The intercept between a given arch and a linear arch at a section is proportional to ............

1. Solution to Provided MCQ

Question: The intercept between a given arch and a linear arch at a section is proportional to ............ A) the horizontal thrust B) the shear force at the section C) the vertical reaction at the support D) the bending moment at the section

• Correct Option: D
• Justification: A "linear arch" (or theoretical line of thrust) represents the shape an arch would take to be subjected purely to compressive thrust with no bending moment. In an actual arch, the Bending Moment ($M$) at any specific section is calculated as the product of the constant Horizontal Thrust ($H$) and the vertical distance (intercept, $z$) between the actual arch axis and the linear arch ($M = H \times z$). Since $H$ is constant for a given loading condition, the intercept $z$ is directly proportional to the Bending Moment $M$ at that section.

2. & 3. Additional High-Yield MCQs

1. A linear arch is subjected only to ............ A) shear force B) thrust (normal thrust) C) bending moment D) normal thrust and shear force

• Correct Answer: B) thrust

2. The moments in an arch will be zero throughout 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

3. The rise of a parabolic arch at quarter points, is equal to ............ A) 1/3 times the rise of the crown B) 1/4 times the rise of the crown C) 1/2 times the rise of the crown D) 3/4 times the rise of the crown

• Correct Answer: D) 3/4 times the rise of the crown

4. A two hinged semi circular arch of radius R carries a concentrated load W at the crown. The horizontal thrust is ............ A) $W/2\pi$ B) $W/\pi$ C) $2W/3\pi$ D) $4W/3\pi$

• Correct Answer: B) $W/\pi$

5. A three hinged arch is a ............ structure. A) statically determinate B) statically indeterminate C) a bent beam D) none of these

• Correct Answer: A) statically determinate

4. Core Theoretical Concepts: Arches and Line of Thrust

• Linear Arch (Line of Thrust): The theoretical geometric shape that the axis of an arch would need to take to support a specific load system purely in compression (normal thrust) without any bending moment or shear force. For a uniform distributed load (UDL) over the entire span, the linear arch is a parabola.
• Bending Moment in Arches ($M = H \cdot z$): The bending moment at any cross-section of an arch is proportional to the vertical deviation (intercept) between the actual centroidal axis of the arch and the theoretical line of thrust.
  • If the actual axis lies above the line of thrust, the moment is sagging (positive).
  • If the actual axis lies below the line of thrust, the moment is hogging (negative).
  • If they coincide, the moment is zero.
• Determinacy:
  • Three-Hinged Arch: Statically determinate (3 equilibrium equations + 1 condition at internal hinge). Temperature changes or support settlement do not induce stresses.
  • Two-Hinged Arch: Statically indeterminate (1st degree). Temperature changes cause internal stress (horizontal thrust).
• Parabolic Geometry: For a parabolic arch with rise $h$, the height at the quarter span ($L/4$) is mathematically calculated as $3/4 \cdot h$.