Ice in te Environment: Proceedings of te 1t IAHR International Symposium on Ice Dunedin, New Zealand, nd t December International Association of Hydraulic Engineering and Researc AN ANALYSIS OF AMPLITUDE AND PERIOD OF ALTERNATING ICE LOADS ON CONICAL STRUCTURES Li Feng 1 and Yue Qianjin 1 ABSTRACT Te paper addresses problems in determining alternating ice forces on conical offsore structures. Full-scale data is used to analyze te amplitude and period of te ice force. Tese parameters are needed in te dynamic analysis for te ice-resistant structure. Teoretical results for te ice force are compared wit results of full-scale tests. A teoretical estimation of te dimensionless vertical component of te ice force is in te range of.7~.7, wile te upper limit of tis parameter was.5 in te tests data. Te ratio between te ice failure lengt and te ice tickness is used as a key parameter in forecasting te period of te alternating ice force. Differences between various metods are discussed, and te influences of te cone angle, friction coefficient and contact widt are examined. INVESTIGATIONS ON THE MECHANISMS AND THE MODELS Te load period and te load amplitude are important parameters to analyze because tey ave major influences on te time-varying ice force. However, unlike te case of static ice force, different components of te ice force must be considered separately because tey ave different periods. It is sown by narrow-cone tests tat te ice-breaking component controls te time function of te ice force. All te oter components may be regarded as low-frequency background (Ota et al., 199). Te breaking lengt l b of te ice seet due to te bending failure of te ice seet acting on a conical structure is a key factor for te period of te ice force. Tis parameter as been investigated earlier by tests (Abdelnour and Sayed, 19; Jin, 193; Tatinclaux, 19; Lau et al., 1999) and by analysis (Frederking, 19; George, 19). A dimensionless parameter, l b / l c, defined as te ratio between te breaking lengt and te caracteristic lengt of te ice seet as been adopted by most researcers. Tis ratio as obtained by teoretical analysis was about. (George, 19) or. (Frederking, 19). However, different values were obtained in te tests. Abdelnour and Sayed (19) obtained a value of about.5, wile Lau et al. (1999) obtained values in te range of.1~.. An alternative dimensionless parameter, l b /, was used by Jin (193) and Tatinclaux (19) by using te ice tickness as te reference. We call tis parameter te 1 Department of Mecanics, Dalian University of Tecnology, Dalian, Cina, 113
lengt-tickness ratio for sort, denoting it as k. Tis parameter is more suitable for in situ researc because of its simplicity. Te ice force and te corresponding ice breaking lengt are independent of te ice velocity in te middle- and low-speed region (Ota, 199; Yue et al., 199). Terefore, te parameter k can be determined in a static mode of interaction. In early studies, a formula for determining te ice force on conical structures was presented by adopting te lengt-tickness ratio, k, as one of te geometry parameters (Jin, 19). Tis metod, owever, was approximate because te effects of friction and buoyancy forces were ignored, and te value of te parameter k ad to be determined experimentally. In our researc, te amplitude and period of te alternating ice force as been determined analytically. Te analytical results are compared wit some model tests, oter analytical solutions and in situ observations. Comparisons and corresponding analysis sow tat tis model is suitable for te narrow-conical problem. Te effects of ride-up and pile-up are not important and may be ignored in te case of interaction between a narrow cone and smoot level ice seet. Terefore, te ice force can be considered as an alternating load tat as a nearly constant period. Yue et al. (199) ave presented a simplified expression of suc an alternating ice force. To determine te alternating ice force, a mecanic model may be selected by considering te following tree scenarios of bending failure (Lau, 1999). (1) A semi-infinite beam wit te longest radial crack; () a cantilever plate wit a very sort radial crack or witout a radial crack, and (3) a wedge cantilever beam wit some radial cracks of te same lengt as te beam. Case (3) was often seen in te observations on a narrow structure in Boai Bay. However, a condition between cases () and (3) as also been seen. In order to simplify te analysis, te cantilever beam model (case 3) is adopted, and te effect of a limited contact widt is considered to account for te trend of transforming from case (3) to case (). Four presumptions are made in developing an analysis model: (a) te conical structure is stiff and te ice may be regarded as elastic-brittle material; (b) te deflections are small wen te ice beam fails in bending against te sloping structure; (c) te amplitude is related only to te breaking component of te ice force, and (d) te ice force is directly proportional to te overall lengt of te circumferential crack. Te sape of te circumferential crack is a semi-circular arc. POINT-CONTACT FAILURE PROBLEM Correlation equation of ice force and its period As sown by Yue (199), te period of te ice force, T, is directly proportional to te breaking lengt l b of te ice seet. In te case of a constant drifting speed, v, te period is given by T = (1) v Te correlation between te ice force and te period can be expressed as a correlation equation of ice force and breaking lengt of ice (or te lengt-tickness ratio). A simplest case to be analyzed is te point-contact failure problem. Tis is relevant to conditions were te contact area between te ice seet and te structure is small at te event of ice failure. Te ice force is ten a concentrated force tat produces a semi-circular-saped circumferential crack. l b
An expression of te ice force describing tis failure pattern was presented by Jin (193). We re-analyzed and revised is equations. A dimensionless ice force is expressed using tree dimensionless parameters: te ice force component ratio ξ = P H /P V ; te lengttickness ratio, k = l b / ; and te ratio l b / l c between te breaking lengt and caracteristic lengt of ice (Li and Yue, 1b): PV π = () σ f lb (1 -.1 ) - ξ lc lb In tese definitions, P H and P V are respectively te orizontal and vertical components of te ice force, is tickness of ice and σ f is te bending strengt of te ice. Analysis of te lengt-tickness ratio A furter simplification is needed to obtain te lengt-tickness ratio from equation (). According to te ice conditions in te Boai Sea, te elastic modulus of te ice is E =.5 GPa under usual ice conditions ( <.5 m), l c 13. Substituting k = l b / in equation () yields a two-parameter equation. If ξ is determined, equation () expresses a continuous and bounded function of k. An approximate value of te parameter k is obtained at te condition were te ice force as its minimum. Te results are given by Li and Yue (1b): 1 3 k = 7.1ξ (3) Te parameter k increases wit te ratio ξ between te orizontal and vertical force components. Tis occurs in conditions were te slope angle of te cone and te friction coefficient increase. Figure 1 sows a comparison of te k values obtained by equation (3) and by tests. Jin (193) carried out tests to study te interaction between a sea ice seet and an inclined pole. Te tickness of te ice seet was ~ cm; te diameter of te pole varied from cm to 3 cm. Te inclination angle of te pole was α = and. Te k values obtained in te tests were in te range of ~. Analysis of te ice force Te ice force is determined by substituting k obtained from (3) into (). A comparison wit te tests mentioned above is sown in Fig.. An analysis of earlier tests results obtained by Jin (193) sows tat te diameter of te pole ad no effect on te ice force and te parameter k. It can be seen from Fig. tat te ice forces obtained by analysis are iger tan te test results, wic is because te buoyancy effect was over-estimated in te analytical model. A conservative result is often necessary for an engineering analysis. THE PROBLEM WITH A LIMITED CONTACT WIDTH Ice force as a function of te lengt-tickness ratio According to observations, te lengt of te circumferential crack increases wit te contact widt, b, between te ice seet and te cone. Tis effect is considered in te geometrical relationsips, as sown in Fig. 3.
k =l b / 9. Upper limit in tests 9. 9. 9. Analysis results 9.... Lower limit in tests....1 Friction coefficient μ PV/σ f.71.7.9..7 Upper limit of te test values..5..3...5..7..9.1 Friction coefficient μ Figure 1: Comparison of k values. Figure : Comparison of ice force D b θ β Cone surface c l b Ice seet Figure 3: Limited-contact widt of ice seet on narrow cone An expression for te ice force as a function of te lengt-tickness ratio was derived by Li and Yue (1a) as n θ P π(1 + tan ) V = k () σ f ξ (1 -. k ) - k A new dimensionless parameter, te contact aspect ratio n = b/, is introduced in Eq. (). Here, b is te contact widt under te condition of ultimate failure between te ice seet and te cone (Fig. 3). As seen in Fig. 3, te geometrical parameter θ is not an independent parameter, but is defined by te geometrical relation θ b tan =. (5) D+ D -b
In tis equation, D is te waterline diameter of te cone. Wen b = D, tan ( θ / ) = 1. Te parameter n is dependent on te crusing strengt and tickness of te ice seet. Tis parameter is defined by experiments. Afanasiev (197) presented a similar equation of ice force wit an experimental coefficient relating to te lengt of circumferential cracking. ANALYSIS AND COMPARISON OF THE LENGTH-THICKNESS RATIO Analysis of te lengt-tickness ratio A searc for te limit value of te ice force by equations () and (5) yields te condition 3 b θ b θ k +.75 tan k 3.33ξ 15 tan = () Tis is a cubic equation of te lengt-tickness ratio k. It can be seen tat for a certain conical structure and a certain ice tickness, k is a function of parameters b and ξ. For a conical structure wit α = and µ =.1, we ave ξ =.. Figure sows for tis condition te parameter k as a function of te contact widt b te ratio D/. It can be seen tat te value of k decreases as te contact widt b decreases. We can also see tat te ice tickness affects te parameter k if te contact widt is larger tan alf of te diameter D. Wen b becomes very small (curve b = D/ in Fig. ), te effect of te ice tickness vanises. Te friction coefficient between te ice and te cone surface is a constant for a steel structure. Figure 5 sows te lengt-tickness ratio k as a function of sloping angle assuming tat te friction coefficient is µ =.1. 1 11 b=d 9 k=lb/ 9 7 5 b=d/ b=d/ k=l b / 7 5 3 o 5 o o 75 o 5 15 5 3 35 5 D/ 1 1 b/ Figure : Lengt-tickness ratio k as a function of contact widt b Figure 5: Lengt-tickness ratio as a function of sloping angle Comparison wit test results Two comparisons of te current analysis wit tests and analogue analysis metods are sown in Fig. and Fig. 7, respectively. In Fig., te data of te wide cone is based on te tests reported by Abdelnour and Sayed (19); te data of te narrow cone model is taken from te report of Jin (193). Te data of present work is taken from Fig., wic pertains to conditions were te contact widt is b = D/.
1 1 1 lower limit up limit 1 lower limit up limit Lengt-tickness ratio k Wide cone model Narrow cone model In-site observation Te current analysis Figure : Comparison of lengt-tickness ratio wit tests Lengt-tickness ratio k George Frederking Te current analysis Figure 7: Comparison of lengt-tickness ratio wit oter analysis models Analysis and comparison of ice force Te ice force is determined by substituting te parameters k, ξ and n into equation (). Te selection of te values of tese parameters depends on te particular problem under consideration. Te platform JZ- is used ere as an example for computation and comparison. Te structure parameters are: α =, D max = m, µ =.1. Te ice parameters are: σ f = 7 kpa, =.1~. m. All te oter metods to be used in te comparison employ te maximum possible ice force. Terefore, a maximum contact widt is assumed ere, by setting b = D max. As a representative value, k = 7.3 is adopted, wic is an average value of te measured data during te winter of 1999~. Figures and 9 sow comparisons between te test results and predictions made by different teoretical models. It can be seen from Fig. tat te results obtained using te present analytical model deviate from te test data, wic is notable only wen te ice tickness is very small. Tis difference can be explained by noting tat te contact area cannot reac te wole of te cone surface wen te ice is tin. Te prediction made by te analysis will be closer to te tests results wen a lower value of b is adopted in te calculation. For te situation of very tin ice, te relatively great difference may be partly caused also because te deflections were assumed to be small. An analysis of te failure of te ice seet under simultaneous longitudinal and transverse loading as given a lower ice force force (Li and Yue, ). Tree analytical metods (Nevel, 197; Frederking, 19; Ralston, 19) were selected for te comparisons in Fig. 9. It can be seen tat te differences are small between te present model and te models proposed by Nevel and Frederking et al. Te metod developed by Ralston gives iger values of te ice force. His metod as been regarded as conservative by oter researcers. CONCLUSIONS Based on a bending teory, an analytical model was developed to determine te alternating ice forces on conical structures. Formulas tat can be used to determine te amplitude and period of te ice force are presented. A main parameter for te analysis, te lengt-tickness ratio of te fractured ice, is obtained analytically. Te rationality of te analyzed results is verified by a comparison wit various oter metods including in situ observations. Te contact aspect ratio, b/, is also an important parameter to identify te
failure mode of te ice seet in cases were te contact widt b is smaller tan te diameter D of te cone. Tis paper sows tat b = D/ is a good approximation of te contact widt for narrow cones wic may be used in engineering analysis.. 3... Nevel Analysis. PV/(σ ft ). 1. H-O Afanasev Kato PV/(σ ft) 1. 1. 1. Te current metod Ralston Frederkin 1......3. /m Figure : Ice force comparison of analysis and tests 1.1..3. /m Figure 9: Ice force comparison wit oter analysis metods REFERENCES Abdelnour, A. and Sayed, W. Ice rider up on a man-made island. In Proceedings of te Offsore Tecnology Conference, OTC (313), Vol. 3 (19) 11 15. Afanasev, V.P., Dolgopoloy, Y.V. and Syeistein, Z.I. Ice pressure on separate supporting structures in te sea. In Proceedings of IAHR Ice Symposium (197). Frederking, R. Dynamic ice force on an inclined structure. In Pysics and Mecanics of Ice, Tryde P. ed., IUTAM Symposium, Copenagen (19) 11. Frederking, R. and Scwarz, J. Model test of ice forces on fixed and oscillating cones. Cold Regions Science and Tecnology : 1 7 (19). George, D.A. River and Lake Ice Engineering. Water Resources Publications OST offices Box : Water Resources Publications (19). Jin, G.L. Current situation of experimental researc of loads of sea ice on offsore structures (in Cinese). Ocean Engineering 1(3): 9 (193). Lau, M. et al. An analysis of ice breaking pattern and ice piece size around sloping structures. In 1 t OMAE Ice Symposium (1999) 199 7. Li, F. and Yue, Q.J. Failure of ice seet under simultaneously longitudinal and transverse load on slope structure. Journal of Hydraulic Engineering 9: 7 (). Li, F. and Yue, Q.J. Analysis of ice force for narrow conical structures (in Cinese), Journal of Dalian University of Tecnology 1(): 3 7 (1a). Li, F. and Yue, Q.J. An analysis for peak value and period of te ice force function on narrow conical structures (in Cinese). Mecanics in Engineering 3(5): (1b). Tatinclaux, J.C. Ice floe distribution in te wake of a simple wedge. In Proceedings of 5 t OMAE, Vol., Tokyo (19) 9. Yue, Q.J., Bi, X.J. and Yu, X.B. Full-scale tests and analysis of dynamic interaction between ice seet and conical structures. In Proceedings of te 1 t IAHR International Symposium on Ice, Potsdam, New York, USA (199) 939 95.