Strain Energy in Linear Elastic Solids
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1 Duke Unverst Department of Cv and Envronmenta Engneerng CEE 41L. Matr Structura Anass Fa, Henr P. Gavn Stran Energ n Lnear Eastc Sods Consder a force, F, apped gradua to a structure. Let D be the resutng dspacement at the ocaton and n the drecton of the force F. If the structure s eastc, the force-dspacement curve foows the same path on oadng and unoadng. F D F j j F D σ ε v() w() F 0 D D+ D D Fgure 1. Forces and dspacements on the surface of an eastc sod. If F s ncreased b F and the correspondng ncrease n the dspacement s D, then as F 0, the ncrementa work, W, done b the oad F passng through a dspacement D s appromate F D, or, more precse, W = D + D D F (D ) dd. (1) When the structure s eastc and near, that s F (D ) = k D, the work of a force ncreasng from 0 to F, movng through correspondng dspacements from 0 to D s W = D 0 F dd = D 0 k D dd = 1 k D = 1 1 F = 1 k F D. ()
2 CEE 41L. Matr Structura Anass Duke Unverst Fa H.P. Gavn If a near eastc structure s subjected to a sstem of pont forces F 1, F,..., F n, R a F 1 D 1 F D D j F j Dn Fn F1 D1 F1 D1 D F F D Dn D j Fj F j D j F n F n D n Rb Rc Fgure. Pont forces and coocated dspacements on near eastc sods and structures. causng dspacements, D 1, D,..., D n, n the drecton of those forces, then the tota eterna work, W, s gven b W = 1 {F 1D 1 + F D + + F n D n } = 1 {F }T {D}. (3) In the absence of an energ dsspaton, ths work s stored n the structure n the form of stran energ. In eastc structures carrng statc oads, the eterna work and stran energ are numerca equa to one another. Eterna Work = Stran Energ W = U (4) Note that forces at fed reacton ponts, R, do no work because the dspacements at the reactons are presumed to be ero. Eampe: Sma eement subjected to norma stress σ
3 Stran Energ n Lnear Eastc Sods 3 Stran Energ n a genera state of stress and stran A three dmensona near eastc sod wth oads supped b eterna forces F 1,..., F n, and through support reactons R, can be consdered to be made up of sma cubc eements as shown beow. R a F 1 D Rb F F j D j D Fn Dn V Rc Fgure 3. Stresses wthn a near eastc sod. σ τ σ τ τ σ The ncrementa stran energ, du, for ths eementa cube of voume dv can be wrtten: du = 1 {σ ɛ + σ ɛ + σ ɛ + τ γ + τ γ + τ γ } dv. Integratng the ncrementa stran energ, du, over an entre voume, V, the tota stran energ, U, s U = 1 {σ ɛ + σ ɛ + σ ɛ + τ γ + τ γ + τ γ } dv. V If the stresses and strans are re-wrtten as vectors, {σ} T = {σ σ σ τ τ τ } {ɛ} T = {ɛ ɛ ɛ γ γ γ }, then the tota stran energ can be wrtten compact as U = 1 V {σ}t {ɛ} dv. (5) Ths equaton s a genera epresson for the nterna stran energ of a near eastc structure of an tpe. It can be smpfed sgnfcant for structures but from a number of prsmatc members, such as trusses and frames.
4 4 CEE 41L. Matr Structura Anass Duke Unverst Fa H.P. Gavn Aa Stran Energ, σ = N /A, ɛ = u () Consder a rod subjected to a norma force, N : N N d ε d σ Fgure 4. Interna aa forces, deformaton, and stresses n an aa-oaded prsmatc bar. The norma stress on an eement da s The correspondng stran s σ = Eɛ = N A. ɛ = σ E = u (). The ncrementa nterna stran energ, du, n an ncrementa voume eement, dv, n terms of aa forces, N, or aa dspacements, u(), s du = 1 σ ɛ dv = 1 σ E dv = 1 N da d EA = 1 Eɛ dv = 1 E(u ()) da d and the tota stran energ n a bar n tenson or compresson s U = 1 EA Snce A = A da, N A da d or U = 1 N E(u ()) A da d. U = 1 EA d or U = 1 EA (u ()) d. (6) A prsmatc bar wth a constant aa force, N, and a constant stran ɛ = /L, aong ts ength, s ke a truss eement, and the stran energ can be epressed as U = 1 NL EA or U = 1 EA L. (7)
5 Stran Energ n Lnear Eastc Sods 5 Bendng Stran Energ, σ = M /I, ɛ = v () Consder a beam subjected to a pure bendng moment about the -as, M : v" d M M σ d Fgure 5. Interna bendng moments, deformaton, and stresses n a prsmatc beam. The norma stress on an eement da at a dstance from the neutra as s The correspondng stran s σ () = Eɛ () = M I. ɛ () = σ E = κ v (). The ncrementa nterna stran energ, du, n a voume eement, dv, n terms of bendng moments, M (), or transverse dspacement, v(), s du = 1 σ ɛ dv = 1 σ E dv = 1 M EI da d = 1 Eɛ dv = 1 E (v () ) da d, and the tota stran energ n a beam under pure bendng moments s U = 1 M EI A da d or U = 1 E(v ()) A da d. Snce the bendng moment of nerta, I, s A da, provded that the orgn of the coordnate sstem es on the neutra as of the beam ( A d d = 0), U = 1 M d or U = 1 EI EI (v ()) d. (8)
6 6 CEE 41L. Matr Structura Anass Duke Unverst Fa H.P. Gavn Shear Stran Energ, τ = V Q()/I t(), γ = v s() Consder a beam subjected to a shear force, V, (and bendng moment): d t() V v s V 01 d τ Fgure 6. Interna shear forces, deformaton, and stresses, of a prsmatc beam. τ () = Gγ () = V Q() I t() Q() = Moment of Area of Cross Secton = du = 1 τ γ dv = 1 τ G da d = 1 d/ V Q() I Gt() t() d da d U = 1 V Q() I G A t() da d = 1 V A Q() GA I A t() da d Ths ast ntegra reduces to a constant that depends on upon the shape of the cross-secton. Ths constant s gven the varabe name α. α = A I A Q() t() Vaues of α for some common cross-secton shapes are gven beow (α > 1). sod crcuar sectons: α 1.08 sod rectanguar sectons: α 1.15 thn-waed crcuar tubes: α 1.95 thn-waed square tubes: α.35 I-sectons n strong-as shear: α A/(td) Wth ths smpfcaton, the nterna stran energ due to shear forces s U = 1 αv GA d = 1 da V G(A/α) d. (9)
7 Stran Energ n Lnear Eastc Sods 7 The term (A/α) s caed the effectve shear area. As a revew of shear stresses n beams, consder the shear stress n a rectanguar secton (wth secton d b). Q() = d/ d/ t() d = b τ = V Q() I t() τ = V I d = b d 4 d/. = b d 8 Ths stress vares paraboca aong the drecton of the apped shear. It s mamum at the centrod of the secton and ero at the ends. The correspondng shear stran energ equaton n terms of dspacements s a bt more subte U = 1 G(A/α)(v s()) d. (10) where the tota transverse dspacement s a combnaston of bendng-reated v b () and shear-reated v s () dspacements, v() = v b () + v s (). For eampe, v b () = M () EI () d and v s() = V () GA()/α d.
8 8 CEE 41L. Matr Structura Anass Duke Unverst Fa H.P. Gavn Torsona Stran Energ, τ θ = T r/j, γ θ = r θ Consder a crcuar shaft subjected to a constant torsona moment, T : r T R θ T τ θ Fgure 7. Interna torsona moments, deformaton, and stresses n a prsmatc rod. d The crcumferenta shear stress τ θ (r) s and the correspondng shear stran s τ θ (r) = Gγ θ (r) = T r J γ θ (r) = τ θ(r) G = r θ. The ncrementa nterna stran energ, du, n terms of torsona moments, T (), or torsona rotatons, θ(), s du = 1 τ θγ θ dv = 1 τθ G dv = 1 Tr da d GJ and the tota stran energ for the shaft s U = 1 = 1 Gγ θ dv = 1 G(r θ ) da d T J G A r da d or U = 1 G(θ ) A r da d. Snce the term A r da s the same as the poar moment of nerta, J, U = 1 T GJ d or U = 1 GJ(θ ). (11)
9 Stran Energ n Lnear Eastc Sods 9 Tota Stran Energ arsng from Combned Aa Stresses As a revew of the matera above, consder a three-dmensona bendng probem wth a super-mposed norma force, N. M M N Fgure 8. Interna aa force and bendng moments n a prsmatc beam. d σ = N A M + M. I I The tota stran energ arsng from aa and pure bendng effects s U n = 1 σ ɛ dv = 1 σ V V E dv = 1 1 E A σ da d. The term σ n the ntegra above can be epanded as foows. N A A σ da = A + M + M N M + N M M M I I AI AI I I But, snce the coordnate aes are assumed to pass through the centrod of the cross-sectona area, A da = A da = A da = 0 Therefore, the tota potenta energ s smp the sum of the potenta energes due to aa and bendng moments ndvdua. U n = 1 N EA d + M d + EI M EI d. da.
10 10 CEE 41L. Matr Structura Anass Duke Unverst Fa H.P. Gavn Tota Stran Energ arsng from Combned Shear Stresses Just as a structura eement can be subjected to combned norma and bendng stresses, combned shear stresses can aso act together. V T V Fgure 9. Interna shear forces and torsona moment n a prsmatc beam. τ = V Q () I t () d τ = V Q () I t () τ θ = T r J Through mathematca manpuatons smar to those above, t can be shown that where U v = 1 V G(A/α ) d + α = A I α = A I A A V G(A/α ) d + Q () t () Q () t () da da T GJ d, Tota Stran Energ The tota stran energ for sods subjected to aa, bendng, shear, and torsona forces s the sum of U n and U v above.
11 Stran Energ n Lnear Eastc Sods 11 Summar Stran energ s a knd of potenta energ arsng from the deformaton of eastc sods. For structura eements (bars, beams, or shafts) stran energ s epressed n terms of the eastct of the matera (E or G), the dmensons of the eement (L, A, I, J, or A/α), and ether the nterna forces (or moments) n the eement (N(), M(), V (), or T ()), or the deformaton of the eement (u (), v (), v s(), θ ()). force deformaton force-based deformaton-based stran-energ stran energ Aa N () u () L =0 N () E()A() d L =0 E()A()(u ()) d Bendng M () v () L =0 M () E()I() d L =0 E()I()(v ()) d Shear V () v s() L =0 V () G()(A()/α) d L =0 G()(A()/α)(v s()) d Torson T () θ () L =0 T () G()J() d L =0 G()J()(θ ()) d where: E() G() A() I() A()/α J() N () M () V () T () u () v () v s() θ () s Young s moduus s the shear moduus s the cross sectona area of a bar s the bendng moment of nerta a beam s the effectve shear area a beam s the torsona moment of nerta of a shaft s the aa force wthn a bar s the bendng moment wthn a beam s the shear force wthn a beam s the torque wthn a shaft s du()/d, the aa stran, u() s the aa dspacement aong the bar s d v()/d, the curvature, v() s the transverse bendng dspacement of the beam s dv s ()/d, the shear deformaton, v s () s the transverse shear dspacement of the beam s dθ()/d, the torsona deformaton, θ() s the torsona rotaton of the shaft
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