This section discusses the ASME Code calculations for the following pressure vessel components:
• Shells • Heads • Conical sections • Flat covers • Nozzles • Nozzle flanges
The calculations required to determine the stresses that result from local loads that are applied to nozzles and attachments to the vessel will also be discussed in general terms.
The ASME Code, Section VIII, Division 1, requires the minimum thickness of shells and heads to be 1.6 mm (0.0625 in.) for most applications, regardless of calculation results. This is the thickness after the vessel components are formed and before corrosion allowance is added. This minimum thickness requirement provides a basic level of mechanical strength for the pressure vessel, even if the calculations indicate that the vessel may be thinner for the design loads that are actually imposed.
Design for Internal Pressure
This section discusses calculation of the wall thickness of shells, heads, and conical sections under internal pressure. Refer to Work Aid 3A for the ASME equations that are required to perform the calculations.
Shells
The idealized equations for the calculation of hoop and longitudinal stresses, respectively, in a cylindrical shell under internal pressure are as follows:
σθ = Prt and σ1 =Pr 2t
These equations assume a uniform stress distribution throughout the thickness of the shell. Since this is an idealized state, the ASME Code formulas have been modified to account for non-ideal behavior. For hoop stress, the formula is:
σθ =Pr tE1+
0.6P E1
Where: P = Internal design pressure, kPa (psig) r = Inside radius of the vessel, mm (in.) t = Vessel thickness, mm (in.)
E1 = Weld joint efficiency for a longitudinal joint
Rearranging this equation and substituting S (allowable stress, kPa [psi]) for σθ yields:
t= Pr SE1− 0. 6P
The formula that follows applies when the thickness required to resist the longitudinal stress due to internal pressure must be calculated:
t = Pr 2SEc + 0.4P
Longitudinal stress can govern the design of particular sections in a pressure vessel. Longitudinal stress can govern when loadings other than internal pressure induce longitudinal stresses that are greater than one half of the hoop stress that is due to internal pressure. In these cases, the longitudinal stress that is due to these other loads is added to the longitudinal stress due to internal pressure. The total combined longitudinal stress is then limited to the maximum allowable stress of the vessel material at the design temperature.
The most common example of where longitudinal stress can govern the design of a vessel component is when wind load on a tall tower causes a bending moment. This bending moment creates a longitudinal bending stress in the cylindrical sections and increases at lower tower elevations because of the greater tower length on which the wind pressure acts. The wind load sometimes requires that the thickness of the lower tower sections be increased beyond the thickness that is required for internal pressure alone, in order for the longitudinal stresses due to wind plus internal pressure to be acceptable.
The thickness of a spherical shell will be approximately half the thickness of a cylindrical shell for the same design conditions, material, and diameter. Refer to Work Aid 3A for the ASME Code shell thickness equations.
When the required thickness for internal pressure must be determined, the specified corrosion allowance must first be added to the new vessel inside radius so that the corroded vessel inside radius is used in the equations. The thickness calculated using these equations must then be increased by the specified corrosion allowance in order to arrive at the minimum required new vessel thickness.
Note that the Pressure Vessel Design Data Sheet has an area where the thickness calculation equations are summarized.
Sample Problem 1 - Cylindrical Shell Thickness Calculation
The pressure vessel described in Figure 6 will be used in this and subsequent Sample Problems. Hand calculations are used in the solution of all Sample Problems, Exercises, and Evaluations in this course to assist in understanding the design concepts and parameters that are involved. However, computer programs are typically used for these calculations on the job. Saudi Aramco engineers often use the CODECALC computer program for these calculations.
The geometry and design data of a vertical cylindrical pressure vessel are specified in Figure 6.
Cost estimates are being prepared for this vessel. It is your job to estimate the required component thicknesses. What are the minimum required thicknesses for the two cylindrical sections?
4'-0" 6'-0" Hemispherical 2:1 Semi-Elliptical 60'-0" 10'-0" 30'-0" DESIGN INFORMATION
Design Pressure = 250 PSIG Design Temperature = 700°F Shell and Head Material is SA-515 Gr. 60
Corrosion Allowance = 0.125" Both Heads are Seamless Shell and Cone Welds are Double Welded and will be Spot Radiographed The Vessel is in All Vapor Service
Cylinder Dimensions Shown are Inside Diameters
MEX 20203.F06
Solution
Since the welds are spot radiographed, E = 0.85. S = 14 400 psi for SA-515/Gr. 60 at 700°F.
Use Work Aid 3A for this solution. 6 ft. - 0 in. Shell r = 0.5D + C = 0.5 × 72 + 0.125 r = 36.125 in. tp = Pr SE1− 0.6P = 250× 36.125 14 400× 0.85 − 0.6 × 250 tp = 0.747 in. t = tp + c t = 0.747 + 0.125
t = 0.872 in. required including corrosion allowance 4 ft. - 0 in. Shell r = 0.5 × 48 + 0.125 r = 24.125 in. tp = 250× 24.125 14 400× 0.85 − 0. 6 × 250 tp = 0.499 in. t = 0.499 + 0.125
Heads
Figure 7 shows typical types of formed closure heads that are used on pressure vessels. Elliptical, hemispherical, and torispherical are the most commonly used head types. Note in Figure 7 that all head types but the conical head have a straight flange (sf) section, which simplifies welding the head to the adjacent cylindrical shell section. The elliptical and torispherical head types have an indicated head depth (h), which is measured from the straight flange to the maximum point of curvature on the inside.
As discussed previously for shells, the internal head dimensions that are used to calculate the required thicknesses must first be increased to account for the specified corrosion allowance. The following procedure is used to adjust the internal head dimensions:
• In equations where the head inside radius is a parameter, the specified corrosion allowance must first be added to the new head inside radius so that the corroded inside radius is used in the equations.
• In equations where the head inside diameter is a parameter, double the specified corrosion allowance and then add this number to the new head inside diameter so that the corroded inside diameter is used in the equations. The pressure vessel corrosion allowance must then be added to the thicknesses that are calculated by the ASME equations for these heads. A different equation is used to calculate the thickness of each head type. Work Aid 3A contains the ASME Code equations that are used to calculate the wall thicknesses of heads.
t t t ID ID ID sf sf sf sf R Flanged Hemispherical h Elliptical Conical t t sf h
Flanged and dished (torispherical) α ID Toriconical t α r MEX 20203.F07
Elliptical Heads - The 2:1 semi-elliptical head is the most
commonly used head type. The thickness of this type of head is normally equal to the thickness of the cylinder it is attached to. Such a head is of semi-elliptical form. Half of its minor axis (that is, the inside depth of the head minus the length of the straight flange section) equals one-fourth of the inside diameter of the head.
To simplify calculations and fabrication, the ASME Code permits an approximation for the actual head geometry. The elliptical head geometry may be assumed to consist of the following: • A spherically dished head with a radius that is equal to
90% of the inside diameter of the shell to which the head is attached, and
• A knuckle radius of 17% of the inside diameter of the shell to which the head is attached. The knuckle is the transition region between the straight flange and the spherically dished area.
The Pressure Vessel Design Data Sheet has a location where elliptical head calculations are done.
Hemispherical Heads - The required thickness of a hemispherical
head is normally one-half the thickness of an elliptical or torispherical head for the same design conditions, material, and diameter. Hemispherical heads are an economical option to consider when expensive alloy material is used. In carbon steel construction, hemispherical heads are generally not as economical as elliptical or torispherical heads because of higher fabrication cost. Hemispherical heads are normally fabricated from segmented sections that are welded together, spun, or pressed. Segmented hemispherical heads may be economical in carbon steel construction for thin, very large-diameter vessels, or in thick, small-diameter vessels.
A hemispherical head is typically half the wall thickness of the cylindrical shell to which it is attached. Therefore, the thickness transition zone between the head and shell must be contoured to minimize the effect of local stress. Figure 8 shows the thickness transition requirements that are contained in the ASME Code.
Thinner part Tangent line l > 3y l > 3y th th ts ts y y Thinner part
Length of required taper, l, may include the width
of the weld
20203.F08
Figure 8: Thickness Transition Between Hemispherical Head and Shell
Note that the equation shown for a torispherical head on the Pressure Vessel Design Data Sheet reduces to the same equation as for a spherical head, since M=1 for a spherical head.
Torispherical Heads - A torispherical (or flanged and dished) head is typically somewhat flatter than an elliptical head and can be the same thickness as an elliptical head for identical design conditions and diameter. The minimum permitted knuckle radius of a torispherical head is 6% of the maximum inside crown radius. The maximum inside crown radius equals the outside diameter of the head.
Sample Problem 2 -
Head Thickness Calculation
For the same vessel described in Sample Problem 1 (See Figure 6), what are the minimum required thicknesses for the top and bottom heads?
Solution
Since both heads are seamless, E = 1.0. Use Work Aid 3A for the solution. Top Head Hemispherical head r = 24 + 0.125 = 24.125 in. tp = Pr 2SE1− 0. 2P = 250× 24.125 2 × 14 400 × 1− 0.2 × 250 tp = 0.21 in. t = tp + c t = 0.21 + 0.125
t = 0.335 in. required including corrosion allowance Bottom Head 2:1 Semi-Elliptical Head D = 72 + 2 × 0.125 D = 72.25 in. tp = PD 2SE− 0. 2P = 250× 72.25 2× 14 400 × 1− 0.2 × 250 tp = 0.628 in. t = 0.628 + 0.125
Conical Sections
Tall towers commonly have sections with different diameters along their length. The transitions between the different diameters are made in conical sections. The most common design for a conical transition does not have formed knuckles at the ends of the cone. The cylindrical sections of different diameter are welded to each end of the cone. The required thickness for internal pressure of a conical shell without transition knuckles is calculated using the equation shown in Work Aid 3A. This equation assumes that half of the cone-apex angle is no greater than 30°.
Formed knuckles are sometimes used at the cone-to-cylinder transition in order to reduce the localized stresses. When knuckles are used, the transition is called toriconical. Knuckles are mandatory when the cone half-apex angle exceeds 30°. Knuckles are also sometimes used for smaller angles when there is concern about potentially high local stresses at the cone-to-cylinder junction. The ASME Code has design procedures for toriconical sections, but these design procedures will not be discussed in this course.
The Pressure Vessel Design Data Sheet does not have a location for the calculation of the thickness of a conical shell section. Therefore, conical shell section calculations must be added by hand when applicable.
Sample Problem 3 - Conical Section Thickness Calculation
For the same vessel described in Sample Problem 1 (See Figure 6), what is the minimum required thickness of the conical section? Assume that the entire cone will be the same thickness.
Solution
E = 0.85 since the welds are spot radiographed. Use Work Aid 3A for the solution.
Determine the cone half-apex angle, α. α = tan−10. 5(DL− DS)
Cone Length
α = tan−10.5(72− 48) 120
α = 5.7° less than 30°, so OK. D = 72 + 2 × 0.125 D = 72.25 in. tp = 250 × 72.25 2 cos5.7°(14 400 × 0.85 − 0.6 × 250) tp = 0.751 in. t = tp + c t = 0.751 + 0.125 t = 0.876 in.
Design for External Pressure and
Compressive Stresses
Pressure vessels are subject to compressive forces such as those caused by dead weight, wind, earthquake, and internal vacuum. Pressure vessel components, such as shells and heads, behave differently under these compressive forces than when they are exposed to internal pressure. This difference in behavior is due to buckling or elastic instability, which make shells weaker in compression than in tension. In failure by elastic instability, the vessel is said to collapse or buckle. The paragraphs that follow discuss buckling of cylindrical shells due to external pressure. These basic principles also apply to other forms of shells as well as to heads and to compressive loads other than external pressure.
The collapse of a pressure vessel due to external pressure normally starts with small irregularities in either the physical properties or the shape of the shell. A small irregularity in the shell produces localized bending moments. These bending moments tend to emphasize the irregularity or to increase the out-of-roundness of the shell. These effects produce an unstable situation where any surface irregularity is increased by the bending moments that are produced. The critical pressure that causes collapse is not a simple function of the stress that is produced in the shell, as is true with tensile loads. The critical pressure is directly proportional to the material's modulus of elasticity (E) and the shell moment of inertia and is inversely proportional to the cube of the radius of curvature.
An ASME Code allowable stress is not used to design pressure vessels that are subject to elastic instability. Instead, the design is based on the prevention of elastic collapse under the applied external pressure. This applied external pressure is normally 103 kPa (ga) (15 psig) for full vacuum conditions.
The maximum allowable external pressure can be increased by welding circumferential stiffener rings (stiffeners) around the vessel shell. The addition of stiffeners reduces the effective buckling length of the shell, and this length reduction increases the allowable buckling pressure. These stiffener rings may be
L L L L L L L L L L h/3 h/3 h/3 h/3
Moment axis of ring
h = Depth of head
MEX 20203.F9
Figure 9: Stiffener Rings on Pressure Vessel Cylinders
Other factors also affect the design of a pressure vessel for external pressure. The relationship between the modulus of elasticity and unit strain is simple for shells that are at room temperature with an applied stress below the yield point. However, temperature effects must be considered in pressure vessel design. When temperature is considered, the material stress-strain curves are nonlinear with no definite yield point and with a variable modulus of elasticity. The temperature relationship between modulus of elasticity and the stress-strain curve must be expressed as a series of curves based on experimental measurements for particular material types. Therefore, the calculation of allowable external pressure considers the unstiffened length of the vessel component, diameter and thickness, and the stress-strain diagram of the material.
Paragraphs UG-28 and UG-33 of the ASME Code contain procedures to calculate the allowable external pressure on cylindrical shells and heads, respectively. These ASME Code external pressure calculation procedures use an iterative approach and are contained in Work Aid 3B. The results of the external pressure calculations must be shown on the Pressure Vessel Design Data Sheet, when applicable.
The maximum allowable compressive stress in a pressure vessel component that is due to loads other than external pressure is limited to the lower of the following:
• The allowable tensile stress; or
• A value determined using the external pressure calculation procedure that is contained in Work Aid 3B.
Shells
The allowable external pressure of a cylindrical shell is a function of material, design temperature, outside diameter, corroded thickness, and unstiffened length. See Work Aid 3B for the ASME Code calculation procedure.
Heads
The allowable external pressure of a head is a function of material, design temperature, outside radius, head depth, and corroded thickness. Stiffening rings are not used to increase the allowable external pressure of heads. The head thickness is increased as required to achieve the required external pressure. Since a head may sometimes be installed inside a pressure vessel to separate two chambers, it may be necessary to design the head for an external pressure that is higher than 103 kPa (ga) (15 psig). Work Aid 3B contains the ASME Code procedures to calculate the allowable external pressure for heads.
Conical Sections
The allowable external pressure of a conical section is a function of material, design temperature, outside diameters at the small and large ends, conical section length, apex angle, and corroded thickness. The allowable external pressure of a conical section may be increased by the addition of stiffener rings to reduce the unstiffened cone length. The allowable external pressure may also be increased by adding to the cone thickness. Work Aid 3B contains the ASME Code procedures to calculate the allowable external pressure for conical sections.
Sample Problem 4 -
External Pressure Calculation
This Sample Problem will provide practice in using the external pressure design procedure for a cylindrical pressure vessel. Use Work Aid 3B to assist in solving this problem.
A tall cylindrical tower is being supplied to Saudi Aramco. The geometry and design conditions are specified in Figure 10. The vendor has proposed that the wall thickness of this tower be 7/16 in., and no stiffener rings have been specified.
4'-0" 150'-0"
2:1 Semi-Elliptical (Typical)