Beam Analysis Calculator

Purpose and model boundary

This calculator analyses one straight, prismatic beam drawn on the page: its length, its supports (free, pin, roller or fixed, at the ends or anywhere along it, up to six) and the loads on it (up to ten point loads, six distributed loads that may vary linearly from start to end, and six applied moments), downward or upward. It returns the support forces and fixed-end moments, the shear, bending moment and deflection along the beam, their largest values and exactly where they occur, the most flexible span against L/360 and L/240 for comparison, and an equilibrium check. Simply supported, overhanging, cantilever, propped, fixed-ended and continuous (statically indeterminate) beams are included. It is a linear-elastic, first-pass analysis for learning and estimating, not a member design, a code check or a certification. It makes no strength, stress or design check.

The spreadsheet is the calculation authority. The page sends the drawn geometry and the values you type to the workbook and draws only what the workbook returns; it computes no engineering value itself.

For a single standard case (a simple span or a cantilever under one load), the Beam Deflection calculator gives the closed-form results.

Inputs and units

• Units. US (feet, kips, ksi, in⁴) or Metric (metres, kilonewtons, MPa, cm⁴). Every length, load, modulus and moment of inertia is read in the selected system's own unit; the workbook never converts a number, so switching systems means entering the values again in the new units. Section dimensions are in inches (US) or millimetres (Metric).
• Beam length. From the left end (x = 0) to the right end. Every support and load position is measured from the left end.
• Supports. Up to 6 rows: an ID, the position and the type. A Pin holds the beam up and sideways, a Roller holds it up only, Fixed also stops it rotating, and Free is no support (an end left free).
• Point loads. Up to 10 rows: an ID, the position and the load (kip or kN), downward positive; a negative load acts upward.
• Distributed loads. Up to 6 rows: an ID, where the load starts and ends, and its intensity at the start and at the end (kip/ft or kN/m), downward positive. The intensity varies in a straight line between the two; a blank end intensity makes the load uniform. Loads may overlap.
• Applied moments. Up to 6 rows: an ID, the position and the moment (kip-ft or kN-m), clockwise positive; a counterclockwise moment is negative.
• Moment of inertia from. Typed I uses the moment of inertia you type, from a section table. Section computes it (and the area) from one row of the Sections table: a profile (I, C, L, HSS, PIPE or RECT) and its outline dimensions, idealized with sharp corners. I and C read d, bf, tf, tw; L and HSS read b, d, t; PIPE reads od, t; RECT reads b, d.
• Modulus of elasticity E, for example about 29,000 ksi (200,000 MPa) for structural steel or 1,600 ksi for sawn lumber. It sets the deflection only.
• Include the self-weight. Yes adds the beam's own weight as a uniform downward load over its whole length; it needs the moment of inertia from Section and a density.
• Density, in lb/ft³ (US) or kg/m³ (Metric), for example 490 lb/ft³ (7,850 kg/m³) for steel. With a section it gives the weight per length; leave it blank for no weight.

A blank row in any table is ignored.

Sign conventions: loads act downward when positive. Applied moments are clockwise when positive. Support forces are upward positive, and a fixed support's moment on the beam is counterclockwise positive.

Governing relationships

The analysis follows linear elastic beam theory for the supports, loads and section entered.

• The total load is the point loads plus the area under each distributed load, plus the self-weight when it is included, upward loads counting negative. The support forces add up to it.
• Shear at a cut is the sum of the vertical forces to the left of it, upward positive. Bending moment is sagging positive: the beam bends concave upward, as a simply supported beam under gravity.
• An applied moment makes the bending moment jump by its size and does not change the shear. A moment at a support, even a fixed or pinned end, is carried by the beam there and taken up by the reactions.
• The bending stiffness EI is E times I, in kip-ft² (US, E in ksi times I in in⁴ divided by 144) or kN-m² (Metric, E in MPa times I in cm⁴ times 0.00001). The reactions, shear and moment of a beam with constant stiffness do not depend on E.
• The self-weight load is the section area times the density, using g = 9.80665 m/s² in Metric.
• The degree of indeterminacy is the number of supports (other than Free) plus the number of Fixed ones, minus 2. A beam of degree 1 or more is solved from its stiffness.
• A span's span-to-deflection ratio is its length divided by its largest deflection (either way), both in the same unit. The L/360 and L/240 figures are that span's length divided by 360 and by 240.

Calculation sequence

1. The workbook checks the inputs in a fixed order: the unit system, the beam length, E, I, each support row, each point load row, each distributed load row, each applied moment row, supports too close together, then stability. The first problem found is named, and nothing is computed.
2. It takes the moment of inertia as typed, or computes it and the area from the section about the horizontal axis through the section's centroid, and adds the self-weight when it is included.
3. It solves the support forces and fixed-support moments by linear elastic beam theory, including statically indeterminate beams.
4. It samples the shear, moment and deflection at 101 evenly spaced stations plus two rows at each end, support, point load, distributed-load start and end and applied moment (the value just left and then just right), at most 173 rows, sorted by position.
5. It finds the extremes. The shear extremes are taken just left and just right of every support and load. The largest sagging and hogging moments are found exactly, at a support or load or where the shear crosses zero. The deflection extremes are refined to the exact point where the slope is zero (or at a free end), not just taken at a sampled station.
6. It finds the span (between two neighbouring supports) or overhang (an end to its nearest support) with the smallest span-to-deflection ratio, gives L/360 and L/240 for it, and adds up the equilibrium checks.

Outputs and interpretation

• Support reactions: one row per support row, in the same order: its ID, position and type as entered, its vertical force and, for a Fixed support, its moment on the beam. The page draws a reaction as a size with an arrow in the direction it acts, and a fixed support's moment as a size with a curved arrow.
• Shear: the largest and the smallest (most negative) shear force anywhere along the beam, either side of every jump, and where each occurs.
• Bending moment: the largest sagging (positive) and the smallest (most negative, hogging) moment and where each occurs, for example hogging over an interior support of a continuous beam or at a fixed end.
• Deflection: the largest downward deflection and where it occurs, and the largest upward deflection (a negative number, for example an overhang lifting up), 0 when no part of the beam rises. Deflection is downward positive and zero at every support; the page draws it downward when the beam sags.
• Diagrams: the sampled shear, moment and deflection along the beam. A jump in the shear or the moment draws as a vertical step because every node carries a just-left and a just-right row.
• The most flexible span: its start and end, its span-to-deflection ratio (412 means the span deflects L/412; a larger number is a stiffer span), and the L/360 and L/240 deflections for comparison only. This calculator gives no pass or fail verdict.
• Totals and section values: the total load, the moment of inertia used, the bending stiffness EI, the degree of indeterminacy, and, with a section, its area, the weight per length and the self-weight load.
• Equilibrium checks: the support forces added up minus the total load, and the moments about the left end of the support forces, fixed-support moments and loads. Both are zero for a balanced beam; a tiny value is rounding.

Validation and status logic

The status reads OK when the beam can be analysed. Otherwise it reads NOT VALID with the first thing to fix, in the fixed order of the calculation sequence, and every result is blank. A beam that can move or rotate freely is unstable and is refused.

Assumptions and limitations

• Linear elastic beam theory; no strength, stress, stability or design check.
• The beam is prismatic: one E and one I along its whole length.
• A section's properties are those of its idealized, sharp-corner outline. An angle (L) is treated as bending about its horizontal centroidal axis only, a simplification of its inclined principal axes.
• The L/360 and L/240 figures are common serviceability limits shown for comparison only.
• When an extreme occurs at several places within a tiny tolerance, the one nearest the left end is given. The ends count as extreme candidates, so a cantilever's largest sagging moment can be 0 at an end.
• A reaction can be negative: an overhang can pull a support down.
• A value that does not apply, such as a Free support's force, a non-Fixed support's moment or the section outputs with Typed I, reads blank, never zero.

Restrictions and non-computing states

• The beam length is from 0.01 to 300 ft (0.01 to 100 m).
• E is from 10 to 100,000 ksi (70 to 700,000 MPa); a typed I is from 0.001 to 1,000,000 in⁴ (0.04 to 40,000,000 cm⁴).
• The density is above 0 and at most 1,600 lb/ft³ (25,000 kg/m³).
• Every support, point load and applied moment must be on the beam, between 0 and the beam length, and a distributed load must end after it starts.
• The beam needs a Fixed support or two supports, and a Pin or Fixed one to hold it sideways, and no two supports closer than 1/1000 of the length.
• A section's thicknesses must fit inside its outline; the self-weight needs the section and a density.

Errors and warnings

Each NOT VALID status names the first input to fix, for example "beam is unstable: it needs a pin or a fixed support to hold it sideways", "two supports are at the same point or closer than 1/1000 of the beam length", "point load P1 must be on the beam, between 0 and the beam length", "distributed load D1 must end after it starts" or "the self-weight needs the section: take the moment of inertia from Section".

The governing method is linear elastic (Euler-Bernoulli) beam theory. The deflected shape is written with singularity (Macaulay) functions, so each load, applied moment and support reaction adds a closed-form term. The unknown reactions of a continuous, propped or fixed beam come from equilibrium together with compatibility: no deflection at any support and no rotation at a fixed support. Shear, bending moment and deflection then follow in closed form along the beam. The L/360 and L/240 figures are shown for comparison only. When the moment of inertia is computed from a section, it comes from the typed dimensions of an idealized sharp-corner outline; no shape table is used.

Sources and further reading

• The original statement of the bracket (singularity function) method for beam deflection: W. H. Macaulay, "Note on the deflection of beams", Messenger of Mathematics 48, 1919, pages 129-130. Scanned volume at the Internet Archive
• Deflection of beams: F. P. Beer, E. R. Johnston, J. T. DeWolf and D. F. Mazurek, Mechanics of Materials, 8th edition, McGraw-Hill Education, 2020, chapter 9, Deflection of Beams.
• Statically indeterminate beams solved from the differential equation of the deflection curve with the support conditions, and the section properties of composite areas: B. J. Goodno and J. M. Gere, Mechanics of Materials, 9th edition, Cengage Learning, 2018, chapter 10, Statically Indeterminate Beams, and appendix D, Review of Centroids and Moments of Inertia. Publisher page
• The force (compatibility) method for statically indeterminate beams: R. C. Hibbeler, Structural Analysis, 11th edition, Pearson, 2024, chapter 9, Analysis of Statically Indeterminate Structures by the Force Method. Publisher page
• Building-code deflection limits for structural members, the context of the L/360 and L/240 figures, which this calculator shows for comparison and does not apply: International Code Council, 2024 International Building Code, section 1604.3, Serviceability, and table 1604.3, Deflection Limits. ICC

No table, chart or text from any standard, manual or commercial publication is reproduced, and none of the authors or publishers named here has reviewed or endorses this calculator.

Applicable design codes and a qualified engineer govern real structures.