Matrix or Vector | Shape Functions | Integration Points | |
---|---|---|---|
Stiffness Matrix and Thermal Load Vector | Equation 11–214, Equation 11–215, and Equation 11–216, or if modified extra shape functions are included (KEYOPT(1) = 0) and element has 8 unique nodes Equation 11–229, Equation 11–230, and Equation 11–231 | 2 x 2 x 2 | |
Mass Matrix | Equation 11–214, Equation 11–215, and Equation 11–216 | 2 x 2 x 2 | |
Pressure Load Vector | Quad | Equation 11–69 and Equation 11–70 | 2 x 2 |
Triangle | Equation 11–49 and Equation 11–50 | 3 |
Load Type | Distribution |
---|---|
Element Temperature | Trilinear thru element |
Nodal Temperature | Trilinear thru element |
Pressure | Bilinear across each face |
References: Willam and Warnke([37]), Wilson([38]), Taylor([49])
Cracking is permitted in three orthogonal directions at each integration point.
If cracking occurs at an integration point, the cracking is modeled through an adjustment of material properties which effectively treats the cracking as a “smeared band” of cracks, rather than discrete cracks.
The concrete material is assumed to be initially isotropic.
Whenever the reinforcement capability of the element is used, the reinforcement is assumed to be “smeared” throughout the element.
In addition to cracking and crushing, the concrete may also undergo plasticity, with the Drucker-Prager failure surface being most commonly used. In this case, the plasticity is done before the cracking and crushing checks.
SOLID65 allows the presence of four different materials within each element; one matrix material (e.g. concrete) and a maximum of three independent reinforcing materials. The concrete material is capable of directional integration point cracking and crushing besides incorporating plastic and creep behavior. The reinforcement (which also incorporates creep and plasticity) has uniaxial stiffness only and is assumed to be smeared throughout the element. Directional orientation is accomplished through user specified angles.
The stress-strain matrix [D] used for this element is defined as:
(13–117) |
where:
N_{r} = number of reinforcing materials (maximum of three, all reinforcement is ignored if M_{1} is zero. Also, if M_{1}, M_{2}, or M_{3} equals the concrete material number, the reinforcement with that material number is ignored) |
[D^{c}] = stress-strain matrix for concrete, defined by Equation 13–118 |
[D^{r}]_{i} = stress-strain matrix for reinforcement i, defined by Equation 13–119 |
M_{1}, M_{2}, M_{3} = material numbers associated of reinforcement (input as MAT1, MAT2, and MAT3 on R command) |
The matrix [D^{c}] is derived by specializing and inverting the orthotropic stress-strain relations defined by Equation 2–4 to the case of an isotropic material or
(13–118) |
where:
E = Young's modulus for concrete (input as EX on MP command) |
ν = Poisson's ratio for concrete (input as PRXY or NUXY on MP command) |
The orientation of the reinforcement i within an element is depicted in Figure 13.15: Reinforcement Orientation. The element coordinate system is denoted by (X, Y, Z) and describes the coordinate system for reinforcement type i. The stress-strain matrix with respect to each coordinate system has the form
(13–119) |
where:
It may be seen that the only nonzero stress component is , the axial stress in the direction of reinforcement type i. The reinforcement direction is related to element coordinates X, Y, Z through
(13–120) |
where:
θ_{i} = angle between the projection of the axis on XY plane and the X axis (input as THETA1, THETA2, and THETA3 on R command) |
φ_{i} = angle between the axis and the XY plane (input as PHI1, PHI2, and PHI3 on R command) |
= direction cosines between axis and element X, Y, Z axes |
Since the reinforcement material matrix is defined in coordinates aligned in the direction of reinforcement orientation, it is necessary to construct a transformation of the form
(13–121) |
in order to express the material behavior of the reinforcement in global coordinates. The form of this transformation by Schnobrich([29]) is
(13–122) |
where the coefficients a_{ij} are defined as
(13–123) |
The vector is defined by Equation 13–120 while and are unit vectors mutually orthogonal to thus defining a Cartesian coordinate referring to reinforcement directions. If the operations presented by Equation 13–121 are performed substituting Equation 13–119 and Equation 13–122, the resulting reinforcement material matrix in element coordinates takes the form
(13–124) |
where:
Therefore, the only direction cosines used in [D^{R}]_{i} involve the uniquely defined unit vector .
The matrix material (concrete) is capable of plasticity, creep, cracking and crushing. The concrete material model with its cracking and crushing capabilities is discussed in Concrete. This material model predicts either elastic behavior, cracking behavior or crushing behavior. If elastic behavior is predicted, the concrete is treated as a linear elastic material (discussed above). If cracking or crushing behavior is predicted, the elastic, stress-strain matrix is adjusted as discussed below for each failure mode.
The presence of a crack at an integration point is represented through modification of the stress-strain relations by introducing a plane of weakness in a direction normal to the crack face. Also, a shear transfer coefficient β_{t} (constant C_{1} with TB,CONCR) is introduced which represents a shear strength reduction factor for those subsequent loads which induce sliding (shear) across the crack face. The stress-strain relations for a material that has cracked in one direction only become:
(13–125) |
where the superscript ck signifies that the stress strain relations refer to a coordinate system parallel to principal stress directions with the x^{ck} axis perpendicular to the crack face. If KEYOPT(7) = 0, R^{t} = 0.0. If KEYOPT(7) = 1, R^{t} is the slope (secant modulus) as defined in the figure below. R^{t} works with adaptive descent and diminishes to 0.0 as the solution converges.
where:
f_{t} = uniaxial tensile cracking stress (input as C_{3} with TB,CONCR) |
T_{c} = multiplier for amount of tensile stress relaxation (input as C_{9} with TB,CONCR, defaults to 0.6) |
If the crack closes, then all compressive stresses normal to the crack plane are transmitted across the crack and only a shear transfer coefficient β_{c} (constant C_{2} with TB,CONCR) for a closed crack is introduced. Then can be expressed as
(13–126) |
The stress-strain relations for concrete that has cracked in two directions are:
(13–127) |
If both directions reclose,
(13–128) |
The stress-strain relations for concrete that has cracked in all three directions are:
(13–129) |
If all three cracks reclose, Equation 13–128 is followed. In total there are 16 possible combinations of crack arrangement and appropriate changes in stress-strain relationships incorporated in SOLID65. A note is output if 1 >β_{c} >β_{t} >0 are not true.
The transformation of to element coordinates has the form
(13–130) |
where [T^{ck}] has a form identical to Equation 13–122 and the three columns of [A] in Equation 13–123 are now the principal direction vectors.
The open or closed status of integration point cracking is based on a strain value called the crack strain. For the case of a possible crack in the x direction, this strain is evaluated as
(13–131) |
where:
The vector {ε^{ck}} is computed by:
(13–132) |
where:
{ε'} = modified total strain (in element coordinates) |
{ε'}, in turn, is defined as:
(13–133) |
where:
n = substep number |
{Δε_{n}} = total strain increment (based on {Δu_{n}}, the displacement increment over the substep) |
If is less than zero, the associated crack is assumed to be closed.
If is greater than or equal to zero, the associated crack is assumed to be open. When cracking first occurs at an integration point, the crack is assumed to be open for the next iteration.
If the material at an integration point fails in uniaxial, biaxial, or triaxial compression, the material is assumed to crush at that point. In SOLID65, crushing is defined as the complete deterioration of the structural integrity of the material (e.g. material spalling). Under conditions where crushing has occurred, material strength is assumed to have degraded to an extent such that the contribution to the stiffness of an element at the integration point in question can be ignored.