notes_15_pure___warp

# notes_15_pure___warp - Differential equation and solution...

This preview shows pages 1–3. Sign up to view the full content.

Differential equation and solution aka Pure and Warping Torsion aka Free and Restrained Warping ref: Hughes 6.1 (eqn 6.1.18) the development of warping torsion up to this point was assumed to be "pure" or "free" i.e. it was the only effect on a beam and it's behavior was unrestrained. this led us to state φ '' and φ ''' were constant. the development of St. Venant's torsion in 13.10 was developed the same way, φ ' constant. we will now address the situation where boundary conditions may affect one or both of these effects. combined torsional resistance determined by: M x = M x St_V + M x w M x St_V is St. Venant's torsion = GK T ⋅φ ' M x w is warping torsion = E I ωω ''' M x is internal or external concentrated torque T ω M x = T ' EI ωω ''' (1) uniform (distributed) torque m x (torque per unit length) is related to Mx equilibrium element => m x = d M x => dx differentiating (1) => m x = ωω ''' T '' (2) solution of (1) has homogeneous and particular solution. rewriting: T M x let λ 2 = T φ ''' ' = ωω ωω ωω mx homogeneous => φ ''' − λ 2 ' = 0 assume solution φ H := e x ( d d x 3 3 φ H − λ 2 d d x φ H m 3 exp(m x) − λ 2 m exp(m x) => m 3 − λ 2 m = mm 2 − λ 2 ) = 0 1 notes_16 _pure_&_warping_torsion.mcd

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
I ωω λ λ λ x λ λ x I ωω I ωω A roots m := 0 m := m := 0 λ⋅ x x homogeneous solution => φ H := c 1 e + c 2 e + c 2 e particular solution assume φ P := Ax φ ''' GK T ⋅φ ' = M x x EI ωω ωω d 3 3 φ P − λ 2 d φ P −λ 2 A is a solution <=> A := M x dx dx λ 2 E −λ 2 A
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 6

notes_15_pure___warp - Differential equation and solution...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online