Computer Aided Engineering Design

338 COMPUTER AIDED ENGINEERING DESIGN

7. Gauss point integration is used extensively in computing the integral for the stiffness matrices of four-
   noded and like finite elements. This problem relates to determining Gauss points and weights. An integral

of the form

–1

1

∫ φ()tdt can be computed as iΣ

n

=1 ()wtiiφ where wi are the non-negative weights and ti are

Gauss points. Let φ(t) = a 0 + a 1 t + a 2 t^2 be exactly integrated using an order 2 Gauss rule, i.e., using 2 Gauss

points such that the points ti,i = 1, 2 are placed symmetrically in – 1 ≤t≤ 1. Also, consider the weights wi,

i = 1, 2 as symmetric (for a two point rule, w 1 = w 2 ). Determine the weights and Gauss points. Next,

determine the weights and Gauss points for an order 3 (or 3 point) rule. (Hint: with weights wi,i = 1, ..., 3

and points ti,i = 1, ..., 3, use symmetry to get w 2 = w 3 and t 2 = –t 3. Also, ti,i = 1, ..., 3 being symmetrically

placed in –1 ≤t≤ 1 would suggest that t 1 = 0).

8. Solve Problem P11.4 using a single bilinear four-node element with nodes 1, ..., 4. Determine the deflections
   at the free nodes.


9. Derive the matrices A and G is Eqs. (11.11i) and (11.11k). Note that we may get different expressions for
   A and G though B = AG will not be altered.