NEED TO ADD ADD_INERTIA_LOAD TO INTERFACE .. SEE EARTHQUAKE_PATTERN CLASS.
BeamWithHinges2D
#include <element/BeamWithHinges2D.h>
class BeamWithHinges2D: public DomainComponent
TaggedObject
MovableObject
DomainComponent
Element
BeamWithHinges2D is a beam-column element which uses the force based formulation for its state determination. This element has material non-linear hinges at both ends and exhibits linear elastic behavior through its interior region, including linear elastic shear effects.
// Constructors
// Destructor
// Public Methods dealing with Nodes and dof
// Public Methods dealing with State
// Public Methods for obtaining Linearized Stiffness, Mass and Damping Matrices
// Public Methods for obtaining Resisting Forces
Constructs a BeamWithHinges2D object with tag tag, unique among all elements in the domain. The end nodes of the element are set to be those with tags nodeI and nodeJ. The elastic properties of the interior beam region are set to be E, the modulus of elasticity; I, the second moment of area of the beam cross-section; A, the beam cross-sectional area; G, the shear modulus; and alpha, the cross-section shape factor for elastic shear effects. Element hinges are created by obtaining copies of sectionI and sectionJ. The hinge lengths are specified as ratios of the total element length, ratioI and ratioJ. The element distributed load (reference value) is set to be distrLoad, and the element mass density per unit length is massDens.
Constructs a BeamWithHinges2D object with tag tag, unique among all elements in the domain. The end nodes of the element are set to be those with tags nodeI and nodeJ. The elastic properties of the interior beam region are set to be E, the modulus of elasticity; I, the second moment of area of the beam cross-section; A, the beam cross-sectional area; G, the shear modulus; and alpha, the cross-section shape factor for elastic shear effects. Element hinges are created by obtaining copies of sectionI and sectionJ. The hinge lengths are specified as ratios of the total element length, ratioI and ratioJ. The element mass density per unit length is massDens.
Constructs a blank BeamWithHinges2D object.
Invokes the section destructors.
Returns 2, the number of external nodes for this element.
Returns an ID containing the tags of the two external nodes for this element.
Returns 6, the number of degrees of freedom for this element.
int commitState(void);
Invokes commitState() on each section and returns the sum of the result of these invocations.
int revertToLastCommit(void);
Invokes revertToLastCommit() on each section and returns the sum of the result of these invocations.
int revertToStart(void);
Invokes revertToStart() on each section and returns the sum of the result of these invocations.
const Matrix &getTangentStiff(void);
Computes the element flexibility matrix, then returns its inverse, the element stiffness matrix. The element flexibility is the sum of the hinge flexibilities, \(\mathbf{f}_I\) and \(\mathbf{f}_J\), and the elastic flexibility of the beam interior, \(\mathbf{f}_{mid}\).
\[\label{eq:fele} \fbas := \int_{0}^{L}{\bint^T \fsec \bint \: dx} = \mathbf{f}_I + \mathbf{f}_{mid} + \mathbf{f}_J\]
The flexibility of the beam interior is obtained in closed form,
\[\mathbf{f}_{mid} = \int_{l_I}^{L-l_J}{\bint^T \fsec_{mid} \bint \: dx}\]
where \(\bint\) is the force interpolation matrix,
\[\bint := \left[ \begin{array}{c c c} 1 & 0 & 0 \\ 0 & \frac{x}{L} & \frac{x}{L} - 1 \\ 0 & \frac{1}{L} & \frac{1}{L} \end{array} \right]\]
and \(\fsec\) is the elastic section flexibility of the beam interior.
\[\fsec_{mid} = \left[ \begin{array}{c c c} \frac{1}{EA} & 0 & 0 \\ 0 & \frac{1}{EI} & 0 \\ 0 & 0 & \frac{1}{\alpha GA} \end{array} \right]\]
The hinge flexibilities, \(\mathbf{f}_I\) and \(\mathbf{f}_J\), are obtained by the midpoint integration rule,
\[{\mathbf{f}}_i = \bint(x_i)^T \fsec_i \bint(x_i) * l_i, \:\: i=I,J\]
where \(x_i\) is the midpoint of hinge \(i\), measured along the length of the beam, and is the point at which the force interpolation matrix, \(\bint\) is evaluated. The flexiblity, \(\fsec_i\), is obtained from the constitutive relation for section \(i\). The element stiffness is then obtained by inversion of the element flexibility, given by Equation [eq:fele].
\[\label{eq:kele} \kbas = \fbas^{-1}\]
The element then obtains the matrix, \(\mathbf{A}\), which transforms the element basic stiffness from its corotating frame to the global frame of reference. The transformed stiffness matrix, \(\kele\), is then assembled into the structural system of equations.
\[\kele = \mathbf{A}^T \kbas \mathbf{A}\]
const Matrix &getSecantStiff(void);
To return the elements secant stiffness matrix. THIS SECANT MAY BE REMOVED.
const Matrix &getDamp(void);
To return the damping matrix.
const Matrix &getMass(void);
Returns the element lumped mass matrix, \(\mele\). It is assumed that the mass density per unit length, \(\rho\), is constant along the entire element, including the hinge regions.
\[\label{eq:mele} \mele = \left[ \begin{array}{c c c c c c} \frac{\rho L}{2} & 0 & 0 & 0 & 0 & 0 \\ 0 & \frac{\rho L}{2} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & \frac{\rho L}{2} & 0 & 0 \\ 0 & 0 & 0 & 0 & \frac{\rho L}{2} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{array} \right]\]
void zeroLoad(void);
This is a method invoked to zero the element load contributions to the residual.
const Vector &getResistingForce(void);
Returns the element resisting force vector. The basic element force vector is obtained as the product of the basic element stiffness, \(\kbas\), given by Equation [eq:kele], and the basic element deformations, \(\vbas\).
\[\qbas = \kbas \vbas\]
The basic element force vector is then transformed from the corotating frame to the global frame of reference. The transformed force vector is then assembled into the structural system of equations.
\[\label{eq:qele}
\qele = \mathbf{A}^T \qbas\]
const Vector &getResistingForceIncInertia(void);
Returns the element resisting force vector, \(\tilde{\qele}\) with inertia forces included,
\[\tilde{\qele} = \qele - \mele \ddot{\mathbf u}\]
where \(\qele\) and \(\mele\) are obtained from Equations [eq:qele] and [eq:mele], respectively, and \(\ddot{\mathbf u}\) is the vector of trial nodal accelerations for the element.