Theory of Elasticity by L.D. Landau & E.M. Lifshitz

Author:L.D. Landau & E.M. Lifshitz
Language: eng
Format: epub
Tags: Physics; Equilibrium; Elastic Waves; Dislocations; Thermal Conduction; Viscosity; Solids; Stress Tensor; Hooke's Law

Fig. 15

Solution. We have F = constant = f everywhere on the rod. At the clamped end (/ = 0), 8 = \it, and at the free end (Z = L, the length of the rod) M = 0, i.e. 8' = 0. Putting 8(L) = 8 0 , we have in (1), Problem 1, Ci — /cos 8 0 and

= VQEiiflj

dO

-v/(cos do — cos 8)

Hence we obtain the equation for 6 0 :

= V(iEIlf)j

dd

\/(cos #o — cos 6)

The shape of the rod is given by

x = ^/(2EIjf)[^/(cos 6 0 )-V (cos do-cos 6) \,

in

y = V(EI/2f)j

cosddd

yYcos do — cos 6) e

Problem 3. The same as Problem 2, but for a force f parallel to the original direction of the rod.

Fig. 16

Solution. We have F = —f; the co-ordinate axes are taken as shown in Fig. 16. The boundary conditions are 8 = 0 for I = 0, 0' = 0 for / = L. Then

vwwj

ie

yYcos 6 —cos #o) o

where

d 0 = 6(L)

is given by

0.

- vmif)j

dd

\/(cOS0—COS0o) 0

For * and y we obtain

x = V(2W)[V(l~ cos 0o)-V(cos0-cos0 o )], e

y=V(EI/2f)j

cos 6 dd

-\/(cos 6 — cos do)

§19

The equations of equilibrium of rods

87

For a small deflection, 0 O <^ 1, and we can write

L

v(£///) !v(^)= w(£///) '

i.e. 0q does not appear. This shows that, in accordance with the result of §21, Problem 3, the solution in question exists only for / 5s ir 2 EI14L 2 , i.e. when the rectilinear shape ceases to be stable.

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