Chapter 12: Beams Flashcards
Discuss the Finite Volume Model
In its linearized form it produces a stiffness matric that is not symmetic, although being positive definite and equivalent to that of finite elements.
Considering the obtained indefinite equilibrium equations of the beam and a set of scalar piecewise-constant weight functions (heaviside functions) defined over the domain of the beam, each associated with a node along the span of the beam.
Consider the weighted residuals weak form of the equilibrium equations (integrals of weights multiplied with the indefinite equilibrium equations)
This method corresponds to splitting the beam in three parts, each associated with one of the three nodes, with two cuts t stations ξ1 and ξ2. The distributed loads acting at each portion are integrating and applied to the corresponding node. The internal forces ad the boundaries are moot: they cancel if the beam continues with another element, or vanish in the case of a free end. The internal forces at stations xi1, xi2 are expressed as functions of the genreralized linear adn angular straints through constitutive properties.
In turb, the generalized linear and angular straints are obtained by parabolic interpolation, based on the kinematics of the three nodes.
This methos is shear lock free.
An interesting aspect of this method is that the forces and moments originating from the element are truly forces and moments energetically conjugated to the virtual displacement and rotation of the nodes they are applied to. They are not generalized forces, namely forces energetically conjugated to some generalized parameter (eg virtual peturbations of rotation parameters).
