Question 1 of 8: Classify each structure — unstable, statically determinate, or statically indeterminate (6 marks)

Question 1: Classify each structure — unstable, statically determinate, or statically 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.

Given. Six structures (a)–(f) with the support and internal-hinge arrangement shown below.

Find. The classification (unstable / determinate / indeterminate, with degree) of each.

[Figure not reproduced: Structures (a)–(f): supports, internal hinges, and truss bracing as scaled from the exam figure. See the official exam paper.]

Approach. For a beam/frame line, count reactions r, members m and joints n, and subtract one condition equation per internal hinge c that connects exactly two members: $$i = (3m+r) - 3n - c$$ For a pin-jointed truss, $$i = m + r - 2n$$ i = 0 is determinate, i > 0 is indeterminate to that degree, and i < 0 (or a mechanism geometry) is unstable.

1. (a) Beam with two rollers, two hinges, fixed end. Members: 4 (each hinge splits the line). Joints: 5. Reactions: roller + roller + fixed $$r = 1+1+3=5$$ Hinges: 2, each joining 2 members, $$c=2$$ $$i=(3\times4+5)-3\times5-2 = 17-15-2=\boxed{0}$$ Statically determinate.
2. (b) Closed rectangular frame, two horizontal members, pin + fixed base. Members: 6 (both verticals split at the mid-height tee); joints: 6; reactions $$r=2(\text{pin})+3(\text{fixed})=5$$ no hinges. $$i=(3\times6+5)-3\times6-0=23-18=\boxed{5}$$ Equivalently: one closed loop (3) plus support redundancy $$(r-3)=2$$ gives the same 5. Statically indeterminate to the 5th degree.
3. (c) Beam on two rollers with a rigid arm up to a pin. This is a single rigid body (no hinge) carried by 3 support points: roller + roller + pin. $$r=1+1+2=4,\qquad i=r-3=\boxed{1}$$ Statically indeterminate to the 1st degree.
4. (d) Beam–column chain: pin–roller–(hinge)–fixed. This is a single open chain (tree, no closed loop). Members 4, joints 5, $$r=2(\text{pin})+1(\text{roller})+3(\text{fixed})=6,\quad c=1$$ $$i=(3\times4+6)-3\times5-1=18-15-1=\boxed{2}$$ Statically indeterminate to the 2nd degree.
5. (e) Truss, pin + roller, no crossing connection. Counting the top chord (4), bottom chord (2), verticals (3) and the four "fan" diagonals (L1U1, U1L3, U2L4, L5U3): $$m=13,\quad n=8,\quad r=2+1=3$$ $$i=m+r-2n=13+3-16=\boxed{0}$$ Statically determinate.
6. (f) Truss with an X-braced square panel and a triangulated apex. Square panel: 4 chords + 2 crossing diagonals (not joined) = 6 members, 4 joints; the apex adds 2 members and 1 joint (a determinate addition). $$m=8,\quad n=5,\quad r=2(\text{pin})+1(\text{roller})=3$$ $$i=8+3-10=\boxed{1}$$ The extra diagonal over-braces the square panel by exactly one member. Statically indeterminate to the 1st degree.