notes_15_pure___warp

notes_15_pure___warp - Differential equation and solution...

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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
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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
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notes_15_pure___warp - Differential equation and solution...

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