HWsolution9 - AE 321 – Homework#9 Solution 1(a For a bar...

Info iconThis preview shows pages 1–5. Sign up to view the full content.

View Full Document Right Arrow Icon

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

View Full Document Right Arrow Icon

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

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: AE 321 – Homework #9 Solution 1. (a) For a bar having a non-circular cross-section subjected to torsion, the stress functions, y x , must satisfy the following relation (obtained from the compatibility equations) 2 x 2 2 y 2 2 (1) where is the material shear modulus and is the angle of twist per unit length of the bar. Then 2 x 2 C 4 x 2 3 h 2 x 2 3 h C 6 x 2 h and 2 y 2 C 6 x 2 h substituting the previous results in (1) we have 2 x 2 2 y 2 C 6 x 2 h C 6 x 2 h 4 hC 2 C 1 2 h (2) substituting (2) in (1) we have 1 2 h x 3 y 2 3 h x 3 y 2 3 h x 1 3 h (3) Now, let’s applied the boundary condition (torque) at the end surface T 2 x , y dA A (4) substituting (3) in (4) we obtain 3 15 3 1 3 2 3 3 2 3 2 2 2 4 3 2 3 9 3 2 3 3 h T dydx h x h y x h y x h T h h h x Therefore 15 3 T h 4 (5) substituting (5) in (3) we obtain the stress function for the torsion of a long cylinder of solid triangular cross section subjected to a torque T at its end face x , y 15 3 2 T h 5 x 3 y 2 3 h x 3 y 2 3 h x 1 3 h (6) (b.1) Stress components xx yy zz xy xz y 15 3 T h 5 y 3 x h and yz x 15 3 T 2 h 5 2 hx 3 x 2 3 y 2 (b.2) Strain components (constitutive equations) ij 1 E 1 ij ij kk xx yy zz xy xz 1 E xz 15 3 T 1 Eh 5 y 3 x h and yz 1 E yz 15 3 T 1 2 Eh 5 2 hx 3 x 2 3 y 2 (b.3) Displacement components xx u x u u y , z (7) yy v y v v x , z (8) zz w z w w x , y (9) xy 1 2 u y v x u y v x (10) substituting (7) and (8) in (10) x z x v y z y u , , z g z yf u and z p z xf v (11) xz 1 2 u z w x A 3 xy hy where A 15 3 T 1 Eh 5 (12) substituting (11) in (12) we have ) 3 ( 2 ) , ( hy xy A x y x w...
View Full Document

{[ snackBarMessage ]}

Page1 / 17

HWsolution9 - AE 321 – Homework#9 Solution 1(a For a bar...

This preview shows document pages 1 - 5. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online