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Kirchhoff–Love Plate Theory
The Kirchhoff–Love theory of plates is a two-dimensional mathematical model that is used to determine the stresses and deformations in thin plates subjected to forces and moments. This theory is an extension of Euler-Bernoulli beam theory and was developed in 1888 by LoveA. E. H. Love, ''On the small free vibrations and deformations of elastic shells'', Philosophical trans. of the Royal Society (London), 1888, Vol. série A, N° 17 p. 491–549. using assumptions proposed by Kirchhoff. The theory assumes that a mid-surface plane can be used to represent a three-dimensional plate in two-dimensional form. The following kinematic assumptions that are made in this theory:Reddy, J. N., 2007, Theory and analysis of elastic plates and shells, CRC Press, Taylor and Francis. * straight lines normal to the mid-surface remain straight after deformation * straight lines normal to the mid-surface remain normal to the mid-surface after deformation * the thickness of the plate does not change ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Plaque Mince Deplacement Element Matiere
Plaque may refer to: Commemorations or awards * Commemorative plaque, a plate or tablet fixed to a wall to mark an event, person, etc. * Memorial Plaque (medallion), issued to next-of-kin of dead British military personnel after World War I * Plaquette, a small plaque in bronze or other materials Science and healthcare * Amyloid plaque * Atheroma or atheromatous plaque, a buildup of deposits within the wall of an artery * Dental plaque, a biofilm that builds up on teeth * A broad papule, a type of cutaneous condition * Pleural plaque, associated with mesothelioma, cancer often caused by exposure to asbestos * Senile plaques, an extracellular protein deposit in the brain implicated in Alzheimer's disease * Skin plaque, a plateau-like lesion that is greater in its diameter than in its depth * Viral plaque, a visible structure formed by virus propagation within a cell culture Other uses * Plaque, a rectangular casino token See also * * * Builder's plate * Plac (other) * ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Theodore Von Kármán
Theodore von Kármán ( hu, ( szőllőskislaki) Kármán Tódor ; born Tivadar Mihály Kármán; 11 May 18816 May 1963) was a Hungarian-American mathematician, aerospace engineer, and physicist who was active primarily in the fields of aeronautics and astronautics. He was responsible for many key advances in aerodynamics, notably on supersonic and hypersonic airflow characterization. He is regarded as an outstanding aerodynamic theoretician of the 20th century. Early life Theodore von Kármán was born into a Jewish family in Budapest, Austria-Hungary, as Kármán Tódor, the son of Helen (Kohn, hu, Kohn Ilka) and Mór Kármán. One of his ancestors was Rabbi Judah Loew ben Bezalel. He studied engineering at the city's Royal Joseph Technical University, known today as Budapest University of Technology and Economics. After graduating in 1902 he moved to the German Empire and joined Ludwig Prandtl at the University of Göttingen, where he received his doctorate in 1908. He taug ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vibration Of Plates
The vibration of plates is a special case of the more general problem of mechanical vibrations. The equations governing the motion of plates are simpler than those for general three-dimensional objects because one of the dimensions of a plate is much smaller than the other two. This suggests that a two-dimensional plate theory will give an excellent approximation to the actual three-dimensional motion of a plate-like object, and indeed that is found to be true.Reddy, J. N., 2007, Theory and analysis of elastic plates and shells, CRC Press, Taylor and Francis. There are several theories that have been developed to describe the motion of plates. The most commonly used are the Kirchhoff-Love theory and the Uflyand-Mindlin. The latter theory is discussed in detail by Elishakoff. Solutions to the governing equations predicted by these theories can give us insight into the behavior of plate-like objects both under free and forced conditions. This includes the propagation of waves ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hamilton's Principle
In physics, Hamilton's principle is William Rowan Hamilton's formulation of the principle of stationary action. It states that the dynamics of a physical system are determined by a variational problem for a functional based on a single function, the Lagrangian, which may contain all physical information concerning the system and the forces acting on it. The variational problem is equivalent to and allows for the derivation of the '' differential'' equations of motion of the physical system. Although formulated originally for classical mechanics, Hamilton's principle also applies to classical fields such as the electromagnetic and gravitational fields, and plays an important role in quantum mechanics, quantum field theory and criticality theories. Mathematical formulation Hamilton's principle states that the true evolution of a system described by generalized coordinates between two specified states and at two specified times and is a stationary point (a point where th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Action (physics)
In physics, action is a scalar quantity describing how a physical system has dynamics (physics), changed over time. Action is significant because the equations of motion of the system can be derived through the principle of stationary action. In the simple case of a single particle moving with a constant velocity (uniform linear motion), the action is the momentum of the particle times the distance it moves, integral (mathematics), added up along its path; equivalently, action is twice the particle's kinetic energy times the duration for which it has that amount of energy. For more complicated systems, all such quantities are combined. More formally, action is a functional (mathematics), mathematical functional which takes the trajectory (also called path or history) of the system as its argument and has a real number as its result. Generally, the action takes different values for different paths. Action has dimensional analysis, dimensions of energy × time or momentu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bending Of Plates
Bending of plates, or plate bending, refers to the deflection of a plate perpendicular to the plane of the plate under the action of external forces and moments. The amount of deflection can be determined by solving the differential equations of an appropriate plate theory. The stresses in the plate can be calculated from these deflections. Once the stresses are known, failure theories can be used to determine whether a plate will fail under a given load. Bending of Kirchhoff-Love plates Definitions For a thin rectangular plate of thickness H, Young's modulus E, and Poisson's ratio \nu, we can define parameters in terms of the plate deflection, w. The flexural rigidity is given by : D = \frac Moments The bending moments per unit length are given by : M_ = -D \left( \frac + \nu \frac \right) : M_ = -D \left( \nu \frac + \frac \right) The twisting moment per unit length is given by : M_ = -D \left( 1 - \nu \right) \frac Forces The shear force ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Biharmonic Equation
In mathematics, the biharmonic equation is a fourth-order partial differential equation which arises in areas of continuum mechanics, including linear elasticity theory and the solution of Stokes flows. Specifically, it is used in the modeling of thin structures that react elastically to external forces. Notation It is written as :\nabla^4\varphi=0 or :\nabla^2\nabla^2\varphi=0 or :\Delta^2\varphi=0 where \nabla^4, which is the fourth power of the del operator and the square of the Laplacian operator \nabla^2 (or \Delta), is known as the biharmonic operator or the bilaplacian operator. In Cartesian coordinates, it can be written in n dimensions as: : \nabla^4\varphi=\sum_^n\sum_^n\partial_i\partial_i\partial_j\partial_j \varphi =\left(\sum_^n\partial_i\partial_i\right)\left(\sum_^n \partial_j\partial_j\right) \varphi. Because the formula here contains a summation of indices, many mathematicians prefer the notation \Delta^2 over \nabla^4 because the former makes clear ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pure Bending
Pure bending ( Theory of simple bending) is a condition of stress where a bending moment is applied to a beam without the simultaneous presence of axial, shear, or torsional forces. Pure bending occurs only under a constant bending moment (M) since the shear force (V), which is equal to \frac = V, has to be equal to zero. In reality, a state of pure bending does not practically exist, because such a state needs an absolutely weightless member. The state of pure bending is an approximation made to derive formulas. Kinematics of pure bending #In pure bending the axial lines bend to form circumferential lines and transverse lines remain straight and become radial lines. #Axial lines that do not extend or contract form a neutral surface. Assumptions made in the theory of Pure Bending #The material of the beam is homogeneous1 and isotropic2. #The value of Young's Modulus of Elasticity An elastic modulus (also known as modulus of elasticity) is the unit of measurement of an object's ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Young's Modulus
Young's modulus E, the Young modulus, or the modulus of elasticity in tension or compression (i.e., negative tension), is a mechanical property that measures the tensile or compressive stiffness of a solid material when the force is applied lengthwise. It quantifies the relationship between tensile/compressive stress \sigma (force per unit area) and axial strain \varepsilon (proportional deformation) in the linear elastic region of a material and is determined using the formula: E = \frac Young's moduli are typically so large that they are expressed not in pascals but in gigapascals (GPa). Example: * Silly Putty (increasing pressure: length increases quickly, meaning tiny E) * Aluminum (increasing pressure: length increases slowly, meaning high E) Higher Young's modulus corresponds to greater (lengthwise) stiffness. Although Young's modulus is named after the 19th-century British scientist Thomas Young, the concept was developed in 1727 by Leonhard Euler. The first experime ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Poisson's Ratio
In materials science and solid mechanics, Poisson's ratio \nu ( nu) is a measure of the Poisson effect, the deformation (expansion or contraction) of a material in directions perpendicular to the specific direction of loading. The value of Poisson's ratio is the negative of the ratio of transverse strain to axial strain. For small values of these changes, \nu is the amount of transversal elongation divided by the amount of axial compression. Most materials have Poisson's ratio values ranging between 0.0 and 0.5. For soft materials, such as rubber, where the bulk modulus is much higher than the shear modulus, Poisson's ratio is near 0.5. For open-cell polymer foams, Poisson's ratio is near zero, since the cells tend to collapse in compression. Many typical solids have Poisson's ratios in the range of 0.2–0.3. The ratio is named after the French mathematician and physicist Siméon Poisson. Origin Poisson's ratio is a measure of the Poisson effect, the phenomenon in which a ma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Plaque Moment Torsion Contrainte New
Plaque may refer to: Commemorations or awards * Commemorative plaque, a plate or tablet fixed to a wall to mark an event, person, etc. * Memorial Plaque (medallion), issued to next-of-kin of dead British military personnel after World War I * Plaquette, a small plaque in bronze or other materials Science and healthcare * Amyloid plaque * Atheroma or atheromatous plaque, a buildup of deposits within the wall of an artery * Dental plaque, a biofilm that builds up on teeth * A broad papule, a type of cutaneous condition * Pleural plaque, associated with mesothelioma, cancer often caused by exposure to asbestos * Senile plaques, an extracellular protein deposit in the brain implicated in Alzheimer's disease * Skin plaque, a plateau-like lesion that is greater in its diameter than in its depth * Viral plaque, a visible structure formed by virus propagation within a cell culture Other uses * Plaque, a rectangular casino token See also * * * Builder's plate * Plac (other) * ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Plaque Moment Flechissant Contrainte New
Plaque may refer to: Commemorations or awards * Commemorative plaque, a plate or tablet fixed to a wall to mark an event, person, etc. * Memorial Plaque (medallion), issued to next-of-kin of dead British military personnel after World War I * Plaquette, a small plaque in bronze or other materials Science and healthcare * Amyloid plaque * Atheroma or atheromatous plaque, a buildup of deposits within the wall of an artery * Dental plaque, a biofilm that builds up on teeth * A broad papule, a type of cutaneous condition * Pleural plaque, associated with mesothelioma, cancer often caused by exposure to asbestos * Senile plaques, an extracellular protein deposit in the brain implicated in Alzheimer's disease * Skin plaque, a plateau-like lesion that is greater in its diameter than in its depth * Viral plaque, a visible structure formed by virus propagation within a cell culture Other uses * Plaque, a rectangular casino token See also * * * Builder's plate * Plac (disambiguation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
