Cumulative Damage Evaluation based on Energy Balance Equation

Transcription

1 Cumulaive Damage Evaluaion based on Energy Balance Equaion K. Minagawa Saiama Insiue of Technology, Saiama S. Fujia Tokyo Denki Universiy, Tokyo! SUMMARY: This paper describes an evaluaion mehod for cumulaive damage by earhquakes based on energy balance equaion. Indusrial faciliies such as power plans require high ani-seismic performance. Generally seismic design of mechanical srucures in indusrial faciliies is based on saic force in elasic region. However dynamic cyclic load may causes cumulaive damage, and such damage induces faigue failure on he srucures. In addiion, evaluaion mehod of soundness afer a huge earhquake is also required. Therefore quaniaive evaluaion of cumulaive damage by earhquakes is required. Auhors have deal wih his issue from he viewpoin of he energy balance equaion. Since energy indicaes cumulaive informaion of moion, he energy balance equaion is suiable o evaluae cumulaive damage and a margin for failure. This paper consiss of inroducion of energy balance equaion, invesigaion ino relaion beween energy and faigue failure, and an applicaion of Miner s law o energy. Keywords: Seismic evaluaion, Energy balance equaion, Faigue failure, Cumulaive damage 1. INTRODUCTION Large earhquakes, such as he Niigaaken Chuesu-oki earhquake in 27 and he grea eas Japan earhquake in 211, aacked indusrial faciliies. Especially he grea eas Japan earhquake had long duraion ime and many afershocks. From hese disasers, some subjecs regarding seismic design and evaluaion of mechanical srucures have been newly recognized. One of he subjecs is raional evaluaion of ulimae srengh, and anoher one is evaluaion of soundness afer an earhquake. In general, seismic design of mechanical srucures in indusrial faciliies is based on saic force in elasic region, because of is easy calculabiliy. However he saic force is a momenary parameer, and i canno consider he duraion ime and he number of afershocks. Thus imporance of seismic margin assessmen (SMA) and damage indicaing parameer (DIP) ha can consider hese subjecs has discussed laely. Auhors focused on an energy balance equaion [1]-[4] for above-menioned subjecs. The energy balance equaion is one of valid mehods for srucural calculaion, and i is easily yielded from he equaion of moion. Therefore he energy balance equaion is suiable for evaluaion and monioring of dynamic response. In addiion, since energy indicaes cumulaive informaion of moion, he energy balance equaion is suiable o evaluae cumulaive damage such as faigue. This paper describes cumulaive damage evaluaion based on he energy balance equaion. A firs, energy balance equaions of boh SDOF and MDOF are inroduced. Nex relaionship beween energy and faigue failure is invesigaed by forced vibraion experimens and heory based on faigue failure field. Finally Miner's law is adaped o energy in order o evaluae seismic damage agains various waves.

2 2. ENERGY BALANCE EQUATION 2.1. Energy Balance Equaion for Single-Degree-of-Freedom model The energy balance equaion is simply derived from he equaion of moion as follows. Therefore he energy balance equaion is able o explain vibraion characerisics dynamically. Equaion (1) shows he equaion of moion of single-degree-of-freedom model. m!!x + c!x + F ( x) =!m!!z H 1 Where x is he relaive displacemen from ground, m is he mass, c is he damping coefficien, F(x) is he resoring force, and z H is he horizonal acceleraion of ground moion. Muliplying Eqn. (1) by displacemen incremen dx (= x d) leads o he work of he ime d, as shown in Eqn. (2). m!!x!xd + c!x 2 d + F ( x)!xd =!m!!z H!xd 2 Finally he energy balance equaion is yielded by he ime inegral of Eqn. (2), as shown in Eqn. (3). m " x x d + c x 2 " d + " F( x) x d = #m " z H x d W k + W d + W e + W p = E 3 Where, W k = m W d = c " " W e + W p = E = #m " x x d x 2 d " ( ) F x z H x d x d 4 In Eqns. (3) and (4), W k is he kineic energy, W d is he dissipaion energy by viscous damping, W e is he elasic srain energy, W p is he cumulaive plasic energy and E is he inpu energy. The energy balance equaion, Eq. (3), shows a sum of he work ill he ime alhough he equaion of moion shows he momenary condiion in a ime. As shown Eqns. (3) and (4), energy is expressed by inegral, so he energy balance is adequae o invesigae he influence of he cumulaive load because i includes he cumulaive informaion. The behavior of he energy is shown in Fig. 1. In he energy balance equaion, only dissipaion energy by viscous damping W d and cumulaive plasic energy W p are cumulaed. Kineic energy W k and elasic srain energy W e converge wih convergence of response. Therefore he sum of he dissipaion energy W d and he cumulaive plasic energy W p is equal o he inpu energy E afer vibraion of sysem finished. Addiionally inpu energy E is easy o calculae because i consiss of parameers easy o measure. Therefore inpu energy E is imporan energy in he energy balance equaion, so inpu energy E is specially focused on in his sudy.

3 Figure 1. Behavior of energy 2.2. Energy Balance Equaion for Muli-Degree-of-Freedom model Some mehods of energy balance equaion for muli-degree-of-freedom model have been proposed. One is based on he equaion of moion expressed in he marix forma [4]. However he mehod based on he marix forma canno express energy of each mass. Thus auhors have proposed anoher mehod ha is focused on each mass poin [5]. The energy balance equaion focused on each mass poin is obained by same procedure as single-degree-of-freedom. In his mehod, relaive relaionship among mass poins is focused on. The relaive displacemen of ih and i-1h mass poin x i! x i!1 replaces wih Y i, he sum of displacemen of ih mass poin and ground x i + z H replaces wih Z i. Equaions (5)-(7) show he energy balance equaion of ih mass poin of N degree-of-freedom model. i =1 :! m 1!!x 1!x 1 d + c 1!x 2! 1 d +! k 1 x 1!x 1 d = "! m 1!!z H!x 1 d "! m k Z!! k 5 N ( )!x 1 d # k=2 i = 2,!, N!1 : m Y i!!!y d! + c i i i!y 2! i d +! k i Y i!y i d = "! m i Z!! i"1!y i d "! # m k Z!! k!y i d 6 N k=i+1 ( ) i = N : m Y! N!!!Y N N d + c N!Y 2! N d +! k N Y N Y! N d = "! m N Z!! N"1!Y N d 7 In he Eqns. (5)-(7), he 1s erm of he lef hand side is a kineic energy, he 2nd erm is a dissipaion energy by viscous damping, he 3rd erm is a sum of a elasic srain energy and he 1s erm of he righ hand side is an inpu energy. In he Eqns. (5) and (6), 2nd erm of he righ hand side can be consrued as an ransferred energy from upper o ih mass poin. Energy of each individual mass poin is clarified by using his mehod Hyseresis Energy Hyseresis energy is known in he radiional faigue failure field. The hyseresis energy is area of he hyseresis loop per cycle. In he elasic deformaion, sress is direcly proporional o srain, so ha

4 he hyseresis loop is no shaped. On he oher hand, in he plasic deformaion, he relaionship beween sress and srain is non-linear, so ha he hyseresis loop is shaped. The area of he hyseresis loop is he oal sum of he produc of he sress and srain per cycle. Therefore hyseresis energy indicaes quaniy of energy absorpion per cycle by plasic deformaion. 3. RELATIONSHIP BETWEEN ENERGY AND FAILURE 3.1. Invesigaion by Vibraion Experimen In order o invesigae relaionships beween energy and faigue failure, a vibraion experimen was carried ou Experimenal procedure A simple single degree of freedom model shown in Fig. 2 was used as an experimenal model. The experimenal model consiss of mass and a pole. Weigh of he mass wih an acceleromeer was.24 kg. The lengh of he pole was.157 m, and cross-secion of he pole was recangle ha has widh of.12 m and breadh of.3 m. The experimenal model was made of sainless seel Japanese Indusrial Sandard (JIS) SUS34. The nominal naural frequency and he damping raio were 16.3 Hz and.73 %, respecively. Random waves having predominan frequency similar o he naural frequency of he experimenal model were inpu. The predominan frequency of he random wave was 13 o 16 Hz. The ime lengh of he random wave was 3seconds, and he random wave was inpu repeaedly. So experimenal models vibrae in resonance condiion. An exciaion was coninued unil faigue failure of he experimenal model. This exciaion was repeaed in various inpu ampliude, ha is 18.8, 21.9, 25.5, 28.8, 31.1 m/s 2. Figure 2. Experimenal model Experimenal resuls Experimenal resuls were arranged from he viewpoin of energy and failure as shown in Fig. 3. The

5 relaionships were pu in order from he viewpoin of ime for failure and inpu energy for failure, and incremen of inpu energy and inpu energy for failure. From Fig. 3, i is confirmed ha inpu energy for failure is proporional o ime for failure, and is inversely proporional o incremen of inpu energy. In oher words, inpu energy for failure is inversely proporional o he maximum inpu acceleraion. In addiion, hey are in good agreemen wih regression curves by power funcion. Therefore relaionships beween inpu energy and faigue failure were confirmed Theoreical Invesigaion Figure 3. Relaionships beween energy and failure Relaionship beween energy and faigue failure is invesigaed from he hyseresis energy. Morrow [6] has repored a relaionship beween he hyseresis energy W h and a faigue life N f as shown in Eqn. (8).!W h = an f d 8 Where, a, d are consans and hese have ranges of a > and -1 < d < [7] [8]. In addiion, he oal hyseresis energy o faigue failure W h is presened as Eqn. (9). W h = N f!w h = an f d+1 9 Eliminaing faigue life N f from Eqns. (8) and (9) lead o he following equaion. W h =!W h a 1 d 1+ 1 d 1 In he above-menioned experimenal resuls, ime for failure f is proporional o faigue life N f, and inpu energy for failure E f is hyseresis energy o faigue failure W h. Therefore a relaion beween inpu energy for failure E f and he ime for failure f is esimaed as following power funcion by referring o Eqn. (11). E f = p f q 11 Where, p, q are consans ha depend on maerial. I is supposed ha a relaion beween an inpu

6 energy E of energy balance equaion and an incremen of inpu energy!e is equivalen o he relaion beween he oal hyseresis energy for faigue failure W h and he hyseresis energy W h shown in Eqn. (1). Thus he relaion beween he inpu energy E and he incremen of inpu energy!e is assumed as following power funcion by referring o Eq. (1). E f = v!e w 12 Where, v is a posiive proporional consan, w is a negaive proporional consan. As shown in Fig. 3, relaionships beween energy and failure ha were confirmed from he experimen were good agreemen wih regression curves calculaed by Eqns. (11) and (12). Therefore relaionships beween energy and failure are expressed by power funcions shown in Eqns. (11) and (12). 4. APPLICATION OF MINER S LAW TO ENERGY Relaionships beween energy and failure were confirmed by vibraion experimen in secion 3.1. However he inpu wave had he consan ampliude unil failure, so ha applicaion mehod for various inpu level is required. Therefore, Miner s law is applied o energy balance equaion in order o esimae he faigue life for various inpu level. The Miner s law is known in he field of faigue srengh Applicaion Mehod In Miner s law, faigue life is esimaed by faigue damage n i / N fi, and faigue failure occurs when oal sum of faigue damage exceeds 1. Where, N fi is a faigue life when sress ampliude is " i, and n i is he number of exciaion when sress ampliude is " i. Tha is o say, he condiion of he faigue failure is he following; n i! "1 13 i N fi For he energy balance equaion, faigue life agains various inpu level is esimaed by energy cumulaive frequency D Ei =!E i i / E fi, and faigue failure occurs when energy cumulaive damage D Ef exceeds 1. Where E fi is inpu energy for failure when incremen of he inpu energy is!e i, and i is ime for failure when incremen of he inpu energy is!e i, and D Ef is oal sum of he energy cumulaive frequency D Ei. Tha is o say, he condiion of he faigue failure is he following; D Ef =! D Ei = "E "E 2 2 E f Vibraion Tes i E f 2 +!+ "E i i E fi +! #1 14 In order o confirm proposed echnique, a vibraion experimen having various inpu level was carried ou. In hese experimens, inpu level was changed on every 36 seconds. The experimen is conduced by wo paerns of combinaions of he inpu level. The maximum inpu acceleraion of paern 1 was changed as ascending order, > > > 31.7 m/s 2, and paern 2 was changed as descending order, >23.6 -> >18.4 -> > 23.7 m/s 2. The same experimenal condiion as secion 3.1 was applied o his experimen. Thus he inpu energy for failure E fi when incremen of he inpu energy!e i is esimaed by he power funcion in he Fig. 6 (b) Figure 4 shows experimenal resuls. The lef verical axis indicaes he maximum inpu acceleraion, and he righ verical axis indicaes he energy cumulaive damage. From Fig. 4, i is confirmed ha he experimenal model fracured when D Ef = 1.16 in he paern 1, and when D Ef = 1.14 in he paern

7 2. These resuls saisfy Eqn. (14). Therefore i was confirmed ha applicaion of Miner s law o energy balance equaion is applicable and faigue life esimaion using energy balance equaion are effecive. In addiion, his echnique is valid for evaluaion of soundness afer an earhquake, because he energy cumulaive damage indicaes a margin o faigue failure. Figure 4. Confirmaion of applicaion of Miner s law 5. CONCLUSION This paper describes cumulaive damage evaluaion based on he energy balance equaion. The resuls are summarized as follows; Energy balance equaion for boh single and muliple degree-of-freedom was inroduced. The energy balance equaion explains cumulaive informaion of moion. Therefore i is suiable for evaluaion of cumulaive damage by earhquakes. Inpu energy for failure is proporional o ime for failure, and is inversely proporional o incremen of inpu energy. These relaionships are expressed by power funcions. Applicaion of Miner s law o energy balance equaion enables esimaion of faigue life for various inpu level. Moreover his echnique enables evaluaion of soundness afer an earhquake, because he energy cumulaive damage indicaes a margin o faigue failure. In order o improve his mehod, quaniaive esimaion of energy for failure and failure poin are required in he fuure. AKCNOWLEDGEMENT The auhors would like o express heir appreciaions o Mr. Kanaeda and Mr. Yamanaka of Graduae School of Tokyo Denki Universiy, Mr. Kiamura and Dr. Waakabe of Japan Aomic Energy Agency for heir devoed assisance. REFERENCES [1] Housner, G. W. (1956), Limi Design of Srucures o Resis Earhquakes, Proceedings of 1s WCEE, pp [2] Housner, G. W. (1959), Behavior of Srucures during Earhquakes, ASCE 85-EM4, pp [3] Kao, B., Akiyama, H. (1975), Energy Inpu and Damages in Srucures Subjeced o Severe Earhquakes, Trans. of he Archiecural Insiue of Japan 235, pp. 9-18, (in Japanese). [4] Akiyama, H. (1999), Earhquake-Resisan Design Mehod for Buildings Based on Energy Balance, Gihodo Shuppan Co., Ld., Chap.1-2, (in Japanese) [5] Minagawa, K., Fujia, S., Kanaeda, S., Endo, R., Amemiya, M. (211), Applicaion of Energy Balance

8 Equaion on Muliple Degree of Freedom Sysems, Trans. of he Japan Sociey of Mechanical Engineers, Series C 77:774, pp , (in Japanese). [6] Morrow, J. (1965), Cyclic Plasic Srain Energy and Faigue of Meals, Inernal Fricion, Damping, and Cyclic Plasiciy, American Sociey for Tesing and Maerials STP-378, pp [7] Dimov, D. M., Andonova, M. M., (1995), Faigue life predicion of pressure vessels using an energy based approach, Srengh of Maerials 27:1-2, pp [8] Saeki, E., Sugisawa, M., Yamaguchi, T., Mochizuki, H., Wada, A. (1995), A Sudy on Hyseresis and Hyseresis Energy Characerisics of Low Yield Srengh Seel, Trans. of Archiecural Insiue of Japan, Journal of srucural and consrucion engineering, 473, pp (in Japanese).