Dynamics of a rectangular foundation contacting a viscoelastic layered medium

Abstract

When constructing structures on a stratum of sedimentary rocks underlain by hard rock or in other special cases it is necessary to take into account the characteristics of dynamic processes in the base. Theoretically, resonance phenomenon affecting oscillations of the structure can occur in the presence of reflecting surfaces in the ground. It is also necessary to take into account the viscoelastic properties of the bases by using various models.The purpose of the work was to develop a method of orthogonal polynomials for solving dynamic contact problems for a rigid plate with a flat rectangular lower surface. An algorithm for numerical modeling of the process of vertical and horizontal-rotational oscillations of rectangular foundations on a viscoelastic medium was developed on the basis of a method presented in [3]. Here are examined such models of a viscoelastic base: 1) half-space -Kelvin body; 2) a layer coupled with a rigid base; 3) a layer on a half-space; 4) two layers on a rigid base (the given enumeration is used below when referring to the type of model),A complex elastic modulus with consideration of its variation in time is used in model 1 for taking into account the viscosity of the medium. For the Kelvin body the function of the time dependence can be represented by the sum of the exponents N Ro(t) = ao + Z an exp (-~nt),n=l where a n and e n are dimensionless constants characterizing the elastic and viscous properties of the medium (a o corresponds N to the prolonged elastic modulus, ~ a n to the instantaneous); t is time. In the limiting cases RO(0 ) = 1.0 and Ro(oO) n=l = a o. When a 0 < 1 the elastic modulus decreases in time from the instantaneous to the prolonged value. In this case movement will increase under a constant load (the creep phenomenon occurs). In models 2, 3, and 4 viscosity is taken into account according to the theory of frequency-~ndependent internal friction. The Lain6 constants X and ~ are complex, do not depend on frequency, and are expressed by the coefficient of losses of a viscoelastic material 3' = const in the following way: X = X exp(i3,), ~ = # exp(i~). The complex modulus E, the real part of which corresponds to the elastic modulus, can be expressed by them, and the Poisson ratio v is considered a real constant. The relation between the logarithmic decrement and loss coefficient has the form 6 = 2~rtg( I ~~ I ) = ~r3,, where ~o k is the phase characteristic of the system during harmonic oscillations [5].Vertical and horizontal-rotational oscillations were examined separately, The calculation schemes are given in Fig. 1. Movement of the foundation is described in rectangular coordinates x, y, z; the ~ axis is directed inside the viscoelastic medium.The initial data are: the size of the area of contact 2a x 2b (the notation b/a = c~ is introduced); height of the plate h; density of the plate material Pl. For the upper part of the base contacting the plate are assigned: vo, the Poisson ratio; Po, the density; Eo, the elastic modulus; viscosity parameters; c 1 and c 2,...