This document presents a closed-form analysis and finite element analysis of stresses in a cylindrical shell connected to a circular plate under internal pressure. It provides equations and calculations for geometry, material properties, stress components, and von Mises stress at the junction. The closed-form von Mises stress of 13.78 ksi compares favorably with the finite element analysis result of 14.20 ksi, with a difference of 3.1%. This validation demonstrates that closed-form solutions can be used to check finite element analysis results for relatively simple configurations.

1. MathCAD - FEA Vallidation by
CAEFEM of Capped-Cylinder
page 1 of 6
Table 31 Formulas for discontinuity stresses and deformations at the junctions
of shells and plates
Case 4a Cylindrical Shell Connected to a Circular
Plate; Internal Pressure
Reference: Roark's Formulas for Stress and Strain 6th Edition
Cylindrical shell connected to a circular plate
Internal pressure q
Notation file Provides a description of Table 31 and the notation used.
Case 4b Note 1: There is no axial load on the left cylinder. An annular line load at
radius R2 on the right shell balances any axial pressure on the joint. For
an enclosed pressure vessel, superpose an axial load P = qb1
2 using
Case 4b.
Julio C. Banks, P.E.
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2. MathCAD - FEA Vallidation by
CAEFEM of Capped-Cylinder
page 2 of 6
Note 2: For external pressure, substitute -q for q, and a1 for b1 in the load terms, which are defined
below.
Cylindrical shell dimensions and properties:
thickness: t1  0.25in
Inner radius: b1  7.075in
Outer radius: a1  b1  t1  7.325in
Mean radius: R1
a1  b1
  7.200in
2
Elastic modulus: E1  9.9106psi
Poisson's ratio: ν1  0.33
Circular-plate dimensions and properties:
Radius of annular line load: R2  b1
Outer radius: a2  a1
Thickness: t2  0.5in
Modulus of elasticity: E2  9.9106psi
Poisson's ratio: ν2  0.33
Internal pressure: q  50psi
Radius of common circumference: RA  R1
Julio C. Banks, P.E.
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3. MathCAD - FEA Vallidation by
CAEFEM of Capped-Cylinder
page 3 of 6
Constants D, ,  and Kpn are preliminary shell constants used in determining boundary values
(below); they are also used in determining other shell constants:
D1
 3
E1 t1
  14.47kipin
2  
12 1 ν1


D2
 3
E2 t2
  115.7kipin
2  
12 1 ν2


λ1
2  
3 1 ν1


R1
2 t1
 2


1
4
0.9531
1
in
  
λ2
2  
3 1 ν2


R2
2 t2
 2


1
4
0.6799
1
in
  
Kp1 1
21  ν2
a2
R1
   1.487
21  ν2
2  
   74.42in2
 2 b1
Kp2 2 R2

Kp1R2  R1

2  
R2
2  b1
2  
2  
Kp1 2 R2
2 R1

 
 2 ln
4 R1
R2
R1


 

 R2  R1
2  
β 12 1 ν2

 

1
2
  3.270
Cn terms are secondary shell constants used in the loading terms for determining shear stresses
and deflections in each member:
Julio C. Banks, P.E.
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4. MathCAD - FEA Vallidation by
CAEFEM of Capped-Cylinder
page 4 of 6
CAA1
E1
  395.3 CAA2
 3
2D1 λ1
 2 R 1Kp1
6D2
E1 t2
  38.15 CAA  CAA1  CAA2  433.4
CAB1
E1t1
2D1 λ1
  94.18 CAB2
 2
E1t1t2R1Kp1
  28.62
4D2
CBB1
E1  t1
2
D1λ1
  44.88 CBB2
 2 R 1Kp1
2D2
E1 t1
  28.62
CAB  CAB1  CAB2  65.56 CBB  CBB1  CBB2  73.49
LTn terms are load constants used in the load terms and in the determination of shear force, moment,
normal force, radial deflection and rotation. For thin-walled members (R/t>10), the load correction terms
LTAC and LTBC should be neglected:
LTA1
b1R1
t1
  815.0 LTA2
2
 2
E1t2 b1
 Kp2  2.766  103
32D2t1R1
LTB1  0 LTB2
 2
E1 b1
 Kp2  2.766  103
16D2R1
LTAC  0 LTBC  0
LTA  LTA1  LTA2  LTAC  3.581  103 LTB  LTB1  LTB2  LTBC  2.766  103
Julio C. Banks, P.E.
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5. MathCAD - FEA Vallidation by
CAEFEM of Capped-Cylinder
page 5 of 6
Loading terms Kn terms are the loading terms used to solve for shear force, moment, normal force,
radial deflection and rotation:
KV1
LTACBB  LTBCAB
CAACBB CAB
  16.13 KM1
 2
LTBCAA  LTACAB
CAACBB CAB
  52.03
 2
General formulas for moment, shear and normal forces, deflection and rotation at the
junction of the two shells
 2 K  M1  162.6 lbf
Unit moment: M1 q t1
Unit meridional shear force: V1 qt1KV1 201.7
lbf
in
  
Unit meridional axial (normal) force: N1 0
lbf
in
 
  0.8355  mil    Radial deflection of common circumference:
ΔRA
qt1
E1
LTA1  KV1CAA1
 KM1CAB1
 
Rotation of the meridian at the common circumference:
ψA
q
E1
KV1CAB1  KM1CBB1 4.119 10 3    rad
Formulas for the stresses in the left cylinder at the junction
Meridional membrane stress: σ1
N1
t1
  0ksi
Circumferential membrane stress: σ2
ΔRAE1
RA
  ν1σ1  1.149ksi
Meridional bending stress: σ'1
6M1
t1
  15.61ksi
2
Circumferential bending stress: σ'2  ν1σ'1  5.151ksi
Julio C. Banks, P.E.
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6. MathCAD - FEA Vallidation by
CAEFEM of Capped-Cylinder
page 6 of 6
Cylinder von Mises Stress Cylinder von Mises Stress at the junction
 2   σ1σ2  1.149ksi σ'VM σ'1
σVM σ1
2
σ2
 2
 σ'2 2
 σ'1σ'2  13.78ksi
The closed-form solution will now be compared to the 2D axisymmetric FEA Solution from CAEFEM
σ'VM_FEA  14.20ksi
%Δσ'VM
σ'VM_FEA  σ'VM
  3.1%
σ'VM
The closed-form von Mises stresses at the junction of the circular-cap disc to the cylindrical shell
compare favorably with the 2D axisymmetric FEA prediction results. This paper demonstrate that
closed-form solutions for relatively simple configurations can be utilize to check the FEA results
as the FEA is often most expedient method of obtaining numerical results.
Julio C. Banks, P.E.
Sell-A-Vision@Outlook.com

7. CAEFEM 2D-Axisymmetric Validation Model
von Mises Stress is 14.20 ksi
by Julio C. Banks, P.E.
Axial direction, Z
Radial direction, r
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