1.054/1.541 Mechanics and Design of Concrete Structures (3-0-9) Outline 3 Failure Theories and Concrete Plasticity

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1 .54/.54 Mehanis and Design of Conrete trutures pring 4 Prof. Oral Buyukozturk Massahusetts Institute of Tehnology Outline.54/.54 Mehanis and Design of Conrete trutures (--9) Outline Failure Theories and Conrete Plastiity Failure of onrete o Conrete is a brittle aterial whih fails through brittle leavage (splitting) at the interfaes and in ortar exept for high triaxial opression where shear slippage ours resulting in a dutile behavior. Failure ours by tensile splitting with the fratured surfae orthogonal to the diretion of the axiu tensile stress or strain. Predition of ultiaxial behavior o In general the aterial properties are known fro siple tests suh as uniaxial loadings giving f and f t. Predition involves strength alulation in ultiaxial situations given the data fro the uniaxial tests. o In the field of onrete researh attepts have been ade to apply soe of the lassial failure theories to onrete. These theories were altered to overoe soe disadvantages or otherwise iprove their agreeent with the phenoenologial behavior of onrete. New failure theories were therefore fored. Prinipal stresses: / 4

2 .54/.54 Mehanis and Design of Conrete trutures pring 4 Prof. Oral Buyukozturk Outline oe lassial failure theories o Maxiu prinipal stress theory o Maxiu prinipal strain theory o Maxiu shear stress theory o Internal frition theory o Maxiu strain energy theory o Distortion energy theory o Frature ehanis based theories stress intensity, toughness frature energy release. These introdue either liitations or ontraditions when applied to onrete. Modifiations to onrete have resulted: o Internal frition-axiu stress theory o Otahedral shear-noral stress theory o Newan s two-part riterion o Loal deforation theories, et. Extensive researh has been onduted to develop better theories: Elasti-plasti, plasti-fraturing, endohroni, bounding surfae et. approahes. Maxiu prinipal stress theory (elasti behavior) o > > o Failure ours when: (Tensile strength) ax t f (Copressive strength) ax It does not reflet splitting nature of failure. Maxiu prinipal strain theory (elasti behavior) o Failure ours when: ax ε εliit εt / 4

3 .54/.54 Mehanis and Design of Conrete trutures pring 4 Prof. Oral Buyukozturk Outline Maxiu shear stress theory o > > o Failure ours when: + λ + s where shear stress, ( ) λ + portion of the voluetri stress, s a ritial shear stress value (e.g. under pure shear) o For etals, λ. For brittle aterials, λ. o For λ, the failure riterion beoes s o The theory gives equal uniaxial tensile and opressive strengths. It is also independent of interediate stress. (pressure sensitivity) Internal frition theory φ n o Consider the effet of noral stress on shear strength: tanφ where n shear strength, ohesive strength, φ angle of internal frition, and n noral stress. o Copression inreases and tension dereases. / 4

4 .54/.54 Mehanis and Design of Conrete trutures pring 4 Prof. Oral Buyukozturk Outline Mohr s theory (generalization of internal frition theory) o τ f ( ) o has no effet on strength. o f ( ) is the envelop of all the irles orresponding to the various states of stress at whih failure takes plae. Otahedral shear and noral stress theory τ o > > n o Failure ours when the ahedral stress exeeds a liiting value. τ + + In uniaxial tension and opression, τ i, i,, o The failure riterion provided by ahedral shear stress theory: τ τ τ τ liit liit tension liit opression This gives the sae ultiate strength for uniaxial tension and opression. It is not valid for onrete. Inlusion of ( + + ) iproves the predition. o Bresler, Pister tests 4 / 4

5 .54/.54 Mehanis and Design of Conrete trutures pring 4 Prof. Oral Buyukozturk Outline τ a a a k + k k + f f f where τ ( ) + ( ) + ( ) a 5 a + + and Invariant forulation o A failure riterion should be based upon an invariant funtion of the state of stress, i.e., independent of the hoie of the oordinate systes. o tress invariants I + +, (ore suitable for applying to onrete) + +, I I where,, are prinipal stresses. eneral stress state representation o τ τ τ τ τzx τzy z x xy xz yx y yz or ij, ij ji,, i, j,, + + i I o i o Average noral stress: a I o Average shear stress: τ a I I 5 5 / 4

6 .54/.54 Mehanis and Design of Conrete trutures pring 4 Prof. Oral Buyukozturk Outline Deviatori stress, i o ij ij δij, δ ij roneker s delta δij, i where,,, are prinipal stresses., o Disussion of physial eaning of deviatori and hydrostati stresses. j j Deviatori stress invariants o J ij + + ( ) J ijji + + J ijjkki + + ( ) Biaxial loading ( ) o ( ) + o I + J ( ) ( ) In general, stress invariants I, J are used to haraterize the behavior of onrete strutures. Invariant forulation of the onrete failure 6 / 4

7 .54/.54 Mehanis and Design of Conrete trutures pring 4 Prof. Oral Buyukozturk Outline F I, J o o Failure riteria J I + I + and 5 9 J I± I+ A I τ 6 u u, where Au, τ u are aterial onstants. Multiaxial failure riterion o Prinipal stresses based F,, o tress invariants based F I, J, J o One odel onsidering the effet of all the three stress invariants and possessing the observed features of the failure surfae suh as soothness, syetry, onvexity, and urved eridians is provided. f,, τ r f ( τ θ) where τ (, θ) τ J, 5 5 I, r r (, θ) r(, θ ) (, θ) osθ r 5 f θ 4( r rt ) os θ + ( r rt ) r r rt os + r rt r 4 r rt os θ + 5rt 4rt r ( ) + ( ) + ( ) t f f f 5 a + a + a at θ o 7 / 4

8 .54/.54 Mehanis and Design of Conrete trutures pring 4 Prof. Oral Buyukozturk Outline r 5 f b + b + b f f o at θ 6 o In the tension-biaxial opression zone, the tensile strength is f t.5f f o In the triaxial tension zone, the failure is defined as i ft i,, o In the opression-biaxial tension zone, the failure is defined as f f t f Daage odel o Inreental daage d dγ F I p, ( θ ) p where γ plasti oponent of ahedral shear strain, o r D R I voluetri stress invariant, and F ( ij, ax ) bounding surfae. where r urrent stress vetor (distane), and R distane to bounding surfae. When D, the aterial is assued to have failed. 8 / 4

9 .54/.54 Mehanis and Design of Conrete trutures pring 4 Prof. Oral Buyukozturk Outline Constitutive odeling of onrete o Approahes for defining stress-strain behavior of onrete: Linear and nonlinear elastiity theories Elasti perfetly plasti odels Elasti strain hardening plastiity odels Plasti daage (fraturing)-type odels Endohroni theory of inelastiity o Isotropi stress odel The stress-strain law ε ε τ γ γ where ahedral noral stress, ε ahedral noral strain, seant bulk odulus, τ ahedral shear stress, γ ahedral shear strain, and seant shear odulus. The nondiensional seant bulk and shear oduli are approxiated by ε ab + d γ r pq sγ + t The tangent bulk and shear oduli are T ( ln b) ε ε a b + d 9 / 4

10 .54/.54 Mehanis and Design of Conrete trutures pring 4 Prof. Oral Buyukozturk Outline T ( ln q) γ γ r p q + t The elasti aterial stiffness atrix: D sy. (Young s odulus: 9 E, Poisson s ratio: ν + + ) o Isotropi strain odel Nonlinear isotropi elasti odel The nondiensional seant bulk and shear oduli: +.5 ( ).9 f +.57 ( τ ).7 f The tangent bulk and shear oduli: T +.8 ( ).9 f T ( τ ).7 f o Orthotropi odel The onept of equivalent uniaxial strains The onstitutive law in ters of the aterial stiffness tensor Dijkl d D dε ij ijkl kl / 4

11 .54/.54 Mehanis and Design of Conrete trutures pring 4 Prof. Oral Buyukozturk Outline where d the tensor of inreental stresses, and ij dε ij the tensor of inreental strains. The atrix of the tangent stiffness tensor: D ( ν ) ν ( + ν) ν ( + ν) ( ν ) E ν ( + ν) ( ν ) E EE EE EE E sy. E E φ φ φ where,, E tangent Young s oduli in diretions,, and, φ ν ν,, inreental shear oduli for planes parallel to oordinates -, -, and -. α E E, α E E, α EE. For unraked +. onrete, α ( ν ) The equivalent uniaxial strain ε iu ε iu εi j + k ν i where ε i prinipal strain in diretion i. o Elasti-hardening plastiity odel It is developed for short-ter onotoni opressive loading of onrete. The onstitutive relationships feature suh harateristis of onrete deforational behavior as inelasti dilatany and fritional effet and inelasti shear aused by hydrostati pressure (hydrostati pressure sensitivity). / 4

12 .54/.54 Mehanis and Design of Conrete trutures pring 4 Prof. Oral Buyukozturk Outline Following the inreental theory of plastiity, the total strain inreents are { } { e p dε dε } { dε } + or dε dε + dε e p ij ij ij where dε C d e ij ijkl kl δ δ + δ δ the elasti opliane tensor 9 6 e C ijkl ik jl ij kl The equation of plasti flow provides e F dεij dkl ξij ξii R kl (as the inreental for of elasti strain) p F dεij dkl ξij (as the inreental for of plasti strain) R kl where R and ξ ij depend on the loading history, and F yield funtion. o Plastiity based odel I f J + I+, where equivalent stress. 5 d H d ε P f f { d} + d d f I { d} + d d f I P { d } H dε o Hardening law (Flow rule) P { dε } P f dε / 4

13 .54/.54 Mehanis and Design of Conrete trutures pring 4 Prof. Oral Buyukozturk Outline If the shape of the urve is assued to expand uniforly in all diretions, the flow rule is referred to as the isotropi hardening law. o tress inreent { d } [ C] { dε e } { dε} { dε e } { dε p } + inreental total strain [ C ] elasti strain to stress transforation atrix { dε e } { dε} { dε p } elasti strain inreent { dε p } plasti strain inreent therefore, f f [ C] [ C] d C d I f f H + [ C] { } [ ] { ε} Note that for perfet plastiity, H, this forulation, leading to a non-singular [ C ], does not ause any nuerial diffiulty. Nonlinear analysis of reinfored onrete o Early studies, by neessity, onentrated on the behavior of isolated eleents suh as beas, oluns, joints, et. As failities developed and oputing apability expanded the sope broadened to inlude entire systes suh as slabs and beas, oupled shear walls, folded plates, and shells oplete with supporting beas for exaple. This broadening stes both fro the desire for better understanding of the behavior of the oplete syste with the possibility of therefore ahieving better strutural effiieny, and / 4

14 .54/.54 Mehanis and Design of Conrete trutures pring 4 Prof. Oral Buyukozturk Outline the need beause of inreased oplexity of the probles requiring solution oupled with ore severe deands being plaed on the struture. Fators whih opliate the analysis of reinforeent onrete strutures o Analytial proedures whih ay aurately deterine stress and deforation states in reinfored onrete ebers and strutures are opliated due to any fators:. Non-linear stress-strain relation. Progressive raking. Consideration of steel reinforeent 4. Creep and shrinkage (tie-dependent behavior) 5. peial probles (shear transfer, yli loading) The developent of finite eleent ethod perits realisti evaluation of internal stresses and displaeents on whih the liit requireents ay be based for iproved strutural effiieny. Furtherore, suh refined analytial solutions help in understanding and interpreting the observed behavior of strutural eleents fro experients. 4 / 4