theorem A.691 Kirchhoff's edge conditions and the corner force

open in the book · appendices/A-long-proofs.tex:33566 · p. 3128

Rests on

depends_on definition A.689 Edge resultants ¶
- depends_on equation A.1110 eq:app-kirchhoff-moments ¶
- depends_on proposition A.682 The plane-stress law ¶
  - depends_on definition 30.63 Flexural rigidity ¶
    - depends_on proposition 30.33 Conversion among the moduli ¶
      - depends_on definition 30.31 The engineering moduli ¶
      - depends_on phenomenon 30.32 Poisson contraction ¶
      - proves proof ch:13-continuum-elasticity@proof-15 ¶
    - depends_on proposition 30.51 Moment and curvature ¶
      - depends_on definition 30.31 The engineering moduli ¶ ↺
      - depends_on equation 30.21 eq:elast-hooke-isotropic ¶
      - proves proof ch:13-continuum-elasticity@proof-27 ¶
  - depends_on equation 30.21 eq:elast-hooke-isotropic ¶ ↺
  - depends_on proposition 30.33 Conversion among the moduli ¶ ↺
  - proves proof app:A-long-proofs@proof-405 ¶
depends_on theorem A.688 The Kirchhoff plate equation ¶
- depends_on lemma A.687 Kinetic energy ¶
  - depends_on definition A.680 Kirchhoff kinematics ¶
    - depends_on definition 30.6 Linear strain and infinitesimal rotation ¶
      - depends_on definition 13.91 Symmetric and antisymmetric parts ¶
      - depends_on equation 30.2 eq:elast-displacement ¶
    - depends_on proposition 30.51 Moment and curvature ¶ ↺
  - proves proof app:A-long-proofs@proof-408 ¶
- depends_on lemma 16.18 Fundamental lemma ¶
  - 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 theorem 7.40 Continuous functions are integrable ¶ ↺
  - proves proof ch:14-calculus-of-variations@proof-8 ¶
- depends_on proposition A.684 The plate energy ¶
  - depends_on definition 30.63 Flexural rigidity ¶ ↺
  - depends_on lemma A.681 Strains under the Kirchhoff hypothesis ¶
    - depends_on definition A.680 Kirchhoff kinematics ¶ ↺
    - depends_on definition 30.6 Linear strain and infinitesimal rotation ¶ ↺
    - proves proof app:A-long-proofs@proof-404 ¶
  - depends_on proposition A.682 The plane-stress law ¶ ↺
  - proves proof app:A-long-proofs@proof-406 ¶
- proves proof app:A-long-proofs@proof-409 ¶
depends_on theorem 16.31 Natural boundary condition ¶
- depends_on equation 16.15 eq:calcvar-first-variation ¶
- depends_on lemma 16.18 Fundamental lemma ¶ ↺
- 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 16.13 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:14-calculus-of-variations@proof-16 ¶
proves proof app:A-long-proofs@proof-410 ¶

Supports

depends_on remark A.692 The corner force is real, and you can feel it ¶

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
`depends_on`	→	Edge resultants	declared	appendices/A-long-proofs.tex:33593
`depends_on`	→	The Kirchhoff plate equation	declared	appendices/A-long-proofs.tex:33593
`depends_on`	→	Natural boundary condition	declared	appendices/A-long-proofs.tex:33593
`depends_on`	←	The corner force is real, and you can feel it	declared	appendices/A-long-proofs.tex:33671
`proves`	←	app:A-long-proofs@proof-410	declared	appendices/A-long-proofs.tex:33597