Question 1 of 8: Classify each structure — unstable / determinate / indeterminate

Question 1: Classify each structure — unstable / determinate / indeterminate (6 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Approach. Count external reaction components r, released internal conditions c (each internal hinge in a beam releases one moment; a pin joining k members releases k−1), and apply the appropriate degree-of-static-indeterminacy (DSI) formula: beams/frames DSI = 3m + r − 3n − c; pin-jointed trusses DSI = m + r − 2n (where m = members, n = joints). DSI > 0 ⇒ indeterminate to that degree; DSI = 0 ⇒ determinate; a negative count or a geometric defect ⇒ unstable.

1. (a) Continuous beam, three rollers + fixed end, two internal hinges. Reactions $r = 3(1)+3 = 6$; conditions $c = 2$ hinges. Treating the run of collinear beam members as one flexural chain, $DSI = r-3-c = 6-3-2 = \boxed{1}$ — indeterminate, degree 1.
2. (b) Continuous beam, two pins + two rollers, one internal hinge. $r = 2+2+1+1 = 6$, $c = 1$. $DSI = 6-3-1 = \boxed{2}$ — indeterminate, degree 2 (the extra horizontal pin restraint is one of the two redundancies).
3. (c) Single-bay, two-storey frame; both beams pin-connected (hinged) to continuous columns fixed at their bases. Model $m = 6$ members, $n = 6$ joints, $r = 3+3 = 6$ (two fixed feet), $c = 4$ (a hinge at each of the four beam ends). $DSI = 3(6)+6-3(6)-4 = \boxed{2}$. Check by closed loops: two rigid ring-circuits close through the foundation, $2\times3 = 6$ redundancies, less the 4 releases $= 2$. Indeterminate, degree 2.
4. (d) Gabled frame: two rafters meeting at a hinged apex, a tie beam, three columns fixed at their bases; hinges at both eaves. $m = 7$, $n = 7$, $r = 3(3) = 9$ (three fixed feet), $c = 5$ (two-member release of 1 at the apex, and a three-member pin release of 2 at each eave). $DSI = 3(7)+9-3(7)-5 = \boxed{4}$ — indeterminate, degree 4.
5. (e) Four-panel parallel truss; the two end panels carry crossed (double) diagonals, pin at one foot and roller at the other. $m = 19$, $r = 3$, $n = 10$. $DSI = m+r-2n = 19+3-20 = \boxed{2}$ — indeterminate, degree 2 (one redundant diagonal in each X-panel).
6. (f) Rotated-square (diamond) truss on two legs, pin + roller feet, one pair of crossed diagonals. $m = 10$, $r = 3$, $n = 6$. $DSI = 10+3-12 = \boxed{1}$ — indeterminate, degree 1 (the square panel carries one redundant diagonal).