The piles are all modeled using OpenSees’ standard beam–column elements with a displacement– based formulation. The OpenSees designation for the beam–column elements used in the model is dispBeamColumn. These elements are able to include distributed plasticity. In- tegration within the elements is based on the Gauss–Legendre quadrature rule (Mazzoni et al., 2007 [17]). For the elastoplastic pile models, the material nonlinearity of each pile is included in the elements through the use of fiber section models, based upon the tem- plate pile cross–sections. The elastic pile models use elastic section models along with the dispBeamColumn elements. These elastic section models are also based off of the template pile cross–sections. No geometric nonlinearity is considered as it is assumed that the defor- mations are not sufficiently large as to introduce any significant geometric effects into the behavior of the piles. All of the pile nodes have six degrees–of–freedom (3 translational, 3 rotational), of which three (1 translational, 2 rotational) are restrained due to symmetry conditions.
3.3.1 Fiber Section Models
Each of the three pile designs display moment–curvature behavior based upon the strength, distribution, and size of the concrete and steel available in their cross–section. In order to incorporate the unique behavior of each pile into a finite element model, fiber section models were created for each pile design. These fiber section models incorporate all of the relevant aspects of the individual pile designs, and define the behavior of each pile model at a cross–sectional level. In order to obtain consistent results for the symmetry conditions imposed upon the soil mesh in the 3D lateral spreading model, only one–half of each pile is incorporated, resulting in semi–circular fiber section models. The reasoning behind this decision is discussed further in Section 3.3.2.
Fiber section models are effective when modeling composite materials, such as reinforced concrete piles, hence their employment in this study. Fiber sections have a geometry de- fined in two levels; an overall geometry, in this case semi–circular, within which exists many smaller subregions of regular shape. A circular shape, or any sector thereof, lends itself
divisions
Angular Radial
divisions
Figure 3.3: Typical fiber discretization for a circular fiber section model.
well to the type of discretization framework shown in Figure 3.3 which includes a set of divisions set at even intervals along the radius of the circle, as well as divisions that are set at equal angular increments. Reinforcement steel can be included in the fiber model as in- dividual fiber regions with appropriate areas and locations. This type of fiber discretization is employed in all of the fiber models created for this research.
Each subregion of the fiber model can be assigned its own unique uniaxial constitutive model. In the case of reinforced concrete, it would follow that the subregions representing the reinforcement are given the material behavior of steel and the surrounding subregions representing the concrete portion of the cross–section are assigned constitutive behavior based upon that of concrete. Through defining the cross–section in this manner, a composite constitutive behavior is achieved for the pile elements. For a given increment of displacement in an element defined using fiber section models, the axial strain and stress in each subregion are calculated based upon location and material properties. These values are combined with the results from the other subregions within the fiber element in order to return the force and moment vectors and axial and rotational tangent stiffness at each Gauss point to the element for use in the next iteration. This modeling technique approximates the three–dimensional behavior of an actual pile while still enabling the use of standard beam elements.
3.3.2 Pile Symmetry and Boundary Conditions
A single, circular, pile embedded in a mass of soil is an inherently symmetric problem. Whenever modeling a problem where symmetry is involved, it is computationally efficient to create a model that takes advantage of this fact. As shown in Figure 2.1, this method is employed in the 3D lateral spreading model. Only one–half of the soil mass surrounding the pile is included in the model, the other half is replaced by appropriate boundary conditions on the soil nodes located where the interface between the two halves would be. This approach greatly reduces the amount of computational effort required to solve the system of equations, without compromising the validity in the results.
In order to create consistency in the results of the model, the fiber section models must account for the symmetry conditions imposed upon the soil. To accomplish this end, each pile model must have one–half of the area and bending stiffness of the corresponding circular pile, assuring that the interaction between the soil and the pile will be appropriate. This is accomplished through the use of semi–circular fiber section models that only consider one half of each pile cross–section. Using this approach, the resulting pile shear and moment data obtained from the simulation correctly reflects the symmetry condition. Results that are applicable to a full, circular, pile can be obtained by simply multiplying the semi–circular pile forces and moments by a factor of two.
As for boundaries, the pile nodes are held fixed against translation perpendicular to the symmetry plane. Additionally, since the semi–circular cross–sections are not doubly– symmetric, the pile nodes must be held fixed with respect to rotations about the pile axis to prevent the sections from twisting during loading. All of the pile nodes are fixed against out–of–plane rotations as well (rotation axis lies in the symmetry plane). These are all required symmetry conditions. The base node of the pile is held fixed against translation in the vertical direction (parallel to the pile axis), a required stability condition. The pile is allowed to rotate in the plane of loading at each node with the exception of cases where a fixed support condition is imposed at the head of the pile. Fixing the pile head is representative of a connection to a pile cap or other structural component. Both fixed and free pile head conditions are considered.