Wuhan University of TechnologyChapter7Response to general dynamic loading:step-by-step methods7-1
7-1 Wuhan University of Technology Chapter 7 Response to general dynamic loading: step-by-step methods
Wuhan University of Technology?Contents7.1 General concepts7.2 Piecewise exact method7.3 Numerical approximation procedures general comments7.4 Second central differenceformulation7.5 Integration methods7.6 Incremental formulation for nonlinear analysis7.7 Summary of the linear acceleration procedure7-2
7-2 Wuhan University of Technology 7.1 General concepts 7.2 Piecewise exact method 7.3 Numerical approximation procedures general comments 7.4 Second central difference formulation 7.5 Integration methods 7.6 Incremental formulation for nonlinear analysis 7.7 Summary of the linear acceleration procedure Contents
Wuhan Universityof Technology7.1 General conceptsThe response analysis procedures whether formulated in the timedomain or in the frequency domain, involve evaluation of manyindependentresponsecontributionsthatarecombinedtoobtainthetotal response.Because superposition is applied to obtain the total result in bothprocedures, neither of these methods is suited for use in analysis ofnonlinearresponse;thereforejudgmentmustbeusedinapplyingtheminearthguakeengineeringwhereitisexpectedthatasevereearthquakewill induce inelastic deformation in a codedesigned structure.7-3
7-3 Wuhan University of Technology 7.1 General concepts The response analysis procedures whether formulated in the time domain or in the frequency domain, involve evaluation of many independent response contributions that are combined to obtain the total response. Because superposition is applied to obtain the total result in both procedures, neither of these methods is suited for use in analysis of nonlinear response; therefore judgment must be used in applying them in earthquake engineering where it is expected that a severe earthquake will induce inelastic deformation in a codedesigned structure
Wuhan University of Technology7.1 General conceptsThe stepbystep procedure is a second general approach to dynamicresponse analysis, and it is well suited to analysis of nonlinear responsebecause it avoids any use of superposition.There are many different stepbystep methods, but in all of them theloading and the responsehistoryaredivided into a sequence of timeintervals or steps.Theresponseduringeachsteptheniscalculatedfromtheinitialconditions existing at the beginning of the step and from the history ofloadingduringthestep.Thustheresponseforeachstepisanindependent analysis problem, and there is no need to combineresponsecontributionswithinthestep.7-4
7-4 Wuhan University of Technology 7.1 General concepts The stepbystep procedure is a second general approach to dynamic response analysis, and it is well suited to analysis of nonlinear response because it avoids any use of superposition. There are many different stepbystep methods, but in all of them the loading and the response history are divided into a sequence of time intervals or steps. The response during each step then is calculated from the initial conditions existing at the beginning of the step and from the history of loading during the step. Thus the response for each step is an independent analysis problem, and there is no need to combine response contributions within the step
WuhanUniversityof Technology7.1 General conceptsNonlinear behavior may be considered easily by this approach merelybyassumingthatthestructuralpropertiesremainconstantduringeachstep, and causing them to change in accordance with any specified formofbehaviorfromonesteptothenext;hencethenonlinearanalysisactually is a sequenceof linear analyses of a changing system.Anydesireddegreeof refinementinthenonlinearbehaviormaybeachieved in this procedure by making the time steps short enough; alsoit can be applied to any type of nonlinearity, including changes of massanddampingpropertiesaswell asthemorecommonnonlinearitiesduetochanges ofstiffness.7-5
7-5 Wuhan University of Technology 7.1 General concepts Nonlinear behavior may be considered easily by this approach merely by assuming that the structural properties remain constant during each step, and causing them to change in accordance with any specified form of behavior from one step to the next; hence the nonlinear analysis actually is a sequence of linear analyses of a changing system. Any desired degree of refinement in the nonlinear behavior may be achieved in this procedure by making the time steps short enough; also it can be applied to any type of nonlinearity, including changes of mass and damping properties as well as the more common nonlinearities due to changes of stiffness