Detailed design procedure of Isolator (lead rubber bearing)

1. ISOLATOR DESIGN
1. Seismic zone factor, z = 0.4 (Table 16 – I)
2. Soil profile type SB (Table 16 – J)
3. Seismic co-efficient, CA = 0.4 (Table 16 – Q)
4. Seismic co-efficient, CV = 0.48 (Table 16 – R)
5. Near source factor, NA = 1 (Table 16 – S)
6. Near source factor, NV = 1.2 (Table 16 – T)
7. MCE Shaking intensity MMZ NA =0.48 , MM = 1.21
8. MCE Shaking intensity MMZ NV =0.58
9. Seismic Source type A (Table 16 – U)
10. Distance of known source (km) = 10 ( from site seismology)
11. MCE Response co-efficient, MM = 1.21 (Table A-16-D)
12. Lateral force co-efficient, R1 = 2 (Table A-16-E)
13. Fixed base Lateral force co-efficient, R = 5.5 (Table 16 – N)
14. Importance factor, I = 1 (Table 16 – K)
15. Seismic co-efficient, CAM = 0.484 (Table A-16-F)
16. Seismic co-efficient, CVM= 0.581 (Table A-16-G)
17. Eccentricity, e =1.8 (5% of d)
18. Shortest building dimension, b = 16 (Building site)
19. Longest building dimension, d = 36
20. Dimension of extreme isolator, y = 18 (from geometry)
21. DTD / DD = DTM / DM = [1 + (y x 12 x e) / (b2 + d2)]
= 1.25
Parameters:
22. Shear modulus (G) = 0.0004 (shear modulus of rubber, Table 5.4)
23. Ultimate elongation , €U = 6.5 (shear modulus of rubber, Table 5.4)
24. Material constant, K = 0.87 (shear modulus of rubber, Table 5.4)
25. Elastic modulus, E = 0.00135 (shear modulus of rubber, Table 5.4)
26. Bulk modulus, E∞ = 1.5 (Typical value for neutral rubber)
27. Damping, β = 0.05 (5% used for plain rubber bearings)
28. Lead yield strength, σy = 0.008 (Usually 7 to 8.5 Mpa)
29. Teflon co-efficient of friction, µ = 0.1 (Use high velocity for design)

2. 30. Gravity, g = 9810
Isolator type & load data:
Number of bearing = 32
Avg DL + LL, Pd = 2593.5
Max DL + LL = 3372.7
Wing load/isolator = 50 50 x 32 = 1600
Seismic weight = 32 x 2593.5 + 1500 = 844.92
Isolator Dimension:
Plan dimension, B = 700mm
Layer thickness, t1 = 10
No of Layers, N = 21
Lead core size, dpl= 175
Shape C
Side cover, tsc = 10
Internal slim thickness, tsl = 3
Load plate thickness, Tpl = 40 [required to get total height]
Total rubber thickness, Tr = 210 [ Nt1 ]
Total height, H = 350 [ Nt1 + (N-1) tsl + 2Tpl]
Total yield level of system = 7.2% [(Qd x No of bearing) / W]
Gross area , Ag = 384650 [пB2 / 4]
Bonding dimension, Bb = 680 [B-2 tsc]
Bonding depth = Nil

3. Bond area, Ab = 322984 [пBb
2 / 4]
Plug area, Apl = 24040 [пdpl
2 / 4]
Net bonding area, Abn = 298944 [Ab - Apl]
Total rubber thickness, Tr = 210 [ Nt1 ]
Bonded perimeter, P = 2135 [пBb]
Shape factor , S =14.00 [Abn / t1p]
Characteristic strength , Qd = 192.4 [σy Apl]
Shear modulus [50%] = 0.0004 [G]
Yielded stiffness, Kr = 0.69 [G (Ag - Apl) / Tr]
For LRB,
C1, Co-efficient on Kr = 6.5 [Typical value]
C2, Co-efficient on Apl/Ab = 12 [Typical value]
Elastic stiffness, Kv = 8.8 [6.5 Kr (1 + (12 Apl / Abn))]
Yield force, Fy = 178.0 [Qd (1- (Kr / Kv))]
Yield displacement, ∆V = 20.20 [Fy / Kv]
Moment of inertia, I = 1.1 x 1010 [пBb
4 / 64] (circular)
Buckling factors
Height free to buckle, Hr = 2.70 [Tr + tsl (N-1)]
Buckling modulus, Eb = 0.20 [E (1+0.742S1
2)]
Constant, T = 2.88 x 1010 [Eb I Hr / Tr]
Constant, R = 186.3 [Kr Tr]
Constant, Q = 0.0116 [п / Hr]
Factor on €U = 0.33 [ factor of safety 3 for gravity]

4. Applied vertical load, PDL+LL = 3372.7
Applied displacement = 0
Applied rotation = 0
Shape factor, S1 = 14.00 [from properties]
Constant, K = 0.87 [from properties]
E= 0.0014 [from properties]
Compressive modulus, Ec = 0.478 [E (1+2KS1
2)]
Reduced area, Ar = 367500 [0.5 {B2 sin-1 (€ / Bb) - ∆€}] where € = (Bb
2 - ∆2
NS)0.5
Vertical stiffness, Kvi = 17566.5 [Ec Ar / t1] per layer
Compressive strain, €c = 0.019 [P / Kvi t1]
Compressive shear strain, €sc = 1.61 [6 S1€c]
Displacement strain, €sl = 0
Rotational strain, €sv = 0
Total strain, € = 1.61 [€sc + €sl + €sv]
Allowable strain = 2.16 [€U / f]
Buckling load, Pcr = 25542
Status Ok Satisfactory
If € < €V / f and Pcr > PDL+LL
Adjusted shear modulus = 0.00045 [G (1+ PD / Pcr)]
Adjusted stiffness, Kr
* = 0.6 [Kr (1- PD / Pcr)]

5. Vertical stiffness calculation
Kvi = 17566.5
Kv = 836.5
Bulk modulus, E∞ = 1.5 [from material properties]
Vertical stiffness, KV = 634 [Kv / (1+ (Ec / E∞))]
DBE: (Performance)
No of Isolators = 32
Elastic stiffness, KU = 8.8
Adjusted stiffness, Kr
* = 0.6
Yield displacement, ∆V = 20.20
Characteristic strength , Qd = 192.4
Iteration 1
Seismic displacement , DD = 180 [Assume a displacement adjust until SD / DD = 1]
Bearing force, F = 300 [Qd + DD Kr
*]
Effective stiffness, Ke = 1.668 [F/DD] Ke x 32 = 53.4
Seismic Weight = 84492 dead load
Seismic Weight = 8.61 W / g
Effective period, TE = 2.52 [2п (M/ Ke)0.5]
Loop area, Ah = 122982 [4 Qd (DD - ∆V)
Damping = 36.49% [(1/2п) x (Ah / Ke DD
2)]
Dampin factor, B = 1.81 [UBC Table A-16-C]
Spectral acceleration, SA = 0.1 [CV / BTE]
Spectral displacement, SD = 166.23 [ (g x CV x TE) / (4 п2B)]

6. Iteration 2
Seismic displacement , DD = 166.23
Bearing force, F = 292.13
Effective stiffness, Ke = 1.73 [F/DD] Ke x 32 = 56.23
Seismic Weight = 84492 dead load
Seismic Weight = 8.61 W / g
Effective period, TE = 2.45
Loop area, Ah = 11236.22
Damping = 37.5%
Dampin factor , B = 1.85
Spectral acceleration , SA = 0.1
Spectral displacement, SD = 158.11
Iteration 3
Seismic displacement , DD = 158.11
Bearing force, F = 287
Effective stiffness, Ke = 1.84 [F/DD] Ke x 32 = 59.19
Seismic Weight = 84492 dead load
Seismic Weight = 8.61 W / g
Effective period, TE = 2.39
Loop area, Ah = 106123.5
Damping = 36.7%
Dampin factor , B = 1.83
Spectral acceleration , SA = 0.1

7. Spectral displacement, SD = 156
Iteration 4
Seismic displacement , DD = 156
Bearing force, F = 286
Effective stiffness, Ke = 1.8 [F/DD] Ke x 32 = 59.2
Seismic Weight = 84492 dead load
Seismic Weight = 8.61 W / g
Effective period, TE = 2.38
Loop area, Ah = 104542
Damping = 36%
Dampin factor , B = 1.82
Spectral acceleration , SA = 0.1
Spectral displacement, SD = 156
Check convergence = 1 [ SD / DD]
MCE performance
No of isolator = 32
Elastic stiffness, KU = 8.8
Adjusted stiffness, Kr
* = 0.6
Yield displacement, ∆V = 20.20
Characteristic strength , Qd = 192.4
Iteration 1
DM = 230
F = 330.4

8. Ke = 1.43
TE = 2.72
Loop area, Ah = 161337
Damping = 33.9%
Dampin factor , B = 1.78
Spectral acceleration , SA = 0.099
Spectral displacement, SD = 182.4
Iteration 2
DM = 182.4
F = 302
Ke = 1.652
TE = 2.52
Loop area, Ah = 124782
Damping = 36%
Dampin factor , B = 1.82
Spectral acceleration , SA = 0.1
Spectral displacement, SD = 165.3
Iteration 3
DM = 165.3
F = 291.1
Ke = 1.76
TE = 2.44
Loop area, Ah = 111546

9. Damping = 37%
Dampin factor , B = 1.84
Spectral acceleration , SA = 0.1
Spectral displacement, SD = 158.3
Iteration 4
DM = 158.3
F = 287.3
Ke = 1.82
TE = 2.41
Loop area, Ah = 106050
Damping = 37%
Dampin factor , B = 1.84
Spectral acceleration , SA = 0.1
Spectral displacement, SD = 156.38
Iteration 5
DM = 156.2
F = 286.7
Ke = 1.83
TE = 2.4
Loop area, Ah = 104066
Damping = 37%
Dampin factor , B = 1.84
Spectral acceleration , SA = 0.1

10. Spectral displacement, SD = 156.2
Check convergence = 1 [ SD / DD]
Earthquake design
Eu = 0.75
Applied vertical load, PDL + SLL + E = 8206 [ Max DL + SLL + E]
DBE displacement = 156 [DBE displacement DD]
Factor on displacement = 1.25 [DTD / DD]
Applied displacement = 195 [DD]
Applied rotation = 0
Shape factor, S1 = 14
Constant, K =0.87 [ from properties]
E = 0.0014
EC = 0.6
Ar = 318500
Kvi = 19110
€c = 0.042
€sc = 3.6
€sh = 0.928
€sr = 0
€ = 4.5, allowable strain = 4.88
Buckling load, Pcr = 19703
Status Ok [€ < €u / f & Pcr > PDL+LL]

11. MCE
Factor of mu€u = 1
Applied vertical load = 8206
MCE displacement = 156.2
Factor on displacement = 195.2
Factor on rotaion = 0
Shape factor = 14
K = 0.87
E = 0.0014
Ec = 0.6
Ar = 318500
Kvi = 19110
€c = 0.042
€sc = 3.6
€sh = 0.928
€ = 4.5, allowable strain = 6.5
Buckling load, Pcr = 19703
Status Ok

12. Gravity strain FS 4.03 €u / € = 6.5 / 1.61
Buckling FS 7.57 Pcr / P = 25542 / 3372.5
DBE strain FS 1.44 €u / € = 6.5 / 4.5
Buckling FS 2.4 Pcr / P = 19703 / 8206
MCE strain FS 1.44 €u / € = 6.5 / 4.5
Buckling FS 2.4 Pcr / P = 19703 / 8206
Reduced area / Gross area 98.6 at MCE = Ar / Ab
Max shear strain at MCE = €sh
DBE MCE Comments
Effective period TD TM 2.38 2.4
From seismic performanceDisplacement DD DM 165 15
Total Displacement DTD DTM 206 206
Force co- efficient Vb / w 0.1 0.1 SA
Force co- efficient Vs / w 0.05 SA / R1
1.5 x yield force / w Fy / w
Wind force / w 0.018 Fw / w
Fixed base v @ TD From UBC
Base shear force
Max (Vs , Vy , Vw , Vf ) x w @
DBE
Damping, Deff 36% 36%
From seismic performance
Damping co-efficient BD BM 1.82 1.84

13. Sl No. Comments
First data line:
1 ID 1 Identification No
2 Type LRB Biaxial Hysteretic
3 KE2 1.76
Spring effective stiffness = Kr
* + Qd / Dd
4 KE3 1.76
5 DE2 0.310
Spring effective damping ratio = β – 0.05
6 DE3 0.310
Second data line:
7 K1 634 Spring stiffness along Axis 1 (axial)
8 K2 8.8
Initial spring constant = Ku
9 K3 8.8
10 FY2 / K11 / CFF2 178
Yield force = (Fy for LRB)
11 FY3 / K22 / CFF3 178
12 RK2 / K33 / CFF2 0.07
Post – yield stiffness ratio = Kr
* / Ku
13 RK3 / CFS3 0.07

14. Hysteresis properties
Displacement Force Comments
Yield displacement 20.25 Bearing properties
Design displacement 165 DBE properties
Yield force, Fy 178 Bearing properties
Origin 0 0 Start of plot
Point A 20.25 178
∆ = ∆Y
F = Fy
Point B 165 291.4
∆ = DD
F = QD + DD Kr
*
Point C 124.5 -6.46
∆ = DD - 2∆Y
F = QD + DD Kr
* - 2Fy
Point D -165 -291.4
∆ = -DD
F = - QD - DD Kr
*
Point E -124.5 6.46
∆ = - DD + 2∆Y
F = - QD - DD Kr
* + 2Fy
Point A 20.25 178
∆ = ∆Y
F = Fy