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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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Chapter 2. Theoretical Manual
Bulk and Shear Moduli
Figure 74. Equation. Default Young's modulus E.
Here, E is Young's modulus and EC = 18.275 MPa (2,651 psi) (which is the value of Young's modulus when f ' c = 10 MPa (1,450 psi)). This value of EC is for simulations that are modeled linear to the peak (no prepeak hardening). Poisson's ratio is typically taken as being between 0.1 and 0.2. A value of η = 0.15 is selected here and is assumed to remain constant with concrete strength. Based on this information, the default bulk and shear moduli (K and G) in Table 1 are derived from the classical relationships between stiffness constants, as shown in Figure 75:
Figure 75. Equation. Shear and bulk moduli, G and K.
The equations in Figure 74 and Figure 75 are implemented in the concrete model initialization routines to set the default moduli of concrete as a function of concrete compressive strength.
Figure 76. Equation. ACI Young's modulus, Ec.
where wc is the density of concrete in kilograms per meters cubed (kg/m3). For normal weight concrete with wc = 2,286 kg/m3 (5,040 pounds per feet cubed (lb/ft3)), this formula reduces to the equation shown in Figure 77:
Figure 77. Equation. Reduced ACI Young's modulus, Ec.
GPa = gigapascals
MPa = megapascals
ksi = kips per square inch
psi = pounds per square inch
Triaxial Compression Surface
The TXC yield surface equation is fit to four strength measurements. For roadside safety applications, the regimes of interest are primarily the tensile and low confining pressure regimes. Hence, the first and most common measurement fitted is unconfined compression, in which the pressure is one-third the strength. The second measurement is uniaxial tension, which is often called direct pull. The third measurement is triaxial tension (equal tension in three directions), which sets the apex of the TXC yield surface. The fourth measurement is TXC at a specified pressure. The pressure selected is 70 MPa (10,153 psi). The fit to this measurement anchors the yield surface at low to moderate pressure.
Strength measurements are given in Table 3. The uniaxial compression and tension measurements are taken from tables and information provided in CEB. The triaxial tension measurement is equal to the uniaxial tension measurement. This choice, along with appropriate selection of the three-invariant scale factors, will model the biaxial tension strength approximately equal to the uniaxial tension strength. This is the recommendation in CEB.
The TXC measurement (principal stress difference) is taken from a review of test data. For example:
Figure 78. Equation. TXC Strength.
At each value of unconfined compressive strength, the four strength parameters (α, λ, β, θ) are simultaneously fit to the four strength values via an iterative procedure. Fitted values at five strengths are given in Table 4.
Obviously, a user may wish to analyze concrete at strengths other than the five listed. To accomplish this, quadratic equations as a function of unconfined compression strength are fit through each parameter, P, as shown in Figure 79:
Figure 79. Equation. Interpolation parameter P.
For the TXC yield surface, the parameter P represents either α, λ, β, or q. The fitted values of AP, BP, and CP are given in Table 5. Fitted values of AP, BP, and CP for all other concrete model input parameters (TOR and TXE yield surfaces, cap, damage, rate effects parameters) are given in subsequent sections.
MPa-1 = 0.006895 psi-1
Triaxial Extension and Torsion Surfaces
Figure 80. Equation. Most general form for Q1, Q2.
where Q1 is the TOR/TXE strength ratio, and Q2 is the TXE/TXE strength ratio. Each ratio may remain constant or vary with pressure. The default fits of these equations to data are given in Table 6 and Table 7, and are based on the following data and assumptions:
MPa-1 = 0.006895 psi-1
MPa-1 = 0.006895 psi-1
Again, because users may want to analyze concrete at a strength other than the five listed, quadratic equations as a function of unconfined compression strength are fit through each set of parameter values for the TOR and TXE surfaces. The quadratic equation coefficients were previously given in Table 5.
Cap Location, Shape, and Hardening Parameters
The cap parameters are selected by fitting pressure-volumetric strain curves measured in hydrostatic compression and uniaxial strain tests. Default fits, given in Table 8, are based on the following data and assumptions:
An example pressure-volumetric strain curve from an isotropic compression simulation is given in Figure 81. This figure demonstrates how each parameter affects the shape of the curve.
The cap initial location varies with compressive strength. The quadratic equation is used to obtain the cap location at a compressive strength other than the five tabulated. The quadratic equation coefficients are: AP = 8.769178e-03 MPa-1, BP = -7.3302306e-02, and CP = 84.85 MPa (12,306 psi).
psi = 145.05 MPa
psi = 145.05 MPa
Figure 81. Graph. This isotropic compression simulation demonstrates how the cap parameters set the shape of the pressure-volumetric strain curve.
Concrete softens in the tensile and low confining pressure regimes. For modeling purposes, fracture energy is defined as the area under the softening portion of a stress-displacement curve from peak stress to complete softening. One equation in CEB relates the measured fracture energy in tension to the unconfined compression strength and the maximum aggregate size, as shown in Figure 82:
Figure 82. Equation. The default fracture energy GF.
KPa-cm = kilopascals-centimeters
1 KPa-cm = 0.05710 Psi-inch
Here GF0 is the fracture energy at f ¢c = 10 MPa (1,450 psi) as a function of the maximum aggregate size. CEB actually lists the value of GF0 as 5.8 for 32-mm (1.26-inch) aggregate, but it has been replaced with 3.8 to make G F consistent with CEB tabulated values. The fit of the quadratic equation to these GF0 values as a function of aggregate size in mm is AP = 0.000520833 cm/KPa, BP = 0.75 cm, and CP = 1.9334 KPa-cm.
Tensile fracture energies calculated from the equation in Figure 82 at five specific concrete strengths are given in Table 10.
1 KPa-cm = 0.05710 Psi-inch
The concrete material model requires specification of the fracture energies in uniaxial tensile stress, uniaxial compressive stress, and pure shear stress. Default values for the tensile fracture energy are given by the equation in Figure 82. Default values for the compressive fracture energy are set at 100 times the tensile fracture energy. Default values for the shear fracture energy are set equal to the tensile fracture energy.
Other input parameters required are the brittle and ductile damage thresholds and the maximum damage levels:
Strain Rate Parameters
Concrete exhibits an increase in strength with increasing strain rate (refer to Figure 13 and Figure 14). Data are typically reported in terms of the ratio of dynamic to static strength, called the dynamic increase factor (DIF). CEB provides specifications for the DIF, as discussed in appendix D. However, the CEB specifications are not a good fit to the tensile data previously shown in Figure 14. Therefore, the DIF used and shown in Figure 83 is based on the developer's experience on various defense contracts, particularly for concrete with strength around f 'c = 45 MPa (6,527 psi). These specifications provide a good fit to both the tension and compression data previously shown in Figure 13 and Figure 14.
DIF specifications are approximately met by running numerous calculations and selecting the viscoplastic rate effects parameters via a trial and error method. The viscoplastic parameters are applied to the plasticity, damage, and fracture energy formulations. These parameters are η0t and nt for fitting uniaxial tensile stress data, and η0c and nc for fitting uniaxial compression data. Quadratic equation coefficients are dependent on the unconfined compression strength, but are independent of the aggregate size.
The default parameters in tension are nt = 0.48, with quadratic equation coefficients for η0t of AP = 8.0614774E-13, BP = −9.77736719E-10, and CP = 5.0752351E-05 for time in seconds and stress in pounds per square inch. The default parameters in compression are nc = 0.78, with quadratic equation coefficients for η0c of AP = 1.2772337-11, BP = −1.0613722E-07, and CP = 3.203497-04. Rate effects parameters in pure shear stress are set equal to those in tension via Srate = 1.
The overstress limits in tension (overt) and compression (overc) limit rate effects at high strain rates ( > 100). The overstress quadratic equation coefficients for overt are AP = 1.309663E-02 MPa-1, BP = −0.3927659, and CP = 21.45 MPa. These provide tensile and compressive overstress limits of 21 MPa (3,046 psi) at an unconfined compression strength of 30 MPa (4,351 psi).
The literature contains conflicting information about whether fracture energy is strain rate dependent. One possibility is to model the fracture energy independent of strain rate (repow = 0). Another possibility is to increase the fracture energy with strain rate by multiplying the static fracture energy by the DIF (repow = 1). The developer's experience has been to increase the value of the fracture energy with strain rate; hence, repow = 1 is the default value. This value provides good correlations with test data for most of the problems analyzed and discussed in the companion concrete model evaluation report.(1) However, the Texas T4 bridge rail simulations correlate best with data if the fracture energy increases with the square root of the strain rate (repow = 0.5).
Figure 83. Graph. Approximate tensile and compressive dynamic increase
factors for default concrete model behavior.
Five systems of units are provided. These are: