EGkS
5.4 STEP-BY-STEP IMPLEMENTATION OF A DESIGN PROCEDURE
5.4.10 Calculate Seismic Performance
139
LRB TFE Comment / Equations Factor f on u 0.33 Factor of safety 3 for gravity Applied Vertical Load, PDL+LL 4948 Maximum DL + LL
Applied Displacement 0.000 Non-Seismic Displacement NS
Applied Rotation 0.000 Non-Seismic Rotation, NS
Shape Factor, Si 20.35 From properties
Constant k 0.87 From material properties
Elastic Modulus, E 0.0014 From material properties Compressive Modulus, Ec 0.974 E (1 + 2kSi2)
Reduced Area, Ar 567450
b
b 1 B
A
NS (Square)
b2 2
b 1 2
B where
sin B B 0.5
NS
(Circular)
Vertical Stiffness, Kvi 55273 EcAr / tI (per layer) Compressive Strain, c 0.009 P / Kviti
Compressive Shear Strain, sc 1.09 6Sic
Displacement Shear Strain, sh 0.00 NS / Tr
Rotational Shear Strain, sr 0.00 Bb2NS / (2 ti Tr ) Total Strain, 1.09 c + sc + sr
Allowable Strain 2.17 u / f
Buckling Load, Pcr 23188
G r
A A
1
R 1 4TQ 2 R
2
Status OK Satisfactory if u/f
and Pcr > PDL + LL
Adjusted Shear Modulus 0.00033 G (1 – PD / Pcr) Adjusted Stiffness Kr* 0.91 Kr (1 – PD / Pcr) Vertical Stiffness Calculation
Kvi 55273 calculated above
Kv 2632 Kvi / N
Bulk Modulus E 1.5 From material properties Vertical Stiffness, Kv 1596 Kv / (1 + Ec/E)
Table 5.12: Gravity Load Capacity
140
3. The effective stiffness of each bearing at this displacement is calculated as F/. The total system effective stiffness is the summation of the individual device stiffness times the number of each type.
4. The effective period is calculated using the total seismic mass and the effective stiffness.
5. The equivalent viscous damping is calculated from the area of the hysteresis loop. For HDR, the damping and shear modulus are interpolated from tabulated values of these quantities versus shear strain.
6. The damping factor, B, is calculated for the calculated level of equivalent viscous damping.
7. The spectral displacement is calculated from the acceleration response spectrum at the effective period, modified by the damping factor B.
8. This displacement is compared with the displacement assumed in Step 1. above. If the difference exceeds a preset tolerance, the calculated displacement defines a new starting displacement and the procedure is repeated until convergence is achieved.
The seismic performance is evaluated for both the design level and the maximum seismic events.
For UBC design the spectral acceleration is calculated as SA =
E V
BT
C and the spectral displacement as SD =
B C T
g E
2 V
4 (which is equivalent to
2 2
4
e A
S gT ). For other codes, there will be different formulas for SA (often defined as the seismic coefficient, C). Provided SA can be defined, SD can be calculated directly from this.
The procedure in Tables 5.13 and 5.14 is developed for LRB, TFE and FPS bearings. For HDR bearings the properties are extracted from tabulated properties of shear modulus and damping versus shear strain as follows:
1. Shear strain is calculated as = / Tr.
2. The effective stiffness is calculated as Ke = GAb/Tr where G isthe shear modulus at strain .
3. The damping, , at strain is used to calculate the equivalent hysteresis loop area as:
Ah = 2Ke2. (5.31)
NOTE: The equivalent hysteresis area is only required if HDR bearings are used with other bearing types. If all bearings are HDR then can be used to calculate the B factor directly.
The iteration procedure can be automated using design office tools such as spreadsheets.
Figure 5.7 provides an example of a subroutine written in VBA which can be used in Excel spreadsheets. This relies on named ranges in the spreadsheet:
1. The cell containing total rubber thickness is named, in this example as Trmax, which is the maximum rubber thickness in any bearing.
141
2. The assumed displacement is named as dbe1 and mce1 for the two levels of earthquake load respectively.
3. The calculated spectral displacement is named as dbe2 and mce2 for the two levels of earthquake load respectively.
A button can be added to the spreadsheet to run the subroutine as a macro. (Named ranges are shown bolded in Tables 5.13 and 5.14).
DBE
Performance LRB TFE Total Comment
Number of Isolators 27 4 31 Number of each type of
isolator Elastic Stiffness, Ku 10.81 0.00
Yielded Stiffness, Kr* 0.91 0.00 Yield Displacement, y 19.78 0.00 Characteristic Strength, Qd 192.4 498.6
Bearing parameters calculated from bearing properties
above
Seismic Displacement, DD 181.3
“dbe1” Assume a displacement, adjust until SD/DD = 1.0 Bearing Force, F 357.2 498.6 Qd+DDKr*
Effective Stiffness, Ke 1.97 2.75 64.22 F/DD
Seismic Weight, W 122058 Sum of dead loads
Seismic Mass, M 12.44 W/g
Effective Period, TE 2.77
Ke
M 2
Loop Area, Ah 124286 361500 4801722 4QD(DD- y)
Damping, 30.55% 63.66% 36.22% 2
2 1
D e
h
D K
A
Damping Factor, B 1.82 UBC Table A-16-C
Spectral Acceleration, SA 0.095
E V
BT C
Spectral Displacement, SD 181.25
“dbe2” B
C T
g E
2 V
4
Check Convergence 1.00 SD/ m
Table 5.13: Seismic Performance for DBE
142 MCE
Performance LRB TFE Total Comment
Number of Isolators 27 4 31
Elastic Stiffness, Ku 10.81 0.00 Yielded Stiffness, Kr* 0.91 0.00 Yield Displacement, y 19.78 0.00 Characteristic Strength, Qd 192.4 498.6
Seismic Displacement, Dm 252.42
“mce1”
Bearing Force, F 421.9 498.6
Effective Stiffness, Ke 1.672 1.975 53.035
Seismic Weight, W 122058
Seismic Mass, M 12.442
Effective Period, TE 3.04
Loop Area, Ah 179058 503423 6848245
Damping, 26.76% 63.66% 32.26%
Damping Factor, B 1.74
Spectral Acceleration, SA 0.110
Spectral Displacement, SD 252.43
“mce2”
Check Convergence 1.00
See Table 5.9
for Formulas
Table 5.14: Seismic Performance for MCE
Sub SolveDisp() tol = 0.0001
ds = Range("Trmax") i = 0
Range("dbe1") = ds
Do Until Abs(1 - (Range("dbe1") / Range("dbe2"))) < tol i = i + 1
ds = Range("dbe2") Range("dbe1") = ds If i > 200 Then
Call MsgBox("Cannot Solve DBE") Exit Sub
End If Loop
ds = Range("Trmax") i = 0
Range("mce1") = ds
Do Until Abs(1 - (Range("mce1") / Range("mce2"))) < tol i = i + 1
ds = Range("mce2") Range("mce1") = ds If i > 200 Then
Call MsgBox("Cannot Solve MCE") Exit Sub
End If Loop End Sub
Figure 5.7: Subroutine to Solve for Displacement
143 5.4.11 Seismic Load Capacity
For elastomeric based bearings the vertical load carrying capacity is a function of applied shear displacement and so the capacity must be checked using the same procedures as were used for gravity loads (Table 5.12) with adjustments to acceptance criteria to reflect the lower frequency of seismic loads.
Table 5.15 lists the calculations for maximum DBE displacements. The center of mass displacements calculated above are factored by the torsional factor (in this example, by 1.248). The equivalent calculations for MCE are listed in Table 5.16. As the MCE seismic load has a long return period the minimum factor of safety is reduced to 1.0 for these displacements.
LRB TFE Comment / Equations
Factor f on Eu 0.75 Factor of safety 1.33 for DBE Applied Vertical Load, PDL+SLL+E 8358 Maximum DL + SLL + E
DBE Displacement 181.3 DBE Displacement DD
Factor on Displacement 1.248 DTD/DD
Applied Displacement 226.2 DTD
Applied Rotation 0.000 Seismic Rotation, E
Shape Factor, SI 20.35 From properties
Constant k 0.87 From material properties
Elastic Modulus, E 0.0014 From material properties Compressive Modulus, Ec 0.974 E (1 + 2kSi2)
Reduced Area, Ar 377451
b
b 1 B
A
DTD (Square)
b2 2
b 1 2
B where
sin B B 0.5
DTD
(Circular)
Vertical Stiffness, Kvi 36766 EcAr / tI (per layer) Compressive Strain, c 0.023 P / Kviti
Compressive Shear Strain, sc 2.78 6Sic
Displacement Shear Strain, sh 1.08 NS / Tr
Rotational Shear Strain, sr 0.00 Bb2NS / (2 ti Tr ) Total Strain, 3.85 c + sc + sr
Allowable Strain 4.88 u / f
Buckling Load, Pcr 15424
G r
A A
1
R 1 4TQ 2 R
2
Status OK Satisfactory if u/f
and Pcr > PDL + LL
Table 5.15: Load Capacity at DBE
144
LRB TFE Comment/Equations
Factor f on Eu 1.00 Factor of safety 1.0 for MCE Applied Vertical Load, PDL+SLL+E 8358 Maximum DL + SLL + E
MCE Displacement 252.4 MCE Displacement DM
Factor on Displacement 1.248 DTM/DM
Applied Displacement 315.1 DTM
Applied Rotation 0 Seismic Rotation, E
Shape Factor, SI 20.35
Constant k 0.87
Elastic Modulus, E 0.0014
Compressive Modulus, Ec 0.974
Reduced Area, Ar 305911
Vertical Stiffness, Kvi 29797
Compressive Strain, c 0.028
Compressive Shear Strain, sc 3.42 Displacement Shear Strain, sh 1.50
Rotational Shear Strain, sr 0.00
Total Strain, 4.93
Allowable Strain 6.50
Buckling Load, Pcr 12501
See Table 5.12
Status OK Satisfactory if u/f
and Pcr > PDL + LL
Table 5.16: Load Capacity at MCE