Wall Thickness Calculation of Ellipsoidal Head 2:1 Under External Pressure
 
  Section VIII Div 1 - UG 33 - Thickness of Formed Heads, Pressure on Convex Side
  Section VIII Div 1 - UG 32 - Thickness of Formed Head, and Sections, Pressure on Concave Side
                                     
    (Enter values in yellow cells for calculations and in blue cells as information)
  DATA INPUT
                                           
    Design Conditions     Material      
    Design Pressure, Pd =  bar Head Material Specification    
    1.67 times Pd ³ =  bar Allowable Stress Head, S  =  MPa  
                                           
    Design Temperature, Td =  °C Dimensions      
                  Inner Diameter, Di =   mm  
    Joint efficiency, E = assuming a joint efficiency E = 1.00 ³                          
          Nominal wall thickness before forming, tn =
 mm  
                                           
          Thickness after forming, tf =  mm  
          Check Thickness after forming as per UG-79(d)(3) ¹    
                                           
          Outside Diameter, Do =  mm  
                                           
          Check If tr ≤ tf    
                                           
                                           
    Notes:                                      
    (1) As per UG-79 (d)(3) The reduction in thickness after forming shall not exceed 1/32 in. (1 mm) or 10% of the nominal thickness of the adjoining surface, whichever is less.  
    (2) As per UG-32(k) All formed heads, thicker than the shell and concave to pressure, intended for butt welded attachment, shall have a skirt length sufficient to meet the requirements of Figure UW-13.1, when a tapered transition is required. All formed heads concave to pressure and intended for butt welded attachment need not have an integral skirt when the thickness of the head is equal to or less than the thickness of the shell. When a skirt is provided, its thickness shall be at least that required for a seamless shell of the same inside diameter.   
    (2) As per UG-32(l) Heads concave to pressure,intended for attachment by brazing, shall have a skirt length sufficient to meet the requirements for circumferential joints in Part UB.   
    (3) As per UG-33 the required thickness for heads due to pressure on the convex side shall be determined by the procedure given in UG-32 for heads with pressure on the concave side (plus heads) using a design pressure 1.67 times the design pressure on the convex side, assuming a joint efficiency E = 1.00 for all cases.  
                                           
                                         
  CALCULATIONS
                                           
    Ellipsoidal Head 2:1                                      
    with tf/L ≥ 0.002                                      
                                           
    Thickness after forming, tf =  in.                                  
                                           
    Inner diameter, Di =  in.                                  
                                           
     Inside depth of the ellipsoidal head with corrosion removed, h  in.   For Ellipsoidal head 2:1 h = D/4                  
    Axis Radius = D/2h  in.                                  
    Factor K1  (The value is obtained from Table UG-37)                        
    Inside spherical or crown radius, L = k1D  in.                                  
    Inside knuckle radius ¹, r =  in.                                  
                                           
    Wall Thickness  [UG-32(c)]                                      
    Minimum required thickness, tr =  in.   tr = P Di                      
            2 S E - 0.2 P                      
                                           
    Check If tr ≤ tf                                      
                                           
    Maximum working pressure  [UG-32(c)]                                      
    Maximum working pressure for Ellipsoidal Head, Pmax =  psig   P = 2 S E tf                      
            Di + 0.2 tf                      
                                           
    Check If Pmax ≥  P                                    
                                           
    Check                                      
    Limitations: tf/L ≥ 0.002 tf/L ≥ 0.002                                  
                                           
    Note:                                      
    (1) As per UG-32(c) An acceptable approximation of a 2:1 ellipsoidal head is one with a knuckle radius of 0.17D and a spherical radius of 0.90D.   
                                           
                                         
  Wall Thickness Calculation of Torispherical Head Under External Pressure
 
  Section VIII Div 1 - UG 33 - Thickness of Formed Heads, Pressure on Convex Side
  Section VIII Div 1 - UG 32 - Thickness of Formed Head, and Sections, Pressure on Concave Side
                                     
    (Enter values in yellow cells for calculations and in blue cells as information)
  DATA INPUT
                                           
    Design Conditions     Material      
    Design Pressure, Pd =  bar Head Material     
    1.67 times Pd ⁴ =  bar Allowable Stress Head, SHt  =  MPa  
                                           
    Design Temperature, Td =  °C Dimensions      
                  Inner Diameter, Di =    mm  
    Joint efficiency, E = assuming a joint efficiency E = 1.00 ⁴                        
          Nominal wall thickness before forming, tn =
 mm  
                                           
          Thickness after forming, tf =  mm  
          Check Thickness after forming as per UG-79(d)(3) ¹    
                                           
          Inside knuckle radius ², r =  mm  
          Check inside knuckle radius    
                                           
          Outside Diameter, Do =  mm  
                                           
          Check If tr ≤ tf    
                                           
                                           
                                           
    Notes:                                      
    (1) As per UG-79 (d)(3) The reduction in thickness after forming shall not exceed 1/32 in. (1 mm) or 10% of the nominal thickness of the adjoining surface, whichever is less.  
    (2) As per UG-32 (i)  The inside crown radius to which an unstayed head is dished shall be not greater than the outside diameter of the skirt of the head. The inside knuckle radius of a torispherical head shall be not less than 6% of the outside diameter of the skirt of the head.  
    (3) As per UG-32(k) All formed heads, thicker than the shell and concave to pressure, intended for butt welded attachment, shall have a skirt length sufficient to meet the requirements of Figure UW-13.1, when a tapered transition is required. All formed heads concave to pressure and intended for butt welded attachment need not have an integral skirt when the thickness of the head is equal to or less than the thickness of the shell. When a skirt is provided, its thickness shall be at least that required for a seamless shell of the same inside diameter.   
    (3) As per UG-32(l) Heads concave to pressure,intended for attachment by brazing, shall have a skirt length sufficient to meet the requirements for circumferential joints in Part UB.   
    (4) the thickness computed by the procedure given in UG-32 for heads with pressure on the concave side (plus heads) using a design pressure 1.67 times the design pressure on the convex side, assuming a joint efficiency E = 1.00 for all cases.  
                                           
                                         
  CALCULATIONS
                                           
    Torispherical Head                                      
    with tf/L ≥ 0.002                                      
    Thickness after forming, tf =  mm                                  
                                           
    Inner diameter of the skirt, L =  mm    L = Di                      
                                           
    Inside knuckle radius, r =  mm                                  
                                           
    L/r = ⁵                                  
                                           
    Factor M =     M = 1/4 (3 + (     L ) 0,5 )                
            r                  
    Wall Thickness [APPENDIX 1-4(d)]                                      
    Minimum required thickness, tr =  mm   tr = P L M                      
            2 S E - 0.2 P                    
                                           
    Check If tr ≤ tf                                      
                                           
    Maximum working pressure  [APPENDIX 1-4(d)]                                      
    Maximum working pressure for Torispherical Head, Pmax =  bar   P = 2 S E tf                      
            L M + 0.2 tf                      
                                           
    Check If Pmax ≥  P                                    
                                           
    Check                                      
    Limitations: tf/L ≥ 0.002 tf/L ≥ 0.002                      
                                           
    Note:                                      
    (5) L/r = 16-2/3 Maximum ratio allowed by UG-32(i) when L equals the outside diameter of the skirt of the head.  
                                           
             
  Wall Thickness Calculation of Head Under External Pressure  
   
  Section VIII Div 1 - UG 33 - Thickness of Formed Heads, Pressure on Convex Side  
  Section VIII Div 1 - UG 32 - Thickness of Formed Head, and Sections, Pressure on Concave Side  
  Section VIII Div 1 - UG-28(d) for determining the thickness for a spherical shell  
    (Enter values in yellow cells for calculations and in blue cells as information)  
  STEP 1  
  Enter the indicated input data:   Calculated Values:    
               
  Head Type =
  Do/2ho =    
               
   Minimum required thickness,  t =   mm  ko =    
  Outside Diameter if Hemisferical head,  Do =  mm For Hemispherical Head, Ro =  mm  
             
  Outside Diameter if Ellipsoidal head,  Do =  mm For Ellipsoidal head, Ro =  mm  
             
  Outside Diameter if Torispherical head,  Do =  mm For Torispherical Head, Ro =  mm  
               
  One‐half of the length of the outside minor axis of the ellipsoidal head, ho=   Factor A =    
  For Ellipsoidal head 2:1 ho = Do/4            
               
               
  STEPS 2 & 3  
  Enter the indicated input data:          
               
  Material =          
               
  Design Temp. =  °C        
               
  With Factor A get Factor B =       From ASME Sec. II D Figs. CS, HA, HT  
        For values of A falling to the left of the material/  
        temperature line, the factor B is not determined.  
        Go to Step 5.  
               
  STEP 4  
    Calculated Value:    
  Using the factor B, calculate the value of the maximum allowable        
  external working pressure Pa Pa =  bar  
               
               
  STEP 5  
  Enter the indicated input data:   Calculated Value:    
               
  Material =          
               
  Design Temp. =  °C        
               
  Modulus of elasticity of Material,  E =  × 10³ MPa Pa =  bar  
  For values of A falling to the left of the applicable material/temperature line.          
               
  STEP 6  
  Compare the calculated value of Pa obtained in Step 4 or Step 5 with P  
  External design pressure, P =   bar Pa =  bar  
   If Pa is smaller than P, select a larger value for t and repeat the design procedure until a value of Pa is obtained that is    
  equal to or greater than P.            
               
   UG 33 - Thickness of Formed Heads, Pressure on Convex Side  
               
  For ellipsoidal and torispherical heads, the required thickness shall be the greater of the following:         
  (a) the thickness computed by the procedure given in UG-32 for heads with pressure on the concave side (plus heads) using a design pressure 1.67 times  
  the design pressure on the convex side, assuming a joint efficiency E = 1.00 for all cases; or         
  (b) the thickness as computed by the appropriate procedure given for spherical shells in UG-28(d).        
  For Hemispherical Heads. The required thickness of a hemispherical head having pressure on the convex side shall be determined in the same manner as  
  outlined in UG-28(d) for determining the thickness for a spherical shell.             
               
  Head Type =          
  (a) the thickness computed by the procedure given in UG-32 =  mm        
  (a) the thickness computed by the procedure given in UG-28(d) =  mm        
   Minimum required thickness,  t =   mm        
               
                 
    Discussion and References  
      Information about Wall thickness calculation of formed heads under external pressure:  
      ASME Section VIII Div 1 paragraphs UG-33, UG-32, UG-28, UG-29 and UG-30   
    Tables and Standards  
  - ASME Section VIII Div 1 paragraph UG-28, Thickness of Shells and Tubes under External Pressure  
  - ASME Section VIII Div 1 paragraph UG-29, Stiffening Rings for Cylindrical Shells Under External Pressure  
  - ASME Section VIII Div 1 paragraph UG-30, Attachment of Stiffening Rings  
  - ASME Section VIII Div 1 paragraph UG-32, Formed Heads, and Sections, Pressure on Concave Side  
  - ASME Section VIII Div 1 paragraph UG-33, Formed Heads, Pressure on Convex Side  
  - ASME Sec. II D Fig. G, Geometric Chart for Components Under External or Compressive Loadings (All Materials)  
  - ASME Sec. II D Fig. CS, Chart for Determining Shell Thickness of Components Under External Pressure Developed for Carbon or Low Alloy Steel  
  - ASME Sec. II D Fig. HA, Chart for Determining Shell Thickness of Components Under External Pressure Developed for Austenitic Steel  
  - ASME Sec. II D Fig. HT, Chart for Determining Shell Thickness of Components Under External Pressure Developed for Quenched and Tempered Low Alloy Steel  
- ASME Sec. II D Table TM-1 ( Modulus of Elasticity,  E )
       
    UG-33 Formed Heads, Pressure on Convex Side      
  (a)  The required thickness for heads due to pressure on the convex side shall be determined as follows.   
    (1) For ellipsoidal and torispherical heads, the required thickness shall be the greater of the following:   
    (-a) the thickness computed by the procedure given in UG-32 for heads with pressure on the concave side  
    (plus heads) using a design pressure 1.67 times the design pressure on the convex side, assuming a joint  
    efficiency E = 1.00 for all cases; or   
    (-b) the thickness as computed by the appropriate procedure given in (d) or (e) below.  
  (c) Hemispherical Heads. The required thickness of a hemispherical head having pressure on the convex side shall be  
    determined in the same manner as outlined in UG-28(d) for determining the thickness for a spherical shell.   
  (d) Ellipsoidal Heads. The required thickness of an ellipsoidal head having pressure on the convex side, either seamless or of  
    built‐up construction with butt joints, shall not be less than that determined by the following the same procedure as that  
    given for spherical shells in UG-28(d).  
  (e) Torispherical Heads. The required thickness of a torispherical head having pressure on the convex side, either seamless  
    or of built‐up construction with butt joints, shall not be less than that determined by the same design procedure as is used for ellipsoidal heads given in (d) above, using the appropriate value for Ro.   
    for ellipsoidal heads given in (d) above, using the appropriate value for Ro.   
    UG-28, Thickness of Shells and Tubes under External Pressure  
  (d) Spherical Shells. The minimum required thickness of a spherical shell under external pressure, either seamless or of  
    built‐up construction with butt joints, shall be determined by the following procedure:  
    Step 1. Assume a value for t and calculate the value of factor A using the following formula:  
    A = 0,125            
    Ro/t            
    Step 2. Using the value of A calculated in Step 1, enter the applicable material chart in Subpart 3 of Section II,   
                 Part D for the material under consideration. Move vertically to an intersection with the material/temperature line   
                 for the design temperature (see UG-20). Interpolation may be made between lines for intermediate temperatures.   
                 If tabular values in Subpart 3 of Section II, Part D are used, linear interpolation or any other rational interpolation   
                 method may be used to determine a B value that lies between two adjacent tabular values for a specific temperature.   
                 Such interpolation may also be used to determine a B value at an intermediate temperature that lies between two   
                 sets of tabular values, after first determining B values for each set of tabular values. In cases where the value at A   
                 falls to the right of the end of the material/temperature line, assume an intersection with the horizontal projection   
                 of the upper end of the material/temperatureline. If tabular values are used,the last (maximum) tabulated value   
                 shall be used. For values at A falling to the left of the material/temperature line, see Step 5.  
    Step 3. From the intersection obtained in Step 2, move horizontally to the right and read the value of factor B.  
    Step 4. Using the value of B obtained in Step 3, calculate the value of the maximum allowable external working pressure  
                 Pa using the following equation:  
    Pa = B            
    Ro/t            
    Step 5. For values of A falling to the left of the applicable material/temperature line, the value of Pa can be calculated using  
                 the following equation:  
    Pa = 0.0625E          
    (Ro/t)2          
    If tabulated values are used, determine B as in Step 2 and apply it to the equation in Step 4.   
    Step 6. Compare Pa obtained in Step 4 or Step 5 with P. If Pa is smaller than P, select a larger value for t and repeat the  
                 design procedure until a value for Pa is obtained that is equal to or greater than P.   
  (e) The external design pressure or maximum allowable external working pressure shall not be less than the maximum  
    expected difference in operating pressure that may exist between the outside and the inside of the vessel at any time.   
    The symbols defined below are used in this procedure:  
  
    Do = outside diameter of the head skirt.  
    Do/2ho = ratio of the major to the minor axis of ellipsoidal heads, which equals the outside diameter  of the head skirt  
                     divided by twice the outside height of the head (see Table UG-33.1).  
    ho = one‐half of the length of the outside minor axis of the ellipsoidal head, or the outside height of  the ellipsoidal head  
            measured from the tangent line (head‐bend line).  
    Ko = factor depending on the ellipsoidal head proportions Do/2ho (see Table UG-33.1).  
    Ro = for hemispherical heads, the outside radius  
         = for ellipsoidal heads, the equivalent outside spherical radius taken as KoDo.  
         = for torispherical heads, the outside radius of the crown portion of the head.  
    t = minimum required thickness of head after forming, in (mm)   
    A = factor determined in step 1 assuming a value for t.  
          Using the value of A calculated in Step 1, enter the applicable material chart in Subpart 3 of Section II, Part D for the  
          material under consideration. For the case of cylinders having  Do/t values less than 10, see (c)(2).  
    B = factor determined from the applicable material chart or table in Subpart 3 of Section II, Part D  (Figures CS, HA, HT)  
           for maximum design metal temperature [see UG-20(c)]  
    E = modulus of elasticity of material at design temperature. For external pressure design in accordance with this Section,   
           the modulus of elasticity to be used shall be taken from the applicable materials chart in Subpart 3 of Section II,  
           Part D. (Interpolation may be made made between lines for intermediate temperatures.)  
    P = external design pressure [see Note in UG-28 (f)].  
    Pa = calculated value of maximum allowable external working pressure for the assumed  
              assumed value of t, [see Note in UG-28(f)].  
    t = minimum required thickness of cylindrical shell or tube, or spherical shell, in (mm)