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Statically Indeterminate
In statics and structural mechanics, a structure is statically indeterminate when the equilibrium equations force and moment equilibrium conditions are insufficient for determining the internal forces and reactions on that structure. Mathematics Based on Newton's laws of motion, the equilibrium equations available for a two-dimensional body are: : \sum \mathbf F = 0 : the vectorial sum of the forces acting on the body equals zero. This translates to: :: \sum \mathbf H = 0 : the sum of the horizontal components of the forces equals zero; :: \sum \mathbf V = 0 : the sum of the vertical components of forces equals zero; : \sum \mathbf M = 0 : the sum of the moments (about an arbitrary point) of all forces equals zero. In the beam construction on the right, the four unknown reactions are , , , and . The equilibrium equations are: : \begin \sum \mathbf V = 0 \quad & \implies \quad \mathbf V_A - \mathbf F_v + \mathbf V_B + \mathbf V_C = 0 \\ \sum \mathbf H = 0 \quad & \implies ... [...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] |
Kinematic Coupling
Kinematic coupling describes fixtures designed to exactly constrain the part in question, providing precision and certainty of location. A canonical example of a kinematic coupling consists of three radial v-grooves in one part that mate with three hemispheres in another part. Each hemisphere has two contact points for a total of six contact points, enough to constrain all six of the part's degrees of freedom. An alternative design consists of three hemispheres on one part that fit respectively into a tetrahedral dent, a v-groove, and a flat. Background Kinematic couplings arose from the need of precision coupling between structural interfaces that were meant to be routinely taken apart and put back together. Kelvin Coupling The Kelvin coupling is named after William Thompson (Lord Kelvin) who published the design in 1868–71. It consists of three spherical surfaces that rest respectively on a concave tetrahedron, a V-groove pointing towards the tetrahedron and a flat plat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Structural Engineering
Structural engineering is a sub-discipline of civil engineering in which structural engineers are trained to design the 'bones and joints' that create the form and shape of human-made Structure#Load-bearing, structures. Structural engineers also must understand and calculate the structural stability, stability, strength, structural rigidity, rigidity and earthquake-susceptibility of built structures for buildings and nonbuilding structures. The structural designs are integrated with those of other designers such as architects and Building services engineering, building services engineer and often supervise the construction of projects by contractors on site. They can also be involved in the design of machinery, medical equipment, and vehicles where structural integrity affects functioning and safety. See glossary of structural engineering. Structural engineering theory is based upon applied physics, physical laws and empirical knowledge of the structural performance of different ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kinematic Determinacy
Kinematic determinacy is a term used in structural mechanics to describe a structure where material compatibility conditions alone can be used to calculate deflections. A kinematically determinate structure can be defined as a structure where, if it is possible to find nodal displacements compatible with member extensions, those nodal displacements are unique. The structure has no possible mechanisms, i.e. nodal displacements, compatible with zero member extensions, at least to a first-order approximation. Mathematically, the mass matrix of the structure must have full rank. Kinematic determinacy can be loosely used to classify an arrangement of structural members as a ''structure'' (stable) instead of a ''mechanism'' (unstable). The principles of kinematic determinacy are used to design precision devices such as mirror mounts for optics, and precision linear motion bearings. See also * Statical determinacy * Precision engineering * Kinematic coupling Kinematic coupling desc ... [...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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Christian Otto Mohr
Christian Otto Mohr (8 October 1835 – 2 October 1918) was a German civil engineer. He is renowned for his contributions to the field of structural engineering, such as Mohr's circle, and for his study of stress. Biography He was born on 8 October 1835 to a landowning family in Wesselburen in the Holstein region. At the age of 16 attended the Polytechnic School in Hannover. Starting in 1855, his early working life was spent in railroad engineering for the Hanover and Oldenburg state railways, designing some famous bridges and making some of the earliest uses of steel trusses. Even during his early railway years, Mohr had developed an interest in the theories of mechanics and the strength of materials. In 1867, he became professor of mechanics at Stuttgart Polytechnic, and in 1873 at Dresden Polytechnic. Mohr had a direct and unpretentious lecturing style that was popular with his students. In addition to a lone textbook, Mohr published many research papers on the theory of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hardy Cross
Hardy Cross (1885–1959) was an American structural engineer and the developer of the moment distribution method for structural analysis of statically indeterminate structures. The method was in general use from c. 1935 until c. 1960 when it was gradually superseded by other methods. Early life Cross was born in Nansemond County, Virginia, to Virginia planter Thomas Hardy Cross and his wife Eleanor Elizabeth Wright. He had an elder brother, Tom Peete Cross, who would later become a Celtic studies scholar. Both studied at Norfolk Academy. Then he attended Hampden-Sydney College where he earned both B.A. and B.S. degrees. He obtained a BS in civil engineering from the Massachusetts Institute of Technology in 1908, and then joined the bridge department of the Missouri Pacific Railroad in St. Louis, where he remained for a year, after which he returned to Norfolk Academy in 1909. After a year of graduate study at Harvard he was awarded the MCE degree in 1911. Hardy Cross developed ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Moment Distribution Method
The moment distribution method is a structural analysis method for statically indeterminate beams and frames developed by Hardy Cross. It was published in 1930 in an ASCE journal. The method only accounts for flexural effects and ignores axial and shear effects. From the 1930s until computers began to be widely used in the design and analysis of structures, the moment distribution method was the most widely practiced method. Introduction In the moment distribution method, every joint of the structure to be analysed is fixed so as to develop the ''fixed-end moments''. Then each fixed joint is sequentially released and the fixed-end moments (which by the time of release are not in equilibrium) are distributed to adjacent members until equilibrium is achieved. The moment distribution method in mathematical terms can be demonstrated as the process of solving a set of simultaneous equations by means of iteration. The moment distribution method falls into the category of displacem ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Finite Element Method
Finite element method (FEM) is a popular method for numerically solving differential equations arising in engineering and mathematical modeling. Typical problem areas of interest include the traditional fields of structural analysis, heat transfer, fluid flow, mass transport, and electromagnetic potential. Computers are usually used to perform the calculations required. With high-speed supercomputers, better solutions can be achieved and are often required to solve the largest and most complex problems. FEM is a general numerical method for solving partial differential equations in two- or three-space variables (i.e., some boundary value problems). There are also studies about using FEM to solve high-dimensional problems. To solve a problem, FEM subdivides a large system into smaller, simpler parts called finite elements. This is achieved by a particular space discretization in the space dimensions, which is implemented by the construction of a mesh of the object: the numer ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stiffness Matrix
In the finite element method for the numerical solution of elliptic partial differential equations, the stiffness matrix is a matrix that represents the system of linear equations that must be solved in order to ascertain an approximate solution to the differential equation. The stiffness matrix for the Poisson problem For simplicity, we will first consider the Poisson problem : -\nabla^2 u = f on some domain , subject to the boundary condition on the boundary of . To discretize this equation by the finite element method, one chooses a set of '' basis functions'' defined on which also vanish on the boundary. One then approximates : u \approx u^h = u_1\varphi_1+\cdots+u_n\varphi_n. The coefficients are determined so that the error in the approximation is orthogonal to each basis function : : \int_ \varphi_i\cdot f \, dx = -\int_ \varphi_i\nabla^2u^h \, dx = -\sum_j\left(\int_ \varphi_i\nabla^2\varphi_j\,dx\right)\, u_j = \sum_j\left(\int_ \nabla\varphi_i\cdot\nabla\varphi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homogeneous System Of Linear Equations
In mathematics, a system of linear equations (or linear system) is a collection of two or more linear equations involving the same variables. For example, : \begin 3x+2y-z=1\\ 2x-2y+4z=-2\\ -x+\fracy-z=0 \end is a system of three equations in the three variables . A '' solution'' to a linear system is an assignment of values to the variables such that all the equations are simultaneously satisfied. In the example above, a solution is given by the ordered triple (x,y,z)=(1,-2,-2), since it makes all three equations valid. Linear systems are a fundamental part of linear algebra, a subject used in most modern mathematics. Computational algorithms for finding the solutions are an important part of numerical linear algebra, and play a prominent role in engineering, physics, chemistry, computer science, and economics. A system of non-linear equations can often be approximated by a linear system (see linearization), a helpful technique when making a mathematical model or computer ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
