Beam with left and right overhangs.

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Build the envelope from equilibrium and shear changes.

The beam is statically determinate once the two support positions are entered. The calculator first replaces the full-length UDL w by W = wT, finds the reactions by force and moment equilibrium, then evaluates the piecewise moment function at supports, load breaks, beam ends, and any interior zero-shear locations.

Implemented equations and signs

Let T be total beam length, x_A and x_B the ordered support coordinates, w the downward UDL, P the downward point load, and x_P its coordinate. With W = wT, R_B = [W(T/2 - x_A) + P(x_P - x_A)]/(x_B - x_A) and R_A = W + P - R_B. Using sagging positive and x from the left end, the sectional moment is M(x) = R_A(x - x_A)H(x - x_A) + R_B(x - x_B)H(x - x_B) - wx²/2 - P(x - x_P)H(x - x_P), where H is 0 before and 1 at/after its event. The displayed range is the largest positive and smallest negative value found from the piecewise candidates; the headline is max(|M|).

Classic load-case variants covered

This parameterized case includes a double overhang when 0 < x_A < x_B < T; a left-only overhang by setting x_A = 0; a right-only overhang by setting x_B = T; and a simply supported span by setting both supports at the physical ends. Set P = 0 for the full-length UDL case, w = 0 for a single point-load case, or place x_P on either overhang or between the supports. No applied end moment is parameterized.

Keep it as a first-order statics screen.

• Use 0 ≤ x_A < x_B ≤ T and 0 ≤ x_P ≤ T; the app rejects positions outside that domain.
• Loads are downward and non-negative in the input form; the returned moment range carries the sagging-positive sign convention.
• Supports are ideal vertical reactions with no support width, stiffness, settlement, friction, or rotational restraint.
• Deflection, shear stress, lateral stability, torsion, load combinations, connection design, and resistance checks are not included.