Vol.3, 09-3, 988. [6] Sreekanth, K. D., Ro Mahapatra, D., and Gopalakrishnan S., A spectral finite element for wave propagation and strctral diagnostic analsis of composite beam with transverse crack, Finite Elements in Analsis and Design, Vol. 40, 79 75, 004 [7] Pande, A. K., Biswas, M. and Samman, M. M., Damage detection from changes in crvatre mode shapes, Jornal of Sond and Vibration, Vol. 45(), 3-33, 99 [8] Krawczk, M., A new finite element for static and dnamic analsis of cracked composite beams, International Jornal of Compters and strctres, Vol. 5(3), 55-6, 994. [9] Amara, K., Tonsi, A., Megeni, A., and Adda-Bedia, E.A., Effect of transverse cracks on the mechanical properties of angle-pl composites laminates, Theoretical and Applied Fractre Mechanics, Vol. 45, 7 78, 006. [0] Ramanamrth, E.V.V., and Chandrasekaran, K., Damage detection in Composite beam sing nmerical modal analsis, International Jornal on Design and Manfactring Technologies, Vol., No., Jl 008. [] Nag, A., Ro Mahapatra, D., Gopalakrishnan, S., and Sankar, T.S., A spectral finite element with embedded delamination for modeling of wave scattering in composite beams, Composites science and Technolog Vol. 63, 87 00,003. [] akshmi, N. K. and Jebaraj, C. Sensitivit analsis of local / global modal parameters for identification of a crack in a beam, Jornal of Sond and Vibration, Vol. 8(5), , 999. [3] Mrat, K., Free vibration analsis of a cantilever composite beam with mltiple cracks, Composites Science and Technolog, Vol. 64, 39 40, 004. [4] Sh, D. and Della Christian, N., Free vibration analsis of composite beams with two Non- over- lapping delaminations, International Jornal of Mechanical Sciences, Vol. 46, , 004. [5] Singh, K. V., i, G. and Pang, S-Seng, Free vibration and phsical parameter identification of non-niform composite beams, Composite Strctres, Vol. 74, 37 50, 006. [6] ANSYS.0 User s manal, 0. ISSN : Vol 6 No Apr-Ma 04 58