For a beam, if fundamental equations of statics are sufficient to determine all the reactive forces at the sup

Question

For a beam, if fundamental equations of statics are sufficient to determine all the reactive forces at the supports, the structure is said to be

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The correct answer is Option B: Statically determinate. A structure is called statically determinate if all its support reactions and internal forces can be uniquely calculated using only the three equations of static equilibrium ($\sum F_x = 0$, $\sum F_y = 0$, $\sum M = 0$ for a 2D system). If the number of unknown reactions exceeds the number of independent equilibrium equations, the structure is classified as statically indeterminate.

Step-by-Step Solution & Concepts:
A structure is called statically determinate if all its support reactions and internal forces can be uniquely calculated using only the three equations of static equilibrium ($\sum F_x = 0$, $\sum F_y = 0$, $\sum M = 0$ for a 2D system). If the number of unknown reactions exceeds the number of independent equilibrium equations, the structure is classified as statically indeterminate.

Working Principle: In a 2D plane structure, there are 3 equilibrium equations available. Let 'r' be the number of unknown reactions and 'c' be the number of condition equations (like internal hinges). If $r = 3 + c$, the structure is stable and determinate. If $r > 3 + c$, the structure is statically indeterminate and requires deformation-based compatibility equations to solve for the extra unknowns.

Key Formulas: $r = 3 + c$ — Determinacy condition for a 2D beam structure; $D_s = r - (3 + c)$ — Degree of static indeterminacy

Key Points: Statically determinate structures do not experience internal stresses due to temperature changes or support settlements.; The stability of a structure must be verified before checking for determinacy.; Indeterminate structures are generally stiffer and have higher redundancy, making them safer against progressive collapse.

Advantages: Simple analysis using basic equilibrium equations.; Internal forces are independent of material properties and support settlements.

Disadvantages: Lower structural redundancy compared to indeterminate structures.; Higher material requirement for long-span conditions compared to continuous (indeterminate) designs.

Applications: Simply supported beams; Cantilever beams; Three-hinged arches

Additional Info: For 3D structures, there are 6 equilibrium equations, so the criteria becomes $r = 6$.. Option C refers to redundant structures where unknowns exceed equilibrium equations, requiring compatibility of deformations.

Answer: B is correct — A structure is statically determinate when the number of independent equilibrium equations equals the number of unknown reactions.

Exam Tip: In competitive exams, always check if the structure is stable first; an unstable structure can have $r=3$ but still not be 'determinate' in the engineering sense.

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Option A: Determinate

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Option C: Statically indeterminate

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Option D: None of these

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