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ASSIGNMENT 2 – TIME-HISTORY ANALYSIS, RESPONSE SPECTRUM CALCULATION NOTES ON SOLUTION Stiffness calculation Various assumptions were made about the trussed frame structure: Assumption Accurate 6m portal 5m portal 7m portal 6m, rigid beam Resulting Stiffness 19,200kN/m 18,640kN/m 31,405kN/m 11,960kN/m 22,222kN/m Period 0.7854s 0.7971s 0.6141s 0.9951s 0.73s Analysis Method (stiffness method, 2m deep beam-col. connection) (moment distribution or stiffness method) (moment distribution or stiffness method) (moment distribution or stiffness method) (moment-area or similar) The 6m simple portal, with the mass also assumed to act at a height of 6m, had an agreeable cancellation of errors leading to a good result. The resulting maximum displacement, u max , was then obtained using the step-by-step integration algorithm, as implemented in spreadsheet or Matlab. Maximum base shear could then be deduced as stiffness x maximum displacement, k u max ; or alternatively, as mass x maximum absolute acceleration m u g u max El Centro Newhall Sylmar Hosp Mexico City . This gave the following results: T = 0.73s max displ. max shear 8.75cm 1944kN 29.0 6456 17.5 3881 0.8 176 T = 0.7971s max displ. max shear 9.5 1771 26.1 4871 18.7 3491 0.9 172 T = 0.7854s max displ. max shear 9.37 1798 26.8 5138 18.6 3570 0.8 158 Response Spectrum The response spectrum was also calculated using the time-history solution – by finding the maximum displacement for a range of single degree of freedom systems with gradually increasing periods. The Excel Data Table facility provided a good way of doing this calculation. The maximum displacement relative to the ground was the spectral displacement, Sd. 2 The spectral pseudo acceleration Sa, could be obtained from Sd by multiplying by 2 2 or . T A common error was to multiply Sd by a particular value of (such as that corresponding to the portal frame) rather than the value corresponding to each period used in the plot. A sample spreadsheet can be seen on the SM3 web site. The true spectral acceleration, u g u max , has also been calculated and plotted. You will see that it is in close agreement with the more usual pseudo spectral acceleration values.