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Unit Dummy Force Method
The Unit dummy force method provides a convenient means for computing displacements in structural systems. It is applicable for both linear and non-linear material behaviours as well as for systems subject to environmental effects, and hence more general than Castigliano's second theorem. Discrete systems Consider a discrete system such as trusses, beams or frames having members interconnected at the nodes. Let the consistent set of members' deformations be given by \mathbf_ , which can be computed using the member flexibility relation. These member deformations give rise to the nodal displacements \mathbf_ , which we want to determine. We start by applying ''N'' virtual nodal forces \mathbf^*_ , one for each wanted ''r'', and find the virtual member forces \mathbf^*_ that are in equilibrium with \mathbf^*_ : In the case of a statically indeterminate system, matrix B is not unique because the set of \mathbf^*_ that satisfies nodal equilibrium is infinite. It can be computed a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Castigliano's Method
Castigliano's method, named after Carlo Alberto Castigliano, is a method for determining the displacements of a linear-elastic system based on the partial derivatives of the energy. He is known for his two theorems. The basic concept may be easy to understand by recalling that a change in energy is equal to the causing force times the resulting displacement. Therefore, the causing force is equal to the change in energy divided by the resulting displacement. Alternatively, the resulting displacement is equal to the change in energy divided by the causing force. Partial derivatives are needed to relate causing forces and resulting displacements to the change in energy. * Castigliano's first theorem – for forces in an elastic structure Castigliano's method for calculating forces is an application of his first theorem, which states: :''If the strain energy of an elastic structure can be expressed as a function of generalised displacement qi then the partial derivative of the s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Flexibility Method
In structural engineering, the flexibility method, also called the method of consistent deformations, is the traditional method for computing member forces and displacements in structural systems. Its modern version formulated in terms of the members' flexibility matrices also has the name the matrix force method due to its use of member forces as the primary unknowns. Member flexibility Flexibility is the inverse of stiffness. For example, consider a spring that has ''Q'' and ''q'' as, respectively, its force and deformation: * The spring stiffness relation is ''Q = k q'' where ''k'' is the spring stiffness. * Its flexibility relation is ''q = f Q'', where ''f'' is the spring flexibility. * Hence, ''f'' = 1/''k''. A typical member flexibility relation has the following general form: where :''m'' = member number ''m''. :\mathbf^m = vector of member's characteristic deformations. :\mathbf^m = member flexibility matrix which characterises the member's susceptibility to deform u ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Virtual Work
In mechanics, virtual work arises in the application of the ''principle of least action'' to the study of forces and movement of a mechanical system. The work of a force acting on a particle as it moves along a displacement is different for different displacements. Among all the possible displacements that a particle may follow, called virtual displacements, one will minimize the action. This displacement is therefore the displacement followed by the particle according to the principle of least action. The work of a force on a particle along a virtual displacement is known as the virtual work. Historically, virtual work and the associated calculus of variations were formulated to analyze systems of rigid bodies, but they have also been developed for the study of the mechanics of deformable bodies. History The principle of virtual work had always been used in some form since antiquity in the study of statics. It was used by the Greeks, medieval Arabs and Latins, and Renais ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |