STRESSSES IN THICK CYLINDER CLASS NOTES FOR MECHANICAL ENGINEERING
STRESSES IN THICK CYLINDER
CLASS NOTES FOR MECHANICAL
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 ( pi > po) 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(σh)
It acts in the tangential direction at the point of consideration
With internal pressure alone, pi , σh 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, po , σh is compression from the inner radius to the outer radius
Both with internal and external pressures with pi > po, σh 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 (σl) THICK CYLINDER
It acts in the longitudinal direction at the point of consideration
Internal pressure alone, pi , σl is tensile from the inner radius to the outer radius
External pressure only, po , σl is compressive from the inner radius to the outer radius
The Internal and external pressures with pi > po, σl is tensile from the inner radius to the outer radius
Radial stress(σr) thick cylinder
It acts in the radial direction at the point of consideration
Only under internal pressure, pi , σr is compressive from the inner radius to the outer radius
External pressure alone, po , σr is compressive from the inner radius to the outer radius
With both internal and external pressure with pi > po, σr is compressive from the inner radius to the outer radius
1. Stresses Only under internal pressure in thick cylinder
pi is there and po =0
σhmax = [pi(ri2+ro2]/ (ro2 –ri2)
stress σhmax > pi
σr max = pi
σhmax > σr max
Both σhmax and σr max occur at the innermost radius and σhmax is always greater than the σr max. Refer Fig. above.
Hydraulic pipes, tanks and steam pipes fall in this category.
Since σhmax > pi, and hence pi 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
pi=0 and po is there. σhmax 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 po.
σh max = –2poro2/ (ro2 –ri2) at r= ri
σr max = p0 at r = ro
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.
PRESTRESSING AND RADIAL DEFLECTION
Fig. Pre-stresses in the Cylinder due to shrinkage pressure
Fig. Prestresses in the jacket due to shrinkage pressure
Fig. Prestresses and resultant stress variations
PRE-STRESSES IN CYLINDER (due to hoop shrinking)
AB stress = –2psr22/ (r22 –r12) compressive
CD stress = ps(r32+r22)/ (r32 –r22) Tensile
STRESSES DUE TO FLUID PRESSURE
AA’ stress= p((r32+r12)/ (r32 –r12)
CC’ stress= pr12((r32+r22)/[r22(r32 –r12)]
FINAL RESULTANT STRESSES
Final stress at A = AA’ – AB=p((r32+r12)/ (r32 –r12) — 2psr22/ (r22 –r12)
Resultant stress at C = CC’ + CD =pr12((r32+r22)/[r22(r32 –r12)]+ ps(r32+r22)/ (r32 –r22)
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 PRE-STRESSING
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((r32+r12)/ (r32 –r12) — 2psr22/ (r22 –r12) = σallow
Resultant stress at C = CC’ + CD
=pr12((r32+r22)/[r22(r32 –r12)]+ p s (r32+r22)/ (r32 –r22)=σallow
Known are p, r1 and σallow determine the proportions. Unknown are r2, r3 and ps. 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 r2 =Cr1 and r3= C2r1 in which C is a constant for a set of values for p and σallow.
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 ps is the shrinkage pressure after hoop shrinking at the common surface. This pressure ps will be external pressure for the cylinder and internal pressure for the jacket.
Let the radii are as under;
Firstly r1 is the inner radius of the cylinder
Secondly r2 is the common radius =outer radius of cylinder
Thirdly r2 is the common radius = inner radius of the jacket
Fourthly r3 is the outer radius of the jacket
The radial displacement is due to change in circumference cylinder and jacket at the common radius.
Let dr2 be the change in radius at radius r2.
Change in circumference = 2 π dr2
Original circumference = 2π r2
Circumferential strain= 2 π dr2/2π r2 = dr2/r2
Consider it in two steps:
(i) first for the cylinder
(ii) Secondly for the jacket
CIRCUMFERENTIAL STRAIN FOR THE CYLINDER (ONLY EXTERNAL PRESSURE)
dr2/r2 = (σh/E –μσr/E) at the radius r2 (7)
σh at r2 due to only external ps
hoop stress σh= (–ps r22 –psr12) /((r22 –r12)
= –ps ((r22+r12]/ (r22 –r12) Compressive
σh = -ps
σr at r2 due to only ps = –ps Compressive
Substituting the values in eq (7), we get
dr2/r2 = (-ps r22/E)[(r22+r12]/ (r22 –r12) –μ]
(dr2)Cyl = (–ps r2/E) [((r22+r12)/((r22 –r12) –μ)] (8)
CIRCUMFERENTIAL STRAIN FOR THE JACKET (ONLY INTERNAL PRESSURE)
Radii are r2 and r3
dr2/r2 =( σh/E –μσr/E) at the radius r2 in the jacket (9)
Hoop stress σh at r2 due to only internal pressure ps will be
σh = ps(r32+r22)/ (r32 –r22) Tensile
Radial stress σr at r2 due to only internal ps = –ps Compression Substituting the values in eq (9), we get
dr2/r2 = [ps(r32+r22/ (r32 –r22)] /E — μ(-ps)/E
(dr2)Jacket = (ps r2/E) [(r32 +r22)/((r32 –r22) +μ)] (10)
Jacket internal radius will increase by dr2.
FINAL RADIAL DEFLECTION
Final change will be the sum of the two changes in dr2. While adding take only with positive sign. It is because dr2 is increase in jacket and decrease for cylinder.
Total dr2 = Initial difference in the radii
=(ps r2/E) [(r32+r22)/((r32 –r22)–μ)]+ (ps r2/E) [(r22 +r12)/((r22 –r12) +μ)]
Rad Def= (ps r2/E) [(r32+r22)/((r32 –r22)+ (r22 +r12)/((r22 –r12) ]
Pre-stresses develop in the vessel during fabrication. These pre-stresses 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 pre-stresses are useful.
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