Which of the following are statically determinate beams?

Question

Examoogle · Accepted Answer

The correct answer is Option D: Cantilever, overhanging and simply supported beam. A beam is classified as statically determinate if all its unknown reaction components can be determined using only the three equations of static equilibrium: $\sum F_x = 0$, $\sum F_y = 0$, and $\sum M = 0$. Cantilever, simply supported, and overhanging beams satisfy this condition because the total number of unknown reactions equals the number of independent equilibrium equations.

Step-by-Step Solution & Concepts:
A beam is classified as statically determinate if all its unknown reaction components can be determined using only the three equations of static equilibrium: $\sum F_x = 0$, $\sum F_y = 0$, and $\sum M = 0$. Cantilever, simply supported, and overhanging beams satisfy this condition because the total number of unknown reactions equals the number of independent equilibrium equations.

Working Principle: The principle of static determinacy relies on the relationship between unknown reactive forces ($R$) and independent equilibrium equations ($E$). For planar structures, $E=3$. When $R = E$, the structure is statically determinate. If $R > E$, the structure is statically indeterminate (hyperstatic), requiring additional compatibility equations based on material deformation to solve for the reactions.

Key Formulas: $R = E$ — Condition for static determinacy where $R$ is total reaction components and $E$ is equilibrium equations; $D_s = R - E$ — Static indeterminacy degree calculation, where $D_s = 0$ for determinate structures

Key Points: A structure is determinate if all internal and external forces can be found using statics alone.; Fixed beams and continuous beams have redundant reactions ($R > 3$), making them statically indeterminate.; The number of independent equilibrium equations for a 2D beam is always 3 (horizontal, vertical, moment).; Overhanging beams are determinate because they function similarly to simply supported beams with added projections.

Advantages: Reactions are independent of material properties and cross-sectional dimensions.; Not affected by temperature changes or support settlements in terms of internal force generation.

Disadvantages: Generally less redundant; failure of a single critical support component can lead to collapse.; Often result in larger member sizes compared to indeterminate structures for the same span.

Applications: Simple architectural structures.; Temporary shoring and structural frames.; Basic bridge girder designs.

Additional Info: Fixed beams have 3 reactions at each end (2 forces, 1 moment) for a total of 6 reactions, exceeding the 3 equations of equilibrium.. Continuous beams are supported by more than two supports, creating redundant reaction forces that require additional deflection analysis.

Answer: D is correct — Cantilever, overhanging, and simply supported beams have reaction forces equal to the available equations of equilibrium, making them statically determinate.

Exam Tip: Always check for redundant reactions ($R > 3$) in beam problems; if you see fixed supports or more than two simple supports, it is almost certainly statically indeterminate.

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Option A: Only simply supported beam

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Option B: Fixed beam

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Option C: Continuous beam

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