Question 1 of 8: Stability and Degree of Static Indeterminacy

Question 1: Stability and Degree of Static Indeterminacy (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.

Given. Six plane structures, drawn below exactly as they appear on the examination paper.

Find. For each structure: unstable, statically determinate, or statically indeterminate — and, when indeterminate, the degree.

Approach. Count first, then look. For a beam or a rigid frame use the general count $i = 3m + r - 3j - c$, where $m$ is the number of members, $j$ the number of joints (supports included), $r$ the number of reaction components and $c$ the number of released conditions (one per internal hinge in a two-member chain); for a chain of members with no closed loop this collapses to $i = r - 3 - c$. For a pin-jointed truss use $i = m + r - 2j$. A non-negative count is necessary but never sufficient, so every count is followed by a check that the restraints are actually arranged so that no part of the structure can move.

1. (a) Continuous beam on four supports with two internal hinges. The supports are one pin and three rollers, so $r = 2 + 1 + 1 + 1 = 5$, and the two hinges supply two condition equations, $c = 2$: $$i = r - 3 - c = 5 - 3 - 2 = \boxed{0}$$ The count says determinate. Checking the arrangement: the left segment carries the pin and one roller and is therefore stable on its own; the suspended middle segment between the two hinges is simply supported by the two outer segments; the right segment carries two rollers and is held horizontally through the hinges by the pin. Nothing can move, so the beam is statically determinate.
2. (b) Two horizontal members joined by a pin-ended riser. The chain is fixed end — roller — hinge — riser — hinge — fixed end, so $r = 3 + 1 + 3 = 7$ and $c = 2$: $$i = 7 - 3 - 2 = \boxed{2}$$ The riser carries a hinge at each end and no load between them, so it is a two-force member and transmits a vertical force only; each horizontal member is independently stable on its own fixed end. The structure is statically indeterminate to the second degree.
3. (c) Rigid A-frame with a horizontal tie, on two pins. Here the closed triangle above the tie matters, so the collapsed formula does not apply. With members apex–left knee, apex–right knee, tie, left knee–support and right knee–support, $m = 5$, $j = 5$, $r = 2 + 2 = 4$ and $c = 0$: $$i = 3m + r - 3j - c = 15 + 4 - 15 - 0 = \boxed{4}$$ Three of those redundancies come from the single closed loop and the fourth from the extra pin. The frame is statically indeterminate to the fourth degree.
4. (d) Portal frame, pinned base and fixed base, hinge at the windward knee. Two columns and a beam give $m = 3$ and $j = 4$; $r = 2 + 3 = 5$ and the knee hinge gives $c = 1$: $$i = 9 + 5 - 12 - 1 = \boxed{1}$$ The frame is statically indeterminate to the first degree. Note how cheap the hinge is: without it the same frame would be indeterminate to the second degree.
5. (e) Trapezoidal truss with crossing diagonals. Counting members off the drawing — two bottom chords, one top chord, two inclined end posts and three diagonals (one of which crosses another without connection) — gives $m = 8$ with $j = 5$ joints and $r = 2 + 1 = 3$: $$i = m + r - 2j = 8 + 3 - 10 = \boxed{1}$$ The extra member is the long diagonal, so the truss is externally determinate but internally indeterminate to the first degree.
6. (f) Roof truss on a pin and a roller. The members are four bottom chords, four rafter segments, two middle-chord segments, three posts and two diagonals, so $m = 15$, with $j = 9$ and $r = 3$: $$i = 15 + 3 - 18 = \boxed{0}$$ The count alone would not settle it, so build the truss up: the triangle formed by the left rafter, the left post and the end bottom chord is rigid, and every remaining joint is then located by exactly two new members, using all fifteen members and leaving no joint free. With a pin and a roller the external restraints are neither parallel nor concurrent, so the truss is statically determinate and stable.