Page 90 - DP-2-2

Design+ Closed-form solution for pressurized obround shells

XXIV, reduce to Lame’s solution. These equations can be internal pressure, the center of section C undergoes a
considered as extensions of the Lame’s solution. significant displacement in the global X direction. δ is
0
used to represent this displacement.
2.3.2. Straight segment AB
In Section 2.1.1, δ was considered as the body motion
0
In the local rectangular coordinates, the stress components of the curved segment and thus neglected, as it did not
are expressed as Timoshenko & Goodier, Xu : affect the load or stress results.
10
11
p
6
M a + ( py
2
2
2
However, δ is the main part of the horizontal
2
σ = I B yp+ t t 3 L − ) + t 5 3 ( 20 y − 3 ) (XXV) displacement at section C, where it maintains the continuity
t
x y
0
x
of displacement at section B.
 y y 3  1
σ =− p 2() 3 − +  (XXVI) u and v can be calculated by Equations XXVIII and

y
 t t 2 2  r t
XXIX:
3 y  x
2
τ =− p  − 6() 2  (XXVII) ap  + (1 b ) 2 
xy
K
−
 2 t  t u =  +(1 r )  − (1 − cos )
r
3
0
2
2
( b − a E   r 0  
)
where t=(b−a) is the wall thickness of the obround shell. 2 pL

2 2
2
+ ( b + ) sin + cos (XXVIII)
a
Here, σ is significantly larger than σ and τ . EN 2 0
y
x
xy
The first three terms in Equation XXV match the results
a r‚ −(
of the beam theory. The linear stress distribution of beam v = 8 M B2 ( b − ) 0 K + K + ) sin + 2 pL ( b + )
2
2
2
2
14
a

4
t
0
3
theory is the consequence of the assumption that “plane EN 1 EN 2
sections remain plane.” cos (XXIX)
The last term of Equation XXV is a nonlinear correction  1+ ( 22
term. Its maximum value is 0.2p at the edge of the section, (  b − )  1−( 1− ) µ lnrr  + ) µ ab 
2
2
a 

0 
y=±t/2. This term is identical for any section within the K = 4 M B2  0 r 0 


straight segment. As the value of this term is relatively small 3 EN 1   b  1−− ( 2 2 
−
compared to the sum of the other terms, the non-linear  ln   + µ )( blnb alna r 
) 0

a 


behavior poses minimal effect on the stress distribution. (XXX)
When the length of the straight segment shrinks to zero,  22 
sections B and D coincide. In contrast to Equation XXII, K = pL (  3 − r ) µ 2 + 1+ ( µ  a )  2 + b − ab    (XXXI)
2
the normal stress distribution of Equation XXV differs 4 EN 2   0  r 0 2  
considerably from Lame’s solution, indicating that the stress
distribution in section B may not be accurately represented. The determination of δ needs to consider the consistent
0
displacement of the straight segment shell at section B in
2.4. Displacements x direction.
Once stress is determined, the corresponding displacements 2.4.2. Straight segment AB
can be derived according to the elastic theory method. 10,11
The displacement components in the local x and y
As the displacement difference between the inner directions are denoted as u and v . They are calculated
x
and outer surfaces is negligible, this paper defines the using Equations XXXII and XXXIII: y
displacements of the obround shell as those at the middle
layer of the shell, that is, the layer of r=r within the curved u = p µ + a x ) (XXXII)
(
0
segment, or y=0 within the straight segment. As a result, x E 2 t
the displacements become functions of θ or x only. p

2
2
2
2
2
2
10
x
x
While the derivation process of displacements v = 20 Et 3   60 L − ( L + ) +(24 +15µ t ) ( L − )
y

is relatively complicated, the derived displacement 6 M
2
2

x
expressions are presented below. + B ( L − )) + v (XXXIII)
Et 3 0
2.4.1. Curved segment BC
In polar coordinates, u and v are used to represent the where, v is the displacement of section B in y direction
t
r
0
radial and tangential displacement components. Under from the curved segment, with the value of (u ) .
p θ =π/2
Volume 2 Issue 2 (2025) 6 doi: 10.36922/DP025060010