Parabolic arch geometry: Write the standard equation of a parabolic arch of span l and rise h with the origin at the left support, x measured along the span, and y the ordinate from the chord.

Introduction / Context:
Parabolic arches are widely used because the funicular shape for uniform loading minimizes bending. Describing the arch geometry with a simple quadratic helps in analysis for thrust and bending under various loadings.

Given Data / Assumptions:

• Span = l (distance between springings).
• Rise = h (ordinate at crown relative to springing chord).
• Coordinate system: x from left springing to right, y vertical ordinate from the chord.

Concept / Approach:
A parabola symmetric about mid-span can be written as y = k x (l − x). Enforcing the crown ordinate y(l/2) = h determines k. Substitute x = l/2 into the general form and solve for k to get the standard parabolic equation.

Step-by-Step Solution:
1) Assume y = k x (l − x).2) At crown x = l/2, y = h ⇒ h = k * (l/2) * (l/2) = k * l^2 / 4.3) Solve for k: k = 4 h / l^2.4) Therefore y = (4 h / l^2) * x (l − x) = 4 h x (l − x) / l^2.

Verification / Alternative check:
At x = 0 and x = l, y = 0; at x = l/2, y = h. The parabola is symmetric and satisfies boundary conditions.

Why Other Options Are Wrong:
Linear or pure quadratic in x/l do not satisfy both end boundary conditions and crown rise simultaneously for an arch spanning 0 to l.

Common Pitfalls:

• Forgetting the factor 4 in 4 h x (l − x) / l^2.
• Using x measured from the crown rather than from a springing without adjusting the equation.

Final Answer:
y = 4 h x (l − x) / l^2