Elastomeric Bearing Stiffness

[samwise753]

Gentlemen,

I am currently in the process of modeling prestressed concrete girder bridges for seismic analysis in WinSeisab. My issue is the modeling of the bearing elements. The project is using elastomeric bearings (plain and laminated), and I am not sure what the appropriate stiffness values should be. I know that they cannot fully restrain the displacements, but they are also not free. Can anyone help with determining stiffness coefficients for the translational and rotational DOF's? Thank you for your time.

 

[asixth]

samwise753,

Good post. Try and find some technical data from a supplier of bearings. In Australia, we have a group called Granor who manufacture laminated elastomeric bearings and have data sheets with the compressive stiffness, shear stiffness and rotational stiffness of bearings.

Otherwise, I will look through AASHTO to see if it provides a procedure for calculating these values from first principles. I will also post excerpts from the Australian codes that you may find helpful.

Some questions:

Is WinSeisab an FE program?
What units do you work with, US customary or SI?

 

[samwise753]

Thanks asixth,

I will try to find some suppliers and see if I can get the info. As for WinSeisab, it is a seismic response analysis program developed by Imbsen Software (Roy Imbsen is a seismic expert out of California). The setup is a lot like StaadPro (or other structural analysis software) in that it uses matrix analysis to set up the compatibility equations for the whole structure. It can run static analyses, but, more importantly, can perform response spectrum analyses for seismic design. The units I am using are US customary. Also, the project is in North Mississippi, US (New Madrid Fault)

 

[asixth]

samwise,

I thought your question was relating to the compressive stiffness before I read again and saw you where after translational stiffness.

The attachment is from the Australian Bridge Code so the units will be in SI. The shear stiffness is:

Ks=A*G/t

where A is the rubber area; G is the shear modulus and t is the thickness that is free to shear. This equation will not need any re-arranging to convert from SI to US.

The rotation stiffness is a little more complicated:

Kr_layer=E*I/t

Where E is adjusted based on the following:

Constant based on Bearing Shape, C2 see 12.7.3(4)
Shear Modulus,G
Shape Factor,S see 12.5.2(1)
Bulk Modulus,B
Aspect Ratio, m

The calculation is performed for all the layers and the total rotation stiffness is given by:

Kr=1/sum(1/Kr_layer)

The rotational stiffness calculation will need some adjustment because of the conversion of units.

These equations are very complicated and I recommend that you put them into a spreadsheet for calculation. Let us know if you have any trouble with them.

If anyone else also has calculations or other technical info for bearings can they please be posted.

Just a question with the program. Are special elements being used for the bearings. When designing bridges, I have been using a general purpose finite element package without any bridge design package. I model the bearings using standard beam elements based on elastic theory, even though their behavior is not truly elastic I make the appropriate adjustments to the stiffness so the attract the correct forces, displace the correct amount and I can then use this to calculate the strains.

 

[samwise753]

Good morning (well, here anyway),

Thank you for looking up those equations, and the attachment is great. I would very much like to use it, but I'm not sure what the ruling is on using the Australian code on an AASHTO job. I looked through AASHTO LRFD Bridge Design Specifications, 4th Ed., 2007, with 2008 Interims Chapter 14, and it doesn't have any set equations for rotational and translational stiffness of elastomeric bearings (it does have rotational stiffness for CDP, but we are not using that). I'm still looking into suppliers for stiffness values.

The program does use special elements. They are zero length, two-node elements with six fully coupled degrees of freedom. They solely add stiffness. The elements are not retained in the final solution as they are statically condensed out.

 

[crossframe]

In the past, I've used "Seismic Design of Bridges Design Example No. 2", Publication No. FHWA-SA-97-007 to determine translational, rotational, and vertical stiffnesses of elastomeric bearings to include in seismic models. The report date is October 1996, and I believe the publication date is 1997. There may be a newer version of this out there.

 

[samwise753]

Hey crossframe,

Thank you for the tip, but I am having trouble finding an online copy of the publication you posted. I entered into the search window at the FHWA website, and nothing came up. I have settled on a procedure to determine the stiffness values, but I always welcome the opportunity for a sanity check. Let me know if you know of a way to get that report. Thank you.

 

[IDS]

Link for the publication mentioned by Crossframe:

http://www.fhwa.dot.gov/engineering/geotech/library_arc.cfm?pub_number=19

Doug Jenkins
Interactive Design Services

http://newtonexcelbach.wordpress.com/

 

[samwise753]

Hey IDS,

Thank you for posting that link, but I'm afraid it does not help for seismic design of bridge superstructure and substructure. Your link did help me get to FHWA's archives where I was able to search for FHWA-SA-097-007, but it does not look like they have it anymore. I have saved a copy of the publications at the link you posted. Seismic design of foundations could come in handy since we will be designing that element as well.

 

[ishvaaag]

See chapter 9 in this nice book.

http://www.scribd.com/doc/16808418/Bridge-Engineering-Seismic-Design

 

[apriley]

Having designed half a dozen precast girder bridges in California, I can safely say that I have never used the lateral stiffness of the bearing pads for seismic design. Typically the bents have diaphragms and act continuously (modeled as such), and the abutment stiffness is calculated based on the passive resistance of the soil. I suppose if you have a low-level earthquake with small displacements, the stiffness might come into play, but for high level earthquakes you'll probably move well beyond your bearing's lateral capacity (ie. you'll destroy the bearing, making its elastic stiffness worthless). Also, typically (not always) the lower your abutment stiffness, the higher your displacements tend to be, so it's usually conservative to ignore.

 

[samwise753]

apriley,

Thank you for the tip on lateral stiffness. For the project I am working on, we are employing shear keys to arrest the transverse motion, so I am modeling the bearing pads as fixed in the transverse at intermediate bents and abutments. At the abutments, I am calculating a shear stiffness in the longitudinal direction of the bridge. Would I be better off not accounting for any stiffness at all?

 

[jrw501]

Farzad Naeim and James Kelly's book might be pretty helpful for you, "Design of Seismic Isolated Structures" or Kelly's book "Earthquake Resistant Design with Rubber" though I don't have either with me right now.

The shear stiffness (usually written as Ks or Kh) is generally taken as GAs/Tr, where G is the shear modulus, As is the effective area in shear (As=A*h/Tr) which accounts for the steel shims being basically rigid, Tr is the total thickness of rubber (n*tr), tr is the thickness of individual layer, h is the bearing height (not including end plates).

The rotational stiffness of the bearings is a bit more complicated but is a function of geometry, shape factor, and compression modulus if I recall. I believe it's generally taken as Ktheta=Pe*h (which you get if you assume infinite shear stiffness, just as you get Kh=GA/Tr if you assume infinite rotational stiffness). Pe is the Euler Buckling Load, pi^2*(EI)eff/h^2. (EI)eff=Er*I*(h/Tr) -- similar modification as done to shear stiffness, Er is the rotational modulus which is the part affected by geometry etc...For a square bearing Er=Ec*(1+0.742S^2), for circular Ec*(1+2/3S^2) where Ec is the compression modulus (usually taken as 3G assuming incompressibility of the elastomer, i.e. G=E/2(1+v), v=0.5. S is the shape factor (defined as the ratio of the loaded area:area free to bulge -- basically plan area/circumferential area) Both equations from Er come from Gent and Meinecke's solution.


Have you checked the AASHTO Guide Specs for Seismic Isolation Design?

 

[jrw501]

I should add that S is for a single rubber layer as well, not the whole bearing.

 

[jrw501]

I should also note that both Kh and Kr reduce nonlinearly with increasing displacement, the provided equations are for the initial stiffness values.

 

[apriley]

Samwise, It depends on what code you're using. In California, we design the transverse shear keys to fail under high seismic loads to protect the abutment foundation, so we use a fictitious spring at the abutments in the transverse direction only to prevent any anomalous results in the model. In the longitudinal direction, it depends on how much the bridge relies on the abutments. We don't use the pads at all, but instead count on passive pressure of soil. You can use zero spring, but it might put too much displacement on your bents. But it's generally recognized that zero spring is conservative (qualify that statement with *usually*).

I assume you're not in CA, otherwise you'd probably know about California's Seismic Design Criteria, and this is all spelled out in there. Since you're probably using a different code, there may be different requirements. I haven't looked at the new LRFD, but I know in Nevada (before the new LRFD) the bridge had to be designed elastically, which is a completely different approach.

In any case, bearing pad resistance will probably not affect your displacements much, so if it works for you, zero spring may be the way to go.

 

[samwise753]

Hey guys and gals,

Sorry about not being here for a while. I will not be counting on the lateral stiffness of the bearing pads at the end bents though the shear keys will be restraining the transverse motion. At the interior bents, the bearing points are being modeled as either pinned or free(simple spans will alternate pinned-free; continous spans will be all pinned.) Shear keys are also present at the interior bents. I am in the process of verifying the WinSeisab analysis with a Staad analysis, though it looks like Staad isn't quite as good as WinSeisab. Also, the substructure columns are being designed for the earthquake forces, but the caps and pile footings are being designed for either the maximum seismic forces or 1.2 of the plastic moment of the column, which ever is smaller. Thank you for all the help, and also the links to resources. It is all information I would have had hard time finding on my own.