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General Design Principles for DuPont Engineering Polymers - Module

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Table 4.03. Formulas <strong>for</strong> Stresses and De<strong>for</strong>mations in Pressure Vessels<br />

Notation <strong>for</strong> thin vessels: p = unit pressure (lb/in2 ); s1 = meridional membrane stress, positive when tensile (lb/in2 );<br />

s2 = hoop membrane stress, positive when tensile (lb/in2 ); s1’ = meridional bending stress, positive when tensile on<br />

convex surface (lb/in2 ); s2’ = hoop bending stress, positive when tensile at convex surface (lb/in2 ); s2’’ = hoop stress<br />

due to discontinuity, positive when tensile (lb/in2 ); ss = shear stress (lb/in2 ); Vo, Vx = transverse shear normal to wall,<br />

positive when acting as shown (lb/linear in); Mo, Mx = bending moment, uni<strong>for</strong>m along circumference, positive when<br />

acting as shown (in·lb/linear in); x = distance measured along meridian from edge of vessel or from discontinuity (in);<br />

R1 = mean radius of curvature of wall along meridian (in); R2 = mean radius of curvature of wall normal to meridian<br />

(in); R = mean radius of circumference (in); t = wall thickness (in); E = modulus of elasticity (lb/in2 ); v = Poisson’s ratio;<br />

D = Et3 ; λ = 3(1 – v 2 ) ; radial displacement positive when outward (in); θ = change in slope of wall at edge of<br />

12(1 – v 2 4<br />

2 2 ) √ R2 t vessel or at discontinuity, positive when outward (radians); y = vertical deflection,<br />

positive when downward (in). Subscripts 1 and 2 refer to parts into which vessel may imagined as divided, e.g.,<br />

cylindrical shell ahd hemispherical head. <strong>General</strong> relations: s1’ = 6M at surface; s s =<br />

t 2 t<br />

Notation <strong>for</strong> thick vessels: s 1 = meridional wall stress, positive when acting as shown (lb/in 2 ); s 2 = hoop wall stress,<br />

positive when acting as shown (lb/in 2 ); a = inner radius of vessel (in); b = outer radius of vessel (in); r = radius from<br />

axis to point where stress is to be found (in); Δa = change in inner radius due to pressure, positive when representing<br />

an increase (in); Δb = change in outer radius due to pressure, positive when representing an increase (in). Other<br />

notation same as that used <strong>for</strong> thin vessels.<br />

Cylindrical<br />

Spherical<br />

Form of vessel<br />

s 2<br />

s 1<br />

s 2<br />

s 1<br />

t<br />

t<br />

R<br />

R<br />

Manner of loading<br />

Thin vessels – membrane stresses s 1 (meridional) and s 2 (hoop)<br />

Uni<strong>for</strong>m internal<br />

(or external)<br />

pressure p, lb/in 2<br />

Uni<strong>for</strong>m internal<br />

(or external)<br />

pressure p, lb/in 2<br />

s 1 = pR<br />

2t<br />

s 2 = pR<br />

t<br />

External collapsing pressure p ′<br />

19<br />

V .<br />

Radial displacement = R (s 2 – vs 1)<br />

E<br />

failure, and holds only when<br />

′ p R > proportional limit.<br />

t<br />

= t sy<br />

R s y<br />

2<br />

1 + 4<br />

R<br />

E t<br />

Internal bursting pressure pu = 2 s<br />

b – a<br />

u<br />

b + a<br />

(Here su = ultimate tensile strength,<br />

a = inner radius, b = outer radius)<br />

where s y = compressive yield point of material. This <strong>for</strong>mula is <strong>for</strong> nonelastic<br />

s 1 = s 2 = pR<br />

2t<br />

Radial displacement = Rs (1 – v )<br />

E<br />

Formulas<br />

(<br />

( ) )

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