Timothy A. Philpot - Mechanics of materials _ an integrated learning system-John Wiley (2017)

thin-walled pressure vessels
Tangential stress and strain in spherical pressure vessel
pr pd pr
σt
= = εt
= (1 −ν)
2t
4t
2tE
Longitudinal and circumferential stresses in cylindrical pressure
vessels
pr pd
pr
σlong
= = εlong
= (1 − 2 ν)
2t
4t
2tE
pr pd
pr
σhoop
= = εhoop
= (2 −ν)
t 2t
2tE
thick-walled pressure vessels
Radial stress in thick-walled cylinder
2 2 2 2
a pi − b po ab( pi − po)
σ r =
−
2 2
2 2 2
b − a ( b − a ) r
or
2
2 2
2
a pi
⎛ b ⎞ b po
a
σ r = 1−
1
2 2 2 2 2 2
b − a ⎝
⎜
r ⎠
⎟ − ⎛ ⎞
−
b − a ⎝
⎜
r ⎠
⎟
Circumferential stress in thick-walled cylinder
2 2 2 2
a pi − b po ab( pi − po)
σ θ =
+
2 2
2 2 2
b − a ( b − a ) r
or
2
2 2
2
a pi
⎛ b ⎞ b po
a
σ = 1+
1
2 2 2 2 2 2
b − a ⎝
⎜
r ⎠
⎟ − ⎛ ⎞
θ
+
b − a ⎝
⎜
r ⎠
⎟
Maximum shear stress
2 2
1
ab p p
2 ( ) ( i − o)
τmax
= σθ
− σ r =
2 2 2
( b − a ) r
Longitudinal normal stress in closed cylinder
2 2
a pi
− b po
σ long =
2 2
b − a
Radial displacement for internal pressure only
2
a pi
δr
=
(1 − ν) r + (1 + ν)
b
2 2
( b − a ) rE
2 2
[ ]
Radial displacement for external pressure only
2
b po
δr
=−
(1 − ν) r + (1 + ν)
a
2 2
( b − a ) rE
2 2
[ ]
Radial displacement for external pressure on solid cylinder
(1 − ν)
pr o
δ r =−
E
Contact pressure for interference fit connection of thick cylinder
onto a thick cylinder
p
c
δ ( 2 2)( 2 2
E c − b b − a )
=
3
2b ( 2 2
c − a )
Contact pressure for interference fit connection of thick cylinder
onto a solid cylinder
p
c
δ ( 2 2
E c − b )
=
2
2bc
Failure theories
Mises equivalent stress for plane stress
2
σM = ⎡⎣ σ p1
− σ p1σ p2 + σ p2
⎤ ⎦ = ⎡⎣ σx − σσ x y + σ y + 3τxy
⎤ ⎦
Column buckling
Euler buckling load
P
cr
2
π EI
=
2
( KL)
Euler buckling stress
σ
cr
2
π E
=
( KL/ r)
Radius of gyration
I
r
2 =
A
Secant formula
P ⎡ ec ⎛ KL
σ max = ⎢1+
sec
2
A r ⎝
⎜
⎣ 2r
2
2 1/2 2 2 2 1/2
P ⎞ ⎤
EA ⎠
⎟ ⎥
⎦
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