Which of the following is not a statically indeterminate beam?

Question

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The correct answer is Option A: Cantilever beam. A cantilever beam is a statically determinate structure because all support reactions can be calculated using only the equations of static equilibrium: $\sum F_x = 0$, $\sum F_y = 0$, and $\sum M = 0$. In contrast, indeterminate structures possess more unknown reaction components than available equilibrium equations, requiring compatibility of displacement conditions for solution.

Step-by-Step Solution & Concepts:
A cantilever beam is a statically determinate structure because all support reactions can be calculated using only the equations of static equilibrium: $\sum F_x = 0$, $\sum F_y = 0$, and $\sum M = 0$. In contrast, indeterminate structures possess more unknown reaction components than available equilibrium equations, requiring compatibility of displacement conditions for solution.

Working Principle: Static determinacy is governed by the relation $D_s = R - E$, where $D_s$ is the degree of static indeterminacy, $R$ is the number of unknown reactions, and $E$ is the number of available equilibrium equations. For a cantilever beam, $R=3$ (vertical, horizontal, and moment reaction at the fixed end) and $E=3$ (for planar structures), resulting in $D_s = 0$. Other options listed have $R > 3$, resulting in $D_s > 0$, making them statically indeterminate.

Key Formulas: $D_s = R - E$ — General equation for degree of static indeterminacy; $D_s = 0$ — Condition for static determinacy

Key Points: A structure is determinate if all reactions can be solved using static equilibrium equations alone.; Indeterminate beams provide higher redundancy and safety in case of local failures.; Propped cantilever beams have 4 unknown reactions ($R=4$) but only 3 equilibrium equations ($E=3$), making $D_s=1$.; Continuous beams are inherently indeterminate due to redundant support reactions.

Advantages: Determinacy allows for easy calculation of bending moments and shear forces.; Cantilever systems are simpler to design for basic architectural requirements.

Disadvantages: Determinate structures do not offer load redistribution if a support fails.; Higher deflections are often observed in simple determinate structures compared to indeterminate counterparts.

Applications: Balcony projections; Simple architectural overhangs; Aviation wings (as cantilever structures)

Additional Info: Fixed beams have 6 unknown reactions in 2D ($3+3=6$) but only 3 equations for each span; they are typically $D_s=3$ for a single span.. Continuous beams have $n$ redundant supports, leading to $D_s = n$.

Answer: A is correct — A cantilever beam is the only structure among the choices that is statically determinate, as its reactions are fully solvable via equilibrium equations.

Exam Tip: Always verify the number of support reactions ($R$) against the available equilibrium equations ($E=3$ for 2D beams) to quickly identify if a structure is indeterminate.

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Option B: Fixed beam

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Option C: Continuous beam

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Option D: Proped Cantilever beam

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