dy ds dz ds dx ds ds ds ds ds ds 4-1 DEFORMATION OF A BODY Let there be a line segment PQ in the body with coordinates as:

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1 5:44 (58:54 ENERGY PRNCPLES N SRUCURAL MECHANCS 4- DEFORMAON OF A BODY Q Q Let there be a ie seget PQ i the bo with cooriates as: P(,,, Q(,, Legth of the ifferetia eeet: P P Uit taget vector aog PQ: e i j k i j k After eforatio, P P' a Q Q' with cooriates as P' ( Q' (, v, w, v v, w w Legth of P'Q': ( ( v ( w SRAN Strai at the poit P is efie as (4- or ; > Sqare both sies a re-arrage ( Lectre #7

2 5:44 (58:54 ENERGY PRNCPLES N SRUCURAL MECHANCS Now, sbstitte for a a epa: ( v v w ( v v ( w v ( w w w Usig the chai re, we get v v v v v v v w w w w w w w Sbstittig i the above, we get (4- where v w ( ( ( v v v w w w Lectre #7

3 5:44 (58:54 ENERGY PRNCPLES N SRUCURAL MECHANCS v w v w v v v v v v w w w w w w (4- the ie otatio, Eq. (4- ca be writte as j i where ( i, j j,i a,i a, j he setric atri, is kow as the strai tesor, sice it obes the tesor aw of trasforatio whe the cooriates are rotate. SHEARNG SRAN Cosier two ie eeets PQ a P that are iitia perpeicar to each other. After eforatio, these ie segets becoe P'Q' a P''. Let θ be the age betwee the after the eforatio. he age [ ( π θ ] is kow as the shearig strai betwee the two ie eeets. Derivatio of Eq. (4-4 Lectre #7

4 5:44 (58:54 ENERGY PRNCPLES N SRUCURAL MECHANCS Let P, P Q, PQ, P P, P Q, PQ, P he iitia irectio cosies of PQ a P Q Q are ( a ( P P θ Sice the two ie eeets are iitia perpeicar to each other, 0 (i.e., 0 he ier proct of two vectors after eforatio isgive as cosθ Diviig both sies b a sig Eq. (4-, cos? ( ( cosθ (a where a are strais of the respective ie eeets. o erive a epressio for the vector after eforatio, cosier a eeet PQ that aps o the eeet P'Q' after eforatio. he cooriates of the poits are give as P(, Q(, P'(, Q'( Now is erive as (, [( (] ( ; [ i, j ] Lectre #7 4

5 5:44 (58:54 ENERGY PRNCPLES N SRUCURAL MECHANCS Lectre #7 5 Foregoig epressio appie to the two ie eeets, Pttig a ito Eq. (a, we get ( ( ( ( cos ( ( θ (4-4 PRNCPAL SRANS hrogh a poit i the efore ei, there are three ta perpeicar ie eeets that reai perpeicar after eforatio. he strais of these three ie eeets are cae the pricipa strais at the give poit. he are eote b,, (.

6 5:44 (58:54 ENERGY PRNCPLES N SRUCURAL MECHANCS he correspoig vaes of the qatit are eote b (,,. he qatities (,, are the three roots of the eteriat eqatio, i δ i 0 (4-5 For a otrivia sotio, δ 0 Sice is setric a rea, the roots i are awas rea. Sice i > -, i -/. he above characteristic eqatio ca be writte as 0 (4-6 where tr( et( ii tr cofactor( (4-9 he sotio i of Eq. (4-5 is the set of irectio cosies of the pricipa irectio i the efore ei correspoig to the pricipa strai i. he three irectios correspoig to the three roots (,, are ta perpeicar. Sice the pricipa strais are iepeet of cooriate sste, Eq. (4-7 shows that the qatities,, are iepeet of the Lectre #7 6

7 5:44 (58:54 ENERGY PRNCPLES N SRUCURAL MECHANCS cooriate sste. hese are cae the first, seco a thir ivariats of the strai tesor. Rotatio. B the theor of rigi-bo ispaceets, there eists for a poit of a eforabe bo, a agar ispaceet that carries the pricipa aes of strai of the efore bo ito the pricipa aes of the efore bo. We a eterie the ais of rotatio a the agar ispaceet of a partice of the ei i ters of the ispaceet vector fie i. Besies the rotatio, the partice receives the trasatio i. Aso, it eperieces the pricipa strais (,, aog the pricipa aes. his eforatio is cae a iatatio. hs the ispaceets i a eighborhoo of a poit are resove ito a trasatio, a rotatio, a a iatatio. he trasatio a the rotatio cotribte othig to the strais. VOLUMERC SRAN Let a eeet of a straie ei have the iitia voe V a the fia voe e V. he voetric strai is efie b V V (4-0 V A voe eeet V i the for of a ifiitesia rectagar paraeepipe with its eges i the pricipa irectios reais a rectagar paraeepipe after eforatio. he strais of the Lectre #7 7

8 5:44 (58:54 ENERGY PRNCPLES N SRUCURAL MECHANCS eges of the paraeepipe are the pricipa strais (e,e,e. Coseqet, herefore, V ( ( ( V e (4- (4- e e 4 (4- SMALL DSPLACEMEN HEORY the sa ispaceet theor, Eq. (4- is approiate b e. he qaratic ters i strai ispaceet reatios are egecte to obtai ( j,i i, j (4-4 Whe the qaratic ters are egecte,,, are the strais of ie eeets that iitia ie parae to the, a aes. Aso,,,, are the shearig strais betwee pairs of the ie eeets that iitia ie parae to the aes iicate b the sbscripts. COMPABLY EQUAONS as ch as there are si eqatios for three kow fctios i, the sste of Eq. (4-4 wi ot have a sige-vae sotio i Lectre #7 8

9 5:44 (58:54 ENERGY PRNCPLES N SRUCURAL MECHANCS geera, if the fctios e were arbitrari assige. Oe st epect that a sotio a eist o if the fctios e satisf certai coitios. t ca be obtaie b eiiatig i fro Eq. (4-4: 0 (4-5, k k, ik, j j,ik Eqs. (4.5 are kow as the copatibiit eqatios i the sa ispaceet theor. t ca be show that if give fctios satisf the copatibiit coitios (4-5, the there eist fctios i that are sotios of the strai-ispaceet reatios (4-4. Lectre #7 9

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