STRESSSES IN THICK CYLINDER CLASS NOTES FOR MECHANICAL ENGINEERING
STRESSES IN THICK CYLINDER
CLASS NOTES FOR MECHANICAL
ENGINEERING
Stresses are uniform in thin shells. Stress
at the inner radius is equal to the stress at
the outer radius. There is no variation of
hoop, longitudinal and radial stresses.
However, when pressure is high, thin
walled vessels are no longer useable.
Use a thick walled vessel. High pressures
exist in hydraulic presses and steam pipes.
A thick vessel is one when D/t <20 i.e.
wall thickness is more. In these vessels,
stresses are not uniform. These vary from
inner radius to the outer radius.
Internal and external pressure
Thick shells may be under both internal and external pressures. Here vessel with higher internal pressure ( p_{i} > p_{o}) is considered. Analysis of the thick walled vessels, consider types of pressures subjected. These are

Thick vessels subjected only internal pressure

Subjected to only external pressure

Vessels subjected to both internal and external pressures such that internal pressure is much greater than the internal pressure
STRESSES AND THEIR NATURE IN THICK CYLINDER
Magnitude and the nature of stresses depend on the magnitude of pressure. The various possibilities are:
Hoop stress or Tangential stress or Circumferential stress(σ_{h})
It acts in the tangential direction at the point of consideration

With internal pressure alone, p_{i} , σ_{h} is tensile from the inner radius to the outer radius
 Fig. Stresses in a Single Wall Thick Shell due to Internal Pressure Only

External pressure only, p_{o} , σ_{h} is compression from the inner radius to the outer radius

Both with internal and external pressures with p_{i} > p_{o}, σ_{h} is tensile from the inner radius to the outer radius

Fig. Stresses in a Single Wall Thick Shell due to Both Internal & External Pressures
Longitudinal stress (σ_{l}) THICK CYLINDER
It acts in the longitudinal direction at the point of consideration

Internal pressure alone, p_{i} , σ_{l} is tensile from the inner radius to the outer radius

External pressure only, p_{o} , σ_{l} is compressive from the inner radius to the outer radius

The Internal and external pressures with p_{i} > p_{o}, σ_{l} is tensile from the inner radius to the outer radius
Radial stress(σ_{r}) thick cylinder
It acts in the radial direction at the point of consideration


Only under internal pressure, p_{i} , σ_{r} is compressive from the inner radius to the outer radius

External pressure alone, p_{o} , σ_{r} is compressive from the inner radius to the outer radius

With both internal and external pressure with p_{i} > p_{o}, σ_{r} is compressive from the inner radius to the outer radius

1. Stresses Only under internal pressure in thick cylinder
p_{i } is there and p_{o} =0
σ_{hmax} = [p_{i}(r_{i}^{2}+r_{o}^{2}]/ (r_{o}^{2} –r_{i}^{2})
stress σ_{hmax} > p_{i}
σ_{r max} = p_{i}
σ_{hmax} > σ_{r max}
Both σ_{hmax} and σ_{r max} occur at the innermost radius and σ_{hmax} is always greater than the σ_{r max}. Refer Fig. above.
Hydraulic pipes, tanks and steam pipes fall in this category.
Since σ_{hmax} > p_{i}, and hence p_{i} can never be greater than the elastic limit stress of the pipe material. Large internal pressure occur as in guns and hydraulic machines. Hoop shrinking produces thick pipes.
2.Case of external pressure only in thick cylinder
p_{i}=0 and p_{o} is there. σ_{hmax} will still be at the innermost radius but will be compressive in nature. Radial stress will be maximum at the outermost radius and will be equal to p_{o}.
σ_{h max} = –2p_{o}r_{o}^{2}/ (r_{o}^{2} –r_{i}^{2}) at r= r_{i}
σ_{r max} = p_{0} at r = r_{o}
Fig..
Hence in this case maximum values occur at separate locations. Only external pressure does not exist in thick walled pressure vessels. It is in case of hoop shrinking. Inner cylinder comes under external pressure. Outer cylinder comes under internal pressure.
NON UNIFORM STRESSES IN A SINGLE WALL Thick CYLINDER
In a thick vessel of single wall, stresses will highly non uniform from the inner radius to the outer radius. It will result in uneconomical use of the material. In engineering, firstly it is desirable that the stresses be uniform. Secondly, it is almost impossible to make a thick walled pressure vessel from a single thick sheet. Two or more thin walled vessels produce a thick vessel. It makes manufacturing easy as well as it results in the uniformity of stress in the wall of the vessel.
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PRESTRESSING AND RADIAL DEFLECTION
Fig. Prestresses in the Cylinder due to shrinkage pressure
Fig. Prestresses in the jacket due to shrinkage pressure
Fig. Prestresses and resultant stress variations
PRESTRESSES IN CYLINDER (due to hoop shrinking)
AB stress = –2p_{s}r_{2}^{2}/ (r_{2}^{2} –r_{1}^{2}) compressive
CD stress = p_{s}(r_{3}^{2}+r_{2}^{2})/ (r_{3}^{2} –r_{2}^{2}) Tensile
STRESSES DUE TO FLUID PRESSURE
AA’ stress= p((r_{3}^{2}+r_{1}^{2})/ (r_{3}^{2} –r_{1}^{2})
CC’ stress= pr_{1}^{2}((r_{3}^{2}+r_{2}^{2})/[r_{2}^{2}(r_{3}^{2} –r_{1}^{2})]
FINAL RESULTANT STRESSES
Final stress at A = AA’ – AB=p((r_{3}^{2}+r_{1}^{2})/ (r_{3}^{2} –r_{1}^{2}) — 2p_{s}r_{2}^{2}/ (r_{2}^{2} –r_{1}^{2})
Resultant stress at C = CC’ + CD =pr_{1}^{2}((r_{3}^{2}+r_{2}^{2})/[r_{2}^{2}(r_{3}^{2} –r_{1}^{2})]+ p_{s}(r_{3}^{2}+r_{2}^{2})/ (r_{3}^{2} –r_{2}^{2})
Final resultant stresses come the combination of prestresses and stresses due to fluid pressure. These are more uniform as compared to stresses in a single walled thick vessel.
ADVANTAGES OF PRESTRESSING

Stresses are more uniform.

Vessel of certain thickness withstands higher fluid pressures.

Requires less thickness for a certain pressure.

Less stress produces due to a certain fluid pressure.
MOST EFFECTIVE PROPORTIONS OF THE CYLINDER AND THE JACKET UNDER PRESTRESSING
Take final stresses at A and C equal to allowable stress. Therefore, the equations become
Final stress at A = AA’ – AB=p((r_{3}^{2}+r_{1}^{2})/ (r_{3}^{2} –r_{1}^{2}) — 2p_{s}r_{2}^{2}/ (r_{2}^{2} –r_{1}^{2}) = σ_{allow}
Resultant stress at C = CC’ + CD
=pr_{1}^{2}((r_{3}^{2}+r_{2}^{2})/[r_{2}^{2}(r_{3}^{2} –r_{1}^{2})]+ p _{s }(r_{3}^{2}+r_{2}^{2})/ (r_{3}^{2} –r_{2}^{2})=σ_{allow}
Known are p, r1 and σ_{allow} determine the proportions. Unknown are r_{2}, r_{3} and p_{s}. Here assume resultant stresses at A and C are equal to the elastic strength of the material. Heating temperatures are such that the material of the vessel does not melt. Now if we take r_{2} =Cr_{1} and r_{3}= C^{2}r_{1} in which C is a constant for a set of values for p and σ_{allow.}
Radial deflection
HOOP SHRINKING THICK CYLINDER
Hoop shrinking produces a thick walled vessel from two or more thin walled vessels. Take two thin walled vessels. one is inner cylinder and other is outer cylinder (jacket). The inner radius of the jacket is slightly smaller than the outer radius of the inner cylinder. Heat the Jacket till its inner radius becomes slightly greater than the outer radius of the cylinder. Then slip the heated jacket (called the HOOP) on to the inner cylinder. Allow the jacket to cool. Jacket grips the cylinder tightly. The cylinder comes in compression. The jacket becomes in tension.
Let p_{s} is the shrinkage pressure after hoop shrinking at the common surface. This pressure p_{s} will be external pressure for the cylinder and internal pressure for the jacket.
Let the radii are as under;
Firstly r_{1} is the inner radius of the cylinder
Secondly r_{2} is the common radius =outer radius of cylinder
Thirdly r_{2} is the common radius = inner radius of the jacket
Fourthly r_{3} is the outer radius of the jacket
The radial displacement is due to change in circumference cylinder and jacket at the common radius.
Let dr_{2} be the change in radius at radius r_{2}.
Change in circumference = 2 π dr_{2}
Original circumference = 2π r_{2}
Circumferential strain= 2 π dr_{2}/2π r_{2} = dr_{2}/r_{2}
Circumferential strain
Consider it in two steps:
(i) first for the cylinder
(ii) Secondly for the jacket
CIRCUMFERENTIAL STRAIN FOR THE CYLINDER (ONLY EXTERNAL PRESSURE)
dr_{2}/r_{2} = (σ_{h}/E –μσ_{r}/E) at the radius r_{2} (7)
σ_{h} at r_{2} due to only external p_{s }
hoop stress σ_{h}= (–p_{s} r_{2}^{2} –p_{s}r_{1}^{2}) /((r_{2}^{2} –r_{1}^{2})
= –p_{s }((r_{2}^{2}+r_{1}^{2}]/ (r_{2}^{2} –r_{1}^{2}) Compressive
σ_{h} = p_{s}
σ_{r} at r_{2} due to only p_{s }= –p_{s } Compressive
Substituting the values in eq (7), we get
dr_{2}/r_{2} = (p_{s} r_{2}^{2}/E)[(r_{2}^{2}+r_{1}^{2}]/ (r_{2}^{2} –r_{1}^{2}) –μ]
(dr_{2})_{Cyl} = (–p_{s} r_{2}/E) [((r_{2}^{2}+r_{1}^{2})/((r_{2}^{2} –r_{1}^{2}) –μ)] (8)
CIRCUMFERENTIAL STRAIN FOR THE JACKET (ONLY INTERNAL PRESSURE)
Radii are r_{2} and r_{3}
dr_{2}/r_{2} =( σ_{h}/E –μσ_{r}/E) at the radius r_{2} in the jacket (9)
Hoop stress σ_{h} at r_{2} due to only internal pressure p_{s} will be
σ_{h} = p_{s}(r3^{2}+r2^{2})/ (r_{3}^{2} –r_{2}^{2}) Tensile
Radial stress σ_{r} at r_{2} due to only internal p_{s }= –p_{s } Compression_{ }Substituting the values in eq (9), we get
dr_{2}/r_{2} = [p_{s}(r3^{2}+r2^{2}/ (r_{3}^{2} –r_{2}^{2})] /E — μ(p_{s})/E
(dr_{2})_{Jacket} = (p_{s} r_{2}/E) [(r_{3}^{2} +r_{2}^{2})/((r_{3}^{2} –r_{2}^{2}) +μ)] (10)
Jacket internal radius will increase by dr_{2}.
FINAL RADIAL DEFLECTION
Final change will be the sum of the two changes in dr_{2}. While adding take only with positive sign. It is because dr_{2} is increase in jacket and decrease for cylinder.
Total dr_{2} = Initial difference in the radii
=(dr_{2})Jacket +(dr_{2})Cylinder
=(p_{s} r_{2}/E) [(r_{3}^{2}+r_{2}^{2})/((r_{3}^{2} –r_{2}^{2})–μ)]+ (p_{s} r_{2}/E) [(r_{2}^{2} +r_{1}^{2})/((r_{2}^{2} –r_{1}^{2}) +μ)]
Rad Def= (p_{s} r_{2}/E) [(r_{3}^{2}+r_{2}^{2})/((r_{3}^{2} –r_{2}^{2})+ (r_{2}^{2} +r_{1}^{2})/((r_{2}^{2} –r_{1}^{2}) ]
CONCLUSION
Prestresses develop in the vessel during fabrication. These prestresses are compressive in the cylinder. These prestresses are tensile stresses in the jacket. This results in uniform stresses in entire wall of the thick shell under fluid pressure. Thus these prestresses are useful.
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