99/105 Comparison of OrcaFlex with standard theoretical results
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1 99/105 Comprison of OrcFlex ith stndrd theoreticl results 1. Introduction A number of stndrd theoreticl results from literture cn be modelled in OrcFlex. Such cses re, by virtue of being theoreticlly solvble, quite simple. ey do, hoever, provide very useful check for the bsic mthemticl model used by OrcFlex.. Anlytic ctenry equtions.1. Introduction It is ell knon tht the sttic equilibrium configurtion of n inextensible, flexible line ithout bend stiffness is ctenry shpe. z A x B Figure 1: Ctenry shpe Such line cn be modelled in OrcFlex nd the configurtion nd tensions predicted by OrcFlex cn be compred ith theory. Extremely close greement is chieved, ith differences reducing s segment length reduces... Ctenry theory e ctenry equtions re s follos: eoreticl results.doc, 4-Sep-07 Pge 1 of 6
2 x = sinh 1 Tv + s sinh 1 Tv z = Tv + s 1+ Tv 1+ here s is the rclength mesured from end A, is the eight per unit length nd Tv re the horizontl nd verticl components of tension t end A. nd ese equtions ssume tht the line is inelstic nd so does not stretch xilly. Hoever, it is strightforrd to modify the equtions ccount for this. If e denote by K the xil stiffness of the line, then the modified ctenry equtions re: x = Tv s Tv + 1 s sinh 1 + sinh (1) K Tv s Tv Tvs s z + = () K K.3. Comprison of OrcFlex ith theory We used these equtions to perform comprison ith OrcFlex using the folloing line dt: Line length (l) 10 m Weight per unit length () kn/m Axil stiffness (K) 500 kn Horizontl spn 55 m Verticl spn 0 m Tble 1: Ctenry dt We rrnged tht both ends of the line ere t the sme verticl position, tht is the verticl spn is 0. is llos us to clculte directly the verticl component of tension t end A, Tv. It is cler tht the sum of verticl tension components t the ends, Tv + Tvb, equls the totl eight of the line, l. Becuse of symmetry e cn see lso tht Tv nd Tv b must be equl, hence Tv = l /. We no see tht there re only to unknons in eqution (1) bove, nmely x nd. If e evlute the eqution t end B, tht is for s = l = 10, then e cn see tht x is simply the horizontl spn, 55m eoreticl results.doc, 4-Sep-07 Pge of 6
3 So, e no hve n eqution ith single unknon,. e eqution cnnot be rerrnged to give direct expression for nd so e solved it using the gol seek functionlity in Microsoft Excel. is results in the theoreticl solution = kN. We modelled the sme ctenry line in OrcFlex nd produced the folloing results. We used progressively finer discretistion of the line nd observed, s expected, tht the difference beteen the OrcFlex results nd the nlytic ctenry equtions reduced. Number of segments OrcFlex (kn) % difference from theory Tble : Comprison of OrcFlex ith theory As finl check e evluted eqution () t the mid-point of the line, tht is for s = l / = 60 to find the z coordinte of the ctenry t its loest point. is produced vlue of z = m. e OrcFlex model ith 10 segments gives corresponding vlue of m hich is difference from the nlytic solution of 0.007%. 3. Nturl frequencies of bem 3.1. Introduction Timoshenko & Gere, eory of Elstic Stbility, nd edition, McGr-Hill, 1961, Section., pp , considers the stbility nd trnsverse vibrtion of pin-ended bem. Weight forces re neglected. Such bem is esily modelled in OrcFlex resulting in close mtch to theoreticl vlues. 3.. Vibrtion theory e nturl frequencies for the first mode of vibrtion re given by: gπ π EI ω = P ql l here q / g is the mss per unit length, l is bem length, EI is bending stiffness nd P is xil compressive lod eoreticl results.doc, 4-Sep-07 Pge 3 of 6
4 3.3. Comprison of OrcFlex ith theory For comprison ith OrcFlex, e tke the folloing rbitrry vlues: q / g 1.0 te/m l 10.0 m EI 1,000 kn.m P Vrious vlues (see belo) Tble 3: Dt for bem We set the line dimeter such tht the line s exctly neutrlly buoynt hich effectively mens tht the eight forces re neglected, s in the theory. Four cses hve been nlysed for to levels of segmenttion. Results re reported from the OrcFlex modl nlysis, nd from time domin nlysis in hich the bem is given smll initil deflection t mid-length. Nturl periods for the first mode of vibrtion re given in the tble belo: P (kn) Nturl period from theory (s) Number of segments in OrcFlex model Nturl period from modl nlysis (s) Nturl period from time domin (s) Tble 4: Comprison of bem nturl periods Both the OrcFlex modl nlysis nd time domin results sho excellent greement for the rnge of P considered. As ould be expected e see closer greement for the models ith more segmenttion. 4. Cntilever bem 4.1. Introduction Formule for deflection, moment, nd slope of cntilever bem re ell knon. For exmple see Rork s Formul s for Stress nd Strin, 7 th edition, McGr-Hill, 00, Tble 8.1-, p191. Cntilever bems re esily modelled in OrcFlex nd e chieve excellent greement ith theoreticl results. 4.. Cntilever bem theory We ssume tht the bem is horizontl, encstré t one end nd free t the other end. We ssume uniform verticl lod, due to the bem s self eight. e bem cn be specified in terms of eight per unit length,, length l nd bend stiffness EI eoreticl results.doc, 4-Sep-07 Pge 4 of 6
5 Stndrd bem theory gives the folloing vlues: Deflection t free end Moment t fixed end Slope t free end 4.3. Comprison of OrcFlex ith theory 4 l 8EI l l 3 6EI Tble 5: Cntilever bem theory For comprison ith OrcFlex, e tke the folloing rbitrry vlues: kn/m l 13.0 m EI 5,000 kn.m Tble 6: Dt for cntilever e theory ssumes tht the bem is inextensible, nd so e used lrge vlue of 1e9kN for. EA For this cse e compred the theoreticl vlues ith OrcFlex using models ith 10, 0 nd 50 segments. e results re tbulted belo: eoreticl results Deflection (m) Slope ( ) Number of segments in OrcFlex model Results from OrcFlex Deflection (m) Slope ( ) Tble 7: Comprison of cntilever theory ith OrcFlex From the bove tble it is cler tht even the 10 segment model gives very good greement ith the theoreticl vlues. As the number of segments is incresed then the difference from theory reduces Discussion of moment results We hve not included the results for moment in the bove comprison becuse they require slightly different tretment. e cntilever bem theory for moment neglects the effect of deflection in effect it ssumes tht the bem is horizontl. OrcFlex s clcultion fully ccounts for the deflection. Becuse of this e ould expect to see difference beteen OrcFlex s reported fixed end moment nd the vlue predicted by theory. e size of this difference ill depend on ho much deflection is present. Cses ith smller deflections ill hve smller differences eoreticl results.doc, 4-Sep-07 Pge 5 of 6
6 To test this e vried the deflection by mens of vrying the bend stiffness vlue EI. For ske of simplicity e did not vry the discretistion of the OrcFlex model for the moment comprisons e used 50 segments throughout. eory (kn.m) EI (kn.m ) OrcFlex (kn.m) Tble 8: Comprison of cntilever end moments As expected, the difference beteen theory nd the OrcFlex result decreses s the cntilever deflection is decresed eoreticl results.doc, 4-Sep-07 Pge 6 of 6
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