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 0011 0011 0011 ε v() 11000 111 110000 1111 110000 1111 01 0011 0011 1110000 1111 1110000 1111 1110000 1111 01 01 01 01 01 01 01 01 01000 111000 111000 111000 111000 0011 0011 0011 w() F 0 D 0 1 000 111 000 111 000 111 000 111 000 111 000 111 000 111 000 111 000 111 000 111 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. ()
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 0 1 0 1 0 1 Fn F1 D1 F1 D1 D F F D Dn D j Fj F j D j F n F n D n 0 1 0 1 0 1 0 1 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
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 1 0011 0011 0011 Rb F F j D j D Fn Dn V 0011 0011 0011 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 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 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 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)
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 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() 0 1 0 1 0 1 0 1 0 1 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)
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 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 111000 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)
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 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.
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