IN-PLANE VIBRATIONS OF PINNED-FIXED HETEROGENEOUS CURVED BEAMS UNDER A CONCENTRATED FORCE

5th International Conference on "Advanced Composite Materials Engineering" COMAT 4 6-7 OCTOBER 4, Braşov, Romania IN-PLANE VIBRATIONS OF PINNED-FIXED HETEROGENEOUS CURVED BEAMS UNDER A CONCENTRATED FORCE György Szeidl, László Kiss Institute of Applied Mechanics, University of Miskolc, Miskolc, HUNGARY gyorgy.szeidl@uni-miskolc.hu, mechkiss@uni-miskolc.hu Astract: The paper deals with the virations of loaded heterogeneous curved eams when a centra load (a constant force) is exerted at the crown point of the eam. The effect of the loading is accounted y the strain it causes. It is assumed that the radius of curvature is constant and the Young s modulus and the Poisson s numer are functions of the cross-sectional coordinates. The paper presents the determination of the Green function matrices for loads directed upwards and downwards. An appropriate numerical model is also provided, which makes possile to determine how the natural frequencies are related to the load. It is also shown that when the strain is zero, the corresponding formulae yield results valid for the free virations of curved eams. Keywords: curved eams, heterogeneous material, natural frequency as a function of the load, Green function matrices. INTRODUCTION Curved eams are widely used in numerous engineering applications let us consider, for instance, arch ridges, roof structures, or stiffeners in aerospace applications. Research into the mechanical ehavior of such structural elements started in the 9 th century see ook y Love for further details. The free virations of curved eams have een under extensive investigation: survey papers were pulished y Markus and Nanasi, Laura and Maurizi as well as Chidampram and Lessia 4. The PhD thesis y Szeidl 5 clarifies how the extensiility of the centerline affects the free virations and staility of circular eams sujected to a constant radial load (dead load) within the frames of the linear theory. The natural frequencies were computed y utilizing different numerical models. One of these relies on the Green function matrix of the corresponding oundary value prolem. Unfortunately, the results of this work have not een pulished in English language. Paper 6 y Huang et al. takes shear deformations into account provided that the eam virates under the action of a constant vertical distriuted load. Lawther 7 investigates how a pre-stressed state of a ody affects the natural frequencies. He studies finite dimensional multiparameter eigenvalue prolems and finds that for multiparameter prolems, the eigenvalue part of the solution is descried y interaction curves in an eigenvalue space, and every such eigenvalue solution has a corresponding eigenvector. If all points on a curve have the same eigenvector, then the curve is necessarily a straight line, ut the converse prolem is far more complex. In the light of Lawther s results, there arises the question: how the frequencies change when a curved eam is sujected to a vertical force at the crown point. We assume that the curved eam is made of heterogeneous, isotropic and linearly elastic material. The cross-section is uniform in terms of oth the geometry and the material composition. The material parameters are functions of the cross-sectional coordinates. Our main ojectives: () derivation of those oundary value prolems, which make possile to clarify how the load affects the natural frequencies; () determination of the Green function matrices, which can e used to reduce the eigenvalue prolems set up for the natural frequencies to eigenvalue prolems governed y systems of Fredholm integral equations; () reduction of the eigenvalue prolems to algeraic ones and (4) the numerical solution of these.. THE PROBLEM FORMULATION Figure (a) shows a portion of the eam with the applied curvilinear coordinate system (ξ s, η, ζ) and () presents a pinned-fixed eam sujected to a load, directed downwards. By assumption, the uniform cross-section is symmetric with respect to the axis ζ. The Young s modulus and the Poisson numer depend on the cross sectional coordinates in such a way that E(η, ζ) E( η, ζ) and ν(η, ζ) ν( η, ζ). Oserve that the coordinate line ξ s coincides with the so-called

Figure. (a) Coordinate system, () Pinned-fixed eam (E-weighted) centerline. This centerline intersects the cross-section at the point C. The location of the latter is otained from the condition that the E-weighted first moment of the cross section with respect to the axis η is zero there: Q eη E(η, ζ) ζ da. () A Let us now separate the load-induced, and otherwise time-independent mechanical quantities from those, which elong to the virations of the loaded eam. The latter ones are the time-dependent increments and are uniformly denoted y a suscript. Let u o, w o and R e the tangential and radial displacements and the radius of the centerline, respectively. Since this 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 a eam under distriuted loading yields that the 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) 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 on the centerline in the tangential and normal directions. Hooke s law expresses the relation etween the inner forces and the deformations 8 in such a way that N I eη R ε oξ M R, A e A M I eη E(η, ζ)da, I eη ( d w o ds A w o R ), N M R I eη R ε oξ, where (4) E(η, ζ)ζ da, m A er I eη (5) A e is referred to as the E-weighted area, I eη is the E-weighted moment of inertia with respect to the ending axis and m is a parameter. For the sake of revity, we introduce dimensionless displacements and a notation for the derivatives: U o u o R, W o w o R ; (...)(n) dn (...) dϕ n, n Z. (6) Upon sustitution of (4) and () into equilirium equations (), we otain the following system of differential equations (DEs): Uo W o (4) m Uo mε oξ W o () m Uo m W o () m ( ε oξ ) Uo If we neglect the effects of the deformations to the equilirium (so we set ε oξ to zero), then we have (4) () () Uo m Uo m Uo Uo W o W o m W o m W o W o R I eη ft f n. (7) R ft. (8) I eη f n With our notational conventions, quantities like the total tangential displacement is equal to the sum u o u o. It turns out that the increments in the axial strain and in the rotation have a very similar structure to equations (): ε m ε oξ ψ oη ψ oη, ψ oη u o R dw o ds, ε oξ du o ds w o R. (9) Based on the principle of virtual work, it can e shown that the equations of equilirium with the increments are ( d N M ) ( N M ) ψ oη f t, (a) ds R R R 4

d M ds N R d ( N M ) ( ψ oη N M ) ψ oη f n. () ds R R Due to the dynamical nature of the prolem, the increments f t and f n are forces of inertia: f t ρ a A u o t, f n ρ a A w o t. () Here A is the area of the cross-section, and ρ a is the averaged density of the cross-section. Application of Hooke s law for the increments in the inner forces yields N I eη R mε oξ M R, M I eη ( d w o ds w ) o R, N M R I eη R mε oξ. A comparison of equations (9), () and () result in the equations of motion: (4) () Uo m Uo W o mε oξ W o () m Uo m W o m ( ε oξ ) Uo W o (a) R ft. () I eη f n We remark that during the formal derivations we have neglected the quadratic term ε oξ ε oξ in (a) and we have used the inequalities ε oξ (ε oξ ψ oη ) () and ε oξ in (), when utilizing Hooke s law. If the virations are assumed to e harmonic with the dimensionless displacement amplitudes Ûo and Ŵo, then we have the following system of DEs Ûo m m Ûo (4) m mε oξ Ûo () m ( ε oξ ) () Ûo Ûo λ ; λ ρ a A R I eη α, (4) where λ and α denote the eigenvalues and eigenfrequencies. For an unloaded eam (ε oξ ), we get ack the equations, which govern the free virations compare equation (4) () () Ûo m Ûo m Ûo Ûo λ m m Ûo to equation () in 9. This system is associated with the oundary conditions valid for pinned-fixed eams and together they constitute an eigenvalue prolem. The left side of equation (4) can e rewritten in the rief form K y (ϕ), ε oξ Py 4 (4) Py () Py () Py (), y T Ûo Ŵo. (5) Oserve that the i-th eigenfrequency α i in the eigenvalue prolem depends on the heterogeneity parameters m and ρ a ; and also, on the magnitude and the direction of the concentrated force. The effects of the latter one are accounted through the axial strain: ε oξ ε oξ (P). Here P is a dimensionless load, defined y P P ζ R ϑ/(i eη ).. THE GREEN FUNCTION MATRIX Differential equations (5) are degenerated, since the matrix P 4 has no inverse. Let r(ϕ) e a prescried inhomogeneity. Consider the oundary value prolems defined y 4 ν K(y) P(ϕ)y (ν) (ϕ) r(ϕ), P(ϕ) (6) ν and the oundary conditions valid for pinned-fixed eams: Û o ( ϑ) ( ϑ) Ŵ () o ( ϑ) Û o (ϑ) (ϑ) Ŵ () o (ϑ). (7) Solution to the homogeneous part of differential equation (5) depends on whether the axial strain ε oξ is positive or negative i.e.: whether the concentrated force is directed upwards or downwards. Let { χ mεoξ ε if oξ < (8) mε oξ ε oξ > and mε oξ >. This solution is of the form 4 y Y ( ) i C i ( ) i e, where (9a) ( ) 5

cos ϕ Y sin ϕ, Y sin ϕ cos ϕ cos χϕ Mϕ, Y, Y χ sin χϕ 4 sin χϕ χ cos χϕ when ε oξ <. However, Y and Y 4 are different when ε oξ > and mε oξ > : cosh χϕ Mϕ sinh χϕ Y, Y χ sinh χϕ 4. (9c) χ cosh χϕ In the aove relations C i are aritrary constant matrices, e are aritrary column matrices and M (m)/m ( ε oξ ). Solutions to the oundary value prolems (6) and (7) are sought in the form y(ϕ) a G(ϕ, ψ)r(ψ)dψ, G(ϕ, ψ) G (ϕ, ψ) G (ϕ, ψ) G (ϕ, ψ) G (ϕ, ψ) (9), () where G(ϕ, ψ) is the Green function matrix, defined y the following properties 5: () The Green function matrix is continuous in ϕ and ψ in each of the triangular ranges ϑ ϕ ψ ϑ and ϑ ξ ϕ ϑ. The elements (G (ϕ, ψ), G (ϕ, ψ)) G (ϕ, ψ), G (ϕ, ψ) are ( times) 4 times differentiale with respect to ϕ, and the following derivatives are continuous in ϕ and ψ: ν G(ϕ, ψ) x ν G (ν) (ϕ, ψ), ν,, ν G i (ϕ, ψ) x ν () Let ψ e fixed in ϑ, ϑ. Although the functions G (ϕ, ψ), G (), are continuous everywhere, the derivatives G () lim ε G () (ϕ ε, ϕ) G() (ϕ ε, ϕ) / P (ϕ), lim ε (ϕ, ψ),g() G (ν) i (ϕ, ψ), ν,..., 4; i,. () G () (ϕ, ψ), G(ν) (ϕ, ψ) have a jump at ϕ ψ: (ϕ, ψ) ν,, ; G(ν) (ϕ, ψ) ν (ϕ ε, ϕ) G() (ϕ ε, ϕ) / 4 P (ϕ). () () Let α denote an aritrary constant vector. For a given ψ ϑ, ϑ, the vector G(ϕ, ψ)α, as a function of ϕ (ϕ ψ) should satisfy the homogeneous differential equation K G(ϕ, ψ)α. (4) The vector G(ϕ, ψ)α, as a function of ϕ, should satisfy the oundary conditions (7). Moreover, there exists only one Green function matrix to each of the oundary value prolems. If the Green function matrix exists then () satisfies the differential equations (6) and the oundary conditions (7). Consider the system of differential equations in the form of Ky λy, where Ky is given y (5) and λ is the unknown eigenvalue. The system of ordinary DEs () and the homogeneous oundary conditions (7) constitute a oundary value prolem, which is, in fact, an eigenvalue prolem with λ as the eigenvalue. Vectors u T u u and v T v v are comparison vectors, if they are different from zero, satisfy the oundary conditions and are differentiale as many times as required. The eigenvalue prolems (), (7) are self-adjoint if the product (u, v) M ϑ ϑ u T Kv dϕ is commutative, i.e., (u, v) M (v, u) M over the set of comparison vectors and it is positive definite for any comparison vector u, if (u, u) M >. If the eigenvalue prolems (), (7) are self-adjoint, then the related Green function matrices are cross-symmetric: G(ϕ, ψ) G T (ψ, ϕ). 4. NUMERICAL SOLUTION TO THE EIGENVALUE PROBLEMS Making use of (), the eigenvalue prolems (), (7) can e replaced y homogeneous integral equation systems: y(ϕ) λ ϑ ϑ G(ϕ, ψ)y(ψ)dψ. Numerical solution to such prolems can e sought e.g., y quadrature methods. Consider the integral formula ϑ n J(φ) φ(ψ) dψ w j φ(ψ j ), ψ j ϑ, ϑ, (6) ϑ j where ψ j (ϕ) is a vector and w j are the known weights. Having utilized the latter equation, we otain from (5) that n w j G(ϕ, ψ j )ỹ(ψ j ) κỹ(ϕ) κ / λ ϑ, ϑ (7) j () (4) (5) 6

is the solution, which yields an approximate eigenvalue λ / κ and the corresponding approximate eigenfunction ỹ(ϕ). After setting ϕ to ψ i (i,,,..., n), we have n w j G(ψ i, ψ j )ỹ(ψ j ) κỹ(ψ i ) κ / λ ψ i, ψ j ϑ, ϑ, or GDỸ κỹ, (8) j where G G(ψ i, ψ j ) for self-adjoint prolems, while D diag(w,..., w... w n,..., w n ) and ỸT ỹ T (ψ ) ỹ T (ψ )... ỹ T (ψ n ). After solving the generalized algeraic eigenvalue prolem (8), we have the approximate eigenvalues λ r and eigenvectors Y r, while the corresponding eigenfunction is otained via sustituting into (7): ỹ r (ϕ) λ r n j w j G(ϕ, ψ j )ỹ r (ψ j ) r,,,..., n. (9) 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 (9), the algeraic eigenvalue prolem otained has the same structure as (9). It is also possile to consider the integral equations (5) 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 diag(n i ), N.5γ(γ ), N γ, N.5γ(γ e ), y i is the value of the eigenfunction y(ϕ) at the left endpoint, the midpoint and the right endpoint of the element, respectively. Upon sustitution of approximation () into (5), we have n ỹ(ϕ) λ e ey G(x, γ)n (η) N (γ) N (γ)dγ y e e T y, () L e e in which, n e is the numer of elements. Using equation () as a point of departure, and repeating the line of thought leading to (8), we again get an algeraic eigenvalue prolem. 5. COMPUTATION OF THE GREEN FUNCTION MATRICES Based on the definition presented in Section, here we show the calculation of the corresponding Green function matrices for the two loading cases of pinned-fixed eams. With regard to property, the Green function can e given in the form G(ϕ, ψ) }{{} ( ) 4 Y j (ϕ) A j (ψ) ± B j (ψ), () j where (a) the sign is positive(negative) if ϕ ψ(ϕ 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,..., 4; () j A j A (c) the coefficients in B j are independent of the oundary conditions. As the matrices Y and Y 4 are different for ε oξ < and for ε oξ >, when mε oξ >, we deal with the two possiilities 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 4 B i, f 4 B i. (4) We note that B B B B see Section. The systems of equations for the unknowns a,..., f can e set up y fulfilling the continuity and discontinuity conditions mentioned in property and for the Green function matrix if ϕ ψ. Therefore, if i, we have j B 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 4 cos (χψ) χ 4 sin (χψ) from where we get the constants as j B a c d e f m, (5) 7

If i, then 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 4 cos χψ χ ( χ, f B 4 ) ( M) m M ψ m ( M). χ 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 4 cos (χψ) χ 4 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 4 sin χψ ( χ ) χ, f B 4 χ. (8) Regarding the unknown scalars A i (ψ), (ψ), (ψ), A i (ψ), matrices A j, property (4), that is the oundary conditions (7) yield 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) cos ϑ sin ϑ χ cos (χϑ) χ sin (χϑ) sin ϑ cos ϑ χ sin (χϑ) χ cos (χϑ) 4 (ψ), (6) (7) 4 A i (ψ), i, ; ψ ϑ, ϑ in the A i 4 4 A i 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 cos ϑ sin ϑ cχ cos (χϑ) eχ sin (χϑ) a sin ϑ cos ϑ cχ sin (χϑ) eχ cos (χϑ). (9) Closed form solution to this system was generated using Maple 5. The unknowns are not detailed here. Calculation of the Green functions matrix if ε oξ > and mε oξ >. For the other loading case, repeating a procedure similar to the procedure leading to (5), we get the following equations if i cos ψ sin ψ cosh (χψ) Mψ sinh (χψ) sin ψ cos ψ χ sinh (χψ) χ cosh (χψ) sin ψ cos ψ χ sinh (χψ) M χ cosh (χψ) cos ψ sin ψ χ cosh (χψ) χ sinh (χψ) sin ψ cos ψ χ sinh (χψ) χ cosh (χψ) cos ψ sin ψ χ 4 cosh (χψ) χ 4 sinh (χψ) The solutions are as follows χ a B ( χ ) ( M) m c B χ ( χ ) ( M) m e 4 B sin ψ a c d e f χ m, B ( χ ) ( M) m, d B ( M) m, sinh χψ cosh χψ χ ( χ, f 4 B ) ( M) m ( M) m Mψ.. (4) cos ψ, (4) 8

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 4 cosh (χψ) χ 4 sinh (χψ) f is the equation system to e solved compare it to (7) 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 4 sinh χψ χ ( χ ), f B 4 χ. For the elements of the matrices A j, oundary conditions (7) yield the equation system 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) cos ϑ sin ϑ χ cosh (χϑ) χ sinh (χϑ) A i sin ϑ cos ϑ χ sinh (χϑ) χ 4 cosh (χϑ) 4 A i 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 cos ϑ sin ϑ cχ cosh (χϑ) eχ sinh (χϑ) a sin ϑ cos ϑ cχ sinh (χϑ) eχ cosh (χϑ) the solutions of which are omitted here. (4) (4). (44) 6. THE LOAD-STRAIN RELATION AND THE CRITICAL STRAIN In practise, the loading is generally the known quantity. However, the formulation has the axial strain ε oξ as a parameter. For a first, linearized model, the effect the deformations have on the equilirium is neglected. We can estalish the loadstrain relationship ε oξ ε oξ (P) on the asis of differential equations (8) if f t f n, and y applying the U o ±ϑ W o ±ϑ M ϑ ψ oη ϑ, U o ϕ U o ϕ, W o ϕ W o ϕ, ψ oη ϕ ψ oη ϕ, N ϕ N ϕ, M ϕ M ϕ, dm/ds ϕ dmds ϕ P ζ (45) oundary and continuity (discontinuity) conditions prescried at the crown point. Here the physical quantities can all e given in terms of the dimensionless displacements U o and W o as ψ oη U o W o (), N A e ε oξ M A e ε oξ, M I ( ) eη ρ o ρ W o () W o. (46) o After solving the oundary value prolem defined y the ODEs (8) (f t f n ) and the aove continuity and discontinuity conditions we get the axial strain in the following form ε(p, m, ϑ) P (ϑ cos ϑ 4 cos ϑ sin ϑ 4 sin ϑ) cos ϑ ( ϑ cos ϑ ) sin ϑ ϑ ( 5 cos ϑ) ϑ m ( ϑ sin ϑ cos ϑ 4ϑ cos ϑ sin ϑ 7 cos ϑ) cos ϑ 4 ϑ ϑ (cos ϑ sin ϑ cos ϑ sin ϑ ϑ). (47) If P is negative (positive), then ε oξ is negative (positive). The critical strain, at which, curved eams lose their staility, can e otained y solving the eigenvalue prolem governed y equations () with the right side set to zero and the corresponding oundary conditions. The eigenvalue sought is χ mε oξ. The general solution for the displacement increments and the oundary conditions to e satisfied are: W o E E cos ϕ E 4 sin ϕ χe 5 cos χϕ χe 6 sin χϕ, (48) (45a) U o E E Mϕ E sin ϕ E 4 cos ϕ E 5 sin χϕ E 6 cos χϕ, (49) 9

U o ±ϑ W o ±ϑ W () W (). (5) ϑ ϑ o o Here E i (i,..., 6) are undetermined constants of integration. The oundary conditions lead to the homogenous equation system Mϑ sin ϑ cos ϑ sin χϑ cos χϑ E Mϑ sin ϑ cos ϑ sin χϑ cos χϑ E cos ϑ sin ϑ χ cos χϑ χ sin χϑ E cos ϑ sin ϑ χ cos χϑ χ sin χϑ E 4. (5) cos ϑ sin ϑ χ cos χϑ χ sin χϑ E 5 sin ϑ cos ϑ χ sin χϑ χ cos χϑ E 6 The vanishing of the determinant results in the following non-linear equation: χ 4 ( cos χϑ ) ( χ ) sin χϑ χϑ ( χ ) cos χϑ sin χϑ sin ϑ χ ϑ ( χ ) cos ϑ sin ϑ cos χϑ ( χϑ ( χ ) cos χϑ sin χϑ ) cos ϑ χ ϑ ( χ ) sin ϑ sin χϑ χ ( χ ) sin ϑ cos χϑ cos ϑ sin χϑ. (5) The lowest reasonale critical value is then approximated y the polynomial χϑ g pf (ϑ).4 875, ϑ 5.78 7 ϑ 4.68 958 ϑ.9 66 ϑ.57 ϑ.749 9, (5) and the critical strain is ε oξ crit ( χ ) m m 7. COMPUTATIONAL RESULTS (gpf ϑ ). (54) Figure. Results for pinned-fixed eams when ε oξ We have solved the eigenvalue prolems governing the virations of curved eams using a Fortran9 code. The numerical results have een compared to those valid for the free virations of curved eams with the same geometry and material. For more details aout the natural frequencies of planar curved eams see 5.

When we set the strain to a very small value, i.e., to ε oξ ε oξ crit 6 for oth loading cases, we get ack the results valid for the free virations of curved eams see 5 and for details. It is known that see e.g.: 9 the i-th eigenfrequency for the free transverse virations of heterogeneous straight eams is otained from the relation α i C i, charπ ρaa I eη l, where the constant C i, char depends on the supports and the ordinal numer of the frequency sought. This time C, char.556, C, char 5.78, C, char.54, C 4, char 7.97 and l is the length of the eam. If we recall Eq.(4) which, for such a small strain considered, expresses the relation etween the eigenvalues λ i and the eigenfrequencies α i α i free for the free virations of curved eams we may write C i,char α i α i λi ρ a A Ieη R π ρ a A Ieη l r ϑ λ i π. (56) This is the connection etween the natural frequencies of curved and straight eams with the same length (l R ϑ), cross-section and material. In Figure, this ratio is plotted against the central angle ϑ of the curved eam. Four different values of the parameter m were picked: (,.4,, ). Oserve that the ratio of the even natural frequencies are independent of m, while the odd ones are not. It is also important to mention that the frequency spectrum changes as ϑ increases e.g.: the first eigenfrequency ecomes the second one in terms of its magnitude if ϑ is sufficiently great. When dealing with the free longitudinal virations of fixed-fixed rods, the natural frequencies assume the form ˆα i K i char l r E ρ a π, where the constant K i char i; (i,,,...) and l r is the length of the rod. If we recall Eq.(4), we can compare the eigenfrequencies of curved eams (given that ε oξ ε oξ crit 6 when calculating λ i ) to those of rods y α i K i char ϑ λi. ˆα i m π These quotients for i, are plotted in Figure. We found that the ratios do not depend on the parameter m and these are equal to and respectively if the central angle is sufficiently small. We remark that these tendencies are the same with a good accuracy for pinned-pinned and for fixed-fixed curved eams as well. (55) (57) (58) Figure. Results for pinned-fixed eams when ε oξ In the forthcoming, the effect of the concentrated load to the length of the centerline is taken into account. In what follows, regarding our notations, α i is i-th eigenfrequency of the loaded curved eam, while the eigenfrequencies that elong to the free virations (the eam is unloaded) are denoted y α i free. Figure 4 represents the quotient α /α free against the quotient ε oξ/ε oξ crit oth for a negative and a positive P ζ. We remark that this time the suscript always refers to the lowest frequencies (which do not coincide with the first one every time see Figure ). The frequencies under compression <tension> decrease <increase> almost linearly and

independently of m and ϑ, given that m > and ϑ >. These relationships can e approximated with a very good accuracy y α. 848 55.98 86 7 ε ( ) oξ εoξ.74 8 54, if ε oξ <, (59) ε oξ crit α free ε oξ crit α α. 98 5.986 64 ε ( ) oξ εoξ.8 7 55, if ε oξ >. (6) free ε oξ crit ε oξ crit We again remark that these results are almost the same, when the curved eam is pinned-pinned or fixed-fixed. Figure 4. Results for the two loading cases of pinned-fixed eams 8. CONCLUDING REMARKS In accordance with our aims, we have investigated the virations of curved eams with cross-sectional heterogeneity under a centra load (a vertical force) exerted at the crown point. We have derived the governing equations of the oundary value prolems, which make it possile to clarify how the radial load affects the natural frequencies. For pinned-fixed eams, we have determined the Green function matrices assuming that the eams are prestressed y a radial load. When computing the corresponding matrices, we had to take into account that the system of differential equations that govern the prolem are degenerated. Making use of the Green function matrices, we have reduced the selfadjoint eigenvalue prolems set up for the eigenfrequencies to eigenvalue prolems governed y homogeneous systems of Fredholm integral equations. These integral equations can e used for those loads, which results in a constant axial strain on the E-weighted centerline. Numerical solutions were provided graphically. For the loaded eams considered, the quotient α/α free depends linearly with a good accuracy on the axial strain ε oξ. With the knowledge of the relationship ε oξ ε oξ (P), we can determine the strain, which elongs to a given load and then the natural frequencies of the loaded structure. Acknowledgements y the second author: This research was supported y the European Union and the State of Hungary, cofinanced y the European Social Fund in the framework of TÁMOP 4..4.A/---- National Excellence Program. REFERENCES A. E. H. Love. Treatese on the mathematical theory of elasticity. New York, Dower, 944. S. Márkus and T. Nánási. Viration of curved eams. The Shock and Viration Digest, (4): 4, 98. P. A. A. Laura and M. J. Maurizi. Recent research on virations of arch-type structures. The Shock and Viration Digest, 9():6 9, 987. 4 P. Chidamparam and A. W. Leissa. Virations of planar curved eams, rings and arches. Applied Mechanis Review, ASME, 46(9):467 48, 99. 5 G. Szeidl. Effect of Change in Length on the Natural Frequencies and Staility of Circular Beams. Ph.D Thesis, Department of Mechanics, University of Miskolc, Hungary, 975. (in Hungarian). 6 C. S. Huang, K. Y. Nieh, and M. C. Yang. In plane free viration and staility of loaded and shear-deformale circular arches. International Journal of Solids and Structures, 4:5865 5886,. 7 Ray Lawther. On the straightness of eigenvalue iterations. Computational Mechanics, 7:6 68, 5. 8 G. Szeidl and L. Kiss. A Nonlinear Mechanical Model For Heterogeneous Curved Beams. In S. Vlase, editor, Proceedings of the 4th International Conference on Advanced Composite Materials Enginnering, COMAT, volume, pages 589 596, 8 - Octoer, Braşov, Romania. 9 L. P. Kiss. Free virations of heterogeneous curved eams. GÉP, LXIV(5):6,. (in Hungarian). Christopher T. H. Baker. The Numerical Treatment of Integral Equations Monographs on Numerical Analysis edited y L. Fox and J. Walsh. Clarendon Press, Oxford, 977. K. Kelemen. Virations of circular arches sujected to hydrostatic follower loads computations y the use of green functions. Journal of Computational and Applied Mechanics, :67 78,.