Strain Energy in Linear Elastic Solids

Size: px

Start display at page:

Download "Strain Energy in Linear Elastic Solids"

Claribel Simpson
6 years ago
Views:

1 Strain Energ in Linear Eastic Soids CEE L. Uncertaint, Design, and Optimiation Department of Civi and Environmenta Engineering Duke Universit Henri P. Gavin Spring, 5 Consider a force, F i, appied gradua to a structure. Let D i be the resuting dispacement at the ocation and in the direction of the force F i. If the structure is eastic, the force-dispacement curve foows the same path on oading and unoading. F i D F j j F i D i σ ε v() w() F i 0 D i D+ D i i D i Figure 1. Forces and dispacements on the surface of an eastic soid. If F i is increased b F i and the corresponding increase in the dispacement is D i, then as F i 0, the incrementa work, W, done b the oad F i passing through a dispacement D i is approimate F i D i, or, more precise, W = Di + D i D i F i (D i ) dd i. (1) When the structure is eastic and inear, that is F i (D i ) = k i D i, the work of a force increasing from 0 to F i, moving through corresponding dispacements from 0 to D i is W = Di 0 F i dd i = Di 0 k i D i dd i = 1 k idi = 1 1 Fi = 1 k i F id i. ()

2 CEE L. Uncertaint, Design, and Optimiation Duke Universit Spring 5 H.P. Gavin If a inear eastic structure is subjected to a sstem of point forces F 1, F,..., F n, F 1 D 1 Fi Di D j F j Dn Fn F1 D1 F1 D1 Di Fi Fi Di Dn D j Fj F j D j F n F n D n Figure. Point forces and coocated dispacements on inear eastic soids and structures. causing dispacements, D 1, D,..., D n, in the direction of those forces, then the tota eterna work, W, is given b W = 1 {F 1D 1 + F D + + F n D n } = 1 {F }T {D}. (3) In the absence of an energ dissipation, this work is stored in the structure in the form of strain energ. In eastic structures carring static oads, the eterna work and strain energ are equa. Interna Strain Energ = Work of Eterna Forces U int = W et (4) Note that forces at fied reaction points, R, do no work because the dispacements at the reactions are presumed to be ero. Eampe: Sma eement subjected to norma stress σ

3 Strain Energ in Linear Eastic Soids 3 Strain Energ in a genera state of stress and strain A three dimensiona inear eastic soid with oads suppied b eterna forces F 1,..., F n, and through support reactions R, can be considered to be made up of sma cubic eements as shown beow. F 1 D 1 Fi F j D j Di Fn Dn V σ τ σ τ τ σ Figure 3. Stresses within a inear eastic soid. The incrementa strain energ, du, for this eementa cube of voume dv can be written: du = 1 {σ ɛ + σ ɛ + σ ɛ + τ γ + τ γ + τ γ } dv. Integrating the incrementa strain energ, du, over an entire voume, V, the tota strain energ, U, is U = 1 {σ ɛ + σ ɛ + σ ɛ + τ γ + τ γ + τ γ } dv. V If the stresses and strains are re-written as vectors, {σ} T = {σ σ σ τ τ τ } {ɛ} T = {ɛ ɛ ɛ γ γ γ }, then the tota strain energ can be written compact as U = 1 V {σ}t {ɛ} dv. (5) This equation is a genera epression for the interna strain energ of a inear eastic structure of an tpe. It can be simpified significant for structures buit from a number of prismatic members, such as trusses and frames.

4 4 CEE L. Uncertaint, Design, and Optimiation Duke Universit Spring 5 H.P. Gavin Aia Strain Energ, σ = N /A, ɛ = u () A short section of a bar subjected to an aia force N stretches b du. da N 00 N d du = (du/d) d = ( ε ) d σ Figure 4. Interna aia forces, deformation, and stresses in a short section of a bar. The strain aong this short section of bar is ɛ = du d = u ()... and... σ = Eɛ. The norma stress on a sma part of the cross section, of area da, is σ = N... and... ɛ = σ /E A The incrementa interna strain energ, du, in an incrementa voume eement, dv, in terms of aia forces N and aia dispacements u() is ( ) N du = 1 σ ɛ dv = 1 (u ()) dv = 1 A EA dv = 1 E(u ()) dv and the tota strain energ in a bar in tension or compression is U = 1 EA Since A = A da, N 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 prismatic bar with a constant aia force, N, and a constant strain ɛ = /L, aong its ength is ike a truss eement, and the strain energ can be epressed as U = 1 NL EA or U = 1 EA L or simp U = 1 N (7)

5 Strain Energ in Linear Eastic Soids 5 Bending Strain Energ, σ = M /I, ɛ v b A short section of a beam subjected to a bending moment M about the -ais bends b an ange dθ. M d θ / d+ ε d = d κ d d θ = (d θ /d) d = κ d d θ / M σ d Figure 5. Interna bending moments, deformation, and stresses in a prismatic beam. The norma stress on a cross-section eement of area da at a distance from the neutra ais is σ () = M I... and... ɛ = σ /E The strain aong this short section of bar at a distance from the neutra ais is ɛ () = κ v b,... and... σ = Eɛ The incrementa interna strain energ, du, in a voume eement, dv, in terms of bending moments M () and transverse dispacement v() is du = 1 σ ɛ dv = 1 ( M I ) ( v b()) dv = 1 M dv = 1 EI E (v b()) dv and the tota strain energ in a beam under pure bending moments is U = 1 M EI A da d or U = 1 E(v b()) A da d. Since the bending moment of inertia, I, is A da, provided that the origin of the coordinate sstem ies on the neutra ais of the beam ( A d d = 0), U = 1 M d or U = 1 EI EI (v b()) d. (8)

6 6 CEE L. Uncertaint, Design, and Optimiation Duke Universit Spring 5 H.P. Gavin Shear Strain Energ, τ = V Q()/I t(), γ = v s() A short section beam subjected to a shear force V defects b an amount dv s. d t() V v s dv = (dv /d)d = v d s s V s 01 d τ Figure 6. Interna shear forces, deformation, and stresses, if a short section of a beam. τ () = V Q() I t()... and... γ = τ /G Q() = Moment of Area of Cross Section = du = 1 τ γ dv = 1 τ G dv = 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 This ast integra reduces to a constant that depends on upon the shape of the cross-section. This constant is given the variabe name α. α = A I A Q() t() Vaues of α for some common cross-section shapes are given beow (α > 1). soid circuar sections: α 1.08 soid rectanguar sections: α 1.15 thin-waed circuar tubes: α 1.95 thin-waed square tubes: α.35 I-sections in strong-ais shear: α A/(td) With this simpification, the interna strain energ due to shear forces is U = 1 αv GA d = 1 da V G(A/α) d. (9)

7 Strain Energ in Linear Eastic Soids 7 The term (A/α) is caed the effective shear area. As a review of shear stresses in beams, consider the shear stress in a rectanguar section (with section d b). Q() = d/ d/ t() d = b τ = V Q() I t() τ = V I d = b d 4 d/. = b d 8 This stress varies paraboica aong the direction of the appied shear. It is maimum at the centroid of the section and ero at the ends. B anaog, the corresponding shear strain energ equation in terms of dispacements is U = 1 G(A/α)(v s()) d (10) where the tota transverse dispacement is a combinastion of bending-reated v b () and shear-reated v s () dispacements, v() = v b () + v s (). For eampe, v b () = M () EI () d and v s() = V () GA()/α d.

8 8 CEE L. Uncertaint, Design, and Optimiation Duke Universit Spring 5 H.P. Gavin Torsiona Strain Energ, τ φ = T r/j, γ φ = rφ A short section of a circuar shaft oaded with a torque T twists b an ange dφ. φ r T R(dφ /d) = R φ dφ T τ φ d 0 1 Figure 7. Interna torsiona moments, deformation, and stresses in a short section of a shaft. d The circumferentia shear stress τ φ (r) is τ φ (r) = T r... and... γ φ = τ φ /G J and the corresponding shear strain is γ φ (r) = φ r... and... τ φ = Gγ φ The incrementa interna strain energ du in terms of torsiona moments T () and torsiona rotations φ() is du = 1 τ φγ φ dv = 1 ( ) T r (φ r) dv = 1 Tr J GJ dv = 1 G(φ r) dv and the tota strain energ for the shaft is U = 1 T J G A r da d or U = 1 G(φ ) A r da d. Since the term A r da is the same as the poar moment of inertia, J, U = 1 T GJ d or U = 1 GJ(φ ()) d. (11) For a prismatic shaft with a constant torque aong its ength T, and a tota twist φ, the strain energ can be epressed as U = 1 TL GJ or U = 1 GJ L φ or simp U = 1 T φ (1)

9 Strain Energ in Linear Eastic Soids 9 Tota Strain Energ arising from Combined Aia Stresses As a review of the materia above, consider a three-dimensiona bending probem with a super-imposed norma force, N. M M N Figure 8. Interna aia force and bending moments in a prismatic beam. d σ = N A M + M. I I The tota strain energ arising from aia and bending effects is U n = 1 σ ɛ dv = 1 σ V V E dv = 1 1 E A σ da d. The term σ in the integra 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, since the coordinate aes are assumed to pass through the centroid of the cross-sectiona area, A da = A da = A da = 0 Therefore, the tota potentia energ is simp the sum of the potentia energies due to aia and bending moments individua. U n = 1 N EA d + 1 M d + 1 EI M EI d da.

10 10 CEE L. Uncertaint, Design, and Optimiation Duke Universit Spring 5 H.P. Gavin Tota Strain Energ arising from Combined Shear Stresses Just as a structura eement can be subjected to combined norma and bending stresses, combined shear stresses can aso act together. V T V d Figure 9. Interna shear forces and torsiona moment in a short section of a beam. τ = V Q () I t () τ = V Q () I t () τ φ = T r J Through mathematica manipuations simiar to those above, it can be shown that where U v = 1 V G(A/α ) d + 1 α = A I α = A I A A V G(A/α ) d + 1 Q () t () Q () t () da da T GJ d, Tota Strain Energ The tota strain energ for soids subjected to aia, bending, shear, and torsiona forces is the sum of U n and U v above.

11 Strain Energ in Linear Eastic Soids 11 Summar Strain energ is a kind of potentia energ arising from stress and deformation of eastic soids. In an eastic soid, the work of eterna forces, W, is stored entire as eastic strain energ, U, within the soid. In inear eastic soids: Dispacements and rotations increase inear with forces and moments. The work of an eterna force F acting through a dispacement D on the soid is given b W = 1 F D. The work of an eterna moment M acting through a rotation Θ on the soid is given b W = 1 MΘ. For sender structura eements (bars, beams, or shafts) the interna forces, moments, shears, and torques var aong the ength of the eement; so do the dispacements and rotations. The strain energ of spatia-varing interna forces F () acting through spatia-varing interna dispacements D() is U = 1 dd() F () d d = 1 F ()D () d The strain energ of spatia-varing interna moments M() acting through spatia-varing interna rotations Θ() is U = 1 dθ() M() d d = 1 M()Θ () d

12 1 CEE L. Uncertaint, Design, and Optimiation Duke Universit Spring 5 H.P. Gavin force deformation strain energ (U) Aia N () u 1 () N ()u 1 ()d Bending M () v b() 1 M ()v b()d 1 Shear V () v s() 1 V ()v s()d 1 Torsion T () φ 1 () T ()φ 1 ()d N () d 1 E()A() M () d 1 E()I() V () G() A() α d 1 T () d 1 G()J() E()A()(u ()) d E()I()(v b()) d G() A() α (v s()) d G()J()(φ ()) d E() G() A() I() A()/α J() N () M () V () T () is Young s moduus is the shear moduus is the cross sectiona area of a bar is the bending moment of inertia of a beam is the effective shear area of a beam is the torsiona moment of inertia of a shaft is the aia force within a bar is the bending moment within a beam is the shear force within a beam is the torque within a shaft u() u () v() v is the aia dispacement aong the bar is the aia dispacement per unit ength, du()/d, the aia strain is the transverse bending dispacement of the beam b() is the bending rotation per unit ength, the curvature, approimate d v()/d v s () is the transverse shear dispacement of the beam v s() φ() φ () is the transverse shear dispacement per unit ength, dv s ()/d is the torsiona rotation (twist) of the shaft is the torsiona rotation per unit ength, dφ()/d

Strain Energy in Linear Elastic Solids

Strain Energy in Linear Elastic Solids 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