Earthquake Resist Design
Earthquake Resist Design CIVL 7119
University of Memphis
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CIE 619 Earthquake Engineering and Structural Dynamics II Instructor Andrew Whittaker SEISMIC PROTECTIVE SYSTEMS This module presents information on seismic protective systems and new and retro t building construction using seismic protective systems The key references for this module are the 2000 NEHRP Recommended Provisions FEMA 2000 FEMA 356 Prestandard and Commentary for the Seismic Rehabilitation of Buildings Naeim and Kelly 1999 and Constantinou Soong and Dargush 1998 This module includes information on 0 Seismic isolation systems and methods of implementation 0 Passive energy dissipation systems and methods for implementation These materials are intended to serve as a springboard for semesterlong presentations in both subject areas 1 SEISMIC ISOLATION SYSTEMS 11 Principles of Seismic Isolation The basic objective with seismic isolation is to introduce horizontally exible components but vertically stiff components at the base of a building to substantially uncouple the superstructure from high frequency earthquake shaking The sketch below is a crosssection through the first seismically isolated building in the United States the Foothill Law and Justice Center in southern California This building was isolated on highdamping rubber bearings in the 1980s 1160 1 t l I GALLERY WALL 16quot SEISMIC ISOLATION GAP BASEMENT WALL ISOLA 1639 SEISMIC ISOLATION UNDER EACH COLUMN Module 05 Page 1 CIE 619 Earthquake Engineering and Structural Dynamics II Instructor Andrew Whittaker The bene ts of adding a horizontally compliant system at the foundation level of a building can be seen below using an acceleration response spectrum 0 What percentage reduction can be achieved b ACCELERATION I PERIOD SHIFT PERIOD Increasing the period of the structure reduces the spectral acceleration for typical earthquake shaking o Is this good 0 Effect on nonstructural components What is the effect on the displacement response of the structure Consider the displacement spectrum below DISPLACEMENT A INCREASING DAMPING p PERIOD For an isolated period of 3 seconds and 5 damping what is the eXpected displacement response of the isolated building Module 05 Page 2 CIE 619 F 39 39 F 39 39 and Structural Dynamics II Instructor39 Andrew Whittaker Does the increase in displacement response lead to more damage in the building superstructure If not why not 0 What is the assumed displaced shape of the building frame below a simple two story twobay frame supported on three isolators Most of the displacement in an isolated building occurs over the height of the isolators and there is generally little deformation in the superstructure o Isolators must be designed to support gravity and earthquakeinduced loads AND accommodate large lateral displacements Displacements in isolated structures are often large and efforts are made to add energy dissipation or damping in the isolation system to reduce displacements What effect does this have on the displacement response Consider the displacement response spectrum of the previous page DISPLACEMENT INCREASING DAMPING PERIOD The addition of energy dissipation to the isolation systems serves to reduce displacements in the seismic isolators which can translate into smaller isolators reduced costs etc Energy dissipation can be added 0 Internal to the isolator lead core specially compounded rubber friction o EXtemal to the isolator viscous dampers lead dampers Japan Module 05 Page 3 CIE 619 Earthquake Engineering and Structural Dynamics II Instructor Andrew Whittaker 12 Types of Seismic Isolation Hardware 12 Introduction There are two common types of largedisplacement capacity base or seismic isolation hardware in the United States at this time o Elastomeric bearings Natural rubber and filled rubber bearings Leadrubber bearings o Sliding bearings 122 Elastomeric Bearings Elastomeric nonleadrubber bearings are available as either lowdamping natural rubber bearings manufactured principally by Andre Bridgestone DIS and SEP or highdamping lled bearings manufactured principally by Andre and Bridgestone A partplan and section through a Bridgestone highdamping rubber bearing is presented in the gure below The maximum shearstrain range for high damping rubber bearings vary as a function of compound and manufacturer but is generally between 200 and 350 HHD060 M 2362 Uml mminch Module 05 Page 4 CIE 619 Earthquake Engineering and Structural Dynamics II Instructor Andrew Whittaker Note that the endplate details in Bridgestone bearings tend to be different from those of Andre and DIS bearings What are the key construction features of this bearing What role to the steel shims play in the performance of elastomeric bearings How are elastomeric bearings constructed Round Relief me 44139 has me awn Se 3m H094 1 WA Conway t Renewing M ma Module 05 39 O t Page 5 CIE 619 Earthquake Engineering and Structural Dynamics II Instructor Andrew Whittaker Energy dissipation in highdamping rubber bearings is achieved by special compounding of the elastomer Damping ratios will generally range between 7 and 14 of critical The shear modulus of highdamping elastomers generally ranges between 50 psi and 200 psi Techniques to engineer mechanical properties are generally considered proprietary The effective stiffness a function of the modulus bonded area and total rubber thickness and damping of an elastomeric bearing will depend on o Elastomer including llers and vulcanization pro le 0 Contact pressure supported load divided by bonded area 0 Velocity of loading 0 Load history 0 Strain history 0 Temperature especially below 20 F The behaviors of many highdamping elastomers are compleX and isolators constructed with these materials are not amenable to simple modeling using standard plasticity or BoucWen formulations Consider the sample data below from the PhD thesis of Clark from tests of smallscale highdamping HD elastomeric bearings Note that these data are not necessarily indicative of the response of all high damping rubber bearings Each elastomer should be carefully studied to assess its dependence on the variables listed above Consider rst data from dynamic cyclic tests of a HD bearing up to shear strains of 250 percent The contact pressure is approximately 500 psi low by current standards 0 Observations Strain Amplitudes 100150200250 percent Force kips Displacement inches Frequency 05 Hz Bearing Number 14 i Axial Load 176 kips Filename 91121921 Module 05 Page 6 CIE 619 Earthquake Engineering and Structural Dynamics II Instructor Andrew Whittaker Now consider tests of the same bearing at shear strain levels between 200 and 350 Strain Amplitudes 200250300350percent g m Tn a 8 m 0 2 63 It o I 3939 m J 5 O 5 Displacement inches Frequency 05 Hz Beating Number 1 4 Axial Load 176 ldps Filename 91121924 0 Observations Below are data from tests of two bearings with properties that are identical to those of the bearing shown above The tests are to a shear strain of 273 Note the signi cant difference between the rst and third cycle response in both cases this phenomenon is known as scragging o Is the rst cycle behavior recoverable with time Must assume yes Data support this conclusion 0 How to account for this wide variation in design of isolation systems gt Bounding analysis I v 39 quot39quot 06 185 2 cycle 1 OStest2cyclea o 12 test 2 cycle 1 quotquot 12est2cyclea A 39 j 0 Il as Mr o U D I f z u If 39 I Strains 273 O r r I I i 4 2 o 2 4 Module 05 Displacement inches Page 7 CIE 619 Earthquake Engineering and Structural Dynamics II Instructor Andrew Whittaker Highdamping HD elastomeric bearings are often characterized as linear viscoelastic components with an effective stiffness and a viscous damping ratio The data below from Clark shows measured variations in material properties as a function of aXial pressure and loading frequency velocity 200 200 E ective Modulus psi 150 Effective Modulus psi 1 50 8 g o 20 4o so 80 100 o 2 a e 48 3 E s m c6 0 D o o 20 40 so so 100 Shear Strain percent Shear Strain percent Axial Loads 5283531758310178 kips Frequency 10 Hz Frequencies 01 03051020 Hz Axial Load 176 kips Figure 320 Variation In keff and 5 as a w Figure 321 Variation in k eff and 5 as a Function ofoial Load Function of Loading Frequency How should an engineer account for these wide variations in material properties 0 Bounding analysis 0 Property modification factor approach to be included in 2003 NEHRP Recommended Provisions 12 3 Lead Rubber Bearings Elastomeric leadrubber bearings are manufactured in the United States by Dynamic Isolation Systems DIS and Seismic Energy Products SEP A cutaway view through a DIS lead rubber LR bearing is presented in the figure below Shown in this figure are the o Alternating layers of elastomer and steel shims 0 Central lead core Leadrubber bearings are generally constructed with lowdamping unfilled elastomers with shear moduli in the range of 85 to 100 psi at 100 shear strain and lead cores with diameters ranging between Module 05 Page 8 CIE 619 39 39 Dvnamic ll Instructor Andrew Whittaker 15 and 33 ofthe bonded diameter of the bearing The elastomer provides the isolation component and the lead core provides the energy dissipation or damping component The maximum shearstrain range for lead rubber bearings varies as a function ofmanufacturer but is generally between 125 and 200 Top Maumhu Inn mt sham EllIvy DIM can Radmsl alnluplka um um Cow Rubber 4mm mol plum steel Wm Plats Plvviduvertiulhd nip6w Camquot h mm w m uqu Mun 113w Mum ma Lawn The steel shims play another important role in leadrubber bearings namely 0 Con nement ofthe lead core Because a leadrubber LR bearings are generally constructed with a lowdamping natural rubber elastomer and b the maximum shear strain rarely exceeds 2000o many of the variations in response that e characteristic of HD rubber bearings scragging load history strain history velocity are not characteristic of LR bearings Note however that if a highly lled elastomer is used in lieu of natural rubber in a LR bearing all of the issues identi ed earlier for HD rubber bearings must be considered in the assessment of the LR hearing The central lead core or plug in a LR bearing will dominate the response of the bearing for typical leadcore diameters especially if the shear strains are small Of importance are the mechanical characteristics of the lead comprising the core The yield point of lead is a function of 0 Temperature gt 18 MPa 10 MPa and 7 MPa at 25 C 25 C and 25 C respectively 0 Con nement that is provided by the steel shims and the top and bottom plate Values ofthe yield point are o en larger in smallscale bearings say 2000 to 2500 psi than in fullsize bearings 1000 psi to 1500 psi The guidance of a manufacturer should be sought prior to sizing lead cores in LR bearings One issue that should be considered in judging the response of LR bearings is the response of the lead core with repeated cycling Consider the HITEC tests of two LR bearings each rated to 750 kips of axial load by the vendors DIS and Skellerup respectively Module 05 Page 9 CIE 619 Earthquake Engineering and Structural Dynamics II Instructor Andrew Whittaker PueJiw 1 39 I u I F v quot1 In I mm MMMBNOI awnnon autumnm 1I 39 vuo r as mm m 4111537 40904 cm 7 f mmsumwmmmm 1 1 11439 mo am we LOAD PLATE 39 12 u rumwaz39 152 2 1 OUTER SHIM HOLES Emu SPACE roe 00115 U I I K E W 235 1 rg I L r DOTTOM 4 1 V FIXING PLATE 211223 4 912 1 1 DIMTCR coumk quotquot WCquot f 39 1 s am 50 T5 mm 395 LOAD 5 CIRCLLAR T55 OW W 59m 25562 DURING comma 5 24 x ona39 ma 5TEEL awe 25 It 03531 THICK WK LAEER5 3631 oarTr g a1 DIS LR bearing Dynamic Isolation Systems mm om Frequency Hz 05000 Test 0039 In 00 Tunadab 750000 Tau 721 150 1oorw w l 1 l z Test mien Shear Load ch3oo8 KIP 100 400 as 10 5 o 5 1o 15 scum195535AM totem thlll l i t t thTB Inches 6PtAvg 7 Flu W a2 lOcycle dynamic response m thn 1200 Dynamic IsolatiOn Systems mm 0135 ma Lou u 150000 Frequency Hz 00500 mu 1231 150 J Test Article Shear Load chaoos Kips 5 myg i 450 p r r t 43 96107 p 33 0 7 5 gt107 3915 1 1 ssmem littoral Dl39sf lquotC l bnffcamg zts metres 5PtAv39g r 9F 18319287 quot r 7 7 7 a3 3cyc1e static response Module 05 339 o Jam quot3 39 39 i I quot 139 THICK 51m 10 oumz sum 0 4 a k 139 1111a 51ch bOTTOM OUTER mm 11231114 man me 519 coves PLAN Vlgw 750 KILQEAEM TYPIEAL CROSSSECTION n I A0 on 759 Ki 39 1 LATERAL nssnan onsvucmsm on 12 noveuw sumo MR 439 Yl LDED snrruess Kr 49 Kin ELASTIC 512535 6 LATERAL so can A I 750 KIP LEADRUBBER BEARING HT3 uunuu rare an quotum Ln 3A0 REERENCF b1 Skellerup LR bearing SKELLERUP IND LTD l requenoMIIza 05000 Tu 0313011 1200 TIAID Bit5 human I 790000 1mll T11 150 a o S 8 2 Test Article Shear Load PCS3004 Kips 5 PLAvg 1 50 40 40 J 0 5 10 16 mum 004014 A thII39II Displacement Intma Itemtor FIT7 Inches 6 mm Flo W2 b2 lOcycle dynamic response SMW 300 All Flo W343 SKEELERUP391NDLTD e FMWrHi quot r 00500 m alumna 1200 muo M Tu L000 Lb I 750000 Test 0 I 7231 100 100 45 107 J 39 o 5 1o 13 mm Dummunnt FIT10mm 5 mm b3 3cyc1e static response Page 10 CIE 619 F 39 39 F 39 39 and Structural Dynamics II Instructor39 Andrew Whittaker Such response as noted above is a function of temperature and con nement Under repeated cycling the temperature of the lead core rises and its yield point diminishes substantially altering the hysteresis of the bearing Although the two bearings shown above have identical rated loads the lead cores in each bearing are quite different in size 475 inches in diameter in the DIS bearing and 65 inches in diameter in the Skellerup bearing The volume of the Skellerup lead core is nearly twice that of the DIS lead core Also the DIS bearing was constructed with 33 internal 012inch thick internal shims whereas the Skellerup bearing was constructed with 24 internal 012inch thick internal shims Consider rst the gures a2 and b2 above The effective yield point of the Skellerup bearing is substantially greater than that of the DIS bearings and the energy dissipated in the Skellerup bearing is much greater than that in the DIS bearing Such differences can be attributed to the larger lead core in the Skellerup bearing The change in hysteresis of the Skellerup bearing is also much grater than that in the DIS bearing with repeated cycling 0 Why Consider now gures a3 and b3 above static tests to 118 and 152 shear strain respectively Figure a3 eXhibits some level of pinching that is likely a function of partial disengagement of the small diameter lead core 16 of the bonded diameter Little change in hysteresis shape can be seen with repeated slow cycling Figure b3 shows little evidence of pinching but a signi cant change in hysteresis that is likely related to the con nement of the lead core 26 of the bonded diameter How should these variations in material properties be accounted for in design 0 Bounding analysis 0 Property modi cation factor approach to be included in 2003 NEHRP Recommended Provisions 12 4 Sliding Bearings Sliding seismic isolation bearings are manufactured in the United States by Earthquake Protection Systems EPS and R J Watson Inc EPS manufacture the Friction PendulumTM FP bearing and RJ Watson manufacture the EradiquakeTM bearing The Eradiquake bearing is widely used on East Coast bridges where the displacement demands are small The FP bearing is used for both buildings and bridges and has large displacement capacity Because the focus of this presentation is on isolation hardware for buildings the discussion below is limited to FP bearings A cutaway view through a FP bearing is presented in the gure below Module 05 Page 11 CIE 619 Earthquake Engineering and Structural Dynamics II Instructor Andrew Whittaker BEARING MATERIAL ARTICULATED FRICTION SLIDER SPMIC CONCAV39E SURFACE Shown in this gure are 0 Concave dish and housing plate that are typical ductile cast iron 0 Articulated slider typically of 304 stainless steel 0 Bearing material high loadlow friction composite Not shown is the stainless steel inlay that is placed and sometimes attached to atop the concave dish The sliding interface plays a crucial role in the response of the FP bearing The frictional resistance of the interface is a function of o Slider diameter effecting the con nement of the composite 0 Contact pressure 0 Sliding velocity 0 Temperature 0 Wear due to extended travel in bridge bearings due to thermal cycling Some general trends can be observed in the gures below Note that these trends apply to all PTFEbase composites similar in part to the composites used in FF bearings Consider rst the general relationship between coef cient of friction sliding velocity and contact pressure from Tsopelas et al 1999 What are the trends evident in this gure 0 Coef cient of friction as a function of velocity Module 05 Page 12 CIE 619 Earthquake Engineering and Structural Dynamics II Instructor Andrew Whittaker 0 Effect of pressure I BREAKAWAY fmax VALUE I Z 9 I48 I 2 n LL LL 0 I z E J 9 g INCREASING I3 NORMAL LOAD gt SLIDING VELOCITY One key issue is the threshold value of the sliding velocity beyond which the coefficient of friction does not substantially change 0 Why Data for one PTFEbased composite sliding on stainless steel are presented below that shows the friction Velocity relationship What conclusions can be drawn for this interface 015 CZ I PTFE COMPOSITE No 1 CONSTANTINOU ET AL 1993 F I 0 172 MPa f 010 E T A I max u 010 V 2 LL 1 SPECIMEN AREA2065 mm 0 I I I lJzl d A 2756 MPa fmax 005 a 03905 SPECIMEN AREA130 mmz E LL 0 IDENTIFICATION TEST 39539 o SEISMIC TEST39 0 I IL r r p 0 200 f 400 600 VELOCITY mms Module 05 Page 13 CIE 619 Earthquake Engineering and Structural Dynamics II Instructor Andrew Whittaker What is response for higher velocities Consider the data below also from Tsopelas et al Similar trends at room temperature were reported by Mosqueda et al following tests on FF bearings at Berkeley At high velocities greater than 20 inches per second the sliding coefficient of friction appears to drop Why A substantial increase in temperature at the sliding interface leads to a drop in the coefficient of friction COMPOSITE No1 z 010 4 9 009 2 a 008 3 E 007 X DNA 2 006 0 g Q E 005 9 50 C U o l 0 H o 4 A 40 00 O 003 f I 30 00 O I 0 3920 OC 9 002 V 10 00 Z 9 0 0C a 001 A 20 00 3 PRESSURE 69 MPa ltgt 50 00 0000JebL39ewiw39HaLanann111D1Din 20 50 100 150 39 200 250 300 350 VELOCITY mmsec What is the effect of low operating temperature and why does the coefficient of friction drop so substantially with an increase in velocity Shown below are data from the HITEC tests of one FP bearing that was rated for an aXial load of 750 kips identical to the rating of the LR bearings described above Figure cl shows a section and a plan of the bearing Data from a highspeed cyclic test are shown in Figure c2 Why is the first cycle response different from that of the second and third cycles 0 Inertial effects in the HITEC testing machine at the start of testing 0 Virgin travel of the composite over stainless steel overlay 0 Little evidence of stickslip behavior at corners of hysteresis loop Data from a lowspeed cyclic test are shown in Figure c3 The inertial effects evident in Figure c2 are not seen in the lowspeed tests Module 05 Page 14 CIE 619 Earthquake Engineering and Structural Dynamics ll Instructor Andrew Whittaker mom4quot 11oquot quot39739 3 mmmv ss gke7 39 R 386quot 39 0 HOUSING PLATE 7 r 2039 e l k CONCAVE PLATE SUDER39 quot cl FP bearing m mm 1200 Earthquake Protection Systems wuo ms Tomcum 750000 FMW H2 39 05000 mu 121 200 mm 1m Earthquak Protection Systems mm mu 7 r rquot Tum1231 or 5 Y n n rrHo quot4 1607 8 13 10 5 7 o 5 1o 15 39 4539 10 39 o 1 39 I 10 16 ttho PM Immraemvg 7 39 f 39 39 7 5 Inches 5mm c2 lOcycle dynamic response c3 3cycle static response How should these variations in material properties be accounted for in design 0 Bounding analysis 0 Property modi cation factor approach to be included in 2003 NEHRP Recommended Provisions 13 Mechanical Characteristics of Seismic Isolation Bearings Seismic isolation bearings are generally characterized as either 0 Linear Viscoelastic components 0 Binlinear components Module 05 Page 15 CIE 619 Earthquake Engineering and Structural Dynamics II Instructor Andrew Whittaker As linear viscoelastic components isolators are characterized by 0 Effective stiffness at selected displacement 0 Equivalent viscous damping ratio 0 FORCE 1 a O PLAS EID DEFORMATION DISPLACEMENT For this calculation the effective stiffness and equivalent viscous damping ratio are calculated as F F ke of d and 1 WD 2 2 7T ke d As bilinear components isolators are characterized by the idealized loop shown below with zero displacement force intercept Q d and secondslope stiffness K d FORCE 1 PLASTIC REFWATION urfv 1 DISPLACEMENT Module 05 Page 16 CIE 619 F 39 39 F 39 39 and Structural Dynamics II Instructor39 Andrew Whittaker Highdamping rubber bearings are generally modeled as linear viscoelastic components where the effective horizontal stiffness of the isolation bearing is given by GAb GAb k 317 T nt r r where G is the shear modulus of the elastomer at the shear strain and aXial pressure under consideration Ab is the bonded area T r is the total thickness of rubber n is the number of individual rubber layers and tr is the thickness of one rubber layer Leadrubber bearings are generally modeled as bilinear components where the zerodisplacement force intercept can be estimated as Qd Alay and the secondslope stiffness can be estimated as K GAb GAb d T ntr r where A is the area of the lead core try is the yield stress of the lead core and all other terms have been de ned previously Friction bearings are generally modeled as bilinear components where the zerodisplacement force intercept can be estimated as Q W and the secondslope stiffness can be estimated as where u is the coef cient of friction W is the supported weight and R is the radius of curvature of the sliding surface The above equations facilitate the preliminary sizing of elastomeric and sliding isolators Many additional checks must be performed by either the manufacturer or the engineer to characterize isolators suf ciently for nal design including checks for isolator buckling and maXimum strain elastomeric bearings and minimum component thickness sliding bearings Such checks are beyond the scope of this class but will be covered in the Fall class on seismic isolation Module 05 Page 17 CIE 619 Earthquake Engineering and Structural Dynamics II Instructor Andrew Whittaker 14 Analysis and Design of Seismic Isolation Systems I 41 Introduction Procedures for the analysis and design of seismic isolation systems are available in the 2000 NEHRP Recommended Provisions for new construction and FEMA 356 for retro t construction The procedures presented in the two documents are most similar because the same key people prepared both documents The presentation that follows draws heavily on FEMA 356 because the twocolumn format in that document is more compact than the onecolumn format of the 2000 NEHRP Recommended Provisions Isolation is used for buildings where the performance level is operational or functional following a major earthquake The procedures of 2000 NEHRP and FEMA 356 assume 0 No damage to the superstructure in the design earthquake o Adequate performance ability to support gravity loads and sustain aftershocks of the seismic isolators in the maXimum earthquake The 2000 NEHRP Recommended Provisions and FEMA 356 support four methods of analysis of isolation systems namely 0 Linear static l 0 Linear responsehistory rarely used o 1ncreas1ng level of rigor o Responsespectrum l 0 Nonlinear response history Linear and nonlinear analysis of isolation systems per 2000 NEHRP or FEMA 356 generally assume that the forcedisplacement response of the isolation system can be characterized by the hysteresis loop shown below FORCE 1 PLASTIC DEFORMATION 390 K DISPLACEMENT Module 05 Page 18 CIE 619 Earthquake Engineering and Structural Dynamics II Instructor Andrew Whittaker 142 Linear Static Analysis The linear static method of analysis should be used for the preliminary design of all isolation systems and can be used for the nal design of isolation systems that can be loosely characterized as linear viscoelastic effective stiffness and equivalent viscous damping Such systems include For linear static analysis the hazard is de ned using the standard NEHRP horizontal response spectrum see Module 02 See the gure below I m a T c 339 sxs Bs 2 Ts O 2 33 sxsBs E s L 3 XSBS o I g I I S SnIBJ I l 3 I I I C l 8 39 39 J 39 39 g 8X1B1 F T i I I 04st I E l I 39 ll I 8 I a a To T3 10 Period T The effect of damping is captured using BS and B1 as discussed in Module 02 For completeness the damping coef cients BS and B1 of FEMA 356 are presented below as a function of the effective damping 8 Effective Viscous Damping B percentage of critical damping 35 B1 5 2 08 08 5 10 10 10 13 12 20 18 15 30 23 17 40 27 19 a 50 30 20 l Damping coef cients shall be based on linear interpolation for effective viscous damping values othet than those given Module 05 Page 19 CIE 619 Earthquake Engineering and Structural Dynamics II Instructor Andrew Whittaker How are the isolators modeled for linear static analysis As linear Viscoelastic components with 0 Effective stiffness at selected displacement 0 Equivalent Viscous damping ratio The key step in the static analysis is the calculation of the displacement of the isolation system in the design and maximum earthquakes Below are excerpts from FEMA 356 gt I m a T a s sxsas 2 T 04 0 2 SggsxsBs E L 8 t 39 a I sslar I 0 to I 39 r l a J g 8X1B1 F T i I I E 04st i U I I 8 I a Q to To Ts 10 Period T Module 05 9242 Deformation Characteristics of the Isolation System The deformation characteristics of the isolation system shall be based on tests performed in accordance with Section 929 39 The deformation characteristics of the isolation system shall explicitly include the effects of the windrestraint and tiedown systems and supplemental energy dissipation devices if such systems and devices are used to meet the design requirements of this standard 9243 Minimum LateralDisplacements 92431 Design Displacement The isolation system shall be designed and constructed to withstand as a minimum lateral earthquake displacements that act in the direction of each of the main horizontal axes of the structure in accordance with Equation 92 92 where SX is evaluated for the Design Earthquake 92432 Effective Period at the Design Displacement The effective period TD of the isolated building at the design displacement shall be determined using the deformation characteristics of the isolation system in accordance with Equation 93 93 92433 Maximum Displacement The maximum displacement of the isolation system BM in the most critical direction of horizontal response shall be calculated in accordance with Equation 94 94 where CIE 619 Earthquake Engineering and Structural Dynamics II S is evaluated for the BSE2 92434 Effective Period at the Maximum Displacement The effective period TM of the isolated building at the maximum displacement shall be determined using the deformation characteristics of the isolation system in accordance with Equation 95 95 92435 Total Displacement The total design displacement D70 and the total maximum displacement Dr of components of the isolation system shall include additional displacement due to actual and accidental torsion calculated considering the spatial distribution of the effective stiffness of the isolation system at the design displacement and the most disadvantageous location of mass eccentricity The total design displacement DTD and the total maximum displacement DTM of components of an isolation system with a uniform spatial distribution of effective stiffness at the design displacement shall be taken as not less than that prescribed by Equations 96 and 97 39 12e D D 1 y 96 TD D b2 d2 b 42 A value for the total maximum displacement DTM less than the value prescribed by Equation 97 but not less than 11 times DM shall be permitted provided the isolation system is shown by calculation to be con gured to resist torsion 9244 92441 Minimum Lateral Forces isolation System and Structural Components and Elements at or below the isolation System The isolation system the foundation and all other structural components and elements below the isolation Module 05 Instructor Andrew Whittaker system shall be designed and constructed to withstand a minimum lateral seismic force Vb prescribed by Equation 98 92442 Structural Components and Elements above the isolation System The components and elements above the isolation system shall be designed and constructed to resist a minimum lateral seismic force Vs equal to the value of Vb prescribed by Equation 98 92443 Limits on V The value of Vs shall be taken as not less than the following 1 The base shear corresponding to the design wind load 2 The lateral seismic force required to fully activate the isolation system factored by 15 92444 Vertical Distribution of Force The total force Vs shall be distributed over the height of the structure above the isolation interface in accordance with Equation 99 V w h Fx 54 99 2 Will At each level designated as x the force Fx shall be applied over the area of the building in accordance with the weight wx distribution at that level hJr Response of structural components and elements shall be calculated as the effect of the force F x applied at the appropriate levels above the base Page 21 CIE 619 P 39 39 P 39 39 and Structural Dynamics II Instructor39 Andrew Whittaker Limitations on the use of the static method are listed in Section 9233 of FEMA 356 and Section 13252 of the 2000 NEHRP Recommended Provisions The method can be used if The isolated building is located at a site with S1 g 06 Isolated building is located on Class A B C or D site Isolated building is low rise less than 65 ft in height and regular Effective period at maXimum displacement is less than 3 seconds Factor of 3 difference between the xed base and isolated superstructure periods Isolation system is not too nonlinear has restoring force has forcede ection properties independent of the rate of loading vertical loading and bilateral loading and does not have a displacement restraint 143 Response Spectrum Analysis Responsespectrum analysis is often used when the limitations on the static method cannot be met namely 0 To capture highermode effects re ected through the 65 ft height limit on the static method 0 If the effective period at the maXimum displacement is greater than 3 seconds 0 If the superstructure framing is irregular Linear viscoelastic characterizations of the isolators are used in the mathematical model of the isolated building The 2000 NEHRP Recommended Provisions and FEMA 356 limit the use of the responsespectrum procedure such that the Isolation system must have a restoring force Isolator behavior is independent of rate of loading and vertical load Isolators have no displacement restraint The isolator not too nonlinear The damping in the isolation system is less than or equal to 30 of critical Module 05 Page 22 CIE 619 F 39 39 F 39 39 and Structural Dynamics II lnstructor39 Andrew Whittaker Unless the isolators and the isolation system can meet all of the above requirements nonlinear response hisiory analysis must be used Further if responsespectrum analysis is used Orthogonal earthquake shaking effects must be considered for displacement and force calculations Total design and total maXimum displacements cannot be less than a selected percentage of those displacements calculated using the linear static method Component actions cannot be less than a selected percentage of those actions calculated using the linear static method 144 Nonlinear Dynamic Analysis Nonlinear dynamic analysis can be used for all isolation systems regardless of height size geometry location and nonlinearity of the isolation system and must be used when the limits set forth on the linear static and responsespectrum procedures cannot be met For such analysis the seismic isolation bearings are modeled eXplicitly as nonlinear components often using the bilinear representation of Section 13 above Bidirectional earthquake inputs should be considered for nonlinear responsehistory analysis and coupled plasticity or BoucWen models should be used to capture the response of the individual isolators 0 Such models work well for leadrubber and FF bearings o Highdamping rubber bearings cannot be modeled well with such models 145 Properly Modification Factors Components of elastomeric and sliding seismic isolation bearings are constructed with materials whose properties can change over the life span of the isolated structure Such variations in material properties should be considered in the design of the isolation system The property modi cation factor approach that was developed by Constantinou can be used for this purpose and such an approach canwill be found in o 1999 AASHTO Guide Speci cation for Seismic Isolation Design 0 2003 NEHRP Recommended Provisions For elastomeric bearings consideration should be given to o Scragging Aging Velocity Temperature For sliding bearings consideration should be given to o Contamination Aging Velocity Temperature Wear Module 05 Page 23 CIE 619 F 39 39 F 39 39 and Structural Dynamics II Instructor39 Andrew Whittaker How are such changes in material properties considered in AASHTO and what is proposed for the 2003 NEHRP Recommended Provisions 0 A H c 139 max 0 15139115 mm i min For linear viscoelastic isolation bearings the effective stiffness would have to be increased and decreased from the target range by 11 and 1111 respectively and bounding analysis undertaken to calculate maXimum displacements and forces For bilinear systems the values of A would have to be calculated for the zerodisplacement force intercept and the secondslope stiffness and then applied to the target ranges for both variables to enable bounding analysis to be undertaken to calculate maXimum displacements and forces 14 6 Additional Design Considerations Superstructure design The design of isolated superstructures is based for new construction on the forces developed in the isolation system during the design earthquake 0 Force distribution over the height of the building 0 Framing must stay essentially elastic strength or LRFD design 0 Ductile detailing is required Why 0 What is the eXpected response of the superstructure in the maXimum earthquake Design and detailing of an isolation system must account for the large displacements that develop across the isolation interface Some of the considerations are 0 Displacement capacity of nonstructural components that cross the isolation plane 0 Displacement capacity around the isolation superstructure moat o Secondorder P 7 A effects on the framing immediately above and below the isolators Module 05 Page 24 CIE 619 F 39 39 F 39 39 and Structural Dynamics II Instructor39 Andrew Whittaker 0 Connection of the isolation bearing to the superstructure and substructure gt A signi cant challenge for seismic retro t using isolation 15 Seismic Isolation for Rehabilitation and Retro t To date seismic isolation has been employed for new and retro t construction For retro t construction isolation has been traditionally limited to historical buildings because conventional retro t would have destroyed too much historic fabric Examples of retro t of historic buildings using base isolation are 0 San Francisco City Hall leadrubber bearings 0 Oakland City Hall leadrubber bearings 0 Los Angeles City Hall highdamping rubber bearings 0 US Court of Appeals Friction Pendulum bearings Other retro t applications include museums data centers and webhosting centers such as the AboveNet building in San Francisco which is discussed below The biggest challenge facing those tasked with retro tting a building using seismic isolation relates to installation of the seismic isolation bearings The AboveNet facility is a lowrise building that was used for production of battle tanks immediately following the Second World War The lateral stiffness and strength of the building were substantial making it an ideal candidate for seismic isolation The client demanded postearthquake functionality from the facility which could only be delivered with seismic isolation The engineerofrecord selected FP bearings for this project Sample information from the retro t construction is presented below Artist s rendering of the building httn39 m erlnrd em hnFlen edn ClassHomePa es cie6l9I 39 39 tif Aerial View of building httn39 m erlnrd em hnFFaln edn ClassHomePa es cie6l9I ectures A 39 39 inc Exterior View of building httpoverlordengbuffaloeduClassHomePagescie6l9LecturesEXteriorjpg Basement construction work showing column reinforcement httn39 m erlnrd em hnFFaln edn ClassHomePa es cie6l9I ectures 39 39 inc Rebar cage around basement column Module 05 Page 25 CIE 619 F 39 39 F 39 39 and Structural Dynamics II Instructor Andrew Whittaker httpoverlordengbuffaloeduClassHomePagescie6l9LecturesRebarCageAroundColumnjpg Cutting column prior to isolator installation httn39 m erlnrd em hnFFaln edn ClassHomePa es cie6l9T 39 39 inc View of isolator before column below cast httn m erlnrd em hnFFaln edn ClassHomePa es cie6l9T Emu Y 39 39 r 39 inc Column jacking and stressing hardware httn39 m erlnrd em hnFFaln edn ClassHomePa es cie6l9I CClulcs 39 39 39 39 39 inc Grouting to underside of isolator httn39 m erlnrd em hnFFaln edn ClassHomePa es cie6l9T echm Y 39 In talk d inc Completed isolator installation atop interior column httn39 m erlnrd em hnFFaln edn ClassHomePa es cie6l9T emu 39 quot quot 39 inc Isolator installed in exterior wall httn39 m erlnrd em hnFFaln edn ClassHomePa es cie6l9T uululu 39 quot39 39 inc Module 05 Page 26 CIE 619 F 39 39 F 39 39 and Structural Dynamics II Instructor Andrew Whittaker 2 PASSIVE ENERGY DISSIPATION SYSTEMS 21 Principles ofPassive Energy Dissipation In conventional construction earthquakeinduced energy is dissipated in components of the gravityload resisting system The action of dissipating energy in framing such as beams in a momentresisting frame produces damage in those components Repair of such damage after an earthquake is typically expensive and often requires evacuation of the building while repair work on the gravity system is undertaken The objective of adding damping hardware to new and existing construction is to dissipate much of the earthquakeinduced energy in disposable elements not forming part of the gravity framing system Key to this philosophy is limiting or eliminating damage to the gravityloadresisting system Although testing and perhaps replacement of all supplemental damping devices in a building should be anticipated after a design earthquake evacuation of the building for repair might not be necessary and the total repair cost will likely be minor compared with the costs associated with repair and business interruption in a conventional building This section of Module 05 introduces the different types of supplemental damping hardware being used or considered for use in the United States at this time describes the new analysis procedures for supplemental dampers in FEMA 356 and presents two new damper con gurations that will facilitate the use of damping technologies in stiff structural framing systems Section 4 22 Types of Passive Energy Dissipation Hardware 22 Introduction Supplemental energy dissipation or damping hardware are divided into three categories hysteretic velocitydependent and other Examples of hysteretic systems include devices based on yielding of metal and friction The gure below presents sample forcedisplacement loops of hysteretic dampers Examples of velocitydependent systems include dampers consisting of viscoelastic solid materials dampers operating by deformation of viscoelastic uids eg viscous shear walls and dampers operating by forcing uid through an ori ce eg viscous uid dampers The behavior of velocitydependent systems is also shown in the gure Other systems have characteristics that cannot be classi ed by one of the basic types depicted in the gure including dampers made of shape memory alloys frictionalspring assemblies with recentering capabilities and uid restoring forcedamping dampers For information on these dampers the reader is referred to ATC 1993 Constantinou et al 1998 EERI 1993 Soong and Constantinou 1994 and Soong and Dargush 1997 Only hysteretic and velocitydependent dampers are discussed in this presentation Module 05 Page 27 CIE 619 F 39 39 F 39 39 and Structural Dynamics II Instructor39 Andrew Whittaker Force A Force Displacement Displacement a Metallicyielding b Friction Force Force Dlsplacement Displacement c Viscoelastic d Viscous Some types of supplemental damping systems can substantially change the forcedisplacement response of a building by adding strength and stiffness Such in uence is demonstrated in the gure below for metallic yielding friction and Viscoelastic dampers Note that these gures are schematic only and that the forcedisplacement relation for the righthand gure assumes that the framing supporting the friction dampers is rigid Viscous damping systems will generally not substantially change the pseudostatic forcedisplacement response of a building The analysis procedures described in the following section account for these effects on framingsystem response With metallicyielding or Viscoelastic With friction dampers dampers Base shear Base shear No dampers No dampers Roof displacement Roof displacemeg t Module 05 age 28 cm 619 Eanhquake Engineering and Structural Dynamics 11 Instructor Andrew Whittaker 222 Hystzrztzc m DzsplaczmemrDependem Dampers Hysteretic dampers exhibit bilinear or trilinear hysteretic elastoplastic or rigidplastic frictional behavior which can be easily captured with structural analysis software currently in the marketplace Details on the modeling of metallicyielding dampers may be found in Whittaker et al 1989 the steel dampers described by Whittaker exhibit stable forcedisplacement response and no temperature dependence Aiken et al 1990 and Nims et al 1993 describe friction devices one type ofmetallic yielding damper the ADAS element is shown below Added Damping and Stiffness ADAS elements have been implemented in one project in the United States j Stiff support framing is required for the metallic yielding dampers such as that shown above to ensure that the displacement across the height ofthe damper is maximized and approximately equal to the interstory displacement for that story in which the damper is installed Such framing is likely more costly than the damper it supports An alternate metallic yielding damper the unbonded brace is shown below to the right This damper was developed in Japan in the mi 1 8 s atanabe et al 1988 and has been used on anumber ofprojects in California This figure illustrates the key components ofthe Nippon Steel brace namely a cruciform cross section of welded steel plate often lowyield steel that is designed to yield in tension and compression and an exterior steel tube of circular or rectangular cross section that is selected such that the buckling capacity of the tube exceeds the squash load of the rucifonn cross section The space between the cruciform cross section and t e steel tube is filled with a concretelike material to delay local buckling of the cruciform cross section outstands proprietary materials are used to debond the cruciform cross section from the concretelike material photographs ofthe unbonded brace an f some sample projects were presented in Module 04 The unbonded brace is designed to have approximately equal strength in tension and compression and is conceptually superior to the concentrically braced frame because the eam at the intersection point ofthe chevron braces does not have to be designed for large outofbalance vertical forces Bruneau et al 1998 Module 05 Page 29 CIE 619 F 39 39 F 39 39 and Structural Dynamics II Instructor39 Andrew Whittaker 22 3 Velocity Dependent Dampers Solid viscoelastic dampers typically consist of constrained layers of viscoelastic polymers They eXhibit viscoelastic solid behavior with mechanical properties dependent on frequency temperature and amplitude of motion A forcedisplacement loop for a viscoelastic solid device under sinusoidal motion of amplitude A0 and frequency a is shown in the gure below The force in the damper may be eXpressed as F KeffA CA where K 619 is the effective stiffness also termed the storage stiffness K 39 as de ned in the gure C is the damping coefficient and A and A are the relative displacement and relative velocity between the ends of the damper respectively Force Fl D39Eplazement 51 W3 The damping coefficient is calculated as WD 7 2 7raJA 0 where WD is the area enclosed within the hysteresis loop and a is the angular frequency of excitation The damping coefficient C is also equal to the loss stiffness K quot divided by a The effective stiffness and damping coefficient are dependent on the frequency temperature and amplitude of motion The frequency and temperature dependence of viscoelastic polymers generally vary as a function of the composition of the polymer The standard linear solid model a spring in series with a Kelvin model which can be implemented in commercially available structural analysis software is capable of modeling behavior over a small range of frequencies which will generally be satisfactory for most projects Fluid viscoelastic devices which operate on the principle of deformation shearing of viscoelastic uids have behavior that resembles a solid viscoelastic device However uid viscoelastic devices have zero effective stiffness under static loading conditions Fluid and solid viscoelastic devices are distinguished Module 05 Page 30 CIE 619 Earthguake Engineering and Structuml Dynamics II Instructor Andrew Whittaker by the mtio of the loss stiffness to the effective or storage stiffness This ratio approaches in nity for uid devices and zero for solid viscoelastic devices as the loading frequency approaches zero Fluid viscoelastic behavior may be modeled with advanced models of viscoelasticity Makris 1993 However for most pmctical purposes the Maxwell model a spring in series with a dashpot can be used to model uid viscoelastic devices The gure to the right presents a photograph of a uid viscoelastic damper that is known by many as a viscous damping wall VDW The VDW is 39 39 composed of a cavityprecast wall that is lled with a viscous uid and attached at its base to the oor fmming A teeshaped paddle is inserted in the uid and is attached to the fmming above the cavity wall Interstory dri in the building frame produces relative movement between the paddle and the cavity wall and dissipates energy The VDW has been proposed for use in the United States but there are no known applications at the time of this writing Pure viscous behavior may be produced by forcing uid through an ori ce Constantinou and Symans 1993 Soong and Constantinou 1994 The force output of a viscous damper has the general form F CO A a sgnA where A is the velocity 1 is an exponent in the range of01 to 20 and sgn is the signum function The simplest form is the linear uid damper for which the exponent is equal to 10 In this presentation the discussion on uid viscous devices is limited to linear uid dampers for a detailed treatment of nonlinear uid viscous dampers the reader is referred to Soong and Constantinou 1994 Force F1 Displace mam A Module 05 Page 31 CIE 619 Earthguake Engineering and Structuml Dynamics II Instructor Andrew Whittaker Fluid viscous dampers are widely used in the United States at this time Much of the technology used in this type of damper was developed for military aerospace and energy applications The gure immediately below shows the internal construction of a singleended uid viscous damper manufactures by Taylor Devices As shown in the gure uid ows from one side of the piston head to the other via annular ori ces and ori ces in the piston head The design of such ori ces is proprietary ACCUMJLATOH SEAL RETAINER HCUSING EH STRENGTH CEYAL RESXN SEAL ch INDEFI CEIMP RES ISLE srchoN FLUID quotSm wastage mm motes To the right is a photogmph of a double acting nonlinear uid viscous dampers used in a new l4story building in San ancisco Such dampers are o en compact because the uid pressure drop across the damper piston head generally mnges between 5000 and 1000 psi 35 to 70 MPa Module 05 Page 32 CIE 619 F 39 39 F 39 39 and Structural Dynamics II Instructor Andrew Whittaker 23 Analysis and Design of Passive Energy Dissipation Systems 2 3 I Introduction The lack of analysis methods guidelines and commentary has been the key impediment to the widespread application of supplemental dampers in buildings and bridges Prior to 1997 seismic design codes and guidelines in the United States focused on designing structures for strength alone where the design forces were set equal to the elastic forces divided by a response reduction factor for buildings Component deformations which are indicators of damage and performance were not checked FEMA 273 entitled Guidelines for the Seismic Rehabilitation of Buildings was published in 1997 after more than ve years of development FEMA 273 represented a paradigm shift is the practice of earthquake engineering in the United States because deformations and not forces were used as the basis for the design of ductile components Performance and damage were characterized in terms of component deformation capacity Recently FEMA 273 was republished as FEMA 356 a Prestandard and Commentaryfor the Seismic Rehabilitation ofBuildings Four methods of seismic analysis are presented in FEMA 356 Linear Static Procedure LSP Linear Dynamic Procedure LDP Nonlinear Static Procedure NSF and Nonlinear Dynamic Procedure NDP All four procedures can be used to implement supplemental dampers in buildings although the limitations on the use of the linear procedures likely will limit their widespread use Of the four only the NDP can eXplicitly capture nonlinear deformations and strain and loadhistory effects The other three procedures are considered to be less precise than the NDP although given the additional uncertainties associated with nonlinear analysis the loss of accuracy might be small The two nonlinear procedures lend themselves to component checking using deformations and displacements component deformation limits are given in FEMA 356 but as discussed in Module 04 most are based on engineering judgment and evaluation of limited test data The two static methods are described below Much additional information on these procedures are available in FEMA 273 274 and FEMA 356 Only implementation of energy dissipation or damping devices using the nonlinear static procedure is discussed below 232 Implementation ofPassive Dampers Using the Nonlinear Static Procedure Two methods of nonlinear static analysis are provided in FEMA 273 for implementing energy dissipation devices or supplemental dampers Method 1 known as the coe icient method and Method 2 known as the capacity spectrum method The two methods are equally precise The use of the coefficient method to implement dampers is described below The reader is encouraged to read Tsopelas et al 1997 and Section C935 of FEMA 274 FEMA 1997 for key information on the use of the capacityspectrum method Only the coefficient method is presented in FEMA 356 Regardless of which method is used to calculate the target displacement and the associated component deformations the nonlinear mathematical model of the building frame must include the nonlinear force Velocitydisplacement relations for the dampers and the mechanical characteristics of the framing supporting the dampers If the stiffness of a damper is dependent upon amplitude frequency or velocity Module 05 Page 33 CIE 619 F 39 39 F 39 39 and Structural Dynamics II Instructor39 Andrew Whittaker the stiffness value used for analysis should be consistent with deformations corresponding to the target quot 39 and f 39 quot to the inverse of the secant period at the maximum 1 1 1 displacement T x As described in Module 04 calculation of the target displacement by the coefficient method is based on the assumption that for periods greater than approximately 05 second for a rock site displacements are preserved in a mean sense that is the mean elastic displacements are approximately equal to the mean inelastic displacements Note that the degree of scatter in the ratio of elastic and inelastic displacements may be substantial and that this assumption is not conservative for buildings with low strength The general form of the target displacement 6 equation from Module 04 is given by T2 S 3 0055T T2 6fC0CCCS CCCCW e 1 2 3 a eff Te472 0 1 2 3 B 4 2C1C2C36e where all terms were de ned previously DisplacementDependent Dampers The bene t of adding displacementdependent dampers to a building frame is recognized in FEMA 273 by the increase in building stiffness afforded by the dampers The increase in stiffness will reduce the effective period T e in the above thereby reducing the maXimum displacement 6 The spectral acceleration in this equation should be calculated using the effective period of the mathematical model that includes the stiffness of the dampers and the value of B assigned to the building frame exclusive of the dampers The effect of adding displacementdependent dampers to a building frame dashed line is indicated in the sketch below 0 Increase in maXimum acceleration Base shear 6 jetro L Pushover curve for eXisting building Pushover curve for retro tted building I Roof quot 139 Module 05 Page 34 CIE 619 F 39 39 F 39 39 and Structural Dynamics II Instructor39 Andrew Whittaker Veloci Dependent Dampers The bene ts of adding velocitydependent dampers to a building frame are recognized in FEMA 273 and FEMA 356 by a the increase in viscous damping and b the increase in building stiffness afforded by the dampers The increase in damping will reduce the spectral acceleration The increase in stiffness will reduce the effective period and the spectral displacement as noted above The gure below shows the effect of adding viscoelastic and viscous dampers to a building frame 0 Is this the complete picture Base shear 6 tretro strawviscous Pushover curve for eXisting building and for retro tted building with viscous dampers Pushover curve for retro tted building with viscoelastic dampers I Roof quot l 39 The effective damping in the building frame at the point of maximum displacement is calculated iteratively using the following four equations The effective damping of the frame incorporating velocity dependent dampers is given by Z W 139 7 ef 477Wk where 3 is the damping in the building frame exclusive of the dampers W J is the work done by device j in one complete cycle corresponding to oor displacements 6 and the summation eXtends over all devices Wk is the maXimum strain energy in the frame that can be calculated as l Wk 7236 Module 05 Page 35 CIE 619 Earthquake Engineering and Structural Dynamics II Instructor Andrew Whittaker where the oor displacements 51 are those that correspond to the target displacement and the forces Fl are the lateral forces on the frame at the target displacement and the summation extends over all oor levels 139 The value of Be is used to calculate a value of B from the table on page 19 The spectral acceleration for use in the target displacement equation on page 34 is calculated using the fundamental period of the framing system whose mathematical model includes the secant stiffness of the velocitydependant dampers The effective damping is calculated using the previous two equations For linear viscous dampers the work done by damper j can be calculated as 2 2 W 7zC 5 LC62 J J 7 7 T J VJ S where C is the damper coefficient for device j 6r is the relative displacement between the ends of device j measured along the aXis of the damper and TS is the secant period of the framing system see below at the maximum displacement of the building frame whose mathematical model includes the stiffness of the dampers This equation assumes harmonic motion of amplitude 6r and periodicity TS As such it may not be applicable if nearfield nonharmonic ground motions are being used for design FEMA 273 and FEMA 356 provide an alternative equation for calculating the effective damping provided by linear viscous devices using first mode data 2 2 TSng cos 6j rj e 47IZ mi i2 l where pl is the first mode displacement of oor level 139 ml is the mass of oor level i 91 is the angle of device j to the horizontal or is first mode relative horizontal displacement between the ends of device j and all other terms are described above The secant period at maXimum displacement can be calculated using the following equation where terms are defined in the Base shear 5 figure to the right K TS Te 6 06Vy 5y 6 Roof displacement Module 05 Page 36