Research Article Exact Static Analysis of In-Plane Curved Timoshenko Beams with Strong Nonlinear Boundary Conditions

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1 Mathematical roblems in Engineering Volume 5 Article ID 439 pages Research Article Exact Static Analysis of In-lane Curved Timoshenko Beams with Strong Nonlinear Boundary Conditions Sen-Yung Lee and Qian-Zhi Yan Department of Mechanical Engineering National Cheng Kung University Tainan 7 Taiwan Correspondence should be addressed to Sen-Yung Lee; sylee@mail.ncku.edu.tw Received April 5; Accepted 8 June 5 Academic Editor: Bo-Qing Dong Copyright 5 S.-Y. Lee and Q.-Z. Yan. This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited. Analytical solutions have been developed for nonlinear boundary problems. In this paper the shifting function method is applied to develop the static deflection of in-plane curved Timoshenko beams with nonlinear boundary conditions. Three coupled governing differential equations are derived via the Hamilton s principle. The mathematical modeling of the curved beam system can be decomposed into a complete sixth-order ordinary differential characteristic equation and the associated boundary conditions. It is shown that the proposed method is valid and performs well for problems with strong nonlinearity.. Introduction Curved beam structures are widely used in engineering fields suchasmechanicalcivilandaerospaceengineering.reviews of research on such structures have been conducted by Henrych [] Markus and Nanasi [] Chidamparam and Leissa [3] and Auciello and De Rosa [4]. The in-plane and out-ofplane problems of plane curved beams have been studied. In most general problems they are coupled. However if the cross section of the curved beam is doubly symmetric and theplaneisaprincipalplaneofthecrosssectionthenthe in-plane and out-of-plane problems are uncoupled. Many investigators have studied linear problems about the static and free vibration behaviors of curved beams. Washizu [5] obtained the equations for the coupled motions of a curved and pretwisted bar. Rao [] usedhamilton s principle to derive the governing equations of motion in consideration of the effects of rotatory inertia and shearing deformation. Fettahlioglu and Mayers [7] studied the static deflection of a ring. Bickford and Maganty [8] used a formulation similar to that of Rao and developed equations of motion for out-of-plane vibrations of symmetrical cross-section thick rings accounting for curvature variation through the thickness. Their frequency predictions were validated with the experimental data of Kuhl [9]. Based on the moment-displacement relationships presented by Rao Silva and Urgueira [] derived the dynamic stiffness matrices for the out-of-plane vibration of curved beams using dynamic equilibrium equations. Lee and Chao [] developed the exact out-of-plane vibration solutions of curved nonuniform beams. In the book by Cook and Young [] the exact static analysis of extensional circular curved Timoshenko beams with some special conditions was revealed. Based on the generalized Green function Lin [3] developed the exact solution of extensible curved Timoshenko beams. Lee and Wu [4] studied the exact in-plane vibration solutions of extensible curved nonuniform Timoshenko beams. For the beams with time dependent boundary conditions Lee and Lin [5] generalized the solution method of Mindlin and Goodman [] and developed the shifting function method to study the problems with general time dependent elastic boundary conditions. Recently Lee et al. [7 8] extend the shifting function method to study the exact large deflection of a Bernoulli-Euler beam and Timoshenko beam with nonlinear boundary conditions. From the existing literatures it shows that exact solutions for curved beam problems with nonlinear boundary conditions are not available. In the previous studies [7 8] the governing differential equations are fourth-order differential equations. In the present study one extends the previous studies and the shifting function method [5] to study the exact large

2 Mathematical roblems in Engineering v (s) ψ z (s) u(s) u (s) M z (s) K θl K NθL r v(s) s= f f s=l KVL K R NVL K VR K NVR f θ 3 f f f 4 5 K θr K NθR r θ K UL K NUL K UR K NUR r da z Figure : Geometry and coordinate system of uniform curved beam with nonlinear boundary conditions subjected to loads V (s) u (s)and M z (s) in r θandz directions respectively. static deflection of in-plane curved Timoshenko beams with nonlinear boundaries. The beam with doubly symmetric cross section is considered. The three coupled governing differential equations for the in-plane curved uniform beams of constant radius are derived via Hamilton s principle. These three coupled governing differential equations are decoupled and reduced into a sixth-order differential equation with nonlinear boundary conditions. Consequently the shifting function method is extended and applied to develop the exact solution of the system. It can be shown that the proposed method is valid for problems with strong nonlinearity.. Mathematical Modeling of the Curved Beam System Consider the static response of an extensional curved uniform Timoshenko beam resting on an elastic foundation with nonlinear boundary conditions subjected to loads V (s) u (s) andm z (s) asshowninfigure. Ifthethicknessof the curved beam is small in comparison with the radius of the curved beam without considering the warping effects the displacement fields of the curved beam in cylindrical coordinates are u r (r s z) = V (s) u θ (r s z) =u(s) rψ z (s) u z (r s z) = where u r u θ andu z denote the displacement of the curved beam in the r θ and z directions respectively and s is the arc length along the neutral axis. R is the constant radius of the curved beam and s = Rθ V and u are the neutral axis displacements of the curved beam in the r and θ directions respectively and ψ z is the angle of rotation due to bendinn the z direction. r is measured inward from the neutral axis in the r direction. () Substituting () into the strain-displacement relations in cylindricalcoordinatesonlytwononzerostrainsnamelyε θθ and γ rθ areobtained: ε θθ = γ rθ = R Rr [( u s V R )r ψ z s ] R Rr ( V s u R ψ z). When r is small in comparison with R thetwostrains reduce to ε θθ =( u s V R )r ψ z s γ rθ =( V s u R ψ z). The strain energy of the curved beam is V = L E[( u A s V R ) r ψ z s ( u s V R ) r ( ψ z s ) ψ z ) da ds ]dads L k z G( V A s u R where L is the length of the neural axis. E G anda denote Young s modulus shear modulus and cross-section area of the curved beam respectively and k z denotes the shear correction factors of the curved beam section about the zaxes. Since the cross section of the curved beam considered is doubly symmetric the integral of the second term in the square brackets vanishes. Equation (4) becomes V = L [EA ( u s V R ) ( ψ z s ) ]ds L k z GA ( V s u R ψ z) ds () (4) (5)

3 Mathematical roblems in Engineering 3 where I z denotes the second area moments of inertia of the curved beam section about the z-axes. Considering the linear and nonlinear spring and moment on the boundary of the curved beam end and the force and moment loads on the curved beam the potential energy and work are respectively U = [K VL V () K NVL V 3 ()]dv () [K VR V (L) K NVR V 3 (L)]dV (L) [K UL u () K NUL u 3 ()]du () [K UR u (L) K NUR u 3 (L)]du (L) [K θl ψ z () K NθL ψ z 3 ()]dψ z () [K θr ψ z (L) K NθR ψ z 3 (L)]dψ z (L) W L = [ V (s) V (s) u (s) u (s) M z (s) ψ z (s)]ds f V () f u () f 3 ψ z () f 4 V (L) f 5 u (L) f ψ z (L) where K VL K UL K θl K VR K UR andk θr are the linear spring and rotational stiffness constants in the r θandz directions attheleftandrightendsofthecurvedbeamrespectively. K NVL K NUL K NθL K NVR K NUR andk NθR are the nonlinear spring and rotational stiffness constants in the r θ and z directions at the left and right ends of the curved beam respectively. V (s) u (s)andm z (s) are the force and moment loads in the r θ andz directions respectively and f f f 3 f 4 f 5 andf are the force and moment loads in the r θ and z directionsattheleftandrightendsofthecurvedbeam respectively. The general form of Hamilton s principle is () (7) t δ (V U W )dt=. (8) t Based on Hamilton s principle the governing differential equations and the associated boundary conditions for the curved uniform Timoshenko beam can be derived. The governing differential equations for the in-plane are three coupled differential equations: k z GA ( V s u R s ψ z s )EA R ( u s V R )= V EA( u s V R s )k zga R ( V s u R ψ z)= u ( ψ z s )k zga ( V s u R ψ z)=m z. (9) The associated boundary conditions are as follows: (i) At s= K VL V K NVL V 3 k z GA ( V s u R ψ z)=f (ii) At s=l K UL uk NUL u 3 EA( u s V R )=f K θl ψ z K NθL ψ z 3 ( ψ z s )=f 3. K VR V K NVR V 3 k z GA ( V s u R ψ z)=f 4 K UR uk NUR u 3 EA( u s V R )=f 5 K θr ψ z K NθR ψ z 3 ( ψ z s )=f. In terms of the following nondimensional quantities V (ξ) = V L U (ξ) = u L Ψ (ξ) =ψ z V (ξ) = VL 3 u (ξ) = ul 3 M z (ξ) = M zl f = f L f = f L f 3 = f 3L f 4 = f 4L f 5 = f 5L f = f L β VL = K VLL 3 () ()

4 4 Mathematical roblems in Engineering β NVL = K NVLL 5 β UL = K ULL 3 β NUL = K NULL 5 β θl = K θll β NθL = K NθLL = K VRL 3 β NVR = K NVRL 5 β UR = K URL 3 β NUR = K NURL 5 β θr = K θrl The associated boundary conditions are as follows: (i) At ξ= β VL Vβ NVL V 3 μ ( V ξ θ UΨ)=f () β UL Uβ NUL U 3 ς( U ξ θ V) = f (7) (ii) At ξ= β θl Ψβ NθL Ψ 3 Ψ ξ =f 3. (8) Vβ NVR V 3 μ ( V ξ θ UΨ)=f 4 (9) β UR Uβ NUR U 3 ς( U ξ θ V) = f 5 () β θr Ψβ NθR Ψ 3 Ψ ξ =f. () The coupled differential equations (5) can be decoupled to get two equations Ψ and V: Ψ= θ δ {δ U (4) δ 3 U () δ θ U β NθR = K NθRL θ = L R μ= k z GAL ς= AL I z ξ= s L () [ ς μθ ( ς μ)] u () δ (δ θ V ς () u θ M z )} V= ςθ 3 δ U (5) ςθ δ U θ U () ςθ V δ () δ ςθ V () δ ςθ δ u ςθ 3 ( δ ς u θ M z () ) () the nondimensional coupled governing characteristic differential equations are where the notation (n) denotes the nth-order differentiation with respect to ξ. Consider δ =θ ( ς μ) V μ ( ξ θ U ξ Ψ ξ )ςθ ( U ξ θ V) = V ς( U ξ θ V ξ )θ μ ( V ξ θ UΨ)= u (4) δ = ς μ δ 3 =θ ( ς μ). (4) Ψ ξ μ ( V ξ θ UΨ)=M z. (5) Substituting () (4) into (5) and () () yields the complete sixth-order ordinary differential characteristic

5 Mathematical roblems in Engineering 5 equation and the associated boundary conditions in the θ direction respectively. Consider U () θ U (4) θ 4 U () (i) At ξ= () =θ V δ θ V θ u μθ () u (5) ς (4) 3 () u θ M z θ M z. β VL ςθ 3 δ U (5) θ δ U (4) β VL ςθ δ U θ δ U () β VL θ U () β NVL V 3 =β VL [ ςθ V δ () ςθ V δ δ () ςθ δ u ςθ 3 ( δ ς () u θ M z )] δ (δ V () ςθ u () δ θ u δ θ M z ) M z f δ θ U(5) δ U β UL Uβ NUL U 3 = δ θ (δ V () ςθ u δ θ u () Mz () ) θ V f δ θ δ U (5) δ β θl θ δ U (4) δ 3 θ δ U δ 3β θl θ δ U () θ U () θ β θl Uβ NθL Ψ 3 () δ (δ V () ςθ u () δ θ u δ θ M z ) M z f 4 δ θ U(5) δ U β UR Uβ NUR U 3 = δ θ (δ V () ςθ u δ θ u () Mz () ) θ V f 5 δ θ δ U (5) δ β θr θ δ U (4) δ 3 θ δ U δ 3β θr θ δ U () θ U () θ β θr Uβ NθR Ψ 3 = β θr θ δ [( ς μδ θ ) u () δ (δ θ V ς () u θ M z )] θ δ [( ς μδ θ ) u () () δ (δ θ V ς () u θ M z )] f. (7) 3. The Shifting Function Method In this paper the solution for the sixth-order differential equation (5) with nonlinear boundary conditions ()-(7) is derived. The shifting function method developed by Lee and Lin [5] given below is used: = β θl θ δ [( ς μδ θ ) u () δ (δ θ V ς () u θ M z )] θ δ [( ς μδ θ ) u () () δ (δ θ V ς () u θ M z )] f 3. (ii) At ξ= ςθ 3 δ U (5) δ θ U (4) ςθ δ U θ δ U () θ U () β NVR V 3 = [ ςθ V δ () ςθ V δ where U (ξ) = a (ξ) i= f =β NVL V 3 () f =β NUL U 3 () f 3 =β NθL Ψ 3 () f 4 =β NVR V 3 () f 5 =β NUR U 3 () f =β NθR Ψ 3 () f i (ξ) (8) (9) δ () ςθ δ u ςθ 3 ( δ ς () u θ M z )] and (ξ) i = aretheshiftingfunctionstobe specified and a(ξ) is the transformed function. Substituting

6 Mathematical roblems in Engineering (8)-(9) into (5) (7) yields the differential equation for a(ξ) and the associated boundary conditions: a () θ a (4) θ 4 a () (ii) At ξ= ςθ 3 δ a (5) δ θ a (4) ςθ δ a θ δ a () θ a () = i= f i ( () θ (4) θ 4 () )θ V () δ θ V θ u μθ () u ς (4) u θ 3 M z θ M z (). (i) At ξ= β VL ςθ 3 δ a (5) θ δ a (4) β VL ςθ δ a θ δ a () β VL θ a () = f 4 i= f i ( ςθ 3 δ (5) δ θ (4) ςθ δ θ δ () θ () ) [ ςθ V δ () δ ςθ V () δ ςθ δ u ςθ 3 ( δ ς u θ M z () )] δ (δ V () ςθ u () δ θ u δ θ M z )M z f 4 = f i= f i ( β VL ςθ 3 δ (5) θ δ (4) β VL ςθ δ θ g () δ i β VL g () θ i )β VL [ ςθ V δ () ςθ V δ δ () ςθ δ u ςθ 3 ( δ ς () u θ M z )] δ (δ V () ςθ u () δ θ u δ θ M z ) δ θ a(5) a β δ UR a=f 5 f i ( δ θ g (5) i i= δ β UR ) δ θ (δ V () ςθ u () () δ θ u Mz ) θ V f 5 δ a (5) δ β θr a (4) δ 3 a δ 3β θr a () θ θ δ θ δ θ δ θ δ a () M z f δ θ a(5) a β δ UL a=f f i ( δ θ g (5) i i= δ β UL ) δ θ (δ V () ςθ u () () δ θ u Mz ) θ V f δ a (5) δ β θl a (4) δ 3 a δ 3β θl a () θ θ δ θ δ θ δ θ δ a () θ β θl a=f 3 i= f i ( δ θ δ (5) δ β θl θ δ (4) δ 3 θ δ δ 3β θl θ δ () θ () θ β θl ) β θl [( θ δ ς μδ θ () ) u δ (δ θ V ς () u θ M z )] θ δ [( ς μδ θ ) u () () δ (δ θ V ς () u θ M z )] f 3. θ β θr a=f i= f i ( δ θ δ (5) δ β θr θ δ (4) δ 3 θ δ δ 3β θr θ δ () θ () θ β θr ) β θr [( θ δ ς μδ θ () ) u δ (δ θ V ς () u θ M z )] θ δ [( ς μδ θ ) u () () δ (δ θ V ς () u θ M z )] f. If the shifting functions (ξ) i = in(8) are chosen to satisfy the differential equation () θ (4) θ 4 () = (33) and the following boundary conditions (i) at ξ= β VL ςθ 3 δ (5) θ δ (4) β VL ςθ δ θ δ () β VL θ () =δ ij j =

7 Mathematical roblems in Engineering 7 δ θ g (5) i g δ i β UL =δ ij j = δ g (5) θ δ i δ β θl g (4) θ δ i δ 3 g θ δ i δ 3β θl () g θ δ i (ii) at ξ= θ () θ β θl =δ ij j = 3 (34) θ β θl a= β θl θ δ [( ς μδ θ ) u () δ (δ θ V ς () u θ M z )] θ δ [( ς μδ θ ) u () () δ (δ θ V ς () u θ M z )]f 3. (37) ςθ 3 δ (5) δ θ (4) ςθ δ θ δ () g () θ i =δ ij j = 4 δ θ g (5) i g δ i β UR =δ ij j = 5 δ g (5) θ δ i δ β θr g (4) θ δ i δ 3 g θ δ i δ 3β θr () g θ δ i θ () θ β θr =δ ij j = (35) where δ ij is a Kronecker symbol then the differential equation and the associated boundary conditions - canbereducedto a () θ a (4) θ 4 a () (ii) At ξ= ςθ 3 δ a (5) δ θ a (4) ςθ δ a θ δ a () θ a () = [ ςθ V δ () δ ςθ V () δ ςθ δ u ςθ 3 ( δ ς () () u θ M z )] (δ δ V () ςθ u δ θ u δ θ M z )M z f 4 δ θ a(5) a β δ UR a= () (δ δ θ V ςθ u δ θ u () Mz () ) θ V f 5 (38) (i) At ξ= () =θ V δ θ V θ u μθ () u ς (4) 3 () u θ M z θ M z. β VL ςθ 3 δ a (5) θ δ a (4) β VL ςθ δ a θ δ a () β VL θ a () =β VL [ ςθ V δ () δ ςθ V () δ ςθ δ u ςθ 3 ( δ ς () () u θ M z )] (δ δ V () ςθ u δ θ u δ θ M z )M z f δ θ a(5) a β δ UL a= () (δ δ θ V () () ςθ u δ θ u Mz ) θ V f δ a (5) δ β θl a (4) δ 3 a δ 3β θl a () θ θ δ θ δ θ δ θ δ a () δ θ δ a (5) δ β θr θ δ a (4) δ 3 θ δ a δ 3β θr θ δ a () θ a () θ β θr a= β θr θ δ [( ς μδ θ ) u () δ (δ θ V ς () u θ M z )] θ δ [( ς μδ θ ) u () () δ (δ θ V ς () u θ M z )] f. Once the transformed function a(ξ) and the shifting functions g (ξ) g (ξ) g 3 (ξ) g 4 (ξ) g 5 (ξ) andg (ξ) are determined they are substituted into (8) yielding U (ξ) = a (ξ) β NVL V 3 () g (ξ) β NUL U 3 () g (ξ) β NθL Ψ 3 () g 3 (ξ) β NVR V 3 () g 4 (ξ) β NUR U 3 () g 5 (ξ) β NθR Ψ 3 () g (ξ). (39) Substituting (39) into () and yields the solutions Ψ(ξ) and V(ξ). It can be observed that final solutions include the superposition of the linear and the nonlinear parts. The shifting function method can deal with the nonlinear parts of the boundary conditions very well.

8 8 Mathematical roblems in Engineering 4. Verification and Examples The previous analysis is illustrated using the following example. Example. Consider the deflection of a beam subjected to uniform distributed load. The curved beam is clamped attheleftendandsupportedattherightendwithlinear and nonlinear springs in the r direction. The corresponding coefficients are u = M z =β NVL =β NUL =β NθL =β UR β UL β VL β θl =β NUR =β θr =β NθR =f =f =f 3 =f 5 =f = (4) Let U (ξ) = a (ξ) f 4 g 4 (ξ) (45) where f 4 =β NVR V 3 (). Here g 4 (ξ) is the shifting function to be specified. a(ξ) is the transformed function which satisfies the differential equation (4) and the associated boundary conditions (43)- (44). Consider a () θ a (4) θ 4 a () =. (4) The boundary conditions are as follows: (i) At ξ= a (5) θ a ςθ δ a () =θ δ V = where is constant. Equations () and become Ψ= θ δ (δ U (4) δ 3 U () δ θ U) V= ςθ 3 U (5) U U () δ ςθ δ θ ςθ. (4) The complete sixth-order ordinary differential characteristic equation in (5) and the associated boundary conditions in ()-(7) become U () θ U (4) θ 4 U () =. (4) (ii) At ξ= a= a (4) δ 3 a () θ δ a=. δ ςθ 3 δ a (5) δ θ a (4) ςθ δ a θ δ a () θ a () = ςθ f 4 δ θ a(5) a = δ θ δ a (5) δ 3 a θ θ δ θ δ a () =. δ (47) (48) (i) At ξ= Itcanbefoundthatthefunctiona(ξ) is U (5) θ U ςθ δ U () =θ δ U= (43) a (ξ) = K K ξk 3 sin (θ ξ) K 4 cos (θ ξ) K 5 ξ sin (θ ξ) K ξ cos (θ ξ) (49) (ii) At ξ= U (4) δ 3 U () θ δ U=. δ ςθ 3 δ U (5) δ θ U (4) ςθ δ U θ δ U () δ U () β θ NVR V 3 = ςθ f 4 δ θ U(5) U = δ θ δ U (5) δ 3 U θ θ δ θ δ U () =. (44) where K K K 3 K 4 K 5 andk are given in Appendix. The shifting function g 4 (ξ) satisfies the following differential equation and boundary conditions: (i) At ξ= g 4 () θ g 4 (4) θ 4 g 4 () =. (5) g 4 (5) θ g 4 ςθ δ g 4 () = g 4 = g (4) 4 δ 3 g () δ 4 θ δ g δ 4 =. (5)

9 Mathematical roblems in Engineering 9 (ii) At ξ= ςθ 3 δ g 4 (5) δ θ g 4 (4) ςθ δ g 4 θ δ g 4 () g () θ 4 = δ θ g (5) 4 g δ 4 = δ g (5) θ δ 4 δ 3 g θ δ 4 θ g () 4 =. Itcanbefoundthatthefunctiong 4 (ξ) is g 4 (ξ) = J J ξj 3 sin (θ ξ) J 4 cos (θ ξ) J 5 ξ sin (θ ξ) J ξ cos (θ ξ) (5) (53) Table : Neutral axis displacements in r direction of cantilever curved beam subjected to unit force in r direction at right end of cantilever curved beam [θ = π/ ς =μ =. =β NVR = f 4 = ]. ξ V(ξ) = = = = = Lin s result [3]. where J J J 3 J 4 J 5 andj are given in Appendix. Substituting (45) (49)and(53) into (4) yields the exact solutions of U(ξ) V(ξ)andΨ(ξ)respectively. Considering uniform straight Timoshenko beams as θ = and f 4 =with (4) (45) (49)and(53)onehas Ψ (ξ) =a ξa ξ ξ3 V (ξ) = (ξ3 ξμ ξa 8ξ a 48μa )ξ 4 where a = 4β NVRV 3 () 8(3 3μ ) a = 44β NVRV 3 () 5 μ. (3 3μ ) (54) (55) With both linear and nonlinear spring stiffness constants being zeros (i.e. β NVR = =) (54) reduce to Ψ (ξ) = ξ3 3ξ 3ξ V (ξ) = ξ4 4ξ 3 ( μ) ξ 4μξ. 4 (5) For a Bernoulli-Euler beam without shear deformation (i.e. μ=) (54) and (55) reduce to Ψ (ξ) =a 3 ξa 4 ξ ξ3 V (ξ) = (ξ3 ξa 3 8ξ a 4 )ξ 4 (57) Table : Neutral axis displacements in r and θ directions and angle of rotation due to bendinn z direction for curved beam with linear and nonlinear boundaries subjected to uniform load in r direction [θ = π/ ς =μ =. =β NVR =f 4 = = ]. ξ U(ξ) Ψ(ξ) V(ξ) where a 3 = 4β NVRV 3 () 8 (3 ) a 4 = 44β NVRV 3 () 5. (3 ) (58) It is the exact nondimensional deflection of cantilevered Timoshenko and Bernoulli-Euler curved beams subjected to uniform nondimensional distributed load.equations(54) (58) are exactly the same as those given by Lee et al. [8]. Table shows that the exact solutions of the shifting function method are the same as Lin s results [3]. With increasing distributed load the amount of deflection in the r direction increases at a given location on the curved beam. Table shows the neutral axis displacements in the r and θ directions and the angle of rotation due to bendinn

10 Mathematical roblems in Engineering V(ξ = ) = β NVR = = β NVR = = β NVR = = β NVR = Figure : Deflections in r direction at right end of curved beam (θ =π/)varying with the uniform load for various boundary conditions and constants [ς = μ =. f 4 =]. V(ξ = ) θ = θ =π/ θ =π Figure 3: Deflections in r direction at right end of curved beam varying with the uniform load for various center angles and constants [ς = μ =. = β NVR = f 4 =]. V(ξ = ) = β NVR = = β NVR = θ = β NVR = = β NVR = Figure 4: Deflections in r direction at right end of curved beam varyingwithcenteranglesforvariousboundaryconditionsand constants [ς = μ =. f 4 = =]. the z direction. The curved beam with linear and nonlinear boundaries has lower deflection in the r direction compared to that of a beam without such boundaries. Of note for the curved beam subjected to uniform load in the r direction the deflections at the right end of the curved beam vary as shown in Figures 4. According to (45) (49) and (53) the nonlinear spring stiffness constant is zero and thus it is reasonable that the deflection at the right end of thecurvedbeamislinearlyrelatedtouniformloadinthe r direction. When the nonlinear spring stiffness constant is nonzero value the relationship can be curvilinear. The spring attherightendofthecurvedbeamisstrongenoughto reduce the deflection variation as shown in Figure. Figure 3 shows the influence of curvature on the static deflection of the beam in the r direction for a given set of boundary conditions. When the center angle is increased the deflection willdecrease.thecurvedbeamitselfcanberegardedasa spring mechanism. Figure 4 shows that the center angle and boundary conditions significantly affect deflection in the r direction.iftheloadonthecurvedbeamisinsufficientand the value of the deflection is less than a linear spring has a greater effect than that of a nonlinear spring. On the contrary if the value of the deflection is over a nonlinear spring has a greater effect than that of a linear spring. 5. Conclusion In this paper the shifting function method is applied to develop the static deflection of in-plane curved Timoshenko beams with nonlinear boundary conditions. Three coupled governing differential equations are derived via Hamilton s principle and decomposed into a complete sixth-order ordinary differential characteristic equation that depends on the neutral axis displacement in the θ direction. The explicit relations among the neutral axis displacement in the r direction the angle of rotation due to bendinn thez direction and the neutral axis displacement in the θ direction are revealed. An example was used to illustrate the analysis. It can be found that the center angle and the linear and nonlinear spring

11 Mathematical roblems in Engineering stiffness constants will have significant influence on the static deflection of the beam. Appendix Coefficients of (49) and (53) are as follows: K = { (ς θ ) cos (θ )[ cos (θ )] ςθ 4 f 4 cos (θ )(δ θ )ςθ sin (θ )} θ 4 Q K = θ 3 K 3 = { (θ ς)[(θ δ )θ sin (θ ) cos (θ )] (ςδ θ ) [δ sin (θ )θ δ cos (θ )θ 4 f 4 sin (θ )θ 3 cos (θ )]} (θ 4 Q) K 4 = { (ς θ ) cos (θ )[ cos (θ )] ςθ 4 f 4 cos (θ )(θ δ )ςθ sin (θ )} θ 4 Q K 5 = (/) δ { (ς θ ) cos (θ )[cos (θ )]ςθ 4 f 4 cos (θ )(θ δ )ςθ sin (θ )} θ 3 Q K = (/) δ [ (ς θ ) sin (θ ) cos (θ )(δ θ 4 f 4 )ςsin (θ )(δ θ )ςθ cos (θ )] θ 3 Q (A.) J = ς cos (θ ) Q J = J 3 = (θ ςδ ) sin (θ ) Q J 4 = ς cos (θ ) Q J 5 = ςθ δ cos (θ ) Q J = ςθ δ sin (θ ) Q where Q=(θ δ )θ ς [ς(δ 3 3δ )θ ] sin (θ ) cos (θ ). Conflict of Interests (A.) The authors declare that there is no conflict of interests regarding the publication of this paper. Acknowledgment It is gratefully acknowledged that this research was supported by the National Science Council of Taiwan under Grant MOST 3-8-E--. References [] J. Henrych The Dynamics of Arches and Frames Elsevier Amsterdam The Netherlands 98. [] S. Markus and T. Nanasi Vibrations of curved beams Shock and Vibration Digestvol.3no.4pp [3]. Chidamparam and A. W. Leissa Vibrations of planar curved beams rings and arches Applied Mechanics Reviews vol. 4 no.9pp [4] N.M.AucielloandM.A.DeRosa Freevibrationsofcircular arches: a review JournalofSoundandVibration vol. 7 no. 4 pp [5] K. Washizu Some considerations on a naturally curved and twisted slender beam Mathematics and hysicsvol. 43no.pp. 94.

12 Mathematical roblems in Engineering [] S. S. Rao Effects of transverse shear and rotatory inertia on the coupled twist-bending vibrations of circular rings Sound and Vibrationvol.no.4pp [7]O.A.FettahliogluandJ.Mayers Consistenttreatmentof extensional deformations for the bending of arches curved beams and rings ASME ressure Vessel Technology vol. 99 no. pp [8] W. B. Bickford and S.. Maganty On the out-of-plane vibrations of thick rings Sound and Vibration vol. 8 no. 3 pp [9] W. Kuhl Messungen zu den Theorien der Eigenschingungen von Kreisringen Beliebiger Wandstarke Akustische Zeitschrift vol. 7 pp [] J. M. M. Silva and A.. V. Urgueira Out-of-plane dynamic response of curved beams an analytical model International Solids and Structures vol. 4 no. 3 pp [] S. Y. Lee and J. C. Chao Out-of-plane vibrations of curved non-uniform beams of constant radius Sound and Vibrationvol.38no.3pp [] R. D. Cook and W. C. Young Advanced Mechanics of Materials Macmillian New York NY USA 985. [3] S. M. Lin Exact solutions for extensible circular curved Timoshenko beams with nonhomogeneous elastic boundary conditions Acta Mechanica vol. 3 no. - pp [4] S. Y. Lee and J. S. Wu Exact solutions for the free vibration of extensional curved non-uniform Timoshenko beams Computer Modelinn Engineering and Sciences vol.4no.pp [5] S. Y. Lee and S. M. Lin Dynamic analysis of nonuniform beams with time-dependent elastic boundary conditions Transactions ASME Applied Mechanicsvol.3no.pp [] R. D. Mindlin and L. E. Goodman Beam vibrations with timedependent boundary conditions ASME Applied Mechanicsvol.7no.4pp [7] S. Y. Lee S. M. Lin C. S. Lee S. Y. Lu and Y. T. Liu Exact large deflection of beams with nonlinear boundary conditions CMES: Computer Modelinn Engineering and Sciencesvol.3 no. pp [8] S. Y. Lee S. Y. Lu Y. T. Liu and H. C. Huang Exact large deflection solutions for Timoshenko beams with nonlinear boundary conditions Computer Modelinn Engineering and Sciences vol. 33 no. 3 pp

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