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.
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/
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)
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
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
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
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
(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
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
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
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)
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
() 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
(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 3. 44 CD, C, D E E Rotaton of bar G: we apply couple forces /L (unt moment), then NL G n D, C, C E
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).
(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
(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 (
(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.