Formulas stress

Stoichiometric compositions of such phases are determined by the valences of the components [2] and, therefore, vacancies in defect phases are known as stoichiometric vacancies. The formulas for defect phases of this type can be written employing Rees s symbol for vacancies (B DX3 A B2 DX4, A B 02X8, etc.). These formulas stress the fact that a stoichiometric vacancy is one of the components in such phases. 

Denote hy W = (w, w ), w horizontal and vertical displacements of the mid-surface points, respectively, and write down the formulae for strain and integrated stress tensor components y(lL), aij W) ... 

When constmction is complete, the pipeline must be tested for leaks and strength before being put into service industry code specifies the test procedures. Water is the test fluid of choice for natural gas pipelines, and hydrostatic testing is often carried out beyond the yield strength in order to reHeve secondary stresses added during constmction or to ensure that all defects are found. Industry code limits on the hoop stress control the test pressures, which are also limited by location classification based on population. Hoop stress is calculated from the formula, S = PD/2t, where S is the hoop stress in kPa (psig) P is the internal pressure in kPa (psig), and D and T are the outside pipe diameter and nominal wall thickness, respectively, in mm (in.). 

The theoretical value of the coefficient Y ranges from 0 to 1. If V = 0, equation 9 reduces to the Barlow or outside diameter formula. The Barlow formula is always conservative and for large D/1 the error is small. The value Y = 0.4 gives the Boardman approximation of the Lame tangential stress (27)... 

The force and moment ia a constrained system can be estimated by the cantilever formula. Leg MB is a cantilever subject to a displacement of and leg CB subject to a displacement Av. Taking leg CB, for example, the task has become the problem of a cantilever beam with length E and displacement of Av. This problem caimot be readily solved, because the end condition at is an unknown quantity. However, it can be conservatively solved by assuming there is no rotation at poiat B. This is equivalent to putting a guide at poiat B, and results ia higher estimate ia force, moment, and stress. The approach is called guided-cantilever method. 

For straight metal pipe under internal pressure the formula for minimum reqiiired w thickness is applicable for D /t ratios greater than 6. Tme more conservative Barlow and Lame equations may also be used. Equation (10-92) includes a factor Y varying with material and temperature to account for the redistribution of circumferential stress which occurs under steady-state creep at high temperature and permits slightly lesser thickness at this range. 

Internal-pressure design rules and formulas are given for cylindrical and spherical shells and for ellipsoidal, torispherical (often called ASME heads), hemispherical, and conical heads. The formulas given assume membrane-stress failure, although the rules for heads include consideration for buckling failure in the transition area from cylinder to head (knuckle area). 

Unstayed flat heads and covers can be designed by very specific rules and formulas given in this subsection. The stresses caused by pressure on these members are bending stresses, and the formulas include an allowance for additional edge moments induced when the head, cover, or blind flange is attached By bolts. Rules are provided for quick-opening closures because of the risk of incomplete attachment or opening while the vessel is pressurized. Rules for braced and stayed surfaces are also provided. 

Roawle s Formulas for Stress and Strain, 6th edn, McGraw-Hill, 1989. 

A metal bar of width w is compressed between two hard anvils as shown in Fig. Al.l. The third dimension of the bar, L, is much greater than zu. Plastic deformation takes place as a result of shearing along planes, defined by the dashed lines in the figure, at a shear stress k. Find an upper bound for the load F when (a) there is no friction between anvils and bar, and (b) there is sufficient friction to effectively weld the anvils to the bar. Show that the solution to case (b) satisfies the general formula... 

As we ve said numerous times before in this chapter, the important thing is that the NPSHa of the system is above the NPSHr of the pump bv a sufficient amount to avoid stress and possible cavitation. If the NPSHa should be inadequate, there are ways to elevate it. Remember from the formula that five elements compose the NPSHa. Two of those elements, the Ha and the Hs, add energy to the fluid. And three elements, the Hvp, the Hf, and the Hi, subtract energ from the fluid. VVe must either increa.se the elements that add energ), or decrease the elements that subtract energy. To increase the NPSHa ... 

Often in stress analysis we may be required to make simplified assumptions, and as a result, uneertainties or loss of aeeuraey are introdueed (Bury, 1975). The aeeuraey of ealeulation deereases as the eomplexity inereases from the simple ease, but ultimately the eomponent part will still break at its weakest seetion. Theoretieal failure formulae are devised under assumptions of ideal material homogeneity and isotropie behaviour. Homogeneous means that the materials properties are uniform throughout isotropie means that the material properties are independent of orientation or direetion. Only in the simplest of eases ean they furnish us with the eomplete solution of the stress distribution problem. In the majority of eases, engineers have to use approximate solutions and any of the real situations that arise are so eomplieated that they eannot be fully represented by a single mathematieal model (Gordon, 1991). 

Natural rubber is composed of polymerized isoprene units. When rubber is under tension, ozone attacks the carbon-carbon double bond, breaking the bond. The broken bond leaves adjacent C = C bonds under additional stress, eventually breaking and placing shll more stress on surrounding C = C bonds. This "domino" effect can be discerned from the structural formulas in Fig. 9-4. The number of cracks and the depth of the cracks in rubber under tension are related to ambient concentrations of ozone. 

Thermal Stress. The formula for thermal stress can be rearranged to calculate the tolerable temperature gradient for keeping defonnation within arbitrary limits. 

The usefulness of this formula is restricted by the difficulty of obtaining good values to substitute in it. They must apply to the alloy selected, and be derived from carefully controlled tests on it. The stress value, S, reflects an engineer s Judgment in the selection of elastic limit or some arbitrary yield strength. The modulus value must match this. The restraint coefficent, K, is seldom known with any precision. 

The restraint coefficient, K, in the thermal stress formula is very potent. It can be varied over a wider range than any of the other parameters. If a designer can build in flexibility, and thus substitute elastic deflection for plastic flow, he can achieve a major safety factor. 

So = Maximum operating hoop stress level as determined by the Barlow formula using the maximum operating pressure. 

The appropriate figure for stress, modulus etc. is taken from the creep curve or a derivative curve. This is then inserted into the formula. 

Buckling is complex to analyse but if we consider the rib as a flat plate, clamped along one edge then Roark gives the formula for the critical buckling stress as... 

The calculations lor recommended makeup torque assume the use ol a thread compound containing 40% to 60% by weight of linety powdered metallic zinc, or 60% by weight of finely powdered metallic lead, applied thoroughly to all threads and shoulders the use of the modified jackscrew formula as shown in the lAOC Drilling Manual and the API Spec RP 7G (latest addition) and a unit stress of 62,500 psi in the box or whichever is weaker. 

Calculations for the minimum performance properties of drill pipe are based on formulas given in Appendix A of API RP 7G. It must be remembered that numbers in Tables 4-80-4-83 have been obtained for the uniaxial state of stress, e.g., torsion only or tension only, etc. The tensile stress resistance is decreased when the drill string is subjected to both axial tension and torque a collapse... 

Consider a case in which the drill pipe is exposed to an axial load (P) and a torque (T). The axial stress (o ) and the shear stress (t) are given by the following formulas ... 

---