Perfect truss (perfect frame) criterion A planar framed structure (truss) is perfect if the number of its members m is _____ (2j − 3), where j is the number of joints.
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Aequal to
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Bless than
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Cgreater than
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Deither (b) or (c)
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Eindependent of
Answer
Correct Answer: equal to
Explanation
Introduction / Context:In planar truss analysis, a “perfect” (or statically determinate and stable) truss has just enough members to maintain shape without redundancy. The classical relation is m = 2j − 3.
Given Data / Assumptions:
- Planar pin-jointed truss.
- No internal mechanisms; joints are ideal pins; members carry only axial force.
- External reactions provide overall stability.
Concept / Approach:The formula m = 2j − 3 comes from counting equations of equilibrium and unknowns for a planar pin-jointed truss with adequate external restraints. If m is less than this number, the frame is deficient (mechanism). If m exceeds, it is redundant (indeterminate by statics alone).
Step-by-Step Solution:
For j internal joints, joint equilibrium gives 2j scalar equations (ΣFx = 0, ΣFy = 0 at each joint).Unknown internal member forces = m; external reaction components = 3 for a stable planar structure.Statical determinacy condition: m + 3 = 2j ⇒ m = 2j − 3.Thus, the perfect truss criterion is satisfied when m equals 2j − 3.Verification / Alternative check:Example: A simple triangle truss has j = 3, m = 3. Formula gives m = 2*3 − 3 = 3, which fits the base case.
Why Other Options Are Wrong:
- Less than: Mechanism — insufficient members to prevent distortion.
- Greater than: Redundant — indeterminate using only statics.
- Either (b) or (c): Contradicts the exact equality needed for a perfect truss.
Common Pitfalls:Applying the formula to space trusses (different relation), or to frames with non-pinned joints and bending members where the count changes.
Final Answer:equal to