Wuhan Universityof Technology7.2 Piecewise exact methodExample E71.tp(t)(a)At=0.0075secm=0.4145kips-sec2/ir(b)4t=0.0225sec(c)=0.045sec26Po=32kips=252.53kips/in=1.4469kips-sec/in(5-5W=mg=160kips2元=0.18sec0(a)SDOFpropertiesT,=T=0.18sec/T,=0.09 sec-(b) Straight line approximations of sine-wave loadingFIGUREE7-1Piecewiseexactexample-SDOFstructureandloading7-11
7-11 Wuhan University of Technology 7.2 Piecewise exact method FIGURE E7-1 Piecewise exact example – SDOF structure and loading. Example E71
Wuhan University of Technology7.2 Piecewise exact methodExample E71.(a)Ar=0.0075sec(b)t=0.0225sed-(c)Af=0.045secotoes0.270.450.090.180.360.540Time,secFIGUREE7-2Piecewiseexactcalculatedresponse7-12
7-12 Wuhan University of Technology 7.2 Piecewise exact method FIGURE E7-2 Piecewise exact calculated response. Example E71
Wuhan University of Technology7.3 Numerical approximation proceduresgeneral commentsBeforethedetails of anyof theproceduresare described,it will be useful tosummarizeafewbasicfacts about the numerical approximationstepbystepmethodsingeneral,asfollows:(1)ThemethodsmaybeclassifiedaseitherexplicitorimplicitAnexplicit(or"open")methodisdefinedasoneinwhichthenewresponsevaluescalculatedineachstepdependonlyonquantitiesobtainedintheprecedingstep,sothattheanalysisproceedsdirectlyfromonesteptothenext.Inan implicitmethod,ontheotherhand,theexpressionsgivingthenewvaluesforagivenstepincludeoneormorevaluespertainingtothatsamestep,sothattrialvaluesofthenecessaryquantitiesmustbeassumedandthenthesearerefinedbysuccessiveiterations.Unlessthecalculationsrequiredforeachstepareverysimple,thecost of iterationwithinastepmaybeprohibitive;thusitoftenisdesirabletoconvertanimplicitmethodtoanexplicit form by a procedure such as that described in Section 75.7-13
7-13 Wuhan University of Technology 7.3 Numerical approximation procedures general comments Before the details of any of the procedures are described, it will be useful to summarize a few basic facts about the numerical approximation stepbystep methods in general, as follows: (1) The methods may be classified as either explicit or implicit. An explicit (or “open”) method is defined as one in which the new response values calculated in each step depend only on quantities obtained in the preceding step, so that the analysis proceeds directly from one step to the next. In an implicit method, on the other hand, the expressions giving the new values for a given step include one or more values pertaining to that same step, so that trial values of the necessary quantities must be assumed and then these are refined by successive iterations. Unless the calculations required for each step are very simple, the cost of iteration within a step may be prohibitive; thus it often is desirable to convert an implicit method to an explicit form by a procedure such as that described in Section 75
Wuhan Universityof Technology7.3 Numerical approximation proceduresgeneral comments(2) The primary factor to be considered in selecting a stepbystep method isefficiency,whichconcernsthecomputationaleffortrequiredtoachievethedesiredlevelofaccuracyovertherangeoftimeforwhichtheresponseisneeded.Accuracy alone cannot be a criterion for method selection because,ingeneral,anydesireddegreeofaccuracycanbeobtainedbyanymethod ifthe time step is made short enough (but with obvious correspondingincreasesofcosts).Inanycase,thetimestepsmustbemadeshortenoughto provide adequate definition of the loading and the response history -ahighfrequency input or response cannot be described by long time steps.7-14
7-14 Wuhan University of Technology 7.3 Numerical approximation procedures general comments (2) The primary factor to be considered in selecting a stepbystep method is efficiency, which concerns the computational effort required to achieve the desired level of accuracy over the range of time for which the response is needed. Accuracy alone cannot be a criterion for method selection because, in general, any desired degree of accuracy can be obtained by any method if the time step is made short enough (but with obvious corresponding increases of costs). In any case, the time steps must be made short enough to provide adequate definition of the loading and the response history –a highfrequency input or response cannot be described by long time steps