COMPUTERS AND STRUCTURES, INC., JULY 215 TECHNICAL NOTE MODIFIED DARWIN-PECKNOLD 2-D REINFORCED CONCRETE MATERIAL MODEL Overview This tehnial note desribes the Modified Darwin-Peknold reinfored onrete aterial odel, a two-diensional onrete aterial odel that an aount diretly for the interation between bending and shear in shear wall strutures. In an atual wall, espeially in a "squat" wall, there an be substantial oupling between axial-bending and shear. In partiular, the shear strength of a wall ay depend substantially on the axial fores and bending oents. The 2D onrete odel attepts to odel this oupling diretly. The odel is a o-axially rotating seared rak onrete odel. It onsiders raking and rushing of the onrete, and when it is obined with a steel aterial it onsiders yield of the steel. Copressive strength redution based on perpendiular tensile strain is aounted for as desribed in Vehio and Collins (1986). The odel is intended for reinfored onrete and does not aount for the tensile strength of onrete. The odel does not onsider bond slip and dowel ation. Uniaxial Conrete Material Behavior The uniaxial stress-strain relationship for the onrete aterial is shown in Figure 1. The aterial has zero strength in tension. The opression behavior onotonially inreases fro point O to point U and has onstant stress between points U and L. The behavior an be trilinear or elasti-perfetlyplasti. Strength loss fro point L to point R is optional. There is onstant stress between points R and X, whih extends indefinitely, represents residual strength in the aterial. In this aterial odel, the unloading and reloading oduli are assued to be equal to E, as shown in Figure 2, and the aterial unloads linearly to zero 1
Uniaxial Conrete Material Behavior 2 stress. The unloading and reloading paths are idential. The odel allows for unloading, reloading, and arbitrary yli loading, as disussed in a later setion. Figure 1 Uniaxial Stress-Strain Relationship Figure 2 Unloading-Reloading Behavior
Uniaxial Conrete Material Behavior 3 Figure 3 Trilinear Approxiation of Conrete Stress-Strain Curve ' If an arbitrary aterial stress-strain urve is speified, a tri-linear approxiation will be onstruted as shown in Figure 3. In Figure 3, f ' is the peak opression stress and ε is the strain orresponding to the peak opression stress. The approxiation is done with the following rules: 1. The initial stiffness E of the tri-linear approxiation is equal to the slope between the origin and losest aterial stress-strain point in opression (the initial stiffness of the user-defined urve). 2. The Y point is deterined so that the area under the trilinear stressstrain relationship up to the point of axiu opression stress f ' is idential to that of the user-defined stress-strain urve. 3. The U point is defined by the point of axiu opression stress losest to the origin. The L point is defined by the point of axiu opression stress furthest fro the origin. If the U and L points oinide (suh as in Figure 3), the U point is plaed at the strain value ' orresponding to.98 f on the defined stress-strain urve. The L point is plaed at the greater of two strain values: (a) the strain orresponding to the axiu opression stress, or (b) 1.5 ties the strain at the U point.
Initial and Priniple Material Axes 4 4. If the defined aterial stress-strain urve has strength loss after the point of axiu opression stress, the R point is defined by the last defined point on the aterial urve. Initial and Priniple Material Axes Figure 3 shows a wall eleent, and the stresses at a point in the wall. The initial aterial axes are fixed relative to the wall eleent. In general there an be noral and shear stresses in these axes, as shown in Figure 4(b). There are also prinipal aterial axes, as shown in Figure 4(). These axes are parallel to the prinipal stress diretions, and thus are only noral stresses. The key assuption of the Darwin odel is that a uniaxial stress-strain relationship an be applied along eah of the prinipal aterial axes. Note that although the shear stress is zero in the prinipal aterial axes, the shear odulus is not zero. Hene, when a strain inreent is applied, the hange in shear stress generally will not be zero. During an analysis, the prinipal stress diretions, and hene the prinipal aterial axes, an rotate progressively. For this aterial odel, the effetive stress-strain relationships in the prinipal aterial axes as they rotate are obtained by interpolating between the relationships in the axes of the previous step. Figure 4 Initial and Priniple Material Axes
Initial Elasti Stress-Strain Relationship and Yield Surfae for Biaxial Stress 5 Initial Elasti Stress-Strain Relationship and Yield Surfae for Biaxial Stress If the aterial has not yet yielded or raked (i.e. the aterial is between the O and Y points as defined in Figure 1), the aterial has a linear elasti relationship with the initial value of Young's odulus is E, and Poisson's ratio is ν, as follows: dσ1 1 dσ 2 = 2 1 ν dτ12 E ν E νe E dε 1 dε 2 1 ν E dτ12 2 (1.1) This equation is independent of the stress and strain diretions, and hene it applies in both the initial and prinipal aterial axes. This aterial odel uses a retangular interation surfae with no expliit stress interation in the two diretions. The effet of biaxial opression stresses on the opression strength of the onrete aterial is not aounted for. The interation between stress and tensile strain is disussed in later setion. Post-yield or Craked Material Behavior After yield or raking, the aterial odulus hanges and the Poisson s ratio is negleted. For exaple, at the Y point in Figure 1 the odulus redues to E h, and at the U point it redues to zero. In general, the stresses, strains and oduli will be different along the two prinipal diretions. Equation 1.1 an be odified for the aterial nonlinearity as follows: or dσ dσ dτ 1 2 12 dσ dε 1 = [ D ep ] dε 2 (1.2a) dτ 12 = D dε (1.2b) where D ep is the elasti-plasti onstitutive atrix in prinipal aterial axes, given by: ep
Strength Redution under Perpendiular Tensile Strain 6 E1 ν E1E2 D 1 ep = ν E E E 2 1 2 2 1 ν (1.3) G The shear odulus in the priniple aterial axes, G, is speified to aintain oaxiality between the prinipal stresses and strains. The orresponding relationship in the initial aterial axes is obtained by applying the rotation between the initial and prinipal aterial axes a rotation by angle θ, as shown in Figure 4. Strength Redution under Perpendiular Tensile Strain When onrete is subjeted to shear stresses, it often raks in one diretion and is in opression in the other diretion. Failure in shear ay our when the onrete rushes in opression. Vehio and Collins (1986) showed that the opression strength of onrete depends on the agnitude of the tensile strain in the perpendiular diretion. The effetive opression strength of onrete in suh situations an be substantially saller than the original Figure 5 shows the relationship between the opression strength and perpendiular tensile strain developed in Vehio and Collins (1986) and ipleented in this aterial odel. f. ' Figure 5 Redution in Copression Strength Due to Tensile Strain in the Perpendiular Diretion
Strength Redution under Perpendiular Tensile Strain 7 The following equation fro Vehio and Collins (1986) is used for the opression strength redution fator, r : 1 r = ε.8.34 ε ' where ε is the instantaneous tension strain (positive) in the perpendiular diretion and ε is the speified uniaxial rushing strain in opression ' 1 (negative). The behavior of the aterial odel is as follows: (1.3) 1. If the onrete is in opression strain along one aterial axis and tensile strain along the other, the opression strength redution fator is alulated using Eq. 1.3. The iniu opression strength redution fator used is based on agnitude of opressive stress, σ, as follows: r in = 1..25 σ <.2 f ' σ >.5 f Linearly interpolated for.2 f ' < σ <.5 f ' (1.4) 2. The strength in the opression diretion ay previously have been redued. If the new redution fator is saller than the old one, the new fator is applied. If not, the new fator is ignored. 3. If the new fator is applied, the stress-strain relationship is odified as indiated in Figure 6. Note that the oduli E and E h do not hange.
Iportant Nuerial Considerations 8 Figure 6 Change in Stress-Strain Relationship to Aount for Strength Redution Iportant Nuerial Considerations General guidelines for working with nonlinear analysis are addressed in Topi Iportant Considerations in Chapter 23 of the CSI Analysis Referene Manual. Speifi nuerial onsiderations for this aterial odel are as follows: 1. Copared to the diretional aterial odels, the Modified Darwin- Peknold odel has a higher degree of nonlinearity and ay require saller tie steps to onverge. 2. This aterial is used in shell eleents, whih use a two-by-two nuerial integration forulation in the plane. Soe refineent of the esh ay be needed to apture varying nonlinear behavior. However are should be taken not to over-refine beause loalized failure ay our for very sall esh sizes. 3. As an alternative to nonlinear stati analysis, using a nonlinear dynai analysis with slowly applied exitation ay result in better onvergene behavior. This applies espeially to ases where signifiant rushing or loss of strength is expeted.
Referenes 9 4. Setting the Poisson s ratio equal to zero in the aterial properties ay iprove onvergene in soe ases beause this dereases the initial oupling between the two aterial axes. Referenes Darwin, D. and Peknold, D.A.W., "Inelasti Model for Cyli Biaxial Loading of Reinfored Conrete", University of Illinois, July 1974. Vehio, F.J. and Collins, M.P., "The Modified Copression-Field Theory for Reinfored Conrete Eleents Subjeted to Shear", Journal of the ACI, Paper No. 83-22, Marh-April 1986.