Stresses in Pressure Vessels and Shells of Revolution - Report Example

Name: Course: Instructor: Date: Stresses in Pressure Vessels and Shells of Revolution Pressure vessels are containers that are mainly designed to hold liquids or gasses at a certain pressure that is mainly different from the ambient pressure. The differences in the pressure is dangerous since over the years it has led to numerous fatal accidents. Thus, the definition of a pressure vessels varies per country. The pressure vessels are cylindrical, spherical, hemispherical ends and others with the semi elliptical ends (Lecture 3). The research paper evaluates the stress equations for thin and thick-walled cylinders, thick and thin walled shells of revolution and discontinuity stresses at the different curvatures. Stress Equations for Thin-Thick Walled Cylinders The stresses of cylinders is distributed through rotational symmetry where the cylinder stress patterns may include the general and standard circumferential stress direction. It is also identified that is corresponding to the tubular symmetry stress direction throughout the pressure vessel. The radial stress of the cylinder level where the distributions of the perpendicular equilibrium axis with that of the coplanar. Thin-Walled Cylinder The thin-walled tube or pressure vessel must atleast contain a thickness of the wall of about a tenth of the cylinders / pressure vessels’ radius, where the concept of young-laplace perception is applied to solve the equivalence through approximating the stress of the thin walled vessel created through either the internal or exterior pressure loaded on the vessel. The stress equation on a thin-walled cylinder-shaped vessel, for a all pressure vessels can be attained through: In this equation, P stipulates the internal prerssure, t is the thickness of the wall, r is the cylinder radius and the the hoop stress. The hoop stress is perceived as the force that is exerted circumferentially in all wall cylinders, which is described as: Where F = the forcethat is circuferentially exterted on a cylinder area with t = refers to the cyinder radial thickness L = is the cylinder axial length Hoop stress can also be described as the wall tension or the stress. It is described as T=F/L The stress equation of the hoop for the thin shells can also be used for the spherical vessels. For closed vessel ends the actso f internal pressures develop an extra forcealong the axis of the cylinder. Thus, while using the axial stress, the stress equation for the cylinder is: Which is ussually approximated to: where in the radial stress, the estimated thin walled stress is presented as Axial stress direction can be expressed in a cylinder or tube in the equation as: Axial direction of the stress of the vessel (σa) = (pi ri2 - po ro2 )/(ro2 - ri2)  σa which is the axial direction of the stress force is given in the MPa, and psi. pi refers to the tube or cylinder pressure that is also given in terms of MPa and psi po is the external cylinder and tube pressure given in MPa and psi ri is given in the mm and in measurements of the cylinder or tube radius internally while ro is the external radius. The hoop stress of the cylinder or the tube is given at the circumferential direction can be provided through the equation: σc = [(pi ri2 - po ro2) / (ro2 - ri2)] - [ri2 ro2 (po - pi) / (r2 (ro2 - ri2))] where, σc = the circumferential direction stress given in the MPa and psi. r = the cylinder wall or tube radius point such as mm, and in. The maximum stress of the cylinder and tube is given in r=ri, which is given internally. Additionally, the radial direction stress is given and expressed through the equation below through using either the cylinder or tube. σr = [(pi ri2 - po ro2) / (ro2 - ri2)] + [ri2 ro2 (po - pi) / (r2 (ro2 - ri2))] Using this equation, supreme stress of the pressure vessels include r = ro, which refers to the external stress pressure. Thick-Walled Vessels Thick wall cylinders with open ends have internal pressure, presented as Pi while the external pressure loaded on a cylinder is presented as Po. The inside radius of the cylinder is given as ri while the outside radius is given as ro. (washington.edu, 1) Open ends of the cylinder lead to the development of the equation σz = σ3 =0, which lead to the stress conditions as explained by the law of Hooke εr = du / dr = I / E (σr - V σθ) εθ = u / r = 1 / E (σθ - σr) The answer given is σr = E / 1 –v2 (du / dr + V u/ r) σθ = E / 1 – V2 (u / r + V du/ dr) When he equation is substituted, it yields: d2u / dr2 + 1 du / r dr – u / r2 = 0 Thus, the solution attained is u = C1r + C2 / r Thus, the stresses will be attained as σr = E / 1 – v2 [c1 (1 + V) – C2 (1 – v/ r2)] σθ = E / 1- V2 [C1 (1 + v) + C2 (1 – v / r2)] (Bennitz, Grip and Schmidt, 963) Consequently, the conditions of the cylinder junctions / boundary occur as σr (ri) = - pi (Ozturk, 97) The stresses of the thick walled equations include the lame equations such as A and B in the equations above are the constants integration discovered through boundary conditions. In the equation, r refers to the interest point of radius. The thick walled cylinder has an inner radius such as a, while the outer radius, which is b (Nejad, Rahimi and Ghannad, 20). The thick-walled cylinders have stresses that are uniformly distributed in the outer and inner surfaces of the cylinder. The thick walled cylinders have a radius of less than 20 times the thickness of the wall. The cylinders are externally or internally pressurized. The pressure stresses of the equations are given in circumfrential, longitudinal and radial. Circumferential equations are also refered to as the hoop equations, which are given in σc, the radial equtions are given in σr (Dornfeld, 2). The longitudinal equations are also called the axial equation given in σl. The dimensions are illustrated in the figure below for easy identification when calculating the stresses of the pressure vessels. Circumferential and Radial Stresses of the Pressure Vessels The radial stresses at R in the pressure vessels for both external and internal pressure are given through the equation: + = the circumferential stress - = the radial stress Additionally, on some instances, the stresses of vessels is considered special for the internal pressure stresses given as po = 0 and the radius is given as r as provided in the equation below (Dornfeld, 3): Longitudinal Stresses The longitudinal pressure vessel stress is calculated through using the equations where force is divided with the area of the vessel. Generally, the force is given by pi and the interior area of the vessel given in Πri2 while the cros section of the annular area is given by Π (ro2 –ri2). It can also be given in the equation below (Dornfeld, 4). Si = pi ri2 / ro2 – ri2. Additionally, the stresses of the cylindersand tubes among other pressure vessels for internal pressurization, the equation to be used is po = 0. The figure above is a representation of the dimensions that are used in calculating the stresses in the pressure vessels. Radial stresses at the internal and external shells are equivalent to subtract from the pressurization (Pietraszkiewicz and Konopinska). On the stipulation, the shells are unpressurized the stress is zero (Pietraszkiewicz and Konopinska). In a pressurized surface, the radial stress is provided as –p due to the availability of compression. Across the internal thick-walled pressurized cylinder, the circumferential stresses will vary at the radius r. The equation is given as: Where the + sign represents circumferential stress and the – sign represents the radial (Dornfeld, 5). Stress equations for thin/thick-walled shells of revolution exposed to internal pressure The following balances are utilized generally to determine stress of thick-walled cylinders under either internal or external pressure (Flügge, 18). The main stresses principal at the given radius r is given through the following equations. When the cylinders have closed ends, the resultant is an axial stress component. σ1 = σθ = - K + C / r2 σ2 = σr = - K + C / r2 But if the cylinder or pressure vessel’s ends are closed, the equation below is used (washington.edu, 5). σ3 = σaxial = - K Where: C = (Po – Pi) [ri2 ro2 / ro2- ri2] K = Po ro2 – Pi ri2 / ro2 – ri2) Under interior pressure the stress equations for thick walled shells of revolution is attained through: Internal pressure stress equation assumes that Po =0 σθ = Pi ri2 / ro2 – ri2 [1 – ro2 / ri2] σr = Pi ri2 / ro2 – ri2 [1 + ro2 / r2] σz = pi ri2 / ro2 – ri2 The inside surface stress level under internal pressure is gained through r = ri σθ = Pi [ro2 + ri2 / ro2 – ri2] σr = Pi σz = pi ri2 / ro2 – ri2 (Gill, 8-9) The stress of the pressure vessel on the outside surface is attained through the r = ro σθ = [2Piri2 / ro2 – ri2] σr = 0 σz = Pi ri 2 / ro2 –ri2 Shells of revolution are mainly pressure vessels such as the cylinders, cones and spheres. The surface of such pressure vessels is attained through the process of rotating a plane curve on the lying curve axis of the plane (Meridian) (Gill, 8). (Gill, 8) The shell element is divided by two meridians and parralles circles. The figure below presents the shear and direct stress resultants of the shell. (Gill, 9) Shells of Revolutions Vectoral Form Equations The equilibrium equations for the stresses of the shells of revolutions with a middle surface equation of r = r (ξ1 , ξ2) with the linear coefficient element of α2j using vectors Nj and Mj for stress and P and q as load intensity vectors. (α2 N1) , 1 + (α1 N2), 2 + α1 α2P = 0 (α2 M1), 1 + (α1 M2), 2 + r, 1 * (α2 N1) + r, 2 * ( α1 N2) + α1 α2 q = 0 (Redekop) Virtual work considerations lead to the introduction of translational and rotational displacement vectors U and ϕ to the strain resultant introduction and couple vectors εj and kj where they are related as presented in the equation below. αj kj = ϕ, j , αj εj = u, j + r, j * ϕ following compatibility equations; (α2 k2), 1 – (α1 k1), 2 = 0 (αε2), 1 – (α1 ε1), 2 + r, 1 * (α2 k2) – r, 2 * (α1 k1) = 0 The equations presented above are homogenous, and imply a static geometric duality expressed as N1 N2 M1 M2 k2 k1 Ε2 ε1 The scalar duality that corresponds to the static duality expressed above is attained through the equation: αj tj = r, j (or) n = t1 * t2, That is; Nj = ∑ N jk tk + Q jn, εj = ∑ε jk tk + γ jn Mj = n * ∑ M jk tk + p jn k j = n * ∑ k jk tk + ʎ jn The above equations can be represented in a static geometric duality expressed as: N1 1 N 1 2 Q 1 N 2 1 N 2 2 Q 2 M1 1 M 1 2 P 1 M 2 1 M2 2 P2 -K 2 2 K 2 1 ʎ2 K 1 2 -K 1 1 -ʎ1 ε 2 2 -ε 2 1 γ 2 -ε 1 2 ε 1 1 -γ1 Pressure vessels structures should not be more than the strength exceeded by the materials. Stress equations thick-walled shells of revolution subjected to internal pressure Thick cylinders have thick walls, leading to a high stress variation across the thickness. The general stress equations can be given without structure forces as follows. For axisymmetry about z-axis ∂ = ∂θ = 0, which gives the following equations. If the cylinder ends can expand,the stress is given through σz = 0, influenced by the radial deformation and symmetry τrz = τθz = τrθ = 0. As a result, the symmetry equation is abridged to ∂r / ∂r + σ r - σ θ/ r = 0. It can also be expressed as r ∂σ/ ∂r + σr = σ θ Strain εr and εθ are calculated using the equations below, for the stresses calculation εr = ∂ur/ ∂ r = 1/ E [σ r – vσθ] since σz = 0 εθ = (r + ur) Δθ – r Δθ/ rΔθ = ur/ r = 1/E [σθ - σ r – vσr] The figure presents the circumferential and radial stresses/ strain. When the εr and εθ equations are combined, the following equations are attained. (r )∂σθ / ∂r - (vr) ∂σr / ∂r + (1 + v) (σθ - σr), which also gives the following equation. ∂σθ / ∂r = r ∂2σr / ∂r2 + 2 ∂σr / ∂r + 2pω2r Combning these two equations leads to the equation r ∂2σr / ∂r2 + ∂σr / ∂r + (3 + v) pω2r when ubected to internal pressure pi for a non-rotating hick cylinder the ω is substituted with 0 in the equation. That is; the equation is r ∂2σr / ∂r2 + 3 ∂σr / ∂r = 0. The solution for this is given in σr = c rn given that c and n are constants. σr = c1 + c2 / r2 σθ = c1 – c2 / r2 Pi represents the internal pressure of the thick-walled cylinder and po is the external pressure (Kharagpur, 3). The borderline or junction conditions for a thick cylinder with interior pressures occur as expressed below. r = ri σr = -pi The negative sign is an outcome of the compressive nature of the pressures. The boundary condition of external pressures occur at r = ro σr = -po Thus, for the constant’s c1 and c2, the equations below are used for determining the stresses of the thick-walled cylinders of shells revolutions. C1 = piri2 – poro2/ ro2 – ri2 C2 = ri2 ro2 (po – pi) / ro2 - ri 2 Thus, the radial stress given as σr and the circumferential stress given as σθ equations are developed as the following; (Radial Stress) σr = pi ri 2 – po ro 2 / ro2 – ri2 + ri2 ro2 (po – pi)/ ro 2 – ri2 1/2 (Circumferential Stress) σθ = pi ri 2 – po ro 2 / ro2 – ri2 – ri2 ro2 (po – pi)/ ro 2 – ri2 ½ When σθ answer is positive, the stress of the cylinder is tensile while a negative answer presents a compressed pressure. σr always is compressed whether the solution is positive or negative. The distribution of stress is always attained through the substitution of the relevant equations. When Po attained from a solution is zero, it simply stipulates the cylinder does not have any external pressure, while when pi is zero, the internal pressure is not available. Without any pressure, the stress is reduced to: σr = pi ri 2/ro2 – ri 2 {- ro2 / r2 + 1} σθ = pi ri2 / ro 2 – ri2 (ro2 / r2 + 1) The distribution of stress through the cylinder is then attained and occurs as: σr and σθ are constants, which stipulates that deformation in the z-direction is uniform. Thus, the perpendicular cross section of the cylinder is constantly at a plane surface (Kharagpur, 5). Thus, the plane stress of the cylinder is justified given that deformation in an element cut by two cross-sections adjacently does not interfere with the element. For instance, if Pi = 0, the cylinder does not have any internal pressure that can reduce the radial and circumferential stresses to σr = po ro 2 / ro2 – ri 2 {ri2 / r2 - 1} σθ = po ro2 / ro 2 – ri2 (ri2 / r2 + 1) Thus, the stress distribution in the thick walled cylinder will occur as follows for external pressures: When the thick walled cylinders are exposed to inner pressure, the maximum stress is perceived as the inside radius of the cylinder, which is attained through the equation below (Kharagpur, 6). σr (max) / r= ri = -pi σθ (max) / r = ri = pi ro2 + ri 2 / ro2 – ri2 The equations stipulate that the pi increase, leads to a σθ exceeding the yield stress. The resultant answer demonstrates that pi < σ yield. For the large internal pressures, the wall thickness must be exceedingly large as presented below. The figure shows the variation of wall thickness and the internal pressure of a thick walled cylinder (Kharagpur, 7). The stresses and pressures of the internal cylinders a the surface contact are given through the equations below (Kharagpur, 8). σr / r= rs = ps rs 2/ ro2 – rs2 (1 – ro 2/ rs2) = - Ps σθ / r= rs = Ps rs 2 / ro 2 – rs2 (1 +ro2 / rs2) Stress equations for thin-walled shells of revolution subjected to internal pressure Stress in thin walled vessels is given by the equation σθ = Pr / 2t where θ represents the hoop stress, p is the gauge pressure and r represents the inner radius, t is a representation of the wall thickness. Thus, the stresses for the thin walled vessels are developed and given in the following equations σθ = (PR/ tc) ; σz = (p R / 2 tc) σr = - (1/2p) Axial stresses in the shells are classified inform of the longitudinal stress of the internal pressure, compressive stress a sum of weights and the bending stress. Thus, the analysis focuses on the longitudinal stress, which is a resultant of internal pressure. When considering the thin-walled thicknesses, the standard equation for the stress created by internal pressure on a thin pressure walled is given through σθ = PDm /2t P denotes the wall’s internal pressure t refers to the thickness of the wall of the cylinder r refers to the cylinder radius Dm denotes the mean diameter, which is attained through the outside diameter subtracting the wall thickness (Engineers Edge). σθ gives the stress of the cylinder, it can also be identified as hoop stress as stated at the beginning of the paper. The pressure vessels with closed ends, leads the internal pressure to acting on t by developing a force along the cylinder axis. This is identified as the longitudinal/ axial stress, which is commonly lower to the hoop stress. The equation below is used to depict the stress of the cylinder when subjected to internal pressure. σz = F / A = Pd2/ (d + 2t)2 – d2 it can also be approximated through σz = Pd / 4t P = the pressure of the cylinder d = the mean diameter of the cylinder (outer diameter of the cylinder – t) t = thickness of the cylinder wall σz = the longitudinal stress (Engineers Edge). Discontinuity stresses at junctions of different curvatures Shell structures today consist regular shell parts that are linked together along common boundaries. Within the linear Kirchhoff-Love type concept of thin curvatures shells, three groups of junction quantities are provided. The boundary/ junction groups influenced the development of the four boundary/ junction settings on the movements and one linear revolution, or three linear cycles and stresses along the edge. Discontinuity stresses occur due to wall thickness changes in the pressure vessels. Cylinders with specials closures have a reduced stress level, and curves. The discontinuity of the pressure vessels occurs in four key shell junctions. Discontinuity first occurs due to the change in the cylinder thickness of the pressure vessel. First, the radial expansion leads to a mismatch of the middles surfaces of the cylinder influenced by the radii difference. The lack of a continuous middle surface, inner surface and outer surface influence the discontinuity due to stress distribution mismatch. Toriconical head is also a key player in influencing the discontinuity stress at its junction since as a cylinder end closure whose head is toriconical and has a sphere portion; the closure has two key discontinuities. That is; the junction between the cylinder and the sphere encounters a discontinuity as well as the junction between the cone and the sphere. The distance between the two curvatures separates them to a point of been evaluated as two different pressure vessels. Thus, the discontinuity occurs at the intersection of two shells openly and by strengthening the ring (Pietraszkiewicz and Konopinska, 23). (shodhganga, 36) Discontinuity stresses also occurs at pressure vessels with hemispherical heads under internal pressure. The shell is mainly in a membrane state of stress where the thickness of the wall is constant. Thus, using the axisymmetric and anti-symmetric outside loadings, the deformations/ discontinuity stress of the vessels occur. It can be attained through: d (Rφ Nφ) / dφ + Nθ Rφ cosφ = 0 Nθ / Rθ + Nφ / Rφ – P = 0 θ and φ represents circumferential and meridional directions Nθ Nφ = stress resultsants where it is attained through multiplying the normal stress with the thickness of the cylinder Rθ Rφ = radii of the curvature where θφ represents the internal pressure directions (shodhganga, 23). In regard to the pressurized thin axisymmetric shell of revolution, the shell is formed conceptually through meridian rotating, which is a curved lined that is usually separated through the meridional plane on the z-axis resulting to a disposed thickness, where the resultant shell is influenced by the internal pressure. The shells are similar to the radial stresses influenced by membrane stress attained from the equilibrium. The following equations provide the discontinuity of stresses among several pressure vessels. Cylinder rθ   =   D/2     where the given value of   rφ   inclines to infinity σθ   =   2σφ   =   pD/2t. it represents and is similar to the thin walled stress equation. Sphere rθ   =   rφ   =   D / 2       σθ   =   σφ   =   pD / 4t Works Cited Bennitz, Anders, Niklas Grip and W, Jacob Schmidt. "Thick-walled cylinder theory applied on a conical wedge anchorage." Meccanica (2011): 959 -977. Dornfeld, W, H. "Thick-Walled Cylinders and Press Fits." 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"Set of field equations for thick shell of revolution made of functionally graded materials in curvilinear coordinate system." Mechanika (2009): 18-26. Ozturk, Fahrettin. "Finite-element modelling of two-disc shrink fit assembly and an evaluation of material pairs of discs." Sage Journals. Journal of Mechanical Engineering (2012): 96-123. Pietraszkiewicz, W and V Konopinska. "Junctions at Shell Structures: A Review. Thin Walled Structures." (2012): 1-46. http://www.imp.gda.pl/files/wp/wppub/2012/Praca10.pdf. Redekop, D. "Vibration analysis of a torus–cylinder shell assembly." Journal of Sound and Vibration (2004): 919 - 930. shodhganga. "Pressure Vessel Heads with Different Geometries." Chapter 3 (2009): 22 - 38. http://shodhganga.inflibnet.ac.in/bitstream/10603/25523/9/09_chapter%203.pdf. washington.edu. "Thick Walled Cylinders." Washington Edu. courses (2002): 1 - 8. http://courses.washington.edu/me354a/Thick%20Walled%20Cylinders.pdf. Read More