Distributed Load Calculator

About the Distributed Load Calculator

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 quickly.

When This Page Helps

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.

How to Use the Inputs

1. Enter the uniform load in pounds per linear foot (plf).
2. Enter the beam span in feet.
3. Enter the modulus of elasticity (E) and moment of inertia (I) for deflection.
4. Read the reactions, moment, shear, and deflection.
5. Compare the moment to the beam's capacity (S × Fb) and deflection to limits.

Example Calculation

Tips & Best Practices

• The maximum moment for a UDL (wL²/8) occurs at mid-span—this is where bending stress is highest.
• Maximum shear occurs at the supports and equals the reaction (wL/2).
• Deflection at mid-span should be compared to L/360 (floors) or L/240 (roofs).
• To convert floor load (psf) to beam load (plf): multiply psf by the tributary width in feet.
• The total load on the beam equals w×L, and each reaction carries half.
• For partial UDLs or loads that don't cover the full span, different formulas apply.

The Four Key Beam Values

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.

Shear and Moment Diagrams

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.

Design Checks Beyond Moment

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.

Frequently Asked Questions

• What is the load per foot for a typical residential floor beam?

  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.

• What if the load is not uniform?

  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.

• How do I find E and I for my beam?

  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.

• What is the L/360 deflection limit?

  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).

• Can this calculator handle continuous beams?

  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.

• What about beam self-weight?

  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.