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#### Year of publication

- 1997 (3)

The problem of the computation of stresses and settlements in the half-space under various types of loads is often presented in geotechnical engineering. In 1885 Boussinesq advanced theoretical expressions to determine stresses at a point within an ideal mass. His equation considers a point load on the surface of a semi-infinite, homogeneous, isotropic, weightless, elastic half-space. Newmark in 1942 performed the integration of Boussinesq's equations for the vertical stress under a corner of a rectangular area loaded with a uniform load. The problem of the determination of vertical stresses under a rectangular shaped footing has been satisfactorily solved with renewal integration of the Boussinesq's equation over the arbitrary rectangle on surface of the half-space, with a non-uniform load represented with piecewise linear interpolation functions. The problem of the determination of stresses in the case when the footing shape is an arbitrary quadrilateral however remains unsolved. The paper discusses an approach to the computation of vertical stresses and settlements in an arbitrary point of the half-space, loaded with a uniform load, which shape in the ground plan can be a general four noded form with straight edges. Since the form is transformed into a biunit square and all integrations are performed over this area, all solutions are valid also for an arbitrary triangle by the implementation of the degeneration rule.

In the design of a structure, the implementation of reliable soil-foundation-structure interaction into the analysis process plays a very important role. The paper presents a determination of parameters of a suitably chosen soil-foundation model and their influence on the structure response. Since the mechanical data for the structure can be determined with satisfactory accuracy, the properties of the soil-foundation model were identified using measured dynamic response of the real structure. A simple model describing soil-foundation structure was incorporated into the classical 3-D finite element analysis of the structure with commercial software. Results obtained from the measured data on the pier were afterwards compared with those obtained with the finite model of the pier-foundation-soil structure. On the basis of this comparison the coefficients describing the properties in the soil-foundation model were adjusted until the calculated dynamic response coincided with the measured ones. In this way, the difference between both results was reduced to 1%. Full-scale tests measuring eigenmotion of the bridge were performed through all erection stages of the new bridge in Maribor. In this way an effective and experimentally verified 3-D model for a complex dynamic analysis of the bridge under the earthquake loading was obtained. The significant advantage of the obtained model is that it was updated on the basis of the dynamic measurements thus improving the model on the basis of in-situ geomechanical measurements. The model is very accurate in describing the upper structure and economical in describing the soil mass thus representing an optimal solution regarding computational efforts.

To fulfil safety requirements the changes in the static and/or dynamic behaviour of the structure must be analysed with great care. These changes are often caused by local reduction of the stiffness of the structure caused by the irregularities in the structure, as for example cracks. In simple structures such analysis can be performed directly, by solving equations of motion, but for more complex structures a different approach, usually numerical, must be applied. The problem of crack implementation into the structure behaviour has been studied by many authors who have usually modelled the crack as a massless rotational spring of suitable stiffness placed at the beam at the location where the crack occurs. Recently, the numerical procedure for the computation of the stiffness matrix for a beam element with a single transverse crack has been replaced with the element stiffness matrix written in fully symbolic form. A detailed comparison of the results obtained by using 200 2D finite elements with those obtained with a single cracked beam element has confirmed the usefulness of such element.