National Academies Press: OpenBook

Rotation Limits for Elastomeric Bearings (2008)

Chapter: Notation

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Page 53
Suggested Citation:"Notation." National Academies of Sciences, Engineering, and Medicine. 2008. Rotation Limits for Elastomeric Bearings. Washington, DC: The National Academies Press. doi: 10.17226/23131.
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Page 53
Page 54
Suggested Citation:"Notation." National Academies of Sciences, Engineering, and Medicine. 2008. Rotation Limits for Elastomeric Bearings. Washington, DC: The National Academies Press. doi: 10.17226/23131.
×
Page 54
Page 55
Suggested Citation:"Notation." National Academies of Sciences, Engineering, and Medicine. 2008. Rotation Limits for Elastomeric Bearings. Washington, DC: The National Academies Press. doi: 10.17226/23131.
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Page 55

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52 N O T A T I O N a = dimensionless coefficient in fatigue model A = area of the bearing = WL Aa = dimensionless coefficient in axial stiffness Aaz = dimensionless coefficient in axial stiffness = Aa (App. E) aij = dimensionless coefficient in FEA error analysis Anet = plan area of bearing based on net dimensions AR = aspect ratio = the smaller of L/W and W/L. Ar = dimensionless coefficient in rotational stiffness Ary = dimensionless coefficient in rotational stiffness = Ar (App. E) b = dimensionless coefficient in fatigue model Ba = dimensionless coefficient in axial stiffness Baz = dimensionless coefficient in axial stiffness (App. E) Br = dimensionless coefficient in rotational stiffness for compressible layers Br0 = dimensionless coefficient in rotational stiffness for incompressible layers Bry = dimensionless coefficient in rotational stiffness = Br C = Right Cauchy-Green strain tensor C10, C20, C30 = Material parameters for Yeoh’s model Ca = dimensionless coefficient in shear strain due to axial load Cazzx = dimensionless coefficient in shear strain due to axial load = Ca (App. E) cn = dimensionless coefficient in fatigue model Cr = dimensionless coefficient in shear strain due to rotational Cryzx = dimensionless coefficient in shear strain due to rotational = Cr (App. E) cS = limiting permissible of S2/n for Method A design cσ = dimensionless stress coefficient (lift-off equations) D = debonding level D = diameter of the bearing (App. G) Da = dimensionless shear strain coefficient for axial load Dr = dimensionless shear strain coefficient for rotation e = Euler’s constant (basis of Napieran logarithm) E = Green-Lagrange strain tensor E = Young’s modulus Eaz = apparent Young’s modulus for axial loading Ery = apparent Young’s modulus for rotational loading Fr = dimensionless coefficient for rotation (uplift equations) G = shear modulus g0, g1 = dimensionless coefficients in fatigue model

Ha = dimensionless coefficient for axial load (uplift equations) Hr = dimensionless coefficient for rotation (uplift equations) hri = thickness of ith interior layer of elastomer hrt = total thickness of all interior layers of elastomer I = moment of inertia (second moment of area) K = bulk modulus Ka = total axial stiffness Kr = total rotational stiffness L = length of bearing based on gross dimensions (= plan dimension of the bearing perpendicular to the axis of rotation under consideration) l = span of a girder Lnet = net length of bearing (average of gross and shim dimensions) M = moment on bearing m = exponent in fatigue model N = number of cycles n = number of interior layers of elastomer Ncr = characteristic number of cycles p = force per unit length P = total axial force Psd = minimum vertical force due to permanent loads S = 2nd Piola-Kirchhoff stress tensor S = shape factor Si = shape factor of instantaneous compressed region (lift-off equations) t = thickness of elastomeric layer W = gross width of elastomeric layer (= plan dimension of the bearing parallel to the axis of rotation under consideration) Wnet = net width of elastomeric layer (average of gross and shim dimensions) x,y,z = coordinates in Cartesian system Δa = axial deflection Δs = maximum total shear displacement of the bearing at the service limit state. Ι1, (Ι1*) = first invariant of C (specialized for uniaxial tension) α = dimensionless load combination parameter = εa/SθL δbottom = vertical displacement of bottom shim δtop = vertical displacement of top shim εa = average axial strain for bearing under axial load εai = axial strain at the middle of the instantaneous compressed region (lift-off equations) εaz = average axial strain = εa εzz = local vertical normal strain in rubber layer γa = shear strain in z-x plane due to axial loading γa,cy = cyclic portion of shear strain in z-x plane due to axial loading γa,max = absolute maximum shear strain in z-x plane due to axial loading γa,st = static portion of shear strain in z-x plane due to axial loading γcap = shear strain capacity γr = shear strain in z-x plane due to rotation loading 53

54 γr,cy = cyclic portion of shear strain in z-x plane due to rotation loading γr,max = absolute maximum shear strain in z-x plane due to rotation loading γr,st = static portion of shear strain in z-x plane due to rotation loading γr0 = shear strain constant in fatigue model γs = shear strain in z-x plane due to shear displacement γs,cy = cyclic portion of shear strain in z-x plane due to shear displacement γs,st = static portion of shear strain in z-x plane due to shear displacement γtot,max = maximum total shear strain in z-x plane γzx = local shear strain in z-x plane η = relative length of the instantaneous compressed region (lift-off equations) λ = compressibility index = λ1, λ2, λ3 = principal stretches (App. E) θ = end rotation of a girder (rotation demand on bearing) θc = characteristic rotation for which the vertical displacement on the “tension” side becomes net upwards θi = rotation of the ith layer of elastomer θL = rotation per layer θx, θy = rotation of whole bearing about x or y axis ρ = dimensionless rotation ratio (lift-off equations) σa = average axial stress σa0 = fictitious average axial stress for entire bearing surface (lift-off equations) σhyd = hydrostatic stress (mean direct stress) σrupture = (hydrostatic) rupture strength of rubber σzz = local vertical normal stress in rubber layer τzx = local shear stress in z-x plane ξ = dimensionless position parameter = 2x/L S G K3

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 Rotation Limits for Elastomeric Bearings
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TRB's National Cooperative Highway Research Program (NCHRP) Report 596: Rotation Limits for Elastomeric Bearings explores the elastomeric bearing design procedures suitable for adoption in the American Association of State Highway and Transportation Officials' load and resistance factor design (LRFD) bridge design specifications.

The appendixes to the report include the following:

Appendix A Test Data

Appendix B Survey of Current Practice

Appendix C Test Apparatus and Procedures

Appendix D Test Results Overview

Appendix E Finite Element Analysis

Appendix F Development of Design Procedures

Appendix G Proposed Design Specifications

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