ghazvin - rashat - anzaliy railway bridge isolation bearings; design of lead rubber bearings,...

8/9/2019 Ghazvin - Rashat - Anzaliy Railway Bridge Isolation Bearings; Design of Lead Rubber Bearings, Structural Calculatio…

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C A L C U L A T I O N S

PROJECT NAME: Ghazvin- Rashat - Anzaliy Railway bridgeIsolation Bearings

PROJECT NUMBER: 107347CALCULATIONS BY: Ernesto De Peralta.DATE: 26 October 2011

PAGE NUMBER: 1

Ghazvin - Rashat - Anzaliy Railway Bridge

DESIGN OF LEAD RUBBER BEARINGS

STRUCTURAL CALCULATIONSRevision 4 (ALP)


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PAGE NUMBER: 2

TABLE OF CONTENTS

1. SUMMARY OF CALCULATIONS........................................................................................... 4

1.1 CHANGES IN REVISION ................................................................................................... 4 1.2 EFFECT OF INCREASED RUBBER SHEAR MODULUS...................................................................... 4

1.2.1 ISOLATION SYSTEM DISPLACEMENTS AND FORCES ............................................................. 4 1.2.2 DYNAMIC PROPERTIES AND DAMPING ........................................................................... 5 1.2.3 TEST CONDITIONS .................................................................................................. 5

2. PROJECT REQUIREMENTS ................................................................................................... 7

2.1 REFERENCES................................................................................................................. 7 2.2 DESIGN CONDITIONS..................................................................................................... 7

3. PROCEDURES USED FOR BEARING DESIGN.......................................................................... 9

3.1 DEFINITIONS ................................................................................................................ 9 3.2 CALCULATION OF EQUIVALENT SHEAR STRAINS ..................................................................... 10 3.3 LIMITING STRAIN CRITERIA............................................................................................... 11

3.3.1 AASHTO LIMITING STRAIN CRITERIA........................................................................... 11 3.4 VERTICAL LOAD STABILITY ................................................................................................ 11

3.4.1 AASHTO VERTICAL LOAD STABILITY ............................................................................ 11 3.5 AASHTO LATERAL RESTORING FORCE ................................................................................ 13 3.6 LATERAL STIFFNESS PARAMETERS FOR BEARING ....................................................................... 13 3.7 SYSTEM PROPERTY MODIFICATION FACTORS ......................................................................... 15

3.7.1 AGE CHANGE IN PROPERTIES ................................................................................... 16 3.7.2 TEMPERATURE CHANGE IN PROPERTIES......................................................................... 17 3.7.3 MODIFICATION FACTORS USED FOR DESIGN ................................................................. 17

4. BEARING DESIGN CALCULATIONS.................................................................................... 19

4.1 ANALYSIS RESULTS USED FOR ISOLATION SYSTEM DESIGN......................................................... 19 4.2 DESIGN LOADS ........................................................................................................... 20 4.3 MATERIAL PROPERTIES .................................................................................................... 20 4.4 BEARING SHAPE .......................................................................................................... 20 4.5 BEARING DIMENSIONS.................................................................................................. 21 4.6 BEARING PROPERTIES..................................................................................................... 22 4.7 LIMITING STRAIN CALCULATIONS ...................................................................................... 23

4.7.1 AASHTO CRITERIA ............................................................................................... 23 4.7.1.1 AASHTO Equation 25 ............................. ................................ .................................. .............. 23 4.7.1.2 AASHTO Equation 26 ............................. ................................ .................................. .............. 24 4.7.1.3 AASHTO Equation 27 ............................. ................................ .................................. .............. 24

4.8 VERTICAL LOAD STABILITY ................................................................................................ 25


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4.8.1 AASHTO REQUIREMENTS....................................................................................... 25 4.9 RESTORING FORCE....................................................................................................... 26 4.10 EFFECTIVE STIFFNESS AND DAMPING .............................................................................. 27


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PAGE NUMBER: 4

1.

SUMMARY OF CALCULATIONS

Holmes Consulting Group have been retained by Robinson Seismic Limited, a manufacturer and supplierof lead rubber bearings, to prepare structural calculations and documentation for supply of isolationbearings for the Ghazvin – Rashat – Anzaliy Railway Bridge, Iran

The preliminary design defines the model properties and the isolators are then checked, and the designadjusted as necessary, using the maximum forces and deformations obtained from the analysis. Thishowever allows for the tuning of the isolator to the response and performance of the sub-structure.

At the time of writing the time history & response spectrum analysis has not been undertaken, so theisolator design has been limited to a preliminary status.

1.1

CHANGES IN REVISION

Highlighted yellowRevised Qd, and Fy to reflect final design

Reduction in Total Displacements to reflect 1.5MCE (Kelly et al) in test procedure.Minor adjustments to report, misc.

1.2 EFFECT OF INCREASED RUBBER SHEAR MODULUS

1.2.1

ISOLATION SYSTEM DISPLACEMENTS AND FORCES

For the design of the isolators we have used a rubber shear stiffness mid-range value of 0.75 MPa. This islower that the original information supplied to us and gives a lower overall displacement, while having anominal increase in force capacity.

Table 1-1 lists the maximum displacements and the maximum force in the isolators. As expected theincreased hardness reduced displacements but increased forces. The reduction in displacements wasgreater at the MDE level but the increase in forces was a maximum under MPE loads. The analysisresults, described in the following section, provide a more accurate assessment of the effects on forces.

T ABLE 1-1 DESIGN DISPLACEMENTS AND FORCES

G= 0.75MPaS800x800 Displacement DBE(mm) 132S800x800 Force (kN) @ DBE disp. 887

C650 Displacement (mm) 134C650 Force (kN) 513


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1.2.2

DYNAMIC PROPERTIES AND D AMPING

Table 1-2 lists the calculated damping and effective period for the two isolator types used in this design.Both Isolators types use a lead core targeting a damping of approx 30%. The core size varies due to theisolator shape, and limiting the core size to approx 10%-11% of the overall isolator area.

T ABLE 1-2 DYNAMIC PROPERTIES

TRANSVERSE LONGITUDINALDisplacement (mm) 134 114.5Damping (% of critical) 35.6% 25.7%Damping B Factor 1.82 1.62Effective Period (Seconds) 1.15 1.34

1.2.3

TEST CONDITIONS

Table 1-3 summarizes the calculation of the isolator capacity under the critical load condition, MPEdisplacement with maximum DL + LLs + E(DBE). This assumes that peak vertical earthquake load andpeak vector displacement occur contemporaneously. :


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T ABLE 1-3 TEST CONDITION C APACITY (UNITS KN, MM)

S800x800 C650Max [DL+1.0LL] 7548 2844Max [DL+1.0LL+E(dbe)] 12703 4592Min [0.85DL-E(dbe)] 259 605Max [DL+1.0LL+E(mce)] 16828 5990Max [DL+1.0LL-E(mce)] 4343 2175MPE (DBE) Displacement 132 134Offset Displacement 0 0Factor on MPE Displacement 1.5 1.5 Applied Displacement (MPE/MCE) 198 201Compression Stiffness 5537 1772Design Shear Force at MPE (DBE) displ 887 513Design Area of Hysteresis Loop at MPE

(DBE) displ.

255377 160856

Shape Factor, Si 19.5 15.4MPE (DBE) Compressive Shear Strain, esc 1.98 2.01MPE (DBE) Displacement Shear Strain, esh 0.81 0.83MPE (DBE) Total Strain 3.83 3.52MPE (DBE) Allowable Strain 4.5 4.5MPE (DBE) Buckling Load, Pcr (reduced) 56886 14191


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2. PROJECT REQUIREMENTS

2.1

REFERENCES

Design parameters were provided for the seismic isolation system. The design followed the procedures

of the AASHTO 1999 Guide Specifications and all of the isolators comply with the AASHTO conditions.

The design was checked to ensure that it also complied with the requirements of this document.

[1] American Association of State Highway Transportation Officials (AASHTO) GuideSpecifications for Seismic Isolation, 1999.

[2] Mondkar, D.P. and Powell, G.H., 1979, ANSR II Analysis of Non-linear Structural ResponseUser's Manual , EERC 79/17, University of California, Berkeley, July.

2.2

DESIGN CONDITIONS

Table 2-1 lists the preliminary bearing design requirements as supplied by the client for the Ghazvin –Rashat – Anzaliy Railway Bridge. The requirements included maximum bearing dimensions, verticalloads and maximum longitudinal displacements in the bearings. Preliminary sizes were provided, along with MDE (DBE) and MPE earthquake accelerations.

1 – LRB 650 Dia (#16)

Total rubber thickness = 126mm (without steel thickness)Each rubber thickness = 9mmG = 14 kg/cm2Damping ratio = 30%

Horizontal displacement = 250mm

2 – LRB 800x800 (#32)

Total rubber thickness = 108mm (without steel thickness)Each rubber thickness = 9mmG = 14 kg/cm2Damping ratio = 30%Horizontal displacement = 250mm

MDEHorizontal Acceleration = 0.37g

Vertical Acceleration = 0.30g


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PAGE NUMBER: 8

This provided recommended, rather than required, stiffness and strength properties. As the computermodel was developed there were some refinements to bridges loads and the actual bearing designconditions, as used in Section 4 of these calculations, vary slightly from the values in Table 2-1.


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3.

PROCEDURES USED FOR BEARING DESIGN

These calculations relate to lead-rubber bearings designed to meet the project specification describedabove in Table 2-1 and Figure 2-1. The bearing design used the definitions and procedures described inthis section. Generally, the procedures follow the guidelines published by the American Association ofState Highway Officials (AASHTO) except where the project specifications are more restrictive.

Some adjustments are made to calculated properties to include empirical adjustments as a result of ourextensive experience in lead-rubber bearing design. These are to ensure that test properties will matchthose developed as part of this design procedure.

3.1 DEFINITIONS

B = Overall plan dimension of square bearing or Overall diameter of circular bearingti = Rubber layer thickness

n = Number of rubber layers Tr = Total rubber thickness

tsh = Thickness of internal shims

Tpl = Thickness of load plates

tsc = Thickness of side cover

Ag = Gross area of bearing, including side cover

Ab = Bonded area of rubber

Ar = Reduced rubber area (overlap area at displaced configuration)

p = Bonded perimeter

Gγ = Shear modulus of rubber (effective modulus at shear strain γ )E = Elastic modulus of rubber (approximately 4G)K = Material constant (varies depending on elastomer used)εu = Minimum elongation at break of rubber

f = Factor applied to elongation for load capacitySi = Shape factor for layer i

P = Applied vertical load

∆ = Applied lateral displacementK r = Lateral stiffness

K v = Vertical stiffness of bearing

K vi = Vertical stiffness of layer i

Fm = Force in bearing

K eff = Effective Stiffnessβ = Equivalent viscous damping


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Ah = Area of hysteresis loop

3.2

CALCULATION OF EQUIVALENT SHEAR STRAINS

The load capacity of elastomeric bearings is governed by the total shear strain in the elastomer due toapplied loads and deformations. The strain for a given load or deformation depends on the internal

construction of the bearing.

The shape factor of an internal layer, Si, is defined as the area free to bulge, that is, the perimeter timesthe layer thickness:

ii

4t

BS = for square and circular bearings

The vertical stiffness of an internal layer is calculated as

[ ]2ii

r vi 2KS1

t

AEK +=

In this equation the reduced area of rubber, Ar, is calculated based on the overlapping areas between the

top and bottom of the bearing at a displacement, ∆, as follows:

∆−=B

1AA br for square bearings

( )22

12r

B

where

BsinB0.5A

∆−=

∆−

= −

ς

ς ς

for circular bearings

When the effective compressive modulus Ec = E[1+2KS2 ] is large compared to the bulk modulus E∞

(between 1000 and 2000 MPa) then the vertical deformation due to the bulk modulus is included bydividing Ec by 1 + (Ec /E∞ ).

AASHTO also requires that the bulk modulus effect be included in strain calculations for bearings of

large shape factor. The shear strain due to compression, εsc, is a function of the maximum shape factor:

)21(2

3

2kS G A

SP

r

c+

=γ For S ≤ 15, or..............................................................(AASHTO Equation 20)


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PAGE NUMBER: 11

r c

GkSA

E GkS P

4

)/81(3 2 ∞+=γ for S > 15. ............................ ................................(AASHTO Equation 21)

If the bearing is subjected to applied rotations the shear strain due to a rotation θ is:

T2t

B

i

2θ γ =r

The shear strain due to a lateral displacement of ∆ is

r T

∆= sγ

3.3

LIMITING STRAIN CRITERIA

3.3.1

AASHTO LIMITING STRAIN CRITERIA

From the AASHTO Guide Specifications, three limiting strain criteria are to be satisfied:

γ c ≤ 2.5 ............................ .................................. .........................(AASHTO Equation 25)

γ c + γ s,s + γ r ≤ 5.0........................ .................................. ............................. .(AASHTO Equation 26)

γ c + γ s,eq + 0.5 γ r ≤ 5.5........................ .................................. ............................. .(AASHTO Equation 27)

The component strains are calculated for compressive loads, lateral displacements and rotations from theequations presented above.

3.4 VERTICAL LOAD STABILITY

3.4.1

AASHTO V ERTICAL LOAD STABILITY

AASHTO also requires that the isolators have a factor of safety of 3.0 under DL+LL and a factor ofsafety of 1.0 under 1.1 times MCE displacements (plus offset displacement). For bearings with a highrubber thickness relative to the plan dimension the elastic buckling load may become critical. Thebuckling load is calculated using the Haringx formula as follows:

Moment of inertia, I is calculated as


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12

BI

4

= for square bearings

64

BI

4π = for circular bearings

The height of the bearing free to buckle, that is the distance between load plates, is

shir 1)t(N)t(NH −+=

An effective buckling modulus of elasticity is defined as a function of the elastic modulus and the shapefactor of the inner layers:

)0.742SE(1E 2i b +=

Constants T, R and Q are calculated as:

r

r bT

HIET =

r

r g

H

TGAR =

r HQ

π =

From which the buckling load at zero displacement is:

−+= 1

R

4TQ1

2

R P

20cr

For an applied shear displacement the critical buckling load at zero displacement is reduced according tothe effective "footprint" of the bearing in a similar fashion to the strain limited load but at a slower rate:

g

A

A

r 0cr cr PP =

γ


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3.5

AASHTO LATERAL RESTORING FORCE

The AASHTO sets to criteria which must be satisfied to demonstrate that the bearings are configured toprovide sufficient restoring force:

1. The period based on the tangent stiffness alone for displacements up to the designdisplacements must be less than 6.0 seconds. For lead rubber bearings the tangent stiffness is

defined as the yielded stiffness and so the period calculated asr

r gK W2T π = must be less than

6.0 seconds.

2. The restoring force at the design displacement must be greater than the force at one-half the

design displacement by at least W/80. That is,80

5.0W

F F >− ∆∆

3.6

LATERAL

STIFFNESS

PARAMETERS

FOR

BEARING

Lead rubber bearings, and elastomeric bearings constructed of high damping rubber, have a nonlinearforce deflection relationship. This relationship, termed the hysteresis loop, defines the effective stiffness(average stiffness at a specified displacement) and the hysteretic damping provided by the system. Atypical hysteresis for a lead-rubber bearing is as shown in Figure 3-1.

FIGURE 3-1 :::: LEAD RUBBER BEARING HYSTERESIS


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For design and analysis this shape is usually represented as a bilinear curve with an elastic (or unloading)stiffness of K u and a yielded (or post-elastic) stiffness of K d. The post-elastic stiffness K d is equal to the

stiffness or the elastomeric bearing alone, K r. The force intercept at zero displacement is termed Qd, the

characteristic strength ply d AQ σ =

The theoretical yield level of lead, σy , is 10.5 MPa but the apparent yield level is generally assumed to be 7

MPa to 8.5 MPa, depending on the vertical load and lead core confinement.

The post-elastic stiffness, K d, is equal to the shear stiffness of the elastomeric bearing alone:

r

r

r T

AGK

γ =

The shear modulus, Gγ , for a high damping rubber bearing is a function of the shear strain γ , but isassumed independent of strain for a lead-rubber bearing manufactured from natural rubber and withstandard cure.

For relatively tall bearings, where the axial load is a significant fraction of the buckling load, the shearstiffness is adjusted based on the ratio of average dead load to the zero displacement buckling load:

−=

2

cr 0r

*r

P

P1K K

The elastic (or unloading) stiffness is defined as:

ru K K = for elastomeric bearings

+=

r

plr u

A

A121K 5.6K for lead-rubber bearings

The shear force in the bearing at a specified displacement is:

∆+= r K dm

QF

An effective stiffness can be calculated as:

∆= meff

FK

The sum of the effective stiffness of all bearings allows the period of response to be calculated as:

eff

eK g

W T

Σ= π 2


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Seismic response is a function of period and damping. High damping and lead rubber bearings providehysteretic damping. For high damping rubber bearings, the hysteresis loop area is measured from tests

for strain levels, γ , and the equivalent viscous damping β calculated as given below. For lead rubber

bearings the hysteresis area is calculated at displacement level ∆m as:

( )y mdh 4Q A ∆−∆=

from which the equivalent viscous damping is calculated as:

∆=

22

1

eff

h

K

A

π β

The isolator displacement can be calculated from the effective period, equivalent viscous damping andspectral acceleration as:

B

T S eam 2

2

4π =∆

where Sa is the spectral acceleration at the effective period Te and B is the damping factor, a function of β which is obtained from AASHTO as listed in Table 3-1.

T ABLE 3-1.D AMPING F ACTORS

Damping (percentage of critical)≤ 2 5 10 20 30 40 50

B 0.80 1.00 1.20 1.50 1.70 1.90 2.00

The formula for ∆m includes Te and B, both of which are a function of ∆m. Therefore, the solution formaximum displacement includes an iterative procedure.

When the time history method of analysis is used the non-linear properties of the bearing are modelledexplicitly and the response of these elements incorporated the hysteretic damping so the effective periodand equivalent viscous damping formulations are not required.

3.7

SYSTEM PROPERTY MODIFICATION FACTORS

AASHTO defines maximum and minimum properties as a function of system modification factors, λ, asfollows:

K d,max=K d x λmax,KdK d,min=K d x λmin,Kd Qd,max=Qd x λmax,Qd


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Qd,max=Qd x λmax,Qd

The lambda factors for the effects of (1) temperature (2) aging (3) change in friction coefficient with velocity (4) travel (wear) (5) contamination in sliding systems and (6) effects of scragging (elastomericsystems).

Effects (3), (4) and (5) apply only to sliding systems and so need not be considered here. Item (6) refersto the change in stiffness and damping in elastomeric bearings after a number of high amplitude cycles.

These changes are significant in high damping rubber bearings (HDRB) which rely on the elastomerproperties to provide system effective stiffness and damping. For lead rubber bearings (LDRB) thescragging effect is considered to be minor and AASHTO recommends a λ factor for scragging of 1.0.

Therefore, the two λ factors which affect LDRB are aging and temperature. The AASHTOrecommendations for these are:

1. For aging, the modification factors are 1.1 for low damping natural rubber (as used in thesebearings) and 1.0 for the lead core.

2. For temperature, modification factors of 1.0 at 21ºC increasing to 1.3 on Qd and 1.1 on K d at0ºC.

The AASHTO recommendations are for the modification factors for the minimum values to be unity. Therefore, the design values effectively form the lower bound value. This is because aging effects tendto increase stiffness, not decrease it, and properties are relatively insensitive to increases in temperature(less than 10% change in effective stiffness at 49ºC).

As the λ factors are equal to or greater than 1.0, the effect is to reduce isolator displacement and increaseisolator forces. Therefore, the modification factors do not govern in isolator design but do governsubstructure design forces.

3.7.1

A GE CHANGE IN PROPERTIES

The rubber tests on compounds used for LDRB show an increase in hardness by up to 3 Shore A afterheat aging. This increase in hardness is equivalent to an increase in shear modulus of 10% which wouldincrease K d by a factor of 1.1.

In service, the change in hardness for bearings would be limited to the outside surface since the coverlayer prevents diffusion of degradants such as oxygen into the interior. Therefore, average effects wouldbe less than the 10% value. For unprotected natural rubber in service over 100 years (for example, Rail Viaduct in Melbourne, Australia) the deterioration was limited to approximately 1.5 mm (0.06 inches)from the exposed surface.

There is not a great database of information on direct measurement of the change in stiffness properties

with time of loaded elastomeric bearings. One example was machine mountings manufactured in 1953and in service continuously in England. In 1983, after 30 years, two test bearings which had been stored


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PAGE NUMBER: 17

with the machine were tested again and were found to have increased in stiffness by 15.5% and 4.5%. Anatural rubber bearing removed from a freeway bridge in Kent showed an increase in shear stiffness ofabout 10% after 20 years service.

Since the time that the bearings above were manufactured, considerable advances have been made inenvironmental protection of the bearings. It is predicted that changes in stiffness of the elastomer will beno more than 10% over the design life of the isolators.

3.7.2

TEMPERATURE CHANGE IN PROPERTIES

Elastomeric bearings are usually compounded from natural rubber and so are subjected to temperatureconstraints typical to this material. The upper operating range of service temperature for natural rubber,

without special compounding, is 60°C and so the upper limit of the design temperatures for most projects will not cause any problems.

The stiffness of natural rubber is a function of temperature but within the range of -20°C to 60°C the

effect is slight and not significant in terms of isolation performance. Below -20°C the stiffness gradually

increases as the temperature is lowered until at about –40°C it is double the value at 20°C. The variationin stiffness is reversible as temperature is increased.

1. The founder of RSL (Dr W Robinson) invented the lead-rubber bearing and as part of thedevelopment tested completed bearings under dynamic displacements. Compared to thebenchmark results at a temperature of 18ºC, at -15ºC the effective stiffness increased by 20%and at -35ºC the effective stiffness increased by 40%. When the temperature was increased to45ºC the effective stiffness decreased but only by -10%.

2. The Hitec test program tested DIS bearings at -26 ºC and recorded a 23% increase in effectivestiffness and Skellerup bearings at -30 ºC and recorded a 56% increase in effective stiffness. Inboth cases, the bearings were maintained at these temperatures for 2 days before testing. TheHitech program also tested the bearings at an elevated temperature of 49ºC and the changes ineffective stiffness were relatively small, -9% for DIS bearing and -5% for the Skellerup bearing.

The AASHTO factors of 1.3 on Qd and 1.1 on K d for 0ºC increase the effective stiffness by 19% at MCEdisplacements. This is seen to be conservative as this increase was measured at a colder temperature of -15ºC

3.7.3

MODIFICATION F ACTORS USED FOR DESIGN

The discussion above shows that the factor of 1.1 on K d is reasonable for LDRB and so this factor isused in the isolation system design and analysis.

The AASHTO λ factors are conservative for the 0ºC condition as the changes in rubber properties aresmall until the temperature is below -20ºC. The average daily temperature at the site is understood to be

greater than 0ºC even in mid-winter and so the bearings are unlikely to be affected as much as suggested


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by AASHTO. Therefore, on the basis that the MCE is a very low probability event, and the probabilitythat this occurs at a time of sustained sub-zero temperatures is negligible:

1. The λ factors for DBE are taken as 1.3 on Qd and 1.21 on K d (combination of aging and lowtemperature).

2. The λ factors for MCE are taken as 1.0 on Qd and 1.1 on K d (aging only).


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4. BEARING DESIGN CALCULATIONS

4.1

ANALYSIS RESULTS USED FOR ISOLATION SYSTEM DESIGN

Table 4-1 summarises the results from the EXCEL spreadsheet based on the design procedures.

T ABLE 4-1 RESULTS

Averageof

IsolatorsMaximum Design EQ (MDE/DBE)

Transverse, ∆ T 114.5Longitudinal, ∆L 134

Maximum Probable EQ (MPE/MCE) Transverse, ∆ T 172

Longitudinal, ∆L 201

Design conditions for the bearings are expanded from Table 4-1 as follows:

The design load conditions (DBE/MCE) on the isolators are summarised in tables: 1-3.

1. The Total design displacement is the Max MDE displacement = 134 mm

2. The maximum displacement is the MPE displacement = 201 mm (1.5 time MDE)

AASHTO does not require that the earthquake records includes vertical components but rather specifiesmaximum loads as 1.20 times dead load , where the 1.20 factor included vertical earthquake effects,uncertainty in DL plus axial loads due to earthquake overturning.


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4.2

DESIGN LOADS

The design conditions for each location are as listed in Table 4-2. These are extracted from the contractdocuments and from Table 4-1 above.

T ABLE 4-2 DESIGN CONDITIONS

Average DL + LL (kN) 5009 From spreadsheetMaximum DL + LL (kN) 7548Design Displacement (mm) 134 MDE (DBE max. disp.)Maximum Displacement (mm) 201 MPE (MCE max. disp.)

4.3 MATERIAL PROPERTIES

The detailed design of the isolation system was performed using an EXCEL spreadsheet based on the

design procedures given in Section 3 of these calculations. The properties of the materials used were aslisted in Table 4-3.

T ABLE 4-3 M ATERIAL PROPERTIES

Shear Modulus (MPa) 0.75

Ultimate Elongation 6.0

Material Constant, k 0.65

Elastic Modulus, E (Mpa) 3.0

Bulk Modulus (Mpa) 287000

Lead Yield Strength (Mpa) 9.0

4.4

BEARING SHAPE

The original lead rubber bearings invented by Dr. Bill Robinson were based on standard elastomericbridge bearings, which were either square or rectangular in plan shape. All the original development andtesting, including low temperature tests and fatigue tests, was performed on square bearings with leadcores inserted. The first isolation projects using lead rubber bearings in New Zealand, for both buildingsand bridges, used a square or rectangular bearing shape. As the number of isolation projects expanded,other manufacturers began manufacturing circular lead rubber bearings and these are now the mostcommon shape for building projects. For bridge projects, both rectangular and circular shapes are used.

For bridges, there are a number of reasons the square or rectangular shape is often preferred over

circular:


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PAGE NUMBER: 21

1. The maximum plan dimension is smaller for the same shear area. This smaller dimension tendsto be more important for bridges than buildings because the bearings are located on piers offixed dimensions.

2. Non-seismic displacements due to thermal movement, creep and shrinkage are applied in thelongitudinal direction and so the bearing is oriented with these displacements applied along aprincipal axis.

3. The shear strain due to rotation is proportional to B2, where B is the plan dimension normal tothe axis of rotation. As rotations are usually uni-directional, rectangular bearings can be used torreduce the strain due to applied rotations by orienting the smaller dimension in the longitudinaldirection.

Because earthquake displacements are omni-directional, there is a perception that a circular shape is moreeffective in resisting these displacements. The stiffness of an elastomeric bearing is based on the sheararea and so is the same in all directions so any advantage of the circular shape compared to rectangular would be due to a more uniform stress distribution. Although intuitively this would appear to be true, weare not aware of any finite element studies which confirm it and extensive testing of square bearings hasnot revealed any failure which could be related to be bearing shape.

One are where a square or rectangular bearing may have disadvantages over circular shapes would be inthat the corners of the internal shims would apply a point load to the elastomer and possible cause astress concentration. To avoid this, the corners of the internal shims are rounded. Robinson Seismicmanufacturing specifications require a minimum radius of 15 mm for all end shims and internal shims.

There are also a few circular bearings used in this project particularly those bridge decks requiring seismicjoints. Circular bearings are ideal for maximizing space.

4.5 BEARING DIMENSIONS

The bearing sizes and construction details which meet the project criteria are listed in Table 4-4. Theseare 800mm square bearings and 650mm circular bearings with a height of 256mm.

T ABLE 4-4 BEARING DIMENSIONS (UNITS MM)

STAGE 1

Overall Bearing Size (Rectangular) 800x800Overall Bearing Size (Circular) 650Layer Thickness 9Number of Layers 18Lead Core Size (for the Rectangular Bearing) 4-140

Lead Core Size (for the Circular Bearing) 4-110Side Cover 10


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Internal Shim Thickness 2Load Plate Thickness (top and bottom) 20.0 Total Height 256

It is preferable to use a square bearing so that the area reduction factor is similar in both directions elsethe bearing will have a much lower load capacity in the shorter direction for a given displacement. Theaverage displacements are similar in both directions so it is preferable to have a similar capacity.

Note that the final shim thickness is 20mm which adjusts this height from previous report. Additionallythe croe has been separated into multiple cores of smaller diameter. The Area of the core is of anequivanlent size so the isolator performance is the same.

4.6

BEARING PROPERTIES

Table 4-4 lists the bearing properties calculated using the procedures described in Section 3 of thesecalculations. These properties are used to calculate the load capacity of the bearings and also to definethe hysteresis shape.

T ABLE 4-5 BEARING PROPERTIES (UNITS KN/ MM)

S800 X 800 RECTANGULAR BEARING

Gross Area, Ag 640000Bonded Dimension 780Bonded Area 608400Plug Area 61575Net Bonded Area 546825 Total Rubber Thickness 162Bonded Perimeter 3120

Shape Factor 19.5Characteristic Strength, Qd 554.2Shear Modulus (50%) 0.000707 Yielded Stiffness K r 2.52Elastic Stiffness K u 38.5 Yield Force 593 Yield Displacement 15.38

C650 CIRCULAR BEARING

Gross Area, Ag 331831Bonded Dimension 630

Bonded Area 311725Plug Area 38013


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Net Bonded Area 273711 Total Rubber Thickness 162Bonded Perimeter 1979Shape Factor 15.4Characteristic Strength, Qd 342.1Shear Modulus (50%) 0.000701 Yielded Stiffness K r 1.27

Elastic Stiffness K u 22.0 Yield Force 363 Yield Displacement 16.47

4.7

LIMITING STRAIN CALCULATIONS

4.7.1

AASHTO CRITERIA

4.7.1.1

AASHTO EQUATION 25

For maximum vertical loads with zero lateral displacement or rotation the total strain is required to be lessthan 2.0. Calculations for this limit state are listed in Table 4-6.

T ABLE 4-6 C APACITY UNDER M AXIMUM V ERTICAL LOADS (ZERO DISPLACEMENT) (UNITS KN, MM)

S800 X 800

Applied Vertical Load 7548Elastic Modulus, E 0.003Compressive Modulus, Ec 1.5Reduced Area 608400

Compressive Shear Strain, esc 0.98 Total Strain 0.98 Allowable Strain 2.00Buckling Load, Pcr (Reduced) 41227 Vertical Stiffness Calculation

Kvi 99668Kv 5537

C650

Applied Vertical Load 2844Elastic Modulus, E 0.003

Compressive Modulus, Ec 0.9Reduced Area 311725


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Compressive Shear Strain, esc 0.81 Total Strain 0.81 Allowable Strain 2.00Buckling Load, Pcr (Reduced) 7766 Vertical Stiffness Calculation

Kvi 31895Kv 1772

4.7.1.2 AASHTO EQUATION 26

Table 4-7 checks the capacity under vertical loads with service load lateral displacements and servicerotations, as required by AASHTO Equation 26 which limits the total strain is required to 3.0.

T ABLE 4-7 C APACITY UNDER NON-SEISMIC DISPLACEMENTS (UNITS KN, MM)

S800 X 800

Max Vertical Load 7548Compressive Shear Strain, esc 0.98Displacement Shear Strain, esh 0.29Rotational Shear Strain, esr 1.04 Total Strain 2.31 Allowable Strain 3.00

C650

Max Vertical Load 2536Compressive Shear Strain, esc 0.81Displacement Shear Strain, esh 0.48Rotational Shear Strain, esr 0.68 Total Strain 1.97 Allowable Strain 3.00

4.7.1.3

AASHTO EQUATION 27

Table 4-8 checks the capacity under vertical loads with MDE (DBE) lateral displacements and one-halfservice rotations, as required by AASHTO Equation 27 which limits the total strain is required to 4.50.

T ABLE 4-8 C APACITY UNDER MDE (DBE) DISPLACEMENTS (UNITS KN, MM)

S800 X 800


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Max Vertical Load 12703MDE Displacement 132Compressive Shear Strain, esc 1.98Displacement Shear Strain, esh 0.81Rotational Shear Strain, esr 1.04 Total Strain 3.83 Allowable Strain 4.50

C650

Max Vertical Load 4592MDE (DBE) Displacement 134Compressive Shear Strain, esc 2.01Displacement Shear Strain, esh 0.83Rotational Shear Strain, esr 0.68 Total Strain 3.52 Allowable Strain 4.50

4.8

VERTICAL LOAD STABILITY

4.8.1

AASHTO REQUIREMENTS

AASHTO Clause 12.3 requires that the bearings be stable at a displacement of 1.1 times the MCEdisplacement plus one-half the non-seismic displacement under a vertical load of 1.2 DL+SLL. Table 4-9 lists the assessment of the bearings under two conditions, (1) maximum displacement with concurrentearthquake load and (2) maximum earthquake load with concurrent displacement (Table 4-2).

For this design we have applied a displacement factor of 1.5 MPE in accordance with recommendationsmade in “isolator design for structural engineers” T.Kelly for locations with a high MPE design

acceleration (>0.19g) Vertical earthquake loads have additionally been factored by 1.7 to representpredicted MPE events. MDE vertical accelerations were supplied see section 2.2

T ABLE 4-9 C APACITY UNDER MPE DISPLACEMENTS (UNITS KN, MM)

S800 X 800

MaximumDisplacement

Max Vertical Load 16828MDE (DBE) Displacement 132Factor on MCE Displacement 1.5

MPE (MCE) Displacement 198Compressive Shear Strain, esc 2.62


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Displacement Shear Strain, esh 1.22Rotational Shear Strain, esr 1.04 Total Strain 4.89 Allowable Strain 6.00

C650

Maximum

DisplacementMax Vertical Load 5990MDE (DBE) Displacement 134Factor on MCE Displacement 1.5MPE (MCE) Displacement 201Compressive Shear Strain, esc 3.19Displacement Shear Strain, esh 1.24Rotational Shear Strain, esr 0.68 Total Strain 5.11 Allowable Strain 6.00

4.9

RESTORING FORCE

AASHTO requires that the isolators meet two criteria related to the minimum restoring force. Thecalculations are listed in Table 4-10:

T ABLE 4-10 RESTORING FORCE C ALCULATIONS (UNITS KN, MM)

S800 X 800

Tangent Stiffness K t 2.52 from Table 4-6Qd 554 from Table 4-5

Seismic Weight 3966Period at Kt 1.38 Calculated, OK < 6 SecondsDisplacement di 132 DBE DisplacementF at di 887 Qd+K tdiF at 0.5di 720 Qd+K tdi/2Difference 167 OK > W/80 W/80 49.6

C650

Tangent Stiffness K t 1.27 from Table 4-6Qd 342.1 from Table 4-5Seismic Weight 1259

Period at Kt 1.38 Calculated, OK < 6 SecondsDisplacement di 134 DBE Displacement


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F at di 513 Qd+K tdiF at 0.5di 427.2 Qd+K tdi/2Difference 85.8 OK > W/80 W/80 15.74

4.10 EFFECTIVE STIFFNESS AND DAMPING

Figure 4-1 shows the hysteresis loop calculated from the properties listed above with the maximumdisplacement equal to the 250 mm.

The equivalent viscous damping plot in included in Figure 4-2 demonstrates how the damping varies withdisplacement amplitude. These particular bearings provide over 30% damping for displacements between30 mm and 201 mm but damping reduces slightly with increased displacements, reaching a value of 26% -30% at the MPE displacement of 201 mm.


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FIGURE 4-1 EQUIVALENT D AMPING

ghazvin - rashat - anzaliy railway bridge isolation bearings; design of lead rubber bearings,...

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