Beam-Column — Axial Load with Transverse Loading Alpha · 2026-09-23

Shear, moment, slope and deflection of a beam that also carries an axial load — compression magnifies bending, tension reduces it; ends held against axial motion also covered

What this calculator does

When a beam also carries a force along its axis, that force acts on the already bent beam and adds a moment P\cdot y. The governing equation gets one more term:

EI\,y'''' \pm P\,y'' = w(x)

With k = \sqrt{P/EI} the solution is built from \cos kx,\ \sin kx (compression) or \cosh kx,\ \sinh kx (tension). The calculator solves it in closed form for six load types and six end restraints, like the straight-beam calculator.

Compression makes it worse, tension makes it better

• Compression magnifies deflection and moment, and the growth becomes very fast near the buckling load P_{cr} = K_1\pi^2EI/l^2. The calculator shows P/P_{cr} and refuses P \ge P_{cr} — then the member buckles and you need the elastic stability calculator.
• Tension straightens the beam and reduces deflection and moment. Ignoring it is safe but can overestimate the deflection a lot.

Every result is compared with the same beam and loads without the axial load. The ratios C_y and C_M tell you how much the axial force changes the maximum deflection and moment.

Several transverse loads

Each transverse load carries the full axial load — the axial force is not shared between them. With that rule the results for several loads can be added, and the calculator does exactly this.

Ends held against axial motion

If both ends are held so that the beam cannot shorten, bending stretches the centre line and creates a tension P by itself. In this mode P is a result: it is found from a cubic equation based on a one-term sine shape, and then the exact tension solution gives the moments. Two cases for a perfectly flexible cable are included as well.

Model limits

• Linear-elastic, small deflections, constant EI, shear deformation ignored.
• Compression must stay below the buckling load.
• Tension is limited to kl \le 25. Beyond that the closed form loses accuracy in double precision; this is a numerical limit, not a physical one.
• Held ends: the axial tension uses a one-term sine approximation of the deflected shape, which is good for small deflections.

Engineering data

• Material Query

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