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337 Chapter 5 Base isolation systems 5.1 Introduction The term isolation refers to the degree of interaction between objects. An object is said to be isolated if it has little interaction with other objects. The act of isolating an object involves providing an interface between the object and its neighbors which minimizes interaction. These definitions apply directly to various physical systems. For example, one speaks of isolating a piece of equipment from its support by mounting the equipment on an isolation system which acts as a buffer between the equipment and the support. The design of isolation systems for vibrating machinery is a typical application. The objective here is to minimize the effect of the machine induced loading on the support. Another application is concerned with minimizing the effect of support motion on the structure. This issue is becoming increasingly more important for structures containing motion sensitive equipment and also for structures located adjacent to railroad tracks or other sources of ground disturbance. Although isolation as a design strategy for mounting mechanical equipment has been employed for over seventy years, only recently has the concept been seriously considered for civil structures, such as buildings and bridges, subjected to ground motion. This type of excitation interacts with the structure at the foundation level, and is transmitted up through the structure. Therefore, it is logical to isolate the structure at its base, and prevent the ground 338 Chapter 5: Base Isolation Systems motion from acting on the structure. The idea of seismic isolation dates back to the late nineteenth century, but the application was delayed by the lack of suitable commercial isolation components. Substantial development has occurred since the mid 1980’s (Naeim and Kelly, 1999), and base isolation for certain types of civil structures is now considered to be a highly viable design option by the seismic engineering community, particularly in Japan (Wada, 1998), for moderate to extreme seismic excitation. A set of simple examples are presented in the next section to identify the key parameters and illustrate the quantitative aspects of base isolation. This material is followed by a discussion of practical aspects of seismic base isolation and a description of some seismically isolated buildings. The remaining sections deal with the behavioral and design issues for base isolated MDOF structural systems. Numerical results illustrating the level of performance feasible with seismic base isolation are included to provide a basis of comparison with the other motion control schemes considered in this text. 5.2 Isolation for SDOF systems The application of base isolation to control the motion of a SDOF system subjected to ground motion was discussed earlier in Section 1.3 as part of a general treatment of design for dynamic excitation. The analytical formulation developed in that section provides the basis for designing an isolation system for simple structures that can be accurately represented with a SDOF model. Examples illustrating the reasoning process one follows are presented below. The formulation is also extended to deal with a modified version of a SDOF model that is appropriate for a low-rise building isolated at its base. This model is useful for preliminary design. SDOF examples The first example considers external periodic forcing of the SDOF system shown in Fig. 5.1. The solution of this problem is contained in Section 1.3. For convenience, the relevant equations are listed below: 5.2 Isolation for SDOF Systems 339 Fig. 5.1: SDOF system. (5.1) (5.2) (5.3) (5.4) (5.5) (5.6) Given and , one can determine for a specific system having mass , stiffness , and damping . With known, the forces in the spring and damper can be evaluated. The reaction can be found by either summing the internal forces, or combining with the inertia force. With the latter approach, one writes (5.7) and expands the various terms using eqns (5.1) through (5.6). The result is expressed as (5.8) (5.9) k c m u R p pp ˆ Ωtsin= uu ˆ Ωt δ–()sin= u ˆ H 1 k p ˆ = H 1 1 1 ρ 2 –[] 2 2ξρ[] 2 + = ρ Ω ω = δtan 2ξρ 1 ρ 2 – = p ˆ Ω u ˆ m kcu ˆ p Rpmu ˙˙ –= RR ˆ Ωt δ r –()sin= R ˆ H 3 p ˆ = 340 Chapter 5: Base Isolation Systems (5.10) (5.11) The function, H 3 , is referred to as the transmissibility of the system. It is a measure of how much of the load p is transmitted to the support. When , and reduces to . Figure 5.2 shows the variation of with and . Fig. 5.2: Plot of versus . The model presented above can be applied to the problem of designing a support system for a machine with an eccentric rotating mass. Here, one wants to minimize the reaction force for a given , i.e. one takes . Noting Fig. 5.2, this constraint requires the frequency ratio, , to be greater than , and it follows that H 3 12ξρ[] 2 + 1 ρ 2 –[] 2 2ξρ[] 2 + = δ r tan ρ 2 H 1 δsin 1 ρ 2 H 1 δcos+ –= ξ 0= δ 0= H 3 H 1 H 3 ρξ 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 ρ Ω ω = H 3 2 ξ 0= ξ 0.2= ξ 0.4= H 3 ρ p ˆ H 3 1< ρ 2 5.2 Isolation for SDOF Systems 341 (5.12) The corresponding periods are related by (5.13) where is the forcing period. For example, taking results in , a reduction of from the static value. The second example illustrates the strategy for isolating a system from support motion. Applying the formulation derived in Section 1.4 to the system shown in Fig. 5.3, the amplitudes of the relative and total displacement of the mass, and , are related to the support displacement by (5.14) (5.15) Taking small with respect to unity reduces the effect of support motion on the position of the mass. The frequency and period criteria are the same as those of the previous example. One takes to reduce . However, since H 2 approaches unity as increases, the magnitude of the relative motion increases and approaches the ground motion, . Therefore, this relative motion needs to be accomodated. Fig. 5.3: SDOF system subjected to support motion. These examples show that isolation is obtained by taking the period of the SDOF system to be large in comparison to the forcing (either external or support) ω Ω 2 < TT f 2> 2 2π Ω   = T f T 3T f = R ˆ 0.125p ˆ = 87.5% u ˆ u ˆ t u ˆ ρ 2 H 1 u ˆ g H 2 u ˆ g == u ˆ t H 3 u ˆ g = H 3 ρ 2> u ˆ t ρ u ˆ g k c m u t u g u+= u g 342 Chapter 5: Base Isolation Systems period. Expressing this requirement as (5.16) where depends on the desired reduction in amplitude, the constraint on the stiffness of the spring is given by (5.17) It should be noted that this derivation assumes that a single periodic excitation is applied. The result is applicable for narrow band excitations which are characterized by a dominant frequency. A more complex analysis involving iteration on the stiffness is required to deal with broad band excitations. One has to ensure that the forcing near the fundamental frequency is adequately controlled by damping in this case. Bearing terminology The spring and damper elements connecting the mass to the support are idealizations of physical objects called bearings. They provide a constraint against motion relative to a support plane, as illustrated in Fig. 5.4. The bearing in Fig. 5.4(a) functions as an axial element and resists the displacement normal to the plane with normal stresses (tension and compression). The bearing shown in Fig. 5.4(b) constrains relative tangential motion through shearing action over the height of the bearing. These elements are usually combined into a single compound bearing, but it is more convenient to view them as being uncoupled when modeling the system. Fig. 5.4: Axial and shear bearings. T ρ ∗ T f ≥⇒ω Ω ρ * < ρ ∗ km Ω ρ ∗ 2 < m 2π ρ ∗ T f 2 = F n , u n axial bearing F t , u t shear bearing (a) (b) 5.2 Isolation for SDOF Systems 343 When applying the formulation developed above, one distinguishes between normal and tangential support motion. For normal motion, axial type bearings such as springs and rubber cushions are used; the defined by eqn (5.17) is the axial stiffness of the bearing . Shear bearings such as laminated rubber cushions and inverted pendulum type sliding devices are used when the induced motion is parallel to the ground surface. In this case, represents the required shearing stiffness of the bearing, . Figure 5.5 shows an air spring/damper scheme used for vertical support. Single and multiple stage laminated rubber bearings are illustrated in Fig 5.6. Rubber bearings used for seismic isolation can range up to 1 m in diameter and are usually inserted between the foundation footings and the base of the structure. A particular installation for a building is shown in Fig 5.7. Fig. 5.5: Air spring bearing. k F n u n ⁄ k F t u t ⁄ 344 Chapter 5: Base Isolation Systems a) Single stage b) multiple stage Fig. 5.6: Laminated rubber bearings. 5.2 Isolation for SDOF Systems 345 Fig. 5.7: Rubber bearing seismic isolation system. Modified SDOF Model In what follows, the support motion is considered to be due to seismic excitation. Although both normal (vertical) and tangential (horizontal) motions occur during a seismic event, the horizontal ground motion is generally more significant for structural systems since it leads to lateral loading. Typical structural systems are designed for vertical loading and then modified for lateral loading. Since the vertical motion is equivalent to additional vertical loading, it is not as critical as the horizontal motion. The model shown in Fig. 5.3 represents a rigid structure supported on flexible shear bearings. To allow for the flexibility of the structure, the structure can be modeled as a MDOF system. Figure 5.8 illustrates a SDOF beam type idealization. One can estimate the equivalent SDOF properties of the structure by assuming that the structural response is dominated by the fundamental mode. The data provided in earlier chapters shows that this assumption is reasonable for low-rise buildings subjected to seismic excitation. An in-depth analysis of low rise buildings modeled as MDOF beams is presented later in this chapter. The objective here is to derive a simple relationship showing the effect of the bearing stiffness on the relative displacement of the 346 Chapter 5: Base Isolation Systems structure, , with respect to the base displacement, . The governing equations for the lumped mass model consist of an equilibrium equation for the mass, and an equation relating the shear forces in the spring and the bearing. (5.18) (5.19) Fig. 5.8: Base isolation models. Neglecting damping, eqn (5.19) can be solved for u b in terms of u. (5.20) Then, substituting for u b in eqn (5.18) leads to (5.21) Equation (5.21) is written in the conventional form for a SDOF system (5.22) where Γ is a participation factor, (5.23) uu b u g + mu ˙˙ cu ˙ ku++ mu ˙˙ b u ˙˙ g +()–= k b u b c b u ˙ b + ku cu ˙ += u g u b u g + uu b u g ++ m k , c k b , c b (a) Actual structure (b) Beam idealization (c) Lumped mass model u b k k b u= m 1 k k b + u ˙˙ ku+ mu ˙˙ g –= u ˙˙ ω eq 2 u+ Γu ˙˙ g –= Γ k b kk b + k b k   1 k b k +   ⁄== [...]... Section 5. 3 deals with this problem 1 0.9 0.8 0.7 k -kf 0.6 0 .5 0.4 0.3 0.2 0.1 0 0 1 2 3 4 5 u b∗ -u∗ 6 7 8 Fig 5. 9: Variation of k ⁄ k f with u b∗ ⁄ u∗ 9 10 5. 2 Isolation for SDOF Systems 353 1 0.9 0.8 0.7 kb -kf 0.6 0 .5 0.4 0.3 0.2 0.1 0 0 1 2 3 4 5 u b∗ -u∗ 6 7 8 9 10 9 10 Fig 5. 10: Variation of k b ⁄ k f with u b∗ ⁄ u∗ 12 10 T eq -Tf 8 6 4 2 0 0 1 2 3 4 5 u b∗ -u∗ 6 7 8 Fig 5. 11:... ∞ reduces eqn (5. 38) to m u max = S v k (5. 44) The fixed base structural stiffness k f follows from eqn (5. 44) 2 mS v k f = k k = ∞ = 2 b [ u∗ ] (5. 45) Using eqn (5. 45) and assuming the value of S v is the same for both cases, the stiffness ratios reduce to, 1 k - = -kf u b∗ 1 + -u∗ u∗ -u b∗ kb - = -kf u b∗ 1 + -u∗ (5. 46) (5. 47) The ratio of the isolated period to... Chapter 5: Base Isolation Systems One needs to take ρ eq > 2 Noting eqn (5. 27), the required value of ρ eq is 1 2 ρ eq = 1 + - (5. 29) Substituting for ρ eq in eqn (5. 28) leads to Ω2 2 ω eq = -1 1 + - (5. 30) Finally, using eqn (5. 23) and (5. 24), the required bearing stiffness is given by 1 k k b = k = -k  k(1 + (1 ⁄ ν))  - – 1 - – 1 ... (5. 54) ˆ F = f d G s u [ sin Ωt + η cos Ωt ] (5. 55) ˆ ˆ ˆ u = γ h = γ nt b (5. 56) A A f d = - = h nt b (5. 57) where Note that f d depends on the bearing geometry whereas η and G s are material properties The standard form of the linearized force-displacement relation is defined 360 Chapter 5: Base Isolation Systems by eqn (3.70) ˙ F = k eq u + c eq u (5. 58) where k eq and c eq are the equivalent linear... stiffness factors are related by νs -k k b = = 0.1k νb (2) Evaluating Γ and ρ eq , using eqns (5. 34) and (5. 35) , νs 0.1 = = 0.0909 νs + νb 1.1 Γ 1 1 1 – - = = = 0.909 2 νs 1.1 ρ eq 2 ρ eq (3) (4) = 11.0011 leads to 2 ω eq = 0.0909Ω 2 and finally to k (5) 350 Chapter 5: Base Isolation Systems m 2 - k = - eq = mΩ 2 Γ (6) Seismic excitation - modified SDOF model An estimate of... relating the stiffness factors It reduces to 1 Γ – 1 + - = -2 νs ρ eq where (5. 34) 5. 2 Isolation for SDOF Systems 349 νs kb ⁄ k Γ = = - s + νb 1 + kb ⁄ k (5. 35) 2 2 Solving eqn (5. 34) for ρ eq leads to ω eq , and then k k Ω2 2 ω eq = - = Γ 2 m ρ eq (5. 36) The following example illustrates the computational steps Example 5. 1: Stiffness factors for prescribed structure and base... are eqns (3.74), (3.76), and (3.77) which are listed below for convenience 1 k eq = f d -N ∑ ) N Gs ( Ωi ) = f d G s (5. 59) i=1 c eq = αk eq N ∑ Gs η  -   Ω i i=1 α = -N ∑ (5. 60) (5. 61) Gs ( Ωi ) i=1 Equation (5. 58) is used in the MDOF analysis presented in a later section Figures 5. 17 and 5. 18 show that the material properties for natural and filled rubber are essentially constant... k2 k s = k 1 + - (5. 70) The equivalent loss factor is defined as 1 W ˜ - η = - 2π E S (5. 71) where W is the hysteretic work per cycle and E S is the maximum strain energy Evaluating the energy terms 2 2 W = 4 ( µ – 1 )k 2 u y + πηk 1 µ 2 u y (5. 72) 2 1 E S = k s [ µu y ] 2 (5. 73) and substituting in eqn (5. 71) leads to k1 4 ( µ – 1 )k 2 ˜ η = - + η -2 ks πk s µ (5. 74) –3 Noting... follows from eqn (5. 20) ub k = u max max kb (5. 39) In this development, the criteria for motion based design of a base isolated structure are expressed as limits on the relative motion terms u max = u∗ ub max = u b∗ (5. 40) (5. 41) The values of k and k b required to satisfy these constraints follow by solving eqns (5. 38) and (5. 39) 5. 2 Isolation for SDOF Systems 351 ku∗ k b = -u b∗ (5. 42) 2 2 mS v... y is about 5 ×10 about 0 .5 , one can estimate µ as and the typical peak response strain is ˆ γ µ = ≈ 100 γy (5. 75) A typical value for the ratio of k 1 to k 2 is k 1 ≈ 0.1k 2 (5. 76) ˜ Then, reasonable estimates for k s and η are k s = 1.1k 1 (5. 77) 0.1 4 ˜ η = + - η = 0.12 + 0.909η 0.11 11π (5. 78) The loss coefficient for high damping rubber can be as high as 0. 15 Combining a 5. 3 Design . For convenience, the relevant equations are listed below: 5. 2 Isolation for SDOF Systems 339 Fig. 5. 1: SDOF system. (5. 1) (5. 2) (5. 3) (5. 4) (5. 5) (5. 6) Given and , one can determine for a specific system. 1.8 2 0 0 .5 1 1 .5 2 2 .5 3 3 .5 4 4 .5 5 ρ Ω ω = H 3 2 ξ 0= ξ 0.2= ξ 0.4= H 3 ρ p ˆ H 3 1< ρ 2 5. 2 Isolation for SDOF Systems 341 (5. 12) The corresponding periods are related by (5. 13) where. section. u ∗ u b ∗ 2 S v Hγ ∗ H γ ∗ H 50 m= γ ∗ 1 200⁄= u ∗ 0.25m= u b u b S v u b ∗ u ∗ 0.25m= u b ∗ 0.3m= k 0. 455 k f = k b 0.833k= T 2.2T f = 5. 3 Design Issues for Structural Isolation Systems 355 5. 3 Design issues