DESIGN SPECTRA FOR BURIED PIPELINES

Transcription

1 th Word Conference on Earthquae Engineering Vancouver, B.C., Canada Augut -6, 4 Paper o. 94 DEIG PECTRA FOR BURIED PIPEIE i-ing HOG and Tzuchien CHE UMMARY For a buried pipeine ytem, the imum repone aong the pipe egment and the imum differentia deformation acro the oint are of deign concern. A impe mode of buried pipeine ytem with uniform propertie i ued to evauate the moda parameter and the tatic deign term. When the moda parameter of the pipeine ytem are combined with the homogeneou oi propertie into the equivaent moda frequencie and damping ratio, the moda dynamic effect can be found from the deign pectra propoed for buiding at the ame ite. Amot a the tatic deign term depend on the rotationa and atera tiffne ratio in the atera vibration, or on the axia tiffne ratio in the axia vibration. A a reut, the firt mode dominate the deign-reated repone in the mot cae of atera and axia vibration. ITRODUCTIO The main concern in the eimic damage evauation of a buried pipeine ytem i probaby the differentia motion acro the oint. eon [] introduced a dicrete mode containing two ong pipe egment connected by a oint to tudy the o-caed interference repone pectrum. Each egment i ined to the ground via a pring and a dahpot. The ame mode wa ued by Hadid [] in the tochatic repone anayi of pipeine. On the other hand, if the eimic repone aong the pipe egment are of maor interet, the pipeine are uuay modeed a uniform beam reted on eatic foundation. The ength of the pipeine coud be finite (Hindy [], Zerva [4]), emi-infinite (ova [5]), or infinite (Ogawa [6]). The buried pipeine mode conidered here i imiar to that propoed by Wang [7], except that the infinitey rigid egment i repaced by a uniform beam and the atera vibration i incuded additionay. In thi buried pipeine mode, the beam egment of equa ength are aigned and connected by identica oint, and the pipe-oi interaction i imuated by an eatic foundation with uniformy ditributed pring and dahpot. evera deign-reated repone, uch a the imum deformation and force aong the egment and the imum differentia deformation acro the oint, are tudied when thi buried pipeine ytem i excited by the axia and atera ground motion. Aociate Profeor, ationa Cheng Kung Univerity, Tainan, Taiwan, Emai:hong@mai.ncu.edu.tw Former Graduate tudent, ationa Cheng Kung Univerity, Tainan, Taiwan

2 PIPEIE MODE In accounting for both the repone aong the pipe egment and the differentia repone acro the oint in a buried pipeine ytem, the anayi mode for axia vibration i hown in Fig.. Uniform pipe egment of equa ength connected by identica oint are excited by the ground motion in the ongitudina direction. Hence, the equation of axia motion are u ( x, t) u( x, t) u( x, t) u( x, t) ug( x, t) m EA c (, ) + + u x t + c = m t x xt t t () where i the number of egment in the range of, i the tota number of the egment ued for anayi, m, E, A and c are the ma per unit ength, Young moduu, cro-ectiona area and damping coefficient of the egment, repectivey, u g i the ground dipacement in the ongitudina direction, u i the axia deformation of the th egment, x i the axia oca coordinate in the range of x, and i the ength of a inge pipe egment. a x E, Amc,,, a x a x, c, c, c, c, c E, Amc,,, a a x E, Amc,,, a Figure Pipeine mode for axia vibration anayi For impicity in the vibration anayi, the effect of pipe-oi interaction i imuated by an eatic foundation with continuou pring of tiffne and continuou dahpot of damping coefficient c, a hown in Figure and Eq. (). On the other hand, the pipeine mode propoed for atera vibration anayi i hown in Figure. Again, the eatic foundation and the uniform propertie in pipe egment and oint are aumed. The equation of atera motion are 4 5 u ( x,) t u( x,) t u( x,) t u( x,) t ug( x,) t m + EI c (, ) u x t + c = m t x xt t t () where EI i the fexura rigidity of pipe egment, u g i the ground dipacement in the atera direction, u i the atera deformation of the th egment. u x EImc,,,, u x u x EImc,,,, u x E, I, m, c,, c, c, c, c, c Figure Pipeine mode for atera vibration anayi

3 MODA AAYI Both Eq. () and Eq. () coud be decompoed into moda equation if the damping ratio of each mode i aumed to diipate the energy. Ony the moda anayi performed to Eq. () i dicued in detai here. For the cae of free vibration, the equation of atera motion of the th pipe egment i 4 5 u( x, t) u( x, t) u( x, t) u( x, t) m + EI c (,) u x t + c = t x xt t () By virtue of the eparation of variabe, i.e., u ( x, t) = φ ( x ) q ( t), Eq. () become = c IV EIφ ( ) () () () x q&& t + q& t + q t = m m mφ ( x ) cq& () t + q () t (4) If the contant obtained from the two expreion in Eq. (4) i ω, then we have two ordinary differentia equation: EIφ ( x ) ω mφ ( x ) = (5) IV c q && () t + + cω q() t + + ω q() t = m & m (6) It i obviou in Eq. (5) and (6) that the moda hape, φ ( x ), i independent of the oi propertie, but the moda dynamic effect, q () t, houd be referred to the compound frequencie and damping coefficient. The th-mode hape of the th egment, φ ( x ), oved in Eq. (5) i φ ( x ) = C in β x + C co β x + C inh β x + C coh β x (7) 4 4 where β = ω m / EI. Furthermore, the moda propertie in Eq. (7), uch a β and determined by the boundary condition at both end: C, can be φ () φ () EI = x x (8) φ () EI = φ () x (9) φ EI x ( ) φ ( ) = x ()

4 φ ( ) EI = φ ( ) x () and the compatibiity condition at oint: EI φ ( ) φ () φ ( ) ( + ) = x x+ x () EI φ () φ () φ ( ) ( + ) ( + ) = x+ x+ x () φ ( ) EI = φ( ) φ ( + ) () x (4) φ () EI = φ( ) φ( + ) () ( + ) x + (5) where and are the rotationa and atera tiffne of a oint, repectivey. It i aumed that the pipeine ytem i connected to two fixed end by the two identica oint. Thi aumption i reaonabe for the repone anayi in the midde pipe egment for a arge pipeine ytem. However, the eection of the number of egment in the pipeine ytem i imited by the aumption of equa ength and traight aignment of egment, and uniform oi propertie. In view of Eq. (8) through (5), the moda parameter depend ony on two tiffne ratio defined a = (6) EI / = (7) EI / The vaue of the rotationa and atera tiffne ratio vary in a wide range and are quite different between DIP and PVC pipe. For comparion with the exact moda frequencie in the pipeine ytem of impe boundary condition, uch a roer, hinge, fixed upport, and even free end, the moda frequencie are non-dimenionized a foow: ω ω % ω = + = + (8) / m EI / m For exampe, the variation of the firt three ω% in a pipeine ytem of three egment i hown in Figure, where the dah ine repreent the exact reut for the cae of impe boundary condition. It i noted that the moda frequencie are o we-eparated that the R rue for the moda combination i expected to provide exceent repone etimate. Obviouy, the we-eparated moda frequencie come from the aumption of uniform propertie of the egment and oint.

5 Moda Frequency (w / ( / m).5 )(+) Mode Mode Mode tiffne Ratio = / (EI / ) Figure on-dimenionized moda frequencie in atera vibration (=, =) imiary, the axia tiffne ratio i defined a a a = (9) EA/ where a i the axia tiffne of a oint. The variation of non-dimenionized moda frequencie in the axia vibration of a pipe ytem of = i hown in Fig. 4. Moda Frequency (w / ( a / m).5 )(+a) Mode Mode Mode π π/ π/... tiffne Ratio a = a / (EA / ) Figure 4 on-dimenionized moda frequencie in axia vibration (=) Once ω i found by a given pair of and interaction are obtained by, the equivaent moda frequencie incuding the pipe-oi

6 ω m = + ω () The compound moda damping ratio can ao be obtained by ζ c / m+ ζ ω = / m+ ω () where ζ i the moda damping ratio for the pipe egment ony. The moda equation of motion i then pecified by ω and ζ, from which the moda dynamic effect coud be found in the deign pectrum at the ite of the buried pipeine ytem. DEIG-REATED REPOE For the buried pipeine ytem under the atera or atera vibration, there are evera imum repone of deign concern, uch a the deformation and tree aong the pipe egment and the differentia motion acro the oint. When the ground motion i invoved and the orthogonaity propertie of mode are appied, the moda equation of motion become φ ( x ) u&& g ( x, t) dx = q&& () t + ζω q& () t + ωq() t = () φ ( x ) dx = Modeing the patia variation of the ground excitation i one of the maor topic in the vibration anayi of ifeine ytem. After the numerica tudy conidering the ampitude decay and phae hift of ground motion, however, the homogeneou ground excitation induce the imum deign-reated repone in mot cae. Therefore, the homogeneou ground motion i aumed for deign purpoe. Then, Eq. () become q&& t + ζωq& t + ωq t = Pu&& t () () () () g() where the moda participation factor P i defined a P = = = φ ( x ) dx φ ( x ) dx (4) Finay, the tota deformation after moda uperpoition i u ( x, t) = φ ( x ) PD ( t) = ψ ( x ) D ( t) = = (5)

7 where ψ ( x ) = P φ ( x ) i the effective participation function, D () t i the deformation time hitory of a inge degree-of-freedom ytem of ω and ζ ubected to the ground acceeration, u&& g () t. ix deign-reated repone in the atera vibration of a pipeine ytem are conidered here. They are the imum deformation, rotation, hear force, and bending moment aong a the pipe egment and the imum differentia deformation and rotation acro the oint. The tatic deign repone of the th mode obviouy depend on the effective participation function and it firt three derivative, and then depend on and a a reut. The dynamic effect in each mode due to u&& g () t, ( D ), coud be found in the deign pectra ued for common buiding deign. It i recommended that the R rue be appied in the moda combination becaue of the aumption of uniform propertie in the pipeine ytem. In fact, the contribution of the even mode to the deign-reated repone in a pipeine ytem of odd number of egment i negigibe becaue of the ymmetry of the pipeine ytem, the uniform oi propertie, and the homogeneou excitation. In mot cae for a pipeine ytem of three egment, ony the firt and the third mode in the atera vibration anayi and the firt mode in the axia vibration are required to get the atifactory reut of the deign-reated repone. For exampe, for a pipeine ytem of three egment and = under the atera vibration, the nondimenionized tatic deign repone of the firt and the third mode for evera typica vaue of are ited in Tabe and Tabe, repectivey. In thoe tabe, ψ, ψ, EIψ, and EIψ are the tatic imum atera deformation, rotation, hear, and bending moment aong the egment, repectivey, and ψ and ψ are the tatic imum atera differentia deformation and rotation acro the oint, repectivey. Probaby, the differentia motion acro the oint i more critica in the eimic damage evauation of a buried pipeine ytem than the other deign-reated repone aong the egment. It i ao noted in Tabe and that the imum atera differentia deformation acro the oint i more enitive to the variation of than the other. Tabe tatic deign vaue of the firt mode in atera vibration (=, =) ω EI m ψ ψ ψ ψ ψ ψ / /.9E+.9E+.58E+ 9.78E-.986E+ 4.98E+.6E-.986E- 6.E+.4E+.E+.5E+.974E+ 6.E+.6E-.974E-.64E+.445E+.9E+.45E+ 4.E+ 6.4E+.65E- 4.E- Tabe tatic deign vaue of the third mode in atera vibration (=, =) ω EI m ψ ψ ψ ψ ψ ψ / /.9E+ 6.9E+ 5.99E- 8.8E-.E+ 9.96E+ 5.9E E- 6.E+.7E+ 5.75E-.884E+ 9.E+.64E+ 6.67E- 9.E-.64E+.E+ 5.55E-.9E+ 9.59E+.59E+.4E- 9.6E- A for the axia vibration of a buried pipeine ytem, the deign-reated repone woud be the imum axia force in the pipe egment and the imum axia deformation acro the oint. For a pipeine

8 ytem of three egment under the axia vibration, the non-dimenionized tatic deign repone of the firt mode for ome typica vaue of a are ited in Tabe. In Tabe, EAψ, and ψ are the tatic imum axia force aong the egment and the tatic imum axia differentia deformation acro the oint, repectivey. Tabe tatic deign vaue of the firt mode in axia vibration (=) a ω / EA / m ψ ψ.e- 7.66E- 8.49E- 8.49E-.E-.64E- 8.E- 8.E-.E+ 6.6E- 5.55E- 5.55E-.E+ 9.68E-.9E+.9E- 6.E+.E+.8E+.8E- On the bai of the above vibration anayi, the procedure for the eimic demand of a buried pipeine ytem i tated a foow.. A pipeine mode of three or five pipe egment i preferred becaue of the aumption of uniform propertie of egment and oi and homogeneou ground excitation ued for impifying the anayi.. The axia, atera, and rotationa tiffne of the oint houd be carefuy evauated in order to etimate the moda parameter and the tatic deign-reated repone.. The oi propertie are then added to obtain the compound moda frequencie and damping ratio in order to find the pectra deformation in the deign pectra propoed for buiding at the ame ite. 4. Each moda deign-reated repone woud be the product of the reut in tep and. The R rue i uggeted to compoe the tota deign-reated repone except that the pipeine are buried in a pretty hard ite. COCUIO A impe mode of buried pipeine ytem with finite boundarie i ued to deveop the procedure for the eimic demand reated to the pipe egment and the oint. The main aumption incude the uniform propertie in pipe, oint and oi, and the homogeneou ground excitation. On the bai of the aumption and numerica reut, ome important concuion are given in the foowing:. A pipeine mode of three or five pipe egment i preferred for the deign purpoe.. The tatic deign-reated repone depend on the rotationa and atera tiffne ratio in the atera vibration, and depend on axia tiffne ratio in the axia vibration.. The contribution of the firt and the third mode to the tatic deign-reated repone are of maor concern in the atera vibration, whie the reut of the firt mode ony are good enough in evauating the tatic deign-reated repone in the axia vibration. 4. After combining the moda parameter of the pipeine ytem and the oi propertie to get the equivaent moda frequencie and damping ratio, the moda dynamic ampification effect can be

9 found in the deign pectra ued for buiding deign at the ame ite. Then, the deign-reated repone are ti dominated by the firt mode even for a pipeine ytem of ma vaue of tiffne ratio. ACKOWEDGMET Thi tudy wa partiay upported by the ationa cience Counci, R.O.C. under Grant C 9-65-Z The author are gratefu to the Centra Weather Bureau, Minitry of Tranportation and Communication, R.O.C. for providing the eimic record in Taiwan. REFERECE. eon I., Weidinger P. Dynamic eimic anayi of ong egmented ifeine. Journa of Preure Vee Technoogy, AME 979; : -.. Hadid M., Afra H. enitivity anayi of ite effect on repone pectra of pipeine. oi Dynamic and Earthquae Engineering ; : Hindy A., ova, M. Pipeine repone to random ground motion. Journa of the Engineering Mechanic Diviion, ACE 98; 6: Zerva A. Pipeine repone to directionay and patiay correated eimic ground motion. Journa of Preure Vee Technoogy, AME 99; 5: ova M., ogami T., Abou-Ea F. Dynamic oi reaction for pane train cae. Journa of the Engineering Mechanic Diviion, ACE 978; 4: Ogawa Y., Koie T. tructura deign of buried pipeine for erve earthquae. oi Dynamic and Earthquae Engineering ; : Wang., Cheng K.-M., eimic repone behavior of buried pipeine. Journa of Preure Vee Technoogy, AME 979; : -.