A horizontal grid is composed of n members meeting at one joint A of which a typical member i is shown in the figure. Write the stiffness matrix corresponding to the coordinates 1, 2, and 3 indicated, with coordinate 1 representing a downward deflection, and 2 and 3 being rotations. The length of the ith member, its torsional rigidity, and its flexural rigidity in the vertical plane through its axis are li, GJi, and EIi respectively.
Hint: The stiffness matrix of the grid can be obtained by the summation of the stiffness matrices of individual members. To obtain the first column of the stiffness matrix of the ith member, a unit downward displacement is introduced and the end-forces are taken from Appendix D. These forces are resolved into components at the three coordinates. For the second and third columns, a unit rotation at coordinate 2 and 3 is resolved into two rotations represented by vectors along and normal to the beam axis and the corresponding end-forces are taken from Appendix D. These forces are then resolved into components at the coordinates to obtain elements of the stiffness matrix.