. The strength of materials; a text-book for engineers and architects. call ;p -^ hp. This assumes that thetube is subjected to pressure on the outside; if it is on theinside the same formulae hold with appropriate change of signas explained later. We may therefore apply to this imaginary hoop the sametreatment as for a thin pipe, the circumferential stress, orhoop stress, being /. Considering a unit length of pipe we have Force tending to cause collapse of ring = {p -{- Sp) X 2 {x + S X) Force resisting collapse of ring =^ 2 f 8 x ^ p x .2 x These must be equal 510 THICK PIPES 511 . •. dividi
. The strength of materials; a text-book for engineers and architects. call ;p -^ hp. This assumes that thetube is subjected to pressure on the outside; if it is on theinside the same formulae hold with appropriate change of signas explained later. We may therefore apply to this imaginary hoop the sametreatment as for a thin pipe, the circumferential stress, orhoop stress, being /. Considering a unit length of pipe we have Force tending to cause collapse of ring = {p -{- Sp) X 2 {x + S X) Force resisting collapse of ring =^ 2 f 8 x ^ p x .2 x These must be equal 510 THICK PIPES 511 . •. dividing by 2 and neglecting the product, S p .Sx,oi twovery small quantities we have {\-pSx ^ f Sx + p . {f — p) h X = X hp (/ - ^) = T^. Fig. 243.—Stresses in Thick Pipes. In the limit when the increments are infinitely small thisgives if-p) = ff w This is one relation between / and p. Now let us assume that the strains along the length of thepipe will be such that a plane section before subjection topressure remains plane after subjection to pressure, thatJongitudinal strain is constant. i. e. ^J^ + ^-J = constant (2) 512 THE STRENGTH OF MATERIALS because both / and p will cause transverse strains in thedirection of the length of the pij)e, and they will have thesame sign. ?. Since -q and E are constant, if our stresses are within theelastic limit we may write f + p — constant = 2 a (say) ••• f = {2a-p) (3) Put this value in (1) and we get 2o X d p a — 2 p =^ , -ax 2a = 2p + =^J (4) Cu tX) (aj JO = x( 2 2> + 1 \ = 2ax. •. d {px^) = 2 ax . dx (5) Integrating we get p x^ = a x^ +6where 6 is a constant -• P -CL + ^2 ••• (6) but / = 2 a - 2> (by 3) f =^a --^ (7) by calculating a an
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