p 1705- behavior of compressor foundation

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Behavior of a Compressor FoundationPredictions and ObservationsShamsher Prakash

Professor in Civil Engineering, University of Missouri-Rolla, Rolla, Missouri, U.S.A.Vijay K. Puri

Assistant Professor, Polytechnic Institute of New York, New York, U.S.A.

SYNOPSIS In a Compressor foundation undergoing excessive vibrations, i ts amplitudes at operating frequency and natu-ra l frequency in free vibrations were monitored. Also in-situ dynamic properties were determined to check design andpredict i ts response. Since the soil constants are strain depondent, two sets of computations were done (1) from thek n ~ w n soil constants and permissible amplitudes and (2) from the known soil constants and the observed amplitudes. Theso1l constants were corrected for confining pressure and relative density of the non-cohesive soil also.Both weightless spring theory (Barkans' Method) and elastic half space theory were used in predicting theresponse. A critical evaluation of these two design approaches has been made and necessity to monitor the performanceof machine foundations is highlighted.INTRODUCTIONA r.eciprocating compressor foundation was vibrating excessively. Its performance was monitored and in-situ soilproperties were determined to check it s design and computeits response.Figure 1 shows a dimensional plan and section of the foundation. The pertinent machine and foundation data are asfollows:Operating speedWeight of compressor and motorHorizontal unbalanced forceVertical unbalanced force PHorizontal moment zVertical moment Mx yyzPermissible vibration amplitude(peak to peak)Area of the foundationWeight of the foundationDepth of foundation

405 RPM11 Ot00.205 t0.185 t-m2.2 t-m

0.025 mmA=7.103m2w= 49.79 t= 2.4 m

Subsequent to the monitoring of the foundation, performance, (Arya et al 1978) and :in-situ dynamic properties determination, the design of foundationby(l)the Barkan'sapproach and (2) elastic half-space approach for 2-caseshave been discussed in this paper; one as for usual designstage as if the monitored performance of the machine isnot known; two, after knowing the monitored performance.The two sets of computation are similar except thatstrains in the soil in two cases are different which af fect the relevant soil properties considerably.The computations by two methods and their comparison withthe monitored performance throw light on the applicability of one method to the analysis of such problems betterthan the other. Remedial measuses are described elsewhere (Arya et a 1 1978)OBSERVATIONS ON THE FOUNDATIONSPmpl itudes Vertical and horizontal amplitudes of vibra-tion were measured at a number of points on the foundation,with the pertinent data as follows:Maximum amplitude of horizontal vibrations at top of block

in Y-direction 0.3156 and in z-direction = 0.1085 mm.Foundation was excited in free vibrations along x-drictionsand the natural frequency of free vibrations was observedto be 17.5Hz. (Fig. 2)In-situ Dynamic Properties The dynamic properties of thesoil used in the analysis of machine foundation may be determined by a number of laboratory or in-situ tests.The most important parameters which affect these propertiesare (1) the mean effective confining pressure (2) the shearstrain amplitude and (3) density in the soil. A good discussion on these corrections has been presented by(Nandakumaran and Puri (1977), Prakash and Puri (1977),Nandakumaran et al (1977), Prakash and Puri (1981) andPrakash (1981) and Indian Standard Code (IS 5249- 1977).In-situ Soil investigations consisted of (1) block vibration tests, (2 ) cyclic plate load tests and (3) standardpenetration tests (Prakash et al 1975). FigureS showsa typical borelog of the area. From the cyclic plateload test data, values of dynamic shear modulus "G" werecomputed. From the uncorrected standard penetration (N)values shear wave velocity "Vs" at a particular depth wasdetermined from equation 1 Ima (1977) and dynamic shearmodulus "G" was computed from equation (2)

0.337Vs 91.0 N . . . . (1)2G Vsxp . . . . . . . . . . . . . . . . . (2)in which p=Y =mass density of soil. Values of "G" fromdifferent ~ t e s t s were corrected for (1) effective confinement in each ca2e computed fo r an effective overburdenpl"essure of 1 kg/em using equation 3.

(3 )={ :vl \"5

\crv2)in which G1 = shear modulus at an effective overburdenpressure of crvland G2 shear modulus at an effective overburden pressureof crv2

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The variation of G with strains from these tests is shownin Fig. 4 curve A. Fig. 4 also shows a plot of_G_ vs. Ye obtained by dividing the ordinates ofGmaxG vs y 8 plot at 4.0 m depth at different strains by thevalue of curve B.Selec tion of Soil Parameters Since the foundation islocated at a depth of 2.4 m below ground level whereasthe earlier tests had been conducted at depth of 4.0 m,following procedure was adopted to determine the valuesof G at this depth using the data at 4.0 m depth-(1) Value of Gmax at 2.4 m depth was computed using equations 1 and 2 and the N-value observed at that depth.(2) The value of was corrected fo r effective overburden pressure and the value of G at an effective overburden pressure of 1 kg;cm2 computed using equation 3.The Values of G vs y8 (curve C) were obrained by multiplying G with ordinates of curve B (Fig. 4). This plotwas s ~ B % e q u e n t l y used to determine the values of "G" at2.4 m depth at the desired strain level for analysis ofthe foundation response.

The strain for two sets of computation (1) for designstage and (2) after monitoring the performance, and thecorresponding soil properties were picked up as follows:1. Design Stage.Permissible amplitude in any mode = 0.0125 mm

Average width of the foundation = 2104.5 mmShear strain (yA) (Prakash = ~ i ~ ! 2 = 5.94x1o- 5and Puri 1981) Gat y8 = 5.94 x 10-6 and av = lkg/cm 2is885 kg/cm 2(Fig . 4 curve C)Effective overburden pressure ov at a depth equal to

one half of the width of the foundation given by

0v= 0vl + 0v2 (4)in which ovl = Overburden due to weight of soiland Vertical stress intensity at a depthequal to 1/2 width due to superimposedload of machine and foundation and maybe computed using Boussinesq theory.

The value o in this case was computed to be 0.8381kg/cm 2. effective value of G = 885( 0 9381 )1121.0= 810 kg/ cm 2Value of the coefficient of elastic uniform compression"Cu" was computed from equations.

C 1 . 13 X 2G (1 + v) . 1 ( 5 )u (1 - v2) 11\in which v = Poissons ration (assumed 0.337).

The value of Cu is computed to be 10.25 kg/cm3

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(2) After monitoring performance.Measured amplitudein y directionand in z directionShear strain Ye induced inthe soil0.3155 mm0.1089 mm0.3156 + 0.1089

2104.5 -42.01 X 10_ 4Value of "G" corresponding to y 8 = 2.01 x 10(Fig. 4) and effective confinementbelow the foundation 415 (0.8381) 0.51400 kg/cm2

The corresponding value of Cu from eqn. 5 is computed tbe 5.130 kg/cm3.PREDICTED RESPONSE OF THE FOUNDATIONThe methods commonly used fo r the analysis and design ofoundations for machines are (1) Barkan's approach and(2) Elastic half space approach. In the Barkan's appro(Barkan, 1952) the foundation soil system is representeas a springmass system, the spring stiffness due to thesoil and mass of the foundation and supported equipmentonly are considered and intertia of the soil and dampinare neglected. In the elastic half space approach thevibrating footing is treated as resting on the surfaceof an elastic, semi-infinite, homogenous, istropic halfspace (Richart 1952). The elasticity of the soil and tenergy carried into the half space by waves travellingaway from the vibrating footing (geometric damping) arethus accounted fo r and the response of such a system rnabe predic ted using a mass-spring-dashpot model (Richartand Whitman 1967a Richart, Hall and Woods (1970).The dynamic response of the foundation was computed usiboth the above methods of analysis.Barkan's Method(a) Vertical vibrations: -Natural frequency of verticavibrations wnz is given by

w = rc;;:A ~ znz j -7-m-- = I_z_m

(6)

and amplitude of vertical vibration Az is given byAz = Pz (7)m(inz - w9

in which m =mass of the foundation and Kz = stiffnessvertical soil spring.w = operating frequency. The computated values of natufrequency of vertical vibrations and amplitudes of vibrtions are listed in Table 1 line 1 and 7 respectively.(b) Simultaenous rocking and sliding: Limiting naturalfrequency of the foundation in sliding wnx is givenby

= A (8)C-r = Coefficient of elastic uniform shear = 1/2 Cuand limiting natural frequency in rocking is givenby J Imo = (9 )


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in whichC ~ = C o e f f i c i e n t of elastic non-uniform compression = 2.CuI =moment of inertia of the foundation contact areaabout the axis of rotation and~ = Mass moment of inertia of the machine and foundation about the area of rotation.

(10)in which Mm =Mass moment of inertia of the system aboutan axis through i ts centre of gravity for the appropriatedirection of vibration. The two natural frequencies ofthe system wnl and w 2 due to combined rocking and sliding are obta1ned inn tenns of wnx and using equation 11.

in wherer = Mm/Mmo

(11)r

The amplitudes of vibration due to combined rockingand sliding due to an exciting moment are given by equations 12 and 13 Horizontal displacement.Xtv


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Ox= Damping ratio in slidingand 04> = Damping ratio in rocking

(30)

Where Bx =modified mass ratio = 7 - 8\l m32n::v-r ---pror(31)

04> = 0.15( 1+B > ),r-s; (32)

in W1ich = inertia ration= 3(1-\l) . I8 pros-Total horizontal and vertical displacements may then beobtained using equations (15) and (16) respectively.Amplitude in yawing may then similarly be computed(line 12 Table 1)FREE VIBRATIONSThe values natural frequencies in x-direction for a sheerstrain of 1x1o6 (free vibration condition) are asfallow;:Barkan's methodfn2 (Hz) 14.75fnl (Hz) 36.77

Elastic Half Space13.0158.07

DISCUSSION AND CONCLUSIONS

Observed

(17. 5)

1. The computed amplitude Ax* (line 10, cols. 3 and 4fo r case I (y = 5.94 x 10-6) is 0.1998 mm and 0.1091mm r e s p e c t i v e ~ y ty Barkan's method and elastic halfspace approach against permissible amplitude of 0.0125 mm.Therefore, the foundation design needed revision. Thishighlights the necessity of proper design using realistic soil parameters in ensuring satrsfactory performanceof machine foundations.2. The computed values of l o ~ ~ e r natural frequency"fn2" in combined rocking and sliding (Line 5, cols.5 &6) fo r case II (y8 = 2.01 x 10" 4 ) are 5.93 and7.03 Hz respectively, (frequency ratios of 0.88 and 1.09)respectively by Barkan's and elastic half space approach.These are too close to the operating frequency of 6.75.Hence large amplitudes should be anticipated, whichactually have been observed to occur.3. Vertical natural frequencies (line 1, cols. 3 &4and 5 &6) show a remarkable agreement with each other.Hoi'.E!ver, the natural frequencies in sliding and rocking(lines 4 & 5) differ from 19% to 31% from each other.The lowest natural frequency in horizontal free vibrations (y8 = 10-6) as computed J:y the Barkan's and elast ic half space approach is 14.75 and 13.01 Hz. The percent error with respect to the smaller natural frequency is 16% and 26% respectively. Thus th e computed na tural frequency by Barkan's approach is closer to theobserved frequency and the error by the elastic halfspace approach is large. No other published data is available on actual phototypes fo r comparison.

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4. Amplitudes of vibration in the vertical directionA; by Barkan's and elastic half space approach are 0.135mm and 0.3031 respectively against the measured value of0.1089 mm. Similarly in the case of horizontal vibrationsth e values of A* are 0.4542 and 0.785 respectively againstth e measured v a ~ u e of 0.3156 mm.The amplitudes computed using elastic half space modeltake into account the geometric damping. Even then thesear e higher than th e undamped amplitudes by Barkan's approach and a 1so much higher than th e observed amp 1 tudes.The observed amplitudes represent the overall effect ofgeometrical as well as material damping. Translationalmodes have a much higher geometrical damping associatedwith them compared to material damping b.Jt in rotationalmodes material damping may be significant since geometricdamping is usually small. Therefore predicted amplitudesusing half space model with geometric damping alone maybe expected to be somewhat higher than ol:served amplitudesIn the present case the difference is much larger than whamay be explained by inclusion of material damping in thesystem also. I t was ol:served earlier (Richart and Whitman(1967 b) , that th e half space approach generally do no tagree with observed amplitudes and may be higher or lowerthan observed amplitudes and in some cases th e differencemay be as large as 100%. However, in thi's case, Barkan'sapproach predicts th e amplitudes W1ich ar e reasonablycloser to observed amplitudes as compared with elastichalf space approach. However a single set of data doesno t warrant a general conclusion.5. There is an urgent necessity to monitor data on performance of machine foundations so that i t may be possible to establish conclusively the superiority of oneapproach over the other in design of machine foundations.Such a data will be meaningful only if sufficient information on dynamic soil properties is also obtained.REFERENCESArya, A.S., Puri, V.K. and Gupta, M.K. (1978), "Evaluation of the performance of a compressor foundation".University of Roorkee, Roorkee.Barkan, D.O. (1962), "Dynamics of bases and foundations".McGraw Hill Book Company.Imai, T. (1977), "Velocities of p- and s- waves in subsurface 1ayers of ground in Japan." Proc. IX Int. Conf.on Soil Mech and Foundation Engr, Vol 2, pp257-260IS: 5249-1977. Code of practice for dertermination ofin-situ dynamic properties of soi l . Inidan StandardsInstitute, New Delhi. First RevisionNandakumaran, P., Puri, V.K., Joshi, V.H. and Mukerjee,S. (1977), "Investigatons for evaluation of dynamic shearmodulus of soils". Bulletin, Indian Society of Earthquake Technology, Vol. 14, No. 2, June pp. 39-54.Prakash, Shamsher and Puri, V.K. (1977). Critical evaluation of IS 5249-1 %9, Indian standard code of practicefo r in-situ dynamic properties of soi l . Indian Geotechnical Journal, Vol. VII, No. 1, January, pp 43-56.Prakash, S. and Puri, V.K. (1981), "Dynamic propertiesof soils from in-situ tests." Paper submitted to ASCEGeotechnical Engr. Division fo r publication.Prakash S., and Puri, V.K. (1981), "Dynamic propertiesof soils from in-situ tests". Proc. ASCE, Journal ofGeotechnical Engr. Division, Vol. 107, No. GT7, July,pp. 943-964.


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rakash, S. and Puri , V.K., (1981), "Observed and Preicted Response of a Machine Foundation", Proc. X Intern.onf. on Sm & FE Stockholm, Vol. 3, pp. 209-273rakash, S. and Puri , V.K. (1985), "Analysis and Designf Machine Foundations" --under publicat ion.' rakash, Shamsher, Ranjan G. , and Sa:ran, S ., (1975). Finaleport on geotechnical investigations fo r Central foundaryorge project , Ranipur hardwar. Department of Civil Engr. 0. R. , Roorkee.

Richart , F.E., Hall, J.R. and Woods, R.D. (1970), "Vibrations of Soils and Foundations". Prentice Hall, Inc. NewYork, NY.Richart, F.E., and Whitman, R.V. (1967a), "Design proceduresfo r dynamically loaded foundations". Journal SMF. Proc.ASCE, Vol 93, SM6, Nov. pp. 168-193.Richart, F.E., and Whitman, R.V. (1967b). Comparison offooting vibrations with theory. Journal of SMF, Proc .ASCE Vol. 93 NOSM6 Nov. pp. 143-167.

TABLE I . Computed Natural Frequencies and AnplitudesQuantity Ye = 5.94 X 10- 6 y6= 2.01 x 10 -4fS.No. (Hz)A(mm) Barkan's Elastic Half Barkan's last ic HalfMethod Space Method Method f>pace Method

1 2 3 4 5 6

1 fn z Hz 19.06 19.05 13.48 13.432 f Hnx z 13.74 17.28 9.53 12.183 fn Hz 9. 95 11.51 7.05 8.124 f nl Hz 26.96 33.26 19.07 23.655 fn 2 Hz 8.38 10.96 5. 93 7.056 f n Hz 16.28 24.30 11.53 17.12

7 Az mm 0.0032 0.0031 0.0075 0.006868 An mm 0,133 0,0071 0.340 0.5239 A rad 6.16 X 10 -5 3.47 X 10-5 1.049 X 10-4 2.42 X 10 -4

10 mm 0.1998 0.10 91 0.4552 0.78511 mm 0.0787 0. 06156 0.135 0.3031 I' I12 A'4J rad 2.78 X 10-61 1.5 X 10- 6 7 X 10 6 2.396 X 10 -4 !I

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L,OO

r-

z

100(a) Secion MM Fig. 2 Typica l Free Vibra t ion Reco:y Retain in Wa

I1129

OL ., 1700 - - - - - - -1 2924Ml fs1s __,...-Ai r Gap

(b) PlanFig. 1 Foundation Details

(A) G - 4.0 m

0 0

24

6y.c: 8-'"'"'=>

10

12Fig. 3 Typical Borelog

LEGENDA Block Vibration Test0 Cyclic Plate Load TestmFrom N - Values

1 .0

0.8

0. 6

0. 4

0. 2

0o 1 L o - - - 6 ~ - - - - ~ ~ - 1 L O _ - s - - ~ - - ~ ~ ~ l ~ 0 - _ 7 4 ~ - - - - ~ - - ~ 1 o i - ~ 3 ~ ~ - - ~ ~ - 1 ~ o ~ - 2

Shear Strain y 8

Fig. 4 G vs Y8 and G vs Y8Gmax1710

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p 1705- behavior of compressor foundation

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