Beam calculator
Work out the deflection, bending moment, shear force and stresses of a simply supported beam, a cantilever or a beam fixed at both ends.
1 Span and loads
A negative load acts upwards. The distributed load w is per length of beam, not per area.
2 Cross-section and material
The 3D view needs WebGL, which is not available in this browser. The results and diagrams below are not affected.
Drag to rotate. Drag the P arrow to move the point load. With the keyboard, the arrow keys turn the view, + and − zoom, and 0 resets it.
Deflection
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Bending stress
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Moment
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Safety factor
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3 Results
Support reactions and internal forces
Deflection and stress
Cross-section
Show the calculation with your numbers
4 Shear force, bending moment and deflection diagrams
Point at or tap a diagram to read the values at that position. Positive shear force and sagging moment are drawn above the axis.
The calculator uses Euler–Bernoulli beam theory for a straight beam with a constant cross-section, a linear elastic material and static loads. It does not check buckling, fatigue, connections or the rules of a design code, and you remain responsible for the design. Volupe provides it free of charge and without warranty.
How to use the beam calculator
Choose the supports at the top: simply supported (a pin at A and a roller at B), a cantilever fixed at A, or a beam fixed at both ends. In step 1, enter the span L, a uniformly distributed load (UDL) w, a point load P and its distance a from A. Either load can be left empty, and a negative value acts upwards.
In step 2, choose the cross-section and its dimensions, or enter the second moment of area I and the distance c to the outermost fibre yourself, in mm⁴ or in cm⁴ as steel tables give it. The material sets E and the density. Add the yield strength to get the safety factor against first yield.
The results update as you type, with diagrams of the shear force, bending moment and deflection along the span. “Show the calculation with your numbers” goes from the section properties to the stresses, so you can check each step by hand.
In the 3D view the colours show the bending stress, blue for compression and red for tension. The calculator runs in your browser and does not send the values you enter anywhere.
Deflection, bending moment and stress in a beam
A load on a beam is carried by shear forces and bending moments that change along the span. The moment curves the beam, and the curvature is M/(EI), where E is the elastic modulus and I the second moment of area of the cross-section, also called the area moment of inertia. The beam bends most sharply where the moment is large, and the deflection builds up from these curvatures along the beam. That is why a cantilever deflects most at its free end, where the moment is zero. For a rectangle of width b and depth h, I = bh³/12, so doubling the depth makes the beam eight times as stiff.
The moment also stresses the beam. The bending stress changes linearly over the depth, from compression on one side to tension on the other, and is zero at the neutral axis, which passes through the centroid of the section when there is no axial force. It is largest in the outermost fibres, σmax = Mc/I = M/W, where c is the distance from the neutral axis to the outermost fibre and W = I/c the elastic section modulus.
The calculator uses one sign convention throughout. Loads and deflections are positive downwards and reactions positive upwards. A negative reaction means that the support holds the beam down. A sagging moment puts the bottom of the beam in tension and is positive, and a hogging moment is negative. At a fixed end the calculator gives the bending moment in the beam. The shear force is the slope of the moment diagram, V = dM/dx. Bending stress is positive in tension.
Simply supported
Cantilever
Fixed at both ends
Supports and load cases
Simply supported beam
A simply supported beam rests on a pin at A and a roller at B. Neither end can move up or down, but both can rotate, and the moment there is zero. The largest moment and deflection are at midspan for a central point load or a uniformly distributed load.
With a distributed load the deflection grows with the fourth power of the span. Twice the span with the same load per metre gives 16 times the deflection, while the moment grows four times. So a long beam that is strong enough can still deflect too much.
Cantilever beam
A cantilever is fixed at A and free at B. The fixed end takes a reaction force and a moment, and the beam can neither move nor rotate there. A point load P at the free end gives a fixed-end moment PL and a tip deflection PL³/(3EI). A distributed load gives wL²/2 and wL⁴/(8EI).
For the same span, section and load, a cantilever loaded at its tip deflects 16 times as much as a simply supported beam loaded at midspan, PL³/(3EI) against PL³/(48EI).
Beam fixed at both ends
A beam fixed at both ends cannot rotate at either end and develops hogging moments at the supports. Under a distributed load its midspan deflection is a fifth of the simply supported value.
Several loads together
Because the theory is linear, the effects of separate loads can be added (superposition). The calculator works out the point load and the distributed load separately and adds them at each point along the span. The two loads do not in general give their largest moment or deflection at the same point, so it searches the combined curves. Adding the two maxima instead can over- or underestimate the result.
Beam deflection formulas
The beam equation
Equilibrium of a short piece of beam and the curvature M/(EI) give the beam deflection equation, EI d⁴v/dx⁴ = w(x), with the deflection v positive downwards like the load. It needs two boundary conditions at each end, such as zero deflection and moment at a pin, or zero deflection and slope at a fixed end. Every formula in the table below solves it for one set of supports and loads.
The six standard cases
The table gives the results for the six cases the calculator combines, with the point load at midspan or at the free end.
| Supports and load | Reactions | Largest moment | Largest deflection |
|---|---|---|---|
| Simply supported, point load P at midspan | P/2 at each end | PL/4 at midspan | PL³/(48EI) at midspan |
| Simply supported, distributed load w | wL/2 at each end | wL²/8 at midspan | 5wL⁴/(384EI) at midspan |
| Cantilever, point load P at the free end | P and PL at the fixed end | PL at the fixed end | PL³/(3EI) at the free end |
| Cantilever, distributed load w | wL and wL²/2 at the fixed end | wL²/2 at the fixed end | wL⁴/(8EI) at the free end |
| Fixed at both ends, point load P at midspan | P/2 and PL/8 at each end | PL/8 at the ends and at midspan | PL³/(192EI) at midspan |
| Fixed at both ends, distributed load w | wL/2 and wL²/12 at each end | wL²/12 at the ends (wL²/24 at midspan) | wL⁴/(384EI) at midspan |
Moments at fixed ends are hogging. EI is the flexural stiffness (elastic modulus E times second moment of area I), P a force, w a force per length and L the span.
A point load anywhere on the span
For a simply supported beam with P at a distance a from A and b = L − a from B, the reactions are RA = Pb/L and RB = Pa/L, the largest moment is Pab/L under the load, and the deflection under the load is Pa²b²/(3EIL). The largest deflection lies between the load and midspan. Setting the slope to zero gives its position, x = √(a(a + 2b)/3) from A when a > b. For a < b, swap a and b and measure x from B.
With both ends fixed, the end moments are Pab²/L² at A and Pa²b/L² at B, and the deflection under the load is Pa³b³/(3EIL³). For a cantilever with the load at a from the fixed end, the tip deflection is Pa²(3L − a)/(6EI).
Slopes
A simply supported beam turns through wL³/(24EI) at each end under a distributed load and PL²/(16EI) under a central point load. The tip of a cantilever turns through PL²/(2EI) under an end load and wL³/(6EI) under a distributed load.
Deflection limits
The deflection limit comes from the design code or the specification for the structure, usually as a fraction of the span. Typical US limits for the deflection under service live load are L/360 for floors with plastered ceilings, L/240 for other floors and L/180 for roofs without plaster. To compare, enter the live load alone and leave the self-weight off. The calculator applies no limit.
Bending stress, shear stress and second moment of area
The dimensions you enter give the section properties, and the largest moment and shear force give the stresses. Every built-in section is symmetric about the axis it bends about, and c is half the depth.
σ = My/I, σmax = Mc/I = M/W (1)
Bending stress at a distance y below the neutral axis, positive in tension, and its largest value.
τ = VQ/(It) (2)
Shear stress. Q is the first moment about the neutral axis of the area beyond the level considered, and t the width of the section at that level.
I = bh³/12, W = bh²/6, τmax = 3V/(2A) (3)
Rectangle, width b and depth h.
I = πd⁴/64, W = πd³/32, τmax = 4V/(3A) (4)
Round bar, diameter d.
I = π(D⁴ − di⁴)/64, di = D − 2t (5)
Tube, outside diameter D and wall thickness t.
I = (BH³ − bihi³)/12, bi = B − 2t, hi = H − 2t (6)
Rectangular hollow section, width B, depth H and wall t.
I = [bh³ − (b − tw)(h − 2tf)³]/12 (7)
Symmetric I‑section, depth h, flange width b, flange thickness tf and web thickness tw.
Notes on the formulas. In (5) to (7) the hole, or the strips beside the web, is subtracted from the full circle or rectangle. The rounded corners of real rectangular hollow sections make their I slightly lower than formula (6) gives, and the root fillets of rolled I‑sections make it slightly higher than (7). For those, enter the tabulated I under own values. Where the web meets the flange of an I‑section, bending and shear stress act at once, and the von Mises stress calculator combines them.
Worked example: deflection and stress of a steel box-section beam
A rectangular hollow steel section, 100 mm deep and 50 mm wide with a 4 mm wall, spans 2.0 m between a pin and a roller. It carries a point load of 10 kN at midspan and a distributed load of 2 kN/m. We take E = 210 GPa and a yield strength of 355 MPa (S355 with a wall under 16 mm). Both loads give their largest moment and deflection at midspan, so here the two results can simply be added. The calculator opens with this example.
| Quantity | Value | How |
|---|---|---|
| Second moment of area I | 1 441 259 mm⁴ | (50 × 100³ − 42 × 92³)/12, formula (6) |
| Section modulus W | 28 825 mm³ | I/c with c = 50 mm |
| Support reactions RA, RB | 7.00 kN | P/2 + wL/2 = 5 + 2 |
| Largest bending moment, at midspan | 6.00 kN·m | PL/4 + wL²/8 = 5.00 + 1.00 |
| Deflection from the point load | 5.51 mm | PL³/(48EI) |
| Deflection from the distributed load | 1.38 mm | 5wL⁴/(384EI) |
| Largest deflection, at midspan | 6.88 mm | the sum, L/290 |
| Largest bending stress σmax | 208.2 MPa | M/W, formula (1) |
| Largest shear stress τmax | 11.0 MPa | VQ/(It) with Q = (BH² − bihi²)/8 = 18 064 mm³ and t = 2 × 4 = 8 mm, formula (2) |
| Safety factor against first yield | 1.71 | 355/208.2 |
Values from the calculator, rounded for display. The shear stress is at the neutral axis next to the supports, where the shear force is largest.
Note that this is not a design check. A code check adds partial factors and its own deflection limit. We checked the example with the formulas above, outside the calculator, and got the same values.
Where beam formulas stop
Short, deep beams
Euler–Bernoulli theory leaves out shear deformation. For a simply supported solid rectangular steel beam with a central point load, shear adds 6Eh²/(5GL²) of the bending deflection, where G is the shear modulus (E/G is about 2.6 for steel). That is about 3 % when the span is ten times the depth and 12 % at five times. With both ends fixed the share is four times as large, and I‑sections and hollow sections are affected two to four times as much as a solid rectangle of the same depth. The calculator estimates the share for your beam, and we add a note from 2 %. For short, deep beams use Timoshenko beam theory or a finite element (FE) model.
Large deflections
The formulas assume small slopes, so that the curvature can be taken as d²v/dx². We set the warning at a deflection of 5 % of the span, as no source gives one number for “small”. If both ends are also held against moving towards each other, the beam picks up axial tension as it bends and becomes stiffer than the formulas predict, from a deflection of about half the depth by a rule of thumb from MIT’s structural mechanics course. For a beam fixed at both ends you get a note when this happens.
Buckling
None of the results cover buckling. An I‑section bent about its strong axis can buckle sideways and twist (lateral–torsional buckling) unless its compression flange is restrained, and so can a deep, narrow rectangle. Thin flanges and webs can buckle locally before the section reaches its full strength. Axial loads are outside the calculator. Viktor Hultgren’s post on setting up buckling solutions in Simcenter 3D shows how to check buckling with an FE model.
First yield and the plastic moment
The safety factor in the calculator is against first yield in the outermost fibre. A ductile section can carry more as the yielding spreads inwards, up to the plastic moment, which is higher by the shape factor, 1.5 for a solid rectangle and about 1.1 for a wide-flange I‑section. Whether a steel section gets there depends on how slender its flanges and web are, which Eurocode 3 sorts into four cross-section classes. The calculator does not classify sections.
Local effects
Beam theory gives the stresses along the span and leaves out local effects where loads and supports act, and at holes, notches and welds. Check those separately, for example with an FE model, and combine the stress components at a point with the von Mises stress calculator.
Partly fixed ends
A fixed end in a model is perfectly rigid, while a real connection turns a little. For a symmetric load, the simply supported and the fixed-end results bracket the real midspan moment and deflection. Where the end stiffness is uncertain, check both and design for the larger of each. With a symmetric load and equally stiff ends, the end moments lie between zero and the fixed-end value. An end that is much stiffer than the other can take more. Check such a beam with an FE model.
From hand calculation to finite element model
Beam formulas cover one straight member with ideal supports. Frames, beams over several supports, openings or changes of section, dynamic loads and the local stresses where parts meet are usually analysed with an FE model. Our guide to finite element analysis explains the method and how to check an FE result.
In Simcenter 3D, beams can be meshed with 1D beam elements (CBEAM) that carry a standard cross-section, such as a tube or a channel, through a beam property. Beam Mesh Automation, introduced in Simcenter 3D 2021.2, creates the beam mesh with its cross-sections and material from the CAD model. Simcenter Femap can combine beam elements with solid elements in one model.
Viktor Hultgren’s post on defining and analysing many load cases in Simcenter 3D shows a steel structure whose beams are joined with RBE2 rigid links and beam elements, analysed for many load cases in one solution and combined afterwards by superposition.
A hand calculation is a good check on such a model. The same span, section, supports and loads in this calculator should give the same reactions and moments as the FE model and close deflections. If the deflections differ much, check the supports, the units and whether the elements include shear deformation. To learn to build and check such models, see Simcenter 3D training.
Sources and how the calculator was checked
Sources
D. Roylance, MIT, Stresses in beams (2000, PDF), Beam displacements (2000, PDF, with the form factor for shear; its shear-deflection ratio is printed with a factor of two too small, used here corrected) and the table of typical material properties.
T. Wierzbicki, MIT OpenCourseWare 2.080J Structural mechanics (2013), lecture notes 4, 5 and 6 (beams, beam deflections and moderately large deflections), 9 (stability) and 11 (buckling of plates and sections).
American Wood Council, Design Aid No. 6: Beam design formulas with shear and moment diagrams (2007), Figure 25 (fixed ends, point load at any point), checked against the calculator. US Forest Products Laboratory, Wood handbook, chapter 9 (2010, PDF).
Missouri S&T, beam deflection table (PDF). D. Baker and W. Haynes, Engineering statics: open and interactive, moments of inertia of common and composite shapes.
S. Sengupta, NJIT MET 301, Transverse shear stress (PDF). A. Rahman, CUNY City Tech MECH 2333, Stresses in beams (presentation, PDF).
SteelConstruction.info, Steel material properties and Member design. The Steel Construction Institute, SCI P363 Steel building design: design data (PDF). A. Varma, Purdue University, CE 470 chapter 5, beam design (PDF).
NIST Special Publication 811 (2008), Guide for the use of the International System of Units, for the US customary conversions.
Siemens, Simcenter 3D tutorial Modeling a beam and cross section (2019.1), and the Simcenter blog posts on Beam Mesh Automation and Femap solid meshes with beam elements.
How the calculator was checked
We compared the calculator with an independent finite element solution (Hermite beam elements, exact at the nodes for this theory) in 1 560 cases: the three supports, each load alone and both together, loads at the supports, upward and downward loads, spans from 0.1 mm to 10 m and stiffnesses over twelve orders of magnitude. At the nodes of the FE model, deflections, slopes, moments and reactions agree to within 10⁻⁸ of a reference value set by each case’s loads, span and stiffness. A second check against exact rational arithmetic in 14 410 cases found the largest moments and deflections to within 10⁻¹² on the same scale. The fixed-end formulas for a point load anywhere on the span were also compared with Figure 25 of Design Aid No. 6.
Changelog
October 2026: first version.
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Setting up buckling solutions in Simcenter 3D
Viktor Hultgren, Volupe · 2025
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Questions about beam calculations
Simcenter 3D
Viktor Hultgren
Application Specialist, M.Sc.
Former Altair tools
Johan Dahlberg
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What is the formula for beam deflection?
For a simply supported beam with a uniformly distributed load w, the largest deflection is 5wL⁴/(384EI), at midspan. With a point load P at midspan it is PL³/(48EI). E is the elastic modulus and I the second moment of area, also called the area moment of inertia. The table under “Beam deflection formulas” gives the six standard cases, and the calculator also handles a point load anywhere on the span and both loads together.
How do you calculate the deflection of a cantilever beam?
With a point load P at the free end, the tip deflects PL³/(3EI). With a uniformly distributed load w it deflects wL⁴/(8EI). With the point load at a distance a from the fixed end, the tip deflects Pa²(3L − a)/(6EI). Choose “Cantilever” in the calculator to get the deflection together with the fixed-end moment and the slope.
How do you calculate the maximum bending moment in a beam?
For a simply supported beam the maximum bending moment is PL/4 under a central point load P and wL²/8 under a uniformly distributed load w, both at midspan. In a cantilever it is PL under an end load and wL²/2 under a distributed load, both at the fixed end. In a beam fixed at both ends under a distributed load it is wL²/12, at the supports. With a point load off centre and a distributed load the two maxima are not at the same point, so the calculator searches the combined moment curve along the span.
What is the difference between the second moment of area and the section modulus?
The second moment of area I describes how the area is spread about the bending axis and sets the stiffness, so it controls the deflection. The section modulus W = I/c sets the largest bending stress, σmax = M/W. For a rectangle, doubling the depth makes I eight times and W four times as large.
Why does a beam fixed at both ends deflect less?
The fixed ends stop the beam from rotating, so hogging moments develop at the supports and reduce the moment in the span. Under a uniformly distributed load the midspan deflection drops to a fifth of the simply supported value, wL⁴/(384EI) instead of 5wL⁴/(384EI), and the largest moment drops from wL²/8 to wL²/12, now at the supports.
Does the calculator include the beam’s self-weight?
Only if you tick “Add the beam’s self-weight”. It then adds ρgA to the distributed load, with ρ the density, g = 9.80665 m/s² and A the area of the section. With own values for I and c, enter the area too.
What deflection limit should I use?
The one your design code or specification sets for the structure. Building codes give limits as a fraction of the span, and typical US limits for the deflection under service live load are L/360, L/240 and L/180, depending on what the beam carries. The calculator shows L/δ for the loads you enter, so enter the live load alone to compare.
When do I need FEA instead of a beam formula?
When the member is not one straight beam on the supports this calculator covers: frames, beams over several supports, openings or changes of section, dynamic loads, or the local stresses at joints. Short, deep beams, where shear deformation matters, call for Timoshenko beam theory or an FE model. Our guide to finite element analysis explains the method. Volupe’s engineers work with Simcenter 3D, Simcenter Femap and Simcenter Nastran.
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