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Publication Number: FHWA-HRT-05-062
Date: May 2007

Users Manual for LS-DYNA Concrete Material Model 159

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Foreword

This report documents a concrete material model that has been implemented into the dynamic finite element code, LS-DYNA, beginning with version 971. This model is in keyword format as MAT_CSCM for Continuous Surface Cap Model. This material model was developed to predict the dynamic performance-both elastic deformation and failure-of concrete used in roadside safety structures when involved in a collision with a motor vehicle. An example of a roadside safety structure is a concrete safety barrier that divides opposing lanes of traffic on a roadway. Default input parameters for concrete are stored in the model and can be accessed for use. This material model only replicates the concrete aggregate. Appropriate reinforcement bars or rods must be included in the structure model separately.

The Users Manual for LS-DYNA Concrete Material Model 159 is the first of two reports that completely document this material model. This report documents the theoretical basis, the required input format, and includes limited hypothetical problems for the user. The second report, Evaluation of LS-DYNA Concrete Material Model 159 (FHWA-HRT-05-063), documents the testing performed to document the model's performance and accuracy of results.

This report will be of interest to research engineers who are associated with the evaluation and crashworthy performance of roadside safety structures, particularly engineers responsible for predicting the crash response of such structures when using the finite element code, LS-DYNA.

Michael Trentacoste

Director, Office of Safety R&D

Notice

This document is disseminated under the sponsorship of the U.S. Department of Transportation in the interest of information exchange. The U.S. Government assumes no liability for the use of the information contained in this document.

The U.S. Government does not endorse products or manufacturers. Trademarks or manufacturers' names appear in this report only because they are considered essential to the objective of the document.

Quality Assurance Statement

The Federal Highway Administration (FHWA) provides high-quality information to serve Government, industry, and the public in a manner that promotes public understanding. Standards and policies are used to ensure and maximize the quality, objectivity, utility, and integrity of its information. FHWA periodically reviews quality issues and adjusts its programs and processes to ensure continuous quality improvement.

Technical Report Documentation Page

1. Report No.

FHWA-HRT-05-062

2. Government Accession No.

3. Recipient's Catalog No.

4. Title and Subtitle

USERS MANUAL FOR LS-DYNA CONCRETE MATERIAL MODEL 159

5. Report Date

May 2007

6. Performing Organization Code

7. Author(s)

Yvonne D. Murray

8. Performing Organization Report No.

9. Performing Organization Name and Address

APTEK, Inc.
1257 Lake Plaza Drive
Colorado Springs, CO 80906

10. Work Unit No. (TRAIS)

11. Contract or Grant No.

DTFH61-01-C-00075

12. Sponsoring Agency Name and Address

Volpe National Transportation Systems Center
55 Broadway, Kendall Square
Cambridge, MA 02142-1093

Federal Highway Administration
6300 Georgetown Pike
McLean, VA 22101-2296

13. Type of Report and Period Covered

Final Report

September 27, 2001 through
September 30, 2004

14. Sponsoring Agency Code

15. Supplementary Notes

The Contracting Officer's Technical Representative (COTR) for this project is Martin Hargrave, Office of Safety Research and Development, HRDS-04, Turner-Fairbank Highway Research Center.

16. Abstract

An elasto-plastic damage model with rate effects was developed for concrete and implemented into LS-DYNA, a commercially available finite element code. This manual documents the theory of the concrete material model, describes the required input format, and includes example problems for use as a learning tool. A default material property input option is provided for normal strength concrete. The model was developed for roadside safety applications, such as concrete bridge rails and portable barriers impacted by vehicles, but it should also be applicable to other dynamic applications.

The companion report to this manual is entitled Evaluation of LS-DYNA Concrete Material Model 159, FHWA-HRT-05-063.

17. Key Word

concrete, LS-DYNA, material model, plasticity, damage, rate effects, reinforced beam

18. Distribution Statement

No restrictions. This document is available through the National Technical Information Service, Springfield, VA 22161.

19. Security Classif. (of this report)

Unclassified

20. Security Classif. (of this page)

Unclassified

21. No. of Pages

89

22. Price

Form DOT F 1700.7 (8-72) Reproduction of completed page authorized

SI (Modern Metric) Conversion Factors


Table of Contents

Chapter 1. Introduction

Chapter 2. Theoretical Manual
   Critical Concrete Behaviors
   Overview of Model Theory
   Elastic Update
   Plastic Update
   Yield Surface
   Damage Formulation
   Rate Effects Formulation
   Kinematic Hardening
   Model Input
   Bulk and Shear Moduli
   Triaxial Compression Surface
   Triaxial Extension and Torsion Surfaces
   Cap Location, Shape, and Hardening Parameters
   Damage Parameters
   Strain Rate Parameters
   Units

Chapter 3. Users Manual
   LS-DYNA Input
   Model Formulation and Input Parameters

Chapter 4. Examples Manual

Appendix A. Modeling Softening

Appendix B. Modeling Rebar

Appendix C. Single Element Input File

Appendix D. CEB Specification for Rate Effects

References


List of Figures

Figure 1. Graph. Example concrete data from Mills and Zimmermann plotted in the meridian plane.(8)

Figure 2. Drawing. Example curves fit by Launay and Gachon to their concrete data and plotted in the deviatoric plane.(10)

Figure 3. Graph. Example plots of the failure surfaces of LS-DYNA Model 159 in the meridian plane

Figure 4. Drawing. Example plots of the failure surfaces of LS-DYNA Model 159 in the deviatoric plane

Figure 5. Graph. Softening response of concrete in uniaxial compression (reprinted from the Comité Euro-Internacional du Béton (CEB) - Federation for Prestressing (FIP) Model Code 1990, courtesy of the International Federation for Structural Concrete (fib)).(11)

Figure 6. Graph. Variation of concrete softening response with confinement. Source: Joy and Moxley.(12)

Figure 7. Graph. The slope during initial loading is steeper than during subsequent loading for this uniaxial tensile stress data. Source: Reprinted with permission from Elsevier.(13)

Figure 8. Graph. The slope during initial loading is steeper than during subsequent loading for this uniaxial compressive stress data. Source: Reprinted with permission from Aedificatio Verlag.(14)

Figure 9. Graph. These loading/unloading data demonstrate that concrete stiffness degrades simultaneously with strength. Source: Reprinted with permission from Aedificatio Verlag.(14)

Figure 10. Graph. Concrete dilates in uniaxial compressive stress. Source: Reprinted from Defense Technical Information Center.(15)

Figure 11. Graph. Concrete dilates in biaxial compression. Source: Data curves scanned from Kupfer et al., American Concrete Institute (ACI).(16)

Figure 12. Graph. The different pressure-volumetric strain behaviors measured in isotropic compression versus uniaxial strain indicate shear enhanced compaction. Source: Data curves scanned from Joy and Moxley.(12)

Figure 13. Graph. A variety of data sources indicates that the compressive strength of concrete increases with increasing strain rate. Source: Reprinted with permission from American Society of Civil Engineers.(17)

Figure 14. Graph. Rate effects are more pronounced in tension than in compression. Source: Reprinted from Ross and Tedesco.(18)

Figure 15. Illustration. General shape of the concrete model yield surface in three dimensions

Figure 16. Illustration. General shape of the concrete model yield surface in two dimensions in the meridonal plane

Figure 17. Equation. Stress invariant J 1, J2, and J3

Figure 18. Equation. Yield function f.

Figure 19. Equation. Shear failure surface function Ff.

Figure 20. Graph. Schematic of shear surface

Figure 21. Graph. Schematic of two-part cap function

Figure 22. Graph. Schematic of multiplicative formulation of the shear and cap surfaces

Figure 23. Equation. Cap failure surface function Fc

Figure 24. Equation. L of kappa

Figure 25. Equation. Simple cap failure surface function Fc

Figure 26. Equation. X as a function of kappa.

Figure 27. Equation. Plastic volume strain ε pv

Figure 28. Illustration. Example two- and three-invariant shapes of the concrete model in the deviatoric plane

Figure 29. Equation. Angle beta hat in the deviatoric plane

Figure 30. Equation. Relationship between beta hat and J hat

Figure 31. Equation. Rubin scaling function ℜ

Figure 32. Equation. Most general form for Q1 and Q2

Figure 33. Equation. Mohr-Coulomb form for Q1, Q2

Figure 34. Equation. Willam-Warnke form for Q1

Figure 35. Equation. Damaged stress σ dij

Figure 36. Graph. This cap model simulation demonstrates strain softening and modulus reduction

Figure 37. Equation. Brittle damage threshold τb

Figure 38. Equation. Ductile damage threshold τd

Figure 39. Equation. Viscoplastic damage threshold r0

Figure 40. Equation. Incremental damage threshold, small rn+1

Figure 41. Equation. Brittle damage small d of tau

Figure 42. Equation. Ductile damage small d of tau

Figure 43. Equation. Variation of dmax with stress invariant ratio

Figure 44. Equation. Variation of dmax with rate effects

Figure 45. Schematic representation of four stress paths and their stress invariant ratios

Figure 46. Equation. Reduction of A with confinement

Figure 47. Equation. Fracture energy integral for Gf

Figure 48. Equation. Brittle damage fracture energy Gf

Figure 49. Equation. Brittle damage threshold difference τ minus small r0b

Figure 50. Equation. Brittle softening parameter C

Figure 51. Equation. Ductile damage fracture energy Gf

Figure 52. Equation. Ductile damage threshold difference τr0d

Figure 53. Equation. Ductile softening parameter A

Figure 54. Equation. Brittle damage threshold Gf Brittle

Figure 55. Equation. Ductile damage threshold Gf Ductile

Figure 56. Equation. The fracture energy with rate effects, Gvpf

Figure 57. Equation. Default damage recovery of d of τt

Figure 58. Equation. Optional damage recovery of d of τt

Figure 59. Equation. Viscoplastic stress update for σvpij

Figure 60. Equation. Two-parameter η

Figure 61. Equation. Dynamic strengths, f ′T dynamic, and f ′C dynamic

EpsilonFigure 62. Equation. Effective strain rate

Figure 63. Equation. Variation of fluidity parameter η in tension

Figure 64. Equation. Variation of fluidity parameter η in compression

Figure 65. Equation. Effective fluidity parameters, ηt, ηc, and η s

Figure 66. Equation. Overstress limit of η

Figure 67. Equation. Back stress α ij n + 1

Figure 68. Equation. Updated stress with hardening, σPij n+1

Figure 69. Equation. Incremental back stress, Δαij

Figure 70. Equation. Brittle rate of translation CH Brittle

Figure 71. Equation. Ductile rate of translation CH Ductile

Figure 72. Equation. The limiting function Gα.

Figure 73. Equation. Modified shear failure surface, Ff

Figure 74. Equation. Default Young's modulus E

Figure 75. Equation. Shear and bulk moduli, G and K

Figure 76. Equation. ACI Young's modulus, Ec

Figure 77. Equation. Reduced ACI Young's modulus, Ec

Figure 78. Equation. TXC Strength

Figure 79. Equation. Interpolation parameter P

Figure 80. Equation. Most general form for Q1, Q2

Figure 81. Graph. This isotropic compression simulation demonstrates how the cap parameters set the shape of the pressure-volumetric strain curve

Figure 82. Equation. The default fracture energy GF

Figure 83. Graph. Approximate tensile and compressive dynamic increase factors for default concrete model behavior

Figure 84. Illustration. General shape of the concrete model yield surface in two dimensions

Figure 85. Equation. Three stress invariants, J1, J2, J3

Figure 86. Equation. Plasticity yield function f

Figure 87. Equation. Shear surface function Ff

Figure 88. Equation. Most general form for scaling functions Q1, Q2

Figure 89. Equation. Cap surface function, Fc

Figure 90. Equation. Definition of L of kappa

Figure 91. Equation. Pressure invariant X as a function of kappa

Figure 92. Equation. Plastic volume strain hardening rule, ε pv

Figure 93. Equation. Transformation of viscoplastic stress to damaged stress, σdij.

Figure 94. Equation. Ductile damage accumulation, τ d

Figure 95. Equation. Brittle damage accumulation, τb

Figure 96. Equation. Brittle damage, d of tb

Figure 97. Equation. Ductile damage, d of τd

Figure 98. Equation. Reduction of A with confinement

Figure 99. Equation. Brittle and ductile damage thresholds, Gf

Figure 100. Equation. Viscoplastic stress, σvpij

Figure 101. Equation. Variation of the fluidity parameter η in tension and compression

Figure 102. Definition of effective strain rate

Fracture energy with rate effects G subscript lowercase F superscript rateFigure 103. Equation. Overstress limit of η

Figure 104. Equation. Fracture energy with rate effects

Figure 105. Computer printout. Example concrete model input for default material property input (option mat_CSCM_concrete)

Figure 106. Computer printout. Example concrete model input for user-specified material property input (option MAT_CSCM)

Figure 107. Graph. Example single element stress-strain results for 30 MPa (4,351 psi) concrete with 19-mm (0.75-inch) maximum aggregate size

Figure 108. Equation. Old generic damage, small d of τ

Figure 109. Equation. New generic damage, small d of τ

Figure 110. Graph. Behavior of the original softening function

Figure 111. Graph. Behavior of the updated softening function

Figure 112. Graph. Rebar yields in a ductile manner at a quasi-static rate of 0.0054/s. Source: U.S. Army Engineer Waterways Experiment Station.(29)

Figure 113. Graph. Rebar exhibits rate effects at a strain rate of 4/s. Source: U.S. Army Engineer Waterways Experiment Station.(29)

Figure 114. Equation. CEB tensile strength dynamic increase factor, DIFten

Figure 115. Equation. CEB compressive strength dynamic increase factor, DIFcomp

Figure 116. Graph. Dynamic increase factors specified in CEB


List of Tables

Table 1. These default bulk and shear moduli of concrete are derived from the formula for Young's modulus provided in CEB.

Table 2. These bulk and shear moduli for concrete are derived from a formula for Young's modulus suggested by ACI Code Committee.

Table 3. Approximate strength measurements used to set default TXC yield surface parameters.

Table 4. TXC yield surface input parameters as a function of unconfined compression strength.

Table 5. Quadratic equation coefficients which set the default TXC, TOR, and TXE yield surface parameters as a function of unconfined compression strength.

Table 6. TOR yield surface input parameters as a function of unconfined compression strength.

Table 7. TXE yield surface input parameters as a function of unconfined compression strength.

Table 8. Cap shape, location, and hardening parameters as a function of unconfined compression strength.

Table 9. Coefficients for the fracture energy equation.

Table 10. Tensile fracture energies tabulated in CEB as a function of concrete strength.

Table 11. Example load curve for modeling rebar strain rate effects with LS-DYNA Material Model #24.


Glossary of Symbols
a a0 a1 a2
Rubin function internal parameters
A B C D
softening parameters (compression and tension)
AP  BP  CP
quadratic equation coefficients
b b0 b1 b2
Rubin function internal parameters
Bs
term used in one rate effects formula
CH
hardening rate parameter
d d b d d
scalar damage parameter (general, brittle, ductile)
dm
maximum of brittle and ductile scalar damage parameters
dmax
maximum damage allowed to accumulate
D1 D2
cap linear and quadratic shape parameters
 
E EcEs
Young's modulus (general, concrete, steel)
f
yield surface function
f*
trial elastic yield surface function
Ff
shear failure surface
Fc
hardening cap surface
G
shear modulus
Gα
hardening model translational limit function for shear surface
Gft Gfc Gfs
fracture energies (tension, compression, shear)
J1
first invariant of the stress tensor
J ¢2 J ¢3
second and third invariants of the deviatoric stress tensor
J1T J ¢2T J ¢3T
trial elastic stress invariants
J1P J ¢2P J ¢3P
inviscid elastic stress invariants
normalized invariant of the deviatoric stress tensor
normalized invariant of the deviatoric stress tensor
K
bulk modulus
L
element length
N
hardening initiation
nt nc
rate effects fluidity parameters (tension, compression)
Nt NC
rate effects power parameters (tension and compression)
P
pressure
Q1 Q2
Rubin scaling functions for torsion and triaxial extension
Rubin strength reduction factor
R
cap aspect ratio
rS
initial damage before activation of rate effects
r0 r0b r0d
initial damage threshold (general, brittle, ductile)
Sij
deviatoric stress tensor
W
maximum plastic volume compaction
x x0
instantaneous displacement and displacement at peak strength
X X0
current cap location and initial cap location
y
integrand of dilogarithm function
αij Δαij
hardening model back stress and incremental back stress tensors
β β1 α2
shear surface constant term (compression, torsion, extension)
βs
term used in one rate effects formula
β β1 β2
shear surface exponent (compression, torsion, extension)
angle in deviatoric plane (invariant)
angle in deviatoric plane (invariant)
plasticity consistency parameter 
plasticity consistency parameter
Δt
time step increment
εij Δ εij
strain tensor and strain increments
effective strain rate and effective strain rate increment effective strain rate and effective strain rate increment 
effective strain rate and effective strain rate increment
term used in one rate effects equation
term used in one rate effects equation
εmax
maximum principal strain
εx εy εz εxy εyz εxz
strain components
εv
volumetric strain
ε pv
plastic volumetric strain
γ
viscoplastic interpolation parameter
γ s
term used in one rate effects formula
ηηt ηcηs
rate effects fluidity parameters (general, tension, compression, shear)
η0t nt
rate effects input parameters in uniaxial tension stress
η0c nc
rate effects input parameters in uniaxial compressive stress
κ κT κP κ0
cap hardening parameters (general, trial elastic, inviscid, initial)
λ1 λ2
shear surface nonlinear term (compression, torsion, extension)
η
Poisson's ratio
θ θ1 θ2
shear surface linear term (compression, torsion, extension)
ρ ρc ρs
density (general, concrete, steel)
ρt ρc ρσ
meridians (tensile, compressive, shear)
σ σT
a stress component (general, trail elastic)
σ x σ r
axial and radial stresses measured in triaxial compression tests
σvp σd
stress components calculated without and with damage
Sigma supescript lowecase v p subscript lowercase i j Sigm superscript uppercase T subscript i j sigma superscirpt rho subscript lowercase i j
stress tensors (viscoplastic, trial elastic, plastic)
σ 1 σ 2 σ 3
principal stress components
τbτd
instantaneous strain energy-type terms for damage accumulation
Acronyms and Abbreviations
ACI
American Concrete Institute
CEB
Comité Euro-Internacional du Béton
CSCM
continuous surface cap model
DIF
dynamic increase factor
FIP
Federation for Prestressing
NCHRP
National Cooperative Highway Research Program
TOR
torsion
TXC
triaxial compression
TXE
triaxial extension

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