HW10_beams_andmembrane

HW10_beams_andmembrane - Labels 1x 1z 2x 2z 3x,3z K1 1.125...

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Unformatted text preview: Labels: 1x, 1z, 2x, 2z, 3x,3z K1 1.125 10 9 × 3.75 − 10 8 × 7.5 − 10 8 × 3.75 − 10 8 × 3.75 10 8 × 3.75 − 10 8 × 1.688 10 9 × 3.75 10 8 × 1.5 − 10 9 × 1.875 − 10 8 × 7.5 − 10 8 × 3.75 10 8 × 7.5 10 8 × 3.75 − 10 8 × 1.5 − 10 9 × 1.5 10 9 × 3.75 − 10 8 × 3.75 10 8 × 3.75 10 8 × 1.875 − 10 8 × 3.75 − 10 8 × 1.875 10 8 × ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ = K1 B T D ⋅ B ⋅ A ⋅ t ⋅ := B 0.5 − 1 1 0.5 − 1 − 1 − 0.5 0.5 ⎛ ⎜ ⎝ ⎞ ⎠ = B 1 2 A ⋅ b1 c1 c1 b1 b2 c2 c2 b2 b3 c3 c3 b3 ⎛ ⎜ ⎝ ⎞ ⎠ ⋅ := c3 x2 x1 − := b3 y1 y2 − := D E 1 υ 2 − 1 υ υ 1 1 υ − 2 ⎛ ⎜ ⎜ ⎜ ⎝ ⎞ ⎟ ⎟ ⎠ ⋅ := c2 x1 x3 − := b2 y3 y1 − := c1 x3 x2 − := b1 y2 y3 − := Element 1 : x3 2 := y3 1 := L3 1 := I 1 12 b ⋅ h 3 ⋅ := x2 := y2 := L2 2 := x1 := y1 1 := h 0.06 := b 0.06 := υ := A 2 1 ⋅ 2 := t 15 10 3 − ⋅ := E 100 10 9 ⋅ := Definition of material properties etc: Coordinate system used: x: to the right, y: into the paper, z: pointing up HW 10 Solution: beams combined with the membrane element inserting zero rows and columns where needed (for the rotational degree of freedom) zero_6_1 0 0 0 0 0 0 ( ) T := zero_1_9 0 0 0 0 0 0 0 0 0 ( ) := inserting zero columns K1_large augment submatrix K1 0 , 5 , , 1 , ( ) zero_6_1 , submatrix K1 0 , 5 , 2 , 3 , ( ) , zero_6_1 , submatrix K1 0 , 5 , 4 , 5 , ( ) , zero_6_1 , ( ) := inserting zero rows K1_large stack submatrix K1_large 0 , 1 , , 8 , ( ) zero_1_9 , submatrix K1_large 2 , 3 , , 8 , ( ) , zero_1_9 , submatrix K1_large 4 , 5 , , 8 , ( ) , zero_1_9 , ( ) := K1_large 1.125 10 9 × 3.75 − 10 8 × 7.5 − 10 8 × 3.75 − 10 8 × 3.75 10 8 × 3.75 − 10 8 × 1.688 10 9 × 3.75 10 8 × 1.5 − 10 9 × 1.875 − 10 8 × 7.5 − 10 8 × 3.75 10 8 × 7.5 10 8 × 3.75 − 10 8 × 1.51....
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HW10_beams_andmembrane - Labels 1x 1z 2x 2z 3x,3z K1 1.125...

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