prob_15_03_02

# prob_15_03_02 - Problem 15.3-2 Show that Eq 15.3-6 follows...

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Problem 15.3-2 Show that Eq. 15.3-6 follows from Eq. 15.3-4 when [B b ] and [Bs] are defined as stated in the text. Solution: ⌠ T k = ⎮ Bm ⋅ Dm⋅ Bm dA = ⎮ ⌡ ⌠ T k = ⎮ Bb ⋅ Dm⋅ Bb dA + ⎮ ⌡ We need to show that: Let: D := 1 G := 2 ν := 0.3 t := 1 ⌠ ⎮ ⎮ ⌡ (Bb + Bs)T ⋅ Dm⋅ (Bb + Bs) dA ⌠ ⎮ B T ⋅ D ⋅ B dA + s mb ⎮ ⌡ ⌠ ⎮ B T ⋅ D ⋅ B dA s ms ⎮ ⌡ ⌠ ⎮ B T ⋅ D ⋅ B dA + b ms ⎮ ⌡ ⌠ ⎮ B T ⋅ D ⋅ B dA + b ms ⎮ ⌡ ν⋅ D D 0 0 0 ... ⎞ ⌠ ⎮ B T ⋅ D ⋅ B dA = 0 s mb ⎮ ⌡ 0 0 0 Let: Ni , x = d Ni dx ⎡D ⎢ν⋅ D k := 1 ⎢ Dm := ⎢ 0 ⎢ ⎢0 ⎢ ⎣0 0 N2 , x 0 0 0 0 0 ⎤ 0 0⎥ 0 ⎥ ( 1 − ν) ⋅ D 0⎥ 0 2 ⎥ k ⋅ G⋅ t 0 ⎥ 0 ⎥ k ⋅ G⋅ t⎦ 0 0 ⎛0 ⎜ ⎜0 Bs = ⎜ 0 ⎜ −N , y ⎜1 ⎜ −N1 , x ⎝ 0 0 0 0 N1 0 0 0 ⎛1 ⎜ 0.3 ⎜ Dm = ⎜ 0 ⎜0 ⎜ ⎝0 0 0 0 0 0 0 0 0.3 1 ⎟ ⎟ 0 0.35 0 0 ⎟ 0 0 2 0⎟ ⎟ 0 0 0 2⎠ 0 00 0 0 0 ... ⎞ ... ⎟ ... ⎟ 0 0 0⎞ ⎛ 0 N1 , x 0 ⎜ ⎜ 0 0 N1 , y Bb = ⎜ 0 N , y N , x 1 1 ⎜ ⎜0 0 0 ⎜ 0 ⎝0 0 For a 3 node element there would be 6 columns in B b and B s. Let each possible nozero term equal 1 ⎟ N2 , y ... ⎟ 0 ⎟ N2 , y N2 , x ... ⎟ ... ⎟ 0 0 ⎟ ... ⎠ 0 0 ⎟ ⎟ ⎠ N1 −N2 , y 0 N2 ... ⎟ 0 ... ⎟ −N2 , x N2 ⎛0 ⎜0 ⎜ Bb := ⎜ 0 ⎜0 ⎜ ⎝0 1 0 0 1 0 0 1 0⎞ ⎟ ⎟ 1 1 0 1 1 0 1 1⎟ 0 0 0 0 0 0 0 0⎟ ⎟ 0 0 0 0 0 0 0 0⎠ 01001001 ⎛0 ⎜0 ⎜ Bs := ⎜ 0 ⎜1 ⎜ ⎝1 0 0 0 0 0 0 0 0⎞ 00000000 ⎟ ⎟ 0 0 0 0 0 0 0 0⎟ 0 1 1 0 1 1 0 1⎟ ⎟ 1 0 1 1 0 1 1 0⎠ ⎛0 ⎜ ⎜0 ⎜0 ⎜ ⎜0 T Bb ⋅ Dm⋅ Bs → ⎜ 0 ⎜0 ⎜ ⎜0 ⎜0 ⎜0 ⎝ 0 0 0 0 0 0 0 0⎞ 0 0 0 0 0 0 0 0⎟ 0 0 0 0 0 0 0 0⎟ 00000000 0000000 0000000 0000000 0000000 0000000 ⎟ ⎟ ⎟ 0⎟ 0⎟ ⎟ 0⎟ 0⎟ 0⎟ ⎠ ⎛0 ⎜ ⎜0 ⎜0 ⎜ ⎜0 T Bs ⋅ Dm⋅ Bb → ⎜ 0 ⎜0 ⎜ ⎜0 ⎜0 ⎜0 ⎝ 0 0 0 0 0 0 0 0⎞ 0 0 0 0 0 0 0 0⎟ 0 0 0 0 0 0 0 0⎟ 00000000 0000000 0000000 0000000 0000000 0000000 ⎟ ⎟ ⎟ 0⎟ 0⎟ ⎟ 0⎟ 0⎟ 0⎟ ⎠ ...
View Full Document

{[ snackBarMessage ]}

Ask a homework question - tutors are online