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

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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⎟ ⎠ ...
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