1. Drucker-Prager & Concrete
Chapter Eight
Drucker-Prager & Concrete

2. 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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3. 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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4. 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 se is the modified equivalent stress, sm is the hydrostatic pressure, and b is a material constant.
Plotted in principal stress space, the yield surface is a cone. s1 s2 s3
s1 =s2 = s3
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5. Drucker-Prager Plasticity and Concrete ... 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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6. Drucker-Prager Plasticity and Concrete ... Drucker-Prager Plasticity
We can write the yield criterion in the following form.
The material parameter b and sy are defined as where f 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 “f” is also required input. It has units of degrees.
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7. 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 st and compressive sc yield stresses as your raw data, they can be converted to material parameters f and c through the following:
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8. 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(f) s1 s2 s3
s1 =s2 = s3
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9. Drucker-Prager Plasticity and Concrete ... Drucker-Prager Plasticity
Besides the two aforementioned parameters f and c, there is an additional parameter called the dilatancy angle ff which is input for DP.
The dilatancy angle ff 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 ff
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10. Drucker-Prager Plasticity and Concrete ... Drucker-Prager Plasticity
On the meridional plane below, ff indicates direction of plastic flow (dilation angle). On the other hand, f describes the direction of the outward normal to the yield surface (angle of internal friction).
If ff=f, then the flow rule is termed associative. Significant volumetric expansion will occur as a result.
If ff<f, then the flow rule is nonassociative. Less volumetric expansion will occur.
If ff=0, then no volumetric expansion will occur (plastic flow is normal to yield surface). This is usually a more conservative approach. p q f ff
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11. Drucker-Prager Plasticity and Concrete ... 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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12. Drucker-Prager Plasticity and Concrete ... 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,cohesion
TBDATA,2,fricangle
TBDATA,3,flowangle
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13. Drucker-Prager Plasticity and Concrete ... 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 spl (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 (n’=0.5). However, if ff0, this is not true (volumetric expansion occurs with yielding). Consider the following case of inelastic strains when ff0, where eeqv should be non-zero:
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14. 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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15. 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 (sc or st), 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 remain the same.
A shear transfer coefficient bt and bc define the shear strength reduction factor for loads which induce sliding across the crack face. e  ft fc
See Section in the Theory manual for details.
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16. 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 associated with any other nonlinear material behavior (i.e., plasticity) must lie inside of the concrete failure surface. Otherwise, the material will completely fail and never yield.
Adjustments to stresses due to plasticity are performed prior to the cracking/crushing checks.
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17. Drucker-Prager Plasticity and Concrete ... Concrete Model
For material failure, there are essentially four domains to consider:
0  s1  s2  s3 (compression - compression - compression)
s1  0  s2  s3 (tension - compression - compression)
s1  s2  0  s3 (tension - tension - compression)
s1  s2  s3  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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18. 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, sah
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.
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19. Drucker-Prager Plasticity and Concrete ... 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 q and f.
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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20. Only SOLID65, 8-node hexahedral element, supports concrete.
Drucker-Prager Plasticity and Concrete 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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21. SOLID65 Real constants refer to rebars (if present):
Drucker-Prager Plasticity and Concrete ANSYS Procedure for Concrete
SOLID65 Real constants refer to rebars (if present):
Main Menu > Real Constants > Add/Edit/Delete …
Orientation angles for q and f shown below:
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22. 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
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23. Up to six sets of temperature-dependent constants may be specified.
Drucker-Prager Plasticity and Concrete 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.)
TB,CONC,1,1,9,
TBTEMP,0
TBDATA,1,ShrCf-Op
TBDATA,2,ShrCf-Cl
TBDATA,3,UnTensSt
TBDATA,4,UnCompSt
TBDATA,5,BiCompSt
TBDATA,6,HydroPrs
TBDATA,7,BiCompSt
TBDATA,8,UnTensSt
TBDATA,9,TenCrFac
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24. If rebars are present, their orientation can be displayed via the GUI:
Drucker-Prager Plasticity and Concrete 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 orientation are shown in red.
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25. After solution, cracks can be plotted:
Drucker-Prager Plasticity and Concrete 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 the status (unfailed, crush, open crack, closed crack), crack orientation angles, and rebar solution, can also be obtained.
In the plot on right, note that crack orientation and plane are plotted per integration point.
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26. Tips for Concrete Analyses:
Drucker-Prager Plasticity and Concrete 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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27. Tips for Concrete Analyses:
Drucker-Prager Plasticity and Concrete 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). e  ft e  ft
Tcft Rt
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28. Tips for Concrete Analyses:
Drucker-Prager Plasticity and Concrete 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.
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29. Drucker-Prager Plasticity and Concrete References for Concrete Model
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, (1975).
ANSYS Theory Manual, Section 4.7 (Concrete)
ANSYS Theory Manual, Section (SOLID65 Element)
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