Simply supported beam of span L with triangular load (zero at one end, maximum at the other) A simply supported prismatic beam of span L is subjected to a linearly varying (triangular) load that is zero at the left support and increases to a maximum at the right support. If W denotes the total load on the span, the maximum sagging bending moment M_max is:

Introduction / Context:Linearly varying (triangular) loads are common idealizations of wind, soil, or hydrostatic pressures. For a simply supported beam, the position of maximum bending moment is not necessarily midspan; it is located where the shear force becomes zero. Expressing the answer in terms of total load W avoids ambiguity between W (total) and w (intensity).

Given Data / Assumptions:

• Span L, triangular load intensity w(x) increasing from 0 at x = 0 to w0 at x = L.
• Total load W = (1/2) w0 L.
• Small deflection linear theory; supports are simple (pin and roller).

Concept / Approach:

Let w(x) = k x where k = w0/L. Determine reactions from equilibrium, write the shear V(x) by integrating the load, find x where V(x) = 0, then evaluate M(x) there to obtain M_max. Finally express the result in terms of W and L.

Step-by-Step Solution:

Verification / Alternative check:

Numerically, (2/(9√3)) ≈ 0.1283, a well-known coefficient for this loading case. The location x = 0.577L matches the zero-shear condition from standard beam tables.

Why Other Options Are Wrong:

• W L / 8 and W L / 6 are for other load patterns (e.g., UDL or central point load, respectively).
• W L / 9 and (W L)/(3√3) do not satisfy equilibrium/compatibility for this load shape.

Common Pitfalls:

• Confusing W (total) with w (intensity). Always convert consistently.
• Assuming maximum moment at midspan; here it occurs at x = L/√3 from the zero-load end.

Final Answer:

M_max = (2 W L) / (9√3).