STIFFNESS AND DAMPING PROPERTIES OF EMBEDDED MACHINE FOUNDATIONS

Transcription

1 Numer Vlume 1 June 006 Jurnal f Engineering Dr. Thamir. Al-Aawi Dr. Raad. Al-Aawi Zuhair. Al-Jaeri Prf. / Civil Eng. Dept. Lecturer/ Civil Eng. Dept. Structural Engineer Cllege f Eng./ Univ. f Baghdad Cllege f Eng./ Univ. f Baghdad Cllege f Eng. / Univ. f Baghdad ABSTRACT In thi tudy, a dynamic analyi f machine fundatin under vertical excitatin i carried ut. The effect f edment and fundatin gemetry ha een taken int accunt. The tiffne and damping f il are cnidered a frequency dependent. A cmputer prgram (CPESP) in FORTRAN POWER STATION ha een cded t evaluate the tiffne and damping cefficient depending n excitatin frequency and edment depth. Reult have hwn that increaing the edment depth lead t increaing the renant frequency and decreaing the amplitude f viratin. %&$!"# +&0+#-13$+,-./0!"*)edment(' )FORTRAN POWER STATION(4#+(CPESP(1+&$(frequency dependent( -+)( #-1"5 +9$)edment('%&6#&)ω(6#&)C( 3<=>/$)equivalent circular apprximatin (:;599' EY WORDS Dynamic, Machine Fundatin, Stiffne, Emedment. $"#9/?63;@'%&? INTRODUCTION Mt f the lutin methd treat the machine fundatin a a lck reting n the urface f an elatic il. The real fting are uually edded and thi cnideraly affect the dynamic repne f fting, Barken.D.D (196). The rigru analytical lutin f edded fting ha many mathematical difficultie. The mt prmiing way f tudying thi prlem i the finite element analyi a had een ued y many reearcher uch a Lymer.J(1979) and y kaldjian.m.j (1969) fr tatic analyi. Neverthele, there i a need fr alternative apprximate lutin that wuld e ale t predict the mtin and t evaluate the tiffne and damping characteritic f edded fting.

2 T.. Al-A awi, Raad. Al-Aawi and Zuhair. Al-Jaeri EQUATION OF MOTION By applying de Alert' principle, the equatin f mtin can e written a; Fig. (1) mu ( t) + C ( ω ) U ( t) + ( ω) U ( t) = P exp( iωt) (1) Z Z Z m = Ttal ma. mu Z (t) = Inertia frce. C ( ω ) U ( t) =Damping frce. Z Z ( ω) U ( t) = Elatic frce. Z Z (ω) =Frequency dependent tiffne. C (ω) =Frequency dependent damping. Z Z O P exp (iω t) m C (ω) (ω) Fig. (1) Fundatin reting n pring and dahpt Fr harmnic lading with an excitatin frequency fω, the teady tate lutin can e aumed a: U ( t) = A exp( iωt) () Sutituting eq. () int eq. (1):- mω A exp( iωt) + + [ ( ω) iωc ( ω) ] A exp( iωt) = P exp( iω t) Dividing th ide f the equatin y exp( iωt) and eparating real and imaginary part, the amplitude f mtin A will e:- P A = ( ( ω) mω ) + iωc ( ω) (3) [ ] Let a = ( ω) mω 1 a = ωc ( ω) Multiplying the numeratr and denminatr f eq. (3) y ( a + ia ), the amplitude can e written 1 a:-

3 Numer ( a + ia ) P 1 exp( i φ) A = = P a + a R 1 R = + And a 1 a Vlume 1 June ω C ( ω) φ = tan ( a a ) = tan 1 ( ω) mω Sutituting A int eq. () the teady tate lutin ecme:- Jurnal f Engineering (4) exp( iφ) U ( t) = P exp( iωt) = R ( P exp ( ω) mω ) [ i( ωt + φ) ] + ω C ( ω) (5) The real part f the amplitude f viratin i:- P AZ = ( Z ( ω) mω ) + ω CZ ( ω) where:- Eq. (5) give the dynamic repne f the fundatin in vertical viratin and fr an exciting frce f cntant amplitude P. The natural frequency f the undamped free viratin i:- ω = (ω m (6) n ) In thi tudy a rigid fundatin will e tudied which i lcated at depth D elw the grund urface. Thi fundatin i ujected t a teady-tate viratin y a harmnic vertical frce, P( t) = P exp( iωt), having an amplitude f P and a circular frequencyω, and acting thrugh the centerid f the ae. Thi dynamic frce i reited y nrmal il tree againt the ae and y hear tree alng the vertical fundatin ide. The rtatinal cillatin that may ccur due t the lack f cmplete ymmetry in the il reactin at the ae and epecially at the fundatin ide are ignred in thi tudy. The teady-tate repne f the fundatin i thu decried y the vertical dynamic ettlement U = U exp( iωt). Due t damping the frce, P (t) i generally ut f phae with the repne U (t). The latter can e divided int tw cmpnent, ne in phae [ U exp( iω t) ] and the ther 90 ut f phae 1 [ U exp( iω ) ] with P. (11) t The crrected dynamic tiffne, (β ) and the dynamic damping cefficient, C (β ) are given y:- ( β ) = ( ω) ωc β (7.a) ( ω) C ( β ) = C + β (7.) ω Where: β = frequency independent damping rati. Fr mt il β range typically frm 0.0 t 0.05, Richart.F.E. (1970).

4 T.. Al-A awi, Raad. Al-Aawi and Zuhair. Al-Jaeri Bth the effective dynamic tiffne and the radiatin damping cefficient f the il fundatin ytem are functin f the frequencyω.it i cnvenient t expre ) a a prduct f the tatic ur dy tiffne, f the ytem time a dynamic tiffne cefficient k (ω) ur ur ) = k( ω) (8) dy ur STATIC STIFFNESS OF SURFACE FOUNDATIONS Fr a urface fundatin f an aritrary hape, the vertical tatic tiffne Dmingue,J(1978): ur, i given y LG = S ur 1 υ L =Semi-length f a rectangle circumcried t ae urface. G =Shear mdulu f il. υ =Pin rati. S =Vertical tatic tiffne parameter. Fr nn-rectangular ae, may e tained a fllw, Prakah,S(1988):- ur (9) ur = 4GR (1 υ) (10) R= Radiu f the equivalent circle = A π The equivalent circle apprximatin predict S a fllw (10) :- S 4 = A 4L (11) π The equivalent circle apprximatin give gd reult fr L/B t 3 a calculated y Dry and Gaeta (1986). Fig () hw that:- S S = 0.8 = ( A L ) 0.75 fr A fr A 4L 0.0 4L 0.0 (1) 1.00 Vertical Static Stiffne Parameter (S) A 4L Fig. () vertical tatic tiffne parameter (S) veru ae hape (10)

5 Numer Vlume 1 June 006 Jurnal f Engineering EFFECT OF EMBEDMENT ON STATTIC STIFFNESS In practice, fundatin are placed at a pecified depth, ay D elw the grund urface and tranmit the lad t il. Uually, increaing the depth D mean increaing the fundatin tiffne. The factr that mdify the fundatin tiffne are the "trench" and "idewall cntact" effect, that tend t increae the tiffne f the edded fundatin. Thee tw effect are t e explained with the aid f Fig. (3). Trench Effect Even in perfectly hmgenu il a rigid fting will ettle le if it i placed at the ttm f an pen trench. The nrmal and hear tree reulting frm the verlying il retrict the vertical mvement and thu reducing the ettlement f the fundatin ae y increaing it vertical tiffne. The trench effect uggeted y Gaeta and Dry (1986) i:- I 1 (13) tre ur = tre i the vertical tatic tiffne f an edded fundatin mat with n idewall cntact. tre Sidewall Effect Part f the applied lad i tranmitted t the grund thrugh hear tree alng the vertical ide f the fting when the ide are in cntact with the urrunding il. A a reult, the verall tiffne f an edded fundatin i larger than tiffne tre crrepnding t a fundatin with the ame depth f edment ut withut ide effect, Ricard.D (1985). tre = I ide 1 (14) P τ = σ = 0 σ τ P σ τ τ P τ (a) I 1 I 1 tre ur = tre () (c) tre = ide Fig. (3) effect f edment n vertical tatic tiffne f fundatin (a) ettlement due t urface fundatin () trench effect (c) cmined trench and idewall effect.

6 T.. Al-A awi, Raad. Al-Aawi and Zuhair. Al-Jaeri Experimental tudie, uch a the f Lymer.j (1969), ffer valuale guidance in thi directin. Cmining eq. 13 and 14 lead t: = I I ur tre ide Baed n tet reult the fllwing empirical equatin had een derived:- D 4 A = + + I 1 1 (15) tre 1B 3 4L I = ( A A ) (16) ide I =Trench factr. tre I =Sidewall factr. ide A =Bae area f fundatin. A =Side area f fundatin. Fig (4) hw that a (D/B) increae the rati f ( ) al increae. Thi trend i mre tre ur prnunced fr the cae f a quare fundatin (L/B=1). The fundatin tatic tiffne ( ) fr a full edment cae i:- LG = S 1 υ D B ( A 4L A + ) ( A ) (17) Fig.(5) hw that a (D/B) increae the rati ( prnunced fr the cae f a quare fundatin (L/B=1) ur ) al increae. Again thi trend i mre 1.1 L/B=1 STATIC RATIO (tre/ur) L/B= L/B=4 L/B= D/B Fig. (4) effect f trench n tatic tiffne

7 Numer Vlume 1 June 006 Jurnal f Engineering 1.60 STATIC RATIO(/ur) L/B=1 L/B= L/B=4 L/B= D/B Fig. (5) effect f edment n tatic tiffne DYNAMIC STIFFNESS COEFFICIENT It had een cncluded empirically y Gerge.G(1986) that the vertical tiffne f elatic fundatin i frequency dependent. The main parameter affecting the dynamic tiffne are a, L B andυ, where:- a =Nrmalied frequency = ω B V V =Shear wave velcity. L B =Fundatin apect rati. υ =Pin rati. The frequency dependent tiffnee are:- Fr Pin rati υ = 0. 33(unaturated il) 0.75 [ ( a ) ( D ) ] ) = k( ω ) B (18.a) dy 0.75 [ ( a ) ( D ) ] ) = k( ω ) B (18.) + tre dy Fr Pin rati υ = 0. 5 (aturated il) 0.5 [ ( a ) ( D ) ] ) = k( ω ) B (18.c) dy

8 T.. Al-A awi, Raad. Al-Aawi and Zuhair. Al-Jaeri ) = k( ω ) ( a ) ( D B) (18.d) + tre dy [ ] Thee equatin were tained y Gaeta and Dry (1986), (ω) :i a dimeninle frequency dependent factr given in Tale (1). Hence the dynamic tiffne f an edded fundatin can e written a:- ) = k( ω ) F = F (19) dy e e 0.75 F = [ ( a ) ( D B) ] e r 0.5 F = [ ( a ) ( D B) ] e a given in eq. (18). The factr F f eq. (19) i the effective edment factr. Fig. (6) Shw the variatin f e thi factr with the nrmalied frequency parameter( a ). The relatinhip have een tained in the preent tudy y cding the ave equatin thrugh a hrt cmputer prgram. Pain rati 0.33 Tale (1) dynamic tiffne factr fr urface fundatin [ (ω) ] Frequency dependent tiffne factr [ (ω) ] ( a ) ( a ) 1 and ( a ) ( a ) ( a ) ( a ) 10 L/B ( a ) ( a ) ( a ) ( a ) ( a ) ( a ) 6 DAMPING COEFFICIENT The cefficient f damping c = c(ω) i a meaure f viratin energy tranmitted int the il and carried away y preading wave. Thee wave are generated at every pint n the il-fundatin interface that in general c (ω) increae with increaing area f cntact. The cntact urface fr a vertically cillating edded fundatin cnit f a hrintal ae and vertical ide. The ae tranmit t the underlying grund cmprein-extenin wave in prpagatin velcity cle t the Lymer (1969) analgy. [ π (1 )] V 3.4 V υ (0) = La Where: V = hear wave velcity V = "Lymer' analg" velcity La On the ther hand the ide tranmit mainly hear wave thrugh the urrunding il. The tw type f wave generated at the ae and at the ide f an edded fundatin are independent. Summing up the repective radiated energie.

9 Numer Vlume 1 June 006 Jurnal f Engineering EFFECTIVE DYNAMIC STIFFNESS FACTOR FOR EMBEDDED FOUNDATION υ = 0.33 L B = 1, D/B=0 D/B=0.5 D/B=1.0 D/B=1.5 D/B=.0 EFFECTIVE DYNAMIC STIFFNESS FACTOR FOR EMBEDDED FOUNDATION υ = 0.33 L B = 4 D/B=0 D/B=0.5 D/B=1.0 D/B=1.5 D/B= NORMALIZED FREQUENCY(a) NORMALIZED FREQUENCY (a) EFFECTIVE DYNAMIC STIFFNESS FACTOR FOR EMBEDDED FOUNDATION υ = 0.33 L B = 6 D/B=0 D/B=0.5 D/B=1.0 D/B=1.5 D/B= NORMALIZED FREQUENCY (a) EFFECTIVE DYNAMIC STIFFNESS FACTOR FOR EMBEDDED FOUNDATIO υ = 0.5 L B = 1 D/B=0.5 D/B=0 D/B=1.0 D/B=.0 D/B= NORMALIZED FREQUENCY(a) EFFECTIVE DYNAMIC STIFFNESS FACTOR FOR EMBEDDED FOUNDATION υ = 0.5 L B = 6 D/B=0 D/B=0.5 D/B=1.0 D/B=1.5 D/B= NORMALIZED FREQUENCY (a) Fig. (6) effective dynamic tiffne factr F fr edded fundatin e

10 T.. Al-A awi, Raad. Al-Aawi and Zuhair. Al-Jaeri C = ( ρ V A ) c( ω) + ρ V A (1) La c (ω) : Cefficient f dynamic damping a given in Tale (). Tale () dynamic damping cefficient [ c (ω)] () Dynamic Damping cefficient c (ω) R= L/B (R a ) exp (R a ) (R a ) (R a ) exp (-R a ) (R a ) (R a ) (R a )+0.004(R a ) exp (-R a ) (R a ) (R a ) exp(-R a ) 10 COMPUTER PROGRAM In thi tudy a cmputer prgram (CPESP) (Cmputer Prgram fr Evaluatin f Sil Prpertie) in Frtran Pwer Statin language ha een cded fr calculating the dynamic tiffne and damping fr urface and edded fundatin. In thi prgram the input data are :- Dimenin B and L f the ae. Side urface area f fundatin A. Sil hear mdulu G Sil pin rati υ Sil denity ρ Sil damping factr β The firt tep i t cmpute the Static Stiffne and damping cefficient. The ecnd tep i t cmpute the dynamic factr fr the tiffne and damping. The effect f edment wa al cnidered in thi prgram. APPLICATIONS Applicatin (1) The develped cefficient f dynamic tiffne and damping are applied t tain the dynamic tiffne and damping uing the (CPESP) prgram fr the edded fundatin hwn in Fig. (7). The reult are hwn in Tale (3)

11 Numer Vlume 1 June 006 Jurnal f Engineering 10 A 10 5 A G =74 MPa ρ =1.85 Mg/m 3 υ = 0.33 V =00 m/ec ω = 30 rad/ec All dimenin in (m) Sec. A-A Fig (7) gemetry and material parameter, Gaeta.G(1979),applicatin (1) Applicatin () The fundatin f applicatin (1) ha een lved uing the equivalent circle apprximatin. The effective radiu f fundatin(r) = A π = GR The equivalent tatic urface tiffne ur = 1 υ =6.47*10 6 kn/m ur Uing eq. (15),(16) the reult are :- I =1.055 tre I =1.11 ide The tatic edment tiffne will e:- =8.68*10 6 kn/m Frm Tale (1) and uing eq. (18a) then the effective edment factr i equal t ) = * F dy e ) =5.895*10 6 kn/m dy Tale (3) hw the final reult fr applicatin (1) and ().

12 T.. Al-A awi, Raad. Al-Aawi and Zuhair. Al-Jaeri Tale (3) dynamic tiffne f edded fundatin uing the preent tudy and apprximate methd. Methd f analyi Preent methd (Applicatin 1) Equivalent Circle Apprximatin (Applicatin ) Surface tatic tiffne (kn/m) Trench factr Sidewall factr Dynamic tiffne cefficient Dynamic edded tiffne (kn/m) 7.333* * * *10 6 Tale (3) cmpare the reult f the preent tudy and the equivalent circle apprximatin and the maximum dicrepancy i aut 3%. The Lymer analg velcity uing eq. (0) i:- V =33.06 m/ec La Frm Tale () and uing ( a =1.5) then the dynamic damping cefficient i :- c (ω) =1.075 The dynamic damping f il uing eq. (1) i:- C = 0.76*10 6 kn.m -1.ec The crrected dynamic tiffne and damping uing eq. (7) are:- (β ) =4.98*10 6 kn/m C (β ) =0.749*10 6 kn.m -1.ec Applicatin (3) The tained cefficient in thi tudy Tale (1 and ) are ued t tudy the dynamic repne f machine fundatin under vertical dynamic lad y uing SAP 000. The analyi parameter are:- Fundatin Parameter Sil Parameter Machine Parameter L=9.6 m υ = 0.33 F v =6.7 in (ω t) B=4.8 m γ =18.75 kn/m 3 ω =61.36 rad/ec D= 1.55 m G = 98 Mpa Machine weight=60.65 kn Fundatin weight=1714 kn The reult tained are ummaried in Tale (4) and Fig. (8).

13 Numer Vlume 1 June 006 Jurnal f Engineering Tale (4) reult tained frm the analyi f SAP000(applicatin 3) D/B Renant Frequency (rad/ec) Max. diplacement (mm) The ame fundatin ha een analyed fr different edment rati (D/B) and the reult fr the diplacement-time utput are hwn in Fig.(8). It i evident that when the depth rati increae the vertical diplacement decreae. A cnvergence in reult i viu when the depth rati will e aut Thi mean that the reductin in dynamic diplacement will e le prnunced when the depth rati i t e increaed higher than D/B= D/B=0.5 D/B=0.38 vertical diplacement (mm) D/B=0.5 D/B= D/B= time (ec) Fig.(8) effect f edment n vertical repne (applicatin 3)

14 T.. Al-A awi, Raad. Al-Aawi and Zuhair. Al-Jaeri Al the increae in edment depth lead t an increae in the renant frequency f machine fundatin, Tale (4) hw thi effect. The reult hw that increaing the edment depth rati (D/B) t increae the renant frequency y %. EFFECTS OF USING A SQUARE FOUNDATION ON THE DYNAMIC RESPONSE It i a matter f interet t tudy the effect f uing a quare fundatin (L=6.8 m) and (B=6.8 m), i.e. B/L=1.0 intead f the rectangular fundatin which ha een tudied in the previu ectin (B/L=0.50). The dimenin f thi fundatin are aed n equal fundatin weight and il preure a cmpared t the cae f the rectangular fundatin. Fig. (9) hw the vertical diplacement-time relatinhip fr different depth rati (D/B) fr the quare fundatin cae. Tale (5) give the rati f diplacement amplitude fr the quare and rectangular fundatin fr different depth rati. The reult indicate a reductin in the dynamic diplacement in a range f (15%-17%) a cmpared t the f rectangular fundatin D/B= D/B=0.38 vertical diplacement(mm) D/B=0.54 D/B= D/B= time (ec) Fig. (9) effect f edment n vertical diplacement fr different depth Rati fr a quare fundatin

15 Numer Vlume 1 June 006 Jurnal f Engineering Tale (5) vertical diplacement amplitude ( ) Depth Rati (D/B) Full edded Rectangular (B/L)= Square (B/L)= S/R * *S=Vertical diplacement amplitude fr a quare fundatin. R=Vertical diplacement amplitude fr a rectangular fundatin CONCUSIONS The effect f edment upn vertical frced viratin f a rigid fting wa invetigated theretically. The cncluin can e ummaried a fllw: 1- The ue f equivalent circle apprach t etimate the dynamic tiffne and damping factr can caue errr a the apect rati f the fundatin (L/B )and the il Pin' rati (υ ) eing increaed. The errr will generally e increaed at higher frequencie. - Emedment f fundatin ha a ignificant effect n the dynamic repne. It caue an increae in the dynamic tiffne and damping cefficient and lead t increae the renant frequency and t decreae the dynamic repne f fundatin. A cnvergence in reult i viu when the depth rati will e aut 0.50.Thi mean that the reductin in dynamic diplacement will e le prnunced when the depth rati i t e increaed higher than The dynamic diplacement in the vertical directin i maller fr the cae f quare fundatin a cmpared t the f rectangular fundatin fr the ame weight and cntact il preure. The reult indicate a reductin in the dynamic diplacement in a range f (15% - 17% )a cmpared t the f the rectangular fundatin. REFERENCES Barken D.D (196), Dynamic f ae and fundatin, Mc Graw-Hill Bk C. New Yurk, N.. (tranlated frm Ruin). Bwel. J. (1988), Fundatin Analyi and Deign, McGraw-Hall k cmpany Dmingue, J. and Ret, J.M (1978)"Dynamic tiffne f rectangular fundatin" Reearch Reprt R 78-0 Dep. f Civil Eng. M.I.T. Dgupta, S.P. and Ra,.N.S.N. (1978), Dynamic f Rectangular fting y finite element, Jurnal f the Getechnical Engineering Diviin ASCE, Vl.104, N.GT5. Gerge Gaeta, Ricard Dry and enneth H. Stke (1986), Dynamic Repne f aritrary haped fundatin: experimental verificatin, Jurnal f Getechnical Engineering Diviin ASCE vl.111, Vl.. Gaeta. G. and Reet. J.M (1979), Vertical viratin f machine fundatin,jurnal f Getechnical Engineering Diviin ASCE, Vl 105, N Gt 1,. PP

16 T.. Al-A awi, Raad. Al-Aawi and Zuhair. Al-Jaeri aldjian M.J. (1969), Dicuin f Deign prcedure fr Dynamically Laded Fundatin,y R.V. Whitman and F.E. Richard. Jr. Jurnal f il Mechanic and Fundatin Diviin ASCE. Vl.95, N.SM1, prc. Paper 634, Jan. Lymer J., and uhlemeyer R.L (1969 ),Finite Dynamic Mdel fr infinite Media,Jurnal f the Engineering Mechanic Diviin ASCE, Vl.95, N.EM4, prc. Paper 6719, Aug. PP Prakah, S. and Puri, V.( (1988), undatin fr Machine: Analyi and Deign,Wiley& Sn Ricard Dry and Gerge Gaeta (1986), Dynamic Repne f Aritrary haped fundatin" Jurnal f Getechnical Engineering Diviin ASCE Vl. 11 N.. Ricard Dry and Gerge Gaeta (1985), Vertical Repne f aritrary haped fundatin, Jurnal f Getechnical Engineering Diviin ASCE Vl.111, N.6. Richart, F, E, Hall. J.R and wd R.D.( 1970), Viratin f il and fundatin, Prentice-Hall. Inc. Englewd cliff N.J Stck,.H. and Richart, F.E. (1974), Dynamic Repne f edded machine fundatin, Jurnal f Getechnical Engneering Diviin ASCE Vl.100, N.GTA, NOTATIONS The fllwing yml are ued in thi paper: A = ae area f fundatin. A = ide area f fundatin. a = nrmalied frequency. B= emi-width f rectangle circumcried t ae urface. C= dynamic damping f il. C =cefficient f dynamic damping. D= trench depth. G=hear mdulu f il. I tre =trench factr. I wall =idewall factr. = tatic edment tiffne f il. ) dy = cefficient f dynamic edment tiffne f il fr trench effect nly. tre ) dy =cefficient f dynamic tiffne fr il fr trench effect nly. ur ) dy =cefficient f dynamic tiffne fr urface fundatin. L= Semi- length f rectangle circumcried t ae urface. S = vertical tatic tiffne parameter. V La = "Lymer' analg" velcity. V =velcity f hear wave. υ = Pin' rati. ρ = ma il denity. ω =circular frequency.