Calculate reactions, maximum moment, shear, and deflection for a uniformly distributed load on a simple beam. Essential for joist and beam design.
A uniformly distributed load (UDL) is the most common loading condition for floor and roof beams. The load is spread evenly along the beam—think of a floor joist supporting an evenly loaded floor, or a beam carrying joists at regular spacing. The UDL formulas for reactions, moment, shear, and deflection are the foundation of beam design.
This distributed load calculator analyzes a simple beam under a uniform load, giving you the support reactions (equal for a UDL), maximum bending moment at mid-span, maximum shear at the supports, and maximum deflection at mid-span. These are the key values needed to size and check any beam.
Enter the load per foot (plf), beam span, and optionally the beam's section properties (E and I) for deflection calculations. The calculator returns all critical design values instantly.
This measurement supports better project estimation, enabling contractors and engineers to deliver accurate bids and avoid costly overruns during the construction process.
UDL analysis is the starting point for nearly every beam design. This calculator puts the four key values (reactions, moment, shear, deflection) at your fingertips for quick design checks and size comparisons. Regular use of this calculation supports compliance with building codes and inspection requirements, helping projects proceed smoothly through the permitting and approval process.
R = w × L / 2 (each reaction) V_max = w × L / 2 (max shear at supports) M_max = w × L² / 8 (max moment at midspan) Δ_max = 5 × w × L⁴ / (384 × E × I)
Result: R = 2,400 lbs, M = 9,600 ft-lbs, Δ = 0.28″
At 300 plf on a 16-ft span: R = 300×16/2 = 2,400 lbs. M = 300×16²/8 = 9,600 ft-lbs. V = 2,400 lbs. Deflection = 5×25×(192)⁴/(384×1.6M×600) = 0.28″ (L/686).
Every simple beam under uniform load is characterized by four values: reactions (R = wL/2), maximum shear (V = wL/2), maximum moment (M = wL²/8), and maximum deflection (Δ = 5wL⁴/384EI). These formulas are the most commonly used in structural engineering and appear on every beam design reference card.
For a UDL on a simple beam, the shear diagram is a straight line decreasing from +wL/2 at the left support to −wL/2 at the right support, passing through zero at mid-span. The moment diagram is a parabola peaking at wL²/8 at mid-span. Understanding these diagrams helps visualize where the beam is most stressed.
After verifying bending stress (M/S ≤ Fb), check: (1) Shear stress: fv = 1.5V/(bd) ≤ Fv. (2) Deflection: Δ ≤ L/360. (3) Bearing at supports: R/(bearing area) ≤ Fc⊥. All three must pass for a safe design.
It depends on the tributary width and total floor load. For a beam supporting 12 ft of floor at 55 psf (15 DL + 40 LL), the load per foot is 12×55 = 660 plf.
Non-uniform loads (triangular, trapezoidal, or partial UDL) require different formulas. You can approximate by using an equivalent UDL that produces the same total load, though this slightly overestimates the moment.
E (modulus of elasticity) depends on wood species: DF-L = 1.6M psi, SPF = 1.2M psi. I (moment of inertia) = b×d³/12 for a rectangle. For a 2-ply 2×12 (3″×11.25″): I = 3×11.25³/12 = 356 in⁴.
L/360 means the maximum allowable deflection is the span divided by 360. For a 16-ft beam: 16×12/360 = 0.53″. If calculated deflection exceeds this, you need a stiffer beam (larger I).
No. Continuous beams (spanning over three or more supports) have different moment and deflection values. They require more complex analysis. This calculator is for simple beams with two end supports only.
Beam self-weight is typically small (2–5 plf for residential wood beams) and is often included in the dead load estimate. For heavy or long beams, add the self-weight to the applied load.