Euler-Bernoulli Beam Theory in the Presence of Fiber Bending Stiffness

Transcription

1 IOSR Journal of Matematics (IOSR-JM) e-issn: , p-issn: X. Volume 13, Issue 3 Ver. V (May - June 017), PP Euler-Bernoulli Beam Teory in te Presence of Fiber Bending Stiffness 1 Ali Farat, NbilaGwila 1 Al-Asmarya Islamic UniversityFaculty of Science-ZlitenMatematics department Al-Asmarya Islamic UniversityMatematics departmentfaculty of Science-Zliten Abstract: in te conventional one-dimensional matematical modelling of fibre-reinforced tin-walled structures it is assumed tat fibres ave negligible tickness and tey are perfectly flexible. Consequently, te constitutive equation forfinitedeformations of fibre-reinforced elastic solids involves no natural lengt. Based on incorporatingfibre bending stiffness into a continuum teory, asymmetric linearised elasticity teory tat takes into consideration te effects of fibres bending stiffness, as been produced in[1]. Relatively, an accurate d tin-walled structures modeling as been presented in []. Based on te aforementioned d-teory and, as a particular case, tis study presents one-dimensional matematical modeling tat takes into consideration te fibres bending stiffness, wit te use of te displacement approximation of te bernoulli-euler beam model. Unlike its conventional one-dimensionalmodelingte solution is obtained tat contains an intrinsic material lengt parameter wic may, for instance, be considered representative of tefibre diameter. Numerical results on te basis of te obtained solution are presented and discussed in tis article for different boundary conditions of omogeneous fibre-reinforced rectangular beams. Keywords:fiber bending stiffness, beam deflection, couple-stress, asymmetric elasticity, intrinsic material area or lengt parameter. I. Introduction Te presence of te fiber bending stiffness as received a significant attention tat appears in te literature for studying te continuum teory of finite deformations of elastic materials. A non-symmetric stress and te couple-stress are requiredinasymmetric elasticity teory presented in [1]. Relatively, a version of asymmetric linear elasticity teory as been produced, as a particular case, by Spencer and Soldatos[1].A generalized, 6-degree-of-freedom, D model for plates wit fibres resistant to bending as been developed by Soldatos in [] were one extra elastic modulus, wic as dimensions of force, is involved in eac layer of te laminate. Consequently, some intrinsic material area or lengt parameter involved in te analysis tat may be associated, for instance, wit fiber tickness of fiber spacing[3]. Wit te use of te plate displacement approximation of Reissner (1944, 1945), plate model tat accounts for fibretickness and, consequently, for te ability of individual fibres to resist bending[4].based on te aforementioned plates model developed in [], different displacement approximations, wic available in te literature, can be used to produce tin elastic plate and beam models tat accounts for te effects of fibre resistant in bending. One of te commune usesof te aforementioned displacement approximations is te Bernoulli-Euler beam teory tat provides a means of calculating te deflection of elastic beams wic deformed in linear scale. As pointed out in [5], because of te simplicity of Euler-Bernoulli beam teory tat provides reasonable engineering approximations wen applied on several problems, it is commonly used. In addition, tis teory appears in te literature wit different names suc as classical beam teory, Bernoulli beam teory, or Euler beam teory. Tis conventional teory was applied to solve problems of small elastic beams deflection were te beams are reinforced by fibres wic assumed to ave negligible tickness and tey are perfectly flexible. In te present study, te fibres are assumed to ave te ability to resist bending. Consequently, an extra elastic modulus as been involved in te teory, wic, unlike its conventional counterparts. Te purpose of te tis article is to present a beam teory wit te use of te displacement approximation of te Bernoulli-Euler beam model in connection wit te generalized, 6-degree-of-freedom, Dplates model tatas been developed by Soldatos in []. A cross-ply laminated beam of transversely isotropic, linearly elastic material tat contains a single, unidirectional family of straigt fibres wic can resist bending is assumed to be subjected to small deformation is considered in te present study. Te Constitutive equations considering fiber bending stiffness ave been produced based on te use aforementioned displacement approximation of te Bernoulli-Euler beam model in section 3. Consequently, section 4 presents te Navier-type differential equations wic ave been solved for te static solution in section 5. Numerical results and relevant discussionbased on te obtained solutions ave been conducted in section 7. Te main conclusions of tis article as well as some researc points for future related work are finally summarized in Section 8. DOI: / Page

2 II. Problem Formulation Consider a cross-ply laminated plate of transversely isotropic, linearly elastic material aving arbitrary constant tickness in te z direction and, orizontal constant lengt L 1 in te x direction. Te considered plate is assumed to be of infinite extent in te y direction. Furtermore, te plate contains fibers wic are lying in parallel to te x-directionand as te ability to resist bending. Different sets of end boundary conditions are applied on te endsx = 0, L 1. Figure1. Co-ordinate system Figure.An example of te beam layer numbering Figure 3.Lateral loading As a result of te plane-strain symmetries involved in tis problem, all quantities involved are independent of y and te displacement function in y direction is equal to zero. According to tis any of te plate cross-sections could alternatively be regarded as a transversely isotropic beam of tickness, lengt L 1 and a unit widt along te y-direction.a given external lateral loadingq x is applied on te beam wic causes small static flexure. Te loading acts normally and downwards on tetop lateral plane of te beam. DOI: / Page

3 III. Constitutive Equations Considering Fiber Bending Stiffness Te elastic beam model begins wit te displacement approximation of te Euler-Bernoulli Beam teory tat as te following form: U x, z, t = u x, t zw,x,(1-a) W x, z, t = w(x, t), (1-b) were t denotes time. In addition, U x, z, t and W x, z, t represent te displacement components of a point x, y, z, t along x and z directions, respectively. Moreover, w(x, t) presents te transverse deflection of te beam. It can be seen tat te transverse deflection is assumed to be independent of te beam tickness. In addition, u x. t represents te in-plane displacement of te beam middle plane. Because of tat, tey ave evidently dimension of lengt. Inserting te displacement approximation (1) into te following well-known linear kinematic relations: ε x = U, γ x xz = U + W, () z x yields te following approximate strain field: ε x = e c c x + zk x, γ xz = 0, (3) were e c x = u,x, andk c x = w,xx, (4) were te prime denotes te ordinary differentiation wit respect to z. Te quantities denoted wit a superscript c are identical wit teir classical beam teory counterparts were te fibers ave no bending stiffness. Te beam is assumed to be composed of an arbitrary number, N, of perfectly bonded transversely isotropic layers. Accordingly, te generalised Hooke s law witin te kt layer of suc a cross-ply laminate is given as follows: ς (k) x = Q (k) (k) 11 ε x, τ (xz ) = Q (k) 55 γ xz = Q (k) 55 0 = 0, (5) wereτ (xz ) denotes to te symmetric part of te stress tensor component and te appearing Q,s are te reduced stiffnesses [6]. Quantities wit superscript k are due to te k t layer of a cross-ply laminate. In te presence of te fiber bending stiffness, te elasticity teory requires to include te couple-stress and consequently asymmetric stress [1]. Te anti-symmetric part of te associated sear stress tensor component is denoted by symbol wit indices in square brackets. Tis part is defined as follows[]: τ (k) [xz ] = τ (k) [zx ] = 1 m (k) xy,x = 1 df (k) K f z,x = 1 df (k) w,xxx. (6) f Here K (k) z represents te fibers curvature.in addition,m (k) xy represents te non-zero couple stress wic is related to te fiber curvature []. Moreover, d f(k) is an elastic modulus tat accounts for te fiber bending stiffness in te k-tlayer. Tis elastic modulus as dimension of force []. In tis regard, te following notation: d f(k) = C 11 l (k) L 1, (7) is employed to include a material intrinsic lengt parameterl (k), wic may, for instance, be considered to be related to te fiber tickness in te k-t layer[3]. Tus, te sear stresses take te following form: τ (k) xz = τ (k) (xz ) + τ (k) [xz ], (8) τ (k) zx = τ (k) (xz ) τ (k) [xz ], Te equations of motion [] take te following form: ς (k) (k) (k) x,x + τ xz,z τ xz,z = ρu (k), (9-a) (k) (k) τ xz,x + τ xz,x + ς (k) z,z = ρw (k). (9-b) Furtermore, wit te use of equations (5), (6) and (8), te sear stresses can be written as follows: τ (k) xz = Q (k) 55 γ xz 1 df (k) w,xxx = 1 df (k) w,xxx, (10-a) τ (k) zx = Q (k) 55 γ xz + 1 df (k) w,xxx = 1 df (k) w,xxx, (10-b) DOI: / Page

4 In addition, te force and moment resultants are as follows: N x c = ς x dz, M x c = ς x zdz, M x f = 1 m xy dz, (11) f Were M x is related to te moment resultants caused by te present of te fiber bending stiffness. In order to obtain te tow unknown degrees of freedom (u, and w), te following relevant equations of motion (Soldatos, 009) will be applied: c N x,x = ρ 0 u ρ 1 w,x, (1-a) c f M x,xx + M x,xx = q x + ρ 0 w + ρ 1 u,x ρ w,xx, (1-b) were te dots stand for ordinary differentiation wit respect to time and, te coefficients wic are in te inertia terms are defined as follows: ρ i = ρ z i dz. (13) Here ρ denotes te material density of te elastic beam considered. Te equations of motion (13) are associated wit te following variationally consistent set of end boundary conditions at x = 0, L 1 []: c eiteru or N x is prescribed, (14-a) c f eiterw or M x,x + M x,x is prescribed, (14-b) eiterw x or M c x + M f x is prescribed, (14-c) IV. Navier-Type Differential Equations Inserting te equations (5-8) and (10) into equations (11) yields te following force and moment resultants in terms of te tow degrees of freedom and teir derivatives: N x c c c B 11 c = A 11 c u,x M x D 11 w,m f x = 1,xx df w,xx, (15) B 11 were te appearing rigidities can be calculated by te use of te following expressions: (A c 11, B c 11 ) = D 11 = k Q 55 1, z dz, (16-a) k Q 11 z dz, (16-b) Te equations of motion (1) wic contain te term tat related to te presence of te fiberbending stiffness can be converted into a set of tow differential equations for te same number of main unknowns as follows: Wit te use of equation (3.13), tese differential equations can be obtained in terms of te displacement field yielding te following Navier-type differential equations system: A c 11 u,xx B c 11 w,xxx = ρ 0 u ρ 1 w,x, (17-a) B c f 11 u,xxx D 11 w,xxxx = q x + ρ 0 w + ρ 1 u,x ρ w,xx, (17-b) wic are called te Navier-type differential equations system were te sown rigidity in equation (3.16-b) is depending on te conventional rigidity,d 11, and te fiber bending elastic modulus d f and defined according to: f = 1 df + D 11 (18) D 11 According to te number of end boundary conditions (14), wit respect to te co-ordinate parameter x, te equations (3.15) would be a sixt order set of ordinary differential equations. Tese can be solved wen a particular set of boundary conditions is specified at eac end of te beam. It can be observed tat, te difference between equations (3.15) and teir counterparts, met in te case perfectly flexible fibers seen in te classical beam teory, is te definition of te rigidityd f 11 sown in equation (18). V. Static Solution In te present section, te considered beam is subjected to under normal static load applied on its top surface. Terefore, te 1D static solution is found for te considered flexure problem. It can be resulted tat te inertia terms appearing in te rigt-and sides of te motion equations (17) are disregarded to yield te following equilibrium equations: DOI: / Page

5 A 11 u,xx B 11 w,xxx = 0, (19-a) B f 11 u,xxx D 11 w,xxxx = q x, (19-b) Te lateral load function can be expressed in te following Fourier series. q x = q m sin p m x, p m = mπ L 1, m = 1,,. (0) In addition, te following boundary conditions can be applied on te endsx = 0, L 1 : at a simply supported end: N c x = 0, w = 0,M c x + M f x = 0, (1-a) at a clamped end: u = 0, w = 0, w,x = 0, (1-b) and at a free end: N c c f x = 0,M x,x + M x,x = 0, M c x + M f x = 0, (1-c) Te general solution of te non-omogeneous ordinary differential system (19) will be te tow degrees of freedomu, andwand as te following form: S = S c + S p. () weres c is te complementary solution and S p is a particular solution of te non-omogeneous system. Te effective way to test te reliability of te presented teory is by performing numerical comparison wit te corresponding results of te static plane strain asymmetric-stress elasticity solution found in[7] and, furter, as been discussed in[3].in tis subsection, te beam is taken as simply supported at te ends x = 0, L 1. Te simply supported boundary conditions (1-a) are satisfied by te following trigonometric displacement coice of S p : u = Acos p m x,w = Csin (p m x). (3) Inserting equations (3) into Navier-type differential equations (1) will convert it into te following system of tow simultaneous linear algebraic equations of te tow unknown constants A and B. p c m A 11 p 3 c m B 11 A p 3 c m B 11 p 4 f m D 11 B = 0, (4) q m In order to find te values of A and B, te algebraic equations system (4) will be solved.for a omogeneous rectangular (N = 1)plate, Eq. (16-a) gives: c B 11 = 0. (5) Terefore, te Navier-type differential equations (19) reduce to te following uncoupled sets of differential equations: A c 11 u,xx = 0, (6-a) f D 11 w,xxxx = q x, (6-b) Te general solution of te sixt order equilibrium differential equations system (19) can be written as follows: u = k 1 Ac x + k 11 Ac +Acos p m x, (7-a) 11 w = k 3 f x 3 k 4 f x k 5 f x k 6 f +Csin (p m x). (7-b) 6D 11 D 11 D 11 D 11 Values of te arbitrary constants (K 1, K,, K 6 ) can be found wen a set of six end boundary conditions is specified, see Eqs.( 1). It sould be noticed tat as in perfectly flexible fibers case, tesearbitrary constants take zero value wen SS boundary conditions are applied. Tis as as well been pointedoutin [8].It is wort mentioning te obtained general solution (7) is influenced te resistance of fibers in bending. Tis general solutioncanbe reduced to te perfectly flexible fibers solution by giving value of zero to te fiber bending elastic modulus d f.consequently, te value of te rigidity D f 11 = D 11 (see equation 18). VI. Numerical Results and Discussion Te following numerical results are for omogeneous transverse fiber-reinforced rectangular beams caracterized by te following material properties[8]: E L E T = 40, G LT E T = 0.5, G TT E T = 0., v LT = v TT = 0.5. (8) Here, te subscript L signifies te longitudinal fibers direction, T denotes te transverse fiber direction, and v LT stands of te Poisson ratio tat measure strain in te transverse direction T under uniaxial normal stress in te L- direction. Te elastic moduli sown in equations (5) and (7) are tus obtained by te use of equations (8) in conjunction wit well-known related formulas (e.g. [6]). As pointed out in [4], te material property (8 a) caracterizes te ig stiffness ratio, suc as, strong grapite-epoxy fibers, an appropriate value to te bending DOI: / Page

6 stiffness modulus of te fibers as to be assigned. In tis regard and, in order to perform an appropriate comparison in te presented numerical results, te additional elastic d f is represented as follows: d f = 1 C 1 11lL 1, (9) wicis employed to include a material intrinsic lengt parameterl, wic may, for instance, be considered related to te tickness or te diameter of te fiber. Furtermore, te following non-dimensional intrinsic material parameter: λ = l (30) wic is related te additional elastic d f, and may be tougt of as a fiber-to- plate or beam tickness.te normalized deflection employed for presentation of numerical results is defined according to: W = E T W q 1 L (31) In te beginning of tis section, te accuracy of te obtained solutions is assessed by comparing te results based on it to teir counterparts based on te exact elasticity solution presented in [7] and [3]. After te reliability of te metod as been cecked, te effects of different sets of endboundary on te deflection of corresponding beams deformed by cylindrical bending. Tese boundary conditions are clamped-clamped end (CC), clamped-free end CF) and clamped-simply end (CS) boundary conditions. Figure 4. Deflection pattern along te x-axis of a SS omogeneous beam Figure 4 depicts and compares non-dimensional SS beam deflection and predicted by bot te solution obtained in te present study and te corresponding exact polar elasticity solution [7] and [3] in wic used as a bencmark in [4]. Te beam is tin ( L = 0.01) were te fiber runs along te x-direction. For comparison reason, te figure describes deflection not only for a case tat fibers wic ave te bending stiffness (λ = 0.03) but also for te case of te perfectly flexible (λ = 0).It is initially observed tat, te differences between te deflection obtained based on te exact and te tin teory were (λ = 0) is very similar to tat obtained based on te present solution presented in tis study and te exact polar elasticity solution [7] and [3] were (λ = 0.03). It can be seen tat, fiber bending resistance (λ = 0.03) reduces te beam deflection sown in te case of teperfectly flexible (λ = 0). Figure 5.Deflection of a SS omogeneous beam at different values of λ DOI: / Page

7 Figure 5 displays te troug lengt distribution of W for te simply-simply supported (SS) beam at different values ofλ. It is observed tat by increasing te values ofλ, te deflection of te beam decreases. Tis, as pointed out in [4], makes evident tat te increase of teλvaluesis equivalent to increasing te fiber bending stiffness. Te curve wic presents te largest values of deflection (λ = 0) is due to te case of perfectly flexible fibers. Figure6.Deflection of a CC omogeneous beam at different values of λ Figure 6 sows te defection of a CC omogeneous beam at different values of λ. It is, again, seen tat increasing te value of λ reduces te beam deflection. Attention is drawn on te sape of te deflection at te fiber ends by presenting aseparate 'window' tat lies underneat of te principal figure. It is observed tat te effect of te clamped-clamped boundary conditions (1-b)becomes more evident were te beam ends are restrained against rotation. Furtermore,regardlessof weter te fibers are assumed perfectly flexible (λ = 0) or ave te ability to resist bending (λ = , λ = andλ = ), te maximum beam deflection of te clamped-clamped case is considerably smaller tan tat in te case of simply-simply supported beam (figure 5). Figure 7. Deflection of a CF omogeneous beam at different values of λ DOI: / Page

8 Figure 7 presents te deflection of a CF omogeneous beam at different values of λ including te case of te perfectly flexible fibers(λ = 0). Te deflection of tis cantilevered beam is clearly seen decreasing wen te value of λ increses. Tis empasise, again, tat te increase of teλ values is equivalent to increasing te fiber bending stiffness or, consequently, increasing te resistance of te beam bending. Figure 8. Deflection of a CS omogeneous beam at different values of λ Te deflection distribution troug te lengt of tecs omogeneous beam is sown in figure 8 for te case of perfectly flexible fiber λ = 0 and different cases of te presence of fiber bending stiffness(λ = , λ = andλ = ). Te difference of te effects of te boundary conditions can be seen clearly at te beamends. Tis is observed were te clamped beam end(x L = 0) is restrained against rotation wile is not at te simply supported end (X L = 1). In addition, te maximum beam deflection of te clamped-simply case is smaller tantat in te case of simply-simply supported beam, and beiger tan tat beam is clamped-clamped ends. It sould be observed tat, te presence of fiber bending stiffness λ = results an approximately 3% reduction of te maximum non-dimensional deflection at te middle beam lengt(x L = 0.5) (see fig.5, 6and 8). VII. Conclusion Fiber bending stiffness is considered in te present study of Euler-Bernoulli beam teory. Te displacement field of te classical beam teory is employed in te equations of motions wic takes into account te ability of fibers to resist bending tat ave been presented in []. Te static solutions of tese equations of motions as been found in te present study were associated wit an appropriate different sets of boundary equations tat consider te fiber bending stiffness. Considerable numerical results ave been presented for te static omogeneous beam bending problem were te beam is reinforced by fibers tat resist bending. An attention is focused on te influence of te resistance of fibers in bending on te beam deflection. It as been evidently, as expected, sown tat te increase of te fiber bending stiffness decreases te deflection of te beam. An attention as been drawn to te effect of te boundary conditions on te sape of deflection practically for te clamped beam ends were compared tat wit te simply-supported ends. Furter study could be conducted on te oter versions of plate teory considering te fiber bending stiffness suc as Kircoff plate teory, Timosenko beam teory and Higer-order teory. References [1] Spencer, A.J.M. and K.P. Soldatos, Finite deformations of fibre-reinforced elastic solids wit fibre bending stiffness. International Journal of Non-Linear Mecanics, (): p [] Soldatos, K.P., Towards a new generation of D matematical models in te mecanics of tin-walled fibre-reinforced structural components. International Journal of Engineering Science, (11-1): p [3] Farat, A.F. and K.P. Soldatos, Cylindrical Bending and Vibration of Polar Material Laminates. Mecanics of Advanced Materials and Structures. (11): p [4] K.P. Soldatos and A.F. Farat, On Reissner s displacement field in modelling tin elastic plates wit embedded fibres resistant in bending. IMA Journal of Applied Matematics, 016. [5] Han, S.M., H. Benaroya, and T. Wei, Dynamics of transversely vibrayion beams using four engineering teories Journal of sound and vibration (5). [6] Jones, R.M., Mecanics of composite materials. Hemispere, New York., [7] Farat, A.F., Basic problems of fibre reinforced structural components wen fibres resist bending. PD tesis, University of Nottingam 013 Web. 4 February [8] Soldatos, K.P. and P. Watson, A metod for improving te stress analysis performance of one- and two-dimensional teories for laminated composites. ActaMecanica, (1): p Ali Farat. "Euler-Bernoulli Beam Teory in te Presence of Fiber Bending Stiffness." IOSR Journal of Matematics (IOSR-JM) 13.3 (017): DOI: / Page