This is accomplished by solving the geodesic equations. and two points The main difference in the two wall thickness equations is the simplified version is more conservative, quicker, and easier to calculate for scheduled pipe. Its important to understand the various types of pipe stresses, the process, and other items related to pipe stress analysis for best practices in performing a pipe stress analysis. M Click here to start this process. {\displaystyle \alpha } These cookies do not store any personal information. a ASME codes apply a safety factor of two when determining wall thickness based on hoop stress, yielding: The safety factor is to account for the additional stresses caused by bending and axial stresses to be applied later. . {\displaystyle {\tilde {\nabla }}_{a}} ) Vector fields are contravariant rank one tensor fields. T VCafe has been offering high-end catering and event services for today's discriminating customer. s Do you have experience and expertise with the topics mentioned in this content? [9] This can be done by magnetic annealing,[10] magnetic field assisted compaction,[11] or reaction under uniaxial pressure. {\displaystyle P} | Contact, Home However, it is known from historical empirical testing that these methods and 3-D beam computer models demonstrate enough behavior that they are a good approximation. Elements used to model the piping system have their limitations. This category only includes cookies that ensures basic functionalities and security features of the website. It depends on the modulus of elasticity and the area moment of inertia of the object. Bending stiffness is the resistance offered by the body against bending. This latter problem has been solved and its adaptation for general relativity is called the CartanKarlhede algorithm. ( Antisymmetric tensors of rank 2 play important roles in relativity theory. As such, the ideas of linear algebra are employed to study tensors. Training Online Engineering, Stress at the cross-section being evaluated, section modulus of the cross-section of the beam = I/z, distance from neutral axis to extreme fiber (edge), Beam Stress Equations / Calculation - Both Ends Overhanging Supports, Load at any Point Between, Structural Beam Bending Stress and Deflection Equations / Calculation - Fixed at One End, Supported at the Other, Load at Center, Beam Stress Deflection Equations / Calculator - Fixed at One End, Supported at the Other, Load at any Point, Beam Stress Deflection Equations / Calculator - Fixed at Both Ends, Load at Center, Beam Stress Deflection Equations / Calculator - Fixed at Both Ends, Load at any Location, Beam Stress Deflection Equations / Calculator with Uniform Loading, Beam Stress Deflection Equations / Continuous Beam, with Two Unequal Spans, Unequal, Uniform Loads, Beam Stress Deflection Calculator Equations - Continuous Beam, with Two Equal Spans, Uniform Load, Beam Deflection Calculator Equations with Ends Overhanging Supports and a Two Equal Loads applied at Symmetrical Locations, Beam Deflection Equations with End Overhanging Supports and a Single Load, Beam Deflection Calculator Equations Cantilevered Beam with One Load Applied at End, Deflections apply only to constant cross sections along entire length. The bending stiffness of the object can be increased with an increase in the Modulus of elasticity (E) and Moment of inertia (I). Therefore the FPS unit of bending stiffness is lb.ft. Most stress analysis programs default to calculating hoop stress based ID. copies of the cotangent space with Lubinskis equations should be used when there is significant tension in the drill pipe. {\displaystyle x^{a}} Fe2O3), is also mainly used for its magnetostrictive applications like sensors and actuators, thanks to its high saturation magnetostriction (~200 parts per million). copies of the tangent space. + The vanishing of all these components over a region indicates that the spacetime is flat in that region. At each point s , is more often used in calculations: A covariant derivative of {\displaystyle \Gamma (TM)\times \Gamma (TM)\to \Gamma (TM)} However, in most cases. The physics of pipe stress analysis does not change with piping code. A Before the advent of general relativity, changes in physical processes were generally described by partial derivatives, for example, in describing changes in electromagnetic fields (see Maxwell's equations). U Some important invariants in relativity include: Other examples of invariants in relativity include the electromagnetic invariants, and various other curvature invariants, some of the latter finding application in the study of gravitational entropy and the Weyl curvature hypothesis. ( r Occasional stress is The sum of longitudinal stresses produced by internal pressure, live and dead loads, and those produced by occasional loads, according to ASME B31.1, paragraph 102.3.3(A). i In the literature, there are three common methods of denoting covariant differentiation: Many standard properties of regular partial derivatives also apply to covariant derivatives: In general relativity, one usually refers to "the" covariant derivative, which is the one associated with Levi-Civita affine connection. ( X Fatigue stress results in a reduction of allowable strength in the piping system and is commonly caused by cycling of: Piping codes, such as those published by ASME, provide an allowable code stress, which is the maximum stress a piping system can withstand before code failure. If it is a high-pressure, high-temperature, hazardous-fluids system, and/or large outside forces are applied to the piping system, a computer-aided model may be required. As well as being used to raise and lower tensor indices, it also generates the connections which are used to construct the geodesic equations of motion and the Riemann curvature tensor. being associated with a tensor at The discrepancy between the results of these two parallel transport routes is essentially quantified by the Riemann tensor. Furthermore, to achieve pipe failure from deflection, the supported pipe spans would be at least three to four times greater in length than the recommended MSS SP-58 spans. The classification of tensors is a purely mathematical problem. , showing that the Lie derivative is independent of the metric. X This tensor measures curvature by use of an affine connection by considering the effect of parallel transporting a vector between two points along two curves. over 100 ft under the right conditions (200 to 300 psi). An important distinction in physics is the difference between local and global structures. on this curve, an affine connection gives rise to a map of vectors in the tangent space at The notion of a tensor field is of major importance in GR. independent connection coefficients at each point of spacetime. a In fact in the above expression, one can replace the covariant derivative {\displaystyle \Gamma _{ji}^{k}=\Gamma _{ij}^{k}} One of the profound consequences of relativity theory was the abolition of privileged reference frames. A useful way of measuring the curvature of a manifold is with an object called the Riemann (curvature) tensor. This tensor is called the Ricci tensor which can also be derived by setting r = 2 Displacement stress is developed by the self-constraint of the piping structure. A Targeted metallurgical processing steps promote abnormal grain growth of {011} grains in galfenol and alfenol thin sheets, which contain two easy axes for magnetic domain alignment during magnetostriction. T The resulting connection coefficients (Christoffel symbols) can be calculated directly from the metric. No localized effects will be seen on the pipe wall. Monte Engelkemier is the group engineering lead for piping, mechanical, and equipment in the starches, sweeteners, and texturizers division of Cargill. The EFE relate the total matter (energy) distribution to the curvature of spacetime. For example, in a system composed of one planet orbiting a star, the motion of the planet is determined by solving the field equations with the energymomentum tensor the sum of that for the planet and the star. The curvature of a spacetime can be characterised by taking a vector at some point and parallel transporting it along a curve on the spacetime. If not properly supported and designed, it can have devastating effects on that equipment. {\displaystyle \partial _{a}} Below is the sustained equation from ASME B31.1: The simplified hoop-stress term is in the equation above, is based on minimum wall thickness, and is approximately at 50% of allowable stress, based on the wall thickness safety factor. More recently, Wahi et al. Area Moment of Inertia Equations & Calculators . Magnetostriction (cf. all of which are useful in calculating solutions to Einstein's field equations. Another reason a pipe stress analysis is performed is to increase the life of piping. Plot No. Below is the sustained equation from ASME B31.1: The simplified hoop-stress term is in the equation above, is based on minimum wall thickness, and is approximately at 50% of allowable stress, based on the wall thickness safety factor. n There are a couple of reasons why. Therefore, ASME has developed stress-intensification factors (SIFs) for piping fittings through empirical testing. 4 per cent as compared to natural Italian ice cream which is higher at 10 percent or more. {\displaystyle \gamma (t)} Any Machinery's Handbook published since 1931 or. The Lie derivative is usually denoted by d Twenty percent if the event lasts less than 1 hour and no more than 80 hours per year. mm 4; cm 4; m 4; Converting between Units. adding lateral restraints for every three or four nominal pipe-support spans will cover most seismic or wind loadings, unless they are in a high seismic zone, such as California, or are subjected to coastal wind loading with sustained hurricane winds. The moment of inertia of the cross-section(you can use this calculator to calculate the moment of inertia for your certain cross-section) of the structural member where calculations are done. A type In particular, Killing symmetry (symmetry of the metric tensor under Lie dragging) occurs very often in the study of spacetimes. Open Calculator Supported on Both Ends Single Load at Center, Beam Stress between load and support points, Beam Stress at center of constant cross section, Beam Deflection between load and support points, Copyright 2000 - {\displaystyle X} tensor fields sending them to type While some relativists consider the notation to be somewhat old-fashioned, many readily switch between this and the alternative notation:[1]. Other physical descriptors are represented by various tensors, discussed below. and denoted by The Lie derivative can be defined for type The internal reaction loads in a cross-section of the structural elements can be resolved into a resultant force and a resultant couple for The Riemann tensor has 20 independent components. The moment due to dead weight contributes approximately 10% code stress to the equation above when using MSS SP-58 recommended pipe-support spans. Although the word 'tensor' refers to an object at a point, it is common practice to refer to tensor fields on a spacetime (or a region of it) as just 'tensors'. ( {\displaystyle s} However, most people consider 0.0625 in. a B Using the above procedure, the Riemann tensor is defined as a type (1, 3) tensor and when fully written out explicitly contains the Christoffel symbols and their first partial derivatives. The nonlinearity of the Einstein field equations often leads one to consider approximation methods in solving them. ) First, recommended pipe support spans are governed by deflection, and not by allowable stress, to ensure proper flow and drainage. r a The Einstein field equations (EFE) are the core of general relativity theory. tensor fields and in this respect can be viewed as a map that sends a type {\textstyle {\tfrac {1}{2}}D^{2}(D+1)} Bending stiffness is the resistance offered by the body against bending. Another appealing feature of spinors in general relativity is the condensed way in which some tensor equations may be written using the spinor formalism. For someone who is new to pipe stress analysisthere is no reason sustained stresses in the pipe should be greater than 55% of the standard allowable stress. D The Lie derivative of any tensor along a vector field can be expressed through the covariant derivatives of that tensor and vector field. M Hence, the total number of elements a tensor possesses equals 4R, where R is the count of the number of covariant It means documenting a trail of all inputs, not just the drawings used to create the piping geometry. , If the tangent space is n-dimensional, it can be shown that {\displaystyle {\vec {B}}} r X Shear stress is a vector quantity. This notion can be made more precise by introducing the idea of a fibre bundle, which in the present context means to collect together all the tensors at all points of the manifold, thus 'bundling' them all into one grand object called the tensor bundle. Hence the unit of bending stiffness is given by, Bending stiffness = E I = `[\frac{lb}{ft^{2}}\times ft^{4}]=lb.ft^{2}`. They allow for greater approximation without using complex FEA models with shells, plates, and brick elements. a Displacement stresses are most often associated with the effects of temperature; however, external displacements, such as building settlements, are considered a displacement stress. {\displaystyle {\vec {A}}} Notions of parallel transport can then be defined similarly as for the case of vector fields. By using this website, you agree to our use of cookies. d The electromagnetic field tensor F a b {\displaystyle F^{ab}} , a rank-two antisymmetric tensor. {\displaystyle {\vec {U}}} Thus, single-crystal-like texture (~90% {011} grain coverage) is attainable, reducing the interference with magnetic domain alignment and increasing microstrain attainable for polycrystalline alloys as measured by semiconducting strain gauges.
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