5.1.3 Rankine Principal Stress

Next: 5.1.4 Egg Cam-clay Up: 5.1 Isotropic Plasticity Previous: 5.1.2 Mohr-Coulomb or Drucker-Prager Contents Index

Subsections

5.1.3.1 User-supplied

5.1.3 Rankine Principal Stress

The commonly used material model for the behavior of concrete combines a Smeared cracking model for tension [Ch.6] with a plasticity model for compression [Ch.5]. In analyses where tension and compression arise simultaneously in one particular stress point, these models may lead to numerical oscillation, especially in plane stress situations.

This section describes the input of alternative models for the behavior of concrete, apt to handle combined tension and compression. DIANA offers three criteria [Fig.5.1]: the single Rankine and two combinations: Rankine/Von Mises and Rankine/Drucker-Prager. See §17.1.5.2 for more background theory.

**Figure 5.1:** Rankine Plasticity models
$\begin{figure}\index{RANKIN input@\texttt{RANKIN} input\vert textbf} \index{RANV... ...enterline{\raise 3.0cm\box\graph} } \end{picture}\end{footnotesize} \end{figure}$

The Rankine plasticity models can only be applied with plane stress, plane strain and axisymmetric elements.

(syntax)

$\begin{figure}\centering \begin{tabbing} \texttt{'MATERI'} \\ [-1.0ex] \rule{14... ...(\cdots\;\){]}\hspace{12ex}\emph{hardening/softening} \end{tabbing} \end{figure}$

YIELD: yldcri is the name of the yield criterion to be used. Each criterion or combination may be combined with a hardening/softening model [Fig.5.2].
YLDVAL: values describe the yield surface, depends on the criterion.
Hardening/softening: are data records to describe the hardening/softening model.

Rankine (syntax)

$\begin{figure}\centering \begin{tabbing} \texttt{'MATERI'} \\ [-1.0ex] \rule{14... ...\>\>\texttt{YLDVAL}\>\texttt{\textit{sigy}}$_{r}\,$ \end{tabbing} \end{figure}$

YIELD: RANKIN specifies that the Rankine yield criterion must be used [Fig.5.1a] [§17.1.5].
YLDVAL: sigy is the yield stress $\sigma_{{\mathrm{y}}}^{}$ .

Rankine/Von Mises (syntax)

$\begin{figure}\centering \begin{tabbing} \texttt{'MATERI'} \\ [-1.0ex] \rule{14... ...extit{fct}}$_{r}\,$ \texttt{\textit{fcc}}$_{r}\,$ \end{tabbing} \end{figure}$

YIELD: RANVMI specifies that the combined Rankine/Von Mises yield criterion must be used [Fig.5.1b]. The Von Mises criterion is applicable in the compressive region, the Rankine criterion bounds the tensile stresses [§17.1.5.2]. For plane strain and axisymmetry, the stresses also in the third direction.
YLDVAL: specifies the yield surface: fct is the Rankine yield stress $\sigma_{{\mathrm{y}}}^{}$ , fcc is the Von Mises yield stress $\sigma_{{\mathrm{y}}}^{}$ .

Rankine/Drucker-Prager (syntax)

$\begin{figure}\centering \begin{tabbing} \texttt{'MATERI'} \\ [-1.0ex] \rule{14... ...sph}}$_{r}\,$ {[}\texttt{\textit{sps}}$_{r}\,${]} \end{tabbing} \end{figure}$

YIELD: RANDRU specifies that the combined Rankine/Drucker-Prager yield criterion must be used [Fig.5.1c]. The Drucker-Prager criterion is applicable in the compressive region, the Rankine criterion bounds the tensile stresses [§17.1.5.2]. For plane strain and axisymmetry, the stresses also in the third direction.
YLDVAL: specifies the yield surface: fct is the Rankine yield stress $\sigma_{{\mathrm{y}}}^{}$ , ch is the cohesion c , sph is sin $\phi$ of friction angle $\phi$ and sps is sin $\psi$ of dilatancy angle $\psi$ . ( $\psi$ $\leq$ $\phi$ )Associated plasticity if $\psi$ = $\phi$ , nonassociated plasticity if $\psi$ < $\phi$ . [ $\psi$ = $\phi$ ]

Hardening/Softening (syntax)

$\begin{figure}\centering \begin{tabbing} \texttt{'MATERI'} \\ [-1.0ex] \rule{14... ...[}\>\texttt{CRACKB}\>\texttt{\textit{h}}$_{r}\,${]} \end{tabbing} \end{figure}$

HARDEN

harhyp specifies the hardening hypothesis: WORK for work hardening or STRAIN for strain hardening. [STRAIN]

HARNAM

tenhar is the name of the hardening/softening model, one of the keywords of Figure 5.2, see §18.1.1 for background theory. For a combined yield criterion this refers to the tensile regime (Rankine). Default is ideal plasticity, no hardening/softening.

**Figure 5.2:** Hardening/softening for concrete
$\begin{figure}\index{LINEAR input@\texttt{LINEAR} input!concrete plasticity\vert... ...enterline{\raise 7.6cm\box\graph} } \end{picture}\end{footnotesize} \end{figure}$

HARVAL

specifies the hardening parameters. Depends on the hardening/softening model.

Hardening property in tension. For linear softening, nonlinear Exponential softening and parabolic hardening/softening tv1 is the fracture energy G_f . For linear hardening, tv1 is the hardening modulus E_har . The ultimate hardening parameter $\kappa$ depends on the `crack bandwidth' h of the element. By default DIANA assumes a value h related to the area or the volume of the element. For models with h in the formulation, it may be useful to overrule the default value and to specify the crack bandwidth explicitly via the CRACKB input data item (see below).

Rankine and combinations. These multilinear diagrams, require pairs of values to be specified: values tv1 to tvn (n $\leq$ 30 )are the equivalent yield stress $\sigma_{{\mathrm{eq}}}^{}$ , k1 to kn are the corresponding hardening parameters $\kappa$ . For strain hardening, these $\kappa$ 's are the equivalent plastic strains.

CMPNAM

comhar is the name of the hardening/softening model in compression, for combined yield criteria only. The compression softening/hardening models are defined similar to the models under tensile loading [Fig.5.2]. Default is ideal plasticity, no hardening/softening.

CMPVAL

specifies the hardening/softening properties in compression and is for combined yield criteria only. For linear softening, nonlinear Exponential softening and parabolic hardening/softening, cv1 is the compressive fracture energy G_c . The ultimate hardening parameter $\kappa$ depends on an equivalent length of the element. DIANA automatically calculates this length as a value related to the area of the element, unless you specify it yourself [§6.3]. For linear hardening, cv1 is the hardening modulus E_har .

Rankine/Von Mises. This multilinear diagram requires pairs of values to be specified: values cv1 to cvn (n $\leq$ 30 )are the equivalent Von Mises yield stresses $\sigma_{{\mathrm{eq}}}^{}$ , k1 to kn are the corresponding compressive hardening parameters $\kappa$ . For strain hardening, these $\kappa$ 's are the equivalent plastic strains.

Rankine/Drucker-Prager. This multilinear diagram requires pairs of values to be specified: values cv1 to cvn are the mobilized cohesion c , k1 to kn are the corresponding compressive hardening parameter $\kappa$ . For strain hardening, $\kappa$ is the equivalent plastic strain.

CRACKB

h is the crack bandwidth h [§6.3].

Concrete (file.dat)

'MATERI'
    1    YOUNG      3.7E+04
         POISON     0.15
         YIELD      RANKIN
         YLDVAL     2.5
         HARDEN     STRAIN
         HARNAM     EXPONE
         HARVAL     0.09

Linear elastic behavior E = 37000 N/mm² , $\nu$ = 0.15 . Rankine plasticity limiting the tensile stresses: f_t = 2.5 N/mm² , G_f = 0.09 N^.mm/mm² with Exponential softening.

Concrete (file.dat)

'MATERI'
    2    YOUNG      3.7E+04
         POISON     0.15
         YIELD      RANVMI
         YLDVAL     2.5 35.0
         HARDEN     STRAIN
         HARNAM     EXPONE
         HARVAL     0.09
         CMPNAM     PARABO
         CMPVAL     5.0

Linear elastic behavior E = 37000 N/mm² , $\nu$ = 0.15 . Rankine plasticity limiting the tensile stresses: f_t = 2.5 N/mm² , G_f = 0.09 N^.mm/mm² with Exponential softening. Von Mises plasticity limiting the compressive stresses f_c = 35 N/mm² , G_c = 5 N^.mm/mm² with parabolic softening.

5.1.3.1 User-supplied

DIANA offers the user-supplied subroutine mechanism for cases where the hardening or the ambient influence on the cohesion or yield stress for the composite yield surface cannot be input as described. The cohesion, respectively the yield stress, can be a function of equivalent plastic strain, temperature, concentration, maturity and time. You may specify the hardening curve for one of three yield criteria: Rankine, Rankine/Von Mises, or Rankine/Drucker-Prager

Hardening - Rankine (syntax)

$\begin{figure}\centering \begin{tabbing} \texttt{'MATERI'} \\ [-1.0ex] \rule{14... ...{USRPAR}\>\texttt{\textit{usrpar}}$_{r\ldots}\,${]} \end{tabbing} \end{figure}$

YIELD: RANKIN specifies the Rankine criterion to be used [§17.1.5].
TENCRV: USRCRV specifies that the tensile strength is determined via the user-supplied subroutine USRCRV [§11.3].
USRPAR: usrpar are the parameters of the hardening curve.

DIANA passes the following information to subroutine USRCRV: the character string 'TENCRV' via argument parnam and parameters usrpar via argument usrpar.

Hardening - Rankine/Von Mises (syntax)

$\begin{figure}\centering \begin{tabbing} \texttt{'MATERI'} \\ [-1.0ex] \rule{14... ...{USRPAR}\>\texttt{\textit{usrpar}}$_{r\ldots}\,${]} \end{tabbing} \end{figure}$

YIELD: RANVMI specifies the Rankine/Von Mises criterion [§17.1.5.2] to be used.
TENCRV: USRCRV specifies that the tensile strength is determined via the user-supplied subroutine USRCRV [§11.3].
COMCRV: USRCRV specifies that the compressive strength is determined via the user-supplied subroutine USRCRV [§11.3].
USRPAR: usrpar are the parameters of the hardening curve.

DIANA passes the following information to subroutine USRCRV: a character string via argument parnam: 'TENCRV' when the tensile regime is evaluated or 'COMCRV' when the hardening curve of the compressive regime is evaluated. Parameters usrpar are passed via argument usrpar.

Hardening - Rankine/Drucker-Prager (syntax)

$\begin{figure}\centering \begin{tabbing} \texttt{'MATERI'} \\ [-1.0ex] \rule{14... ...{USRPAR}\>\texttt{\textit{usrpar}}$_{r\ldots}\,${]} \end{tabbing} \end{figure}$

YIELD: RANDRU specifies the Rankine/Drucker-Prager criterion [§17.1.5.2] to be used.
SINPHI: sphi specifies sin $\phi$ , the sine of friction angle $\phi$ .
SINPSI: spsi specifies sin $\psi$ , the sine of dilatancy angle $\psi$ . ( $\psi$ $\leq$ $\phi$ )If you don't specify sin $\psi$ then DIANA assumes associated plasticity. [ $\psi$ = $\phi$ ]
TENCRV: USRCRV specifies that the tensile strength is determined via the user-supplied subroutine USRCRV [§11.3].
COHCRV: USRCRV specifies that the cohesion is determined via the user-supplied subroutine USRCRV [§11.3].
USRPAR: usrpar are the parameters of the hardening curve.

DIANA passes the following information to subroutine USRCRV: a character string via argument parnam: 'TENCRV' when the tensile regime is evaluated or 'COHCRV' when the hardening curve of the cohesion is evaluated. Parameters usrpar are passed via argument usrpar.

Next: 5.1.4 Egg Cam-clay Up: 5.1 Isotropic Plasticity Previous: 5.1.2 Mohr-Coulomb or Drucker-Prager Contents Index

DIANA-9.3 User's Manual - Material Library
First ed.

Copyright (c) 2008 by TNO DIANA BV.