Location of maximum deflection for a simply supported beam with an off-center point load\n\nA simply supported beam of span l carries a single point load W at an intermediate point C (not necessarily at midspan). Where does the maximum deflection occur?

Introduction / Context:
Deflection shapes of beams depend on load positions. For a simply supported beam with a single concentrated load, designers need the location of the maximum deflection to check serviceability limits and camber.

Given Data / Assumptions:

• Simply supported beam of length l with a single point load W at position C.
• Small deflections, linear elastic material behavior.
• Supports at A and B provide vertical reactions only.

Concept / Approach:
The curvature of a beam is proportional to bending moment: 1/R ∝ M/EI. For a single point load, the bending moment diagram peaks under the load. Since deflection is the double integral of curvature, the maximum deflection aligns with the maximum “area effect” of curvature, which is at the load point.

Step-by-Step Solution:
For a point load at x = a from the left: reactions RA = W(1 − a/l), RB = W a/l.Moment is piecewise linear, peaking at x = a (load location).Slope changes most rapidly where moment is largest; integrating twice leads to maximum y at x = a.Closed-form deflection formulae confirm ymax occurs at the load position for any a ∈ (0, l).

Verification / Alternative check:
Special case a = l/2 (midspan) produces well-known maximum deflection at midspan. Off-center loads shift the maximum to the load.

Why Other Options Are Wrong:
Supports (A or B) are deflection nodes (y = 0). Regions other than the load do not attain the maximum for a single point load.

Common Pitfalls:
Assuming maximum deflection always occurs at midspan regardless of load position; confusing maximum moment with maximum slope.

Final Answer:
point C (under the load)