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Variational Principles and Analytical Dynamics
Published in Haym Benaroya, Mark Nagurka, Seon Han, Mechanical Vibration, 2017
Haym Benaroya, Mark Nagurka, Seon Han
The linking of Newton’s second law of motion with the principle of virtual work means that the principle is applicable to particles at rest and to particles in motion. The virtual displacement involves a possible but purely mathematical (virtual) experiment that can be applied at any specific time. At that instant, the actual motion of the body is not at issue as the dynamic problem is reduced to a static one.
Variational and Related Methods
Published in K.T. Chau, Theory of Differential Equations in Engineering and Mechanics, 2017
This is called the principle ofvirtual displacement. When the body is elastic, (14.7) is related to Castigliano’s first theorem. In other words, if a body is in equilibrium, the total virtual work done is zero. In structural mechanics, the principle of virtual displacement is normally used to find the real force or stress of a body.
Flexural modeling and design
Published in Alva Peled, Barzin Mobasher, Arnon Bentur, Textile Reinforced Concrete, 2017
Alva Peled, Barzin Mobasher, Arnon Bentur
Using the principle of virtual work, it is stated that if the system is in equilibrium, and a kinematically admissible virtual displacement is imposed on the system, the work by the external forces on the virtual displacement is equal to the work done by the real internal stresses on the virtual strains. The virtual work expression can be written as a work balance equation using the virtual and real parameters. Indeed, one can use the principle of virtual work to equate the external and internal work measures. In the case of a simple beam subjected to force F, the internal plastic moment M developed can be used, as F and M represent the force and plastic bending moment, while δΔ and δn represent the virtual displacement and associated rotation as shown in Figure 7.23a: Wint=Wext∑FδΔ=∫0LMPδϕdx Flexural capacity of a simply supported beam subjected to concentrated load as shown in Figure 7.16 is developed first and then extended to a distributed load case. A plastic analysis methodology uses the principle of virtual work to equate the internal and external dissipated work to obtain the collapse load. The work equations are derived based on the concept presented in Figure 7.16: ∑F×Δ=∑M×L×θPuΔ=MP2θ=MP2Δ0.5LPu=4MPL
Real-time Structural Stability of Domes through Limit Analysis: Application to St. Peter’s Dome
Published in International Journal of Architectural Heritage, 2023
Marco Francesco Funari, Luis Carlos Silva, Elham Mousavian, Paulo B. Lourenço
A formulation based on the static theorem of limit analysis allows tracing the statically admissible funicular thrust network of the load. In the incipient collapse state, the thrust line passes by the extrados of the key section and a rotational hinge is expected to appear. The point at which the thrust line intersects the dome’s intrados defines the second hinge location. From a computational implementation standpoint, a kinematic theorem of limit analysis was, however, assumed in this study. In particular, the theoretical background of the developed digital analysis tool is based on Como 2013. According to Como (2013), the failure mechanism that allows describing the kinematic problem is presented in Figure 8a. The principle of virtual work is, therefore, adopted and the mechanism can be kinematically described through a unique parameter, the virtual displacement . Multiple failure mechanisms — defined by the rotational hinge position along the dome’s intrados — need to be considered to evaluate the minimum of the kinematically compatible load multipliers, for which an optimization routine is used. The equation of the virtual work for the considered failure mechanism reads as:
Non-polynomial hybrid models for the bending of magneto-electro-elastic shells
Published in Mechanics of Advanced Materials and Structures, 2023
Joao C. Monge, Jose Luis Mantari, Miguel A. Hinostroza
A very simple manner for evaluating different 2D models for shells and selecting different through-the-thickness functions is achieved by Carrera’s Unified Formulation [14, 22]. The mechanical displacements are described in terms of an Equivalent Single Layer, in which all the layers are considered as one. The electric and magnetic potentials are described in terms of a Layerwise approach in which each layer is considered separately from the other. The mechanical displacements are defined as a vector and its variational form Einstein’s notation is used for writing the mechanical displacement as: The corresponding thickness functions are described as The order of the expansion is The summation notation according to Einstein’s notation is considered as and The super index represents each layer of the shell. The virtual variational is used to formulate the Principle of Virtual Displacement. For an order of expansion equal to the displacements written in the framework of CUF are expressed as:
Modeling of Formation and Evolution of Cracks in Zirconium-Based Claddings of Nuclear Fuel Rods Within DIONISIO 3.0
Published in Nuclear Science and Engineering, 2021
Ezequiel Goldberg, Alejandro Soba
In this work, we present a new fracture mechanics module based on the CZM that was developed for the DIONISIO nuclear fuel code. As a brief description, the model stems from the weak formulation in which both the cohesive and equilibrium expressions are derived from the principle of virtual work. If a virtual displacement field is applied, the external virtual work must be equal to the internal one, the former being the product of the external forces and the virtual displacements and the latter being the product of the stress and the strain. The left side of the equation, which encompasses the internal work, comprises the sum between the strain energy and the cohesive energy of fracture, as deduced by Eq. (1):