7 Esimaion of Kineic Fricion Coefficien for Sliding Rigid Block Nonsrucural Componens Cagdas Kafali Ph.D. Candidae, School of Civil and Environmenal Engineering, Cornell Universiy Research Supervisor: Mircea Grigoriu, Professor of Srucural Engineering Summary The kineic fricion coefficien beween a sliding rigid block and is supporing floor is esimaed. The block can be used o model a broad range of nonsrucural componens such as free sanding and unresrained building conens, and mechanical and elecrical equipmens. Experimenal resuls and heoreical consideraions are used o esimae his coefficien, since he coefficien of kineic fricion canno be measured direcly. Esimaes of kineic fricion coefficien are obained using acceleraion and displacemen-based mehods for carpe on seel inerface. These esimaes accoun implicily for he uncerainy in experimenal errors, imperfecions in block-floor inerfaces, and he relaionship beween he kineic fricion coefficien and he loading, and he block size. I is shown ha mos of he pairs of acceleraion and displacemen-based esimaes of he kineic coefficien of fricion are included in he 9% probabiliy conour of hese parameers assumed o be Gaussian. Inroducion In recen sudies i has been shown ha, for a faciliy, he financial consequences of seismic occurrences resul mainly from he poor performance of is nonsrucural componens and sysems (Gould and Griffin 23; Kircher 23). The seismic performance of free sanding and unresrained componens, for example, mechanical and elecrical equipmens, depends srongly on he kineic fricion coefficien beween he componens and he supporing floors. However, he kineic fricion coefficiens for common componen-floor inerfaces are no available in he lieraure (Soler and Singh 1984; Garcia and Soong 23; Casiglioni 23). The objecive of his sudy is o esimae he kineic fricion coefficien beween a sliding rigid block and is supporing floor. The block can be used o model a broad range of nonsrucural componens such as mechanical and elecrical equipmens, free sanding and unresrained building conens. Since he coefficien of kineic fricion canno be measured direcly, experimenal resuls and heoreical consideraions are needed o esimae his coefficien. Esimaes of kineic fricion coefficien are obained for carpe on seel inerface, using acceleraion and displacemen-based mehods. I is shown ha mos of he pairs of acceleraion and displacemen-based esimaes of he kineic coefficien of fricion are included in he 9% probabiliy conour of hese parameers assumed o be Gaussian random variables wih he second momen properies esimaed from experimens. Esimaion of Kineic Fricion Coefficien for Sliding Rigid Block Nonsrucural Componens 43
Dynamic Analysis of Block-Table Sysem Consider a free sanding rigid block of mass m, which can represen a sliding nonsrucural componen, siing on a shake able. Le z() and y() be he displacemen of he shake able and of he block relaive o an absolue frame whose origin corresponds o he verical posiion of he block cenroid, respecively, be he coefficien of kineic fricion beween he block and he surface of he shake able, and g be he acceleraion of he graviy as shown in Figure 1. Denoe he displacemen of he block relaive o he shake able by x(), so ha y()=z()+x(). block m absolue frame, fricion g, graviy able + z() y() x() Figure 1. Sysem of sliding block and shaking able The equaion of moion for he block is z( ) g sign( x( )), sliding condiion (1) x( ) =, sicking condiion where sign(b)=-1,, 1, for b<, b= and b>, respecively. The shake able acceleraion used in he experimens performed a he Universiy a Buffalo has he form z ) = w( ) αgsin( υ), [, ] (2) ( f where w() is a modulaion funcion increasing o 1 and saring a w()=. A compuer algorihm for calculaing he block relaive acceleraion, velociy and displacemen is presened in Kafali and Grigoriu (25). Figure 2 shows he calculaed oal and relaive acceleraion and velociy responses of he block, and he acceleraion and he velociy of he shake able for =.2, α=.8, υ=2πt, T=.5 sec and f =3.1 T, in Eqs. 1 and 2. The calculaed seady sae oal block acceleraion, velociy and displacemen responses have zero emporal mean, and are symmeric abou he ime axis, under he sinusoidal exciaion defined in Eq.-2 because of he symmery of he equaion of moion. On he oher hand, recorded block responses exhibi drifs since he properies of he block-able inerface exhibi spaial variaion. To relae calculaed resuls o experimenal resuls, block response records need o be correced, as shown in he following secions. 44 Seismic Rerofi of Acue Care Faciliies
acc. acceleraions (in g unis) 1.5 - -.5 z () zdd y () -1.5 1 1.5 ydd x () xdd ime (sec) vel. (in velociy g sec unis).1.5 -.5 z () zd x () xd y yd () -.1.5 1 1.5 ime (sec) Figure 2. Algorihm Esimaion of Kineic Fricion Coefficien The coefficien of kineic fricion is esimaed using he maximum responses of he block obained hrough experimens a he Universiy a Buffalo, and a relaionship beween he maximum responses and he kineic fricion coefficien, obained a Cornell Universiy using heoreical consideraions. The maximum absolue oal acceleraion response, max y ( ), of a block subjeced o he inpu acceleraion defined by Eq. 2, is obained using Eq. 1 max g, y( ) = max z( ) = α g, sliding condiion sicking condiion (3) I is difficul o obain a relaionship in closed form beween he maximum absolue displacemen response max y() and he kineic fricion coefficien. The algorihm defined in Kafali and Grigoriu (25) is used o relae max y() and. Figure 3 shows an example of maximum oal displacemen versus kineic coefficien fricion curve for [.1,.9], α=.8, υ=2πt, T=.5 sec, f =4.2 T. 6 x 1-3 max y() 5 4 3 2 1.1.2.3.4.5.6.7.8.9 Figure 3. Relaion beween max y() and Esimaion of Kineic Fricion Coefficien for Sliding Rigid Block Nonsrucural Componens 45
Daa Analysis A series of shake able experimens on rigid blocks have been performed a he Universiy a Buffalo o characerize he kineic fricion coefficien carpe on seel inerface (explained in deail in Fahali 25). The surface of he shake able is seel and he carpe is aached o he boom of he blocks. The exciaions are modulaed unidirecional sine waves wih differen ampliudes and frequencies o accoun for he uncerainy relaed o he dependence of he kineic fricion coefficien on he loading. To accoun for he uncerainy relaed o he experimenal errors, six nominally idenical blocks are esed simulaneously for given ampliude, period pair. Two mehods are used o esimae he kineic coefficien of fricion. The mehods are based on acceleraion and displacemen records of blocks obained in Fahali (25). Acceleraion-Based Esimaes of Kineic Fricion Coefficien: Le a(), [, f ], be he recorded acceleraion ime hisory of a block in a given es, wih respec o a fixed frame. The corresponding block acceleraion in calculaions is denoed by y (). The following 4-sep procedure was used o find esimaes acc of based on displacemen records (deails can be found in Kafali and Grigoriu 25). Sep-1: The acceleraion record a() is correced by subracing is emporal mean. Sep-2: The seady sae par of he correced acceleraion response record a ss,c () is obained. Sep-3: The maximum absolue acceleraion max a ss,c (), is esimaed by is mos likely value from he hisogram of a ss,c (). Sep-4: The coefficien of kineic fricion is obained using Eq. 3. acc. (in assc g unis).5 -.5 seady sae correced acceleraion zdd z () a ss,c () 1 2 3 4 5 ime (sec) max maxaassc ass,c() (a) 1.5 ydd 65b1maxaassc from he hisogram his1 of a ss,c () max ( ) p z =.474g muacc.2.4.6.8 1 mu (c) 15 1 5.1.2.3.4.5.6 aassc acc =.4 hisogram 65b1aasschis of a ss,c () max bin a ss,c () =.4 a ss,c () (in g unis) (b) Figure 4. Acceleraion-based esimaion (es 65, block 1) 46 Seismic Rerofi of Acue Care Faciliies
Displacemen-Based Esimaes of Kineic Fricion Coefficien: Le d(), [, f ], be he recorded displacemen ime hisory of a block in a given es, wih respec o a fixed frame. The corresponding block displacemen in calculaions is denoed by y(). The following 4-sep procedure was used o find esimaes disp of based on displacemen records (deails can be found in Kafali and Grigoriu 25). Sep-1: The displacemen record d() is correced by subracing is drif. Sep-2: The seady sae par of he correced displacemen response record d ss,c () is obained. Sep-3: The maximum absolue acceleraion max d ss,c (), is esimaed by is mos likely value from he hisogram of d ss,c (). Sep-4: The kineic fricion coefficien is obained using he relaion beween max d ss,c () and. disp. (in g sec 2 unis) dssc.1.5 -.5 seady sae correced 65b1dssc displacemen z() z -.1 1 2 3 4 5 max dss,c() maxadssc y ime (sec) (a).12.9.6.3 d ss,c () n 1 8 6 4 2 65b1maxadssc from he hisogram his1 of d ss,c ().1.2.3.4.5 mu (c) hisogram 65b1adsschis of d ss,c () max d ss,c () =6.262 1-3 bin 1 2 3 4 5 6 7 d adssc x 1 x1-3 ss,c () (in g sec 2 unis) -3 disp =.37 mudisp (b) Figure 5. Displacemen-based esimaion (es 65, block 1) Resuls For a given inerface ype, he coefficien of kineic fricions are obained using he acceleraion and displacemen responses of all he blocks in each es for ha inerface ype following he procedures described above. Figure 4 shows he ( acc, disp ) pairs obained for carpe-seel inerface. Saisics of acc and disp are calculaed using (i) only he small blocks, (ii) only he large blocks, and (iii) all he blocks, are shown in Table 1 for carpe-seel inerface. Esimaion of Kineic Fricion Coefficien for Sliding Rigid Block Nonsrucural Componens 47
.6 small blocks large blocks.5 disp.4.3.2.2.3.4.5.6 acc Figure 6. Fricion coefficiens I is observed ha he correlaion beween acc and disp obained using he large blocks is significanly higher han he correlaion obained using he small blocks. This suggess ha he esimaes of based on small blocks have more noise han hose corresponding o large blocks. However, he esimaed means and he coefficiens of variaion are insensiive o block size. I is concluded ha here is no apparen size effec in he esimaes of acc and disp. Table 1. Saisics of kineic fricion coefficien Small blocks Large blocks All blocks acc disp acc disp acc disp Mean.374.379.41.394.393.387 Sandard Deviaion.51.38.63.62.6.53 Coefficien of variaion.137.11.154.157.153.136 Correlaion coefficien -.1.752.59 Assuming ha acc and disp are correlaed Gaussian random variables wih means, sandard deviaions and correlaion coefficiens given in Table 1, conour lines of he join probabiliy densiy funcion of acc and disp corresponding o 9% probabiliy are shown in Figure 4. Mos of he daa poins are in he 9% probabiliy conour. Concluding Remarks The kineic fricion coefficien beween a sliding rigid block and is supporing floor for carpe on seel inerface is esimaed using experimenal resuls and heoreical consideraions. Esimaes of he kineic fricion coefficien are obained using acceleraion and displacemen-based mehods, and 48 Seismic Rerofi of Acue Care Faciliies
accoun implicily for he uncerainy in experimenal errors, imperfecions in block-floor inerfaces, and he relaionship beween he kineic fricion coefficien and he loading, and he block size. I is shown ha mos of he pairs of acceleraion and displacemen-based esimaes of he kineic coefficien of fricion are included in he 9% probabiliy conour of hese parameers assumed o be Gaussian random variables wih he second momen properies esimaed from experimens. Acknowledgemens This research was carried ou under he supervision of Dr. Mircea Grigoriu and primarily suppored by he Earhquake Engineering Research Ceners Program of he Naional Science Foundaion, under award number EEC-971471 o he Mulidisciplinary Cener for Earhquake Engineering Research. The auhor also would like o hank Prof. Harry E. Sewar of Cornell Universiy for he commens and suggesions, and Prof. Andrew Whiaker and Saeed Fahali of Universiy a Buffalo for providing he daa. References Gould MJ, Griffin NC (23): The value of seismically insalling and srenghening nonsrucural equipmen and sysems o significanly reduce business inerrupion losses. In Proceedings of Seminar on Seismic Design, Performance, and Rerofi of Nonsrucural Componens in Criical Faciliies (ATC-29-2), pages 215-226, Newpor Beach, CA. Kircher CA (23): Make dollars and sense o improve nonsrucural sysem performance. In Proceedings of Seminar on Seismic Design, Performance, and Rerofi of Nonsrucural Componens in Criical Faciliies (ATC-29-2), pages 19-119, Newpor Beach, CA. Soler AI, Singh KP (1984): Seismic response of a free sanding fuel rack consrucion o 3D floor moion. Nuclear Engineering and Design, volume 8, pages 315-329. Garcia DL, Soong TT (23): Sliding fragiliy of block-ype nonsrucural componens. Par 1: Unresrained componens. Earhquake Engineering and Srucural Dynamics, volume 32, pages 111-129. Casiglioni CA (23): Seismic behaviour of seel sorage racks. In Proceedings of he Fourh Inernaional Conference on he Behavior of Seel Srucures in Seismic Areas (STESSA 23), Naples, Ialy. Kafali C, Grigoriu M (25): Esimaion of kineic fricion coefficien for sliding rigid block nonsrucural componens. Repor (in preparaion), Mulidisciplinary Cener for Earhquake Engineering Research, Buffalo, NY. Fahali S (25): Fragiliy evaluaion of sliding blocks. Repor (in preparaion), Mulidisciplinary Cener for Earhquake Engineering Research, Buffalo, NY. Esimaion of Kineic Fricion Coefficien for Sliding Rigid Block Nonsrucural Componens 49