A parabolic cable is subjected to a rise of t °C in temperature. Then increase in dip of cable is proportional to ............

1. Solution to Provided MCQ

Question: A parabolic cable is subjected to a rise of $t$ °C in temperature. Then the increase in dip of the 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 $t$, the change in length is $dS = S \alpha t$. Differentiating the length equation with respect to dip $h$ gives the relationship $dh = \frac{3l^2 \alpha t}{16h}$ (approximating $S \approx l$). Thus, the change in dip ($dh$) is directly proportional to the square of the 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 the center B) minimum at the center C) zero at all points D) none of these

• 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 these

• Correct Answer: C) parabola

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

• Correct Answer: A) at the highest point in the cable

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

• Correct Answer: C) the arch axis coincides with the 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 these

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

4. Core Theoretical Concepts: Cables & Arches

• Temperature Effects on Cables:

  • An increase in temperature causes the 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 the relation $dh = \frac{3l^2 \alpha t}{16h}$, making it proportional to $l^2$.

• Cable Mechanics:

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

• Arches:

  • Line of Thrust: This is the theoretical path of the compressive forces. If the physical axis of the arch aligns exactly with the line of thrust, the 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 thermal expansion and support settlement through rotation at the hinges without generating additional internal stresses.