Beam Deflection Calculator

The Beam Deflection Calculator estimates the deflection, bending moment and shear in a single span under a standard loading arrangement, and checks the deflection against the two span-over ratios most often used for serviceability. It covers six arrangements: a simply supported beam under a uniform load, a point load at midspan, a point load at any distance along the span, or two equal loads at the third points, and a cantilever under a uniform load or a point load at the free end.

It works in United States customary units: feet for the span, kips and kips per foot for loads, kips per square inch for the modulus of elasticity, and inches to the fourth power for the moment of inertia. Deflections come back in inches and moments in kip-feet.

Use it for a first-pass serviceability check, not as a design. It is an elastic analysis of one span in bending, and it checks no strength limit state at all.

Purpose and model boundary

This model calculates reactions, maximum bending moment, maximum shear, and maximum transverse deflection for six classical prismatic-beam load cases. It also compares deflection with span-over-360 and span-over-240 serviceability limits. It is a first-pass linear-elastic analysis, not member design or certification.

The spreadsheet remains the calculation authority. The page sends the named inputs to the workbook and presents the returned values, diagrams, and status without recreating the equations in browser code.

Inputs and units

The workbook converts span and load locations to inches where required by the deflection equations and returns moment in kip-feet, shear and reactions in kips, and deflection in inches.

Governing relationships

The selected load case chooses one closed-form Euler-Bernoulli elastic solution. Representative governing values are:

• simple span, uniform load: Mmax = wL²/8, Vmax = wL/2, δmax = 5wL⁴/(384EI);
• simple span, midpoint point load: Mmax = PL/4, Vmax = P/2, δmax = PL³/(48EI);
• simple span, eccentric point load: with b = L − a and d = min(a,b), Mmax = Pab/L and δmax = P d (L² − d²)^(3/2) / (9√3 EIL);
• cantilever, uniform load: Mmax = wL²/2, Vmax = wL, δmax = wL⁴/(8EI);
• cantilever, tip point load: Mmax = PL, Vmax = P, δmax = PL³/(3EI);
• two equal third-point loads of magnitude P each: Mmax = PL/3, Vmax = P, δmax = 23PL³/(648EI).

Allowable deflections are L/360 and L/240, with L expressed in inches. When deflection is positive, the span-over-deflection ratio is L / δmax.

Calculation sequence

1. The load-case selection controls which load inputs are relevant and which closed-form row is selected.
2. The workbook converts the entered geometry and evaluates all six formula rows.
3. The selected row supplies governing moment, shear, deflection, reactions, and total applied load.
4. The deflection is compared with L/360 and L/240 and plotted alongside the published moment and shear diagrams.
5. The workbook evaluates Model_Status in the precedence shown below.

Outputs and interpretation

Maximum_Moment, Maximum_Shear, and Maximum_Deflection are magnitudes for the selected idealized case. Reactions and total load support equilibrium review. Span_Over_Deflection is blank when deflection is nonpositive; otherwise, a larger denominator indicates a stiffer response. Passes_L360 and Passes_L240 compare only against those generic limits and do not select the code limit appropriate to a particular occupancy, finish, or load combination.

Validation and status logic

The workbook evaluates status in this order:

The L/240 comparison remains visible but does not have a separate status branch.

Assumptions and limitations

• The member is straight, prismatic, linearly elastic, and adequately represented by Euler-Bernoulli small-deflection theory.
• Supports, restraint, and loading match one of the six idealized cases exactly.
• E and I are constant; shear deformation, residual stress, cracking, composite action, support settlement, and geometric nonlinearity are omitted.
• Self-weight is included only when the user incorporates it in the distributed load.
• The model does not check flexure, shear strength, lateral-torsional buckling, vibration, bearing, connections, stability, load combinations, or code-specific serviceability requirements.
• The charts are formula-backed analysis diagrams, not a finite-element solution.

Restrictions and non-computing states

This calculator restricts the load case to six exact choices, enforces nonnegative loads, and supplies formula-backed visibility for mode-specific inputs. The workbook requires positive span, modulus, and inertia, and an eccentric-load distance strictly between zero and span. A zero active load produces zero response and no span-over-deflection value by design. Protected formulas may display zeros in invalid states; they are not analysis results.

Errors and warnings

A rejected entry means the request did not meet the published input rules and no workbook analysis was made. A workbook NOT VALID status identifies geometry, stiffness, or loading that prevents the selected equation from being used. CHECK means the solution is calculated but exceeds the workbook's L/360 reference. A connection or calculation-service failure is an availability problem, not a structural conclusion.

The workbook derives its relations rather than reproducing any table, chart or figure from a specification, standard or agency publication. The six load cases are the classical closed-form solutions of elastic beam theory, obtained by integrating the Euler-Bernoulli equation for a prismatic member, and they appear in substantially the same form in every structural engineering text and handbook.

• Wikipedia. Deflection (engineering), which tabulates the standard cases. https://en.wikipedia.org/wiki/Deflection_(engineering)
• Wikipedia. Euler-Bernoulli beam theory, for the assumptions behind the closed forms and where they stop holding. https://en.wikipedia.org/wiki/Euler%E2%80%93Bernoulli_beam_theory

The span-over-360 and span-over-240 ratios are conventional serviceability targets in United States practice rather than provisions reproduced here. The limit that actually governs your member comes from the building code adopted where the work is built and from what the member supports.

Material and section values are supplied by you and carry no authority from this tool. Values for design must come from the governing specification, the manufacturer, or local design data.

Additional source notes migrated from Methodology

The workbook uses classical elastic-beam relationships. The delivered audit reconciles the supported-beam cases with Engineering ToolBox beam stress and deflection formulas; the reviewer packet also lists SkyCiv's beam calculator and ClearCalcs beam analysis as independent comparators. Applicable design codes and a qualified engineer govern real structures.