Beam calculator

A beam carries its loads to its supports by bending. How far it bends and how highly it is stressed depend on its section, its length, its supports and its loads. The calculator computes the section's properties from its exact outline and solves the beam for any arrangement of supports and loads along it.

The section

The section is a standard profile drawn from its sizes, with its root radii, or any outline from a DXF, SVG, STEP or BREP file, read as in the calculator. The standard profiles are the I-section, the channel, the angle, the rectangle, the round bar and the round and rectangular tube. The beam bends about the horizontal axis through the section's centroid, so the second moment of area I_x about that axis governs it. For an I-section drawn upright, as in the default, that is the strong axis, which European steel tables call y.

Supports

The supports are a table of positions along the beam, each pinned or fixed. A pinned support, a pin or a roller, holds the beam up but lets it turn, and a fixed support also stops it from turning. Two pinned supports at the ends make a simply supported beam, one fixed support at an end a cantilever, and a support inside the beam a second span or an overhang. A beam on more supports than it needs to stand, such as one fixed at both ends or one continuous over three supports, is statically indeterminate, and its reactions follow from its deformation as well as from equilibrium. The calculator solves such beams as well. A beam that could still turn or move, such as one on a single pinned support, is refused.

Loads

A point load acts at one position, a uniform load along the whole beam and a partial load along a stretch of it. Loads point down, in kN for a point load and in kN/m for the others, and a negative value points up. With its own weight switched on, the beam also carries ρ g A along its length, from the density ρ and the section's area A.

Bending stress

For a section that is symmetric about a vertical axis, the bending stress is σ = M y / I_x, with y measured from the centroid. Any other section, such as an angle, also bends sideways under a vertical load, and the stress is σ = M (I_y y − I_xy x) / (I_x I_y − I_xy²), with the product of inertia I_xy. The calculator evaluates the stress along the whole outline of the section and reports the largest tensile and compressive values.

Units

Positions and the beam's length are in meters, the section's sizes in millimeters, loads in kN and kN/m, and Young's modulus in GPa. The results give forces in kN, moments in kNm, stresses in MPa and deflections in millimeters. A section file is read in the unit it names and converted to millimeters.

1. Load

   The page loads Python and planex2d into this tab. Your browser keeps both in its cache, so later visits start faster.

2. Solve

   planex2d computes the section, and the beam is solved for the settings, again whenever one changes.

3. Save

   The results are saved as a PDF report or a CSV table, made in the tab as well.

The bending line

The bending moment is written with Macaulay brackets as one expression along the beam, with an unknown reaction at every support and an unknown moment at every fixed one. Integrating EI w″ = −M twice gives the deflection with two constants. No deflection at every support, no rotation at every fixed one and the equilibrium of forces and moments give a small linear system for the reactions, the moments and the constants. The result is exact for every arrangement of supports and loads. The tests compare it with the textbook formulas, such as P L³/(48 EI) for a central load on a simply supported beam, w L⁴/(384 EI) for a uniform load on a beam fixed at both ends, and the reactions 3 w l/8, 5 w l/4 and 3 w l/8 of a beam continuous over two equal spans.

Supported files

Run it in Python

The section's properties are one call in the library, and the beam's formulas a few lines more. The example reads the calculator's IPE 200 sample in its own millimeters and computes the default beam, 4 m long and simply supported with 20 kN at midspan, from the textbook formulas. See Measuring in the docs for the section's properties.

$ pip install planex2d

import planex as px

ipe = px.Face(px.FromDXF("ipe-200.dxf", scale=1))
Ix = ipe.Measure()["Ih"]
P, L, E = 20e3, 4000.0, 210e3
print("Ix", Ix, "mm⁴")
print("deflection", P * L**3 / (48 * E * Ix), "mm")
print("stress", P * L / 4 * 100 / Ix, "MPa")

It prints an I_x of 19.43 × 10⁶ mm⁴, a deflection of 6.535 mm and a stress of 102.92 MPa, as the calculator shows for the default beam. The units follow from the inputs, newtons and millimeters, with E = 210 000 N/mm² and the outer fiber 100 mm from the centroid.

Privacy

Limits

• The calculator uses Euler-Bernoulli beam theory, for a linear elastic material and small deflections.
• Shear deformation is not included, which matters for short, deep beams.
• Loads act through the shear center, so the beam does not twist, and torsion is not computed.
• Lateral-torsional buckling, local buckling of thin walls, stress concentrations and the checks of a design code are not included. The results give a beam's stiffness and stresses, not a design to a standard.
• A section file must be planar, and its text, dimensions and hatches are not read.
• A section that fails or a result that looks wrong? Open an issue on GitHub and attach the file if you can.