is). Pressure vessels are very ante y a common in many industries. The purpose of this exercise is to examine how stresses evolve within the walls of a pressure vessel as a function of position within the wall and wall thickness. Recall the thick-walled cylinder solution developed in class. Re-write the solution to the case of a thick-walled cylinder with inner radius a, outer radius b, internal pressure p₁, and external pressure, p2. Orr = 000= + a²p₁-b²p₂ (P2 − P₁)a²b² 1 (b²-a²) (6²-a²) 2¹ a²p₁-b²p2 (6²-a²) - 1 (p2 - P₁)a²b² 1 (b²-a²) r2 (a) Draw a diagram showing the evolution of orr and one from a to b when a/b-0.5 and P2 = 0. (b) Make the assumption that p2 = 0, a + b 2r, b-a≈t, and a b≈r to derive the expression for Orr and one for a thin-walled pressure vessel. For a sanity check, your solution should be orr = 0 and 000 = pir/t. (c) Now consider the classical strength of materials approach to pressure vessels. Imagine the pressure vessel shown in Fig. 2 (note, the radius of the thin-walled cylinder is r). Taking a section cut through the cylinder parallel to its axis and examining the balance of forces acting per unit length, L, in one direction that is balanced by the force generated by e acting through a section of length L and thickness t, verify your calculation from (b). Repeat this for to balance forces through a cut made perpendicular to the cylinder axis to find the axial stress in the thin-walled pressure vessel, da = (pir)/(2t). (d) Consider the combination of conditions that are possible in the axi-symmetric thick- walled cylinder problem: imposed displacements at r = a and/or r = b and imposed tractions at r = a and/or r = b. List (in words) all combinations of boundary conditions that are possible and the general equations you would use to solve for constants. You do not need to solve for the constants. You may assume plane strain or plane stress. Hint: there are four possible combinations.
is). Pressure vessels are very ante y a common in many industries. The purpose of this exercise is to examine how stresses evolve within the walls of a pressure vessel as a function of position within the wall and wall thickness. Recall the thick-walled cylinder solution developed in class. Re-write the solution to the case of a thick-walled cylinder with inner radius a, outer radius b, internal pressure p₁, and external pressure, p2. Orr = 000= + a²p₁-b²p₂ (P2 − P₁)a²b² 1 (b²-a²) (6²-a²) 2¹ a²p₁-b²p2 (6²-a²) - 1 (p2 - P₁)a²b² 1 (b²-a²) r2 (a) Draw a diagram showing the evolution of orr and one from a to b when a/b-0.5 and P2 = 0. (b) Make the assumption that p2 = 0, a + b 2r, b-a≈t, and a b≈r to derive the expression for Orr and one for a thin-walled pressure vessel. For a sanity check, your solution should be orr = 0 and 000 = pir/t. (c) Now consider the classical strength of materials approach to pressure vessels. Imagine the pressure vessel shown in Fig. 2 (note, the radius of the thin-walled cylinder is r). Taking a section cut through the cylinder parallel to its axis and examining the balance of forces acting per unit length, L, in one direction that is balanced by the force generated by e acting through a section of length L and thickness t, verify your calculation from (b). Repeat this for to balance forces through a cut made perpendicular to the cylinder axis to find the axial stress in the thin-walled pressure vessel, da = (pir)/(2t). (d) Consider the combination of conditions that are possible in the axi-symmetric thick- walled cylinder problem: imposed displacements at r = a and/or r = b and imposed tractions at r = a and/or r = b. List (in words) all combinations of boundary conditions that are possible and the general equations you would use to solve for constants. You do not need to solve for the constants. You may assume plane strain or plane stress. Hint: there are four possible combinations.
Elements Of Electromagnetics
7th Edition
ISBN:9780190698614
Author:Sadiku, Matthew N. O.
Publisher:Sadiku, Matthew N. O.
ChapterMA: Math Assessment
Section: Chapter Questions
Problem 1.1MA
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Transcribed Image Text:ante y se IN..
is). Pressure vessels are very
common in many industries. The purpose of this exercise is to examine how stresses evolve
within the walls of a pressure vessel as a function of position within the wall and wall thickness.
Recall the thick-walled cylinder solution developed in class. Re-write the solution to the case
of a thick-walled cylinder with inner radius a, outer radius b, internal pressure p₁, and external
pressure, p2.
Orr
σαθ
a²p₁b²p2
(6² - a²)
a²p₁-b²p2
(6²-a²)
axis
+
1
(a) Draw a diagram showing the evolution of orr and σ00 from a to b when a/b=0.5 and
P2 = 0.
(P2 - P₁)a²6² 1
(6²-a²) 2¹
(P2 - P₁)a²6² 1
(b²-a²) 2
(b) Make the assumption that p2 = 0, a + b 2r, bat, and abr to derive the
expression for rr and 000 for a thin-walled pressure vessel. For a sanity check, your
solution should be orr = 0 and 000 = pir/t.
t
(c) Now consider the classical strength of materials approach to pressure vessels. Imagine
the pressure vessel shown in Fig. 2 (note, the radius of the thin-walled cylinder is r).
Taking a section cut through the cylinder parallel to its axis and examining the balance of
forces acting per unit length, L, in one direction that is balanced by the force generated
by e acting through a section of length L and thickness t, verify your calculation from
(b). Repeat this for to balance forces through a cut made perpendicular to the cylinder
axis to find the axial stress in the thin-walled pressure vessel, σa = (p₁r)/(2t).
(d) Consider the combination of conditions that are possible in the axi-symmetric thick-
walled cylinder problem: imposed displacements at r = a and/or r = b and imposed
tractions at r = a and/or r = b. List (in words) all combinations of boundary conditions
that are possible and the general equations you would use to solve for constants. You do
not need to solve for the constants. You may assume plane strain or plane stress. Hint:
there are four possible combinations.
р
000
000
Cut parallel to cylinder axis
L
σα
Figure 2: Diagram of thin-walled cylinder / pressure vessel.
p
Cut perpendicular to cylinder axis
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