# 2-D RC Constitutive Relationship Subroutine RCFER2002

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```					                                       Important Reference Document
2-D RC Constitutive Relationship       and Program
Subroutine                     Important Reference Document
“RCFER2002”                       “Nonlinear FEA for RC Structure”, JIANG Jianjing
“Principle of Reinforced Concrete” , GUO Zhenhai
“FEA of RC and Ultimate analysis for Plate and Shell”,
J.M. SHEN, C.Z. WANG, J.J. JIANG
LU Xinzheng
“FEA for Reinforced Concrete”, Q. L. KANG
Dept. Civil Engineering
“Finite Element Analysis of Reinforced Concrete”,
Tsinghua University, Beijing      Bazant
Dec. 2002.                “Finite Element Method for Nonlinear Problems”, Bathe

Important Reference Document
and Program                            Basic Information of RCFER2002
Important Reference Program£º          Fortran 77+Fortran 90
RCFER, Coded by J.J. JIANG           Double Precision
NONSAP, Coded by Bathe
RCNFEA, Coded by C.C. LI
More than 60 subroutine
Application of RCFER2002                         Basic Request for RCFER2002
Courses Education                                Can simulate the stress and deformation of
“RC Structure”(for Undergraduate Students)     concrete under planar compression, tension,
Can simulate the cracking and cracking surface
Research                                         effect of concrete
RCPEFG (Code by LU)                            Can be embedded into Msc.MARC with user
RCFER (Code by JIANG)                          subroutine function of Msc.MARC
RC Element analysis                            Can simulate the behavior of concrete under cycle

Strength Criteria                                Strength Criteria (T/T)
Using the simplified planar strength criteria
ft
based on Li-Guo and Tasuji-Slate-Nilson           fc                         Both Tension
criteria
σ1>0 & σ2>0
Tension Failure
Strength Criteria (C/T)                    Strength Criteria (C/T)
ft                                         ft
Compression/Tension                        Compression/Tension
fc                                         fc
σ1>0 , σ2<0                                σ1>0 , σ2<0
&, σ1/ σ2<-0.05                            & σ1/ σ2>-0.05
Uniaxial Compression
Tension Failure                            Failure

Strength Criteria (C/C)                    Strength Criteria (C/C)
ft                                         ft
fc
Both Compression         fc
Both Compression

σ1<0 , σ2<0                                σ1<0 , σ2<0
& σ1/ σ2<0.2                               & σ1/ σ2>0.2
Uniaxial Compression                       Biaxial Compression
Failure                                    Failure
Equivalent Uniaxial Stress-Strain
Summary of Strength Criteria                                  Relationship
Uniaxial Compression Stress-strain
ft
fc
Divide the failure of     Relationship
concrete in planar                                F
Developed by Saenzc
conditions into 3
  ε        2

failure model, 5                      1 − 

        E0                   Fu
 ε f                                     Es
stress combination.     Et =                        
2                  Eu=γσ/ε<=E0
 E                               2

Hence, the result                           
1 +  0 − 2  ε
  ε
+

       
  Es

 ε
 f
 ε              
has obvious                                         f             

physical meanings.
εc            εu

Equivalent Uniaxial Stress-Strain
Relationship                                                 when the concrete is inside the failure surface
Developed by J.J. JIANG
when the concrete is outside the failure surface
σ = f t e − a (ε −ε )  cr
6 nonlinear index is used Evv(1~6)
Stiffness is used*
β = (β 0 − β 1 )e − a (ε −ε ) + β 1
Eu
cr
εcr
Nonlinear Index                                               Unify for Nonlinear Index
Evv(1): Nonlinear index for biaxial failure               Evv(1):
Evv(2): Nonlinear index for uniaxial failure                       (1.2ε c + dε * ) / ε c
2

Evv(3): Nonlinear index when no failure happens
Evv(4): Nonlinear index for crack, only one crack                  dε 2 = (dε 2 − 0.2dε 1 ) / 0.96
*

Evv(5): Nonlinear index for the 1st crack, two
crack conditions                                          Evv(2) (ε + dε ) / ε
c    2      c
Evv(6): Nonlinear index for the 2nd crack, two
crack conditions

Unify for Nonlinear Index                                     Unify for Nonlinear Index
Evv(3):                                                   Evv(4):
ft
σ1                             1 + dε 1* E 0 / f t
fc                                          ft
fc       Evv(5)
σ2
σ2 −
ft
σ1              1 + dε 1* E 0 / f t
− fc           − fc
Evv(6) 1 + dε * E / f
2 0     t
σ 2 − 0.2σ 1
σ2             − fc
− 1.2 f c
Crack Opening and Closing                              Independent Variable List (1)
Once the stress in crack direction σ<0, we             SIG(3),EPS(3),dEPS(3),dSIG(3): tσij, tεij, dσij, dεij
consider the crack is closed                           Stress(3),Strain(3) t+dtσij, t+dtεij
Once the crack has closed, the crack can               E0,MU0,Fc,Ft,SIGMAU,EPSC,EPSU
not bearing any tensional                                E, µ,fc,ft,σu,εc, εu

force=>CrRelease(i)=1                                  N(3,3), T(3,3) Coordination transform matrix
D(3,3) constitutive matrix
SIGP(3), EPSP(3) tσi, tεi principle stress/strain

Independent Variable List (2)                          Independent Variable List (3)
dSIGP(3), dEPSP(3) dσi, dεi Principle stress/strain    ANGLE Cracking Angle
increment
ANG, Alpha£¬ Principle stress angle,
dEPSP1(3),equivalent principle strain increment
principle stress ratio
Evv(6), Evmax(6) nonlinear index
A1,A2,Beta0,Beta1, cracking surface
parameter
Crack =-2 Biaxial Compression Failure, -1 Uniaxial
Tension Failure, 0 No Failure, 1 One Crack 2 Two       Inc, Ncycle, increment, sub-increment
Crack, 100 Entire Failure                              ITEM, State index
Subroutine List (1)                                Subroutine List (2)
Con_N2Crack01(C) biaxial compression
Con_GetD(Lxz_Con): General Control Subroutine       failure, under tensile stress, entire failure
Con_Crack1(C)         when Crack==-2 biaxial
compression failure
Con_N2Crack02(C) biaxial compression
Con_Crack4(C) when Crack==1 one crack
Con_Crack5(C) when Crack==2 two crack
Con_Crack100(C) when Crack==100 entire failure

Subroutine List (3)                                Subroutine List (4)
Con_N1Crack01(C) uniaxial compression             Con_NoCrack01(C) no crack,
failure, under tensile stress, entire failure
Con_NoCrack02(C) no crack, C/C,
Con_N1UnLoad(C) uniaxial compression                Con_NoCrack04(C) no crack, C/T, ¦Á >-0.05,
Subroutine List (5)                                               Subroutine List (6)
Con_NoUnLoad01(C) no crack,                                       One crack, closed, just like no crack
Con_NoUnLoad04(C) no crack, C/T, ¦Á >-0.05,

Subroutine List (7)                                               Subroutine List (8)
One crack,opened, if there is crush in the vertical direction,    Both cracks are closed ,treated as no crack condtions
concrete is entire failed, if there is crack in the vertical      Con_P2Crack01(C): 2 Cracks, Both Closed, C/C, >0.2, L/L
direction, change to 2 cracks condition
Con_P1Crack06(C): 1 Crack, Open, T/C, L/L(L: Loading, U:          Con_P2Crack02(C) : 2 Cracks, Both Closed, C/C, <0.2, L/L
Con_P1Crack07(C) : 1 Crack, Open, T/C, L/U                        L/L
Con_P1Crack08(C) : 1 Crack, Open, T/C, U/L                        Con_P2Crack04(C) : 2 Cracks, Both Closed, C/C, >-0.05,
Con_P1Crack09(C) : 1 Crack, Open, T/C, U/U                        L/L
Con_P1Crack10(C) : 1 Crack, Open, T/T, L/L                        Con_P2Crack05(C) : 2 Cracks, Both Closed, T/T, L/L
Con_P1Crack11(C) : 1 Crack, Open, T/T, L/U
Con_P1Crack12(C) : 1 Crack, Open, T/T, U/L
Con_P1Crack13(C) : 1 Crack, Open, T/T, U/U
Subroutine List (9)                                        º¯ÊýÁÐ±í                      (10)
1 of 2 Cracks is closed, treat as 1 crack conditions                 Both Cracks are open, it is assumed that the concrete can
Con_P2Crack06(C) 2 cracks, 1 close, 2 open, C, T, U,                 go on bearing shear force, that is β= min(βcrack1 βcrack2)
U                                                                    Con_P2Crack14(C) 2 Cracks, Both opend, T, T, L, L
Con_P2Crack07(C) 2 cracks, 1 close, 2 open, C, T, U,                 Con_P2Crack15(C) 2 Cracks, Both opend, T, T, U, L
L                                                                    Con_P2Crack16(C) 2 Cracks, Both opend, T, T, L, U
Con_P2Crack08(C) 2 cracks, 1 close, 2 open, C, T, L,                 Con_P2Crack17(C) 2 Cracks, Both opend, T, T, U, U
U
Con_P2Crack09(C) 2 cracks, 1 close, 2 open, C, T, L,
L
Con_P2Crack10(C) 2 cracks, 1 open, 2 close, C, T, U,
U

Connection with Msc.Marc                                   Example 1
Use the user subroutine for Hyperelastic                             Balance reinforced cantilever beam
material of Msc.Marc                                                 Compare of Msc.Marc and RCFER2002 (Red:
SUBROUTINE HYPELA (D,G,E,DE,S,TEMP,DTEMP,NGENS,N,NN,KC,
MATS,NDI,NSHEAR)
Msc.Marc, Blue: RCFER2002)
250000
Post Process: Crack, Load conditions                                   200000

Use the user subroutine of PLOTV                                       150000

100000

50000

0
0   0.002 0.004 0.006 0.008   0.01   0.012 0.014
Example 2                                                                  Example 3
4 nodes element under pure shear force                                     Tension-shear combination force
Reaction Force Y Node 2 (x10e6)                               3.50E+06

1200000                                                                     3.00E+06

1000000                                                                     2.50E+06

800000                                                                     2.00E+06

600000                                                                     1.50E+06

400000                                                                     1.00E+06

200000                                                                     5.00E+05

0                                                                     0.00E+00
0    0.0005      0.001   0.0015     0.002   0.0025                           0   0.000 0.001 0.001 0.002 0.002 0.003 0.003
5           5           5           5

Example 4                                                                  Example 5
0.00E+00                                       5.00E+06
-0.006   -0.005      -0.004       -0.003       -0.002    -0.001      0
-5.00E+06
-0.003     -0.002         -0.001        0            0.001     0.002
-1.00E+07                                     -5.00E+06

-1.50E+07
-1.50E+07
-2.00E+07

-2.50E+07
-2.50E+07
-3.00E+07

-3.50E+07
-3.50E+07
150000

Example 6                                            100000                       Example 7
50000

Cantilever beam                   -0.01   -0.005
0
0   0.005   0.01
Shear test for FRP sheet
-50000
9.00E+08

-150000                          8.00E+08

250000                                      200000                          7.00E+08

200000                                      150000
6.00E+08
150000                                      100000
5.00E+08
100000                                       50000                                                                                         100
200
50000                                            0                          4.00E+08                                                       300
400
0                    -0.01   -0.005    -50000 0     0.005   0.01       3.00E+08
500

-0.01   -0.005    -50000 0   0.005   0.01                    -100000                          2.00E+08
-100000                                     -150000
1.00E+08
-150000                                     -200000
0.00E+00
-200000                                     -250000                                     0   0.02   0.04   0.06   0.08   0.1   0.12   0.14

-1.00E+08

Thank you!

```
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