Beam Calculator: Shear, Moment, Deflection
Draw the shear force and bending moment diagrams of a beam and find its deflection. Simply supported, cantilever, overhanging, fixed–fixed or propped, with point loads, distributed loads and a point moment, plus a bending stress and deflection check.
How to Use
- Pick the beam type: simply supported, cantilever, overhanging, fixed at both ends, or propped cantilever.
- Enter the span and its unit, then the loads: up to three point loads, two distributed loads (over all or part of the span) and a point moment, each with its position measured from the left end.
- Choose the material (steel, aluminium, timber or your own E) and the section: a rectangle, round bar, tube, box, I-beam, or I and c typed in directly.
- Enter an allowable bending stress and pick a deflection limit such as L/360 to get a pass or fail on each.
- Read the shear force, bending moment and deflection diagrams, with the peaks and zero crossings marked, and the four headline results beside them.
- Open Show Work for the reactions, the peak moments and where they occur, the stiffness, and a check against the textbook formula for standard cases.
Worked Example
A steel beam with a central load. A 4 m simply supported I-beam (200 × 100 mm, 8.5 mm flanges, 5.6 mm web, so I = 18,455,902 mm⁴) carries 10 kN at mid-span. Each support takes 5 kN, the shear changes sign under the load, and the peak moment is PL/4 = 10 × 4 ÷ 4 = 10 kN·m. The bending stress is σ = Mc/I = 10,000,000 × 100 ÷ 18,455,902 = 54.18 MPa, and with E = 200 GPa the deflection is PL³/48EI = 3.61 mm, or L/1,107.
A timber floor joist. A 47 × 195 mm C24 joist (E = 11 GPa, I = 29,041,594 mm⁴) spans 3.6 m and carries 1.2 kN/m. The reactions are 2.16 kN each, the moment is wL²/8 = 1.2 × 3.6² ÷ 8 = 1.944 kN·m (6.53 MPa), and the deflection 5wL⁴/384EI = 8.22 mm, L/438, inside the 10 mm that L/360 allows.
The common mistake: putting the total load into wL²/8. The joist above carries 1.2 kN/m × 3.6 m = 4.32 kN in all. Using that total as w gives 4.32 × 3.6² ÷ 8 = 7.0 kN·m, 3.6 times too much. Either use w per metre in wL²/8 (1.944 kN·m) or the total W in WL/8 = 4.32 × 3.6 ÷ 8 = 1.944 kN·m; they are the same formula.
Show Work
Formulas
How Beam Theory Was Worked Out
Galileo posed the question in Two New Sciences (1638) with a cantilever sticking out of a wall and a weight on its end. He got the right idea, that strength grows with the square of the depth, but put the neutral axis at the bottom face. Leonhard Euler, building on Jacob and Daniel Bernoulli’s work, described the bent shape of an elastic strip in 1744, the curvature-proportional-to-moment idea now called Euler–Bernoulli beam theory, which is the EI·y″ = M this calculator integrates.
Claude-Louis Navier pulled the pieces together in his 1826 lectures at the École des Ponts et Chaussées: he placed the neutral axis through the centroid and solved beams with more supports than statics alone can handle, the fixed–fixed and propped cases here. In 1919 W. H. Macaulay published the bracket notation that lets one equation cover a beam with many loads, the same singularity-function idea this calculator uses to build M(x).
The simple formulas still carry the theory’s assumptions: the material stays elastic, deflections are small compared with the span, and plane sections stay plane. For deep, short beams shear deformation adds to the deflection, and for slender ones lateral-torsional buckling can govern before the bending stress does.
About This Tool
This calculator works out a straight beam of one section and one material under any mix of point loads, distributed loads over all or part of the span, and a point moment. It finds the support reactions, draws the shear force and bending moment diagrams with their peaks and zero crossings marked, and integrates M/EI twice for the slope and deflection. The integration runs on a grid with a node at every load and support and uses an end-corrected rule that is exact for these load shapes, so for the standard cases it agrees with PL³/48EI, 5wL⁴/384EI, PL³/3EI and wL⁴/8EI to the last digit shown, and Show Work prints that comparison. The same method handles statically indeterminate fixed–fixed and propped beams, solving the reactions and the end conditions together.
Material stiffness values are typical: structural steel 200 GPa (29,000 ksi, as used by AISC), aluminium 6061-T6 69 GPa (ASM handbook value), and C24 softwood 11 GPa (the EN 338 mean). The beam’s own weight is not added automatically; enter it as a distributed load. The section shapes use the same geometry as the Moment of Inertia Calculator. This is a study and checking aid, not a substitute for a design to your local code. Everything runs in your browser; nothing you enter is sent anywhere.
Related tools: Moment of Inertia & Centroid Calculator, Force Calculator, and Torque Calculator.
Frequently Asked Questions
How do you calculate the deflection of a simply supported beam?
For a point load P at mid-span, δ = PL³ ÷ 48EI; for a uniform load w over the whole span, δ = 5wL⁴ ÷ 384EI. A 4 m steel I-beam (E = 200 GPa, I = 1,845.6 cm⁴) with 10 kN at the middle deflects 10,000 × 4³ ÷ (48 × 200×10⁹ × 1.84559×10⁻⁵) = 3.61 mm, which is L/1,107.
What is the maximum bending moment in a simply supported beam?
PL/4 for a central point load and wL²/8 for a full-length uniform load, both at mid-span where the shear crosses zero. 10 kN at the middle of a 4 m span gives 10 kN·m; 1.2 kN/m on a 3.6 m joist gives 1.2 × 3.6² ÷ 8 = 1.944 kN·m.
How do I check the bending stress in a beam?
Use σ = M·c ÷ I, where c is the distance from the neutral axis to the outer face. 10 kN·m in a 200 mm deep I-beam (c = 100 mm, I = 18,455,902 mm⁴) gives 10,000,000 × 100 ÷ 18,455,902 = 54.18 MPa, 32.8% of a 165 MPa allowable stress.
What deflection limit should I use?
Building codes commonly use L/360 for floors under live load and L/240 for total load, with L/180 for some roofs; the International Building Code lists these in Table 1604.3. On a 3.6 m joist L/360 allows 10 mm, so the 8.22 mm this calculator finds for 1.2 kN/m on a 47 × 195 mm C24 joist passes at L/438.
Why does a beam fixed at both ends deflect so much less?
The walls stop the ends rotating, which puts hogging moments of wL²/12 at the supports and cuts the mid-span moment to wL²/24. The deflection drops from 5wL⁴/384EI to wL⁴/384EI, one fifth. 10 kN/m on a 5 m fixed–fixed 300 mm I-beam gives 20.83 kN·m at the walls, 10.42 kN·m mid-span and just 1.02 mm of sag.
How do I use the Beam Calculator: Shear, Moment, Deflection?
Simply type your numbers and read the result, which refreshes the instant you change something. There is nothing to submit and nothing to wait for.
Is it free? Does it work without internet?
Yes to both. It is free with no sign-up, and once the page has loaded it keeps working even with no internet.
Where does my data go?
Nowhere — every calculation runs on your own device. Nothing you enter is uploaded, logged, or stored.
Common Use Cases
Floor joists
A 47 × 195 mm C24 joist spanning 3.6 m under 1.2 kN/m carries 1.944 kN·m at 6.53 MPa and sags 8.22 mm, L/438.
Steel beams and lintels
Two 20 kN loads at the third points of a 6 m, 300 mm I-beam give 40 kN·m, 75.0 MPa and 9.59 mm of deflection (L/626).
Shelves and brackets
A 200 × 20 mm timber shelf sticking out 300 mm with 200 N at its tip takes 60 N·m at the wall, 4.5 MPa, and dips 1.23 mm.
Overhangs and balconies
A 5 m beam on supports at 0 and 4 m with 5 kN/m and 8 kN at the tip has 10.5 kN·m of hogging over the support and a contraflexure point at 2.95 m.
Choosing a material
The same section in aluminium (69 GPa) deflects 200 ÷ 69 = 2.9 times as much as in steel, so it usually needs a deeper section.
Homework and checking hand calculations
Show Work gives the reactions, the peak moments and their positions, and compares the result with PL³/48EI, 5wL⁴/384EI, PL³/3EI or wL⁴/8EI where they apply.
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