Answer Exercise 6.3.8 for the illustrated two- and three-dimensional houses. In the two-dimensional case, the width and total height of the vertical bars is 2 units, and the peak is an additional .5 unit higher. In the three-dimensional house, the width and vertical heights are equal to 1 unit, the length is 3 units, while the peaks are 1.5 units above the base.

Exercise 6.3.8
(a) Determine the reduced incidence matrix A for this structure.
(b) How many distinct modes of instability are there? Describe them geometrically, and indicate whether they are mechanisms or rigid motions.
(c) Suppose we apply a combination of forces to each non-fixed node in the structure. Determine conditions such that the structure can support the forces. Write down an explicit nonzero set of external forces that satisfy these conditions, and compute the corresponding elongations of the individual bars. Which bar is under the most stress?
(d) Add in a minimal number of bars so that the resulting structure can support any force. Before starting, decide, from general principles, how many bars you need to add.
(e) With your new stable configuration, use the same force as before, and recomputed the forces on the individual bars. Which bar now has the most stress? How much have you reduced the maximal stress in your reinforced building?