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
Wuhan University of Technology7.2 Piecewise exact methodThesimpleststepbystepmethodforanalysisofSDOFsystemsisthesocalled"piecewiseexact"method,whichisbasedontheexactsolutionoftheequationofmotionforresponseofa linearstructuretoaloadingthatvarieslinearlyduringadiscretetimeinterval.In using this method,the loading historyis divided into time intervals, usuallydefinedbysignificant changes of slope in the actual loading history;betweenthesepoints,it isassumedthattheslopeoftheloadcurveremainsconstantAlthoughtheresponseexpressionderivedfortheselinearlyvaryingload stepsis exact, it must be recognized that the actual loading history is onlyapproximated by the constant slope steps.Thusthecalculatedresponsegenerallyisnotanexactrepresentationofthetrue response to the real loading; however, the error can be reduced to anyacceptablevaluemerelybyreducingthelengthofthetimestepsandthusbetterapproximatingtheloading.Ifdesired,thelengthofthetimestepscanbevariedfromoneintervaltothenextinordertoachievethebestpossibletoftheloadinghistorybythesequenceofstraightlinesegments;however,forreasonsofcomputationalefficiencyitiscustomarytouseaconstanttimestep7-6
7-6 Wuhan University of Technology 7.2 Piecewise exact method The simplest stepbystep method for analysis of SDOF systems is the socalled “piecewise exact” method, which is based on the exact solution of the equation of motion for response of a linear structure to a loading that varies linearly during a discrete time interval. In using this method, the loading history is divided into time intervals, usually defined by significant changes of slope in the actual loading history; between these points, it is assumed that the slope of the load curve remains constant. Although the response expression derived for these linearly varying load steps is exact, it must be recognized that the actual loading history is only approximated by the constant slope steps. Thus the calculated response generally is not an exact representation of the true response to the real loading; however, the error can be reduced to any acceptable value merely by reducing the length of the time steps and thus better approximating the loading. If desired, the length of the time steps can be varied from one interval to the next in order to achieve the best possible t of the loading history by the sequence of straight line segments; however, for reasons of computational efficiency it is customary to use a constant time step
Wuhan University of Technology7.2 Piecewise exact methodUi+14 U(r)tpt)Actualα=Po-PiAssumed00p(n)= Po+αThUUj+1totit1to→t=1-1oT=1-10(b)Responsehistory(a)LoadinghistoryFIGURE 7-1 Notation for piecewise exact analysis.7-7
7-7 Wuhan University of Technology 7.2 Piecewise exact method FIGURE 7-1 Notation for piecewise exact analysis
Wuhan University of Technology17.2 Piecewise exact methodThe assumed linearly varying loading during the time step is given byp(T)=Po+Q Tmi+ci+ku=po+αT(T) = Uh(T) +Up(T)Then the damped freevibration response, as shown byEq.(248),is givenbyVh(T)=exp(-&WT)[AcOsWpT+BsinWpTac(PO+αT)Up(T) =7-8
7-8 Wuhan University of Technology 7.2 Piecewise exact method The assumed linearly varying loading during the time step is given by Then the damped freevibration response, as shown by Eq. (248), is given by
Wuhan University of Technology7.2 Piecewise exact methodThe displacement during the time step is(T)=Ao+AT+A2exp(wT) cosWpT+Aexp(-SwT)sinWpTIn which250VoAow203aA1 =w2A2 = uo - AoA3=+swA2Uoww27-9
7-9 Wuhan University of Technology 7.2 Piecewise exact method The displacement during the time step is In which
Wuhan University of Technology7.2 Piecewise exact methodSimilarly, the velocity during the time step is found to bei(T)=A+(WpA3-&wA2)exp(-wT)cOSWpT-(WDA2+wA3)exp(-SwT)sinWpTFor situations where the applied loading may be approximated wellby a series of straight line segments, this piecewise exact methodundoubtedly is the most efficient means of calculating the response ofaSDOFsystem.However,itmustalwaysberememberedthattheloadingbeingconsidered is only an approximation of the true loading history, whichusuallyisasmoothlyvaryingcurve,andthatthesteplengthsmustbechosensoastoachieveanacceptableapproximationofthetrueresponse history.7-10
7-10 Wuhan University of Technology 7.2 Piecewise exact method Similarly, the velocity during the time step is found to be For situations where the applied loading may be approximated well by a series of straight line segments, this piecewise exact method undoubtedly is the most efficient means of calculating the response of a SDOF system. However, it must always be remembered that the loading being considered is only an approximation of the true loading history, which usually is a smoothly varying curve, and that the step lengths must be chosen so as to achieve an acceptable approximation of the true response history