Question 4: 3-D 8-node Linear Brick Element Method (20 marks) The 8-node isoparametric brick element shown...

   

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Question 4: 3-D 8-node Linear Brick Element Method (20 marks) The 8-node isoparametric brick element shown in Cartesian coordinate system (x.y.z) below is mapped to the natural coordinate (. n. 9). The coordinates of each node in mm are: (x₁.₁.₁)=(0,0,0) (x₂.₂.2₂)=(2,0,0) (x3.3.23)=(2.1.0) (x4Y424) = (0,3,0) (X5. 5. 25) = (0.0.4+0.q) (x6.6.26)=(3,0.3) (x7.1₁.2₁) = (3.3.3) (xg. Jg. Zg) = (0, 4,4+0.q) You are required to use shape functions to determine: (a) the Cartesian co-ordinate of centroid (= 0 . n = 0,=0); (b) the displacements at the centroid (= 0, n = 0, = 0) if the nodal displacement is: q=111 d¹ = [00 d3d4 0 0 ₁ ds]x24 The non-zero nodal displacements are: d₂ = {1.2.-1}, d₁ = {-2.0.-2), d, = {2.2.2), dg = {-1.1.2} Z (0,4,4)+ 0.q (3,3,3) (0,0,4)+q (3,0,3) ty 1 (0,3,0) (2,1,0) (2,0,0) 2 X mapping (-1,1,1) (-1,-1,1) (-1,1,-1) 7 55 (-1,-1,-1) Figure 4: 8-node Isoparametric Brick Element. (1,1,-1) (1,1,1) 6 (1,-1,1) ६ (1,-1,-1) Question 4: 3-D 8-node Linear Brick Element Method (20 marks) The 8-node isoparametric brick element shown in Cartesian coordinate system (x.y.z) below is mapped to the natural coordinate (. n. 9). The coordinates of each node in mm are: (x₁.₁.₁)=(0,0,0) (x₂.₂.2₂)=(2,0,0) (x3.3.23)=(2.1.0) (x4Y424) = (0,3,0) (X5. 5. 25) = (0.0.4+0.q) (x6.6.26)=(3,0.3) (x7.1₁.2₁) = (3.3.3) (xg. Jg. Zg) = (0, 4,4+0.q) You are required to use shape functions to determine: (a) the Cartesian co-ordinate of centroid (= 0 . n = 0,=0); (b) the displacements at the centroid (= 0, n = 0, = 0) if the nodal displacement is: q=111 d¹ = [00 d3d4 0 0 ₁ ds]x24 The non-zero nodal displacements are: d₂ = {1.2.-1}, d₁ = {-2.0.-2), d, = {2.2.2), dg = {-1.1.2} Z (0,4,4)+ 0.q (3,3,3) (0,0,4)+q (3,0,3) ty 1 (0,3,0) (2,1,0) (2,0,0) 2 X mapping (-1,1,1) (-1,-1,1) (-1,1,-1) 7 55 (-1,-1,-1) Figure 4: 8-node Isoparametric Brick Element. (1,1,-1) (1,1,1) 6 (1,-1,1) ६ (1,-1,-1) Question 4: 3-D 8-node Linear Brick Element Method (20 marks) The 8-node isoparametric brick element shown in Cartesian coordinate system (x.y.z) below is mapped to the natural coordinate (. n. 9). The coordinates of each node in mm are: (x₁.₁.₁)=(0,0,0) (x₂.₂.2₂)=(2,0,0) (x3.3.23)=(2.1.0) (x4Y424) = (0,3,0) (X5. 5. 25) = (0.0.4+0.q) (x6.6.26)=(3,0.3) (x7.1₁.2₁) = (3.3.3) (xg. Jg. Zg) = (0, 4,4+0.q) You are required to use shape functions to determine: (a) the Cartesian co-ordinate of centroid (= 0 . n = 0,=0); (b) the displacements at the centroid (= 0, n = 0, = 0) if the nodal displacement is: q=111 d¹ = [00 d3d4 0 0 ₁ ds]x24 The non-zero nodal displacements are: d₂ = {1.2.-1}, d₁ = {-2.0.-2), d, = {2.2.2), dg = {-1.1.2} Z (0,4,4)+ 0.q (3,3,3) (0,0,4)+q (3,0,3) ty 1 (0,3,0) (2,1,0) (2,0,0) 2 X mapping (-1,1,1) (-1,-1,1) (-1,1,-1) 7 55 (-1,-1,-1) Figure 4: 8-node Isoparametric Brick Element. (1,1,-1) (1,1,1) 6 (1,-1,1) ६ (1,-1,-1) Question 4: 3-D 8-node Linear Brick Element Method (20 marks) The 8-node isoparametric brick element shown in Cartesian coordinate system (x.y.z) below is mapped to the natural coordinate (. n. 9). The coordinates of each node in mm are: (x₁.₁.₁)=(0,0,0) (x₂.₂.2₂)=(2,0,0) (x3.3.23)=(2.1.0) (x4Y424) = (0,3,0) (X5. 5. 25) = (0.0.4+0.q) (x6.6.26)=(3,0.3) (x7.1₁.2₁) = (3.3.3) (xg. Jg. Zg) = (0, 4,4+0.q) You are required to use shape functions to determine: (a) the Cartesian co-ordinate of centroid (= 0 . n = 0,=0); (b) the displacements at the centroid (= 0, n = 0, = 0) if the nodal displacement is: q=111 d¹ = [00 d3d4 0 0 ₁ ds]x24 The non-zero nodal displacements are: d₂ = {1.2.-1}, d₁ = {-2.0.-2), d, = {2.2.2), dg = {-1.1.2} Z (0,4,4)+ 0.q (3,3,3) (0,0,4)+q (3,0,3) ty 1 (0,3,0) (2,1,0) (2,0,0) 2 X mapping (-1,1,1) (-1,-1,1) (-1,1,-1) 7 55 (-1,-1,-1) Figure 4: 8-node Isoparametric Brick Element. (1,1,-1) (1,1,1) 6 (1,-1,1) ६ (1,-1,-1)

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