THİN WALLED PRESSURE VESSELS

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(tvid- 8.b.)

8.2

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• Our aim in this section is to derive formulas by which we can calculate the stresses caused by internal pressure in thin-walled, internally pressurized Cylindrical or Spherical Vessels and determine safe dimensions.
• In the last part, examples that reinforce the subject will be solved.

8.2 Thin Walled Pressure Vessels

8.2.1 Importance of the Topic

• However, due to internal pressure, stresses occur on the walls of the vessel, and if these stresses exceed a certain level, it causes damage to the container.
• For these reasons, it is extremely important to determine the safe dimensions of these vessels.
• Since tensile stresses occur in the axial and radial directions on the outer wall, it can be said that there is actually a combined loading situation. In fact, a 3-axis normal stress condition occurs on the inner surface.

• Thin-walled vessels are frequently used in industry and contain liquid or gas filled under pressure.

• The wall thickness of these vessels, which are generally cylindrical and spherical, is quite small compared to their diameter.

Figure 8.2.1.a

Figure 8.2.1.b

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8.2.2 Cylindrical Vessels

• We want to calculate the stresses occurring at a point on the wall due to internal pressure. Here;
• r:average wall radius, t: wall thickness.
• The wall thickness is very small compared to the radius. (t<< r).
• The hollow cross-sectional area is considered to be A = 2𝜋𝑟.𝑡 on average.

8.2 Thin Walled Pressure Vessels

The external force F_ext.-1 arising from the internal pressure p in the cover section is balanced by the axial internal force F_int-1 in the section. From this internal force the axial stress can be calculated. (It should not be overlooked that the middle of the section is empty.)

(Cut I –left part )

(8.2.1)

Figure 8.2.2

Figure 8.2.3

Strength of Materials - Lecture Notes / Mehmet Zor

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Mohr’s Circle

(External Surface)

(Cut II )

We cut the cylindrical container longitudinally (parallel to its axis) with the II cut and remove the part Δx.

(side view)

8.2 Thin Walled Pressure Vessels

(8.2.2)

Also, if we consider equation 8.2.1..>>

(8.2.3)

Δx

Since the vertical components of the pressures of symmetrical points such as a and b are in opposite directions and of equal intensity, these components balance each other (p_ya=p_yb=p.sinα). Since the horizontal components of the pressures (p_xa=p_xb=p.cosα) are in the same direction, their sum, the external force (F_ext.-2), is balanced by the internal force F_int.-2 in the radial direction. From here radial stress is obtained.

Figure 8.2.4

Figure 8.2.4

Figure 8.2.5

Figure 8.2.6

Figure 8.2.7

Strength of Materials - Lecture Notes / Mehmet Zor

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8.2.3 Spherical Vessel

Mohr’s Circle

The same result is found for horizontal cutting with similar operations.

8.2 Thin Walled Pressure Vessels

(8.2.4)

Figure 8.2.8

Figure 8.2.10

Figure 8.2.11

Figure 8.2.9

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At some point on the surface

Stresses

Mohr Circles

Solution:

The material is ductile. If we choose the Tresca criterion: the maximum shear stresses arising at a pressure p must be the same in both vessels so that they have the same safety.

2-) It is desired that the capacity of the cylindrical container be the same. Therefore, its volume must be equal to that of the spherical container.

1-) In terms of safety, it is desirable that the cylindrical container has the same risk as the spherical container..

Spherical

Cylindrical

The concavity in the cover parts is neglected.

8.2 Thin Walled Pressure Vessels

Figure 8.2.12

Figure 8.2.13

Strength of Materials - Lecture Notes / Mehmet Zor

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Example 8.2.2 :A cylindrical pressure vessel will be fabricated by wrapping a long, narrow steel plate around a mandrel and then welding it along the edges of the plate to form a helical joint. Helical welding will make an angle of α = 55° with the cylinder axis. Internal pressure is 800kPa, section radius is 1.8m, Elasticity modulus for steel material is E=200GPa, Poisson ratio ν=0.3, yield strength σ_yield=180MPa, safety coefficient of the system is n=2.5. Accordingly, calculate a-) safe (allowable) wall thickness, b-) stress components that will occur in the weld seam at this time (at the safety limit).

Solution:

Stresses for a point A on the surface:

8.2 Thin Walled Pressure Vessels

(θ= 55°)

Since steel is a ductile material:

Figure 8.2.14

Figure 8.2.15

Figure 8.2.16

Strength of Materials - Lecture Notes / Mehmet Zor

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At the safety limit:

We see from the 3D Mohr circle that; Maximum shear stress:

a-)

8.2 Thin Walled Pressure Vessels

τ

σ₃=0

σ

σ₁=σ_r

σ₂

C

k

O

H

R

D₁

3D Mohr’s circle

At point B, the stress state is the same for the element parallel to A. :.>>

b-)

The x-y axes coincide with the 1-2 principal axes.

The Mohr circle in the x-y plane is the median (blue) circle with center C. The weld seam surface is the k plane and the stress components can be found from the Mohr circle as follows:

allowable

Figure 8.2.16

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Example 8.2.3 : It is desired to produce an internally pressurized vessel in the largest possible volume, without exceeding the boundaries of a 5m x 5m area. A type of steel with a yield stress of 240MPa is used as the material. Taking the factor of safety as n=2 and the wall thickness as 4cm, determine which of the spherical or cylindrical vessels is more advantageous in terms of a-) volumetric and b-) safety.

Answers: a-) cylindrical vessel is more advantageous, b-) spherical vessel is more advantageous

8.2 Thin Walled Pressure Vessels

Figure 8.2.17

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Strain-gage

8.2 Thin Walled Pressure Vessels

Figure 8.2.18