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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 Yeohâs 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 cÏ = 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 = Eulerâs constant (basis of Napieran logarithm) E = Green-Lagrange strain tensor E = Youngâs modulus Eaz = apparent Youngâs modulus for axial loading Ery = apparent Youngâs 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 Îa = axial deflection Îs = maximum total shear displacement of the bearing at the service limit state. Î1, (Î1*) = 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 âtensionâ 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 Ï = dimensionless rotation ratio (lift-off equations) Ïa = average axial stress Ïa0 = fictitious average axial stress for entire bearing surface (lift-off equations) Ïhyd = hydrostatic stress (mean direct stress) Ïrupture = (hydrostatic) rupture strength of rubber Ïzz = local vertical normal stress in rubber layer Ïzx = local shear stress in z-x plane ξ = dimensionless position parameter = 2x/L S G K3