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1 Seoul Nat l Unersty Conrete Plastty Hong Sung Gul Chapter 1 Theory of Plastty 1-1 Hstory of truss model Rtter & Morsh s 45 degree truss model Franz Leonhardt - Use of truss model for detalng of renforement. B. Thurlmann (ETH), P. Mart, P. Mueller Collns & Mthell deformaton of truss model (ompresson feld theory) Veho (modfed ompresson feld theory) onsderaton of tensle strength of onretee Shlah - Strut-and-te models Pratal applaton of strut-and-te model for D-regons 1. Consttute equatons 1--1 on Mse ss flow rule In general mehans requres three - stress feld(stats) : equlbrum ondton - dsplaement feld : ompatblty ondton - stress-stran relatonshp : ex)hook s law exat soluton 1

2 Relatonshp between strength and plast deformaton etors Ex) nteraton ure for olumns 1-- on Msess flow rule Consder a rgd plast materal for dsspaton energy : rreersble energy For gen stresses and stran defnton of yeld surfaes - generalzed stresses : M, V, P (,τ ) - generalzed strans : To explan yeld ondtons for a unaxal state and ombned stresses f y (for a unaxal stress state) f ( Q 0) (generalzed form) For ombned stress states a boundary represents Q = 0 : yeld surfae f ( Q1, Q,..., ) n q Φ, Δ (ε, γ..) f < 0 safe On the yeld surfae we hae the dsspaton work f 0 yeldng

3 ( n n) D= Qq + + Q q d= wd Von Mses s hypothess on Maxmum work The stresses orrespondng to a gen stran feld assume suh alues that W beomes large as possble ε = = W ( q1, q,..., qn ) ( Q Q Q ),,..., n 1 = ε If ε s assumed s to be determned so that W beomes as large as possble f s onex A losed surfae ontans Q( 0, 0, 0,..) We need to proe the onexty and the normalty rule W = ε =Σ ε osθ f ε s assumed gen s to be determned suh that W s as large as possble f ( ) = 0 ( δ ) ( ) δw = W Q + Q W Q ( ) w w = W Q + Q + Q + + W Q δ ( δ )... ( ( )) θ θ = δqq= 0 a ( ) ( ) δ f = f θ + δθ f θ f = δθ = 0 θ From Eq s a and b f = q λ θ b (normalty rule) Q ΔQ Q 1 3

4 ( ) W Q+Δ Q = W +Δ ε ( ) = W + Δ ε osθ W Q+ΔQ W = Δ ε osθ < 0 θ > 90 ε ' Q Δ ' Q 1 ' w w onex Plast stran nrements or equalently, plast stran rates are of prmary nterest at ollapse. A falure mehansm s determned by the ratos of the plast stran nrements; ther absolute alues are rreleant sne only nfntesmal deformatons are onsdered. In order to aod omplex formulatons, the notaton wthout the supersrpt for the plast stran rates wll be used. The stran rates do not represent dfferentaton of the strans wth respet to the physal tme t; rather, t s just a salar, and produts are thus termed work or dsspaton rather than power or rate of dsspaton. Fg. 3.1 (a) shows a yeld surfae determned by the yeld ondton enlosng an a plast doman. For states of stress below the yeld lmt, the body remans rgd, whle for stress ombnatons at the yeld surfae,, plast flow may our. Assumng that the a plast doman s onex and that the plast stran nrements at falure are perpendular to the yeld surfae,.e., that the assoated flow or normalty rule s ald, ε = κ gradφ (3.1) where κ denotes an arbtrary non-negate fator, one obtans * ( )εε 0 (3.) where s the atual stress state at the yeld surfae orrespondng to, and s any other stress state at or wthn the yeld surfae, Fg. 3.1 (a). Rearrangng Eq. (3.) one obtans (3.3) whh s the prnple of maxmum energy dsspaton postulated by on Mses [10]. Aordng to ths prnple, the (rtual) dsspaton per unt olume done on a gen plast stran rate assumes a maxmum for the assoated (or ompatble) state of 1-3 Extreme prnples for rgd-plast materals 4

5 Q q = f λ Q on f = 0 f < 0 Q The lower bound theorem If an equlbrum dstrbuton of stress and the appled load E f ( j ) < 0, then T on stress boundares the body at load E j T, F an be found whh balane the body fore A and s eerywhere below the yeld surfae T F wll not ollapse. F n (proof) If the body at T, ε j, and dsplaement rate F ollapse, then a u exst. A T ollapse pattern wth atual stresses T = T j, stran rates F = F u = o on A A u two equlbrum systems exst. j, ε j, u j prnple of rtual work TudA+ t A Fud= = ε d j j 5

6 ) E ( ε j j j (*) annot be true Q,,E,D w w ' > 0 for E j (*) 1-3- the upper bound theorem theorem A load Σ p > wd = p for whh ε d that s a load performng work greater than P, annot be arred by the body. (proof) Assume the load an be arred by the body. If so a statally admssble dstrbuton orrespondng to stress on or wthn the yeld surfae an be found for the load P Q ' Q Q Q 1 f ( Q ) < 0 equlbrum ondton By the prnple of rtual work Σ p = It s not ertan thatt > p ε d Σ p > ' ε d ε d ' ε d ' ε d Σ p ' ε d Q ' orre ontradt spondng to the flow rule orresponds to the stran ε Test shows 6

7 = γ 0 = x 1 For pure shear and normal strength onrete f = γ 0 = Hgher strengths lead to hgher senstty to load ndued mro-rakng Effet of maro-raks on strength - Crak dretons may hange seeral tmes - Sldng reduton fator γ s - - Coheson : 50% reduton Tensle strength s zero when w = 0.1mm ϕ = the unqueness theorem Between lower and upper bounds we hae defntely an exat soluton that s omplete soluton Q Q by lower soluton s R by upper bound soluton 7

8 1-4 the solutons of plastty problems lower bound soluton (dsontnuous stress feld) upper bound soluton (dsontnuous dsplaement feld) The stat or lower-bound method of the theory of plastty s based on the lower bound theorem. Startng from statally admssble states of stress or stress felds eerywhere at or below yeld, one attempts to maxmze the assoated ultmate load. Aordng to the lower-bound theorem, the ultmate load s equal to or hgher than the hghest load found n ths way and hene, the stat method yelds safe or lower-bound solutons for the atual ultmate load. Note that a state of stress obtaned from a lnear elast analyss represents a statally admssble stress feld sne equlbrum and stat boundary ondtons are satsfed. Hene, although the elastally determned state of stress normally deates from the atual state of stress n the struture, Chapter 3.1, desgn based on the theory of elastty an be justfed by the lower-bound theorem of lmt analyss. The knemat or upper-bound method of the theory of plastty s based on the upper-bound theorem. Startng from knematally admssble states of deformaton or falure mehansms, one attempts to mnmze the assoated ultmate load. Aordng to the upper-bound theorem, the ultmate load s equal to or lower than the lowest load found n ths way and hene, the knemat method yelds unsafe or upper-bound solutons for the atual ultmate load. No stress Dsontnuous stress feld unaxal Lower bound soluton reep upper bound soluton + τ + τ + = τ + t t + = + t t Stress free unaxal 1-5 renfored onrete strutures Lower bound theorem : desgn ex) η moment or fores may be hosen how to selet the redundant moment mnmzng the total amount of bendng renforement M dx mn EI 8

9 For a fully raked member M y - 1 ha s As h ϕ ϕ Curature k E h A From 1 s s f Y M M, As ha hf s M Es M h EI Esh = hf Y fy M Es M h f M dx M Y = = = EI f E h Y ε s Y s 9