Statically determinate beam with three reaction unknowns If a beam is supported such that there are exactly three independent reaction components, which equilibrium equations are sufficient to determine them?
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A∑H = 0 only
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B∑V = 0 only
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C∑H = 0; ∑H = 0 (duplicate)
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D∑H = 0; ∑V = 0; ∑M = 0
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E∑M = 0; ∑H = 0 (missing vertical equilibrium)
Answer
Correct Answer: ∑H = 0; ∑V = 0; ∑M = 0
Explanation
Introduction / Context:A planar, statically determinate problem can be solved using only the equations of static equilibrium. For beams in the plane, there are three independent equations available.
Given Data / Assumptions:
- Planar system (2D), rigid body equilibrium.
- No additional redundants or internal releases.
- Supports provide exactly three independent reaction components in total.
Concept / Approach:The independent equilibrium equations in a plane are:∑H = 0∑V = 0∑M = 0These three equations can determine three unknown reaction components if the structure is statically determinate.
Step-by-Step Solution:
Count unknown reactions = 3.Provide three independent equations of equilibrium.Solve the linear system to obtain reactions.Verification / Alternative check:If more than three unknowns exist, the structure is statically indeterminate to that degree and requires compatibility (deformations) to solve.
Why Other Options Are Wrong:
- Options a, b: insufficient equations.
- Option c: duplicates ∑H and omits the others.
- Option e: omits ∑V = 0; system remains underdetermined.
Common Pitfalls:Counting dependent reaction components or missing the moment equation about a convenient point.
Final Answer:∑H = 0; ∑V = 0; ∑M = 0