Page 557 - Aircraft Stuctures for Engineering Student

538 Matrix methods of structural analysis

Fig. P.12.7
Ans. Fy,l =Fy,5 = -PI16

Fy,z = Fy,4 = 9P/ 16
MZ1. =M45 = -P1/16 (hogging)
MZ3 =M43 = -PI112 (hogging)
Twisting moment in 62, 82, 74 and 94 is P1/96.

P.12.8 It is required to formulate the stiffness of a triangular element 123 with
coordinates (0, 0), (a, 0) and (0, a) respectively, to be used for 'plane stress' problems.
(a) Form the [B] matrix.
(b) Obtain the stiffness matrix [K'].
Why, in general, is a finite element solution not an exact solution?

P.12.9 It is required to form the stiffness matrix of a triangular element 123 for
use in stress analysis problems. The coordinates of the element are (1, l), (2,l) and
(2,2) respectively.
(a) Assume a suitable displacement field explaining the reasons for your choice.
(b) Form the [B] matrix.
(c) Form the matrix which gives, when multiplied by the element nodal displace-
ments, the stresses in the element. Assume a general [D] matrix.

P.12.10 It is required to form the stiffness matrix for a rectangular element of side
2a x 2b and thickness t for use in 'plane stress' problems.
(a) Assume a suitable displacement field.
(b) Form the [q matrix.
(c) Obtain Jvol [qT [Dl [q d v.
Note that the stiffness matrix may be expressed as