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Timoshenko–Ehrenfest Beam Theory
The Timoshenko–Ehrenfest beam theory was developed by Stephen Timoshenko and Paul EhrenfestIsaac Elishakoff (2020) "Who developed the so-called Timoshenko beam theory?", ''Mathematics and Mechanics of Solids'' 25(1): 97–116 early in the 20th century.Timoshenko, S. P. (1921) "LXVI. On the correction for shear of the differential equation for transverse vibrations of prismatic bars", ''The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science'' 41(245): 744–746 Timoshenko, S. P. (1922) "X. On the transverse vibrations of bars of uniform cross-section", ''The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science'' 43(253): 125–131 The model takes into account shearing (physics), shear deformation and rotational bending effects, making it suitable for describing the behaviour of thick beams, sandwich-structured composite, sandwich composite beams, or beams subject to high-frequency excitation when the wavelength approaches the thick ... [...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] |
Statics
Statics is the branch of classical mechanics that is concerned with the analysis of force and torque acting on a physical system that does not experience an acceleration, but rather is in mechanical equilibrium, equilibrium with its environment. If \textbf F is the total of the forces acting on the system, m is the mass of the system and \textbf a is the acceleration of the system, Newton's second law states that \textbf F = m \textbf a \, (the bold font indicates a Euclidean vector, vector quantity, i.e. one with both Magnitude (mathematics), magnitude and Direction (geometry), direction). If \textbf a =0, then \textbf F = 0. As for a system in static equilibrium, the acceleration equals zero, the system is either at rest, or its center of mass moves at constant velocity. The application of the assumption of zero acceleration to the summation of Moment (physics), moments acting on the system leads to \textbf M = I \alpha = 0, where \textbf M is the summation of all momen ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Density
Density (volumetric mass density or specific mass) is the ratio of a substance's mass to its volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' (or ''d'') can also be used: \rho = \frac, where ''ρ'' is the density, ''m'' is the mass, and ''V'' is the volume. In some cases (for instance, in the United States oil and gas industry), density is loosely defined as its weight per unit volume, although this is scientifically inaccurate this quantity is more specifically called specific weight. For a pure substance, the density is equal to its mass concentration. Different materials usually have different densities, and density may be relevant to buoyancy, purity and packaging. Osmium is the densest known element at standard conditions for temperature and pressure. To simplify comparisons of density across different systems of units, it is sometimes replaced by the dimensionless quantity "relative den ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Partial Differential Equations
In mathematics, a partial differential equation (PDE) is an equation which involves a multivariable function and one or more of its partial derivatives. The function is often thought of as an "unknown" that solves the equation, similar to how is thought of as an unknown number solving, e.g., an algebraic equation like . However, it is usually impossible to write down explicit formulae for solutions of partial differential equations. There is correspondingly a vast amount of modern mathematical and scientific research on methods to numerically approximate solutions of certain partial differential equations using computers. Partial differential equations also occupy a large sector of pure mathematical research, in which the usual questions are, broadly speaking, on the identification of general qualitative features of solutions of various partial differential equations, such as existence, uniqueness, regularity and stability. Among the many open questions are the existence an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Free Body Diagram
In physics and engineering, a free body diagram (FBD; also called a force diagram) is a graphical illustration used to visualize the applied forces, moments, and resulting reactions on a free body in a given condition. It depicts a body or connected bodies with all the applied forces and moments, and reactions, which act on the body(ies). The body may consist of multiple internal members (such as a truss), or be a compact body (such as a beam). A series of free bodies and other diagrams may be necessary to solve complex problems. Sometimes in order to calculate the resultant force graphically the applied forces are arranged as the edges of a polygon of forces or force polygon (see ). Free body A body is said to be "free" when it is singled out from other bodies for the purposes of dynamic or static analysis. The object does not have to be "free" in the sense of being unforced, and it may or may not be in a state of equilibrium; rather, it is not fixed in place and is thus " ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stress Resultants
Stress resultants are simplified representations of the stress state in structural elements such as beams, plates, or shells. The geometry of typical structural elements allows the internal stress state to be simplified because of the existence of a "thickness'" direction in which the size of the element is much smaller than in other directions. As a consequence the three traction components that vary from point to point in a cross-section can be replaced with a set of resultant forces and resultant moments. These are the stress resultants (also called '' membrane forces'', ''shear forces'', and ''bending moment'') that may be used to determine the detailed stress state in the structural element. A three-dimensional problem can then be reduced to a one-dimensional problem (for beams) or a two-dimensional problem (for plates and shells). Stress resultants are defined as integrals of stress over the thickness of a structural element. The integrals are weighted by integer pow ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Orientation (vector Space)
The orientation of a real vector space or simply orientation of a vector space is the arbitrary choice of which ordered bases are "positively" oriented and which are "negatively" oriented. In the three-dimensional Euclidean space, right-handed bases are typically declared to be positively oriented, but the choice is arbitrary, as they may also be assigned a negative orientation. A vector space with an orientation selected is called an oriented vector space, while one not having an orientation selected, is called . In mathematics, '' orientability'' is a broader notion that, in two dimensions, allows one to say when a cycle goes around clockwise or counterclockwise, and in three dimensions when a figure is left-handed or right-handed. In linear algebra over the real numbers, the notion of orientation makes sense in arbitrary finite dimension, and is a kind of asymmetry that makes a reflection impossible to replicate by means of a simple displacement. Thus, in three dimensi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cantilever Beam
A cantilever is a rigid structural element that extends horizontally and is unsupported at one end. Typically it extends from a flat vertical surface such as a wall, to which it must be firmly attached. Like other structural elements, a cantilever can be formed as a beam, plate, truss, or slab. When subjected to a structural load at its far, unsupported end, the cantilever carries the load to the support where it applies a shear stress and a bending moment. Cantilever construction allows overhanging structures without additional support. In bridges, towers, and buildings Cantilevers are widely found in construction, notably in cantilever bridges and balconies (see corbel). In cantilever bridges, the cantilevers are usually built as pairs, with each cantilever used to support one end of a central section. The Forth Bridge in Scotland is an example of a cantilever truss bridge. A cantilever in a traditionally timber framed building is called a jetty or forebay. In the southern ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bending Moment
In solid mechanics, a bending moment is the Reaction (physics), reaction induced in a structural element when an external force or Moment of force, moment is applied to the element, causing the element to bending, bend. The most common or simplest structural element subjected to bending moments is the Beam (structure), beam. The diagram shows a beam which is simply supported (free to rotate and therefore lacking bending moments) at both ends; the ends can only react to the Shear stress, shear loads. Other beams can have both ends fixed (known as encastre beam); therefore each end support has both bending moments and shear reaction loads. Beams can also have one end fixed and one end simply supported. The simplest type of beam is the cantilever, which is fixed at one end and is free at the other end (neither simple nor fixed). In reality, beam supports are usually neither absolutely fixed nor absolutely rotating freely. The internal reaction loads in a cross section (geometry), cr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Boundary Condition
In the study of differential equations, a boundary-value problem is a differential equation subjected to constraints called boundary conditions. A solution to a boundary value problem is a solution to the differential equation which also satisfies the boundary conditions. Boundary value problems arise in several branches of physics as any physical differential equation will have them. Problems involving the wave equation, such as the determination of normal modes, are often stated as boundary value problems. A large class of important boundary value problems are the Sturm–Liouville problems. The analysis of these problems, in the linear case, involves the eigenfunctions of a differential operator. To be useful in applications, a boundary value problem should be well posed. This means that given the input to the problem there exists a unique solution, which depends continuously on the input. Much theoretical work in the field of partial differential equations is devote ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
