Transcription of Useful Equations
1 Useful Equations Bending deformation: dV dM. = w( x) = V ( x) M = EIv ''. dx dx M = M 0. Ey M zz y ( x, y ) = =. I zz (rectangle), (circle). , 3V. max =. 2 A (rectangle), 4V. max =. 3 A (circle). Strain energy density: 1. = [ ( ) + + ( ) + + + ]. 2. Energy methods: 1 F ( x) 1 f sV ( x ) 1 M ( x) 1 T ( x). L 2 L 2 L 2 L 2. 2 0 EA 2 0 GA 2 0 EI 2 0 GI p U= dx U= dx U= dx U= dx Work-energy principle: U = W. Castigliano's 2nd theorem: U U U. Pi = Mi = Ti =. Pi M i Ti L L L L. M ( x) M ( x) F ( x) F ( x) T ( x) T ( x) f V ( x) V ( x). Pi = dx + dx + dx + s dx 0. EI Pi 0. EA P i 0. GI p P i 0. AG P i L L L L. M ( x) M ( x) F ( x) F ( x) T ( x) T ( x) f V ( x) V ( x). Mi = dx + dx + dx + s dx 0. EI M i 0. EA M i 0. GI p M i 0. AG M i L L L L. M ( x) M ( x) F ( x) F ( x) T ( x) T ( x) f V ( x) V ( x).
2 Ti = dx + dx + dx + s dx 0. EI Ti 0. EA Ti 0. GI p Ti 0. AG Ti fs=6/5 (rectangular cross section), fs=10/9 (circular cross section). Page 1 of 1.
