1. The Thick Walled Cylinder
EGM 5653 Advanced Mechanics of Materials

2. 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

3. 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..

4. 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

5. 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)

6. 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.

7. 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 .

8. 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

9. 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

10. 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. 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 .

12. 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
Radial Displacement for an Open Cylinder

13. 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

14. 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 mm

15. 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
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

16. 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 .

17. 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

18. 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 %
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)

19. 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

20. 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

21. 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= 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,

22. 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.