Response of tower structures to earthquake perturbations

Gopal Madabhushi, Cambridge University, Geotechnical Engineering Group

**Abstract**

The response of tower structures founded on saturated sand beds which are subjected to earthquake perturbations was investigated. The dynamic behaviour of this tower-soil system was studied by conducting centrifuge modelling of towers and by carrying out numerical analysis using finite element methods. Dynamic centrifuge modelling offers a method by which scaled down models can be tested at correct prototype stresses and the tower-soil interaction effects are accounted for automatically. The use of physical and numerical modelling techniques to model the tower-soil problem leads to satisfactory comparison of these techniques, so that a practicing engineer may employ these techniques with confidence in design. In the case of important structures the cost of conducting a centrifuge test may be justified. Design of smaller structures can be undertaken using numerical methods, if such methods are validated using the centrifuge test data.

Eleven centrifuge tests were conducted on eight different structures and over 60 earthquakes were fired. The rocking mode of vibration was excited in the model towers. As the saturated sand bed was subjected to cyclic shear stresses, excess pore pressures were generated, which had two distinct effects depending on the initial natural frequency of the tower-soil system. When the natural frequency was above the earthquake driving frequency, the generation of excess pore pressure (onset of partial liquefaction) resulted in a degradation of soil stiffness and a lowering of the natural frequency. A jump in the natural frequency of the system to a discrete frequency at which the earthquake's energy is concentrated was observed. However, when the initial natural frequency was either equal to or less than the driving frequency of the earthquake the increase in excess pore pressures did not result in any change of the natural frequency. The centrifuge tests on low frequency structures indicated a prolonged response of the structures which extends much beyond the duration of the earthquake itself. The effects of embedment of the structure into the sand bed were investigated.

The important dynamic parameters like the natural frequency of the tower-soil system can be predicted by using a simple discrete model with springs and dashpots. For any given class of soil-structure problem an optimum number of basic elements are required in the discrete model to predict the natural frequency of the system. For the tower-soil problem it was shown that such a discrete model must contain at least two rotational springs to predict the rocking behaviour of the tower satisfactorily. This model was employed to predict the dynamic response of the towers to the model earthquakes. The results from the discrete model and the centrifuge tests were compared directly as well as by constructing Lissajous figures. Provided that the shear modulus of sand was corrected for shear strain amplitude in the sand the degradation of soil stiffness and corresponding reduction in the natural frequency could be accommodated in the discrete model. The shear strain amplitude was estimated by assuming that plastic mechanisms of a given shape develop under the rocking towers. The shear strain amplitude estimated at one characteristic point compared satisfactorily with the shear strain obtained from a more rigorous Finite Element analyses.

In this thesis, a special emphasis was laid on accurate simulation of the semi infinite extent of the soil medium both in centrifuge modelling and in the numerical modelling. In the centrifuge tests a non reflecting material, duxseal was used to simulate the free field conditions along the transverse boundaries of the physical model. The use of duxseal was justified by a series of 162 experiments carried out under 1-g conditions in which P and S wave propagation in a vertical sand column was studied. A method of analysis using correlation functions of the acceleration time histories observed in the sand column was developed to evaluate the performance of a duxseal boundary on a quantitative basis. The results from the test series indicated that at least 65% of the incident P waves and at least 60% of the incident S waves were absorbed by the duxseal even under 1g conditions. It was expected that the performance of the duxseal would be improved in the high 'g' environment.

In parallel with this work on the physical model, two alternative schemes of non reflecting boundaries were suggested for the numerical models, which can be used in conjunction with any finite element program. The existing Smith-Cundall boundary scheme was improved and a condition for complete wave transmission at an interface joining three media that arises in this boundary scheme was suggested. An alternative scheme which uses compound parabolic collectors to hold the incident waves in the boundary region during the time of analyses was developed. The performance of this boundary was found to be satisfactory for a wide range of incident waves.

Several centrifuge tests were conducted on the same model at different 'g' levels. The natural frequency of the tower-soil system varies as the fourth root of the centrifugal acceleration. A general set of scaling relations for models at different 'g' levels were derived from basic equations using conditions of similitude. Based on these relations, it is possible to relate the centrifuge models tests to lg shaking table tests. Conjugate centrifuge tests with similar configuration and different pore fluids were conducted to isolate the effects of pore fluid on damping. It was observed that the damping associated with models saturated with water was slightly higher than the models saturated with high viscosity oil.

Finally, the numerical analyses of the tower-soil problem were carried out using finite element methods. An F.E. program called SWANDYNE for dynamic problems in Geomechanics was chosen. In carrying out the F.E. analyses it was shown that accurate simulation of the initial conditions of velocity and displacement is important. Further, non reflecting boundaries based on compound parabolic collectors were used in these analyses. The results from the F.E. analyses compared well with the centrifuge test data.