**Answer to Provided MCQ:**  

**Correct Option:** C) 3  

**Justification:** In a general coplanar force system, all forces lie in same plane but can act in any direction. For static equilibrium, three independent equations must be satisfied: sum of forces in x-direction equals zero (ΣFx = 0), sum of forces in y-direction equals zero (ΣFy = 0), and sum of moments about any point equals zero (ΣM = 0). Therefore, a maximum of three unknown forces can be determined.

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### Additional High-Yield MCQs on Equilibrium & Static Determinacy

1. **In a co planar parallel force system, number of unknown forces that can be found by principle of statics ......**  

   a) 3  

   b) 2  

   c) 1  

   d) 0  

   **Correct Answer:** b) 2

2. **The number of independent equations to be satisfied for static equilibrium of a plane structure is ......**  

   a) 1  

   b) 2  

   c) 3  

   d) 6  

   **Correct Answer:** c) 3

3. **The number of independent equations to be satisfied for static equilibrium in a space structure is ......**  

   a) 2  

   b) 3  

   c) 4  

   d) 6  

   **Correct Answer:** d) 6

4. **What is maximum number of unknown reaction components that can be determined using only statics?**  

   a) 0  

   b) 1  

   c) 2  

   d) 3  

   **Correct Answer:** d) 3

5. **If beam is supported so that re are only three unknown reactive elements at supports. These can be determined by using following fundamental equation of statics ......**  

   a) ΣH = 0  

   b) ΣV = 0  

   c) ΣH = 0, ΣV = 0  

   d) ΣH = 0, ΣV = 0, ΣM = 0  

   **Correct Answer:** d) ΣH = 0, ΣV = 0, ΣM = 0

6. **For a beam, if fundamental equations of statics are not sufficient to determine all reactive forces at supports, structure is said to be ......**  

   a) determinate  

   b) statically determinate  

   c) statically indeterminate  

   d) none of se  

   **Correct Answer:** c) statically indeterminate

7. **A beam is said to be in equilibrium if ......**  

   a) it moves horizontally  

   b) it moves vertically  

   c) it rotates about its CG  

   d) none of se  

   **Correct Answer:** d) none of se

8. **If two forces acting at a joint are not along straight line, n for equilibrium of joint ......**  

   a) one of forces must be zero  

   b) each force must be zero  

   c) forces must be equal and of same sign  

   d) forces must be equal in magnitude but opposite in sign  

   **Correct Answer:** b) each force must be zero

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### Core Theoretical Concepts (General Coplanar Force System & Equilibrium)

- A general coplanar force system consists of forces that all lie in same plane but can have any direction.

- For a body to be in static equilibrium, net force and net moment acting on it must be zero.

- Three independent equations of equilibrium exist for a 2D (coplanar) system:

  1. ΣFx = 0 (Sum of horizontal forces equals zero)

  2. ΣFy = 0 (Sum of vertical forces equals zero)

  3. ΣM = 0 (Sum of moments about any point equals zero)

- These three equations can be used to solve for a maximum of three unknown quantities (e.g., support reactions, internal forces).

- If number of unknowns exceeds number of available equilibrium equations, structure is **statically indeterminate** and requires compatibility conditions for analysis.

- The choice of point for taking moments is arbitrary; a strategically chosen point can simplify calculations by eliminating unknowns.

- For a **coplanar parallel force system** (a special case where all forces are parallel), equilibrium equations reduce to two: ΣF (in direction of forces) = 0 and ΣM = 0.

- For a 3D (space) structure, six equilibrium equations exist: ΣFx=0, ΣFy=0, ΣFz=0, ΣMx=0, ΣMy=0, ΣMz=0.

1. Solution to Provided MCQ

Question: A parabolic cable is subjected to a rise of $t$ °C in temperature. Then increase in dip of cable is proportional to ............ A) span length B) square of span length C) cube of span length D) reciprocal of span length

• Correct Option: B) square of span length
• Justification: The approximate length $S$ of a parabolic cable with span $l$ and dip $h$ is given by $S = l + \frac{8h^2}{3l}$. When temperature rises by $t$, change in length is $dS = S \alpha t$. Differentiating length equation with respect to dip $h$ gives relationship $dh = \frac{3l^2 \alpha t}{16h}$ (approximating $S \approx l$). Thus, change in dip ($dh$) is directly proportional to square of span length ($l^2$).

2. & 3. Additional High-Yield MCQs

1. The bending moment in a cable carrying a system of loads will be: A) maximum at center B) minimum at center C) zero at all points D) none of se

• Correct Answer: C) zero at all points

2. A cable subjected to a uniformly distributed load (UDL) over its entire span assumes a shape of: A) semi circle B) an isosceles triangle C) parabola D) none of se

• Correct Answer: C) parabola

3. The maximum tension in a cable occurs: A) at highest point in cable B) at lowest point in cable C) at center of cable D) at all points in cable

• Correct Answer: A) at highest point in cable

4. The moment in an arch will be zero if: A) ends are hinged B) ends are fixed C) arch axis coincides with line of thrust D) arch axis is parallel to line of thrust

• Correct Answer: C) arch axis coincides with line of thrust

5. Due to rise in temperature, in a three hinged arch, induced stress is: A) direct compressive B) direct tensile C) shear D) none of se

• Correct Answer: D) none of se (Note: A three-hinged arch is statically determinate, so support displacements or rmal expansion result in geometric changes without inducing internal stress.)

4. Core Theoretical Concepts: Cables & Arches

• Temperature Effects on Cables:

  • An increase in temperature causes cable length to expand.
  • This expansion leads to an increase in dip (sag) and a decrease in tension (horizontal thrust).
  • The increase in dip is governed by relation $dh = \frac{3l^2 \alpha t}{16h}$, making it proportional to $l^2$.

• Cable Mechanics:

  • Flexibility: Cables are assumed perfectly flexible, meaning y possess zero flexural rigidity ($EI=0$). Consequently, bending moment is zero at all points along cable.
  • Shape: The shape depends on applied load type. A cable supporting its own weight forms a catenary, while a cable supporting a UDL across horizontal span forms a parabola.
  • Tension: Tension varies along cable, being maximum at supports (highest vertical reaction component) and minimum at lowest point (where vertical shear is zero and tension equals horizontal thrust).

• Arches:

  • Line of Thrust: This is oretical path of compressive forces. If physical axis of arch aligns exactly with line of thrust, arch is in pure compression with zero bending moment.
  • Three-Hinged Arch: This is a statically determinate structure. Unlike fixed or two-hinged arches, it accommodates rmal expansion and support settlement through rotation at hinges without generating additional internal stresses.