Assessment of the Bill Emerson Memorial Bridge - FTP Directory ...

9. Appendix A: Stiffness and Damping CoefficientsThe stiffness and damping of a single pile and a pile group withappropriate interaction factors are evaluated for all the piers in the Illinoisapproach of the Bill Emerson Memorial Cable-stayed Bridge. The Novak’sformulations (1974) were followed with the sign convention as depicted in FigureA.1 and the following assumptions:1. Each pile behaves linear elastically and has a round cross section. For othershapes, an equivalent radius r is determined in each mode of vibration.o2. There is no separation between soil and pile during vibration.(a) Translational and coupled constraintsFigure A.1 Sign conventionA.1 Stiffness and damping factors of a single pile(b) Rotational constantsFor vertical vibration, the stiffness (k z ) and damping factors (c z ) arekz⎡EpA⎤= ⎢ ⎥ f⎣ ro⎦w1⎡E pA ⎤, c z = ⎢ ⎥ f⎣ vs ⎦w 2(A.1)in which E p is the modulus of elasticity of pile, A is the cross section of a singlepile, r o is the radius of a solid pile or an equivalent pile radius, V s is the shearwave velocity of soils along the floating pile, and f w1 and f w2 are the twoparameters that can be obtained from Figure A.2. The torsional stiffness (k ψ ) anddamping factors (c ψ ) can be expressed into⎡GJ⎤k = p pψ ⎢ ⎥ f⎣ ro⎦T ,1⎡GJp p⎤, c = ψ ⎢ ⎥ f (A.2)T ,2⎣ V s ⎦where G p is the shear modulus of the pile, J p is the polar moment of inertia of asingle pile about z axis, and f T,1 and f T,2 are the parameters from Figure A.3.100