NCHRP FY 2004 Rotational Limits for Elastomeric Bearings. Final Report. Appendix I. John F. Stanton Charles W. Roeder Peter Mackenzie-Helnwein

Transcription

1 NCHRP FY 2004 Rotational Limits for Elastomeric Bearings Final Report Appendix I John F. Stanton Charles W. Roeder Peter Mackenzie-Helnwein Department of Civil and Environmental Engineering University of Washington Seattle, WA i -

2 TABLE OF CONTENTS APPENDIX I NOTATION I-1 - ii -

3 APPENDIX I Notation a = dimensionless coefficient in fatigue model A = area of the bearing = WL A a = dimensionless coefficient in axial stiffness A az = dimensionless coefficient in axial stiffness = A a (App. E) a ij = dimensionless coefficient in FEA error analysis A net = plan area of bearing based on net dimensions AR = aspect ratio = the smaller of L/W and W/L. A r = dimensionless coefficient in rotational stiffness A ry = dimensionless coefficient in rotational stiffness = A r (App. E) b = dimensionless coefficient in fatigue model B a = dimensionless coefficient in axial stiffness B az = dimensionless coefficient in axial stiffness (App. E) B r = dimensionless coefficient in rotational stiffness for compressible layers B r0 = dimensionless coefficient in rotational stiffness for incompressible layers B ry = dimensionless coefficient in rotational stiffness = B r C C 10, C 20, C 30 = Right Cauchy-Green strain tensor = Material parameters for Yeoh s model C a = dimensionless coefficient in shear strain due to axial load C azzx = dimensionless coefficient in shear strain due to axial load = C a (App. E) c n = dimensionless coefficient in fatigue model C r = dimensionless coefficient in shear strain due to rotational C ryzx = dimensionless coefficient in shear strain due to rotational = C r (App. E) = limiting permissible of S 2 /n for Method A design c S c σ = dimensionless stress coefficient (lift-off equations) D = debonding level D = diameter of the bearing (App. G) D a D r e = dimensionless shear strain coefficient for axial load = dimensionless shear strain coefficient for rotation = Euler s constant (basis of Napieran logarithm) - I-1 -

4 E E E az E ry F r G g 0, g 1 H a H r h ri h rt I K K a K r L l L net M m N n N cr p P P sd S S S i t W = Green-Lagrange strain tensor = Young s modulus = apparent Young s modulus for axial loading = apparent Young s modulus for rotational loading = dimensionless coefficient for rotation (uplift equations) = shear modulus = dimensionless coefficients in fatigue model = dimensionless coefficient for axial load (uplift equations) = dimensionless coefficient for rotation (uplift equations) = thickness of i th interior layer of elastomer = total thickness of all interior layers of elastomer = moment of inertia (second moment of area) = bulk modulus = total axial stiffness = total rotational stiffness = length of bearing based on gross dimensions (= plan dimension of the bearing perpendicular to the axis of rotation under consideration) = span of a girder = net length of bearing (average of gross and shim dimensions) = moment on bearing = exponent in fatigue model = number of cycles = number of interior layers of elastomer = characteristic number of cycles = force per unit length = total axial force = minimum vertical force due to permanent loads = 2 nd Piola-Kirchhoff stress tensor = shape factor = shape factor of instantaneous compressed region (lift-off equations) = thickness of elastomeric layer = gross width of elastomeric layer (= plan dimension of the bearing parallel to the axis of rotation under consideration) - I-2 -

5 W net x,y,z Δ a Δ s 1 Ι, (Ι * 1 ) α δ bottom δ top ε a ε ai = net width of elastomeric layer (average of gross and shim dimensions) = coordinates in Cartesian system = axial deflection = maximum total shear displacement of the bearing at the service limit state. = first invariant of C (specialized for uniaxial tension) = dimensionless load combination parameter =ε a /Sθ L = vertical displacement of bottom shim = vertical displacement of top shim = average axial strain for bearing under axial load = axial strain at the middle of the instantaneous compressed region (lift-off equations) ε az ε zz γ a γ a,cy γ a,max γ a,st γ cap γ r γ r,cy γ r,max γ r,st γ r0 γ s γ s,cy γ s,st γ tot,max γ zx η = average axial strain = ε a = local vertical normal strain in rubber layer = shear strain in z-x plane due to axial loading = cyclic portion of shear strain in z-x plane due to axial loading = absolute maximum shear strain in z-x plane due to axial loading = static portion of shear strain in z-x plane due to axial loading = shear strain capacity = shear strain in z-x plane due to rotation loading = cyclic portion of shear strain in z-x plane due to rotation loading = absolute maximum shear strain in z-x plane due to rotation loading = static portion of shear strain in z-x plane due to rotation loading = shear strain constant in fatigue model = shear strain in z-x plane due to shear displacement = cyclic portion of shear strain in z-x plane due to shear displacement = static portion of shear strain in z-x plane due to shear displacement = maximum total shear strain in z-x plane = local shear strain in z-x plane = relative length of the instantaneous compressed region (lift-off equations) λ = compressibility index = S 3 G / K - I-3 -

6 λ 1, λ 2, λ 3 = principal stretches (App. E) θ θ c θ i θ L θ x, θ y ρ σ a σ a0 σ hyd σ rupture σ zz τ zx = end rotation of a girder (rotation demand on bearing) = characteristic rotation for which the vertical displacement on the tension side becomes net upwards = rotation of the i th layer of elastomer = rotation per layer = rotation of whole bearing about x or y axis = dimensionless rotation ratio (lift-off equations) = average axial stress = fictitious average axial stress for entire bearing surface (lift-off equations) = hydrostatic stress (mean direct stress) = (hydrostatic) rupture strength of rubber = local vertical normal stress in rubber layer = local shear stress in z-x plane ξ = dimensionless position parameter = 2 x / L - I-4 -