# Chapter Eight. Review and Summary. Two methods in solid mechanics ---- vectorial methods and energy methods or variational methods

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1 Chapter Eght Energy Method 8. Introducton 8. Stran energy expressons 8.3 Prncpal of statonary potental energy; several degrees of freedom Castglano s frst theorem ---- Examples 8.4 Prncpal of statonary complementary energy Castglano;s second theorem ----Examples 8.5 Statcally ndetermnate problems ----Examples 8. Introducton Revew and Summary Two methods n sold mechancs ---- vectoral methods and energy methods or varatonal methods () vectoral methods ---- emphaszed n elementary courses and are formulated n terms of vector quanttes such as forces and dsplacement () Energy methods---- formulated n terms of scalar quanttes such as wor and energy. dvantages o f energy methods ablty to avod some extraneous detal and to yeld approxmate soluton for complcated problems. Degree of freedom (d.o.f.) the number of ndependent quanttes needed to defne unquely the confguraton of a system ----generalzed coordnates.

2 8. Stran energy expresson Wor s done by a force as t moves through a dstance, and by a moment as t turns through an angle. dw ( f cos ) du W (the wor done by a moment : f cos du mcos d ) If the effect of force s to dstort an elastc body (such as a lnear sprng), wor done by f d stored as stran energy U (expressed n terms of dsplacement). Complementary stran energy U * (expressed n terms of force). or a lnear elastc materal U and U * are numercally equal. () The stran energy U=U() U fdu udu m d f=u () The complementary stran energy U * =U * () * f U udf df U=f/

3 unt column of lnear elastc materal can be vewed as a lnear sprng, so the complementary stran energy U * s The complementary stran energy U * (expressed n terms of stress or force) for a bar of length L under general complex loadng s (by volume ntegral)

4 Expresson for the stran energy U (n terms of dsplacements) of a slender straght bar s (replacng M y by EI y ( d w/ dx ),) M z EI z ( d w/ dx ), T GJ ( d / dx) Slender crcular rngs U EI R Moment-curvature relatons for the rng M d v ( v) d v EI R d Rd d v ( R, U stran energy for straght bar) Energy of transverse shear 6V z h Ezx ( z ), bh h 4 L h / * U zxbdzdx h / G L Vz. dx G

5 8.3 Prncpal of statonary potental energy; several degrees of freedom ---- Castglano s frst theorem ---- Examples dmssble or nematcally admssble confguratons Potental energy of a structure ---- =U+ U ---- the stran energy ---- the potental of the loads or lnear sprng, d.o.f.= U /, U / The prncpal of statonary potental energy: mong all admssble confguraton (that satsfed statc equlbrum condtons) maes the potental energy statonary wth respect to small admssble varatons of dsplacement. If the statonary condton s a mnmum, the equlbrum state s stable. d ( ) d d ( ) d

6 Several degrees of freedom ---- Castglano s frst theorem () the potental energy for n degree of freedom D U U U ( D, D, D n P n ) () then prncpal of statonary potental energy gves d dd dd ddn D D D or any and all of these dd, d must vansh, ths s possble only f (,,, n) D (3) Castglano s frst theorem: If stran energy U s expressed n terms of ndependent dsplacement d.o.f., then the load P that corresponds to d.o.f. D s gven by the partal dervatve of U wth respect to D n rom above D (for,,, n) We have U P D or example : (for,,, n) U U D, D then, M

7 Examples (Determne the dsplacement d.o.f. that defne the statc equlbrum confguraton) () Two-ar Lnage The only d.o.f. s, that = as reference state, we have (neglectng stran energy U) L W ( cos ) (Lsn ) d tan d W ( s the value for statc equlbrum.) () Rgd ar The d.o.f. s ( C = ), the rotaton of the rgd bar s = /b E E U ( ) M C L L b E 5 M C L b d L M d 5Eb the forces n the wres E M C P, L 5b C P C E L ( ) M 5b C

8 (3) Sprng n Seres Two d.o.f. are needed to defne the confguraton:,. Intally, = =. ) ( U Usng Castglano s rst theorem, we have, U U

9 8.4 Prncpal of statonary complementary energy castglano s second theorem ---- Examples Complementary energy of a structure ---- U * + or lnear sprng U / The prncpal of statonary complementary energy: among all statcally admssble stress felds, the actual stress feld (that yelds nematcally admssble dsplacements) maes the complementary energy statonary wth respect small statcally varatons of stress. d (), d () If there are several forces, we have * P for =,,,n In prevous example of two sprngs n seres, we have

10 The complementary energy of a structure loaded by concentrated forces and/or moments s n U P D P U D (,,..., n) P Castglaon s second theorem: the partal dervatve of complementary energy U * wth respect to a load yelds the dsplacement component of the loaded pont n the drecton of that load. * U D (,, n) P P can be a force or moment M. * * U U, M

11 Unt load method ( convent format of Castglano s second theorem) Usng complementary stran energy expresson U * for curved U bar and D P, we fnd D L M EI y y M P y M EI z z M P z zvz G V z P dx ntroducng m y m z V z M ym y M zmz Tt dx EIy EIz GJ M y my P s a moment produced by a unt load P (a unt u force or a unt moment)

12 Examples () Cantlever beam (a) rst determne the vertcal deflecton of. The bendng moment s (neglect shearng) M( x) x qx usng unt load methods, M x 3 4 qx L ql ( x )( x) dx EI 3EI 8EI (b) nd the deflecton at due to q along (=), two method: () smply set = n above expresson for, () temporarly apply a load (such as unt load) n the desred dorecton, after usng unt load method, then set ths load to zero: ((case (b)) 4 qx ql ( x) ( x) dx EI 8EI my Set ths load= (c) the horzontal deflecton at C h L qx H ()( s) ds EI EI 3 ql h hdx EI 6

13 () Splt Rng Determne the Z-drecton dsplacement of the loaded end and ts rotaton component about the y axs (a) endng and twstng moments (n rng) M R and T are: R sn C sn M R T R( cos) C cos here C s the unt couple for calculaton of rotaton (b) Calculate deflecton, we set C= M R M R T T z dx EI GJ Rsn ( R sn ) Rd EI R( cos ) R( cos ) Rd GJ 3 3 R 3R EI GJ (c) Calculate the rotaton, C= m R =-sn and T=-cos, n M the above equaton, we use M R, T R T C C,,, we C C R R obtan y EI GJ

14 (3) Truss nalyss (a) (b) (c) Calculate the vertcal deflecton at C by unt force method: frst calculate the nternal force N n each bar due to Q; then the force n due to unt force. Nn s nonzero only bars D and DC, so C NL n E NL n E QL E NL n E DC QL E D Relatve moton of ponts C. G. : we apply collnear force (unt load) as shown. Then NL QL CG n CD, C, D E E Rotaton of bar G: we apply couple forces /L (unt moment), then NL G n D, C, C E

15 8.5 Statcally ndetermnate problems ---- Examples Determne the bendng moment n () propped cantlever beam the statcally ndetermnate beam (a) (b) (c) nd the redundant and regard t as a nown load on the structure. The bendng moment s M Rx qx / * U L M M R or dx R EI R.e., L qx Rx xdx R 3qL / 8 EI we can choose M L as redundant, then by * U L M M or ds M L EI M we L Obtan the same result as n (b).

16 (3) Elastcally support cantlever beam We use two methods to solve ths problem: (a) Consder the U * of the beam only, then L M qx U dx EI where M Rx U R U ( ) or L M EI mdx when s nown, then R= (b) Consder the U * of whole system (beam and sprng). R L M U dx where M Rx qx EI R s the reacton at the base of the sprng, by Castglano s second theorem or unt load method: U R L M or mdx R EI -x we can get R R M R x

17 (3) Semcrcular arch nd the support reacton of the arch Ths s the statcally ndetermnate to the thrd degree. There are several methods to solve the problem: (a) nd U * of the arch as U * = U * (V, h, M ), then by usng U U U,, V H M we obtan V, H, M. (b) y usng symmetry, only have unnows, H, M as shown n (c), then U H U U, M U H, M, by, we solve the problem. (c) s shown n (d), U U H, M ), by U H, U M (

18 (4) Internally ndetermnate truss nd the forces n all bars we elect to use forces n bars D and C as redundants. These forces D and C are exposed f bars D and C are cut. Then the truss s rendered as statcally determnate,.e., all forces n bars can be wrtten n terms of Q, Q, D and C. We can wrte U * as N L U E D and C can be obtan by U U, D, C Snce n the magned cut n each bar (before and after loads D, C ), the relatve approach or separaton of the cut ends s zero.

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