phenomenon 30.52 Euler buckling

open in the book · parts/03-classical-mechanics/13-continuum-elasticity.tex:1914 · p. 1028

phenomenon-tested: no experiment declares a \tests edge to it and it is not listed in untested.txt

Rests on

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 definition 30.17 Traction ¶
      - depends_on theorem 30.18 Cauchy: the stress tensor exists ¶
      - proves proof ch:13-continuum-elasticity@proof-8 ¶
    - depends_on theorem 5.79 Spectral theorem for a self-adjoint operator ¶
      - depends_on corollary 5.72 Existence of an eigenvalue over $\C$ ¶
      - depends_on definition 5.41 Adjoint ¶
      - depends_on theorem 5.40 Rank–nullity ¶
      - proves proof ch:03-linear-algebra-representations@proof-31 ¶
  - 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 definition 7.13 Composite function ¶
      - depends_on proposition 7.27 Differentiable implies continuous ¶
      - depends_on proposition 7.6 Algebra of limits ¶
      - proves proof ch:05-real-analysis@proof-15 ¶
    - depends_on proposition 7.30 Leibniz rule ¶
      - depends_on definition 7.12 Product of functions ¶
      - depends_on equation 7.14 eq:ana-derivh ¶
      - depends_on proposition 7.27 Differentiable implies continuous ¶ ↺
      - depends_on proposition 7.6 Algebra of limits ¶ ↺
      - proves proof ch:05-real-analysis@proof-14 ¶
    - depends_on theorem 7.43 Fundamental theorem of calculus, II ¶
      - depends_on definition 7.41 Antiderivative ¶
      - depends_on theorem 7.42 Fundamental theorem of calculus, I ¶
      - proves proof ch:05-real-analysis@proof-25 ¶
    - 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 definition 7.98 Functions of class $C^{1}$ ¶
      - depends_on theorem 7.40 Continuous functions are integrable ¶
    - 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 definition 16.11 Admissible class; functional ¶ ↺
      - depends_on definition 6.24 Metric ¶
  - depends_on lemma 7.33 Fermat: interior extremum ¶
    - depends_on definition 7.26 Derivative of a function at a point ¶
      - depends_on definition 7.16 Limit ¶
      - depends_on definition 7.9 Real function ¶
    - depends_on proposition 7.18 Two-sided limit from one-sided limits ¶
      - depends_on definition 7.16 Limit ¶ ↺
      - depends_on equation 7.7 eq:ana-limit-left ¶
      - depends_on equation 7.5 eq:ana-limit-right ¶
      - proves proof ch:05-real-analysis@proof-6 ¶
    - 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 ¶

Supports

depends_on example 30.53 A steel rod buckles at a twentieth of its crushing load ¶
depends_on remark 30.54 Other end conditions, and what buckling is variationally ¶
depends_on remark 30.55 Beyond the critical load: the elastica ¶
depends_on remark 30.68 Shells carry load in a different way ¶

Neighborhood

Every logical edge within two steps of this node.

declared and complete
partly declared
a check failed
not graded
declared in the source
inferred from structure

Edges

type	direction	node	provenance	where
`cites`	→	Methodus inveniendi lineas curvas maximi minimive proprietate gaudentes, sive solutio problematis isoperimetrici latissimo sensu accepti	derived	parts/03-classical-mechanics/13-continuum-elasticity.tex:1926
`depends_on`	→	Moment and curvature	declared	parts/03-classical-mechanics/13-continuum-elasticity.tex:1927
`depends_on`	→	Euler–Lagrange	declared	parts/03-classical-mechanics/13-continuum-elasticity.tex:1927
`depends_on`	←	A steel rod buckles at a twentieth of its crushing load	declared	parts/03-classical-mechanics/13-continuum-elasticity.tex:1983
`depends_on`	←	Other end conditions, and what buckling is variationally	declared	parts/03-classical-mechanics/13-continuum-elasticity.tex:2013
`depends_on`	←	Beyond the critical load: the elastica	declared	parts/03-classical-mechanics/13-continuum-elasticity.tex:2038
`depends_on`	←	Shells carry load in a different way	declared	parts/03-classical-mechanics/13-continuum-elasticity.tex:2431
`proves`	←	ch:13-continuum-elasticity@proof-28	declared	parts/03-classical-mechanics/13-continuum-elasticity.tex:1930