Dynamic equations for curved submerged floating tunnel

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1 Appied Mathematics and Mechanics Engish Edition, 7, 8:99 38 c Editoria Committee of App. Math. Mech., ISSN Dynamic equations for curved submerged foating tunne DONG Man-sheng, GE Fei, ZHANG Shuang-yin, HONG You-shi State Key Laboratory of Noninear Mechanics, Institute of Mechanics, Chinese Academy of Sciences, Beijing 8, P. R. China Contributed by HONG You-shi Abstract In virtue of reference Cartesian coordinates, geometrica reations of spatia curved structure are presented in orthogona curviinear coordinates. Dynamic equations for heica girder are derived by Hamiton principe. These equations indicate that four generaized dispacements are couped with each other. When spatia structure degenerates into panar curviinear structure, two generaized dispacements in two perpendicuar panes are couped with each other. Dynamic equations for arbitrary curviinear structure may be obtained by the method used in this paper. Key words submerged foating tunne SFT, dynamic equations, Hamiton principe, curved girder Chinese Library Cassification U459.9, TB Mathematics Subject Cassification 74K, 7H5 Digita Object IdentifierDOI.7/s z Introduction Submerged foating tunne SFT, aso caed Archimedes bridge, is a potentia traffic channe foating between water surface and water bed, which is a soid structure made of metas or reinforced concrete or mixed. In some water areas of environmenta restriction or protection, it may be impossibe to set up a traditiona crossing. For such cases, SFT coud be a promising soution. However, ti now, there is not an SFT constructed in the word. This is a great chaenge to engineering science community and has attracted preiminary investigations [ 6]. Due to oca geographica and environmenta conditions, a curved sect at the end of an SFT to connect the water bank sometimes is more feasibe so as to decrease tunne ength and/or reduce tunne sope. Doube wa structure is often used in the conceptua design of SFT for which warp effect coud be ignored. Thus the SFT structure may be regarded as a spatia girder without warping. Severa research resuts are avaiabe on dynamic behavior of curved girder [7 ], which mosty are reated to panar curved bridge structure, especiay about panar circuar structure. However, there are few reports reated to dynamic behavior of spatia curved structure. The dynamic behavior of SFT subjected to water wave and current is a major probem in the SFT reaization. This paper considers the dynamic behavior of curved girder used in civi engineering especiay in SFT structure. Based on reference Cartesian coordinates and orthogona curviinear coordinates, dynamic equations for heica girder are derived by means of Hamiton principe. Received Jan. 7, 6; Revised Apr. 6, 7 Project supported by the Nationa Natura Science Foundation of China No. 537 Corresponding author HONG You-shi, Professor, Doctor, E-mai: hongys@imech.ac.cn

2 3 DONG Man-sheng, GE Fei, ZHANG Shuang-yin, et a. Hamiton principe [3] Hamiton principe is written as δt + δπ dt =. Equation means that for an actua movement, the tota integra of δt + δπ is zero in any time interva, where δt is the variation of dynamic energy and δπ is the virtua work due to a effective forces. δπ may be expressed as δπ = δv + δw, where δv is the virtua work due to potentia forces, and δw is the virtua work due to non-potentia forces. Thus we have the genera form of Hamiton principe: δt δv + δw dt =. When the forces appied on the system are ony potentia forces, i.e., δw =,wehave δt δv dt =. The above equation of Hamiton principe may aso be expressed by Lagrangian function of L = T V,then δldt =. For hoonomic system, variation symbo and integra symbo are substitutabe each other. So that δldt = δ Ldt. By introducing the expression of Hamiton s action, I = t Ldt, thus Hamiton principe for hoonomic system may be written as variation form: δi = δ Ldt =. 3 Equation 3 suggests that for a hoonomic system appied by potentia forces, Hamiton s action of actua movement takes stagnation vaue at any time interva, compared with possibe movement for the same boundary condition. In short, the variation of Hamiton s action is zero. Hamiton principe invoves ony two dynamic quantities of dynamic energy and work or potentia energy, and there is no any restrictions for system geometry. For straight girder, the configuration function represents defection. When the girder is spatiay curviinear, torsion ange and axia dispacement need to be considered. Geometrica reations of curviinear girder It is convenient to anayze the movement of spatia curviinear girder by introducing reference Cartesian coordinates xyz, in which x and z axes are in horizonta pane, and y axis points verticay and downward. Anaysis coordinates take the form of orthogona curviinear coordinates with the origin overapping that of reference coordinates, as shown in Fig..

3 Dynamic equations for curved submerged foating tunne 3 z y x Fig. Reference coordinates and curviinear coordinates The curviinear structure is of spira shape. The parametric equations in reference coordinates are x = R R cos t, y = kt, k >, 4 z = R sin t, where t is a parameter and k is screw-pitch. The reation between curvature radius and defective radius of the curve is = R + k R, = R + k. 5 k The direction of axis in curviinear coordinates is the tangent of the curve, i.e., = R sin ti kj + R cos tk. 6 R + k The direction of axis in curviinear coordinates is the principa norma of the curve, i.e., =costi sin tk. 7 The direction of axis is perpendicuar to both axes of and, and satisfies right hand principe, i.e., = = k sin ti + Rj + k cos tk. R + k 8 Suppose that the eft end of the girder be ocated at the origin of coordinates and that it extends aong the spira ine as shown in Fig.. Letting the parameter t equa t at the vicinity of eft end, the axis directions in curviinear coordinates are = R + k R sin t i kj + R cos t k, =cost i sin t j, 9 = R + k k sin t i + Rj + k cos t k. For the increment of the parameter dt, the axis directions for an arbitrary sma portion

4 3 DONG Man-sheng, GE Fei, ZHANG Shuang-yin, et a. corresponding to t + dt in curviinear coordinates are = R + k R sin t + dt i kj + R cos t + dt k, =cost + dt i sin t + dt k, = R + k k sin t + dt i + Rj + k cos t + dt k. So the orientation correations of the axes at the two ends for the sma portion in curviinear coordinates are as foows: cos, = R cos dt + k R sin dt R + k, cos, = R + k, cos kr cos dt, = R + k, R sin dt cos, = R + k, cos, =cosdt, cos, = k sin dt R + k, kr cos dt k sin dt cos, = R + k, cos, = R + k, cos, = k cos dt + R R + k. There are four independent generaized dispacements for curved girder, incuding three dispacements u, u,and u aong curviinear axes and the torsion ange φ with respect to axis. There exist foowing geometrica reations: d = R + k dt, φ = du d, φ = du d, where φ is the torsion ange with respect to axis and φ is the torsion ange with respect to axis. The spatia curviinear axia strain ε is ε = im dt u + du cos, +u + du cos, +u + du cos, u d = u u u. 3 With the hep of Fig., we may describe the meaning of ε. First the projection of the dispacements at the right end to the tangent of the eft end subtracts the tangentia dispacement at the eft end. Then the difference is divided by the ength of sma curve portion to give the vaue of ε. The expression of origina curvature for the curved girder with respect to axis is k/r + k. The increment of the defection curvature with respect to axis due to the dispacement in direction is χ = u u. 4 The expression of origina principa curvature for curved girder with respect to axis is R/R + k. The increment of curvature with respect to axis due to the dispacement in direction is χ = u. u 5

5 Dynamic equations for curved submerged foating tunne 33 Deformation curvatures of curved girder χ and χ are defined as the increment of the curvature due to deformation. In the cacuation of χ, the curvature increments χ and χ and the increment induced by the girder rotation shoud be considered. Referring to Fig. 3, deformation curvature χ with respect to axis is χ = u φ + dφ cos, +φ + dφ cos, +φ + dφ cos, φ + im dt d = u u + φ. 6 Simiary, deformation curvature χ with respect to axis is χ = u φ + dφ cos, +φ + dφ cos, +φ + dφ cos, φ + im dt d = u + u + φ. 7 Torsion curvature τ is φ + dφ cos, +φ + dφ cos, +φ + dφ cos, φ τ = im dt d = φ + u + u. 8 u +du φ +dφ u u +du u +du φ φ +dφ φ +dφ Fig. u u A sect of girder schematicay showing the dispacements and their increments φ φ Fig. 3 Schematic showing φ, φ, φ and their increments 3 Dynamic equations of curved girders The system energy of curved girder consists of system dynamic energy, eastic strain energy and work done by externa forces. Dynamic energies due to rotation with respect to and axes are φ da = da = ρi φ = ρi u, s s ρ φ s ρ φ da = ρ φ s da = ρi φ = ρi u.

6 34 DONG Man-sheng, GE Fei, ZHANG Shuang-yin, et a. Hence the dynamic energy of the structure is T = [ ρ A u + u + u +a φ ] + I u + I u d, 9 where a is the radius of gyration of girder cross-section, A is the cross-section area, ρ is the density of structure, the head dot indicates the derivative with respect to time, and the use of prime denotes the deferentia with respect to. Strain energy is V = EI χ + EI χ + GI d τ + EAε d, where E is eastic moduus, I and I are the moments of inertia with respect to and, respectivey, and I d is the torsion moment. Work done by externa forces is W = f u + f u + f u + m φ + m φ + m φ d, where f,f,f are forces in directions,,, respectivey, and m,m,m are the moments with respect to the unit axis of,,. It is convenient to perform the variation anaysis of dynamic energy T,strainenergy V and work W separatey. The variation of dynamic energy T is δ Tdt = δ = The variation of strain energy V δ Vdt= EA GI d EI + m u + u + u + a φ + ρi u + ρi u ddt ρ Aü I ü δu ρ Aü I ü δu ρaü δu ρaa φ δφ ddt. is EI u 4 u u + u u u + u + u + φ u δu + EA + EI + u ρ 4 φ φ + EI u 4 u u u The variation of work done by externa forces is u + u + φ + u GI d + u ρ 4 δu EA + u + u + φ u u u δu + φ GI d + u + u δφ ddt. δ Wdt = f δu + f δu + f δu + m δφ + m δφ + m δφ ddt. Considering the geometrica reation of Eq., one may write the above equation as δ Wdt = 3 f m δu + f + m δu + f δu + m δφ ddt. 4

7 Dynamic equations for curved submerged foating tunne 35 In a word, based on the genera form of Hamiton principe Eq., one is abe to write the dynamic equations of curved SFT without energy dissipation, such that ρ Aü I ü + EI u 4 + u + φ + u ρ 4 + φ GI d = f m, + u + u EA ρ Aü I ü + EI u 4 GI d = f + m, ρaü EA + u + u u u u EA u = f, u u u φ + u ρ 4 + φ u u u a ρa φ + EI u + u + φ + EI u + u + φ GI d + u + u = m. 5 When k = and, spatia curve degenerates into pane curve. So dynamic differentia equations are ρ Aü I ü + EI u 4 + u + u EA u u = f m, ρ Aü I ü + EI u 4 ρaü EA a ρa φ + EI u u = f, u + φ φ ρ 4 GI d GI d + u + u = m. = f + m, For the soution of Eqs. 5and 6, it is necessary to convert the oading in the reference coordinate system into that in the curviinear coordinate system. Thus one may sove them in the curviinear coordinate system. 4 Intrinsic characteristics of curved structure vibration The boundary conditions are given in terms of vibration function. i Fixed end: dispacement and anguar rotation equa zero, i.e., Ux, t Ux, t =, =, 7 where U = qtv =qtu,u,u,φ T and x =. Equation 7 can be expressed in terms of vibration function: V x V x =, =. 8 ii Simpy supported end: dispacement and bending moment equa zero, i.e., Ux, t =, 6 Ux, t =. 9

8 36 DONG Man-sheng, GE Fei, ZHANG Shuang-yin, et a. It can be expressed in terms of vibration function: V x V x =, =. 3 iii Free end: bending moment and shear force equa zero, i.e., Ux, t =, 3 Ux, t 3 =. 3 It can be expressed in terms of vibration function: V x 3 V x =, 3 =. 3 iv Ends with rotation resistance by spring matrix k α and with dispacement restraint by spring matrix k d : bending moment Mx is the product of mutipying k α and rotationa ange, and shear force Qx is the product of mutipying k d and end dispacement, i.e., Mx =kα T Ux, t, Qx =kd T Ux, t. 33 v Ends with concentrated mass: bending moment is zero, and shear force equas inertia force, i.e., M =, Q = m Ux, t t. 34 Here, the intrinsic characteristics of panar curved structure due to bending vibration and torsiona vibration are anayzed by the use of second and fourth formuas in Eq. 5. Suppose that the boundary conditions be: simpy supported at both ends and torsion of ends restricted, i.e., V = V =, φ = φ =. 35 where is the arc ength of structure. According to the boundary conditions, generaized dispacements u and φ are u = φ = j= j= U j, tsin jπ, Φ j, tsin jπ, where u and φ are Fourier-Sine transformations of U and Φ, i.e., U = u,tsin jπ d, Φ = φ,tsin jπ d, where j is the number of vibration mode, with j =,,,. Suppose that the right sides of the second and the fourth formuas of Eq. 5 equa zero. According to the Fourier transformation and orthogona property of vibration mode, we have 4 + GI d jπ EI jπ + GI d ρ A+I jπ a ρa Φ+ EI R Ü + EI jπ jπ + GI d R jπ R U + U + EI R + GI d R jπ R Φ =. jπ Φ =, 38

9 Dynamic equations for curved submerged foating tunne 37 Let 4 jπ ω = EI + GI d jπ / jπ R ρ A + I, ω φ = EI jπ / a R + GI d ρa, ω EI jπ = + GI d jπ / jπ ρ A + I, R R ωφ EI jπ = + GI d jπ / a ρa. R R 39 Then Eq. 38 can be simpified as { Ü + ωu + ωφ =, Φ + ωφ U + ωφ Φ =, 4 where ω and ω φ are the circuar frequencies of bending vibration and torsion vibration for curved girder, respectivey, and ω and ω φ are the couping items due to curvature. When R tends to infinity, we have ω = and ω φ =. If rotation energy is not considered, ω and ω φ are frequencies of bending vibration and torsiona vibration for straight beams, respectivey. In the design of potentia SFT with thousands of meters in ength, the ength of one span modue is of the order of magnitude of meters. For the mode of ow frequency vibration with jπ/, the vaue of rotationa energy is sma enough compared with dynamic energy and has itte effect on frequencies. So ow frequencies may be written as 4 jπ ω = EI + GI d jπ R /m, 5 Concusions ω φ = EI R + GI d ω = EI R ω φ = EI R jπ /a m, jπ + GI d jπ /m, R jπ + GI d jπ /a m. R The paper gives dynamic equations of spatia curved SFT with Hamiton principe. In this system, four generaized dispacements are couped with each other. When spatia curviinear structure degenerates into panar curviinear structure, defection is couped with torsion ange and axia dispacement is couped with radia dispacement. In other words, generaized dispacements in a pair of perpendicuar panes are couped. Dynamic equations of arbitrary curviinear structures may be written by the present method, i.e., with a reference coordinate system. 3 When ow frequencies of curved SFT are anayzed, rotationa energy coud be negected. 4

10 38 DONG Man-sheng, GE Fei, ZHANG Shuang-yin, et a. References [] Dong M S, Ge F, Hong Y S. Anaysis of therma forces for curved submerged foating tunnes[j]. Engineering Mechanics, 6, 3Supp.: 4 in Chinese. [] Ge F, Dong M S, Hui L, Hong Y S. Vortex-induced vibration of submerged foating tunne tethers underwaveand currenteffects[j]. Engineering Mechanics, 6, 3Supp.:7 in Chinese. [3] Tveit P. Ideas on downward arched and other underwater concrete tunnes[j]. Tunneing and Underground Space Technoogy,, 5: [4] Brancaeoni F, Casteani A, D Asdia P. The response of submerged tunnes to their environment[j]. Eng Struct, 989, : [5] Remseth S, Leira B J, Okstad K M, Mathisen K M, et a. Dynamic response and fuid/structure interaction of submerged foating tunnes[j].computers and Structures, 999, 74: [6] Fogazzi P, Perotti F. Dynamic response of seabed anchored foating tunnes under seismic excitation[j]. Earthquake Engineering and Structura Dynamics,, 93: [7] Chai H Y, Fehrenbach Jon P. Natura frequency of curved girder[j]. Journa of the Engineering Mechanics Division, ASCE, 98, 74: [8] Scheing D R, Gados N H, Sahin M A. Evauation of impact factors for horizontay curved stee box bridges[j]. Journa of Structura Engineering, ASCE, 99, 8:33 3. [9] Gados N H, Scheing D R, Sahin M A. Methodoogy for impact factor for horizontay curved stee box bridge[j]. Journa of Structura Engineering, ASCE, 993, 96: [] Snyder J M, Wison J F. Free vibrations of continuous horizontay curved beams[j]. Journa of Sound and Vibration, 99, 57: [] Fam A R M, Turkstra C. Mode study of horizontay curved box girder[j ].Journa of the Structura Division, ASCE, 976, 5:97 8. [] Muppidi N R. Latera vibrations of pane curved bars[j]. Journa of the Structura Division, ASCE, 968, 94:97. [3] Ye M, Xiao L X. Anaytica mechanics[m]. Tianjin: Tianjin University Press,.

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