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<br />(3) Dynamic Analyses Methods <br /> <br />These methods include any analyses which determines the <br />structural response based upon the dynamic characteristics <br />of the structures and of the earthquake ground motions <br />established for the site. The dynamic method currently <br />used most often is the modal analysis method. This method <br />is based on the fact that, for certain forms of damping <br />that are reasonable models for many structures, the <br />response in each natural mode of vibration can be computed <br />independently of the others and the modal responses can be <br />combined to determine the total response. Either a <br />response spectrum or an acceleration-time record can be <br />used with the modal analysis technique. Dynamic methods <br />should be used whenever a stress analysis of a concrete <br />dam is required. <br /> <br />(a) Response Spectrum Method <br /> <br />In this method the maximum response in each mode, in <br />the form of equivalent lateral loads, is directly <br />computed from the earthquake response spectrum and <br />the structure's dynamic charscteristics. These modal <br />responses are then combined to obtain estimates (but <br />not the exact value) of the maxinum total response. <br />Adding the absolute values of each mode gives an <br />upper bound for the total response, while using the <br />square root of the sum of the squares yields a more <br />probable total re sponse. The internal stresses are <br />then computed by a static analysis of the structure <br />subjected to the equivalent lateral loads and added <br />algebraically to the static load stresses. Either <br />a simple beam analysis (P/A :t Me/I) or finite element <br />analysis can be used to compute the stresses. <br /> <br />(b) Time-History Me thod <br /> <br />In this method the response of each mode, in the form <br />of equivalent lateral loads, is calculated for the <br />entire duration of an earthquake acceleration-time <br />record or synthetic acce1erogram, starting with <br />initial conditions, taking a small time interval, and <br />computing the response at the end of each time <br />interval. The responses of each mode are added <br />together for each time interval to yield the total <br />time-history response. The internal stresses are <br />then computed by a static analysis, normally using <br />the finite element method of analysis, and added <br />algebraically to the static load stresses. Stress <br />histories of horizontal and vertical stresses on each <br />face show the maxi1llUms, the number and duration of <br />excursions beyond the tension or compression limits, <br />and the area of the dam in distress. <br /> <br />-20- <br />