The Thick Walled Cylinder CHAPTER 11 The Thick Walled Cylinder EGM 5653 Advanced Mechanics of Materials

Introduction This chapter deals with the basic relations for axisymmetric deformation of a thick walled cylinder In most applications cylinder wall thickness is constant, and is subjected to uniform internal pressure p1, uniform external pressure p2, and a temperature change ΔT The practical application of mathematics and science to create, design, test, improve, and develop knowledge, research, money, business, economics, and technology. This is why engineering is such a challenging and demanding field of study: It involves areas of expertise that continue to evolve independently yet are required to perform together as part of the engineering process. Thus, an engineer must be expert in many areas, must know how to communicate knowledge between those areas, and must apply that knowledge to create, design, study, research, and invent all kinds of things. It is not uncommon for engineers to begin their careers as mathematicians, applied scientists, or even economists

Introduction contd. Solutions are derived for open cylinders or for cylinders with negligible “end cap” effects. The solutions are axisymmetrical- function of only radial coordinate r Thick walled cylinders are used in industry as pressure vessels, pipes, gun tubes etc..

11.1 Basic Relations Equations of equilibrium derived neglecting the body force Strain- Displacement Relations and Compatibility Condition Three relations for extensional strain are where u= u(r,z) and w= w(r,z) are displacement components in the r and z directions

11.1 Basic Relations Contd. At sections far from the end shear stress components = 0 and we assume εzz = constant. Therefore, by eliminating u = u(r)

11.1 Basic Relations Contd. Stress Strain Temperature Relations The Cylinder material is assumed to be Isotropic and linearly elastic The Stress- Strain temperature relations are: The practical application of mathematics and science to create, design, test, improve, and develop knowledge, research, money, business, economics, and technology. This is why engineering is such a challenging and demanding field of study: It involves areas of expertise that continue to evolve independently yet are required to perform together as part of the engineering process. Thus, an engineer must be expert in many areas, must know how to communicate knowledge between those areas, and must apply that knowledge to create, design, study, research, and invent all kinds of things. It is not uncommon for engineers to begin their careers as mathematicians, applied scientists, or even economists Where, E is Modulus of Elasticity ν is Poisson’s ratio α is the Coefficient of linear thermal expansion ΔT is the change in the temp. from the uniform reference temp.

11.2 Closed End Cylinders Stress Components at sections far from the ends The expressions for the stress components σrr,σθθ, σzz for a cylinder with closed ends and subjected to internal pressure p1, external pressure p2, axial load P and temperature change ΔT. From the equation of equilibrium, the strain compatibility equation and the stress- strain temperature relations we get the differential expression, The practical application of mathematics and science to create, design, test, improve, and develop knowledge, research, money, business, economics, and technology. This is why engineering is such a challenging and demanding field of study: It involves areas of expertise that continue to evolve independently yet are required to perform together as part of the engineering process. Thus, an engineer must be expert in many areas, must know how to communicate knowledge between those areas, and must apply that knowledge to create, design, study, research, and invent all kinds of things. It is not uncommon for engineers to begin their careers as mathematicians, applied scientists, or even economists .

11.2 Closed End Cylinders Stress Components at sections far from the ends contd. Eliminating the stress component σθθ and integrating, we get Using this in the previous expression and evaluating σθθ, we get The practical application of mathematics and science to create, design, test, improve, and develop knowledge, research, money, business, economics, and technology. This is why engineering is such a challenging and demanding field of study: It involves areas of expertise that continue to evolve independently yet are required to perform together as part of the engineering process. Thus, an engineer must be expert in many areas, must know how to communicate knowledge between those areas, and must apply that knowledge to create, design, study, research, and invent all kinds of things. It is not uncommon for engineers to begin their careers as mathematicians, applied scientists, or even economists

11.2 Closed End Cylinders Stress Components at sections far from the ends contd. The effects of temperature are self- equilibrating. The expression for εzz at section far away from the closed ends of the cylinder can be written in the form The expression for σzz at section far away from the ends of the cylinder can be written in the form

Open Cylinder No axial loads applied on its ends. The equilibrium equation of an axial portion of the cylinder is: The expression for εzz and σzz may be written in the form

11.3 Stress Components and Radial Displacement for Constant Temperature For a closed cylinder (with end caps) in the absence of temperature change ΔT = 0 the stress components are obtained as The practical application of mathematics and science to create, design, test, improve, and develop knowledge, research, money, business, economics, and technology. This is why engineering is such a challenging and demanding field of study: It involves areas of expertise that continue to evolve independently yet are required to perform together as part of the engineering process. Thus, an engineer must be expert in many areas, must know how to communicate knowledge between those areas, and must apply that knowledge to create, design, study, research, and invent all kinds of things. It is not uncommon for engineers to begin their careers as mathematicians, applied scientists, or even economists .

11.3.2 Radial Displacement for an Closed Cylinder The radial displacement u for a point in a thick wall closed cylinder may be written as 11.3.3 Radial Displacement for an Open Cylinder

Example 11.2 Stresses and Deformations in Hollow cylinder A thick walled closed-end cylinder is made of an aluminum alloy, E = 72 GPa, ν = 0.33,Inner Dia. = 200mm, Outer Dia. = 800 mm, Internal Pressure = 150 MPa. Determine the Principal stresses, Maximum shear stress at the inner radius (r= a = 100 mm), and the increase in the inside diameter caused by the internal pressure Solution: The Principal stresses for the conditions that

Example 11.2 Stresses and Deformations in Hollow cylinder contd. The maximum shear stress is given by the equation The increase in the inner diameter caused by the internal pressure is equal to twice the radial displacement for the conditions The increase in the internal pressure caused by the internal pressure is 0.6006 mm

11.4.1 Failure of Brittle Materials 11.4 Criteria of Failure Recap Maximum Principal stress criterion – Design of Brittle isotropic materials – If the principal stress of largest magnitude is the tensile stress Maximum shear stress or the Octahedral shear- stress criterion – Design of Ductile isotropic materials 11.4.1 Failure of Brittle Materials Condition for Failure Maximum principal stress = Ultimate tensile Strength σu At sections far removed from the ends Maximum Principal stress = Circumferential stress σθθ(r=a) or axial stress σzz

11.4.2 Failure of Ductile Materials General Yielding Failure Yielding at sections other than the points of stress concentration Thick walled cylinders occasionally subjected to static loads or peak loads Member has yielded over a considerable region as with fully plastic loads Fatigue failure Subjected to repeated pressurizations (loading and unloading) Found predominantly around the region of stress concentration Maximum shear stress or maximum octahedral shear stress to be determined at the regions of stress concentration The practical application of mathematics and science to create, design, test, improve, and develop knowledge, research, money, business, economics, and technology. This is why engineering is such a challenging and demanding field of study: It involves areas of expertise that continue to evolve independently yet are required to perform together as part of the engineering process. Thus, an engineer must be expert in many areas, must know how to communicate knowledge between those areas, and must apply that knowledge to create, design, study, research, and invent all kinds of things. It is not uncommon for engineers to begin their careers as mathematicians, applied scientists, or even economists .

11.4.3 Material Response Data for Design General Yielding Property : Yield stress Criteria : Maximum shear stress or Octahedral shear stress Fatigue Failure Property : Fatigue strength Criteria : Maximum shear stress and Octahedral shear stress in conjunction Values from tensile test specimen and hollow thin-wall tube Values obtained from tests of either a tension specimen or hollow thin walled cylinder in torsion The hollow thin walled cylinder specimen values led to more accurate prediction of thick walled cylinders than the tension specimen The critical state of stress is usually at the inner wall , for a pressure loading it is only pure shear in addition to hydrostatic state of stress The practical application of mathematics and science to create, design, test, improve, and develop knowledge, research, money, business, economics, and technology. This is why engineering is such a challenging and demanding field of study: It involves areas of expertise that continue to evolve independently yet are required to perform together as part of the engineering process. Thus, an engineer must be expert in many areas, must know how to communicate knowledge between those areas, and must apply that knowledge to create, design, study, research, and invent all kinds of things. It is not uncommon for engineers to begin their careers as mathematicians, applied scientists, or even economists

11.4.3 Material Response Data for Design contd. Since for most materials the hydrostatic stress does not affect yielding yielding is caused by the pure shear Hence maximum shear- stress criterion and octahedral shear-stress criterion predict with errors of <1% in comparison with 15.5 % 11.4.4 Ideal Residual Stress Distributions for Composite Open Cylinders The practical application of mathematics and science to create, design, test, improve, and develop knowledge, research, money, business, economics, and technology. This is why engineering is such a challenging and demanding field of study: It involves areas of expertise that continue to evolve independently yet are required to perform together as part of the engineering process. Thus, an engineer must be expert in many areas, must know how to communicate knowledge between those areas, and must apply that knowledge to create, design, study, research, and invent all kinds of things. It is not uncommon for engineers to begin their careers as mathematicians, applied scientists, or even economists Stress distributions in a closed cylinder at initiation of yielding (b=2a)

11.4.4 Ideal Residual Stress Distributions for Composite Open Cylinders contd. The practical application of mathematics and science to create, design, test, improve, and develop knowledge, research, money, business, economics, and technology. This is why engineering is such a challenging and demanding field of study: It involves areas of expertise that continue to evolve independently yet are required to perform together as part of the engineering process. Thus, an engineer must be expert in many areas, must know how to communicate knowledge between those areas, and must apply that knowledge to create, design, study, research, and invent all kinds of things. It is not uncommon for engineers to begin their careers as mathematicians, applied scientists, or even economists Stress distributions in composite cylinder made of brittle material that fails at inner radius of both cylinders simultaneously (a) Residual stress distributions (b) Total stress distributions

11.4.4 Ideal Residual Stress Distributions for Composite Open Cylinders contd. Stress distributions in composite cylinder made of ductile material that fails at inner radius of both cylinders simultaneously (a) Residual stress distributions (b) Total Stress distributions

Example 11.5 Yield of a Composite Thick Wall Cylinder Problem: Consider a Composite cylinder, Inner cylinder: Inner Radii a =10 mm and outer radii ci=25.072 mm Outer cylinder: Inner Radii co =25 mm and outer radii b= 50 mm Ductile steel ( E=200 GPa and ν=0.29) Determine minimum yield stress for a factor of safety SF=1.75 Solution: It is necessary to consider the initiation of yielding for the inside of both the cylinders.σzz =0 for both the cylinders At the inside of the inner cylinder, the radial and circumferential stresses for a pressure (SF)p1 are,

Example 11.5 Yield of a Composite Thick Wall Cylinder contd. At the inside of the outer cylinder, the radial and circumferential stresses for a pressure (SF)p1 are, In an ideal design, the required yield stress will be the same for the inner and the outer cylinders.