Mechanics

Rectangular Beam Stress and Deflection

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Analyze a solid rectangular beam under a downward point load plus a uniform full-span load. Choose simple supports with a central point load, or a left-fixed cantilever with a tip load.
Dimensions define the bending axis: I = bd³/12, S = bd²/6, A = bd. Depth is the dimension in the direction of transverse deflection.
For simple supports: each reaction = (P + qL)/2; Mmax = PL/4 + qL²/8; δmax = PL³/(48EI) + 5qL⁴/(384EI).
For a cantilever: upward reaction = P + qL; external reaction moment = PL + qL²/2; δmax = PL³/(3EI) + qL⁴/(8EI).
Bending stress magnitude = Mmax/S; maximum rectangular-section shear stress = 1.5Vmax/A. These maxima occur at different section locations and are not a combined failure criterion.
Assumes a straight, prismatic, homogeneous beam with linear elasticity, small deflections and ideal supports. Deflection includes bending only; shear deformation, torsion, axial force and support settlement are excluded.
Enter total service loads, including self-weight in the uniform load if needed. Both loads are downward and may be zero. No material strength, buckling, fatigue, connection, deflection-limit or building-code check is performed.
With the illustrative defaults: each support carries 1250 N, maximum moment is 1312.5 N·m, and maximum deflection is 0.0000817383 m. The modulus is an example, not a verified material selection.