CP 2 Properties of polygonal plane areas
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1 CP Propertes of polgoal plae areas Ket D. Hjelmstad Scool for Sustaale Egeerg ad te Bult Evromet Ira. Fulto Scools of Egeerg roa State Uverst
2 Te propertes of cross sectos Cross Secto Cross Secto Logtudal s Te propertes of cross sectos. I te stud of eams (cludg te aal ar ad torso) t ecomes evdet tat muc of te eavor s dctated te propertes of te cross sectoal geometr. I fact, te resstace to deformato s alwas a fucto of te dstruto of materal a cross secto. For te aal ar te cross sectoal area s ke; for torso te polar momet of erta sows up; for fleure te momet of erta aout te as of edg s a mportat propert. Tese propertes are essetal to determg te deformato uder load. Most of te classcal trcks of te trade tat ca e used servce of te computato of geometrc propertes amout to makg use of te propertes of smple geometrc peces (ofte rectagles) to uld up te overall cross sectoal propertes. Wat we sow ts set of otes s tat t s possle to derve te tegrals eeded to compute te cross sectoal propertes for a geeral tragle ad te use tat asc foudato to create a metod to compute te cross sectoal propertes of a closed polgoal rego. Ts approac s ased o two ke oservatos: () all tegrals ca e dvded to peces ad () t s possle to add a tegral tat s ot te pscal rego as log as ou sutract t rgt ack out.
3 Te ke cross sectoal propertes c e 3 C e r ρ Te tesor (or outer) product of vectors v v v v T v v vv v v v vv v Te ke cross sectoal propertes. Cosder te geeral cross secto defed at left. We estals a (, ) coordate sstem wt org, ad mage aoter coordate sstem (, ) passg troug te cetrod C. For eam teor we eed to compute area ad momet of erta (aout te cetrod). To compute te momet of erta we eed te locato of te cetrod. Te area s smpl defed as te sum of te ftesmal areas tat make up te cross secto: J ρρ Te cetrod of a area s defed as follows: c r I oter words, t s te costat vector c tat s te average over te area of te posto r of te ftesmal areas. Fall, te momet of erta (tesor) s defed as Were te vector r s te dstace from te cetrod C to te ftesmal area of tegrato.
4 Te momet of erta tesor Te momet of erta tesor. Te prolem wt computg J s tat t s defed terms of te cetrodal coordate sstem. We ca compute t te regular coordates otg tat ( vector addto) Tus, we ca compute ρ r c c e 3 C e r ρ J ρρ r c r c r r c r r c c c r r c r r c c c r r c c c c c c r r c c c c cc T c c c c c c c c c c r r rr T Terefore, we ca compute te momet of erta tesor te orgal coordate sstem as J r r c c We eed te cetrodal dstace c, ad we eed to fgure out ow to compute te tegral of rr.
5 Te geeral tragular rego c Te asc tragle e 3 j e ll of tese ke quattes ca e computed from te vectors ad j j j m Te geeral traglular rego. We ca specale te oto of our cross sectoal propertes to a geeral tragular rego (sow at left). We wll eed to get specfc aout ow to do te tegrals over ts rego. ce we ave foud te results, we ca use tem as a asc uldg lock a algortm to compute te tree asc propertes of a polgoal cross secto (ad, takg pots close togeter we ca appromate cross sectos wt curved edges, too). We wll costruct our cocept of te tragle from te posto vectors of te two vertces ad j (wc wll le o a edge of te polgo later). Tese vectors are ad j. From tese vectors we ca compute,, ad te ut vectors ad m (as sow te o at left). j j m j
6 Itegrato over te tragle Itegrato over te tragular rego. It wll e coveet to set up te area tegrato te coordate sstem ad m (wc appes ot to e rectagular). m d m d w( ) Tus, te elemet of tegrato s m e d d e d m d s d d s m e m m 3 3 Te varales of tegrato are (measures dstace drecto) ad (measures dstace m drecto), wc are te rage Note ow te lmts of tegrato are doe te o at rgt. d te tegrato over te area s doe as w( ) ( ) s ( ) d d s ( ) d d
7 rea of te geeral tragle rea of te geeral tragle. We ca compute area of te tragle as follows: m w( ) s s d s d s s d d Note ow oe alf ase tmes egt stll seems to e part of te pcture. Te factor s takes care of two tgs: () tat te tragle s ot a rgt tragle ad () f t turs out to e egatve te te area s egatve. Te area of te geeral tragle s For eam teor we eed to compute area, cetrodal locato, ad momet of erta (aout te cetrod): s
8 Te cetrod of te tragle m w( ) We wll sum te cotrutos for eac tragle so we wll ot compute c (te locato of te cetrod) for te cross secto utl we add up all of te p cotrutos. Te cetrod of te tragle. We ca compute te tegral of r over te tragle as follows: p s m d d s r m d s md 3 3 s 3 6 s 3 6 m 3 3 m p r m 3 3 m Note tat p s oe trd of te dstace from te edges, ut adjusted te geeral area ad measured te drectos ad m. Te locato of te cetrod of te geeral tragle s
9 Te tegral of r r Te vector r to te elemetal pot of tegrato s r m It s coveet to defe tesors B m m C m m Tese tesors are costat over te tragle ad terefore ca e pulled out of te tegral. Note tat we wll ot compute te actual momet of erta aout te cetrod at ts pot ecause we pla to add togeter multple peces to costruct te momet of erta for polgoal regos. So we wll smpl compute (ad accumulate) te tegrals of te outer product. Te tegral of r r. We ca compute te tegral of te outer product of r wt tself for te tragular rego as follows: r r s m m d d d d B C s 3 s B 3 C d s B 3 Cd s 4 8 B s B C B C 4 6 Hece, we ave te followg result (wc we wll evetuall use to compute te momet of erta) R r r B C C
10 Postve ad egatve regos We ca compute tegrals over a cross secto cosderg postve ad egatve areas. I regos were tere are ot, te cacel eac oter out. Te I-eam, for eample, ca e costructed from te large (postve) rectagle ad te two smaller (egatve) rectagles. (+) (-) (-) w w w w w (+) + (-) = () ctual cross secto Postve rego Negatve rego Zero rego We wll use ts geeral dea to accumulate te cotrutos of te tragles tat defe te sdes of te polgo. If s s greater ta ero te area wll e postve ad f t s less ta ero te te area wll e egatve. For a rego tat as ot a postve ad egatve cotruto te et effect ( all of our tegrals) wll e ero. Hece, ts s a good wa to la out te calculato.
11 ccumulato of propertes Cosder te cross secto sow at rgt. Takg te pots sequetall we see te cotrutos of four tragles. If m s to te paper te te area s egatve (red), oterwse t s postve (lue). Notce ow te areas outsde te cross secto get eactl oe postve ad oe egatve cotruto. I te ed, ol te lue rego remas. 4 3 m Te Cross Secto Tragle (sde -) 4 4 m 3 3 m m Tragle (sde -3) Tragle 3 (sde 3-4) Tragle 4 (sde 4-)
12 Postve regos ad egatve regos m Postve regos ad egatve regos. Te power of te approac we are takg s te alt to dstgus etwee postve regos ad egatve regos. We we use te geeral tragle as te uldg lock for computg tegrals over a polgoal sape we must e ale to assg a postve or egatve value to eac pece accord wt te eed to eter add tat porto to te sum or sutract t from te sum. Te term s takes care of ts ssue. Recall te defto e 3 e me 3 3 s m m ccordg to te defto of cross product, te agle s clockwse from te vector to te vector m. You ca see ow ts plas out te two eamples elow (te tragles for sdes - ad -3 for our eample). Te se fucto s postve te frst ad secod quadrat ad egatve te trd ad fourt, ad ts gves te tegrals ter algerac sg. lso, as te vectors ted to pot alog te same le te magtude gets smaller (to reflect tat te rego of te tragle gets smaller). m s s m
13 lgortm 4 Te Cross Secto Te process ca e put to a program wt te steps outled at rgt. I te loop we carr out te computatos assocated wt te tragle wose edge s defed te curret vertces. We te accumulate tose to sums for eac of te propertes (.e., we are addg up all of te cotrutos of all tragles). ce te loop s complete we ca fs te computatos of c ad J, fd te prcpal values, ad prt ad/or plot te results. 3 lgortm. We ca orgae te computato as follows:. Store coordates of vertces arra N N. Itale propertes, p, R 3. Loop over te sdes, : N a. Compute te ut vectors ad m from. Compute s m e c. Compute te tragle cotrutos, p r, ad R r r d. dd te cotrutos to te wole, p p p, R R R 4. Fs te computato of c ad J c p /, J R c c 5. Compute prcpal values of J 6. Prt ad plot results
14 Prcpal values of J e 3 e C Te mamum ad mmum values of te erta tesor J are ts egevalues. Tese are te values of J tat we would compute te (',') coordate sstem (f we kew wat t was advace). For a smmetrc cross secto J s dagoal ad te te ma/m values are equal to te dagoal elemets of J. Prcpal values of J. We te off-dagoal elemets of te tesor J are ot ero tat mples tat te J ad J values are ot te mamum or mmum momets of erta of te cross secto. Te wa we computed J ad te frst step of lag dow a coordate sstem (, ). We foud te locato of te cetrod C ad tat allowed us to al dow te coordate sstem (,), wc s smpl a traslato of te orgal coordate sstem to pass troug te cetrod. Tese two coordate sstems were proal cose ecause of some oter aspect of te prolem we are trg to solve (e.g., edg aout te -as). But, as far as te propertes of te cross secto are cocered, te coce s artrar. terestg questo to ask s ts: Could we pck a coordate sstem (',') suc a wa tat te values of J ad J are te ggest or smallest possle? Te aswer s es ad te coordate sstem we eed pots te drecto of te egevectors of J. Smlarl, te actual ma/m values are te egevalues of J. MTLB gves us a ver eas wa to compute te egevalues of a matr: Jma = eg(j)
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