REVIEW

Membrane Theory

Calculation of R 1 and R 2 1.Cylindrate shell R 1 = ∞ R 2 = D / 2 δ D p

Spherical shell subjected to uniform gas internal pressure: uniform gas internal pressure: R 1 = R 2 = D / 2 R 1 = R 2 = D / 2 D δ

3.Elliptical shell Known: Major semiaxis – a Short semiaxis - b y x a b A(x,y)...  k2k2 k1k1 Special Points: x=0: R 1 =R 2 =a 2 /b x=a : R 1 =b 2 /a, R 2 =a Special Points: x=0: R 1 =R 2 =a 2 /b x=a : R 1 =b 2 /a, R 2 =a

Conical shell Find R 1 and R 2 of point A: R 1 = ∞ R 2 = A K 2 = r / cos Small end: R 1 = ∞ R 1 = ∞ R 2 = 0 R 2 = 0 D  r. A k2k2

For sphericalsegment: (Point C and B) (Point C and B) R1=R2=R R1=R2=R For knuckle segment of transition section : R 1 = r 1, R 2 = Point B: R1= r 1, R 2 = R Point A: R1= r 1, R 2 = r For cylindrical shell: Point A and D R 1 =∞, R 2 = r Dished shell r r r1r1 r1r1

Basic calculation equation of membrane stress :

Standard elliptical head cylinder Conical head P=2MPa ， D=1000mm ， c=500mm ， S=10mm ， α=45° 1)σm,σθ at Points A, B and C 2)Stress distribution Example 1

Point A Point A

Point B Point B

Point C

Stress Distribution

Example 2 Stresses at points A, B and C Point A

Point B

Point C

Standard elliptical heads:  The elliptical heads whose ratio of major and short semiaxis a / b = 2 are called standard elliptical heads.  a / b = 2 —— x=0 (Top): x=0 (Top): x=a (Boundary): x=a (Boundary):  mm a / b = 2

Conical shell subjected to uniform gas internal pressure D  r. A k2k2 mm 

Strength Design of Cylinders and Heads subjected to Internal-Pressure Se=Sn-C1-C2

Equation of strength verification:

Equation of [p w ] —— the maximum allowable working pressure

Design 1 A pressure vessel of 15MnVR with an inside diameter 2000mm is subjected to an internal pressure of 1.8MPa,and it was equipped with rupture disks. For 15MnVR, allowable stress is 171 MPa at design temperature (300 ℃ ), yield stress and tensile strength are respectively 390MPa and 530MPa at normal temperature. Assume all seams are double-welded butt joints and are full NDE, C1=0.8mm and C2=1mm. Determine the thickness of the cylindrical shell and the standard elliptical heads, and verify the strength in hydrostatic test.

Design 2 Determine the maximum working pressure in a cylindrical shell subjected to an internal working pressure. Let DN = 1100mm, Sn =11mm, C=2mm, Φ=0.9, [σ]t=147MPa. Determine the maximum working pressure in a cylindrical shell subjected to an internal working pressure. Let DN = 1100mm, Sn =11mm, C=2mm, Φ=0.9, [σ]t=147MPa.

Cylinders and Formed Heads subjected to External-Pressure Factors affect the critical pressure Factors affect the critical pressure Long, short and rigid cylinders Long, short and rigid cylinders Critical Length

Pcr max~Pcr min ?? Pcr max~Pcr min ??

Design of External-P Vessels Design of External-P Vessels

Head

Two situations maybe encountered: (1)If point A is at the right of the curve, the value of B can be found from the figure directly. (2)If point A is at the left of the curve, directly calculating:

( 10 points)Examine the stability of the vacuum vessel at 200 ℃.Assume C1=0.8mm and C2=1mm.The other conditions are given in Fig.3. Material 16MnR, Standard elliptical heads. ( 10 points)Examine the stability of the vacuum vessel at 200 ℃.Assume C1=0.8mm and C2=1mm.The other conditions are given in Fig.3. Material 16MnR, Standard elliptical heads.

Flange Connection

Support for vessels Positon Positon Dangerous Section Dangerous Section

Shearing Force Diagram Bending Moment Diagram M1 M2 M3 FF AA q

Reinforcement for opening of vessels A1A1 A3A3 A4A4 A2A2 A B h1h1 h2h2 d

A = S×d A1 = (B – d) (Se – S) – 2 (Sn.t – C) (Se – S) (1 – fr) A2 = 2 h1 ( Sn.t – St – C ) fr + 2 h2 ( Sn.t – C – C2 ) fr A3 = according to the actual dimension

ii. Designing Steps in Reinforcement for openings (1)Getting the following data from the strength calculation: calculation: Calculating wall thickness of cylinders or heads S Nominal wall thickness of cylinders or heads S n Calculating wall thickness of nozzles S t Nominal wall thickness of nozzles S n.t Additional value of wall thickness C = C 1 + C 2

(2)Calculating the effective reinforcement range B, h 1, h 2 B, h 1, h 2 (3)Calculating the necessary reinforcement area A according to P183 Table 6-17 A according to P183 Table 6-17 (4)Calculating the available reinforcement area A 1, A 2, A 3 A 1, A 2, A 3

(5)Judging whether it is necessary to add some reinforcement area some reinforcement area If A 1 + A 2 + A 3 ≥ A If A 1 + A 2 + A 3 ≥ A reinforcement not required reinforcement not required If A 1 + A 2 + A 3 < A If A 1 + A 2 + A 3 < A reinforcement required reinforcement required

(6)If reinforcement is required, calculating the added reinforcement area A 4 added reinforcement area A 4 A 4 = A －（ A 1 + A 2 + A 3 ） A 4 = A －（ A 1 + A 2 + A 3 ）(7)Comparison Finally getting A 1 + A 2 + A 3 + A 4 ≥ A Finally getting A 1 + A 2 + A 3 + A 4 ≥ A