In arch geometry (civil engineering): For a parabolic arch of span L and crown rise h, what is the rise (ordinate from the springing line) at the quarter points relative to the crown? State it as a multiple of the crown rise h.

Introduction / Context:
Parabolic arches are widely used in bridges and roofs because a parabola closely matches the bending-moment line of a uniformly distributed load. Questions on ordinates at specific fractions of the span (such as quarter points) test understanding of the arch geometry and the standard parabola equation used in structural analysis.

Given Data / Assumptions:

• Parabolic arch with span L.
• Crown rise (maximum ordinate at midspan) = h.
• Quarter point is located at x = L/4 from either support, measured horizontally along the span.
• Springings (supports) are at ordinate y = 0.

Concept / Approach:
The standard equation for the extrados/intrados of a symmetric parabolic arch referred to the springing line is:
y(x) = 4h * x * (L - x) / L^2The ordinate y is the rise above the springing line at distance x. Evaluating this at special positions (midspan, quarter points) gives simple fractions of h.

Step-by-Step Solution:
At midspan: x = L/2 → y = 4h * (L/2) * (L/2) / L^2 = h (checks the crown rise).At quarter point: x = L/4 → y = 4h * (L/4) * (3L/4) / L^2Simplify numerator: 4h * (3L^2/16) = 12hL^2/16 = 3hL^2/4Divide by L^2: y = (3h/4)Therefore, rise at quarter = 0.75 * h = three-fourths of crown rise.

Verification / Alternative check:
Because a parabola is symmetric and zero at supports, the ordinate curve is concave down. At one-quarter span, the ordinate must be less than h but greater than h/2. The value 3h/4 satisfies this monotonic behavior and symmetry.

Why Other Options Are Wrong:

• One-half of the crown rise (1/2 h): Too small; occurs nearer to the supports than quarter points.
• One-third of the crown rise (1/3 h): Far too small for a parabola at x = L/4.
• One-fourth of the crown rise (1/4 h): Incorrect by derivation; much lower than expected at quarter span.
• None of these: Incorrect because 3/4 h is exact.

Common Pitfalls:

• Using a linear instead of parabolic relation for ordinates.
• Confusing rise at quarter points with one-half rise due to intuition rather than formula.
• Mixing up x with fractional span (using x = 1/4 without multiplying by L).

Final Answer:
Three-fourths of the crown rise (3/4 h)