example 30.53 A steel rod buckles at a twentieth of its crushing load

open in the book · parts/03-classical-mechanics/13-continuum-elasticity.tex:1957 · p. 1029

example -- no derivation owed by its kind

Rests on

depends_on equation 30.50 eq:elast-euler-load ¶
depends_on phenomenon 30.52 Euler buckling ¶
- depends_on proposition 30.51 Moment and curvature ¶
  - depends_on definition 30.31 The engineering moduli ¶
    - depends_on definition 30.20 Pressure and deviatoric stress ¶
      - depends_on proposition 30.19 The stress tensor is symmetric ¶
      - depends_on theorem 5.79 Spectral theorem for a self-adjoint operator ¶
    - depends_on equation 30.21 eq:elast-hooke-isotropic ¶
  - depends_on equation 30.21 eq:elast-hooke-isotropic ¶ ↺
  - proves proof ch:13-continuum-elasticity@proof-27 ¶
- depends_on theorem 16.22 Euler–Lagrange ¶
  - depends_on equation 16.15 eq:calcvar-first-variation ¶
  - depends_on lemma 16.20 Mixed form ¶
    - depends_on corollary 7.44 Substitution and integration by parts ¶
      - depends_on proposition 7.31 Chain rule ¶
      - depends_on proposition 7.30 Leibniz rule ¶
      - depends_on theorem 7.43 Fundamental theorem of calculus, II ¶
      - proves proof ch:05-real-analysis@proof-26 ¶
    - depends_on lemma 16.19 du Bois-Reymond ¶
      - depends_on equation 16.18 eq:calcvar-dbr-hyp ¶
      - depends_on theorem 7.42 Fundamental theorem of calculus, I ¶
      - proves proof ch:14-calculus-of-variations@proof-9 ¶
    - proves proof ch:14-calculus-of-variations@proof-10 ¶
  - depends_on proposition 16.16 Stationarity is necessary ¶
    - depends_on definition 16.15 Variation; the first variation ¶
      - depends_on definition 16.11 Admissible class; functional ¶
      - depends_on equation 7.177 eq:ana-functional-derivative ¶
    - depends_on definition 16.13 Weak and strong extrema ¶
      - depends_on definition 16.12 The two norms; weak and strong neighbourhoods ¶
    - depends_on lemma 7.33 Fermat: interior extremum ¶
      - depends_on definition 7.26 Derivative of a function at a point ¶
      - depends_on proposition 7.18 Two-sided limit from one-sided limits ¶
      - proves proof ch:05-real-analysis@proof-17 ¶
    - proves proof ch:14-calculus-of-variations@proof-6 ¶
  - proves proof ch:14-calculus-of-variations@proof-11 ¶
- proves proof ch:13-continuum-elasticity@proof-28 ¶

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type	direction	node	provenance	where
`depends_on`	→	eq:elast-euler-load	declared	parts/03-classical-mechanics/13-continuum-elasticity.tex:1983
`depends_on`	→	Euler buckling	declared	parts/03-classical-mechanics/13-continuum-elasticity.tex:1983