By Weltzien C.
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E. , x2, x1 by back substitution Step 1: To eliminate x1 in the lower equations: (i) First equation is maintained as it is (ii) For equations below 1, aij(1) = aij − and bij(1) = bi − ai1 aij a11 ai1 b1 a11 At the end of this, the equations will be LMa MM 0 MM 0M MM M MN 0 11 a12 (1) a22 a13 ... a1k ... a1n (1) ... a2(1k) ... a2(1n) a23 ak(12) (1) (1) ... akn ak(13) ... akk an(12) (1) (1) ... ann an(13) ... ank OP PP PP PP PQ R| x U| R| b U| ||x || ||b || S|xM V| = S|bM V| || M || || M || |Tx |W |Tb |W 1 2 1 (1) 2 k (1) k n (1) n The above process is called pivotal operation on a11.
Give strain displacement relations in case of a three dimensional elasticity problem upto (i) accuracy of linear terms only (ii) accuracy of quadratic terms. 5. Explain the terms, ‘Anisotropic’, ‘Orthotropic’ and ‘Isotropic’ as applied to material properties. 6. Give constitutive laws for three dimensional problems of (i) orthotropic materials (ii) isotropic materials. 7. Explain the terms ‘Plane stress’ and ‘Plane strain’ problems. Give constitutive laws for these cases. 8. Explain the term ‘Axi-symmetric problems’ and give constitutive law for such problems.
7), it may be observed that axial force do not affect values of bending moment and shear force and vice versa is also true. Hence stiffness matrix for the element shown in Fig. 8 is obtained by combining the stiffness matrices of bar element and beam element and arranging in proper locations. 8) 2 L2 (b) (a) Fig. 7 2 5 1 4 6 3 Fig. 8 28 Finite Element Analysis The following special features of matrix displacement equations are worth noting: (i) The matrix is having diagonal dominance and is positive definite.