Question

Consider the following three structures (shown): \includegraphics[width=0.5\linewidth]{image33.png} Which of the following statements is/are TRUE?

• A. Structure I is unstable
• B. Structure II is unstable
• C. Structure III is unstable
• D. All three structures are stable

Hint:

For planar trusses, first check $m+r \;\overset{?}{=}\; 2j$ for determinacy, but remember: $m+r=2j$ is necessary—not sufficient—for stability. Always combine the count check with geometry (triangulation) and support restraints.

Answer 1

The Correct Option is D

Structure I (beam with internal hinges).
Reactions: A (2), C (1), E (1) $\Rightarrow$ total $=4$. Two internal hinges (at B and D) split the beam into three segments, supplying two extra independent equilibrium relations. The system is statically determinate and stable. Hence (A) is false.
Structure II (4–joint truss with both diagonals).
Joints $j=4$, members $m=6$ (4 sides $+$ 2 diagonals), reactions $r=4$ (A: 2, B:1, D:1). $m+r=10>2j=8$ $\Rightarrow$ externally indeterminate of degree 2, but supports provide sufficient restraints and the triangulated geometry is rigid. Stable, not a mechanism. So (B) is false.
Structure III (6–joint truss with braced right panel).
Joints $j=6$, members $m=9$, reactions $r=3$ (A:2, C:1). $m+r=12=2j$ $\Rightarrow$ statically determinate. The right bay is triangulated (X-braced) fixing joints B and E; with supports at A and C, the left bay is restrained through AB, AD, DE and EB. Hence stable. So (C) is false.
\[ \boxed{\text{All three structures are stable (Option D).}} \]