Global numbering of nodes italics elements and degrees of freedom numbers on vectors 6 4 5 3 2 2 2 3 1 1 1 Form element stiffiness matrix from column vectors see text p 56 with linalg a1 matrix 4 1 c s c s a2 matrix 1 4 c s c s c s a1 c s a2 c s c Multiply these to get element stiffness matrix of Eq 2 10 s k evalm A E L a1 a2 A E c 2 L A E c s L k A E c 2 L A E c s L AEcs L A E s2 L AEcs L A E s2 L A E c2 L AEcs L A E c2 L AEcs L Trigonometric relations A E c s L A E s 2 L A E c s L 2 A E s L c cos theta s sin theta theta arctan y 2 y 1 x 2 x 1 c cos s sin y2 y1 arctan x x 1 2 Nodal coordinates of element 1 x 1 0 y 1 0 x 2 1 5 y 2 25 Set precision get length Digits 4 L sqrt x 2 x 1 2 y 2 y 1 2 L 1 521 Define area and modulus unprotecting E this way is dangerous A 3 142e 4 unprotect E E 210e9 A 0003142 E 210 10 12 Evaluate stiffness matrix save as k1 k1 map eval k 4221 10 8 7035 10 7 k1 4221 10 8 7035 10 7 7035 10 7 1173 10 7 7035 10 7 1173 10 7 4221 10 8 7035 10 7 4221 10 8 7035 10 7 7035 10 7 1173 10 7 7035 10 7 1173 10 7 4221 10 8 7035 10 7 4221 10 8 7035 10 7 7035 10 7 1173 10 7 7035 10 7 1173 10 7 Redefine nodal coordinates for element 2 x 1 1 5 y 1 25 x 2 0 y 2 5 Reevaluate stiffness matrix save as k2 k2 map eval k 4221 10 8 7035 10 7 k2 4221 10 8 7035 10 7 7035 10 7 1173 10 7 7035 10 7 1173 10 7 Define global stiffness matrix K matrix 6 6 k1 1 1 k1 1 2 k1 1 3 k1 1 4 0 0 k1 2 1 k1 2 2 k1 2 3 k1 2 4 0 0 k1 3 1 k1 3 2 k1 3 3 k2 1 1 k1 3 4 k2 1 2 k2 1 3 k2 1 4 k1 4 1 k1 4 2 k1 4 3 k2 2 1 k1 4 4 k2 2 2 k2 2 3 k2 2 4 0 0 k2 3 1 k2 3 2 k2 3 3 k2 3 4 0 0 k2 4 1 k2 4 2 k2 4 3 k2 4 4 4221 10 8 7035 10 7 4221 10 8 7035 10 7 0 0 7035 10 7 1173 10 7 7035 10 7 1173 10 7 0 0 4221 10 8 7035 10 7 8442 10 8 0 4221 10 8 7035 10 7 K 7035 10 7 1173 10 7 0 2346 10 7 7035 10 7 1173 10 7 0 0 4221 10 8 7035 10 7 4221 10 8 7035 10 7 0 0 7035 10 7 1173 10 7 7035 10 7 1173 10 7 Solve for unknown displacements expand rows 3 and 4 of system dof 1 2 5 6 known to have zero displacement row3 K 3 3 u 3 K 3 4 u 4 0 row4 K 4 3 u 3 K 4 4 u 4 2000 row3 8442 10 8 u3 0 row4 2346 10 7 u4 2000 Solve for unknown displacements solve row3 row4 u 3 u 4 Solve for unknown forces from f KU Left hand side displacement vector u3 0 u4 0008526 U matrix 6 1 0 0 0 0008526 0 0 0 0 0 U 0008526 0 0 Form KU product to get right hand side force vector f evalm K U 5998 1000 0 f 2000 5998 1000
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