ELASTIC-PLASTIC ANALYSIS FOR CRACK AT THE HORIZONTAL BRACE OF SEMI-SUBMERSIBLE PLATFORM LOADED BY TENSION

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1 HE PUBLISHING HOUSE PROEEDINGS OF HE ROMANIAN AADEMY, Serie A, OF HE ROMANIAN AADEMY Volume 18, Number 3/017, pp ELASI-PLASI ANALYSIS FOR RAK A HE HORIZONAL BRAE OF SEMI-SUBMERSIBLE PLAFORM LOADED BY ENSION Fei WANG 1, Zheng LIANG 1, Xiong DENG 1 Southwet Petroleum Univerity, School of Mechatronic Engineering, hengdu, , RP hina Southwet Petroleum Univerity, School of Petroleum and Natural Ga Engineering, hengdu, , RP hina hanhuichun1@16.com Abtract. he article preent an elatic-platic theoretical analye method to calculate the mechanical characteritic of emi-ubmerible platform horizontal brace with a circumferential through-crack lie at the boundary between the horizontal brace and column loaded by tenion. he olution i clear and cloed form olution i found, which i epecially uitable olving problem with complicated boundary condition and could give atifactory preciion in practical engineering application. he variation tendency of the horizontal brace cracked ection which i divided into crack zone, tenile platic zone, elatic zone and compreive platic zone a well a the influence of different tenion and angle of the crack to them are analyed in thi article with a practical engineering application example which could give good uggetion to emi-ubmerible platform deigner and manager. Key word: elatic-platic analyi, horizontal brace, circumferential through-crack, emi-ubmerible platform. 1. INRODUION Semi-ubmerible platform, one of the mot widely ued reuable exploitation platform due to their mobility and ability to operate in deep-water, have gained popularity in recent decade with on-going development of deep-water oil and ga exploitation. he horizontal brace i one of the main tructure in emi-ubmerible platform which erve a the upporting tructure epecially when the platform encounter horizontal tenion load in ocean engineering. Although the afety deign tandard for thi kind of tructure of the emi-ubmerible are quit trict, hitorical record [1, ] how that diatrou event epecially the crack-induced total loe of the emiubmerible uch a Sedco in 1967, Alexander L. Kielland in 1980, Ocean Ranger in 198, repectively, cannot be completely avoid. o enure the afety and reliability of the emi-ubmerible platform which would be ubject to very harh marine environment during their ervice life, analyi of the mechanical characteritic of a cracked horizontal brace i neceary. he problem of brace tructure having a crack ha been invetigated by a number of author [3, 4, 5]. In mot of the cited article the crack ha been aumed to be urface crack or lie far away from boundary. However, a circumferential through-crack in a horizontal brace uually initial near a joint or imilar dicontinuity uch a in the diatrou event of emi-ubmerible platform Alexander L. Kielland caued by the failure of a horizontal brace (D-6) having a circumferential through-crack near the joint of the brace and then the remaining brace failed by overloading rapidly. he preence of uch crack at critical location of horizontal brace can compromie the afety of the whole emi-ubmerible platform. A for the method of computational analyi, the mot popularly applied method to analye the characteritic of cracked brace tructure i the finite element method whoe effectivene ha been accepted by the engineering community. Neverthele, the finite element method will be carried out for every pecific and tructure ytem with ome local defective element or nonlinear calculation will be inefficient and pend ignificant reource which hould not be neglected. In thi ene, the theoretical analyi i till neceary. he preent article i aimed to invetigate an elatic-platic theoretical analyi method calculating the mechanical characteritic of emi-ubmerible platform horizontal brace having a circumferential throughcrack which lie at the boundary between the horizontal brace and column by tenion.

2 40 Fei WANG, Zheng LIANG, Xiong DENG. GOVERNING EQUAIONS A horizontal brace of emi-ubmerible platform with a circumferential through-crack which lie at the boundary between the horizontal brace and pontoon or column of the emi-ubmerible platform, which can be aumed to be coniderably tiffer than the horizontal brace, ubjected to tenion load i illutrated in Fig. 1, where a coordinate ytem and direction of the load are hown. Fig. 1 he horizontal brace of the emi-ubmerible platform loaded by tenion. he horizontal brace of the emi-ubmerible platform are of medium length which mean that the radiu of the cro ection i ignificantly maller than the length of the horizontal brace. From the viewpoint of the hell theory, the emi-ubmerible platform horizontal brace belong to the mid-long cylindrical hell category. Regarding the length of circumferential through-crack on the brace tructure, hitorical record [4 8] how that the crack are of very long. ake the diatrou even of emi-ubmerible platform Alexander L. Kielland a an example, the circumferential through-crack near the joint of the horizontal brace have propagated to almot 67% of circumferential length of the brace before fracture [9]. hen, the characteritic equation of the horizontal brace could be expreed under a emi-membrane tate [3], in which characteritic function varie lowly in the z direction but not too rapidly in the direction, for a complex-valued variable Ω=w+iζ a follow, Ω Ω iε 0. + Ω = omplex characteritic function Φ and φ both atify Eq. (1) here and are related to each other by (1) Φ = ε ϕ. () Here, ε i a mall parameter given by ε = (h/r) [1(1 ν )] -1/ where h and R i the thickne and radiu of the horizontal brace of the emi-ubmerible platform, ν i Poion ratio. hen, the expreion of dimenionle complex diplacement u, v, and w, tre function χ z, χ, and ζ, membrane tree N z, N, and N z and bending tree M z, M, and M z can be given in term of Φ and φ by 3 Φ Φ Φ ε u= ; ε v = ; w= + iφ 3 3 Φ Φ Φ εχz= i ; εχ i ; ζ i φ = = + 3 φ φ φ Nz = ; N ; N = z = φ φ φ φ φ Mz = i + μ ; M = i + μ ; M z = i( 1 μ). Moreover, the dimenionle load parameter correponding to tenion load i defined a σ = /(Rhσ F ) and the dimenionle characteritic function mentioned above are given a follow, (3)

3 3 Elatic-platic analyi for crack at the horizontal brace of emi-ubmerible platform loaded by tenion 41 σ R E F ( u, v, w) = ( u, v, ε w ); ( χ χ ζ) ε ( χ χ ε - z,, =σfr h z,, ζ) ( z z) = F ( z z) ( z z) =σf ε ( z z) N, N, N σ h N, N, N ; M, M, M Rh M, M, M. (4) Here, E i Young modulu, σ F i yield tre of the material and denote dimenional quantity. he complete olution Φ c to the problem preented in thi article can be expreed a Φ = Φ + Φ. (5) c Here, Φ b mean the elementary olution and Φ can be thought of a the olution incurred by the exitence of the crack. he elementary olution which i compoed of imple axial tenion olution, rigidbody motion olution (for which tree diappear) and null olution (for which diplacement diappear) can be expreed a Φb { } b = 0.5i 1+ iε + μ + 1+ iεμ 0.5ε i σ i ε ico ε co. z + a bz + c + dz Here, a, b, c, d are unknown contant. A there i a tatic-geometric analogy between diplacement and tre function, if the boundary condition on the tre meaure can be equivalently expreed in term of condition on tre function, the treatment of the boundary condition will be implified. Expreion for the tre function in term of integral of the effective Kirchhoff edge reultant wa derived by Sander [10]. Expreion for the boundary value of the tre function in term of precribed edge load z,, V and M n acting on the edge z = 0 are a follow z = + V + V ε χ in inη ε coη dη co coη ε inη dη dη (6) = z + Mn + z + Mn z εχ in ε inηdη co ε coηdη dη ( z Mn) ( z Mn) ζ = in + ε coηdη + co + ε inηdη (7) ζ = in ( inη + ε Vcoη) dη co ( εinη) dη. Here, η i infiniteimal angle of the cracked ection. he cracked ection i at the boundary between the horizontal brace and pontoon or column of the emi-ubmerible platform, illutrated a in Fig.. From the condition of ymmetry, the cro ection z = 0 i the border. Fig. ircumferential through-cracked ection of the horizontal brace.

4 4 Fei WANG, Zheng LIANG, Xiong DENG4 4 On the cracked ection, hown a in Fig., the opening angle of the circumferential through-crack i α, while β α and γ are the tenile and compreive platic zone repectively. he boundary condition can be obtained by mean of method given by Sander [11]. hoe and the diplacement condition are given a follow, z = = V = Mn = 0 ( 0 < α) v = 0 ; z = 1 ; V = Mn = 0 ( α < β) w u = v = w = = 0 ( β < γ) v = 0 ; z = 1 ; V = Mn = 0 ( γ ). he boundary condition mentioned above can be obtained in term of the characteritic function Φ and φ following Eq. (3) and Eq. (7) a Φ Φ R i 0; i 0 ( 0 ) = R = < α Φ R 0; i co α 0.5 α 1 α β = R = + < Φ R 0; 0 (β γ) = R = < Φ R = 0; i = R σ co G1co G ( γ ). he ymbol R { } denote the real part of the expreion in bracket and the ubcript c mean that the expreion are in term of the complete characteritic function. With Eq. (5), Eq. (6) and Eq. (9), the particular integral Φ at the boundary incurred by the exitence of the crack turn into Φ Φ R i = εbi + εdico; R i co = ar + cr + ( ) σ ( 0 < α) Φ R a co; i co = I + ci R a = R + cr + ( ) σ co( α) 0.5( α) + 1 (α < β) (10) Φ R = a co ; I + ci R = εbr εdr co; (β < γ) Φ R co; i = ai + ci R a = R + G + ( cr G1) co+ ( ) ( 0.5 1) σ + co (γ ). Here, the ubcript R and I refer to the real and imaginary part of thee contant. Furthermore, any olution to the Eq. (1) atify the condition [10] (8) (9)

5 5 Elatic-platic analyi for crack at the horizontal brace of emi-ubmerible platform loaded by tenion 43 Φ Φ d cod = 0 = 0 0 z = 0 z = 0 (11) 3 Φ Φ 1 i ε. = + Φ (1) Now put and ue Eq. (11) and Eq. (1) to get Φ z = 0 = 0 0 F. (13) ( F' ' + F) d = 0; F ''co d = 0 (14) Φ i ε ''. 3 1 F F = + z z = 0 Additionally, by uing Eq. (15) and Eq. (15), Eq. (10) can be expreed in the equivalent form a follow, F R ( a b ) ( c d ) S R + I R + I co + co σ 0 < α = ai + cico α ( a c ) R + Rco σ 0 < α ( ar + crco) ( 0.5 1) σ + co( α) + 0.5( α) 1 α β FI = β β. (17) br ai + ( br ci) co + Pco + Qco β < γ ar + G + ( cr G1) co ( ) ( 0.5 1) + σ + co 0.5( ) + 1 γ And S, P, Q, G 1 and G are real contant of integration. Adding the undetermined β and γ there exit totally 15 unknown contant in the above equation. he continuity of the diplacement and tre function at the cracked ection implie that F R, F R, F R, at α and F I, F I, F I, F I at β and γ hould be continuou (F I, F I, F I at α are continuou from imply checking), from which 11 equation can be got. Four more equation can be found for the real part and the imaginary part equal to zero in Eq. (14). here are thu 15 condition to determine the 15 contant and all the contant can be finally determined by mean of algebraic method. And reult for the 15 contant follow from (15) (16) α α α 1 3 ai = S co in ασ 3 α bi = ar ai + 0.5Sco 0.5α σ α ci coa= Sco ( α inα coα ) ( α co α α inα coα) σ α di coα = cr coα ci coα + 0.5Sco + σ (18)

6 44 Fei WANG, Zheng LIANG, Xiong DENG6 6 γ β γ β ar = ( β) A+ in + ( γ) 1 co P+ Q ( γ β ) γ( γ) σ + ( β α) + ( γ) 6 6 br = ( A ar + ai) 4 cr = ( β γ+ inβ coβ inγ coγ) + ( γ inγ coγ) G1 + γ β γ β + 4 inγ inβ co coβ in P + ( inα inβ inγ) + γ β γ β 4 inγ + inβ in coβ co Q ( inγ inβ) σ + ( β α) coα + + coβ in β α inγ coγ+ γ dr = ( cr + ci) 4 A= γ β γ β P + Q σ + inβ= βσ + in β α + β α 1 ( γ) σ γ inα ( ) co in 0.5β 0.5 β α Gcoγ = coγ + P σ coγ 1 G = A P γ+ 0.5γ σ γ γ β γ β Pin = Qco βσ + β α Q = + αcoα σ S =. α α co αin inα co he following imultaneou trancendental equation are ued to determine the extenion of the platic zone β and γ with given α and σ. Here certain numerical technique are required where B1σ N1 = 0 Bσ N = 0 γ β γ β B1 = ( inβ+ βcoβ) in + γ+ βcoβco inβ γ β γ β B = ( γ) coγ inγ in ( γ) co + β inγ γ β γ β N1 = inβ inα+ ( β α) coβ in γ ( β α) co inβ γ β γ β N = inγ ( γ) coγ in + ( γ) co β+ α inγ (19) (0) (1)

7 7 Elatic-platic analyi for crack at the horizontal brace of emi-ubmerible platform loaded by tenion 45 From the above reult diplacement on the crack are obtained and the dimenionle crack tip opening diplacement (OD) ubjected to the tenion load i 1 δ= ε A+ coα ( 0.5α + 1) σ. 4 () Further, for the preent load circumtance the full platic condition on the cracked ection i σ = arc co ( 0.5inα ) 0.5α and β= γ= 0.5( + α+ σ ). (3) 3. SOLUIONS he olution proce of the analyi method, propoed to calculate the mechanical characteritic of emi-ubmerible platform horizontal brace having a circumferential through-crack which lie at the boundary between the horizontal brace and column or pontoon loaded by tenion in thi article, i imple and the cloed form olution i found. At the beginning of the olution, baic parameter uch a parameter of the horizontal brace, angle of the crack and the tenion load hould be input for the olution for platic and OD. Numerical method hould be ued to calculate of the platic zone of the horizontal brace becaue it hard to find explicit olution from Eq. (1). o tart the calculation, auming an initial β 0 and initial γ 0 i neceary. And then, define two more parameter F and λ according to Eq. (1) following from ( β, γ ) σ ( β, γ ) ( β, γ ) σ ( β, γ ) F= B N + B N F ( β, γ ) λ =. 0 0 F ( β , γ 0) F ( β 0, γ 0) F ( β 0, γ ) F ( β 0, γ0) 0.01β γ 0 o guarantee to preciion of the olution, verification i needed to ee if F < ς i atified, where ς i a mall pecified quantity. If F < ς doe not appear, the aumed initial β 0 and initial γ 0 hould be modified a ( β , γ ) ( β, γ ) ( β, γ ) ( β, γ ) F F F F β= β λ γ= γ λ β γ 0 After F < ς i atified, the olution for platic zone of the cracked horizontal brace i done. he output of β and γ can then input into Eq. (18), Eq. (19) and Eq. (0) to determine the 15 unknown parameter a I, b I, c I, d I, a R, b R, c R, d R, A,, G 1, G, P, Q, S and the crack tip opening diplacement(od) can be obtained from Eq. () too. ill now, the olution of the mechanical characteritic of emi-ubmerible platform horizontal brace having a circumferential through-crack which lie at the boundary are done. 4. EXAMPLES o illutrate the previou model with ome example, a typical emi-ubmerible platform with circumferential through-crack at the boundary between the horizontal brace and column i elected with parameter of the horizontal brace including the radiu R = m, thickne h = m, Poion ratio ν = 0.3, Young modulu E = 10 GPa and yield tre of the material σ F = 400 MPa. According to the tranvere force tranfer function and long term repone of tranvere force of longitudinal ection in center plane of the choen emi-ubmerible platform, hown a in Fig. 3, the larget tenion load which the horizontal brace could be loaded i N. Different tenion load and angle of the circumferential through-crack which lie at the boundary between horizontal brace and column have been choen to analye the characteritic function of the cracked ection.

8 46 Fei WANG, Zheng LIANG, Xiong DENG8 8 Fig. 3 a) Semi-ubmerible platform; b) tranvere force tranfer function of longitudinal ection in centre plane; c) long term repone of tranvere force of longitudinal ection in centre plane. he tenile platic zone (β - α) on the circumferential through-cracked ection of emi-ubmerible platform horizontal brace vary with tenion load and angle of the crack (Fig. 4a). he tenile platic zone of the cracked ection increae harply after a moothly increae with the tenion load. Additionally, the cracked ection of the horizontal brace with greater circumferential through-crack ha a larger tenile platic zone while loaded a ame tenion and the compreive platic zone appear (parameter γ < ) earlier. Same a the variation tendency of the tenile platic zone, the crack tip opening diplacement (OD) on the cracked ection increae moothly while the horizontal brace of the emi-ubmerible platform loaded by mall tenion load and the crack tip opening diplacement (OD) increae dramatically when loaded by larger tenion (Fig. 4b. Alo, the cracked ection ha a larger OD while the angle of the crack lie at the boundary between the horizontal brace and column of the emi-ubmerible platform i greater. Fig. 4 a) he tenile platic zone on the cracked ection under different tenion load with different half angle of the crack; b) the crack tip opening diplacement (OD) on the cracked ection under different tenion load with different half angle of the crack. he elatic zone of the horizontal brace cracked ection, whoe variation tendency i contrary to the variation tendency of platic zone (hown a in Fig. 5a, decreae with the tenion load until the cracked ection i in the full platic condition while contain cracked zone, tenile platic zone and compreive platic zone. Shown a in Fig. 5b, the tenile platic zone decreae with the angle of the cracked ection during the compreive platic zone increae with the angle of the cracked ection and the tenile platic zone i alway larger than the compreive platic zone while the cracked ection i in the full platic condition. he load condition vary obviouly with the angle of the cracked ection, hown a in Fig. 5c, the tenion load decreae with the angle of the circumferential through crack i almot linear. Fig. 5 a) endencie of elatic zone and platic zone of the cracked ection; b) the platic zone of the cracked ection in different angle of the crack; c) the variation tendency of tenion load for different angle of the crack while cracked ection i in the full platic condition.

9 9 Elatic-platic analyi for crack at the horizontal brace of emi-ubmerible platform loaded by tenion ONLUSIONS A imple elatic-platic theoretical analyi method calculating the mechanical characteritic of emiubmerible platform horizontal brace having a circumferential through-crack which lie at the boundary between the horizontal brace and the column loaded by tenion i propoed in thi article. he olution proce i clear and cloed form olution i found, which i epecially uitable to olve problem with complicated boundary condition. In practical Engineering application, it could give atifactory preciion. he preented model i alo applied to analye the cracked ection, which i divided into crack zone, tenile platic zone, elatic zone and compreive platic zone in thi article, a well a the influence of different tenion load and angle of the crack to them with a practical engineering application example in thi paper and ome variation tendency ha been got: (1) he tenile platic zone and the crack tip opening diplacement on the cracked ection of the horizontal brace increae dramatically after a moothly increae with the tenion load and the horizontal brace with greater circumferential through-crack ha a larger tenile platic zone and crack tip opening diplacement. Additionally, the cracked ection of the horizontal brace with greater circumferential through-crack ha a larger tenile platic zone while loaded a ame tenion and the compreive platic zone appear earlier. () he elatic zone of the horizontal brace cracked ection, whoe variation tendency i contrary to the variation tendency of platic zone, decreae with the tenion load until the cracked ection i in the full platic condition, in which the tenile platic zone decreae with the angle of the cracked ection during the compreive platic zone increae with the angle of the cracked ection and the tenile platic zone i alway larger than the compreive platic zone. he load condition vary obviouly with the angle of the cracked ection, the tenion load decreae with the angle of the circumferential through crack i almot linear. AKNOWLEDGMENS he author would like to acknowledge the upport of Grant No. 01AA09A03 from the National High echnology Reearch and Development Program of hina (863 Program) and Grant No. 011SS04 from the Key Laboratory of Oil and Ga Equipment from Minitry of Education, hina. REFERENES 1. olin H., Epen F., Martyn., Worldwide Offhore Accident Databank, Det Norke Verita, Olo, Giovani D., Mariana S., Rik Baed in Inpection Applied to a Semi-Submerible Platform, Offhore echnology onference, Rio de Janeiro, Brazil, Sander JL., Dugdale model for circumferential through-crack in pipe loaded by bending, International Journal of Fracture, 34, 1, pp , Andrea., Roberto B., Sabrina V., Notched hell with urface crack under complex loading, International Journal of Mechanical Science, 48, 6, pp , Brighenti R., Surface crack in hell under different hoop tre ditribution, International Journal of Preure Veel and Piping, 77, 9, pp , Inge L., Odd O., Rik aement of lo of tructural integrity of a floating production platform due to gro error, Marine Structure, 17, 7, pp Maier G., ae Hitorie in Offhore Engineering, Springer Vienna Publiher, Reaon J., Managing the Rik of Organizational Accident, Ahgate Publiher, Moan., Berge S., Holthe K., Analyi of the fatigue failure of the Alexander L. Kielland, ASME Winter Annual Meeting, Wahington, D, Sander JL., loed form olution to the emi-infinite cylindrical hell problem. Rotterdam Dam, Delft Univerity Pre, Sander JL., On tre boundary condition in hell theory, Journal of Applied Mechanic, 47, 1, pp. 0 04, Received January 13, 015