Theoretical Study On The Vertical Dynamic Response Of Rigid Strip Footings On Soil Layer Overlying Rigid Bedrock

The study of “Machine-footing-foundation” system caused by the unbalanced disturbance or impact of a dynamic machine on its foundation in the “Dynamic Soil-Structure Interaction”(DSSI)problem,satisfying the demand for structural seismic design and foundation vibration research in the industrialization of construction industry,and providing a reliable theoretical reference for the design of dynamic footings.Most previous studies about the dynamic response of strip footings have assumed that the soil layer is an elastic half-space,which can simplify the solution of the governing equations,but this assumption ignores the propagation and reflection of waves generated by footing vibrations in the soil layer,and cannot accurately quantify the dynamic response when the thickness of soil layer is thin.In this paper,the governing equations of motion of the soil layer are formulated based on the theory of elastic dynamics and the theory of saturated porous media,and are solved by using the Fourier integral transform.By combining the mixed boundary conditions of the displacement and stress,and the rigid bedrock underneath the soil layer,the dynamic response of rigid strip footings subjected to vertical harmonic loads is investigated.The main investigations and conclusions are as follows:(1)Considering the influence of soil thickness,the vertical vibration response of rigid strip footings on viscoelastic soil is theoretically investigated.Based on the assumptions for footing and foundation with rigid bedrock boundary condition,employing the method of Fourier integral transform,this mixed-boundary value problem is cast as a pair of dual integral equations,which are transformed into a set of linear equations by means of Jacobi orthogonal polynomials and solved numerically.The solution is compared with the solution of elastic half-space to verify its validity.Parametric application of the solution demonstrates that attenuation of ground waves generated by the vibration of the footing,and more so the dynamic compliance coefficient of the foundation,are sensitive to the thickness of the foundation soil layer.The obtained results suggest that considering the soil layer as infinite half-space,as in existing solutions,is reasonable only in cases where the thickness of the top soil layer exceeds approximately fifty times the width of the footing.(2)Considering the influence of soil thickness,the vertical vibration response of rigid strip footing on poroelastic soil is theoretically investigated.Based on the Biot’s poroelastodynamic theory,the governing equations of motion of saturated soil are established and solved by using Fourier integral transform.To verify its validity,the proposed solution was degenerated to the case of half-space and compared with the existing half-space solutions for single-phase and saturated soils.The dynamic response of the rigid footing is analyzed in terms of the soil thickness and the parameters of saturated soil.(3)Based on the Biot’s poroelastodynamic theory,the vertical dynamic response of a rigid strip footing on the soil layer overlying rigid bedrock is studied by considering the presence of the water table and its variation.The soil layer above the water table is treated as a single-phase elastic soil and the soil layer below the water table is treated as saturated soil.The resulting solution,which is degenerated to the case of half-space,is compared with the half-space solution with a water table to verify its validity,focusing on the effect of varying the water table and parameters of soil layer on the dynamic response of rigid strip footings overlying rigid bedrock.