General Design Principles for DuPont Engineering Polymers - Module

Table 4.03. Formulas for Stresses and Deformations in Pressure Vessels
Notation for thin vessels: p = unit pressure (lb/in2 ); s1 = meridional membrane stress, positive when tensile (lb/in2 );
s2 = hoop membrane stress, positive when tensile (lb/in2 ); s1’ = meridional bending stress, positive when tensile on
convex surface (lb/in2 ); s2’ = hoop bending stress, positive when tensile at convex surface (lb/in2 ); s2’’ = hoop stress
due to discontinuity, positive when tensile (lb/in2 ); ss = shear stress (lb/in2 ); Vo, Vx = transverse shear normal to wall,
positive when acting as shown (lb/linear in); Mo, Mx = bending moment, uniform along circumference, positive when
acting as shown (in·lb/linear in); x = distance measured along meridian from edge of vessel or from discontinuity (in);
R1 = mean radius of curvature of wall along meridian (in); R2 = mean radius of curvature of wall normal to meridian
(in); R = mean radius of circumference (in); t = wall thickness (in); E = modulus of elasticity (lb/in2 ); v = Poisson’s ratio;
D = Et3 ; λ = 3(1 – v 2 ) ; radial displacement positive when outward (in); θ = change in slope of wall at edge of
12(1 – v 2 4
2 2 ) √ R2 t vessel or at discontinuity, positive when outward (radians); y = vertical deflection,
positive when downward (in). Subscripts 1 and 2 refer to parts into which vessel may imagined as divided, e.g.,
cylindrical shell ahd hemispherical head. General relations: s1’ = 6M at surface; s s =
t 2 t
Notation for thick vessels: s 1 = meridional wall stress, positive when acting as shown (lb/in 2 ); s 2 = hoop wall stress,
positive when acting as shown (lb/in 2 ); a = inner radius of vessel (in); b = outer radius of vessel (in); r = radius from
axis to point where stress is to be found (in); Δa = change in inner radius due to pressure, positive when representing
an increase (in); Δb = change in outer radius due to pressure, positive when representing an increase (in). Other
notation same as that used for thin vessels.
Cylindrical
Spherical
Form of vessel
s 2
s 1
s 2
s 1
t
t
R
R
Manner of loading
Thin vessels – membrane stresses s 1 (meridional) and s 2 (hoop)
Uniform internal
(or external)
pressure p, lb/in 2
Uniform internal
(or external)
pressure p, lb/in 2
s 1 = pR
2t
s 2 = pR
t
External collapsing pressure p ′
19
V .
Radial displacement = R (s 2 – vs 1)
E
failure, and holds only when
′ p R > proportional limit.
t
= t sy
R s y
2
1 + 4
R
E t
Internal bursting pressure pu = 2 s
b – a
u
b + a
(Here su = ultimate tensile strength,
a = inner radius, b = outer radius)
where s y = compressive yield point of material. This formula is for nonelastic
s 1 = s 2 = pR
2t
Radial displacement = Rs (1 – v )
E
Formulas
(
( ) )