The th International Conference on Mechanics of Solids, Acoustics and Virations & The 6 th International Conference on "Advanced Composite Materials Engineering" ICMSAV6& COMAT6 Brasov, ROMANIA, -5 Novemer 6 GREEN FUNCTION MATRICES FOR ELASTICALLY RESTRAINED HETEROGENEOUS CURVED BEAMS György Szeidl, László Kiss Institute of Applied Mechanics, University of Miskolc, Miskolc, HUNGARY gyorgy.szeidl@uni-miskolc.hu, mechkiss@uni-miskolc.hu Astract: Virations of heterogeneous curved eams have recently een investigated,,. The thesis and papers cited deal with, among others, the issue what effect the central load has on the viration of pinned-pinned and fixed-fixed curved eams made of heterogeneous material. To find numerical solution, the authors determine the Green function matrices for pre-loaded eams. Then they reduce the eigenvalue prolems, which yield the eigenfrequencies as a function of the load, to eigenvalue prolems governed y a system of Fredholm integral equations. A similar investigation for elastically restrained heterogeneous curved eams is provided in this article. The end-restraints are modeled y linear volute springs. Keywords: Curved eams, functionally graded materials FGM, rotational restraint, natural frequency as a function of the load, Green function matrix. INTRODUCTION Curved eams are widely used in engineering applications let us consider, for instance, arch ridges or roof structures. Research concerning the mechanical ehavior of such structural elements egan in the 9 th century. The free virations of curved eams have een under extensive investigation see, e.g.:, 5 for more details. Considering the virations of pre-loaded circular eams, the numer of the availale articles is much less than for the free virations. Wasserman 6, for example, investigates the load-frequency relationship for spring supported inextensile arches. The load can e dead or follower. Here, similarly as in 7, the Galerkin method was presented as an effective way to get solutions. Chidamparam and Leissa 7 investigate the virations of pinned-pinned and fixed-fixed prestressed homogeneous circular arches under distriuted loads. The extensiility of the centerline is taken into account. We intend to contriute to the literature y improving the mechanical model, extending it for heterogeneous materials and using a numerical technique ased on the Green function matrix.. KINEMATICAL ASSUMPTIONS & GOVERNING EQUATIONS Figure. a Coordinate system D model of the curved eam We have developed a D eam model to investigate the viratory prolem. The curvilinear coordinate system ξ s, η, ζ is attached to the E weighted centerline as shown in Figure. The radius of curvature R is constant there and, moreover, the cross-section geometry and material distriution are uniform. However, the material composition and thus the material parameters can vary over the cross-sectional coordinates η, ζ as long as axis ζ is a symmetry axis oth for the geometry and for the material distriution. Therefore, it is possile to model homogeneous, functionally graded FG and even multi-layered eams, considering each material component to e linearly elastic and isotropic.
The centerline intersects the shown cross-section at C, whose location can e found y fulfilling the definition that the E-weighted first moment of the cross-section with respect to the axis η is zero there: Q eη Eη, ζ ζ da. A We consider the validity of the Euler-Bernoulli eam theory for the investigations, i.e., the cross-sections rotate as if they were rigid odies and remain perpendicular to the deformed centerline. Let u o, w o and ϑ e the tangential, radial displacement coordinates and the semi-vertex angle of the eam. Since the radius is constant the coordinate line s and the angle coordinate ϕ are related to each other y s Rϕ. The axial strain ε oξ and the rigid ody rotation ψ oη on the centerline can e expressed in terms of the displacements as ε oξ du o ds w o R, ψ oη u o R dw o ds. The principle of virtual work for the eam shown in Figure yields equilirium equations dn ds dm R ds N M d dm ψ oη f t, R ds ds N M ψ oη N R R f n a which should e fulfilled y the axial force N and the ending moment M. Here f t and f n denote the intensity of the distriuted loads in the tangential and normal directions f f t e t f n e n. Recalling Hooke s law 8, the relation etween the strains and inner forces ecome N I eη R ε oξ M R, A e A M I eη Eη, ζda, I eη d w o ds A w o R, N M R I eη R ε oξ, where Eη, ζζ da, m A er I eη 5 A e is the E-weighted area of the cross-section, I eη is the E-weighted moment of inertia with respect to the ending axis while m is a geometry-heterogeneity parameter the effect of the material distriution is incorporated into the model through the latter one. For simplicity reasons, we introduce dimensionless displacements and a notational convention for the derivatives taken with respect to the angle coordinate: U o u o R, W o w o R ;...n dn... dϕ n, n Z. 6 If we plug equations and into and perform some manipulations these are detailed in we get: Uo m Uo W o mε oξ W o Uo Uo R ft m m W o m ε oξ W o I eη f n. 7 Within the framework of the linear theory, we can freely neglect the effect of the deformations on the equilirium i.e., ε oξ. In the sequel, the increments which occur ecause of the viratory nature of the prolem in the typical quantities are identified y a suscript. Each physical quantity can e given in a form similar to the total tangential displacement which is equal to the sum u o u o. Here u o is the static displacement caused y the pre-load, and u o is the dynamic displacement increment. It turns out that the increments in the axial strain and in the rotation have a similar structure to equations : ε m ε oξ ψ oη ψ oη ε oξ,, ψ oη u o R dw o ds, ε oξ du o ds w o R. 8 The principle of virtual work for the increments yields the equilirium equations see for details : d N M N M ψ oη f t, ds R R R d M ds N R d ds where f t and f n are forces of inertia: N M ψ oη N M ψ oη f n, 9 R R f t ρ a A u o t, f n ρ a A w o t, 9a Thesis is downloadale from the url address http://www.siphd.uni-miskolc.hu/ertekezesek/5/kisslaszlopeter_phd.pdf
in which A as the area of the cross-section and ρ a as the average density on the cross-section. The increments of the inner forces can e given in terms of the displacement increments via Hooke s law: N I eη R mε oξ M d R, M w o I eη ds w o R, N M R I eη R mε oξ. a Sustituting 8 and into 9, we get the equilirium equations in the following form: Uo m Uo W o mε oξ W o m Uo Uo m m ε oξ W o W o R ft I eη f n. As regards the details we refer the reader to. For harmonic virations the amplitudes and Ŵo should satisfy the following differential equations: m m m mε oξ m ε oξ λ ; λ ρ a A R I eη α, where λ and α denote the eigenvalues and eigenfrequencies. The left side of can e rewritten as K y ϕ, ε oξ Py Py Py Py, y T Ŵo. Let rϕ e a dimensionless load: r T r r. The differential equation K y ϕ, ε oξ r descries the ehavior of the pre-loaded eam if the eam is sujected to a further load r given here in a dimensionless form. The free virations ε oξ of heterogeneous circular eams are governed y the differential equations m m m m λ For rotationally restrained eams differential equations, 5 and 6 are associated with the following oundary conditions: ϕ±ϑ ϕ±ϑ Û o,, Ŵ o ± K γŵ o 7 ϕ±ϑ where K γ k γ R/I eη is a dimensionless spring constant given in terms of k γ the spring constant. Equations, 7 and 7 determine an eigenvalue prolem. The i-th eigenfrequency α i depends on the heterogeneity parameters m and ρ a ; and also on the magnitude and the direction of the concentrated force P ζ. The effect of the latter one is accounted through the axial strain it causes: ε oξ ε oξ P, m, ϑ. Here P is a dimensionless load: P P ζ R ϑ/i eη.. NUMERICAL SOLUTION ALGORITHM The definition and the most important properties of the Green function matrix can e found in,. Solution to the inhomogeneous oundary value prolem 5, 7 is sought in the form ϑ G ϕ, ψ G yϕ Gϕ, ψrψdψ, Gϕ, ψ ϕ, ψ, 8 G ϕ, ψ G ϕ, ψ ϑ where G is the Green function matrix and ϕ, ψ are angle coordinates. If we write λy for r in 8, the eigenvalue prolem, 7 is replaced y a homogeneous integral equation system: ϑ yϕ λ Gϕ, ψyψdψ. 9 ϑ Numerical solution to such prolems can e sought e.g., y quadrature methods 9. Consider the integral formula ϑ n Jφ φψ dψ w j φψ j, ψ j ϑ, ϑ, ϑ j 5. 6
where ψ j ϕ is a vector and w j are the known weights. Having utilized the latter equation, we otain from 9 that n w j Gϕ, ψ j ỹψ j κỹϕ κ / λ ϑ, ϑ j is the solution, which yields an approximate eigenvalue λ / κ and the corresponding approximate eigenfunction ỹϕ. After setting ϕ to ψ i i,,,..., n, we have n w j Gψ i, ψ j ỹψ j κỹψ i κ / λ ψ i, ψ j ϑ, ϑ, or GDỸ κỹ, j where G Gψ i, ψ j for self-adjoint prolems, while D diagw,..., w... w n,..., w n and ỸT ỹ T ψ ỹ T ψ... ỹ T ψ n. After solving the generalized algeraic eigenvalue prolem, we have the approximate eigenvalues λ r and eigenvectors Y r, while the corresponding eigenfunction is otained via sustituting into : ỹ r ϕ λ r n j w j Gϕ, ψ j ỹ r ψ j r,,,..., n. Divide the interval ϑ, ϑ into equidistant suintervals of length h and apply the integration formula to each suinterval. By repeating the line of thought leading to, the algeraic eigenvalue prolem otained has the same structure as. It is also possile to consider the integral equations 9 as if they were oundary integral equations and apply isoparametric approximation on the suintervals elements. If this is the case, one can approximate the eigenfunction on the e-th element the e-th suinterval which is mapped onto the interval γ, and is denoted y L e y e y N γ e y N γ e y N γ e y, where quadratic local approximation is assumed: N i diagn i, N.5γγ, N γ, N.5γγ e, y i is the value of the eigenfunction yϕ at the left endpoint, the midpoint and the right endpoint of the element, respectively. Upon sustitution of approximation into 9, we have n ỹϕ λ e ey Gx, γn η N γ N γdγ y e e T y, 5 L e e in which, n e is the numer of elements. Using equation 5 as a point of departure, and repeating the line of thought leading to, we get again an algeraic eigenvalue prolem.. THE GREEN FUNCTION MATRICES Based on theses,, the Green function can e given in the form Gϕ, ψ Y j ϕ A j ψ ± B j ψ, 6 }{{} j where a the sign is positivenegative if ϕ ψϕ ψ; the matrices A j and B j have the following structure j j A j A A j j A j A j, Bj B B B j B j j,..., ; 7 j A j A c the coefficients in B j are independent of the oundary conditions. The columns in the matrices Y i are solutions to the homogeneous differential equations K y ϕ, ε oξ. If ε oξ < the concentrated force is compressive, then Y cos ϕ sin ϕ sin ϕ, Y cos ϕ j B j B cos χϕ Mϕ, Y χ sin χϕ However, Y and Y are different when mε oξ > : cosh χϕ Mϕ sinh χϕ Y, Y χ sinh χϕ χ cosh χϕ sin χϕ, Y χ cos χϕ. 8, M m m ε oξ. 9 Consequently we should deal with these two loading cases separately. The Green functions matrix if ε oξ <. Let us now introduce the following notational conventions a B i, B i, c B i, d B i, e B i, f B i.
It follows from the structure of the solutions Y l, l,..., that B i B i A i A i. The systems of equations for the unknowns a,..., f can e set up y fulfilling the continuity and discontinuity conditions the Green function matrix, should meet if ϕ ψ. Therefore, if i, we have cos ψ sin ψ cos χψ Mψ sin χψ sin ψ cos ψ χ sin χψ χ cos χψ sin ψ cos ψ χ sin χψ M χ cos χψ cos ψ sin ψ χ cos χψ χ sin χψ sin ψ cos ψ χ sin χψ χ cos χψ cos ψ sin ψ χ cos χψ χ sin χψ from where we get the constants as a c d e f m, a B χ sin ψ χ, χ cos ψ B M m χ, M m c B sin χψ χ M m χ, d B M m, e B cos χψ χ χ, f B M m M ψ m M. χ If i, then cos ψ sin ψ cos χψ Mψ sin χψ sin ψ cos ψ χ sin χψ χ cos χψ sin ψ cos ψ χ sin χψ M χ cos χψ cos ψ sin ψ χ cos χψ χ sin χψ sin ψ cos ψ χ sin χψ χ cos χψ cos ψ sin ψ χ cos χψ χ sin χψ is the equation system, the solution of which assumes the form a c d e f a B cos ψ χ, B sin ψ χ, c B cos χψ χ χ, d B, e B sin χψ χ χ, f B χ. Let α e an aritrary column matrix of size The unknown scalars A i ψ, A i ψ, A i ψ, A i ψ, A i ψ, A i ψ, i, ; ψ ϑ, ϑ in the matrices A j can e determined from the condition that the product Gϕ, ψα should satisfy the oundary conditions 7. This leads to the equation system cos ϑ sin ϑ cos χϑ Mϑ sin χϑ cos ϑ sin ϑ cos χϑ Mϑ sin χϑ sin ϑ cos ϑ χ sin χϑ χ cos χϑ sin ϑ cos ϑ χ sin χϑ χ cos χϑ sin ϑ K γ cos ϑ cos ϑ K γ sin ϑ χ sin χϑ K γχ cos χϑ χ cos χϑ K γχ sin χϑ K γ cos ϑ sin ϑ cos ϑ K γ sin ϑ K γχ cos χϑ χ sin χϑ χ cos χϑ K γχ sin χϑ a cos ϑ sin ϑ c cos χϑ dmϑ e sin χϑ f a cos ϑ sin ϑ c cos χϑ dmϑ e sin χϑ f a sin ϑ cos ϑ cχ sin χϑ d eχ cos χϑ a sin ϑ cos ϑ cχ sin χϑ d eχ cos χϑ a sin ϑ K γ cos ϑ cos ϑ K γ sin ϑ c χ sin χϑ χ cos χϑ e χ cos χϑ K γχ sin χϑ a sin ϑ K γ cos ϑ cos ϑ K γ sin ϑ c χ sin χϑ χ cos χϑ e χ cos χϑ K γχ sin χϑ The closed form solutions are presented in Appendix A. A i A i A i A i A i A i. 5 5
Calculation of the Green functions matrix if ε oξ > and mε oξ >. Similarly as aove, we get the following equations if i cos ψ sin ψ cosh χψ Mψ sinh χψ a sin ψ cos ψ χ sinh χψ χ cosh χψ sin ψ cos ψ χ sinh χψ M χ cosh χψ c cos ψ sin ψ χ cosh χψ χ sinh χψ d m. 6 sin ψ cos ψ χ sinh χψ χ cosh χψ e cos ψ sin ψ χ cosh χψ χ sinh χψ f The solutions are as follows χ a B χ M m c B χ χ M m e B sin ψ χ, B χ M m, d B M m, sinh χψ cosh χψ χ χ, f B M m M m Mψ. cos ψ, 7 If i cos ψ sin ψ cosh χψ Mψ sinh χψ a sin ψ cos ψ χ sinh χψ χ cosh χψ sin ψ cos ψ χ sinh χψ M χ cosh χψ c cos ψ sin ψ χ cosh χψ χ sinh χψ d sin ψ cos ψ χ sinh χψ χ cosh χψ e cos ψ sin ψ χ cosh χψ χ sinh χψ f is the equation system to e solved compare it to and the solutions we have otained are: a B cos ψ χ, B sin ψ χ, c B cosh χψ χ χ d B, e B sinh χψ χ χ, f B χ. 8 9 By repeating the steps that resulted in 5 we otain the following equation system for A i,..., A i : cos ϑ sin ϑ cosh χϑ Mϑ sinh χϑ cos ϑ sin ϑ cosh χϑ Mϑ sinh χϑ sin ϑ cos ϑ χ sinh χϑ χ cosh χϑ sin ϑ cos ϑ χ sinh χϑ χ cosh χϑ sin ϑ K γ cos ϑ cos ϑ K γ sin ϑ χ sinh χϑ K γχ cosh χϑ χ cosh χϑ K γχ sinh χϑ K γ cos ϑ sin ϑ cos ϑ K γ sin ϑ χ sinh χϑ K γχ cosh χϑ χ cosh χϑ K γχ sinh χϑ a cos ϑ sin ϑ c cosh χϑ dmϑ e sinh χϑ f a cos ϑ sin ϑ c cosh χϑ dmϑ e sinh χϑ f a sin ϑ cos ϑ cχ sinh χϑ d eχ cosh χϑ a sin ϑ cos ϑ cχ sinh χϑ d eχ cosh χϑ a sin ϑ K γ cos ϑ cos ϑ K γ sin ϑ c χ sinh χϑ K γχ cosh χϑ e χ cosh χϑ K γχ sinh χϑ a sin ϑ K γ cos ϑ cos ϑ K γ sin ϑ c χ sinh χϑ K γχ cosh χϑ e χ cosh χϑ K γχ sinh χϑ The closed form solutions are presented in Appendix B. Assume that K γ. Then the limit lim Kγ Gϕ, ψ yields the Green function matrix for pre-loaded pinned-pinned eams. When K γ, the limit lim Kγ Gϕ, ψ results in the Green function matrix for pre-loaded fixed-fixed eams. 5. COMPUTATIONAL RESULTS In this section we shall present the most important result of the computations only. Let ε oξ crit e the axial strain that elongs to the load P ζ P crit where P crit is the critical load that causes the staility loss of the eam. Further let α i e the i-th eigenfrequency of the loaded eam, while the eigenfrequencies that elong to the free virations then the eam is unloaded are denoted y α i free. Figure represents the quotient α /α free against the quotient ε oξ/ε oξ crit oth for a negative and for a positive P ζ. We remark that this time the suscript always refers to the lowest eigenfrequency for small ϑ the order of the eigenfrequencies will change. The frequencies under compression <tension> decrease <increase> almost linearly A A A A A A 6
and independently of m, ϑ and K γ given that m > and ϑ >. These relationships can e approximated with a very good accuracy y the linear functions α α free α α free..98 ε oξ ε oξ crit, if ε oξ <,..986 ε oξ ε oξ crit, if ε oξ >. Note that these results are almost the same as those valid for pinned-pinned or fixed-fixed curved eams. Figure. Results for the two loading cases of elastically restrained curved eams 6. CONCLUDING REMARKS We have presented a new model to clarify the viratory ehavior of circular eams pre-loaded y central load a vertical force at the crown point. The model is ased on the Euler-Bernoulli eam theory and is applicale for heterogeneous materials. The eam-end supports are rotationally restrained pins, which are modeled y linear volute springs having the same spring constant. The effect of the pre-load is incorporated into the model via the strain it causes. An eigenvalue prolem was estalished y using the principle of virtual work. This eigenvalue prolem was transformed to an eigenvalue prolem governed y Fredholm integral equations. The Green function matrix is given in a closed form oth for compressive load and for tensile load. A numerical algorithm is proposed for the solution. Though some computational results are provided, further computations are needed to refine the results and to clarify how the higher eigenfrequencies depend on the load. Acknowledgements y the second author: This research was supported y the National Research, Development and Innovation Office - NKFIH, K57 and y the Zénó Terplán Program. APPENDIX A. SOLUTIONS FOR Let us introduce the following constants: and A i,..., A i IF ε oξ < D sin χϑ sin ϑ sin χϑ K cos ϑ χ sin χϑ sin ϑ Kχ cos χϑ sin ϑ χ sin ϑ sin χϑ K γ cos ϑ sin χϑ χ sin ϑ cos χϑ, D cos ϑ sin χϑ χ sin ϑ cos χϑ Mχϑ χ cos ϑ cos χϑ K γ χ sin ϑ sin χϑ Mχϑ χ cos ϑ sin χϑ sin ϑ cos χϑ. Making use of the constants introduced the solutions sought can e given in the following forms: A i A in /D, A in χ cos ϑ sin χϑ dχ sin χϑ K γ sin ϑ sin χϑ χ cos ϑ cos χϑ d cos χϑ eχ, a A i A in /D, A in aχ cos ϑ cos χϑ amχϑ χ sin ϑ cos χϑ a sin ϑ sin χϑ cχ fχ cos χϑ K γ amχϑ χ sin ϑ sin χϑ cos ϑ cos χϑ a χ cos ϑ sin χϑ cmχ ϑ fχ sin χϑ, A i A in /χd, A in d sin ϑ eχ χ sin ϑ cos χϑ K γ eχ χ sin χϑ sin ϑ cos ϑ cos χϑ d cos ϑ, c A i A in /D, A in χ χ a cos χϑ c cos ϑ f cos ϑ cos χϑ K γ a χ sin χϑ cχ χ sin ϑ fχ sin ϑ cos χϑ χ cos ϑ sin χϑ, d 7
A i A in /D, A in a cmϑχ χ cos ϑ sin χϑ c χ sin ϑ sin χϑ cos χϑ cos ϑ f cos ϑ K γ amϑ cmϑχ χ cos ϑ cos χϑ sin ϑ sin χϑ c χ sin ϑ cos χϑ f sin ϑ, e A i A in /χd, A in χ χ sin χϑ dmϑχ χ sin ϑ sin χϑ dχ cos ϑ sin χϑ d sin ϑ cos χϑ eχ χ sin ϑ K γ χ cos χϑ d χ cos ϑ cos χϑ dmϑχ χ sin ϑ cos χϑ cos ϑ sin χϑ eχ χ cos ϑ. f APPENDIX B. SOLUTIONS FOR Let and A i,..., A i IF ε oξ > AND mε oξ >. D χ sin ϑ sinh χϑ K γ χ sin ϑ cosh χϑ cos ϑ sinh χϑ 5 D cos ϑ sinh χϑ χ sin ϑ cosh χϑmϑχ χ cos ϑ cosh χϑk γ χ sin ϑ sinh χϑ Mϑχ χ cos ϑ sinh χϑ sin ϑ cosh χϑ. With the two constants introduced A i A in /D, A in χ cos ϑ sinh χϑ dχ sinh χϑ K γ sin ϑ sinh χϑ χ cos ϑ cosh χϑ dχ cosh χϑ eχ, 6 7a A i A in /D, A in a sin ϑ sinh χϑ aχ cos ϑ cosh χϑ amϑχ χ sin ϑ cosh χϑ fχ cosh χϑ cχ K γ amϑχ χ sin ϑ sinh χϑ cos ϑ cosh χϑ cχ Mϑ a χ cos ϑ sinh χϑ fχ sinh χϑ, 7 A i A in /χd, A in d sin ϑ eχ χ sin ϑ cosh χϑ K γ eχ χ sin ϑ sinh χϑ cos ϑ cosh χϑ d cos ϑ, 7c A i A in /D, A in aχ χ cosh χϑ cχ χ K γ a χ cos ϑ fχ χ sinh χϑ cχ cos ϑ cosh χϑ sin ϑ fχ χ cos ϑ sinh χϑ sin ϑ cosh χϑ, 7d χ A i A in /D, A in a cχ sin ϑ sinh χϑ c cos ϑ cosh χϑ cmϑχ χ cos ϑ sinh χϑ f cos ϑ K γ amϑ c χ sin ϑ cosh χϑ cmϑχ χ cos ϑ cosh χϑ sin ϑ sinh χϑ f sin ϑ, 7e A i A in /χd, are the solutions sought. REFERENCES A in χ χ K γ χ cosh χϑ d sinh χϑdmϑχ χ χ sin ϑ sinh χϑd χ cos ϑ sinh χϑ sin ϑ cosh χϑ eχ χ sin ϑ cos ϑ cosh χϑ dmϑχ χ sin ϑ cosh χϑ cos ϑ sinh χϑ eχ χ cos ϑ. 7f L. P. Kiss. Virations and Staility of Heterogeneous Curved Beams. PhD thesis, Institute of Applied Mechanics, University of Miskolc, Hungary, 5. pp.. L. Kiss, Gy. Szeidl, S. Vlase, B. P. Gálfi, P. Dani, I. R. Munteanu, R. D. Ionescu, and J. Száva. Virations of fixed-fixed heterogeneous curved eams loaded y a central force at the crown point. International Journal for Engineering Modelling, 7-:85,. L. Kiss and Gy. Szeidl. Virations of pinned-pinned heterogeneous circular eams sujected to a radial force at the crown point. Mechanics Based Design of Structures and Machines, : 9, 5. Gy. Szeidl. The Effect of Change in Length on the Natural Frequencies and Staility of Circular Beams. PhD thesis, Department of Mechanics, University of Miskolc, Hungary, 975. in Hungarian. 5 P. Chidamparam and A. W. Leissa. Virations of planar curved eams, rings and arches. Applied Mechanis Review, ASME, 69:67 8, 99. 6 Y. Wasserman. The influence of the ehaviour of the load on the frequencies and critical loads of arches with flexily supported ends. Journal of Sound and Virations, 5:55 56, 977. 7 P. Chidamparam and A. W. Leissa. Influence of centerline extensiility on the in-plane free virations of loaded circular arches. Journal of Sound and Virations, 85:779 795, 995. 8 Gy. Szeidl and L. Kiss. A Nonlinear Mechanical Model For Heterogeneous Curved Beams. In S. Vlase, editor, Proceedings of the th International Conference on Advanced Composite Materials Enginnering, COMAT, volume, pages 589 596,. 8 - Octoer, Braşov, Romania. 9 C. T. H. Baker. The Numerical Treatment of Integral Equations Monographs on Numerical Analysis edited y L. Fox and J. Walsh. Clarendon Press, Oxford, 977. 8