Structural behavior of circular slabs: Under symmetric external loading (e.g., uniform load), a circular slab deflects into which surface shape?

Introduction / Context:
Circular slabs in floors, water tanks, and rafts often experience axisymmetric loading. The resulting deflected surface provides insight into bending moments (radial and circumferential), reinforcement orientation, and serviceability. Recognizing the characteristic shape aids intuition and preliminary checks.

Given Data / Assumptions:

• Thin circular plate (slab) with small deflection theory applicable.
• Axisymmetric loading such as self-weight or uniform pressure.
• Material obeys linear-elastic assumptions within service levels.

Concept / Approach:

For axisymmetric loading on a circular plate, the governing differential equation yields a deflection surface that is parabolic in the radial coordinate. In three dimensions, the deflection surface is a paraboloid. This result underpins standard closed-form expressions for bending moments M_r and M_θ and for reinforcement detailing in circular slabs.

Step-by-Step Solution:

Verification / Alternative check:

Classic plate theory solutions (e.g., clamped or simply supported edges) demonstrate quadratic radial deflection and corresponding linear moment distributions, confirming the paraboloid shape under uniformly distributed load.

Why Other Options Are Wrong:

Semi-hemisphere and ellipsoid imply different curvature relationships not produced by linear plate bending. A hyperbolic paraboloid is a saddle surface requiring different boundary/loading conditions. A cone would imply piecewise linear deflection, not typical of elastic plates.

Common Pitfalls:

Assuming a spherical cap shape due to symmetry; ignoring boundary conditions which affect magnitude, not the basic parabolic character for uniform loads; confusing membrane behavior with bending-dominated response.

Final Answer:

Paraboloid