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Plate Theory
In continuum mechanics, plate theories are mathematical descriptions of the mechanics of flat plates that draw on the theory of beams. Plates are defined as plane structural elements with a small thickness compared to the planar dimensions.Timoshenko, S. and Woinowsky-Krieger, S. "Theory of plates and shells". McGraw–Hill New York, 1959. The typical thickness to width ratio of a plate structure is less than 0.1. A plate theory takes advantage of this disparity in length scale to reduce the full three-dimensional solid mechanics problem to a two-dimensional problem. The aim of plate theory is to calculate the deformation and stresses in a plate subjected to loads. Of the numerous plate theories that have been developed since the late 19th century, two are widely accepted and used in engineering. These are * the Kirchhoff–Love theory of plates (classical plate theory) * The Reissner-Mindlin theory of plates (first-order shear plate theory) Kirchhoff–Love theory ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Solid Mechanics
Solid mechanics (also known as mechanics of solids) is the branch of continuum mechanics that studies the behavior of solid materials, especially their motion and deformation (mechanics), deformation under the action of forces, temperature changes, phase (chemistry), phase changes, and other external or internal agents. Solid mechanics is fundamental for civil engineering, civil, Aerospace engineering, aerospace, nuclear engineering, nuclear, Biomedical engineering, biomedical and mechanical engineering, for geology, and for many branches of physics and chemistry such as materials science. It has specific applications in many other areas, such as understanding the anatomy of living beings, and the design of dental prosthesis, dental prostheses and surgical implants. One of the most common practical applications of solid mechanics is the Euler–Bernoulli beam theory, Euler–Bernoulli beam equation. Solid mechanics extensively uses tensors to describe stresses, strains, and the r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Plaque Mince Deplacement Rotation Fibre Neutre New
Plaque may refer to: Commemorations or awards * Commemorative plaque, a plate, usually fixed to a wall or other vertical surface, meant 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Infinitesimal Strain Theory
In continuum mechanics, the infinitesimal strain theory is a mathematical approach to the description of the deformation of a solid body in which the displacements of the material particles are assumed to be much smaller (indeed, infinitesimally smaller) than any relevant dimension of the body; so that its geometry and the constitutive properties of the material (such as density and stiffness) at each point of space can be assumed to be unchanged by the deformation. With this assumption, the equations of continuum mechanics are considerably simplified. This approach may also be called small deformation theory, small displacement theory, or small displacement-gradient theory. It is contrasted with the finite strain theory where the opposite assumption is made. The infinitesimal strain theory is commonly adopted in civil and mechanical engineering for the stress analysis of structures built from relatively stiff elastic materials like concrete and steel, since a common goal in ... [...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 permits a two-dimensional plate theory to give an excellent approximation to the actual three-dimensional motion of a plate-like object.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 and the study of standing waves and vibration ... [...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 fo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bending Rigidity
Flexural rigidity is defined as the force couple required to bend a fixed non- rigid structure by one unit of curvature, or as the resistance offered by a structure while undergoing bending. Flexural rigidity of a beam Although the moment M(x) and displacement y generally result from external loads and may vary along the length of the beam or rod, the flexural rigidity (defined as EI) is a property of the beam itself and is generally constant for prismatic members. However, in cases of non-prismatic members, such as the case of the tapered beams or columns or notched stair stringers, the flexural rigidity will vary along the length of the beam as well. The flexural rigidity, moment, and transverse displacement are related by the following equation along the length of the rod, x: :\ EI \ = \int_^ M(x) dx + C_1 where E is the flexural modulus (in Pa), I is the second moment of area (in m4), y is the transverse displacement of the beam at x, and M(x) is the bending moment at ''x'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Shear Modulus
In materials science, shear modulus or modulus of rigidity, denoted by ''G'', or sometimes ''S'' or ''μ'', is a measure of the Elasticity (physics), elastic shear stiffness of a material and is defined as the ratio of shear stress to the shear strain: :G \ \stackrel\ \frac = \frac = \frac where :\tau_ = F/A \, = shear stress :F is the force which acts :A is the area on which the force acts :\gamma_ = shear strain. In engineering :=\Delta x/l = \tan \theta , elsewhere := \theta :\Delta x is the transverse displacement :l is the initial length of the area. The derived SI unit of shear modulus is the Pascal (unit), pascal (Pa), although it is usually expressed in Pascal (unit), gigapascals (GPa) or in thousand pounds per square inch (ksi). Its dimensional analysis, dimensional form is M1L−1T−2, replacing ''force'' by ''mass'' times ''acceleration''. Explanation The shear modulus is one of several quantities for measuring the stiffness of materials. All of them arise in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Shear Stress
Shear stress (often denoted by , Greek alphabet, Greek: tau) is the component of stress (physics), stress coplanar with a material cross section. It arises from the shear force, the component of force vector parallel to the material cross section. ''Normal stress'', on the other hand, arises from the force vector component perpendicular to the material cross section on which it acts. General shear stress The formula to calculate average shear stress or force per unit area is: \tau = ,where is the force applied and is the cross-sectional area. The area involved corresponds to the material face (geometry), face parallel to the applied force vector, i.e., with surface normal vector perpendicular to the force. Other forms Wall shear stress Wall shear stress expresses the retarding force (per unit area) from a wall in the layers of a fluid flowing next to the wall. It is defined as:\tau_w := \mu\left.\frac\_,where is the dynamic viscosity, is the flow velocity, and is the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Raymond Mindlin
Raymond David Mindlin (New York City, 17 September 1906 – 22 November 1987) was an American mechanical engineer, Professor of Applied Science at Columbia University, and recipient of the 1946 Presidential Medal for Merit and many other awards and honours.IEEE UFFCIn Memoriam: Raymond D. Mindlin" at ''ieee-uffc.org'', 2014. Accessed 2017-07-19. He is known as mechanician, who made seminal contributions to many branches of applied mechanics, applied physics, and engineering sciences. Biography Education In 1924 he enrolled at Columbia University, where he received a B.A. in 1928, followed by a B.S. in 1931, and in 1932 by a C.E. and the Illig medal for "proficiency in scholarship." During his graduate study, Mindlin attended a series of summer courses organized by Stephen Timoshenko in 1933, '34, and '35, and there is no doubt that the experience at the University of Michigan served to confirm him in his choice of his life's work. Career For his doctoral research Mindlin set ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Eric Reissner
Max Erich (Eric) Reissner (January 5, 1913 – November 1, 1996) was a German-American civil engineer and mathematician, and Professor of Mathematics at the Massachusetts Institute of Technology. He was recipient of the Theodore von Karman Medal in 1964, and the ASME Medal in 1988. Reissner is known as co-developer of the Mindlin–Reissner plate theory. He is remembered by ''The New York Times'' (1996) as the "mathematician whose work in applied mechanics helped broaden the theoretical understanding of how solid objects react under stress and led to advances in both civil and aerospace engineering."Tim Hilchey.Eric Reissner, 83, Well-Known Math Scholar, Dies" ''The New York Times'', Nov. 11, 1996. Accessed 2017-07-19. Biography Reissner was born in Aachen, Germany, son of Hans Jacob Reissner, an aeronautical engineer, and Josefine (Reichenberger) Reissner. At the Technische Hochschule Berlin-Charlottenburg he obtained dregrees in Applied Mathematics in 1935, and in Civil E ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Orthotropic Material
In material science and solid mechanics, orthotropic materials have material properties at a particular point which differ along three orthogonal axes, where each axis has twofold rotational symmetry. These directional differences in strength can be quantified with Hankinson's equation. They are a subset of anisotropic materials, because their properties change when measured from different directions. A familiar example of an orthotropic material is wood. In wood, one can define three mutually perpendicular directions at each point in which the properties are different. It is most stiff (and strong) along the grain (axial direction), because most cellulose fibrils are aligned that way. It is usually least stiff in the radial direction (between the growth rings), and is intermediate in the circumferential direction. This anisotropy was provided by evolution, as it best enables the tree to remain upright. Because the preferred coordinate system is cylindrical-polar, this type of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pure Bending
In solid mechanics Solid mechanics (also known as mechanics of solids) is the branch of continuum mechanics that studies the behavior of solid materials, especially their motion and deformation (mechanics), deformation under the action of forces, temperature chang ..., pure bending (also known as the theory of simple bending) is a condition of Stress (mechanics), stress where a bending moment is applied to a Beam (structure), beam without the simultaneous presence of Cylinder stress, axial, Shear stress, shear, or Torsion (mechanics), torsional forces. Pure bending occurs only under a constant bending moment () since the shear force (), which is equal to \tfrac, has to be equal to zero. In reality, a state of pure bending does Idealization (philosophy of science), 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 ben ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
