EN0175-10

EN0175-10 - EN0175 10 / 05 / 06 Maximum and minimum shear...

Info iconThis preview shows pages 1–3. 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 DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: EN0175 10 / 05 / 06 Maximum and minimum shear stresses in a solid An general stress state ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ = 33 32 31 23 22 21 13 12 11 σ σ σ σ σ σ σ σ σ σ , when expressed in the principal directions, becomes diagonalized as ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ = III II I σ σ σ σ 1 e v 2 e v 3 e v I II III t v n σ s σ n v The traction on an arbitrary plane with normal n v is 3 3 2 2 1 1 e n e n e n n t III II I v v v v v σ σ σ σ + + = = The magnitude of is therefore t v ( ) 2 1 2 3 2 2 2 2 2 1 2 n n n t III II I σ σ σ + + = v The normal stress on the plane is 2 3 2 2 2 1 n n n n n t n III II I n σ σ σ σ σ + + = ⋅ = ⋅ = v v v v Note that 2 2 2 t n s v = + σ σ , therefore ( ) 2 2 3 2 2 2 1 2 3 2 2 2 2 2 1 2 2 n n n n n n III II I III II I s σ σ σ σ σ σ σ + + − + + = We wish to find the maximum/minimum values of s σ subject to constraint (because is a unit vector). 1 2 3 2 2 2 1 = + + n n n n v Introduce Lagrangian multiplier λ , 1 EN0175 10 / 05 / 06 i i S n n F λ σ − = 2 i.e. ( ) ( ) 2 3 2 2 2 1 2 2 3 2 2 2 1 2 3 2 2 2 2 2 1 2 n n n n n n n n n F III II I III II I + + − + + − + + = λ σ σ σ σ σ σ Letting ( 3 , 2 , 1 = = ∂ ∂ j n F j ) , we have ( )...
View Full Document

This note was uploaded on 01/10/2010 for the course EN 0175 taught by Professor Huajiangao during the Spring '06 term at Brown.

Page1 / 7

EN0175-10 - EN0175 10 / 05 / 06 Maximum and minimum shear...

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

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