Page 91 - DP-2-2

Design+ Closed-form solution for pressurized obround shells

2
ap 1 b 2 For cylindrical shells, the maximum hoop stress is one
v 1 ¼r of the most critical factors in the design process. Similar
0
0
2
2
)
( b a E r 0 to a cylinder shell, the hoop stress of an obround shell is
pL defined as the tangential stress of the curved segment σ
K a (XXXIV) θ
2
2
b
3
EN 2 and axial stress of straight segment σ . This implies that
x
the direction of hoop stress is parallel to the edge of the
At section B, the positive directions of v and u are opposite. obround section.
x
t
However, the displacement of section B should be identical
from both sides of the section, that is, (u ) =−(v ) . As there are no geometric and load discontinuities,
t θ=π/2
x x=L
Therefore, the value of δ can be determined as: the stress distribution on the section can be assumed to
0
vary smoothly along θ (curved segment) or x (straight
4 M pL a segment). The section with the greatest bending moment
2
= EN B 2 ( b 2 − a r 0 ) −(K 3 + ) + E ( 2 + ) (XXXV) should be the section with the greatest hoop stress.
K
0
4
t
1
Therefore, the maximum stress should occur at sections C
Meanwhile, the maximum displacements of an obround or D or locations C or D . For design purposes, only the
i
o
shell in global X and Y directions occur on the symmetrical stresses at C and D need to be concerned. These sections
i
o
planes. They are defined as δ =(v ) and δ =(u ) . These are located far from junction or section B; hence, the
y x=0
D
r θ=0
C
are given by: calculated stresses at C and D are expected to be accurate.
i
o
ML 2 pL 2 The ASME Code considers hoop stress in obround
2
t  +
= 2 B EI + 240 EI  50 L + 24( +15 ) 2  v (XXXVI) shells to consist of bending stress and membrane stress.

D
0
The stresses at the inner and outer surfaces of sections B,
2
ap  + (1 b ) 2  C, and D need to be calculated to compare with allowable
= ( b − )   r 0 +(1 − r )  + 0 (XXXVII) limits. From the above analysis, the stress at section B
1,2
C
0
2
2

a E 

section B can be safely disregarded.
The term v represents the vertical displacement was lower than at sections C and D. Therefore, stress at
0
in section B. It is the result of internal pressure and the Due to tensile membrane stress, the maximum stress
corresponding end loads of the curved segment. always occurs at the tensile side. Thus, only the stresses at
Both δ and δ describe horizontal displacements at the C and D should be concerned.
o
i
C
0
center of section C. δ is the real displacement, whereas In addition, the shear stress in an obround shell is
C
δ is the virtual one used to maintain the displacement relatively small and is therefore ignored in the following
0
continuity at the junction. discussion.
The first term of Equation XXXVII (from the pressure The expressions of displacement, Equations XXVIII –
ring) is much smaller than δ . It can be considered as δ ≈δ . XXXVII, are discussed and compared with FEA results.
0
C
0
For plane strain issue, E and μ in Section 2.4 should be 3.1. Stress distributions
replaced by E and µ , respectively.
1− µ 2 1− µ An example of FEA was analyzed. In this example, a thick-
walled obround shell was chosen to exaggerate possible
3. Discussion stress errors and nonlinear stress behavior.
The stress distribution expressions in Equations XXII – In this paper, the two-dimensional element of plane183
XXVII resulted from superposing the stress distributions of Ansys was used to perform FEA. A square element shape
of the existing closed-form solutions in elasticity theory. was used. The mesh size was 1/20 of the wall thickness.
An additional deformation condition introduced in the The parameters of the obround shell used in FEA were
analysis was that, at the junction of the curved and straight a=300 mm, b=600 mm, L=380 mm, and p=8 MPa.
segments, the displacement and rotation angle at the Material properties used were E=210 GPa and μ=0.3.
section center are continuous. Based on this continuity The hoop stress and radial stress are defined as σ and σ
condition, the bending moment at the junction was for the curved segment BC, and σ and σ for the straight
θ
r
obtained. With this value, the bending moments at any segment AB. x y
other section can then be calculated. The displacement
components of the middle layer of the shell are derived Figures 5 and 6 are the hoop and radial stress distribution
from stress distributions. The maximum displacements in curves at section C, θ=0, and at section D, x=0. The linear
the global X and Y directions, δ and δ , were also obtained. curves in Figure 5 are the sum of membrane and bending
D
c
Volume 2 Issue 2 (2025) 7 doi: 10.36922/DP025060010