Background

The HDRBYamamoto3DLink class creates a bi-axial high-damping rubber bearing element with two-nodes in three-dimensions. The number of degree-of-freedom depend on the node. Currently node with 3 (three translation) and 6 (three translation and three rotation) dofs are supported. The main purpose of this element in 3D is to force a bi-axial behavior along the orthogonal direction to local axis 1. Figure provides a simple representation of this element where (1) and (2) represent the start and end nodes.

The linkHDRBYamamoto3DLink mass matrix in global coordinates is \(\textbf{M}^\textrm{e} = 0\). On the other hand, the linkHDRBYamamoto3DLink solves the differential equation:

\[ \dot{\textbf{q}} = \frac{\dot{\textbf{p}}}{\alpha} \left( \hat{\dot{\textbf{p}}} - \|\textbf{q}\|^n \, \bar{\textbf{q}} \right) \,, \]

and the force is computed as

\[F(\textbf{p},\textbf{q}) = -\hat{\textbf{p}} \, F_r - \textbf{q} \, F_s\,, \]

where \(F_r = \tau_r\,A\), \(F_s = \tau_s\,A\), \(A\) is the rubber area, and

\[ \tau_r = \begin{cases} 0.22\,\gamma,& 0\leq \gamma \leq 1.8 \\ 0.22\,\gamma + 0.20(\gamma-1.8)^2, & \gamma > 1.8 \end{cases} \]

\[ \tau_s = 0.25 + 0.02\,\gamma + 0.016\,\gamma^3\,. \]

Note that the coefficients provided in previous equation are such that units in \(\tau_r\) and \(\tau_s\) are [MPa]. The local to global transformation matrix is

\[ \textbf{T}_\textrm{e} = \begin{bmatrix} \textbf{N} & 0 \\ 0 & \textbf{N} \end{bmatrix} \,, \; \text{ with } \textbf{N} = \begin{bmatrix} \hat{\textbf{N}} & 0 \\ 0 & \hat{\textbf{N}} \end{bmatrix} \,, \]

and \(\hat{\textbf{N}} = [\hat{\textrm{n}}_1, \hat{\textrm{n}}_2, \hat{\textrm{n}}_3]^\top\) is the unit matrix of local coordinate axes, and \(\hat{\textrm{n}}_\textrm{i}\) is the unit vector of the i-th local coordinate axis. Note that \(\hat{\textbf{N}} \in \mathbb{R}^{3 \times 3}\), and \(\hat{\textrm{n}}_\textrm{i} \in \mathbb{R}^3\). Note that \(\textbf{M}^\textrm{e}, \textbf{K}^\textrm{e} \in \mathbb{R}^{\textrm{N}_\textrm{dof}^\textrm{e} \times \textrm{N}_\textrm{dof}^\textrm{e}}\) with \(\textrm{N}_\textrm{dof}^\textrm{e} = 12\) in this case.

The stiffness matrices as well as the force vector in global coordinates are defined as:

\[ \textbf{K}^\textrm{e} = \textbf{T}_\textrm{e}^\top \, \mathcal{K}^\textrm{e} \, \textbf{T}_\textrm{e} \,, \; \textbf{F}^\textrm{e} = \textbf{T}_\textrm{e}^\top \, \mathcal{F}^\textrm{e} \,, \]

and \(\hat{\textbf{N}} = [\hat{\textrm{n}}_1, \hat{\textrm{n}}_2, \hat{\textrm{n}}_3]^\top\) is the unit matrix of local coordinate axes, and \(\hat{\textrm{n}}_\textrm{i}\) is the unit vector of the i-th local coordinate axis. Note that \(\hat{\textbf{N}} \in \mathbb{R}^{3 \times 3}\), and \(\hat{\textrm{n}}_\textrm{i} \in \mathbb{R}^3\). Note that \(\textbf{M}^\textrm{e}, \textbf{K}^\textrm{e} \in \mathbb{R}^{\textrm{N}_\textrm{dof}^\textrm{e} \times \textrm{N}_\textrm{dof}^\textrm{e}}\) with \(\textrm{N}_\textrm{dof}^\textrm{e} = 6, \text{ or } 12\) (solid or structural).

REFERENCE:

Yamamoto, Minewaki, Yoneda, and Higashino (2012), "Nonlinear behavior of high-damping rubber bearings under horizontal bidirectional loading: full-scale tests and analytical modeling". Earthquake Engng Struct. Dyn

Pre-Analysis

The python Pre-Analysis in the 01-Pre_Process/Method/Attach.py file provides with an interface to populate the Entities dictionary. This file contains several functions to populate specific fields. For example, to create a HDRBYamamoto3DLink, using json format, use:

addElement(tag, name='HDRBYamamoto3DLink', conn, attributes):
- tag : The identifier of this element, i.e., tag > -1
- name : Seismo-VLAB element class name
- conn : Connectivity array of this element
- attributes : Specific properties for the created element, for example
  - 'De' External rubber diameter.
  - 'Di' Internal rubber diameter.
  - 'Hr' Rubber bearing height.
Example

A HDRBYAMAMOTO3DLINK element can be defined using the python interface as follows:
SVL.addElement(tag=1, name='HDRBYamamoto3DLink', conn=[1,2], attributes={'De': 1.3,'Di': 0.3,'Hr': 0.216})

Application
Please refer to the A13-DY_3D_UniAxial_YamamotoHDRB_Link.py file located at 03-Validations/01-Debugging/ to see an example on how to define a HDRBYamamoto3DLink element.

On the contrary, the 01-Pre_Process/Method/Remove.py file provides with an interface to depopulate the Entities dictionary. For example, to remove an already define Element, use:

delElement(tag):
- tag : The identifier of the element to be removed, i.e., tag > -1

Run-Analysis

The C++ Run-Analysis in the 02-Run_Process/04-Elements/02-Link/HDRBYamamoto3DLink.cpp file provides the class implementation. A HDRBYamamoto3DLink element is created using the built-in json parse-structure provided in the Driver.hpp. A HDRBYamamoto3DLink is defined inside the "Elements" json field indicating its "Tag" as follows,

{
    "Elements": {
        "Tag": {
            "name" : "HDRBYAMAMOTO3DLINK",
            "conn" : [ ],
            "attributes": {
                "De": double,
                "Di": double,
                "Hr": double,
                "dim": int
            }
        }
    }
}

Variable	Description
`Tag`	Unique positive element identifier.
`conn`	The element connectivity node array.
`De`	External diameter of rubber.
`Di`	Internal diameter of rubber.
`Hr`	Total rubber height.
`dim`	The number of degree of freedom of the nodes (2, or 3).

Attention: The element acts only in two direction (perpendicular to local axis 1).; The element provides no-contribution to the mass matrix.; The element only provides contribution to the stiffness matrix.; The length of the link is not used.; The units uses the international system (IS).; Internal force responses are provided in global coordinates.; Strain and Stress link responses are relative deformation and non-linear force and in local coordinates respectively.

Example

A HDRBYAMAMOTO3DLINK element between nodes 1 and 2 with 2 dofs in a 3D problem is constructed as:
{ "Elements": { "1": { "name" : "HDRBYAMAMOTO3DLINK", "conn" : [1,2], "attributes": { "De": 1.3, "Di": 0.3, "Hr": 0.216, "dim": 3 } } } }