STEEL-E STEEL -EDUCA DUCATIONAL COUNCIL
TECHNICN TECH NICN.. INFORMATiON INFORMATiON & PRODUCT SERVICE
SEPTEMBER, 1991
Design Des ign Practice to Pre Pr eve vent nt Flo Floor or Vi Vibr brat atio ions ns
by Farzad Naeim
Desi De sign gn Practice to Prev Pr even entt Flo Floor or Vi Vibr brat atio ions ns
About the Auth Author: or: Farzad Nae Naeim, im, Ph.D., S.E., is Direct Director or of Re Rese sear arch ch and and De Deve velo lopm pmen entt fo forr the Stru Struct ctur ural al Engineering Engineeri ng firm of John A. Martin and Associates, Inc., Inc ., Los Angeles, California. Calif ornia. He has been in charge of design review and analysis of numerous complex projects across the United States for Vibra Vibratio tion n as wel welll as other ser servi vicea ceabil bilit ity y con concer cerns. ns. Dr. Nae Naeim im has reg regula ularl rly y lec lectur tured ed on variou var iouss aspe aspects cts of str structu uctural ral desi design gn and ear earthq thquak uake e eng engine ineeri ering ng at Univer University sity of Sou Southe thern rn California Calif ornia and California California Stat State e University, University, Nor Northr thridg idge. e. He is an active member member of sev severa erall profes pro fessio sional nal org organi anizat zation ionss and has more tha than n 30 pub public licati ations ons cov cover ering ing a wid wide e spec spectrum trum of structural and earthquake engineering applications.
INTRODUCTION The current trend trend towards lon longer ger spans spans and and lig light hter er floor sys system tems, s, com combin bined ed with reduc red uced ed damp damping ing and new acti activitie vities, s, such as aerobics aerobics exercise exercises, s, ha have ve resulted resulted in a signif sig nifica icant nt in incr crea ease se in the nu numbe mberr of flo floor or vibration com compla plaint ints s by bui buildi lding ng owners an and d occupa occ upants nts.. Thi This s ha has s in incr crea ease sed d the de degr gree ee of att attent ention ion paid du durin ring g the design design pr proc oces ess, s, to preventing preve nting,, or redu reducing cing flo or vibra vibration tion prob problems lems.. The purpose purpose of this pub public licati ation on is to pro provid vide e de desi sign gn en engi gine neer ers s with a pra practi ctical cal yet comprehe comp rehensive nsive review o f the crit criteria eria and and metho methods ds av avai aila labl ble e t o preve prevent nt floor vibra vibration tion problems. Becaus Beca use e of the comp complexi lexities ties invol involved ved in human respon response se to vibra vibration tion an and d the differe different nt objective obje ctives s pe pers rsue ued d by vario various us inves investigat tigators, ors, the pred predictio ictions ns of the metho methods ds pr pres esen ente ted d here he re are are not always con consis sisten tent. t. Unf Unfort ortuna unatel tely, y, a ge gene nera rall con consen sensus sus on the rel relati ative ve accura acc uracy cy and rel reliab iabili ility ty o f these these met method hods s do does es not yet exist. exist. How Howeve ever, r, it is ho hope ped d that collec col lectiv tive e review, ap appli plicat cation ion,, an and d com compar pariso ison n of these met metho hods ds will he help lp to form this seriously serio usly ne need eded ed cons consensu ensus s in the ne near ar future future.. Annoyi Anno ying ng floor vi vibr brat atio ions ns ma may y be ca caus used ed by occu occupant pant ac acti tivi viti ties es.. Wal Walkin king, g, da danc ncin ing, g, jumping jum ping,, aero aerobics, bics, and audience audience parti participa cipation tion at musi music c conc concerts erts an and d spor sporting ting events are some so me pri prime me examples examples o f occupant act activi ivitie ties s which cre create ate floor vib vibrat ration ions. s. Operation of mech Operation mechanica anicall equi equipment pment is anot another her ca caus use e for conce concern. rn. Heati Heating, ng, vent ventilati ilation, on, and air air-co -cond nditi ition oning ing sys system tems s (HV (HVAC AC)) as well as was washin hing g and dry drying ing mac machin hines, es, if not properly prop erly isola isolated, ted, ca can n ca caus use e serio serious us vibra vibration tion prob problems lems.. Most of the sources sources contributing to rep report orted ed hum human an discomfort res restt on the floor system system itself its elf.. However, hum human an act activi ivitie ties s or mac machin hinery ery of f a floor can cause cause sig signif nifica icant nt floor vibrat vib ration ions. s. On more than one occasio occasion, n, aer aerobi obics cs on on one e flo floor or of a hig high-r h-rise ise building building ha has s been be en reporte reported d t o ca caus use e vib vibrat ration ion discomfort at ano anothe therr le leve vell in the bui buildi lding. ng. Th The e vibrat vib ration ions s ca caus used ed by aut automo omobil biles es on par parkin king g le leve vels ls below ha have ve been reporte reported d to disrupt sensit sen sitive ive lab labora orator tory y work on upper floors. floors. Oth Other er equ equipm ipment ent and ac activ tiviti ities es off the floor that ca can n cont contribut ribute e to a floor vi vibr brat atio ion n pr prob oble lem m ar are e gr grou ound nd or air traffic, traffic, dr dril illi ling ng,, impa impact ct of fal fallin ling g obj object ects, s, an and d oth other er construct construction ion re rela late ted d eve events nts..
FACTORS INFLUENCING VIBRATION VIBRATION PER PERCEP CEPTIB TIBILIT ILITY Y Severa Seve rall factors inf influe luence nce the le leve vell of per percep ceptio tion n an and d the de degr gree ee o f sensitiv sensitivity ity of pe peop ople le to vibrat vib ration ions. s. Amo Among ng them ar are: e: (a)
.Position of the human body body..
Consider the human body coord Consider coordinat inate e syst system em de defin fined ed in Fi Figu gure re 1. He Here re,, the x-a x-axis xis defines defines the bac back-t k-to-c o-che hest st dir direct ection ion,, the y-axis y-axis def define ines s the rig right ht si side de to lef leftt si side de dir direct ection ion,, an and d the z-axis z-axis de defi fine nes s the foot -(o -(or-b r-but uttoc tocksks)to)t o-he head ad dir direct ection ion.. According to ISO9 O9,,•o, the fre freque quency ncy ra rang nge e of max maximu imum m sensitivity sensitiv ity to acc accele elerat ration ion for hum humans ans is bet betwee ween n 4 to 8 Hz for vib vibrat ration ion al alon ong g the z-a zaxi xis s an and d 0 to 2 Hz for vi vibr brat atio ion n al alon ong g the x- or y- ax axes es.. Whi While le z-axi z-axis s vi vibr brat atio ion n is most important in the de desi sign gn o f off office ices s an and d other wor workpl kplace aces, s, al alll three ax axes es become beco me i mportant in the de desig sign n of re resi side denc nces es an and d hotels where sle sleep epin ing g comfort should shou ld be cons considere idered. d.
Z
Y
x X surface Suppo•ng surface
X
Z Sup0orting surface
•X,, Qvo Q= •X
-- accele acceleration ration in the directions of the x-, .v-, .v-, z-exes z-exes
x-axiS
=
y-axis
= right side side to left side side
z-axis
= f oot-(or buttock buttocks-ito-h s-ito-head ead
Figure 1.
Y
Directions of basicen basicentric tric coord coordinate inate system systems s for vibrat vibrations ions influencing humans. TM 2
(bi
Excitation source characteristics such as amplitude, amplitude, frequ frequency ency content an and d duration.
(c)
Exoosure time. As sh own in Figure Figures s 2 and and 3, hu huma man n toler tolerance ance of vibra vibration tion decre de creas ases es in a char characte acterist ristic ic way wit h in incre creas asin ing g expo exposure sure time9.
(d)
Floor system characteristi¢• such as natural frequency frequency (stiff (stiffness; ness; ma mass ss), ), and damping.
(e)
Level of exoectancv. Th The e mo more re one expects expects vibra vibration tion an and d knows about its so sour urce ce the le less ss start startling ling the vibra vibration tion be beco come mes. s. Beca Because use peop people le expe expect ct mo more re vibra vibration tion in workshops workshop s tha than n in ho hote tell lo lobb bbie ies, s, they can put up with mo more re in the for former mer than than in the lat latter ter.. Anxiety and discomfort discomfort ca can n be redu reduce ced d if occ occupa upants nts are made made aware aware of the nat nature ure of vib vibrat ration ions s an and d ar are e as assu sure red d that they ar are e not a threat to the their ir saf safety ety and an d well be being ing..
(f)
Tvoe of act/v/tv engaged in. Th The e le leve vell of pe perce rcepti ption on var varie ies s with the na natu ture re o f activity that on one e is engag engaged ed in such as office work, din dinnin ning, g, wal walkin king, g, or da danc ncin ing. g.
CATEGORIES OF HUM UMAN AN RESPONSE
ISO9 IS O9 class classifi ifies es hum human an res respon ponse se to vibrations into three categories: (a)
limi li mitt beyond wh which ich the the co comfo mfort rt is red reduced (" ("re redu duce ced d co comfo mfort rt bo boun unda dary ry") ")
(b)
limit be limit beyo yond nd which th the e work working ing ef effi fici cien ency cy is imp impai aire red d (" fa fati tigu guee-de decr cre eas ased ed proficien prof iciency cy boun boundary" dary"))
(c)) (c
limit lim it be beyo yond nd which the hea ealt lth h or sa safe fety ty is enda endang nger ered ed (" ("ex expo posu sure re limit") limit")
These Thes e cate categorie gories s were de deriv rived ed from vario various us studies conducted for trans transporta portation tion industrie indu stries s an and d gene generally rally reflect a much highe higherr le leve vell of tole tolerance rance than w hat would be acceptable accep table in a bu build ildin ing g envi environme ronment. nt. Acco According rding to IS ISO O 26 2631 31-2 -2•o •o:: "Experience "Experie nce has has shown in many countries that com complai plaints nts rega regardin rding g build building ing vibratio vibr ations ns in resi resident dential ial situations are likely to arise from occu occupant pants s of buildings buildings when whe n the vib vibrat ration ion ma magni gnitud tudes es are only slightly in in ex exces cess s of perceptio perception n levels. levels. In general, gene ral, the satisfactory magn magnitud itudes es ar are e related to the minimum adverse adverse comment level by the occ occupa upants nts and are not de deter termi mined ned by any other factors, suc such h as short- short- term te rm health health ha haza zard rd and working efficiency. efficiency. Indeed, in practically practically all cases cases the the magni ma gnitud tudes es ar are e such that the there re is no possibi possibility lity of fat fatigu igue e or other vib vibra ratio tion- n- induced symptoms." Murray's •3 cate categoriz gorizatio ation n of hu huma man n re resp spon onse se is more de desig sign n orien oriented ted an and d he henc nce e mo more re usefu use ful. l. He defin defines es four re resp spon onse se categories, categories, among which the firs t t wo ar are e ac acce cept ptab able le as far as de desi sign gn is concerned: concerned: (a) Vibra Vibration tion,, thoug though h pre prese sent nt,, is not pe perc rcei eive ved d by the occu occupants pants.. (b)) Vibra (b Vibration tion is pe perc rcei eive ved d but does not anno annoy. y. (c)) Vibra (c Vibration tion annoys an and d dist disturbs. urbs. (d)) Vibra (d Vibration tion is so severe severe that ma make kes s occup occupants ants il ill. l.
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ISO IS O INT INTERN ERNAT ATION IONAL AL STANDARD 2631 PRO PROVI VISIO SIONS NS ISO 2631-2 TM pro ISO provid vides es a num numbe berr of hu huma man n per percep ceptib tibili ility ty ba base se curves curves for flo floor or velocity and acc accele elerat ration ion.. According to ISO, at vibratio vibration n mag magnit nitude udes s below the base base cu curve rves, s, adve ad vers rse e comments, sen sensat sation ions, s, or co compl mplain aints ts ar are e very ra rare re.. The They y not note e how howeve ever, r, that this thi s do does es not me mean an tha t the val values ues ab abov ove e the base curve curves s will gi give ve ri rise se to ad adve vers rse e comments comm ents or dissa dissatisfa tisfaction ction.. Si Sinc nce e the magn magnitude itude which is consid considered ered to be satisfactory satisfactory depe de pend nds s on the cir circum cumsta stance nces, s, ISO suggest suggests s spe specif cifyin ying g satisfactory vibrati vibration on le leve vels ls in terms te rms of mul multip tiple les s of the these se base curves curves.. Base curve curves s for foo foot-t t-to-h o-head ead,, ba backck-toto-che chest, st, and an d side side-to-s -to-side ide acce accelera lerations tions ar are e shown in Fi Figu gure res s 4 and 5. In terms terms of hum human an re resp spon onse se,, ISO divides divides vib vibrat ration ions s into tw o cl clas asse ses: s: (a)transient (also called impulsive) and (b)continuous or intermittent. Trans Transient ient vibra vibration tion is define defined d as a rapi ra pid d bui buildld-up up t o a pea peak k followed by a da damp mped ed dec decay, ay, suc such h as vib vibrat ration ion ca caus used ed by the impact of a single single heavy object on a flo floor or sys syste tem. m. It can can al also so consist consist of se seve vera rall cyc cycles les of vibrat vib ration ion at approximately the sa same me amp amplit litude ude,, pro provid viding ing that the dur durati ation on is sho short rt ( less less than tha n abo about ut 2 sec second onds). s). Continuou Contin uous s vibration on the oth other er ha hand nd is vib vibrat ration ion which re rema main ins s uni uninte nterru rrupte pted d ove overr the time pe peri riod od und under er con consid sidera eratio tion. n. Intermit Intermittent tent vib vibrat ration ion is defined defined as a str string ing of vib vibrat ration ion incide inc idents nts,, ea each ch of short dur durati ation, on, se sepa para rate ted d by int interv ervals als of muc much h lower vib vibrat ration ion magnit mag nitude ude (fo (forr exa exampl mple e vib vibrat ration ion ca caus used ed by a gro group up of pe peop ople le walking or ele elevat vators ors operating). In an an appendix appendix to ISO 2631-2, a set of sta state te of the art mul multip tiplic licat ation ion fac factor tors s fre freque quentl ntly y used us ed with the ISO ba base se curves curves are pr pres esen ente ted. d. Th Thes ese e fac factor tors s which lead to magnitud magnitudes es of vib vibrat ration ion below which the pro proba babil bility ity of rea reacti ction on is Iow are su summ mmar ariz ized ed in Table Table 1. In many sit situat uation ions s the same same bui buildi lding ng sp spac ace, e, re resi side denc nces es an and d hot hotel el gue guest st roo rooms, ms, for exampl exa mple, e, may be us used ed in both sta standi nding ng an and d lyi lying ng pos positi itions ons.. Fo Forr these cases cases,, ISO 2631-2 sugges sug gests ts us usin ing g a com combin bined ed sta standa ndard rd that re repr pres esen ents ts the w orst ca case se com combin binati ation on of zaxis axi s an and d x/y ax axes es con condit dition ions. s. The combine combined d sta standa ndard rd cur curves ves for acc accele elerat ration ion re resp spon onse se are ar e pr pres esen ente ted d in Figur Figure e 6. Not Notice ice that the mul multip tiplic licat ation ion fac factor tors s in Tabl Table e I ha have ve alrea already dy been be en ap appl plied ied t o these curv curves. es. COMPUTI COMP UTING NG FLOOR SYSTEM SYSTEM CHAR CHARACTE ACTERIST RISTICS ICS Unless Unle ss otherwise not noted ed the following ass assump umptio tions ns are used in this this pub public licati ation on for calculatin calcu lating g floor system vibr ation char characte acteristi ristics: cs: (1)) (1
Full compo Full composi site te ac acti tion on is as assu sume med d to exist betwe between en th the e co conc ncre rete te slab slab and and st stee eell beam be am re rega gard rdle less ss of the numb number er of sh shea earr studs prese present• nt•2 2.
(2)
The Th e beam is modele led d as a single degre ree e of fr free eedo dom m (SDOF (SDOF)) syste system. m.
(3)
The tr The tran ansf sfor orme med d mom momen entt of in iner erti tia a (It) is cal calcu cula late ted d usin ing g Mu Murra rray' y's s assu as sump mpti tion ons• s• 2,• 3. 3.14 14..
As pointe pointed d out by All Allen3 en3..4.s, it is better better to cal calcul culate ate the first na natur tural al fre freque quency ncy,, f, on defle deflection: ction: [1]
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TABLE 1 --- Ran Range ges s of mul multip tiplyi lying ng fac factor tors s us used ed in se sever veral al countr countries ies to spe specif cify y sa satis tisfac factor tory y magnitudes magnitu des of bui buildi lding ng vib vibrat ration ions s with respect to human res respon ponse{ se{1 1) (from (fr om ISO 26 2631 31-2 -2:: 19 1989 89))
PLACE Critical Critic al Work Working ing Areas (forr exam (fo example ple some hospital operatingth eatr es, some precision la laboratories)
TIME
Co n ti nu o u s o r I nt nt e r m ittent V i brati on
T r an si en t V ib ib r ati o n (Excitation w i t h several occurrences occurre nces per day day))
1
1 (2,3)
2 to 4
30 to 90 (4,5,6,7!
D ay Ni ght Day
Residential Nig ht
1.4
(4!
1.4 to 20
Day Da y Office
Workshop(9)
Night Day Ni ght
4 (a)
60 t o 128 (s)
8 (8.1o)
90 t o 128 (8.1°)
1) Tab Table le leads to magmtudas of vibration below below whG whGch ch the probability probability of rea reacti ction on s Iow. (Any acoustic acoustic noes noese e caused by vibrating walls is not consid considered.) ered.)
2) Also includes quasi quasi-statio -stationary nary vibra vibrations tions caused by repet repetitive itive shocks. Shoc k is definecl in ISO 2041: 1975. 1975. clause 3, 3, and is sometimes sometimes refer referred red to as transient transient (imp (impulsiv ulsive) e) vibra vibration. tion. 3) Magnit Magnitudes udes of transie transient nt vibration i n hospital hospital opera operating-t ting-theatre heatres s and critica criticall working place places s pert pertam amn n to periods periods of t,me when ooeratqon ooeratqona a are in progress or critical wor k is being performed. At other times, magni magnitudes tudes as h•g h•gh IS thoGe for reside residence nce are satisfactory prov provided ided that there is due agreement agreement and warning. 4) With in resid residentia entiall areas there there are wide varia variations tions in vibration tolera tolerance. nce. Speci fic values are are dependent dependent upon social and cultural cultural factors, psycho psychologic logical al attitude attitudes s and expe expected cted interf interferenc erence e w ith privac privacy. y. 5) The "tra de-of f" between number of events events per day and and mag magnit nitude udes s is not well establis established hed.. The following prowsiona prows ionall relat relationsh ionship ip shall be used for cases of mo re than three events events a day pending fur ther resea research rch into human vibration vibration tolerance tolerance.. Thi This s inv invol olves ves further multiplying by a number factor Fn = 1,7 N -0 5 where where .•/i .•/is s the number of events per day. day. This "tr ade -of f" equa equation tion does not apply when values are are lower than those those given by the factors for continuou s vibration. vibration. When the range of event me gni tud # is small small (wnthin (wnthin I haft amplitude of the blrgest), blrges t), the arith arithmetic metic mean can be used. OthenA OthenAtie tiee e only the largest need be consid considered. ered. 6) For di discre screte te events events wit h durations exceedi exceeding ng 1 s, the factors can be adjusted by further multiplying by a durati dur ation on factor, Fd: Fd = T - 1.22fo r con concre crete te floo • and T is between 1 end 20 20 Fo ,, T-0.
for wood wooden en floo floors rs and and Tis betwee between n I en end d 60 60
where T is th e duration of the event, in seconds, seconds, and can be estimate estimated d fro m the 10 percentag percentage e ( -20 dB) point s of the motion time his histori tories. es. 7) In hard rock exc excava avatio tion, n, where underground underground dis distur turban bances ces cause higher frequency frequency vib vibrat ration ion,, a factor of up to 128 has has been found t o be satisfac satisfactory tory f or reside residential ntial properties properties in some countri countries. es. 8) The mag magnit nitude udes s for tran transie sient nt vibration in offices end workshop area areas s should not be increas increased ed without con con-sideri sid ering ng the Ix•sibility of significant disruption of working activity. activity. 9) Vibration acting o n operators of certa certamn mn process processes, es, such as drop forges or crushers whi ch vibrate working places,, may be in a separate category fro m the wor kshop areas comddered places comddered here. here. Vib ration magni magnitudes tudes,, for the operators opera tors of the exciting processes, w hich are speci specified fied in ISO 2631-1, 2631-1, will then apply. 10)) Doubling the suggest 10 suggested ed vibration magnitudes for continuous or intermittent vibration and repe repeate ated d tra trannsiam vibration (fourth column) may result in adve adveme me • n tand a nd tt • may incre increase ase •gn'•,antfy if the lev levels els are quadruple quad rupled d (where avai availabl lable, e, do dose se/re /re•k •koo oonse nse curves can be consul consulted). ted).
7
where whe re A is the mi midd-sp span an def deflec lection tion of an equivale equivalent nt SDOF SDOF syste system m du due e to its own weight and g is the gra and gravita vitation tional al acc accele elerati ration on (386.4 in./ in./sec2 sec2). ). For a floor sys system, tem, & ma y be approximate appro ximated d by (A8 + AG AG)) [2]
=
1.3
+ &S
where whe re AB is def deflec lectio tion n of floor be beam am due due to flex flexure ure and sh shea ear, r,AG AG is the defl deflect ection ion of the gird gi rder er at the be beam am support support du due e to flex flexure ure and sh shea ear, r, an and d AS is the sho shorten rtening ing of the column colu mn or wall sup suppo port. rt. Th The e constant 1. 1.3 3 in the the ab abov ove e eq equa uati tion on ap appl plie ies s to bo both th si simp mply ly Supp Su ppor orte ted d an and d fixe fixed-e d-end nd be beam ams. s. For fixed-can fixed-cantile tilevers vers a va valu lue e of 1.5 should be us used ed.. In the cal calcul culati ation on of A, con continu tinuous ous be beam ams s on pi pin n supp supports orts should should be treated treated as sim simpl ply y support sup ported, ed, si sinc nce e vibr vibratio ation n no node des s exis existt at the supp supports orts.. If sh shea eari ring ng defo deforma rmatio tions ns ar are e neg neglig ligibl ible, e, t hen the tran transfor sformed med moment of ine inertia rtia of the floor flo or be beam am,, It, ma may y be used to est estima imate te it its s nat natura urall freq frequen uency: cy: [3]
.x /gEI /gEItt f = K • W L3
whe wh ere K = •
for sim simpl ply y su supp ppor orte ted d beams. Va Vallue ues s of K for var vario ious us end end condi conditi tion ons s are
read re adil ily y av avail ailab able le from tab tables les suc such h as those contained contained in Referen Reference ce [7]. It is the transforme transfo rmed d moment of ine inertia rtia of the com compos posite ite be beam am section, section, E is is the mod modulu ulus s of elastic ela sticity ity of st stee eell (29000 ks ksi) i) and and W is total weight sup support ported ed by the be beam am.. Us Usua uall lly ya sust su stai aine ned d por portion tion of the live load load (about 10% 10% to 25% of the total de desi sign gn li live ve load) load) is incl in clud uded ed in this weig weight ht es esti tima mate. te. Fi Fina nall lly, y, L is the span span length length of the bea eam m. For comp co mput utat atio ion n of It, the eff effec ectiv tive e sl slab ab depth depth (de de)) is assum assumed ed eq equa uall to the de dept pth h of a rectangular rectang ular slab slab ha havi ving ng the sa same me weight as the act actual ual sl slab ab,, inc includ luding ing the conc concrete rete in vall va lley eys s of the decking decking an and d the weight of the metal metal deck (see Fig Figur ure e 7) 7).. The effect of gi The gird rder er and and col column umn sup support port fle flexib xibili ilities ties on the first nat natura urall freq frequen uency cy of the system, syst em, ma may y al also so be app approx roxima imated ted by: [ 4]
1 f 2
-
I (f b) 2
+
1 1 • + ( f g) 2 (fs)2
where whe re fb, fg, an and d fs ar are e the na natu tura rall freq frequen uencies cies of the be beam am,, gi gird rder er,, an and d col column umn supp supports orts each ea ch compu computed ted indiv individuall idually. y. The re The read ader er sh shou ould ld not note e that floor sys systems tems are com comple plex x an and d ha have ve mul multip tiple le natu natural ral freq fr eque uenc ncie ies. s. Th The e ab abov ove e sim simpli plifie fied d pr proc oced edur ures es usu usuall ally y pr prov ovid ide e a good good est estima imate te of the first natu na tura rall fre frequ quen ency cy.. Ho Howe wever ver,, de depe pend ndin ing g on the activity of con concer cern, n, thi this s mig might ht or migh mightt not be the na natu tura rall freq frequen uency cy of gre greates atestt concern. concern. Fo Forr examp example le,, for most non non-rh -rhythm ythmic ic activiti acti vities es (i (i.e .e.. wa walki lking) ng) it is very unl unlike ikely ly that the col column umn supports supports will ha have ve a sig signifi nificant cant partici par ticipat pation ion in the re resp spon onse se.. Fo Forr these ca case ses, s, the nat natura urall freq frequen uencie cies s of gre great at inte interest rest are those those of the floor be beam am alone alone,, the gi gird rder er al alon one, e, an and d the combine combined d be beam am and and gird girder er system sys tem.. On the oth other er ha hand nd,, al alll three na natu tura rall freq frequen uencie cies s (i (i.e .e.. be beam am;; be beam am +girder; beam be am + girder-i-s girder-i-support) upport) shoul should d be con consid sidere ered d in design design f or rhythm rhythmic ic activities.
• . • See Tabl Table !
0.63
•c!• -•
I
0,4 0,25
.-
- 6 0 ' •
32
0,16
L
0,1
;
%.• 0 ,0 63
t
H16
0,04 -'• 0 , 0 2 5 0,016
2 1,
001 0,0063 0.004 0,002 5 0,0016 0.001
1,6
1
2.5
/+
6.3
10
16
i
t
25
40
63
10 0
Frequency •' centre frequency of one one-thi -third rd octave ban band, d, Hz
F ig igu re re 6 . C om om b in in ed ed d i r ec ec t i o n c ri ri te te ri ri a c ur ur ve ve s f o r v i br br a t io io n in b ui ui ld ld in in gs gs l o 1, _3L._
Spacing S
· ·
· ·
b
· ·
1 ·
·
e
L
e
..... I
Z]
Beam Spacing S ·
e
e
i*
·
·
l i, ·
I
T
ACTUAL
MODEL
, r e
:,
F ig ig ur ur e 7 . T ee ee -B -B ea ea m m o de de l f o r c o m p u titi n g t r an an s fo fo r me me d m o m e n t of i ne ne rt rt ia ia TM. 9
EXAMPLE EXAMP LE 16: 16: Est Estima imate te the na natu tura rall freque frequency ncy of the following floor be beam am.. The girder and and column colum n support motio motions ns are small small and can be ig igno nored red.. GIVEN:
BEAM: W21x44
, ,
S LA B :
2 in i n. m et etal de d ec k + 3
slab w e i g h t = 41 41 ps p sf f 'c ' c = 3 0 00 00 psi
light lig ht weight conc concret rete e fi
Conc Co ncre rete te weigh weightt = 115 pc SPAN -- 4 1' 1 ' - 0"
SPACING = 1 0' 0' -0 -0 "
LIVE LOADS:
Offi Of fice ce ........ .......... 50 ps psff Part Pa rtit itio ions ns ..... 20 psf Misc Mi sc.. -.... -......... ..... 10 psf
- - - > T ot otal L iv ive Lo Load = 8 0 ps ps f
SOLUTION:
Support motio Support motions ns are negligibl negligible e an and d the be beam am is not de deep ep.. Hence, Hence, the she sheari aring ng deforma def ormatio tions ns may be ignored ignored as well an and d we ca can n use the It for formul mula a to calc calcula ulate te f. de
S 1•4
i
vi
S / n i
525", .j.--W21 x44 As =13.0 =13.0in in 20.66"
Is =84 =843 3
.
d = 20.66in 20.66in..
BEAM MODEL
EC = ( W c ) l ' 5 • c = (115 p cf c f )l ) l .5 . 5 •• - 3k 3 k si si Es n - Ec
=
2 13 1 3 6k 6 k si si
41 ps f in actual sl actual slab ab weight 115 5 pcf (12•) (12•) d e -- conc concrete rete weight = 11
2 9 0 0 0 ks ksi 2 1 3 6 ksi - 13 1 3 .6
Dist Di stan ance ce from c.g. to sl slab ab top, Yt is calcul calculate ated d as as:: I 10'x12 ,,,, . 20 . 66 " (2)(' 1• 1•-.6 ) ( 4 . 3 ) " + ( 1 3 . 0 in in 2 ) t • + 5 .2 5 " ) Yt =
( 1 0 'x 1 2 ) 13 .6 (4.3") + 1 3 . 0 i n 2
= 5.6"
The transfo transformed rmed moment o f ine inerti rtia a is: __1_1 It = (
1 0 'x 1 2 ) ( 4 . 3 " ) 3 )( 1 3 . 6 )(
+ ( 1 0 ' x 1 2 ) ( 4 . 3 ) ( 5 64-. 3 " , 2 1 3.6 ' 2 I
,20.66" + ( 1 3 . 0 i n 2 ) (· • +
5.25"-5.6") 2 =
10
+ 8 4 3 in 4 +
2648in 4
=
4 .3 .3 "
Assuming Assumi ng that 10% of the des design ign live load acts acts as a sustain sustained ed load during during vibration, the partic par ticipa ipatin ting g weig ht is calculat calculated ed as as:: WDL = WS WSllab + WB WBe eam = (0.041 ks ksf) f)(4 (41' 1')( )(10 10') ') + (0 (0.0 .044 44 k/ft k/ft)(4 )(41') 1') = 18 18.6 .6 k WLL WL L = (0 (0.1 .10) 0)(0 (0.0 .080 80 ksf)(41') ksf)(41')(10' (10')) = 3.3 k W = WDL + WLL = 18.6 + 3 . 3 = 21.9 k Hence, Henc e, the nat natura urall frequency is: _ ! gE gEIt It
_
.
,
=
5 .3.z
EXAMPL EXAM PLE E 2: Fo Forr the typical int interi erior or beam shown bel below, ow, est estima imate te the first nat natura urall frequency by: (all usi (a using ng [3] and [4 ]; (b) using [1] assuming colu column mn sho shorte rtenin ning g is inconseq inconsequen uentia tial; l; (c) usin us ing g [1] as assu sumi ming ng column column sh shor orte teni ning ng o f AS= 0.5 0• inche inches s shoul should d be con consi side dere red. d. Assume the beam self-weight an Assume and d 10% o f the design live load are are inc includ luded ed in the 80 psf estimate of floor weight. ,
4 0 ' - 0 "
_ • j :
,
W21x50 (It =3533 in )
(
z
I•om under DO '
....
-
(• 3' MLr'ALZX• DCSIGH LIllE
•
W21x50 (It (It=3 =353 533 3 in4)
SOLUTION:
Since both beams and gird girder er are shall shallow, ow, shear shearing ing defor deformatio mations ns may be be ignor ignored. ed. (a)
For the be beam: JgEIt Forr the gird Fo girder: er: fg = K
W = (8 (80 0 ps psf) f)(1 (10' 0')( )(40 40') ') = 32, 32,000 000 lb lb = 32 ki kips
(2(2)) " V j13 j1386. 86.4)( 4)(290 29000) 00)(35 (3533) 33)
W = 2(32 k i p s )+ )+ ( 0. 0. 05 05 5 )( )( 30 30 ') ') = 6 5. 5. 65 65 k ip ips
(2) WL3
=
'•
/i380 4)(29000)(4485)
(65.6•2"•
=
6.36
Hz
* Cal Calcul culate ated d bas based ed on a total column height of 130 f t. and an average susta sustaine ined d axi axial al stress of 12 ksi. A
LG ( 1 3 0 ft f t ) ( 1 2 ) ( 1 2 ks k s i) = --"- = -- 0 . 6 4 i n . E 29000 The factor 1.30 is appl applica icable ble to A for frequency cal calcul culati ations ons sin since ce uni unifor form m mas mass s distribut distribution ion alo along ng the A 0.64 c ol ol um um n h ei ei gh gh t is a ss ssum ed ed: A s - 1.--3 - 1 .3 .3 - 0 . 5 0 in in . 11
1
Fr o m [ 4 ] ' (b)
f2
-
1
(5.2 5)2
+
1
(6.3 6)2
- - - >
f = 4. 4.05 05 Hz
The Th e beam deflection deflection at midspan is: is: 5wL4 5(32)(40 5(32 )(40x12) x12) 3 = AB = 38 384E 4Eit it 384(29000) (353 (3533) 3)
= 0 . 4 5 in.
The gi girde rderr defle deflection ction at the beam support support (1 (1/3r /3rd d poi point) nt) is is:: 5PL 3
5 ( 3 2 ) ( 3 0 x l 2) 3
AG - 162EIt
16 2( 29 0 00 ) ( 44 85 )
0 . 3 5 in in.
The Th e nat natura urall freque frequency ncy of the syste system m is then det determ ermin ined: ed: A =
f
lc)
-
I
•
(AB + AG AG) (0. 45 + 0.3 5) = 1. 3 1. 3 =
1
/386.4
=
0 . 6 2 in.
- 3. 97 Hz
Addin Add ing g co colu lumn mn shorteni shortening ng t o the nat natura urall fr frequ equenc ency y calcu calculatio lation: n: A =
0.62" + 0.50" = 1.12"
I • 1 38J--•.4 - 2 . 9 6 Hz Hz f - 2t 2to ' • = 2--• '• 1•2
FLOOR FLO OR VIBR VIBRATIO ATION N DU DUE E TO WALKING
To mod model el the im impul pulse se caused by a person walk walking, ing, a sta standa ndard rd heel drop impact hasbeen defin de fined2, ed2,•. •. Th This is is the im impu puls lse e in init itia iate ted d by a person weighing weighing 170 pou pounds nds who sup suppor ports ts his hi s weig weight ht on his his toe toes s with the hee heels ls rais raised ed about 2.5 in inche ches, s, an and d the then n sud sudden denly ly dro drops ps his wei weight ght through his heels heels to th the e fl floo oor. r. A plo plott of the re resu sult ltin ing g hee heell drop drop im impa pact ct and a typical typi cal floor response response to suc such h im impac pactt are shown in Fi Figu gures res 8 an and d 9, res respec pecti tive vely ly.. Several investigat Several investigators ors have sugg suggested ested metho methods ds to eval evaluate uate and and desi design gn for floor vibr vibratio ations ns caused by hee heell drop d rop impacts 2.8,• 1.13,14.17018. Among the them, m, Mur Murray' ray's s acc accepta eptabilit bility y criterion TM enj enjoy oys s the mos mostt wi wide de-sp -spre read ad use by structu structural ral des desig igner ners s in United United Sta States tes.. In this sec secti tion on,, si six x suc such h met metho hods ds are in intro troduc duced ed and applied to a sample sample floor vibrati vibration on design example. Murray's Accept Acceptability ability Cr CritE itErio rion n
Murray TM provi provides des a step step-by-by-step step proc procedure edure for eval evaluati uating ng potential floor vibr vibratio ation n problems in residential and office envi environm ronments. ents. Design tables tables have been publ publishe ished d which simplify appl applicat ication ion of this tech techniq nique ue 6. The The method is based based on field field measu measuremen rements ts and an d human respons response e st studi udies es per perfor formed med on app appro roxi ximat matel ely y 100 floor sy syst stems ems.. Fo Forr envir ironm onment ents, s, the use of the cr crit iter eria ia sug sugges gested ted by an ASCE Ad Ho Hoc c commercial env committee committ ee chaired by El Elli ling ngwo wood od [1986] and cov covere ered d la later ter in this pub publi licat catio ion n is recommended.
1 2
FIO FI O
I
J
I
I
%1
600
SUIIO Slit
d01 ii d
0
O ·
I
I
I
I
lO
!'0
310
40
SO
,lO
yI&I{. MI
Figu Fi gure re 8. Averag Average e pl plot ot of for force ce ver versus sus tim time e for heel heel imp impact4 act4
I I-
I
INITIAL INITIA L r i a lA C C l t l l A l l O t t
i-i--/'
!: /r
.•...•
O A & I P l N GI A I l O
t *
t!^t•
A
,
^
-
-
Z
o
f f
I
6
I
!
I . it
3 .O
ti&Il
Figu Fi gure re 9.
e.
S
Typi Ty pica call fl floo oorr response to hee heell imp impact act (High fre freque quenci ncies es fil ilte tere red d out out).4 ).4
TABLE 2 ----- S uggeste d ranges for av avail ailabl ablee fl floor oor sy syst steem damp damping ing
(after Murra Murray y 12.13.14) So ur c e
Da mpi ng
C omme nts
Bare Fl Floor
1 % - 3%
'Ceiling
1% - 3%
Me c ha n i c a l ....Systems Partitions
1% - 10%
Lower li l imit fo fo r th thin sl slab of of li l i ght w e i g ht concre con crete; te; up upper per li limi mitt for thi thick ck sl slab ab of regu re gular lar weight conc concrete rete Lowe r li limit fo fo r hu hung ce ceiling; up u pper li limit fo fo r sheetr she etrock ock on fur furrin ring g att attach ached ed to bea beams ms Depends on on am a m ou nt an and at a t t a c h m e nt
10% 20%
I f a t t a c h e d to to tth he fl floor a t th three po points o orr more a nd nd n ot ot s pa paced m or ore t ha han e v e r y fi fi v e floorr beam floo beams. s. 13
The Th e pr proc oced edur ure e for appl applying ying Murray's acce acceptabi ptability lity criterion criterion is as follow follows6 s6:: (1)
Esti Es tima mate te the to tota tall am amou ount nt of da dam mpi ping ng that wi will ll be av avai aila labl ble, e, Dav avaiI aiI.. Mur Murray ray's 's estimate estim ates s of av avai aila labl ble e da damp mpin ing g which ar are e ba base sed d on obs obser erva vati tion on onl only y ar are e shown in Tabl Ta ble e 2. If the total av avai aila labl ble e da damp mpin ing g is gr great eater er than 8 to 10%, the beam is is satisfacto satis factory ry an and d furth further er inves investigat tigation ion is not ne nece cess ssar ary. y.
(2)
Comput Comp ute e co comp mpos osit ite e se sect ctio ion n pr prop oper erti ties es an and d th the e fi firs rstt na natu turral fr freq eque uenc ncy y of the be beam am,, f. If f is gr grea eate terr than 10 Hz, the beam beam is satisfac satisfactor tory y re rega gard rdle less ss of the da damp mpin ing g provided.
(3)
Comp Co mput ute e th the e in init itiial maxim maximum um am ampl plit itud ude e of the beam beam,, Ao Aot, t, du due e to a st stan anda dard rd heeldrop impac impactt as as:: [5]
L3 A o t = (DLF)ma x x (8-•-•t )
where al alll uni units ts ar are e in ki kips ps and in inch ches es an and d (D (DLF LF)m )ma a x is the'dy namic lo load ad factor factor.. Value Val ues s of DLF for vari various ous na natu tura rall fr frequ equen enci cies es ar are e li list sted ed in Table Table 3. (4)
Account for the Account the st stif iffn fness ess co cont ntri ribu buti tion on of ad adja jace cent nt be beam ams s by estima estimati ting ng the tota totall effecti effe ctive ve num number ber of be beam ams, s, Ne Neff, ff, whe where: re: [6 ]
Neff=
2.97-0.0578Idle]
+ 2.56x10-8[L-•-tl
where wher e S is beam spacin spacing g an and d de is the effecti effective ve sl slab ab thickness thickness,, both in in inch ches es (s (see ee Figure Figu re 7). (5)
Divi Di vide de Aot by Neff to obtai obtain n a mod odiifi fied ed ini niti tial al maxim maximum um am ampl pliitu tude de,, Ao Ao,, wh whic ich h accounts accoun ts for the stiffne stiffness ss of adjac adjacent ent beams: beams: [ 7]
(6)
Esti Es tim mat ate e the req equi uirred level level of dampi damping ng,, Dr Dre eqd as: as: [8 ]
(7)
Ao -
Aot Neff
D r e q d = 3 5A o f + 2 . 5
Compare valu es es o f Davai I an and Dreqd: [9 ]
If
Dreqd---< Davail
If
Dreqd > D av av a i l
- -
- > Th The beam is s a t i s f a c t o r y
- - -> Redesign is is recommend ed ed
If the av avai aila labl ble e damping damping canno cannott be es esti tima mated ted,, Murray sug sugges gests ts the co comp mpar aris ison on summar sum marize ized d in Table Table 4.
14
TABL TA BLE E 3 ----- Dy Dyna namic mic Id Idad ad fac factor tors s for heel-dr heel-drop op Impa Impact. ct. 14
f, Hz 1.00 1.10 1.20 1.30 1.40 1,50 1.60 1.70 1.80 1.90 2.00 2. 10 2. 20 2. 30 2. 40 2. 50 2. 60 2. 70 2. 80 2. 90 3. 00 3.10 3.20 3.30 · 3.40 3.50 3.60 3.70 3.80 3.90 4.00 4.10 4.20 4.30 4.40 4.50 4.60 4.70 4.80 4.90 5.00 5.01 5.20 5.30 5.40
DLF
F, Hz
DLF
F, Hz
DLF
0.1541 0.1695 0.1847 0.2000 0.2152 0.2304 0.2456 0.2607 0.2758 0.2908 0.3058 0.3207 0.3356 0.3504 0.3651 0.3798 0.3945 0.4091 0.4236 0.4380 0.4524 0.4667 0.4809 0.4950 0.5091 0.5231 0.5369 0.5507 0.5645 0.5781 0.5916 0.6050 0.6184 0.6316 0.6448 0.6578 0.6707 0.6635 0.6962 0.7088 0.7213 0.7337 0.7459 0.7580 0.7700
5.50 5.60 5.70 5.80 5.90 6.00 6.10 6.20 6.30 6.40 6.50 6.60 6.70 6.80 6.90 7.00 7.10 7.20 7.30 7.40 7.50 7.60 7.70 7.80 7.90 8.00 8.10 8.20 8.30 8.40 8.50 8.60 8.70 8.80 8.90 9.00 9.10 9.20 9.30 9.40 9.50 9.60 9.70 9.80 9.90
0.7819 0.7937 0.8053 0.8168 0.8282 0.8394 0.8505 0.8615 0.8723 0.8830 0,8936 0.9040 0,9143 0.9244 0.9344 0,9443 0,9540 0,9635 0,9729 0,9821 0.9912 1,0002 1.0090 1.0176 1.0261 1.0345 1.0428 1.0509 1.0588 1.0667 1.0744 1.0820 1.0895 1.0969 1.1041 1.1113 1.1183 1.1252 1.1321 1.1388 1.1434 1.1519 1.1583 1.1647 1.1709
10.00 10.10 10.20 10.30 10.40 10.50 10.60 10.70 10.80 10.90 11.00 11.10 11.20 11.30 11.40 11.50 11.60 11.70 11.80 11.90 12.00 12.10 12.20 12.30 12.40 12.50 12.60 12.70 12.80 12.90 13.00 13.10 13.20 13.30 13.40 13.50 13.60 13.70 13.80 13.90 14.00 14.10 14.20 14.30 14.40
1.1770 1.1831 1.1891 1.1949 1.2007 1.2065 1.2121 1.2177 1.2231 1.2285 1.2339 1.2391 1.2443 1.2494 1.2545 1.2594 1.2643 1.2692 1.2740 1.2787 1.2834 1.2879 1.2925 1.2970 1.3014 1.3058 1.3101 1.3143 1.3185 1.3227 1.3268 1.3308 1.3348 1.3388 1.3427 1.3466 1.3504 1.3541 1.3579 1.3615 1.3652 1.3688 1.3723 1.3758 1.3793
15
TABLE 4 --- Required Required damping com parison chart (aft (a fter er Mu Murr rray ay 12.13,•4)
Computed Comp uted Require Required d Damping Damping Ra Range nge
Comments System will be sat System satisfa isfacto ctory ry ev even en if suppor sup porte ted d areas are are comple completely tely free of fixed parti partitions. tions. Desig Des igner ner must carefu carefully lly consider the the office environment enviro nment an and d the inten intended ded use use..
Dreq Dr eqd d <• 3.5%
3.5% <
D re q d < 4 . 2 %
Design Desi gner er mus mustt be ab able le to identify an exac exactt source sou rce of dampi damping ng or artif artificiall icially y pro provid vide e additi add itiona onall da dampi mping ng to be su sure re the flo floor or syst sy stem em will be sat satisf isfac acto tory. ry. If this this ca can n not be accompl accomplished, ished, red redesi esign gn is nec necess essary ary..
Dreqd > 4 . 2% 2%
EXAMPL EXAM PLE E 36: Use M urray's acc accept eptabi abilit lity y crit criteri erion on to inve investi stigat gate e the ade adequa quacy cy of the floor be beam am of Ex Exam ampl ple1 e1 for walk walking ing in indu duce ced d vi vibra brati tion on.. Th This is is a floor be beam am in an an offi office ce buildi bui lding ng whe where re the gird girder er and col column umn supp support ort mot motion ions s are small small and ca can n be ign ignore ored. d. SOLUTION:
(1 )
Estimat Est imate e avai availab lable le damping: (Flloo (F oorr at 1%) 1%) + (Ce Ceiiling at 1% 1%)) + (Me Mec cha han nica call at 3%) > D a v ai I = 5 % < 8 % > C ontinue t h e an analysis
(2)
Calcu Cal cula late te natura naturall freque frequency, ncy, f: f r o m EXAMPLE 1: 1 : f = 5 . 3 Hz <: 10. Hz Since Sin ce f is less than 10 Hz, the ana analy lysis sis pro proced cedure ure is continued.
(3)
Comput Comp ute e the ini initia tiall maximum amplitu amplitude de of the be beam am,, Aot: For f =
5 . 3 Hz Hz f r o m Ta T able 3
( D L F ) m ax
=
0.7580
L3 (41 (4 1 'xl 2)3 Aot -- (D (DLF LF)m )max ax x (8(8---0-•t ) = (0.7580) (0.7580) x I- 80 80(2 (290 9000 00)( )(26 2648 48))-]] = 0.015 in. (4)
Calcul Cal culat ate e Neff: Nef f = 2.97- 0 . 0 5 7 8
; (5)
oo
+ 2. 2 . 5 6 x 1 0 -8
[l4O.3x"l Jl
=
+ 2' 5 6 x1 0- 8 [
2'• §
Calculate Calculat e the modifie modified d initial initial maximum maximum amplitude, Ao Ao:: ao -
Aot N ef f
-
0 . 0 1 5 in in 1 .9 2
- 0.0078"
16
J = 1 .9 2
(6)
Estimat Est imate e Dreq Dreqd: d: Oreqd = 3 5A 5A o f + 2 . 5 = 3 5 ( 0 . 0 0 7 8 ) ( 5 . 3 ) + 2 . 5 - 3 .9 .9 %
(7)
Compa Com pare re the val values ues of Dav DavaiI aiI and and Dre Dreqd qd:: D req d
= 3 .9 .9 % <: Da D a v a iI = 5 . 0
- - - > The be beam is is s at atisfac to ry
Elling Elli ngwo wood od et et.. al Recomme Recommend ndat atio ions ns for C0 C0mm mmer er••ia iall En Envi viron ronmen ments ts As a part of a rep report ort issu is sued ed by an Ad Ho Hoc c ASCE committee committee on serv servicea iceabil bility ity re rese sear arch ch,, a vibration vibration crit criterio erion n for commercial commerc ial f loor systems systems,, for ex examp ample le in shopping centers, centers, was re reco comme mmend nded ed l. l.s. s. The criteri crit erion on is co cons nsid ider ered ed sat satisf isfied ied if the max maximu imum m def deflec lectio tion n for a 45 0 lb force ap appl plie ied d anywhe any where re on the floor do does es not excee exceed d 0.02 in inch ches es.. Bo Both th the Ca Cana nadi dian an St Stan anda dard rds s Associations Associa tions and and Murray •4 recom recommend mend that the natural frequ frequency ency o f commercial floor systems syst ems be kep keptt gr grea eate terr tha than n 8 Hz in order to min minimi imize ze the pos possib sibili ility ty of re reso sona nanc nce e du due e to walk walking ing..
EXAMPLE EXA MPLE 4: Det Deter ermin mine e if the floor syste system m of Ex Examp ample le 2(b) satisfies Ellingwood Ellingwood et. al. recommenda recomme ndatio tions ns as a part part of a shoppi shopping ng cen center ter floor syst system. em. As Assu sume me the num number ber of effec ef fectiv tive e te teee-be beam ams, s, Neff = 1. 1.96 96..
SOLUTION: Exam Ex amin ine e the max maximu imum m def deflec lectio tion n du due e to a 450 lb load load on the be beam am:: * . 4 4''girder 4'support max = " b e a m t 2 + 4 0.4 5 0L 3 1 Abeam = (. 44--•-•t )(N--•)
, 0.450(40'x12) 3) ( 1 , = t 48 48 x2 x2 90 90 00 00 x3 x3 53 53 3 1-•-•1 = 0 .0 .0 05 0 5 2 in .
0. 45 0L 3 ( 0 .4 . 4 50 50 (3 ( 3 0' 0 ' xl xl 2) 3 , 4.girde r = (' 4•-•t ) = 4 8x 8x 29 29 0 00 00 x 44 44 8 5J 5J = 4.ma x = 0 . 0 0 5 2 +
0 . 0 03 0 3 4 in .
0.0034 0.00 2 + 4 - 0 . 0 0 6 9 in. < 0. 0 . 0 2 0 i n. "
> O.K.
However, si However, sinc nce e the floor floor sys system tem natu natura rall fre freque quency ncy of 3.2 3.21 1 Hz is sig signif nifica icantl ntly y less than than Murray's su sugg gges este ted d va valu lue e of 8.0 Hz Hz,, redesign is i s recom recommende mended. d.
wi88-Parm wi88 -Parmelee elee Ra Rati ting ng Fact Factor or Cr Crite iterio rion n Wiss an and d Pa Parme rmele lee• e•s conduct conducted ed a laboratory stu dy t o investi investigate gate hu huma man n percept perception ion o f transi tra nsient ent flo floor or vibrations. vibrations. 40 volu voluntee nteers rs wer were e sub subjec jected ted to pla platfo tform rm mot motion ions s de desi sign gned ed to simulate simula te fl oor vibratio vibrations ns du due e to hee heel-d l-drop rop impact impact.. An empirical formula was dev develo eloped ped which re relat lated ed huma human n res respo ponse nse to the floor system' system's s maxi maximum mum displ displacemen acementt ampli amplitude tude A0, the firs t na natu tura rall fre freque quency ncy,, f, an and d av avai aila labl ble e dam dampi ping, ng, Da Dava vaiI, iI, such such that: [ 10 ]
R = 5 .0 8 [
fao
0.217]
0.265
(Davail) where whe re R is the me mean an re resp spon onse se rat rating ing,, int interp erpret reted ed as follows:
17
1] [11]
R
=
3
•
i m p er c e p t i b l e
/ b a r e ly p e r c e p t i b l e = / d i s t i n c t l y pe perceptible
strongly perceptible severe The Wiss-P Wiss-Parm armele elee e rating fact or was ad adop opted ted by the Un Unite ited d Sta States tes Depa Department rtment of Hous Ho usin ing g an and d Ur Urba ban n Dev Develo elopme pment nt as a criteria criteria for acc accepta eptabil bility ity of floor syst systems ems whe where re a limit of R<__ 2,5 was establ established, ished, The Wiss-P Wiss-Parm armel elee ee rating metho method, d, which is also also refe re ferr rred ed to as the GS GSA/ A/PB PBS S acc accept eptabi abilit lity y crit criteri erion on ha has s been crit critici icized zed for not be bein ing g suffic suf ficient ient ly sensitive tofl tofloor oor system damping 13,•7.
EXAMPL EXAM PLE E 5: Det Determ ermine ine i f the floor be beam am of Ex Exam ampl ple e 3 is acceptabl acceptable e acco accordi rding ng to the GSA/PBS GSA /PBS criterion. crit erion. SOLUTION: With f, A0, an and d Da Dava vaiI iI alrea already dy known from Ex Exam ampl ple e 3, we ca can n dir directl ectly y
proc pr ocee eed d with cal calcul culatio ation n of the Wi Wiss ss-P -Par arme mele lee e rati rating ng factor:
R = 5.08 [
fao
0.217 ]
0.265
(Davai I)
>
= 5. 08
[ ( 5 . 3 Hz Hz)(0.0078")•10.265 0.217 " ( 0. 05)
= 2. 59 > 2.50
Beam not accep accepta tabl ble e ac acco cord rdin ing g to GSA/P GSA/PBS BS prov provis isio ions ns..
Modifie Mod ified d Reih Reiher-M er-Meis eister ter Sca Scale le
As ear early ly as 1931, Rei Reiher her and and Mei Meister ster re repo port rted ed re resu sult lts s of the their ir inve investig stigatio ation n of hum human an percep per ceptibi tibility lity to stea steady dy stat state e vib vibrati ration on •s. The Their ir stud studies ies co cove vere red d a forc forcing ing freq frequen uency cy ra rang nge e of 3 to 100 Hz and and a di disp spla lace ceme ment nt am ampl plit itud ude e ra rang nge e of 0.0004 in inch ches es to 0.40 in inch ches es.. In the ear early ly 1960's, Le Lenz nzen en su sugg gges este ted d that if the Re Reih iher er-M -Mei eist ster er amp amplit litude ude sc scal ale e was mult mu ltip ipli lied ed by a factor of 10 10,, the re resu sulti lting ng sc scal ale e would be ap appl plic icab able le to ligh lightly tly dampe damped d floor sys systems tems (da (dampi mping ng less than 5% of crit critical ical). ). Th The e res resulti ulting ng sc scal ale e which corr correlat elates es huma hu man n per percep ceptib tibili ility ty with nat natura urall freq frequen uency cy an and d dis displa placem cement ent amp amplit litude ude,, is cal called led the Modifi Mod ified ed Re Reih iher er-M -Mei eist ster er Sc Scal ale e an and d is shown in Figur Figure e 10 10.. As a res result ult of st stud udie ies s conducte cond ucted d on num numero erous us be beam ams, s, Murr Murray ay in a 197 1975 5 pa pape perr 12 sugg suggeste ested d that "steel be "steel beam am-c -con oncr cret ete e s/ s/ab ab floors floors w/t w/th h 4% to 10 10% % cri critic tica/ a/ da damp mpin ing g whi which ch plo t above abo ve the upper one-ha/f of the dis tinctl y percepti perceptible ble range range w/I/resu lt in complaints compla ints from the occupants; and systems systems in the strong/) strong/)/percept /perceptible ible ran range ge will be una unacce ccepta ptable ble to both occupant occupants s and owners owners." ."
The Modif Modified ied Re Reihe iher-M r-Mei eiste sterr sca scale le is freque frequently ntly used by des design igners ers along wi th an additi add itiona onall met method hod (for example example Murray's acc accept eptabi abilit lity y crit criterio erion) n) to pas s judgemen judgementt on bord bo rder er-l -lin ine e situ situati ations. ons. Th The e ma main in cri critici ticism sm of this scale scale is its lac lack k o f exp explic licit it con consid sidera eration tion of da damp mpin ing, g, whic which h is consid considere ered d to be the the most important important factor in invol volved•. ved•.
18
EXAMPLE EXAM PLE 6: Us Use e the Modi Modifie fied d Reih Reiherer-Meis Meister ter scale to dete determin rmine e a vibration perceptibili perceptibility ty
level le vel for the floor beam of Exa Exampl mple e 3.
SOLUTION: With f - 5.3 Hz and and A0 = 0.0078 inc inches hes enter enter the Mod Modifi ified ed Re Reih iher er-M -Mei eist ster er chart of Fig Figure ure 10. Th The e beam pl plot ots s bel below ow the distinct distinctly ly perc perceptibl eptible e range and hence is acceptable.
Canadian Canad ian Standards Asso Associati ciation on Scale Sc ale (CSA (CSA))
Based Base d on the exten extensive sive rese research arch wo rk by All Allen en an and d Ra Rain iner er= = an an anno annoyanc yance e cri criter teria ia for floor vibrat vibrations ions in re resi siden denti tial al,, offic office e an and d sch schoo ooll roo room m env envir ironm onment ents s was ado adopte pted d by the Canadian Canad ian Stand Standards ards Association (CSA (CSA)) and was incl included uded as A ppendix G to CS CSA A Standa Sta ndard rd S16.1-1974 (Steel Structu Structure res s for Bui Build ldin ings gs -- Limi Limitt Sta States tes Design). Design). Th This is cri criter teria ia sets set s limi limits ts on peak acc accel elera erati tion on exp exper erien ienced ced by the floor syste system m in terms of its nat natur ural al frequency an and d avai availabl lable e damp damping ing (see Figure 11). For design For design purp purposes, oses, the peak acce acceler lerati ation, on, 3' 3' may be esti estimate mated d from the no w fami familia liarr maximum maxi mum dis displa placemen cementt ampl amplitu itude, de, A0, assu assuming ming a harmonic harmonic floor respo response nse at the floor's first na natur tural al frequen frequency: cy: 7 = (2tt (2tt f)2 (A0) The chart in Figure The Figure 11 11 co consi nsists sts of a base base curve curve for continuous vibration, an and d thr three ee limi limitt curv cu rves es for wal walki king ng vib vibrat ration ion,, for 3%, 6%, an and d 12% av avai aila labl ble e da damp mpin ing. g. A flo floor or sy syst stem em plotting plotti ng belo below w the cor corre respo spondi nding ng limi limitt cur curve ve is consider considered ed sati satisfacto sfactory. ry.
EXAMPLE EXAMP LE 7: Use the CS CSA A scale as devised devised by Al Alle len n an and d Reiner Reiner• to de deter termin mine e acceptabil accept ability ity of the floor beam in Example Example 3. 5.3 3 Hz and and A0 = 0.0078 inc nche hes, s, es esti tima mate te the peak acc accel eler erat atio ion: n: SOLUTION: For f = 5. y = (2• f ) 2 ( A 0 ) = (2 • x 5.3 )2 )2(0 .0 .00 78 78) = 8 .6 .6 4 i n/ n/sec 2 = 2. 2 % g Enter the chart o f Figure 11 with the Enter these se val values ues.. The The req requir uired ed da dampi mping ng sug sugges gested ted by the chart is less tha than n the 5% pro provid vided. ed. Hence, the beam is satisfactory. satisfactory.
Tolaymat's Tolaymat' s Cr Crite iterio rion n
Tolaymat •7 re Tolaymat revi viewe ewed d •esu •esult lts s of 96 com compo posi site te floor sys syste tems ms stu studie died d by Murra Murray y as a bas basis is for hi his s accep acceptabil tability ity crite criterion• rion•3 3, an and d sug sugges gested ted a new ra rati ting ng syste system m that is claimed claimed to provid pro vide e a better cor correl relati ation on betwe between en test res resul ults ts an and d rep report orted ed hum human an perce perceptibi ptibility lity le leve vels. ls. In contrast to most other met method hods s cov covere ered d in this this se secti ction on,, which are based based on study of a sing si ngle le heel dro drop p imp impac act, t, Tola Tolaymat ymat used a series of im impa pacts cts to si simul mulate ate excita excitation tion caused caused by walk walking ing hum humans ans.. According Accordi ng to this app approa roach, ch, a floor system is rated accepta acceptabl ble e if it sa sati tisf sfie ies s on one e of the following two con following condit dition ions: s: 19
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IQ
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100
F r e q u e n c y , CP S
F ig ig ur ur e 1 0 . Modif Modified ied Reiher Reiher*Meis *Meister ter perce perceptib ptibility ility chart chart.. 10g
I
I
I
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/ / /
20 -
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/ DAMPING IATIO 12 %J
31.
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/ /
0 tt S
0AMPING lAl lC
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2
IIAIlG 3% •
--------C!ITE TEIIIA FOI WALKING VIIIAT VIIIATION$ ION$ AS GIVEN IY #ILE LEI,, IMPACT TEST
" CIIIT II IIIIA PO POI CONTINUOUS VIIR VIIRATIO ATION N (10 (1 0 TG 30 CYCLIS)
1.0
AVE' AV E'AGE 2/t,• ,•X IC--• VVV•
0.$ 0. $
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FREQUENCY, N4
Figu Fi gure re 1 1. 1. CSA annoy annoyance ance cr crit iter eria ia cha chart rt for floor floor vib vibrat ration ionss 2. 20
[1 3]
(1 )
A2 A 0 __< 1. 1 .1 5
[1 4]
(2 )
(Amax)X(f) < 0. 0 . 05 0
wher wh ere e A0
w i t h Am Ama x <__. 0 . 0 1 5 in in.
and f are are as defi define ned d pr pre evi viou ousl sly, y, A2 is the se seco cond nd heel heel drop drop maxim maximum um
amplitude ampli tude an and d Amax is the ab abso solu lute te maxi maximum mum heel drop drop ampli amplitude tude,, both in inches inches.. While on the surface the appli While applicatio cation n of this ap appr proa oach ch se seem ems s si simp mple le,, the re read ader er should should be remin re minde ded d that dete determina rmination tion of A 2 and and Am Amax ax ,i ,in n ge gene nera ral, l, re requ quir ires es calculation calculation o f the dynamic res dynamic respo pons nse e of a SDOF SDOF system system (i (i.e .e.. floor be beam am)) to a ge gene nera rall excit excitatio ation n (i (i.e .e.. a serie series s of heel dro drop p imp impact acts). s). A pr proc oced edur ure e not suita suitabl ble e for hand cal calcul culati ations ons.. A ra rath ther er si simp mple le computer prog program, ram, howe however, ver, ca can n do the job an and d a diskette diskette conta containing ining on one e su such ch pr prog ogra ram m accompanies Ref Refere erence nce 15. FLOOR FLO OR VIBRATION VIBRATION FROM RHYTHMIC RHYT HMIC ACTIVITIES Coordinated Coordinat ed rhythmic acti activitie vities s su such ch as da danc ncin ing, g, au audi dien ence ce part participa icipation tion in arena arenas s an and d concert ha hall lls, s, an and d most impo importan rtantly tly ae aero robi bics cs ca can n resu result lt in un unde desir sirab able le le leve vels ls o f vibra vibration tion.. Forr rhythmic acti Fo activitie vities, s, it is re reso sona nant nt or ne near ar resonant resonant behavior behavior that re resu sult lts s in significant significant dynamic dyna mic ampli amplificat fication ion an and d he henc nce e hu huma man n disco discomfort mfort.. The most ra rati tion onal al de desi sign gn strat strategy egy is to pro provid vide e eno enough ugh of a ga gap p bet betwee ween n the na natu tura rall fre freque quency ncy of the flo floor or sys system tem,, and the dominant domin ant frequ frequencie encies s excit excited ed by pl plan anne ned d hu human man acti activitie vities s to re reas ason onab ably ly as assu sure re that reso re sona nanc nce e will not occu occur. r. Mult Multi-pu i-purpos rpose e facil facilities ities,, su such ch as floor systems in ae aero robi bics cs gyms and an d office sp spac ace e on the sa same me floor, pose the the most difficu lt vibra vibration tion de desi sign gn task. Allen3.4.s ha Allen3. has s reported the most com compre prehen hensiv sive e des design ign gui guidel deline ines s on this subject. Hi His s recommend recom mendation ations s ha have ve been re refle flect cted ed in the recen recentt serv servicea iceabilit bility y crite criteria ria supplemen supplementt to the Natio National nal Bu Buil ildi ding ng Co Code de of Canada. Not surp surprisin risingly, gly, the ma mate teri rial al pre prese sent nted ed in this section secti on is mainly mainly base based d on infor informatio mation n co cont ntai aine ned d in Re Refe fere renc nces es 3,4, an and d 5. While for most rhy thmic acti activitie vities, s, cons considera ideration tion of the first harm harmonic onic (main (main frequency frequency)) of the activity is suf suffici ficient ent,, for ae aero robi bics cs an and d other coordin coordinate ated d jum jumpin ping g ex exer erci cise ses, s, the seco se cond nd an and d thir third d harmo harmonics nics ca can n ma make ke signi significan ficantt cont contribut ributions ions an and d sh shou ould ld be co cons nsid ider ered ed in the an anal alys ysis is.. Fi Figu gure re 12 shows su such ch a third harm harmonic onic re reso sona nanc nce e which was caused caused by aero ae robi bics cs activity at 2.25 Hz on a 6.7 Hz floor sys system4 tem4.. TIME RECORD 5.19 - -
-
I
I ·
FOURIER FOUR IER TRANSFORM I
•
2.87
3. 32
2.39
1.46
1.91
I I 11'IIRO HARMONIC RESONANCE
I
I
B
, . .
-2. 28
0.96
- 4. 14
V
0
0.4
' [
0,8
0.48
I•
1.6
2.0
TIME TI ME,, s
2
4 6 FREQUENCY FREQU ENCY.. Hz
8
Figure Figur e 1 2. Vib Vibrat ration ion of a 6.7 Hz floor due to ae aero robi bics cs at 2.25 Hz4 z4.. 21
10
Accordin Acco rding g to AlienS: "Reson "Res onan ance ce is the most impor tant factor affect affecting ing aer aerobi obics cs vib vibrat ratio ion, n, hen hence ce natural frequency frequenc y is the most important structural design design param paramet eter. er. The problem problem is to get the natural natural frequency away from the three three harmo harmonics." nics." Design Desi gn steps to prev prevent ent floor vibr vibratio ation n from rhythmic acti activitie vities s ma may y be su summ mmar ariz ized ed as follows: (1)) (1
Forr eac Fo each type of act activit ivity, y, de dete term rmin ine e the do domi mina nant nt ran ang ge of fo forc rcin ing g fr freq eque uenc ncy, y, ff (see Ta (see Table ble 5). Noti Notice ce that for ae aero robi bics cs an and d ju jumpi mping ng ex exer erci cise ses, s, the first thre three e harmonics should be considered considered..
(2)
Sele Se lect ct a maxim maximum um ac acce cept ptab able le limi limitt for flo floor or accel acceler erati ation on,, a0 a0.. Use the valu values es recommend recom mended ed in Table Table 6, or IS ISO O charts as dis discus cussed sed previous previously ly
(3)
Select Sele ct a dy dyna nami mic c load fa fact ctor or,, (x. See See Tab Table le 5 for guida guidanc nce. e. Es Esti tima mate te the the distrib dist ributed uted weight o f the part particip icipants ants,, Wp. When When onl only y a portion portion of sp span an is used for the activity the lo load ad Wp can be estima estimate ted d by taking taking the total lo load ad on the pa parti rtial ally ly loaded load ed sp span an and dis distrib tributi uting ng it uniformly over the ent entire ire span. span. Tab Table le 5 may be used used to ar arri rive ve at a re reas ason onab able le est estima imate te for Wp Wp..
(4)
Comp Co mput ute e the total total floor loa load, wt by add addin ing g the norma normall lly y su sust stai aine ned, d, no nonn-ac acti tive ve load loa d and Wp Wp..
(5)
Comput Comp ute e the na natu tura rall fr freq eque uenc ncy y of th the e flo floor or sy syst stem em,, f, usi sin ng an ap appr prop opri riat ate e me meth thod od such as one one of the met method hods s di disc scus usse sed d in this pub public licatio ation. n.
(6)
Check Chec k the followi following ng cr crit iter erio ion n for the the mi mini nimu mum m na natu tura rall fr freq eque uenc ncy y of the floor floor system: , • [15]
f _•> ff
1.3 ( ZWp I + ao/g wt
where ao/g is the acc accele elerati ration on limit di disc scus usse sed d in step 2 ab abov ove, e, ex expr pres esse sed d in percent of gravita percent gravitation tional al acce acceler leratio ation. n. The factor 1.3 in [15] is subject subject to the same same discussi disc ussion on pr prov ovid ided ed for [2]. For ae For aero robi bics cs an and d jum jumpin ping g ex exer erci cise ses, s, the f irst thre three e har harmon monics ics o f the forc forcing ing frequency frequen cy sho shoul uld d be consid considere ered. d. However However,, sin since ce these harmonics add add togethe together, r, the factor 1.3 in [15] sh shou ould ld be incr increa ease sed d to 2.0. He Henc nce, e, the gov govern ernin ing g cri criter terio ion n for aero aerobics bics beco becomes mes:: , • [ 16 ]
f •
( i)(ff) (i
2 . 0 (x Wp I + a0/g wt
where i= 1,2,3 is the the har harmon monic ic num number ber.. Con Condit dition ion [16] sho should uld be satisfie satisfied d for each each of the thre three e har harmon monics. ics. Furthermor Furthe rmore, e, All Allen en3 3 recomme recommends nds that floor syst systems ems in assembly assembly occu occupanc pancies ies that do not meet the minimum natural frequencies frequencies o f Table 7 sho shoul uld d be evaluat evaluated ed more carefully. 22
TABLE TABL E 5 --- Sugg Suggeste ested d desi design gn para paramete meters rs for rhythmic events3. events3.4 4,s.
Activity
Forcing frequency f f , Hz
Dynamic load factore.
Weight of participants* Wp, psf
Dynamic load OrWp, psf
(z I
12.5 (27 ft2/couple)
0.5
6.25
1.5 1. 5 - 3.0
31.3 (5 ft2/person) ft2/person)
0.25
7.83
2 - 2.75 4 - 5.50 6 - 8.25
4.2 (42 ft2/person) ***
1.5 0. 6 0.1
6.30 2.52 0.42
Dancing
1 .5 .5 - 3 . 0
Lively concert Lively or sport event
Aerobics 1st Harmonic 2nd Harmonic 3rd Harmo Harmonic nic
* **
4.2 (42 ft2/person) *** 4.2 (42 ft2/pe ft2/person) rson) *** Density Den sity of particip participants ants is for commonl commonly y encou encounte ntered red conditions. Fo Forr spec special ial even events ts the density of participants can be greater.
Values Value s of o f (x ar are e based based on on commonly encoun encountered tered event events s involving a minimum minimum of about 20 particip participants. ants. Values Val ues of = should be increased increased for well-c well-coordin oordinated ated events (e.g. jum p dances) dances) or for fewer than th an 20 participants.
*** Sug Sugge geste sted d revision to the 1985 1 985 supplement supplement of CS CSA A codes. codes.
activities ies 4 TABLE TAB LE 6 ----- Rec Recom omme mende nded d acc accele elerat ration ion lim limits its for vib vibrat ration ion du due e to rhythmi rhythmic c activit Occupanci Occupa ncies es affected by the v ibrat ion ion
Acceleration Accelerati on limit limit,, per cen t •l •lrav ravity ity
Office and resid residenti ential al Dining Din ing,, Dan Dancin cing, g, Weight-lifting
0.4 t o 0.7 5 1.5 t o 2 ..5 4to 7
Aerobi Aer obics, cs, rhythmic act activit ivities ies only Mixed us use e occ occup upanc ancies ies hou housin sing g aerobics
2 m
TABLE TAB LE 7 --- Minimu Minimum m recomme recommended nded nat natura urall ass assemb embly ly floor fre freque quenci ncies, es, Hz3 Hz3.. Da nce f l o o r s * , gy mnasia**
stadia, ar enas**
C om pos it e ( s t e e l - c onc r ete) S ol i d C o n c r e t e
9 7
6 5
Wood
12
8
T yp e o f floor construction
* **
L im im itit in in g p ea ea k a ccele rat ion 0 . 02 02 g, L im im itit in in g p e ea ak ac cc cele ra ra tition 0 . 0 5 g .
EXAMPL EXAM PLE E 8: De Dete term rmin ine e the min minimum imum na natu tura rall freq frequen uency cy nee eed ded for a composite composite floor system in a gymnasium to be used used exc exclus lusive ively ly for aer aerob obics ics and other similar similar exercise exercises. s. 23
Deter termi mine ne the min minimu imum m na natur tural al frequen frequency cy ne need eded ed for a compo composite site floor EXAMPLE 8: De system in a gymn gymnasiu asium m to be used exclusiv exclusively ely f or aer aerob obics ics and other similar exercis exercises. es. The Th e tot total al nor normal mally ly su sust stai aine ned d lo load ad on the floo floorr inc includ luding ing the de dead ad weight an and d th'e weight weight of non-pa non-participa rticipating ting au audi dien ence ce is estimated estimated at 80 pou pound nds s pe perr square square foot .
SOLUTION:
Followin Foll owing g the foreme foremention ntioned ed step-by-ste step-by-step p pro proced cedure ure::
(1)
From Fr om Table Table 5, sel select ect a re reas ason onab able le va valu lue e for forc forcing ing freq frequen uency, cy, say 2.5 Hz.
(2)
Since Sinc e the floor is to be used for ae aero robi bics cs and rhythmic acti activiti vities es only, from Ta Tabl ble e 6 an accele accelerati ration on limit of 4% to 7%g is reaso reasona nabl ble. e. Fo Forr this exa exampl mple e we sele select ct an acce ac cele lerat ratio ion n lim limit it of a0 = 0.05g 0.05g..
(3)
We use use the su sugg gges este ted d va valu lues es from Ta Tabl ble e 5 for weight of the par partici ticipan pants, ts, dyn dynami amic c load lo ad facto factors, rs, an and d dyn dynami amic c lo load ads. s. He Henc nce, e, dyn dynami amic c lo load ads s for t he three ha harm rmon onic ics s are ar e 6.30, 2.52, an and d 0.42 ps psf, f, res respe pecti ctivel vely. y.
(4)
wt
(5)) (5
This Thi s step does does not app apply ly to this pr prob oble lem. m.
(6)
Chec Ch eck k [16] for ea each ch of the three three ha harm rmon onic ics s
= WD.L.
+ Wp = 8 0 + 4. 2 -- 8 4. 2 p sf sf
1st harmo harmonic: nic: f
> (1 )( 2. 50)
,•/
2.0 6 6.. 3 0 5.00 0 Hz I + 0. 0. 0 5 84 84. 2 - 5.0
2nd harmon harmonic: ic: f •
(2)(2.50) . • 1 + 02_.•5 2•..•52
7.41 7. 41 Hz
3rd harmo harmonic: nic: f >. (3 ( 3) (2 . 50)
. •
2.0 0 . 4 2 I + 0 0.. 0 5 84 8 4 . 2 - 8.21 Hz
Cont rol s
The floor sys The system tem should should be de desi sign gned ed to ha have ve a first nat natura urall freq frequen uency cy la larg rger er than 8.21 8.2 1 Hz. No Notic tice e that Ta Tabl ble e 7 su sugg gges ests ts a mi mini nimu mum m na natur tural al frequency frequency of 9 Hz for this case.
ACKNOWLEDGMENTS
The author's work on this pub public licati ation on was sp spon onso sore red d in part by •J •John A. Mar Martin tin and Assoc As sociat iates es,, Inc Inc.. Cr Crit itica icall re revie view w of th the e ma manu nusc scri ript pt by Dr. Ro Roge gerr M. Di DiJu Juli lio o an and d Mr. Jame Ja mes s Mar Marsh sh is gratef gratefully ully appreciated.
24
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25
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