Assessment of the Bill Emerson Memorial Bridge - FTP Directory ...

knω = (5.4)nm nSimilarly, when damping is present, the n th modal equation of motion can be derived asm nq&& n() t + c nq& n+ k nq n() t = pn () t(5.5)where c is the damping coefficient of the n mode of vibration. Eq. (5.5) indicates thatnthe n th modal displacement q tn ( ) depends on its corresponding natural frequencyω n ,damping ratio cnth/2 m k n, and the frequency content of external excitation. After thenmodal displacement q (t) has been determined, the contribution of the n thnmode to thedisplacement {U( t)} can be evaluated by Eq. (5.2).5.2.2. Modal analysis of Bill Emerson Memorial cable-stayedbridgeThe natural frequencies and mode vectors of the cable-stayed bridge were determined byan eigensolution method. For time-history analyses, the so-called Ritz-vector method isused. This is because Ritz-vectors can provide a better basis than eigenvectors do whenused for response-spectrum or time-history analysis (Wilson, 1982). The Ritz vectors aregenerated by taking into account the spatial distribution of the dynamic loading, whereasthe direct use of natural mode shapes neglects this very important information.How many modes of vibration must be included in analysis is a practical question. Inbuilding design, a rule of thumb is to have accumulated modal participating mass factorsin all directions of over 90%. For complex 3-D cable-stayed bridge structures, it isextremely difficult to achieve that level of mass participation in all directions. In thisstudy, an effort is made to include an accumulated modal participating mass factor ofover 70% in every direction. To this endeavor, a total of 4000 modes are specified in theinitial analysis of the cable-stayed bridge. The relationship between natural frequency andmode number is depicted in Figure 5.1. It was observed that the natural frequencies of thebridge up to 45.54 Hz were covered. The model was calibrated by slightly increasing themass density of concrete so that the first several natural frequencies can match theexperimental frequencies obtained from the measured responses. This modificationmainly represents the uncertainty of material density and geometry and the effects ofnon-structural components such as bracings, railing supports, and other permanentattachments.53