If the beam is supported so that there are only three unknown reactive elements at the supports, these can be

Question

If the beam is supported so that there are only three unknown reactive elements at the supports, these can be determined by using the following fundamental equation of statics

Examoogle · Accepted Answer

The correct answer is Option D: $\sum H = 0, \sum V = 0, \sum M = 0$. A beam is considered statically determinate when the number of unknown reactive forces equals the number of independent equilibrium equations available. For a general coplanar force system, three independent equations of statics exist to ensure the beam remains in a state of equilibrium.

Step-by-Step Solution & Concepts:
A beam is considered statically determinate when the number of unknown reactive forces equals the number of independent equilibrium equations available. For a general coplanar force system, three independent equations of statics exist to ensure the beam remains in a state of equilibrium.

Working Principle: According to Newton's First Law and the conditions for static equilibrium, a rigid body subjected to a coplanar force system is in equilibrium if the sum of all horizontal forces, vertical forces, and moments about any point in the plane are zero. When a beam has three unknown reaction components, these three linear equations provide a sufficient system of algebraic equations to uniquely solve for all unknowns.

Key Formulas: $\sum F_x = 0$ — Summation of horizontal forces equals zero; $\sum F_y = 0$ — Summation of vertical forces equals zero; $\sum M_A = 0$ — Summation of moments about point A equals zero

Key Points: Equilibrium is reached when the resultant force and resultant moment are zero.; A structure with three unknown reactions is statically determinate.; If unknowns > 3, the beam is statically indeterminate and requires deformation-based methods (e.g., Slope Deflection or Moment Distribution) for solution.

Advantages: Solvable using basic algebraic methods; Independent of material properties and cross-sectional stiffness

Disadvantages: Cannot determine reactions if structure is statically indeterminate; Assumes the structure is rigid

Applications: Simply supported beams; Overhanging beams; Cantilever beams

Additional Info: For a coplanar system, the maximum number of independent equilibrium equations is limited to three.. Option A, B, and C provide partial conditions and are insufficient for solving three independent unknowns.

Answer: D is correct — The three standard equations of static equilibrium are necessary and sufficient to determine three unknown reactions in a 2D beam structure.

Exam Tip: Always verify the degree of static indeterminacy ($D_s$) using $D_s = r - 3$ before attempting to solve beam reactions.

Examoogle · Answer

Option A: $\sum H = 0$

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Option B: $\sum V = 0$

Examoogle · Answer

Option C: $\sum H = 0, \sum V = 0$

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