Drucker-Prager & Concrete

Chapter Eight

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Training Manual

Drucker-Prager Plasticity and Concrete

Chapter Overview

• This chapter will cover some geotechnical constitutive models, namely Drucker-Prager plasticity and concrete, which are available in ANSYS.

– Drucker-Prager is useful for granular materials such as soils, rock, concrete.

– The concrete model allows for characterization of brittle materials, including rock and certain ceramics. Cracking and crushing options are present.

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Drucker-Prager Plasticity and Concrete

... Chapter Overview

• In this chapter, the following will be discussed:

A. Drucker-Prager plasticity

B. Concrete model

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Training Manual

Drucker-Prager Plasticity and Concrete

A. Drucker-Prager Plasticity

• Drucker-Prager (DP) plasticity is applicable to granular (frictional) materials such as soils, rock, and concrete.

• Unlike metal plasticity, the yield surface is a pressure-dependent von Mises surface for DP:

where e is the modified equivalent stress, m is the hydrostatic pressure, and is a material constant.

• Plotted in principal stressspace, the yield surface isa cone.

2

1

2

13

sMs T

me

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... Drucker-Prager Plasticity

• Some important points to note:

– In compression, an increase in hydrostatic pressure produces an increase in the yield strength.

– Because volumetric strain is associated with hydrostatic pressure, volumetric expansion of the material due to yielding can be accounted for.

– No hardening is assumed, so the material behavior is elasto-perfectly plastic.

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Drucker-Prager Plasticity and Concrete

... Drucker-Prager Plasticity

• We can write the yield criterion in the following form.

• The material parameter and y are defined as

where is the angle of internal friction and c is the cohesion value.

– The cohesion value (shear yield stress) “c” is required input for the DP model. It has units of stress

– The angle of internal friction “” is also required input. It has units of degrees.

yT

m sMsF

2

1

2

13

sin33

sin2

sin33

cos6

cy

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Drucker-Prager Plasticity and Concrete

... Drucker-Prager Plasticity

• Note that the yield stress in compression is greater than the yield stress in tension.

• If you have uniaxial tensile t and compressive c yield stresses as your raw data, they can be converted to material parameters and c through the following:

tc

tcy

tc

tc

3

2

3

cos6

sin33

32

33sin 1

yc

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Drucker-Prager Plasticity and Concrete

... Drucker-Prager Plasticity

• Simple manipulation of the equations show that the distance between the origin and yield in tension in principal stress space is equal to (c)cot()

cotc

cot

sin33

cos6

sin33

sin23

3

c

c

m

m

ym

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Drucker-Prager Plasticity and Concrete

... Drucker-Prager Plasticity

• Besides the two aforementioned parameters and c, there is an additional parameter called the dilatancy angle f which is input for DP.

– The dilatancy angle f controls the amount of volumetric expansion which will occur.

– An example for compacted granular materials is when the grains “rise” over each other as the material shears.

– One way to picture this is plotting the yield surface on the meridional plane. “p” is hydrostatic pressure, and “q” is the modified equivalent stress.

p

q

f

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... Drucker-Prager Plasticity

– On the meridional plane below, f indicates direction of plastic flow (dilation angle). On the other hand, describes the direction of the outward normal to the yield surface (angle of internal friction).

– If f=, then the flow rule is termed associative. Significant volumetric expansion will occur as a result.

– If f<, then the flow rule is nonassociative. Less volumetric expansion will occur.

– If f=0, then no volumetric expansion will occur (plastic flow is normal to yield surface). This is usually a more conservative approach.

p

q

f

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... ANSYS Procedure for DP

• Only certain elements support DP plasticity:

– ‘Core’ elements: PLANE42, SOLID45, PLANE82, SOLID92, ahd SOLID95

– Other elements: LINK1, PLANE2, LINK8, PIPE20, BEAM23, BEAM24, SHELL43, SHELL51, PIPE60, SOLID62, SOLID65, SHELL91, SHELL93, and SHELL143

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... ANSYS Procedure for DP

• The DP material parameters can be input via the Materials GUI or the TB commands:

– Main Menu > Preprocessor > Material Props > Material Models…

• Structural > Nonlinear > Inelastic > Non-metal Plasticity

All constants should be input (i.e., cohesion value cannot be zero). Note that elastic material properties (Young’s Modulus EX) also needs to be input. No temperature-dependency is allowed for this material law.

TB,DP,1,,,0 TBDATA,1,cohesionTBDATA,2,fricangleTBDATA,3,flowangle

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... ANSYS Procedure for DP

• Drucker-Prager is a rate-independent plasticity model. For solution options, the same considerations apply for models with other rate-independent plasticity.

– Specify nonlinear geometric effects (NLGEOM,ON), as needed.

– Specify an adequate number of substeps to capture the path-dependent response.

• Postprocessing considerations:

– If the material has yielded, equivalent plastic strain (NL,EPEQ) will be non-zero

– The equivalent stress parameter pl (NL,SEPL) is the von Mises equivalent stress at the current hydrostatic stress level:

– Note that for equivalent strains (EPPL,EQV), ANSYS assumes incompressible inelastic strains (’=0.5). However, if f0, this is not true (volumetric expansion occurs with yielding). Consider the following case of inelastic strains when f0, where eqv should be non-zero:

mypl

e 33

0.0 , , pleqv

plz

ply

plx AAA

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Drucker-Prager Plasticity and Concrete

B. Concrete Model

• The concrete material model in ANSYS can be used to model brittle materials, such as concrete, rock and ceramics.

– Both cracking and crushing failure modes are included.

– Prior to failure, behavior is assumed to be linear elastic. However, plasticity and/or creep may be combined with concrete to provide nonlinear behavior prior to failure.

– This constitutive model is meant for low tensile strength but high compressive load carrying capability.

– A “smeared” reinforcement can be specified via real constants along three element coordinate directions, or discrete reinforcements can be separately added via LINK or COMBIN elements.

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Drucker-Prager Plasticity and Concrete

... Concrete Model

• The concrete material has the following characteristics:

– Material calculations are performed at each integration point of the element.

– The concrete model assumes linear elastic behavior up until the failure point. At the failure load (c or t), crushing or cracking can occur, and the material completely fails at that point.

– In the case of crushing (compression), the material completely fails.

– Cracking (tension) is permitted in three orthogonal directions at each integration point. Cracking can occur in one or several directions. For the direction in which cracking occurs, tensile strength essentially becomes zero, although, if the crack closes, compressive stresses normal to the crack can be transferred. Material properties for directions in which crack has not occurred remainthe same.

– A shear transfer coefficient t and c define the shear strength reduction factor for loads which induce sliding across the crack face.

ft

fc

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Drucker-Prager Plasticity and Concrete

... Concrete Model

• The concrete material can be combined with other nonlinearities:

– Plasticity and creep may be included with concrete. Usually, multilinear elastic or Drucker-Prager plasticity is used for concrete. Note that the plasticity yield surface must lie inside the concrete failure surface, otherwise no yielding will occur.

– The concrete failure surface as plotted in principal stress space is shown on right. Hence, the yield surface associatedwith any other nonlinearmaterial behavior (i.e.,plasticity) must lie insideof the concrete failuresurface. Otherwise, thematerial will completelyfail and never yield.

– Adjustments to stressesdue to plasticity are performed prior to thecracking/crushing checks.

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Drucker-Prager Plasticity and Concrete

... Concrete Model

• For material failure, there are essentially four domains to consider:

– 0 1 2 3 (compression - compression - compression)

– 1 0 2 3 (tension - compression - compression)

– 1 2 0 3 (tension - tension - compression)

– 1 2 3 0 (tension - tension - tension )

• For three-dimensional stress state, the failure surface is a function of the principal stresses and five input parameters discussed next. The failure surface is different for each of the four domains noted above.

– The user is referred to the ANSYS Theory Manual, Section 4.7, for details on the equations.

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Drucker-Prager Plasticity and Concrete

... Concrete Model

• The required concrete material constants are as follows:

– Ultimate tensile strength, ft

– Ultimate compressive strength, fc

– Ultimate biaxial compressive strength, fcb

– Ambient hydrostatic stress state, ah

– Ultimate compressive strength for a state of biaxial compression superimposed on hydrostatic stress state, f1

– Ultimate compressive strength for a state of uniaxial compression superimposed on hydrostatic stress state, f2

• The first two constants ft and fc are required. The others default to

This approximation is only valid for situations with a low hydrostatic stress component, or . Otherwise, the user should supply all of the above values. ch f3

c

c

ccb

ff

ff

ff

725.1

45.1

2.1

2

1

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... Concrete Model

• Through the real constants, reinforcement may also be specified as a volume ratio (VR).

– The reinforcements are “smeared”. If discrete reinforcements need to be modeled, use of LINK or COMBIN elements are suggested.

– The required input consists of the rebar material number, volume ratio, and orientation angles and .

– The reinforcement may also include plasticity and creep. Generally, one of the more common kinematic hardening laws is used for the rebars.

– Up to three rebars can be specified.

– The sum of the volume ratios of the rebars cannot exceed 1.0.

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... ANSYS Procedure for Concrete

• Only SOLID65, 8-node hexahedral element, supports concrete.

– Main Menu > Preprocessor > Element Type > Add/Edit/Delete …

– SOLID65 also includes extra displacement shapes by default (KEYOPT(1))

– Stress relaxation for cracking option will be discussed later.

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... ANSYS Procedure for Concrete

• SOLID65 Real constants refer to rebars (if present):

– Main Menu > Real Constants > Add/Edit/Delete …

– Orientation angles for and shown below:

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Drucker-Prager Plasticity and Concrete

... ANSYS Procedure for Concrete

• The Concrete material parameters can be input via the Materials GUI or the TB commands:

– Main Menu > Preprocessor > Material Props > Material Models…

• Structural > Nonlinear > Inelastic > Non-metal Plasticity

c

ah

cb

c

t

c

t

T

f

f

f

f

f

2

1

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... ANSYS Procedure for Concrete

• Up to six sets of temperature-dependent constants may be specified.

– Constants 1-8 discussed previously. Constant 9 will be covered later. A value of “-1” for constants 3 or 4 removes cracking or crushing behavior, respectively. Constants 5-8 default as discussed in previous slides. (Command input shown on left.)

Constant Symbol Meaning1

t Shear transfer coefficients for an open crack. (defaults to 1e-6)

2 c Shear transfer coefficients for a closed

crack. (defaults to 1e-6)3 ft Uniaxial tensile cracking stress.4 fc Uniaxial crushing stress (positive).5 fcb Biaxial crushing stress (positive).6 a

h Ambient hydrostatic stress state for use with constants 7 and 8. (default is 0.0)

7 f1 Biaxial crushing stress (positive) under the ambient hydrostatic stress state (constant 6).

8 f2 Uniaxial crushing stress (positive) under the ambient hydrostatic stress state (constant 6).

9 Tc Stiffness multiplier for cracked tensile condition, used if KEYOPT(7) = 1 (defaults to 0.6).

TB,CONC,1,1,9, TBTEMP,0TBDATA,1,ShrCf-OpTBDATA,2,ShrCf-ClTBDATA,3,UnTensStTBDATA,4,UnCompStTBDATA,5,BiCompStTBDATA,6,HydroPrsTBDATA,7,BiCompStTBDATA,8,UnTensStTBDATA,9,TenCrFac

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... ANSYS Procedure for Concrete

• If rebars are present, their orientation can be displayed via the GUI:

– Utility Menu > PlotCtrls > Device Options > Vector mode [ON]

– Utility Menu > PlotCtrls > Style > Size and Shape > Display of element shapes based on real constant descriptions [ON]

– Utility Menu > Plot > Elements

Or via commands:

– /DEV,VECTOR,1

– /ESHAPE,1

– EPLOT

The rebar orientationare shown in red.

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... ANSYS Procedure for Concrete

• After solution, cracks can be plotted:

– Main Menu > General Postproc > Plot Results > -Concrete Plot- Crack/Crush …

or via command:

– PLCRACK

Other items such as thestatus (unfailed, crush,open crack, closedcrack), crack orientationangles, and rebarsolution, can also beobtained.

In the plot on right, notethat crack orientationand plane are plottedper integration point.

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... ANSYS Procedure for Concrete

Tips for Concrete Analyses:

• Ensure that the concrete constants are valid for your particular application. For example, if there is large hydrostatic stress component (typical of large concrete structures such as dams), the default calculated values for fcb, f1, and f2 will not be suitable.

• Similarly, a tensile strength of zero would cause convergence difficulties and is not physically realistic.

• If the concrete structure completely fails in a load carrying area, non-convergence is expected because the stiffness would be zero. This would represent a geometric instability (see Chapter 9).

• Use of large number of substeps may be required if the material cracks or crushes, in order to ensure proper load transfer.

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... ANSYS Procedure for Concrete

Tips for Concrete Analyses:

• For cracking problems, use of KEYOPT(7)=1 may aid convergence. This is stress relaxation after cracking.

– After failure, the material stiffness suddenly drops to zero (figure on left).

– Constant Tc is input as the ninth constant for the concrete material property. This is a factor (default=0.6) which acts as a multiplier for the stress relaxation.

– When using adaptive descent on, the secant modulus Rt will be used. Rt will slowly be decreased to zero during the equilibrium iterations per substep (figure on right). This gradual change in slope to zero will aid convergence.

– This is for cracking situations only (not crushing).

ft

ft

Tcft

Rt

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... ANSYS Procedure for Concrete

Tips for Concrete Analyses:

• You can include the effects of pressure load stiffness in a geometric nonlinear analysis using SOLCON,,,INCP.

– The pressure load stiffness matrix aids convergence by providing a fully consistent tangent stiffness matrix.

• At those integration points where crushing has occurred, the output plastic and creep strains are from the previous converged substep.

• When cracking has occurred, the elastic strain output includes the cracking strain.

• The lost shear resistance of cracked and/or crushed elements cannot be transferred to the rebars, which have no shear stiffness.

• It is not recommended to have large-rotation effects when cracking/crushing is present since crack angles are based on rotation, so results could be in error in this situation.

September 30, 2001

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Training Manual

Drucker-Prager Plasticity and Concrete

References for Concrete Model

References for Concrete:

• Willam, K. J., and Warnke, E. D., “Constitutive Model for the Triaxial Behavior of Concrete”, Proceedings, International Association for Bridge and Structural Engineering, Vol. 19, ISMES, Bergamo, Italy, . 174 (1975).

• ANSYS Theory Manual, Section 4.7 (Concrete)

• ANSYS Theory Manual, Section 14.65 (SOLID65 Element)