tailieunhanh - dohrmann Episode 1 Part 7

Tham khảo tài liệu 'dohrmann episode 1 part 7', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Given a uniform strain element with volume V the so-called B matrix is defined as dV Bn Trf dxn 1 where XịỊ z 1 2 3 and I are the coordinates of node I. Following the development in Ref. 1 the nodal forces associated with the element stresses are given by fil TjjBji j i 2 where Tij are elements of the Cauchy stress tensor assumed constant throughout the element . All the elements discussed in this section use the same formulation. The only differences are the volume expressions and the number of nodes per element. The uniform strain hexahedron and the transition element each have eight nodes. The uniform strain tetrahedron has four vertex nodes and from one to four mid-face nodes. Uniform Strain Hexahedron Consider a hexahedral element with nodal coordinates xu X2I x3f for I 1 . 8. Spatial coordinates coordinates Xỵ x2 and x3 are related to isoparametric coordinates 771 TỊ2 and 7 3 by the equations Xi t i 772 773 52 xưự i 7 i 772 773 1 3 where ự i 1 771 1 - 772X1 - 773 8 Ộ3 1 7 1 1 1 - 8 05 1 - m 1 - 72 1 8 07 1 771 1 772 1 T73 8 02 1 T71 1 - 772X1 - 8 04 1 - 771X1 1 - 8 ộĩ 1 771 1 772X1 T73 8 f 8 1 - 771 1 772X1 773 8 4 5 6 7 The Jacobian determinant J of the element is given by dxi 9x2 Ỡ7Ị1 0771 0X1 0X2 TÌ2 ậm 0X1 ỜX2 . Ô773 ỔT73 Ô773 8 The volume of the hexahedron can be expressed in terms of J by 9 2 The B matrix of the hexahedron is defined as T hex _ dvhex dxu 10 Equations for BịỊX are provided in Reference 1. In addition one has yhex _ 1 1 1 1 8 8 52 52 x2IB x 11 Transition Element Consider a polyhedron with fourteen vertices and twenty four triangular faces. Eight of the vertices are the nodes of the hexahedron. The remaining vertices are located at the geometric centers of the six faces of the hexahedron. The coordinates of these vertices are given by Xia xn Xi2 Xie 4- zi5 4 xib Xie Zi5 Xie xi7 4- xi8 4 xid Xie 2-14 4 Xil 4 Xie 4 Xịè Ị Á. Xịf Xị3 4- Xi4 4- Xis 4- Xít 4 12 Xi2 4- 4- Xj4 4- xi3 4 13 Xi2 4- Xị3 4- Xi7 4- Xiè 4 14 The triangular faces

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